International Congress in Honour of. Professor Ravi P. Agarwal

Transcription

1 International Congress in Honour of Professor Ravi P. Agarwal June 23 26, 2014 Uludag University, Bursa Turkey

2 2 International Congress in Honour of Professor Ravi P. Agarwal Scientific Committee R. P. Agarwal (USA) J. L. Bona (USA) A. Bruno (Russia) I. N. Cangul (Turkey) A. S. Cevik (Turkey) E. Di Benedetto (USA) R. Finn (USA) R. P. Gilbert (USA) W. M. Haddad (USA) I. Kiguradze (Georgia) W. A. Kirk (USA) I. Lasiecka (USA) V. Lokesha (India) A To-Ming Lau (Canada) J. Mawhin (Belgium) G. Milovanovic (Serbia) B. Mordukhovich (USA) J. Neuberger (USA) J. Nieto (Spain) D. Motreanu (France) D. O Regan (Ireland) [email protected] S. Park (Korea) [email protected] A. Peterson (USA) [email protected] V. Radulescu (Romania) [email protected] S. Reich (Israel) [email protected] E. Savas (Turkey) [email protected] M. D. Schechter (USA) [email protected] Y. Simsek (Turkey) [email protected] W. Takahashi (Japan) [email protected] D. Taşçı (Turkey) [email protected] R. Triggiani (USA) [email protected] J. R. L. Webb (UK) [email protected] H.-K. Xu (Taiwan) [email protected]

3 International Congress in Honour of Professor Ravi P. Agarwal 3 Local Organizing Committee Ismail Naci Cangul (Uludag University, Turkey) Ahmet Sinan Cevik (Selcuk University, Turkey) Nihat Akgunes (Selcuk University, Turkey) Elvan Akın (Missouri University, USA) Firat Ates (Balikesir University, Turkey) Muge Capkin (Uludag University, Turkey) Elif Cetin (Uludag University, Turkey) Musa Demirci (Uludag University, Turkey) Ayşe Feza Güvenilir (Ankara University, Turkey) Sebahattin Ikikardes (Balikesir University, Turkey) Nazlı Yildiz Ikikardes (Balikesir University, Turkey) Ilker Inam (Uludag University, Turkey) Erdal Karapınar (Atlm University, Turkey) Billur Kaymakçalan (Çankaya University, Turkey) Hacer Ozden (Uludag University, Turkey) Birsen Ozgur (Uludag University, Turkey) Metin Ozturk (Uludag University, Turkey) Recep Sahin (Balikesir University, Turkey) Umit Sarp (Balikesir University, Turkey) Ekrem Savas (Istanbul Commerce University, Turkey) Gokhan Soydan (Uludag University, Turkey) Kenan Taş (Çankaya University, Turkey) Dursun Taşçı (Gazi University, Turkey) Ahmet Tekcan (Uludag University, Turkey) Sibel Yalcin Tokgoz (Uludag University, Turkey) Elif Yasar (Uludag University, Turkey) Emrullah Yasar (Uludag University, Turkey) Aysun Yurttas (Uludag University, Turkey)

4 4 International Congress in Honour of Professor Ravi P. Agarwal Preface On behalf of the Scientific and Organising Committees, I would like to welcome you all to Bursa for this International Congress. First of all, I would like to mention the willingness and capacity of the Mathematics Department at Uludag University to organize and actively take part in Mathematical events. We have been organizing numerous national and international congresses, conferences, workshops and seminars with the help of our colleagues at other universities. The following are just a few examples of the events in last years. On 21 st -23 rd August, 2008, we organized The Twentieth International Congress of the Jangjeon Mathematical Society, of which I have been honoured to be a member, in Karinna Hotel at Mount Uludag. The refereed proceedings of this congress were published in Advanced Studies in Contemporary Mathematics and Proceedings of the Jangjeon Mathematical Society. Following this event, we organized The International Congress in Honour of Professor H. M. Srivastava on his 70 th Birth Anniversary, again in Karinna Hotel on 18th-21st August, The duly-refereed prooceedings of this congress were published as a special volume of the Elsevier journal Applied Mathematics and Computation was the year that we hosted the 24 th National Mathematics Symposium at Uludag University. Finally The International Congress in Honour of Professor Hari M. Srivastava was held at the Auditorium at the Campus of Uludag University, Bursa, Turkey on 23 rd -26 th August, The duly-refereed prooceedings of this congress were published in special volumes of the four open access Springer journals Advances in Difference Equations, Boundary Value Problems, Fixed Point Theory and Applications and Journal of Inequalities and Applications. Prof. Dr. Ravi P. Agarwal has been coworking with many Turkish mathematicians in a wide range of topics and his contributions, in particular to Turkish mathematics and mathematicians, are endless. This is one of the reasons that made me proud to organize The International Congress in Honour of Professor Ravi P. Agarwal on 23 rd -26 th June, We hoped to thank him, at least partially, for his support and contributions to Turkish mathematicians. It is my great pleasure to welcome you all to The International Congress in Honour of Professor Ravi P. Agarwal and to Bursa. The duly refereed proceedings of this congress will be published in two special issues in open access Springer journals Advances in Difference Equations and Applications and Journal of Inequalities and Applications. I especially thank in advance to the editor Prof. Dr. Ravi P. Agarwal, and to the guest editors Prof. Dr. Billur Kaymakçalan, Prof. Dr. Elvan Akın and Prof. Dr. Erdal Karapınar who spend a lot of time and effort with me to produce the best possible special issues to be remembered for many years. Please allow me to thank all my colleagues and students who worked with me for months to make this congress a success. One particular mathematician needs to be mentioned especially: My good friend and coworker Prof. Dr. Ahmet Sinan Cevik, who is the co-chair of this congress and helped me in many aspects. I am proud to organize all these meetings together with this special people and I wish our cowork and friendship will go on forever. My final thanks go to Prof. Dr. Ahmet Tekcan who had spent serious amount of time to produce this booklet as nicely as it is. Finally, on behalf of all the friends and colleagues, I take this opportunity to wish Prof. Dr. Agarwal a happy life together with all his beloved ones and continuation of his contributions to Mathematics and Mathematicians. Prof. Dr. Ismail Naci Cangul Chair of the Congress, Dean of the Faculty of Arts and Science Uludag University, Gorukle Campus, Bursa, Turkey [email protected], [email protected]

5 International Congress in Honour of Professor Ravi P. Agarwal 5 About Prof. Dr. Ravi P. Agarwal Age and Date of Birth: 66 years, 10th July, 1947 Present Position: Professor & Chairman, Department of Mathematics Texas A&M University- Kingsville Kingsville, TX 78363, U.S.A. [email protected] Telephone Numbers: 1(361) (office) 1(361) (personal) Degrees: Master in Science (1969) Agra Univ., 1st class, 2nd position Ph.D. (1973) Indian Institute of Technology, Madras, India Field of Research: Numerical Analysis, Differential and Difference Equations, Inequalities, Fixed Point Theorems Research Experience: 44 years Research Publications: Over 1175 research papers in the following Journals and Series: 1. Acta Applicandae Mathematicae 2. Acta Mathematica Hungarica 3. Advances in Difference Equations 4. Advances in Mathematical Sciences and Application 5. Aequationes Mathematicae 6. Analele Stiintifice ale Universitatii. Al. I. Cuza din Iasi 7. Annales Polonici Mathematici 8. Applied Mathematics and Computation 9. Applied Mathematics Letters 10. Applicable Analysis 11. Archivum Mathematicum (Brnö) 12. Atti della Accad. Nazionale Dei Lincei 13. BIT 14. Boundary Value Problems 15. Bulletin of the Institute of Mathematics, Academia Sinica 16. Bulletin UMI 17. Chinese Journal of Mathematics 18. Communications in Applied Analysis 19. Communications in Applied Numerical Methods 20. Computers and Mathematics with Applications 21. Differential and Integral Equations 22. Dynamic Systems and Applications 23. *Dynamic Systems and Applications, Dynamic Publishers 24. Dynamics of Continuous, Discrete and Impulsive Systems 25. Fixed Point Theory and Applications 26. Fluid Dynamics Research 27. Functional Differential Equations 28. Funkcialaj Ekvacioj 29. Georgian Mathematical Journal 30. Hiroshima Mathematical Journal 31. IMA Journal of Applied Mathematics 32. Indian Journal of Pure and Applied Mathematics 33. International Journal of Computer Mathematics 34. *International Series of Numerical Mathematics, Birkhäuser 35. Japan Journal of Industrial and Applied Mathematics 36. Journal of Applied Mathematics and Stochastic Analysis

6 6 International Congress in Honour of Professor Ravi P. Agarwal 37. Journal of Approximation Theory 38. Journal of the Australian Mathematical Society. Series A 39. Journal of the Australian Mathematical Society. Series B 40. Journal of Computational and Applied Mathematics 41. Journal of Difference Equations and Applications 42. Journal of Differential Equations 43. Journal of Inequalities and Applications 44. Journal of the Korean Mathematical Society 45. Journal of the London Mathematical Society 46. Journal of Mathematical Analysis and Applications 47. Journal of Mathematical and Physical Sciences 48. Journal of Nonlinear and Convex Analysis 49. Journal of Optimization Theory and Applications 50. Korean Journal of Computational and Applied Mathematics 51. *Lecture Notes in Mathematics, Springer-Verlag 52. Mathematica Slovaca 53. Mathematical Inequalities and Applications 54. Mathematical Methods in the Applied Sciences 55. Mathematical and Computer Modelling 56. Mathematical Problems in Engineering: Theory, Methods and Applications 57. Mathematics Seminar Notes, Kobe University 58. Mathematika 59. Mathematische Nachrichten 60. *Matscience Reports 61. Neural, Parallel and Scientific Computations 62. Nonlinear Analysis Forum 63. Nonlinear Analysis : Theory, Methods and Applications 64. Nonlinear Functional Analysis and Applications 65. Nonlinear World 66. *North-Holland Mathematics Studies 67. PanAmerican Mathematical Journal 68. Proceedings of the American Mathematical Society 69. *Proceedings of the Conference of ISTAM 70. *Proceedings of the International Conference on Difference Equations and Applications, Gordon and Breach 71. *Proceedings of the First World Congress of Nonlinear Analysts, Walter de Gruyter 72. Proceedings of the Indian Academy of Sciences 73. Proceedings of the Royal Society of Edinburgh 74. Proceedings of the Edinburgh Mathematical Society 75. *Proceedings of Symposia in Applied Mathematics, American Math. Soc. 76. Proceedings of the Tamil Nadu Acad. Sci. 77. Publications of the Research Institute for Mathematical Sciences 78. Results in Mathematics 79. Rivista di Math. della Univ. Parma 80. Rocky Mountain Journal of Mathematics 81. Series in Mathematical Analysis and Applications, Gordon and Breach 82. *Stability and Control: Theory, Methods and Applications, Gordon and Breach 83. Studies in Applied Mathematics 84. Tamkang Journal of Mathematics 85. Tohuku Mathematical Journal 86. Topological Methods in Nonlinear Analysis 87. Utilitas Mathematica

7 International Congress in Honour of Professor Ravi P. Agarwal ZAA 89. ZAMM (* Conference Proceedings/ Special Volumes) Monographs and Books: (1) R.P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore, Philadelphia, 1986, p This comprehensive monograph provides an exhaustive state of the art coverage of basic results on boundary value problems associated with higher order differential equations. It is without question one of the most through reviews I have seen, on any subject. Those doing research in this field would be well advised to refer to this work. The author consistently poses questions to researchers who are looking for open problems. (Mathematical Reviews) The monograph is an excellent account of the various techniques available in the literature to prove existence and uniqueness of various boundary value problems which occur in applications. Graduate students and research mathematicians will find it very useful. (Zentralblatt für Mathematik) (2) R.P. Agarwal and R.C. Gupta, Essentials of Ordinary Differential Equations, McGraw-Hill Book Co., Singapore, New York, 1991, p.467. (3) R.P. Agarwal, Difference Equations and Inequalities : Theory, Methods and Applications, Marcel Dekker, Inc., New York, 1992, p.777. This book is a virtual encyclopedia of results concerning difference equations. It is well written and is easy to read. This book covers over 400 recent publications. This book should not only be of interest to mathematicians and statisticians but also to electrical engineers, biologists, economists, psychologists, and sociologists to name a few. This indeed is a very good book to have in ones own personal library. (Mathematical Reviews) This new monograph combines all aspects of the theory and methods of solutions of difference equations and their applications in real world problems providing in depth coverage of more than 400 recent publications. This monograph with the wealth of information it contains is very well come. (Newsletter on Computational and Applied Mathematics) This book contains a complete account of standard results concerning difference equations, as well as an extensive discussion of recent papers concerning the theory and practice of their solutions. This book should be useful both as textbook and for reference. (Mathematika) This book is essential for the enrichment of knowledge in mathematics, physics and statistics. The comprehensive compilation of the book is useful for researchers of natural philosophy. (Indian J. Physics) Comprehensive treatment develops discrete versions of Rolle s, mean value, Kneser s theorems (The American Mathematical Monthly) This excellent monograph combines all aspects of the theory and methods of solutions of difference equations and their applications providing in depth coverage of more than 400 recent publications. It serves as a basic reference for mathematicians and users of mathematics interested in differential and difference equations and their applications. (Acta Sci. Math. Szeged) It is a definite reference for applied mathematicians, numerical analysts, physicists, engineers, and graduate level students in courses on difference equations. (INSPEC The Institute of Electrical Engineers) Focusing on a wide range of possible mathematical uses, the book offers various methods of solving linear and nonlinear difference equations. (Bulletin Bibliographique) Deals with the many aspects of difference equations including theory, methods of solutions, and applications. Reviews more than 400 recent related publications. (The New York Public Library) (4) R.P. Agarwal and V. Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, World Scientific, Singapore, 1993, p The book is devoted to a branch of the theory of differential equations that is classical on the one hand but still alive and developing on the other hand. The book is very interesting and well written. It is warmly recommended to any student in analysis and to any specialist in the theory of differential equations. (Mathematical Reviews and Zentralblatt für Mathematik)

8 8 International Congress in Honour of Professor Ravi P. Agarwal (5) R.P. Agarwal and P.J.Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993, p.365. A main theme of this book lies behind the selection and organization of the material in it is the use of interpolation in the theory of ordinary differential equations. It will no doubt find uses among specialists in differential equations. Otherwise, the wealth of detail and the precision of the error estimates in it go beyond what is generally available in book or monograph form and commend the work to a more general audience. (Journal of Approximation Theory) (6) R.P. Agarwal and R.C. Gupta, Solutions Manual to Accompany Essentials of Ordinary Differential Equations, McGraw-Hill Book Co., Singapore, New York, 1993, p.209. (7) R.P. Agarwal and P.Y.H. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht, 1995, p.393. The monograph under review presents a complete survey of results related to the Opial s inequality developed over the last three decades. The book under review is very well written and most of the material is presented with detailed proofs. The book can be warmly recommended not only to specialist working in the area of mathematical analysis and applications but also to graduate students, engineers and researchers in the applied sciences. (Zentralblatt für Mathematik) (8) R.P. Agarwal and P.J.Y. Wong, Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht, 1997, p.507. One of the specialists in the field is without doubt Ravi P Agarwal. His previous book Difference Equations and Inequalities (1992) is a survey of the theory of difference equations and contains a wealth of information for the researchers. This new book, co authored by Patricia J. Y. Wong, can be seen as an update of the first one. The results in this book are of great interest to other specialists in the field. This book offer an easy way to get access to them. (Mathematical Reviews) The book contains a collection of recent results and it will serve as a reference book for researchers in discrete dynamical systems and their applications and reader will also find material, which is not available in other books on difference equations. It will also be of interest to graduate students interested in the theory of finite difference equations and their applications. The presentation is clear and it is a welcome addition to the literature. (Zentralblatt für Mathematik) (9) R.P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht, 1998, p.289. Agarwal s great knowledge of the literature in this area makes this book very appealing. The book will be useful for a graduate course concerned with boundary value problems for either differential equations or difference equations. It also would be an excellent book for mathematicians doing research in this area. (Mathematical Reviews) (10) R.P. Agarwal, D. O Regan and P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999, p.417. The majority of the book is devoted to some of the recent developments by the authors. The book should be a good reference book and the extensive bibliography could prove to be very helpful. In addition, the examples at the end of each chapter are a good source of illustrative material. (Mathematical Reviews) (11) R.P. Agarwal, Difference Equations and Inequalities: Second Edition, Revised and Expended, Marcel Dekker, New York, 2000, xv+980pp. (12) R.P. Agarwal, M. Meehan and D. O Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001, 170pp. (13) R.P. Agarwal, S.R. Grace and D. O Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 2000, 337pp. This good monograph contains some of the recent developments in the oscillation theory of difference and functional-differential equations (FDEs). It provides an excellent reference to the recent work for research workers in this interesting field. (Mathematical Reviews) (14) R.P. Agarwal and D. O Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 2001, 341pp. This book develops the basic ideas used in proving the existence of solutions to boundary value problems on infinite intervals and it mainly contains the results which the authors have obtained in their

9 International Congress in Honour of Professor Ravi P. Agarwal 9 research during the last decade. Mathematical Reviews) (15) R.P. Agarwal, M. Meehan and D. O Regan, Nonlinear Integral Equations and Inclusions, Nova Science Publishers, New York, 2001, 362pp. (16) R.P. Agarwal, S.R. Grace and D. O Regan, Oscillation Theory for Second Order Linear, Half linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic Publishers, The Netherlands, 2002, 672pp. Those who are already in the field will welcome the systematic organization of the material and find the book to be a valuable reference. (Mathematical Reviews) (17) R.P. Agarwal, S.R. Grace and D. O Regan, Oscillation Theory for Second Order Dynamic Equations, Taylor & Francis, U.K., 2003, 404pp. The authors study systematically various techniques about oscillation and nonoscillation of each type of equations. There are numerous examples in each chapter and each chapter ends with detailed historical notes and an extensive list of references. The book is very readable and it is a valuable source and an important contribution to oscillation theory. (Mathematical Reviews) (18) R.P. Agarwal and D. O Regan, Singular Differential and Integral Equations with Applications, Kluwer Academic Publishers, Dordrecht, 2003, 402pp. The authors have produced a monograph in which they present some of the recent development in existence of solutions theory of nonlinear singular integral and differential equations. In addition to theory, the monograph focuses on applications. Much of the material focuses on recent developments of the authors. A primary purpose of the monograph is to provide a readable account and introduce the material to a broader audience. (Mathematical Reviews) (19) R.P. Agarwal, M. Bohner and W.-T. Li, Nonoscillation and Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 2004, 376pp. (20) R.P. Agarwal, M. Bohner, S.R. Grace and D. O Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, 2005, 1000pp. This is truly a compendium of many different results, all having a relation in some way to results which may or may not be fairly well known for the continuous case. One of the very useful features of this book is the discussion at the end of each chapter of the results presented and references to the original sources, as far as the authors are aware. Moreover, the authors have included a large number of examples throughout which serve to illustrate the many and varied results which are obtained. All of the authors are very well known in oscillation theory and have all contributed a great deal to this area. It is indeed a useful addition to the literature to have such a comprehensive survey of the area and to point the direction to new results. It will serve as a valuable reference in the area for many years to come. (Mathematical Reviews) (21) R.P. Agarwal and D. O Regan, An Introduction to Ordinary Differential Equations, Springer, New York, (22) R.P. Agarwal and D. O Regan, Ordinary and Partial Differential Equations with Special Functions, Fourier Series and Boundary Value Problems, Springer, New York, (23) R.P. Agarwal, D. O Regan and D.R. Sahu, Fixed Point Theory for Lipschitzian type Mappings with Applications, Springer, New York, 2009 (24) R.P. Agarwal, S. Ding and C.A. Nolder, Inequalities for Differential Forms, Springer, New York, (25) K. Perera, R.P. Agarwal and D. O Regan, Morse Theoretic Aspects of p Laplacian Type Operators, Mathematical Surveys and Monographs, Volume 161, American Mathematical Society, Providence Island, (26) R.P. Agarwal, K. Perera and S. Pinelas, An Introduction to Complex Analysis, Springer, New York, (27) S.K. Sen and R.P. Agarwal,, e, with MATLAB: Random and Rational Sequences with Scope in Supercomputing Era, Cambridge Scientific Publishers, Cambridge, (28) R.P. Agarwal, L. Berezansky, E. Braverman and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, 2012.

10 10 International Congress in Honour of Professor Ravi P. Agarwal (29) A. Aral, V. Gupta and R. P. Agarwal, Applications of q Calculus in Operator Theory, Springer, New York, (30) R.P. Agarwal, D. O Regan and P.J.Y. Wong, Constant Sign Solutions of Systems of Integral Equations, Springer, in press. Teaching and Other Experiences: years, various courses for B.Sc., M.Sc., B.E. and M.E. 2. U.G.C. Visiting Professor, Marathwada University, Aurangabad (February, 1979). 3. Visiting Scientist, Indian Institute of Science, Bangalore (September 1979). 4. Alexander Von Humboldt Foundation Fellow at der Ludwig -Maximilians Universität, München, with Prof. Dr. G. Hämmerlin. ( ) 5. Visiting Professor, Instituto Matematico, Firenze, Italy ( ), with Prof. Roberto Conti. 6. Visiting Scientist, International Center for Theoretical Physics, Trieste (April 1982, April 1983). 7. Visiting Scientist, The University of Manitoba, Winnipeg, Canada (April-May 1983, April 1986). 8. Visiting Scientist, The University of Western Australia (April 1989). 9. Visiting Professor, University of Saskatchewan, Canada (April 1991). 10. Visiting Scientist, JSPS Cooperation Programmes, Japan (June 1991). 11. Visiting Professor, Politecnico di Milano, Milano, Italy (June 1995). 12. Visiting Professor, University of Delaware, USA (June 1997 May 1998). 13. Visiting Professor, Politecnico di Milano, Milano, Italy (December 2007). 14. Visiting Professor, University of Roma, Italy (May 2007). 15. Visiting Professor, Politecnico di Milano, Milano, Italy (June 2008). 16. KFUPM Chair Professor, King Fahd Univ. Petro. Minerals, Saudi Arabia (June 2010). 17. KFUPM Chair Professor, King Fahd Univ. Petro. Minerals, Saudi Arabia (May-July 2011). 18. Honorary Distinguished Professor, King Abdual Aziz University, Saudi Arabia (2011 ). Thesis Direction: 1. P.R. Krishnamoorthy, Boundary Value Problems for Higher Order Differential Equations, Ph.D. thesis, University of Madras, E. Thandapani, On Continuous and Discrete Inequalities, Ph.D. thesis, University of Madras, P.J.Y. Wong, On Two-Point Boundary Value Problems, Honours Project, National University of Singapore, F.C. Weng, On the Oscillatory Behaviour of Second Order Delay Differential Equations, Honours Project, National University of Singapore, G.L. Meng, Maximum Principles for Higher Order Differential Inequalities, Honours Project, National University of Singapore, 1987.

11 International Congress in Honour of Professor Ravi P. Agarwal P.J.Y. Wong, Error Bounds for Quintic and Biquintic Spline Interpolation, Masters thesis, National University of Singapore, Ng Bee Cheow, On Gronwall s Inequality and its Applications, Honours Project, National University of Singapore, Goh Lee Leng, Uniqueness of Initial Value Problems, Honours thesis, National University of Singapore, P.J.Y. Wong, Sharp Polynomial Interpolation Error Bounds for Derivatives and their Applications, Ph.D. thesis, National University of Singapore, Lim Ee Tuo, Nonlinear Variation of Parameters for Differential and Difference Equations, Honours Project, National University of Singapore, Chan Kwok Leong, Opial Type Inequalities, Honours Project, National University of Singapore, Ngan Ngiap Teng, Gram Matrices, Inequalities and Applications, Honours Project, National University of Singapore, T.A. Smith, On Periodic Solutions of Nonlinear Hyperbolic Equations of the Fourth Order, Ph.D. thesis, Florida Institute of Technology, U.S.A Citations : Over 7000 in the following Journals and Series are known. 1. Acta Math. Hungar. 2. Advances in Computational Mathematics 3. Advances in Difference Equations 4. Annales Polonici Mathematici 5. Appl. Math. Comp. 6. Appl. Math. Letters 7. Applicable Analysis 8. Archiv der Mathematik 9. Arch. Math. (Brnö) 10. Astrophysics and Space Science 11. Boundary Value Problems 12. Bull. Austral. Math. Soc. 13. Comm. Appl. Numer. Methods 14. Computers Math. Applic. 15. Computing 16. CWI Monograph, North-Holland 17. Czech. Math. Jour. 18. de Gruyter Series in Nonlinear Analysis and Applications 19. Differential and Integral Equations 20. Dynamic Systems and Applications 21. Fixed Point Theory and Applications 22. Funkcialaj Ekvacioj 23. IEEE Trans. on Automatic Control 24. Int. Jour. Comp. Math. 25. Int. Series of Numer. Math. 26. Izv. Akad. Nauk. Arm. SSR, Mathematika 27. Jour. Approximation Theory 28. Jour. Comp. Appl. Math.

12 12 International Congress in Honour of Professor Ravi P. Agarwal 29. Jour. Comp. Physics 30. Jour. Difference Equations and Appl. 31. Jour. Differential Equations 32. Jour. Inequalities and Applications 33. Jour. Math. Anal. Appl. 34. Jour. Mathl. Phyl. Sci. 35. Mathematical and Computer Modelling 36. Mathematics and its Applications, Kluwer Academic Publishers 37. Mathematics in Science and Engineering, Academic Press, Inc. 38. Mathematics Studies, North Holland 39. Mathematika 40. Nonlinear Analysis : TMA 41. Nonlinear Times and Digest 42. Numerische Mathematik 43. Pitman Advanced Publishing Program 44. Prentice Hall Series in Computational Mathematics 45. Proc. Amer. Math. Soc. 46. Proc. R. Soc. London 47. Proc. Royal Society of Edinburgh 48. Rocky Mountain J. Math. 49. SIAM J. Math. Anal. 50. SIAM Review 51. Trans. Amer. Math. Soc. 52. World Scientific Series in Applicable Analysis 53. ZAA 54. ZAMM Refereed more than 5000 papers for the following Journals: 1. Journal of Differential Equations 2. Journal Approximation Theory 3. Journal of Mathematical Analysis and Applications 4. Nonlinear Analysis 5. Applicable Analysis 6. Applied Mathematics Letters 7. Applied Mathematics & Optimization 8. Journal of Computational and Applied Mathematics 9. Communications in Applied Numerical Methods 10. Communications in Numerical Methods in Engineering 11. Computers & Mathematics with Applications 12. Advances in Computational Mathematics 13. Dynamic Systems and Applications 14. Journal of Difference Equations and Applications 15. Archivum Mathematicum 16. Mathematical and Computer Modelling 17. Mathematische Nachrichten 18. Japan Jour. Indusl. Appl. Math. 19. International Journal of Math. and Mathl. Sciences 20. International Journal of Numer. Methods Engg. 21. Jour. Appl. Math. Simulation 22. IEEE Trans. Automatic Control 23. Proc. Edinburgh Math. Soc.

13 International Congress in Honour of Professor Ravi P. Agarwal Numer. Methods Partial Diff. Equns. 25. Jour. Austral. Math. Soc. 26. Trans. Amer. Math. Soc. Service as a reviewer of research monographs: Refereed several research monographs for Kluwer Academic, Springer Verlag and World Scientific publishers. I have also written reviews for several monographs in the journal SIAM Reviews. Member of the Editorial Boards: 1. Editor-in-Chief, Journal of Inequalities and Applications, Springer, U.S.A. 2. Editor-in-Chief, Advances in Difference Equations, Springer, U.S.A. 3. Editor-in-Chief, Boundary Value Problems, Springer, U.S.A. 4. Editor-in-Chief, Fixed Point Theory and Applications, Springer, U.S.A. 5. Editor, Nonlinear Analysis: Theory, Methods and Applications, Elesiver, The Netherlands (till 2012) 6. Editor, Nonlinear Analysis: Real World Applications, Elesiver, The Netherlands (till 2012) 7. Senior Editor, Applied Mathematics and Computation, Elsevier, The Netherlands. 8. Editor, Series in Mathematical Analysis and Applications, Gordon and Breach, U.K. 9. Editor, World Scientific Series in Applicable Analysis, World Scientific, Singapore. 10. Editor, Far East Journal of Mathematical Sciences, Pushpa Publishing House, India.(till 2008) 11. Associate Editor, Advances in Mathematical Sciences and Application, Japan. 12. Honorary Editor, Applicable Analysis, Gordon and Breach, U.K. (till 2012) 13. Associate Editor, Applied Mathematics Letters, Elsevier, The Netherlands.(till 2012) 14. Associate Editor, Archivum mathematicum, Masaryk Univ., Brno, Czech Rep. (till 2007) 15. Associate Editor, Communications in Applied Analysis, Dynamic Publishers, U.S.A. 16. Associate Editor, Communications in Applied Nonlinear Analysis, International Publications, U.S.A. (till 2012) 17. Associate Editor, Communications of the Korean Mathematical Society, Korea.(till 2010) 18. Associate Editor, Computers and Mathematics with Applications, Elsevier, The Netherlands.(till 2012) 19. Associate Editor, Dynamics of Continuous, Discrete and Impulsive Systems, University of Waterloo, Canada. 20. Associate Editor, Dynamics of Continuous, Discrete and Impulsive Systems (series B, Applied Mathematics), University of Waterloo, Canada.(till 2008) 21. Associate Editor, Facta Universitatis: Mathematics and Informatics, University of Nis, Yugoslavia. 22. Associate Editor, Functional Differential Equations, The Research Institute, College of Judea and Samaria, Israel.

14 14 International Congress in Honour of Professor Ravi P. Agarwal 23. Associate editor, International Journal of Applied Mathematics, Academic Publications, Bulgaria. 24. Associate Editor, International Journal of Computational and Numerical Analysis and Applications, Academic Publishers, Bulgaria 25. Associate Editor, International Journal of Computer Mathematics, Gordon and Breach, U.K.(till 2009) 26. Associate Editor, International Journal of Differential Equations and Applications, Academic Publications, Bulgaria. 27. Associate Editor, Journal of Inequalities in Pure and Applied Mathematics, Australia 28. Associate Editor, Journal of Mathematical Analysis and Applications, Academic Press, U.S.A (till 2007) 29. Associate Editor, Journal of Nonlinear and Convex Analysis, Yokohama Publishers, Japan 30. Associate Editor, The Korean Journal of Computational and Applied Mathematics, Korea (till 2010) 31. Associate Editor, Mathematical and Computer Modelling, Elsevier, The Netherlands. (till 2012) 32. Associate Editor, Mathematical Inequalities and Applications, Zagreb, Croatia. 33. Associate Editor, Mathematical Sciences Research Hot-Line, International Publications, U.S.A. (till 2009) 34. Associate Editor, Memoirs on Differential Equations and Mathematical Physics, Publishing House GCI, Tiblisi, Republic of Georgia. 35. Associate Editor, Neural, Parallel and Scientific Computations, Dynamic Publishers, U.S.A. 36. Associate Editor, Nonlinear Differential Equations: Theory, Methods and Applications, Andhra University, India. 37. Asociate Editor, Nonlinear Analysis Forum, Korea. 38. Associate Editor, Nonlinear Functional Analysis and Applications, Kyungnam University Press, Korea. 39. Associate Editor, Nonlinear Oscillations, The Publication of the Institute of Mathematics, National Academy of Sciences of Ukraine, Ukraine 40. Associate Editor, PanAmerican Mathematical Journal, International Publications, U.S.A. (till, 2012) 41. Associate Editor, Journal of Mathematical Control Science and Applications, International Science Press, India. 42. Associate Editor, East Asian Mathematical Journal, The Busan Gyeongnam Mathematical Society, Korea. Editorial Work: 1. Numerical Mathematics, Singapore, (with Y. M. Chow and S. J. Wilson) International Series of Numerical Mathematics, Volume 86. Birkhäuser Verlag, Basel, 1988, p. 526.

15 International Congress in Honour of Professor Ravi P. Agarwal Recent Trends in Differential Equations, World Scientific Series in Applicable Analysis, Volume 1, 1992, p Contributions in Numerical Analysis, World Scientific Series in Applicable Analysis, Volume 2, 1993, p Inequalities and Applications, World Scientific Series in Applicable Analysis, Volume 3, 1994, p Advances in Difference Equations, Special issue: Computers and Mathematics with Applications, Pergaman - Press, Volume 28 Numbers 1-3 (1994), pp Dynamical Systems and Applications, World Scientific Series in Applicable Analysis, Volume 4, 1995, p Recent Trends in Optimization Theory and Applications, World Scientific Series in Applicable Analysis, Volume 5, 1995, p Advances in Differential and Integral Inequalities, Special issue: Nonlinear Analysis: Theory, Methods and Applications, Pergaman - Press, Volume 25 Numbers 9-10 (1995), pp Computer Aided Geometric Design (with Ruibin Qu), Special issue: Neural, Perallel & Scientific Computations, Dynamic Publishers, Volume 5 Numbers 1-2 (1997), Positive Solutions of Nonlinear Problems, Special issue: Journal of Computational and Applied Mathematics, Elsevier, Volume 88 Number 1 (1998), pp Proceedings of Equadiff 9 (with J. Vosmansk y), Special issue: Archivum mathematicum, Masaryk University, Volume 34 (1998), pp Proceedings of Equadiff 9 (with F. Neuman and J. Vosmanský), Stony Brook: Electronic Publishing House, 1998, p Advances in Difference Equations II, Special issue: Computers and Mathematics with Applications, Pergaman Press, Volume 36 Numbers (1998), Proceedings of the International Workshop on Difference and Differential Inequalities (with L. E. Persson and A. Zafer), Special issue: Mathematical Inequalities and Applications, Volume 1 Number 3 (1998), Discrete and Continuous Hamiltonian Systems (with M. Bohner), Special issue: Dynamic Systems and Applications, Volume 8 Numbers 3-4 (1999), Fixed Point Theory with Applications in Nonlinear Analysis (with Donal O Regan), Special issue: Journal of Computational and Applied Mathematics, Elsevier, Volume 113 Numbers 1-2 (2000), Integral and Integrodifferential Equations (with Donal O Regan), Series in Mathematical Analysis and Applications, Gordon & Breach, Amesterdam, Volume 2, 2000, p Lakshmikantham s Legacy: A Tribute on his 75th Birthday, Special issue: Nonlinear Analysis: Theory, Methods and Applications, Pergaman - Press, Volume 40 Numbers 1-8 (2000), pp Nonlinear Operator Theory (with Donal O Regan), Special issue: Mathematical and Computational Modelling, Pergamon Press, Volume 32 Numbers (2000), Advances in Difference Equations III (with Donal O Regan), Special issue: Computers and Mathematics with Applications, Pergaman Press, Volume 42 Numbers 3-5 (2001),

16 16 International Congress in Honour of Professor Ravi P. Agarwal 21. Orthogonal Systems and Applications (with G.V. Milovanovic), Special issue: Applied Mathematics and Computation, Elsevier, Volume 128 Issues 2-3 (2002), Advances in Difference Equations IV (with Martin Bohner and Donal O Regan), Special issue: Computers and Mathematics with Applications, Pergaman Press, Volume 45 Numbers 6-9 (2003), Advances in Integral Equations (with Donal O Regan), Special issue: Dynamic Systems and Applications, Dynamic Publishers, Volume 14 Number 1 (2005), Proceedings of the Conference Differential and Difference Equations and Applications (with K. Perera), Hindawi, 2006, p International Conferences: Participated and gave invited lectures in the following conferences 1. Approximate Methods for Navier - Stokes Problems (Paderborn 1979, Germany) 2. General Inequalities 3 (Oberwolfach 1981, Germany) 3. Operator Inequalities (Oberwolfach 1981, Germany) 4. Ordinary Differential Equations (Oberwolfach 1983, Germany) 5. General Inequalities 4 (Oberwolfach 1983, Germany) 6. International Conference on Qualitative Theory of Differential Equations (Edmonton 1984, Canada) 7. Trends in the Theory and Practice of Nonlinear Analysis (Texas 1984, U.S.A.) 8. EQUADIFF 6 (Brnö 1985, Czechoslovakia) 9. General Inequalities 5 (Oberwolfach 1986, Germany) 10. International Conference on Optimization : Techniques and Applications (1987, Singapore) 11. International Conference on Functional Equations and Inequalities (Szczawnica, 1987, Poland) 12. International Conference on Numerical Mathematics (1988, Singapore) 13. International Symposiumon Asymptotic and Computational Analysis (Winnipeg 1989, Canada) 14. General Inequalities 6 (Oberwolfach 1990, Germany) 15. First World Congress of Nonlinear Analysts (Tampa, 1992, U.S.A.) 16. Second International Conference on Dynamic Systems and Applications (Atlanta 1995, U.S.A.) 17. First International Conference on Neural, Parallel and Scientific Computations (Atlanta 1995, U.S.A.) 18. Second International Conference on Difference Equations and Applications (Vesprem 1995, Hungary) 19. General Inequalities 7 (Oberwolfach 1995, Germany) 20. International Workshop on Difference and Differential Inequalities (Gebze, 1996, Turkey) 21. Second World Congress of Nonlinear Analysts (Athens, 1996, Greece) 22. Modelling and System Stability Investigations (Kiev, 1997, Ukraine)

17 International Congress in Honour of Professor Ravi P. Agarwal EQUADIFF 9 (Brnö 1997, Czechoslovakia) 24. Third Midwest-Southeastern Atlantic Joint Regional Conference on Differential Equations (Nashville, TN 1997, U.S.A) 25. The Centennial Celebration: A Century of mathematics and Statistics at Nebraska (Lincoln 1998, U.S.A) 26. Third International Conference on Dynamic Systems and Applications (Atlanta 1999, U.S.A.) 27. Third World Congress of Nonlinear Analysts (Catania, 2000, Italy) 28. Sixth International Conference on Difference Equations and Applications (Augsburg, 2001, Germany) 29. International Conference on Differential, Difference Equations and their Applications (Patras 2002, Greece) 30. Fourth International Conference on Dynamic Systems and Applications (Atlanta 2003, U.S.A.) 31. Fourth World Congress of Nonlinear Analysts (Orlando, 2004, USA) 32. The 24th Annual Southeastern-Atlantic Regional Conference on Differential Equations (University of Tennessee at Chattanooga, 2004, USA) 33. Fifth International Conference on Dynamic Systems and Applications (Atlanta 2007, U.S.A.) 34. Fifth World Congress of Nonlinear Analysts (Orlando, 2008, USA) 35. Boundary Value Problems (Santiago de Compostela, 2008, Spain) 36. Fourteenth International Conference on Difference Equations and Applications (Istanbul, 2008, Turkey) 37. EQUADIFF 12 (Brnö 2009, Czech Republic) 38. International Conference on Differential and Difference Equations and Applications (Ponta Delgada 2011, Portugal) 39. International Conference on Applied Analysis and Algebra (Istanbul 2012, Turkey) th International Workshop on Dynamical Systems and Applications (Ankara 2012, Turkey) 41. The International Conference on Mathematical Inequalities and Nonlinear Functional Analysis with Applications (Cinju 2012, Korea) 42. Southeastern Atlantic Regional Conference on Differential Equations (Georgia Southern University, USA) 43. International Conference on Applied Analysis and Mathematical Modelling (Istanbul 2013, Turkey) 44. International Conference on Anatolian Communications in Nonlinear Analysis (Bolu 2013, Turkey) Colloquium Talks: Several Colloquium talks delivered at the following centers 1. Universität Karlsruhe (Germany, 1979) 2. der Universität München (Germany, 1979) 3. Georg - August - Universität Göttingen (Germany, 1980) 4. Universität Stuttgart (Germany, 1980)

18 18 International Congress in Honour of Professor Ravi P. Agarwal 5. Mathematisch Centrum Amsterdam (Holland, 1980) 6. University Van Amsterdam (Holland, 1980) 7. Universita Degli Studi Di Parma (Italy, 1980) 8. Universita Degli Studi Di Firenze (Italy, 1980) 9. University of Ioannina (Greece, 1980) 10. Universität Karlsruhe (Germany, 1981) 11. der Universität München (Germany, 1981) 12. Technische Hochschule Darmstadt (Germany, 1981) 13. Universität Osnabrück (Germany, 1981) 14. Universität Hannover (Germany, 1981) 15. Universita Degli Studi Di Firenze (Italy, 1982) 16. Georg - August - Universität Göttingen (Germany, 1983) 17. Johann Wolfgang Goethe - Universität Frankfurt (Germany, 1983) 18. Albert - Ludwigs - Universität Freiburg (Germany, 1983) 19. der Universität Tübingen (Germany, 1983) 20. Universita Degli Studi Di Trieste (Italy, 1983) 21. Universita Degli Studi Di Trento (Italy, 1983) 22. J. E. Purkne University Brnö (Czechoslovakia, 1983) 23. Comenius University Bratislava (Czechoslovakia, 1983) 24. The University of Manitoba (Canada, 1983) 25. The University of Manitoba (Canada, 1986) 26. Scuola Normale Superiore, Pisa (Italy, 1987) 27. Politechnica di Milano (Italy, 1987) 28. Rheinisch - Westfälische Technische Hochschule Aachen (Germany, 1989) 29. Universität Karlsruhe (Germany, 1989) 30. der Universität München (Germany, 1989) 31. Georg - August - Universität Göttingen (Germany, 1989) 32. Johann Wolfgang Goethe - Universität Frankfurt (Germany, 1989) 33. University of Western Australia (Australia, 1989) 34. Murdoch University (Australia, 1989) 35. University of Dundee (U.K. 1989) 36. Brunel, The University of West London (U.K. 1989) 37. The University of Sussex (U.K. 1989) 38. The University of Liverpool (U.K. 1989) 39. University of Manchester (U.K. 1989) 40. University of Cambridge (U.K. 1989) 41. Oxford University (U.K. 1989) 42. University of Saskatchewan (Canada, 1991) 43. The University of Tokyo (Japan, 1991) 44. Tohoku University (Japan, 1991) 45. Nagoya University (Japan, 1991) 46. Ehime University (Japan, 1991) 47. Okayama University (Japan, 1991) 48. Kagoshima University (Japan, 1991) 49. Hiroshima University (Japan, 1991) 50. RIMS, Kyoto University (Japan, 1991) 51. der Universität München (Germany, 1995) 52. Universita Degli Studi Di Firenze (Italy, 1995) 53. Universita Degli Studi Di Trieste (Italy, 1995) 54. Scuola Normale Superiore, Pisa (Italy, 1995) 55. Politechnica di Milano (Italy, 1995) 56. Universita Degli Studi Di Roma (Italy, 1995)

19 International Congress in Honour of Professor Ravi P. Agarwal Universita Degli Studi Di Bologna (Italy, 1995) 58. SISSA, Trieste (Italy, 1995) 59. Universität Augsburg (Germany, 1995) 60. Institute of Mathematics, Ukrainian Acad. Sci, Kiev (Ukraine, 1997) 61. University of Nebraska, Lincoln, Nebraska (USA, 1997) 62. Auburn University, Auburn, Alabama (USA, 1997) 63. Washington University, St. Louis (USA, 1997) 64. Wake Forest University, Winston Salem (USA, 1998) 65. Florida Institute of Technology, Melbourne (USA, 1998) 66. University of Central Florida, Orlando (USA, 1998) 67. North Carolina State University, Raleigh (USA, 1998) 68. San Diego State University, San Diego (USA, 1998) 69. University of Southern California, Los Angeles (USA, 1998) 70. University of Missori, Rolla (USA, 2000) 71. The University of Queensland, (Australia, 2000) 72. University of Delaware, (USA 2000) 73. Geogia Institute of technology, (USA, 2000) 74. Auburn University, Auburn, Alabama (USA, 2005) 75. The Hong Kong Polytechnic University, (Hong Kong, 2006) 76. The University of Hong Kong, (Hong Kong, 2006) 77. City University of Hong Kong, (Hong Kong, 2006) 78. Howard University, (USA, 2006) 79. Georgetown University, (USA, 2006) 80. Western Kentucky University, (USA, 2006) 81. Michigan Technological University, (USA, 2007) 82. University of Rome, (Italy, 2008) 83. Politechnica di Milano (Italy, 2008) 84. Seattle University (USA, 2009) 85. King Fahd University of Petroleum and Minerals (Saudi Arabia, 2009) 86. Istanbul Technical University (Turkey, 2009) 87. Middle East Technical University (Turkey, 2009) 88. Cankaya University (Turkey, 2009) 89. Osmangazi University (Turkey, 2009) 90. Izmir University (Turkey, 2009) 91. Universidade De Santiago De Compostela (Spain, 2010) 92. King Abdulaziz University (Saudi Arabia, 2010) 93. American University of Sharjah (Sharjah, 2010) 94. United Arab Emirates University (Al-Ain, 2010) 95. King Fahd University of Petroleum and Minerals (Saudi Arabia, 2010) 96. King Abdulaziz University (Saudi Arabia, 2011) 97. King Fahd University of Petroleum and Minerals (Saudi Arabia, 2011) 98. Texas AM University-Corpus Christi (USA, 2012)

20 20 International Congress in Honour of Professor Ravi P. Agarwal Contents 1 A New Class of Riemannian Manifolds Yavuz Selim Balkan 35 2 Estimating M S ARCH Models Using Recursive Method Ahmed Ghezal 36 3 Approximation by q Durrmeyer type Polynomials in Compact Disks in the Case q > 1 Nazim Idrisoğlu Mahmudov 37 4 Finite Groups Whose Intersection Graphs are Planar Selçuk Kayacan 38 5 On Solving Some Functional Equations Dmitry V.Kruchinin and Vladimir V.Kruchinin 39 6 On Computing Some Topological Indices Mohamed Amine Boutiche 40 7 History Slip-Dependent Evolutionary Quasi-Variational Inequalities with Volterra Integral Term Nouiri Brahim 41 8 Approximation Properties of Weighted Kantorovich Type Operators in a Compact Disks Mustafa Kara and Nazm I.Mahmudov 42 9 A Derivative Formula Associated with Eisenstein Series Aykut Ahmet Aygüneş General Result on Viscoelastic Wave Equation with Degenerate Laplace Operator of Kirchhoff-Type in R n Zennir Khaled Moore-Penrose Inverse and Partial Isometries Safa Menkad Multiplicity Result for the Hamiltonian System TacksunJung and Q-Heung Choi One-Parameter Apostol-Bernoulli Polynomials and Apostol-Euler Polynomials Veli Kurt On p-bernoulli Numbers Mourad Rahmani The Roots of a Dual Split Quaternion Hesna Kabadayı On Some Inequalities for Hadamard Product of Special Types of Matrices Seyda Ildan and Hasan Köse 50

21 International Congress in Honour of Professor Ravi P. Agarwal Generalized Newton Transformation and its Application to Transversal Submanifolds Abdelmalek Mohammed Multiple Positive Solutions for Elliptic Singular Systems with Hardy Sobolev Exponents Benmansour Safia A Collocation Method for Solution of the Nonlinear Lane-Emden type Equations in Terms of Generalized Bernstein Polynomials Ayşegül Akyüz-Daşcıoğlu and Neşe Işler Acar GALA and GADP2 Comparison for the Scheduling Problem of R m /S ijk /C max Duygu Yilmaz Eroğlu and H.Cenk Özmutlu Improved MIP Model for Parallel Machines Scheduling Problem Duygu Yilmaz Eroğlu and H.Cenk Özmutlu Multiple Positive Solutions for Elliptic Singular Systems with Cafarelli Kohn Niremberg Exponents Matallah Atika Positive Solution for a Singular Second-Order Discrete Three-Point Boundary Value Problem Noor Halimatus Sa diah Ismail and Mesliza Mohamed Positive Solution to Fourth Order Three-Point Boundary Value Problem M.Mohamed, M.S.M.Noorani, M.S.Jusoh, M.N.M.Fadzil and R.Saian Existence of Positive Solutions for Non-Homogeneous BVPs of p-laplacian Difference Equations Fatma Tokmak and Ilkay Yaslan Karaca L2 Norm Deconvolution Algorithm Applied to Ultrasonic Phased Array Signal Processing Abdessalem Benammar, Redouane Drai and Ahmed Khechida Multi-Soliton Solutions for Non-Integrable Equations: Asymptotic Approach Georgy A.Omel yanov On Interpolation Functions for the q-analogue of the Eulerian Numbers Associated with any Character Mustafa Alkan and Yilmaz Simsek Computation of p-values for Mixtures of Gaussians Burcin Simsek and Satish Iyengar Nonclassical Appell Polynomials Rahime Dere and Yilmaz Simsek Remarks on the Central Factorial Numbers Yilmaz Simsek 65

22 22 International Congress in Honour of Professor Ravi P. Agarwal 32 Nodals Solutions of the Fourth Order Equations Involving Paneitz-Branson Operator with Critical Sobolev Exponent Boughazi Hichem Bilinear Multipliers of Weighted Wiener Amalgam Spaces and Variable Exponent Wiener Amalgam Spaces Öznur Kulak and A.Turan Gürkanlı Global Optimization Problem of Lipschitz Functions Using α-dense Curves Djaouida Guettal and Mohamed Rahal Estimating 2-D GARCH Models by Quasi-Maximum Likelihood Soumia Kharfouchi An Approach Using Stream Ciphers Algorithm for Speech Encryption and Decryption Belmeguenai Aissa, Mansouri Khaled and Lashab Mohamed A Generalized Statistical Convergence for Sequences of Sets via Ideals Ömer Kişi and Ekrem Savaş Some Embedding Questions for Weighted Difference Spaces Leili Kussainova and Ademi Ospanova On (λ, I) Statistical Convergence of Order α of Sequences of Function Hacer Şengül and Mikail Et Range Kernel Orthogonality of Generalized Derivations Messaoudene Hadia On Stancu Variant of q-baskakov-durrmeyer Type Operators P.N.Agrawal and A.Sathish Kumar Generalised Baskakov Kantorovich Operators P.N.Agrawal and Meenu Goyal Approximate Solutions of Fractional Order Boundary Value Problems by a Novel Method Ali Akgul Some Power Series on Archimedean and Non-Archimedean Fields Fatma Çalışkan Existence and Monotone Iteration of Symmetric Positive Solutions for Integral Boundary- Value Problems with φ-laplacian Operator Tugba Senlik and Nuket Aykut Hamal Analytical Calculation of Partial Differential Equations Applied to Electrical Machines With Ideal Halbach Permanent Magnets Mourad Mordjaoui, Ibtissam Bouloukza and Dib Djalel Principal Functions of Differential Operators with Spectral Parameter in Boundary Conditions Nihal Yokuş 81

23 International Congress in Honour of Professor Ravi P. Agarwal Generalized Typically Real Functions S.Kanas and A.Tatarczak The Abel-Poisson Summability of Fourier Series in a Banach Space with Respect to a Continuous Linear Representation Seda Öztürk Existence of Solutions for Integral Boundary Value Problems in Banach Spaces Fulya Yoruk Deren and Nuket Aykut Hamal Existence and Uniqueness Solution of Electro-Elastic Antiplane Contact Problem with Friction Mohamed Dalah, Khoudir Kibeche, Amar Megrous, Ammar Derbazi and Soumia Ahmed Chaouache Almost Convex Valued Perturbation to Time Optimal Control Sweeping Processes Doria Affane and Dalila Azzam-Laouir Evolution Problem Governed by Subdifferential Operator Mustapha Yarou Nonlinear Elliptic Problem Related to the Hardy Inequality with Singular Term at the Boundary B.Abdellaoui, K.Biroud, J.Davila and F.Mahmoudi On Periodic Solutions of Nonlinear Differential Equations in Banach Spaces Abdullah Çavuş, Djavvat Khadjiev and Seda Öztürk Generalized α-ψ-contractive type M Mappings of Integral Type Erdal Karapinar, P.Shahi and Kenan Tas Caristi s Fixed Point Theorem in Fuzzy Metric Spaces Hamid Mottaghi Golshan Determination of the Unknown Coefficient in Time Fractional Parabolic Equation with Dirichlet Boundary Conditions Ebru Ozbilge and Ali Demir On p-adic Ising Model with Competing Interactions on the Cayley Tree Farrukh Mukhamedov, Hasan Akın and Mutlay Dogan A Spectral Domain Computational Technique Dedicated to Fault Detection in Induction Machine A.Medoued, A.Lebaroud, O.Boudebbouz and D.Sayad Some Results on Double Fuzzy Topogenous Orders Vildan Çetkin and Halis Aygün Finding Fixed Points of Firmly Nonexpansive-Like Mappings in Banach Spaces Fumiaki Kohsaka A Fourth Order Accurate Approximation of the First and Pure Second Derivatives of the Laplace Equation on a Rectangle A.A.Dosiyev and H.M.Sadeghi 97

24 24 International Congress in Honour of Professor Ravi P. Agarwal 64 On the Positive Solutions for the Boundary Value Problems at Resonance Ummahan Akcan and Nüket Aykut Hamal On Weighted Approximation of Multidimensional Singular Integrals Gümrah Uysal and Ertan Ibikli On Hermite-Hadamard Type Inequalities for ϕ Convex Functions via Fractional Integrals Mehmet Zeki Sarıkaya and Hatice Yaldız Behavior of Positive Solutions of a Multiplicative Difference Equation Durhasan Turgut Tollu, Yasin Yazlık and Necati Taşkara A New Generalization of the Midpoint Formula for n Time Differentiable Mappings which are Convex Çetin Yıldız and M.Emin Özdemir Global Bifurcations of Limit Cycles in the Classical Lorenz System Valery Gaiko Curvature of Curves Parameterized by a Time Scale Sibel Paşalı Atmaca and Ömer Akgüller Essential Norms of Products of Weighted Composition Operators and Differentiation Operators Between Banach Spaces of Analytic Functions Jasbir S.Manhas and Ruhan Zhao On the Null Forms, Integrating Factors and First Integrals to Path Equations Ilker Burak Giresunlu and Emrullah Yaşar Commutativity of Lommel and Halm Differential Equations Mehmet Emir Koksal Equivalence Between Some Iterations in CAT (0) Spaces Kyung Soo Kim On Certain Combinatoric Convolution Sums of Divisor Functions Daeyeoul Kim and Nazli Yildiz Ikikardes Some Properties of the Genocchi Polynomials with the Variable [x] q J.Y.Kang and C.S.Ryoo Boundedness of Localization Operators on Lorentz Mixed Normed Modulation Spaces Ayşe Sandıkçı p Summable Sequence Spaces with Inner Products Şükran Konca, Hendra Gunawan and Mochammad Idris An Alternative Proof of a Tauberian Theorem for Abel Summability Method Ibrahim Çanak and Ümit Totur Positive Periodic Solutions for a Nonlinear First Order Functional Dynamic Equation by a New Periodicity Concept on Time Scales Erbil Çetin and F.Serap Topal 114

25 International Congress in Honour of Professor Ravi P. Agarwal Potential Flow Field Around a Torus Rajai Alassar On B 1 -Convex Functions and Some Inequalities Gabil Adilov and Ilknur Yesilce On the Global Behaviour of a Higher Order Difference Equation Yasin Yazlik, D.Turgut Tollu and Necati Taskara Identifying an Unknown Time Dependent Coefficient for Quasilinear Parabolic Equations Fatma Kanca and Irem Baglan On Special Semigroup Classes and Congruences on Some Semigroup Constructions Seda Oğuz and Eylem Güzel Karpuz The Rate of Pointwise Convergence of q Szasz Operators Tuncer Acar Some Properties of Cohomology Groups for Graphs Özgür Ege and Ismet Karaca Stability with Respect to Initial Time Difference for Generalized Delay Differential Equations Ravi Agarwal and Snezhana Hristova On Ramanujan s Summation Formula, his General Theta Function and a Generalization of the Borweins Cubic Theta Functions Chandrashekar Adiga L Error Estimate of Parabolic Variational Inequality Arising of the Pricing of American Option S.Madi, M.Hariour and M.C.Bouras The Smoothness of Convolutions of Zonal Measures on Compact Symmetric Spaces Sanjiv Kumar Gupta and Kathryn Hare A Tauberian Theorem for the Weighted Mean Method of Summability of Sequences of Fuzzy Numbers Zerrin Önder, Sefa Anıl Sezer and Ibrahim Çanak Asymptotic Constancy for a System of Impulsive Delay Differential Equations Fatma Karakoç and Hüseyin Bereketoğlu An Extension w with rankw = 3 of a Valuation v on a Field K with rankv = 2 to K(x) Figen Öke Inclusions Between Weighted Orlicz Space Alen Osançlıol On the Some Graph Parameters for Special Graphs Nihat Akgüneş, Ahmet Sinan Çevik and Ismail Naci Cangül 130

26 26 International Congress in Honour of Professor Ravi P. Agarwal 97 A Note on the Dirichlet-Neumann First Eigenvalue of a Family of Polygonal Domains in R 2 A.R.Aithal and Acushla Sarswat An Approach to the Numerical Verification of Solutions for Variational Inequalities C.S.Ryoo Local Rings and Projective Coordinate Spaces Fatma Özen Erdoğan and Süleyman Çiftçi An Improved Numerical Solution of the Singular Boundary Integral Equation of the Compressible Fluid Flow Around Obstacles Using Modified Shape Functions Luminita Grecu New Aspects of Calculating Volumes in E n Daniela Bittnerová and Daniela Bímová Applications of an Alternative Methods for Volumes of Solids of Revolution Daniela Bímová and Daniela Bittnerová On Certain Sums of Fibonomial Coefficients Emrah Kılıç and Aynur Yalçıner Null Generalized Helices of a Null Frenet Curve in L 4 Esen Iyigün Geometrical Methods and Numerical Computations for Prey-Predator Lotka-Volterra Systems Adela Ionescu, Romulus Militaru and Florian Munteanu Fractional Calculus Model of Dengue Epidemic Moustafa El-Shahed Zagreb Co Indices and Augmented Zagreb Index and its Polynomials of Phenylene and Hexagonal Squeeze P.S.Ranjini, V.Lokesha and Usha.A A Note on Class Numbers of Real Quadratic Fields with Certain Fundamental Discriminants Ayten Pekin On Three Dimensional Dynamical Systems on Time Scales Elvan Akın On the Difference Equation System x n+1 = 1+yn y n, y n+1 = 1+yn x n Necati Taskara, Durhasan Turgut Tollu and Yasin Yazlik The Binomial Transforms of Tribonacci and Tribonacci-Lucas Sequences Nazmiye Yilmaz and Necati Taskara On the Random Functional Central Limit Theorems with Almost Sure Convergence for Subsequences Zdzislaw Rychlik 146

27 International Congress in Honour of Professor Ravi P. Agarwal Some Fixed Point Theorems for a Pair of Mappings in Complex Valued b-metric Spaces Aiman Mukheimer Some Characterizations of Slant Curves on Unit Dual Sphere S 2 Seda Oral and Mustafa Kazaz On Solving Some Partial Differential Equations Ümit Sarp and Sebahattin Ikikardes Some Spectrum Properties in C - Algebras Nilay Sager and Hakan Avcı On Function Spaces with Fractional Fourier Transform in the Weighted Lebesgue Spaces Erdem Toksoy and Ayşe Sandıkçı Some Convergence Results for Modified SP-Iteration Scheme in Hyperbolic Spaces Aynur Şahin and Metin Başarır Characterization of W p type of Spaces Involving Fractional Fourier Transform S.K.Upadhyay and Anuj Kumar Rates of Convergence for an Estimator of a Density Function Based on Hermite Polynomials Elif Erçelik and Mustafa Nadar Estimation of Reliability in Multicomponent Stress-Strength Model Based on Marshall Olkin Weibull Distribution Mustafa Nadar and Fatih Kızılaslan Some New Results on The Π Regularity of Some Monoids Ahmet Emin and Fırat Ateş On Traveling Wave Solutions of Fractional Differential Equations Şerife Müge Ege and Emine Mısırlı On the Oscillation of Second Order Nonlinear Neutral Dynamic Equations on Time Scales Elvan Akın, Can Murat Dikmen and Said Grace A Collocation Approach to Parabolic Partial Differential Equations Kubra Erdem Biçer and Salih Yalçınbaş From Simplicial Homotopy to Crossed Module Homotopy I.Ilker Akça and Kadir Emir On Algebraic Semigroup and Graph-Theoretic Properties of a New Graph Ahmet Sinan Çevik, Eylem Güzel Karpuz and I.Naci Cangül Embeddability and Gröbner-Shirshov Basis Theory Eylem Güzel Karpuz 162

28 28 International Congress in Honour of Professor Ravi P. Agarwal 129 An Application of Fixed Point Theorems to a Problem for the Existence of Solutions of a Nonlinear Ordinary Differential Equations of Fractional Order Masashi Toyoda A Numerical Solution for Vibrations of an Axially Moving Beam Duygu Dönmez Demir and Erhan Koca Some Principal Congruence Subgroups of the Extended Hecke Groups and Relations with Pell-Lucas Numbers Zehra Sarıgedik, Sebahattin Ikikardeş and Recep Şahin On the Metric Geometry and Regular Polyhedrons Temel Ermiş and Rüstem Kaya On the Addition of Collinear Points in Some PK-Planes Basri Celik and Abdurrahman Dayioglu Local Stability Analysis of Strogatz Model with Two Delays Sertaç Erman and Ali Demir Weighted Statistical Convergence in Intuitionistic Fuzzy Normed Spaces Selma Altundağ and Esra Kamber Sturm Comparison Theorems for Some Elliptic Type Equations with Damping and External Forcing Terms Sinem Şahiner, Emine Mısırlı and Aydın Tiryaki A Note on Solutions of the Nonlinear Fractional Differential Equations via the Extended Trial Equation Method Meryem Odabasi and Emine Misirli On Quantum Codes Obtained From Cyclic Codes Over F 2 + uf 2 + u 2 F u m F 2 Abdullah Dertli, Yasemin Cengellenmiş and Şenol Eren On Some Functions Mapping the Zeros of L n (x) to the Zeros of L n(x) Nihal Yılmaz Özgür and Öznur Öztunç Finite Blaschke Products and R-Bonacci Polynomials Nihal Yılmaz Özgür, Öznur Öztunç and Sümeyra Uçar Convergence of Nonlinear Singular Integral Operators to the Borel Differentiable Functions Harun Karsli and Ismail U.Tiryaki Regularization of an Abstract Class of Ill-Posed Problems Djezzar Salah and Benmerai Romaissa Decompositions of Soft Continuity Ahu Açıkgöz and Nihal Taş Lacunary Statistical Convergence of Double Sequences in Topological Groups Ekrem Savaş 178

29 International Congress in Honour of Professor Ravi P. Agarwal On Fuzzy Pseudometric Spaces Elif Aydın and Servet Kütükçü On Fixed Points of Extended Hecke Groups Bilal Demir and Özden Koruoğlu New Lagrangian Forms of Modified Emden Equation by Jacobi Method Gülden Gün Polat and Teoman Özer Fixed Point Theorems for ψ-contractive Mappings on Modular Space Ekber Girgin and Mahpeyker Öztürk Convexity and Schur Convexity on New Means V.Lokesha, U.K.Misra and Sandeep Kumar On Radial Signed Graphs Gurunath Rao Vaidya, P.S.K.Reddy and V.Lokesha Delta and Nabla Discrete Fractional Grüss Type Inequality A.Feza Güvenilir On Tame Extensions and Residual Transcendental Extensions of a Valuation with rankv = n Burcu Öztürk and Figen Öke Time Series Forecasting with Grey Modelling Seval Ene and Nursel Öztürk Periodic Solution of Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response and Impulses Ayşe Feza Güvenilir, Billur Kaymakçalan and Neslihan Nesliye Pelen Approximation Properties of Kantorovich-Stancu Type Generalization of q-bernstein- Schurer-Chlodowsky Operators on Unbounded Domain Tuba Vedi and Mehmet Ali Özarslan Use of Golden Section in Music Sümeyye Bakım On Analysis of Mathews-Lakshmanan Oscillator Equation via Nonlocal Transformation and Lagrangian-Hamiltonian Description Özlem Orhan and Teoman Özer On Singularities of the Galilean Spherical Darboux Ruled Surface of a Space Curve in the Pseudo-Galilean Space G 1 3 Tevfik Şahin and Murteza Yılmaz Existence of Positive Solutions for Second Order Semipositone Boundary Value Problems on the Half-Line F.Serap Topal and Gülşah Yeni Some Congruent Number Families Refik Keskin and Ümmügülsüm Öğüt 194

30 30 International Congress in Honour of Professor Ravi P. Agarwal 161 On Some Fourth-Order Diophantine Equations Merve Güney Duman and Refik Keskin Characteristic Subspaces of Finite Rank Operators Mohamed Najib Ellouze Fixed Point Theory in WC-Banach Algebras Bilel Mefteh Oscillation and Nonoscillation Criteria for Second Order Generalized Difference Equations Yaşar Bolat On Generalizations of Some Inequalities Containing Diamond-Alpha Integrals and Applications Billur Kaymakçalan On Reciprocity Law of the Y (h, k) Sums Associated with PDE s of the Three-Term Polynomial Relations Elif Cetin, Yilmaz Simsek and Ismail Naci Cangul Permutation Method for a Class of Singularly Perturbed Discrete Systems with Time-Delay Tahia Zerizer Existence of Minimal and Maximal Solutions for Quasilinear Elliptic Equation with Nonlocal Boundary Conditions on Time-Scales Mohammed Derhab and Mohammed Nehari Application of Filled Function Method in Chemical Control of Pests Ahmet Şahiner, Meryem Öztop, Gülden Kapusuz and Ozan Demirözer A New Approach to the Filled Function Method for Non-smooth Problems Nurullah Yilmaz and Ahmet Sahiner Determining of the Achievement of Students by Using Classical and Modern Optimization Techniques Ahmet Şahiner and Raziye Akbay Fuzzy Logic Approach to an UH-1 Helicopter Fuel Consumption and Calculation of Power Problem Ahmet Şahiner and Reyhane Ercan Determination of Effects of Brassinosteroid Applications on Secondary Metabolite Accumulation in Salt Stressed Peppermint (Mentha piperita L.) by Modern Optimization Tecniques Ahmet Sahiner, Tuba Yigit, Ozkan Coban and Nilgun Gokturk Baydar On a Completeness Property of C(X) Equipped with a Set-Open Topology Smail Kelaiaia Existence of Solutions of a Class of Second Order Differential Inclusions D.Azzam-Laouir and F.Aliouane 209

31 International Congress in Honour of Professor Ravi P. Agarwal Applications of Generalized Fibonacci Autocorrelation Sequences {Γ k,n (τ)} τ Sibel Koparal and Neşe Ömür The Computer Simulation of Nuclear Magnetic Resonance Hyperfine Structure Constant for AB 2, A 2 B 2 and A 2 B 3 Systems Containing Some Organic Molecules with Spin 1 2 Using Jacobi Programme Hüseyin Ovalıoğlu, Adnan Kılıç and Handan Engin Kırımlı The Computer Simulation of Nuclear Magnetic Resonance Hyperfine Structure Constant for ANX, ABC and A 3 BC Systems Containing Some Organic Molecules with Spin 1 2 Using Jacobi Programme Hüseyin Ovalıoğlu, Handan E.Kırımlı, Cengiz Akay and Adnan Kılıç Necessary and Sufficient Conditions for First Order Differential Operators to be Associated with a Disturbed Dirac Operator in Quaternionic Analysis Uğur Yüksel Theoretical Investigation of Substituent Effect on the Carbonyl Stretching Vibration Ilhan Küçük and Aslı Ayten Kaya Modeling of the Optical Properties of the CdS Thin Films by Using Artificial Neural Network Aslı Ayten Kaya, Kadir Ertürk, Nil Küçük and Ilker Küçük Nonprinciple Solutions and Extensions of Wong s Oscillation Criteria to Forced Second-Order Impulsive and Delay Differential Equations Abdullah Özbekler and Ağacık Zafer Modeling of Exposure Buildup Factors for Concrete Shielding Materials up to 10 mfp Using Generalized Feed-Forward Neural Network Nil Kucuk, Vishwanath P.Singh and N.M.Badiger Calculation of Gamma-Ray Exposure Buildup Factors for Some Biological Samples Nil Kucuk, Vishwanath P.Singh and N.M.Badiger Determination of Thermoluminescence Kinetic Parameters of ZnB 2 O 4 : La Phosphors Nil Kucuk, A.Halit Gozel, Mustafa Topaksu and Mehmet Yüksel Improved Numerical Radius and Spectral Radius Inequalities for Operators Fuad Kittaneh and Amer Abu-Omar n-dimensional Sobolev type spaces involving Chebli-Trimeche Transform Mourad Jelassi A Fixed Point Theorem for Multivalued Mappings with δ-distance on Complete Metric Space Özlem Acar and Ishak Altun Existence of Solutions of α (2, 3] Order Fractional Three Point Boundary Value Problems with Integral Conditions Sinem Unul and N.I.Mahmudov Vector-Valued Variable Exponent Amalgam Spaces Ismail Aydın 224

32 32 International Congress in Honour of Professor Ravi P. Agarwal 191 Soliton Solutions of Sawada Kotera Equation by Hirota Method Esra Karataş and Mustafa Inc Certain Quasi-Cyclic Codes which are Hadamard Codes Mustafa Özkan and Figen Öke Pointwise Convergence of Derivatives of New Baskakov-Durrmeyer-Kantorovich Type Operators Gulsum Ulusoy, Ali Aral and Emre Deniz On the High Order Lipschitz Stability of Inverse Nodal Problem for String Equation Emrah Yılmaz and Hikmet Koyunbakan Positive Solutions of a Boundary Value Problem with Derivatives in the Nonlinear Term Patricia J.Y.Wong One Step Iteration Scheme for Two Multivalued Mappings in CAT(0) Spaces Izhar Uddin and M.Imdad A Variant Akaike Information Criterion for Mixture Autoregressive Model Selection Fayçal Hamdi Zagreb Polynomials of Three graph Operators A.R.Bindusree, V.Lokesha, I.Naci Cangul and P.S.Ranjini A Note on the Moment Estimate for Stochastic Functional Differential Equations Young-Ho Kim Issues Optimization of Public Administration Canybec Sulayman and Gulnar Suleymanova Jacobi Orthogonal Approximation with Negative Integer and its Application Zhang Xiao-yong and Wan Zheng-su Existence Results for Nonlinear Impulsive Fractional Differential Equations with p Laplacian Operator Ilkay Yaslan Karaca and Fatma Tokmak A Relation Between the Lefschetz Fixed Point Theorem and the Nielsen Fixed Point Theorem in Digital Images Ismet Karaca Second Order Nonlinear Boundary Value Problems with Integral Boundary Conditions on Time Scales F.Serap Topal and Arzu Denk Oguz Existence of a Solution of Integral Equations via Fixed Point Theorem Selma Gülyaz Triangular and Square Triangular Numbers Arzu Özkoç 240

33 International Congress in Honour of Professor Ravi P. Agarwal Approximation Methods on a Complete Geodesic Space Yasunori Kimura Fixed Point Results for α-admissible Multivalued F Contractions Gonca Durmaz and Ishak Altun Advances on Fixed Point Theory Erdal Karapınar Fixed Point Theorems for a Class of α-admissible Contractions and Applications to Boundary Value Problem Inci M.Erhan Feng-Liu Type Fixed Point Theorems for Multivalued Mappings Gülhan Mınak and Ishak Altun Qualitative Analysis for the Differential Equation Associated to the Dynamic Model for an Access Control Structure Daniela Coman, Adela Ionescu and Sonia Degeratu Zagreb Indices of Double Graphs Aysun Yurttas, Muge Togan and Ismail Naci Cangul Several Zagreb Indices of Subdivision Graphs of Double Graphs Muge Togan, Aysun Yurttas and Ismail Naci Cangul On the Solutions of the Diophantine Equation x n + p y n = p 2 z n Caner Ağaoğlu and Musa Demirci A Weak Contraction Principle in Partially Ordered Cone Metric Space with Three Control Functions Binayak S.Choudhury, L.Kumar, T.Som and N.Metiya On the Diophantine Equation (20n) x + (99n) y = (101n) z Gokhan Soydan, Musa Demirci and Ismail Naci Cangul Halpern Type Iteration with Multiple Anchor Points in a Hadamard Space Yasunori Kimura and Hideyuki Wada Multimaps in Fixed Point Theorems in Terms of Measure of Noncompactness Mehdi Asadi Pointwise Approximation in Lp Space by Double Singular Integral Operators Mine Menekşe Yılmaz, Gümrah Uysal and Ertan Ibikli Some Tauberian Remainder Theorems for Iterations of Weighted Mean Methods of Summability Sefa Anıl Sezer and Ibrahim Çanak On The Semi-Fredholm Spectrum Arzu Akgül Critical Fixed Point Theorems in Banach Algebras Under Weak Topology Features A.Ben Amar and A.Tlili 257

34 34 International Congress in Honour of Professor Ravi P. Agarwal 224 Modeling of Effect of the Components of Distance Education in Achievement of Students Hamit Armagan, Tuncay Yigit and Ahmet Sahiner On the Weighted Integral Inequalities for Convex Function Mehmet Zeki Sarıkaya and Samet Erden 259

35 International Congress in Honour of Professor Ravi P. Agarwal 35 1 A New Class of Riemannian Manifolds Yavuz Selim Balkan In this study, we introduce a new class of (2n + 1) dimensional Riemannian manifolds. Such type manifolds are called almost contact metric manifolds which have ϕ recurrent τ curvature tensor. We investigate some curvature properties of this type manifold. We obtain that these manifolds are η Einstein manifolds under some algebraic conditions. [1] De U. C., Yıldız A. and Yalınız A. F., On ϕ recurrent Kenmotsu manifolds, Turkish J. Math. 32 (2008), [2] De U. C. and Guha N., On generalized recurrent manifolds, J. Nat. Acad. Math. India 9 (1991), [3] Tripathi M. M. and Gupta P., τ curvature tensor on a semi-riemannian manifold, J. Adv. Math. Stud., 4 (2011), no. 1, Duzce University, Faculty of Arts and Sciences, Department of Mathematics, Düzce, Turkiye, [email protected]

36 36 International Congress in Honour of Professor Ravi P. Agarwal 2 Estimating M S ARCH Models Using Recursive Method Ahmed Ghezal In this note we offer model more realistically the variability of financial time series. Markov-switching autoregressive conditional heteroskedasticity (M S ARCH) model introduced by Cai that incorporates the features of both Hamilton and Engle ARCH model to study the matter of volatility persistence in the monthly excess revenues. The matter can be resolved by taking into account occasional transformations in the asymptotic variance of the MS ARCH process that cause the Pseudomonas persistence of the volatility process. One of the interesting issues of financial time series volatility relates to the persistence of shocks to the variance. A common finding using high-frequency financial data concerns the apparent persistence implied by the estimates for the conditional variance functions. In these models, the parameters are allowed to depend on an unobservable time-homogeneous and stationary Markov chain with finite state space. The statistical inference for these models is rather difficult due to the dependence to the whole regime path. We propose a recursive algorithm for parameter estimation in M S ARCH. The proposed method which is useful for long time series as well as for data available in real time. The main idea is to use the maximum likelihood estimation (M LE) method and from this develop a recursive Expectation-Maximization (EM) algorithm. [1] A. Aknouche, Recursive online EM estimation of mixture autoregressions. Journal of Statistical Computation and Simulation, Vol. 83, No. 2 (2013), , [2] J. Cai, A Markov model of switching regime ARCH. J. Bus. Econ. Stat. 12 (1994), , [3] O. Cappé, E. Moulines, Online expectation maximization algorithm for latent data models, J. R. Statist. Soc. B 71 (2009), , [4] I. Conllings, V. Krishnamurthy, J. B. Moore, Online identification of hidden Markov models via recursive prediction error techniques, IEEE Trans. Signal Process. 42 (1994), , [5] J. D. Hamilton, R. Susmel, Autoregressive conditional heteroskedasticity and changes in regime. J. Econ 64 (1994), , [6] U. Holst, G. Lindgren, Recursive estimation in mixture models with Markov regime, IEEE Trans. Inform. Theory 37 (1991), , [7] U. Holst, G. Lindgren, J. Holst, M. Thuvesholmen, Recursive estimation in switching autoregressions with a Markov regime, J. Time Ser. Anal. 15 (1994), , University of Constantine 1, Algeria, Faculty of Science, Department of Mathematics, [email protected]

37 International Congress in Honour of Professor Ravi P. Agarwal 37 3 Approximation by q Durrmeyer type Polynomials in Compact Disks in the Case q > 1 Nazim Idrisoğlu Mahmudov In this talk, we discuss approximation properties of the complex q-durrmeyer type operators in the case q > 1. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex q-durrmeyer type polynomials attached to analytic functions in compact disks will be given. In particular, we show that for functions analytic in {z C : z < R}, R > q, the rate of approximation by the q-durrmeyer type polynomials (q > 1) is of order q n versus 1/n for the classical (q = 1) Durrmeyer type polynomials. Explicit formulas of Voronovskaya type for the q-durrmeyer type operators for q > 1 are also given. [1] R. P Agarwal and V. Gupta, On q-analogue of a complex summation-integral type operators in compact disks, Journal of Inequalities and Applications, 2012, 2012:111. [2] G. A. Anastassiou and S.G. Gal, Approximation by Complex Bernstein-Durrmeyer Polynomials in Compact Disks, Mediterr. J. Math., 7 (2010), No. 4, [3] Andrews G E, Askey R, Roy R. Special functions. Cambridge: Cambridge University Press; [4] A. Aral, V. Gupta, R.P. Agarwal, Applications of q-calculus in Operator Theory, Springer, New York, [5] M-M. Derriennic, Modified Bernstein polynomials and Jacobi polynomials in q-calculus, Rendiconti Del Circolo Matematico Di Palermo, Serie II 2005; 76(Suppl.): [6] S. G. Gal, Overconvergence in Complex Approximation, Springer New York Heidelberg Dordrecht London, [7] S. G. Gal, Approximation by complex genuine Durrmeyer type polynomials in compact disks, Appl. Math. Comput., 217(2010), [8] S. G. Gal, V. Gupta, N. I. Mahmudov, Approximation by a complex q-durrmeyer type operator, Ann Univ Ferrara, (2012) 58: [9] N. I. Mahmudov, Approximation by Genuine q-bernstein-durrmeyer Polynomials in Compact Disks in the case q > 1. Abstract and Applied Analysis. [10] N. I. Mahmudov, Approximation properties of complex q-szász-mirakjan operators in compact disks. Comput. Math. Appl. 60, (2010) [11] N. I. Mahmudov, Approximation by genuine q -Bernstein-Durrmeyer polynomials in compact disks, Hacettepe Journal of Mathematics and Statistics, 40 (1) (2011), [12] N. I. Mahmudov, Approximation by Bernstein Durrmeyer-type operators in compact disks, Applied Mathematics Letters, 24 (7) (2011), [13] S. Ostrovska, q-bernstein polynomials and their iterates,. J. Approx. Theory, 123 (2003), Eastern Mediterranean University, Department of Mathematics, Gazimagusa, TRNC, Mersin 10, Turkey, nazim. [email protected]

38 38 International Congress in Honour of Professor Ravi P. Agarwal 4 Finite Groups Whose Intersection Graphs are Planar Selçuk Kayacan The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if H K 1 where 1 denotes the trivial subgroup of G. In this talk we characterize all finite groups whose intersection graphs are planar. Our methods are elementary. Among the graphs similar to the intersection graphs, we may count the subgroup lattice and the subgroup graph of a group, each of whose planarity was already considered before in [2, 10, 11, 12, 13]. [1] Alonso, J.: Groups of order pq m with elementary abelian Sylow q-subgroups. Proc. Amer. Math. Soc. 65(1), (1977) [2] Bohanon, J.P., Reid, L.: Finite groups with planar subgroup lattices. J. Algebraic Combin. 23(3), (2006) [3] Burnside, W.: Theory of groups of finite order. Dover Publications Inc., New York (1955). 2d ed [4] Cole, F.N., Glover, J.W.: On Groups Whose Orders are Products of Three Prime Factors. Amer. J. Math. 15(3), (1893) [5] Gorenstein, D.: Finite groups, second edn. Chelsea Publishing Co., New York (1980) [6] Hölder, O.: Die Gruppen der Ordnungen p 3, pq 2, pqr, p 4. Math. Ann. 43(2-3), (1893) [7] Le Vavasseur, R.: Les groupes d ordre p 2 q 2, p étant un nombre premier plus grand que le nombre premier q. Ann. Sci. École Norm. Sup. (3) 19, (1902) [8] Miller, G.A.: Groups having a small number of subgroups. Proc. Nat. Acad. Sci. U. S. A. 25, (1939) [9] Rotman, J.J.: An introduction to the theory of groups, Graduate Texts in Mathematics, vol. 148, fourth edn. Springer-Verlag, New York (1995) [10] Schmidt, R.: On the occurrence of the complete graph K 5 in the Hasse graph of a finite group. Rend. Sem. Mat. Univ. Padova 115, (2006) [11] Schmidt, R.: Planar subgroup lattices. Algebra Universalis 55(1), 3 12 (2006) [12] Starr, C.L., Turner III, G.E.: Planar groups. J. Algebraic Combin. 19(3), (2004) [13] Yaraneri, E.: Intersection graph of a module. J. Algebra Appl. 12(5), 1250,218, 30 pp. (2013) Department of Mathematics, Istanbul Technical University, Maslak, Istanbul, Turkey, [email protected]

39 International Congress in Honour of Professor Ravi P. Agarwal 39 5 On Solving Some Functional Equations Dmitry V.Kruchinin and Vladimir V.Kruchinin In this talk, we discuss some methods for solving functional equations based on generating functions. Particularly, using the notion of the composita and Lagrange inversion theorem, we present techniques for solving the following functional equation A(x) = G(x A(x) α ), where A(x), G(x) are generating functions with G(0) 0, and α is any real number. Also we give some examples. [1] D. V. Kruchinin and V. V. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, J. Math. Anal. Appl., 404 (2013), , [2] D. V. Kruchinin and V. V. Kruchinin, A method for obtaining generating functions for central coefficients of triangles, Journal of Integer Sequences., textbf15(12.9.3) (2012), 1 10, [3] D. V. Kruchinin and V. V. Kruchinin, Explicit formulas for some generalized polynomials, Applied Mathematics and Information Sciences, 7(5) (2013), , [4] R. P. Stanley, Enumerative Combinatorics 2, vol. 62 of Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Cambridge, 1999, [5] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1989, [6] G. P. Egorichev, Integral Representation and the Computation of Combinatorial Sums, Amer. Math. Soc. 1984, [7] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Dmitry V. Kruchinin: Department of Complex Information Security, Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia, [email protected] Vladimir V. Kruchinin: Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia, kru@ie. tusur.ru The reported study was partially supported by the Ministry of education and science of Russia, government order No 1220 Theoretical bases of designing informational safe systems.

40 40 International Congress in Honour of Professor Ravi P. Agarwal 6 On Computing Some Topological Indices Mohamed Amine Boutiche The Wiener index of a graph G = (V, E) defined as W (G) = u,v V (G) d G (u, v) where d G (u, v) is a distance between two vertices u, v V (G) (the minimum number of edges on a path in G between u and v), was introduced by Harold Wiener in In this talk, we show how to compute some of well-known topological indices; the Wiener and the Wiener Polarity Index for Sun Graphs. [1] H. Deng, On the extremal Wiener polarity index of chemical trees, MATCH Commun. Math. Comput. Chem. (in press). [2] H. Deng, H. Xiao, F. Tang, On the extremal Wiener polarity index of trees with a given diameter, MATCH Commun. Math. Comput. Chem. 63 (2010) [3] H. Deng, H. Xiao, The maximum Wiener polarity index of trees with k pendants, Appl. Math. Lett. 23 (2010) [4] W. Du, X. Li, Y. Shi, Algorithms and extremal problem on Wiener polarity index, MATCH Commun. Math. Comput. Chem. 62 (2009) [5] I. Gutman, E. Estrada, Topological indices based on the line graph of the molecular graph, J. Chem. Inf. Comput. Sci. 36 (1996) [6] I. Gutman, L. Popovic, B.K. Mishra, M. Kaunar, E. Estrada, N. Guevara, Application of line graphs in physical chemistry. Predicting surface tension of alkanes, J. Serb. Chem. Soc. 62 (1997) [7] H. Hosoya, Mathematical and chemical analysis of Wiener s polarity number, in: D.H. Rouvray, R.B. King (Eds.), Topology in Chemistry-Discrete Mathematics of Molecules, Horwood, Chichester, 2002, p. 57. [8] B. Liu, H. Hou and Y. Huang, On the Wiener polarity index of trees with maximum degree or given number of leaves, Comput. Math. Appl. 60 (2010) [9] I. Lukovits, W. Linert, Polarity-numbers of cycle-containing structures, J. Chem. Inf. Comput. Sci. 38 (1998) [10] N. Trinajstic, Chemical Graph Theory, CRC Press, Boca Raton, FL, [11] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) Université des sciences et de la Technologie Houari Boumediene, BP 32, El Alia 16111, Bab Ezzouar, Faculty of Mathematics, Department of Operations research, Algiers, Algeria, [email protected] or [email protected] This work was supported by Scientific Research Project CNEPRU (OTMADS 3I), Project number B

41 International Congress in Honour of Professor Ravi P. Agarwal 41 7 History Slip-Dependent Evolutionary Quasi-Variational Inequalities with Volterra Integral Term Nouiri Brahim In this talk, we present and analyze a class of history slip-dependent evolutionary quasi-variational inequalities with Volterra integal term. We prove the existence and uniqueness result, by using arguments of evolutionary variational inequalities with voscosity and Banach s fixed-point theorem. Next, we study the dependence of the solution on the long-term memory and derive a convergence result. Finally, we present a number of concrete examples of frictional contact problems for which our results apply. [1] N. Brahim. Étude théorique et numérique des phénomènes vibratoires lies au frottement sec des solides déformables. Thèse de Doctorat en Sciences, Université Hadj Lakhdar de Batna, Algérie, [2] M. Sofonea and A. Matei. Variational inequalities with applications, A Study of Antiplane Frictional Contact Problems. Springer: New York, [3] M. Sofonea and A. Rodriguez-Arós and J. M. Viaño. A Class of Integro-Differential Variational Inequalities with Applications to Viscoelastic Contact. Mathematical and Computer Modelling, 41: , Laboratory of Computer Science and Mathematics, University of Laghouat, Raod of Ghardaïa, BP 37G, Laghouat (03000), Algeria, [email protected] This work was supported by Thematic Agency of Rresearch in Sciences and Technology (ATRST) within the framework of the national projects of researches: 8-Sciences fundamental.

42 42 International Congress in Honour of Professor Ravi P. Agarwal 8 Approximation Properties of Weighted Kantorovich Type Operators in a Compact Disks Mustafa Kara and Nazm I.Mahmudov In this talk, we discuss approximation properties of the complex weighted Kantorovich Type operators. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex weighted Kantorovich polynomials attached to analytic functions in compact disks will be given. In particular, we show that for functions analytic in {z C : z < R}, the rate of approximation by the weighted complex Kantorovich type operators is 1/n. [1] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, [2] B. Della Vecchia, G. Mastroianni and J. Szabados, A weighted generalization of the classical Kantorovich operator, Rend. Circ. Mat. Palermo (2), 82 (2010), 1-27 [3] B. Della Vecchia, G. Mastroianni and J. Szabados, A weighted generalization of the classical Kantorovich operator.ii Saturation, Mediter. J. Math., to appear. [4] D. S. YU, Weighted approximation by modified Kantorovich-Bernstein operators, Acta Math. hungar., 141 (1-2) (2013), [5] Bernstein, S. N., Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités, Commun. Soc. Math. Kharkow, 13. No. 2, 1-2, ( ). [6] S.G. Gal, Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, Vol. 8, World Scientific Publishing Co, Eastern Mediterranean University, Department of Mathematics, Gazimagusa, TRNC, Mersin 10, Turkey, [email protected]

43 International Congress in Honour of Professor Ravi P. Agarwal 43 9 A Derivative Formula Associated with Eisenstein Series Aykut Ahmet Aygüneş In this talk, we construct a new formula which derives the modular functions of weight 8k + 12 by using the modular functions of weight 4k + 4. Then we substitute Eisenstein series into our formula and we obtain some results. Also we investigate some properties of operators related to our derivative formula. [1] T. M. Apostol, Modular functions and Dirichlet series in Number Theory, (Berlin, Heidelberg and New York) Springer-Verlag (1976). [2] A. A. Aygunes, A new operator related to generating modular forms and their applications, preprint. [3] A. A. Aygunes, Derivative formulae for modular forms and their properties, preprint. [4] A. A. Aygunes, Y. Simsek, H. M. Srivastava, A sequence of modular forms associated with higher order derivative Weierstrass-type functions, preprint. [5] E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht in Göttingen, [6] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York (1993). [7] S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), [8] A. Sebbar, A. Sebbar, Eisenstein Series and Modular Differential Equations, Canad. Math. Bull. 55 (2011), [9] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, New York, Heidelberg and Berlin, Springer-Verlag (1994). Department of Mathematics, Faculty of Art and Science University of Akdeniz TR Antalya, Turkey, [email protected] The authors are supported by the research fund of Akdeniz University.

44 44 International Congress in Honour of Professor Ravi P. Agarwal 10 General Result on Viscoelastic Wave Equation with Degenerate Laplace Operator of Kirchhoff-Type in R n Zennir Khaled We shall give general energy decay of solutions to viscoelastic wave equations of p Laplacian in Kichhoff type. In order to compensate the lack of Poincare s inequality in R n and for wider class of relaxation functions, we are going to use weighted spaces. [1] M. Abdelli and A. Benaissa, Energy decay of solutions of degenerate Kirchhoff equation with a weak nonlinear dissipation, Nonlinear Analysis, 69 (2008), [2] Alabau-Boussouira, F. and Cannarsa, P., A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, Ser. I 347, (2009). [3] Arnold, V. I., Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, [4] A. Benaissa, S. Mokeddem, Global existence and energy decay of solutions to the Cauchy problem for a wave equation with a weakly nonlinear dissipation, Abstr. Appl. Anal, 11(2004) [5] Brown, K.J.; Stavrakakis, N. M, Global bifurcation results for semilinear elliptic equations on all of R n, Duke Math Journ, 85 (1996), 77?94. [6] M.M. Cavalcanti, H.P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42(4)(2003) [7] Irena Lasiecka, Salim A. Messaoudi, and Muhammad I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, JOURNAL OF MATHEMATICAL PHYSICS 54, (2013). [8] M. Kafini, uniforme decay of solutions to Cauchy viscoelastic problems with density, Elec. J. Diff. Equ Vol.2011 (2011)No. 93, pp [9] M. Kafini and S. A. Messaoudi, On the uniform decay in viscoelastic problem in R n, Applied Mathematics and Computation 215 (2009) [10] M. Kafini, S. A. Messaoudi and Nasser-eddine Tatar, Decay rate of solutions for a Cauchy viscoelastic evolution equation, Indagationes Mathematicae 22 (2011) [11] G. Kirchhoff, Vorlesungen uber Mechanik,3rd ed., Teubner, Leipzig, (1983). [12] karachalios, N.I; Stavrakakis, N.M, Existence of global attractor for semilinear dissipative wave equations on R n, J. Diff. Equ 157 (1999) [13] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optimal. Calc. Var. 4(1999) [14] Muhammad I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, JOURNAL OF MATHEMATICAL PHYSICS 53, (2012). [15] J. E. Munoz Rivera, Global solution on a quasilinear wave equation with memory, Boll. Un. Mat. Ital. B (7) 8 (1994), no. 2, University 20 Aout Skikda, Algeria, [email protected]

45 International Congress in Honour of Professor Ravi P. Agarwal Moore-Penrose Inverse and Partial Isometries Safa Menkad In this talk, we shall give a characterization of the class of all normal partial isometries, using a version of Corach-Porta-Recht inequality for Moore-Penrose invertible operators. [1] G. Corach, R. Porta, and L. Recht, An operator inequality, Linear Algebra Appl. 142(1990), , [2] R. Hart and M. Mbekhta, on generalized inverse in C -algebra, Studia mathematica, 103 (1992), [3] S. Menkad and A Seddik, operator inequalities and normal operators, Banach J. Math. Ana.,6 (2012), Cremona, Algorithms for Modular Elliptic Curves, 2nd Edition, Cambridge Univ. Press, Cambridge, 1997, [4] A. Seddik, Some results related to Corach-Porta-Recht inequality, Proc. Amer. Math. Soc. 129(2001), Department of Mathematics, Faculty of Science, Hadj Lakhdar University, Batna, Algeria, menkad [email protected]

46 46 International Congress in Honour of Professor Ravi P. Agarwal 12 Multiplicity Result for the Hamiltonian System TacksunJung and Q-Heung Choi We get a theorem which shows the multiple weak solutions for the bifurcation problem of the superquadratic nonlinear Hamiltonian system. We obtain this result by using the variational method, the critical point theory in terms of the S 1 -invariant functions and the S 1 -invariant linear subspaces. [1] K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser, (1993), [2] M. Degiovanni, L. Olian Fannio, Multiple periodic solutions of asymptotically linear Hamiltonian systems, Quaderni Sem. Mat.Brescia 8/93, (1993), [3] T. Jung, Q. H. Choi, On the number of the periodic solutions of the nonlinear Hamiltonian system, Nonlinear Analysis TMA, 71, No. 12 e1100-e1108 (2009), [4] T. Jung, Q. H. Choi, Periodic solutions for the nonlinear Hamiltonian systems, Korean Journal of Mathematics 17, No (2009), [5] T. Jung, Q. H. Choi, Existence of four solutions of the nonlinear Hamiltonian system with nonlinearity crossing two eigenvalues, Boundary Value Problems, 2008, Kunsan National University, Department of Mathematics, Kunsan , Republic of Korea, Q- Heung Choi: Inha University, Department of Mathematics Education, Incheon , Republic of Korea, This work(tacksun Jung) was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (KRF ).

47 International Congress in Honour of Professor Ravi P. Agarwal One-Parameter Apostol-Bernoulli Polynomials and Apostol- Euler Polynomials Veli Kurt In this work, we define one-parameter Apostol-Bernoulli polynomials B n (β) (x; α, λ) of order β and oneparameter Apostol-Euleri polynomials E n (β) (x; α, λ) of order β, β N. We prove some identities and relations between these polynomials. Also, we give different form analogue of the Srivastava-Pintér additional theorem for these polynomials. [1] T. M. Apostol, On the Lerch zeta function, Pasific J. Math., 1 (1951), [2] J. Choi, P. J. Aderson and H. M. Srivastava, Some q-extensions of the Apostol-Bernoulli and Apostol- Euler polynomials of order n and the multiple Hurwitz zeta functions, Appl. Math. and Comp., 199 (2008), [3] M. Garg, K. Jain and H. M. Srivastava, Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions, Integral Trans. and Special Function, 17 (2007), [4] B.-N. Guo and F.Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. of Comp. and Appl. Math., 255 (2014), [5] D. S. Kim and T. Kim, A study on the integral of the product of several Bernoulli polynomials, Rocky Mountain J. Math., (2014), (Submitted). [6] D.-Q. Lu and H. M. Srivastava, Some series identities involving the generalized Apostol type and related polynomials, Comp. and Math. with Appl., 62 (2011), [7] Q.-M Luo, Multiplication formulas for Apostol-type polynomials and Multiple Alternating sums, Mathematical Notes, 91 (2012), [8] Q.-M Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Analy. Appl., 308 (2005), [9] Q.-M Luo and H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Computers and MAth. with Appl., 51 (2006), [10] Q.-M Luo, B. Guo, F. Qi and L. Debnath, Generalizations of Bernoulli numbers and polynomials, Int. J. Math. and Math. Sciences, 59 (2003), [11] F. Qi, Explicit formulas for computing Euler polynomials in terms of the second kind Stirling number, arxiv: [v], 19 oct [12] H. M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, London [13] H. M. Srivastava and A. Pintér, A remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Letters, 17 (2004), Department of Mathematics, Faculty of Sciences, University of Akdeniz, TR Antalya, Turkey, vkurt@akdeniz. edu.tr

48 48 International Congress in Honour of Professor Ravi P. Agarwal 14 On p-bernoulli Numbers Mourad Rahmani In this talk, we define a new family of p-bernoulli numbers which are derived from the Gaussian hypergeometric function, and we establish some basic properties. Furthermore, an algorithm for computing Bernoulli numbers based on three-term recurrence relation is given. A similar algorithm for Bernoulli polynomials is also presented. [1] S. Akiyama, Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J., 5 (4) (2001) [2] A. Z. Broder, The r-stirling numbers, Discrete Math., 49 (3) (1984) [3] K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Seq., 4 (1) (2001) Article [4] D. Dumont, Matrices d Euler-Seidel, Sémin. Lothar. Comb. 5, B05c (1981), 25 p. [5] M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Seq., 3 (2000) Article [6] D. E. Knuth, T. J. Buckholtz, Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 (1967), [7] M. Rahmani, The Akiyama Tanigawa matrix and related combinatorial identities, Linear Algebra Appl., 438 (2013), pp [8] M. Rahmani, Generalized Stirling transform, preprint, (2012), available electronically at USTHB, Faculty of Mathematics, P. O. Box 32, El Alia, Bab Ezzouar, 16111, Algiers, Algeria, [email protected]

49 International Congress in Honour of Professor Ravi P. Agarwal The Roots of a Dual Split Quaternion Hesna Kabadayı In this paper, we express De Moivre s formula for dual split quaternions and find roots of a dual split quaternion using this formula. [1] R. Ablamowicz and G. Sobczyky, Lectures on clifford (Geometric) Algebras and Applications Birkhäuser, Boston [2] L. Brand,. The Roots of a Quaternion, American Mathematical Monthly 49 (8) (1942) [3] E. Cho,. De Moivre s Formula for Quaternions, Applied Mathematics Letters 11(6) (1998) [4] W. K. Clifford Preliminary sketch of bi-quaternions, Proc. of London Math. Soc. 4 n. 64, 65 (1873), [5] H. Kabadayi. and Y. Yayli. De Moivre s formula for Dual Quaternions, Kuwait journal of Science and Engineering, 38 (1A) pp , [6] S. Li - Q. J. Ge, Rational Bezier Line Symmetric Motions, ASME J. of Mechanical Design, 127 (2) (2005), [7] C. Mladenova, Robot problems over configurational manifold of vector-parameters and dual vectorparametes J. Intelligent and Robotic systems 11 (1994) [8] I. Niven, The Roots of a Quaternion, American Mathematical Monthly 49 (6) (1942) [9] M. Özdemir,. Roots of a Split Quaternion, Applied mathematics letters 22(2009) [10] E. Study, Geometrie der Dynamen, Leipzig, [11] R. Wald, Class. Quant. Gravit. 4, 1279, Ankara University, Science Faculty, Mathematics Department, Tandoğan, Ankara-Türkiye, [email protected]. edu.tr, [email protected]

50 50 International Congress in Honour of Professor Ravi P. Agarwal 16 On Some Inequalities for Hadamard Product of Special Types of Matrices Seyda Ildan and Hasan Köse In this paper, we review some determinantal inequalities for Hadamard product of positive definite matrices, M matrices and inverse M matrices. Than we improve these inequalities for some special types of matrices. [1] S. Chen, İnequalities for M matrices and inverse M matrices, Linear Algebra and Its Applications, 426(2007) [2] J. Liu,L. Zhu, Some İmprovement Of Oppenheim s Inequality For M Matrices, SIAM J. MATRIX ANAL., APPL., Vol. 18, No. 2, pp [3] B.-Y. Wang, X. Zhang, F. Zhang, On The Hadamard Product Of İnverse M Matrices, Linear Algebra and Its Applications, 305(2000) [4] S. Chen, Some determinantal Inequalities for Hadamard Product of Matrices, Linear Algebra and Its Applıcations, 368(2003) [5] K. Fan, Inequalities for M matrices, Indag. Math., 26 (1964), pp [6] T. Ando, Inequalities for M matrices, Linear and Multilinear Algebra, 8 (1980), pp [7] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1. Selcuk University, Konya, Turkey, [email protected]

51 International Congress in Honour of Professor Ravi P. Agarwal Generalized Newton Transformation and its Application to Transversal Submanifolds Abdelmalek Mohammed In this paper, we study some properties of generalized Newton transformation T U of a family of endomorphisms. As application we establish a relation between the transversality of two submanifolds and ellipticity of T U. [1] M. P. do Carmo, Riemannian Geometry, Birkhauser, 1979 first edition. [2] K. Andrzejeweski, The Newton transformation and new integral formulae for foliated manifolds, Ann Glob Anal Geom, 37 (2010), [3] L. J. Alias, J. H. S. De Lira, J. M. Malacarne. Constant higher order mean curvature hypersurfaces in Riemannian spaces, J. Inst. Math. Jussieu 5 (2006), no, 4, [4] W. Kozlowski, Generalized Newton transformation and its applications to extrinsic geometry, preprint. Ecole préparatoire en sciences économiques, commerciales et sciences de gestion, Département de mathématiques, Tlemcen -Algérie, [email protected]

52 52 International Congress in Honour of Professor Ravi P. Agarwal 18 Multiple Positive Solutions for Elliptic Singular Systems with Hardy Sobolev Exponents Benmansour Safia In this work, we prove the existence of at least two positive solutions for an elliptic singular system of two weakly coupled equations with singular weights and critical Hardy Sobolev exponents. We use Mountain Pass theorem and Eukland s variationnal principle. [1] M. Bouchekif, A. Matallah, On singular elliptic equations involving a concave term and critical Cafarelli-Kohn-Nirenberg exponent. Math. Nachr. 284, (2011). [2] H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of funct ionals, Proc. Amer. Math. Soc. 88, (1983 ). [3] L. Cafarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53, (1984). [4] F. Catrina, Z. Wang, On the Cafarelli-Kohn -Nirenberg inequalities: sharp constants, existence (and non existence) and symmetry of extremal func tions, C omm. Pure Appl. Math. 54, (2001). [5] J. Chen, Multiple positive solutions for a class of non linear elliptic equations, J. Math. Anal. Appl. 295, (2004 ). [6] J. Chen, E. M. Rocha, Four solutions of an inhomogeneous elliptic equation with critical exponent and singulareties, Non linear Anal. 71, (2009 ). [7] K.S. Chou, C.W. Chu, On the best constant for a weighted Sobolev-Hardy Inequality J. London Math. Soc. 2, (1993). Ecole préparatoire en sciences économiques, commerciales et sciences de gestion, Département de mathématiques, Tlemcen-Algérie, [email protected]

53 International Congress in Honour of Professor Ravi P. Agarwal A Collocation Method for Solution of the Nonlinear Lane- Emden type Equations in Terms of Generalized Bernstein Polynomials Ayşegül Akyüz-Daşcıoğlu and Neşe Işler Acar In this talk, a collocation method based on Bernstein polynomials defined on the interval [a, b] is presented for approximate solution of the nonlinear Lane-Emden type equations that have an important place in astrophysics and mathematical physics. The proposed method reduces the solution of nonlinear problem to the solution of a system of linear algebraic equations iteratively by using quasilinearization technique and collocation points. Some numerical examples are given to illustrate the efficiency, validity and applicability of the method. [1] C. M. Bender, K. A. Milton, S. S. Pinsky and L. M. Simmons, A new perturbation approach to nonlinear problems, J. Math. Phys. 30 (1989) [2] A. M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Comput. 118 (2001) [3] A. M. Wazwaz, A new method for solving singular value problems in the second order ordinary differential equations, Appl. Math. Comput. 128 (2001) [4] J. I. Ramos, techniques for singular initial-value problems of ordinary differential equations, Appl. Math. Comput. 161 (2005) [5] H. Aminikhah, S. Moradian, Numerical Solution of Singular Lane-Emden Equation, Math. Phys. doi: /2013/ [6] A. Yıldırım, T. Ozis, Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method, Phys. Lett. A 369 (2007) [7] M. S. H. Chowdhury, I. Hashim, Solutions of a class of singular second-order IVPs by homotopyperturbation method, Phys. Lett. A. 368 (2007) [8] G. Hojjeti, K. Parand, An efficient computational algorithm for solving the nonlinear Lane-Emden type equations, World Academy of Science, Engineering and Technology, 56 (2011) [9] R. K. Pandey, N. Kumar, Solution of Lane Emden type equations using Bernstein operational matrix of differentiation, New Astron. 17 (2012) [10] D. G. Wang, W. Y. Song, P. Shi and H. R. Karimi, Approximate Analytic and Numerical Solutions to Lane-Emden Equation via Fuzzy Modeling Method, Math. Probl. Eng. doi: /2012/ [11] K. Parand, M. Dehghan, A. R. Rezaei and S. M. Ghaderi, An approximation algorithm for the solution of the nonlinear Lane Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun. 181 (2010) Ayşegül Akyüz-Daşcıoğlu: Pamukkale University, Faculty of Arts&Sciences, Department of Mathematics, Kınıklı, Denizli-Turkey, [email protected] Neşe İşler Acar: Mehmet Akif Ersoy University, Faculty of Arts&Sciences, Department of Mathematics, İstiklal, Burdur- Turkey, [email protected] This work is supported by Scientific Research Project coordination Unit of Pamukkale University, No:2012FBE036.

54 54 International Congress in Honour of Professor Ravi P. Agarwal 20 GALA and GADP2 Comparison for the Scheduling Problem of R m /S ijk /C max Duygu Yilmaz Eroğlu and H.Cenk Özmutlu We presented GALA, which is hybrid local search algorithm for the scheduling problem with setup times that includes non-identical parallel machines. This problem type is studied by many of researcher and most of them compare their algorithms via datasets from literature. In this study, we will compare our study s results with GADP2 (integrating the dominance properties with a genetic algorithm which is proposed by Chang and Chen (2011)) algorithm s results using the same datasets. In spite of GALA s local search is inspired from dominant properties method of GADP2, the results of GALA gives better result. This is probably caused by chromosome structure of GALA which is constituted by random numbers that are generated between 0 and 1. [1] Arnaout, J.P., Rabadi, G., & Musa, R. (2010) A two-stage Ant Colony Optimization algorithm to minimize the makespan on unrelated parallel machines with sequence-dependent setup times. Journal of Intelligent Manufacturing, 21, [2] Chang, P.C., & Chen, S.H. (2011). Integrating dominance properties with genetic algorithms for parallel machine scheduling problems with setup times. Applied Soft Computing, 11, [3] Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison- Wesley, Reading. [4] Rabadi, G., Moraga, R., & Al-Salem, A. (2006). Heuristics for the Unrelated Parallel Machine Scheduling Problem with Setup Times. Journal of Intelligent Manufacturing, 17, [5] Vallada, E., & Ruiz, R. (2011). A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times. European Journal of Operational Research, 211, [6] SchedulingResearch. (2005). Accessed June 07, 2012 from Duygu Yilmaz Eroglu (Corresponding Author): Department of Industrial Engineering, Uludag University, Gorukle Campus, Bursa 16059, Turkey, [email protected] H.Cenk Ozmutlu: Department of Industrial Engineering, Uludag University, Gorukle Campus, Bursa 16059, Turkey

55 International Congress in Honour of Professor Ravi P. Agarwal Improved MIP Model for Parallel Machines Scheduling Problem Duygu Yilmaz Eroğlu and H.Cenk Özmutlu In this study, MIP model, which is developed for unrelated parallel machines, is improved and additional constraints that satisfies equal sub orders are added into the formulation. This research is motivated by a practical need at a loom scheduling. MIP helps us in this problem to validate the developed heuristics methods, using small scale data sets from literature. The comparison of the results of developed MIP model and heuristic algorithm shows effectiveness of presented algorithms. Job splitting is scarcely studied in the literature but needs more attention because of possible flexibility effects also on the other sectors (i.e logistics) besides scheduling. [1] Ruiz, R., Maroto, C. (2006). A genetic algorithm for hybrid flowshops with sequence dependent setup times and machine eligibility. European Journal of Operational Research, 169, [2] Ruiz, R., Serifoglu, F.S., Urlings, T. (2008). Modeling realistic hybrid flexible flowshop scheduling problems. Computers & Operations Research, 35, [3] Lin, Y., Li, W. (2004). Parallel machine scheduling of machine-dependent jobs with unit-length. European Journal of Operational Research, 156, [4] Liao, L.W., Sheen, G.J. (2008). Parallel machine scheduling with machine availability and eligibility constraints. European Journal of Operational Research, [5] Centeno, G., Armacost, R.L. (2004). Minimizing makespan on parallel machines with release time and machine eligibility restrictions. International Journal of Production Research, 42, 6, Duygu Yilmaz Eroglu (Corresponding Author): Department of Industrial Engineering, Uludag University, Gorukle Campus, Bursa 16059, Turkey, [email protected] H.Cenk Ozmutlu: Department of Industrial Engineering, Uludag University, Gorukle Campus, Bursa 16059, Turkey

56 56 International Congress in Honour of Professor Ravi P. Agarwal 22 Multiple Positive Solutions for Elliptic Singular Systems with Cafarelli Kohn Niremberg Exponents Matallah Atika In this work, we prove the existence of at least two positive solutions for an elliptic singular system of two weakly coupled equations with singular weights and critical Cafarelli Kohn Niremberg exponents. We use Mountain Pass theorem and Eukland s variationnal principle. [1] H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of funct ionals, Proc. Amer. Math. Soc. 88, (1983). [2] L. Cafarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53, (1984). [3] F. Catrina, Z. Wang, On the Cafarelli-Kohn -Nirenberg inequalities: sharp constants, existence (and non existence), and symmetry of extremal func tions, Comm. Pure Appl. Math. 54, (2001). [4] J. Chen, Multiple positive solutions for a class of non linear elliptic equations, J. Math. Anal. Appl. 295, (2004). [5] J. Chen, E. M. Rocha, Four solutions of an inhomogeneous elliptic equation with critical exponent and singulareties, Non linear Anal. 71, (2009). [6] K.S. Chou, C.W. Chu, On the best constant for a weighted Sobolev-Hardy Inequality J. London Math. Soc. 2, (1993). Ecole préparatoire en sciences économiques, commerciales et sciences de gestion, Département de mathématiques, Tlemcen-Algérie, atika [email protected]

57 International Congress in Honour of Professor Ravi P. Agarwal Positive Solution for a Singular Second-Order Discrete Three- Point Boundary Value Problem Noor Halimatus Sa diah Ismail and Mesliza Mohamed Using the Krasnoselskii fixed point theorem, we prove the existence and multiplicity of positive solution for a singular three point boundary value problem 2 y(k 1) + λh(k)f(y(k)) = 0, k {1,..., T }, y(0) α y(0), y(t + 1) = βy(n). where f is singular at y = 0, λ > 0 and T 3 is a fixed positive integer, n {2,..., T 1}, constant α, β > 0 such that H := T + 1 βn + α(1 β) > 0, and T + 1 βn > 0. [1] X. Lin and W. Liu, Three positive solutions for a second order diference equation with three-point boundary value problem, J. Appl. Math. Comput., 31 (2009), [2] M. Krasnoselskii, Positive solutions of operator equations, Noordhoff, Groningen, [3] M. Mohamed and O. Omar, Positive periodic solutions of singular first order functional difference equation, Int. J. of Math Analysis., vol 6, 54(2012), [4] F. M. Atici and A. Cabada, Existence and uniqueness result for the second-order periodic boundary value problems, Comp. Math. Appl., 45(2003), [5] W. A. W. Azmi, M. Mohamed, Existence and multiplicity of positive solutions for singular second order Dirichlet boundary value problem, Proceedings of the International Conference on Mathematical Sciences and Statistics (ICMSS2013) 2013, Vol 1557, Isue 1, 66 71, AIP Publishing. Noor Halimatus Sa diah Ismail: Fakulti Sains Komputer & Matematik, Universiti Teknologi MARA (Pahang), Bandar Tun Abdul Razak, Jengka, Malaysia., [email protected] Mesliza Mohamed (Corresponding author): Fakulti Sains Komputer & Matematik, Universiti Teknologi MARA (Pahang), Bandar Tun Abdul Razak, Jengka, Malaysia, [email protected] The authors thank to Ministry of Higher Education for Fundamental Research of Grant Sciences (600-RMI/ FRGS5/3/Fst(9/2012))

58 58 International Congress in Honour of Professor Ravi P. Agarwal 24 Positive Solution to Fourth Order Three-Point Boundary Value Problem M.Mohamed, M.S.M.Noorani, M.S.Jusoh, M.N.M.Fadzil and R.Saian This work concerned with the fourth order boundary value problem u 4 (t) + f(t, u(t), u (t)) = 0, 0 < t < 1, subject to boundary conditions u(0) = u (0) = u (0) = 0 and u (1) αu (η) = λ where 0 < η < 1 and α [0, 1 η ] are constant and λ [0, + ) is a parameter. By imposing a sufficient structure on the nonlinearity f(t, u, u ), we deduce the existence of at least one positive solution to the problem by applying the Krasnosel skii fixed point theorem. [1] R. Ma, J. Zhang, S. Fu, The method of upper and lower solutions for fourth-order two point boundary value problems, J. Math. Anal. Appl, (1997), 215, [2] R. Ma, H. Wang, On The Existence Of Positive Solutions Of Fourth-Order Ordinary Differential Equation, Appl. Anal, (1995), 59, [3] Q. Yao, Existence And Multiplicity Of Positive Solutions To A Class Of Nonlinear Cantilever Beam Equations, J. Syst. Sci. Math. Sci, (2009), 1, [4] Z. Bai, H. Wang, On The Positive Solutions Of Some Nonlinear Fourth-Order Beam Equations, J. Math. Anal. Appl, (2002), 270, [5] JR. Graef, C. Qian, B. Yang, A Three Point Boundary Value Problem For Nonlinear Fourth-Order Differential Equations, J. Math. Anal. Appl, (2003), 287, [6] Y. Sun and C Zhu, Existence of positive solutions for singular fourth order three-point boundary value problems, Advances in Difference Equations, (2013) 2013:51, pp 13. M. Mohamed (Corresponding author), Fakulti Sains Komputer & Matematik, Universiti Teknologi MARA (Pahang), Bandar Tun Abdul Razak, Jengka, Pahang, Malaysia, [email protected] M. S. M. Noorani: Faculty of Science & Technology, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, [email protected] M. S. Jusoh, Faculty of Civil Engineering, Universiti Teknologi MARA (Pahang), Bandar Tun Abdul Razak, Jengka, Pahang, Malaysia, [email protected] M. N. M. Fadzil, Fakulti Sains Komputer & Matematik, Universiti Teknologi MARA (Perlis), Arau, Perlis, Malaysia, [email protected] R. Saian:Fakulti Sains Komputer & Matematik, Universiti Teknologi MARA (Perlis), Arau, Perlis, Malaysia, [email protected] The authors thank to Ministry of Higher Education for Fundamental Research of Grant Sciences (600-RMI/ FRGS5/3/Fst(9/2012))

59 International Congress in Honour of Professor Ravi P. Agarwal Existence of Positive Solutions for Non-Homogeneous BVPs of p-laplacian Difference Equations Fatma Tokmak and Ilkay Yaslan Karaca In this talk, by using Avery-Peterson fixed point theorem, we investigate the existence of at least three positive solutions for a third order p-laplacian difference equation. As an application, an example is given to illustrate our main results. [1] D. R. Anderson, Discrete third-order three-point right-focal boundary value problems, Comput. Math. Appl., 45 (2003), , [2] R. Avery and A. Peterson, Three positive fixed points of nonlinear operators on an ordered Banach space, Comput Math Appl., 208 (2001), , [3] Z. He, On the existence of positive solutions of p-laplacian difference equations, J. Comput. Appl. Math., 161 (2003), , [4] I. Y. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math., 205 (2007), , [5] Y. Liu, Studies on nonhomogeneous multi-point BVPs of difference equations with one-dimensional p-laplacian, Mediterr. J. Math., 8 (2011), , [6] Y. Liu, Three positive solutions of multi-point BVPs for difference equations with the nonlinearity depending on -operator, An. Ştiint. Univ. Ovidius Constanta Ser. Mat., 20 (2012), 65-81, [7] J. Xia, L. Debnath, H. Jiang and Y. Liu, Three positive solutions of Sturm-Liouville type multi-point BVPs for second order p-laplacian difference equations, Bull. Pure Appl. Math., 4 (2010), Fatma Tokmak: Gazi University, Faculty of Science, Department of Mathematics, Teknikokullar, Ankara-Turkey and Ege University, Faculty of Science, Department of Mathematics, Bornova, İzmir-Turkey, [email protected] or [email protected] İlkay Yaslan Karaca: Ege University, Faculty of Science, Department of Mathematics, Bornova, İzmir-Turkey, [email protected]

60 60 International Congress in Honour of Professor Ravi P. Agarwal 26 L2 Norm Deconvolution Algorithm Applied to Ultrasonic Phased Array Signal Processing Abdessalem Benammar, Redouane Drai and Ahmed Khechida Detection of failure in laminate composites is complicated compared with ordinary non-destructive testing for metal materials as they are sensitive to echoes drown in noise due to the properties of the constituent materials and the multi-layered structure of the composites. In recent years, rapid development in the fields of microelectronics and computer engineering lead to wide application of phased array systems. Different signal processing and image reconstruction techniques are applied in ultrasonic testing. In this work, the objective is to improve the time resolution of signals obtained from inspection of CFRP sample. The signal processing scheme used is based on L2 Norm deconvolution of the measured signal by fast sequential algorithm. This algorithm performs a search of events by increasing order of importance with respect to a criterion which is described in detail. It gives good results over a wide range of applications. The experimental results show that the L2 Norm deconvolution can enhance the time resolution of the CFRP ultrasonic phased array inspection effectively and help identify the location of defects. Keywords: Ultrasonic Phased Array, signal processing, L2 Norm deconvolution. [1] Reza Bohlouli, Babak Rostami, and Jafar Keighobadi, Application of Neuro-Wavelet Algorithm in Ultrasonic-Phased Array Nondestructive Testing of Polyethylene Pipelines, Journal of Control Science and Engineering, (2012), [2] S.C. Ng, N. Ismail, Aidy Ali, Barkawi Sahari, J.M. Yusof, B.W. Chu. Non-destructive Inspection of Multi-layered Composites Using Ultrasonic Signal Processing, Materials Science and Engineering 17 (2011), [3] Zhang Yicheng, Li Xiaohong, Zhang Jun, Ding Hui, Model based reliability analysis of PA ultrasonic testing for weld of hydro turbine runner, Procedia Engineering 16 (2011) , [4] R. J. Ditchburn and M. E. Ibrahim, Ultrasonic Phased Arrays for the Inspection of Thick-Section Weldsâ, DSTO Defence Science and Technology Organisation 506 Lorimer St Fishermans Bend Victoria 3207 Australia, Abdessalem Benammar: Welding and NDT Research Center (CSC), BP 64, Cheraga, Algeria, Abs [email protected] Redouane Drai: Welding and NDT Research Center (CSC), BP 64, Cheraga, Algeria. Ahmed Khechida: Welding and NDT Research Center (CSC), BP 64, Cheraga, Algeria.

61 International Congress in Honour of Professor Ravi P. Agarwal Multi-Soliton Solutions for Non-Integrable Equations: Asymptotic Approach Georgy A.Omel yanov We describe an approach to construct multi-soliton asymptotic solutions for essentially non-integrable equations. The general idea is realized for the GKdV-4 equation: u t + u4 x + ε2 3 u x 3 = 0, x R1, t > 0, where the dispersion parameter ε is assumed to be small. It has been proved that two and three solitons interact preserving in the leading term the KdV-type scenario of collision: they pass through each other almost without deformation. At the same time, a small radiation tail appears on the left of the solitons. Our main tool is the Weak Asymptotics Method [1, 2]. We indicate also how to modify this approach in order to construct N-soliton asymptotic solutions for N 3. A brief review of asymptotic methods as well as results of numerical simulation are included. [1] V. G. Danilov, G. A. Omel yanov, Weak asymptotics method and the interaction of infinitely narrow delta-solitons, Nonlinear Analysis: Theory, Methods and Applications, 54 (2003), [2] V. G. Danilov, G. A. Omel yanov, V. M. Shelkovich, Weak asymptotics method and interaction of nonlinear waves, in: M.V. Karasev (Ed.), Asymptotic methods for wave and quantum problems, AMS Trans., Ser. 2, 208, AMS, Providence, RI, 2003, Universidad de Sonora, Departamento de Matematicas, calle Rosales y Blvd. Encinas, s/n, 83000, Hermosillo, Sonora, Mexico, [email protected] This work was supported by the SEP-CONACYT under grant (Mexico)

62 62 International Congress in Honour of Professor Ravi P. Agarwal 28 On Interpolation Functions for the q-analogue of the Eulerian Numbers Associated with any Character Mustafa Alkan and Yilmaz Simsek Simsek [1] defined generating functions for the Eulerian numbers and polynomials. In this paper, we study on these generating functions and their properties. The aim of this paper is to construct q-interpolation functions of the generalized Eulerian type numbers attached to any characters. We give some results, remarks and identities related to these functions and characters. [1] Y. Simsek, interpolation function of the Eulerian type polynomils and numbers, Adv. Studies Contemp. Math 23 (2013)2, [2] M. Alkan and Y. Simsek, Generating function for q-eulerian polynomials and their decomposition and applications, Fixed Point Theory Appl. 2013, 72. [3] Y. Simsek, On q-analogue of the twisted L-functions and q-twisted Bernoulli numbers, J. Korean Math. Soc. 40(6) (2003), [4] 78. Y. Simsek, Generating Functions for q-apostol Type Frobenius-Euler Numbers and Polynomials, Axioms 1(2012) [5] J. Choi P. J. Anderson and H. M. Srivastava, Some q -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function, Appl. Math. Comput. 2008, 199: Y. Simsek, On q-analogue of the twisted L-functions and q-twisted Bernoulli numbers, J. Korean Math. Soc. 40(6) (2003), [6] H. M. Srivastava, T. Kim and Y. Simsek, q-bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russian J. Math Phys. 2005, 12: Mustafa Alkan: University of Akdeniz, Faculty of Science, Department of Mathematics, TR Antalya, Turkey, [email protected] Yilmaz Simsek: University of Akdeniz, Faculty of Science, Department of Mathematics, TR Antalya, Turkey, [email protected]

63 International Congress in Honour of Professor Ravi P. Agarwal Computation of p-values for Mixtures of Gaussians Burcin Simsek and Satish Iyengar For unimodal distributions, p-values are typically tail probabilities. In this paper, we address the problem of computing p-values for mixtures of the Gaussian distributions. The tail regions are those that have small probability under each component of the mixture. We compare the use of moment methods and exponential tilting to estimate the probabilities of such tail regions. Burcin Simsek: Statistics Department University of Pittsburgh, Pittsburgh, PA-USA, com Satish Iyengar: Statistics Department University of Pittsburgh, Pittsburgh, PA-USA,

64 64 International Congress in Honour of Professor Ravi P. Agarwal 30 Nonclassical Appell Polynomials Rahime Dere and Yilmaz Simsek In this paper we study on the nonclassical Appell polynomials associated with umbral calculus. We introduce nonclassical Bernoulli polynomials and nonclassical Euler polynomials, which are the Appell polynomials. Furthermore, we give some identities of these polynomials by using nonclassical umbral calculus methods. [1] R. Dere and Y. Simsek, Genocchi polynomials associated with the Umbral algebra, Appl. Math. Comput. 218(3) (2011) [2] R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Studies Contemp. Math. 22 (2012) [3] R. Dere and Y. Simsek, Remarks on the Frobenius-Euler Polynomials on the Umbral Algebra, Numerical Analysis and Applied Mathematics ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics. AIP Conference Proceedings (2012), [4] R. Dere and Y. Simsek, Normalized polynomials and their multiplication formulas, Advances in Difference Equations. (2013), 2013:31. [5] R. Dere, Y. Simsek and H. M. Srivastava, A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra, Journal of Number Theory 133 (2013), [6] E. C. Ihrig and M. E. H. Ismail, A q-umbral calculus, J. Math. Anal. Appl. 84 (1981) [7] V. Kac and P. Cheung, Quantum Calculus, Springer, [8] S. Roman, More on the Umbral Calculus, with Emphasis on the q-umbral Calculus, J. Math. Anal. Appl. 107(1) (1985) [9] S. Roman, The Umbral Calculus, Dover Publ. Inc. New York, Rahime Dere: University of Akdeniz, Faculty of Science, Department of Mathematics, TR Antalya, Turkey, [email protected] Yilmaz Simsek: University of Akdeniz, Faculty of Science, Department of Mathematics, TR Antalya, Turkey, [email protected]

65 International Congress in Honour of Professor Ravi P. Agarwal Remarks on the Central Factorial Numbers Yilmaz Simsek In [3], we gave some analytic functions which are related to the generating functions for the central factorial numbers. By using these functions, we derive identities-some old and some new-for the central factorial numbers, the Stirling numbers and special numbers. [1] J. Cigler, Fibonacci Polynomials and Central Factorial Numbers, Preprint. [2] S. Roman, The Umbral Calculus, Dover Publ. Inc. New York, [3] Y. Simsek, Special Numbers on Analytic Functions, Applied Mathematics [4] Y. Simsek, On q-deformed Stirling numbers, International Journal of Computer Mathematics, 15, 70-80; [5] Y. Simsek, Generating Functions for Generalized Stirling type Numbers, Array Type Polynomials, Eulerian Type Polynomials and Their Applications, Fixed Point Theory and Applications, 87, ; [6] H. M. Srivastava and G.-D. Liu, Some Identities and Congruences Involving a Certain Family of Numbers, Russian Journal of Mathematical Physics, 16, ; University of Akdeniz, Faculty of Science, Department of Mathematics, TR Antalya, Turkey, [email protected]

66 66 International Congress in Honour of Professor Ravi P. Agarwal 32 Nodals Solutions of the Fourth Order Equations Involving Paneitz-Branson Operator with Critical Sobolev Exponent Boughazi Hichem Given (M, g) a smooth compact Einstein manifold of dimension n 5, with negative scalar curvature S g, for u C (M), the geometric Paneitz-Branson operator P g is reduced to P g u = 2 gu + a n S g g u + b n Sgu.M.Benalili 2 and H.Boughazi defined the k th Paneitz-Branson invariant by µ k (M, g) = inf λ k( g)[vol(m, g)] 4 n, where the λ k (g) is the k th eigenvalue when the scalar curvature S g is negative, g [g] the Paneitz-Branson operator is non necessary coercive and we give a new technic for study the standard Paneitz-Branson invariant µ(m, g), the first Paneitz-Branson invariant µ 1 (M, g) and the second Paneitz- Branson invariant µ 2 (M, g). The main point of this work is to complete the results of [1] M.Benalili, H.Boughazi. We recall that study the standard Paneitz-Branson invariant is a challenging open problem, we find nodals solutions in few cas, study the positivite of solutions it seem to be impossible, we have always µ(m, g) > contrary to µ 1 (M, g) and µ(m, g) is always attained, the nodals solutions w of the fourth order equations involving Paneitz-Branson operator with critical Sobolev exponent i.e, P g v = µ 2 (M, g) w N 2 w. [1] M.Benalili, H.Boughazi. Nodals solutions of the fourth order equations involving Paneitz-Branson operator with critical Sobolev exponent (submit). [2] M.Benalili, H.Boughazi. On the second Paneitz Branson invariant, Houston J. Math. 36 (2010),no. 2, MR (2011h:58047) [3] M.Benalili, H.Boughazi. The second Yamabe invariant with singularities, Annales mathématique Blaise Pascal.Volume 19, no.1, (2012),p Preparatory School in Economics, Business and Management Sciences, Department of Mathematics, Tlemcen, Algeria, [email protected]

67 International Congress in Honour of Professor Ravi P. Agarwal Bilinear Multipliers of Weighted Wiener Amalgam Spaces and Variable Exponent Wiener Amalgam Spaces Öznur Kulak and A.Turan Gürkanlı Let ω 1, ω 2 be slowly increasing weight function and let ω 3 be any weight function on R n. Assume that m (ξ, η) is a bounded function on R n R n. We define B m (f, g) (x) = f (ξ) g (η) m (ξ, η) e 2πi ξ+η,x dξdη R n R n for all f, g ɛcc (R n ), where Cc (R n ) denotes the space of infinitely differentiable complex-valued functions with compact support on differentiable R n. Also let W ( ) ( ) ( ) L p1, L q1 ω 1, W L p 2, L q2 ω 2 and W L p 3, L q3 ω 3 be Wiener amalgam spaces. We say that m (ξ, η) is a bilinear multiplier on R n of type (W (p 1, q 1, ω 1 ; p 2, q 2, ω 2 ; p 3, q 3, ω 3 )) if B m is bounded operator from W ( ) ( ) ( ) L p1, L q1 ω 1 W L p 2, L q2 ω 2 to W L p 3, L q3 ω 3 where 1 p 1 q 1 <, 1 p 2 q 2 <, 0 < p 3, q 3. We denote by BM (W (p 1, q 1, ω 1 ; p 2, q 2, ω 2 ; p 3, q 3, ω 3 )) the vector space of bilinear multipliers of type (W (p 1, q 1, ω 1 ; p 2, q 2, ω 2 ; p 3, q 3, ω 3 )). In the First Section of this work, we investigate some properties of this space and we give some examples of these bilinear multipliers. In the Second Section, by using variable exponent Wiener amalgam spaces we define the bilinear multipliers of type (W (p 1 (x), q 1, ω 1 ; p 2 (x), q 2, ω 2 ; p 3 (x), q 3, ω 3 )) from W ( ) ( ) L p1(x), L q1 ω 1 W L p 2(x), L q2 ω 2 to W ( ) L p3(x), L q3 ω 3 where p1 (x) q 1, p 2 (x) q 2, p 1, p 2, p 3 < for all p 1 (x), p 2 (x), p 3 (x) ɛp (R n ). We denote by BM (W (p 1 (x), q 1, ω 1 ; p 2 (x), q 2, ω 2 ; p 3 (x), q 3, ω 3 )) the vector space of bilinear multipliers of type (W (p 1 (x), q 1, ω 1 ; p 2 (x), q 2, ω 2 ; p 3 (x), q 3, ω 3 )). Similarly, we discuss some properties of this space. Some key references are given below. [1] Aydın, İ, Gürkanlı, A. T: Weighted variable exponent amalgam spaces. Glasnik Mathematicki. 47(67), (2012) [2] Bennett, C, Sharpley, R: Interpolation of Operators. Academic Press (1988) [3] Blasco, O: Notes on the spaces of bilinear multipliers. Rev. Un. Mat. Argentina. 50(2), (2009) [4] Dobler, T: Wiener Amalgam Spaces on Locally Compact Groups, Master s Thesis, University of Viena (1989). [5] Feichtinger, H. G: A characterization of Wiener s algebra on locally compact groups, Arch. Math. (Basel), 29, (1977) [6] Feichtinger, H. G: Banach convolution algebras of Wiener type functions, series, operators, Proc. Conf. Budapest 38, Colloq. Math. Soc. Janos. Boyai., (1980) [7] Feichtinger, H. G: Banach spaces of distribution of Wiener s type interpolation, Proc. Conf. Funct. Anal. Approx., Oberwolfach, 60, Birkhaüser, Basel, (1981) [8] Kulak, Öznur, Gürkanlı A.Turan: Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces, Journal of Inequalities and Applications 2013, 2013:259, 1-21 (2013). Öznur Kulak: Ondokuz Mayıs University, Faculty of Arts and Science, Department of Mathematics, Kurupelit, Atakum, Samsun-Turkey, [email protected] A.Turan Gürkanlı: İstanbul Arel University, Faculty of Science and Letters, Department of Mathematics and computer Sciences, Tepekent, Büyükçekmece-İstanbul, [email protected] This work was supported by the Ondokuz Mayıs University, Project number PYO.FEN

68 68 International Congress in Honour of Professor Ravi P. Agarwal 34 Global Optimization Problem of Lipschitz Functions Using α-dense Curves Djaouida Guettal and Mohamed Rahal In this paper, we study a coupling of the Alienor method with the algorithm of Piyavskii-Shubert. The classical multidimensional global optimization methods involves great difficulties for their implementation to high dimensions. The Alienor method allows to transform a multivariable function into a function of a single variable for which it is possible to use efficient and rapid method for calculating the the global optimum. This simplification is based on the using of a reducing transformation called Alienor. [1] Y. Cherruault, Optimisation: Méthodes locales et globales, Presses Universitaire de France, [2] R. Horst and H. Tuy, Global Optimization, Deterministic Approach, Springer-Verlag, Berlin, [3] G. Mora and Y. Cherrualult, Characterization and Generation of α-dense Curves, Comp. and Math. with Applic. Vol. 33, No.9, p.p , [4] G. Mora, Y. Cherrualult and A. Ziadai, Functional Equations Generating Space-Densifing Curves, Comp. and Math. with Applic. 39, p.p , [5] A. Torn and A. Silinkas, Global Optimization, Springer-Verlag, New york, [6] A. Ziadi and Y. Cherruault, Generation of α-dense curves and application to global optimization, Kybernetes, Vol. 29 No.1, pp.71-82,2000. [7] A. Ziadi, Y. Cherruault and G. Mora, Global Optimization, a New Variant of the Alienor Method, Comp. and math. with Applic. 41, p.p , [8] S. A. Piyavsky, An algorithm for finding the absolute extremum for a function. USSR Comput. Mathem. and Mathem. Phys., 12, No.4, [9] A. Ziadi and Y. Cherruault, Generation of α-dense Curves in a cube of R n, Kybernetes Vol. 27 No.4, pp , Guettal Djaouida and Rahal Mohamed: Laboratory of Fundamental and Numerical Mathematics Department of Mathematics, University Ferhat Abbas of Setif 1, Algeria, [email protected], mrahal [email protected]

69 International Congress in Honour of Professor Ravi P. Agarwal Estimating 2-D GARCH Models by Quasi-Maximum Likelihood Soumia Kharfouchi The introduction of the Autoregressive Conditional Heteroscedasticity (ARCH) model in the famous paper of Engel (1982) was a natural starting point in modeling the temporal dependencies in the conditional variance of financial time series. This model allow the variance to depend on the past of the random process. Since, numerous variants and extensions of this model have been proposed. Generalized ARCH (GARCH) model is the main natural extension of this model, the passage has been done in a way that is similar to the passage from the AR model to the ARMA one. A large strand of the financial literature is devoted to one-dimensional GARCH model; see for example Bellerslev (1986), Bellerslev, Engle and Nelson (1994), Palm (1996), Shephard (1996). Next, this GARCH model has seen many extensions with the introduction of lagged values of the variance or models allowing to take into account the phenomena of asymmetry such as EGARCH models (Exponential GARCH) proposed by Nelson (1991), TGARCH models (Threshold GARCH) proposed by Zakoian (1991), or again DCC-MVGARCH models (Multivariate GARCH with Dynamical Conditional Correlation) proposed by Engle and Sheppard (2001). The treatment of spatial interaction (dependence) and spatial structure (heterogeneity) in practice may be modeled by some random fields (X t) t Z d. Noiboar and Cohen (2005) had the idea of extending the one-dimensional GARCH model into two-dimensions in order to take into account the variability of the variance trough the space. They could also show that the two-dimensional GARCH model generalizes the causal Gauss Markov Random Field (GMRF), largely used in clutter modeling with the disadvantage of having a constant conditional variance trough the space which makes the use of a GARCH clutter modeling better than the use of a GMRF one. This phenomena is often found on natural images because they are corrupted due to several factors, such as performance of imaging sensors and characteristics of the transmission channel (Amirmazlaghani and Amindavar (2010)). Furthermore, data of textural information such as images of geographical regions that allow the production of some maps, and in general, a lot of images of the earth are characterized by a behavior in cluster of the space variability (clustering of innovations) i.e. significant changes tend to follow big changes small changes tend to follow small changes; it is clearly seen in the image it self where the decrease of the level of gray calls a decrease. On the other hand, research on statistical properties of images wavelet coefficients have shown that the marginal distribution of wavelet coefficients are highly kurtotic, and can be described using suitable heavy-tailed distribution (cf. Achim et al 2003). Indeed, Amirmazlaghani and Amindavar (2009) shown that the subband decomposition of SAR images has significantly non-gaussian statistics that are best described by the 2-D GARCH model. It should be noted that statistical and probabilistic properties as well as building the parameter estimates have been gained more attention for the spatial linear models than the nonlinear one, despite of the well known nonlinearity structure of many spatial series, this is partly due to the fact that the existence of spatial dependence creates difficulties for building such estimates. So, the purpose of this paper is to present the quasi-maximum likelihood (QML) method which provides, for GARCH models, theoretical framework for proving efficiency of estimators under mild regularity conditions, but with no moment assumptions on the observed process. Consistency and the asymptotic normality of the QML estimators of coeffients of 2-D GARCH are derived under optimal conditions. [1] Achim, A., Bezerianos, A., and Tsakalides, P. (2003). SAR image denoising via Bayesian wavelet shrinkage based on heavy tailed modeling. IEEE Transaction on Geoscience and Remote Sensing, 41 (8), [2] Amirmazlaghani, M., Amindavar, H. (2010). Image denoising using two-dimensional GARCH model. Systems, Signals and image processing, [3] Amirmazlaghani, M., Amindavar, H., and Moghaddamjoo A. (2009). Speckle Suppression in SAR Images Using 2-D GARCH Model. IEEE Trans. Image Processing, 18 (2), [4] Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. in: Cox, D. R., O. E. Barndor -Nielsen & D. V. Hinkley, eds., Statistical Models in Econometrics, Finance and other fields, [5] Tjostheim, D. (1978). Statistical spatial series modelling I. Adv. App. Prob. 10, [6] Tjostheim, D. (1983). Statistical spatial series modelling II. Adv. App. Prob. 15, [7] Whittle, P. (1954) On stationary process in the plane. Biometrika 41, Département de Médecine, Université 3 Constantine, Algeria, s [email protected]

70 70 International Congress in Honour of Professor Ravi P. Agarwal 36 An Approach Using Stream Ciphers Algorithm for Speech Encryption and Decryption Belmeguenai Aissa, Mansouri Khaled and Lashab Mohamed In this work, we have done an e cient implementation of stream ciphers algorithm for speech data encryption and decryption. The stream cipher algorithm is proposed. The design based on linear feedback shift register (LFSR) whose polynomial is primitive and nonlinear Boolean function. At first three speech signal were recorded from different speakers and were saved as wav file format. Then our developed program was used to transform the original speech signal wav file into positive signal data, and transform the positive data signal into positive digital signal file. Finally, we used our implemented program to encrypt and decrypt speech data. We conclude the paper by showing that The design can resist to certain known attacks. [1] Md. M. Rahman, T. K. Saha and Md.A. Bhuiyan, Implementation of RSA Algorithm for Speech Data Encryption and Decryption, International Journal of Computer Science and Network Security IJCSNS, vol. 12, no. 3, 2012, pp [2] K. Merit1 and A. Ouamri, Securing Speech in GSM Networks using DES with Random Permutation and Inversion Algorithm, International Journal of Distributed and Parallel Systems (IJDPS) Vol.3, No.4, [3] A. Musheer, A. Bashir and F. Omar, Chaos Based Mixed Key Stream Genrator for Voice Data Encryption, International Journal on Cryptography and Information Security (IJCIS),Vol.2, No.1,March [4] H. Kohad, V.R.Ingle and M.A.Gaikwad, Security Level Enhancement In Speech Encryption Using Kasami Sequence, International Journal of Engineering Research and Applications (IJERA), V 2, 2012, pp [5] M. Ashtiyani, P. Moradi Birgani, S. S. Karimi Madahi, Speech Signal Encryption Using Chaotic Symmetric Cryptography, Journal of Basic and Applied Scienti c Research, pp , [6] C. Carlet, On the cost weight divisibility and non linearity of resilient and correlation immune functions, Proceeding of SETA01 (Sequences and their applications 2001), Discrete Mathematics, Theoretical Computer Science, Springer p , [7] P.Van Oorschot A. Menezes and S. Vantome, Handbook of Applied Cryptography, hac /,1996. [8] G.Ars, Une application des bases de Grobner en Cryptographie, DEA de Rennes I, [9] Y. V. Tarannikov, On resilient Boolean functions with maximum possible nonlinearity, Proceedings of INDOCRYPT 2000, lecture Notes in Computer Science 1977, pp19-30, [10] E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean function with a good correlation-immunity, In: Advances in cryptology- EUROCRYPT98, lecture Notes in Computer Science, N Pp , Springer-Verlag, [11] D. K. Dalai, On Some Necessary Conditions of Boolean Functions to Resist Algebraic Attacks, thesis, [12] E.R Berlekamp, Algebraic Coding Theory, Mc Grow- Hill, New- York, Laboratoire de Recherche en Electronique de Skikda, Universite 20 Aout 1955-Skikda, BP 26 Route d El-hadaeik Skikda, Algeria

71 International Congress in Honour of Professor Ravi P. Agarwal A Generalized Statistical Convergence for Sequences of Sets via Ideals Ömer Kişi and Ekrem Savaş The notion of statistical convergence of sequences of numbers was introduced by Fast [1] and Schoenberg [5] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, Number Theory. Later on, statistical convergence turned out to be one of the most active ares of research in summability theory after the works of Fridy [2] and Salat [4]. In last few years, many generalization of statistical convergence have appeared. The concept of I convergence of real sequences is a generalization of statistical convergence which is based on the structure of the ideal I of subsets of the set of natural numbers. P. Kostyrko et al. [3] introduced the concept of I convergence of sequences in a metric space and studied some properties of this convergence. In this study, we make a new approach to the notions of [V, λ] summability and λ statistical convergence of sequence of sets by using ideals and introduce new notions, namely, I [V, λ] summability, I λ statistical convergence of sequence of sets. We mainly examine the relation between these two methods and also the relation between I [V, λ] summability, I λ statistical convergence of sequence of sets are introduced by the authors recently. [1] H. Fast, Sur la convergence statistique, Collog. Math. 2 (1951) [2] J. A. Fridy, On statistical convergence, Analysis, 5 (1985) [3] P. Kostyrko, T. Salat, W.Wilczynski, I convergence, Real Anal. Exchange, 26(2) (2000/2001), [4] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980), [5] Schoenberg, I.J. (1959). The integrability of certain functions and related summability methods. Amer. Math. Monthly, 66: Ömer Kişi: Faculty Of Education, Mathematics Education Depertmant, Cumhuriyet University, Sivas, Turkey, okisi@ cumhuriyet.edu.tr, [email protected] Ekrem Savaş: Istanbul Ticaret University, Deparment of Mathematics, Üsküdar, İstanbul, Turkey, ekremsavas@yahoo

72 72 International Congress in Honour of Professor Ravi P. Agarwal 38 Some Embedding Questions for Weighted Difference Spaces Leili Kussainova and Ademi Ospanova We introduce weighted space w 2 p (υ) which is a difference analogue of weighted Sobolev space W 2 p (υ) (1 p < ). A compactness question for operator A : w 2 p (υ) l q (u) acting from w 2 p (υ) into the space of sequences l q (u) (1 < p q < ) is investigated. Also estimates of approximation numbers for the embedding operator A are considered. Besides, it is possible to apply these research methods to the spectral theory of difference analogues of differential operators. [1] B. Musilimov, M. Otelbaev, Estimate of the least eigenvalue of a class of matrices that corresponds to the Sturm- Liouville difference equation, Zh. Vychisl. Mat. i Mat. Fiz., 21 (1981), no. 6, (Russian), [2] E. Z. Grinshpun, M. Otelbaev, Smoothness of the solution to the Sturm-Liouville equation in L 1 (, ), Izv. AN KazSSR, Ser. phys.-mat, no 5 (1984), (Russian), [3] K. T. Mynbayev, M. O. Otelbayev, Weighted functional spaces and differential operators spectrum, Moscow, Nauka, 1988 (Russian), [4] E. S. Smailov, Difference embedding theorems for weighted Sobolev spaces and their applications, Soviet Math. Dokl., vol. 270, no 1 (1983), (Russian, English), [5] G. Muchamediev, Spectrum of a difference operator and some embedding theorems, Kraevye zadachi dlya dif. ur. i ich prilozh. v mechanike i technike, Alma-Ata, Nauka (1983), (Russian), [6] R. Oinarov, A. P. Stikharnyi, Criteria for the boundedness and compactness of a difference inclusion, Mat. Zametki, 50 (1991), no. 5, 54-60; translation in Math. Notes 50 (1991), no. 5-6, (Russian, English), [7] A. T. Bulabaev, A. T. Muchambetzhanov, Embedding theorems for some multi-dimensional spaces, Izv. AN RK, ser. phys.-mat., no 3 (1992) (Russian), [8] A. T. Bulabaev, A. T. Muchambetzhanov, On some difference embedding theorems, Sbornik KazGNU (1993) (Russian), [9] M. S. Bichegkuev, On the spectrum of difference and differential operators in weighted spaces, Funktsional. Anal. i Prilozhen., 44 (2010), no. 1, 80-83; translation in Funct. Anal. Appl. 44 (2010), no. 1, (Russian, English), [10] A. G. Baskakov, Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations, and semigroups of difference relations, Izv. Ross. Akad. Nauk Ser. Mat. 73 (2009), no. 2, 3-68; translation in Izv. Math. 73 (2009), no. 2, (Russian, English), [11] M. Otelbaev, L. K. Kussainova, Spectrum estimates for one class of differential operators, Sbornik trudov Ins-ta matem. NAN Ukraine Operators theory, differential equation and function theory, vol. 6, no. 1 (2009), (Russian), [12] L. Kussainova, A. Ospanova, An Embedding Theorem for Difference Weighted Spaces, to appear in Proceedings of The World Congress on Engineering, Leili Kussainova: L. N. Gumilyov Eurasian National University, Department of Mechanics and Mathematics, Astana, Kazakhstan, [email protected] Ademi Ospanova: L. N. Gumilyov Eurasian National University, Department of Theoretical Informatics, Astana, Kazakhstan, [email protected]

73 International Congress in Honour of Professor Ravi P. Agarwal On (λ, I) Statistical Convergence of Order α of Sequences of Function Hacer Şengül and Mikail Et In this talk, we introduce and examine the concepts of pointwise (λ, I) statistical convergence of order α and pointwise w p (f, λ, I) summability of order α of sequences of real valued functions and we investigated between their relationship. We aim some notions and results from the statistical convergence of order α of sequences of function are extended to the I convergence of order α of sequences of function. [1] Connor, J. S. The Statistical and strong p Cesàro convergence of sequences, Analysis 8 (1988), 47-63, [2] Çolak, R. Statistical convergence of order α, Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub, 2010: , [3] Savas, Ekrem; Das, Pratulananda. A generalized statistical convergence via ideals, Appl. Math. Lett. 24 (2011), no. 6, , [4] Šalát, T. ; Tripathy, B. C. and Ziman M. On I convergence field, Ital. J. Pure Appl. Math. No. 17 (2005), 45 54, [5] Kostyrko, P. ; Šalát, T. and Wilczyński, W. I convergence, Real Anal. Exchange 26 (2000/2001), , [6] Et, M. ; Altınok, H. and Altın, Y. On generalized statistical convergence of order α of difference sequences, J. Funct. Spaces Appl. 2013, Art. ID , 7 pp, [7] Et, M. ; Çınar, M. and Karakaş, M. On λ statistical convergence of order α of sequences of function, J. Inequal. Appl. 2013, 2013:204, 8 pp, [8] Mursaleen, M. λ statistical convergence, Math. Slovaca, 50(1) (2000), , [9] Çolak, R. and Bektaş Ç. A. λ statistical convergence of order α, Acta Mathematica Scientia 31(3) (2011), Hacer Şengül: Siirt University, Faculty of Science, Department of Mathematics, Siirt-Turkiye, [email protected] Mikail Et: Firat University, Faculty of Science, Department of Mathematics, Elazig-Turkiye, [email protected]

74 74 International Congress in Honour of Professor Ravi P. Agarwal 40 Range Kernel Orthogonality of Generalized Derivations Messaoudene Hadia Let L (H) be the algebra of all bounded linear operators acting on a complex separable and infinite dimensional Hilbert space H. For operators A, B, X L(H), we define the generalized derivation δ A,B associated with (A, B) by δ A,B (X) = AX XB for X L(H). The purpose of this work is to find for wich operators A, B L(H) we have: for all X L(H) and for all T kerδ A,B. T (AX XB) T Faculty of Economics sciences and Management, University of Tebessa-Algeria, [email protected]

75 International Congress in Honour of Professor Ravi P. Agarwal On Stancu Variant of q-baskakov-durrmeyer Type Operators P.N.Agrawal and A.Sathish Kumar In recent years, one of the most interesting areas of research in approximation theory is the application of q- calculus. Phillips [8], first introduced the q-analogue of well known Bernstein polynomials. Subsequently, several researchers proposed the q- analogues of exponential, Kantorovich and Durrmeyer type operators. Recently q-baskakov operators and their Kantorovich and Durrmeyer variants have been studied in [3, 1] and [2] respectively. Stancu [?] introduced a generalization of Bernstein polynomials by defining the positive linear operators P (α,β) n : C[0, 1] C[0, 1] by P n (α,β) (f, x) = n k=0 b n,k(x)f where b n,k (x) = ( n k) x k (1 x) n k and α, β are any two real numbers which satisfy the condition 0 α β. If α = β = 0, the above sequence of operators reduces to Bernstein polynomials. His work led many researchers to consider similar type of modification of various sequences of operators. In 2012, the authors [?] studied some approximation properties of the Baskakov-Durrmeyer-Stancu operators. Recently, we [1] introduced the q-analogue of Bernstein-Schurer-Stancu operators and discussed the local and global approximation results for these operators. To approximate Lebesgue integrable functions on the interval [0, ), Agrawal and Thamer [2] introduced the following operators: M n(f(t); x) = (n 1) p n,k (x) p n,k 1 (t)f(t)dt + (1 + x) n f(0), (41.1) k=1 0 ( n + k 1 ) where p n,k (x) = x k (1 + x) (n+k), x [0, ). The rate of pointwise approximation by the operators (41.1) k for functions of bounded variations was considered by Gupta in [5]. Later on, Gupta and Abel [6] proposed the Bezier- Durrmeyer integral variant of the operators (41.1) and studied the rate of convergence for functions of bounded variation. In the present paper, we propose to study the approximation properties for the Stancu type modification of Baskakov- Durrmeyer operators given by (41.1) based on q-integers. Let α, β be any two real numbers such that 0 α β, q (0, 1), n N and f C γ[0, ) = {f C[0, ) : f(t) = O(t γ )as t, for some γ > 0}, the q-analogue of the Stancu variant of the operators (41.1) is defined as follows: Mn,q α,β (f, x) = [n /A 1]q p q n,k (x) ( where p q n + k 1 ) n,k (x) = q k(k 1) 2 k q k=1 x k (1 + x) (n+k) q 0 ( ( q k 1 p q [n]qt n,k 1 (t)f + α )d qt + p qn,0 [n] q + β (x)f α [n] q + β ( k+α n+β ), ), (41.2), x [0, ). In the case, α = β = 0 and q 1, the above operators (41.2) reduce to (41.1). The purpose of this paper is to study some approximation properties of the operators defined in (41.2). First, we give the basic convergence theorem and then obtain Voronovskaja type theorem. Subsequently, we study the local approximation results and then obtain the rate of convergence in terms of the weighted modulus of continuity. Also, we study the A-statistical convergence of these operators. Finally, we consider a modification of the operators (41.2), following King s approach to get a better approximation. [1] P. N. Agrawal, V. Gupta and A. Sathish Kumar, On q-analogue of Bernstein-Schurer-Stancu operators, Appl. Math. Comput. 219 (2013) [2] P. N. Agrawal and K. Thamer, Approximation of unbounded functions by a new sequence of linear positive operators, J. Math. Anal. Appl. 225 (1998) [3] A. Aral and V. Gupta, Generalized q-baskakov operators, Math. Slovaca. 61 (4) (2011) [4] A. Aral and V. Gupta, On the Durrmeyer type modification of the q Baskakov type operators, Nonlinear Anal.: Theory Methods Appl. 72 (2010) [5] V. Gupta, Rate of approximation by a new sequence of linear positive operators, Comput. Math. Appl. 45 (12) (2003) [6] V. Gupta and U. Abel, Rate of convergence of bounded variation functions by a Bezier-Durrmeyer variant of the Baskakov operators, IJMMS. 9 (2004) [7] V. Gupta and C. Radu, Statistical approximation properties of q-baskakov-kantorovich operators. Cent. Eur. J. Math. 8 (1) (2009) [8] G. M. Phillips, Bernstein polynomials based on the q integers, The heritage of P.L. Chebyshev: A Festschrift in honor of the 70th-birthday of Professor T.J. Rivlin., Ann. Numer. Math. 4 (1997) Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee , India, pna [email protected], [email protected] Dedicated to Prof. R. P. Agarwal on his 67 th birthday.

76 76 International Congress in Honour of Professor Ravi P. Agarwal 42 Generalised Baskakov Kantorovich Operators P.N.Agrawal and Meenu Goyal For f L 1 [0, 1] (class of Lebesgue integrable functions on [0, 1]), Kantorovich introduced the operators n 1 K n(f; x) = (n + 1) p n,ν(x) χ(t)f(t)dt, ν=0 0 where p n,ν(x) = ( n ν) x ν (1 x) n ν, x [0, 1] is the Bernstein basis function and χ(t) is the characteristic function of the [ ] ν interval n+1, ν+1. n+1 Many authors have studied the approximation properties of these operators. subsequently, several authors have proposed the Kantorovich-type modification of different linear positive operators and studied their approximation properties. Recently in [1], Erencin defined the Durrmeyer type modification of generalised Baskakov operators introduced by Mihesan [2], as L n(f; x) = Wn,k a (x) 1 t k f(t)dt, x 0, B(k + 1, n) k=0 0 (1 + t) n+k+1 where Wn,k a ax (x) = e 1+x P k(n,a) k! x k (1+x), p k(n, a) = n+k k ( n k) (x) i a k i, and (x) 0 = 1, (x) i = x(x + 1)...(x + i 1) for i 1. i=0 Inspired by the above work, we consider the Kantorovich type modification of generalised Baskakov operators for the function f defined on C γ[0, ) := {f C[0, ) : f(t) M(1 + t) γ for some M > 0, γ > 0} as follows : K a n (f; x) = (n + 1) k=0 W a n,k (x) k+1 n+1 k n+1 f(t)dt, a 0. (42.1) The purpose of this paper is to study some local direct results, degree of approximation for a Lipschitz type space, approximation of continuous functions with polynomial growth, simultaneous approximation properties for the operators defined in (42.1). In the last section, we construct the bivariate case for these operators and then discuss the rate of convergence in terms of the modulus of continuity. [1] A. Erencin, Durrmeyer type modification of generalized operators, Appl. math. comput. (218) (2011) [2] V. Mihesan, Uniform approximation with positive linear operators generated by generalised Baskakov method, Automat. Comput. Appl. Math. (1) (1998) Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee , India, pna [email protected], [email protected]

77 International Congress in Honour of Professor Ravi P. Agarwal Approximate Solutions of Fractional Order Boundary Value Problems by a Novel Method Ali Akgul An approximate solution of a fractional order two-point boundary value problem (FBVP) is given in this work. We use the reproducing kernel Hilbert space method. In order to illustrate the applicability and accuracy of the present method, the method is applied to some examples. The results are compared with the ones obtained by the Cubic splines and sinc-galerkin methods. There are only a few studies regarding the application of reproducing kernel method to fractional order differential equations. Therefore, this study is going to be a new contribution and highly useful for the researchers in fractional calculus area of scientific research. Results of numerical examples show that the presented method is very effective. [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68: , 1950, [2] M. Cui, Y. Lin, Nonlinear numerical analysis in the reproducing kernel space. Nova Science Publishers Inc.,New York, 2009, [3] M. Inc, A. Akgul, The reproducing kernel Hilbert space method for solving Troeschs problem, Journal of the Association of Arab Universities for Basic and Applied Sciences, (2013) 14, 19-27, [4] I. Podlubny, Fractional di erential equations, volume 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999, [5] A. Secer, S. Alkan, M. A. Akinlar, and M. Bayram, Sinc- Galerkin method for approximate solutions of fractional order boundary value problems., Boundary Value Problems, 2013: Dicle University, Faculty of Education, Department of Mathematics, Diyarbakir-Turkiye, [email protected] This work was supported by Dicle University

78 78 International Congress in Honour of Professor Ravi P. Agarwal 44 Some Power Series on Archimedean and Non-Archimedean Fields Fatma Çalışkan In the present study, we proved that the theorem which was established in complex (Archimedean) field and p-adic (non- Archimedean) field has an analogue in the formal Laurent series (non-archimedean) field over a finite field. Hence we show that some power series in the formal Laurent series field on the finite field F take values either Liouville or from F(x) for Liouville arguments under certain conditions, where F(x) is the quotient field of the polynomial ring F[x] on the finite field F. [1] P. Bundschuh,Transzendenzmasse in Körpern formaler Laurenteihen, J. Reine Angew Math. 299/300 (1978), , [2] E. Dubois,On Mahler s Classification in Laurent Series Fields, Rocky Mt. J. of Math. 26 (1996), , [3] K. Mahler,Zur Approximation der Exponantialfunktion und des Logarithmus I, J. Reine Angew Math. 166 (1932), , [4] K. Mahler,Über eine Klassen-Einteilung der p-adischen Zahlen,Mathematica Leiden 3 (1961), , [5] M. H. Oryan, Über gewisse Potenzreihen deren Funktionswerte für Argumente aus der Menge der Liouvilleschen zahlen U-zahlen vom Grad m sind, İstanbul Üniv. Fen Fak. Mec. Seri A 47 ( ), 15-34, [6] M. H. Oryan, Über gewisse Potenzreihen deren Funktionswerte für Argumente aus der Menge der p-adischen Liouvilleschen zahlen p-adische U-zahlen vom Grad m sind, İstanbul Üniv Fen Fak. Mecm. Seri A, 47 ( ), Istanbul University, Faculty of Science, Department of Mathematics, Vezneciler/Istanbul, Turkey, fatmac@ istanbul.edu.tr

79 International Congress in Honour of Professor Ravi P. Agarwal Existence and Monotone Iteration of Symmetric Positive Solutions for Integral Boundary-Value Problems with φ-laplacian Operator Tugba Senlik and Nuket Aykut Hamal The purpose of this talk is to investigate the existence and iteration of symmetric positive solutions for integral boundaryvalue problem. An existence result of positive, concave and symmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. An example is worked out to demonstrate the main result. [1] H. Feng, Triple symmetric positive solutions for multipoint boundary-value problem with one-dimensional p-laplacian, Math. Comput. Model. 47 (2008), , [2] Y. Ding, Monote Iterative Method for Obtaining Positive Ssolutions of Integral Boundary-Value Problems with φ- Laplacian Operator, Electron. J. Dif. Equ., 219 (2012), 1-9, [3] M. Pei, S.K. Chang, Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem, Math. Comput. Model. 51 (2010), , [4] H. Pang, Y. Tong, Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions, Bound. Value Probl., 2013:150, [5] Y. Cui, Y. Zou, Monotone iterative method for differential systems with coupled integral boundary value problems, Bound. Value Probl., 2013:245, Tugba Senlik: Ege University, Faculty of Science, Department of Mathematics, Bornova, Izmir- Turkey, Nuket Aykut Hamal: Ege University, Faculty of Science, Department of Mathematics, Bornova, Izmir-Turkey, nuket. [email protected]

80 80 International Congress in Honour of Professor Ravi P. Agarwal 46 Analytical Calculation of Partial Differential Equations Applied to Electrical Machines With Ideal Halbach Permanent Magnets Mourad Mordjaoui, Ibtissam Bouloukza and Dib Djalel Recently several electrical machines and devices use high energy permanent magnets with different direction of flux penetration. For the design and dimensioning of these electromechanical systems, we must know the distribution of the magnetic field in each part of the magnetic system and in particular at the air gap in which the energy conversion takes place. Generally, Maxwell s partial differential equations supplemented by material s law are used to describe the magnetic field problems. However, a numerical calculation is necessary, especially with the complex geometry of these devices. This paper deals with an analytical calculation of magnetic field distribution of iron-cored internal rotor of surface mounted permanent magnetic synchronous motor with ideal Halbach magnetization. A Halbach array is a special arrangement of permanent magnets that concentrates the magnetic flux lines on one side while reducing the flux lines on the other side to nearly zero. The model is based on evaluation and calculation of governing partial differential equations at no load conditions. Both field and magnetic induction in airgap and magnet are presented. Results obtained are compared with those obtained by finite-element analysis. [1] K. Halbach, Design of permanent multipole magnets with oriented rare earth cobalt material, Nucl. Instrum. Methods, vol. 169, pp. 1 10, [2] Y. N. Zhilichev, Analytic solutions of magnetic field problems in slotless permanent magnet machines, Int. J. Comput. Math. Elect. Electron. Eng., vol. 19, no. 4, pp , [3] A. Rahideh, and T. Korakianitis, Analytical Magnetic Field Distribution of Slotless Brushless Machines with Inset Permanent Magnets, IEEE Trans. Magn., vol. 47, no. 06, pp , Jun [4] P-D. Pfister,.and Y. Perriard Slotless Permanent-Magnet Machines: General Analytical Magnetic Field Calculation, IEEE Trans. Magn., vol. 47, no. 06, pp , Jun [5] A. Rahideh, and T. Korakianitis, Analytical calculation of open-circuit magnetic field distribution of slotless brushless PM machines, Electrical Power and Energy Systems 44 (2013) [6] M. Marinescu and N. Marinescu, New concept of permanent magnet excitation for electrical machines Analytical and numerical computation, IEEE Trans. Magn., vol. 28, pp , [7] Z. P. Xia, Z. Q. Zhu, and D. Howe, Analytical Magnetic Field Analysis of Halbach Magnetized Permanent-Magnet Machines, IEEE Trans. Magn., vol. 44, no. 04, pp , Jul Mourad Mordjaoui: Electrical Engineering Department, University of 20 August Skikda Algeria, mordjaoui Ibtissam Bouloukza: Electrical Engineering Department, University of 20 August Skikda Algeria, Boulekza [email protected] Dib Djalel: Electrical Engineering Department, University of Tebessa. Tebessa, Algeria, [email protected]

81 International Congress in Honour of Professor Ravi P. Agarwal Principal Functions of Differential Operators with Spectral Parameter in Boundary Conditions Nihal Yokuş In this talk, we investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem y + q(x)y = λ 2 y, x R + = [0, ), and ( α0 + α 1 λ + α 2 λ 2) y (0) ( β 0 + β 1 λ + β 2 λ 2) y (0) = 0, where q is a complex valued function and α i, β i C, i = 0, 1, 2 with α 2, β 2 0. [1] M. A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis, Amer. Math. Soc. Trans. Ser.2. Vol.16 (1960), [2] E.P.Dolzhenko, Boundary value uniqueness theorems for analytic functions, Math. Notes 26 (1979), [3] V. A. Marchenko, Expansion in eigenfunctions of non-selfsdjoint singular second order differential operators, Amer. Math. Soc. Transl. Ser. 2 25(1963) [4] V. E. Lyance, A differential operator with spectral singularities I,II, Amer. Math. Soc. Trans., Ser.2, Vol.60 (1967), , Karamanoglu Mehmetbey University, Faculty of Kamil Özdag Science, Department of Mathematics, Karaman-Turkiye, [email protected] This is joint work with Turhan Köprübaşı.

82 82 International Congress in Honour of Professor Ravi P. Agarwal 48 Generalized Typically Real Functions S.Kanas and A.Tatarczak Let f(z) = z + a 2 z 2 + be regular in the unit disk and real valued if and only if z is real and z < 1. Then f is said to be typically real function. Rogosinski found the necessary and sufficient condition for a regular function to be typically-real. The main purpose of the presented paper is a consideration of the generalized typically-real functions defined via the generating function of the generalized Chebyshev polynomials of the second kind where 1 p, q 1, θ 0, 2π, z < 1. Ψ p,q(e iθ 1 ; z) = (1 pze iθ )(1 qze iθ ) = U n(p, q; e iθ )z n, n=0 [1] W. Fenchel, Bemerkungen über die in Einheitskreis meromorphen schlichten Funktionen, Preuss. Akad. Wiss. Phys. - Math. Kl. 22/23(1931), [2] W. Janowski, Extremal problem for a family of functions with positive real part and for some related families, Ann. Polon. Math 23(1970), [3] J. A. Jenkins, On a conjecture of Goodman concerning meromorphic univalent functions, Michigan Math. J. 9 (1962), [4] G. Loria,Spezielle Algebraische und Transzendente Ebene Kurven, Tjeorie und Geschichte, Vol. I, transl. F. Schütte, Teubner, Leipzig, [5] I. Naraniecka, J. Szynal, A. Tatarczak, The generalized Koebe function, Trudy Petrozawodskogo Universiteta, Matematika 17 (2010), [6] I. Naraniecka, J. Szynal, A. Tatarczak An extension of typically-real functions and associated orthogonal polynomials, Ann. UMCS, Mathematica 65(2011), [7] Ch. Pommerenke, Linear-invariant Familien analytischer Funktionen, Mat. Ann. 155 (1964), [8] S. Richardson, Some Hele-Shaw flows with time-dependent free boundaries, J. Fluid Mech. 102 (1981), [9] M. S. Robertson, On the coefficients of typically real functions, Bull. Amer. Math. Soc. 41(1935), [10] W. W. Rogosinski, Über positive harmonische Entwicklungen und typische-reelle Potenzreihen, Math.Z. 35(1932), [11] C. Zwikker, The Advanced Geometry of Plane Curves and their Applications, Dover, New York, S.Kanas: University of Rzeszow, Faculty of Mathematics and Natural Sciences, ul. S. Pigonia 1, Rzeszow, Poland, [email protected] A.Tatarczak: Maria Curie-Sklodowska University in Lublin, Department of Mathematics, Poland, [email protected] This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, Faculty of Mathematics and Natural Sciences, University of Rzeszow.

83 International Congress in Honour of Professor Ravi P. Agarwal The Abel-Poisson Summability of Fourier Series in a Banach Space with Respect to a Continuous Linear Representation Seda Öztürk Let (C, +,.) denote the field of the complex numbers, T be the topological group of the unit circle with respect to the Euclidian topology, H a complex Banach space, α a continuous isometric linear representation of T in H and x an element of H. In [2,3] a fourier series of x with respect to a continuous isometric linear representation α of the form + F k (α, x) k= is defined where +π F k (α, x) := 1 2π e ikt α(t)(x)dt π for every k Z, and it is proved that this series converges to x in sense of Cesàro summability method. In this work, it is directly proved that the series + k= F k (α, x) is Abel-Poisson summable. [1] Lybich,Y.I, Introduction to the Theory of Banach Representations of Groups, Birkhäuser,Berlin (1988), [2] Khadjiev.D, Çavuş.A, Fourier series in Banach spaces, in: M.M. Lavrentyev (Ed.), Ill-posed and Non-classical Problems of Mathematical Physics and Analysis, Proc. of the Internat. Conf., Samarkand, Uzbekistan, in: Inverse Ill-posed Probl. Ser., VSP, Utrehct/Boston, 2003, pp , [3] Khadjiev, D., The widest continuous integral. J. Math. Anal. Appl. 326, (2007) [4] Vretblad, A., Fourier Analysis and Its Applications,Springer-Verlag,New York,Inc., Karadeniz Technical University, Faculty of Science, Department of Mathematics, Trabzon-Turkey, [email protected]

84 84 International Congress in Honour of Professor Ravi P. Agarwal 50 Existence of Solutions for Integral Boundary Value Problems in Banach Spaces Fulya Yoruk Deren and Nuket Aykut Hamal In this talk, by using the Sadovski fixed point theorem, we establish the existence results of solutions for nonlinear boundary value problems of second order differential equations with integral boundary conditions in Banach spaces. [1] R. P. Agarwal, D. O Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Acad., Dordrecht (2001). [2] D. Guo, V. Lakshmikantham, X. Liu; Nonlinear Integral Equation in Abstract Spaces, Kluwer Academic publishers, Dordrecht (1996). [3] Y. Liu, Boundary Value Problems for Second Order Differential Equations on Unbounded Domains in a Banach Space, Appl. Math. Comput. 135, (2003), [4] Y. Liu, Multiple Bounded Positive Solutions to Integral Type BVPs for Singular Second Order ODEs on the Whole Line, Abstract and Applied Analysis, Volume 2012, Article ID [5] M. Feng, D. Ji, W. Ge, Positive Solutions for a Class of Boundary Value Problem with Integral Boundary Conditions in Banach Spaces, Journal of Computational and Applied Mathematics 222, (2008), [6] F. Yoruk Deren, N. Aykut Hamal, Second Order Boundary Value Problems with Integral Boundary Conditions on the Real Line, Electronic Journal of Differential Equations, Vol (2014), No. 19, [7] K. Demling; Ordinary Differential Equations in Banach Spaces, Springer-Verlag, Berlin (1977). Fulya Yoruk Deren: Ege University, Faculty of Science, Department of Mathematics, 35100, Bornova, Izmir-Turkey, [email protected] Nuket Aykut Hamal: Ege University, Faculty of Science, Department of Mathematics, 35100, Bornova, Izmir-Turkey, [email protected]

85 International Congress in Honour of Professor Ravi P. Agarwal Existence and Uniqueness Solution of Electro-Elastic Antiplane Contact Problem with Friction Mohamed Dalah, Khoudir Kibeche, Amar Megrous, Ammar Derbazi and Soumia Ahmed Chaouache We study electro-mechanical problem modeling the antiplane shear deformation of a cylinder in frictional contact with a rigid foundation. The material is assumed to be electro-elastic and the foundation is assumed to be electrically conductive and the friction is modeled with Tresca s law. For each problem we present the mathematical model, its variational formulation, and state an existence and uniqueness result. [1] Borelli, A., Horgan C.O., Patria, M. C., Saint-Venant s principal for antiplane shear deformations of linear piezoelectric materials. SIAMJ. Appl. Math., 62, (2002) [2] Chau, O., Dynamic contact problems for viscoelastic materials. Proceedings of Fourteenth International Symposium on Mathematical theory of networks and systems., (MTNS 2000), Perpignan (2000). [3] M. Fémond, Adhérence des Solides, J. Mécanique Théorique et Appliquée., 6, (1987). [4] Shillor, M., Sofonea, M., Telega, J. J., Models and Analysis of Quasistatic Contact. Lect. Notes Phys., 655, Springer, Berlin Heidelberg, (2004). [5] Sofonea, M., Essoufi, El H., Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl., 14, (2004). [6] Sofonea, M., Dalah, M., Antiplane Frictional Contact of Electro-Viscoelastic Cylinders. Electronic Journal of Differential Equations. no. 161, (2007) [7] Sofonea, M., Dalah, M., Ayadi, A., Analysis of an antiplane electro-elastic contact problem. Adv. Math. Sci. Appl., 17, (2007) [8] Dalah, M., Analysis of electro-viscoelastic antiplane contact problem with total slip rate dependent friction. Electronic Journal of Differential Equations., no. 118, (2009) Mohamed Dalah: University of Constantine 1, Faculty of Sciences, Department of Mathematics, B.P. 325 Route Ain El Bey, Constantine 25017, Algeria, [email protected] This work is supported in part by la Direction Générale de la Recherche Scientifique et du Développement Technologique CNEPRU project & PNR Project, , CODE (valeur) : 8/u250/4506 registered in University Constantine 1, Algeria, under grant B , Title: Modélisation mathématiques pour les problèmes Electro-Elastique et Visco-Elastique : analyse, optimisation et approche numérique des modã ès. number UAP(F)-2012/15.

86 86 International Congress in Honour of Professor Ravi P. Agarwal 52 Almost Convex Valued Perturbation to Time Optimal Control Sweeping Processes Doria Affane and Dalila Azzam-Laouir We prove existence of solution for first order differential inclusion governed by the sweeping process of the form u(t) N K(t) u(t) + F (u(t)) u(t ) K(t) (52.1) u(0) = u 0 where the perturbation F is an upper semicontinuous multifunction with compact almost convex values. Moreover, we prove the existence of solutions to an associate time optimal control problem. Laboratoire LMPA, Université de Jijel, [email protected]

87 International Congress in Honour of Professor Ravi P. Agarwal Evolution Problem Governed by Subdifferential Operator Mustapha Yarou In the present talk we consider the Cauchy problem for first order differential inclusion of the form ẋ(t) F (x(t)) + f(t, x(t)), x(0) = x 0 (53.1) where F is a given set-valued map with nonconvex values and f is a Carathéodory function. The nonconvexity of the values of F do not permit the use of classical technique of convex analysis to obtain the existence of solution to this problem (see for instance [2]). One way to overcome this fact is to suppose F upper semicontinuous cyclically monotone, ie. the values of F are contained in the subdifferential of a proper convex lower semicontinuous function. The first result is du to [6] when f 0 and [1] for the problem (53.1) in the finite dimensional setting. An extension of [6] is obtained by [3] and [4] in the finite and infinite dimensional setting, under the assumption that F (x) is contained in the subdifferential of a locally Lipschitz and regular function. A different class of function has been used in [5] to solve the same problem, namely the authors take F (x) in the proximal subdifferential of a locally Lipschitz uniformly regular function and proved that any convex lower semicontinuous function is uniformly regular. We prove that, for locally Lipschitz functions, the class of convex functions, the class of lower-c 2 functions and the class of uniformly regular functions are strictly contained within the class of regular functions and we present existence results to problem (1.1) in R n and in an infinite dimensional Hilbert space by replacing the additional assumptions in [3] and [4] by a weaker and more natural condition. [1] F. Ancona; G. Colombo, Existence of solutions for a class of nonconvex differential Inclusions, Rend. Sem. Mat. Univ. Padova, Vol. 83, 71-76, (1990). [2] J. P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, (1984). [3] H. Benabdellah; Sur une classe d équations différentielles multivoques semi-continue superieurement á valeurs non convexes, Séminaire d Analyse Convexe, Montpellier, Exposé No. 6, [4] H. Benabdellah; C. Castaing and A. Salvadori Compactness and Discretization Methods for differential Inclusions and Evolution Problems, Atti. Sem. Mat. Univ. Modena, XLV, 9-51, (1997). [5] M. Bounkhel; Existence Results of Nonconvex Differential Inclusions, J. Portugaliae Mathematica, Vol. 59 (2002), No. 3, pp [6] A. Bressan; A. Cellina; G. Colombo, Upper semicontinuous differential Inclusions without convexity, Proc. Amer. Math. Soc., 106, , (1989). Laboratoire de Mathématiques Pures et Appliquées, Jijel University, Algeria, Laboratoire de Mathématiques Pures et Appliquées, Département de Mathématiques, Jijel University, Algeria, [email protected]

88 88 International Congress in Honour of Professor Ravi P. Agarwal 54 Nonlinear Elliptic Problem Related to the Hardy Inequality with Singular Term at the Boundary B.Abdellaoui, K.Biroud, J.Davila and F.Mahmoudi Let Ω R N be a bounded regular domain of R N and 1 < p <. The paper is divided in two main parts. In the first part we prove the following improved Hardy Inequality for convex domains. Namely, for all φ W 1,p 0 (Ω), we have φ p dx ( p 1 φ p p )p d p dx C φ (log p Ω Ω Ω ( )) D p dx, d where d(x) = dist(x, Ω), D > sup d(x) and C is a positive constant depending only on p, N and Ω. The optimality of the x Ø exponent of the logarithmic term is also proved. In the second part we consider the following class of elliptic problem u = uq d 2 in Ω, u > 0 in Ω, u = 0 on Ω, where 0 < q 2 1. We investigate the question of existence and nonexistence of positive solutions depending on the range of the exponent q. Boumediene Abdellaoui: Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen 13000, Algeria, [email protected] K. Biroud: Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen 13000, Algeria, kh J. Davila: Departamento de Ingenieria Matematica, CMM, Universidad de Chile, Casilla Correo 3, Santiago, Chile, [email protected] F. Mahmoudi: Departamento de Ingenieria Matematica, CMM, Universidad de Chile, Casilla Correo 3, Santiago, Chile, [email protected]

89 International Congress in Honour of Professor Ravi P. Agarwal On Periodic Solutions of Nonlinear Differential Equations in Banach Spaces Abdullah Çavuş, Djavvat Khadjiev and Seda Öztürk Let A denote the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a complex Banach space H. In this work, an analog of the resolvent operator which is called quasi-resolvent operator R λ is defined for points of the spectrum, some equivalent conditions for compactness of the quasi-resolvent operators R λ are given. Then using these and some results obtained in [1-7], some theorems on existence of periodic solutions to the non-linear equations Φ(A)x = f(x) are given,where Φ(A) is a polynomial of A with complex coefficients and f is a continuous mapping of H into itself. [1] Y.L. Lybich, Introduction to the Theory of Banach Representations of Groups, Birkhäuser,Berlin (1988), [2] J. Andres,B. Krajc, Periodic solutions in a given set of differential systems J. Math.Anal. Appl., 264 (2001) [3] H. Bart, Periodic strongly continuous semigroups, Ann. Mat. Pura. Appl. 115(1977) [4] M. Bartha, Periodic solutions for differential equations with state-dependent delay and positive feedback, Nonlinear Analysis 53 (2003) [5] A. Çavuş, D. Khadjiev, M. Kunt,On periodic one-parameter groups of linear operators in a Banach space and applications, Journal of Inequalities and Applications, 288(2013),1-20. [6] D. Khadjiev, A.Çavuş, Fourier series in Banach spaces, Inverse and Ill-Posed Problems Series, Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis, [7] D. Khadjiev,The widest continuous integral, J. Math. Anal. Appl., 326 (2007), Abdullah Çavuş: Karadeniz Technical University, Faculty of Science,Department of Mathematics,Trabzon-Turkey, [email protected] Djavvat Khadjiev: Karadeniz Technical University, Faculty of Science,Department of Mathematics,Trabzon-Turkey, [email protected] Seda Öztürk: Karadeniz Technical University, Faculty of Science,Department of Mathematics,Trabzon-Turkey, seda. [email protected]

90 90 International Congress in Honour of Professor Ravi P. Agarwal 56 Generalized α-ψ-contractive type M Mappings of Integral Type Erdal Karapinar, P.Shahi and Kenan Tas In this talk, we introduce the two classes of generalized α-ψ-contractive type mappings of integral type and to analyze the existence of fixed points for these mappings in complete metric spaces. Our results are improved versions of a multitude of relevant fixed point theorems of the existing literature. Examples are provided to support the results and concepts presented herein. [1] Samet, B, Vetro, C. and Vetro, P., Fixed point theorem for α-ψ contractive type mappings, Nonlinear Anal. 75 (2012) [2] Banach, S., Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fundamenta Mathematicae 3 (1922) [3] Ali, M. U. and Kamran, T.: On (α, ψ)-contractive multi-valued mappings, Fixed Point Theory Appl., :137 doi: / [4] Berzig, M. and Rus, M., Fixed point theorems for α-contractive mappings of Meir-Keeler type and applications, math.gn/ [5] Jleli, M., Karapinar, E. and Samet, B., Best proximity points for generalized alpha-psi-proximal contractive type mappings, Journal of Applied Mathematics, Article No: (2013) [6] Jleli, M., Karapinar, E. and Samet, B., Fixed point results for α ψ λ contractions on gauge spaces and applications, Abstract and Applied Analysis, (2013) Article Id: [7] Karapinar, E. and Samet, B., Generalized α-ψ-contractive type mappings and related fixed point theorems with applications, Abstract and Applied Analysis 2012 Article ID , 17 pages doi: /2012/ [8] Mohammadi, B. Rezapour, S. Shahzad, N.: Some results on fixed points of α-ψ-ciric generalized multifunctions. Fixed Point Theory Appl., :24 doi: / [9] Rus, I. A.: Generalized contractions and applications, Cluj University Press, Cluj-Napoca, [10] Bianchini R.M., Grandolfi, M.: Transformazioni di tipo contracttivo generalizzato in uno spazio metrico, Atti Acad. Naz. Lincei, VII. Ser., Rend., Cl. Sci. Fis. Mat. Natur. 45 (1968), [11] Proinov, P.D. : A generalization of the Banach contraction principle with high order of convergence of successive approximations Nonlinear Analysis (TMA) 67 (2007), [12] Proinov, P.D. : New general convergence theory for iterative processes and its applications to Newton Kantorovich type theorems, J. Complexity 26 (2010), [13] Shahi, P., Kaur, J. and Bhatia, S. S., Fixed point theorems for α-ψ-contractive type mappings of integral type with applications, accepted for publication in Journal of Nonlinear and Convex Analysis. [14] Agarwal, R. P., El-Gebeily, M.A. and Regan, D. O, Generalized contractions in partially ordered metric spaces, Applicable Analysis 87 (2008) 1 8. [15] Aliouche, A., A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type, Journal of Mathematical Analysis and Applications 322, no. 2, (2006), [16] Arandjelović, I., Kadelburg, Z. and Radenović, S., Boyd-Wong-type common fixed point results in cone metric spaces, appl. Math. Comput. 217, (2011), [17] Berinde, V., Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, [18] Berinde, V., Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces, math.fa/ [19] Berinde, V. and Borcut, M., Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Analysis 74 (2011) Erdal Karapinar: Department of Mathematics, Atilim University 06836, Incek, Ankara, Turkey, ekarapinar@atilim. edu.tr P. Shahi: School of Mathematics and Computer Apps, Thapar University, Patiala , Punjab, India, [email protected] Kenan Tas: Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey, [email protected]

91 International Congress in Honour of Professor Ravi P. Agarwal Caristi s Fixed Point Theorem in Fuzzy Metric Spaces Hamid Mottaghi Golshan In the present work, we extend Caristi s Fixed Point theorem, Ekeland s variational principle and Takahashi s maximization theorem to fuzzy metric spaces in the sense of George and Veeramani [A. George, P Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems. 64 (1994) ]. Further, a direct simple proof of the equivalences between these theorems is provided. [1] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions,. Amer. Math. Soc. 215 (1976) [2] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994) [3] I. Ekeland, Remarques sur les probléms variationnels 1, C. R. Acad. Sci., Paris Sér. A B, 275 (1972) [4] W. Takahashi, Existence theorems generalizing fixed point theorems for multivalued mappings, in: Fixed Point Theory and Applications, Marseille, 1989, in: Pitman Res. Notes Math. Ser., vol. 252, Longman Sci. Tech., Harlow, 1991, pp Department of Mathematics, Islamic Azad University, Ashtian Branch, Ashtian, Iran, Tel,Fax: [email protected] or [email protected] This research was partially supported by Islamic Azad University, Ashtian branch.

92 92 International Congress in Honour of Professor Ravi P. Agarwal 58 Determination of the Unknown Coefficient in Time Fractional Parabolic Equation with Dirichlet Boundary Conditions Ebru Ozbilge and Ali Demir In this talk the mathematical analysis of the inverse coefficient problem of identifying the unknown coefficient k(x) in the linear time fractional parabolic equation D α t u(x, t) = (k(x)ux)x 0 < α 1, with Dirichlet boundary conditions u(0, t) = ψ 0 (t), u(1, t) = ψ 1 (t) was discussed. By defining the input-output mappings Φ[ ] : K C 1 [0, T ] and Ψ[ ] : K C 1 [0, T ] the inverse problem is reduced to the problem of their invertibility. Hence the main purpose of this study is to investigate the distinguishability of the input-output mappings Φ[ ] and Ψ[ ]. This work shows that the input-output mappings Φ[ ] and Ψ[ ] have distinguishability property. Moreover, the value k(0) of the unknown diffusion coefficient k(x) at x = 0 and the value k(1) of the unknown diffusion coefficient k(x) at x = 1 can be determined explicitly by making use of measured output data (boundary observation) k(0)u x(0, t) = f(t) and k(1)u x(1, t) = h(t) respectively, which brings greater restriction on the set of admissible coefficients. It is also shown that the measured output data f(t) and h(t) can be determined analytically by a series representation, which implies that the input-output mappings Φ[ ] : K C 1 [0, T ] and Ψ[ ] : K C 1 [0, T ] can be described explicitly. [1] J.Canon and Y.Lin, An inverse problem for finding a parameter in a semi-linear heat equation, J. Math. Anal.Appl., 145 (1990), , [2] J.Canon and Y.Lin, Determination of a parameter p(t) in some quasi-linear parabolic differential equations, Inv.Prob., 4 (1998), 35-44, (3-4): , [3] J.Canon and Y.Lin, Determination of source parameter in parabolic equations, Mechanica, 27 (1992), 85-94, [4] M.Dehgan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. Meth. Part. Diff. Eq., 21 (2004), , [5] A.Demir and E. Ozbilge, Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient, Math. Meth. Appl.Sci.,31 (2008), , [6] A.Demir and E.Ozbilge,Semigroup approach for identification of the unknown coefficient in a quasi-linear parabolic equation, Math. Meth. Appl.Sci.,30 (2007), , [7] A.Demir and A.Hasanov, Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach, J. Math. Anal. Appl.,340 (2008),5-15, [8] A. Fatullayev, Numerical procedure for the simultaneous determination of unknown coefficients in a parabolic equation, Appl.Math.Comp.,164 (2005), , [9] V.Isakov, Inverse problems for partial differential equations, Springer-Verlag, 1998, [10] R.E. Showalter, Monotone operators in Banach Spaces and nonlinear partial differential equations, United States of America: American Mathematical Society, 1997, [11] Y.Luchko, Initial boundary value problems for the one dimensional time-fractional diffusion equation,frac.calc.appl.anal.,15 (2012), Ebru Ozbilge: Izmir University of Economics, Department of Mathematics, Faculty of Science and Literature, Sakarya Caddesi, No.156, 35330, Balcova, Izmir, Turkey, [email protected] Ali Demir: Kocaeli University, Department of Mathematics, Umuttepe, 41380, Izmit, Kocaeli, Turkey, [email protected] The research was supported by parts by the Scientific and Technical Research Council (TUBITAK) of Turkey and Izmir University of Economics.

93 International Congress in Honour of Professor Ravi P. Agarwal On p-adic Ising Model with Competing Interactions on the Cayley Tree Farrukh Mukhamedov, Hasan Akın and Mutlay Dogan It is known that the Ising model is one of the most studied models in statistical mechanics. Since, this model is related to a number of outstanding problems in statistical and mathematical physics, and in graph theory. On the other hand, that most of modern science is based on mathematical analysis over real and complex numbers. However, it is turned out that for exploring complex hierarchical systems it is sometimes more fruitful to use analysis over p-adic numbers and ultrametric spaces. Therefore, in this direction a lot of investigations are devoted to the mathematical physics models over p-adic field. In the present paper, we further develop the theory of statistical mechanics. Namely, we consider p-adic Ising model with competing next-nearest-neighbor interactions on the Cayley tree of order two. Note that usual p-adic Ising model on the tree was earlier studied by the first author. A main aim of this work is the establishment of a phase transition phenomena for the mentioned model. Here the phase transition means the existence of two nontrivial p-adic Gibbs measures. To prove the occurrence of the phase transition we reduce the problem to the existence of at leat two solutions of nonlinear difference equations. Farrukh Mukhamedov: Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box, 141, 25710, Kuantan, Pahang, Malaysia, [email protected], darrukh [email protected] Hasan Akın: Department of Mathematics, Faculty of Education, Zirve University, Kizilhisar Campus, Gaziantep, 27260, Turkey, [email protected] Mutlay Dogan: Department of Mathematics, Faculty of Education, Zirve University, Kizilhisar Campus, Gaziantep, 27260, Turkey, [email protected]

94 94 International Congress in Honour of Professor Ravi P. Agarwal 60 A Spectral Domain Computational Technique Dedicated to Fault Detection in Induction Machine A.Medoued, A.Lebaroud, O.Boudebbouz and D.Sayad This paper presents a computational technique in the spectral domain based on the analysis of three parameters of any signal issued from experimental measurements. This analysis is applied for the purpose of a better detection and characterization of defects in induction machine. This technique concerns the analysis of the power spectrum signal, the stator current signal decomposed in terms of the instantaneous phase and instantaneous frequency using Hilbert transform of the current signal absorbed by the induction machine. The advantage of this technique is that it makes it possible to highlight the defects of the machine components independently of the amplitude of the measured signals and regardless of load level. [1] G. B. Kliman, W. J. Premerlani, R. A. Koegl, and D. Hoeweler, A new approach to on-line turn fault detection in AC motors, in Conf. Rec. IEEE, IAS Annu. Meeting, San Diego, CA, (1996), p [2] Abdesselam Lebaroud, Guy Clerc, Abdelmalek Khezzar, Ammar Bentounsi, Comparison of the Induction Motors Stator Fault Monitoring Methods Based on Current Negative Symmetrical Component, EPE Journal, Vol. 17, no 1 March (2007). [3] Abdesselam Lebaroud, Guy Clerc, Classification of Induction Machine Faults by Optimal Time Frequency Representations, IEEE Trans. on Industrial Electronics, vol. 55, no. 12, Dec (2008), p [4] Ibrahim Ali, El Badaoui Mohamed, Guillet François, Bonnardot Frédéric, A New Bearing Fault Detection Method in Induction Machines Based on Instantaneous Power Factor, IEEE Transactions on Industrial Electronics, Vol 55, No. 12, December (2008), p [5] Zwe-Lee Gaing; Neural Network induction machine Fault classification, IEEE Transactions on Power Delivery, Vol. 19, Issue: 4, (2004), p [6] F. Zidani, M. E. H. Benbouzid, D. Diallo, and M. S. Nait-Said, Induction motor stator faults diagnosis by a current Concordia pattern-based fuzzy decision system, IEEE Transactions on Energy Conversion, vol. 18, (2003), p [7] C. Concari, G. Franceschini, C. Tassoni, Differential Diagnosis Based on Multivariable Monitoring to Assess Induction Machine Rotor Conditions, IEEE Trans. on Industrial Electronics, vol. 55, no. 12, Dec (2008), p [8] A. Khezzar, El Kamel Oumaamar, M. Hadjami, M. Boucherma, M. Razik, H. Induction Motor Diagnosis Using Line Neutral Voltage Signatures, IEEE Trans. on Industrial Electronics, vol. 56, no. 11, Nov (2009), p [9] P. J Moore, Frequency relaying based on instantaneous frequency measurement, IEEE, Transaction on power delivery, vol. 11, No. 4, (1996). [10] G. Didier, Modélisation et diagnostic de la machine asynchrone en présence de défaillances, doctoral thesis, France, (2004). [11] A.Medoued, A.Lebaroud, A.Boukadoum, T.Boukra, G. Clerc, Back Propagation Neural Network for Classification of Induction Machine Faults, 8th SDEMPED, IEEE Symposium on Diagnostics for Electrical Machines, Power Electronics & Drives September 5-8, 2011, Bologna, Italy, pp , [12] Medoued Ammar, A Lebaroud and D Sayad, Application of Hilbert transform to fault detection in electric machines, Journal advances indifference equations, Springer Open Journal Volume A. Medoued, A. Lebaroud, A. Laifa, D. Sayad: Université du 20 août 1955-Skikda, Faculté de Technologie, Département de génie électrique, Alegria, [email protected], [email protected].

95 International Congress in Honour of Professor Ravi P. Agarwal Some Results on Double Fuzzy Topogenous Orders Vildan Çetkin and Halis Aygün The first aim of this talk is to introduce the concept of lattice valued double fuzzy topogenous structure. The second is to investigate the connections between the double fuzzy topogenous order, double fuzzy topology, double fuzzy interior operator and also double fuzzy proximity order. So, we have some results on lattice valued double fuzzy topogenous structure. [1] S. E. Abbas, A. A. Abd-Allah, Lattice valued double syntopogenous structure, Journal of the Association of Arab Universities for Basic and Applied Sciences, 10 (2011): 33-41, [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986): [3] G. Birkhoff, Lattice Theory, Ams Providence, 1995, [4] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968): , [5] V. Çetkin and H. Aygün, On (L, M)-Fuzzy Interior Spaces, Advances in Theoretical and Applied Mathematics, 5 (2010): , [6] V. Çetkin and H. Aygün, Lattice valued double fuzzy preproximity spaces,computers and Mathematics with Applications, 60 (2010): , [7] D. Çoker, An Introduction to Fuzzy Subspaces in Intuitionistic Fuzzy Topological Spaces,J. Fuzzy Math. 4 (1996): , [8] J. G. Garcia, S.E. Rodabaugh, Order-theoretic, topological, categorical redundancides of interval-valued sets, grey sets, vague sets, interval-valued intuitionistic sets, intuitionistic fuzzy sets and topologies, Fuzzy Sets and Systems, 156 (2005): , [9] U. Höhle, Upper Semicontinuous Fuzzy Sets and Applications, J. Math. Anall. Appl. 78 (1980): , [10] U. Höhle and E. P. Klement, Non-Classical Logic and Their Applications to Fuzzy Subsets, Kluwer Academic Publisher, Dordrecht, 1995, [11] B. Hutton, Normality in Fuzzy Topological Spaces, J. Math. Anal. Appl., 50 (1975): 74-79, [12] S. Jenei, Structure of Girard Monoids on [0,1] Chapter 10 in: Topological and Algebraic Structures in Fuzzy Sets, S.E.Rodabaugh, E.P.Klement eds., Kluwer Acad. Publ., 2003, [13] A.K.Katsaras, On fuzzy syntopogenous structures, Rev. Roum. Math. Pure Appl., 30 (1985): , [14] T. Kubiak, On Fuzzy Topolgies, Ph.D Thesis, A. Mickiewicz, Poznan (1985), [15] Y. M. Liu, M. K. Luo, Fuzzy Topology, World Scientific Publishing, Singapore, (1997): , [16] S. K. Samanta, T. K. Mondal, On Intuitionistic Gradation of Openness,Fuzzy Sets and Systems, 131 (2002): , [17] A. P. Šostak, On a Fuzzy Topological Structure, Suppl. Rend. Circ. Matem. Palermo ser. II, 11 (1985): Vildan Çetkin: Kocaeli University, Faculty of Arts and Science, Department of Mathematics, Umuttepe Campus, Kocaeli-Turkiye, [email protected], [email protected] Halis Aygün: Kocaeli University, Faculty of Arts and Science, Department of Mathematics, Umuttepe Campus, Kocaeli- Turkiye, [email protected]

96 96 International Congress in Honour of Professor Ravi P. Agarwal 62 Finding Fixed Points of Firmly Nonexpansive-Like Mappings in Banach Spaces Fumiaki Kohsaka We construct a strongly relatively nonexpansive sequence from a given sequence of firmly nonexpansive-like mappings with a common fixed point in Banach spaces. Using this construction, we next obtain two convergence theorems for firmly nonexpansive-like mappings in Banach spaces and discuss their applications to a zero point problem for maximal monotone operators and a convex feasibility problem. Let C be a nonempty subset of a real smooth Banach space X and J : X X the normalized duality mapping. A mapping T : C X is said to be firmly nonexpansive-like [2, 4] if for all x, y C. T x T y, J(x T x) J(y T y) 0 [1] K. Aoyama, Y. Kimura, and F. Kohsaka, Strong convergence theorems for strongly relatively nonexpansive sequences and applications, J. Nonlinear Anal. Optim. 3 (2012), [2] K. Aoyama and F. Kohsaka, Existence of fixed points of firmly nonexpansive-like mappings in Banach spaces, Fixed Point Theory Appl. 2010, Art. ID , 15 pages. [3] K. Aoyama and F. Kohsaka, Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings, Fixed Point Theory Appl. 2014, 2014: 95, 13 pages. [4] K. Aoyama, F. Kohsaka, and W. Takahashi, Three generalizations of firmly nonexpansive mappings: Their relations and continuity properties, J. Nonlinear Convex Anal. 10 (2009), [5] K. Aoyama, F. Kohsaka, and W. Takahashi, Strongly relatively nonexpansive sequences in Banach spaces and applications, J. Fixed Point Theory Appl. 5 (2009), [6] K. Ball, E. A. Carlen, and E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math. 115 (1994), [7] B. Beauzamy, Introduction to Banach spaces and their geometry, North-Holland Publishing Co., Amsterdam, [8] L. M. Brègman, The method of successive projection for finding a common point of convex sets, Soviet Math. Dokl. 6 (1965), [9] G. Crombez, Image recovery by convex combinations of projections, J. Math. Anal. Appl. 155 (1991), [10] Y. Kimura and K. Nakajo, The problem of image recovery by the metric projections in Banach spaces, Abstr. Appl. Anal. 2013, Art. ID , 6 pages. [11] B. Martinet, Détermination approchée d un point fixe d une application pseudo-contractante. Cas de l application prox, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A163 A165. [12] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization 14 (1976), [13] Y. Takahashi, K. Hashimoto, and M. Kato, On sharp uniform convexity, smoothness, and strong type, cotype inequalities, J. Nonlinear Convex Anal. 3 (2002), Department of Computer Science and Intelligent Systems, Oita University, Japan, [email protected]

97 International Congress in Honour of Professor Ravi P. Agarwal A Fourth Order Accurate Approximation of the First and Pure Second Derivatives of the Laplace Equation on a Rectangle A.A.Dosiyev and H.M.Sadeghi In this talk, we discuss an approximation of the first and pure second order derivatives of a solution of the Dirichlet problem on a rectangular domain. The boundary values on the sides of the rectangle are supposed to have the sixth derivatives satisfying the Hölder condition. On the vertices, besides the continuity condition, the compatibility conditions, which result from the Laplace equation for the second and fourth derivatives of the boundary values, given on the adjacent sides, are also satisfied. Under these conditions uniform approximation of order O(h 4 ) (h is the grid size), is obtained for the solution of the Dirichlet problem on a square grid, its first and pure second derivatives, by a simple difference schemes. A. A. Dosiyev: Eastern Mediterranean University, Department of Mathematics, Gazimagusa, Cyprus, Mersin 10, Turkey, [email protected] H. M. Sadeghi: Eastern Mediterranean University, Department of Mathematics, Gazimagusa, Cyprus, Mersin 10, Turkey, [email protected]

98 98 International Congress in Honour of Professor Ravi P. Agarwal 64 On the Positive Solutions for the Boundary Value Problems at Resonance Ummahan Akcan and Nüket Aykut Hamal In this study, we investigate the existence of two positive concave solutions to the second-order three-point boundary value η problems with integral boundary conditions, u (x) + f(x, u(x)) = 0, u (0) = u(0), u(1) = α u(s)ds, where 0 < η < 1 and 0 f C((0, 1) [0, + ), [0, + )). The interesting point here is that we consider the BVP to the resonance case αη(2 + η) = 4 to find a new existence result. The proof is based upon the Monoton Iterative Technique. [1] H. Liu, Z. Ouyang, Existence of solutions for second-order three-point integral boundary value problems at resonance, Boundary Value problem., (2013):197 doi: / [3] J. J. Nieto, Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance, Boundary Value Problem., (2013):130 doi: / , [4] A. Boucherif, Second-order boundary value problems with integral boundary conditions, Nonlinear Anal., 70 (2009) , [5] T. Jankowski, Differantial equations with integral boundary conditions, Journal of Computational and Applied Mathematics, 147 (2002) 1-8, [6] X. Zhang, Existence and iteration of monotone positive solutions for an elastic beam equation with a corner, Nonlinear Analysis: Real World Applications, 10 (2009) , [7] H. Pang, M. Feng, W. Ge, Existence and monotone iteration of positive solutions for a three-point boundary value problem, Applied Mathematics Letters, 21 (2008) , [8] Q. Yao, Successive Iteration and Positive Solutions for Nonlinear Second-Order Three-Point Boundary Value Problems, Computer and Mathematics with Applications, 50 (2005) , Ummahan Akcan: Anadolu University, Faculty of Science, Department of Mathematics, Eskisehir-Turkey, [email protected] Nüket Aykut Hamal: Ege University, Faculty of Science, Department of Mathematics, İzmir-Turkey, nuket.aykut@ege. edu.tr

99 International Congress in Honour of Professor Ravi P. Agarwal On Weighted Approximation of Multidimensional Singular Integrals Gümrah Uysal and Ertan Ibikli In this talk, we give some theorems about pointwise approximation to the functions belong to weighted Lebesgue space L 1,w (R n ), by family of convolution type singular integral operators. Moreover, we will verify the theoretical results with some graphical illustrations. [1] R. Taberski, Singular integrals depending on two parameters, Rocznicki Polskiego towarzystwa matematycznego, Seria I. Prace matematyczne, VII (1962), pp [2] A.D. Gadjiev, On convergence of integral operators depending on two parameters, Dokl. Acad. Nauk. Azerb. SSR, XIX (1963) No. 12, pp [3] A.D. Gadjiev, On the order of convergence of singular integrals depending on two parameters, Special questions of functional analysis and its applications to the theory of differential equations and functions theory, Baku, (1968), pp [4] P.L. Butzer and R.J.Nessel, Fourier Analysis and Approximation, Academic Press, New York, London, (1971). [5] H. Karslı and E. İbikli, On convergence of convolution type singular integral operators depending on two parameters, Fasc. Math., no.38, (2007), pp [6] G. Folland, Real Analysis: Modern Techniques, John Wiley & Sons, Second Edition, (1999). [7] G.A. Anastassiou and S.G. Gal, Approximation Theory: Moduli of Continuity and Global Smoothness Preservation. Birkhauser, Boston, (2000). [8] S.E. Almali, Approximation of non-convolution type of integral operators at Lebesgue point for non-integrable of function, International Congress On Computational and Applied Maths., Ghent, Belgium, (2012), pp. 27. Gümrah Uysal: Karabuk University, Faculty of Science, Department of Mathematics, Balıklarkayası Mevkii, Karabuk, Turkey, [email protected] Ertan İbikli: Ankara University, Faculty of Science, Department of Mathematics, Tandogan, Ankara, Turkey, Ertan. [email protected]

100 100 International Congress in Honour of Professor Ravi P. Agarwal 66 On Hermite-Hadamard Type Inequalities for ϕ Convex Functions via Fractional Integrals Mehmet Zeki Sarıkaya and Hatice Yaldız In this talk, we establish integral inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals for ϕ-convex functions and some new inequalities of right-hand side of Hermite-Hadamard type are given for functions whose first derivatives absolute values ϕ convex functions via Riemann-Liouville fractional integrals. [1] A.G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Math., 28 (1994), [2] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3) (2009), Art. 86. [3] Z. Dahmani, New inequalities in fractional integrals, International Journal of Nonlinear Scinece, 9(4) (2010), [4] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1) (2010), [5] Z. Dahmani, L. Tabharit, S. Taf, Some fractional integral inequalities, Nonl. Sci. Lett. A, 1(2) (2010), [6] Z. Dahmani, L. Tabharit, S. Taf, New generalizations of Gruss inequality usin Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3) (2010), [7] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, [8] S. S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett., 11(5) (1998), [9] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Springer Verlag, Wien (1997), [10] S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993, p.2. [11] J.E. Pečarić, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, [12] M.Z. Sarikaya and H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration, Abstract and Applied Analysis, Volume 2012 (2012), Article ID , 10 pages. [13] M. Z. Sarikaya, M. Buyukeken and M. E. Kiris, On some generalized integral inequalities for ϕ-convex functions, Studia Universitatis Babeş-Bolyai Mathematica, (2014)accepted. [14] M.Z. Sarikaya, E. Set, H. Yaldiz and N., Basak, Hermite -Hadamard s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, DOI: /j.mcm , 57 (2013) [15] M. Tunc, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat 27:4 (2013), [16] E. A. Youness, E-convex sets, E-convex functions and E-convex programming, J. of Optim. Theory and Appl., 102, No. 2 (1999), Düzce University, Faculty of Science and Arts, Department of Mathematics, Düzce-Turkiye, [email protected], [email protected]

101 International Congress in Honour of Professor Ravi P. Agarwal Behavior of Positive Solutions of a Multiplicative Difference Equation Durhasan Turgut Tollu, Yasin Yazlık and Necati Taşkara In this talk, we deal with the positive solutions of the multiplicative difference equation ay n 1 y n+1 = by ny n 1 + cy n 1 y n 2 + d, n N 0, where the coefficients a, b, c, d are positive real numbers and the initial conditions y 2, y 1, y 0 are nonnegative real numbers. Here, we investigate global character, periodicity, boundedness and oscillation of positive solutions of the above equation. [1] E. Camouzis and G. Ladas. Dynamics of Third-order Rational Difference Equations with Open Problems and Conjectures, Volume 5 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, FL, [2] C. Çinar, On the positive solutions of the difference equation x n+1 = x n 1 1+x nx n 1, Applied Mathematics and Computation, 150 (2004), [3] S. Stević, More on a rational recurrence relation, Appl. Math. E-Notes 4 (2004), [4] A. Andruch-Sobilo, M. Migda, Further properties of the rational recursive sequence x n+1 = Mathematica 26(3)(2006) ax n 1 b+cx nx n 1, Opuscula [5] I. Bajo, E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Differ. Equations Appl. 17 (10) (2011), x [6] S. Stević, On the difference equation x n = n k, Applied Mathematics and Computation 218 (2012), b+cx n 1 x n k [7] R. Abu-Saris, C. Cinar, I. Yalcinkaya, On the asymptotic stability of x n+1 = a+xnx n k, Computers & Mathematics x n+x n k with Applications, 56(5) (2008) [8] H.M. El-Owaidy, A.M. Ahmet, A.M. Youssef, On the dynamics of the recursive sequence x n+1 = αx n 1 β+γx p, Applied n 2 Mathematics Letters, 18 (9)(2005), [9] R. Karatas, Global behaviour of a higher order difference equation, Computers & Mathematics with Applications 60 (2010), [10] M. A. Obaid, E. M. Elsayed and M. M. El-Dessoky, Global Attractivity and Periodic Character of Difference Equation of Order Four, Discrete Dynamics in Nature and Society, (2012), Article ID: [11] M. E. Erdogan, Cengiz Çinar, Ibrahim Yalcinkaya, On the dynamics of the recursive sequence x n+1 = αx n 1, Computers & Mathematics with Applications, 61 (2011), β+γx 2 n 2 x n 4+γx n 2 x 2 n 4 [12] Y. Yazlik, On the solutions and behavior of rational difference equations, Journal of Computational Analysis and Applications, 17(3)(2014), [13] T.F. Ibrahim, On the third order rational difference equation x n+1 = x nx n 2, Int. J. Contemp. Math. x n 1(a+bx nx n 2) Sciences 4(27)(2009), [14] X. Yang, W. Su, B. Chen, G. M. Megson, and D. J. Evans, On the recursive sequence x n+1 = ax n 1+bx n 2 c+dx n 1 x n 2, Applied Mathematics and Computation, 162 (2005) [15] H. El-Metwally, E. M. Elsayed, Qualitative study of solutions of some difference equations, Abstract and Applied Analysis, Article ID , (2012), 16 pages. x [16] D. Simsek, C. Cinar and I. Yalcinkaya, On the recursive sequence x n+1 = n (5k+9), Taiwanese Journal 1+x n 4 x n 9...x n (5k+4) of Mathematics 12(5)(2008), [17] D. T. Tollu, Y. Yazlik, N. Taskara, On the Solutions of two special types of Riccati Difference Equation via Fibonacci Numbers, Advances in Difference Equations, 2013, 2013:174. Durhasan Turgut Tollu: Department of Mathematics-Computer Sciences, Faculty of Science, Necmettin Erbakan University, 42090, Konya-Türkiye, [email protected] Yasin Yazlik: Department of Mathematics, Faculty of Science and Art, Nevsehir Haci Bektas Veli University, Nevsehir-Türkiye, [email protected] Necati Taskara: Department of Mathematics, Faculty of Science, Selcuk University, Campus, 42075, Konya- Türkiye, [email protected]

102 102 International Congress in Honour of Professor Ravi P. Agarwal 68 A New Generalization of the Midpoint Formula for n Time Differentiable Mappings which are Convex Çetin Yıldız and M.Emin Özdemir Let f : I R R be a convex mapping defined on the interval I of real numbers and a, b I, with a < b. The following double inequality is well known in the literature as the Hermite-Hadamard inequality: ( ) a + b f 1 b f(a) + f(b) f(x)dx. 2 b a a 2 A function f : [a, b] R R is said to be convex if whenever x, y [a, b] and t [0, 1], the following inequality holds: f(tx + (1 t)y) tf(x) + (1 t)f(y). In this paper, a new identity for n time differentiable functions is established and by using the obtained identity, some new inequalities Hermite-Hadamard type are obtained for functions whose nth derivatives in absolute value are convex and concave functions. [1] M. Alomari, M. Darus and S.S. Dragomir, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex. Tamkang Journal of Mathematics, 41(4), 2010, [2] S.-P. Bai, S.-H. Wang and F. Qi, Some Hermite-Hadamard type inequalities for n-time differentiable (α, m)-convex functions, Jour. of Ineq. and Appl., 2012, 2012:267. [3] P. Cerone, S.S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math., 32 (4) (1999), [4] P. Cerone, S.S. Dragomir and J. Roumeliotis and J. Šunde, A new generalization of the trapezoid formula for n-time differentiable mappings and applications, Demonstratio Math., 33 (4) (2000), [5] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, Online:[ hadamard.html]. [6] D.-Y. Hwang, Some Inequalities for n-time Differentiable Mappings and Applications, Kyung. Math. Jour., 43 (2003), [7] J. L. W. V. Jensen, On konvexe funktioner og uligheder mellem middlvaerdier, Nyt. Tidsskr. Math. B., 16, 49-69, [8] W.-D. Jiang, D.-W. Niu, Y. Hua and F. Qi, Generalizations of Hermite-Hadamard inequality to n-time differentiable function which are s-convex in the second sense, Analysis (Munich), 32 (2012), [9] M.E. Özdemir, Ç. Yıldız, New Inequalities for n-time differentiable functions, Arxiv: v1. Çetin Yıldız: Atatürk University, K. K. Education Faculty, Department of Mathematics, 25240, Campus, Erzurum- Turkey, [email protected] M. Emin Özdemir: Atatürk University, K. K. Education Faculty, Department of Mathematics, 25240, Campus, Erzurum- Turkey, [email protected]

103 International Congress in Honour of Professor Ravi P. Agarwal Global Bifurcations of Limit Cycles in the Classical Lorenz System Valery Gaiko We consider a three-dimensional polynomial dynamical system ẋ = σ(y x), ẏ = x(r z) y, ż = xy bz (69.1) known as the Lorenz system. Historically, (69.1) was the first dynamical system for which the existence of an irregular attractor (chaos) was proved for σ = 10, b = 8/3, and 24,06 < r < 28. The Lorenz system (69.1) is dissipative and symmetric with respect to the z-axis. The origin O(0, 0, 0) is a singular point of system (69.1) for any σ, b, and r. It is a stable node for r < 1. For r = 1, the origin becomes a triple singular point, and then, for r > 1, there are two more singular points in the system: O 1 ( b(r 1), b(r 1), r 1) and O 2 ( b(r 1), b(r 1), r 1) which are stable up to the parameter value r a = σ(σ + b + 3)/(σ b 1) (r a 24,74 for σ = 10 and b = 8/3). For all r > 1, the point O is a saddle-node. For many years, the Lorenz system (69.1) has been the subject of study by numerous authors; see, e. g., [1] [5]. However, until now the structure of the Lorenz attractor is not clear completely yet, and the most important question at present is to understand the bifurcation scenario of chaos transition in system (69.1) which is related to Smale s Fourteenth Problem [4]. In this talk, we present a new bifurcation scenario for system (69.1), where σ = 10, b = 8/3, and r > 0, using numerical results of [5] and a bifurcational geometric approach to the global qualitative analysis of three-dimensional dynamical systems which was applied earlier in the two-dimensional case [6] [8]. This scenario connects globally the homoclinic, period-doubling, Andronov Shilnikov, and period-halving bifurcations of limit cycles in the Lorenz system (69.1) [9]. [1] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, [2] N. A Magnitskii, S. V. Sidorov, New Methods for Chaotic Dynamics, World Scientific, New Jersey, [3] L. P. Shilnikov et al., Methods of Qualitative Theory in Nonlinear Dynamics. I, II, World Scientific, New Jersey, 1998, [4] S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), [5] C. Sparrou, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer, New York, [6] V. A. Gaiko, Global Bifurcation Theory and Hilbert s Sixteenth Problem, Kluwer, Boston, [7] V. A. Gaiko, On limit cycles surrounding a singular point, Differ. Equ. Dyn. Syst., 20 (2012), [8] V. A. Gaiko, The applied geometry of a general Liénard polynomial system, Appl. Math. Letters, 25 (2012), [9] V. A. Gaiko, Chaos transition in the Lorenz system, Herald Odesa Nation. Univ. Ser. Math. Mech., 18 (2013), United Institute of Informatics Problems, National Academy of Sciences of Belarus, Minsk, Belarus,

104 104 International Congress in Honour of Professor Ravi P. Agarwal 70 Curvature of Curves Parameterized by a Time Scale Sibel Paşalı Atmaca and Ömer Akgüller Geometric aspect of the theory of time scales is extensively studied afterwards the introduction of partial derivatives on time scales. However, an intrinsic characteristic such as curvature of a curve parameterized by a time scale is still an open question. In this talk, we present the concept of curvature via symmetric derivative on time scales. This approach involves both characteristics of discrete and classical differential geometry, and accurately applicable to globally discrete settings. By the help of this definition of curvature, we also present the bending energy for a curve parameterized by a time scale. [11] A.M.C. Brito Da Cruz,N. Martins,D.F.M. Torres, Symmetric differentiation on time scales, Applied Mathematics Letters 26 (2), (2013), pp [6] C. Dinu, Diamond-α tangent lines of time scales parametrized regular curves, Carpathian Journal of Mathematics, 25 (1), (2009), pp [8] D. Uçar, M.S. Seyyidoǧlu, Y. Tunçer,M.K. Berktaş, V.F. Hatipoǧlu, V.F., Forward curvatures on time scales, Abstract and Applied Analysis, (2011), art. no [3] E. Özyılmaz, Directional derivative of vector field and regular curves on time scales, Applied Mathematics and Mechanics (English Edition), 27 (10), (2006), pp [2] G. Guseinov, E. Özyılmaz, Tangent lines of generalized regular curves parametrized by time scales, Turkish Journal of Mathematics, 25 (4), (2001), pp [9] J.L. Cieśliński, Pseudospherical surfaces on time scales: A geometric definition and the spectral approach, Journal of Physics A: Mathematical and Theoretical, 40 (42), (2007), pp [1] M. Bohner, G. Guseinov, Partial differentiation on time scales, Dynamic Systems and Applications, 13, (2004), [10] M. Bohner, G. Guseinov, Surface areas and surface integrals on time scales, Dynamic Systems and Applications, 19 (3-4), (2010), pp [4] S. P. Atmaca, Normal and osculating planes of -regular curves, Abstract and Applied Analysis, (2010), art. no [5] S. P. Atmaca, Ö. Akgüller, Surfaces on time scales and their metric properties, Advances in Difference Equations, (2013), Volume 2013, June 2013, Article number 170. [7] S. P. Atmaca, Ö. Akgüller, The time scale calculus approach to the geodesic problem in 3D dynamic data sets, Mathematical and Computational Applications, 18 (3), (2013), pp Sibel Paşalı Atmaca: Muğla Sıtkı Koçman University, Faculty of Science, Department of Mathematics, Menteşe, Muğla- Turkiye, [email protected] Ömer Akgüller: Muğla Sıtkı Koçman University, Faculty of Science, Department of Mathematics, Menteşe, Muğla- Turkiye, [email protected]

105 International Congress in Honour of Professor Ravi P. Agarwal Essential Norms of Products of Weighted Composition Operators and Differentiation Operators Between Banach Spaces of Analytic Functions Jasbir S.Manhas and Ruhan Zhao We obtain several estimates of the essential norms of the products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions with general weights. Jasbir S. Manhas: Sultan Qaboos University, Department of Mathematics & Statistics, Muscat, Oman, edu.om Ruhan Zhao: State University of New York (SUNY), Department of Mathematics, Brockport, U.S.A., edu

106 106 International Congress in Honour of Professor Ravi P. Agarwal 72 On the Null Forms, Integrating Factors and First Integrals to Path Equations Ilker Burak Giresunlu and Emrullah Yaşar In this work, we consider the path equation y f (y) f(y) (y ) 2 f (y) f(y) = 0 which modeling the drag forces [1]. The drag forces are the major source of energy loss for objects moving in a fluid medium. Using the relationship between semi-algorithmic Prelle-Singer method [6] and λ-symmetry approach [6], we obtained nullforms, integrating factors, first integrals and general solutions [4]. Nevertheless exploiting the Lie-point type symmetries we constructed systematically the Jacobi Last Multiplier s (JLM) [5]. After yielding the these multipliers we obtained Darboux polynomials of the under considered equation. The results was tabulated and compared with those gained by the other methods [6, 7]. [1] M. Pakdemirli, The drag work minimization path for a flying object with altitude-dependent drag parametres, Proceedings of the Institution of Mechanical Engineers C, Journal of Mechanical Engineering Science, 223(5) (2009), , [2] V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations, Proc. R. Soc. A., 461 (2005), [3] C. Muriel and J. L. Romero, Integrating Factors and λ-symmetries, JNMP, 15 (2008), , [4] R. Mohanasubha, V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second order differential equations, Proc. R. Soc. A, 470 (2014), [5] M. Nucci, Jacobi last multiplier and Lie symmetries: a novel application of an old relationship, J. Nonlinear Math. Phys., 12 (2005), , [6] M. Pakdemirli and Y. Aksoy, Group classification for path equation describing minimum drag work and symmetry reductions, Applied Mathematics and Mechanics (English Edition), 31(7) (2010), , [7] G. Gün and T. Özer, First Integrals, Integrating Factors, and Invariant Solutions of the Path Equation Based on Noether and λ-symmetries, Abstract and Applied Analysis, 15 (2013). Uludag University, Faculty of Arts and Science, Department of Mathematics, Görükle, Bursa-Turkiye, [email protected], [email protected]

107 International Congress in Honour of Professor Ravi P. Agarwal Commutativity of Lommel and Halm Differential Equations Mehmet Emir Koksal Many engineering systems are composed of cascade connection of subsystems of simple orders. This is very important in design of electrical and electronic systems. Hence, the commutativity, which is the functional invariance under the change of the connection order, is very important from the practical point of view. This presentation introduces the commutativity of systems defined by Lommel and Halm type differential equations. These differential equations have been described in the literature and they represents some particular physical systems. The explicate requirements for commutativity of these systems are derived. Under certain circumstances, these systems have commutative pairs some of which have explicit analytical solutions. The outcomes of the presentation is expected to lead new design trends in engineering as to improve the total system performance covering sensitivity, stability, disturbance and robustness. [1] E. Marshall, Commutativity of time varying systems. Electronic Letters, 18 (1977), [2] M. Koksal, Commutativity of second order time-varying systems, Int. J. of Cont., 3 (1982), , [3] M. Koksal, Commutativity of 4th order systems and Euler systems, Proceeding of National Congress of Electrical Engineers (in Turkish), Paper no:bi-6, Adana, Turkey, 1985, [4] M. Koksal and M. E. Koksal, Commutativity of linear time-varying differential systems with non-zero initial conditions: a review and some new extensions, Mathematical Problems in Engineering, 2011 (2011), 1-25, [5] M. E. Koksal, The Second Order Commutative Pairs of a First Order Linear Time-Varying System, Applied Mathematics and Information Sciences, 9 (2015), 1-6. Mevlana University, Department of Primary Mathematics Education, Selcuklu, Konya, Turkey, [email protected]

108 108 International Congress in Honour of Professor Ravi P. Agarwal 74 Equivalence Between Some Iterations in CAT (0) Spaces Kyung Soo Kim In this talk, we obtain some equivalence conditions for the convergence of iterative sequences for set-valued contraction mapping in CAT (0) spaces are obtained. [1] M. Bridson and A. Haefliger, Metric spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, [2] F. Bruhat and J. Tits, Groups réductifss sur un corps local. I. Données radicielles valuées, Publ. Math. Inst. Hautes Études Sci., 41 (1972), [3] P. Chaoha and A. Phon-on, A note on fixed point sets in CAT (0) spaces, J. Math. Anal. Appl., 320 (2006), [4] S. Dhompongsa and B. Panyanak, On triangle-convergence theorems in CAT (0) spaces, Comput. Math. Anal., 56 (2008), [5] J.C. Dunn, Iterative construction of fixed points for multivalued operators of the monotone type, J. Funct. Anal., 27(1) (1978), [6] R. Espínola and B. Pi atek, The fixed point property and unbounded sets in CAT (0) spaces, J. Math. Anal. Appl., 408 (2013), [7] S. Ishikawa, Fixed point by a new iteration, Proc. Amer. Math. Soc., 44 (1974), [8] M.A. Khamsi and W.A. Kirk, On uniformly Lipschitzian multivalued mappings in Banach and metric spaces, Nonlinear Anal., 72 (2010), [9] J.K. Kim, K.S. Kim and Y.M. Nam, Convergence and stability of iterative processes for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in convex metric spaces, J. of Compu. Anal. Appl., 9(2) (2007), [10] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68(12) (2008), [11] L. Leustean, A quadratic rate of asymptotic regularity for CAT (0)-spaces, J. Math. Anal. Appl., 325 (2007), [12] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), [13] S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30 (1969), [14] E. Picard, Sur les groupes de transformation des équations différentielles linéaires, Comptes Rendus Acad. Sci. Paris, 96 (1883), [15] S. Saejung, Halpern s iteration in CAT (0) spaces, Fixed Point Theory Appl., 2010, Article ID , 13 pages. Kyungnam University, Department of Education, Mathematics Education Major, Changwon, Gyeongnam, , Korea, [email protected] This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2012R1A1A4A )

109 International Congress in Honour of Professor Ravi P. Agarwal On Certain Combinatoric Convolution Sums of Divisor Functions Daeyeoul Kim and Nazli Yildiz Ikikardes In this talk, we study certain combinatorial convolution sums involving divisor functions and their relations to Bernoulli polynomials. We establish two explicit formulas for certain combinatoric convolution sums of divisor functions derived from Bernoulli polynomials. [1] D. Kim and A. Bayad, Convolution identities for twisted Eisenstein series and twisted divisor functions, Fixed Point Theory and Applications, 2013, 81, [2] D. Kim and N.Y. Ikikardes, Certain combinatoric Bernoulli polynomials and convolution sums of divisor functions, Advance Difference Equations, 2013, 2013:310, 11pp., [3] K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, Daeyeoul Kim: National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon , South Korea, Nazli Yildiz Ikikardes: Department of Elementary Mathematics Education, Necatibey Faculty of Education, Balikesir University, Balikesir, Turkey,

110 110 International Congress in Honour of Professor Ravi P. Agarwal 76 Some Properties of the Genocchi Polynomials with the Variable [x] q J.Y.Kang and C.S.Ryoo We introduce the Genocchi polynomials with the variable [x] q and we get some relations of their polynomials by the p-adic integral on Z p. We also observe an interesting phenomenon of scattering of the zeros of the Genocchi polynomials with the variable [x] q in comlex plane. [1] Cangul, I. N., Ozden, H., Simsek, Y., A new approach to q-genocchi numbers and their interpolation functions, Nonlinear Analysis Series A: Theory, Methods and Applications, , [2] B. Kupershmidt, Reflection symmetries of q-bernoulli polynomials, J. Nonlinear Math. Phys., 12 (2005), suppl. 1, [3] D. S. Kim, T. Kim, Y. H. Kim, S. H. Lee, Some arithmetic properties of Bernoulli and Euler numbers, Adv. Stud. Contemp. Math., 22 (4) (2012), [4] Veli Kurt and Mehmet Cenkci, A nes approach to q-genocchi numbers and polynomials, Bull. Korean Math. Soc., 47 (3) (2010), [5] Min-Soo Kim, On Euler numbers, polynomials and related p-adic integrals, Journal of Number Theory, 129 (2009), [6] T. Kim, q-volkenborn integration, Russian Journal of Mathematical Physics, 9 (2002), [7] T. Kim, On the q-extension of Euler and Genocchi numbers, Journal of Mathematical Analysis and Applications, 326 (2007), [8] T. Kim, An invariant p-adic q-integrals on Z p, Applied Mathematics Letters, 21 (2008), [9] S. H. Rim, K. H. Park and E. J. Moon, On Genocchi Numbers and Polynomials, Abstract and Applied Analysis, 2008, Article ID , 7 pages. [10] C. S. Ryoo, A numerical computation on the structure of the roots of q-extension of Genocchi polynomials, Applied Mathematics Letters, 21 (4) (2008), [11] Y. Simsek, I. N. Cangul, V. Kurt and D. Kim, q-genocchi numbers and polynomials associated with q-genocchi-type L-Functions, Advances in Difference Equations, (2008), Article ID: , 12 pages. J.Y. Kang: Department of Mathematics, Hannam University, Daejeon , Korea, [email protected] C.S. Ryoo: Department of Mathematics, Hannam University, Daejeon , Korea, [email protected] This work was supported by NRF(National Research Foundation of Korea) Grant funded by the Korean Government (NRF-2013-Fostering Core Leaders of the Future Basic Science Program).

111 International Congress in Honour of Professor Ravi P. Agarwal Boundedness of Localization Operators on Lorentz Mixed Normed Modulation Spaces Ayşe Sandıkçı The localization operator A ϕ 1,ϕ 2 a with symbol a S ( R d) and windows ϕ 1, ϕ 2 is defined to be A ϕ 1,ϕ 2 a f (t) = a (x, w) V ϕ1 f (x, w) M wt xϕ 2 dxdw. R 2d In this work we study certain boundedness properties for localization operators on Lorentz mixed normed modulation spaces, when the operator symbols belong to appropriate Wiener amalgam spaces and Lorentz spaces with mixed norms. Some key references are given below. [1] P. Boggiatto, Localization operators with L p symbols on the modulation spaces, In Advances in Pseudo-Differential Operators, vol. 155 of Oper. Theory Adv. Appl., , Birkhäuser, Basel, [2] E. Cordero, K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal., 205(1) (2003), [3] H.G. Feichtinger, F. Luef, Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis, Collect. Math., Vol.57 No.Extra Volume (2006), [4] D.L. Fernandez, Lorentz spaces, with mixed norms, J. Funct. Anal., 25 (1977), [5] A.T. Gürkanlı, Time-frequency analysis and multipliers of the spaces M (p, q) ( R d) and S (p, q) ( R d), J. Math. Kyoto Univ., 46-3 (2006), [6] K. Gröchenig, Foundation of Time-Frequency Analysis. Birkhäuser, Boston 2001, ISBN [7] R.A. Hunt, On L (p, q) spaces, Extrait de L Enseignement Mathematique, T.XII, fasc.4 (1966), [8] A. Sandıkçı, On Lorentz mixed normed modulation spaces, J. Pseudo-Differ. Oper. Appl., 3 (2012), [9] A. Sandıkçı, A.T. Gürkanlı, Gabor Analysis of the spaces M (p, q, w) ( R d) and S (p, q, r, w, ω) ( R d), Acta Math. Sci., 31B (2011), [10] A. Sandıkçı, A.T. Gürkanlı, Generalized Sobolev-Shubin spaces, boundedness and Schatten class properties of Toeplitz operators, Turk J. Math., 37 (2013), [11] A. Sandıkçı, Continuity of Wigner-type operators on Lorentz spaces and Lorentz mixed normed modulation spaces, Turk J. Math., 38 (2014), Ondokuz Mayıs University, Faculty of Arts and Science, Department of Mathematics, Atakum, Samsun-Turkey,

112 112 International Congress in Honour of Professor Ravi P. Agarwal 78 p Summable Sequence Spaces with Inner Products Şükran Konca, Hendra Gunawan and Mochammad Idris We revisit the space l p of p-summable sequences of real numbers. In particular, we show that this space is actually contained in a (weighted) inner product space. The relationship between l p and the (weighted) inner product space that contains l p is studied. For p > 2, we also obtain a result which describe how the weighted inner product space is associated to the weights. [1] S.K. Berberian, Introduction to Hilbert Space, Oxford University Press, New York, [2] H. Gunawan, W. Setya-Budhi, Mashadi, S. Gemawati, On volumes of n-dimensional parallelepipeds on l p spaces, Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika, 16 (2005): [3] D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Reprinted 1953, New York, Chelsea, [4] M. Idris, S. Ekariani, H. Gunawan, On the space of p-summable sequences, Matematiqki Vesnik. 65 (2013), (1): [5] E. Kreyszig, Introductory Functional Analysis with Applications, New York, John Wiley & Sons, [6] P.M. Miličić, Une généralisation naturelle du produit scaleaire dans un espace normé et son utilisation, Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika (Beograd), 42 (1987), (56): Şükran Konca: Sakarya University, Faculty of Science, Department of Mathematics, 54187, Sakarya-Turkey, Bitlis Eren University, Department of Mathematics, 13000, Bitlis-Turkey, [email protected] and [email protected] Hendra Gunawan: Institute of Technology Bandung, Department of Mathematics, 40132, Bandung-Indonesia, Mochammad Idris: Institute of Technology Bandung, Department of Mathematics, 40132, Bandung-Indonesia, Lambung Mangkurat University, Department of Mathematics, 70711, Banjarbaru-Indonesia [email protected]

113 International Congress in Honour of Professor Ravi P. Agarwal An Alternative Proof of a Tauberian Theorem for Abel Summability Method Ibrahim Çanak and Ümit Totur Using a corollary to Karamata s main theorem [Math. Z. 32 (1930), ], we prove that if a slowly decreasing sequence of real numbers is Abel summable, then it is convergent in the ordinary sense. [1] R. Schmidt, Über divergente Folgen und lineare Mittelbildungen, Math. Z., 22 (1925), , [2] I. J. Maddox, A Tauberian theorem for ordered spaces, Analysis, 9 (1989), , [3] F. Móricz, Ordinary convergence follows from statistical summability (C, 1) in the case of slowly decreasing or oscillating sequences, Colloq. Math., 99 (2004), , [4] Ö. Talo, F. Başar, On the slowly decreasing sequences of fuzzy numbers, Abstr. Appl. Anal., (2013), Art. ID , 7 pp, [5] G. H. Hardy, Divergent series, Oxford University Press, 1956, [6] J. Karamata, Über die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z., 32 (1930), , [7] K. Knopp, Theory and application of infinite series, Dover Publications, 1990, [8] G. A. Mikhalin, Theorem of Tauberian type for (J, p n) summation methods, Ukrain. Mat. Zh., 29 (1977), , [9] Č. V. Stanojević, V. B. Stanojević, Tauberian retrieval theory, Publ. Inst. Math. (Beograd) (N.S.), 71 (2002), , [10] A. Tauber, Ein satz aus der theorie der unendlichen reihen, Monatsh. f. Math. u. Phys., 7 (1897), , İbrahim Çanak: Ege University, Faculty of Science, Department of Mathematics, Bornova, Izmir-Turkey, Ümit Totur: Adnan Menderes University, Faculty of Science, Department of Mathematics, Aydin-Turkey,

114 114 International Congress in Honour of Professor Ravi P. Agarwal 80 Positive Periodic Solutions for a Nonlinear First Order Functional Dynamic Equation by a New Periodicity Concept on Time Scales Erbil Çetin and F.Serap Topal In this talk, we consider the existence, multiplicity and nonexistence of positive periodic solutions in shifts δ ± for the nonlinear functional dynamic equation on a periodic time scale in shifts δ ± with period P [t 0, ) T. By using the Krasnosel skii fixed point theorem and Leggett-Williams multiple fixed point theorem, we present different sufficient conditions for the existence of at least one, two or three positive solutions in shifts δ ± of the problem on time scales. We extend and unify periodic differential, difference, h-difference and q-difference equations and more by a new periodicity concept on time scales. [1] E. Cetin, F.S. Topal, Periodic solutions in shifts δ ± for a nonlinear dynamical equation on time scales, Abstract and Applied Analaysis, Volume 2012, Article ID707319, 17 pages.2 [2] E.R. Kaufmann, Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl. 319 (2006) [3] M. Adıvar, A new periodicity concept for time scales, Math. Slovaca 63 (2013), No [4] M. Bohner and A. Peterson, Dynamic Equations on time scales, An Introduction with Applications, Birkhäuser, Boston, [5] M. Bohner, A. Peterson,(Eds) Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, [6] S. Padhi, S. Srivastava, S. Pati, Three periodic solutions for a nonlinear first order functional differential equation, Applied Mathematics and Computation 216 (2010) [7] S. Hilger, Ein Masskettenkalkà 1 4 l mit Anwendug auf Zentrumsmanningfaltigkeiten, Phd Thesis, Universität Würzburg, [8] M.A. Krasnoselsel skiĭ, Positive Solutions of Operator Equations Noordhoff, Groningen, [9] X. L. Liu, W.T. Li, Periodic solution for dynamic equations on time scales, Nonlinear Analysis 67, no.5 (2007) [10] H. Wang, Positive periodic solutions of functional differential equations, J. Differential Equations, 202, (2004) [11] S. Padhi, S. Srivastava, Existence of three periodic solutions for a nonlinear first order functional differential equation, Journal of the Franklin Institute 346 (2009) [12] Y. Li, L. Zhu, P. Liu, Positive Periodic Solutions of Nonlinear Functional Difference Equations Depending on a Parameter, Computers and Mathematics with Applications 48 (2004) [13] A. Weng, J. Sun, Positive Periodic Solutions of first-order Functional Difference Equations with Parameter, Journal of Computational and Applied Mathematics 229 (2009) [14] F. Qiuxiang, Y. Rong, On the Lasota-Wazewska model with piecewise constant arguments, Acta. Math. Sci. 26B(2)(2006) [15] D. Jiang, J. Wei, B. Jhang, Positive periodic solutions of functional differential equations and population models, Electron. J. Differential Equations 71 (2002) [16] Y. Luo, W. Wang, J. Shen, Existence of positive periodic solutions for two kinds of neutral functional differential equations, Appl. Math. Lett. 21(6) (2008) [17] W.S.C Gurney, S.P. Blathe and R.M. Nishet, Nicholson s blowflies revisited, Nature 287 (1980) [18] Y. Kuang, Delay Differential equations with Applications in Population Dynamics, Academic Press, New York, [19] A. Wan, D. Jiang, A new existence theory for positive periodic solutions to functional differential equations, Comput. Math. Appl. 47 (2004) [20] H.I. Freedman, J. Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal. 23 (1992) [21] K.P. Hadeler, J. Tomiuk, Periodic solutions of difference differential equations, Arch. Rational Mech. Anal. 65 (1977) [22] J.R. Graef, L. Kong, Existence of multiple periodic solutions for first order functional differential equations, Mathematical and Computer Modelling 54 (2011) Erbil Çetin: Ege University, Faculty of Science, Department of Mathematics, Bornova Izmir-Turkiye, erbil.cetin@ege. edu.tr Fatma Serap Topal: Ege University, Faculty of Science, Department of Mathematics, Bornova Izmir-Turkiye, f.serap. [email protected]

115 International Congress in Honour of Professor Ravi P. Agarwal Potential Flow Field Around a Torus Rajai Alassar In this note, we present the potential flow field around a torus. We use the naturally fit toroidal coordinates system to recast the governing equation. We show that the governing equation has a series solution in terms of toroidal functions with coefficients that satisfy a three-term recurrence relation. King Fahd University of Petroleum & Minerals (KFUPM), Department of Mathematics and Statistics, KFUPM Box 1620, Dhahran 31261, Saudi Arabia, [email protected]

116 116 International Congress in Honour of Professor Ravi P. Agarwal 82 On B 1 -Convex Functions and Some Inequalities Gabil Adilov and Ilknur Yesilce A subset U of R n ++ is B 1 -convex if for all x, y U and all λ [1, ) one has λx y = (min {λx 1, y 1 }, min {λx 2, y 2 },..., min {λx n, y n}) U. For each kind of convex functions, some inequalities such as Hermite-Hadamard inequalities, Jensen inequalities, etc., are obtained by many authors ([1], [2], [3], etc.). In this work, similar inequalities are analyzed for B 1 -convex functions. [1] G. Adilov, Increasing Co-radiant Functions and Hermite-Hadamard Type Inequalities, Mathematical Inequalities and Applications, 14(1) (2011), 45-60, [2] G. Adilov and S. Kemali, Abstract Convexity and Hermite-Hadamard Type Inequalities, Journal of Inequalities and Applications, 2009, Article ID , doi: /953534, (2009), 13 pages, [3] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000, (ONLINE: Gabil Adilov: Akdeniz University, Faculty of Education, Department of Mathematics, Antalya-Turkey, [email protected] or [email protected] Ilknur Yesilce: Mersin University, Faculty of Science and Letters, Department of Mathematics, Mersin-Turkey, [email protected]

117 International Congress in Honour of Professor Ravi P. Agarwal On the Global Behaviour of a Higher Order Difference Equation Yasin Yazlik, D.Turgut Tollu and Necati Taskara In this paper, we deal with the behavior well-defined solutions of the difference equation x n = ax n 1 + b + x n m ax n m 1 c + x n m ax n m 1, n N 0, where N 0 = N {0}, the parameters a, b and c and the initial conditions x m 1, x m,..., x 1, x 0 are real numbers. [1] R. P. Agarwal, Difference Equations and Inequalities, 1 st edition, Marcel Dekker, New York, 1992, 2 nd edition, [2] V. L. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, The Netherlands. [3] R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Adv. Stud. Contem. Math. 17(2) (2008), [4] L. Brand, A sequence defined by a difference equation. Am. Math. Mon. 62 (1955), [5] L. Berg, S. Stević, On some systems of difference equations, Applied Mathematics and Computation 218 (2011), [6] S. Stević, On some solvable systems of difference equations, Applied Mathematics and Computation 218 (2012), [7] S. Stevic, M. A. Alghamdi, N. Shahzad and D. A. Maturi, On a class of solvable difference equations, Abstract Applied Analysis (2013), Article ID: , 7 pages. [8] I. Yalcinkaya, On the difference equation x n+1 = α + x n m, Discrete Dynamics in Nature and Society (2008), Article x k n ID: , 8 pages. [9] S. E. Das and M. Bayram, Dynamics of a higher-order nonlinear rational difference equation, International Journal of Phsical Sciences 6(12) (2011), [10] E.M.E. Zayed and M.A. El-Moneam, On the rational recursive sequence x n+1 = γx n k + axn+bx n k cx n dx n k, Bulletin of the Iranian Mathematical Society 36(1) (2010), [11] H. El-Metwally, E. M. Elsayed and E. M. Elabbasy, On the solutions of difference equations of order four, Rocky Mountain Journal of Mathematics 43(3), (2013), [12] S. Ozen, I. Ozturk and F. Bozkurt, On the recursive sequence y n+1 = α+y n 1 β+y n Computation 188 (2007), y n 1, Applied Mathematics and y n [13] C. Cinar, M. Toufik and I. Yalcinkaya, On the difference equation of higher order, Utilitas Mathematica 92 (2013), [14] D. T. Tollu, Y. Yazlik and N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Advances in Difference Equations 2013, 2013:174. [15] I. Bajo, E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Differ. Equations Appl. 17(10) (2011), , [16] E.A. Grove, Y. Kostrov, G. Ladas, and S. W. Schultz, Riccati difference equations with real period-2 coeficients, Commun. Appl. Nonlinear Anal. 14 (2007), [17] G. Papaschinopoulos and B.K. Papadopoulos, On the Fuzzy Difference Equation x n+1 = A + B/x n, Soft Computing 6 (2002), [18] D. T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Applied Mathematics and Computation 233 (2014), Yasin Yazlık: Nevsehir Hacı Bektaş Veli University, Faculty of Science and Art, Department of Mathematics, Nevsehir- Turkey, [email protected] D. Turgut Tollu: Necmettin Erbakan University, Faculty of Science, Department of Mathematics-Computer Sciences, Konya-Turkey, [email protected] Necati Taşkara: Selcuk University, Science Faculty, Department of Mathematics, Konya-Turkey, [email protected]

118 118 International Congress in Honour of Professor Ravi P. Agarwal 84 Identifying an Unknown Time Dependent Coefficient for Quasilinear Parabolic Equations Fatma Kanca and Irem Baglan This talk deals with the mathematical analysis of the inverse problem of identifying the unknown time-dependent coefficient in the quasilinear parabolic equation with the nonlocal boundary and integral overdetermination conditions. The existence, uniqueness and continuously dependence upon the data of the solution are proved by iteration method in addition to the numerical solution of this problem is considered with an example. [1] F. Kanca, The inverse coefficient problem of the heat equation with periodic boundary and integral overdetermination conditions, Journal of Inequalities and Applications,, 108 (2013), 1-9. [2] F. Kanca, Inverse Coefficient Problem of the Parabolic Equation with Periodic Boundary and Integral Overdetermination Conditions, Abstract and Applied Analysis, 2013 (2013) 1-7. [3] M. Ismailov, F. Kanca, An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions, Mathematical Methods in the Applied Science, 34 (2011), [4] F. Kanca F.,Baglan I.,Continuous dependence on data for a solution of the quasilinear parabolic equation with a periodic boundary condition, Boundary Value Problems, 28 (2013). [5] F. Kanca, I. Baglan, An inverse coefficient problem for a quasilinear parabolic equation with nonlocal boundary conditions, Boundary Value Problems, 213 (2013). [6] F. Kanca, I. Baglan,An inverse problem for a quasilinear parabolic equation with nonlocal boundary and overdetermination conditions, Journal of inequalities and applications, 76 (2014). [7] I. Sakınc(Baglan), Numerical Solution of a Quasilinear Parabolic Problem with Periodic Boundary Condition, Hacettepe, Journal of Mathematics and Statistics, 39 (2010), [8] A. M. Nakhushev, Equations of Mathematical Biology, Moscow, 1995 (in Russian). [9] N. I. Ionkin, Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition. Differential Equations, 13 (1977), Fatma Kanca: Department of Management Information Systems, Kadir Has University, 34083, Istanbul, Turkey, fatma. [email protected] İrem Baglan: Department of Mathematics, Kocaeli University, Kocaeli 41380, Turkey,[email protected]

119 International Congress in Honour of Professor Ravi P. Agarwal On Special Semigroup Classes and Congruences on Some Semigroup Constructions Seda Oğuz and Eylem Güzel Karpuz The purpose of this study is to consider under which conditions Bruck-Reilly and generalized Bruck-Reilly*-extensions might belong to some special classes of semigroups such as regular, unit regular, completely regular, inverse, orthodox and E-unitary inverse. In addition to this we qualify the general types of congruences on generalized Bruck-Reilly*-extension. [1] J. M. Howie, Fundamentals of Semigroup Theory, Clarendon Press-Oxford, (1995), [2] S. Oğuz, E. G. Karpuz, Semigroup Properties on Bruck-Reilly and Generalized Bruck-Reilly -Extensions of Monoids, in preparation, [3] B. Piochi, Congruences on Bruck-Reilly extensions of monoids, Semigroup Forum 50 (1995), , [4] Y. Shang, L. M. Wang, -Bisimple Type A w 2 -Semigroups as Generalized Bruck-Reilly -Extensions, Southeast Asian Bulletin of Mathematics 32 (2008), Seda Oğuz: Cumhuriyet University, Education Faculty, Department of Secondary School Science and Mathematics Education, Sivas-Turkiye, [email protected] Eylem Güzel Karpuz: Karamanoğlu Mehmetbey University, Kamil Özdağ Science Faculty, Department of Mathematics, Karaman-Turkiye, [email protected]

120 120 International Congress in Honour of Professor Ravi P. Agarwal 86 The Rate of Pointwise Convergence of q Szasz Operators Tuncer Acar In this talk, we mainly study on Voronovskaya type theorems for q Szasz operators, defined in [Mahmudov, N. I., On q parametric Szasz-Mirakjan Operators, Mediterr. J. Math., 7 (2010), ], and their q derivatives. To do this, we consider the weighted spaces of approximation functions and related weighted modulus of continuity and we obtain quantitative Voronovskaya type theorem in terms of weighted modulus of continuity of q derivatives of approximating function. By this way, we either obtain the rate of pointwise convergence of q Szasz operators and their derivatives or we present these results for continuous functions although classical ones are valid for differentiable functions. [1] Aral, A, Acar, T., Voronovskaya type result for q-derivative of q-baskakov operators, J. Applied Func. Anal., 7 (4), (2012), [2] Aral, A., A generalization of Szász-Mirakyan operators based on q-integers, Math. Comput. Modelling, 47 (2008), no. 9-10, [3] Aral, A., Gupta, V., The q-derivative and applications to q-szász Mirakyan operators, Calcolo 43 (2006), no. 3, [4] Aral, A., Gupta,V. and Agarwal, R. P., Applications of q-calculus in Operator Theory, Springer, (2013). [5] Finta, Z., Remark on Voronovskaja theorem for q-bernstein operators, Stud. Univ. Bab es-bolyai Math. 56(2011), No. 2, [6] Gasper, G., Rahman, M., Basic Hypergeometrik Series. Encyclopedia of Mathematics and Its Applications, vol. 35, Cambridge University Press, Cambridge, (1990). [7] Kac, V., Cheung, P., Quantum Calculus, Springer, NewYork, (2002). [8] Mahmudov, N. I., On q-parametric Szasz-Mirakjan Operators, Mediterr. J. Math., 7 (2010), [9] Mahmudov, N. I., q-szász-mirakjan operators which preserve x2, J. Comput. Appl. Math. 235 (2011), no. 16, [10] Mahmudov, N. I., On q-parametric Szász-Mirakjan operators, Mediterr. J. Math., 7 (2010), no. 3, [11] Philips, G. M., Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), [12] Rajkovic, Predrag M., Stankovic, Miomir S., Marinkovic, Sla ana D., Mean value theorems in q-calculus, Proceedings of the 5th International Symposium on Mathematical Analysis and its Applications (Niška Banja, 2002), Mat. Vesnik 54 (3-4) (2002), [13] Videnskii, V. S., On some classes of q-parametric positive linear operators, Operator Theory: Adv. and Appl., 158 (2005), [14] Voronovskaya, E.V., Determination of the asymptotic form of approximation of functions by the polynomials of S.N. Bernstein, Dokl. Akad. Nauk SSSR, A, (1932), Kirikkale University, Faculty of Science and Arts, Department of Mathematics, Yahsihan, 71450, Kirikkale-Turkey, [email protected]

121 International Congress in Honour of Professor Ravi P. Agarwal Some Properties of Cohomology Groups for Graphs Özgür Ege and Ismet Karaca In this work, we would like to construct cohomology theory for graphs. For this purpose, we deal with the singular homology of graphs and take its dual structure. We then give the Universal coefficient theorem for singular cohomology in graphs. We show that the Künneth theorem doesn t yield for graphs. We also give explanatory examples on the topic. Lastly, we deal with fixed point properties of graphs using singular cohomology groups. [1] E. Babson, H. Barcelo, M. Longueville and R. Laubenbacher, Homotopy theory of graphs, Journal of Algebraic Combinatorics, 24 (2006), [2] H. Barcelo, X. Kramer, R. Laubenbacher and C. Weaver, Foundations of a connectivity theory for simplicial complexes, Advances in Applied Mathematics, 26 (2001), [3] H. Barcelo and R. Laubenbacher, Perspectives on A-homotopy theory and its applications, Discrete Mathematics, 298 (1-3) (2005), [4] A. Dochtermann, Hom complexes and homotopy theory in the category of graphs, European Journal of Combinatorics, 30 (2009), [5] A. Dochtermann, Homotopy groups of Hom complexes of graphs, Journal of Combinatorial Theory, Series A 116 (2009), [6] O. Ege and I. Karaca, The Lefschetz Fixed Point Theorem for Digital Images, Fixed Point Theory and Applications, doi: / / , (2013). [7] A. Granas and J. Dugundji, Fixed Point Theory, Springer, [8] V.V. Prasolov, Elements of Combinatorial and Differential Topology, American Mathematical Society, [9] E. Spanier, Algebraic Topology, McGraw-Hill, New York, [10] M.E. Talbi and D. Benayat, The homotopy exact sequence of a pair of graphs, Acta Scientiarum Technology, (2013). [11] M.E. Talbi and D. Benayat, Homology theory of graphs, Mediterranean Journal of Mathematics, (2013). Özgür Ege: Celal Bayar University, Faculty of Science and Letters, Department of Mathematics, Muradiye Campus, Manisa-Turkey, [email protected] İsmet Karaca: Ege University, Faculty of Science, Department of Mathematics, Bornova, İzmir-Turkey,

122 122 International Congress in Honour of Professor Ravi P. Agarwal 88 Stability with Respect to Initial Time Difference for Generalized Delay Differential Equations Ravi Agarwal and Snezhana Hristova Stability with initial data difference of nonlinear delay differential equations is introduced and studied. This type of stability generalizes the known in the literature concept of stability. It gives us the opportunity to compare the behavior of two nonzero solutions which initial values as well as initial intervals are different. Lyapunov functions as well as comparison results for scalar ordinary differential equations have been employed. Several examples will be given to illustrate both concepts and obtained results. [1] Agarwal R., Hristova S., Strict stability in terms of two measures for impulsive di erential equations with supremum, Appl. Analysis, 91, 7, 2012, [2] Bainov D., Hristova S., Di erential Equations with Maxima, Francis& Taylor, CRC Press, [3] Jankowski T., Delay integro-di erential inequalities with initial time di erence and applications, J. Math. Anal. Appl. 291 (2004) [4] Lakshmikantham V., Leela S., Devi J.V., Stability criteria for solutions of di erential equations relative to initial time di erence, Int. J. Nonlinear Di. Eqns. 5 (1999) [5] Li A, Wei L, Ye J.,Exponential and global stability of nonlinear dynamical systems relative to initial time difference, Appl. Math. Comput. 217 (2011) [6] Li A., Feng E., Li S., Stability and boundedness criteria for nonlinear di erential systems relative to initial time difference and applications, Nonlinear Anal.: Real World Appl., 10 (2009) [7] Shaw M.D., Yakar C., Stability criteria and slowly growing motions with initial time di erence, Probl. Nonl. Anal. Eng. Sys. 1 (2000) [8] Song X., Li A., Wang Zh., Study on the stability of nonlinear differential equations with initial time difference, Nonlinear Anal.: Real World Appl. 11 (2010) [9] Song X., Li S., Li A., Practical stability of nonlinear differential equation with initial time difference, Appl. Math. Comput. 203 (2008) [10] Yakar C., Shaw M., A Comparison result and Lyapunov stability criteria with initial time difference, Dynam. Cont. Discr. Impul. Sys.: Math. Anal. 12 (2005) [11] Yakar C., Shaw M.D., Initial time di erence stability in terms of two measures and variational comparison result, Dynam. Cont. Discr. Impul. Sys.: Math. Anal. 15 (3) (2008) [12] J. Henderson, S. Hristova, Eventual Practical Stability and cone valued Lyapunov functions for differential equations with Maxima, Commun. Appl. Anal., 14, 4, 2010, Ravi Agarwal: Texas A&M University-Kingsville, Department of Mathematics, Kingsville, TX 78363, USA, agarwal@ tamuk.edu Snezhana Hristova: Plovdiv University, Faculty of Mathematics and Informatics, Department of Applied Mathematics, Bulgaria, [email protected] Research was partially supported by Fund Scientific Research MU13FMI002, Plovdiv University.

123 International Congress in Honour of Professor Ravi P. Agarwal On Ramanujan s Summation Formula, his General Theta Function and a Generalization of the Borweins Cubic Theta Functions Chandrashekar Adiga In Chapter 16 of his second notebook, Ramanujan develops two closely related topics, q-series and theta- functions. In the first part of the talk, we discuss about Ramunujan s summation formula and his general theta function. The Borwein brothers have introduced and studied three cubic theta functions. Many generalizations of these functions have been studied as well. In the second part of the talk, we introduce a new generalization of these functions and establish general formulas that are connecting our functions and Ramanujan s general theta function. Many identities found in the literature follow as a special case of our identities. We further derive general formulas for certain products of theta functions. [1] C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson, Chapter 16 of Ramanujan s second notebook: Theta functions and q-series, Mem. Amer. Math. Soc., 315 (1985), 1-91, [2] B. C. Berdnt, Ramanujan s Notebooks, Part III, Springer-Verlag, New York, 1991, [3] S. Bhargava and S. N. Fathima, Laurent coefficients for cubic theta functions, South East Asian J. Math. Soc., 1 (2) (2003), 27-31, [4] S. Bhargava and S. N. Fathima, Unification of modular transformations for cubic theta functions, New Zealand J. Math., 33 (2004), , [5] J. M. Borwein and P. B. Borwein, Pi and AGM, Wiley, New York, 1987, [6] J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi s identity and the AGM, Trans. Amer. Math. Soc, 323 (2) (1991), , [7] S. Ramanujan, Notebooks (2 Volumes), Tata Institute of Fundamental Research, Bombay, Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore , INDIA, c adiga@ hotmail.com

124 124 International Congress in Honour of Professor Ravi P. Agarwal 90 L Error Estimate of Parabolic Variational Inequality Arising of the Pricing of American Option S.Madi, M.Hariour and M.C.Bouras This contribution deals with the numerical analysis using the semi-implicit scheme with respect to t-variable combined with finite element spacial approximation applied in parabolic variational inequalities arising from pricing of American option. Where the presented numerical result is efficient. [1] A. Bensoussan, J.L. Lions, Impulse control and quasi-variational inequalities, Gauthier Villars, Paris, (1984), [2] P. Jaillet, D. Lamberton, and B. Lapeyre, Variational Inequalities and the Pricing of American Options, Acta Applicandae Mathematicae 21: , 1990, [3] M. Boulbrachene, M. Haiour, The finite element approximation of Hamilton Jacobi Bellman equations, Computers and Mathematics with Applications, 41(2001), , [4] Y. Achdou, F. Hecht and D. PommierA Posteriori Error Estimates for Parabolic Variational Inequalities, J Sci Comput (2008), DOI /s , [6] S. Boulaaras, M. Haiour, A new approach to asymptotic behavior for a finite element approximation in parabolic variational inequalities, Hindawi Publishing Corporation, J. Mathematical Analysis, doi: /2011/ LANOS Laboratory, Badji Mokthtar-Annaba University, P.O. Box 12, Annaba, Algeria, [email protected] This work was supported by LANOS Laboratory of Annaba university-algeria

125 International Congress in Honour of Professor Ravi P. Agarwal The Smoothness of Convolutions of Zonal Measures on Compact Symmetric Spaces Sanjiv Kumar Gupta and Kathryn Hare We prove that for every compact symmetric space, G c/k, of rank r, the convolution of any (2r + 1) continuous, K-biinvariant measures is absolutely continuous with respect to the Haar measure on G c. We also prove that the convolution of (r + 1) continuous, K-invariant measures on the 1 eigenspace in the Cartan decomposition of the Lie algebra of G c is absolutely continuous with respect to Lebesgue measure. These results are nearly sharp. [1] Boudjemaa Anchouch, Sanjiv Kumar Gupta, Convolution of Orbital Mesaures in Symmetric Spaces, Bull. Aust. Math. Soc. 83(2011), [2] Bump, D., Lie Groups, Graduate texts in mathematics 225, Springer, New York, [3] Dunkl, C., Operators and harmonic analysis on the sphere, Trans. Amer. Math. Soc. 125(1966), [4] Gupta, S.K. and Hare, K.E., Singularity of orbits in SU(n), Israel J. Math. 130(2002), [5] Gupta, S.K. and Hare, K.E., Singularity of orbits in classical Lie algebras, Geom. Func. Anal. 13(2003), [6] Gupta, S.K. and Hare, K.E., Convolutions of generic orbital measures in compact symmetric spaces, Bull. Aust. Math. Soc. 79(2009), [7] Gupta, S.K. and Hare, K.E., L 2 -singular dichotomy for orbital measures of classical compact Lie groups, Adv. Math. 222(2009), [8] Gupta, S.K. and Hare, K.E., Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n), Monatsch. Math 159(2010), [9] Gupta, S.K., Hare, K.E. and Seyfaddini, S., L 2 -singular dichotomy for orbital measures of classical simple Lie algebras, Math. Zeit. 262(2009), [10] Humphreys, J., Introduction to Lie algebras and representation theory, Springer Verlag, New York, [11] Helgason, S., Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, [12] Kane, R., Reflection groups and invariant theory, Canadian Math. Soc., Springer, N.Y., [13] Knapp, A., Lie groups beyond an introduction, Birkhauser, Verlag AG (2002). [14] Ragozin, D., Zonal measure algebras on isotropy irreducible homogeneous spaces, J. Func. Anal. 17(1974), [15] Ragozin, D., Central measures on compact simple Lie groups, J. Func. Anal. 10(1972), [16] F. Ricci and E. Stein, Harmonic analysis on nilpotent groups and singular integrals. II. Singular kernels supported on submanifolds, J. Func. Anal. 78(1988), [17] Wright, A., Sums of adjoint orbits and L 2 -singular dichotomy for SU(m), Adv. Math. 227(2011), Sanjiv K. Gupta: Dept. of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36 Al Khodh 123, Sultanate of Oman, gupta.squ.edu.om Kathryn E. Hare: Dept. of Pure Mathematics, University of Waterloo, Waterloo, Ont, Canada, N2L 3G1, kehare@ uwaterloo.ca The first author would like to thank the Dept. of Pure Mathematics for their hospitality while some of this research was done. This research was supported in part by NSERC and the Sultan Qaboos University.

126 126 International Congress in Honour of Professor Ravi P. Agarwal 92 A Tauberian Theorem for the Weighted Mean Method of Summability of Sequences of Fuzzy Numbers Zerrin Önder, Sefa Anıl Sezer and Ibrahim Çanak The notion of fuzzy set was realized by many researchers who are interested in Mathematics, Computer Science and Engineering and the idea was applied for studies in di?erent branches of science from different aspects. One of the areas which was applied it is the summability theory as well. In this talk, we focus on the weighted mean method of summability of sequences of fuzzy numbers and present a Tauberian theorem of slowly decreasing type. [1] Y. Altin, M. Mursaleen and H. Altinok. Statistical summability (C,1) for sequences of fuzzy real numbers and a Tauberian theorem. J. Intell. Fuzzy Systems, 21(6), [2] S. Aytar, M. A. Mammadov and S. Pehlivan. Statistical limit inferior and limit superior for sequences of fuzzy numbers. Fuzzy Sets and Systems, 157(7): , [3] B. Bede. Mathematics of fuzzy sets and fuzzy logic. Springer, Berlin, [4] I. Canak. On the Riesz mean of sequences of fuzzy real numbers. J. Intell. Fuzzy Systems, DOI: /IFS [5] I. Canak. Tauberian theorems for Cesaro summability of sequences of fuzzy numbers. J. Intell. Fuzzy Systems, DOI: /IFS [6] D. Dubois and H. Prade. Operations on fuzzy numbers. Internat. J. Systems Sci., 9(6): , [7] D. Dubois and H. Prade. Fuzzy sets and systems. Mathematics in Science and Engineering. Academic Press, New York-London, [8] R. Goetschel and W. Voxman. Elementary fuzzy calculus. Fuzzy Sets and Systems, 18(1):31-43, [9] M. Matloka. Sequences of fuzzy numbers. Busefal, 28:28-37, [10] F. Moricz and B.E. Rhoades. Necessary and su?cient Tauberian conditions for certain weighted mean methods of summability. II. Acta Math. Hung., 102(4): , [11] S. Nanda. On sequences of fuzzy numbers. Fuzzy Sets and Systems, 33(1): , [12] P. V. Subrahmanyam. Cesaro summability for fuzzy real numbers. J. Anal., 7: , [13] O. Talo and F. Basar. On the slowly decreasing sequences of fuzzy numbers. Abstr. Appl. Anal., pp. Art. ID , 7, [14] O. Talo and C. C akan. On the Cesaro convergence of sequences of fuzzy numbers. Appl. Math. Lett., 25(4): , [15] B. C. Tripathy and A. Baruah. Norlund and Riesz mean of sequences of fuzzy real numbers. Appl. Math. Lett., 23(5): , [16] L. A. Zadeh. Fuzzy sets. Information and Control, 8: , Zerrin Önder: Ege University, Department of Mathematics, 35100, Izmir, Turkey, [email protected] Sefa Anıl Sezer: Ege University, Department of Mathematics, 35100, Izmir, Turkey, [email protected], [email protected] İbrahim Çanak: Ege University, Department of Mathematics, 35100, Izmir, Turkey, [email protected],

127 International Congress in Honour of Professor Ravi P. Agarwal Asymptotic Constancy for a System of Impulsive Delay Differential Equations Fatma Karakoç and Hüseyin Bereketoğlu In this talk, we investigate a class of impulsive delay differential equations system. First, convergence of solution is proved. Then a formula is obtained for the limit of the solution. [1] F.V. Atkinson, J.R. Haddock, Criteria for asymptotic constancy of solutions of functional-differential equations. JMAA 91 (2) (1983). [2] H. Bereketoğlu and M. Pituk, Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete and Continuous Dynamical Systems, Supp. Vol., (2003) [3] F. Karakoç and H. Bereketoğlu, Some results for linear impulsive delay differential equations. Dynamics of Continuous, Discrete and Impulsive Systems Series A: 16, , (2009). Fatma Karakoç: Ankara University, Faculty of Science, Department of Mathematics, Ankara-Turkey, [email protected] Hüseyin Bereketoğlu: Ankara University, Faculty of Science, Department of Mathematics, Ankara-Turkey, [email protected]

128 128 International Congress in Honour of Professor Ravi P. Agarwal 94 An Extension w with rankw = 3 of a Valuation v on a Field K with rankv = 2 to K(x) Figen Öke Let v = v 1 ov 2 be a valuation on a field K with rankv = 2. In this study an extension w = w 1 ow 2 ow 3 of v such that rankw = 3 and w 2 is trivial over the residue field k v1 is defined and its properties are investigated. [1] V. Alexandru, N. Popescu, A. Zaharescu, A theorem of characterization of residual transcendental extension of a valuation, J. Math. Kyoto Univ. 28 (1988), [2] V. Alexandru, N. Popescu, A. Zaharescu, Minimal pair of definition of a residual transcendental extension of a valuation, J. Math. Kyoto Univ., 30 (1990), no. 2, [3] V. Alexandru, N. Popescu, A. Zaharescu, All valuations on K(X), J. Math. Kyoto Univ., 3 (1990), no. 2, [4] N. Bourbaki, Algebre Commutative, Ch. V: Entiers, Ch. VI: Valuations, Hermann, Paris (1964). [5] O. Endler, Valuation Theory, Springer, Berlin -Heidelberg-New York (1972). [6] N. Popescu, C. Vraciu, On the extension of valuations on a field K to K(x)-I, Ren. Sem. Mat. Univ. Padova, 87 (1992), [7] N. Popescu, C. Vraciu On the extension of valuations on a field K to K(x)-II, Ren. Sem. Mat. Univ. Padova, 96 (1996), 1-14 [8] O.F.G. Schilling, The Theory of Valuations, A.M.S. Surveys, no. 4, Providence, Rhode Island (1950). Trakya University Department of Mathematics, Edirne, Turkey, [email protected]

129 International Congress in Honour of Professor Ravi P. Agarwal Inclusions Between Weighted Orlicz Space Alen Osançlıol Let (X, Σ, µ) be a measure space and Φ be a Young function. The weighted Orlicz space with weight ω is denoted by L Φ ω (X) and it is a natural generalization of the weighted Lebesgue space in which characterization of inclusion is well known. In this talk, we investigate the inclusion between weighted Orlicz spaces L Φ 1 w 1 (X) and L Φ 2 w 2 (X) with respect to Young functions Φ 1, Φ 2 and weights w 1, w 2. We also define the weighted Orlicz norm and show that the inclusion map is continuous. Moreover, in case of X = R n with Lebesgue measure on R n, we give a necessary and sufficient conditions on weights for the equality of two weighted Orlicz spaces when Φ 1 = Φ 2. [1] H.G. Feichtinger, Gewichtsfunktione nauf lokal kompakten Gruppen, Sitzungsberichte der Österr. Akad d Wissenchaften, Mathem-naturw, Klasse, AbteilungII, (1979), 188, Bd, 8. bis 10, [2] M. M. Rao and Z.D. Ren, Theory of Orlicz Spaces, CRC Press, 1 edition, 1991, [3] A. Villani, A Note on the Inclusion L p (µ) L q (µ), The American Mathematical Monthly, Vol.92 (1985), No.7, , Sabancı University, Faculty of Engineering and Natural Sciences, Tuzla, Istanbul-Turkey, [email protected] This work was supported by the Scientific Research Projects Coordination Unit of Istanbul University, Project number

130 130 International Congress in Honour of Professor Ravi P. Agarwal 96 On the Some Graph Parameters for Special Graphs Nihat Akgüneş, Ahmet Sinan Çevik and Ismail Naci Cangül In this talk, we discuss some graph parameters for important special graphs, for instance, ladder graph. Actually, the n-ladder graph can be defined as P 2 P n, where P n is a path graph. We aim to implement some algorithms for computing the some important graph parameters for that graphs. We will investigate some good result for some parameters of that graphs. [1] H. Hosoya, and F. Harary, On the Matching Properties of Three Fence Graphs. J. Math. Chem., 12 (1993), [2] M. Noy, and A. Ribó, Recursively Constructible Families of Graphs. Adv. Appl. Math. 32 (2004), [3] C. P. Mooney, Generalized Irreducible Divisor Graphs, Communications in Algebra, (2014), , DOI: / [4] R. Chang, and Z. Yan, On The Harmonic Index and The Minimum Degree of A Graph, Romanian Journal of Information Science And Technology 15-4 (2012), [5] L. Zhong, The harmonic index for graphs, Applied Mathematics Letters 25-3 (2012), [6] H. Deng, S. Balachandran, S. K. Ayyaswamy and Y. B. Venkatakrishnan, On the harmonic index and the chromatic number of a graph. Discrete Applied Mathematics, (2013)., [7] N. Akgunes, K.C. Das, A.S. Cevik, Topological indices on a graph of monogenic semigroups, in: Ivan Gutman (Ed.), Topics in Chemical Graph Theory, Mathematical Chemistry Monographs, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac,( 2014), ISBN No.16a, IV+262 pp Nihat Akgüneş: Necmettin Erbakan University, Faculty of Science, Department of Mathematics-Computer Sciences, Konya- Turkiye, [email protected] Ahmet Sinan Çevik: Selcuk University, Faculty of Science, Department of Mathematics, Konya-Turkiye, İsmail Naci Cangül: Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkiye, [email protected] This work was partially supported by the Necmettin Erbakan University, Selcuk University, Uludag University.

131 International Congress in Honour of Professor Ravi P. Agarwal A Note on the Dirichlet-Neumann First Eigenvalue of a Family of Polygonal Domains in R 2 A.R.Aithal and Acushla Sarswat Let 1 and 0 be closed, regular, convex, concentric polygons having n sides in R 2 such that the circumradius of 0 is strictly less than the inradius of 1. We fix 1 and vary 0 by rotating it about its center. Let Ω be the interior of 1 \ 0. In this paper we examine the behaviour of the first Dirichlet-Neumann eigenvalue λ 1 (Ω) through a variation of the domain. [1] A. E. Soufi and R. Kiwan, Extremal First Dirichlet Eigenvalue of Doubly Connected Plane Domains and Dihedral Symmetry, SIAM J. Math. Anal., 39(4)(2007), pp , [2] A. R. Aithal and R. Raut, On the Extrema of Dirichlet s First Eigenvalue of a Family of Punctured Regular Polygons in Two Dimensional Space Forms, Proc. Indian Acad. Sci. Math. Sci., 122(2012), pp , [3] G. B. Folland, Introduction to Partial Differential Equations, Second Edition, Prentice-Hall of India, New Delhi, 2001, [4] P. Grisvard, Singularities in Boundary Value Problems,Recherches en Mathématiques Appliqués 22, Masson, Paris; Springer-Verlag, Berlin, 1992, [5] J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization-Shape Sensitivity Analysis, Springer-Verlag, A.R. Aithal: Department of Mathematics, University of Mumbai, Vidyanagari, Mumbai , India, [email protected] Acushla Sarswat: Department of Mathematics, University of Mumbai, Vidyanagari, Mumbai , India, [email protected]

132 132 International Congress in Honour of Professor Ravi P. Agarwal 98 An Approach to the Numerical Verification of Solutions for Variational Inequalities C.S.Ryoo In this talk, we describe a numerical method to verify the existence of solutions for a unilateral boundary value problems for second order equation governed by the variational inequalities. It is based on Nakao s method by using finite element approximation and its explicit error estimates for the problem. Using the Riesz present theory in Hilbert space, we first transform the iterative procedure of variational inequalities into a fixed point form. Then, using the Schauder fixed point theory, we construct a high efficiency numerical verification method that through numerical computation generates a bounded, closed, convex set in which includes the approximate solution. Finally, a numerical example is illustrated. [1] N. W. Bazley, D. W. Fox, Comparision operators for lower bounds to eigenvalues, J. reine angew. Math., 223(1966) [2] J. Cea, Optimisation, théorie et algorithmes, Paris: Dunod, [3] X. Chen, A verification method for solutions of nonsmooth equations, Computing, 58(1997) [4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, [5] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York, [6] I. Hlaváček, J. Haslinger, J. Nečas, J. Lovíšek, Solution of Variational Inequalities in Mechanics, Springer Ser. Appl. Math. Sci., 66, [7] T. Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan, 4(1949) [8] N. J. Lehmann, Optimale Eigenwerteinschließungen, Numer. Math., 5(1963) [9] U. Mosco, Approximation of the solutions of some variational inequalities, Roma: Press of Roma University, [10] M. T. Nakao, A numerical verification method for the existence of weak solutions for nonlinear boundary value problems. Journal of Math. Analysis and Appl.164(1992) [11] Y. Watanabe, N. Yamamoto, M. T. Nakao, Verified computations of solutions for nondifferential elliptic equations related to MHD equilibria. Nonlinear Analysis, Theory, Mathods & Applications, 28(1997) [12] F. Natterer, Optimale L 2 -Konvergenz finiter Elemente bei Variationsungleichungen. Bonn. Math. Schr.89(1976) 1-12 [13] M. Plum, Explict H 2 -Estimates and Pointwise Bounds for Solutions of Second-Order Elliptic Bounday Value Problems. Journal of Mathematical Analysis and Appl.165(1992)36-61 [14] M. Plum, Enclosures for weak solutions of nonlinear elliptic boundary value problems, WSSIAA, World Scientific Publishing Company, 3(1994) [15] M. Plum, Existence and Enclosure Results for Continua of Solutions of Parameter -Dependent Nonlinear Boundary Value Problems, Journal of Computational and Applied Mathematics, 60(1995) [16] S. M. Rump, Solving algebraic problems with high accuracy, A new approach to scientific computation ( Edited by U. W. KULISH and W. L. MIRANKER ), Academic Press, New York, 1983, pp [17] C. S. Ryoo, Numerical verification of solutions for a simplified Signorini problem, Comput. Math. Appl., 40(2000) [18] C.S. Ryoo, An approach to the numerical verification of solutions for obstacle problems, Computers and Mathematics with Applications, 53 (2007), [19] C.S. Ryoo, Numerical verification of solutions for Signorini problems using Newton-like mathod, International Journal for Numerical Methods in Engineering, 73 (2008), [20] C.S. Ryoo and R.P. Agarwal, Numerical inclusion methods of solutions for variational inequalities, International Journal for Numerical Methods in Engineering, 54 (2002), [21] C.S. Ryoo and R.P. Agarwal, Numerical verification of solutions for generalized obstacle problems, Neural, Parallel & Scientific Compuataions, 11 (2003), [22] C.S. Ryoo and M.T. Nakao, Numerical verification of solutions for variational inequalities, Numerische Mathematik, 81 (1998), [23] F. Scarpini, M. A. Vivaldi, Error estimates for the approximation of some unilateral problems. R.A.I.R.O. Numerical Analysis, 11(1977) [24] G. Strang, G. Fix, An analysis of the finite element method. Englewood Cliffs, New Jersey, Prentice-Hall, Department of Mathematics, Hannam University, Daejeon , Korea, [email protected]

133 International Congress in Honour of Professor Ravi P. Agarwal Local Rings and Projective Coordinate Spaces Fatma Özen Erdoğan and Süleyman Çiftçi In this paper; some properties of modules constructed over the real plural algebra A are investigated and also a module over the linear algebra of matrix K = M mm(r) is constructed. Then, projective coordinate spaces over a local ring R are addressed. Finally, the concept of a projective space over a vector space is generalized to a space over a module by the help of equivalence classes. [1] B. R. McDonald, Geometric algebra over local rings, New York: Marcel Dekker, 1976, [2] F. Machala, Fundamentalsatze der projektiven Geometrie mit Homomorphismus, Rozpravy CSAV, Rada Mat. Prirod. Ved, 90, 5, 1980, [3] H. Lenz, Vorlesungen über projektive Geometrie I. Leipzig, 1965, [4] J.W.P. Hirschfeld, Projective Geometries over Finite Fields. Oxford Science Publications, New York, 1998, [5] K. Nomizu, Fundamentals of Linear Algebra, New York, McGraw-Hill, 1966, [6] M. Jukl, Linear forms on free modules over certain local ring, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 32 (1993), 49-62, [7] M. Jukl, Grassmann formula for certain type of modules, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 34 (1995), 69-74, [8] M. Jukl, V. Snasel, Projective equivalence of quadrics in Klingenberg Projective Spaces over a special local ring, International Journal of Geometry, 2 (2009), 34-38, [9] R. Lingenberg, Grundlagen der Geometrie I. Bibliographisches Institut Mannheim (Wien) Zürich, 1969, [10] T. W. Hungerford, Algebra, New York, Holt, Rinehart and Winston, Fatma Özen Erdoğan: Uludag University, Faculty of Science, Department of Mathematics, Görükle, Bursa-Turkiye, [email protected] Süleyman Çiftçi: Uludag University, Faculty of Science, Department of Mathematics, Görükle, Bursa-Turkiye, This work is supported by The Research Fund of University of Uludag Project number KUAP(F)-2012/56.

134 134 International Congress in Honour of Professor Ravi P. Agarwal 100 An Improved Numerical Solution of the Singular Boundary Integral Equation of the Compressible Fluid Flow Around Obstacles Using Modified Shape Functions Luminita Grecu In this work an improved numerical solution of the singular boundary integral equation of the 2D copressible fluid flow around obstacles is obtained by a boundary element method based on modified shape functions and cubic boundary elements. The singular boundary integral equation with sources distribution is considered in this paper, and, for its discretization, cubic boundary elements are used. The integrals of singular kernels are evaluated using modified shape functions which are deduced by using series expansions for the basis functions choose for the local approximation models. A computer code is made using Mathcad programming language, and based on it some particular cases are solved. In order to validate the proposed method, comparisons between numerical solutions and exact ones are performed for the considered test problems. A comparison between the numerical solution obtained by the method proposed and the one obtained by a method that also uses cubic boundary elements but doesn t use modified shape functions for evaluating the singularities is also made. [1] H.M.Antia, Numerical Methods for Scientists and Engineers, Birkhausen, 2002 [2] M. Bonne, Bounndary integral equation methods for solids and fluids, John Wiley and Sons, [3] C.A. Brebbia, J.C.F.Telles, L.C.Wobel, Boundary Element Theory and Application in Engineering, Springer-Verlag, Berlin,1984. [4] L. Dragoş, Mathematical Methods in Aerodinamics, Editura Academiei Române, Bucureşti, [5] L. Dragoş, Fluid Mechanics I, Editura Academiei Române, Bucureşti, [6] L. Grecu, A solution of the boundary integral equation of the theory of infinite span airfoil in subsonic flow with linear boundary elements, Annals of the University of Bucharest, Year LII, Nr. 2(2003), [7] L. Grecu, A solution with cubic boundary elements for the compressible fluid flow around obstacles, Boundary Value Problems :78 [8] IK. Lifanov, Singular integral equations and discrete vortices, VSP, Utrecht, The Netherlands, 1996.A. Carabineanu, A boundary element approach to the 2D potential flow problem around airfoils with cusped trailing edge, Computer Methods in Applied. Mechanics and Engineering. 129(1996), [9] I. Vladimirescu, L. Grecu, Weakening the Singularities when Applying the BEM for 2D Compressible Fluid Flow, Lecture Notes in Engineering and Computer Science: Proc. of The International MultiConference of Engineers and Computer Scientists 2010, Hong Kong, University of Craiova, Department of Applied Mathematics, Craiova, Romania, [email protected]

135 International Congress in Honour of Professor Ravi P. Agarwal New Aspects of Calculating Volumes in E n Daniela Bittnerová and Daniela Bímová The talk presents an alternative approach to the calculation of the volumes of solids in n-dimensional Euclidean space and shows some applications of that theory, including the proof of the formula for it. The mentioned method uses basic topological properties, among others. To solve volumes of solids, we must find suitable parametric descriptions of surface areas of given solids. These surface areas must be smooth or piecewise smooth areas in Euclidean space of the corresponding dimensions. The advantage of the theory could be in the using of integrals of the dimension less then n. This contribution also refers to correspondence between the curvilinear and surface integral theory for calculations of areas of closed figures, respectively volumes of solids, and the results of the alternative theory. However, it is kept generally in n-dimensional space for the alternative theory. [1] D. Bittnerová, Alternative Method for Calculations of Volumes by Using Parameterizations Surfaces Areas, AIP Conf. Proc. 1570, 3 (2013), [2] D. Bittnerová, P. Červenková, The Corollary of Gauss-Ostrogradsky Theorem, Proceedings of XXX. International Colloquium. Brno, 2012, [3] S. Huggett, D. Jordan, A Topological Aperitiv, Springer-Verlang, 2001, ISBN , [4] S. Dineen, Multivariate Calculus and Geometry, Springer-Verlag, 2001, ISBN X. Daniela Bittnerová: Technical University of Liberec, Faculty of Science, Humanities and Education, Department of Mathematics and Didactics of Mathematics, Liberec - Czech Republic, [email protected] Daniela Bímová: Technical University of Liberec, Faculty of Science, Humanities and Education, Department of Mathematics and Didactics of Mathematics, Liberec - Czech Republic, [email protected] The paper was supported by the project SGS - FP - TUL 2014 Nonlinear Parameterization - applications using graphic software. Special thanks to students H. Vacatová, P. Vurmová, and D. Vacata for their help with solving examples.

136 136 International Congress in Honour of Professor Ravi P. Agarwal 102 Applications of an Alternative Methods for Volumes of Solids of Revolution Daniela Bímová and Daniela Bittnerová In this talk, we connect the contribution New Aspects of Calculating Volumes in E n in this congress where an alternative theory for calculations of volumes of solids in the n-dimensional Euclidean space is presented. Now we discuss the application of that theory to volumes of solids of revolution with circular and elliptical perpendicular sections (sphere, toroid, axoid, horn toroid, melanoid). The proofs of the formulas of the alternative theory mentioned above are given for that type of solids. Using the mentioned theory, the formulas of volumes computed solids of revolution are simpler than the general ones are. [1] D. Bittnerová, Alternative Method for Calculations of Volumes by Using Parameterizations Surfaces Areas, AIP Conf. Proc. 1570, 3 (2013), [2] D. Bittnerová, Parameterization and Volume of a Torus, ICPM 07 Conf. Proc., Liberec (2007), [3] S. Huggett, D. Jordan, A Topological Aperitiv, Springer-Verlang, 2001, ISBN , [4] S. Dineen, Multivariate Calculus and Geometry, Springer-Verlag, 2001, ISBN X. Daniela Bímová: Technical University of Liberec, Faculty of Science, Humanities and Education, Department of Mathematics and Didactics of Mathematics, Liberec - Czech Republic, [email protected] Daniela Bittnerová: Technical University of Liberec, Faculty of Science, Humanities and Education, Department of Mathematics and Didactics of Mathematics, Liberec - Czech Republic, [email protected] The paper was supported by the project SGS - FP - TUL 2014 Nonlinear Parameterization - applications using graphic software. Special thanks to students H. Vacatová, P. Vurmová, and D. Vacata for their help with solving examples and drawing figures.

137 International Congress in Honour of Professor Ravi P. Agarwal On Certain Sums of Fibonomial Coefficients Emrah Kılıç and Aynur Yalçıner In this talk, we present some classes of sums formulas including Fibonomial coefficients with finite product of generalized Fibonacci and Lucas numbers as coefficients. We translate everything into q-notation and then use generating function and Rothe s identity to prove them. [1] G. E. Andrews, R. Askey, R. Roy, Special functions, Cambridge University Press, 2000, [2] H. W. Gould, The bracket function and Fontene-Ward generalized binomial coefficients with application to Fibonomial coefficients, The Fibonacci Quarterly, 7 (1969), 23-40, [3] V. E. Hoggatt Jr., Fibonacci numbers and generalized binomial coefficients, The Fibonacci Quarterly, 5 (1967), , [4] A. F. Horadam, Generating functions for powers of a certain generalized sequence of numbers, Duke Math. J., 32 (1965), , [5] E. Kılıç, The generalized Fibonomial matrix, European J. Combinatorics, 31 (2010), , [6] E. Kılıç, H. Prodinger, I. Akkus, H. Ohtsuka, Formulas for Fibonomial Sums with generalized Fibonacci and Lucas coefficients, The Fibonacci Quarterly, 49:4 (2011), , [7] E. Kılıç, H. Prodinger, Closed form evaluation of sums containing squares of Fibonomial coefficients, accepted in Mathematica Slovaca. [8] E. Kılıç, H. Ohtsuka, I. Akkus, Some generalized Fibonomial sums related with the Gaussian q-binomial sums, Bull. Math. Soc. Sci. Math. Roumanie, 55:103, No. 1, (2012), 51-61, [9] D. Marques, P. Trojovsky, On Some New Sums of Fibonomial Coefficients, The Fibonacci Quarterly, 50:2 (2012), , [10] P. Trojovsky, On some identities for the Fibonomial coefficients via generating function, Discrete Appl. Math., 155:15, (2007), Emrah Kılıç: TOBB University of Economics and Technology, Mathematics Department, Ankara, Turkey, Aynur Yalçıner: Selçuk University, Faculty of Science, Department of Mathematics, Campus 42075, Konya, Turkey, [email protected].

138 138 International Congress in Honour of Professor Ravi P. Agarwal 104 Null Generalized Helices of a Null Frenet Curve in L 4 Esen Iyigün In this paper; we study the null generalized helices in view of curvature functions and harmonic curvatures in 4-dimensional Lorentzian space for two time-like and two null vectors by using the Frenet frame in [2] for a null curve. [1] B. O Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983, [2] K. L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1996, [3] F. A. Yalınız, H. H. Hacısalihoğlu, Null Generalized Helices in L m+2, Bull. Malays. Math. Sci. Soc. (II), 30(1) (2007), 74-85, [4] N. Ekmekçi, H. H. Hacısalihoğlu and K. İlarslan, Harmonic Curvatures in Lorentzian Space, Bull. Malays. Math. Sci. Soc. (Second Series), 23 (2000), , [5] R. Aslaner, A. İ. Boran, On The Geometry of Null Curves in The Minkowski 4-space, Turk. J. Math., 33 (2009), , [6] A. Altın, Harmonic Curvatures of Null Curves and The Null Helix in R m+2 1, International Mathematical Forum, 2(23) (2007), , Uludag University, Faculty of Science, Department of Mathematics, Görükle, Bursa-Turkiye, [email protected]

139 International Congress in Honour of Professor Ravi P. Agarwal Geometrical Methods and Numerical Computations for Prey-Predator Lotka-Volterra Systems Adela Ionescu, Romulus Militaru and Florian Munteanu The purpose of this paper is to study symmetries and conservation laws for the mathematical model of the multi-species interactions, given by Volterra-Lotka equations. We will recall the Hamilton-Poisson realisations of 2D and 3D Volterra- Lotka systems and we use the geometric formalism to obtain new conservation laws starting from symmetries. Our study is a interplay between dynamical systems geometrical theory and computational calculus of dynamical systems, knowing that the theory provides a framework for interpreting numerical observations and foundations for algorithms. [1] P. Gao, Hamiltonian structure and first integrals for the Lotkaa Volterra systems, Physics Letters A, 273 (2000), [2] B. Grammatjcos, J. Moulin-Ollagnier, A. Ramani, J. M. Strelcyn and S. Wojciechowski, Integrals of quadratic ordinary differential equations in R 3 : the Lotka-Volterra system, Physica A 163 (1990) [3] R. Militaru, F. Munteanu, Geometric Methods for the Study of Biodynamical Systems. Symmetries and Conservation Laws, Proc. 9th Int. Conf. Cellular and Molecular Biology, Biophysics and Bioengineering(BIOa) Chania, Crete, Greece, August 27-29, 2013, pp [4] R. Militaru, F. Munteanu, Symmetries and Conservation Laws for Biodynamical Systems, Int. J. of Math. Models and Methods in Applied Sciences, Issue 12, Vol. 7 (2013), pp [5] Y. Nutku, Hamiltonian structure of the Lotka-Volterra equations, Physics Letters A, 145 (1) (1990), [6] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, [7] R. Tudoran, A. Gîrban, On a Hamiltonian version of a three-dimensional Lotka-Volterra system, Nonlinear Analysis: Real Word Applications, 13 (2012), Adela Ionescu: Department of Applied Mathematics, University of Craiova, Al. I. Cuza 13, Craiova , Romania, [email protected] Romulus Militaru: Department of Applied Mathematics, University of Craiova, Al. I. Cuza 13, Craiova , Romania, [email protected] Florian Munteanu: Department of Applied Mathematics, University of Craiova, Al. I. Cuza 13, Craiova , Romania, [email protected] This work was partially supported by the grant number 19C/2014, awarded in the internal grant competition of the University of Craiova.

140 140 International Congress in Honour of Professor Ravi P. Agarwal 106 Fractional Calculus Model of Dengue Epidemic Moustafa El-Shahed This paper deals with the fractional order dengue epidemic model. The stability of disease free and positive fixed points is studied. Adams Bashforth Moulton algorithm have been used to solve and simulate the system of differential equations. Department of Mathematics, Faculty of Art and Sciences, P.O. Box 3771, Unizah-Qassim, Qassim University, Saudi Arabia,

141 International Congress in Honour of Professor Ravi P. Agarwal Zagreb Co Indices and Augmented Zagreb Index and its Polynomials of Phenylene and Hexagonal Squeeze P.S.Ranjini, V.Lokesha and Usha.A The topological indices are the graph invariants obtained from the molecular graphs corresponding to the structural features of organic molecules. A topological index of a chemical compound characterizes the compound and obeys a particular rule. In this paper, we find the Augmented Zagreb index, Zagreb Co indices, Zagreb Co indices polynomials and Augmented Zagreb polynomials for Phenylene and Hexagonal Squeeze. Ranjini P.S: Department of Mathematics, Don Bosco Institute Of Technology,Bangalore-74, India, ranjini p s@yahoo. com V. Lokesha: Department of Mathematics, Vijayanagara Sri Krishnadevaraya University, Bellary, India, Usha.A: Department of Mathematics, Alliance College of Engineering and Design, Alliance University, Anekal- Chandapura Road, Bangalore, India, [email protected]

142 142 International Congress in Honour of Professor Ravi P. Agarwal 108 A Note on Class Numbers of Real Quadratic Fields with Certain Fundamental Discriminants Ayten Pekin ONO, proved a theorem in by applying Sturm s Theorem on the congruence of modular form to Cohen s half integral weight modular forms. Later, Dongho Byeon proved a theorem and corollory by refining Ono methods. In this paper, we will give a theorem for certain real quadratic fields by considering above mentioned studies. To do this, we shall obtain an upper bound different from current bounds for L(1, χ D ) and use Dirichlet s class number formula. Department of Mathematics, Faculty of Science, Istanbul University, Istanbul-Turkey, [email protected]

143 International Congress in Honour of Professor Ravi P. Agarwal On Three Dimensional Dynamical Systems on Time Scales Elvan Akın In this talk, motivated by Thandapani and Ponnamal [6], we investigate oscillation and asymptotic properties of solutions of three dimensional systems of first order dynamic equations on a time scale, nonempty closed subset of real numbers. The theory of dynamic equations on time scales has been created in order to unify continuous and discrete analysis, see books by Bohner and Peterson [4] and [5]. We also refer the readers to manuscripts by Akin, Dosla, Lawrence [2], [3] and Akgul and Akin [1]. [1] Akgul, A. and Akin, E., Almost oscillatory three dimensional dynamical systems of first order delay dynamic equations, Nonlinear Dynamics and Systems Theory. To appear, [2] Akin-Bohner, E., Dosla, Z., and Lawrence, B. Oscillatory properties for three dimensional dynamic systems. Nonlinear Anal., 69 (2) (2008) [3] Akin-Bohner, E., Dosla, Z., and Lawrence, B. Almost oscillatory three-dimensional dynamical system. Adv. Difference Equ., 46 (2012) 14 pages. [4] Bohner, M. and Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, Boston, [5] Bohner, M. and Peterson, A., Advances in Dynamic Equations on Time Scales. Birkhauser, Boston, [6] Thandapani, E. and Ponnammal, B. Oscillatory properties of solutions of three-dimensional difference systems. Math. Comput. Modelling, 45 (5-6) (2005) Missouri University Science Technology 400 W 12th Street Rolla, MO, , USA

144 144 International Congress in Honour of Professor Ravi P. Agarwal 110 On the Difference Equation System x n+1 = 1+y n y n, y n+1 = 1+y n x n Necati Taskara, Durhasan Turgut Tollu and Yasin Yazlik In this paper, we mainly consider the system of difference equations x n+1 = 1+yn, y y n+1 = 1+yn, n N n x 0 where initial n conditions x 0 and y 0 are real numbers such that the denominators are always nonzero. We give exact information about the behavior of solutions of the system. [1] R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York (1992), [2] M. R. S. Kulenović and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman&Hall/CRC Press, (2002), [3] L. Brand, A sequence defined by a difference equation. Am. Math. Mon. 62 (1955), , [4] D. T. Tollu, Y. Yazlik, N. Taskara, On the Solutions of two special types of Riccati Difference Equation via Fibonacci Numbers, Advances in Difference Equations, (2013), 2013:174, [5] E.A. Grove, Y. Kostrov, G. Ladas, and S. W. Schultz, Riccati difference equations with real period-2 coeficients, Commun. Appl. Nonlinear Anal., 14 (2007), 33-56, [6] G. Papaschinopouls and B.K. Papadopoulos, On the Fuzzy Difference Equation, Soft Computing, 6 (2002), , [7] I. Yalçinkaya, C. Çinar, M. Atalay, On the Solutions of Systems of Difference Equations, Advances in Difference Equations, Vol. (2008), Article ID , [8] D. T. Tollu, Y. Yazlik, N. Taskara, On fourteen solvable systems of difference equations, Applied Mathematics and Computation, 233 (2014), , [9] L. Berg, S. Stevice, On some systems of difference equations, Appl. Math. Comput. 218 (2011) , [10] S. Stevice. On some solvable systems of difference equations, Appl. Math. Comput. 218 (2012) , [11] N. Taskara, K. Uslu and D.T. Tollu, The periodicity and solutions of the rational difference equation with periodic coefficients, Computers & Mathematics with Applications, 62 (2011), Necati Taskara: Selcuk University, Faculty of Science, Department of Mathematics, Selçuklu, Konya-Turkiye, D.T. Tollu: Necmettin Erbakan University, Faculty of Science, Department of Mathematics-Computer Sciences, Meram, Konya-Türkiye, [email protected] Y. Yazlik: Nevsehir Haci Bektas Veli University, Faculty of Science and Art, Department of Mathematics, 50300, Nevsehir-Türkiye, [email protected]

145 International Congress in Honour of Professor Ravi P. Agarwal The Binomial Transforms of Tribonacci and Tribonacci- Lucas Sequences Nazmiye Yilmaz and Necati Taskara In this study, we apply the binomial transforms to Tribonacci and Tribonacci-Lucas sequences. Also, the Binet formulas, summations, generating functions of these transforms are found using recurrence relations. Finally, we illustrate the relation between these transforms by deriving new formulas. [1] K.W. Chen, Identities from the binomial transform, Journal of Number Theory 124, (2007), , [2] S. Falcon, A. Plaza, Binomial transforms of k-fibonacci sequence, International Journal of Nonlinear Sciences and Numerical Simulation 10(11-12), (2009), , [3] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY (2001), [4] H. Prodinger, Some information about the binomial transform, The Fibonacci Quarterly 32 (5), (1994), , [5] N. Yilmaz, N. Taskara, Binomial transforms of the Padovan and Perrin matrix sequences, Abstract and Applied Analysis, Article Number: Published: 2013, [6] M. Elia, Derived sequences, the Tribonacci recurrence and cubic forms, The Fibonacci Quarterly 39(2), (2001), , [7] Y. Yazlik, N. Yilmaz, N. Taskara, The Binomial Transforms of the generalized (s,t)-matrix sequence, 4th International Conference of Matrix Analysis and Applications (ICMAA2013), Konya 2013, [8] M. Feinberg, Fibonacci-Tribonacci, The Fibonacci Quarterly 1(3), (1963), 70-74, [9] A.N. Philippou, A. A. Muwafi, Waiting for the Kth consecutive success and the Fibonacci sequence of order K, The Fibonacci Quarterly 20(1), (1982), 28-32, [10] W.R. Spickerman, Binet s formula for the Tribonacci sequence, The Fibonacci Quarterly 20(2), (1982), , [11] M. Catalani, Identities for Tribonacci-related sequences, Cornell University Library, (2002), avaliable at [12] L. Marek-Crnjac, On the mass spectrum of the elementary particles of the standard model using El Naschie s golden field theory, Chaos, Solutions & Fractals, 15(4) (2003), [13] L. Marek-Crnjac, The mass spectrum of high energy elementary particles via El Naschie s golden mean nested oscillators, the Dunkerly-Southwell eigenvalue theorems and KAM, Chaos, Solutions & Fractals, 18(1) (2003), [14] P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-lucas Sequences and its Properties, Journal of mathematics and computer science, 8, (2014), Nazmiye Yilmaz: Selcuk University, Faculty of Science, Department of Mathematics, Selçuklu, Konya-Turkiye, [email protected] Necati Taskara: Selcuk University, Faculty of Science, Department of Mathematics, Selçuklu, Konya-Turkiye,

146 146 International Congress in Honour of Professor Ravi P. Agarwal 112 On the Random Functional Central Limit Theorems with Almost Sure Convergence for Subsequences Zdzislaw Rychlik In this talk, we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related logarithmic limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems for subsequences. [1] I. Berkes, E. Csáki, A universal result in almost sure central limit theory, Stochastic Process. Appl., 94 (2001), , [2] I. Berkes, E. Csáki, S. Csörgő, Almost sure limit theorems for the St. Petersburg game, Statist. Probab. Lett., 45 (1999), 23-3, [3] M. Csörgő, L. Horváth, Invariance principles for logarithmic averages, Math. Proc. CambridgePhilos. Soc., 112 (1992), , [4] I. Fazekas, Z. Rychlik, Almost sure functional limit theorems, Ann. Univ. Mariae Curie-Sk lodowska, 56(1) (2002), 1-18, [5] M. T. Lacey, W. Philipp, A note on the almost sure central limit theorem, Statist. Probab. Lett., 9 (1990), , [6] B. Rodzik, Z. Rychlik, An almost sure central limit theorem for independent random variables, Ann. Inst. H. Poincaré, 30 (1994), 1-11, [7] Z. Rychlik, K. S. Szuster, On the strong versions of the central limit theorem, Statist. Probab. Lett., 61 (2003), , [8] Z. Rychlik, K. S. Szuster, On the random functional central limit theorems with almost sure convergence for subsequences, Demonstratio Mathematica, XLV(2) (2012), , [9] P. Schatte, On the central limit theorems with almost sure convergence, Probab. Math. Statist., 11 (1991), , Maria Curie-Sk lodowska University, Faculty of Mathematics, Physics and Computer Science, Lublin-Poland and State School of Higher Education, Che lm-poland, [email protected]

147 International Congress in Honour of Professor Ravi P. Agarwal Some Fixed Point Theorems for a Pair of Mappings in Complex Valued b-metric Spaces Aiman Mukheimer In this paper, we generalize and study the results of M. Kutbi et al, by improving the conditions of the contraction which is the product and the quotient of metrics, and we establish the existence and uniqueness of common coupled fixed points for a pair of mappings on complex valued b-metric spaces. Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia

148 148 International Congress in Honour of Professor Ravi P. Agarwal 114 Some Characterizations of Slant Curves on Unit Dual Sphere S 2 Seda Oral and Mustafa Kazaz In this paper, we consider the dual Darboux frame {ẽ, t, g} of a ruled surface in Euclidean 3-space E 3. By the aid of the E.Study Mapping, a ruled surface can be consider as a dual spherical curve. Then, we define some new types of curves on unit dual sphere S 2, called slant dual curves that each vector of Darboux frame makes a constant dual angle with some fixed directions in dual 3-space D 3. Furthermore, we give some characterizations for a curve to be a slant dual curve which is important for differential geometry, surface geometry and especially surface design theory. [1] Ali, A., T., Position vectors of slant helices in Euclidean Space E 3. Journal of Egyptian Mathematical Society, V. 20(1), 2012, 1-6. [2] Barros, M., General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125, no.5, (1997) [3] Blaschke, W., Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie Dover, New York, (1945). [4] Karger, A., Novak, J., Space Kinematics and Lie Groups. STNL Publishers of Technical Lit., Prague, Czechoslovakia (1978). [5] Kula, L., Ekmeki, N., Yayl, Y., larslan, K., Characterizations of Slant Helices in Euclidean 3-Space, Turk J Math., 33 (2009) [6] Struik, D., J., Lectures on Classical Differential Geometry, 2nd ed. Addison Wesley, Dover, (1988). [7] O Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, (1983). [8] Önder, M., Slant ruled surfaces in Euclidean 3-space E 3. [9] Kula, L. and Yayli, Y., On slant helix and its spherical indicatrix, Applied Mathematics and Computation. 169, (2005), Seda Oral: Celal Bayar University, Muradiye, Manisa, Turkey, [email protected] Mustafa Kazaz: Celal Bayar University, Faculty of Science and Arts, Department of Mathematics, Muradiye Campus, Manisa, Turkey, [email protected]

149 International Congress in Honour of Professor Ravi P. Agarwal On Solving Some Partial Differential Equations Ümit Sarp and Sebahattin Ikikardes In this talk, the numerical solutions of some partial differential equations have been analyzed by using Differential Transform Method and the obtained results are compared with other numerical methods. The study show us that the results obtained by using Differential Transform Method are compatible with the existing solutions. Also Differential Transform Method can easily be adapted to many computer programs. [1] Arikoglu, A. and Ozkol, I., (2006). Solution of differential-difference equations by using differential transform method. Applied Mathematics and Computation, Issue 181(1), pp [2] Ayaz, F., (2003). On the two-dimensional differential transform method. Applied Mathematics and Computation, Issue 143, pp [3] Ayaz, F., (2004). Applications of differential transform method to differential-algebraic equations. Applied Mathematics and Computation, Issue 152, no. 3, p [4] Ayaz, F., (2004). Solutions of the system of differential equations by differential transform Method. Applied Mathematics and Computation, Cilt 147, p [5] Chen, C. K. and Ho, S. H., (1996). Application of differential transformation to eigenvalue problems. Applied Mathematics and Computation, pp [6] Chen, C. K. and Ho, S. H., (1999). Solving partial differential equations by two-dimensional differential transform method. Applied Mathematics and Computation, Issue 106, p [7] Hassan, A. H. and I., H., (2002). Different applications for the differential transformation in the differential equations. Applied Mathematics and Computation, 2-3(129), p [8] Hassan, A. H. and I., H., (2002). On solving some eigenvalue problems by using a differential Transformation. Applied Mathematics and Computation, Issue 127, p [9] Jang, M. J., Chen, C. L. and Liu, Y. C., (2001). Two-dimensional differential transform for partial differential equations. Applied Mathematics and Computation, Issue 121, pp [10] Keskin, Y. and Oturanç, G., (2009). Reduced differential transform method for partial differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 6(10), pp [11] Kurnaz, A., Oturanç, G. and Kiriş,. E. M., (2005). n-dimensional differential transformation method for solving PDE. International Journal of Computer Mathematics, 3(82), pp [12] Zhou, J. K., (1986). Differential Transformation and Its Applications for Electrical Circuits. Huazhong University Press. Ümit Sarp: Balikesir University, Faculty of Arts and Science Department of Mathematics, Cagis Campus Balikesir- Turkey, [email protected], [email protected] Sebahattin Ikikardes: Balikesir University Faculty of Arts and Science Department of Mathematics, Cagis Campus Balikesir-Turkey [email protected], [email protected] This work was supported by the Scientific Research Projects Unit of Balikesir University, Project number 2014/155.

150 150 International Congress in Honour of Professor Ravi P. Agarwal 116 Some Spectrum Properties in C - Algebras Nilay Sager and Hakan Avcı We show that if ϕ is a - homomorphism between unital commutative C - algebras A and B with A 1 = ϕ 1 ( B 1), then property of mapping of spectrum is satisfied and adjoint mapping ϕ : (B) (A) is surjective, that is, maximal ideal space of B maps to maximal ideal space of A. [1] E. Kaniuth, A Course in Commutative Banach Algebras, Springer - Verlag, New York, 2009, [2] J. Dixmier, C - Algebras, Elsevier North - Holland Publishing Company, Inc., 1977, [3] M. Takesaki, Theory of Operator Algebras I, Springer - Verlag, New York, 1979, [4] W. Rudin, Functional Analysis, Second Edition, McGraw - Hill, Inc., 1991, [5] R. Larsen, Banach Algebras : An Introduction, Marcel Dekker, 1973, [6] M. Bresar, S. Spenko, Determining Elements in Banach Algebras Through Spectral Properties, Journal of Mathematical Analysis and Applications, 393 (2012), , [7] B. Russo, Linear Mappings of Operator Algebras, Proceedings of the American Mathematical Society, 17 (1966), , [8] T. W. Palmer, Jordan - Homomorphisms Between Reduced Banach - Algebras, Pacific Journal of Mathematics, 58 (1975), , [9] I. Kovacz, Invertibility - Preserving Maps of C - Algebras With Real Rank Zero, Abstract and Applied Analysis, 6 (2005), Nilay Sager: Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Kurupelit, Samsun- Turkey, [email protected] Hakan Avcı: Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Kurupelit, Samsun- Turkey, [email protected]

151 International Congress in Honour of Professor Ravi P. Agarwal On Function Spaces with Fractional Fourier Transform in the Weighted Lebesgue Spaces Erdem Toksoy and Ayşe Sandıkçı Let w and ω be weight functions on R d. In this work, we define A w,ω ( α,p R d ) to be the vector space of f L 1 ( w R d ) such that the fractional Fourier transform belongs to L p ( ω R d ) for 1 p <. We endow this space the sum norm f w,ω = f A α,p 1,w + Fαf ( p,ω and show that Aw,ω α,p R d ) becomes a Banach space and invariant under time frequence shifts. Further we show that the mapping y T yf is continuous from R d into A w,ω ( α,p R d ) and the mapping z M zf is continuous from d into A w,ω ( α,p R d ) and A w,ω ( α,p R d ) is a Banach Module over L 1 ( w R d ) with Θ convolution operation. At the end of this work, we discuss inclusion properties of these spaces. Some key references are given below. [1] A. Bultheel, H. Martinez, A Shattered Survey of the Fractional Fourier Transform, Department of Computer Science, K.U.Leuveven, Report TW337, 2002, [2] H. G. Feichtinger, A. T. Gürkanlı, On a Family of Weighted Convolution Algebras, Internat. J. Math. Sci, 13 (1990), , [3] R. H. Fischer, A. T. Gürkanlı and T. S. Liu, On a Family of Weighted Spaces, Math. Slovaca, 46 (1) (1996), 71-82, [4] V. Namias, The Fractional Order of Fourier Transform and its Application in Quantum Mechanics, Journal of the Institute of Mathematics and its Applications, 25 (1980), , [5] H. M. Ozaktas, M. A. Kutay and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing, John Wiley & Sons Ltd, England, 2001, [6] A. Sahin, H. M. Ozaktas and D. Mendlovic, Optical Implementations of Two-Dimensional Fractional Fourier Transforms and Linear Canonical Transforms with Arbitrary Parameters, Applied Optics, 37 (11) (1998), , [7] A. K. Singh, R. Saxena, On Convolution and Product Theorems for FRFT, Wireless Pers Commun, 65 (2012), , Erdem Toksoy: Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Kurupelit, Samsun-Turkey, [email protected] Ayşe Sandıkçı: Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Kurupelit, Samsun-Turkey, [email protected]

152 152 International Congress in Honour of Professor Ravi P. Agarwal 118 Some Convergence Results for Modified SP-Iteration Scheme in Hyperbolic Spaces Aynur Şahin and Metin Başarır In this study, we prove some strong and -convergence theorems for a modified SP-iteration scheme for total asymptotically nonexpansive mappings in hyperbolic spaces by employing recent technical results of Khan et. al. [An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. (2012) 2012:54]. The results presented here extend and improve some well known results in the current literature. [1] Kohlenbach, U: Some logical metatheorems with applications in functional analysis. Trans. Amer. Math. Soc. 357(1), (2004) [2] Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Marcel Dekker, New York (1984) [3] Bridson, M, Haefliger, A: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999) [4] Chang, SS, Wang, L, Joesph Lee, HW, Chan, CK, Yang, L: Demiclosed principle and -convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces. Appl. Math. Comput. 219, (2012) [5] Fukhar-ud-din, H, Kalsoom, A, Khan, MAA: Existence and higher arity iteration for total asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. arxiv: v2 [math.fa] (2013) [6] Phuengrattana, W, Suantai, S: On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 235, (2011) [7] Khan, AR, Fukhar-ud-din, H, Khan, MAA: An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Fixed Point Theory Appl. (2012) 2012:54, doi: / [8] Zhao, LC, Chang, SS, Wang, XR: Convergence theorems for total asymptotically nonexpansive mappings in hyperbolic spaces. J. Appl. Math. vol. 2013, Article ID , 5 pages (2013) [9] Şahin, A, Başarır, M: On the strong and -convergence of SP-iteration on a CAT(0) space. J. Inequal. Appl. (2013) 2013:311, doi: / X [10] Dhompongsa S, Panyanak, B: On -convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56(10), (2008) [11] Leustean, L: Nonexpansive iterations in uniformly convex W-hyperbolic spaces. In: Leizarowitz, A, Mordukhovich, BS, Shafrir, I, Zaslavski, A (eds.), Nonlinear Analysis and Optimization I: Nonlinear Analysis, Contemp. Math., Am. Math. Soc. AMS 513, (2010) Aynur Şahin: Sakarya University, Faculty of Sciences and Arts, Department of Mathematics, Sakarya-Turkey, Metin Başarır: Sakarya University, Faculty of Sciences and Arts, Department of Mathematics, Sakarya-Turkey, [email protected] This work was supported by the Commission of Scientific Research Projects of Sakarya University, Project number

153 International Congress in Honour of Professor Ravi P. Agarwal Characterization of W p type of Spaces Involving Fractional Fourier Transform S.K.Upadhyay and Anuj Kumar The characterizations of W p type of spaces and mapping relations between W and W p type of spaces are discussed by using fractional Fourier transform. The uniqueness of Cauchy problems is also investigated by using the same transform. S.K. Upadhyay: DST-CIMS and Department of Mathematical Sciences Indian Institute of Technology (BHU), Varanasi , India, [email protected] Anuj Kumar: ST-CIMS, Faculty of Science, Banaras Hindu University, Varanasi , India, [email protected]

154 154 International Congress in Honour of Professor Ravi P. Agarwal 120 Rates of Convergence for an Estimator of a Density Function Based on Hermite Polynomials Elif Erçelik and Mustafa Nadar Let X 1, X 2,... be a sequence of i.i.d random variables with unknown density function f. We investigate the mean integrated square error convergency rate of an estimator based on Hermite polynomials for an unknown density function f which incorporate certain delta sequences. Walter [8] and Greblicki and Pawlak [1] studied the mean integrated square error (MISE) convergency rate of the estimator based on Hermite series method. Later on, Letellier [2] studied MISE convergency rate of the estimator based on delta sequences by using Jakobi polynomials. In this ( work we ) obtained MISE convergency rate of the estimator based on delta sequences by using Hermite polynomials as O N r r+1 which is faster than that of Walter and Letellier and slower than of Greblicki and Pawlak. [1] Greblicki W., Pawlak M., Hermite Series Estimates of a Probability Density and Its Derivatives, Journal of Multivariate Analysis 15, , [2] Letellier J. A., Rates of convergence for an estimator of a density function based on Jakobi polynomials, Communication in Statistics- Theory and Methods, 26:1, , [3] Nadar M., Local convergence rate of mean squared error in density estimation, Communication in Statistics- Theory and Methods, Vol 40, pp , [4] Susarla V., Walter G., Estimation of a multivariate density function using delta sequences, Annals of Statistics, Vol.9, pp , [5] Szegö G., Orthogonal Polynomials, American Mathematical Society, [6] Timan A. F., Theory of Approximation of Functions of a Real Variable, Oxford, England: Pergammon Press, [7] Walter G., Blum.J., Probability density estimation using delta sequences, Annals of Statistics, Vol.7, pp , [8] Walter G., Properties of Hermite Series Estimation of Probability Density, Annals of Statistics, Vol.5.No. 6, , [9] Zygmund A., Trigonometric Series, New York: Chelsea Publishing Company, Elif Erçelik: Gebze Institute of Technology, Department of Mathematics, Kocaeli, Turkey and İstanbul Technical University, Department of Mathematical Engineering, İstanbul, Turkey, [email protected] or [email protected] Mustafa Nadar: İstanbul Technical University, Department of Mathematical Engineering, İstanbul, Turkey,

155 International Congress in Honour of Professor Ravi P. Agarwal Estimation of Reliability in Multicomponent Stress-Strength Model Based on Marshall Olkin Weibull Distribution Mustafa Nadar and Fatih Kızılaslan We consider a system which have k identical strength components and each component is a random vector (X 11, X 12 ), (X 21, X 22 ),..., (X k1, X k2 ) following Marshall Olkin Bivariate Weibull distribution with Parameters ( σ, θ 1, θ 2, θ 3 ). Let Z i = min(x i1, X i2 ), i = 1,..., k.the system is regarded as operating only if at least s out of k (1 s k) strength variables exceeds a random stress Y which has Weibull distribution with parameters ( σ, α). Then, when σ is known, the reliability R s,k in the described multicomponent stress-strength model is obtained as R s,k = k k i ( k i i=s j=0 ) ( k i j ) ( 1) j α [θ(i + j) + α] where θ = θ 1 + θ 2 + θ 3. Finally, a Monte Carlo Simulation study is performed to compare the reliability using both maximum likelihood and Bayesian estimation. [1] Bhattacharyya, G.K., Johonson, R.A Estimation of reliability in multicomponent stress-strength model, Journal of the American Statistical Association, 69, [2] G. Srinivasa Rao, Muhammad Aslam, Debasis Kundu Burr-XII Distribution parametric estimation and estimation of reliability of multicomponent stress-strength, Comm. Statist.-Theory and Meth., [3] Davarzani, N., Haghighi, F., Parsian, A Estimation of P (X Y ) for a Bivariate Weibull Distribution, Journal of Applied Probability & Statistics, 4(2): [4] Marshall, A.W., Olkin, I., A multivariate exponential distribution. Journal of the American Statistical Association 62, Mustafa Nadar: İstanbul Technical University, Department of Mathematical Engineering, İstanbul, Turkey, Fatih Kızılaslan: Gebze Institute of Technology, Department of Mathematics, Kocaeli, Turkey, [email protected]

156 156 International Congress in Honour of Professor Ravi P. Agarwal 122 Some New Results on The Π Regularity of Some Monoids Ahmet Emin and Fırat Ateş In this talk we give some new results on the regularity and Π -Regularity of Schetzenberger and Crossed products of monoids. [1] ] A. Emin, F.Ateş, S.Ikikardeş, İ.N.Cangul, A New Monoid Construction Under Crossed Product, Journal of Inequalities and Applivations,2013:244 [2] Y.Zhang, S.Li, D.Wang, semidirect products and wreath products of strongly - inverse monoids, Georgian Math.Journal,3(3) (1996), [3] F.Ateş, Some New Monoid And Group Constructions Under Semidirect Products., Ars.Comb.91, ,2009 [4] E.G.Karpuz, F.Ateş, S.Çevik, Regular and -Inverse Monoids Under Schetzenberger Products, Algebras Groups And Geometries 27, ,2010 [5] J.M.Howie, N.Ruskuc, Construction and Presentations For Monoids, Communications In Algebra, 22(15), ,1994 [6] E.G.Karpuz, A.S.Cevik, A New Example Of Strongly -inverse monoids,hacettepe Journal Of Mathematics and Statistics, 40(3), ,2011 Ahmet Emin: Balikesir University, Faculty of Art and Science, Department of Mathematics, Cagıs Campus, Balıkesir- Turkiye, [email protected] Fırat Ateş: Balikesir University, Faculty of Art and Science, Department of Mathematics, Cagıs Campus, Balıkesir- Turkiye, [email protected]

157 International Congress in Honour of Professor Ravi P. Agarwal On Traveling Wave Solutions of Fractional Differential Equations Şerife Müge Ege and Emine Mısırlı In this work, the space-time fractional Hirota-Satsuma-Coupled KdV equation and the space-time fractional Fokas equation are handled by using the modified Kudryashov method. Consequentially, many analytical exact solutions are obtained including rational solutions and symmetrical Fibonacci function solutions. This method is powerful, effectual and can be used as an alternative to constitute new solutions of various types of fractional differential equations applied in scientific fields. [1] N. A. Kudryashov, One method for finding exact solutions of nonlinear differential equations, Commun. Nonl. Sci. Numer. Simul., 17 (2012), , [2] S. M. Ege, E. Misirli, The modified Kudryashov method for solving some evolution equations, AIP Conf. Proc., 1470 (2012), , [3] S. M. Ege, E. Misirli, The modified Kudryashov method for solving some fractional-order nonlinear equations, Advances in Difference Equations, 135 (2014), 1-13, [4] S. M. Ege, E. Misirli, Solutions of the space-time fractional foam-drainage equation and rhe fractional Klein-Gordon equation by use of modified Kudryashov method, International Journal of Research in Advent Technology, 2321(9637) (2014), , [5] M. M. Kabir, Modified Kudryashov method for generalized forms of the nonlinear heat conduction equation, Int. J. Phys. Sci., 6 (2011), , [6] M. M. Kabir, A. Khajeh, E. A. Aghdam, A. Y. Koma, Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations, Math. Meth. Appl. Sci., 34 (2011), , [7] A. Stakhov, B. Rozin, On a new class of hyperbolic functions, Chaos, Solitons Fract., 23 (2005), , [8] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Compt. Math. Appl., 51, (2006) , [9] G. Jumarie, Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, J. Appl. Math. Compt., 24, (2007) Şerife Müge Ege: Ege University, Faculty of Science, Department of Mathematics, Bornova, İzmir-Turkiye, serife. [email protected] Emine Mısırlı: Ege University, Faculty of Science, Department of Mathematics, Bornova, İzmir-Turkiye, emine. [email protected] This research is supported by Ege University, Scientific Research Project (BAP), Project Number: 2012FEN037.

158 158 International Congress in Honour of Professor Ravi P. Agarwal 124 On the Oscillation of Second Order Nonlinear Neutral Dynamic Equations on Time Scales Elvan Akın, Can Murat Dikmen and Said Grace In this talk, we investigate some new oscillation criteria and give sufficient conditions to ensure that all solutions of second order nonlinear neutral dynamic equations with distributed deviating arguments are oscillatory on a time-scale T, via comparison with second order nonlinear dynamic equations whose oscillatory character are known and extensively studied in the literature. [1] E. Akın-Bohner, Z. Došlá, B. Lawrence, Oscillatory properties for three-dimensional dynamic systems, Nonlinear Analysis, 69 : , [2] M. Bohner and A. Peterson Dynamic equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, [3] T. Candan, Oscillation of second order nonlinear neutral dynamic equations on time scales with distributed deviating arguments, Comput. Math. Appl., 62 : , [4] S.R. Grace, R.P. Agarwal, B. Kaymakcalan and W. Sae-Jie, On the oscillation of certain second order nonlinear dynamic equations, Math. Comput. Modelling, 50 : , [5] S.R. Grace, R.P. Agarwal, M. Bohner and D. O Regan, Oscillation of second order strongly superlinear and strongly sublinear dynamic equations, Commun. Nonlin. Sces. Numer. Simulat., 14 : , Elvan Akın: Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA, [email protected] Can Murat Dikmen: Bulent Ecevit Universitesi, Fen-Edebiyat Fakultesi, Matematik Bolumu, Zonguldak, Turkey, [email protected] Said Grace: Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Gizza 12221, Egypt, [email protected]

159 International Congress in Honour of Professor Ravi P. Agarwal A Collocation Approach to Parabolic Partial Differential Equations Kubra Erdem Biçer and Salih Yalçınbaş In this study, a collocation approach is presented to solve parabolic partial differential equations. For this, an approximate method based on Bernoulli polynomials is developed. The method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations. The obtained matrix equation is based on the matrix forms of Bernoulli polynomials and their derivatives by means of collocations. Also error analysis and a numerical example are presented to demonstrate the validity and applicability of the technique. [1] K. Erdem, S.Yalçınbaş, Bernoulli Polynomial Approach to High-Order Linear Differential-Difference Equations, AIP Conf. Proc (2012) [2] K. Erdem, S.Yalçınbaş, Numerical approach of linear delay difference equations with variable coefficients in terms of Bernoulli polynomials, AIP Conf. Proc (2012) [3] K. Erdem, S. Yalçınbaş.,M. Sezer, A Bernoulli Polynomial Approach with Residual Correction for Solving Mixed Linear Fredholm Integro-Differential-Difference Equations, Journal of Difference Equations and Applications, vol:19, s: [4] Paul M.N. Feehan, Camelia A. Pop, A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients, Journal of Differential Equations, 254 (2013) [5] Yuan-Ming Wang, Error and extrapolation of a compact LOD method for parabolic differential equations, Journal of Computational and Applied Mathematics 235 (2011) [6] Suayip Yüzbasi, Niyazi Şahin, Numerical solutions of singularly perturbed one-dimensional parabolic convection diffusion problems by the Bessel collocation method, Applied Mathematics and Computation 220 (2013) Kubra Erdem Bicer: Celal Bayar University, Faculty of Science, Department of Mathematics, Manisa, Turkey, kubra. [email protected] Salih Yalcinbas: Celal Bayar University, Faculty of Science, Department of Mathematics, Manisa, Turkey,

160 160 International Congress in Honour of Professor Ravi P. Agarwal 126 From Simplicial Homotopy to Crossed Module Homotopy I.Ilker Akça and Kadir Emir As is known from [1, 3, 5], simplicial algebras with Moore complex of length 1 (2) lead to crossed (2-crossed) modules that are related to Kozsul complex and Andre-Quillen homology constructions for use in homotopical and homological algebra. Homotopy of crossed complex morphisms on groupoids was first introduced by Brown and Higgins in [2]. Then Martins clearly defined and formulated the homotopy of crossed module morphisms on groups in [4]. In this study, we will define the homotopy of crossed module morphisms on commutative algebras. Upon this, we will define a map that carries the homotopy from simplicial algebras to crossed modules, and vice versa; as a part of the functors between them. [1] Arvasi Z., Porter T., Higher Dimensional Peiffer Elements in Simplicial Commutative Algebras, TAC, [2] Brown R., Higgins P.J., Tensor Products and Homotopies for ω-groupoids and Crossed Complexes, Journal of Pure and Applied Algebra, [3] Grandjean A.R., Vale M.L., 2-Modulos Cruzados en la Cohomologia de André-Quillen, Memorias de la Real Academia de Ciencias, [4] Martins J.F., The Fundamental 2-Crossed Complex of a Reduced CW-Complex, Homology, Homotopy and Applications, [5] Porter T., Homology of Commutative Algebras and an Invariant of Simis and Vasconceles, Journal of Algebra, Eskişehir Osmangazi University, Faculty of Science and Arts, Department of Mathematics and Computer Science, Eskisehir-Turkey.

161 International Congress in Honour of Professor Ravi P. Agarwal On Algebraic Semigroup and Graph-Theoretic Properties of a New Graph Ahmet Sinan Çevik, Eylem Güzel Karpuz and I.Naci Cangül In this talk, firstly, we define a new graph based on the semi-direct product of some monoids, and then investigate the interplay between the semi-direct product over monoids and the graph-theoretic properties of this product in terms of its relations. [1] A. S. Çevik, K. C. Das, I. N. Cangul, A. D. Maden, Minimality over Free Monoid Presentations, Hacettepe J. Math. Stat., (accepted). [2] E. G. Karpuz, K. C. Das, I. N. Cangul, A. S. Çevik, A New Graph Based on the Semi-Direct Product of Some Monoids, J. Inequalities and Appl., doi: / x Ahmet Sinan Çevik: Selçuk University, Faculty of Science, Department of Mathematics, Alaaddin Keykubat Campus, Konya-Türkiye, [email protected] or [email protected] Eylem Güzel Karpuz: Karamanoglu Mehmetbey University, Kamil Ozdag Science Faculty, Department of Mathematics, Karaman-Türkiye, [email protected] İsmail Naci Cangül: Uludag University, Faculty of Science, Department of Mathematics, Görükle, Bursa-Türkiye, [email protected]

162 162 International Congress in Honour of Professor Ravi P. Agarwal 128 Embeddability and Gröbner-Shirshov Basis Theory Eylem Güzel Karpuz A semigroup P embeds in a group G if there exists a monomorphism from P into G, and then a semigroup P is embeddable into group, or is group-embeddable, if there exists some group G into which P embeds. In this talk, we discuss embeddability of a semigroup in a group via the Gröbner-Shirshov basis theory. Then by considering braid groups, we give some examples. Some parts of this talk have been prepared from the joint work [3]. [1] S. I. Adjan, Defining Relations and Algorithmic Problems for Groups and Semigroups, Proceedings of the Steklov Institute of Mathematics 85 (1966). [2] L. A. Bokut, Y. Chen, Q. Mo, Gröbner-Shirshov Bases and Embeddings of Algebras, Inter. Journal of Algebra and Comp. 20(7) (2010) [3] E. G. Karpuz, A. S. Çevik, Jörg Koppitz, Gröbner-Shirshov Bases and Embedding of a Semigroup in a Group, in prep. Karamanoglu Mehmetbey University, Kamil Özdag Science Faculty, Department of Mathematics, Karaman-Türkiye, [email protected]

163 International Congress in Honour of Professor Ravi P. Agarwal An Application of Fixed Point Theorems to a Problem for the Existence of Solutions of a Nonlinear Ordinary Differential Equations of Fractional Order Masashi Toyoda In this talk, we consider the Cauchy problem in a class of fractional differential equations. Let 1 < α 2. We consider the Cauchy problem D0+ α u(t) = p(t)ta u(t) σ, u (t) lim t 0+ u(t) = 0, lim t 0+ = (α 1)λ tα 2 where p is continuous, a, σ, λ R with σ < 0, λ > 0 and D0+ α is the Riemann-Liouville fractional derivative. If α = 2, then this problem is the problem in [8]. This is a joint work of Professor Toshiharu Kawasaki. [1] Z. Bai and H. Lü, Positive solutions for boundary value problem on nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 311 (2005), [2] T. Kawasaki and M. Toyoda, Existence of positive solution for the Cauchy problem for an ordinary differential equation, Nonlinear Mathematics for Uncertainly and its Applications, Advances in Intelligent and Soft Computing, 100, Springer- Verlag, Berlin and New York, 2011, [3] T. Kawasaki and M. Toyoda, On the existence of solutions of second order ordinary differential equations, Proceedings of the International Symposium on Banach and Function Spaces IV, Kitakyushu, Japan, 2012, [4] T. Kawasaki and M. Toyoda, Positive solutions of initial value problems of negative exponent Emden-Fowler equations, Memoris of the Faculty of Engineering, Tamagawa University, 48 (2013), [5] T. Kawasaki and M. Toyoda, Existence of positive solutions of the Cauchy problem for a second-order differential equation, Journal of Inequalities and Applications 2013, 2013:465 (7 November 2013). [6] T. Kawasaki and M. Toyoda, Existence of positive solution for the Cauchy problem for an ordinary differential equation, RIMS Kôkyˆroku (2014), No.1821, [7] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, In North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, [8] J. Knežević-Miljanović, On the Cauchy problem for an Emden-Fowler equation, Differential Equations, 45(2009), Faculty of Engineering, Tamagawa University, Tamagawa-gakuen Machida-shi, Tokyo, , Japan, [email protected]

164 164 International Congress in Honour of Professor Ravi P. Agarwal 130 A Numerical Solution for Vibrations of an Axially Moving Beam Duygu Dönmez Demir and Erhan Koca The dynamic responce of an axially elastic, tensioned beam with constant velocity is considered. The equation of motion is solved by using Adomian Decomposition Method (ADM) and Method of Multiple Time Scales. We obtain displacement one at a time. Also, the comparison of (ADM) with the perturbation method for this model is presented. The computed results are indicated numerically. [1] G. Adomian, Stochastic Systems, Academic Press, London, 1983 [2] A. M. Wazwaz, Partial Differantial Equations, A. A. Belkema Publishers, Tokyo, 2002, [3] J. A. Wickert, Non-Linear Vibration of a Travelling Tensioned Beam, International Journal of Non-Linear Mechanics, 27 (1992), [4] H. R. Oz, M. Pakdemirli, Vibrations of an Axially Moving Beam With Time-Dependent Velocity, Journal of Sound and Vibration, 227(2) (1999), [5] G. Yigit, Wave Propagation in Composite Materials With the Help of Adomian Decomposition Method, 2002 [6] I. Karagoz, Numerical Analysis and Engineering Applications, Third Edition, Nobel Publication. [7] K. Abbaoui, Y. Cherruault, Converge of Adomians Method Applied to Differential Equations, Comp. Math. Appl., 28 (1994), Duygu Dönmez Demir: Celal Bayar University, Faculty of Art & Science, Department of Mathematics, Muradiye, Manisa-Turkiye, [email protected] Erhan Koca: Celal Bayar University, Faculty of Art & Science, Department of Mathematics, Muradiye, Manisa-Turkiye

165 International Congress in Honour of Professor Ravi P. Agarwal Some Principal Congruence Subgroups of the Extended Hecke Groups and Relations with Pell-Lucas Numbers Zehra Sarıgedik, Sebahattin Ikikardeş and Recep Şahin In this talk, we consider the Hecke groups H( m) and the extended Hecke groups H( m) for m = 2 or 3. Firstly, we give the generators of the principal congruence subgroups H 2 ( m) and H 2 ( m) of H( m) and H( m), respectively. Then, using some of these generators, we define a sequence V k which is generalized version of the Pell-Lucas numbers sequence Q k given in [9] for the modular group, in the extended Hecke groups H( m) for m = 1, 2 and 3. [1] Hecke, E. Über die bestimmung dirichletscher reichen durch ihre funktionalgleichungen, Math. Ann., 112, , (1936). [2] Lang, M. L. Normalizers of the congruence subgroups of the Hecke groups G 4 and G 6, J. Number Theory 90, no. 1, 31 43, (2001). [3] Sahin, R.; Bizim, O. Some subgroups of the extended Hecke groups H( λ q), Acta Math. Sci., Ser. B, Engl. Ed., Vol.23, No.4, , (2003). [4] Sahin, R.; Bizim, O.; Cangul, I. N. Commutator subgroups of the extended Hecke groups H( λ q), Czechoslovak Math. J. 54(129), no. 1, , (2004). [5] Ikikardes, S.; Sahin, R.; Cangul, I. N. Principal congruence subgroups of the Hecke groups and related results, Bull. Braz. Math. Soc. (N.S.), 40, No. 4, , (2009). [6] Sahin, R.; Ikikardes, S.; Koruoglu, O. Extended Hecke groups H( λ q) and their fundamental regions, Adv. Stud. Contemp. Math. (Kyungshang), 15, no. 1, 87 94, (2007). [7] Cangul, I. N.; Bizim, O. Congruence subgroups of some Hecke groups, Bull. Inst. Math. Acad. Sinica, 30 (2002), no. 2, [8] Alperin, R. C. The modular tree of Pythagoras, Amer. Math. Monthly, 112, no. 9, , (2005). [9] Mushtaq, Q.; Hayat, U. Pell numbers, Pell-Lucas numbers and modular group, Algebra Colloq., 14, no.1, , (2007). [10] Goldman, W. M.; Neumann, W D. Homological action of the modular group on some cubic moduli spaces, Math. Res. Lett., no. 4, , (2005). [11] Kock, B.; Singerman, D Real Belyi theory, Q. J. Math. 58, no. 4, , (2007). Zehra Sarıgedik: Celal Bayar University, Koprubasi Vocational High School, Koprubasi/Manisa-Turkiye, Sebahattin İkikardeş: Balikesir University, Faculty of Science, Department of Mathematics, Balikesir-Turkiye, Recep Şahin: Balikesir University, Faculty of Science, Department of Mathematics, Balikesir-Turkiye,

166 166 International Congress in Honour of Professor Ravi P. Agarwal 132 On the Metric Geometry and Regular Polyhedrons Temel Ermiş and Rüstem Kaya The Platonic solids known as the regular polyhedrons, all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex. The regular polyhedrons were first described by Plato. That is the reason why they called as Platonic solids. In the previous studies, hexahedron and octahedron associated to maximum and taxicab metrics, respectively. In this work, we find two new metrics of which unit spheres are the dodecahedron and icosahedron, and study the structure of related spaces. [1] A. C. Thompson, Minkowski Geometry, Cambridge University Press, (1996), [2] G. E. Martin, Transformation Geometry, Springer - Verlag New York Inc., (1987), [3] M. Özcan, Ö. Gelişgen and R. Kaya, Distance Formulae in Chinese Checker Space, International Journal of Pure and Applied Mathematics, 26-1, (2006), [4] Ö. Gelişgen and R. Kaya, On α Distance in Three Dimensional Space, Applied Science (APPS), Vol.8, No 1. (2006), [5] Ö. Gelişgen and R. Kaya, Generalization of α-distance to n-dimensional Space, Scientific and Professional Journal of the Croatian Society for Geometry and Graphics (KoG), Vol.10, 33-35, (2006), [6] Ö. Gelişgen and R. Kaya, The Taxicab Space Group, Acta Mathematica Hungarica, Acta Mathematica Hungarica, Vol. 122, No.1-2, , (2009), [7] S. Ekmekçi, A. Bayar, Ö.Gelişgen and R. Kaya, On The Group of Isometries of The Plane with Generalized Absolute Value Metric, Rocky Mountain Journal of Mathematics, Vol.39, No.2, , (2009), [8] R. S. Millman, G. D. Parker, Geometry, A Metric Approach with Models, Undergraduate Texts in Mathematics, Springer-Verlag, (1981). Eskişehir Osmangazi University, Faculty of Science and Arts, Department of Mathematics and Computer Science, Eskişehir, Turkey

167 International Congress in Honour of Professor Ravi P. Agarwal On the Addition of Collinear Points in Some PK-Planes Basri Celik and Abdurrahman Dayioglu In this study, we extend the addition of points which is defined in [3] for the points of the special line OU = [0, 1, 0], to the points of the lines [m,1,k],[1,n,p] in any PK-plane coordinated with the dual local ring of quaternion Q(ε) = Q + Qε, where m,k and p Q(ε), n Qε. Also some geometric and algebraic properties of the addition has examined. [1] Baker CA, Lane ND and Lorimer JW, A coordinatization for Moufang-Klingenberg planes, Simon Stevin, 1991, 65,3 22. [2] Celik B. and Dayioglu A., The collineations which act as addition and multiplication on points in a certain class of projective Klingenberg planes, Journal of Inequalities and Applications, 2013:193, [3] Celik B. and Erdogan F. O., On addition and multiplication of points in a certain class of projective Klingenberg planes, Journal of Inequalities and Applications, 2013:230, [4] Conway J. H. and Smith D.A., On Quaternions and Octonions, AK Peters, Massachusetts, [5] Hughes D.R. and Piper F.C., Projective Planes, Springer, New York, [6] Keppens D., Coordinazation of Projective Klingenberg Planes, Simon Stevin, 1988,62, Basri Celik: Uludağ University, Faculty of Art and Science, Department of Mathematics, Görükle, Bursa-Turkiye Abdurrahman Dayioglu: Uludağ University, Faculty of Art and Science, Department of Mathematics, Görükle, Bursa- Turkiye

168 168 International Congress in Honour of Professor Ravi P. Agarwal 134 Local Stability Analysis of Strogatz Model with Two Delays Sertaç Erman and Ali Demir In this talk, we consider the model of interpersonal interactions with two delays which is a direct extension of Strogatz model. We attempt to show stability regions of the model in various parameter spaces by using D-partition method. The stability of model is investigated for various values of delays. We conclude that delays effect the stability of the model. [1] S. Strogatz, Love affairs and differential equations, Math. Mag. 65 (1) (1988) 35, [2] S. Strogatz, Nonlinear Dynamics and Chaos, Westwiev Press, 1994, [3] N. Bielczyk, U. Forys, T. P latkowski, Dynamical models of dyadic interactions with delay, J. Math. Sociology (2013), [4] N. Bielczyk, M. Bodnar, U. Forys, Delay can stabilize: Love affairs dynamics, App. Math. and Comp. 219 (2012) , [5] A.M. Krall, Stability Techniques for Continuos Linear System, Gordon and Breach Science Publishers, London, 1967, [6] L. E. El sgol ts, S. B. Norkin, Introduction to the Theory andapplication of Differential Equations with Deviating Arguments, Academic Press, London, 1973 [7] T. Insperger, G. Stépán,Semi-Discretization Stability and Engineering Applications for Time-Delay Systems, Springer, Newyork, 2011 [8] V. B. Kolmanıvskii, V. R. Nosov, Stability of Functional Differential Equations, Academic Press, London, 1986 Sertaç Erman: Kocaeli University, Faculty of Science and Art, Department of Mathematics, Umuttepe, Kocaeli-Turkiye, @kocaeli.edu.tr Ali Demir: Kocaeli University, Faculty of Science and Art, Department of Mathematics, Umuttepe, Kocaeli-Turkiye, [email protected]

169 International Congress in Honour of Professor Ravi P. Agarwal Weighted Statistical Convergence in Intuitionistic Fuzzy Normed Spaces Selma Altundağ and Esra Kamber In this talk, we define the concepts of weighted statistical convergence, ( N, p n ) statistical summability and strong ( N, pn ) - summability in intuitionistic fuzzy normed spaces. We also establish relations between these concepts. [1] K.Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems, 20 (1986), 87-96, [2] R. Saadati, J.H. Park, Intuitionistic fuzzy euclidean normed spaces, Commun. Math. Anal., 12 (2006), 85-90, [3] S. Vijayabalaji, N. Thillaigovindan, Y.B. Jun, Intuitionistic fuzzy n-normed linear space, Bull. Korean. Math. Soc., 44 (2007), , [4] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73-74, [5] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), , [6] C.Şençimen, S.Pehlivan, Statistical convergence in fuzzy normed linear spaces, Fuzzy sets and systems, 159 (2008), , [7] A.Alotaibi, A.M.Alroqi, Statistical convergence in a paranormed space, Journal of inequalities and applications, 39 (2012), 1-6, [8] M.Mursaleen, λ-statistical convergence, Math. Slovaca, 50 (2000), , [9] S. Karakuş, K. Demirci, O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals, 35 (2008), , [10] S.A.Mohiuddine,Q.M.Danish Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos, Solitons and Fractals, 42(2009), , [11] V.Karakaya, N.Şimşek, M.Ertürk, F.Gürsoy, λ-statistical convergence of seque, Journal of function spaces and applications, 2012(2012), 1-14, [12] J.A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160 (1993), 43-51, [13] M.Mursaleen, Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Comput. Appl. Math., 233 (2009), , [14] V.Karakaya, N.Şimşek, M.Ertürk, F.Gürsoy, Lacunary statistical convergence of sequences of functions in intuitionistic fuzzy normed space, Journ. of intelligent and fuzzy systems, 26(2014), , [15] M. Sen, P. Debnath, Lacunary statistical convergence in intuitionistic fuzzy n-normed spaces, Math. and Comp. Modelling, 54 (2011), , [16] S. Altundag, E. Kamber, Lacunary -statistical convergence in intuitionistic fuzzy n-normed spaces, Math. and Comp. Modelling, 40 (2014), 1-12, [17] F. Moricz, Tauberian conditions, under which statistical convergence follows from statistical summability (C, 1), J. Math. Anal. Appl., 275 (2002), , [18] F.Moricz, C.Orhan, Tauberian conditions, under which statistical convergence follows from statistical summability by weighted means, Studia Sci.Math. Hung, 41(2004), , [19] V. Karakaya, T.A. Chishti, Weighted statistical convergence, Iran. J. Sci. Technol. Trans. A Sci., 33 (2009), , [20] M.Mursaleen,V. Karakaya, M.Ertürk, F.Gürsoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. and Comput., 218 (2012), , [21] C.Belen,S.A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math.and Comput. 219 (2012), , [22] R.Saadati, J.H. Park, On the intuitionistic fuzzy topological spaces., Chaos, Solitons and Fractals, 27(2006), , [23] T.K.Samanta, I.Jebril, Finite dimensional intuitionistic fuzzy normed linear space, Int. J. Open Problems Compt. Math, 2 (2009), , [24] L. Hong, J.Q. Sun, Bifurcations of fuzzy nonlinear dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 1 (2006), 1-12, [25] M.A.Alghamdi, A.Alotaibi,Q.M.D.Lahani, Statistical limit superior and limit inferior in intuitionistic fuzzy normed spaces Journal of inequalities and applications, 96 (2012), Selma Altundağ: Sakarya University, Faculty of Science, Department of Mathematics, Sakarya-Turkey, [email protected] Esra Kamber: Sakarya University, Faculty of Science, Department of Mathematics, Sakarya-Turkey, [email protected]

170 170 International Congress in Honour of Professor Ravi P. Agarwal 136 Sturm Comparison Theorems for Some Elliptic Type Equations with Damping and External Forcing Terms Sinem Şahiner, Emine Mısırlı and Aydın Tiryaki After the Picone s significant work in 1909, numerous authors extended the Picone type identity for differential equations of various types. In this talk, we will give a Picone-type inequality for a class of some nonlinear elliptic type equations with damping and external forcing terms, and establish Sturmian comparison theorems using the Picone-type inequality. [1] Adams, R. A. and Fournier, J. J. F., Sobolev Spaces, Second Edition, Academic Press, 2003, [2] Allegretto, W. and Huang, Y. X., A Picone s identity for the p-laplacian and applications, Nonlinear Anal., 32, (1998), , [3] Bognár, G. and Dosly, O., The application of Picone-type identity for some nonlinear elliptic differential equations, Acta Math. Univ. Comenian. 72 (2003), 45-57, [4] Clark, C., Swanson, C., A., Comparison theorems for elliptic differential equations, Proc. Amer. Math. Soc. 16 (1965) , [5] Hardy G. H., Littlewood J. E. and Polya G., Inequalities, Cambridge Univ. Press, 1988, [6] Jaroš, J. and Kusano, T., A Picone type identity for second order half-linear differential equations, Acta Marth. Univ. Comenian. 68 (1999), , [7] Jaroš, J., Kusano, T. and Yoshida, N., Picone-type inequalities for nonlinear elliptic equations and their applications, J. Inequal. Appl. 6 (2001), , [8] Jaroš, J., Kusano, T. and Yoshida, N., Picone-type inequalities for elliptic equations with first order terms and their applications, J. Inequal. Appl. (2006), 1-17, [9] Kusano, T., Jaroš, J. and Yoshida, N., A Picone-type identity and Sturmain comparison and oscillation theorems for a class of half-linear partial differential equations of second order, Nonlinear Analysis, Theory, Methods and Applications 40 (2000), , [10] Picone M., Sui valori eccezionali di un parametro da cui dipende un equazione differenziale lineare ordinaria del second ordine, Ann. Scuola Norm. Sup. Pisa 11(1909), 1-141, [11] Swanson C. A. A comparison theorem for elliptic differential equations, Proc. Amer. Math. Soc. 17 (1966), , [12] Yoshida, N., Sturmian comparison and oscillation theorems for a class of half-linear elliptic equations, Nonlinear Analysis, Theory, Methods and Applications, 71(2009) e , [13] Yoshida, N., A Picone identity for half-linear elliptic equations and its applications to oscillatory theory, Nonlinear Anal. 71 (2009), Sinem Şahiner: Izmir University, Faculty of Arts and Sciences, Department of Mathematics and Computer Science, Izmir-Turkiye, [email protected] Emine Mısırlı: Ege University, Faculty of Science, Department of Mathematics, Izmir-Turkiye, [email protected] Aydın Tiryaki: Izmir University, Faculty of Arts and Sciences, Department of Mathematics and Computer Science, Izmir-Turkiye, [email protected]

171 International Congress in Honour of Professor Ravi P. Agarwal A Note on Solutions of the Nonlinear Fractional Differential Equations via the Extended Trial Equation Method Meryem Odabasi and Emine Misirli In this study, we investigate the solutions of nonlinear fractional differential equations that have many advantages in physical sciences and dynamic systems. By using the extended trial equation method we have successfully obtained analytical solutions of some nonlinear fractional differential equations. The results show that extended trial equation method is an effective and powerful mathematical tool for solving nonlinear fractional differential equations arising in mathematical physics. [1] M. Dalir and M. Bashour, Applications of Fractional Calculus, Applied Mathematical Sciences 4 (2010), no , [2] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198 (1999), [3] C. S. Liu,, A New Trial Equation Method and Its Applications, Communications in Theoretical Physics, 45 (2006), , [4] H. Bulut, H. M. Baskonus and Y. Pandir, The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation, Abstract and Applied Analysis, (2013), Article ID , [5] Y. Pandir, Y. Gurefe, E. Mısırlı, The Extended Trial Equation Method for Some Time Fractional Differential Equations,Discrete Dynamics in Nature and Society, (2013) Article ID , [6] G. Jumarie, Modified Riemann-Liouville Derivative and Fractional Taylor Series of Nondifferentiable Functions Further Results, Computers and Mathematics with Applications, 51 (2006), no. 9-10, , [7] Y. Gurefe, E. Misirli, A. Sonmezoglu and M. Ekici, Extended Trial Equation Method to Generalized nonlinear Partial Differential Equations, Applied Mathematics and Computation 219 (2013), Meryem Odabasi: Ege University, Tire Kutsan Vocational School, Tire, Izmir-Turkiye, [email protected] Emine Misirli: Ege University, Faculty of Science, Department of Mathematics, Bornova, Izmir-Turkiye, [email protected]

172 172 International Congress in Honour of Professor Ravi P. Agarwal 138 On Quantum Codes Obtained From Cyclic Codes Over F 2 + uf 2 + u 2 F u m F 2 Abdullah Dertli, Yasemin Cengellenmiş and Şenol Eren A method to obtain self orthogonal codes over finite fields F 2 is given and the parameters of quantum codes which are obtained from cyclic codes over R = F 2 + uf 2 + u 2 F u m F 2 are determined. [1] A.R.Calderbank, E.M.Rains, P.M.Shor, N.J.A.Sloane, Quantum error correction via codes over GF (4), IEEE Trans. Inf. Theory, 44 (1998), [2] J.Qian, Quantum codes from cyclic codes over F 2 + vf 2, Journal of Inform.& Computational Science 10:6(2013), [3] J.Qian, L.Zhang, S.Zhu, Cyclic Codes Over F p + uf p u k 1 F p, IEICE Trans. Fundamentals, Vol. E 88-A, No. 3,2005. [4] J.Qian, W.Ma, W.Gou, Quantum codes from cyclic codes over finite ring, Int. J. Quantum Inform., 7(2009), [5] M.Mehrdad, Torsion codes Over a finite chain rings, Second Workshop on Algebra and its Applications, , , [6] P.W.Shor,Scheme for reducing decoherence in quantum memory, Phys. Rev. A,52(1995), [7] X.Kai,S.Zhu, Quaternary construction of quantum codes from cyclic codes over F 4 + uf 4, Int. J. Quantum Inform., 9(2011), [8] X.Yin, W.Ma, Gray map and quantum codes over the ring F 2 + uf 2 + u 2 F 2, International Joint Conferences of IEEE Trust Com11,2011. [9] Y.Cengellenmis, On (1 u m ) Cyclic codes over F 2 + uf 2 + u 2 F u m F 2, Int. J. Contemp. Math. Sciences, Vol.4, 2009, no.20, Abdullah Dertli: Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Samsun, Turkey, [email protected] Yasemin Cengellenmiş: Trakya University, Faculty of Arts and Sciences, Department of Mathematics, Edirne, Turkey, [email protected] Şenol Eren: Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Samsun, Turkey, [email protected]

173 International Congress in Honour of Professor Ravi P. Agarwal On Some Functions Mapping the Zeros of L n (x) to the Zeros of L n(x) Nihal Yılmaz Özgür and Öznur Öztunç As it is well known, studying zeros of polynomials plays an increasingly important role in Mathematical research. Fibonacci polynomials F n (x) are defined recursively by F n (x) = xf n 1 (x) + F n 2 (x), by initial conditions F 1 (x) = 1, F 2 (x) = x. Similarly Lucas polynomials L n (x) are defined by L n (x) = xl n 1 (x) + L n 2 (x), with the initial values L 1 (x) = x and L 2 (x) = x (see [2]). In this study, we give some functions which map the modulus of the zeros of Lucas polynomials to the modulus of the zeros of the derivative of Lucas polynomials. Also we examine the roots of first order derivatives of these polynomials. [1] P. Filipponi, A. Horadam, Derivative sequences of Fibonacci and Lucas polynomials, Applications of Fibonacci numbers, Vol. 4 (Winston-Salem, NC, 1990), , Kluwer Acad. Publ., Dordrecht, 1991, [2] P. Filipponi, A. Horadam, Second derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quart. 31 (1993), no. 3, , [3] M. X. He, D. Simon, P. E. Ricci, Dynamics of the zeros of Fibonacci Polynomials, Fibonacci Quarterly, 35 (1997), no.2, , [4] M. X. He, P. E. Ricci, D. Simon, Numerical results on the zeros of generalized Fibonacci Polynomials, Calcolo, 34 (1998), no.1-4, 25-40, [5] V. E. Hoggat, M. Bicknell, Generalized Fibonacci Polynomials, Fibonacci Quart. 11 (1973), no. 5, , [6] V. E. Hoggat, M. Bicknell, Roots of Fibonacci Polynomials, Fibonacci Quart. 11 (1973), no. 3, , [7] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley 2001, [8] Y. Yuan, Z. Wenpeng, Some identities involving the Fibonacci Polynomials, Fibonacci Quart. 40 (2002), no. 4, , [9] J. Wang, On the k th derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quart. 33 (1995), no. 2, , [10] C. Zhou, On the k th-order derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quart. 34 (1996), no. 5, Nihal Yılmaz Özgür: Balıkesir University, Department of Mathematics, Balıkesir, Turkiye, [email protected] Öznur Öztunç: Balıkesir University, Balıkesir, Turkiye, [email protected] Both authors are supported by the Scientific Research Projects Unit of Balıkesir University under the project number 2013/02.

174 174 International Congress in Honour of Professor Ravi P. Agarwal 140 Finite Blaschke Products and R-Bonacci Polynomials Nihal Yılmaz Özgür, Öznur Öztunç and Sümeyra Uçar A Blaschke product of degree n is a function defined by n 1 z a j B(z) = β 1 a j=1 j z where β = 1 and a j are in the unit disc. We know that every Blaschke product B, B(0) = 0 degree n, is associated with a unique Poncelet curve, B identifies the vertices of the n gon. In this study, we give some examples of Poncelet curves of finite Blaschke products using the zeros of the derivatives of the r Bonacci polynomials. In these cases the Poncelet curves are precisely n ellipses. [1] U. Daepp, P. Gorkin and R. Mortini, Ellipses and finite Blaschke products, Amer. Math. Monthly 109 (2002), no. 9, , [2] U. Daepp, P. Gorkin, K. Voss, Poncelet s theorem, Sendov s conjecture, and Blaschke products, J. Math. Anal. Appl. 365 (2010), no. 1, , [3] M. Frantz, How conics govern Möbius transformations, Amer. Math. Monthly 111 (2004), no. 9, , [4] T. Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New York). Wiley- Interscience, New York, 2001, [5] N. Yılmaz Özgür, Finite Blaschke Products and Circles that Pass Through the Origin, Bull. Math. Anal. Appl. 3 (2011), no. 3, 64-72, [6] N. Yılmaz Özgür, Some Geometric Properties of finite Blaschke Products, Proceedings of the Conference RIGA 2011, (2011), , [7] M. Fujimura, Inscribed Ellipses and Blaschke Products, Comput. Methods Funct. Theory 2013, [8] H. W. Gau, P. Y. Wu, Numerical Range and Poncelet Property, Taiwanese J. Math. 7 (2003), no 2, [9] M. Bicknell, V.E. Hoggatt, Roots of Fibonacci polynomials, Fibonacci Quarterly, 11, (1973), , [10] M. Bicknell, V.E. Hoggatt Generalized Fibonacci polynomials, Fibonacci Quarterly, 16, (1978), , [11] M. X. He, P. E. Ricci and D. Simon, Numerical results on the zeros of generalized Fibonacci polynomials, Calcolo, 34, (1997), 25 40, [12] M. X. He, D. Simon, P. E. Ricci, Dynamics of the zeros Fibonacci Polynomials, Fibonacci Quarterly, 35 no.2, (1997), Nihal Yılmaz Özgür: Balıkesir University, Department of Mathematics, Balıkesir, Turkiye, [email protected] Öznur Öztunç: Balıkesir University, Balıkesir, Turkiye, [email protected] Sümeyra Uçar: Balıkesir University, Department of Mathematics, Balıkesir, Turkiye, sumeyraucar@balikesir. edu.tr

175 International Congress in Honour of Professor Ravi P. Agarwal Convergence of Nonlinear Singular Integral Operators to the Borel Differentiable Functions Harun Karsli and Ismail U.Tiryaki In this paper convolution type nonlinear singular integral operators of the form b T λ (f; x) = K λ (t x, f(t)) dt, a where < a, b > is an arbitrary interval in R, λ Λ, f L 1 < a, b > and K λ is a family of kernels satisfying suitable properties. We give some approximation results about the concergence of the operators T λ to right, left and symmetric Borel differentiable functions. We note that our results are an extention of the classical ones, namely, the results dealing with the linear singular integral operators [9] and Poisson integrals [10]. [1] R. Taberski, Singular integrals depending on two parameters, Rocznicki Polskiego towarzystwa matematycznego, Seria I. Prace matematyczne, VII, (1962), [2] P.L. Butzer and R.J. Nessel, Fourier Analysis and Approximation, V.1, Academic Press, New York, London, [3] C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 9, xii pp., [4] E. Ibikli, Approximation of Borel derivatives of functions by singular integrals. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 21 (2001), no. 4, Math. Mech., 72 76, 238. [5] T. Ikegami, On Poisson integrals. Proc. Japan Acad Abant Izzet Baysal University, karsli [email protected], [email protected]

176 176 International Congress in Honour of Professor Ravi P. Agarwal 142 Regularization of an Abstract Class of Ill-Posed Problems Djezzar Salah and Benmerai Romaissa In this talk, we present an abstract class of ill-posed problems described by a differential equation with a self-adjoint unbounded operator coefficient on a Hilbert space. The class under study is regularized using a new modified quasiboundary value method to obtain an approximate family of well-posed problems. As a result of this regularization, an approximate family of regularized solutions is obtained. Moreover, some results concerning the stabilty estimates for these regularized solutions as well as some convergences results are provided. [1] Ames, K. A., Payne, L. E., Schaefer, P. W., Energy and pointwwise bounds in some non-standard parabolic problems. Proc. Roy. Soc. Edinburgh Sect. A 134, No.1, 1-9, [2] Clark, G. W., Oppenheimer, S. F., Quasireversibility Methods for Non-Well-Posed Problems, Electronic Journal of Differential Equations, No. 8, 1-9, [3] M. Denech and S. Djezzar, (2006), A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems Boundary-Value Problems, Volume Article ID 37524, 8 pages. [4] S. Djezzar and N. Teniou, (2011),. Improved regularization method for backward Cauchy problems associated with continuous spectrum operator International Journal of Differential Equations, Volume 2011, Article ID 93125, 11 pages. [5] Lettes, R. and Lions, J. L., Methode de Quasi-Reversibilit et Applications, Dunond, Paris, [6] Payne, L. E., Some general remarks on improperly posed problems for partial differential equations, Symposium on Non-Well-Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, 316, 1-30, Springer Verlag, Berlin, [7] Showalter, R. E., The final value problem for evolution equations, J. Math. Anal. Appl., 47, , Djezzar Salah: University of Constantine 1, Faculty of Exact Sciences, Department of Mathematics, Constantine, Algeria, [email protected] Benmerai Romaissa: University of Constantine 1, Faculty of Exact Sciences, Department of Mathematics, Constantine, Algeria, [email protected]

177 International Congress in Honour of Professor Ravi P. Agarwal Decompositions of Soft Continuity Ahu Açıkgöz and Nihal Taş In this talk, we introduce soft θ - open, soft θ - preopen, soft θ - semiopen, soft θ - β - open and soft θ - α - open sets in soft topological spaces and show the relationships between defined new soft sets and other soft sets using diagram. We investigate some properties of these soft sets. Also we define the concepts of θ - pre - soft continuity, θ - β - soft continuity, θ (A,E) - soft continuity, θ pre - B - soft continuity and θ β - B - soft continuity. Finally, we obtain decompositions of soft continuity. [1] M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl., 57 (2009) , [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986) 87-96, [3] W. L. Gau and D. J. Buehrer, Vague sets, IEEE Trans. System Man Cybernet, 23(2) (1993) , [4] S. Hussain and B. Ahmad, Some properties of soft topological space, Comput. Math. Appl., 62 (2011) , [5] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and A. M. ABD El-Latif, γ - operation and decompositions of some forms of soft continuity in soft topological spaces, Ann. Fuzzy Math. Inform. 7(2) (2014) , [6] P. K. Maji, R. Biswas and A. R. Roy, Intuitionistic fuzzy soft sets, The J. of F. Math. 9 (2001) , [7] P. K. Maji, R. Biswas and A. R. Roy, Fuzzy soft sets, The J. of F. Math. 9 (2001) , [8] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) , [9] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19-31, [10] M. Shabir and M. Naz, On soft topological spaces, Comput. Math. Appl. 61 (2011) , [11] L. A. Zadeh, Fuzzy sets, Infor. and Control 8 (1965) , [12] I. Zorlutuna, M. Akdag, W. K. Min and S. Atmaca, Remarks on soft topological spaces, Ann. Fuzzy Math. Inform. 3 (2012) Ahu Açıkgöz: Department of Mathematics, Balikesir University, Balikesir, Turkey, [email protected] Nihal Taş: Department of Mathematics, Balikesir University, Balikesir, Turkey, [email protected]

178 178 International Congress in Honour of Professor Ravi P. Agarwal 144 Lacunary Statistical Convergence of Double Sequences in Topological Groups Ekrem Savaş By X, we will denote an abelian topological Hausdorff group, written additively, which satisfies the first axiom of countability. The double sequence θ = {(k r, l s)} is called double lacunary if there exist two increasing of integers such that and k 0 = 0, h r = k r k k 1 as r l 0 = 0, h s = l s l s 1 as s. Notations: k r,s = k rl s, h r,s = h rh s,θ is determine by I r = {(k) : k r 1 < k k r}, I s = {(l) : l s 1 < l l s}, I r,s = {(k, l) : k r 1 < k k r & l s 1 < l l s}, q r = kr, q k s = ls, and q r 1 l r,s = q r q s 1 s. We will denote the set of all double lacunary sequences by N θr,s. In 2005, R. F. Patterson and E. Savas [1] studied double lacunary statistically convergence by giving the definition for complex sequences as follows: Tanım Let θ be a double lacunary sequence; the double number sequence x is st 2 θ convergent to L provided that for every ɛ > 0, 1 P lim {(k, l) I r,s : x(k, l) L ɛ} = 0. r,s h r,s In this case write st 2 θ lim x = L or x(k, l) L(S2 θ ). In this paper we introduce and study lacunary statistical convergence for double sequences in topological groups and we shall also present some inclusion theorems. [1] R. F. Patterson and Ekrem Savaş, Lacunary statistical convergence of double sequences,j. Math. Commun., 10 (2000), İstanbul Commerce University, Department of Mathematics, Uskudar, İstanbul, TURKEY, [email protected]

179 International Congress in Honour of Professor Ravi P. Agarwal On Fuzzy Pseudometric Spaces Elif Aydın and Servet Kütükçü In this paper, we introduce two classifications in fuzzy pseudometric spaces and examine relationships between them illustrating with examples. [1] A. George, P. Veeramani, On Some Results in Fuzzy Metric Spaces, Fuzzy Sets and Systems, 64(3)(1994), , [2] B. Schweizer, A. Sklar, Statistical Metric Spaces, Pacific Journal of Mathematics, 10(1)(1960), , [3] L. A. Zadeh, Fuzzy Sets, Information and Control, 8(3)(1965), , [4] M. Grabiec, Fixed Points in Fuzzy Metric Spaces, Fuzzy Sets and Systems, 27(1)(1989), , [5] M. A. Erceg, Metric Spaces in Fuzzy Set Theory, Journal Mathematical Analysis Applications, 69 (1979), , [6] O. Kaleva and S. Seikkala, On Fuzzy Metric Spaces, Fuzzy Sets and Systems 12 (1984), , [7] Z. Deng, Fuzzy Pseudometric Spaces, Journal Mathematical Analysis Applications, 86(1982), Elif Aydın: Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Kurupelit, Samsun- Turkey, elif Servet Kütükçü: Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Mathematics, Kurupelit, Samsun-Turkey,

180 180 International Congress in Honour of Professor Ravi P. Agarwal 146 On Fixed Points of Extended Hecke Groups Bilal Demir and Özden Koruoğlu Hecke groups H(λ q) are Fuchsian groups of the first kind and generated by two linear fractional transformations; T (z) = (z) 1 and S(z) = (z + λ q) 1, where λ q = 2 cos(π/q), q 3 integer[1]. Then H(λ q) has a presentation; H(λ q) =< T, S : T 2 = S q = I > C 2 C q The extended Hecke groups have been defined in [7], [8] by adding the reflection R(z) = 1/z to the generators of Hecke groups H(λ q). H(λ q) =< T, S, R : T 2 = S q = R 2 = I, T R = RT, SR = RS q 1 > D 2 C2 D q Let W (z) be an arbitrary element of H(λ q). The solutions of the equation W (z) = z are called fixed points of the element W (z).each transformation in H(λ q) has at most two fixed points except for identity. In extended Hecke groups case a whole circle can be fixed by a transformation. In this talk we give some results about fixed points of the elements in extended Hecke groups H(λ q). [1] Hecke E, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 112 (1936), [2] Cangül İN, Normal Subgroups and Elements of H (λ q), Tr. J. of Mathematics 23 (1999), [3] Cangül İN, The Group Structure of H(λq), Tr. J. of Mathematics 20 (1996), [4] Cangül İN and Singerman D, Normal Subgroups of Hecke Groups and Regular Maps, Math. Proc. Camb. Phil. Soc,. 123 (1998), [5] Ikikardes S, Sahin R and Cangül İN, Principal congruence subgroups of the Hecke groups and related results, Bull. Braz. Math. Soc. (N.S.) 40, No. 4, (2009). [6] Sahin R and Koruoğlu Ö, Commutator subgroups of the power subgroups of some Hecke groups, Ramanujan J. 24 (2011), no. 2, [7] Sahin R and Bizim O, Some Subgroups of Extended Hecke Groups H(λ q), Actua Math. Sci. 23 (4) (2003), [8] Sahin R, Bizim O and Cangül İN, Commutator Subgroups of the Extended Hecke Groups, Czech. Math..28 (2004), Bilal Demir: Balikesir University Necatibey Faculty of Education Department of Mathematics Education, Balıkesir- Türkiye, [email protected] Özden Koruoğlu: Balikesir University Necatibey Faculty of Education Department of Primary Mathematics Education, Balıkesir-Türkiye, [email protected] This work was supported by the Commission of Scientific Research Projects of Balikesir University, Project number 2014/99.

181 International Congress in Honour of Professor Ravi P. Agarwal New Lagrangian Forms of Modified Emden Equation by Jacobi Method Gülden Gün Polat and Teoman Özer The aim of the our work determination of Lagrangians and first integrals of modified Emden equation with respect to the Jacobi method. This novel approach enable to us to obtain Jacobi last multiplier s by means of known Lie symmetries of governing equation. Ratio of two Jacobi last multiplier corresponds to first integral. Based on this fact we present different first integrals of modified Emden equation. Furthermore some Hamiltonians and explicit solutions are derived. [1] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989 [2] L.V. Ovsiannikov, Group Analysis of Differential Equations, Moscow, Nauka, 1978 [3] N.H. Ibragimov, editor. CRC Handbook of Lie Group Analysis of Differential Equations vols, I-III, 1994 [4] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986 [5] T. Özer, Symmetry group classification for one-dimensional elastodynamics problems in nonlocal elasticity, Mechanics Research Communications, 30(6), , 2003 [6] T. Özer, On symmetry group properties and general similarity forms of the Benney equations in the Lagrangian variables, Journal of Computational and Applied Mathematics, 169 (2), , 2004 [7] T. Özer, Symmetry group classification for two-dimensional elastodynamics problems in nonlocal elasticity, International Journal of Engineering Science, 41(18), , 2003 [8] M. N. Nucci, K. M. Tamizhmani, Using an old method of Jacobi to derive Lagrangians: a nonlinear dynamical system with variable coefficients, arxiv: v1, 2008 [9] M. N. Nucci, P.G.L. Leach, An old method of Jacobi to find Lagrangians, J. Nonlinear Msth. Phys. 16, , 2009 [10] M. N. Nucci, P.G.L. Leach, The Jacobi Last Multiplier and its applications in mechanics, Physica scripta, 78, 2008 [11] A. Bhuvaneswari, R. Kraenkel, M. Senthilvelan, Application of the lambda-symmetries approach and time independent integral of the modified Emden equation, Nonlinear Analysis: Real World Applications. 13(2), , 2012 [12] E. Yaşar, M. Reis, Application of Jacobi Method and integrating factors to a class of Painlevé-Gambier equations, J. Phys. A: Math. Theor. 43, 2010 [13] M. N. Nucci, P.G.L. Leach, Jacobi s last multiplier and symmetries for the Kepler problem plus a lineal story, J. Phys. A: Math. Gen. 37, , 2004 [14] M. N. Nucci, Jacobi last multiplier and Lie symmetries: a novel application of an old relationship, J. Nonlinear Msth. Phys. 12, , 2005 [15] M. N. Nucci, P.G.L. Leach, Lagrangians galore, J. Math. Phys. 48, , 2007 Gülden Gün Polat: İstanbul Technical University, Faculty of Science and Letters, Department of Mathematics, Maslak, İstanbul-Turkey Teoman Özer: İstanbul Technical University, Division of Mechanics, Faculty of Civil Engineering, Maslak, İstanbul-Turkey, [email protected] Corresponding author: Teoman Özer.

182 182 International Congress in Honour of Professor Ravi P. Agarwal 148 Fixed Point Theorems for ψ-contractive Mappings on Modular Space Ekber Girgin and Mahpeyker Öztürk We introduce ψ ρ-contractive mappings, then we establish fixed point theorem for such mappings on modular spaces. As a consequences of this theorems, we obtain fixed point theorems on modular space with a graph. In addition, we present an example to illustrate the usability of the main results. [1] H. Nakano, Modulared Semi-Ordered Linear Spaces, In Tokyo Math Book Ser, (1), Maruzen Co., Tokyo (1950). [2] S. Koshi, T. Shimogaki, On F-norms of quasi-modular spaces, J Fac Sci Hokkaido Univ Ser I, (15) (3-4), (1961), [3] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans Amer Math Soc., (90), (1959), [4] WAJ. Luxemburg, Banach function spaces, Thesis, Delft, Inst of Techn Asser, The Netherlands, (1955). [5] T. Musielak, W. Orlicz, On Modular spaces, Studia Math, (18), (1959), [6] C.M. Chen, Fixed Point Theorems for Weak ψ-contractive Mappings in Ordered Metric Spaces, Journal of APplied Mathematics, (2012), (2012), 10, Article ID [7] H. K. NAshine, Z. Golubovic, Z. Kadelburg, Modified ψ-contractive mappings in ordered metric spaces with applications, Fixed Point Theory and Applications, (2013), (2013), doi: / [8] S. Chandok, S. Dinu, Common Fixed Points for Weak ψ-contractive Mappings in Ordered Metric Spaces with Applications, Abstract and Applied Analsis, (2013), (2013), 7, Article ID [9] J. Jachymski, The contraction principle for mappings on a metric space endowed with a graph, Proc. Amer. Math. Soc., 136(2008), [10] M. Abbas, T. Nazir, Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph, Fixed Point Theory and Applications, (2013), (2013), 20. [11] M. Öztürk, E. Girgin, On Some Fixed Point Theorems with ϕ-contractions in Cone Metric Spaces Involving a Graph, Int. Journal of Math. Analysis., 7(2013), [12] M. Öztürk, E. Girgin, On some fixed-point theorems for ψ-contraction on metric space involving a graph, Journal of Inequalities and Applications, 2014 (2014):39, doi: / X [13] M. Öztürk, M. Abbas, E. Girgin, Fixed Points of Mappings Satisfying Contractive Condition of Integral Type in Modular Spaces endowed with a Graph, Submitted Mahpeyker Öztürk: Department of Mathematics, Sakarya University, 54187, Sakarya, Turkey, [email protected] Ekber Girgin: Department of Mathematics, Sakarya University, 54187, Sakarya, Turkey, [email protected]

183 International Congress in Honour of Professor Ravi P. Agarwal Convexity and Schur Convexity on New Means V.Lokesha, U.K.Misra and Sandeep Kumar In this talk, we discuss some Convexity and Schur harmonic convexity of the Gnan mean, HP-mean and its dual forms are discussed. One can go further investigations on the geometrical aspects of Convexities. [1] P. S. Bullen, Handbook of Means and Their Inequalities, Kluwer Acad. Publ., Dordrecht, [2] V. Lokesha, Zhi-Hua Zhang and K. M. Nagaraja, Gnan mean for two variables, Far East Journal of Mathematics, 31(2) (2008), [3] V. Lokesh, K. M. Nagaraja, B. Naveen Kumar and Y-.D. Wu, Shur convexity of Gnan mean for positive arguments, Notes on Number Theory and Discrete Mathematics, 17(4) (2011), [4] K. M. Nagaraja and P. Siva Kota Reddy, Logarithmic convexity and concavity of some Double sequences, Scientia Magna, 7(2) (2011), [5] B. Naveenkumar, Sandeepkumar, V. Lokesha and K. M. Nagaraja, Ratio of difference of means and its convexity, International ejournsl Mathematics and Engineeting, 2(2) (2011), [6] H. N. Shi, Y. M. Jiang and W. D. Jiang, Schur-Geometrically concavity of Gini Mean, Comp. Math. Appl., 57(2009), [7] R. Webster, Convexity, Oxford University Press, Oxford, New York, Tokyo, [8] X. M. Zang, Geometrically Convex Functions, AnHui University Press, Hefei, 2004(in Chi- nese). V.Lokesha: Department of Mathematics, V. S.K. University, Bellary, India, U.K.Misra: Department of Mathematics, Berhampur University, Berhampur, Orissa, India Sandeep Kumar: Department of Mathematics, Acharya Institute of technology, Bangalore, The First author thankful to Authorities of the V. S. K. University, Bellary

184 184 International Congress in Honour of Professor Ravi P. Agarwal 150 On Radial Signed Graphs Gurunath Rao Vaidya, P.S.K.Reddy and V.Lokesha In this paper we introduced a new notion radial signed graph of a signed graph and its properties are obtained. Also, we obtained the structural characterization of radial signed graphs. Further, we presented some switching equivalent characterizations. Gurunath Rao Vaidya: Department of Mathematics, Acharya Institute of Graduate Studies, Bangalore , India, [email protected] P.S.K. Reddy: Department of Mathematics, S.I.T, Tumkur, India, [email protected] V. Lokesha: Department of Mathematics, V.S.K. University, Bellary, India, [email protected] The first author thankful to The Chairman, Acharaya Instiutes, Bangalore

185 International Congress in Honour of Professor Ravi P. Agarwal Delta and Nabla Discrete Fractional Grüss Type Inequality A.Feza Güvenilir Properties of the discrete fractional calculus in the sense of a backward and forward difference are introduced and developed. Here, we prove a more general version of the Grüss type inequality for the delta and nabla fractional case. An example of our main result is given. Ankara University, Ankara, [email protected]

186 186 International Congress in Honour of Professor Ravi P. Agarwal 152 On Tame Extensions and Residual Transcendental Extensions of a Valuation with rankv = n Burcu Öztürk and Figen Öke Let (K, v 1 ) be a henselian valued field, v i be a valuation of residue field k vi and v = v 1 v 2 /circ... v n be a composite of valuations v 1, v 2,..., v n for i = 2,..., n. Let L/K be a finite extension, z 1 be an extension of v 1 to L, z i be an extension of v i to the residue field k zi and z = z 1 z 2... z n be an extension of v to L which is a composite of valuations z 1, z 2,..., z n for i = 2,...n. In this paper it is shown that if (L, z)/(k, v)) is a tame extension then (L, z 1 )/(K, v 1 )) and (k zi 1, z i 1 )/(k vi 1, v i )) are tame extensions for i = 2,..., n. Also in this paper a residual transcendental extension w = w 1 w 2... w n of v to K(x) is studied where w 1 is a residual transcendental extension of v 1 to the rational function field K(x) defined by minimal pair (a 1, δ 1 ) and w i be a residual transcendental extension of v i to the residue field k wi 1 defined by minimal pair (a i, δ i ) for i = 2,...n. [1] O. Endler, Valuation Theory, Springer-Verlag, 1972 [2] O. Zariski, P. Samuel, Commutative Algebra Volume II, D. Von. Nostrand, Princeton, 1960 [3] N. Popescu, C. Vraciu,On the Extension of a valuation on a field K to K(x).-II, Rend. Sem. Mat. Univ. Padova, Vol 96, (1996), 1-14 [4] N. Popescu, A. Zaharescu, On the Structure of the Irreducible Polynomials over Local Fields, J. Number Theory, 52, No.1, (1995), [5] V. Alexandru, N. Popescu, A. Zaharescu, A Theorem of Characterization of Residual Transcendental Extensions of a Valuation, J. Math. Kyoto Univ., 28-4, (1988), [6] K. Aghigh, S. K. Khanduja, On Chains Associated with Elements Algebraic over a Henselian Valued Field, Algebra Colloquim, 12:4 (2005), [7] K. Aghigh, S. K. Khanduja, A Note on Tame Fields, Valuation Theory and its Applications, Vol II, Fields Institute Communications, 33, 1-6, 2003B. [8] Ozturk, F. OKE, On Residual Transcendental Extensions of a Valuation with rankv=2, Selcuk J. Appl. Math. Vol.12 No.2. pp , 2011B. [9] Ozturk, F. OKE, Some Constants and Tame Extensions According to a Valuation of a Field with rankv=2, Proc. Jangjeon Math. Soc., 15, No. 4, , 2012 Burcu Öztürk: Trakya University, Faculty of Science, Department of Mathematics, Edirne-Turkiye, Figen Öke: Trakya University, Faculty of Science, Department of Mathematics, Edirne-Turkiye, [email protected]

187 International Congress in Honour of Professor Ravi P. Agarwal Time Series Forecasting with Grey Modelling Seval Ene and Nursel Öztürk Time series is a collection of data points measured over a period of time. Time series forecasting defines the process of predicting future value based on previously observed data by using a mathematical model. Time series forecasting methods have a wide application field as in engineering, social science, economics etc. However in some real world applications, we have limited and uncertain data. Grey models forecast the future values of a time series based on most recent data other than classical forecasting models. Grey system theory was first introduced by Deng (1982). The theory is an interdisciplinary scientific research area. In this paper grey modeling is proposed for forecasting time series characterized as uncertain and small sized. To test the performance of the proposed grey model, data sets from literature are used. Obtained results showed the performance and applicability of the model particularly for forecasting data sets with small size and uncertainty. [1] [1] J. Deng, Introduction to grey system theory, J. Grey Syst., 1 (1989), [2] E. Kayacan, B. Uluta, O. Kaynak, Grey system theory-based models in time series prediction. Expert Syst. Appl., 37 (2010), [3] E. Kse, S. Erol,. Temiz, Grey system approach for EOQ models. Sigma, 28 (2010), [4] X. Wang, L. Qi, C. Chen, J.Tang, M. Jiang, Grey system theory based prediction for topic trend on Internet. Eng. Appl. Artif. Intel., 29 (2014), [5] A.W.L. Yao, S.C. Chi, J.H. Chen, An improved Grey-based approach for electricity demand forecasting, Electr. Pow. Syst. Res., 67 (2003), [6] M. Shah, Fuzzy based trend mapping and forecasting for time series data, Expert Syst. Appl., 39 (2012), Seval Ene: Uludag University, Faculty of Engineering, Industrial Engineering Department, Gorukle Campus, 16059, Bursa, Turkey, [email protected] Nursel Öztürk: Uludag University, Faculty of Engineering, Industrial Engineering, Department, Gorukle Campus, 16059, Bursa, Turkey, [email protected]

188 188 International Congress in Honour of Professor Ravi P. Agarwal 154 Periodic Solution of Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response and Impulses Ayşe Feza Güvenilir, Billur Kaymakçalan and Neslihan Nesliye Pelen We consider two dimensional predator-prey system with Beddington-DeAngelis type functional response and impulses on Time Scales. For this special case we try to find under which conditions the system has periodic solution. Our study is mainly based on continuation theorem in coincidence degree theory. This study will also give beneficial results for continuous and discrete case. [1] D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, Harlow, UK, vol. 66,1993. [2] J. R. Beddington Mutual interference between parasites or predators and its effect on searching efficency Journal of Animal Ecology,vol. 44, pp , [3] Martin Bohner,Meng Fan,Jimin Zhang. Existence of periodic solutions in predatorprey and competition dynamic systems. Nonlinear Analysis: RealWorld Applications 7 (2006) [4] D. L. DeAngelis, R. A. Goldstein, and R. V. ONeill, A model for trophic interaction, Ecology, vol.56, pp , [5] M. Fan, S. Agarwal, Periodic solutions for a class of discrete time competition systems, Nonlinear Stud. 9 (3) (2002) [6] M. Fan, K. Wang, Global periodic solutions of a generalized n-species GilpinAyala competition model, Comput. Math. Appl. 40 (1011) (2000) [7] M. Fan, K.Wang, Periodicity in a delayed ratio-dependent predatorprey system, J. Math. Anal. Appl. 262 (1) (2001) [8] M. Fan, Q.Wang, Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predatorprey systems, Discrete Contin.Dynam. Syst. Ser. B 4 (3) (2004) [9] H.F. Huo, Periodic solutions for a semi-ratio-dependent predatorprey system with functional responses, Appl. Math. Lett. 18 (2005) [10] Y.K. Li, Periodic solutions of a periodic delay predatorprey system, Proc. Amer. Math. Soc. 127 (5) (1999) [11] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A:Monographs and Treatises,World Scientific, River Edge,NJ, USA,vol. 14, [12] S. Tang, Y. Xiao, L. Chen, and R. A. Cheke, Integrated pest management models and their dynamical behaviour, Bulletin of Mathematical Biology, vol. 67, no. 1, pp , [13] Q. Wang, M. Fan, K. Wang, Dynamics of a class of nonautonomous semi-ratio-dependent predatorprey systems with functional responses,j. Math. Anal. Appl. 278 (2) (2003) [14] Peiguang Wang. Boundary Value Problems for First Order Impulsive Difference Equations. International Journal of Difference Equations. Volume 1 Number 2 (2006), pp [15] Weibing Wang, Jianhua Shen,Juan J. NietoPermanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses Discrete Dynamics in Nature and Society Volume 2007, Article ID 81756, 15 pages. [16] Chunjin Wei and Lansun Chen Periodic Solution of Prey-Predator Model with Beddington-DeAngelis Functional Response and Impulsive State Feedback Control Journal of Applied Mathematics Volume 2012, 2012, 17 pages. [17] Z. Xiang, Y. Li, and X. Song, Dynamic analysis of a pest management SEI model with saturation incidence concerning impulsive control strategy, Nonlinear Analysis, vol. 10, no. 4, pp , [18] R. Xu, M.A.J. Chaplain, F.A. Davidson, Periodic solutions for a predatorprey model with Holling-type functional response and time delays, Appl. Math. Comput. 161 (2) (2005) Ayşe Feza Güvenilir: Ankara University, Faculty of Science, Department of Mathematics, Ankara, Turkey, Billur Kaymakçalan: Çankaya University, Department of Mathematics and Computer Science, 06810, Ankara, Turkey, [email protected], billur Neslihan Nesliye Pelen: Middle East Technical University, Department of Mathematics, Ankara, Turkey, [email protected]

189 International Congress in Honour of Professor Ravi P. Agarwal Approximation Properties of Kantorovich-Stancu Type Generalization of q-bernstein-schurer-chlodowsky Operators on Unbounded Domain Tuba Vedi and Mehmet Ali Özarslan In this paper, we introduce the Kantorovich-Stancu type generalization of q-bernstein-chlodowsky operators on the unbounded domain. We should note that this generalization include various kind of operators which are not introduced earlier. We calculate the error of approximation of these operators by using modulus of continuity and Lipschitz-type functionals. Finally, we give generalization of the operators and investigate its approximations. [1] Agrawal, PN, Gupta, V, Kumar, SA: On a q-analogue of Bernstein-Schurer-Stancu operators, Applied Mathematics and Computation, 219, (2013). [2] Barbosu, D: Schurer-Stancu Type Operators, Babeş-Bolyai Math., XLVIII (3), (2003). [3] Barbosu, D: A survey on the approximation properties of Schurer-Stancu operators, Carpatian J. Math., 20, 1-5 (2004). [4] Büyükyazıcı, İ: On the approximation properties of two dimensional q-bernstein-chlodowsky polynomials, Math Commun., 14 (2), (2009). [5] Büyükyazıcı, İ, Sharma, H: Approximation properties of two-dimensional q-bernstein-chlodowsky-durrmeyer operators, Numer. Funct. Anal. Optim., 33 (2), (2012). [6] İbikli, E: On Stancu type generalization of Bernstein-Chlodowsky polynomials, Mathematica, 42 (65), (2000). [7] Chlodowsky, I: Sur le development des fonctions defines dans un interval infini en series de polynomes de M. S. Bernstein, Compositio Math., 4, (1937). [8] DeVore, RA, Lorentz, GG: Constructive Approximation, Springer-Verlag, Berlin (1993). [9] Gadjiev, AD: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of P. P. Korovkin, Dokl. Akad. Nauk SSSR 218 (5), English Translation in Soviet Math. Dokl. 15 (5), (1974). [10] İbikli, E: Approximation by Bernstein-Chlodowsky polynomials, Hacettepe Journal of Mathematics and Statics, 32, 1-5 (2003). [11] Kac, V, Cheung, P: Quantum Calculus, Springer, [12] Karslı, H, Gupta, V: Some approximation properties of q-chlodowsky operators, Applied Mathematics and Computation, 195, (2008). [13] Muraru, CV: Note on q-bernstein-schurer operators, Babeş-Bolyaj Math., 56, (2011). [14] Özarslan, MA: q-szasz Schurer operators, Miscolc Mathematical Notes, 12, (2011). [15] Özarslan, MA, Vedi, T: q- Bernstein-Schurer-Kantorovich Operators, J. of Ineq. and Appl., 2013, 2013:444 doi: / x [16] Phillips, GM: On Generalized Bernstein polynomials, Numerical analysis, World Sci. Publ., River Edge, 98, (1996). [17] Phillips, GM: Interpolation and Approximation by Polynomials, Newyork, (2003). [18] Schurer, F: Linear Positive Operators in Approximation Theory, Math. Inst., Techn. Univ. Delf Report, [19] Stancu, DD: Asupra unei generalizari a polinoamelor lui Bernstein [ On generalization of the Bernstein polynomials], Studia Univ. Babeş-Bolyai Ser. Math.-Phys., 14 (2), (1969). [20] Vedi, T, Özarslan, MA: Some Properties of q-bernstein-schurer operators, J. Applied Functional Analysis, 8 (1), (2013). [21] Vedi, T, Özarslan, MA: Chlodowky variant of q-bernstein-schurer-stancu operators, J. of Ineq. and Appl., 2014, / X Tuba Vedi: Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey, [email protected] Mehmet Ali Özarslan: Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey, [email protected]

190 190 International Congress in Honour of Professor Ravi P. Agarwal 156 Use of Golden Section in Music Sümeyye Bakım In this study the relationship of Fibocacci Sequence and Golden Ratio with music questioned. Pre acceptances on the studies applied to the chosen examples of some European art music/multi- vocal composers up to now have been discussed within the framework of mathematical and musical. [1] Koshy, T., 2001, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience Publication, Canada. [2] Lehmann, I., Posamentier, Alfred S., 2007, The (Fabulous) Fibonacci Numbers, Alfred S. Posamentier, Ingmar, Lehmann, Prometheus Books, 2007, [3] Power, T., 2001, J. S. Bach and the Divine Proportion, Doctoral Thesis, Department of Music, Duke University. KTO Karatay Universite,

191 International Congress in Honour of Professor Ravi P. Agarwal On Analysis of Mathews-Lakshmanan Oscillator Equation via Nonlocal Transformation and Lagrangian-Hamiltonian Description Özlem Orhan and Teoman Özer In this study, we consider Mathews-Lakshmanan Oscillator Equation which possesses exact periodic solution, exhibiting the characteristic amplitude-dependent frequency of nonlinear oscillator in spite of the sinusoidal nature of the solution of equation. Mathews-Lakshmanan Oscillator Equation has a natural generalization in three dimensions and these systems can be also quantized exhibiting many interesting features and can be interpreted as an oscillator constrained to move on a three-sphere. Firstly, we examine the first integral in the form A(t, x)ẋ + B(t, x) and then, we consider other first integrals of the equation via this finding λ-symmetry. Using the coefficients of the equation, we characterize this equation that can be linearized by means of nonlocal transformation that is called Sundman transformation. In addition, the time independent integrals for Mathews-Lakshmanan Oscillator Equation are obtained by using modified Prelle-Singer procedure. We demonstrate that the equation is integrable by these first integrals. Further, the Lagrangian- Hamiltonian forms are investigated using this time-independent first integral. Finally, we compare results obtained by two different methods. [1] G.W. Bluman and S. Kumei, Symmetries and Differential Equations, 1989, [2] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1993, [3] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations vols. I-III, CRC Press, [4] H. Stephani, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge,1989. [5] C. Muriel, and J. L. Romero, Nonlocal transformation of linearization of second-order Ordinary Differential Equations, Journal of Physics A: Mathematical and Theoretical 43 (2010), , [6] C. Muriel, and J. L. Romero, Second-Order Ordinary Differential Equations and First Integrals of The Form A(t, x)ẋ + B(t, x), Journal of Nonlinear Mathematical Physics 16(2009), , [6] T. Özer, Symmetry group classification for two-dimensional elastodynamics problems in nonlocal elasticity, International Journal of Engineering Science 41(18)(2003), , [8] L. G. S. Duarte, S. E. S. Duarte, L. A. C. P. da Mota and J. E. F. Skea, Solving second-order ordinary differential equations by extending the Prelle-Singer method, J. Phys. A:Math. Gen. 34(2001), [9] E. Yasar, λ-symmetries, nonlocal transformations and first integrals to a class of Painleve-Gambier equations, Mathematical Methods in the Applied Sciences 13 (2012), ,13 (2012), , [10] Ajey K. Tiwari, S. N. PANDEY, M. Senthilvelan and M. Lakshmanan, Classification of Lie point symmetries for quadratic Lienard type equation x+f(x)x2+g(x)=0, Journal of Mathematical Physics, 54 (2013), , [11] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator, Journal of Mathematical Physics, 48 (2007), Özlem Orhan: İstanbul Technical University, Faculty of Science and Letters, Department of Mathematics Engineering, Maslak, İstanbul-Turkiye, [email protected] Teoman Özer: İstanbul Technical University, Division of Mechanics, Faculty of Civil Engineering, Maslak, İstanbul-Turkiye, [email protected]

192 192 International Congress in Honour of Professor Ravi P. Agarwal 158 On Singularities of the Galilean Spherical Darboux Ruled Surface of a Space Curve in the Pseudo-Galilean Space G 1 3 Tevfik Şahin and Murteza Yılmaz In this paper, we study the singularity theory in pseudo-galilean space, a special type of Cayley-Klein spaces. In particular, we investigate the singularities of pseudo-galilean height functions intrinsically related to the Frenet frame along a curve embedded into pseudo-galilean space. We also establish relationships between singularities of discriminant, bifurcation sets of the function, and geometric invariants under the action of pseudo-galilean group of curves in pseudo-galilean space. [1] Arnol d, V.I., Gusein-Zade, S.M. and Varchenko, A.N. Singularities of Differentiable Maps vol. I (Birkhäuser, 1986). [2] Bruce, J.W. On Singularities, envelopes and elementary differential geometry, Math.Proc.Cambridge Phil.Soc. 89, 43-48, [3] Bruce, J.W. and Giblin, P.J. Generic geometry, Amer.Math.Monthly. 90, , [4] Bruce, J.W. and Giblin, P.J. Curves and Singularities, second edition (Cambridge Univ.Press, 1992). [5] Che, M.G., Jiang, Y. and Pei, D. The hyperbolic Darboux image and rektifying Gaussian surface of nonlightlike curve in Minkowski 3-space, Journal of Math. Research & Exposition. 28, , [6] Divjak, B. Geometrija psudogalilejevih prostora, Ph.D. Thesis, (University of Zagreb, 1997). [7] Divjak, B. and Šipuš, Ž. M. Some special surfaces in the pseudo-galilean space, Acta Math. Hungar. 118(3), , [8] Cox, D., Little, J. and O shea, D. Ideals,varieties, and algorithms. Second edition. (New York Springer, 1977). [9] Casse, R. Projective geometry on introduction (Oxford Univ. Press, 2006). [10] Erjavec, Z. and Divjak, B. The general solution of the Frenet system of differential equations for curves in the pseudo- Galilean space G 1 3, Mathematical Communications. 2, , [11] Izumiya, S., Katsumi, H. and Yamasaki, T. The Rectifying developable and the spherical darboux image of a space curve, Geometry and topology of caustics 98 Banach Center Publications. 50, , [12] Izumiya, S. and Takeuchi, N. Generic properties of helices and Bertrand curves, J.Geom. 74, , [13] Izumiya, S. and Sano, T. Generic affine differential geometry of space curves, Proc.Royal soc. Edinburgh. 128A, , [14] Molnar, E. The projective interpretation of the eight 3-dimensional homogeneous geometries, Beitr. Algebra Geom. 38, , [15] Mond, D. Singularities of the tangent developable surface of a space curve, Quart. J. Math. Oxford. 40, 79-91, [16] Montaldi, J.A. Surfaces in 3-space and their contact with circles, J.Differential Geom. 23(2), , [17] O Neill, B. Elementary Differential Geometry (Academic Press, Inc., 1966). [18] Pei, D. and Sano, T. The focal developable and the binormal indicatrix of a nonlightlike curve in Minkowski 3-space, Tokyo J. Math. 23(1), , [19] Röschel, O. Die Geometrie des Galileischen Raumes, Habilitationssch (Inst. für Mat. und Angew. Geometrie, 1984). [20] Sahin, T. and Yılmaz, M. On singularities of the Galilean spherical darboux ruled surface of a space curve in G3, Ukranian Math. Journal, 62(10), , [21] Sahin, T. and Yılmaz, M. The rectifying developable and the tangent indicatrix of a curve in Galilean 3-space, Acta Math. Hungar. 132(1-2) , [22] Šipuš, Ž. M. and Divjak, B. Surface of constant curvature in the pseudo-galilean space, International J. of Math. and Math. Sci. Vol 2012, 28 p., [23] Yaglom, I.M. A Simple Non-Euclidean Geometry and Physical Basis (Sprınger-Verlag Newyork, 1979). Tevfik Şahin: Amasya University, Faculty of Sciences and Arts, Department of Mathematics, Amasya-Turkey, [email protected] Murteza Yılmaz: TOBB University of Economics & Technology, [email protected]

193 International Congress in Honour of Professor Ravi P. Agarwal Existence of Positive Solutions for Second Order Semipositone Boundary Value Problems on the Half-Line F.Serap Topal and Gülşah Yeni In this talk, we aim to establish a sufficient condition for the existence of positive solution for semipositone singular Sturm- Liouville boundary value problems on the half-line of an unbounded time scale by using Guo-Krasnosels kii fixed point theorem. F.Serap Topal: Ege University, Faculty of Science, Department of Mathematics, [email protected] Gülşah Yeni: Missouri University of Science and Technology, Department of Mathematics and Statistics, [email protected]

194 194 International Congress in Honour of Professor Ravi P. Agarwal 160 Some Congruent Number Families Refik Keskin and Ümmügülsüm Öğüt In this study, we give some congruent number families concerning generalized Lucas sequences (V n). [1] J. Coates, Congruent Number Problem, Pure and Appl. Math. Quaterly, Volume1, Number1, 14-27, [2] J. S. Chahal, Congruent Numbers and Elliptic Curves, Amer. Math. Monthly, Vol. 113, No. 4, , [3] K. Conrad, The Congruent Number Problem, 2/congruent number.pdf. [4] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Second Edition, Springer Verlag, [5] A. Lozano-Robledo, Elliptic Curves, Modular Forms and their L-Functions, Student Mathematical Library, [6] S. Zhang, Congruent Numbers and Heegner Points, Asia Pacific Newsletter, Vol 3 No 2, 12-15, [7] A. Wiles, Modular Elliptic Curves and Fermat s Last Theorem, Ann. of Math. (2) 141, , [8] R. Keskin, M. G. Duman, Positive integer solutions of some Pell equations (submitted). [9] F. Izadi, Congruent numbers via the Pell equations and its analogous counterpart, arxiv: v4[math,HO] 30 Dec [10] P. Serf, Congruent numbers and elliptic curves, Computational Number Theory, Walter de Gruyter and Co., Berlin, New York, , [11] G. Kramarz, All congruent numbers less than 2000, Mathematische Annalen, Berlin, Heidelberg, , [12] R. Alter, T.B. Curtz, A note on congruent numbers, Math. Comp., , Refik Keskin: Sakarya University, Faculty of Sciences and Arts, Department of Mathematics, Sakarya-Turkiye, Ümmügülsüm Öğüt: Sakarya University, Faculty of Sciences and Arts, Department of Mathematics, Sakarya-Turkiye, [email protected]

195 International Congress in Honour of Professor Ravi P. Agarwal On Some Fourth-Order Diophantine Equations Merve Güney Duman and Refik Keskin Let k 3 be an odd integer. In this paper, we show that the equations x 4 (k 2 4)y 2 = 4(k 2), x 4 (k 2 4)y 2 = 4(k + 2), x 2 (k 2 4)y 4 = 4(k + 2) and x 4 kx 2 y + y 2 = (k + 2) have no positive integer solutions. Moreover, we show that if k 1(mod8), then the equation x 4 (k 2 4)y 2 = 4(k 2) has no positive integer solutions, if k 3(mod8), then the equation x 4 (k 2 4)y 2 = 4(k + 2) has no positive integer solutions and if k 2 4 is a squarefree integer, then the equations x 4 kx 2 y + y 2 = (k 2)(k 2 4) and x 4 kx 2 y + y 2 = (k + 2)(k 2 4) have no positive integer solutions. In addition, we define all positive integer solutions of the some fourth-order diophantine equations. [1] LeVeque J. W., Topics in Number Theory, Volume 1 and 2, Dover Publications 2002, [2] Lucas E., Theorie des Fonctions, Numeriques Simplement Periodiques, American Journal of Mathematics, 1,2, , 1878, [3] Ribenboim P., McDaniel W. L., The square terms in Lucas sequences, J. Number Theory, 58, , 1996, [4] Ribenboim P., My Numbers, My Friends, Springer-Verlag New York, Inc., 2000, [5] Kalman D., Mena R., The Fibonacci Numbers exposed, Mathematics Magazine 76, , 2003, [6] McDaniel W. L., Diophantine Representation of Lucas Sequences, The Fibonacci Quarterly 33, 58-63, 1995, [7] Şiar Z., Keskin R., Some new identities concerning generalized Fibonacci and Lucas numbers, Hacettepe Journal of Mathematics and Statistics, 42, 3, , 2013, [8] Şiar Z., Keskin R., The Square Terms in Generalized Lucas Sequences, Mathematika 60, 1, , 2014, [9] Keskin R., G. Duman M., Positive integer solutions of some second order Diophantine equations (submitted), [10] Keskin R., Solutions of Some Quadratics Diophantine Equations, Computers and Mathematics With Applications, 60, 8, , 2010, [11] Melham R., Conics Which Characterize Certain Lucas Sequences, The Fibonacci Quarterly 35, , 1997, [12] Nagell T., Introduction to Number Theory, Chelsea Publishing Company, New York, 1981, [13] Robinowitz S., Algorithmic Manipulation of Fibonacci Identities, in: Application of Fibonacci Numbers, vol. 6, Kluwer Academic Pub, Dordrect, The Netherlands, , 1996, [14] Jones J.P., Representation of Solutions of Pell equations Using Lucas Sequences, Acta Academia Pead. Agr., Sectio Mathematicae, 30, 75-86, 2003, [15] Keskin R., Yosma Z., On Fibonacci and Lucas numbers of the form cx 2, Journal of Integer Sequences, Vol 14, 1-12, 2011, [16] Keskin R., Generalized Fibonacci and Lucas Numbers of the form wx 2 and wx 2 ±1, Bulletin of the Korean Mathematical Society (accepted), [17] Nakamula K., Pethö A., Squares in Binary Recurrence Sequences, In: Number Theory, Walter de Gruyter GmbH& Co., Berlin-New York, 1998, pp , [18] Mignotte M., Pethö A., Sur les carrés dans certaines suites de Lucas, Journal de Théorie des Nombres de Bordeaux, 5 no. 2 (1993), , [19] Cohn J. H. E., Lucas and Fib. Numbers Some Diophantine Equation, Proc. Glasgow Math. assoc., 7(1965), Merve Güney Duman: Sakarya University, Faculty of Science and Arts, Department of Mathematics, Adapazarı, Sakarya-Turkiye [email protected] Refik Keskin: Sakarya University, Faculty of Sciences and Arts, Department of Mathematics, Sakarya-Turkiye,

196 196 International Congress in Honour of Professor Ravi P. Agarwal 162 Characteristic Subspaces of Finite Rank Operators Mohamed Najib Ellouze Recently, Uffe Haagerup and Hanne Schultz proved in [1] that for any operator T in M, where M is a factor of type II 1 such that the spectral measure of Brown is not concentrated in a singleton (cf. [2]), has a non-trivial closed T -hyperinvariant subspace, constructed by spectral projectors of the limit A := lim n + (T n T n ) 2n 1 (cf. [3]). In this paper, we prove the existence of this limit for all finite rank operators, and we define the corresponding characteristic subspaces. [1] U.Haagerup and H. Schultz. Invariant subspaces for operators in a general II 1 -factor, Publications mathã c matiques July 2009, Vol 109, Issue 1, pp [2] L.G.Brown, Lidskii s theorem in the type II Case, Pitman Res. Notes in Math. Ser 123, Longman Sci. Tech 1986, pp [3] R.V.Kadison and J.R.Ringrose, Fundamentals of the Theory of Operator Algebras, Vol 1. Academic Press, New York, Sfax University, Faculty of Sciences, Departement of Mathematics, Route de Soukra 3018 Sfax BP 802, Tunisia, [email protected]

197 International Congress in Honour of Professor Ravi P. Agarwal Fixed Point Theory in WC-Banach Algebras Bilel Mefteh In this paper, we will prove some fixed point theorems for the sum and the product of nonlinear weakly sequentially continuous operators acting on a WC-Banach algebra. Our results improve and correct some results of the recent paper of Banas and Taoudi [1], and extend some several earlier works using the condition P where [1] is the following reference: [1] J. Banas, M. A. Taoudi, Fixed points and solutions of operator equations for the weak topology in Banach algebras, Taiwanese Journal of Mathematics. DOI: /tjm , (2014). Department of Mathematics, Faculty of Sciences of Sfax, Road Soukra Km 3.5, B.P , Sfax - Tunisia, [email protected]

198 198 International Congress in Honour of Professor Ravi P. Agarwal 164 Oscillation and Nonoscillation Criteria for Second Order Generalized Difference Equations Yaşar Bolat In this talk, we discuss some new oscilation and nonoscillation criteria for second order nonlinear difference equation with generalized difference operators which generalize and improve some results in the literatures. Also, some examples illustrating the results are included. [1] Y. Bolat and Ö. Akın, Oscillation criteria for higher order half linear delay difference equations involving generalized difference, Mathematica Slovaca, in press. [2] S. Chen and L. H. Erbe, Riccati Techniques and Discrete Oscillations, J. Math. Anal. Appl., 142, (1989). [3] X.Z. He, Oscillatory and Asymptotic Behavior of Second Order Nonlinear Difference Equations, J. Math. Anal. Appl., 175(2), (1993). [4] M. M. S. Manuel and at all, Asymptotic behavior of solutions of generalized nonlinear difference equations of second order, Communications in Differential and Difference quations, 3 (1), 13-21, (2012). [5] M. M. S. Manuel and at all, Oscillation, nonoscillation and growth of solutions of generalized second order nonlinear α- difference equations, Global Journal of Mathematical Science: Theory and Pratical, 4(3), (2012). [6] J. Popenda, Oscillation and nonoscillation theorems for second order difference equations, J. Math. Anal. Appl., 123, (1987). [7] Z. Szafranski and B. Szmanda, Oscillatory Behavior of Difference Equations of Second Order, J. Math. Anal. Appl., 150, (1990). [8] M-C. Tan and E-H. Yang, Oscillation and nonoscillation theorems for second order difference equations, J. Math. Anal. Appl., 276, , (2002). [9] E. Thandapani, Oscillation theorems for perturbed nonlinear second order difference equations, Computers & Mathematics with Applications, 28(1 3), (1994). [10] E. Thandapani, Oscillation criteria for a second order damped difference equation, Applied Mathematics Letters, 8(1), 1 6 (1995). [11] E. Thandapani, Tamilnadu, Oscillation theorems for a second order damped nonlinear difference equation, Czechoslovak Mathematical Journal, 45 (120), (1995). [12] P.J. Y. Wong, Oscillation theorems and existence of positive monotone solutions for second order nonlinear difference equations, Mathematical and Computer Modelling, 21 (3), (1995). [13] B.G. Zhang, Oscillation and Asymptotic Behavior of Second Order Difference Equations, J. Math. Anal. Appl., 173(1), (1993). Kastamonu University, Faculty of Science and Arts, Department of Mathematics, Kastamonu-Turkey,

199 International Congress in Honour of Professor Ravi P. Agarwal On Generalizations of Some Inequalities Containing Diamond-Alpha Integrals and Applications Billur Kaymakçalan We present a survey of some generalizations and refinements of the Opial, Hlder, Hardy and Constantin type Inequalities containing the diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals. Some related applications are also given. Çankaya University, Department of Mathematics and Computer Science, Ankara, Turkey, [email protected]

200 200 International Congress in Honour of Professor Ravi P. Agarwal 166 On Reciprocity Law of the Y (h, k) Sums Associated with PDE s of the Three-Term Polynomial Relations Elif Cetin, Yilmaz Simsek and Ismail Naci Cangul By using PDE s of the three-term polynomial relations, we find a new finite sum which is related to the Hardy-Berndt sums and the Simsek s sum Y (h, k). By using PDEs, we give another poof of reciprocity law of this sum. Our method is different from that of Simsek s (On Analytic properties and character analogs of Hardy Sums, Taiwanese J. Math. 13 (1) (2009), ). We also give some relations and remarks on these sums. [1] Apostol, T. M., Modular functions and Dirichlet Series in Number Theory, Springer-Verlag(1976). [2] Apostol, T. M., and Vu, T. H., Elementary Proofs of Berndt s Reciprocity Laws, Pasific J. Math. 98 (1982), [3] Beck, M., Geometric proofs of polynomial reciprocity laws of Carlitz, Berndt, and Dieter, M. Beck, in Diophantine analysis and related fields 2006, Sem. Math. Sci. 35, Keio Univ., Yokohama, 2006, pp [4] Berndt, B. C., Analytic Eisenstein Series, Theta-functions, and Series relations in the spirit of Ramanujan, J. Reine Angew. Math. 303/304(1978), [5] Berndt, B., and Dieter, U., Sums involving the greatest integer function and Riemann Stieltwes integration, J. Reine Angew. Math. 337, (1982). [6] Berndt, B. C., and Goldberg, L. A., Analytic Properties of Arithmetic Sums arising in the theory of the classical Theta-functions, SIAM., J. Math. Anal. 15(1984), [7] M. Can and V. Kurt, CHARACTER ANALOGUES OF CERTAIN HARDY-BERNDT SUMS, International Journal of Number Theory 10 (3) (2014), [8] Carlitz, L., Some polynomials associated with Dedekind Sums, Acta Math. Sci. Hungar, 26 (1975), [9] E. Cetin, Y. Simsek and I. N. Cangul, Some special finite sums related to PDE s of the three-term polynomials relations and their applications, preprint. [10] Goldberg, L. A., Transformation of Theta-functions and analogues of Dedekind sums, Thesis, University of Illinois Urbana(1981). [11] Hardy, G. H., On certain series of discontinues functions, connected with the modular functions, Quart. J. Math. 36(1905), pp (= Collected papers, vol.iv, pp Clarendon Press Oxford (1969)). [12] Pettet, M. R., and Sitaramachandraro, R., Three-Term relations for Hardy sums, J. Number Theory 25(1989), [13] Simsek, Y., On Generalized Hardy Sums s 5 (h, k), Ukrainian Math. J., 56(10) (2004), [14] Simsek, Y., Theorems on Three-Term Relations for Hardy sums, Turkish J. Math. 22(1998), [15] Simsek, Y., A note on Dedekind sums, Bull. Cal. Math. Soci. 85(1993) [16] Simsek, Y., On Analytic properties and character analogs of Hardy Sums, Taiwanese J. Math. 13 (1) (2009), [17] Sitaramachandrarao, R., Dedekind and Hardy sums, Acta Arith. XLVIII (1978). Elif Cetin: Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa-Turkey, Celal Bayar University, Faculty of Arts and Science, Department of Mathematics, Manisa, Turkey, [email protected] Yilmaz Simsek: Akdeniz University, Faculty of Science, Department of Mathematics, Antalya, Turkey, İsmail Naci Cangul: Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa-Turkey, The authors are supported by the research fund of Akdeniz University and Uludag University.

201 International Congress in Honour of Professor Ravi P. Agarwal Permutation Method for a Class of Singularly Perturbed Discrete Systems with Time-Delay Tahia Zerizer Discrete-time systems with state delay have strong background in engineering applications. However, the singularly perturbed discrete system with time-delay has not been fully investigated. In this paper, we develop the perturbation method for a class of linear singularly perturbed discrete systems with time delay. Convergent algorithms are provided showing the steps of the method. Scientific Classes, Faculty of Sciences, Jazan University, Jazan, Saudi Arabia, [email protected]

202 202 International Congress in Honour of Professor Ravi P. Agarwal 168 Existence of Minimal and Maximal Solutions for Quasilinear Elliptic Equation with Nonlocal Boundary Conditions on Time-Scales Mohammed Derhab and Mohammed Nehari The purpose of this work is the construction of minimal and maximal solutions for a class of second order quasilinear elliptic equation subject to nonlocal boundary conditions. More specifically, we consider the following nonlinear boundary value problem ( ϕ p ( u )) = f (x, u), in (a, b)t, u (a) a 0 u (a) = g 0 (u), u (σ (b)) + a 1 u (σ (b)) = g 1 (u), where p > 1, ϕ p (y) = y p 2 y, ( ϕ p ( u )) is the one-dimensional p Laplacian, f : [a, b]t R R is a rd-continuous function, g i : C rd ( [a, b]t ) Crd ( [a, b]t ) R (i = 0 and 1) are rd-continuous and a0 and a 1 are a positive real numbers. Mohammed Derhab: Department of Mathematics, Faculty of Sciences, University Abou-Bekr Belkaid Tlemcen, B.P.119, Tlemcen, 13000, Algeria, [email protected] Mohammed Nehari: Department of Mathematics, Faculty of Sciences, University Ibn Khaldoun Tiaret, Algeria, nehari [email protected]

203 International Congress in Honour of Professor Ravi P. Agarwal Application of Filled Function Method in Chemical Control of Pests Ahmet Şahiner, Meryem Öztop, Gülden Kapusuz and Ozan Demirözer Rhodococcus perornatus (Cockerell & Parrott) is an important pest on oil-bearing rose(rosa damascena Mill.) and the methidation has been used for a long time in order to control of the pest. As usual determining the effects of plant protection products on the organism is very important in pest management practices. In this study, the effect of the Methidathion E.C. 426 g/l on R. perornatus is modeled by using fuzzy logic approach. To obtain the maximum effect on the pest, how long the pesticide should be applied is determined by using the Filled Function Method. [1] Zadeh LA Fuzzy sets, Inform. Control, 8, [2] Sakawa M Fuzzy sets and interactive multi objective optimiztion, With 1 IBM-PC floppy disk, Appl. Info. T., (Plenum Press). [3] Kosko B Neural network and Fuzzy systems. A dynamical systems approach to machine intelligence With 1 IBM-PC floppy disk (Prentice Hall, Inc.-1992). [4] Mamdani EH Application of fuzzy algorithms for simple dynamic plant, Proc. IEE 121, [5] Saltan M, Saltan S, Sahiner A Fuzzy logic modeling of deflection behavior aganist dynamic loading in flexible pavements, Construction and Building Materials, 21, [6] Mamdani EH, Assilian S An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man-Machine Studies 7 (1), [7] Ge RP A filled function method for finding global minimizer of a function os several variables, Mathematical programming, 46, [8] Ge RP The theory of filled function method for finding global minimizer of a nonlinearly constrained Minimization problem, J. Comput. Math. 5 (1), 1-9. [9] Ge RP The globally convexized filled functions for global optimzation, Appl. Math. Comput. 35, [10] Liu X Several filled functions with mitigators, Appl. Math. Comput. 133 (2002), [11] Liu, X Finding global minima eith computable filled function, J. Global Optim. 19, [12] Sahiner A, Gokkaya H An application of filled function method to the hardness property of Fe-Mn Binary Alloys, Uncertainty Modeling in Knowledge Engineering and Decision Making, 10th International FLINS Conference, İstanbul. Ahmet Şahiner: Suleyman Demirel University, Department of Mathematics, Isparta, Turkiye, [email protected] Meryem Öztop: Suleyman Demirel University, Department of Mathematics, Isparta, Turkiye, [email protected] Gülden Kapusuz: Suleyman Demirel University, Department of Mathematics, Isparta, Turkiye, Ozan Demirözer: Suleyman Demirel University, Department of Plant Protection, Isparta, Turkiye, ozandemirozer@sdu. edu.tr

204 204 International Congress in Honour of Professor Ravi P. Agarwal 170 A New Approach to the Filled Function Method for Nonsmooth Problems Nurullah Yilmaz and Ahmet Sahiner The filled function method (FFM) is one of the effective method for smooth problems. Recently, studies on FFM have been concentrated on non-smooth problems. In this study, we present a new algorithm which hold some properties of the FFM, to find the global minimizer of the non-smooth but continuous functions. [1] W. X. Wang, Y. L.Shang, L. S. Zhang, Y. Zhang, Global optimization of non-smooth unconstrained problems with filled functions, Optim. Lett. 7, , (2013). [2] W. X. Wang, Y. L.Shang, Y. Zhang, Finding Global Minima with a Novel Filled Function for Non-smooth Unconstrained Optimisation, Int. J. Syst. Sci. 4, , (2012). [3] Z.Y. Wu, H. W. J.Lee, L.S. Zhang, X. M. Yang, A Novel Filled Function Method and Quasi-Filled Function Method for Global Optimization, Comput. Optim. Appl., 34, (2005). [4] W.X. Wang, Y.L. Shang, Y. Zhang, A Filled Function Approach for Nonsmooth Constrained Global Optimization, Math. Probl. Eng., 2010, Article ID , 9 pages (2010). [5] Z. Xu,H.X. Huang, P. M. Pardalos, C. X. Xu,Filled functions for unconstrained global optimization, J. Glob. Opt., 20, (2001). [6] Y. Zhang, L.Zhang, Y. Xu, New filled functions for non-smooth global optimization, App. Math. Model., 33, (2009). [7] N. Karmitsa, Nonsmooth Optimization in 30 minutes, 2013,web-page: Erişim: [8] Y. Lin, Y. Yang, L. Zhang, A Novel Filled Function Method for Global Optimization, J. Korean Math. Soc. 47, 6, (2010). [9] N. Mahdavi-Amiri, R. Yousefpour, An Effective Optimization Algorithm for Locally Nonconvex Lipschitz Functions Based on Mollifier Subgradients, Bull. Iranian Math. Soc., 31 (1), , (2011). [10] A. Sahiner, H. Gokkaya, T. Yigit, A new filled function for non-smooth global optimization, AIP Conf. Proc. 1479, (2012). Nurullah Yilmaz: Suleyman Demirel University, Department of Mathematics, Cunur, Isparta, Turkiye, Ahmet Sahiner: Suleyman Demirel University, Department of Mathematics, Cunur, Isparta, Turkiye, [email protected]

205 International Congress in Honour of Professor Ravi P. Agarwal Determining of the Achievement of Students by Using Classical and Modern Optimization Techniques Ahmet Şahiner and Raziye Akbay The purpose of this study is to investigate effects of sleeping hours and study time to students achievement and find out in which case minimum and maximum achievement level occurs by using global optimization methods. [1] E. H. Mamdani, Application of Fuzzy Logic to Approximate Reasoning Using Linguistic Synthesis, IEEE Trans. Computers, 26(12): , [2] F. Esragh and E.H. Mamdani, A general approach to linguistic approximation,fuzzy Reasoning and Its Applications, Academic Press,1981. [3] Mamdani, E.H., Application of fuzzy algorithms for simple dynamic plant. Proc. IEE 121, [4] Sahiner, A., Gokkaya, H., Ucar, Nazım., Nonlinear Modelling of the Layer Microhardnes of Fe-Mn Binory Allays, Journal of Balkan Tribological Association, (2013) 4, [5] Sahiner, A., Uney İ., Gurbuz M. F., An Application of Fuzzy Logic in Entomology: Estimating the Egg Production and Opening of Pimpla Turianellae L., IWBCMS-2013 Procoeding Book, [6] Saltan M., Saltan S., Sahiner, A., Fuzzy logic modeling of de.ection behavior aganist dynamic loading in. exible pavements, Construction and Building Materials, 21, [7] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst., Man, Cybern, 15: , [8] Zadeh, L., A., Fuzzy Sets, Inf. Control, 8, [9] Zadeh, L., A., 1978.Fuzzy Sets as a Basis for a Theory of Posibility.Fuzzy Sets Syst, 1, [10] Zadeh,L., The consept of a linguistic variable and its application to approximate reasoning I, Information Sciences, 8(3), , (1975) [11] Karagöz S.,Zülfikar H.,Kalaycı T.,2014.öğrenme sürecine ilişkin değerlendirmeler ve fuzzy karar verme tekniği ile sürece dair bir uygulama [12] Ari E., Vatansever F.,2009,Bulanık mantık tabanlı mesleki yönlendirme vocatıonal guidance based on fuzzy logic, Ahmet Şahiner: Suleyman Demirel University, Faculty of Art and Science, Department of Mathematics, Isparta, Turkiye, [email protected] Raziye Akbay: Suleyman Demirel University, Faculty of Art and Science, Department of Mathematics, Isparta, Turkiye, [email protected]

206 206 International Congress in Honour of Professor Ravi P. Agarwal 172 Fuzzy Logic Approach to an UH-1 Helicopter Fuel Consumption and Calculation of Power Problem Ahmet Şahiner and Reyhane Ercan The fuel consumption of UH-1 Helicopter is related with air temperature,altitude,speed and weight. Before the flight, pilots spend many times to calculate estimated fuel consumption. Aim of this work is shorten the time of pilots spend for calculation and find the fuel consumption which can be minimum in which conditions by using fuzzy logic in filled function. [1] Allahviranloo, T., Successive Over Relaxation Iterative Method for Fuzzy System of Linear Equations. ApplMath Comput, 162(1), [2] B. Kosko, Fuzzy thinking: The new science of fuzzy logic, Hyperion, New York, [3] Elsalamony, G., A note on fuzzy neighbourhood base spaces. Fuzzy Sets Syst, 157(20), [4] E. H. Mamdani, Application of Fuzzy Logic to Approximate Reasoning Using Linguistic Synthesis, IEEE Trans. Computers, 26(12): , [5] F. Esragh and E.H. Mamdani, A general approach to linguistic approximation,fuzzy Reasoning and Its Applications, Academic Press,1981. [6] Mamdani, E.H., Application of fuzzy algorithms for simple dynamic plant. Proc. IEE 121, [8] Sahiner, A., Gokkaya, H., Ucar, Nazım., Nonlinear Modelling of the Layer Microhardnes of Fe-Mn Binory Allays, Journal of Balkan Tribological Association, (2013) 4, [9] Sahiner, A., Uney İ., Gurbuz M. F., An Application of Fuzzy Logic in Entomology: Estimating the Egg Production and Opening of Pimpla Turianellae L., IWBCMS-2013 Procoeding Book, [10] Saltan M., Saltan S., Sahiner, A., Fuzzy logic modeling of de.ection behavior aganist dynamic loading in. exible pavements, Construction and Building Materials, 21, [11] UH-1 Genel Maksat Helikopteri Operatör Talimnamesi Tamamlayıcı Ders Notları. Kr.Hvcl. K.lığı Matbaası,2009, 9-11, Ankara. [12] T. Takagi and M. Sugeno, Fuzzy identi.cation of systems and its applications to modeling and control, IEEE Trans. Syst., Man, Cybern, 15: , [13] Zadeh, L., A., Fuzzy Sets.Inf Control, 8, [14] Zadeh, L., A., 1978.Fuzzy Sets as a Basis for a Theory of Posibility.Fuzzy Sets Syst, 1, [15] Zadeh,L., The consept of a linguistic varia,ble and its application to approximate reasoning. I.Information Sciences, 8(3), , (1975) Ahmet Şahiner: Suleyman Demirel University, Faculty Art and Science of Science, Department of Mathematics, Isparta, Turkiye, [email protected] Reyhane Ercan: Suleyman Demirel University, Faculty of Art and Science, Department of Mathematics, Isparta, Turkiye, [email protected]

207 International Congress in Honour of Professor Ravi P. Agarwal Determination of Effects of Brassinosteroid Applications on Secondary Metabolite Accumulation in Salt Stressed Peppermint (Mentha piperita L.) by Modern Optimization Tecniques Ahmet Sahiner, Tuba Yigit, Ozkan Coban and Nilgun Gokturk Baydar Some classical methods remain incapable for the modeling of complex systems. Application of these methods can be costly and time consuming to regulate tha data due to the excess of variable especially, in other disciplines such as medicine, agriculture, biology, econometrics. The fuzzy logic approach is a useful mathematical tool to eliminate these troubles. By using this method the new data is optained for untested conditions under the NaCl stress and comments In this study, the effect of different levels (0, 05, 1,5 and 2,5 mg/l) of 24 epibrassinolidone, an active form of brassinosteroid, on the accumulation of essential oil yield and total phenolic content in peppermint (Mentha piperita L.) plants grown in the media containing 0,100 and 150 mm NaCl was modelled by the fuzzy logic approach. [1] E.E. Aziz, H. Al-Amier, L.E. Craker, Influence of Salt Stress on Growth and Essential Oil Production in Peppermint, Pennyroyal, and Apple Mint. Journal of Herbs, Spices and Medicinal Plants, (2008), 14 (1-2), [2] S. Khorasaninejad, A. Mousavi, H. Soltanloo, K. Hemmati, A. Khalighi, The Effect of Salinity Stress on Growth Parameters, Essential oil Yield and Constituent of Peppermint (Mentha piperita L.). World Applied Sciences Journal, 11 (11), , (2010). [3] S. Queslati, N. Karray-Bouraoui, H. Attia, M. Rabhi, R. Ksouri, M. Lachaal, Physiological and Antioxidant Responses of Mentha pulegium (Pennyroyal) to Salt Stress. Acta Physiologiae Plantarum, (2010), 32(2), [4] V.L. Singleton, J.R. Rossi, Colorimetry of Total Phenolics with Phosphomolybdic-Phosphotungstic Acid, American Journal of Enology and Viticulture, 16, , (1965). [5] S.J. Tabatabaie, J. Nazari, Influence of Nutrient Concentration and NaCl Salinity on Growth, Photosynthesis and Essential Oil Content of Peppermint and Lemon verbena. Turkish Journal of Agriculture, (2007), 31, [6] L.A. Zadeh Fuzzy sets, Inform. Control,8, , (1965). [7] M. Saltan, S. Saltan, A. Sahiner, Fuzzy logic modeling of deflection behavior aganist dynamic loading in flexible pavements, Construction and Building Materials, 21, (2007). [8] E. H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man-Machine Studies 7 (1), 1-13 (1975). [9] Sahiner, A., Gokkaya, H., Ucar, Nazım., Nonlinear Modelling of the Layer Microhardnes of Fe-Mn Binory Allays, Journal of Balkan Tribological Association, 4, , (2013). Ahmet Sahiner: Suleyman Demirel University, Department of Mathematics, Cunur, Isparta, Turkiye, Tuba Yigit: Suleyman Demirel University, Department of Mathematics, Cunur, Isparta, Turkiye, [email protected] Ozkan Coban: Suleyman Demirel University, Department of Agricultural Biotechnology, Cunur, Isparta, Turkiye, [email protected] Nilgun G.Baydar: Suleyman Demirel University, Department of Agricultural Biotechnology, Cunur, Isparta, Turkiye, [email protected]

208 208 International Congress in Honour of Professor Ravi P. Agarwal 174 On a Completeness Property of C(X) Equipped with a Set- Open Topology Smail Kelaiaia Let C(X) be the set of all real-valued continuous functions on a topological space X. We give, in the framework of a particular set-open topology defined on C(X) and by using a topological game, some conditions for C(X) to be weakly α favorable. favorable. This generalize some results obtained by R.A McCoy and I. Ntantu for the compact-open topology. Department of Mathematics University of Annaba, Algeria, [email protected]

209 International Congress in Honour of Professor Ravi P. Agarwal Existence of Solutions of a Class of Second Order Differential Inclusions D.Azzam-Laouir and F.Aliouane In the present paper we prove, in a separable Banach space, the existence of solutions for the second order sweeping process of the form x.. N K(t) (ẋ(t)) + F (t, x(t), ẋ(t)), a.e.t [0, T ], where F is an upper semicontinuous set-valued mapping with nonempty closed convex values, K a nonempty ball compact and r prox-regular E set-valued mapping and N K(t) (.) the proximal normal cone of K(t). [1] F. Bernicot, J. Venel, Existence of sweeping process in Banach spaces under directional prox-regularity. J. Convex Anal. 17 (2010), [2] M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space. J. Nonlinear Convex Anal. 6 (2001), [3] L. Thibault, Sweeping process with regular and nonregular sets. J. Diff. Equations 193 no. 1 (2003), Laboratory of Pure and Applied Mathematics, University of Jijel, Algeria, [email protected]

210 210 International Congress in Honour of Professor Ravi P. Agarwal 176 Applications of Generalized Fibonacci Autocorrelation Sequences {Γ k,n (τ)} τ Sibel Koparal and Neşe Ömür In this study, we give the elements of the generalized Fibonacci Autocorrelation sequences { Γ k,n (τ) } τ Γ k,n (τ) def = Γ n (U ki, τ). defined as and some interesting sums involving the numbers Γ k,n (τ), where odd integer number k and nonnegative integers τ, n. For example, we show that n [ ] Uk(n+1) + U kn U 2 k Γ k,n (τ) =, V τ=0 k ( ) n 2 Uk(n+1) U kn + U k V k ( 1) τ Γ k,n (τ) =, if n is odd V τ=0 k U k(n+1) U kn, if n is even. [1] P. Filipponi and H. T. Freitag, Autocorrelation Sequences, The Fibonacci Quarterly, Vol.32, No.4, pp , [2] T. Koshy, Fibonacci and Lucas Numbers with Applications. Pure and Applied Mathematics, Wiley-Interscience, New York, [3] S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section. John Wiley & Sons, Inc., New York, [4] E.Kılıç and P. Stanica, Factorizations and Representations of Second Order Linear Recurrences with Indices in Arithmetic Progressions, Bol. Soc. Mat. Mex. III. Ser., Vol.15, No.1, pp.23-35, Sibel Koparal: Kocaeli University, Faculty of Arts and Sciences, Department of Mathematics, İzmit, Kocaeli-Turkiye, [email protected] Neşe Ömür: Kocaeli Üniversitesi, Faculty of Arts and Sciences, Department of Mathematics, İzmit, Kocaeli-Turkiye, [email protected]

211 International Congress in Honour of Professor Ravi P. Agarwal The Computer Simulation of Nuclear Magnetic Resonance Hyperfine Structure Constant for AB 2, A 2 B 2 and A 2 B 3 Systems Containing Some Organic Molecules with Spin 1 2 Using Jacobi Programme Hüseyin Ovalıoğlu, Adnan Kılıç and Handan Engin Kırımlı The energy matrices of molecules of AB 2, A 2 B 2 and A 2 B 3 type have been calculated for three different chemical shifts and several indirect spin-spin coupling coefficients (Jij) to obtain Nuclear Magnetic Resonance (NMR) hyperfine structure of such systems. The JACOBI programme were used to calculate eigenvalues and eigenvectors of these systems. We have developed a programme to calculate the transition probabilities and the transition energies. It is observed that the theoretically calculated spectra is in agreement with the experimental spectra. Keyword: Nuclear Magnetic Resonance (NMR) [1] A. Abragam, The Principles of Nuclear Magnetism, Oxford, (1973). [2] J. W. Akitt, NMR and Chemistry, An Introduction to Modern NMR Spectroscopy, Chapman & Hall, London, (1992). [3] F. Apaydın, Magnetik Rezonans, Temel İlkeler, Deney Düzenekleri, Ölçüm Yöntemleri, H.Ü. Müh. Fak. Ders Kit. No:3, Ankara, (1991), [4] P. L. Corio, Structure of High-Resolution NMR Spectra., Academic Press New York, (1966), p202. Hüseyin Ovalıoğlu: Uludağ University, Faculty of Science, Department of Physics, Görükle, Bursa-Türkiye, [email protected] Adnan Kılıç: Uludağ University, Faculty of Science, Department of Physics, Görükle, Bursa-Türkiye, [email protected] Handan Engin Kırımlı: Uludağ University, Faculty of Science, DEpartment of Physics, Görükle, Bursa-Türkiye, This work was supported by the Commission of Scientific Research Projects of Uludag University, Project number OUAP(F)-2012/30.

212 212 International Congress in Honour of Professor Ravi P. Agarwal 178 The Computer Simulation of Nuclear Magnetic Resonance Hyperfine Structure Constant for ANX, ABC and A 3 BC Systems Containing Some Organic Molecules with Spin 1 2 Using Jacobi Programme Hüseyin Ovalıoğlu, Handan E.Kırımlı, Cengiz Akay and Adnan Kılıç The energy matrices of molecules of ANX, ABC and A3BC type have been calculated values of four different chemical shifts and several indirect spin-spin coupling coefficients (Jij) to obtain Nuclear Magnetic Resonance (NMR) hyperfine structure of such systems. The JACOBI programme were used to calculate eigenvalues and eigenvectors of these systems. We have developed a programme to calculate the transition probabilities and the transition energies. Also, it has been observed that the theoretically calculated spectra is in agreement with the experimental spectra for molecule of ANX. Keyword: Nuclear Magnetic Resonance (NMR) [1] A. Abragam, The Principles of Nuclear Magnetism, Oxford, (1973). [2] J. W. Akitt, NMR and Chemistry, An Introduction to Modern NMR Spectroscopy, Chapman & Hall, London, (1992). [3] F. Apaydın, Magnetik Rezonans, Temel İlkeler, Deney Düzenekleri, Ölçüm Yöntemleri, H.Ü. Müh. Fak. Ders Kit. No:3, Ankara, (1991), [4] D. W. Mathieson, NMR For Organic Chemists, Academic Press, London, (1967), [5] P. L. Corio, Structure of High-Resolution NMR Spectra, Academic Press, New York, (1966). Hüseyin Ovalıoğlu: Uludağ University, Faculty of Science, DEpartment of Physics, Görükle, Bursa-Türkiye, [email protected] Handan Engin Kırımlı: Uludağ University, Faculty of Science, Department of Physics, Görükle, Bursa-Türkiye, [email protected] Cengiz Akay: Uludağ University, Faculty of Science, Department of Physics, Görükle, Bursa-Türkiye, [email protected] Adnan Kılıç: Uludağ University, Faculty of Science, Department of Physics, Görükle, Bursa-Türkiye, [email protected] This work was supported by the Commission of Scientific Research Projects of Uludag University, Project number OUAP(F)-2012/13.

213 International Congress in Honour of Professor Ravi P. Agarwal Necessary and Sufficient Conditions for First Order Differential Operators to be Associated with a Disturbed Dirac Operator in Quaternionic Analysis Uğur Yüksel Recently the initial value problem tu = Lu := i=1 u (0, x) = u 0 (x) 3 A (i) (t, x) xi u + B(t, x)u + C(t, x) has been solved uniquely by N. Q. Hung [1] using the method of associated spaces constructed by W. Tutschke [2] in the space of generalized regular functions in the sense of quaternionic analysis satisfying the equation D αu := Du + αu = 0, where D = 3 e j xj is the Dirac operator, and t is the time variable. Only sufficient conditions has been obtained in [1] j=1 for the operators L and D α to be associated. In the present talk we will prove necessary and sufficient conditions for the underlined operators to be associated. This criterion makes it possible to construct all linear operators L for which the initial value problem with an arbitrary initial generalized regular function is always solvable. Further we will correct a mistake made in the calculation of the interior estimate in [1]. [1] N. Q. Hung, Initial Value Problems in Quaternionic Analysis with a Disturbed Dirac Operator, Adv. appl. Clifford alg., Vol. 22, Issue 4 (2012), pp [2] W. Tutschke, Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer Verlag, α R Atilim University, Ankara, Turkey, [email protected]

214 214 International Congress in Honour of Professor Ravi P. Agarwal 180 Theoretical Investigation of Substituent Effect on the Carbonyl Stretching Vibration Ilhan Küçük and Aslı Ayten Kaya Gaussian 03 is the electronic structure program, is used by chemists, chemical engineers, biochemists, physicists and others for research in established and emerging areas of chemical interest. Starting from the basic laws of quantum mechanics, Gaussian predicts the energies, molecular structures, and vibrational frequencies of molecular systems, along with numerous molecular properties derived from these basic computation types. In this study, the molecular geometry and vibrational frequencies of substitute isonitrosoacetophenone (inaph) molecules in the ground state have been calculated using density functional method (B3LYP) with the G(d, p) basis set. The values calculated by the Gaussian 03 program were used to the artificial neural network. The developed neural network, which has four input neurons, one output neuron, four hidden layers, five, six, seven and eight neurons of hidden layers and full connectivity between neurons. The input parameters were electronegativity, dipole moment, C and O Mulliken charges. A total of 240 input vectors obtained from varied samples were available in the training data. The results show that the ANN model has a 99% correlation with Gaussian program data. All statistical values prove that the proposed ANN model is suitable to predict the vibration frequency values very close to the results of the calculated values. İlhan Küçük: Uludag University, Faculty of Science, Department of Chemistry, Görükle, Bursa-Turkiye, [email protected] Aslı Ayten Kaya: Uludag University, Faculty of Science, Department of Physics, Görükle, Bursa-Turkiye, [email protected]

215 International Congress in Honour of Professor Ravi P. Agarwal Modeling of the Optical Properties of the CdS Thin Films by Using Artificial Neural Network Aslı Ayten Kaya, Kadir Ertürk, Nil Küçük and Ilker Küçük In this study, CdS thin films were produced by electro-deposition method. Optical properties of thin films were investigated UV-Vis. Spectrophotometer. A new model was developed with experimental data using by Artificial Neural Network (ANN). This model has three hidden layers with twenty-one neurons and full connectivity between them. The input parameters were sample number (n), absorbance (A), thickness (d) and absorbance constant (α). A total of 2400 input vectors were available in the training and testing data. The number of hidden layers and neurons in each layer were determined through trial and error to be optimal including different transfer functions such as hyperbolic tangent, sigmoid and hybrid. After the network was trained, a better result was obtained from the network formed by the hyperbolic tangent transfer function in the hidden and output layers. The number of epochs was 10 6 for training. Aslı Ayten Kaya: Uludag University, Faculty of Science, Department of Physics, Görükle, Bursa-Turkiye, Kadir Ertürk: Namik Kemal University, Faculty of Science, Department of Physics, Merkez, Tekirdag-Turkiye, [email protected] Nil Küçük: Uludag University, Faculty of Science, Department of Physics, Görükle, Bursa-Turkiye,[email protected] İlker Küçük: Uludag University, Faculty of Science, Department of Physics, Görükle, Bursa-Turkiye, [email protected] This work was supported by the Commission of Scientific Research Projects of Uludag University, Project number OUAP(F)-2013/14.

216 216 International Congress in Honour of Professor Ravi P. Agarwal 182 Nonprinciple Solutions and Extensions of Wong s Oscillation Criteria to Forced Second-Order Impulsive and Delay Differential Equations Abdullah Özbekler and Ağacık Zafer Wong s well-known oscillation theorem states that if z is a positive nonprincipal solution of (r(t)x ) + q(t)x = 0, t a satisfying where then every solution of lim H(t) = lim H(t) =, t t t ( 1 s ) H(t) := a r(s)z 2 z(σ)f(σ)dσ ds, (s) a (r(t)x ) + q(t)x = f(t) is oscillatory. In this talk, we give some extentions of above result to impulsive and delay differential equations. It is shown that the oscillation behavior may be altered due to presence of the delay and impulse action. Extensions to Emden-Fowler type impulsive and delay equations are also provided. [1] M. Morse and W. Leighton, Singular quadratic functionals, Trans. Amer. Math. Soc. 40, , (1936). [2] James S.W. Wong, Oscillation criteria for forced second-order linear differential equation, J. Math. Anal. Appl. 231, (1999). [3] A. Özbekler, James S.W. Wong and A. Zafer, Forced oscillation of second-order nonlinear differential equations with positive and negative coefficients, Appl. Math. Lett. 24, (2011). Atilim University, Ankara, Turkey, [email protected]

217 International Congress in Honour of Professor Ravi P. Agarwal Modeling of Exposure Buildup Factors for Concrete Shielding Materials up to 10 mfp Using Generalized Feed-Forward Neural Network Nil Kucuk, Vishwanath P.Singh and N.M.Badiger In this work, generalized feed-forward neural network (GF F NN) was presented for the computation of the gamma-ray exposure buildup factors (B D ) of the seven concrete shielding materials [ordinary (OR), hematite-serpentine (HS), ilmenitelimonite (IL), basalt-magnetite (BM), ilmenite ((IT), steel-scrap (SS), steel-magnetite (SM)] in the energy region MeV, and for penetration depths up to 10 mean free path (mfp). The GF F NN has been trained by a Levenberg- Marquardt learning algorithm. The developed model is in 99% agreement with the ANSI/ANS standard data set. Furthermore, the model is fast and does not require tremendous computational efforts. The estimated B D data for concrete shielding materials have been given with penetration depth and incident photon energy as comparative to the results of the interpolation method using the Geometrical Progression (G-P) fitting formula. Nil Kucuk: Uludag University, Faculty of Art and Sciences, Department of Physics, Gorukle Campus, Bursa, Turkey, [email protected] Vishwanath P.Singh: Karnatak University, Department of Physics, Dharwad, , India and Health Physics Section, Kaiga Atomic Power Station-3&4, NPCIL, Karwar , India, [email protected] N.M.Badiger: Karnatak University, Department of Physics, Dharwad, , India, [email protected]

218 218 International Congress in Honour of Professor Ravi P. Agarwal 184 Calculation of Gamma-Ray Exposure Buildup Factors for Some Biological Samples Nil Kucuk, Vishwanath P.Singh and N.M.Badiger Gamma-ray exposure buildup factors (EBF) have been calculated for some biological samples (viz. lungs, pancreas, and ovaries) in the energy region MeV, up to penetration depths of 40 mean free paths (mfp). The five-parameter geometric progression (G-P) fitting approximation and ANSI/ANS (American National Standard) library have been used to calculate EBF. The EBF have been studied as functions of incident photon energy and penetration depth. The variations in the EBF, for all the biological samples, in different energy regions, have been presented in the form of graphs. Buildup factors of these biological samples cannot be found in any standard database, so these studies will help in estimating safe dose levels for radiotherapy patients. Nil Kucuk: Uludag University, Faculty of Art and Sciences, Department of Physics, Gorukle Campus, Bursa, Turkey, Vishwanath P.Singh: Karnatak University, Department of Physics, Dharwad, , India and Health Physics Section, Kaiga Atomic Power Station-3&4, NPCIL, Karwar , India, N.M.Badiger: Karnatak University, Department of Physics, Dharwad, , India,

219 International Congress in Honour of Professor Ravi P. Agarwal Determination of Thermoluminescence Kinetic Parameters of ZnB 2 O 4 : La Phosphors Nil Kucuk, A.Halit Gozel, Mustafa Topaksu and Mehmet Yüksel Thermoluminescence (TL) glow curves of 1%, 2%, 3% and 4% ZnB 2 O 4 : La phosphors synthesized by nitric acid method were obtained by irradiation at the dose range of 143 mgy - 60 Gy with 90 Sr/ 90 Y beta source, which has 40 mci activity, included in the Risø TL/OSL DA-20 reader system. TL glow curves were recorded after pre-heating process at 140 C and then heating up to 450 C in nitrogen atmosphere at a constant heating rate of 5 C/s. In this study, with the help of glow curve readings, kinetic parameters of the main TL glow peaks of ZnB 2 O 4 : La phosphors (i.e. activation energies and frequency factors) were determined and evaluated by the method of Computerized Glow Curve Deconvolution (CGCD), Peak Shape (P S) method and Initial Rise (IR) method. In conclusion, kinetic parameters found in this study by the methods applied to ZnB 2 O 4 : La phosphors were consistent with each other. Nil Kucuk: Uludag University, Faculty of Art and Sciences, Department of Physics, Gorukle Campus, Bursa, Turkey, [email protected] Aziz Halit Gozel: Adiyaman University, Faculty of Art and Sciences, Department of Physics, Adiyaman, Turkey, [email protected] Mustafa Topaksu: Cukurova University, Faculty of Art and Sciences, Department of Physics, Adana, Turkey, [email protected] Mehmet Yüksel: Cukurova University, Faculty of Art and Sciences, Department of Physics, Adana, Turkey, [email protected], [email protected]

220 220 International Congress in Honour of Professor Ravi P. Agarwal 186 Improved Numerical Radius and Spectral Radius Inequalities for Operators Fuad Kittaneh and Amer Abu-Omar We establish an improvement of the triangle inequality for the numerical radius and give necessary and sufficient conditions for the equality case. New numerical radius and spectral radius inequalities are also given. Our results include an improvement of a well-known spectral radius inequality concerning the subadditivity property for commuting operators. Department of Mathematics, The University of Jordan, Amman, Jordan and Al-Ahliyya Amman University, Deanship of Graduate Studies and Scientific Research, Amman, Jordan, [email protected]

221 International Congress in Honour of Professor Ravi P. Agarwal n-dimensional Sobolev type spaces involving Chebli-Trimeche Transform Mourad Jelassi Using Chébli Trimèche transform, we define and study n-dimensional Sobolev type spaces. In particular, we give some properties including completeness and boundedness of convolution product in these spaces. Next, a Titchmarch type theorem for the Chébli Trimèche transform is investigate. Carthage University, ISSAT Mateur, Department of Mathematics, 7030 Mateur-Bizerte, Tunisia, [email protected]

222 222 International Congress in Honour of Professor Ravi P. Agarwal 188 A Fixed Point Theorem for Multivalued Mappings with δ-distance on Complete Metric Space Özlem Acar and Ishak Altun In this talk, we mainly study on fixed point theorem for multivalued mappings with δ-distance using Wardowski s technique on complete metric space. Let (X, d) be a metric space and B(X) be family of all nonempty bounded subsets of X. Define δ : B(X) B(X) R by δ(a, B) = sup {d(a, b) : a A, b B}. Considering δ-distance, it is proved that if (X, d) be a complete metric space and T : X B(X) be a multivalued certain contraction, then T has a fixed point. [1] Fisher, B., Common fixed points of mappings and set-valued mappings. Rostock. Math. Kolloq. No. 18 (1981), [2] Fisher, B., Fixed points for set-valued mappings on metric spaces. Bull. Malaysian Math. Soc. (2) 4 (1981), no. 2, [3] Fisher, B., Set-valued mappings on metric spaces. Fund. Math. 112 (1981), no. 2, [4] Fisher, B., Common fixed points of set-valued mappings. Punjab Univ. J. Math. (Lahore) 14/15 (1981/82), [5] Hicks, T. L., Fixed point theorems for multivalued mappings. Indian J. Pure Appl. Math. 20 (1989), no. 11, [6] Hicks, T. L., Set-valued mappings on metric spaces. Indian J. Pure Appl. Math. 22 (1991), no. 4, [7] Altun, I., Fixed point theorems for generalized ϕ-weak contractive multivalued maps on metric and ordered metric spaces. Arab. J. Sci. Eng. 36 (2011), no. 8, [8] Mınak, G.; Acar, Ö.; Altun, I., Multivalued pseudo-picard operators and fixed point results. J. Funct. Spaces Appl. 2013, Art. ID , 7 pp. Özlem Acar: Kirikkale University, Faculty of Science and Arts, Department of Mathematics, Yahsihan, 71450, Kirikkale- Turkey, [email protected] İshak Altun: Kirikkale University, Faculty of Science and Arts, Department of Mathematics, Yahsihan, 71450, Kirikkale- Turkey, [email protected]

223 International Congress in Honour of Professor Ravi P. Agarwal Existence of Solutions of α (2, 3] Order Fractional Three Point Boundary Value Problems with Integral Conditions Sinem Unul and N.I.Mahmudov In this talk, existence of solutions for α (2, 3] order fractional differential equations with three point fractional boundary and integral conditions will be discussed: ( ) D α 0 + u(t) = f t, u(t), D β 1 0 +u(t), Dβ 2 0 +u(t) ; 0 t T ; 2 < α 3 with the two point and integral boundary conditions T a 0 u(0) + b 0 u(t ) = λ 0 g 0 (s, u (s))ds, 0 T a 1 D β 1 0 +u(η) + b 1D β 1 0 +u(t ) = λ 1 g 1 (s, u (s))ds, 0 < β 1 1, 0 < η < T, 0 T a 2 D β 2 0 +u(η) + b 2D β 2 0 +u(t ) = λ 2 g 2 (s, u (s))ds, 1 < β 2 2, 0 where D α 0 + denotes the Caputo fractional derivative of order α. [1] Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, (2010) [2] Bai, ZB: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, (2010) [3] Ahmad, B, Nieto, JJ, Alsaedi, A: Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions. Acta Math. Sci. 31, (2011) [4] Ahmad, B, Ntouyas, SK: Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions. Electron. J. Differ. Equ. 2012, Article ID 98 (2012) [5] Bai, ZB, Sun, W: Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 63, (2012) [6] Ahmad, B, Ntouyas, SK: A boundary value problem of fractional differential equations with anti-periodic type integral boundary conditions. J. Comput. Anal. Appl. 15, (2013) 23. [7] Ahmad B., Ntouyas S. K and Alsaedi A., On fractional differential inclusions with anti-periodic type integral boundary conditions, Boundary Value Problems 2013, 2013:82. [8] M. Aitaliobrahim, Neumann boundary-value problems for differential inclusions in banach spaces, Electronic Journal of Differential Equations, vol. 2010, no. 104, pp. 1 5, Sinem Unul: Eastern Mediterranean University, Department of Mathematics, Gazimagusa, TRNC, Mersin 10, Turkey, [email protected] N.I.Mahmudov: Eastern Mediterranean University, Department of Mathematics, Gazimagusa, TRNC, Mersin 10, Turkey, nazim. [email protected]

224 224 International Congress in Honour of Professor Ravi P. Agarwal 190 Vector-Valued Variable Exponent Amalgam Spaces Ismail Aydın In this talk, we define the vector-valued (Banach space valued) variable exponent amalgam spaces and discuss the basic properties, the dual space, the reflexivity and some embedding properties of these spaces. [1] İ. Aydın and A.T. Gürkanlı, Weighted Variable Exponent Amalgam spaces W (L p(x), L q w), Glasnik Matematicki, Vol. 47(67), (2012), [2] İ. Aydın, On Variable Exponent Amalgam Spaces, Analele Stiintifice Ale Universitatii Ovidius Constanta-Seria Matematica, Vol. 20(3),(2012), [3] A. T. Gürkanlı and İ. Aydın, On The Weighted Variable Exponent Amalgam Space W (Lp(x), L q w), Acta Mathematica Scientia, 34B(4),(2014), [4] V. Kokilashvili, A. Meskhi and M. A. Zaighum, Weighted Kernel Operators in Variable Exponent Amalgam Spaces, Journal of Inequalities and Applications, (2013), 2013:173. [5] A. T. Gürkanlı, The Amalgam Space W (L p(x), L {pn} ) and Boundedness of Hardy-Littlewood Maximal Operators, ISAAC 2013 Proceeding (Accepted for publication). [6] A. Meskhi and M. A. Zaighum, On The Boundedness Of Maximal And Potential Operators In Variable Exponent Amalgam Spaces, Journal of Mathematical Inequalities, Volume 8(1), (2014), [7] C. Cheng and J. Xu, Geometric Properties of banach Space Valued Bochner-Lebesgue Spaces With Variable Exponent, Journal of Mathematical Inequalities, Vol. 7(3), (2013), [8] D. V. Lakshmi and S. K. Ray, Vector-valued Amalgam Spaces, International Journal of Computational Cognition ( VOL. 7(4), (2009), [9] D. V. Lakshmi and S. K. Ray, Convolution Product on Vector-valued Amalgam Spaces, International Journal of Computational Cognition ( VOL. 8(3), (2010), Sinop University, Faculty of Science and Letters, Department of Mathematics, Sinop-Türkiye, [email protected]

225 International Congress in Honour of Professor Ravi P. Agarwal Soliton Solutions of Sawada Kotera Equation by Hirota Method Esra Karataş and Mustafa Inc In this work the Sawada Kotera equation is studied. The Hirota Bilinear Method is used to determine multiple-soliton solutions for this equation. By means of this method, three soliton solutions for fifth order nonlinear partial differential equation is formally obtained. Esra Karataş: Çanakkale Onsekiz Mart University, Gelibolu Piri Reis Vocational School, Department of, Mathematics, Çanakkale-Turkiye, Mustafa Inc: Firat University, Science Education, Department of Mathematics, Elazg-Turkiye, This work was supported by Çanakkale Onsekiz Mart University

226 226 International Congress in Honour of Professor Ravi P. Agarwal 192 Certain Quasi-Cyclic Codes which are Hadamard Codes Mustafa Özkan and Figen Öke A n n matrix such that all components are 1 or 1 and M.M t = n.i is called Hadamard matrix. A code obtained by using a Hadamard matrix is called Hadamard code. In this study it is shown that Hadamard codes which have codewords in the ring F 2 + vf 2 can be obtained by some special matrices lexicographically ordered. This relation is obtained by using two different Gray maps from (F 2 + vf 2 ) n to F 2n 2. [1] Krotov, D. S. Z4-linear perfect codes,diskretn. Anal. Issled. Oper. Ser.1.Vol.7, 4,2000 P [2] Krotov,D. S. Z4-linear Hadamard and extended perfect codes, Procs. of the International Workshop on Coding and Cryptography,Paris, 2001, pp [3] Jian-Fa,Q., Zhang L.N. and Zhu S.X.,(1 + u)- cyclic and cyclic codes over the ring F 2 + uf 2,Applied Mathematics Letters,19,2006, [4] Jian-Fa Qian, Li-Na Zhang and Shi-Xin Zhu, Constacyclic and cyclic codes over F 2 + uf 2 + u 2 F 2, IEICE Trans. Fundamentals, E89-A, No 6, 2006, [5] Vermani, L. R., Elements of Algebraic Coding Theory, Chapman Hall, India., Mustafa Özkan: Trakya University, Faculty of Science, Department of Mathematics, Edirne-Turkiye, mustafaozkan [email protected] Figen Öke: Trakya University, Faculty of Science, Department of Mathematics, Edirne-Turkiye, [email protected]

227 International Congress in Honour of Professor Ravi P. Agarwal Pointwise Convergence of Derivatives of New Baskakov- Durrmeyer-Kantorovich Type Operators Gulsum Ulusoy, Ali Aral and Emre Deniz Recently in [1], we have constructed a new sequence of integral type operators which contain characteristic properties of Baskakov Durrmeyer and Baskakov Kantorovich opearators. In this talk, we continue focus on pointwise convergence of derivatives of these operators by the means of Voronoskaya type asymptotic formula. Moreover, to describe the rate of convergence and an estimate of error in terms of modulus of continuity in simultaneous approximation (approximation of derivatives of functions by the corresponding order derivatives of operators) by this new durrmeyer operators, we present Voronovskaya type asymtotic formula in quantitative form. [1] Deniz, E., Aral, A. and Ulusoy, G., New Integral Type Operators, (submitted) [2] Deo, N., Direct result on exponential-type operators, Appl. Math. Comput., 204 (2008), 109ñ115. [3] Gadjiev, A.D., Ibragimov, I.I., On a sequence of linear positive operators, Soviet Math. Dokl., 11 (1970), [4] Kasana, H.S., Agrawal, P.N., Gupta, Vijay: Inverse and saturation theorems for linear combination of modifed Baskakov operators, Approx. Theory Appl., 7(2) (1991), [5] Stan, I. G., On the Durrmeyer-Kantorovich type operator, Bulletin of the Transilvania University of Brasov, Vol 6 (55), No. 2, Gülsum Ulusoy: Kirikkale University, Faculty of Science and Arts, Department of Mathematics, Yahsihan, 71450, Kirikkale-Turkey, [email protected] Ali Aral: Kirikkale University, Faculty of Science and Arts, Department of Mathematics, Yahsihan, 71450, Kirikkale- Turkey, [email protected] Emre Deniz: Kirikkale University, Faculty of Science and Arts, Department of Mathematics, Yahsihan, 71450, Kirikkale- Turkey, emredenizñ@hotmail.com

228 228 International Congress in Honour of Professor Ravi P. Agarwal 194 On the High Order Lipschitz Stability of Inverse Nodal Problem for String Equation Emrah Yılmaz and Hikmet Koyunbakan Inverse nodal problem on the string operator is the finding the density function using nodal sequence {z (n) k }. In this paper, we solve a stability problem using nodal set of eigenfunctions and show that the space of high order density functions is homeomorphic to the partition set of the space of quasinodal sequences. Basically, this method is similar to [1] and [2] which is given for Sturm-Liouville and Hill operators, respectively. [1] C. K. Law and J. Tsay, On the well-posedness of the inverse nodal problem, Inverse Problems, 2001, 17: [2] Y. H. Cheng and C. K. Law, The inverse nodal problem for Hill s equation, Inverse Problems, 2006, 22: [3] V. A. Ambartsumyan, Über eine frage der eigenwerttheorie, Zeitschrift für Physik, 1929, 53: [4] B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: Self adjoint ordinary differential operators, American Mathematical Society, Providence, Rhode Island, [5] J. R. McLaughlin, Analytic methods for recovering coefficients in differential equations from spectral data, SIAM, 1986, 28: [6] J. Pöschel and E. Trubowitz, Inverse spectral theory, volume 130 of Pure and Applied Mathematics, Academic Press, Inc, Boston, MA, [7] C. F. Yang and A. Zettl, Half inverse problems for quadratic pencils of Sturm-Liouville operators, Taiwanese Journal of Mathematics, 2012, 16(5): Emrah Yılmaz: Department of Mathematics, Firat University, Elazığ-Turkiye, [email protected] Hikmet Koyunbakan: Department of Mathematics, Firat University, Elazığ-Turkiye, [email protected]

229 International Congress in Honour of Professor Ravi P. Agarwal Positive Solutions of a Boundary Value Problem with Derivatives in the Nonlinear Term Patricia J.Y.Wong We consider the Sturm-Liouville boundary value problem y (m) (t) + F ( t, y(t), y (t),, y (q) (t) ) = 0, t [0, 1] y (k) (0) = 0, 0 k m 3 ζy (m 2) (0) θy (m 1) (0) = 0, ρy (m 2) (1) + δy (m 1) (1) = 0 where m 3, 1 q m 2, λ > 0 and F is continuous at least in the domain of interest. It is noted that boundary value problems with derivative-dependent nonlinear terms are seldom investigated in the literature due to technical difficulty. In this talk, we employ a new technique to establish existence of positive solutions of the boundary value problem. Nanyang Technological University, School of Electrical and Electronic Engineering, 50 Nanyang Avenue, Singapore , Singapore, [email protected]

230 230 International Congress in Honour of Professor Ravi P. Agarwal 196 One Step Iteration Scheme for Two Multivalued Mappings in CAT(0) Spaces Izhar Uddin and M.Imdad In this paper, we study the one step iteration scheme for two multivalued nonexpansive mappigs in CAT(0) spaces and prove convergence as well as strong convergence theorems. Thus, our results generalize and extend many relevant results in Abbas et al. (Appl. Math. Lett. 24 (2011), no. 2, ), Khan (Bull. Belg. Math. Soc. Simon Stevin 17 (2010) ), Khan (Nonlinear Anal. 8 (2005) ) and Fukhar-ud-din (J. Math. Anal. Appl. 328 (2007) ) and references cite therein. Department of Mathematics, Aligarh Muslim University, Aligarh , Uttar Pradesh, India, izharuddin [email protected], [email protected]

231 International Congress in Honour of Professor Ravi P. Agarwal A Variant Akaike Information Criterion for Mixture Autoregressive Model Selection Fayçal Hamdi In this talk, we consider the problem of order selection of Mixture Autoregressive (MAR) models. These models are among the most powerful tools for modelling some stylized features exhibited by many time series such as multimodality, tail heaviness, change in regime and asymmetry. We aim to present a variant of the Akaike information criterion (AIC), for MAR model selection, based on complete-data rather than incomplete-data and which different from the standard criteria. We compare the performance of our proposed criterion to that of the traditional AIC criterion and certain other competitors in a simulation study. [1] Akaike, H. (1973). Infonuation theory and an extension of the maximum likelihood principle. In: Petrov, B.N., Csaki, F. (Eds.), Proe. 2nd Internat. Syrup. on Inform. Theory. Akademia Kiado, Budapest, [2] Cavanaugh, J. E. and Shumway, R. H. (1998). An Akaike information criterion for model selection in the presence of incomplete data. J. Stat. Plan. Infer., 67, [3] Dempster, A.P., Laird, N.M., Rubin and D.B., (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J.Roy. Statist. Soc. B 39, [4] Wong, C. and Li,W., (2000). On a mixture autoregressive model. J. Roy. Statist. Soc. Ser. B, 62, RECITS Laboratory, Faculty of Mathematics, University of Science and Technology Houari Boumediene (USTHB), Po Box 32. El Alia, 16111, Bab Ezzouar, Algiers, Algeria, [email protected] or hamdi [email protected]

232 232 International Congress in Honour of Professor Ravi P. Agarwal 198 Zagreb Polynomials of Three graph Operators A.R.Bindusree, V.Lokesha, I.Naci Cangul and P.S.Ranjini A topological index is a graph invariant applicable in chemistry.the first and second Zagreb indices are amongst the oldest and best known topological indices defined in 1972 by Gutman and are given different names in the literature, such as the Zagreb group indices, the Zagreb group parameters and most often, the Zagreb indices. Zagreb indices were among the first indices introduced,and has been used to study molecular complexity, chirality,ze-isomerism and hetero-systems. Zagreb indices exhibited a potential applicability for deriving multi-linear regression models. Let G be a connected graph with n vertices and m edges.the vertex set and edge set are denoted by V (G) and E(G) respectively.for every vertex v i ɛ V (G),where i = 1, 2,...n, the edge connecting v i and v j is denoted by (v i, v j ) and d(v i ) denotes the degree of vertex v i in G. The first and the second zagreb indices are defined as follows. M 1 (G) = [d(v i ) 2 ] M 2 (G) = v i ɛv (G) (v i,v j )ɛe(g) [d(v i ).d(v j )]. First and Second zagreb polynomials are derived from First and Second Zagreb indices respectively. They are defined as follows. M 1 (G, x) = M 2 (G, x) = v i,v j ɛe(g) v i,v j ɛe(g) x d(v i)+d(v j ) x d(v i).d(v j ) where x is a dummy variable. Moreover, the First and Second zagreb indices can be obtained from its polynomial.because for i = 1, 2 M i (G) = M i(g, x) x The third Zagreb index, M 3 (G) and third zagreb polynomial M 3 (G, x)are deined respectively as, M 3 (G) = [ d(v i ) d(v j ) ]. M 3 (G, x) = (v i,v j )ɛe(g) (v i,v j )ɛe(g) x d(v i) d(v j ). In this paper,the relation between Zagreb polynomials on three graph operators is discussed.we investigates the relation between Zagreb polynomial of a graph G and a graph obtained by applying the operators S(G),R(G) and Q(G).Moreover, relation between Zagreb polynomial of a graph G and its corona is also described. [1] Asadpour,Jafar. Some Topological Indices of nanostructures, Optoelectronics And Advanced Materials- Rapid Communications, 5(2011): [2] Astaneh-Asl, A., and Gholam Hossein Fath-Tabar. Computing the first and third Zagreb polynomials of Cartesian product of graphs. Iranian J. Math. Chem 2.2 (2011): [3] Farahani,Mohammed Reza. First and second Zagreb Polynomials of V C 5 C 7 [p, q] and HC 5 C 7 [p, q] nanotubes, Int.Letters of Chemistry,Physics and Astronomy,12(2014): [4] Farahani,Mohammed Reza. Zagreb index, Zagreb Polynomial of Circumcoronene Series of Benezoid. Advances in materials and Corrosion 2.1(2013): [5] Farahani,Mohammed Reza. Zagreb indices and Zagreb Polynomials of Polycyclic Aromatic Hydrocarbons PAHs.Journal of chemica Acta 2.2(2013): [6] Fath-Tabar, Gholam hossein. Zagreb Polynomial And Pi Indices Of Some Nano Structures. Digest Journal of Nanomaterials and Biostructures (DJNB) 4.1 (2009). A.R.Bindusree: Sree Narayana Gurukulam College of Engineering V.Lokesha: Department of Mathematics, Vijayanagara Sri Krishnadevaraya University, Bellary, India, İsmail Naci Cangül: Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkiye, [email protected] Ranjini P.S: Department of Mathematics, Don Bosco Institute Of Technology,Bangalore-74, India, ranjini p s@yahoo. com

233 International Congress in Honour of Professor Ravi P. Agarwal A Note on the Moment Estimate for Stochastic Functional Differential Equations Young-Ho Kim In this talk, we are consider a stochastic functional differential equation with initial value under non-lipschitz condition and a weakened linear growth condition. By applying the Itô formula, a class of moment estimates of the solution of stochastic differential equations is studied. [1] Y. El. Boukfaoui, M. Erraoui, Remarks on the existence and approximation for semilinear stochastic differential in Hilbert spaces, Stochastic Anal. Appl. 20 (2002), [2] Y.J. Cho, S.S. Dragomir, Y-H. Kim: A note on the existence and uniqueness of the solutions to SFDEs. J. Inequal. Appl. 2012:126 (2012), pp [3] T.E. Govindan, Stability of mild solution of stochastic evolution equations with variable delay, Stochastic Anal. Appl. 21 (2003), [4] D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific Publishing Co [5] K. Liu, Lyapunov functionals and asymptotic of stochastic delay evolution equations, Stochastics and Stochastic Rep. 63 (1998) [6] X. Mao, Stochastic Differential Equations and Applications, Horwood Publication Chichester, UK, [7] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, London, UK: Imperial College Press, [8] Y. Ren and N. Xia, Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput. 210 (2009) [9] Y. Ren and N. Xia, A note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput. 214 (2009) [10] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differential Equations 96 (1992), [11] F. Wei and K. Wang, The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl. 331 (2007), [12] Y. -H. Kim, An estimate on the solutions for stochastic functional differential equations, J. Appl. Math. and Informatics, 29 (2011) no.5-5, pp [13] Y. -H. Kim, A note on the solutions of neutral SFDEs with infinite delay, J. Ineq. Appl., 2013 (2013) no.181, doi / x Department of Mathematics Changwon National University, Changwon, , Korea [email protected]

234 234 International Congress in Honour of Professor Ravi P. Agarwal 200 Issues Optimization of Public Administration Canybec Sulayman and Gulnar Suleymanova In this talk, we discuss optimization of computational aspects in public administration with example of Kyrgyzstan. Our work analyzes the inefficiencies in public administration in Kyrgyzstan and uses mathematical models to provide, in our opinion, decision-making insight on how to reduce or completely eliminate effect of these inefficiencies. We will conclude with common challenges encountered by our researchers in application of these mathematical techniques. [1] Public Foundation Media Consulting Foundation, Analytical report on the research results of the level of access to information and discussion of problems on the possible solutions in local communities. This research produced with support of the German Federal Ministry of Economic Cooperation and Development. (2014) [2] European Bank for Reconstruction and Development, TRANSITION REPORT (2013) [3] Kydyraliev S., Sulayman C., Suleymanova G., New taxation policy in Kyrgyzstan:Theory and practice. The 18th NISPAcee Annual Conference. Public Administration in Times of Crisis. Warsaw,Poland. [4] Scott, Foresman, Functions, Statistics, and Trigonometry, The University of Chicago School Mathematics Project, Teacher s Edition (1992) p.19. [5] David M Diaz, Christopher D Barr, Mine Çetinkaya-Rundel, OpenIntro Statistics, Second Edition (2013) Canybec Sulayman: University of California Los Angeles (UCLA), Anderson School of Management, Candidate of Masters in Business Administration, Los Angeles, CA, USA, [email protected] Gülnar Suleymanova: Kyrgyzstan-Turkey Manas University, Institute of Natural and Applied Sciences, Candidate of Master s Program in Mathematics, Bishkek Kyrgyzstan, gulnara [email protected] This work was not supported by any grant or university program.

235 International Congress in Honour of Professor Ravi P. Agarwal Jacobi Orthogonal Approximation with Negative Integer and its Application Zhang Xiao-yong and Wan Zheng-su In this paper, the Jacobi spectral method for ordinary differential equation is proposed, which is based on the Jacobi approximation with negative integer. This method is very efficient for the initial value problem of the ordinary differential equations. The global convergence of proposed algorithm is proved. Numerical results demonstrate the spectral accuracy of this new approach and coincide well with theoretical analysis. [1] A.M.Stuart and A.R.Humphries,Dynamical systems and Numerical Analysis, Cambridge University Press, Cambridge, [2] Ben-yu Guo, Jacobi Approximation in Certain Hilbert Spaces and Their Applications to Singualr Differential Equations, J.Math.Anal.Appl. (2000), [3] Chao Zhang, Ben-Yu Guo and Tao Sun, Laguerre Spectral Method for High Order Problems,Numer.Math.Theor. Meth.Appl.Vol.6, No.3, pp (2013). [4] D. J. Higham, Analysis of the Enright-Kamel partitioning method for stiff ordinary differential equations,ima J.Numer.Anal.,9(1989), [5] E. Hairer, C.Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Diffrential Equations,Springer Series in Comput.Mathmatics, Vol.31, Springer-Verlag, Berlin, [6] E.Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential Equation :Nonstiff Problems, Springer-Verlag, Berlin, Zhang Xiao-yong: Department of Mathematics, Shanghai Maritime University, Haigang Avenue, 1550, Shanghai, , China Wan Zheng-su: Department of Mathematics, Hunan Institute of Science and Technology, Yueyang , China

236 236 International Congress in Honour of Professor Ravi P. Agarwal 202 Existence Results for Nonlinear Impulsive Fractional Differential Equations with p Laplacian Operator Ilkay Yaslan Karaca and Fatma Tokmak This paper is concerned with the existence of solutions for a nonlinear boundary value problem of impulsive fractional differential equations with p-laplacian operator. By applying some standard fixed point theorems, obtain sufficient conditions for the existence of solutions of the problem at hand. Examples are presented to demonstrate the applicability of our results. [1] R. P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Difference Equ Art. ID (2009) 47 pp. [2] R. P. Agarwal and B. Ahmad, Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations, Dyna. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 18 (2011) [3] B. Ahmad and S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal.: Hybrid Syst. 4 (2010) [4] I. Y. Karaca, On positive solutions for fourth-order boundary value problem with impulse, J. Comput. Appl. Math. 225 (2009) [5] I. Y. Karaca and F. Tokmak, Existence of solutions for nonlinear impulsive fractional differential equations with p- Laplacian operator, Mathematical Problems in Engineering 2014 Art. ID (2014) 11 pp. [6] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, [7] G. Wang, B. Ahmad and L. Zhang, Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Comput. Math. Appl. 62 (2011) [8] G. Wang, W. Liu and C. Ren, Existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions, Electron. J. Differential Equations 54 (2012) 10 pp. [9] G. Wang, B. Ahmad and L. Zhang, New existence results for nonlinear impulsive integro-differential equations of fractional order with nonlocal boundary conditions, Nonlinear Stud. 20 (2013) Ilkay Yaslan Karaca: Department of Mathematics, Ege University, Bornova, Izmir, Turkey, [email protected] Fatma Tokmak: Department of Mathematics, Gazi University, Teknikokullar, Ankara, Turkey,

237 International Congress in Honour of Professor Ravi P. Agarwal A Relation Between the Lefschetz Fixed Point Theorem and the Nielsen Fixed Point Theorem in Digital Images Ismet Karaca We present the Nielsen fixed point theorem for digital images. We deal with some important properties of the Nielsen number and calculate the Nielsen number for some digital images. Finally, we give a relation between the Lefschetz fixed point theorem and the Nielsen fixed point theorem in digital images. [1] H. Arslan, I. Karaca, and A. Oztel, Homology groups of n-dimensional digital images XXI. Turkish National Mathematics Symposium 2008, B1-13. [2] L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15(1994), [3] L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vis. 10(1999), [4] L. Boxer, Properties of digital homotopy, Journal of Mathematical Imaging and Vision 22(2005), [5] L. Boxer, Homotopy properties of sphere-like digital images, Journal of Mathematical Imaging and Vision 24(2006), [6] L. Boxer, Digital products, wedges and covering spaces, Journal of Mathematical Imaging and Vision 25(2006), [7] L. Boxer and I. Karaca, The classification of digital covering spaces. J. Math. Imaging Vis.32, (2008). [8] L. Boxer and I. Karaca, Some properties of digital covering spaces. J. Math. Imaging Vis.37, (2010). [9] L. Boxer, I. Karaca, and A. Oztel, Topological invariants in digital images, Journal of Mathematical Sciences: Advances and Applications 11(2011)(2), [10] O. Ege and I. Karaca, Fundamental properties of digital homology groups, American Journal of Computer Technology and Application, 1(2), pp (2013). [11] O. Ege and I. Karaca, the Lefschetz Fixed Point Theorem for Digital Images, Preprint (2013). [12] S.E. Han, Non-product property of the digital fundamental group. Inf. Sci. 171, 7-91 (2005). [13] G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55(1993), [14] I. Karaca and O. Ege, Some results on simplicial homology groups of 2D digital images, International Journal of Information and Computer Science 1(2012) no.8, [15] E. Khalimsky, Motion, deformation, and homotopy in finite spaces. In: Proceedings IEEE International Conference on Systems, Man, and Cybernetics, pp (1987). [16] J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, (1984). [17] E. Spanier, Algebraic Topology, McGraw-Hill, New York (1966). Departments of Mathematics, Ege University Bornova Izmir, Turkey, [email protected]

238 238 International Congress in Honour of Professor Ravi P. Agarwal 204 Second Order Nonlinear Boundary Value Problems with Integral Boundary Conditions on Time Scales F.Serap Topal and Arzu Denk Oguz This study investigates the existence of symmetric positive solutions for a class of nonlinear boundary value problem of second order dynamic equations with integral boundary conditions on time scales. Under suitable conditions, the existence of symmetric positive solutions are established by using monotone iterative technique. An example is presented to demonstrate the application of our main result. [1] Y. Li, T. Zhang, Multiple positive solutions for second-order p-laplacian dynamic equations with integral boundary conditions. Bound. Value Probl., Article ID (2011) [2] A.Boucherif, Second-order boundary value problems with integral boundary conditions, Non-linear Analysis, 70, (2009) [3] M. Benchohra, J.J. Nieto, Abdelghani Ouahab, Second-Order Boundary Value Problem with Integral Boundary Conditions, Boundary Value Problems, Article ID (2011) Ege University, Faculty of Science, Department of Mathematics, Bornova Izmir-Turkiye, [email protected]

239 International Congress in Honour of Professor Ravi P. Agarwal Existence of a Solution of Integral Equations via Fixed Point Theorem Selma Gülyaz In this talk, we establish a solution to the following integral equation T u(t) = G(t, s)f(s, u(s)) ds, for all t [0, T ], 0 where T > 0, f : [0, T ] R R and G : [0, T ] [0, T ] [0, ) are continuous functions. For this purpose, we also obtain some auxiliary fixed point results which generalize, improve and unify some fixed point theorems in the literature. [1] Alghamdi, MA, Hussain, N, Salimi, P: Fixed point and coupled fixed point theorems on b-metric-like spaces. J. Inequal. Appl. 2013, 402 (2013) [2] Harandi, AA, Emami, H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 72, (2010) [3] Nieto, JJ, Lopez, RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, (2005) [4] Harjani, J, Sadarangani, K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 71, (2009) [5] Moradi, S, Karapinar, E, Aydi, H: Existence of solutions for a periodic boundary value problem via generalized weakly contractions. Abstr. Appl. Anal. 2013, Article ID (2013) [6] Nahsine, HK: Cyclic generalized ψ-weakly contractive mappings and fixed point results with applications to integral equations. Nonlinear Anal. 75, (2012) [7] Karapinar, E, Shatanawi, W: On weakly (C, ψ, φ)-contractive mappings in partially ordered metric spaces. Abstr. Appl.Anal. 2012, Article ID (2012) [8] Karapinar, E, Yuce, IS: Fixed point theory for cyclic generalized weak φ-contraction on partial metric spaces. Abstr.Appl. Anal. 2012, Article ID (2012) [9] Karapinar, E: Best proximity points of Kannan type cyclic weak φ-contractions in ordered metric spaces. An. Univ. govidiush ConstanCta, Ser. Mat. 20(3), (2012) [10] Karapinar, E: Best proximity points of cyclic mappings. Appl. Math. Lett. 25(11), (2012) Department of Mathematics, Cumhuriyet University, Sivas, Turkey, [email protected]

240 240 International Congress in Honour of Professor Ravi P. Agarwal 206 Triangular and Square Triangular Numbers Arzu Özkoç In this work, we obtain some algebraic identities on triangular numbers denoted by T n and square triangular numbers denoted by S n. And also we construct a connection between triangular and square triangular numbers. We determine when the equality T m = S n holds by using s n and t n denote the sides of the corresponding square and triangle respectively. We derive some formulas on perfect squares, divisibility properties, sums of s n, t n, S n, T n and Pythagorean triples. [1] A. Behera and G.K. Panda. On the Square Roots of Triangular Numbers. The Fibonacci Quarterly, 37(2)(1999), [2] G.K. Panda. Some Fascinating Properties of Balancing Numbers. Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications, Cong. Numer. 194(2009), [3] G.K. Panda and P.K. Ray. Some Links of Balancing and Cobalancing Numbers with Pell and Associated Pell Numbers. Bul. of Inst. of Math. Acad. Sinica 6(1)(2011), [4] P.K. Ray. Balancing and Cobalancing Numbers. PhD thesis, Department of Mathematics, National Institute of Technology, Rourkela, India, [5] S.F. Santana and J.L. Diaz Barrero. Some Properties of Sums Involving Pell Numbers. Missouri Journal of Mathematical Science 18(1)(2006), Düzce University, Faculty of Arts and Science, Department of Mathematics, Konuralp, Düzce - Turkiye, arzuozkoc@ duzce.edu.tr

241 International Congress in Honour of Professor Ravi P. Agarwal Approximation Methods on a Complete Geodesic Space Yasunori Kimura In this talk, we propose iterative methods to approximate a common fixed point of mappings defined on a complete geodesic space with curvature bounded above. We also consider calculation error when generating an iterative sequence and we observe its convergence property. [1] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, [2] Y. Kimura, Approximation of a common fixed point of a finite family of nonexpansive mappings with nonsummable errors in a Hilbert space, J. Nonlinear Convex Anal. 15 (2014), [3] Y. Kimura, A shrinking projection method for nonexpansive mappings with nonsummable errors in a Hadamard space, Ann. Oper. Res., to appear. [4] Y. Kimura and K. Satô, Two convergence theorems to a fixed point of a nonexpansive mapping on the unit sphere of a Hilbert space, Filomat 26 (2012), [5] W. Takahashi, Y. Takeuchi, and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008), Toho University, Faculty of Science, Department of Information Science, 2-2-1, Miyama, Funabashi, Chiba , Japan, [email protected]

242 242 International Congress in Honour of Professor Ravi P. Agarwal 208 Fixed Point Results for α-admissible Multivalued F Contractions Gonca Durmaz and Ishak Altun In this study, we give some fixed point results for multivalued mappings using Pompeiu-Hausdorff distance on compete metric space. For this, we consider the α-admissibility of multivalued mappings. Our results are real generalizations of Mizoguchi-Takahashi fixed point theorem. We also provide an example showing this fact. [1] R. P. Agarwal, D. O Regan and D. R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, New York, [2] I. Altun, G. Mınak and H. Dağ, Multivalued F -Contractions On Complete Metric Space, Journal of Nonlinear and Convex Analysis, In press. [3] V. Berinde and M. Păcurar, The role of the Pompeiu-Hausdorff metric in fixed point theory, Creat. Math. Inform., 22 (2) (2013), [4] Lj. B. Ćirić, Multi-valued nonlinear contraction mappings, Nonlinear Anal., 71 (2009), [5] D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl., 334 (2007), [6] G. Mınak and I. Altun, Some new generalizations of Mizoguchi-Takahashi type fixed point theorem, Journal of Inequalities and Applications, 2013, 2013:493. [7] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl.,141 (1989), [8] S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital., 4 (5) (1972), [9] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012, 2012:94, 6 pp. [10] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Analysis 75 (2012), [11] E. Karapınar and B. Samet, Generalized α-ψ-contractive type mappings and related fixed point theorems with applications, Abstract and Applied Analysis 2012 (2012), Article ID , 17 pages. [12] J.H. Asl, S. Rezapour and N. Shahzad, On fixed points of α-ψ-contractive multifunctions, Fixed Point Theory and Applications 212 (2012), 6 pages, doi: / [13] H. Nawab, E. Karapınar, P. Salimi and F. Akbar, α-admissible mappings and related fixed point theorems, Journal of Inequalities and Applications 114 (2013), 11 pages. Gonca Durmaz: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, Kirikkale, Turkey, [email protected] Ishak Altun: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, Kirikkale, Turkey, [email protected]

243 International Congress in Honour of Professor Ravi P. Agarwal Advances on Fixed Point Theory Erdal Karapınar In this talk, we discuss on the advances on metric fixed point theory and some other abstract spaces via the recent publications on the topics. In particular, we point out the extension and improvement in various abstract spaces, such as generalized metric space. [1] R. Agarwal and E. Karapinar, Remarks on some coupled fixed point theorems in G-metric spaces, Fixed Point Theory and Applications (2013), 2013:2 [2] R.P.Agarwal, E. Karapinar, A.Roldan, Fixed point theorems in quasi-metric spaces and applications to multidimensional fixed point theorems on G-metric spaces, Journal of Nonlinear and Convex Analysis [3] E. Karapinar and R.P. Agarwal, Further fixed point results on G-metric spaces, Fixed Point Theory and Applications, 1 (2013) 2013:154 Atilim University, Department of Mathematics, Incek,Ankara Turkey [email protected],

244 244 International Congress in Honour of Professor Ravi P. Agarwal 210 Fixed Point Theorems for a Class of α-admissible Contractions and Applications to Boundary Value Problem Inci M.Erhan In this talk, we introduce a class of α-admissible contraction mappings defined via altering distance functions and acting on complete metric spaces. We investigate conditions for the existence and uniqueness of fixed points for these contractions and discuss the results in partially ordered spaces. As an application, we consider boundary value problems for a first order differential equations with periodic boundary conditions. [1] R. P. Agarwal, M. A. El- Gebeily and D. O Regan Generalized contractions in partially ordered metric spaces, Appl. Anal., 87, 1 8, (2008). [2] T. G. Bhaskar and V. Lakshmikantham, Fixed point theory in partially ordered metric spaces and applicaitons, Nonlinear Analysis, 65, , (2006). [3] V. Lakshmikantham and Lj. B. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Analysis, 70, , (2009). [4] H. K. Nashine and B. Samet, Fixed point results for mappings satisfying (ψ, φ) weakly contractive condition in partially ordered metric spaces, Nonlinear Analysis, 74, , (2011). Atılım University, Department of Mathematics, 06530, İncek Ankara-Turkiye, [email protected]

245 International Congress in Honour of Professor Ravi P. Agarwal Feng-Liu Type Fixed Point Theorems for Multivalued Mappings Gülhan Mınak and Ishak Altun In this talk, considering the recent technique, which is used by Jleli and Samet for fixed points of single valued mappings, we give some results of fixed points for multivalued mappings on complete metric space. Our results are proper generalizations of some related fixed point theorems including the famous Feng-Liu s result in the literature. We also give some examples to both illustrate and show that our results are real generalizations of mentioned theorems. [1] R. P. Agarwal, D. O Regan and D. R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, New York, [2] M. Berinde and V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl., 326 (2007), [3] Lj. B. Ćirić, Multi-valued nonlinear contraction mappings, Nonlinear Anal., 71 (2009), [4] Y. Feng and S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl., 317 (2006), [5] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl.,141 (1989), [6] S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30 (1969), [7] S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital., 4 (5) (1972), [8] T. Suzuki, Mizoguchi-Takahashi s fixed point theorem is a real generalization of Nadler s, J. Math. Anal. Appl., 340 (2008), [9] M., Jleli and B. Samet, A new generalization of the Banach contraction principle, Journal of Inequalities and Applications 2014, 2014:38 8 pp. Ishak Altun: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, Kirikkale, Turkey, [email protected] Gülhan Mınak: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, Kirikkale, Turkey, [email protected]

246 246 International Congress in Honour of Professor Ravi P. Agarwal 212 Qualitative Analysis for the Differential Equation Associated to the Dynamic Model for an Access Control Structure Daniela Coman, Adela Ionescu and Sonia Degeratu This paper presents some analytical considerations regarding the dynamical behavior of an access control structure, based on the mathematical model associated to this structure. This structure type is large analyzed in the literature. A modern approach of this structure based on SMA (shape memory alloy) is taken into account, because of some particular advantages due: unique characteristics (superelastic effect, as well as the single and double shape memory effects), damping capacity of noise and vibration, simplify and lower weight structure, high resistance to corrosion and wear, resistance to fatigue (which can occur even after hundreds of thousands of cycles), diversification of the control and command possibilities. The qualitative analysis of the mathematical model associated to this structure is taken into account. Namely, the differential equation associated to the variation of the angle describing the position of the access control structure is analyzed from the influence of parameters standpoint. The MAPLE11 soft is used in order to evaluate the behavior of the equation solution with respect to the parameters variation. This analysis produces a data collection which is useful both for further developing a fuzzy logic controller for the active control of this access structure and for further refinements of the mathematical model associated to this structure type [1] M.L. Abell, J.P. Braselton,Maple by Example, 3rd edition. Elsevier Academic Press, San Diego, California (2005) [2] S. Degeratu, P. Rotaru, S. Rizescu, N.G. Bizdoaca, Thermal study of a shape memory alloy (SMA) spring actuator designed to insure the motion of a barrier structure, Journal of Thermal Analysis and Calorimetry, 111 (2013), [3] S. Degeratu, N.G. Bizdoaca, S. Rizescu, P. Rotaru, V. Degeratu, G. Tont, Barrier structures using shape memory alloy springs, Proceedings of the International Conference on Development, Energy, Enviroment, Economics (DEEE 10), [4] N.G. Bizdoaca, A. Petrisor, S. Degeratu, P. Rotaru, E. Bizdoaca, Fuzzy logic controller for shape memory alloy tendons actuated biomimetic robotic structure, International Journal on Automation, Robotics and Autonomous Systems (ARAS), Issue II, vol. 09 (2009), Daniela Coman: Department of Engineering and Management of Technological Systems, Faculty of Mechanics, University of Craiova, Calugareni Str no1, , Romania, [email protected] Adela Ionescu: Department of Applied Mathematics, University of Craiova, Al. I. Cuza 13, Craiova , Romania, [email protected] Sonia Degeratu: Faculty of Automatics, Department of Electromechanics, 107 Decebal Blvd., , University of Craiova, Romania, [email protected] This work was partially supported by the grant number 7C/2014, awarded in the internal grant competition of the University of Craiova.

247 International Congress in Honour of Professor Ravi P. Agarwal Zagreb Indices of Double Graphs Aysun Yurttas, Muge Togan and Ismail Naci Cangul In this presentation, authors will give some new results and inequalities on several types of Zagreb indices for double graphs. [1] K. C. Das, N. Trinajstić, Relationship Between the Eccentric Connectivity Index and Zagreb Indices, Comp. Math. Appl., 62 (4) (2011), [2] I. Gutman, K. C. Das, The First Zagreb Index 30 Years After, MATCH Commun. Math. Comput. Chem. 50 (2004), [3] P. S. Ranjini, V. Lokesha, I. N. Cangul, On the Zagreb Indices of the Line Graphs of the Subdivision Graphs, Appl. Math. Comput., 218 (2011), [4] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, The First and Second Zagreb Indices of some Graph Operations, Discrete Appl. Math., 157 (2009), [5] K. Ch. Das, A. Yurttas, M. Togan, I. N. Cangul, A. S. Cevik, The multiplicative Zagreb indices of graph operations, Journal of Inequalities and Applications, 90, (2013) [6] I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of Society of Mathematicians Banja Luka, 18 (2011), [7] M. Togan, A. Yurttas, I. Naci Cangul, All versions of Zagreb indices and coindices of subdivision graphs of certain graph types (submitted) [8] M. Togan, A. Yurttas, I. Naci Cangul, All versions of Zagreb indices and coindices of r-subdivision graphs of certain graph types (submitted) [9] M. Togan, A. Yurttas, I. Naci Cangul, Zagreb indices and multiplicative Zagreb indices of subdivision graphs of double graphs for several graph types (submitted) Uludag University, Department of Mathematics, Gorukle Bursa, Turkey, The authors are supported by the Commission of Scientific Research Projects of Uludag University, project numbers 2012/15, 2012/19, 2012/20 and 2013/23.

248 248 International Congress in Honour of Professor Ravi P. Agarwal 214 Several Zagreb Indices of Subdivision Graphs of Double Graphs Muge Togan, Aysun Yurttas and Ismail Naci Cangul In this presentation, authors study the subdivision graphs of the double graphs of certain graph types and give some new results and inequalities on several types of Zagreb indices for subdivision graphs of double graphs. [1] K. C. Das, N. Trinajstić, Relationship Between the Eccentric Connectivity Index and Zagreb Indices, Comp. Math. Appl., 62 (4) (2011), [2] I. Gutman, K. C. Das, The First Zagreb Index 30 Years After, MATCH Commun. Math. Comput. Chem. 50 (2004), [3] P. S. Ranjini, V. Lokesha, I. N. Cangul, On the Zagreb Indices of the Line Graphs of the Subdivision Graphs, Appl. Math. Comput., 218 (2011), [4] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, The First and Second Zagreb Indices of some Graph Operations, Discrete Appl. Math., 157 (2009), [5] K. Ch. Das, A. Yurttas, M. Togan, I. N. Cangul, A. S. Cevik, The multiplicative Zagreb indices of graph operations, Journal of Inequalities and Applications, 90, (2013) [6] I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of Society of Mathematicians Banja Luka, 18 (2011), [7] M. Togan, A. Yurttas, I. Naci Cangul, All versions of Zagreb indices and coindices of subdivision graphs of certain graph types (submitted) [8] M. Togan, A. Yurttas, I. Naci Cangul, All versions of Zagreb indices and coindices of r-subdivision graphs of certain graph types (submitted) [9] M. Togan, A. Yurttas, I. Naci Cangul, Zagreb indices and multiplicative Zagreb indices of subdivision graphs of double graphs for several graph types (submitted) Uludag University, Department of Mathematics, Gorukle Bursa, Turkey, The authors are supported by the Commission of Scientific Research Projects of Uludag University, project numbers 2012/15, 2012/19, 2012/20 and 2013/23.

249 International Congress in Honour of Professor Ravi P. Agarwal On the Solutions of the Diophantine Equation x n + p y n = p 2 z n Caner Ağaoğlu and Musa Demirci In this paper we considered the Diophantine equation x n + p y n = p 2 z n (215.1) when n 2 and x, y, z are positive integers. Some special cases of (215.1) was already undertaken in the literature. In general form (215.1) we used Fermat s Method of Infinite Descent (FMID) to determine the existence of solutions. [1] Andreescu T., Andrica D., Cucurezeanu I., An introduction to Diophantine Equations, Springer, 2010, ISBN [2] Powell B. J., Proof of the Impossibility of the Fermat Equation x p + y p = z P for Special Values of p and of More General Equation b x n + c y n = d z n, Journal of Number Theory 18 (1984), [3] Manley S., On Quadratic Solutions of x 4 + p y 4 = z 4, Rocky Mountain Journal of Mathematics, 36-3 (2006), Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkey, [email protected]

250 250 International Congress in Honour of Professor Ravi P. Agarwal 216 A Weak Contraction Principle in Partially Ordered Cone Metric Space with Three Control Functions Binayak S.Choudhury, L.Kumar, T.Som and N.Metiya In this paper we utilize three functions to define a weak contraction in a cone metric space with a partial order and establish that this contraction has necessarily a fixed point either under the continuity assumption or an order condition which we state here. The uniqueness of the fixed point is also derived under some additional assumptions. The result is supported with an example. The methodology used is a combination of order theoretic and analytic approaches. Binayak S.Choudhury: Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah , West Bengal, India, [email protected], [email protected] L.Kumar: Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi , India, [email protected] T.Som: Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi , India, [email protected] N.Metiya: Department of Mathematics, Bengal Institute of Technology, Kolkata , West Bengal, India, [email protected]

251 International Congress in Honour of Professor Ravi P. Agarwal On the Diophantine Equation (20n) x + (99n) y = (101n) z Gokhan Soydan, Musa Demirci and Ismail Naci Cangul For a positive integer n, the triple (a, b, c) with a = u 2 v 2, b = 2uv, c = u 2 + v 2, u > v > 0, 2 uv, (u, v) = 1 satisfies a 2 + b 2 = c 2. There are conjectures and results on (an) x + (bn) y = (cn) z (217.1) with x, y, z Z +. (x, y, z) = (2, 2, 2) satisfies (217.1). In 1956, Sierpinśki, [6], showed that (217.1) has no other solution when n = 1 and (a, b, c) = (3, 4, 5) and Jeśmanowicz, [3], proved that when n = 1 and (a, b, c) = (5, 12, 13), (7, 24, 25), (9, 40, 41), (11, 60, 61), only solution is (x, y, z) = (2, 2, 2). He conjectured that (217.1) has no positive integer solutions other than (x, y, z) = (2, 2, 2). In 1959, Lu, [5], proved that (217.1) has the unique solution (x, y, z) = (2, 2, 2) if n = 1 and (a, b, c) = (4k 2 1, 4k, 4k 2 + 1). In 1998, Deng and Cohen, [1], proved that Jeśmanowicz conjecture is true for (a, b, c) = (3, 4, 5). In 1999, Le, [4], gave certain conditions for (217.1) to have positive integer solutions (x, y, z) with (x, y, z) (2, 2, 2). Recently several authors showed that Jeśmanowicz conjecture is true with 2 k 4 and k = 8. In 2013, Tang and Yang, [7], dealt with the case k = 2 and Deng, [2], also wrote a general paper covering this case. In 2012, Zhijuan and Jianxin, [9], discussed the case k = 3. Deng, [2], studied the case k = 2 s, 1 s 4 and this covers the case k = 4. Finally the case k = 8 is covered by Tang and Weng, [8], where the authors considered a special case (n = 3) that c is a Fermat number c = F n = 2 2n + 1, a = F n 2 and b = 2 2n 1 +1 for n 1. Next we consider (217.1) with k = 5. We consider (217.1) with (a, b, c) = (20, 99, 101) and conclude that (217.1) has no solution other than (x, y, z) = (2, 2, 2). [1] M. J. Deng, G. L. Cohen, On the conjecture of Jeśmanowicz concerning Pythagorean triples, Bull. Aust. Math. Soc. 57 (1998), [2] M. J. Deng, A note on the Diophantine equation (na) x + (nb) y = (nc) z, Bull. Aust. Math. Soc. (2013), to appear. [3] L. Jeśmanowicz, Several remarks on Pythagorean numbers, Wiadom. Math. 1 (1955/56), [4] M. H. Le, A note on Jeśmanowicz conjecture concerning Pythagorean triples, Bull. Aust. Math. Soc. 59 (1999), [5] W. D. Lu, On the Pythagorean numbers 4n 2 1, 4n and 4n 2 + 1, Acta Sci. Natur. Univ. Szechuan 2 (1959), [6] W. Sierpinśki, On the equation 3 x + 4 y = 5 z, Wiadom. Math. 1 (1955/56), [7] M. Tang, Z. J. Yang, Jeśmanowicz conjecture revisited, Bull. Aust. Math. Soc. 88 (2013), [8] M. Tang, J. X. Weng, Jeśmanowicz conjecture revisited-ii, Bull. Aust. Math. Soc. (2013), to appear. [9] Y. Zhijuan, W. Jianxin, On the Diophantine equation (12n) x + (35n) y = (37n) z, Pure and App. Math.(Chinese) 28 (2012), Department of Mathematics, Uludağ University, Bursa, Turkey, [email protected], [email protected], [email protected]

252 252 International Congress in Honour of Professor Ravi P. Agarwal 218 Halpern Type Iteration with Multiple Anchor Points in a Hadamard Space Yasunori Kimura and Hideyuki Wada In this talk, we consider an approximation theorem of common fixed points of nonexpansive mappings in a Hadamard space. Saejung [2] obtained that a Halpern type iteration with a nonexpansive mapping converges strongly to the fixed point in a Hadamard space. We introduce that another style of Halpern type iteration with multiple nonexpansive mappings converges strongly to the common fixed point in a Hadamard space. Kimura, Takahashi and Toyoda [1] proved the approximation of common fixed points of a finite family of nonexpansive mappings in a uniformly convex Banach space whose norm is Gâteaux differentiable. We obtain the main result under similar conditions of theirs. In the known results, the anchor point of Halpern type iteration is single, however the anchor points of our iterative sequence are multiple. [1] Y. Kimura, W. Takahashi, M, Toyoda, Convergence to common fixed points of a finite family of nonexpansive mappings, Arch. Math. (Basel) 84 (2005), [2] S. Saejung, Halpern s Iteration in CAT(0) Spaces, Fixed Point Theory and Appl (2010), 13pp. [3] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel) 58 (1992), Yasunori Kimura: Toho University, Department of Information Science, 2-2-1, Miyama, Funabashi, Chiba , Japan, [email protected] Hideyuki Wada: Toho University, Graduate School of Science, Department of Information Science, 2-2-1, Miyama, Funabashi, Chiba , Japan, [email protected]

253 International Congress in Honour of Professor Ravi P. Agarwal Multimaps in Fixed Point Theorems in Terms of Measure of Noncompactness Mehdi Asadi We present some of fixed point theorems for multimaps in fixed point theory and applications on measure of noncompactness. The main results are formulated in terms of definition of measure of noncompactness. Our theorems extend in a broad sense some new and classical results. Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran, [email protected]

254 254 International Congress in Honour of Professor Ravi P. Agarwal 220 Pointwise Approximation in Lp Space by Double Singular Integral Operators Mine Menekşe Yılmaz, Gümrah Uysal and Ertan Ibikli In this talk, we will prove the pointwise approximation of L λ (f, x, y) to f (x 0, y 0 ), as (x, y, λ) tends to (x 0, y 0, λ 0 ) small in the space L p by double singular integral operators at the characteristic point. [1] A. D.Gadjiev, On the order of convergence of singular integrals which depending on two parameters. Special Prob. of Funct. Analysis and its Appl. to the Theory of D. E. and the Theory of Funct. Izdat. Akad. Nauk Azerbaĭdažan, Baku, (1968), [2] H. Karsli, and E. Ibikli, On convergence of convolution type singular integral operators depending on two parameters, Fasc. Math. 38 (2007), [3] R.J. Nessel, Contributions to the theory saturation for singular integrals in several variables, III, radial kernels. Indag. Math. 29. Ser. A. (1965), [4] S. A. Stanislaw, Theorem of Romanovski type for double singular integrals. Comment. Math. 29.(1986), [5] R. Taberski, On double integrals and Fourier Series. Ann. Pol. Math. (1964), [6] M.M. Yilmaz, On convergence of Singular Integral Operators Depending on Three Parameters with Radial Kernels, Int. Journal of Math. Analysis, 4, (2010), no 39, Mine Menekşe: Gaziantep University, Faculty of Arts and Science, Department of Mathematics, Gaziantep, Turkey, [email protected] Gümrah Uysal: Karabuk University, Faculty of Science, Department of Mathematics, Balıklarkayası Mevkii, Karabuk, Turkey, [email protected] Ertan İbikli: Ankara University, Faculty of Science, Department of Mathematics, Tandogan, Ankara, Turkey, Ertan. [email protected]

255 International Congress in Honour of Professor Ravi P. Agarwal Some Tauberian Remainder Theorems for Iterations of Weighted Mean Methods of Summability Sefa Anıl Sezer and Ibrahim Çanak In this study, our aim is to retrieve λ boundedness of a real sequence from its λ boundedness by (N, p, k) summability method. To that end, we provide several Tauberian remainder theorems for the (N, p, k) summability method using the general control modulo of the oscillatory behavior given by Dik [2]. [1] İ. Çanak and Ü. Totur. Some Tauberian theorems for the weighted mean methods of summability. Comput. Math. Appl., 62(6) (2011), , [2] M. Dik. Tauberian theorems for sequences with moderately oscillatory control moduli. Math. Morav., 5 (2001), 57 94, [3] G. Kangro. A Tauberian remainder theorem for the Riesz method. Tartu Riikl. Ül. Toimetised, 277 (1971), , [4] O. Meronen and I. Tammeraid. Generalized Nörlund method and convergence acceleration. Math. Model. Anal., 12(2) (2007), , [5] O. Meronen and I. Tammeraid. Several theorems on λ-summable series. Math. Model. Anal., 15(1) (2010), ,. [6] O. Meronen and I. Tammeraid. General control modulo and Tauberian remainder theorems for (C, 1) summability. Math. Model. Anal., 18(1) (2013), , [7] A. Šeletski and A. Tali. Comparison of speeds of convergence in Riesz-type families of summability methods. II. Math. Model. Anal., 15(1) (2010), , [8] S. A. Sezer and İ. Çanak. Tauberian Remainder Theorems for the Weighted Mean Method of Summability. Math. Model. Anal., 19(2) (2014), , [9] Ü. Totur and İ. Çanak. Some general Tauberian conditions for the weighted mean summability method. Comput. Math. Appl., 63(5) (2012), Sefa Anıl Sezer: Ege University, Faculty of Science, Department of Mathematics, İzmir-Turkey and İstanbul Medeniyet University, Faculty of Science, Department of Mathematics, İstanbul-Turkey, [email protected] or [email protected] İbrahim Çanak: Ege University, Faculty of Science, Department of Mathematics, İzmir-Turkey, ibrahim.canak@ ege.edu.tr

256 256 International Congress in Honour of Professor Ravi P. Agarwal 222 On The Semi-Fredholm Spectrum Arzu Akgül In this talk, a version of semi Fredholm joint spectrum for families of noncommuting operators is defined. Moreover, by using homological methods and the connections between Fredholm joint spectrum and upper-semi Fredholm and lower semi Fredholm spectrum, spectral mapping theorem is proved and some propertiesof semi Fredholm spectrum are investigated. [1] Atkinson, Multiparameter Eingen Value Problems, New York, 1968 [2] Dosiyev, A. Spectra of İnfinte Parametrized Banach Complexes, Journal of Operator Theory, 48, 2002, no.3, [3] Fainshtein A.S., Taylor Joint Spectrum for Families of Operators Generating Nilpotent Lie algebras. Journal of Operator Theory, , 2-27 [4] Taylor, J.L. A joint Spectrum for Several Commuting Operators, Journal of Functional Analysis, v.26,1970,, Kocaeli University, Faculty of Science and Arts, Department of Mathematics, Kocaeli, [email protected]

257 International Congress in Honour of Professor Ravi P. Agarwal Critical Fixed Point Theorems in Banach Algebras Under Weak Topology Features A.Ben Amar and A.Tlili In this paper we prove some new fixed point theorems for weakly sequentially continuous operators of type x = AxBx+Cx in a Banach algebra. For this purpose, we introduce the concept of multi-valued mappings under conditions of weak topology. We also provide some new results concerning the sum and the product of nonlinear weakly sequentially continuous mappings in a Banach algebra satisfying a certain sequential condition (P). [1] R. E. Edwards, Functional analysis, Theory and applications. Holt. Rein. Wins. New York, (1965). [2] A. Ben Amar, A. Jeribi and M. Mnif, On a generalization of the Schauder and Krasnoselskii fixed point theorems and application to biological model. Number. Funct. Anal. Optim., 29 (1), 1-23, (2008). [3] A. Ben Amar and M. Mnif, Leary-Schauder alternatives for weakly sequentially continuous mappings and application to transport equation. Math. Methods Appl. Sci., 33, 80-90, (2010). [4] A. Ben Amar and A. Sikorska-Nowak, On some fixed point theorems for 1-weakly contractive multi-valued mappings with weakly sequentially closed graph. preprint, (2010). [5] D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) [6] I. Dobrakov, On representation of linear operators on C Math. J. 21 (96) (1971) [7] B. C. Dhage, Multi-valued operators and fixed point theorems in Banach algebras I, Nonlinear functional anal and appl., 10(4) (2006), [8] E.Zeidler, Non linear functional analysis and applications. Vol. 1. Springer. New York (1986).

258 258 International Congress in Honour of Professor Ravi P. Agarwal 224 Modeling of Effect of the Components of Distance Education in Achievement of Students Hamit Armagan, Tuncay Yigit and Ahmet Sahiner Distance education is a kind of education that brought together course advisor,student and educational materials in a different time and place through commenicational technologies. In this educational system the success of education is directly related to audio, video and interaction. In this study,a model is created by using fuzzy logic with the success of distance education students and the components of distance education. In addition, with a global optimization method it is determined which are the highest student achievement points. [1] E. H. Mamdani, Application of Fuzzy Logic to Approximate Reasoning Using Linguistic Synthesis, IEEE Trans. Computers, 26(12) (1977), [2] F. Esragh, E.H. Mamdani, A general approach to linguistic approximation, Fuzzy Reasoning and Its Applications, Academic Press,1981. [3] E.H. Mamdani, Application of fuzzy algorithms for simple dynamic plant. Proc. IEE 121 (1974), [4] A. Sahiner, H. Gokkaya, N. Ucar, Nonlinear Modelling of the Layer Microhardnes of Fe-Mn Binory Allays, Journal of Balkan Tribological Association, 4 (2013), [5] A. Sahiner, I. Uney, M. F. Gurbuz, An Application of Fuzzy Logic in Entomology: Estimating the Egg Production and Opening of Pimpla Turianellae L., IWBCMS-2013 Procoeding Book, [6] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst., Man, Cybern, 15 (1985), [7] L. A. Zadeh, Fuzzy Sets, Inf. Control, 8 (1965), [8] Z.Y. Wu, H. W. J.Lee, L.S. Zhang, X. M. Yang, A Novel Filled Function Method and Quasi-Filled Function Method for Global Optimization, Comput. Optim. Appl., 34 (2005), [9] L. A. Zadeh, Fuzzy Sets as a Basis for a Theory of Posibility, Fuzzy Sets Syst, 1 (1978), [10] L. A. Zadeh, The consept of a linguistic variable and its application to approximate reasoning I, Information Sciences, 8(3) (1975), , (1975) [11] Y. Lin, Y. Yang, L. Zhang, A Novel Filled Function Method for Global Optimization, J. Korean Math. Soc. 47(6) (2010), [12] A. Sahiner, H. Gokkaya, T. Yigit, A new filled function for non-smooth global optimization, AIP Conf. Proc (2012), [13] S. Karagöz, H. Zülfikar, T. Kalayci, Ogrenme surecine ilisskin degerlendirmeler ve fuzzy karar verme teknigi ile surece dair bir uygulama, Istanbul Universitesi Sosyal Bilimler Dergisi, (1) (2014), Hamit Armagan: Suleyman Demirel University, Department of Information, Isparta/Turkiye, [email protected] Tuncay Yigit: Suleyman Demirel University, Faculty of Engineering, Department of Computer Engineering, Isparta/Turkiye, [email protected] Ahmet Sahiner: Suleyman Demirel University, Faculty of Art and Science, Department of Mathematics, Isparta/Turkiye, [email protected]

259 International Congress in Honour of Professor Ravi P. Agarwal On the Weighted Integral Inequalities for Convex Function Mehmet Zeki Sarıkaya and Samet Erden In this talk, we establish several weighted inequalities for some differantiable mappings that are connected with the celebrated Hermite-Hadamard-Fejér type and Ostrowski type integral inequalities. The results presented here would provide extensions of those given in earlier works. [1] F. Ahmad, N. S. Barnett and S. S. Dragomir, New Weighted Ostrowski and Cebysev Type Inequalities, Nonlinear Analysis: Theory, Methods & Appl., 71 (12), (2009), [2] F. Ahmad, A. Rafiq, N. A. Mir, Weighted Ostrowski type inequality for twice differentiable mappings, Global Journal of Research in Pure and Applied Math., 2 (2) (2006), [3] N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27(1), (2001), [4] N. S. Barnett, S. S. Dragomir and C.E.M. Pearce, A Quasi-trapezoid inequality for double integrals, ANZIAM J., 44(2003), [5] S. S. Dragomir, P. Cerone, N. S. Barnett and J. Roumeliotis, An inequlity of the Ostrowski type for double integrals and applications for cubature formulae, Tamsui Oxf. J. Math., 16(1), (2000), [6] S. Hussain, M.A.Latif and M. Alomari, Generalized duble-integral Ostrowski type inequalities on time scales, Appl. Math. Letters, 24(2011), [7] M. E. Kiris and M. Z. Sarikaya, On the new generalization of Ostrowski type inequality for double integrals, International Journal of Modern Mathematical Sciences, 2014, 9(3): [8] L. Fejer, Über die Fourierreihen, II. Math. Naturwiss. Anz Ungar. Akad. Wiss., 24 (1906), (Hungarian). [9] U.S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), [10] A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10(1938), [11] J. Pečarić, F. Proschan and Y.L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, [12] A. Qayyum, A weighted Ostrowski-Grüss type inequality and applications, Proceeding of the World Cong. on Engineering, Vol:2, 2009, 1-9. [13] A. Rafiq and F. Ahmad, Another weighted Ostrowski-Gr üss type inequality for twice differentiable mappings, Kragujevac Journal of Mathematics, 31 (2008), [14] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, Vol. LXXIX, 1(2010), pp [15] M. Z. Sarikaya On the Ostrowski type integral inequality for double integrals, Demonstratio Mathematica, Vol. XLV, No 3, pp: , Mehmet Zeki Sarıkaya: Düzce University, Faculty of Science and Arts, Department of Mathematics, Düzce-Turkiye, [email protected] Samet Erden: Bartın University, Faculty of Science, Department of Mathematics, Bartın-Turkiye, erdem1627@gmail. com

260 260 International Congress in Honour of Professor Ravi P. Agarwal [16] M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, The Arabian Journal for Science and Engineering (AJSE)-Mathematics, (2011) 36: [17] M. Z. Sarikaya, On the generalized weighted integral inequality for double integrals, Annals of the Alexandru Ioan Cuza University-Mathematics, accepted. [18] M. Z. Sarikaya, On new Hermite Hadamard Fejer Type integral inequalities, Studia Universitatis Babes-Bolyai Mathematica., 57(2012), No. 3, [19] M. Z. Sarikaya and H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Submited [20] K-L. Tseng, G-S. Yang and K-C. Hsu, Some inequalities for differentiable mappings and applications to Fejer inequality and weighted trapozidal formula, Taiwanese J. Math. 15(4), pp: , [21] C.-L. Wang, X.-H. Wang, On an extension of Hadamard inequality for convex functions, Chin. Ann. Math. 3 (1982) [22] S.-H. Wu, On the weighted generalization of the Hermite-Hadamard inequality and its applications, The Rocky Mountain J. of Math., vol. 39, no. 5, pp , 2009.

261 International Congress in Honour of Professor Ravi P. Agarwal 261 List of Participants 1. A.A.Dosiyev: Eastern Mediterranean University 2. A.Laifa: Université du 20 août 1955-Skikdav 3. A.Lebaroud: Université du 20 août 1955-Skikda 4. A.Medoued: Université du 20 août 1955-Skikda 5. A.R.Aithal: University of Mumbai 6. A.R.Bindusree: Sree Narayana Gurukulam College of Engineering 7. A.Tatarczak: Maria Curie-Sklodowska University in Lublin 8. A.Turan Gürkanlı: İstanbul Arel University 9. Abdelmalek Mohammed: Ecole préparatoire en sciences économiques 10. Abdessalem Benammar: Welding and NDT Research Center (CSC) 11. Abdullah Çavuş: Karadeniz Technical University 12. Abdullah Dertli: Ondokuz Mayıs University 13. Abdullah Özbekler: Atilim University 14. Abdurrahman Dayioglu: Uludağ University 15. Acushla Sarswat: University of Mumbai 16. Adela Ionescu: University of Craiova 17. Ademi Ospanova: L.N.Gumilyov Eurasian National University 18. Adnan Kılıç: Uludağ University 19. Ahmed GhezalUniversity of Constantine Ahmed Khechida: Welding and NDT Research Center (CSC) 21. Ahmet Emin: Balikesir University 22. Ahmet Sinan Çevik: Selcuk University 23. Ahmet Şahiner: Suleyman Demirel University 24. Ahu Açıkgöz: Balikesir University 25. Aiman Mukheimer: Prince Sultan University 26. Alen Osançlıol : SabancıUniversity 27. Ali Akgul: Dicle University 28. Ali Aral: Kirikkale University 29. Ali Demir: Kocaeli University 30. Anuj Kumar: Banaras Hindu University 31. Arzu Denk Oguz: Ege University 32. Arzu Akgül: Kocaeli University 33. Arzu Özkoç: Düzce University 34. Aslı Ayten Kaya: Uludag University 35. Aydın Tiryaki: Izmir University 36. Aykut Ahmet Aygüneş: University of Akdeniz 37. Aynur Şahin: Sakarya University 38. Aynur Yalçıner: Selçuk University 39. Aysun Yurttas: Uludag University 40. Ayşe Feza Güvenilir: Ankara University 41. Ayşe Sandıkçı: Ondokuz Mayıs University 42. Ayşegül Akyüz-Daşcıoğlu: Pamukkale University 43. Ayten Pekin: Istanbul University 44. Aziz Halit Gozel: Adiyaman University 45. Basri Celik: Uludağ University 46. Belmeguenai Aissa: Universite 20 Aout 1955-Skikda

262 262 International Congress in Honour of Professor Ravi P. Agarwal 47. Benmansour Safia: Ecole préparatoire en sciences économiques 48. Benmerai Romaissa: University of Constantine Bilal Demir: Balikesir University 50. Bilel Mefteh: Sfax University 51. Billur Kaymakçalan: Çankaya University 52. Binayak S.Choudhury: Bengal Engineering and Science University 53. Boughazi Hichem: Preparatory School in Economics 54. Boumediene Abdellaoui: Université Abou Bakr Belkaïd 55. Burcin Simsek: University of Pittsburgh 56. Burcu Öztürk: Trakya University 57. C.S.Ryoo: Hannam University 58. Can Murat Dikmen: Bulent Ecevit Universitesi 59. Caner Ağoğlu: Uludag University 60. Canybec Sulayman: University of California 61. Cengiz Akay: Uludağ University 62. Chandrashekar Adiga: University of Mysore 63. Çetin Yıldız: Atatürk University 64. D.Azzam-Laouir: University of Jijel 65. D.Sayad: Université du 20 août 1955-Skikda 66. Daeyeoul Kim: National Institute for Mathematical Sciences 67. Dalila Azzam-Laouir: Université de Jijel 68. Daniela Bímová: Technical University of Liberec 69. Daniela Bittnerová: Technical University of Liberec 70. Daniela Coman: University of Craiova 71. Dib Djalel: University of Tebessa 72. Djavvat Khadjiev: Karadeniz Technical University 73. Djezzar Salah: University of Constantine Dmitry V.Kruchinin: Tomsk State University of Control Systems and Radioelectronics 75. Doria Affane: Université de Jijel 76. Durhasan Turgut Tollu: Necmettin Erbakan University 77. Duygu Dönmez Demir: Celal Bayar University 78. Duygu Yilmaz Eroglu: Uludag University 79. Ebru Ozbilge: Izmir University of Economics 80. Ekber Girgin: Sakarya University 81. Ekrem Savaş: Istanbul Ticaret University 82. Elif Aydın: Ondokuz Mayıs University 83. Elif Cetin: Uludag University, Celal Bayar University 84. Elif Erçelik: Gebze Institute of Technology, İstanbul Technical University 85. Elvan Akın: Missouri University Science Technology 86. Emine Mısırlı: Ege University 87. Emrah Kılıç: TOBB University of Economics and Technology 88. Emrah Yılmaz: Firat University 89. Emre Deniz: Kirikkale University 90. Emrullah Yaşar: Uludag University 91. Erbil Çetin: Ege University 92. Erdal Karapinar: Atilim University 93. Erdem Toksoy: Ondokuz Mayıs University

263 International Congress in Honour of Professor Ravi P. Agarwal Erhan Koca: Celal Bayar University 95. Ertan İbikli: Ankara University 96. Esen İyigün: Uludag University 97. Esra Kamber: Sakarya University 98. Esra Karataş: Çanakkale Onsekiz Mart University 99. Eylem Güzel Karpuz: Karamanoğlu Mehmetbey University 100. F.Aliouane: University of Jijel 101. F.Mahmoudi: Universidad de Chile 102. Farrukh Mukhamedov: International Islamic University Malaysia 103. Fatih Kızılaslan: Gebze Institute of Technology 104. Fatma Çalışkan: Istanbul University 105. Fatma Kanca: Kadir Has University 106. Fatma Karakoç: Ankara University 107. Fatma Özen Erdoğan: Uludag University 108. Fatma Serap Topal: Ege University 109. Fatma Tokmak: Gazi University and Ege University 110. Fayçal Hamdi: RECITS Laboratory 111. Fırat Ateş: Balikesir University 112. Figen Öke: Trakya University 113. Florian Munteanu: University of Craiova 114. Fuad Kittaneh: The University of Jordan & Jordan and Al-Ahliyya Amman University 115. Fulya Yoruk Deren: Ege University 116. Fumiaki Kohsaka: Oita University 117. Gabil Adilov: Akdeniz University 118. Georgy A. Omel yanov: Universidad de Sonora 119. Gonca Durmaz: Kirikkale University 120. Gokhan Soydan: Uludağ University 121. Guettal Djaouida: University Ferhat Abbas of Setif Gurunath Rao Vaidya: Acharya Institute of Graduate Studies 123. Gülden Gün Polat: İstanbul Technical University 124. Gülden Kapusuz: Suleyman Demirel University 125. Gülnar Suleymanova: Kyrgyzstan-Turkey Manas University 126. Gülşah Yeni: Missouri University of Science and Technology 127. Gümrah Uysal: Karabuk University 128. Gülsum Ulusoy: Kirikkale University 129. H.Cenk Ozmutlu: Uludag University 130. H.M.Sadeghi: Eastern Mediterranean University 131. Hacer Şengül: Siirt University 132. Hakan Avcı: Ondokuz Mayıs University 133. Halis Aygün: Kocaeli University 134. Hamid Mottaghi Golshan: Islamic Azad University 135. Hamit Armagan: Suleyman Demirel University 136. Handan Engin Kırımlı: Uludağ University 137. Harun Karsli: Abant Izzet Baysal University 138. Hasan Akın: Zirve University 139. Hasan Köse: Selcuk University 140. Hatice Yaldız: Düzce University

264 264 International Congress in Honour of Professor Ravi P. Agarwal 141. Hendra Gunawan: Institute of Technology Bandung 142. Hesna Kabadayı: Ankara University 143. Hideyuki Wada: Toho University 144. Hikmet Koyunbakan: Firat University 145. Hüseyin Bereketoğlu: Ankara University 146. Hüseyin Ovalıoğlu: Uludağ University 147. Ibtissam Bouloukza: University of 20 August Ilknur Yesilce: Mersin University 149. Ismail U.Tiryaki: Abant Izzet Baysal University 150. Izhar Uddin: Aligarh Muslim University 151. İ.İlker Akça: Eskişehir Osmangazi University 152. İbrahim Çanak: Ege University 153. İlhan Küçük: Uludag University 154. İlkay Yaslan Karaca: Ege University 155. İlker Küçük: Uludag University 156. İlker Burak Giresunlu: Uludag University 157. İnci M.Erhan: Atılım University 158. İrem Baglan: Kocaeli University 159. İshak Altun: Kirikkale University 160. İsmail Aydın: Sinop University 161. İsmail Naci Cangül: Uludag University 162. İsmet Karaca: Ege University 163. J.Davila: Universidad de Chile 164. J.Y.Kang: Hannam University 165. Jasbir S. Manhas: Sultan Qaboos University 166. K.Biroud: Université Abou Bakr Belkaïd 167. Kadir Emir: Eskişehir Osmangazi University 168. Kadir Ertürk: Namik Kemal University 169. Kathryn E.Hare: University of Waterloo 170. Kenan Tas: Cankaya University 171. Kubra Erdem Bicer: Celal Bayar University 172. Kyung Soo Kim: Kyungnam University 173. L.Kumar: Banaras Hindu University 174. Lashab Mohamed: Universite 20 Aout 1955-Skikda 175. Leili Kussainova: L.N. Gumilyov Eurasian National University 176. Luminita Grecu: University of Craiova 177. M.Emin Özdemir: Atatürk University 178. M.Hariour: Badji Mokthtar-Annaba University 179. M.Imdad: Aligarh Muslim University 180. M.C.Bouras: Badji Mokthtar-Annaba University 181. M.S.Jusoh: Universiti Teknologi MARA 182. M.N.M.Fadzil: Universiti Teknologi MARA 183. M.S.M.Noorani: Universiti Kebangsaan Malaysia 184. Mahpeyker Öztürk: Sakarya University 185. Mansouri Khaled: Universite 20 Aout 1955-Skikda 186. Masashi Toyoda: Tamagawa University 187. Matallah Atika: Ecole préparatoire en sciences économiques

265 International Congress in Honour of Professor Ravi P. Agarwal Meenu Goyal: Indian Institute of Technology Roorkee 189. Mehdi Asadi: Islamic Azad University 190. Mehmet Ali Özarslan: Eastern Mediterranean University 191. Mehmet Emir Koksal: Mevlana University 192. Mehmet Yüksel: Cukurova University 193. Mehmet Zeki Sarıkaya: Düzce University 194. Merve Güney Duman: Sakarya University 195. Meryem Odabasi: Ege University 196. Meryem Öztop: Suleyman Demirel University 197. Mesliza Mohamed: Universiti Teknologi MARA 198. Messaoudene Hadia: University of Tebessa 199. Metin Başarır: Sakarya University 200. Mikail Et: Firat University 201. Mine Menekşe: Gaziantep University 202. Mochammad Idris: Institute of Technology Bandung 203. Mohamed Amine Boutiche: Université des sciences et de la Technologie Houari Boumediene 204. Mohamed Dalah: University of Constantine Mohammed Derhab: University Abou-Bekr Belkaid Tlemcen 206. Mohamed Najib Ellouze: Sfax University 207. Mohammed Nehari: University Ibn Khaldoun Tiaret 208. Mourad Jelassi: Carthage University 209. Mourad Mordjaoui: University of 20 August Mourad Rahmani: USTHB 211. Moustafa El-Shahed: Qassim University 212. Muge Togan: Uludag University 213. Murteza Yılmaz: TOBB University of Economics & Technology 214. Musa Demirci: Uludag University 215. Mustafa Alkan: University of Akdeniz 216. Mustafa Inc: Firat University 217. Mustafa Kara: Eastern Mediterranean University 218. Mustafa Kazaz: Celal Bayar University 219. Mustafa Nadar: İstanbul Technical University 220. Mustafa Özkan: Trakya University 221. Mustafa Topaksu: Cukurova University 222. Mustapha Yarou: Jijel University 223. Mutlay Dogan: Zirve University 224. N.Metiya: Bengal Institute of Technology 225. N.M.Badiger: Karnatak University 226. Nazim Idrisoğlu Mahmudov: Eastern Mediterranean University 227. Nazli Yildiz Ikikardes: Balikesir University 228. Nazmiye Yilmaz: Selcuk University 229. Necati Taskara: Selcuk University 230. Neslihan Nesliye Pelen: Middle East Technical University 231. Neşe İşler Acar: Mehmet Akif Ersoy University 232. Neşe Ömür: Kocaeli Üniversitesi 233. Nihal Yılmaz Özgür: Balıkesir University 234. Nihal Taş: Balikesir University

266 266 International Congress in Honour of Professor Ravi P. Agarwal 235. Nihal Yokuş: Karamanoglu Mehmetbey University 236. Nihat Akgüneş: Necmettin Erbakan University 237. Nil Küçük: Uludag University 238. Nilay Sager: Ondokuz Mayıs University 239. Nilgun G.Baydar: Suleyman Demirel University 240. Nouiri Brahim: University of Laghouat 241. Noor Halimatus Sa diah Ismail: Universiti Teknologi MARA 242. Nuket Aykut Hamal: Ege University 243. Nurullah Yilmaz: Suleyman Demirel University 244. Nursel Öztürk: Uludag University 245. Ozan Demirözer: Suleyman Demirel University 246. Ozkan Coban: Suleyman Demirel University 247. Ömer Akgüller: Muğla Sıtkı Koçman University 248. Ömer Kişi: Cumhuriyet University 249. Özden Koruoğlu: Balikesir University 250. Özgür Ege: Celal Bayar University 251. Özlem Acar: Kirikkale University 252. Özlem Orhan: İstanbul Technical University 253. Öznur Kulak: Ondokuz Mayıs University 254. Öznur Öztunç: Balıkesir University 255. P.N.Agrawal: Indian Institute of Technology Roorkee 256. P.S.K.Reddy: S.I.T 257. R.Saian: Universiti Teknologi MARA 258. P.Shahi: Thapar University 259. Patricia J.Y.Wong: Nanyang Technological University 260. Rahal Mohamed: University Ferhat Abbas of Setif Rahime Dere: University of Akdeniz 262. Rajai Alassar: King Fahd University of Petroleum & Minerals (KFUPM) 263. Ranjini P.S: Don Bosco Institute of Technology 264. Ravi Agarwal: Texas A&M University-Kingsville 265. Raziye Akbay: Suleyman Demirel University 266. Recep Şahin: Balikesir University 267. Redouane Drai: Welding and NDT Research Center (CSC) 268. Refik Keskin: Sakarya University 269. Reyhane Ercan: Suleyman Demirel University 270. Romulus Militaru: University of Craiova 271. Ruhan Zhao: State University of New York (SUNY) 272. Rüstem Kaya: Eskişehir Osmangazi University 273. S.Kanas: University of Rzeszow 274. S.K.Upadhyay: Indian Institute of Technology 275. S.Madi: Badji Mokthtar-Annaba University 276. Said Grace: Cairo University 277. Salih Yalcinbas: Celal Bayar University 278. Safa Menkad: Hadj Lakhdar University 279. Safia Benmansour: Preparatory School of Economics of Tlemcen 280. Samet Erden: Bartın University 281. Sandeep Kumar: Acharya Institute of Technology

267 International Congress in Honour of Professor Ravi P. Agarwal Sanjiv K.Gupta: Sultan Qaboos University 283. Sathish Kumar: Indian Institute of Technology Roorkee 284. Satish Iyengar: University of Pittsburgh 285. Sebahattin Ikikardes: Balikesir University 286. Seda Oğuz: Cumhuriyet University 287. Seda Oral: Celal Bayar University 288. Seda Öztürk: Karadeniz Technical University 289. Sefa Anıl Sezer: Ege University 290. Selçuk Kayacan: Istanbul Technical University 291. Selma Altundağ: Sakarya University 292. Selma Gülyaz: Cumhuriyet University 293. Sertaç Erman: Kocaeli University 294. Servet Kütükçü: Ondokuz Mayıs University 295. Seval Ene: Uludag University 296. Seyda Ildan: Selcuk University 297. Sibel Koparal: Kocaeli University 298. Sibel Paşalı Atmaca: Muğla Sıtkı Koçman University 299. Sinem Şahiner: Izmir University 300. Sinem Unul: Eastern Mediterranean University 301. Smail Kelaiaia: University of Annaba 302. Snezhana Hristova: Plovdiv University 303. Sonia Degeratu: University of Craiova 304. Soumia Kharfouchi: Université 3 Constantine 305. Süleyman Çiftçi: Uludag University 306. Sümeyra Uçar: Balıkesir University 307. Sümeyye Bakım: KTO Karatay University 308. Şenol Eren: Ondokuz Mayıs University 309. Şerife Müge Ege: Ege University 310. Şükran Konca: Sakarya University, Bitlis Eren University 311. T.Som: Banaras Hindu University 312. Ṫacksun Jung: Kunsan National University 313. Tahia Zerizer: Jazan University 314. Temel Ermiş: Eskişehir Osmangazi University 315. Teoman Özer: İstanbul Technical University 316. Tevfik Şahin: Amasya University 317. Tuba Vedi: Eastern Mediterranean University 318. Tuba Yigit: Suleyman Demirel University 319. Tugba Senlik: Ege University 320. Tuncay Yigit: Suleyman Demirel University 321. Tuncer Acar: Kirikkale University 322. U.K.Misra: Berhampur University 323. Uğur Yüksel: Atilim University 324. Ummahan Akcan: Anadolu University 325. Usha A.: Alliance University 326. Ümit Sarp: Balikesir University 327. Ümit Totur: Adnan Menderes University 328. Ümmügülsüm Öğüt: Sakarya University

268 268 International Congress in Honour of Professor Ravi P. Agarwal 329. V.Lokesha: Vijayanagara Sri Krishnadevaraya University 330. Valery Gaiko: National Academy of Sciences of Belarus 331. Veli Kurt: University of Akdeniz 332. Vildan Çetkin: Kocaeli University 333. Vishwanath P.Singh: Karnatak University 334. Vladimir V.Kruchinin: Tomsk State University of Control Systems and Radioelectronics 335. Wan Zheng-su: Hunan Institute of Science and Technology 336. Yasemin Cengellenmiş: Trakya University 337. Yasin Yazlik: Nevsehir Haci Bektas Veli University 338. Yasunori Kimura: Toho University 339. Yaşar Bolat: Kastamonu University 340. Yavuz Selim Balkan: Duzce University 341. Yilmaz Simsek: University of Akdeniz 342. Young-Ho Kim: Changwon National University 343. Zdzislaw Rychlik: Maria Curie-Sk lodowska University 344. Zehra Sarıgedik: Celal Bayar University 345. Zennir Khaled: University 20 Aout Zerrin Önder: Ege University 347. Zhang Xiao-yong: Shanghai Maritime University