# Quasi Contraction and Fixed Points

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Available online at Volume 2012, Year 2012 Article ID jnaa-00168, 6 Pages doi: /2012/jnaa Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady 2 (1) Department of Mathematics, Faculty of Sciences, Golestan University, P.O.Box. 155, Gorgan, Iran. (2) Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. Copyright 2012 c Mehdi Roohi and Mohsen Alimohammady. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this note, we establish and improve some results on fixed point theory in topological vector spaces. As a generalization of contraction maps, the concept of quasi contraction multivalued maps on a topological vector space will be defined. Further, it is shown that a quasi contraction and closed multivalued map on a topological vector space has a unique fixed point if it is bounded value. Keywords: : Topological vector space, Quasi contraction, Fixed point, Multi-valued map. 1 Introduction Fixed point theory has many applications in almost all branches of mathematics. The Banach contraction principles occupies a central position in fixed point theory. Banach s original theorem is expressed in metric spaces and some authors have extended this result in some other versions [1], [4], [6] and [7]. Kirk [5] and Edelstien [3] studied and achieved some basic results in fixed point theory. Here, we would improve their results for multivalued maps in topological vector spaces. For two sets X and Y and each element x of X we associate a nonempty subset F (x) of Y and this correspondence x F (x) is called a multivalued mapping or a multifunction from X into Y ; i.e., F is a function from X to P (Y ), where P (Y ) is the set of all nonempty subsets of Y. The lower inverse of a multivalued mapping F is the multi-valued mapping F l of Y into X defined by F l (y) = {x X : y F (x)}, Corresponding author. address: 1

2 also for any nonempty subset B of Y we have, F l (B) = {x X : F (x) B }, finally it is understood that F l ( ) =. The set {x X : F (x) B} is the upper inverse of B and is denoted by F u (B). F is lower semicontinuous (upper semicontinuous), if for every open set U Y, F l (U)(F u (U)) is open in X. It is well known [2] that F is upper semi-continuous at x 0 if for each open set V containing F (x 0 ) there exists a neighborhood U of x 0 such that x U implies that F (x) V. For two multivalued maps F : X P (Y ) and G : Y P (Z) the composition GoF : X P (Z) is defined by GoF (x) = G(y). 2 Main Results y F (x) The following definition of quasi contraction is an extended definition of contraction in metric spaces and we achieve some results which they extend some results in invariant metric spaces. Definition 2.1. Suppose (X, τ) is a topological vector space. A multi-valued map F : X P (X) is said to be : (a) quasi contraction map if for all x, y X and any open neighborhood U of 0 there is a constant 0 c < 1 such that x y U implies that F (x) F (y) cu. (b) closed map if x n x, y n y and y n F (x n ) imply that y F (x). (c) bounded valued map if F (x) is a bounded set in X for all x X. (d) upper semi-continuous at x 0 if for each open set V containing F (x 0 ) there exists a neighborhood U(x 0 ) such that x U(x 0 ) implies that F (x) V. It is well known that any contraction multivalued map between metric spaces is continuous. Extending this fact is our next aim. Theorem 2.1. Suppose (X, τ) is a topological vector space and F : X P (X) is quasi contraction. If 0 F (0) then F is upper semicontinuous at 0. Proof. Suppose V is any open neighborhood of F (0). Consider open neighborhood U of 0 for which cu V. Then U = U {0} U. Therefore, F (U) F (U) F (0) cu V. Corollary 2.1. Suppose (X, τ) is a topological vector space and F : X P (X) is quasi contraction. Then F is upper semicontinuous. Proof. Consider x 0 X, y 0 F (x 0 ) and V an open neighborhood of F (x 0 ). Then V y 0 = U is an open neighborhood of 0. Define G(x) = F (x + x 0 ) y 0. We claim that g is contraction, too. To see this, if W is an open neighborhood of 0 and x y W, then G(x) G(y) = (F (x + x 0 ) y 0 ) (F (y + x 0 ) y 0 ) cw. Also, G(0) = F (x 0 ) y 0 contains 0, so from Theorem 2.1 for V y 0, there is open neighborhood W of 0 such that G(W ) V y 0, so F (W + x 0 ) V. 2 ISPACS GmbH

3 Lemma 2.1. Suppose (X, τ) is a locally convex space and x X. If F : X P (X) is a contraction bounded valued map, then any sequence {y n } is a Cauchy sequence, where y n F n (x) for all n N. Proof. Let U be an open convex neighborhood of 0 which is also balanced. From the assumption F (x) and so F (x) x are bounded sets. Since U is absorbent, there is α 0 > 0 such that F (x) x αu, for all α with α α 0, so F (x) x α 0 U. Then F n+1 (x) F n (x) c n (α 0 )U. Since 0 c < 1, there is N N such that (c m + + c n )α 0 < 1 for all m, n N. Therefore, if m > n N we have This completes the proof. F m+1 (x) F n+1 (x) c m α 0 U + + c n α 0 U = (c m + + c n )α 0 U U. The following result is other version of Banach contraction Theorem. First we need to the next lemma. Theorem 2.2. Suppose (X, τ) is a sequentially complete locally convex space and F : X P (X) is a quasi contraction bounded valued map. If F is closed multi-valued map, then F has a unique fixed point. Proof. Suppose U is any open neighborhood of 0 and x is any element in X. Make a sequence {y n } in Y by induction, where y 1 F (x) and y n+1 F (y n ) for all n N. Applying Lemma 2.1, {y n } is a Cauchy sequence. Since X is sequentially complete, so {y n } converges to an element y X. That y F (y) follows from closedness of F, y n F (y n 1 ) and y n 1 y. For the uniqueness, suppose x, y X are two distinct fixed points of F. Then there is a convex open neighborhood U of 0 such that x y / U. Since U is absorbent so there is α 0 > 0 such that x y α 0 U. Therefore, F (x) F (y) cα 0 U and so x y c n α 0 U for each n N. Since 0 c < 1, we can assume that c n α 0 < 1 for some n N. On the other hand, U is convex so c n α 0 U U. Consequently, x y U which is a contradiction. Remark 2.1. It should be noticed that closedness in Theorem 2.2 could be reduced to the following condition y n y and y n F (y n 1 ) = y F (y). As a special case of quasi contraction multi-valued maps, we introduce the quasi contraction maps. Definition 2.2. Suppose (X, τ) is a topological vector space. A function f : X X is said to be quasi contraction if for all x, y X and any open neighborhood U of 0 there is a constant 0 c < 1 such that x y U implies that f(x) f(y) cu. 3 ISPACS GmbH

4 Corollary 2.2. Suppose (X, τ) is a sequentially complete locally convex space, also suppose f : X X is quasi contraction. Then f has a unique fixed point. Proof. It is a direct result of Theorem 2.2. Theorem 2.3. Let (X, τ) be a locally convex space and f : X X be a quasi contraction map. If for some x o X there exists a convergence subsequence f n i (x 0 ) to an element u X, then u is a fixed point for f. Proof. F is quasi contractive, so (f n (x 0 )) n is a Cauchy sequence from the proof of Lemma 2.1. Hence from the assumption f n (x 0 ) u. From Theorem 2.1, f is continuous, so f(u) = f(lim f n (x 0 )) = lim f n+1 (x 0 ) = u. Definition 2.3. A family {A j : j J} of sets in X has finite intersection property if each finite subfamily of it, has nonempty intersection. For a multi-valued map F : X P (X), set O(F n (x)) = m n F m (x), where it is understood that F 0 (x) = {x}. The following result would improve a result of Ciric [4]. First we need to the following lemma. Lemma 2.2. [1] Suppose F : X P (X) is a multi-valued map and there is x 0 X such that O(x 0 ) has finite intersection property. Then F has a fixed point if O(F 2 (x)) F (x) for all x X. Proof. It is easy to see that F (O(x 0 )) O(x 0 ). Set K = {A O(x 0 ) : A, F (A) A}. Then partially ordered K by inclusion. Since O(x 0 ) has finite intersection property, so from Zorns lemma K has minimal element, say C. Then F (C) C, but F (F (C)) F (C) implies that F (C) = C. Now, if u / F (u) for each u C, then from assumption u / O(F 2 (u)). Since u C, so F (u) F (C) = C, therefore F k (u) C for any nonnegative integer k. Now O(F 2 (u)) = C follows from minimality of C. Consequently, u O(F 2 (u)) which is a contradiction. Theorem 2.4. Suppose X is a topological vector space, O(x) has the finite intersection property for each x X and F : X P (X) is a multi-valued map. Then F has a fixed point if x / F (x) implies that x / O(F m (x)) for all m 2. Proof. Assume x / F (x). We claim that x / O(F 2(x)). On the contrary there are two cases : (a) x = lim i y ni where n i 2, y ni F n i (x) (b) x F m (x) for some m 2. If (a) satisfies, then x (O(F m(x))) O(F m (x)) hence, from the assumption x F (x), which is a contradiction. Assume (b) satisfies, then x O(F m (x)) which is impossible again. Therefore, x / O(F 2 (x)) and so O(F 2 (x)) F (x). That F has a fixed point follows from Lemma ISPACS GmbH

5 References [1] M. Alimohammady and M. Roohi, Fixed point in minimal spaces, Nonlinear Analysis : Moddeling and Control, 10 (4)(2005), [2] J.P. Aubin and J. Siegel, Fixed point and stationary points of dissipative multi-valued maps, Proc. Amer. Math. Soc., 78(1980), [3] C. Berge, Espaces topologiques, Dunod, Paris, (1959). [4] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29(2002), [5] A. Cataldo, E. Lee, X. Liu, E. Matsikoudis and H. Zheng, Discrete-Event Systems: Generalizing metric spaces and fixed point semantics, to appear. [6] H. Covitz and S.B. Nadler Jr., Multi-valued contraction mappings in generalized metric space, Israel J. Math. 8 (1970), [7] Lj. B. Ciric, Fixed point theorems in topological spaces, Fund. Math. 87 (1975), 1-5. [8] M. Edelstien, On fixed and periodic points under contractive, J. London Math. Soc. 37 (1962), [9] 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), [10] P. Gerhardy, A quantitative version of Kirks fixed point theorem for asymptotic contractions, J. Math. Anal. Appl. (in press). [11] D. Gopal, M. Imdad, C. Vetro and M. Hasan, Fixed point theory for cyclic weak ϕ- contraction in fuzzy metric spaces, s, Volume 2012 (2012) [12] J. R. Jachymski, Converses to fixed point theorems of Zeremlo and Caristi, Nonlinear Anal. 52 (2003), [13] S.-Y. Jang, Ch. Park and H. Azadi Kenary, Fixed points and fuzzy stability of functional equations related to inner product, s, Volume 2012 (2012), [14] W. A. Kirk, On mappings with diminishing orbital diameters, J. London Math. Soc. 44 (1969), ISPACS GmbH

6 [15] J. Merryfield and Jr. J. D. Steein, A generalization of the Banach contraction principle, J. Math. Anal. Appl. 273 (2002), [16] S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), [17] E. Schorner, Ultrametric fixed point theorems and applications, International Conference and Workshop on Valuation Theory, University of Saskatchewan, Canada, [18] N. Singh, R. Jain and H. Dubey, Common fixed point theorems in non-archimedean fuzzy metric spaces, s, Volume 2102 (2012), [19] T. Wang, Fixed point theorems and fixed point stability for multivalued mappings on metric spaces, J. Nanjing Univ. Math. Baq. 6 (1989), ISPACS GmbH

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

### Convergence and stability results for a class of asymptotically quasi-nonexpansive mappings in the intermediate sense

Functional Analysis, Approximation and Computation 7 1) 2015), 57 66 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac Convergence and

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction

italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### TRIPLE FIXED POINTS IN ORDERED METRIC SPACES

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197-207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON

### Compactness in metric spaces

MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### Convergent Sequence. Definition A sequence {p n } in a metric space (X, d) is said to converge if there is a point p X with the following property:

Convergent Sequence Definition A sequence {p n } in a metric space (X, d) is said to converge if there is a point p X with the following property: ( ɛ > 0)( N)( n > N)d(p n, p) < ɛ In this case we also

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Mila Stojaković 1. Finite pure exchange economy, Equilibrium,

Novi Sad J. Math. Vol. 35, No. 1, 2005, 103-112 FUZZY RANDOM VARIABLE IN MATHEMATICAL ECONOMICS Mila Stojaković 1 Abstract. This paper deals with a new concept introducing notion of fuzzy set to one mathematical

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### A Note on the Powers of Bazilevič Functions

International Journal of Mathematical Analysis Vol. 9, 015, no. 4, 061-067 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.015.56169 A Note on the Powers of Bailevič Functions Marjono Faculty

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### CHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES

CHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES DEFINITION A Banach space is a real normed linear space that is a complete metric space in the metric defined by its norm. A complex Banach space is a

### The Real Numbers. Here we show one way to explicitly construct the real numbers R. First we need a definition.

The Real Numbers Here we show one way to explicitly construct the real numbers R. First we need a definition. Definitions/Notation: A sequence of rational numbers is a funtion f : N Q. Rather than write

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### Continuity of the Perron Root

Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### On Fixed Point Theorems in D-Metric Spaces

Int. Journal of Math. Analysis, Vol. 1, 2007, no. 22, 1059-1065 On Fixed Point Theorems in D-Metric Spaces Seong-Hoon Cho Department of Mathematics Hanseo University Chungnam, 356-706 South Korea shcho@hanseo.ac.kr

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.

MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.

### TOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS. Contents

TOPOLOGICAL PROOFS OF THE EXTREME AND INTERMEDIATE VALUE THEOREMS JAMES MURPHY Abstract. In this paper, I will present some elementary definitions in Topology. In particular, I will explain topological

### COMMON FIXED POINTS OF FOUR MAPS USING GENERALIZED WEAK CONTRACTIVITY AND WELL-POSEDNESS

Int. J. Nonlinear Anal. Appl. (011) No.1, 73 81 ISSN: 008-68 (electronic) http://www.ijnaa.com COMMON FIXED POINTS OF FOUR MAPS USING GENERALIZED WEAK CONTRACTIVITY AND WELL-POSEDNESS MOHAMED AKKOUCHI

### A NOTE ON CLOSED GRAPH THEOREMS. 1. Introduction

Acta Math. Univ. Comenianae Vol. LXXV, 2(2006), pp. 209 218 209 A NOTE ON CLOSED GRAPH THEOREMS F. GACH Abstract. We give a common generalisation of the closed graph theorems of De Wilde and of Popa. 1.

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

### Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations

Wang and Agarwal Advances in Difference Equations (2015) 2015:312 DOI 10.1186/s13662-015-0650-0 R E S E A R C H Open Access Relatively dense sets, corrected uniformly almost periodic functions on time

### Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg

### CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOM-VALUED VECTOR FUNCTIONS

C O L L O Q U I U M M A T H E M A T I C U M VOL. 97 2003 NO. 1 CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOM-VALUED VECTOR FUNCTIONS BY RAFAŁ KAPICA (Katowice) Abstract. Given a probability space (Ω,

### Available online at Math. Finance Lett. 2014, 2014:2 ISSN A RULE OF THUMB FOR REJECT INFERENCE IN CREDIT SCORING

Available online at http://scik.org Math. Finance Lett. 2014, 2014:2 ISSN 2051-2929 A RULE OF THUMB FOR REJECT INFERENCE IN CREDIT SCORING GUOPING ZENG AND QI ZHAO Think Finance, 4150 International Plaza,

### TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh

Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp -. TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties

Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### On the Exponential Diophantine Equation (12m 2 + 1) x + (13m 2 1) y = (5m) z

International Journal of Algebra, Vol. 9, 2015, no. 6, 261-272 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.5529 On the Exponential Diophantine Equation (12m 2 + 1) x + (13m 2 1) y

### ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### BOREL SETS, WELL-ORDERINGS OF R AND THE CONTINUUM HYPOTHESIS. Proof. We shall define inductively a decreasing sequence of infinite subsets of N

BOREL SETS, WELL-ORDERINGS OF R AND THE CONTINUUM HYPOTHESIS SIMON THOMAS 1. The Finite Basis Problem Definition 1.1. Let C be a class of structures. Then a basis for C is a collection B C such that for

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### CHAPTER III - MARKOV CHAINS

CHAPTER III - MARKOV CHAINS JOSEPH G. CONLON 1. General Theory of Markov Chains We have already discussed the standard random walk on the integers Z. A Markov Chain can be viewed as a generalization of

### A CHARACTERIZATION OF QUASICONVEX VECTOR-VALUED FUNCTIONS. 1. Introduction

1 2 A CHARACTERIZATION OF QUASICONVEX VECTOR-VALUED FUNCTIONS Abstract. The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are K-quasiconvex with

### v 1. v n R n we have for each 1 j n that v j v n max 1 j n v j. i=1

1. Limits and Continuity It is often the case that a non-linear function of n-variables x = (x 1,..., x n ) is not really defined on all of R n. For instance f(x 1, x 2 ) = x 1x 2 is not defined when x

### Convex analysis and profit/cost/support functions

CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

### ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS

ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute

### On the Algebraic Structures of Soft Sets in Logic

Applied Mathematical Sciences, Vol. 8, 2014, no. 38, 1873-1881 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43127 On the Algebraic Structures of Soft Sets in Logic Burak Kurt Department

### Pooja Sharma and R. S. Chandel FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN FUZZY METRIC SPACES. 1. Introduction

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Pooja Sharma and R. S. Chandel FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN FUZZY METRIC SPACES Abstract. This paper presents some fixed point

### PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.

### BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam

Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad

### be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that

Problem 1A. Let... X 2 X 1 be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that i=1 X i is nonempty and connected. Since X i is closed in X, it is compact.

### 1 Fixed Point Iteration and Contraction Mapping Theorem

1 Fixed Point Iteration and Contraction Mapping Theorem Notation: For two sets A,B we write A B iff x A = x B. So A A is true. Some people use the notation instead. 1.1 Introduction Consider a function

### EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

### Pacific Journal of Mathematics

Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000

### CHARLES D. HORVATH AND MARC LASSONDE. (Communicated by Peter Li)

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 4, April 1997, Pages 1209 1214 S 0002-9939(97)03622-8 INTERSECTION OF SETS WITH n-connected UNIONS CHARLES D. HORVATH AND MARC LASSONDE

### On the convergence of modified S-iteration process for generalized asymptotically quasi-nonexpansive mappings in CAT(0) spaces

Functional Analysis, Approximation and Computation 6 (2) (2014), 29 38 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac On the convergence

### EXTENSIONS OF MAPS IN SPACES WITH PERIODIC HOMEOMORPHISMS

BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 4, July 1972 EXTENSIONS OF MAPS IN SPACES WITH PERIODIC HOMEOMORPHISMS BY JAN W. JAWOROWSKI Communicated by Victor Klee, November 29, 1971

### Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008

Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008 David R. Wilkins Copyright c David R. Wilkins 1989 2008 Contents 1 Basic Theorems of Complex Analysis 1 1.1 The Complex Plane........................

### The Second Hankel Determinant of Functions Convex in One Direction

International Journal of Mathematical Analysis Vol. 10, 2016, no. 9, 423-428 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2016.619 The Second Hankel Determinant of Functions Convex in One

### Andrew McLennan January 19, Winter Lecture 5. A. Two of the most fundamental notions of the dierential calculus (recall that

Andrew McLennan January 19, 1999 Economics 5113 Introduction to Mathematical Economics Winter 1999 Lecture 5 Convergence, Continuity, Compactness I. Introduction A. Two of the most fundamental notions

### Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### Convexity in R N Main Notes 1

John Nachbar Washington University December 16, 2016 Convexity in R N Main Notes 1 1 Introduction. These notes establish basic versions of the Supporting Hyperplane Theorem (Theorem 5) and the Separating

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

### ON PARAMETRIC LIMIT SUPERIOR OF A SEQUENCE OF ANALYTIC SETS

Real Analysis Exchange ISSN:0147-1937 Vol. 31(1), 2005/2006, pp. 1 5 Szymon G l ab, Mathematical Institute, Polish Academy of Science, Śniadeckich 8, 00-956 Warszawa, Poland.email: szymon glab@yahoo.com

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### POSITIVE INTEGERS, INTEGERS AND RATIONAL NUMBERS OBTAINED FROM THE AXIOMS OF THE REAL NUMBER SYSTEM

MAT 1011 TECHNICAL ENGLISH I 03.11.2016 Dokuz Eylül University Faculty of Science Department of Mathematics Instructor: Engin Mermut Course assistant: Zübeyir Türkoğlu web: http://kisi.deu.edu.tr/engin.mermut/

### f(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write

Math 525 Chapter 1 Stuff If A and B are sets, then A B = {(x,y) x A, y B} denotes the product set. If S A B, then S is called a relation from A to B or a relation between A and B. If B = A, S A A is called

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Double Sequences and Double Series

Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions,

### k=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =

Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem

### THE REGULAR OPEN CONTINUOUS IMAGES OF COMPLETE METRIC SPACES

PACIFIC JOURNAL OF MATHEMATICS Vol 23 No 3, 1967 THE REGULAR OPEN CONTINUOUS IMAGES OF COMPLETE METRIC SPACES HOWARD H WICKE This article characterizes the regular T o open continuous images of complete

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### Topics in weak convergence of probability measures

This version: February 24, 1999 To be cited as: M. Merkle, Topics in weak convergence of probability measures, Zb. radova Mat. Inst. Beograd, 9(17) (2000), 235-274 Topics in weak convergence of probability

### ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### A Simple Proof of Duality Theorem for Monge-Kantorovich Problem

A Simple Proof of Duality Theorem for Monge-Kantorovich Problem Toshio Mikami Hokkaido University December 14, 2004 Abstract We give a simple proof of the duality theorem for the Monge- Kantorovich problem

### Cylinder Maps and the Schwarzian

Cylinder Maps and the Schwarzian John Milnor Stony Brook University (www.math.sunysb.edu) Bremen June 16, 2008 Cylinder Maps 1 work with Araceli Bonifant Let C denote the cylinder (R/Z) I. (.5, 1) fixed