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


 Nickolas Lewis
 2 years ago
 Views:
Transcription
1 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* Communicated by Mohammad Asadzadeh Abstract. In this paper we generalize the main results of Bai, Wang and Ge (Electron J. Diff. Eqns. 6 (2), 8) by considering a general type of boundary value problem associated with a nonlinear fractional differential equation. We obtain sufficient conditions for the existence of at least three positive solution with corresponding upper and lower bounds.. Introduction Fractional differential equations have been studied by many authors caused by its applications in solving ordinary differential equations and also many applications in physics, mechanics, chemistry, and engineering. Positive solutions for ordinary differential equations and difference equations also have been considered by many authors, e.g. [, 6, 8]. The major tool in finding positive solutions for both fractional and ordinary differential equations have been fixed point theorems and Leray Schauder theory. In this paper we consider a typical fractional differential equations and improve the results obtained in [, 5] using fixed MSC(2): Primary: 3B5; Secondary: 3B8 Keywords: Fractional differential equation, Boundary value problem, Positive solution, Green s function, Fixedpoint theorem Received: 3 January 27, Revised: 7 June 27, Accepted: 25 June 27 Corresponding author c 27 Iranian Mathematical Society.
2 2 Dehghani and Ghanbari point theorems. We consider a nonlinear fractional differential equation of the following type: { D α x(t) + q(t)f(t, x(t), x (t)) ( < t < ) (.) x() x(), (.2) where x(t) and q(t) are nonnegative in (, ), f : [, ] [, ) R [, ) is continuous, 2 α < 3 is a real number, and D α is the standard RiemannLioville fractional differentiation (see definition 2.2). Furthermore q(t) does not identically vanish on any subinterval of (, ) and q(t) satisfies the following inequality: < [t( t)] α q(t) dt <. The motivation for considering 2 α < 3 is the importance of second order differential equations, which is equivalent to α 2 in this case. We obtain our main result by using the fixed point theorem of a cone preserving operator on an ordered Banach space that will be defined in Section 2. First we obtain an integral representation of the solution by the corresponding Green s function. 2. Background materials and definitions First we present the necessary definitions from fractional calculus theory and cone theory in Banach spaces, e.g. [2, 9]. Definition 2.. The fractional integral of order α > of a function y : (, ) R is given by I α y(t) t (t s) α y(s) ds, provided the right side is pointwise defined on (, ). Definition 2.2. The fractional derivative of order α > of a continuous function y : (, ) R is given by D α y(t) Γ(n α) ( d dt )n t y(s) ds, (t s) α n+
3 Nonlinear fractional differential equations 3 where n [α] +. Remark 2.3. As a basic example we quote for λ >, D α t λ Γ(λ + ) Γ(λ α + ) tλ α, giving in particular D α t α m, m, 2,..., N. Where N is the smallest integer greater than or equal to α. From Definition 2.2 and Remark 2.3, we obtain the following lemma. Lemma 2.. Let α >. If we assume u C(, ) L (, ), then the fractional differential equation D α u(t), has u(t) C t α + C 2 t α C N t α N, C i R, i,..., N as unique solutions. As D α I α u u for all u C(, ) L(, ), from Lemma 2. we deduce the following law of composition. Lemma 2.5. Assume that u C(, ) L (, ) with a fractional derivative of order α > that belongs to u C(, ) L (, ). Then I α D α u(t) u(t) + C t α + C 2t α C Nt α N, for some C i R, i,..., N. Now we find the integral representation of the solution and specify the corresponding Green s function. Lemma 2.6. Given y C[, ] and 2 α < 3, the unique solution of D α u(t) + y(t) ( < t < ) (2.) u() u(), (2.2)
4 Dehghani and Ghanbari is where G(t, s) u(t) G(t, s)y(s) ds, [t( s)] α (t s) α ( s t ) [t( s)] α. ( t s ) (2.3) Proof. We may apply Lemma 2.2 to reduce Eq.(2.) to an equivalent equation u(t) I α y(t) C t α C 2t α 2 C 3t α 3, for some C, C 2, C 3 R. Consequently, the general solution of Eq.(2.) is t (t s) α u(t) y(s) ds C t α C 2t α 2 C 3t α 3. By (2.2), therefore, C 2 C 3 and C ( s) α y(s) ds. Therefore, the unique solution of the differential equation (2.) subject to (2.2) is u(t) t t (t s) α ( s) α t α y(s) ds + y(s) ds [t( s)] α (t s) α [t( s)] α y(s) ds + y(s) ds G(t, s)y(s) ds. The proof is now completed. Lemma 2.7. The function G(t, s) defined by Eq.(2.3) satisfies the following conditions: (i) G(t, s) >, for t, s (, ) (ii) There exists a positive function γ C(, ) such that min G(t, s) γ(s) max G(t, s) γ(s)g(s, s), for < s <. t 3 t t
5 Nonlinear fractional differential equations 5 Proof. From (2.3) it is clear that G(t, s) > for s, t (, ). In the following, we consider the existence of γ(s). For s (, ), G(t, s) is s increasing with respect to t for t, with respect to t for t s ( ( s) α α 2 ), ( ( s) α α 2 ) and decreasing, and for s (, ), G(t, s) is decreasing with respect to t for s t and increasing with respect to t for s t. If we define then g (t, s) [t( s)]α (t s) α, g 2 (t, s) min t 3 G(t, s) min{g (, s), g (, s)}, s (, ) min{g ( 3, s), g 2(, s)}, s (, 3 ) g 2 (, s), s ( 3, ) min{g (, s), g (, s)}, s (, ) g ( 3, s), s (, r) g 2 (, s), s (r, ) min{ ( s)α, [t( s)]α, {[ ( s)]α ( s)α }}, s (, ) {[ 3 ( s)]α ( 3 s)α }, s (, r) ( s) α, α s (r, ) where < r < 3 is the unique solution of the equation [ 3 ( r)]α ( 3 r)α ( r)α α ; in particular r.5 as α 2 and r.26 as α 3. Using the monotonicity of G(t, s), we have max G(t, s) G(s, s) t [s( s)]α, s (, ).
6 6 Dehghani and Ghanbari Consequently, the desired function γ is γ(s) This completes the proof. min{ ( s)α, [ [s( s)] α ( s)]α ( s)α }, s (, [s( s)] α ) [ 3 ( s)]α ( 3 s)α, s ( [s( s)] α, r). s (r, ) (s) α Definition 2.8. Let E be a real Banach space over R. A nonempty convex closed set P E is said to be a cone provided that (i) au P for all u P and all a and (ii) u, u P implies u. Note that every cone P E induces an ordering in E given by x y if y x P. Let β and θ be nonnegative continuous convex functionals on P, ϕ be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P. Then for positive real numbers a, b, c and d, we define the following convex sets: P (β, d) {x P β(x) < d}, P (β, ϕ, b, d) {x P b ϕ(x), β(x) d}, P (β, θ, ϕ, b, c, d) {x P b ϕ(x), θ(x) c, β(x) d}, and a closed set R(β, ψ, a, d) {x P a ψ(x), β(x) d}. The following fixed point theorem due to Avery and Peterson is fundamental in the proof of our main results. Theorem 2.9. [2] Let P be a cone in a real Banach space E. Let β and θ be nonnegative continuous convex functionals on P, ϕ be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P satisfying ψ(λx) λψ(x) for λ such that for some positive numbers M and d, ϕ(x) ψ(x) and x M β(x) (2.) for all x P (β, d). Suppose that T : P (β, d) P (β, d) is completely continuous and there exist positive numbers a, b, and c with a < b such
7 Nonlinear fractional differential equations 7 that (s) {x P (β, θ, ϕ, b, c, d) ϕ(x) > b}, and ϕ(t x) > b for x P (β, θ, ϕ, b, c, d); (s2) ϕ(t x) > b for x P (β, ϕ, b, d) with θ(t x) > c; (s3) R(β, ψ, a, d) and ψ(t x) < a for x R(β, ψ, a, d) with ψ(x) a. Then T has at least three fixed points x, x 2, x 3 P (β, d) such that β(x i ) d for i, 2, 3; b < ϕ(x ); a < ψ(x 2 ) with ϕ(x 2 ) < b; ψ(x 3 ) < a. 3. Main results In this section, we impose growth conditions on f which allow us to apply Theorem 2. to establish the existence of triple positive solutions of the boundary value problem (.) with initial conditions (.2). Let X C [, ] be endowed with the ordering x y if x(t) y(t) for all t [, ], and the maximum norm Define the cone P X by x max{ max x(t), max t t x (t) }. P {x X x(t) on [, ], and x() x() }. Let the nonnegative continuous concave functional ϕ, the nonnegative continuous convex functionals θ, β, and the nonnegative continuous functional ψ be defined on the cone P by β(x) max t x (t), The following is wellknown. ψ(x) θ(x) max t x(t), ϕ(x) min x(t). t 3 Lemma 3.. [5]. If x P, then max t x(t) 2 max t x (t). By Lemma 3., the functionals defined above satisfy ϕ(x) θ(x) ψ(x), x max{θ(x), β(x)} β(x) (3.)
8 8 Dehghani and Ghanbari for all x P (β, d) P. Therefore, condition (2.) is satisfied. Lemma 3.2. Let T : P P be the operator defined by T x(t) G(t, s)q(s)f(s, x(s), x (s)) ds Then T : P P is completely continuous. Proof. The operator T : P P is continuous in view of nonnegativeness and continuity of G(t, s) and q(t)f(t, x, x ). Let Ω P be bounded, i.e., there exists a positive constant M > such that x M for all x Ω. Let L max q(t)f(t, x(t), t, x M x (t)) +. Then for x Ω, we have T x(t) G(t, s)q(s)f(s, x(s), x ) ds L G(s, s) ds. Hence, T (Ω) is bounded. On the other hand, given ɛ >, we set δ 2 ( ɛ L ) α. Then, for each x Ω, t, t 2 [, ], t < t 2, and t 2 t < δ one has T x(t 2 ) T x(t ) < ɛ. That is to say, T (Ω) is equicontinuous.
9 Nonlinear fractional differential equations 9 In fact, we have T x(t 2 ) T x(t ) G(t 2, s)q(s)f(s, x(s), x (s)) ds G(t, s)q(s)f(s, x(s), x (s)) ds + t [G(t 2, s) G(t, s)]q(s)f(s, x(s), x (s)) ds t 2 [G(t 2, s) G(t, s)]q(s)f(s, x(s), x (s)) ds t2 + [G(t 2, s) G(t, s)]q(s)f(s, x(s), x (s)) ds t < L [ t ( s) α (t α 2 t α ) ds + + ( s) α (t α 2 t α ) ds t 2 t2 t ( s) α (t α 2 t α ) ds In the following, we divide the proof into two cases. CASE. δ t < t 2 <. ] < L (tα 2 t α ). T x(t 2 ) T x(t ) < L (tα 2 t α ) L α δ 2 α (t 2 t ) CASE 2. t < δ, t 2 < 2δ. L (α )δα ɛ. T x(t 2 ) T x(t ) < L (tα 2 t α ) L tα 2 L (2δ)α ɛ. By the means of the ArzelaAscoli theorem, we have T : P P is completely continuous. The proof is now completed.
10 Dehghani and Ghanbari Let M N δ 3 α ( s)α s α 2 q(s) ds, G(s, s)q(s) ds, γ(s)g(s, s)q(s) ds. We assume there exist constants < a < b d 8 such that (A) f(t, u, v) d M, for (t, u, v) [, ] [, d 2 ] [ d, d]; (A2) f(t, u, v) > b δ, for (t, u, v) [, 3 ] [b, b] [ d, d]; (A3) f(t, u, v) < a N, for (t, u, v) [, ] [, a] [ d, d]. Theorem 3.3. Under the assumptions (A)(A3), the boundaryvalue problem (.)(.2) has at least three positive solutions x, x 2, and x 3 satisfying max t x i(t) d, for i, 2, 3; b < min x (t) ; (3.2) t 3 a < max x 2(t), with min x t 2 (t) < b; t 3 max x 3(t) < a. t Proof. The boundary value problem (.) with initial conditions (.2) has a solution x x(t) if and only if x solves the operator equation x(t) T x(t) G(t, s)q(s)f(s, x(s), x (s)) ds where T : P P, is completely continuous (Lemma 3.2). We now show that all the conditions of Theorem 2. are satisfied. If x P (β, d), then β(x) max t x (t) d. By Lemma 3. and max x(t) d t 2, the assumption (A) implies f(t, x(t), x (t)) d M.
11 Nonlinear fractional differential equations Define H(t, s) (α ) [( s)α t α 2 (t s) α 2 ], ( s t ) (α ) ( s)α t α 2, ( t s ) where 2 α < 3. For given s (, ), H(t, s) is decreasing with respect to t for s t and increasing with respect to t for t s. Using monotonicity of H(t, s), we have Consequently, (α ) max H(t, s) H(s, s) t ( s)α s α 2. β(t x) max (T t x) (t) t max (α ) t [( s)α t α 2 (t s) α 2 ]q(s)f(s, x(s), x (s)) ds (α ) + t ( s)α t α 2 q(s)f(s, x(s), x (s)) ds max H(t, s)q(s)f(s, x(s), x (s)) ds t (α ) d M (α ) ( s) α s α 2 q(s)f(s, x(s), x (s)) ds Hence, T : P (β, d) P (β, d). ( s) α s α 2 q(s) ds d M M d. To check the condition (s) of Theorem 2.9, we choose x(t) b, t. It is easy to see that x(t) b P (β, θ, ϕ, b, b, d) and ϕ(x) ϕ(b) > b, and so x {P (β, θ, ϕ, b, b, d) ϕ(x) > b}. Hence, if x P (β, θ, ϕ, b, b, d), then b x(t) b, x (t) d for t 3. From assumption (A2), we have f(t, x(t), x (t)) b δ for t 3, and by Lemma 2. ϕ(t x) min (T x)(t) t 3 b δ 3 γ(s)g(s, s)q(s) ds b δ δ b. γ(s)g(s, s)q(s)f(s, x(s), x (s)) ds
12 2 Dehghani and Ghanbari Thus, ϕ(t x) > b, for all x P (β, θ, ϕ, b, b, d). This shows that the condition (s) of Theorem 2.9 is satisfied. Notice that the condition (s) of Theorem 2.9 implies the condition (s2) of Theorem 2.9. Clearly, as ψ() < a, there holds that R(β, ψ, a, d). Suppose that x R(β, ψ, a, d) with ψ(x) a. Then, by the assumption (A3), ψ(t x) max (T x)(t) max t t < a N max t G(t, s)q(s) ds a N G(t, s)q(s)f(s, x(s), x (s)) ds G(s, s)q(s) ds a. So, the condition (s3) of Theorem 2.9 is satisfied. Therefore, an application of Theorem 2. implies that the boundary value problem (.) with initial conditions (.2) has at least three positive solutions x, x 2 and x 3 satisfying (3.). Example 3.. Consider the boundaryvalue problem where D 5 2 x(t) + f(t, x(t), x (t)) < t <, (3.3) x() x(), (3.) { e f(t, u, v) t u3 + ( v 6 )3 for u 2, e t ( v 6 )3 for u 2. Here we have α 5/2, q(t). By definition of M, N, and δ we have M α ( s)α s α 2 q(s) ds, (α )Γ(α ) π/8.22. Γ(2α ) Similarly we have N π/32.5. Using Lemma 2., r is the root of the equation [ 3 ( r)]α ( 3 r)α ( r)α α,
13 Nonlinear fractional differential equations 3 which is r.53 in this example. Substituting γ(s) given by Lemma 2. and observing that G(s, s) [s( s)]α, we find δ 3 γ(s)g(s, s)q(s) ds.. Choosing a, b 3, d 6, we have f(t, u, v) < a N 2 for t, u, 6 v 6; f(t, u, v) > b δ 3 for t 3, 3 u 2, 6 v 6; f(t, u, v) < d M for t, u 3, 6 v 6. Then all assumptions of Theorem 3. hold. Thus, by Theorem 3., the boundary value problem (3.3) with initial conditions (3.) has at least three positive solutions x, x 2 and x 3 such that max t x i(t) 6, for i, 2, 3 ; 3 < min x (t) ; t 3 < max x 2(t), with min x t 2 (t) < 3 ; t 3 max x 3(t) <. t Concluding Remark. As we mentioned in the abstract, we considered a general boundary values problem for a nonlinear fractional differential equation of the form (.). The particular case α 2 has been studied by [5] for an ordinary differential equation of the form x + q(t)f(t, x(t), x (t)) with the same boundary conditions as (.2). Our paper in fact generalizes the main results of [5] for a more general boundary value problem. Acknowledgment The authors would like to thank the anonymous referee for careful reading of the manuscript and valuable comments.
14 Dehghani and Ghanbari References [] R. P. Agarwal, D. O Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference, and Integral Equation, Kluwer Academic Publishers, Boston, 999. [2] R. I. Avery and A. C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 2 (2), [3] A. Babakhani and V. D. Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (23), 3 2. [] Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 3 (25), [5] Z. Bai, Y. Wang and W. Ge, Triple positive solutions for a class of twopoint boundaryvalue problems, Electron. J. Diff. Eqns. Vol. 2 (6)(2), pp. 8. [6] J. Henderson and Y. Wang, Positive solutions for nonlinear eigenvalue problems J. Math. Anal. Appl. 28 (997), [7] M. A. Krasnosel skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 96. [8] W. C. Lian, F. H. Wong and C. C. Yeh, On the existence of positive solutions of nonlinear second order differential equations Proc. Amer. Math. Soc. 2 (996), [9] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York, 993. [] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 999. [] J. Spanier and K. Oldham, The Fractional Calculus, Theory and Applications of Differentiation and Integration to Arbitrary Order, Academid Press, New York, 97. [2] S. Q. Zhang, Existence of positive solution for some class of a nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (23), [3] S. Q. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252 (2), R. Dehghani and K. Ghanbari Department of Mathematics Sahand University of Technology Tabriz, Iran
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
More informationMultiple Solutions to a Boundary Value Problem for an nth Order Nonlinear Difference Equation
Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni, & J. Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp 129 136. ISSN: 10726691.
More informationSolutions series for some nonharmonic motion equations
Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp 115 122. http://ejde.math.swt.edu or http://ejde.math.unt.edu
More informationOn construction of converging sequences to solutions of boundary value problems
Mathematical Modelling and Analysis ISSN: 13926292 (Print) 16483510 (Online) Journal homepage: http://www.tandfonline.com/loi/tmma20 On construction of converging sequences to solutions of boundary value
More informationOscillation of Higher Order Fractional Nonlinear Difference Equations
International Journal of Difference Equations ISSN 09736069, Volume 10, Number 2, pp. 201 212 (2015) http://campus.mst.edu/ijde Oscillation of Higher Order Fractional Nonlinear Difference Equations Senem
More informationOn the Fractional Derivatives at Extreme Points
Electronic Journal of Qualitative Theory of Differential Equations 212, No. 55, 15; http://www.math.uszeged.hu/ejqtde/ On the Fractional Derivatives at Extreme Points Mohammed AlRefai Department of
More informationProperties 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 nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationEXPLICIT ABS SOLUTION OF A CLASS OF LINEAR INEQUALITY SYSTEMS AND LP PROBLEMS. Communicated by Mohammad Asadzadeh. 1. Introduction
Bulletin of the Iranian Mathematical Society Vol. 30 No. 2 (2004), pp 2138. EXPLICIT ABS SOLUTION OF A CLASS OF LINEAR INEQUALITY SYSTEMS AND LP PROBLEMS H. ESMAEILI, N. MAHDAVIAMIRI AND E. SPEDICATO
More informationNotes 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
More informationTRIPLE FIXED POINTS IN ORDERED METRIC SPACES
Bulletin of Mathematical Analysis and Applications ISSN: 18211291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON
More informationON TOPOLOGICAL SEQUENCE ENTROPY AND CHAOTIC MAPS ON INVERSE LIMIT SPACES. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXVIII, 2(1999), pp. 205 211 205 ON TOPOLOGICAL SEQUENCE ENTROPY AND CHAOTIC MAPS ON INVERSE LIMIT SPACES J. S. CANOVAS Abstract. The aim of this paper is to prove the
More informationM. Abu Hammad 1, R. Khalil 2. University of Jordan
International Journal of Differential Equations and Applications Volume 13 No. 3 014, 177183 ISSN: 131187 url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijdea.v13i3.1753 PA acadpubl.eu ABEL S
More informationSOME TYPES OF CONVERGENCE OF SEQUENCES OF REAL VALUED FUNCTIONS
Real Analysis Exchange Vol. 28(2), 2002/2003, pp. 43 59 Ruchi Das, Department of Mathematics, Faculty of Science, The M.S. University Of Baroda, Vadodara  390002, India. email: ruchid99@yahoo.com Nikolaos
More informationARTICLE IN PRESS Nonlinear Analysis ( )
Nonlinear Analysis ) Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A question concerning a polynomial inequality and an answer M.A. Qazi a,,
More informationRAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER
Volume 9 (008), Issue 3, Article 89, pp. RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER MARK B. VILLARINO DEPTO. DE MATEMÁTICA, UNIVERSIDAD DE COSTA RICA, 060 SAN JOSÉ,
More informationLINEAR SEARCH FOR A BROWNIAN TARGET MOTION 1
23,23B(3):321327 LINEAR SEARCH FOR A BROWNIAN TARGET MOTION 1 A. B. ElRayes 1 Abd ElMoneim A. Mohamed 1 Hamdy M. Abou Gabal 2 1.Military Technical College, Cairo, Egypt 2.Department of Mathematics,
More informationApplications of Fourier series
Chapter Applications of Fourier series One of the applications of Fourier series is the evaluation of certain infinite sums. For example, n= n,, are computed in Chapter (see for example, Remark.4.). n=
More informationRemark on Rotation of Bilinear Vector Fields
Remark on Rotation of Bilinear Vector Fields A.M. Krasnosel skii Institute for Information Transmission Problems, Moscow, Russia Email: amk@iitp.ru A.V. Pokrovskii Department of Applied Mathematics and
More informationThe distance of a curve to its control polygon J. M. Carnicer, M. S. Floater and J. M. Peña
The distance of a curve to its control polygon J. M. Carnicer, M. S. Floater and J. M. Peña Abstract. Recently, Nairn, Peters, and Lutterkort bounded the distance between a Bézier curve and its control
More informationSequences of Functions
Sequences of Functions Uniform convergence 9. Assume that f n f uniformly on S and that each f n is bounded on S. Prove that {f n } is uniformly bounded on S. Proof: Since f n f uniformly on S, then given
More informationMath 497C Sep 17, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 17, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 4 1.9 Curves of Constant Curvature Here we show that the only curves in the plane with constant curvature are lines and circles.
More informationRemark on Rotation of Bilinear Vector Fields
Remark on Rotation of Bilinear Vector Fields A.M. Krasnosel skii Institute for Information Transmission Problems, Moscow, Russia Email: Sashaamk@iitp.ru A.V. Pokrovskii National University of Ireland
More informationProblem 1 (10 pts) Find the radius of convergence and interval of convergence of the series
1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,
More informationQuasi Contraction and Fixed Points
Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa00168, 6 Pages doi:10.5899/2012/jnaa00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady
More informationFurther 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
More informationMonotone maps of R n are quasiconformal
Monotone maps of R n are quasiconformal K. Astala, T. Iwaniec and G. Martin For Neil Trudinger Abstract We give a new and completely elementary proof showing that a δ monotone mapping of R n, n is K quasiconformal
More informationTHE RICCI FLOW ON RADIALLY SYMMETRIC R 3. Thomas A. Ivey. Department of Mathematics UC San Diego La Jolla CA
THE RICCI FLOW ON RADIALLY SYMMETRIC R 3 Thomas A. Ivey Department of Mathematics UC San Diego La Jolla CA 920930112 tivey@euclid.ucsd.edu Introduction The Ricci flow g/ t = 2Ric(g) is a way of evolving
More informationPacific 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
More informationLegendre fractional differential equation and Legender fractional polynomials
International Journal of Applied Mathematical Research, 3 (3) (2014) 214219 c Science Publishing Corporation www.sciencepubco.com/index.php/ijamr doi: 10.14419/ijamr.v3i3.2747 Research Paper Legendre
More informationCourse 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...............
More informationDynamical Systems Method for solving nonlinear operator equations
Dynamical Systems Method for solving nonlinear operator equations A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 665062602, USA E:mail: ramm@math.ksu.edu Abstract Consider an
More informationHow many miles to βx? d miles, or just one foot
How many miles to βx? d miles, or just one foot Masaru Kada Kazuo Tomoyasu Yasuo Yoshinobu Abstract It is known that the Stone Čech compactification βx of a metrizable space X is approximated by the collection
More informationCharacterizing the Sum of Two Cubes
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 6 (003), Article 03.4.6 Characterizing the Sum of Two Cubes Kevin A. Broughan University of Waikato Hamilton 001 New Zealand kab@waikato.ac.nz Abstract
More information1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.
More informationOn the greatest and least prime factors of n! + 1, II
Publ. Math. Debrecen Manuscript May 7, 004 On the greatest and least prime factors of n! + 1, II By C.L. Stewart In memory of Béla Brindza Abstract. Let ε be a positive real number. We prove that 145 1
More informationSolvability of Fractional Dirichlet Problems with Supercritical Gradient Terms.
Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms. Erwin Topp P. Universidad de Santiago de Chile Conference HJ2016, Rennes, France May 31th, 2016 joint work with Gonzalo Dávila
More informationSinguläre Phänomene und Skalierung in mathematischen Modellen
RHEINISCHE FRIDRICHWILHELMUNIVERSITÄT BONN Diffusion Problem With Boundary Conditions of Hysteresis Type N.D.Botkin, K.H.Hoffmann, A.M.Meirmanov, V.N.Starovoitov no. Sonderforschungsbereich 6 Singuläre
More informationON NONARCHIMEDEAN MODULAR METRIC SPACES AND SOME NONLINEAR CONTRACTION MAPPINGS
ON NONARCHIMEDEAN MODULAR METRIC SPACES AND SOME NONLINEAR CONTRACTION MAPPINGS M.PAKNAZAR, M.A.KUTBI, M.DEMMA, P.SALIMI Abstract. Proving fixed point theorems in modular metric spaces are not possible
More information7. Operator Theory on Hilbert spaces
7. Operator Theory on Hilbert spaces In this section we take a closer look at linear continuous maps between Hilbert spaces. These are often called bounded operators, and the branch of Functional Analysis
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationA MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASEFIELD MODEL WITH MEMORY
Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASEFIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,
More informationLecture 3: Fourier Series: pointwise and uniform convergence.
Lecture 3: Fourier Series: pointwise and uniform convergence. 1. Introduction. At the end of the second lecture we saw that we had for each function f L ([, π]) a Fourier series f a + (a k cos kx + b k
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS
ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute
More informationRIGIDITY OF MANIFOLDS WITH BOUNDARY UNDER A LOWER RICCI CURVATURE BOUND
RIGIDITY OF MANIFOLDS WITH BOUNDARY UNDER A LOWER RICCI CURVATURE BOUND YOHEI SAKURAI Abstract. We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature
More informationFixed 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
More informationOn the Oscillation of Nonlinear Fractional Difference Equations
Mathematica Aeterna, Vol. 4, 2014, no. 1, 9199 On the Oscillation of Nonlinear Fractional Difference Equations M. Reni Sagayaraj, A.George Maria Selvam Sacred Heart College, Tirupattur  635 601, S.India
More informationA Game Theoretic Approach for Solving Multiobjective Linear Programming Problems
LIBERTAS MATHEMATICA, vol XXX (2010) A Game Theoretic Approach for Solving Multiobjective Linear Programming Problems Irinel DRAGAN Abstract. The nucleolus is one of the important concepts of solution
More informationA new continuous dependence result for impulsive retarded functional differential equations
CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationParabolic surfaces in hyperbolic space with constant Gaussian curvature
Parabolic surfaces in hyperbolic space with constant Gaussian curvature Rafael López Departamento de Geometría y Topología Universidad de Granada 807 Granada (Spain) email: rcamino@ugr.es eywords: hyperbolic
More informationOn generalized gradients and optimization
On generalized gradients and optimization Erik J. Balder 1 Introduction There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus
More informationDouble Sequences and Double Series
Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine Email: habil@iugaza.edu Abstract This research considers two traditional important questions,
More informationAcknowledgements: I would like to thank Alekos Kechris, who told me about the problem studied here.
COMPACT METRIZABLE GROUPS ARE ISOMETRY GROUPS OF COMPACT METRIC SPACES JULIEN MELLERAY Abstract. This note is devoted to proving the following result: given a compact metrizable group G, there is a compact
More informationIntroduction to Green s Functions: Lecture notes 1
October 18, 26 Introduction to Green s Functions: Lecture notes 1 Edwin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE16 91 Stockholm, Sweden Abstract In the present notes I try to give a better
More informationWeak topologies. David Lecomte. May 23, 2006
Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such
More informationHOLOMORPHIC FRAMES FOR WEAKLY CONVERGING HOLOMORPHIC VECTOR BUNDLES
HOLOMORPHIC FRAMES FOR WEAKLY CONVERGING HOLOMORPHIC VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Perhaps the most useful analytic tool in gauge theory is the Uhlenbeck compactness
More informationSumit 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
More information1 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
More informationAdvanced results on variational inequality formulation in oligopolistic market equilibrium problem
Filomat 26:5 (202), 935 947 DOI 0.2298/FIL205935B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Advanced results on variational
More informationMath 504, Fall 2013 HW 3
Math 504, Fall 013 HW 3 1. Let F = F (x) be the field of rational functions over the field of order. Show that the extension K = F(x 1/6 ) of F is equal to F( x, x 1/3 ). Show that F(x 1/3 ) is separable
More information5. Convergence of sequences of random variables
5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,
More informationResolution of Singularity
Resolution of Singularity Ka Laam Chan November, 2011 Abstract We will follow the first two sections of Kirwan Chapter 7 to give us a description of the resolution of singularity C of a sinuglar curve
More informationMultiple positive solutions of a nonlinear fourth order periodic boundary value problem
ANNALES POLONICI MATHEMATICI LXIX.3(1998) Multiple positive solutions of a nonlinear fourth order periodic boundary value problem by Lingbin Kong (Anda) and Daqing Jiang (Changchun) Abstract.Thefourthorderperiodicboundaryvalueproblemu
More information4.3 Limit of a Sequence: Theorems
4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,
More informationRegularity of solution maps of differential inclusions under state constraints
Regularity of solution maps of differential inclusions under state constraints Piernicola Bettiol, Hélène Frankowska Abstract Consider a differential inclusion under state constraints x (t) F (t, x(t)),
More informationRelatively 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/s1366201506500 R E S E A R C H Open Access Relatively dense sets, corrected uniformly almost periodic functions on time
More informationF. 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
More informationON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. GacovskaBarandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. GacovskaBarandovska Smirnov (1949) derived four
More informationMath 421, Homework #5 Solutions
Math 421, Homework #5 Solutions (1) (8.3.6) Suppose that E R n and C is a subset of E. (a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as defined in Definition
More informationA Formalization of the Turing Test Evgeny Chutchev
A Formalization of the Turing Test Evgeny Chutchev 1. Introduction The Turing test was described by A. Turing in his paper [4] as follows: An interrogator questions both Turing machine and second participant
More informationFractional Calculus Model for Damped Mathieu Equation: Approximate Analytical Solution
Applied Mathematical Sciences, Vol. 6, 2012, no. 82, 40754080 Fractional Calculus Model for Damped Mathieu Equation: Approximate Analytical Solution Abdelhalim Ebaid 1, Doaa M. M. ElSayed 2 and Mona D.
More informationBOUNDARY VALUE PROBLEMS OF NABLA FRACTIONAL DIFFERENCE EQUATIONS. Abigail Brackins A DISSERTATION. Presented to the Faculty of
BOUNDARY VALUE PROBLEMS OF NABLA FRACTIONAL DIFFERENCE EQUATIONS by Abigail Brackins A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment
More informationA NEW APPROACH TO THE CORONA THEOREM FOR DOMAINS BOUNDED BY A C 1+α CURVE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 40, 2015, 767 772 A NEW APPROACH TO THE CORONA THEOREM FOR DOMAINS BOUNDED BY A C 1+α CURVE José Manuel EnríquezSalamanca and María José González
More information2. Functions and limits: Analyticity and Harmonic Functions
2. Functions and limits: Analyticity and Harmonic Functions Let S be a set of complex numbers in the complex plane. For every point z = x + iy S, we specific the rule to assign a corresponding complex
More informationMTH6109: Combinatorics. Professor R. A. Wilson
MTH6109: Combinatorics Professor R. A. Wilson Autumn Semester 2011 2 Lecture 1, 26/09/11 Combinatorics is the branch of mathematics that looks at combining objects according to certain rules (for example:
More informationDISTRIBUTIONS AND FOURIER TRANSFORM
DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth
More informationarxiv: v1 [math.oa] 17 Dec 2016
arxiv:161.05683v1 [math.oa] 17 Dec 016 COMPLETE SPECTRAL SETS AND NUMERICAL RANGE KENNETH R. DAVIDSON, VERN I. PAULSEN, AND HUGO J. WOERDEMAN Abstract. We define the complete numerical radius norm for
More information3.7 Nonautonomous linear systems of ODE. General theory
3.7 Nonautonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some
More information7. The GaussBonnet theorem
7. The GaussBonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an nsided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
More informationRieszFredhölm Theory
RieszFredhölm 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 AscoliArzelá Result 18 B Normed Spaces
More informationContinuous random variables
Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the socalled continuous random variables. A random
More informationFourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks
Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results
More informationSufficient Statistics and Exponential Family. 1 Statistics and Sufficient Statistics. Math 541: Statistical Theory II. Lecturer: Songfeng Zheng
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Sufficient Statistics and Exponential Family 1 Statistics and Sufficient Statistics Suppose we have a random sample X 1,, X n taken from a distribution
More informationA NOTE ON THE RATES OF CONVERGENCE OF DOUBLE SEQUENCES
SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22) (2014), 87 92 DOI: 10.5644/SJM.10.1.11 A NOTE ON THE RATES OF CONVERGENCE OF DOUBLE SEQUENCES METIN BAŞARIR Abstract. In this paper, we define the rates of convergence
More informationGeometry of Linear Programming
Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms
More informationThe idea to study a boundary value problem associated to the scalar equation
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 23, 2004, 73 88 DEGREE COMPUTATIONS FOR POSITIVELY HOMOGENEOUS DIFFERENTIAL EQUATIONS Christian Fabry Patrick Habets
More informationTHE PRIME NUMBER THEOREM
THE PRIME NUMBER THEOREM NIKOLAOS PATTAKOS. introduction In number theory, this Theorem describes the asymptotic distribution of the prime numbers. The Prime Number Theorem gives a general description
More informationEXISTENCE AND NONEXISTENCE 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: 7669. UL: http://ejde.math.txstate.edu
More informationFourier Series Representations
Fourier Series Representations Introduction Before we discuss the technical aspects of Fourier series representations, it might be well to discuss the broader question of why they are needed We ll begin
More informationAN ITERATIVE METHOD FOR THE HERMITIANGENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT
Bulletin of the Iranian Mathematical Society Vol. 39 No. 6 (2013), pp 1291260. AN ITERATIVE METHOD FOR THE HERMITIANGENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT
More informationAN IMPROVED EXPONENTIAL REGULA FALSI METHODS WITH CUBIC CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS
J. Appl. Math. & Informatics Vol. 8(010), No. 56, pp. 14671476 Website: http://www.kcam.biz AN IMPROVED EXPONENTIAL REGULA FALSI METHODS WITH CUBIC CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS S. A. HODA
More informationOn the communication and streaming complexity of maximum bipartite matching
On the communication and streaming complexity of maximum bipartite matching Ashish Goel Michael Kapralov Sanjeev Khanna November 27, 20 Abstract Consider the following communication problem. Alice holds
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationHarmonic Oscillator Physics
Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the timeindependent Schrödinger equation: d ψx
More informationNo: 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
More informationChapter 5: Application: Fourier Series
321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1
More informationPART 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
More informationOptimal investment with derivative securities
Finance Stochast. 9, 585 595 (25) DOI: 1.17/s785154y c SpringerVerlag 25 Optimal investment with derivative securities Aytaç İlhan1, Mattias Jonsson 2, Ronnie Sircar 3 1 Mathematical Institute, University
More informationNecessary Optimality Condition for Vector Optimization Problems in Infinitedimensional Linear Spaces Xuanwei Zhou
International Conference on Education, Management and Computing Technology (ICEMCT 25) Necessary Optimality Condition for Vector Optimization Problems in Infinitedimensional Linear Spaces Xuanwei Zhou
More information