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MATHEMATICS RESEARCH DEVELOPMENTS SERIES FUNCTIONAL EQUATIONS, DIFFERENCE INEQUALITIES AND ULAM STABILITY NOTIONS (F.U.N.) No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.
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MATHEMATICS RESEARCH DEVELOPMENTS SERIES Boundary Properties and Geometric Properties and Problems Applications of the Differentiated of Thick Knots Poisson Integral for Different Yuanan Diao and Claus Ernst Domains 2009. ISBN: 978-1-60741-070-6 Sergo Topuria 2009. ISBN: 978-1-60692-704-5 Lie Groups: New Research Altos B. Canterra (Editor) Quasi-Invariant and Pseudo- 2009. ISBN: 978-1-60692-389-4 Differentiable Measures in Banach Spaces Lie Groups: New Research Sergey Ludkovsky Altos B. Canterra (Editor) 2009. ISBN: 978-1-60692-734-2 2009. ISBN: 978-1-61668-164-7 Online Book Operator Splittings and their Applications Emerging Topics on Differential Istvan Farago and Agnes Havasiy Geometry and Graph Theory 2009. ISBN: 978-1-60741-776-7 Lucas Bernard and Francois Roux (Editors) Measure of Non-Compactness for 2009. ISBN: 978-1-60741-011-9 Integral Operators in Weighted Lebesgue Spaces Weighted Norm Inequalities for Alexander Meskhi Integral Transforms 2009. ISBN: 978-1-60692-886-8 with Product Kernals Vakhtang Kokilashvili, Alexander Mathematics and Mathematical Meskh and Lars-Erik Persson Logic: New Research 2009. ISBN: 978-1-60741-591-6 Peter Milosav and Irene Ercegovaca (Editors) Group Theory: Classes, 2009. ISBN: 978-1-60692-862-2 Representation and Connections, and Applications Role of Nonlinear Dynamics in Charles W. Danellis (Editor) Endocrine Feedback 2010. ISBN: 978-1-60876-175-3 Chinmoy K. Bose 2009. ISBN: 978-1-60741-948-8 Functional Equations, Difference Inequalities and Ulam Stability Notions (F.U.N.) John Michael Rassias (Editor) 2010. ISBN: 978-1-60876-461-7
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MATHEMATICS RESEARCH DEVELOPMENTS SERIES FUNCTIONAL EQUATIONS, DIFFERENCE INEQUALITIES AND ULAM STABILITY NOTIONS (F.U.N.) JOHN MICHAEL RASSIAS EDITOR Nova Science Publishers, Inc. New York
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Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Functional equations, difference inequalities, and Ulam stability notions (F.U.N.) / [edited by] John Michael Rassias. p. cm. Includes index. ISBN 978-1-61122-575-4 (eBook) Published by Nova Science Publishers, Inc. New York
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CONTENTS Preface vii Chapter 1 Ulam's Stability of a Class of Linear Cauchy Functional Equations 1 Ahmed Charifi, Mohamed Ait Sibaha and Belaid Bouikhalene Chapter 2 Sequential Antagonistic Games with an Auxiliary Initial Phase 15 Jewgeni H. Dshalalow and Weijun Huang Chapter 3 Some Stability Results for Equations and Inequalities Connected 37 with the Exponential Function Włodzimierz Fechner and Roman Ger Chapter 4 On a Problem of John M. Rassias Concerning the Stability 47 in Ulam Sense of Euler–Lagrange Equation L. Găvruţa and P. Găvruţa Chapter 5 Hyers–Ulam–Aoki–Rassias Sstability and Ulam–Gavruta–Rassias 55 Stability of Quadratic Homomorphisms and Quadratic Derivations on Banach Algebras M. Eshaghi Gordji and N. Ghobadipour Chapter 6 Fundamental Solutions for the Generalized Elliptic Gellerstedt 73 Equation Anvar Hasanov, John Michael Rassias and Mamasali Turaev Chapter 7 Pointwise Superstability and Superstability of the Jordan Equation 85 Ji-Rong Lv, Huai-Xin Cao and J.M. Rassias Chapter 8 A Problem with Non-Local Conditions on the Line of Degeneracy 95 and Parallel Characteristics for a Mixed Type Equation with Singular Coefficient M. Mirsaburov and M. Kh. Ruziev Chapter 9 On the Stability of AN Additive Functional Inequality 107 in Normed Modules Choonkil Park
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vi Contents Chapter 10 Cubic Derivations and Quartic Derivations on Banach Modules 119 Choonkil Parkand John Michael Rassias Chapter 11 Tetrahedron Isometry Ulam Stability Problem 131 John Michael Rassias Chapter 12 Hyers–Ulam Stability of Cauchy Type Additive Functional 143 Equations Matina J. Rassias Chapter 13 Solution and Ulam Stability of a Mixed Type Cubic and Additive 149 Functional Equation J.M. Rassias, K. Ravi, M. Arunkumar and B.V. Senth. Kumar Chapter 14 Stability of Mappings Approximately Preserving Orthogonality 177 and Related Topics Aleksej Turnšek Chapter 15 The Frankl Problem for Second Order Nonlinear Equations 191 of Mixed Type with Non-smooth Degenerate Curve Guo Chun Wen Index 209
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PREFACE 1. Ulam’s Biography: S. M. ULAM was born in Lwow, Poland on April 3, 1909 and died in Santa Fe, U.S.A. on May 13, 1984. He graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwow in 1933. Ulam worked at: The Institute for Advanced Study, Princeton (1936), Harvard University (1939-40), University of Wisconsin (1941-43), Los Alamos Scientific Laboratory (1943-65), University of Colorado (1965-76), and University of Florida (1974-). He was a member of the American Academy of Arts and Sciences and the National Academy of Sciences. He made fundamental contributions in mathematics, physics, biology, computer science, and the design of nuclear weapons. His early mathematical work was in set theory, topology, group theory, and measure theory. While still a schoolboy in Lwow, Ulam signed his notebook "S. Ulam, astronomer, physicist and mathematician". As Ulam notes, "the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation." Ulam worked with Stefan Banach, Kazimir Kuratowski, Karol Borsuk, Stanislaw Mazur, Hugo Steinhaus, John von Neumann, Garrett Birkhoff, Cornelius Everett, Dan Mauldin, D. H. Hyers, Mark Kac, P. R. Stein, Enrico Fermi, John Pasta, Richard Feynman, Ernest Lawrence, J. Robert Oppenheimer, Teller, and many other people of applied and exact sciences. Ulam was invited to Los Alamos by his friend John von Neumann, one of the most influential mathematicians of the twentieth century. Ulam's most remarkable achievement at Los Alamos was his contribution to the postwar development of the thermonuclear or hydrogen (H-) bomb in which nuclear energy is released when two hydrogen or deuterium nuclei fuse together. One of Ulam's early insights was to use the fast computers at Los Alamos to solve a wide variety of problems in a statistical manner using random numbers. This method has become appropriately known as the Monte Carlo method. One example that may have biological relevance is the subfield of cellular automata founded by Ulam and von Neumann. Finally Ulam had a unique ability to raise important unsolved problems. One of these problems was solved by the editor of this F.U.N. volume (J. Approx. Th., Vol. 57, 268-273, 1989).
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viii John Michael Rassias 2. Ulam’s volume F. U. N. : Functional Equations and Difference Inequalities and Ulam Stability Notions, is a forum for exchanging ideas among eminent mathematicians and physicists, from many parts of the world, as a tribute to the first centennial birthday anniversary of Stanislaw Marcin ULAM. This collection is composed of outstanding contributions in mathematical and physical equations and inequalities and other fields of mathematical and physical sciences. It is intended to boost the cooperation among mathematicians and physicists working on a broad variety of pure and applied mathematical areas. This transatlantic collection of mathematical ideas and methods comprises a wide area of applications in which equations, inequalities and computational techniques pertinent to their solutions play a core role. Ulam's influence has been tremendous on our everyday life, because new tools have been developed, and revolutionary research results have been achieved , bringing scientists of exact sciences even closer, by fostering the emergence of new approaches, techniques and perspectives. The central scope of this commemorating 100birthday anniversary volume is broad, by deeper looking at the impact and the ultimate role of mathematical and physical challenges, both inside and outside research institutes, scientific foundations and organizations. We have recently observed a more rapid development in the areas of research of Ulam worldwide. This F.U.N. volume is suitable for graduate students and researchers interested in functional equations, and differential equations and would make an ideal supplementary reading or independent study research text. This item will also be of interest to those working in other areas of mathematics and physics. It is a work of great interest and enjoyable read as well as unique in market. This Ulam's volume (F.U.N.) consists of research papers containing various parts of contemporary pure and applied mathematics with emphasis to Ulam's mathematics. It contains various parts of Functional Equations and Difference Inequalities as well as related topics in Mathematical Analysis, namely: Ulam’s stability of a class of linear Cauchy functional equations; Sequential antagonistic games with an auxiliary initial phase; Some stability results for equations and inequalities connected with the exponential function; On a problem of John M. Rassias concerning the stability in Ulam sense of Euler-Lagrange equation; Hyers-Ulam- Aoki-Rassias stability and Ulam-Gavruta-Rassias stability of quadratic homomorphisms and quadratic derivations on Banach algebras; Fundamental solutions for the generalized elliptic Gellerstedt equation; Pointwise superstability and superstability of the Jordan equation; A problem with non-local conditions on the line of degeneracy and parallel characteristics for a mixed type equation with singular coefficient; On the stability of an additive functional inequality in normed modules; Cubic derivations and quartic derivations on Banach modules; Tetrahedron isometry Ulam stability problem; Hyers-Ulam stability of Cauchy type additive functional equations; Solution and Ulam stability of a mixed type cubic and additive
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Preface ix functional equation; Stability of mappings approximately preserving orthogonality and related topics; The Frank problem for second order nonlinear equations of mixed type with non- smooth degenerate curve. John Michael RASSIAS Editor of Ulam’ s volume F. U. N. National and Capodistrian University of Athens 4, Agamemnonos Str., Aghia Paraskevi, Attikis 15342 Athens, GREECE E-mail: jrassias@primedu.uoa.gr Web: http://www.primedu.uoa.gr/~jrassias
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In: Functional Equations, Difference Inequalities... ISBN 978-1-60876-461-7 Editor: John Michael Rassias, pp. 1-14 ©c 2010 Nova Science Publishers, Inc. Chapter 1 ULAM’S STABILITY OF A CLASS OF LINEAR CAUCHY FUNCTIONAL EQUATIONS Ahmed Charifia, Mohamed Ait Sibahaa, Belaid Bouikhaleneb,∗ and Samir Kabbaja a Laboratory: L. A. M. A., Department of Mathematics University of Ibn Tofail, Faculty of Sciences BP 133 K´enitra 14000, Morocco b University Sultan Moulay Slimane Polydisciplinaire Faculty Beni Mellal Morocco Abstract In this work, we describe the solution of (1.1) connected with additive functions and we study the Ulam’s problem of this equation. Some applications deal with new equations of type linear Cauchy in Banach spaces, are given. 2000 Mathematics Subject Classifications: 39B32, 39B42, 39B72. Key words: Functional equation, Ulam problem, stability. 1. Introduction Let E and F be Banach spaces with norms ‖ · ‖ and ‖ · ‖ respectively. A mapping f : E −→ F is called, additive function, if it satisfies the Cauchy functional equation f(x + y) = f(x) + f(y) for all x, y ∈ E. In 1940, S. M. Ulam (see [22]) raised the question concerning the stability of group homomorphisms: “when is it true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation?”. The first answer to Ulam’s question, concerning the Cauchy equation, was given by D. H. Hyers [8]. Thus we speak about the Hyers–Ulam stability. This terminology is also applied to the case of other functional equations. Th. M. Rassias [19] generalized the theorem of ∗E-mail address: bbouikhalene@yahoo.fr
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2 A. Charifi, M. Ait Sibaha, B. Bouikhalene et al. Hyers for approximately linear mappings [19]. The stability phenomena that was proved by Th. M. Rassias [19] is called the Hyers–Ulam–Rassias stability. The modified Ulam’s stability problem with the generalization control function was proved by P. Gˇavruta [6] in the following way Theorem 1.1. Let E be a vector space, E be a Banach space and let ϕ : E×E −→ [0,+∞[ be a function satisfying ∑+∞ 1 ψ(x, y) = ϕ(2kx,2ky) < +∞ 2k+1 k=0 for all x, y ∈ E. If a function f : E −→ F satisfies the functional inequality ‖f(x + y) − f(x) − f(y)‖ < ϕ(x, y) for all x, y ∈ E. Then there exists a unique additive function T : E −→ F which satisfies ‖f(x) − T(y)‖ ≤ ψ(x, y) for all x ∈ E. J. M. Rassias [14]–[18] solved the Ulam’s problem for different mappings, in the fol- lowing way Theorem 1.2. Let X be a real normed linear space and let Y be a real normed linear space. Assume in addition that if f : X −→ Y is a mapping for which there exist constant δ > 0 and p, q ∈ R such that r = p + q 6= 1 and f satisfies the inequality ‖f(x + y) − f(x) − f(y)‖ ≤ δ‖x‖p‖y‖q for all x, y ∈ X. Then there exists a unique additive mapping L : X −→ Y satisfying δ r ‖f(x) − L(x)‖ ≤ ‖x‖ ‖2r − 1‖ for all x ∈ X. As the words “differing slightly” and “be close” in Ulam’s problem have various mean- ings, different kinds of stability can be dealt with. It may happen that all approximate solution are in fact exact solutions. Then we speak about superstability. To get acquainted with the theory of the stability of functional equation we refer to papers [8]–[15]. In this paper, we introduce the new general linear Cauchy type functional equation of the form ( ) ∑M ∑n ∑n xi + aj = M f(xi) for all x1, . . . , xn ∈ E (1.1) j=1 i=1 i=1 and for any arbitrary fixed elements a1, . . . , aM in E. When aj = 0 for all j ∈ {1, . . . , M}, the equation (1.1) reduces to the equation ( ) ∑n ∑n f xi = f(xi) for all x1, . . . , xn ∈ E. (1.2) i=1 i=1
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Ulam’s Stability 3 The Ulam’s problem of the question (1.2) was studied by J. M. Rassias in [18 Theorem 2]. If n = 2 and M = 2, the equation (1.1) becomes f(x + y + a) + f(x + y + b) = 2f(x) + 2f(y). (1.3) The Ulam’s problem of this equation was studied by authors in [1]. Following this investi- gation, we study here the Ulam’s problem of the functional equation (1.1) and a bouquet of special cases, namely f(x + y + z + a) + f(x + y + z + b) = 2f(x) + 2f(y) + 2f(z) (1.4) and f(x + y + a) + f(x + y + b) + f(x + y + c) = 3f(x) + 3f(y), (1.5) for all x, y, z ∈ E and any arbitrary a, b, c ∈ E. This paper is organized as follows: in the first section, after this introduction we gave the general solution of (1.1). In the second section we investigate the Ulam’s problem for the general linear Cauchy equation (1.1). In Corrollaries (5.5) and (5.6) we deduce the Ulam’s stability for equation (1.4) and (1.5). 2. Solution of (1.1) In this section we give the general solution of the functional equation (1.1). Theorem 2.1. Let M, N be integers, M > 0 and N > 1. Let E and F be vectors spaces. A function f : E −→ F satisfies the functional equation (1.1) if and only if there exists an additive function g : E −→ F such that ∑M g((N − 1)x + a ) f(x) = j for all x ∈ E. M(N − 1) j=1 Proof. If f is a solution of (1.1), then by substituting x1, . . . , xN−1 by 0 and xN by x1 + · · · + xN in (1.1) we have ( ) ( ) ∑M ∑N ∑N f xi + aj = M(N −1)f(0)+M f xi for all x1, . . . , xN ∈ E. (2.6) j=1 i=1 i=1 By taking x1 = · · · = xN = 0 in (1.1) we get ∑M (f − f(0))(aj) = M(N − 1)f(0). j=1 From (1.1) and (2.1) we deduce that ( ) ∑N ∑N f(xi) = (N − 1)f(0) + f xi i=1 i=1
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4 A. Charifi, M. Ait Sibaha, B. Bouikhalene et al. so ( ) ∑N ∑N (f − f(0))(xi) = (f − f(0)) xi . i=1 i=1 We pose g = f − f(0), then ∑M ∑M g(aj) g((N − 1)x + aj) f(x) = g(x) + j=1 = j=1 . M(N − 1) M(N − 1) Conversely, let g : E −→ F be an additive function. It’s elementary to verify that f(x) = ∑M g((N−1)x+aj) j=1 is a solution of (1.1). This ends the proof. M(N−1) 3. Ulam’s Stability for the Functional Equation (1.1) In this section we establish the Ulam’s stability for equation (1.1). Theorem 3.1. Let E be a vector space, F a Banach space and δ > 0. Suppose that the function f : E −→ F satisfies the inequality ∥ ( ) ∥∥ ∥∥∑M ∑N ∑N ∥ ∥ f x + a − M f(x )∥∥ < δ for all x , . . . , x ∈ E. (3.7) ∥∥ i j i ∥ 1 N j=1 i=1 i=1 Then, there exists a unique function S : E −→ F solution of (1.1) such that δ ‖f(x) − S(x)‖ ≤ for all x ∈ E. M(N − 1) Proof. Assume that f : E −→ F satisfies the inequality (3.1), we use induction on n to ∑M prove that the sequence functions f0(x) = f(x) and fn(x) = fn−1(N x + aj) for all j=1 x ∈ E, n ∈ N∗ satisfy the following statements ‖f (x) − M N f (x)‖ < Mn−1δ for all x ∈ E, n ∈ N∗, (3.8) n n−1 Nn − 1 ‖f (x) − MnNnf (x)‖ < Mn−1 δ for all x ∈ E, n ∈ N∗, (3.9) n 0 N − 1 ∥ ( ) ∥∥ ∥∥∑M ∑N ∑N ∥ ∥ f x + a − M f (x )∥∥ < Mnδ for all x , . . . , x ∈ E, n ∈ N∗. ∥∥ n i j n i ∥ 1 N j=1 i=1 i=1 (3.10) By taking x1 = · · · = xN = x in (3.1), we get that ‖f1(x) − M N f0(x)‖ < δ.
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Ulam’s Stability 5 We have for all x , . . . , x ∈ E and n ∈ N∗ 1 n ∥ ( ) ∥∥ ∥∑M ∑N ∑N ∥ ∥∥ ∥ ∥ f1 xi + aj − M f1(xi)∥ ∥j=1 i=1 i=1 ∥ ∥ ( ) ∥ ∥∥∑M ∑M ∑N ∑M ∑N ∥ ∥ ∥∥ = ∥ f0 N xi + N aj + al − M f0(N xi + aj)∥ ∥j=1 l=1 i=1 j=1 i=1 ∥ ∥ ( ) ∥ ∑M ∥∑M ∑N ∑N ∥∥ ≤ ∥∥ f (N x + a ) + a − M f (N x + a )∥ < M δ. ∥ 0 i j l 0 i j ∥ j=1 l=1 i=1 i=1 Consequently, the assertions (3.2), (3.3) and (3.4) are trues for n = 1. Assuming that the assertions are trues for all integers i, 1 ≤ i ≤ n. It follows from the induction assumption that ∥ ∥∥ ∥∥∑M ∑M ∥ ‖f (x) − M N f (x)‖ = ∥ f (N x + a ) − M N f (N x + a )∥ n+1 n ∥ n j n−1 j ∥∥ ∥j=1 j=1 ∑M ≤ ‖f (N x + a ) − M N f (N x + a )‖ < M(Mn−1δ) = Mnδ n j n−1 j j=1 for all x ∈ E. We have for all x ∈ E that ∥∥ ∥∥ ∥ ∥ ∥∑n ∥ ∥f (x) − Mn+1Nn+1f (x)∥ = ∥ MiNif (x) − Mi+1Ni+1f (x)∥ n+1 0 ∥ n+1−i n−i ∥ i=0 ∑n ∑n Nn+1 − 1 ≤ MiNi ‖f (x) − M N f (x)‖ ≤ MiNiMn−iδ = Mn δ. n+1−i n−i N − 1 i=0 i=0 Now, for all xi, 1 ≤ i ≤ n, we get ∥∥ ( ) ∥ ∥∑M ∑N ∑N ∥ ∥ ∥∥ ∥ fn+1 xi + aj − M fn+1(xi)∥ ∥j=1 i=1 i=1 ∥ ∥∥ ( ) ∥ ∑M ∑M ∑N ∑M ∑N ∥ ∥∥ ∥ = ∥ f N x + N a + a − M f (N x + a )∥∥ n i j l n i j ∥j=1 l=1 i=1 j=1 i=1 ∥ ∥∥ ( ) ∥ ∑M ∥∑M ∑N ∑N ∥∥ ≤ ∥ f (N x + a ) + a − M f (N x + a )∥ < M(Mnδ) = Mn+1δ, ∥ n i j l n i j ∥ j=1 l=1 i=1 i=1 (3.11) ( f (x) ) which gives the sought results. From (3.2), it follows that the sequence functions nn n M N n is a Cauchy sequence. Since F is complete, the above sequence has a limit in F. We define S : E −→ F by fn(x) S(x) = lim . n→+∞ MnNn
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6 A. Charifi, M. Ait Sibaha, B. Bouikhalene et al. From (3.3), it follows that δ ‖f(x) − S(x)‖ ≤ for all x ∈ E. M(N − 1) From (3.4), we deduce that S : E −→ F satisfies the functional equation (1.1). In the next we will show the uniqueness of S. Let S′ : E −→ F be another solution of the functional equation (1.1) which satisfies ′ δ ‖f(x) − S (x)‖ ≤ for all x ∈ E. M(N − 1) We will prove by induction that ∥ ∥ Mn−1 ∥f (x) − MnNnS′(x)∥ ≤ δ for all x ∈ E. (3.12) n N − 1 Since S′ satisfies (1.1), we get ∥∥ ∥∥ ∥ ∥ ∥∑M ∑M ∥ ∥f (x) − M N S′(x)∥ = ∥ f (N x + a ) − S′(N x + a )∥ 1 ∥∥ 0 j j ∥∥ j=1 j=1 ∑M ∥ ∥ M δ δ ≤ ∥f(N x + a ) − S′(N x + a )∥ ≤ = . j j M(N − 1) N − 1 j=1 Assuming that (3.6) is true for all integers i, 1 ≤ i ≤ n, hence we have ∥ ∥ ∥ ∥ ∥∥∑M ∑M ∥ ∥ ∥ ∥ ∥∥ f (x) − Mn+1Nn+1S′(x) = f (N x + a ) − MnNn S′(N x + a ) n+1 ∥ n j j ∥ ∥j=1 j=1 ∥ ∑M ∥ ∥ Mn ≤ ∥f (N x + a ) − MnNnS′(N x + a )∥ ≤ δ. n j j N − 1 j=1 By letting n −→ +∞ we get from inequality ∥ ∥ ∥ fn(x) ∥∥ δ ∥∥ − S′(x)∥ ≤ MnNn M(N − 1)Nn that S = S′. This ends the proof of theorem (3.1). Theorem 3.2. Let E be a vector space, let F be a Banach space and let ϕ : EN −→ [0,+∞[ be a function. We define ϕ0 = ϕ and ∑M ϕ (x , . . . , x ) = ϕ (N x +a , . . . , N x +a ) for all x , . . . , x ∈ E, n ∈ N∗. n 1 N n−1 1 j N j 1 N j=1 (3.13)
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Ulam’s Stability 7 Suppose that ∑+∞ 1 ϕ˜(x1, . . . , xN) = ϕn(x1, . . . , xN) < +∞ for all x1, . . . , xN ∈ E. Mn+1Nn+1 n=0 (3.14) Assume that a function f : E −→ F satisfies the inequality ∑M ∑N ∑N ‖ f( xi + aj) − M f(xi)‖ < ϕ(x1, . . . , xN) f or all x1, . . . , xN ∈ E. (3.15) j=1 i=1 i=1 Then, there exists a unique function S : E −→ F solution of (1.1) such that ‖f(x) − S(x)‖ ≤ ϕ˜(x, . . . , x) for all x ∈ E. (3.16) Proof. We use induction on n to prove that the sequence functions f0(x) = f(x) and ∑M f (x) = f (N x + a ) for all x ∈ E, n ∈ N∗, satisfy the following statements n n−1 j j=1 ‖f (x) − M N f (x)‖ < ϕ (x, . . . , x) for all x ∈ E, n ∈ N∗, (3.17) n n−1 n−1 n∑−1 ‖f (x) − MnNnf (x)‖ < MiNiϕ (x, . . . , x) for all x ∈ E, n ∈ N∗, n 0 n−1−i i=0 (3.18) ∥ ( ) ∥∥ ∥∥∑M ∑N ∑N ∥ ∥ f x +a −M f (x )∥∥< ϕ (x , . . . , x ) for all x , . . . , x ∈E, n∈N. ∥∥ n i j n i ∥ n 1 N 1 N j=1 i=1 i=1 (3.19) By using the definition of fn and the inequality (3.9) we have ∥∥ ∥∥ ∥∑M ∥ ‖f (x) − M N f (x)‖ = ∥∥ f(N x + a ) − M N f(x)∥∥ < ϕ(x, . . . , x). 1 0 j ∥j=1 ∥ We have from (3.5) ∥ ( ) ∥ ∥∥∑M ∑N ∑N ∥ ∥ ∥∥ ∥ f1 xi + aj − M f1(xi)∥ ∥j=1 i=1 i=1 ∥ ∥ ( ) ∥∥ ∑M ∥∑M ∑N ∑N ∥ ≤ ∥∥ f (N x + a ) + a − M f (N x + a )∥ ∥ 0 i j l 0 i j ∥ j=1 l=1 i=1 i=1 ∑M < ϕ(N x1 + aj, . . . , N xn + aj) = ϕ1(x1, . . . , xN). j=1
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8 A. Charifi, M. Ait Sibaha, B. Bouikhalene et al. The assertions (3.11), (3.12) and (3.13) are now trues for n = 1. Assuming that the as- sertions are trues for all integers i, 1 ≤ i ≤ n. It follows from (3.9) and the induction assumption that ∥ ∥ ∥∥∑M ∑M ∥ ∥ ∥∥ ‖fn+1(x) − M N fn(x)‖ = ∥ fn(N x + aj) − M N fn−1(N x + aj)∥ ∥j=1 j=1 ∥ ∑M ≤ ‖fn(N x + aj) − M N fn−1(N x + aj)‖ j=1 ∑M < ϕn−1(N x + aj, . . . , N x + aj) = ϕn(x, . . . , x). j=1 We have from (3.5) ∥ ( ) ∥∥ ∥∥∑M ∑N ∑N ∥ ∥ f x + a − M f (x )∥∥ ∥∥ n+1 i j n+1 i ∥ j=1 i=1 i=1 ∥ ( ) ∥ ∑M ∥∑M ∑N ∑M ∥∥ ≤ ∥∥ f (N x + a ) + a − M f (N x + a )∥ ∥ n i j l n i l ∥ j=1 l=1 i=1 l=1 ∑M < ϕn(N x1 + aj, . . . , N xN + aj) = ϕn+1(x1, . . . , xN). j=1 Also we have ∥∥ ∥∥ ∥ ∥ ∥∑n ∥ ∥f (x) − Mn+1Nn+1f (x)∥ = ∥ MiNi [f (x) − M N f (x)]∥ n+1 0 ∥ n+1−i n−i ∥ i=0 ∑n < MiNiϕ (x, . . . , x). n−i i=0 ( f (x) ) This gives the sought results. It follows from (3.12) that the sequence nn n is a Cauchy M N n sequence. Since F is complete, the sequence has a limit in F. We define S : E −→ F, by fn(x) S(x) = lim . n→+∞ MnNn It follows from (3.12) that ‖f(x) − S(x)‖ ≤ ϕ˜(x, . . . , x) for all x ∈ E. From (3.13), it follows that the function S : E −→ F satisfies the functional equation (1.1). In the next we will show the uniqueness of S. Let S′ : E −→ F be another solution of the functional equation (1.1) which satisfies ‖f(x) − S′(x)‖ ≤ ϕ˜(x, . . . , x) for all x ∈ E.
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