Stability of Functional Equations in Banach Algebras

Yeol Je Cho · Choonkil Park Themistocles M. Rassias · Reza Saadati

Stability of Functional Equations in Banach Algebras

Stability of Functional Equations in Banach Algebras

Yeol Je Cho • Choonkil Park Themistocles M. Rassias • Reza Saadati

Stability of Functional Equations in Banach Algebras

123

Yeol Je Cho Department of Mathematics Education and the RINS Gyeongsang National University College of Education Jinju, Korea, Republic of South Korea Themistocles M. Rassias Department of Mathematics National Technical University of Athens Athens, Greece

ISBN 978-3-319-18707-5 DOI 10.1007/978-3-319-18708-2

Choonkil Park Department of Mathematics Hanyang University Seoul, Korea, Republic of South Korea Reza Saadati Department of Mathematics Iran University of Science and Technology Tehran, Iran

ISBN 978-3-319-18708-2 (eBook)

Library of Congress Control Number: 2015939079 Mathematics Subject Classification (2010): 39-XX, 26-XX, 41-XX, 46-XX, 47-XX Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)

To S.M. Ulam for the 75th anniversary of his stability problem for approximate homomorphisms

Preface

The main purpose of this book is to present some of the old and recent results on homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C -algebras, C -ternary algebras, non-Archimedean Banach algebras, and multinormed algebras. In 1940, S. M. Ulam [321] proposed a stability problem on group homomorphisms in metric groups. In 1941, D. H. Hyers [133] proved the stability of additive mappings in Banach spaces associated with the Cauchy equation. In 1978, Th. M. Rassias [267] proved the stability of R-linear mappings associated with the Cauchy equation, and in 2002 C. Park [220] proved the stability of C-linear mappings in the spirit of Hyers, Ulam, and Rassias in Banach modules. Homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C -algebras, C ternary algebras, non-Archimedean Banach algebras and multi-normed algebras are additive and R-linear or C-linear, and so we study the stability problems for additive functional equations and additive mappings. Using the direct method and the fixed point method, the authors have studied the stability and the superstability of homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C -algebras, C -ternary algebras, non-Archimedean Banach algebras, and multinormed algebras, which are also associated with additive functional equations and additive functional inequalities. The book provides a survey of both the latest and new results especially on the following topics: (1) Stability theory for several new functional equations in Banach algebras and C -algebras via fixed point method and direct method. (2) Stability theory for several new functional inequalities in Banach algebras and C -algebras via fixed point method and direct method. (3) Stability theory of well-known new functional equations in non-Archimedean Banach algebras and non-Archimedean C -algebras. (4) Stability theory for several new functional equations and functional inequalities in multi-Banach algebras and multi-C -algebras via fixed point method and direct method. vii

viii

Preface

The book is intended to be accessible especially to graduate students who have a basic background with operator theory, functional analysis, functional equations, and analytic inequalities including an introduction to Banach algebras, quasi-Banach algebras, C -algebras, C -ternary algebras, non-Archimedean Banach algebras, and multi-normed algebras. In Chap. 1, we provide a brief introduction to concepts with historic remarks for functional equations and their stability and the definitions of Banach algebras, quasi-Banach algebras, C -algebras, C -ternary algebras, non-Archimedean Banach algebras, and multi-normed algebras. In Chap. 2, we study the stability of additive functional equations in Banach spaces as well as the stability and the superstability of isomorphisms, homomorphisms, derivations, and generalized derivations in Banach algebras and quasiBanach algebras associated with additive functional equations. In Chap. 3, we study the stability and the superstability of isomorphisms, homomorphisms, and derivations in C -algebras, Lie C -algebras, and JC -algebras, as well as the stability and the superstability of linear mappings in Banach modules over unital C -algebras. Moreover, we study Jordan -derivations, quadratic Jordan -derivations, .˛; ˇ;  /-derivations on Lie C -algebras, square root functional equations, 3rd root functional equations, and positive-additive functional equations. In Chap. 4, we study the stability of C-linear mappings in Banach spaces and linear mappings in normed modules over a C -algebra as well as the stability of homomorphisms and derivations in proper CQ -algebras associated with functional inequalities. In Chap. 5, we study the stability and the superstability of C -ternary homomorphisms, C -ternary derivations, C -ternary 3-homomorphisms, and C -ternary 3-derivations in C -ternary algebras as well as investigate the stability of JB -triple homomorphisms and JB -triple derivations in JB -triples by using the direct method and the fixed point method. In Chap. 6, we study the stability of linear mappings in multi-Banach spaces as well as the stability and the superstability of isomorphisms, homomorphisms, and derivations in multi-Banach algebras, multi-C -algebras, proper multi-CQ algebras, and multi-C -ternary algebras. Moreover, we study the stability of ternary Jordan homomorphisms and ternary Jordan derivations in multi-C -ternary algebras. Finally, in Chap. 7, we study the stability of additive functional equations in non-Archimedean Banach spaces as well as the stability of homomorphisms and derivations in non-Archimedean C -algebras and non-Archimedean Lie C algebras. Jinju, South Korea Seoul, South Korea Athens, Greece Tehran, Iran March 2015

Yeol Je Cho Choonkil Park Themistocles M. Rassias Reza Saadati

Acknowledgments

We would like to express our thanks to the referees for reading the manuscript and providing valuable suggestions and comments which have helped to improve the presentation of the book. This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology. Last but not least, it is our pleasure to acknowledge the superb assistance provided by the staff of Springer for the publication of the book. Yeol Je Cho Choonkil Park Themistocles M. Rassias Reza Saadati

ix

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Quasi-Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 C -Algebras.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 C -Ternary Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Non-Archimedean Normed Algebras .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Multi–normed Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 7 9 10 12 13 14

2 Stability of Functional Equations in Banach Algebras . . . . . . . . . . . . . . . . . . 2.1 Stability of 1q f .qx C qy C qz/ D f .x/ C f .y/ C f .z/ . . . . . . . . . . . . . . . . . 2.1.1 Isomorphisms in Banach Algebras .. . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Derivations in Banach Algebras .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Stability of m-Variable Functional Equations . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Stability of Homomorphisms in Banach Algebras . . . . . . . . . 2.2.2 Stability of Derivations in Banach Algebras .. . . . . . . . . . . . . . . 2.3 Stability in Quasi-Banach Algebras . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Stability of Homomorphisms in Quasi-Banach Algebras .. 2.3.2 Isomorphisms in Quasi-Banach Algebras . . . . . . . . . . . . . . . . . . 2.3.3 Stability of Generalized Derivations in Quasi-Banach Algebras .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Stability of Cauchy–Jensen Functional Equations . . . . . . . . . . . . . . . . . . . 2.4.1 Stability of Homomorphisms in Real Banach Algebras . . . 2.4.2 Stability of Generalized Derivations in Real Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

19 20 20 23 24 24 29 31 32 35

3 Stability of Functional Equations in C -Algebras .. . .. . . . . . . . . . . . . . . . . . . . 3.1 Isomorphisms in Unital C -Algebras .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 C -Algebra Isomorphisms in Unital C -Algebras . . . . . . . . . 3.1.2 On the Mazur-Ulam Theorem in Modules over C -Algebras .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Apollonius Type Additive Functional Equations .. . . . . . . . . . . . . . . . . . . . 3.2.1 Homomorphisms and Derivations on C -Algebras . . . . . . . .

51 53 54

36 38 38 44

61 63 64 xi

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3.2.2 Homomorphisms and Derivations in Lie C -Algebras.. . . . 3.2.3 Homomorphisms and Derivations in JC -Algebras .. . . . . . . 3.3 Stability of Jensen Type Functional Equations in C -Algebras . . . . . 3.3.1 Stability of Homomorphisms in C -Algebras . . . . . . . . . . . . . . 3.3.2 Stability of Derivations in C -Algebras . . . . . . . . . . . . . . . . . . . . 3.3.3 Stability of Homomorphisms in Lie C -Algebras.. . . . . . . . . 3.3.4 Stability of Lie Derivations in C -Algebras . . . . . . . . . . . . . . . . 3.4 Generalized Additive Mapping . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.1 Hyers–Ulam Stability of Functional Equations in Banach Modules over a C -Algebra .. . . . . . . . . . . . . . . . . . . . 3.4.2 Homomorphisms in Unital C -Algebras . . . . . . . . . . . . . . . . . . . 3.5 Generalized Additive Mappings in Banach Modules . . . . . . . . . . . . . . . . 3.5.1 Odd Functional Equations in d–Variables . . . . . . . . . . . . . . . . . . 3.5.2 Stability of Odd Functional Equations in Banach Modules over a C -Algebra . . . .. . . . . . . . . . . . . . . . . . . . 3.5.3 Isomorphisms in Unital C -Algebras . . .. . . . . . . . . . . . . . . . . . . . 3.6 Jordan -Derivations and Quadratic Jordan -Derivations . . . . . . . . . . 3.6.1 Stability of Jordan -Derivations.. . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.2 Stability of Quadratic Jordan -Derivations . . . . . . . . . . . . . . . . 3.6.3 Stability of Jordan -Derivations: The Fixed Point Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 .˛; ˇ;  /-Derivations on Lie C -Algebras: The Direct Method . . . . . 3.8 Square Roots and 3rd Root Functional Equations: The Direct Method.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.1 Stability of the Square Root Functional Equation .. . . . . . . . . 3.8.2 Stability of the 3rd Root Functional Equation .. . . . . . . . . . . . . 3.9 Square Root and 3rd Root Functional Equations: The Fixed Point Method .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.1 Stability of the Square Root Functional Equation .. . . . . . . . . 3.9.2 Stability of the 3rd Root Functional Equation .. . . . . . . . . . . . . 3.10 Positive-Additive Functional Equation . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.10.1 Stability of the Positive-Additive Functional Equations: The Fixed Point Method . . . .. . . . . . . . . . . . . . . . . . . . 3.10.2 Stability of the Positive-Additive Functional Equations: The Direct Method . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.11 Stability of -Homomorphisms in JC -Algebras .. . . . . . . . . . . . . . . . . . . 3.11.1 -Homomorphisms in JC -Algebras . . .. . . . . . . . . . . . . . . . . . . . 3.11.2 Stability of -Homomorphisms in JC -Algebras . . . . . . . . . . 4 Stability of Functional Inequalities in Banach Algebras . . . . . . . . . . . . . . . . 4.1 Stability of Additive Functional Inequalities in Banach Algebras . . 4.1.1 Stability of C-Linear Mappings in Banach Spaces . . . . . . . . . 4.1.2 Stability of Homomorphisms in Proper CQ -Algebras . . . . 4.1.3 Stability of Derivations in Proper CQ -Algebras . . . . . . . . . .

67 69 70 70 79 81 83 85 85 92 98 99 100 106 110 110 114 119 124 130 131 135 137 137 141 143 144 148 151 153 160 165 166 166 171 178

Contents

4.2

xiii

Stability of Functional Inequalities over C -Algebras .. . . . . . . . . . . . . . 4.2.1 Functional Inequalities in Normed Modules over C -Algebras . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 On Additive Functional Inequalities in Normed Modules over C -Algebras .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.3 Generalization of Cauchy-Rassias Inequalities via the Fixed Point Method .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 

184 185 192 196

5 Stability of Functional Equations in C -Ternary Algebras . . . . . . . . . . . . . 5.1 C -Ternary 3-Homomorphism and C -Ternary 3-Derivations .. . . . . 5.1.1 C -Ternary 3-Homomorphisms in C -Ternary Algebras . . 5.1.2 C -Ternary 3-Derivations in C -Ternary Algebras . . . . . . . . 5.2 Apollonius Type Additive Functional Equations .. . . . . . . . . . . . . . . . . . . . 5.2.1 Homomorphisms in C -Ternary Algebras .. . . . . . . . . . . . . . . . . 5.2.2 Derivations in C -Ternary Algebras . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 Homomorphisms in JB -Triples . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.4 Derivations in JB -Triples .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.5 C -Ternary Homomorphisms: Fixed Point Method .. . . . . . . 5.2.6 C -Ternary Derivations: The Fixed Point Method . . . . . . . . . 5.2.7 JB -Triple Homomorphisms: The Fixed Point Method .. . . 5.2.8 JB -Triple Derivations: Fixed Point Method . . . . . . . . . . . . . . . 5.3 Bi--Derivations in JB -Triples . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

201 202 203 207 209 210 212 213 214 215 216 217 218 219

6 Stability of Functional Equations in Multi-Banach Algebras .. . . . . . . . . . 6.1 Stability of m-Variable Additive Mappings . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.1 Stability of Homomorphisms in Multi-Banach Algebras . . 6.1.2 Stability of Derivations in Multi-Banach Algebras.. . . . . . . . 6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C -Ternary Algebras . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 Stability of Homomorphisms in Multi-C -Ternary Algebras . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 Stability of Derivations in Multi-C -Ternary Algebras . . . . 6.3 Generalized Additive Mappings and Isomorphisms in Multi-C -Algebras.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 Stability of Odd Functional Equations in Multi-Banach Modules over a Multi-C -Algebra .. . . . . . . . . 6.3.2 Isomorphisms in Unital Multi-C -Algebras .. . . . . . . . . . . . . . . 6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras . . . 6.4.1 Stability of C-Linear Mappings in Multi-Banach Spaces . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.2 Stability of Homomorphisms in Proper Multi-CQ -Algebras .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.3 Stability of Derivations in Proper CQ -Algebras . . . . . . . . . .

229 230 230 238 241 242 249 252 252 262 264 264 269 277

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6.5

Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 282 6.5.1 Stability of Homomorphisms .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 282 6.5.2 Stability of Derivations in Multi-C -Ternary Algebras . . . . 297

7 Stability of Functional Equations in Non-Archimedean Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.1 Stability of Homomorphisms in Non-Archimedean C -Algebras.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.2 Stability of Derivations in Non-Archimedean C -Algebras .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.3 Stability of Homomorphisms in Non-Archimedean Lie C -Algebras . . . .. . . . . . . . . . . . . . . . . . . . 7.1.4 Stability of Non-Archimedean Lie Derivations in C -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Stability for m-Variable Additive Functional Equations . . . . . . . . . . . . . 7.2.1 Stability of Homomorphisms and Derivations in Non-Archimedean C -Algebras .. . . . .. . . . . . . . . . . . . . . . . . . . 7.2.2 Stability of Homomorphisms and Derivations in Non-Archimedean Lie C -Algebras .. . . . . . . . . . . . . . . . . . . .

303 304 304 314 316 318 320 320 324

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 327 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 341

Chapter 1

Introduction

In this chapter, we recall some definitions and results which will be used later on in the book. The study of functional equations has a long history. In 1791 and 1809, Legendre [184] and Gauss [121] attempted to provide a solution of the following functional equation: f .x C y/ D f .x/ C f .y/ for all x; y 2 R, which is called the Cauchy functional equation. A function f W R ! R is called an additive function if it satisfies the Cauchy functional equation. In 1821, Cauchy [67] first found the general solution of the Cauchy functional equation, that is, if f W R ! R is a continuous additive function, then f is R–linear, that is, f .x/ D mx, where m is a constant. Further, we can consider the biadditive function on R  R as follows: A function f W R  R ! R is called a biadditive function if it is additive in each variable, that is, f .x C y; z/ D f .x; z/ C f .y; z/ and f .x; y C z/ D f .x; y/ C f .x; z/ for all x; y; z 2 R. It is well–known that every continuous biadditive function f W R  R ! R is of the form f .x; y/ D mxy for all x; y 2 R, where m is a constant. © Springer International Publishing Switzerland 2015 Y.J. Cho et al., Stability of Functional Equations in Banach Algebras, DOI 10.1007/978-3-319-18708-2_1

1

2

1 Introduction

Since the time of Legendre and Gauss, several mathematicians had dealt with additive functional equations in their books [4–6, 178, 313] and a number of them have studied Lagrange’s mean value theorem and related functional equations, Pompeiu’s mean value theorem and associated functional equations, two-dimensional mean value theorem and functional equations as well as several kinds of functional equations. We know that the mean value theorems have been motivated to study the functional equations (see the book “Mean Value Theorems and Functional Equations” by Sahoo and Riedel [305]) in 1998. In 1940, Ulam [321] proposed the following stability problem of functional equations: Given a group G1 , a metric group G2 with the metric d.; / and a positive number ", does there exist ı > 0 such that, if a mapping f W G1 ! G2 satisfies d.f .xy/; f .x/f .y//  ı for all x; y 2 G1 , then there exists a homomorphism h W G1 ! G2 such that d.f .x/; h.x//  " for all x 2 G1 ? Since then, several mathematicians have dealt with special cases as well as generalizations of Ulam’s problem. In fact, in 1941, Hyers [133] provided a partial solution to Ulam’s problem for the case of approximately additive mappings in which G1 and G2 are Banach spaces with ı D " as follows: Let X and Y be Banach spaces and let " > 0. Then, for all g W X ! Y with sup kg.x C y/  g.x/  g.y/k  "; x;y2X

there exists a unique mapping f W X ! Y such that sup kg.x/  f .x/k  " x2X

and f .x C y/ D f .x/ C f .y/ for all x; y 2 X. This proof remains unchanged if G1 is an Abelian semigroup. Particularly, in 1968, it was proved by Forti ([115], Proposition 1) that the following theorem can be proved:

1 Introduction

3

Theorem F (Forti). Let .S; C/ be an arbitrary semigroup and E be a Banach space. Assume that f W S ! E satisfies kf .x C y/  f .x/  f .y/k  ":

(A)

Then the limit f .2n x/ n!1 2n

g.x/ D lim

(B)

exists for all x 2 S and g W S ! E is the unique function satisfying kf .x/  g.x/k  ";

g.2x/ D 2g.x/:

Finally, if the semigroup S is Abelian, then G is additive. Here, the proof method which generates the solution g by the formula like (B) is called the direct method. If f is a mapping of a group or a semigroup .S; / into a vector space E, then we call the following expression: Cf .x; y/ D f .x  y/  f .x/  f .y/ the Cauchy difference of f on S  S. In the case that E is a topological vector space, we call the equation of homomorphism stable if, whenever the Cauchy difference Cf is bounded on S  S, there exists a homomorphism g W S ! E such that f  g is bounded on S. In 1980, Rätz [298] generalized Theorem F as follows: Let .X; / be a powerassociative groupoid, i.e., X is a nonempty set with a binary relation x1  x2 2 X such that the left powers satisfy xmCn D xm  xn for all m; n  1 and x 2 X. Let .Y; j  j/ be a topological vector space over the field Q of rational numbers with Q topologized by its usual absolute value j  j. Theorem R (Rätz). Let V be a nonempty bounded Q-convex subset of Y containing the origin and assume that Y is sequentially complete. Let f W X ! Y satisfy the following conditions: for all x1 ; x2 2 X, there exist k  2 such that n

n

n

f ..x1  x2 /k / D f .xk1  xk2 /

(C)

f .x1 / C f .x2 /  f .x1  x2 / 2 V:

(D)

for all n  1 and

Then there exists a function g W X ! Y such that g.x1 / C g.x2 / D g.x1  x2 / and f .x/  g.x/ 2 V, where V is the sequential closure of V for all x 2 X. When Y is a Hausdorff space, then g is uniquely determined.

4

1 Introduction

Note that the condition (C) is satisfied when X is commutative and it takes the place of the commutativity in proving the additivity of g. However, as Rätz pointed out in his paper, the condition n

n

n

.x1  x2 /k D xk1  xk2

for all x1 ; x2 2 X, where X is a semigroup, and, for all k  1, does not imply the commutativity. In the proofs of Theorems F and R, the completeness of the image space E and the sequential completeness of Y, respectively, were essential in proving the existence of the limit which defined the additive function g. The question arises whether the completeness is necessary for the existence of an odd additive function g such that f  g is uniformly bounded, given that the Cauchy difference is bounded. For this problem, in 1988, Schwaiger [306] proved the following: Theorem S (Schwaiger). Let E be a normed space with the property that, for each function f W Z ! E, whose Cauchy difference Cf D f .xCy/f .x/f .y/ is bounded for all x; y 2 Z and there exists an additive mapping g W Z ! E such that f .x/g.x/ is bounded for all x 2 Z. Then E is complete. Corollary 1. The statement of Theorem S remains true if Z is replaced by any vector space over Q. In 1950, Aoki [17] generalized Hyers’ theorem as follows: Theorem A (Aoki). Let E1 and E2 be two Banach spaces. If there exist K > 0 and 0  p < 1 such that kf .x C y/  f .x/  f .y/k  K.kxkp C kykp / for all x; y 2 E1 , then there exists a unique additive mapping g W E1 ! E2 such that kf .x/  g.x/k 

2K kxkp 2  2p

for all x 2 E1 . In 1978, Th. M. Rassias [267] formulated and proved the stability theorem for the linear mapping between Banach spaces E1 and E2 subject to the continuity of f .tx/ with respect to t 2 R for each fixed x 2 E1 . Thus Rassias’ Theorem implies Aoki’s Theorem as a special case. Later, in 1990, Th. M. Rassias [274] observed that the proof of his stability theorem also holds true for p < 0. In 1991, Gajda [119] showed that the proof of Rassias’ Theorem can be proved also for the case p > 1 by just replacing n by n in (B). These results are stated in a generalized form as follows (see Rassias and Šemrl [293]):

1 Introduction

5

Theorem RS (Th.M. Rassias and P. Semrl). Let ˇ.s; t/ be nonnegative function for all nonnegative real numbers s; t and positive homogeneous of degree p, where p is real and p ¤ 1, i.e., ˇ.s; t/ D p ˇ.s; t/ for all nonnegative ; s; t. Given a normed space E1 and a Banach space E2 , assume that f W E1 ! E2 satisfies the inequality kf .x C y/  f .x/  f .y/k  ˇ.kxk; kyk/ for all x; y 2 E1 . Then there exists a unique additive mapping g W E1 ! E2 such that kf .x/  g.x/k  ıkxkp for all x 2 E1 , where ( ı WD

ˇ.1;1/ ; 22p ˇ.1;1/ ; 22p

p < 1; p > 1:

The proofs for the cases p < 1 and p > 1 were provided by applying the direct methods. For p < 1, the additive mapping g is given by (B), while, in case p > 1, the formula is g.x/ D lim 2n f n!1

x : 2n

Corollary 2. Let f W E1 ! E2 be a mapping satisfying the hypotheses of Theorem RS and suppose that f is continuous at a single point y 2 E1 . Then the additive mapping g is continuous. Corollary 3. If, under the hypotheses of Theorem RS, we assume that, for each fixed x 2 E1 , the mapping t ! f .tx/ from R to E2 is continuous, then the additive mapping g is R–linear. Remark 4. (1) For p D 0, Theorem RS, Corollaries 2 and 3 reduce to the results of Hyers in 1941. If we put ˇ.s; t/ D ".sp C tp /, then we obtain the results of Rassias [267] in 1978 and Gajda [119] in 1991. (2) The case p D 1 was excluded in Theorem RS. Simple counterexamples prove that one can not extend Rassias’ Theorem when p takes the value one (see Z. Gajda [119], Rassias and Šemrl [293] and Hyers and Rassias [135] in 1992). A further generalization of the Hyers-Ulam stability for a large class of mappings was obtained by Isac and Rassias [139] by introducing the following: Definition 5. A mapping f W E1 ! E2 is said to be -additive if there exist ˚  0 and a function  W RC ! RC satisfying .t/ D0 t!C1 t lim

6

1 Introduction

such that kf .x C y/  f .x/  f .y/k  ˚Œ.kxk/ C .kyk/ for all x; y 2 E1 . In [139], Isac and Rassias proved the following: Theorem IR (Isac and Rassias). Let E1 be a real normed vector space and E2 be a real Banach space. Let f W E1 ! E2 be a mapping such that f .tx/ is continuous in t for each fixed x 2 E1 . If f is -additive and  satisfies the following conditions: (1) .ts/  .t/.s/ for all s; t 2 R; (2) .t/ < t for all t > 1, then there exists a unique R–linear mapping T W E1 ! E2 such that kf .x/  T.x/k 

2 .kxk/ 2  .2/

for all x 2 E1 . Remark 6. (1) If .t/ D tp with p < 1, then, from Theorem IR, we obtain Rassias’ Theorem [267]. (2) If p < 0 and .t/ D tp with t > 0, then Theorem IR is implied by the result of Gajda in 1991. Since the time the above stated results have been proven, several mathematicians (cf. [1, 3, 14, 44, 46, 49–65, 69–80, 82–85, 87–90, 95–99, 101–107, 109, 116– 118, 120, 122, 124, 125, 129–134, 136, 137, 140–149, 151–153, 156–158, 160– 162, 164, 173–179, 187, 189, 190, 195–201, 207, 208, 212, 214, 217, 219, 221– 223, 226, 228, 230, 236, 239–241, 262, 266, 268, 269, 275–288, 296–303, 309, 311–322, 324, 330, 331] and also very recent survey papers [42, 60, 61]) have extensively studied stability theorems for several kinds of functional equations in various spaces, for example, Banach spaces, 2-Banach spaces, Banach nLie algebras, quasi-Banach spaces, Banach ternary algebras, non-Archimedean normed and Banach spaces, metric and ultra metric spaces, Menger probabilistic normed spaces, probabilistic normed space, p-2-normed spaces, C -algebras, C ternary algebras, Banach ternary algebras, Banach modules, inner product spaces, Heisenberg groups and others. Further, we have to pay attention to applications of the Hyers-Ulam-Rassias stability problems, for example, (partial) differential equations, Fréchet functional equations, Riccati differential equations, Volterra integral equations, group and ring theory and some kinds of equations (see [66, 142, 150, 154, 159, 176, 185, 186, 192–194, 259–261, 327, 329]). For more details on recent development in Ulam’s type stability and its applications, see the papers of Brillouët-Belluot et al. [53] and Ciepli´nski [86] in 2012 (see also [3–22, 47, 78, 79, 81, 110–112, 138, 148–150, 154, 157–163, 165, 166, 183, 204, 205, 213, 215, 216, 245, 246, 249, 255, 257, 263, 264, 270–295, 302, 304, 328]).

1.1 Fixed Point Theorems

7

A functional equation is called stable if any function satisfying the functional equation “approximately” is near to a true solution of the functional equation. We say that a functional equation is superstable if every approximate solution is an exact solution of it (see some recent papers [40, 41, 55, 59]).

1.1 Fixed Point Theorems In this section, we present some fixed point theorems which will play important roles in proving our main theorems. All stability results for functional equations were proved by applying direct method. Since the direct method sometimes does not work. In consequence, the fixed point method for studying the stability of functional equations was used for the first time by Baker in 1991 (see [43]). Next, in 2003, Radu [265] gave a lecture at Seminar on Fixed Point Theory Cluj-Napoca and proved a stability of functional equation by fixed method. Then, in 2003, C˘adariu and Radu [62, 64] considered Jensen functional equation and proved a stability result via fixed point method. Jung and Chang [155] proved the stability of a cubic type functional equation with the fixed point alternative. Since then, some authors [151–153, 156, 157, 162, 164, 191, 211, 234, 251, 256] considered some important functional equations and proved the stability results via fixed point method introduced by Baker and Radu. The Banach fixed point theorem [45] (also known as the Banach contraction principle) is an important tool in the theory of metric spaces because it guarantees the existence and uniqueness of fixed points of certain self mappings of metric spaces and provides a constructive method to find those fixed points. The theorem is named after Banach (1892–1945) and was first stated by him in 1922. Theorem 1.1 (Banach [45]). Let .X; d/ be a complete metric space and T W X!X be a contraction, i.e., there exists ˛ 2 Œ0; 1/ such that d.Tx; Ty/  ˛d.x; y/ for all x; y 2 X. Then there exists a unique a 2 X such that Ta D a. Moreover, for all x 2 X, lim T n x D a

n!1

and, in fact, for all x 2 X, d.x; a/ 

1 d.x; Tx/: 1˛

8

1 Introduction

Theorem 1.2 ([62, 265]). Let .X; d/ be a complete metric space and J W X ! X be a strictly contractive mapping, i.e., there exists a Lipschitz constant L < 1 such that d.Jx; Jy/  Ld.x; y/ for all x; y 2 X. Then we have (1) The mapping J has a unique fixed point x 2 X; (2) The fixed point x is globally attractive, i.e., lim J n x D x

n!1

for all x 2 X; (3) The following inequalities hold: d.J n x; x /  Ln d.x; x /; 1 d.J n x; J nC1 x/; 1L 1 d.x; Jx/ d.x; x /  1L

d.J n x; x / 

for all x 2 X and n  1. Following Luxemburg [188], the concept of a generalized complete metric space may be introduced as in this quotation: Let X be a nonempty set. A function d W X  X ! Œ0; 1 is called a generalized metric on X if, for any x; y; z 2 X, (1) d.x; y/ D 0 if and only if x D y; (2) d.x; y/ D d.y; x/; (3) d.x; z/  d.x; y/ C d.y; z/. This concept differs from the usual concept of a complete metric space by the fact that not every two points in X have necessarily a finite distance. One might call such a space a generalized complete metric space. Next, Diaz and Margolis [95] proved a theorem of the alternative for any contraction mapping T on a generalized complete metric space X. The conclusion of the theorem, speaking in general terms, asserts that: either all consecutive pairs of the sequence of successive approximations (starting from an element x0 of X) are infinitely far apart or the sequence of successive approximations, with initial element x0 converges to a fixed point of T (what particular fixed point depends, in general, on the initial element x0 ). Theorem 1.3 ([62, 95]). Let .X; d/ be a complete generalized metric space and J W X ! X be a strictly contractive mapping with a Lipschitz constant L < 1. Then, for each x 2 X, either

1.2 Quasi-Banach Algebras

9

d.J n x; J nC1 x/ D 1 for all n  0 or there exists a positive integer n0 such that (1) (2) (3) (4)

d.J n x; J nC1 x/ < 1 for all n  n0 ; The sequence .J n x/ converges to a fixed point y of J; y is the unique fixed point of J in the set Y D fy 2 X W d.J n0 x; y/ < 1g; 1 d.y; y /  1L d.y; Jy/ for all y 2 Y.

1.2 Quasi-Banach Algebras Let X be a vector space on field C. A normed space X in which, for all x; y 2 X, xy 2 X and kxyk  kxkkyk is called a complex normed algebra. A complete normed algebra is called a Banach algebra. Moreover, if there exists a unit element e such that ex D xe D x for all x 2 X, then kek D 1 and X is called a unital Banach algebra. Let X; Y be Banach algebras. A C-linear mapping H W X ! Y is called a homomorphism in Banach algebras if H satisfies H.xy/ D H.x/H.y/ for all x; y 2 X. A C-linear mapping ı W X ! X is called a derivation on X if ı satisfies ı.xy/ D ı.x/y C xı.y/ for all x; y 2 X. We recall some basic facts concerning quasi-Banach spaces and some preliminary results. Definition 1.4 ([48, 300]). Let X be a real linear space. A quasi-norm is a realvalued function on X satisfying the following: (1) kxk  0 for all x 2 X and kxk D 0 if and only if x D 0; (2) kxk D jj  kxk for all  2 R and x 2 X; (3) There is a constant K  1 such that kx C yk  K.kxk C kyk/ for all x; y 2 X. The pair .X; k  k/ is called a quasi-normed space if k  k is a quasi-norm on X. The smallest possible K is called the modulus of concavity of k  k. Obviously, the balls with respect to k  k define a linear topology on X. By a quasi-Banach space we mean a complete quasi-normed space, i.e., a quasi-normed space in which every k  k-Cauchy sequence in X converges. This class includes Banach spaces and the most significant class of quasi-Banach spaces which are not Banach spaces are the Lp spaces for 0 < p < 1 with the quasi-norm k  kp .

10

1 Introduction

A quasi-norm k  k is called a p-norm .0 < p  1/ if kx C ykp  kxkp C kykp for all x; y 2 X. In this case, a quasi-Banach space is called a p-Banach space. For any p-norm, the formula d.x; y/ WD kx  ykp gives us a translation invariant metric on X. By the Aoki–Rolewicz theorem [300] (see also [48]), each quasi-norm is equivalent to some p-norm. Since it is much easier to work with p-norms than quasi-norms, henceforth we restrict our attention mainly to p-norms. Definition 1.5 ([10]). Let .A; k  k/ be a quasi-normed space. The quasi-normed space .A; k  k/ is called a quasi-normed algebra if A is an algebra and there exists a constant C > 0 such that kxyk  Ckxk  kyk for all x; y 2 A. A quasi-Banach algebra is a complete quasi-normed algebra. If the quasi-norm k  k is a p-norm, then the quasi-Banach algebra is called a p-Banach algebra.

1.3 C -Algebras Let U be a Banach algebra. Then an involution on U is a mapping u ! u from U into U which satisfies the following conditions: (1) u D u for all u 2 U; (2) .˛u C ˇv/ D ˛u C ˇv  ; (3) .uv/ D v  u for all u; v 2 U. If, in addition, ku uk D kuk2 for all u 2 U, then U is a C -algebra. Let U; V be C -algebras. A C-linear mapping H W U ! V is called a homomorphism in C -algebras if H satisfies H.xy/ D H.x/H.y/;

H.x / D H.x/

for all x; y 2 U. A C-linear mapping ı W U ! U is called a derivation on U if ı satisfies ı.xy/ D ı.x/y C xı.y/ for all x; y 2 U.

1.3 C -Algebras

11

Suppose that A is a complex Banach -algebra. Let C-linear mapping ı W D.ı/ ! A be a derivation on A, where D.ı/ is the domain of ı and D.ı/ is dense in A. If ı satisfies the additional condition ı.a / D ı.a/ for all a 2 A, then ı is called a -derivation on A. It is well-known that, if A is a C -algebra and D.ı/ is A, then the derivation ı is bounded. Now, we consider proper CQ -algebras, which arise as completions of  C -algebras (see [15–39]) as follows: Let A be a Banach module over the C -algebra A0 with an involution  and C -norm k  k0 such that A0  A. We say that .A; A0 / is a proper CQ algebra if (1) A0 is dense in A with respect to its norm k  k; (2) An involution , which extends the involution of A0 , is defined in A with the property .xy/ D y x for all x; y 2 A whenever the multiplication is defined; (3) kyk0 D supx2A;kxk1 kxyk for all y 2 A0 . Definition 1.6. Let .A; A0 / and .B; B0 / be proper CQ -algebras. (1) A C-linear mapping h W A ! B is called a proper CQ -algebra homomorphism if h.xy/ D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined; (2) A C-linear mapping ı W A ! A is called a derivation on A if ı.xy/ D ı.x/y C xı.y/ for all x; y 2 A whenever the multiplication is defined. A C -algebra C endowed with the Lie product Œx; y WD C -algebra (see [224, 225, 227]). 

xyyx 2

on C is called a Lie

Definition 1.7. Let A and B be Lie C -algebras. A C-linear mapping H W A ! B is called a Lie C -algebra homomorphism if H.Œx; y/ D ŒH.x/; H.y/ for all x; y 2 A. Definition 1.8. Let A be a Lie C -algebra. A C-linear mapping ı W A ! A is called a Lie derivation if ı.Œx; y/ D Œı.x/; y C Œx; ı.y/ for all x; y 2 A.

12

1 Introduction

1.4 C -Ternary Algebras Ternary algebraic structures appear more or less naturally in various domains of theoretical and mathematical physics, for example, the quark model inspired a particular brand of ternary algebraic system. One such attempt has been proposed by Nambu in 1973 and is now known under the name of “Nambu mechanics” [316] (see also [332]). A C -ternary algebra is a complex Banach space A, equipped with a ternary product .x; y; z/ 7! Œx; y; z of A3 into A, which is C-linear in the outer variables, conjugate C-linear in the middle variable and associative in the sense that Œx; y; Œz; w; v D Œx; Œw; z; y; v D ŒŒx; y; z; w; v and satisfies kŒx; y; zk  kxk  kyk  kzk;

kŒx; x; xk D kxk3

(see [332]). If a C -ternary algebra .A; Œ; ; / has the identity, i.e., an element e 2 A such that x D Œx; e; e D Œe; e; x for all x 2 A, then it is routine to verify that A, endowed with x ı y WD Œx; e; y and x WD Œe; x; e, is a unital C -algebra. Conversely, if .A; ı/ is a unital C -algebra, then Œx; y; z WD x ı y ı z makes A into a C -ternary algebra. A C-linear mapping H W A ! B is called a C -ternary algebra homomorphism if H.Œx; y; z/ D ŒH.x/; H.y/; H.z/ for all x; y; z 2 A. A C-linear mapping ı W A ! A is called a C -ternary derivation if ı.Œx; y; z/ D Œı.x/; y; z C Œx; ı.y/; z C Œx; y; ı.z/ for all x; y; z 2 A (see [231]). Ternary structures and their generalization, the so-called n-ary structures, are important in view of their applications in physics (see [171]). Suppose that J is a complex vector space endowed with a real trilinear composition J  J  J 3 .x; y; z/ 7! fxy zg 2 J which is complex bilinear in .x; z/ and conjugate linear in y. Then J is called a Jordan triple system if fxy zg D fzy xg and ffxy zgu vg C ffxy vgu zg  fxy fzu vgg D fzfyx ug vg:

1.5 Non-Archimedean Normed Algebras

13

We are interested in Jordan triple systems having a Banach space structure. A complex Jordan triple system J with a Banach space norm k  k is called a J  triple if, for all x 2 J , the operator xx is hermitian in the sense of Banach algebra theory. Here the operator xx on J is defined by .xx /y WD fxx yg. This implies that xx has the real spectrum .xx /  R. A J  -triple J is called a JB -triple if every x 2 J satisfies .xx /  0 and kxx k D kxk2 . A C-linear mapping H W J ! L is called a JB -triple homomorphism if H.fxyzg/ D fH.x/H.y/H.z/g for all x; y; z 2 J . A C-linear mapping ı W J ! J is called a JB -triple derivation if ı.fxyzg/ D fı.x/yzg C fxı.y/zg C fxyı.z/g for all x; y; z 2 J (see [225]).

1.5 Non-Archimedean Normed Algebras By a non-Archimedean field we mean a field K equipped with a function (valuation) j  j from K into Œ0; 1/ such that (1) jrj D 0 if and only if r D 0; (2) jrsj D jrj jsj; (3) jr C sj  maxfjrj; jsjg for all r; s 2 K. Clearly, j1j D j  1j D 1 and jnj  1 for all n  1. By the trivial valuation we mean the mapping j  j taking everything but 0 into 1 and j0j D 0. Let X be a vector space over a field K with a non-Archimedean non-trivial valuation j  j. A function k  k W X ! Œ0; 1/ is called a non-Archimedean norm if it satisfies the following conditions: (1) kxk D 0 for all x 2 X if and only if x D 0; (2) For all r 2 K; x 2 X, krxk D jrjkxk; (3) The strong triangle inequality (ultrametric) holds, i.e., kx C yk  maxfkxk; kykg for all x; y 2 X. Then .X; k  k/ is called a non-Archimedean normed space. From the fact that kxn  xm k  maxfkxjC1  xj k W m  j  n  1g

14

1 Introduction

for all n  1 with n > m, a sequence fxn g is a Cauchy sequence if and only if fxnC1  xn g converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent. For any nonzero rational number x, there exists a unique integer nx 2 Z such that xD

a nx p ; b

where a and b are integers not divisible by p. Then jxjp WD pnx defines a nonArchimedean norm on Q. The completion of Q with respect to the metric d.x; y/ D jx  yjp is denoted by Qp , which is called the p-adic number field. A non-Archimedean Banach algebra is a complete non-Archimedaen algebra A which satisfies kabk  kak  kbk for all a; b 2 A. For more detailed definitions of non-Archimedean Banach algebras, the readers refer to [310]. If U is a non-Archimedean Banach algebra, then an involution on U is a mapping t ! t from U into U satisfying the following: (1) t D t for all t 2 U; (2) .˛s C ˇt/ D ˛s C ˇt ; (3) .st/ D t s for all s; t 2 U. If, in addition kt tk D ktk2 for all t 2 U, then U is a non-Archimedean C –algebra. 

1.6 Multi–normed Algebras The notion of multi-normed space was introduced by Dales and Polyakov in [92]. This concept is somewhat similar to the operator sequence space and has some connections with the operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples are given in [91, 92, 206]. Let .E; k  k/ be a complex normed space and let k 2 N. We denote by E k the linear space E ˚    ˚ E consisting of k-tuples .x1 ;    ; xk /, where x1 ;    ; xk 2 E. The linear operations on E k are defined coordinate-wise. The zero element of either E or E k is denoted by 0. We denote by Nk the set f1; 2;    ; kg and by ˙k the group of permutations on k symbols. Definition 1.9. A multi-norm on fE k W k 2 Ng is a sequence .k  kk / D .k  kk W k 2 N/ such that k  kk is a norm on E k for each k 2 N: (A1) k.x .1/ ;    ; x .k/ /kk D k.x1 ;    ; xk /kk for all  2 ˙k and x1 ;    ; xk 2 E; (A2) k.˛1 x1 ;    ; ˛k xk /kk  .maxi2Nk j˛i j/ kx1 ;    ; xk kk for all ˛1 ;    ; ˛k 2 C and x1 ;    ; xk 2 E;

1.6 Multi–normed Algebras

15

(A3) k.x1 ;    ; xk1 ; 0/kk D k.x1 ;    ; xk1 /kk1 for all x1 ;    ; xk1 2 E; (A4) k.x1 ;    ; xk1 ; xk1 /kk D k.x1 ;    ; xk1 /kk1 for all x1 ;    ; xk1 2 E. In this case, we say that ..E k ; k  kk / W k 2 N/ is a multi-normed space. Lemma 1.10 ([206]). Suppose that ..E k ; k  kk / W k 2 N/ is a multi-normed space and let k 2 N. Then (1) k.x;    ; x/kk D kxk for all x 2 E; Pk (2) maxi2Nk kxi k  k.x1 ;    ; xk /kk  iD1 kxi k  k maxi2Nk kxi k for all x1 ;    ; xk 2 E. It follows from (2) that, if .E; k  k/ is a Banach space, then .E k ; k  kk / is a Banach space for each k 2 N. In this case, ..E k ; k  kk / W k 2 N/ is a multi-Banach space. Now, we state two important examples of multi-norms for an arbitrary normed space E (see [92]). Example 1.11. The sequence .k  kk W k 2 N/ on fE k W k 2 Ng defined by kx1 ;    ; xk kk WD max kxi k i2Nk

for all x1 ;    ; xk 2 E is a multi-norm called the minimum multi-norm. The terminology “minimum” is justified by the property (2). Example 1.12. Let f.k  k˛k W k 2 N/ W ˛ 2 Ag be the (non-empty) family of all multi-norms on fE k W k 2 Ng. For k 2 N, set kjx1 ;    ; xk kjk WD sup k.x1 ;    ; xk /k˛k ˛2A

for all x1 ;    ; xk 2 E. Then .kj  kjk W k 2 N/ is a multi-norm on fE k W k 2 Ng, which is called the maximum multi-norm. We need the following observation which can be easily deduced from the triangle inequality for the norm k  kk and the property (2) of multi-norms. Lemma 1.13. Suppose that k 2 N and .x1 ;    ; xk / 2 E k . For each j 2 f1;    ; kg, let .xjn /n1 be a sequence in E such that limn!1 xjn D xj . Then, for each .y1 ;    ; yk / 2 E k , we have lim .x1n  y1 ;    ; xkn  yk / D .x1  y1 ;    ; xk  yk /:

n!1

Definition 1.14. Let ..E k ; k  kk / W k 2 N/ be a multi-normed space. A sequence .xn / in E is a multi-null sequence if, for any " > 0, there exists n0 2 N such that sup k.xn ;    ; xnCk1 /kk < " k2N

for each n  n0 . We say that the sequence .xn / is multi-convergent to a point x 2 E .write limn!1 xn D x/ if .xn  x/ is a multi-null sequence.

16

1 Introduction

Definition 1.15 ([92, 170, 202]). Let .A; k  k/ be a normed algebra such that ..Ak ; k  kk / W k 2 N/ is a multi-normed space. Then ..Ak ; k  kk / W k 2 N/ is called a multi-normed algebra if k.a1 b1 ;    ; ak bk /kk  k.a1 ;    ; ak /kk  k.b1 ;    ; bk /kk for all k 2 N and a1 ;    ; ak ; b1 ;    ; bk 2 A. Further, the multi-normed algebra ..Ak ; k  kk / W k 2 N/ is called a multi-Banach algebra if ..Ak ; k  kk / W k 2 N/ is a multi-Banach space. Example 1.16 ([92, 202, 252]). Let p; q with 1  p  q < 1 and A D `p . The algebra A is a Banach sequence algebra with respect to coordinatewise multiplication of sequences. Let .k  kk W k 2 N/ be the standard .p; q/-multi-norm on fAk W k 2 Ng. Then ..Ak ; k  kk / W k 2 N/ is a multi-Banach algebra. Definition 1.17. Let .A; kk/ be a Banach -algebra with the involution . A multiC -algebra is a multi-Banach algebra such that k.a1 a1 ;    ; ak ak /k D k.a1 ;    ; ak /k2 : In a series of the papers [15–33, 36–38] and [318–320], many authors have considered a special class of quasi -algebras, called proper CQ -algebras, which arise as completions of C -algebras. They can be introduced in the following way: Let A be a linear space and A be a -algebra contained in A. We say that A is a quasi--algebra over A if the right and left multiplications of an element of A and an element of A are always defined and linear. An involution  which extends the involution of A is defined in A with the property .ab/ D b a whenever the multiplication is defined. A quasi--algebra .A; A/ is said to be topological if there exists a locally convex topology  on A such that (Q1) The involution a 7! a is continuous; (Q2) The mappings a 7! ab and a 7! ba are continuous for each b 2 A; (Q3) A is dense in A with topology . In a topological quasi--algebra, the associative law holds in the following two formulations: a.bc/ D .ab/c;

b.ac/ D .ba/c

for all b; c 2 A and a 2 A. A CQ -algebra is a topological quasi--algebra .A; A/ with the following properties: (CQ1) .A; kk / is a C -algebra with respect to the norm kk and the involution ; (CQ2) .A; k  k/ is a Banach space and ka k D kak for all a 2 A;

1.6 Multi–normed Algebras

17

(CQ3) For all b 2 A, we have kbk D max

n

o sup kabk; sup kbak :

kak1

kak1

Bagarello and Trapani [34] showed that both .Lp .X; /; C0 .X// and .Lp .X; /; L .X// are CQ -algebras. Now, we define the multi-CQ -algebra. Let .A; A/ be a CQ -algebra. We say that f.Ak ; Ak / W k 2 Ng is a multi-CQ algebra if, for each k 2 N, the couple .Ak ; Ak / is a CQ -algebra, where fAk W k 2 Ng and fAk W k 2 Ng are a multi-Banach algebra and a multi-C -algebra, respectively. 1

Example 1.18. In [34], the authors showed that the couple .A; A/ is a CQ -algebra, where A D `p and A D c0 . Now, consider Example 1.16. Then f.Ak ; Ak / W k 2 Ng is a multi-CQ -algebra. Definition 1.19. Let ..Ak ; k  kk / W k 2 N/ be a multi-Banach space. A multi-C ternary algebra is a complex multi-Banach space ..Ak ; k  kk / W k 2 N/ equipped with a ternary product.

Chapter 2

Stability of Functional Equations in Banach Algebras

Beginning around the year 1980, the topic of approximate homomorphisms and derivations and their stability theory in the field of functional equations and inequalities was taken up by several mathematicians (see Hyers and Rassias [135], Rassias [285] and the references therein). In this chapter, in the first section, we show that, if X and Y are normed spaces with the norms k  kX and k  kY , respectively, and f W X ! Y is a mapping such that  1   kf .x/ C f .y/ C f .z/kY   f .qx C qy C qz/ Y q for all x; y; z 2 X and for a fixed nonzero rational number q, then f is Cauchy additive. Next, we approximate isomorphisms and derivations in Banach algebras by the direct method. In Sect. 2.2, we consider the m-variable additive functional equation: m  m m m  X  X  X X f mxi C xj C f xi D 2f mxi iD1

iD1

jD1; j¤i

iD1

for all m 2 N and m  2 and, by the fixed point method, we approximate homomorphisms and derivations in Banach algebras. In Sect. 2.3, we prove the Hyers-Ulam stability of homomorphisms in quasiBanach algebras and generalized derivations on quasi-Banach algebras for the following functional equation: n X iD1

0 f@

n X jD1

1 q.xi  xj /A C nf

n X

! qxi

iD1

© Springer International Publishing Switzerland 2015 Y.J. Cho et al., Stability of Functional Equations in Banach Algebras, DOI 10.1007/978-3-319-18708-2_2

D nq

n X

f .xi /:

iD1

19

20

2 Stability of Functional Equations in Banach Algebras

In Sect. 2.4, we approximate homomorphisms in real Banach algebras and generalized derivations on real Banach algebras for the following Cauchy-Jensen functional equations f

x C y 2

 x  y  Cz Cf C z D f .x/ C 2f .z/ 2

and 2f

x C y 2

 C z D f .x/ C f .y/ C 2f .z/:

2.1 Stability of 1q f .qx C qy C qz/ D f .x/ C f .y/ C f .z/ Using the direct method, we investigate isomorphisms in Banach algebras and derivations on Banach algebras associated with the following functional equation 1 f .qx C qy C qz/ D f .x/ C f .y/ C f .z/ q

(2.1)

for a fixed nonzero rational number q.

2.1.1 Isomorphisms in Banach Algebras Here we consider isomorphisms in Banach algebras associated with the functional equation (2.1). Lemma 2.1. Let X and Y be normed spaces with norms kkX and kkY , respectively. Let f W X ! Y be a mapping with f .0/ D 0 such that  1   kf .x/ C f .y/ C f .z/kY   f .qx C qy C qz/ Y q for all x; y; z 2 X, then f is Cauchy additive, i.e., f .x C y/ D f .x/ C f .y/: Proof. Letting z D 0 and y D x in (2.2), we get  1   kf .x/ C f .x/kY   f .0/ D 0 Y q

(2.2)

2.1 Stability of 1q f .qx C qy C qz/ D f .x/ C f .y/ C f .z/

21

for all x 2 X. Hence f .x/ D f .x/ for all x 2 X. Letting z D x  y in (2.2), we get kf .x/ C f .y/  f .x C y/kY D kf .x/ C f .y/ C f .x  y/kY  1     f .0/ Y q D0 for all x; y 2 X. Thus f .x C y/ D f .x/ C f .y/ for all x; y 2 X. This completes the proof.



Here, we assume that A is a Banach algebra with the norm kkA and B is a Banach algebra with the norm k  kB . Theorem 2.2. Let r ¤ 1,  be nonnegative real numbers and f W A ! B be a bijective mapping with f .0/ D 0 such that  1   k f .x/ C f .y/ C f .z/kB   f .q x C qy C qz/ B q

(2.3)

and 2r kf .xy/  f .x/f .y/kB  .kxk2r A C kykA /

(2.4)

for all 2 T1 WD f 2 C W jj D 1g and x; y; z 2 A, then the bijective mapping f W A ! B is an isomorphism. Proof. Let D 1 in (2.3). By Lemma 2.1, the mapping f W A ! B is Cauchy additive. Letting z D 0 and y D  x in (2.3), we get f .x/  f . x/ D f .x/ C f . x/ D 0 for all x 2 A and so f . x/ D f .x/ for all x 2 A. By (2.3), the mapping f W A ! B is C-linear. (i) Assume that r < 1. By (2.4), we have 1 kf .4n xy/  f .2n x/f .2n y/kB 4n 4nr 2r  lim n .kxk2r A C kykA / n!1 4 D0

kf .xy/  f .x/f .y/kB D lim

n!1

22

2 Stability of Functional Equations in Banach Algebras

for all x; y 2 A and so f .xy/ D f .x/f .y/ for all x; y 2 A. (ii) Assume that r > 1. By a similar method to the proof of the case (i), one can prove that the mapping f W A ! B satisfies f .xy/ D f .x/f .y/ for all x; y 2 A. Therefore, the bijective mapping f W A ! B is an isomorphism in Banach algebras. This completes the proof.  Theorem 2.3. Let r ¤ 1,  be nonnegative real numbers and f W A ! B be a bijective mapping satisfying f .0/ D 0 and (2.3) such that kf .xy/  f .x/f .y/kB    kwkrA  kxkrA

(2.5)

for all x; y 2 A, then the bijective mapping f W A ! B is an isomorphism. Proof. By (2.3), the mapping f W A ! B is C-linear. (i) Assume that r < 1. By (2.5), we have 1 kf .4n xy/  f .2n x/f .2n y/kB 4n 4nr  lim n   kwkrA  kxkrA n!1 4 D0

kf .xy/  f .x/f .y/kB D lim

n!1

for all x; y 2 A and so f .xy/ D f .x/f .y/ for all x; y 2 A. (ii) Assume that r > 1. By a similar method to the proof of the case (i), one can prove that the mapping f W A ! B satisfies f .xy/ D f .x/f .y/ for all x; y 2 A. Therefore, the bijective mapping f W A ! B is an isomorphism. This completes the proof. 

2.1 Stability of 1q f .qx C qy C qz/ D f .x/ C f .y/ C f .z/

23

2.1.2 Derivations in Banach Algebras We consider derivations on Banach algebras associated with the functional equation (2.1). Theorem 2.4. Let r ¤ 1,  be nonnegative real numbers and f W A ! A be a mapping with f .0/ D 0 such that  1   k f .x/ C f .y/ C f .z/kA   f .q x C qy C qz/ A q

(2.6)

and 2r kf .xy/  f .x/y  xf .y/kA  .kxk2r A C kykA /

(2.7)

for all 2 T1 and x; y; z 2 A, then the mapping f W A ! A is a derivation on A. Proof. By (2.7), the mapping f W A ! A is C-linear. (i) Assume that r < 1. By (2.7), kf .xy/  f .x/y  xf .y/kA 1 kf .4n xy/  f .2n x/  2n y  2n xf .2n y/kA n!1 4n 4nr 2r  lim n .kxk2r A C kykA / n!1 4 D0

D lim

for all x; y 2 A and so f .xy/ D f .x/y C xf .y/ for all x; y 2 A. (ii) Assume that r > 1. By a similar method to the proof of the case (i), one can prove that the mapping f W A ! A satisfies f .xy/ D f .x/y C xf .y/ for all x; y 2 A. Therefore, the mapping f W A ! A is a derivation on A. This completes the proof.  Theorem 2.5. Let r ¤ 1,  be nonnegative real numbers and f W A ! A be a mapping satisfying f .0/ D 0 and (2.6) such that kf .xy/  f .x/y  xf .y/kA    kxkrA  kykrA for all x; y 2 A, then the mapping f W A ! A is a derivation on A.

24

2 Stability of Functional Equations in Banach Algebras

Proof. The proof is similar to the proofs of Theorems 2.2 and 2.4.



2.2 Stability of m-Variable Functional Equations In this section, using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms and of derivations on Banach algebras for the following additive functional equation (see [108]): m  m m m  X  X  X X f mxi C xj C f xi D 2f mxi iD1

jD1; j¤i

iD1

iD1

for all m 2 N with m  2.

2.2.1 Stability of Homomorphisms in Banach Algebras For any mapping f W A ! B, we define D f .x1 ;    ; xm / WD

m X iD1

m m m   X   X  X f mxi C xj C f xi  2f mxi jD1; j¤i

iD1

iD1

for all 2 T1 WD f 2 C W j j D 1g and x1 ;    ; xm 2 A. Now, we prove the Hyers-Ulam stability of homomorphisms in Banach algebras for the functional equation D f .x1 ;    ; xm / D 0. Assume that A is a complex Banach algebra with norm k  kA and B is a complex Banach algebra with norm k  kB . Theorem 2.6. Let f W A ! B be a mapping for which there are functions ' W Am ! Œ0; 1/ and W A2 ! Œ0; 1/ such that limj!1 mj '.mj x1 ;    ; mj xm / D 0;

(2.8)

kD f .x1 ;    ; xm /kB  '.x1 ;    ; xm /;

(2.9)

kf .xy/  f .x/f .y/kB 

.x; y/;

(2.10)

limj!1 m2j .mj x; mj y/ D 0

(2.11)

2.2 Stability of m-Variable Functional Equations

25

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. If there exists L < 1 such that '.mx; 0;    ; 0/  mL'.x; 0;    ; 0/ for all x 2 A, then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

1 '.x; 0;    ; 0/ m  mL

(2.12)

for all x 2 A. Proof. Consider the set X WD fg W A ! Bg and introduce the generalized metric on X: d.g; h/ D inffC 2 RC W kg.x/  h.x/kB  C'.x; 0;    ; 0/; 8x 2 Ag; which .X; d/ is complete. Now, we consider the linear mapping J W X ! X such that Jg.x/ WD m1 g.mx/ for all x 2 A. Now, we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting D 1, x D x1 and x2 D    D xm D 0 in (2.9), we get kf .mx/  mf .x/kB  '.x; 0;    ; 0/

(2.13)

for all x 2 A and so   1 1   f .x/  f .mx/  '.x; 0;    ; 0/ B m m for all x 2 A. Hence d.f ; Jf /  m1 . By Theorem 1.3, there exists a mapping H W A ! B such that (1) H is a fixed point of J, i.e., H.mx/ D mH.x/

(2.14)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (2.13) such that there exists C 2 .0; 1/ satisfying kH.x/  f .x/kB  C'.x; 0;    ; 0/ for all x 2 A;

26

2 Stability of Functional Equations in Banach Algebras

(2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality f .mn x/ D H.x/ n!1 mn

(2.15)

lim

for all x 2 A; 1 d.f ; Jf /, which implies the inequality d.f ; H/  (3) d.f ; H/  1L implies that the inequality (2.12) holds.

1 mmL :

This

Thus it follows from (2.8), (2.9) and (2.14) that m m m m X  X  X   X   H mxi C xj C H xi  2H mxi   iD1

iD1

jD1; j¤i

iD1

B

m  m  X 1  X nC1 n f m x C m x  i j n!1 mn iD1

D lim

jD1; j¤i

Cf

m X

m  X   mn xi  2f mnC1 xi 

iD1

B

iD1

1 '.mn x1 ;    ; mn xm / D 0 n!1 mn

 lim

for all x1 ;    ; xm 2 A and so m m m m  X  X   X X H mxi C xj C H xi D 2H mxi iD1

iD1

jD1; j¤i

(2.16)

iD1

for all x1 ;    ; xm 2 A. By a similar method given in above, we get H.mx/ D H.m x/ for all 2 T1 and x 2 A. Thus one can show that the mapping H W A ! B is C-linear. It follows from (2.10) that 1 kf .mn xy/  f .mn x/f .mn y/kB mn 1  lim n .mn x; mn y/ n!1 m D0

kH.xy/  H.x/H.y/kB D lim

n!1

for all x; y 2 A and so H.xy/ D H.x/H.y/ for all x; y 2 A. Thus H W A ! B is a homomorphism satisfying (2.12). This completes the proof.  Corollary 2.7. Let r < 1,  be nonnegative real numbers and f W A ! B be a mapping such that kD f .x1 ;    ; xm /kB    .kx1 krA C kx2 krA C    C kxm krA /

(2.17)

2.2 Stability of m-Variable Functional Equations

27

and kf .xy/  f .x/f .y/kB    .kxkrA  kykrA /

(2.18)

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. Then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

 kxkrA m  mr

for all x 2 A. Proof. The proof follows from Theorem 2.6 by taking '.x1 ;    ; xm / D   .kx1 krA C kx2 krA C    C kxm krA /; .x; y/ WD   .kxkrA  kykrA / for all x1 ;    ; xm ; x; y 2 A and L D mr1 .



Theorem 2.8. Let f W A ! B be a mapping for which there exist the functions ' W Am ! Œ0; 1/ and W A2 ! Œ0; 1/ such that lim mj '.mj x1 ;    ; mj xm / D 0;

(2.19)

kD f .x1 ;    ; xm /kB  '.x1 ;    ; xm /;

(2.20)

j!1

kf .xy/  f .x/f .y/kB 

.x; y/;

lim m2j .mj x; mj y/ D 0

j!1

(2.21) (2.22)

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. If there exists L < 1 such that '.x; 0;    ; 0/  mL '.mx; 0;    ; 0/ for all x 2 A, then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

L '.x; 0;    ; 0/ m  mL

for all x 2 A. Proof. We consider the linear mapping J W X ! X such that Jg.x/ WD mg

x m

(2.23)

28

2 Stability of Functional Equations in Banach Algebras

for all x 2 A. It follows from (2.13) that   x  x  L   ; 0;    ; 0  '.x; 0;    ; 0/  ' f .x/  mf m B m m for all x 2 A and so d.f ; Jf /  H W A ! B such that

L . m

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.mx/ D mH.x/

(2.24)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (2.20) such that there exists C 2 .0; 1/ satisfying kH.x/  f .x/kB  C'.x; 0;    ; 0/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality lim mn f

n!1

 x  D H.x/ mn

for all x 2 A; 1 (3) d.f ; H/  1L d.f ; Jf /, which implies the inequality d.f ; H/ 

L ; m  mL

which implies that the inequality (2.23) holds. The rest of the proof is similar to the proof of Theorem 2.6.



Corollary 2.9. Let r > 1,  be nonnegative real numbers and let f W A ! B be a mapping such that kD f .x1 ;    ; xm /kB    .kx1 krA C kx2 krA C    C kxm krA /

(2.25)

kf .xy/  f .x/f .y/kB    .kxkrA  kykrA /

(2.26)

and

2.2 Stability of m-Variable Functional Equations

29

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. Then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

mr

 kxkrA m

for all x 2 A. Proof. The proof follows from Theorem 2.6 by taking '.x1 ;    ; xm / D   .kx1 krA C kx2 krA C    C kxm krA /; .x; y/ WD   .kxkrA  kykrA / for all x1 ;    ; xm ; x; y 2 A and L D m1r .



2.2.2 Stability of Derivations in Banach Algebras Now, we prove the Hyers-Ulam stability of derivations on Banach algebras for the functional equation D f .x1 ;    ; xm / D 0. Theorem 2.10. Let f W A ! A be a mapping for which there exist the functions ' W Am ! Œ0; 1/ and W A2 ! Œ0; 1/ such that lim mj '.mj x1 ;    ; mj xm / D 0;

(2.27)

kD f .x1 ;    ; xm /kA  '.x1 ;    ; xm /;

(2.28)

j!1

kf .xy/  f .x/y  xf .y/kA  lim m

2j

j!1

.x; y/;

.m x; m y/ D 0 j

j

(2.29) (2.30)

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. If there exists L < 1 such that '.mx; 0;    ; 0/  mL'.x; 0;    ; 0/ for all x 2 A. Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

1 '.x; 0;    ; 0/ m  mL

(2.31)

for all x 2 A. Proof. By the same reasoning as in the proof of Theorem 2.6, there exists a unique C-linear mapping ı W A ! A satisfying (2.31). The mapping ı W A ! A is given by f .mn x/ n!1 mn

ı.x/ D lim

(2.32)

30

2 Stability of Functional Equations in Banach Algebras

for all x 2 A. It follows from (2.29), (2.30) and (2.32) that kı.xy/  ı.x/y  xı.y/kA 1 kf .m2n xy/  f .mn x/  mn y  mn xf .mn y/kA m2n 1  lim 2n .mn x; mn y/ n!1 m D0

D lim

n!1

for all x; y 2 A and so ı.xy/ D ı.x/y C xı.y/ for all x; y 2 A. Thus ı W A ! A is a derivation satisfying (2.31). This completes the proof.  Corollary 2.11. Let r < 1,  be nonnegative real numbers and f W A ! A be a mapping such that kD f .x1 ;    ; xm /kA    .kx1 krA C    C kxm krA /

(2.33)

kf .xy/  f .x/y  xf .y/kA    .kxkrA  kykrA /

(2.34)

and

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

 kxkrA m  mr

for all x 2 A. Proof. The proof follows from Theorem 2.10 by taking '.x1 ;    ; xm / WD   .kx1 krA C    kxm krA /; .x; y/ WD   .kxkrA  kykrA / for all x1 ;    ; xm ; x; y 2 A and L D mr1 .



Remark 2.12. Let f W A ! B be a mapping for which there exist the functions ' W Am ! Œ0; 1/ and W A2 ! Œ0; 1/ such that lim mj '.mj x1 ;    ; mj xm / D 0;

j!1

(2.35)

2.3 Stability in Quasi-Banach Algebras

31

kD f .x1 ;    ; xm /kA  '.x1 ;    ; xm /; kf .xy/  f .x/y  xf .y/kA  lim m

j!1

2j

j

.x; y/;

j

.m x; m y/ D 0

(2.36) (2.37) (2.38)

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. If there exists L < 1 such that '.mx; 0;    ; 0/  mL '.x; 0;    ; 0/ for all x 2 A. Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

L '.x; 0;    ; 0/ m  mL

(2.39)

for all x 2 A. Corollary 2.13. Let r > 1,  be nonnegative real numbers and f W A ! A be a mapping such that kD f .x1 ;    ; xm /kA    .kx1 krA C    kxm krA /

(2.40)

kf .xy/  f .x/y  xf .y/kA    .kxkrA  kykrA /

(2.41)

and

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

 kxkrA mr  m

for all x 2 A. Proof. Consider Remark 2.12 and take '.x1 ;    ; xm / WD   .kx1 krA C    kxm krA /; .x; y/ WD   .kxkrA  kykrA / for all x1 ;    ; xm ; x; y 2 A and L D m1r .



2.3 Stability in Quasi-Banach Algebras Let q be a positive rational number and n be a nonnegative integer. We consider the Hyers-Ulam stability of homomorphisms in quasi-Banach algebras and generalized derivations on quasi-Banach algebras for the following functional equation:

32

2 Stability of Functional Equations in Banach Algebras n n n n  X  X X X f q.xi  xj / C nf qxi D nq f .xi /: iD1

jD1

iD1

(2.42)

iD1

This is applied to investigate some isomorphisms in quasi-Banach algebras (see [180, 233, 238]).

2.3.1 Stability of Homomorphisms in Quasi-Banach Algebras Let q be a positive rational number. For any mapping f W A ! B, we define Df W An ! B by Df .x1 ;    ; xn / n n n n  X  X X X f q.xi  xj / C nf qxi  nq f .xi / WD iD1

jD1

iD1

iD1

for all x1 ;    ; xn 2 X. Lemma 2.14. Let f W A ! B be a mapping satisfies the functional equation (2.42). Then the mapping f is Cauchy additive and R-linear. 

Proof. The proof is easy (see also [229, 267]).

Now, we prove the Hyers-Ulam stability of homomorphisms in quasi-Banach algebras. Theorem 2.15. Assume that r > 2 if nq > 1 and 0 < r < 1 if nq < 1. Let  be a positive real number and f W A ! B be an odd mapping such that kDf .x1 ;    ; xn /kB  

n X

kxj krA

(2.43)

jD1

and kf .xy/  f .x/f .y/kB  .kxkrA C kykrA /

(2.44)

for all x; y; x1 ;    ; xn 2 A. If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB  for all x 2 A.

 ..nq/pr



1

.nq/p / p

kxkrA

(2.45)

2.3 Stability in Quasi-Banach Algebras

33

Proof. Letting x1 D    D xn D x in (2.43), we get knf .nqx/  n2 qf .x/kB  nkxkrA

(2.46)

for all x 2 A and so   x     kxkrA   f .x/  nqf nq B .nq/r for all x 2 A. Since B is a p-Banach algebra, we have   x   x p   m  .nq/ f  .nq/l f .nq/l .nq/m B 

m1 X

 .nq/j f

jDl



 x   x p  jC1  .nq/ f  .nq/j .nq/jC1 B

(2.47)

m1  p X .nq/pj pr kxkA .nq/pr jDl .nq/prj

for all m  1, l with m > l and x 2 A. It follows from (2.47) that the sequence x f.nq/d f . .nq/ d /g is a Cauchy sequence for all x 2 A. Since B is complete, the sequence x d f.nq/ f . .nq/ d /g converges. So one can define a mapping H W A ! B by H.x/ WD lim .nq/d f . d!1

x / .nq/d

for all x 2 A. Moreover, letting l D 0 and m ! 1 in (2.47), we get (2.46). It follows from (2.43) that   x xn   1  kDH.x1 ;    ; xn /kB D lim .nq/d Df ;    ;  d!1 .nq/d .nq/d B n .nq/d  X kxj krA d!1 .nq/dr jD1

 lim D0 for all x1 ;    ; xn 2 A. Thus we have

DH.x1 ;    ; xn / D 0 for all x1 ;    ; xn 2 A. By (2.43), the mapping H W A ! B is Cauchy additive and R-linear. It follows from (2.45) that

34

2 Stability of Functional Equations in Banach Algebras

kH.xy/  H.x/H.y/kB    D lim .nq/2d f d!1

  x   y  xy   f f  .nq/d .nq/d .nq/d .nq/d B

.nq/2d  .kxkrA C kykrA / d!1 .nq/dr

 lim D0 for all x; y 2 A and so

H.xy/ D H.x/H.y/ for all x; y 2 A. Now, let T W A ! B be another mapping satisfying (2.46). Then we have kH.x/  T.x/kB   x   x     T D .nq/d H  .nq/d .nq/d B   x   x   x     x       f  f  .nq/d K H C   T .nq/d .nq/d B .nq/d .nq/d B 

2  .nq/d K ..nq/pr



1

.nq/p / p .nq/dr

kxkrA ;

which tends to zero as n ! 1 for all x 2 A. So we can conclude that H.x/ D T.x/ for all x 2 A. This proves the uniqueness of H. Thus the mapping H W A ! B is a unique homomorphism satisfying (2.46). This completes the proof.  Theorem 2.16. Assume that 0 < r < 1 if nq > 1 and that r > 2 if nq < 1. Let  be a positive real number, and let f W A ! B be an odd mapping satisfying (2.43) and (2.45). If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

 ..nq/p



1

.nq/pr / p

for all x 2 A. Proof. It follows from (2.46) that kf .x/ 

 1 f .nqx/kB  kxkrA nq nq

for all x 2 A. Since B is a p-Banach algebra, we have

kxkrA

(2.48)

2.3 Stability in Quasi-Banach Algebras

35

 1 p 1  l m  f ..nq/ x/  f ..nq/ x/   B .nq/l .nq/m 

jDl



p 1  1 j jC1  f ..nq/ x/  f ..nq/ x/   B .nq/j .nq/jC1

m1 X

(2.49)

m1  p X .nq/prj pr kxkA .nq/p jDl .nq/pj

for all m  1, l with m > l and x 2 A. It follows from (2.49) that the sequence 1 d f .nq/ d f ..nq/ x/g is a Cauchy sequence for all x 2 A. Since B is complete, the 1 d sequence f .nq/ d f ..nq/ x/g converges. So one can define a mapping H W A ! B by

H.x/ WD lim

d!1

1 f ..nq/d x/ .nq/d

for all x 2 A. Moreover, letting l D 0 and m ! 1 in (2.49), we get (2.48). The rest of the proof is similar to the proof of Theorem 2.15.



2.3.2 Isomorphisms in Quasi-Banach Algebras Assume that A is a quasi-Banach algebra with the quasi-norm k  kA and the unit e and B is a p-Banach algebra with the p-norm k  kB and the unit e0 . Let K be the modulus of concavity of k  kB . Now, we consider isomorphisms in quasi-Banach algebras. Theorem 2.17. Assume that r > 2 if nq > 1 and that 0 < r < 1 if nq < 1. Let  be a positive real number and f W A ! B be an odd bijective mapping satisfying (2.43) such that f .xy/ D f .x/f .y/

(2.50)

e 0 for all x; y 2 A. If limd!1 .nq/d f . .nq/ d / D e and f .tx/ is continuous in t 2 R for each fixed x 2 A, then the mapping f W A ! B is an isomorphism.

Proof. The condition (2.50) implies that f W A ! B satisfies (2.45). By the same reasoning as in the proof of Theorem 2.15, there exists a unique homomorphism H W A ! B, which is defined by H.x/ WD lim .nq/d f d!1

for all x 2 A. Thus we have

 x  .nq/d

36

2 Stability of Functional Equations in Banach Algebras

H.x/ D H.ex/ D lim .nq/d f d!1

D lim .nq/d f d!1

 ex   e  d D lim .nq/ f  x d!1 .nq/d .nq/d

 e  f .x/ D e0 f .x/ D f .x/ .nq/d

for all x 2 A. So the bijective mapping f W A ! B is an isomorphism. This completes the proof.  Remark 2.18. Assume that 0 < r < 1 if nq > 1 and that r > 2 if nq < 1. Let  be a positive real number and f W A ! B be an odd bijective mapping satisfying (2.43) and (2.50). If f .tx/ is continuous in t 2 R for each fixed x 2 A and lim

d!1

1 f ..nq/d e/ D e0 ; .nq/d

then the mapping f W A ! B is an isomorphism.

2.3.3 Stability of Generalized Derivations in Quasi-Banach Algebras Assume that A is a p-Banach algebra with the p-norm k  kA . Let K be the modulus of concavity of k  kA . Definition 2.19 ([18]). A generalized derivation ı W A ! A is R-linear and fulfills the generalized Leibniz rule: ı.xyz/ D ı.xy/z  xı.y/z C xı.yz/ for all x; y; z 2 A. Now, we prove the Hyers-Ulam stability of generalized derivations on quasiBanach algebras. Theorem 2.20. Assume that r > 3 if nq > 1 and that 0 < r < 1 if nq < 1. Let  be a positive real number and f W A ! A be an odd mapping satisfying (2.43) such that kf .xyz/  f .xy/z C xf .y/z  xf .yz/kA  .kxkrA C kykrA C kzkrA /

(2.51)

for all x; y; z 2 A. If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique generalized derivation ı W A ! A such that

2.3 Stability in Quasi-Banach Algebras

37

kf .x/  ı.x/kA 

 ..nq/pr

1

 .nq/p / p

kxkrA

(2.52)

for all x 2 A. Proof. By the same reasoning as in the proof of Theorem 2.15, there exists a unique R-linear mapping ı W A ! A satisfying (2.52). The mapping ı W A ! A is defined by ı.x/ WD lim .nq/d f d!1

 x  .nq/d

for all x 2 A. It follows from (2.51) that kı.xyz/  ı.xy/z C xı.y/z  xı.yz/kA   xyz   xy  z   f D lim .nq/3d f d!1 .nq/3d .nq/2d .nq/d   y x y x  yz   C f  f  .nq/d .nq/d .nq/d .nq/d .nq/2d A .nq/3d  .kxkrA C kykrA C kzkrA / d!1 .nq/dr

 lim D0 for all x; y; z 2 A and so

ı.xyz/ D ı.xy/z  xı.y/z C xı.yz/ for all x; y; z 2 A. Thus the mapping ı W A ! A is a unique generalized derivation satisfying (2.52). This completes the proof.  Theorem 2.21. Assume that 0 < r < 1 if nq > 1 and that r > 3 if nq < 1. Let  be a positive real number and f W A ! A be an odd mapping satisfying (2.43) and (2.51). If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique generalized derivation ı W A ! A such that kf .x/  ı.x/kA 

 ..nq/p



1

.nq/pr / p

kxkrA

(2.53)

for all x 2 A. Proof. By the same reasoning as in the proof of Theorem 2.16, there exists a unique R-linear mapping ı W A ! A satisfying (2.53). The mapping ı W A ! A is defined by 1 f ..nq/d x/ d!1 .nq/d

ı.x/ WD lim

for all x 2 A. The rest of the proof is similar to the proof of Theorem 2.20.



38

2 Stability of Functional Equations in Banach Algebras

2.4 Stability of Cauchy–Jensen Functional Equations In this section, we prove the Hyers-Ulam stability of homomorphisms in real Banach algebras and generalized derivations on real Banach algebras for the following Cauchy-Jensen functional equations(see also [19, 147, 232]): f

x C y 2

 x  y  Cz Cf C z D f .x/ C 2f .z/ 2

and 2f

x C y 2

 C z D f .x/ C f .y/ C 2f .z/:

2.4.1 Stability of Homomorphisms in Real Banach Algebras Assume that A is a real Banach algebra with the norm k  kA and B is a real Banach algebra with the norm k  kB . For any mapping f W A ! B, we define Cf .x; y; z/ WD f

x C y 2

 x  y  Cz Cf C z  f .x/  2f .z/ 2

for all x; y; z 2 A. Now, we prove the Hyers-Ulam stability of homomorphisms in real Banach algebras for the functional equation Cf .x; y; z/ D 0. Lemma 2.22. Let X and Y be vector spaces. If a mapping f W X ! Y satisfies f

x C y 2

f

 x  y  Cz Cf C z D f .x/ C 2f .z/; 2

x C y 2

 x  y  Cz f C z D f .y/ 2

(2.54)

(2.55)

or 2f

x C y 2

 C z D f .x/ C f .y/ C 2f .z/

for all x; y; z 2 X, then the mapping f W X ! Y is Cauchy additive. Proof. Letting x D y in (2.54), we get f .x C z/ C f .z/ D f .x/ C 2f .z/

(2.56)

2.4 Stability of Cauchy–Jensen Functional Equations

39

for all x; z 2 X and so f .x C z/ D f .x/ C f .z/ for all x; z 2 X. Hence f W X ! Y is Cauchy additive. Letting x D y in (2.55), we get f .x C z/  f .z/ D f .x/ for all x; z 2 X and so f .x C z/ D f .x/ C f .z/ for all x; z 2 X. Hence f W X ! Y is Cauchy additive. Letting x D y in (2.56), we get 2f .x C z/ D 2f .x/ C 2f .z/ for all x; z 2 X and so f .x C z/ D f .x/ C f .z/ for all x; z 2 X. Hence f W X ! Y is Cauchy additive. This completes the proof.  The mappings f W X ! Y given in the statement of Lemma 2.22 are called Cauchy–Jensen type additive mappings. Putting z D 0 in (2.56), we get the Jensen additive mapping 2f . xCy 2 / D f .x/ C f .y/ and, putting x D y in (2.56), we get the Cauchy additive mapping f .x C z/ D f .x/ C f .z/. Theorem 2.23. Let f W A ! B be a mapping for which there exists a function ' W A3 ! Œ0; 1/ such that 1 X 1 '.2j x; 2j y; 2j z/ < 1; j 2 jD0

(2.57)

kCf .x; y; z/kB  '.x; y; z/;

(2.58)

kf .xy/  f .x/f .y/kB  '.x; y; 0/

(2.59)

for all x; y; z 2 A. If there exists L < 1 such that x x x '.x; x; x/  2L'. ; ; / 2 2 2 for all x 2 A and f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

1 '.x; x; x/ 2  2L

for all x 2 A. Proof. Consider the set X WD fg W A ! Bg

(2.60)

40

2 Stability of Functional Equations in Banach Algebras

and introduce the generalized metric on X defined by d.g; h/ D inffC 2 RC W kg.x/  h.x/kB  C'.x; x; x/; 8x 2 Ag; which .X; d/ is complete. Now, we consider the linear mapping J W X ! X such that Jg.x/ WD

1 g.2x/ 2

for all x 2 A. Note that d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting y D z D x in (2.58), we get kf .2x/  2f .x/kB  '.x; x; x/

(2.61)

for all x 2 A and so   1 1   f .x/  f .2x/  '.x; x; x/ B 2 2 for all x 2 A. Hence d.f ; Jf /  H W A ! B such that

1 2.

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(2.62)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (2.62) such that there exists C 2 .0; 1/ satisfying kH.x/  f .x/kB  C'.x; x; x/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality f .2n x/ D H.x/ n!1 2n lim

for all x 2 A;

(2.63)

2.4 Stability of Cauchy–Jensen Functional Equations

(3) d.f ; H/ 

1 1L d.f ; Jf /,

41

which implies the inequality d.f ; H/ 

1 : 2  2L

This implies that the inequality (2.60) holds. It follows from (2.57), (2.58) and (2.63) that  x C y   x  y    Cz CH C z  H.x/  2H.z/ H B 2 2 1 D lim n kf .2n1 .x C y/ C 2n z/ C f .2n1 .x  y/ C 2n z/ n!1 2 f .2n x/  2f .2n z/kB  lim

n!1

1 '.2n x; 2n y; 2n z/ 2n

D0 for all x; y; z 2 A and so H

x C y 2

 x  y  Cz CH C z D H.x/ C 2H.z/ 2

for all x; y; z 2 A. By Lemma 2.22, the mapping H W A ! B is Cauchy additive. By (2.58), the mapping H W A ! B is R-linear. It follows from (2.59) that 1 kf .4n xy/  f .2n x/f .2n y/kB 4n 1  lim n '.2n x; 2n y; 0/ n!1 4 1  lim n '.2n x; 2n y; 0/ n!1 2 D0

kH.xy/  H.x/H.y/kB D lim

n!1

for all x; y 2 A and so H.xy/ D H.x/H.y/ for all x; y 2 A. Thus H W A ! B is a homomorphism satisfying (2.60). This completes the proof.  Corollary 2.24. Let r < 1,  be nonnegative real numbers and f W A ! B be a mapping such that kCf .x; y; z/kB  .kxkrA C kykrA C kzkrA /

(2.64)

42

2 Stability of Functional Equations in Banach Algebras

and kf .xy/  f .x/f .y/kB  .kxkrA C kykrA /

(2.65)

for all x; y; z 2 A. If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

3 kxkrA 2  2r

for all x 2 A. Proof. The proof follows from Theorem 2.23 by taking '.x; y; z/ WD .kxkrA C kykrA C kzkrA / for all x; y; z 2 A and L D 2r1 . We get the desired result.



Theorem 2.25. Let f W A ! B be a mapping for which there exists a function ' W A3 ! Œ0; 1/ satisfying (2.58) and (2.59) such that 1 X jD0

4j '

x y z <1 ; ; 2j 2j 2j

(2.66)

for all x; y; z 2 A. If there exists L < 1 such that '.x; x; x/ 

1 L'.2x; 2x; 2x/ 2

for all x 2 A and f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

L '.x; x; x/ 2  2L

for all x 2 A. Proof. We consider the linear mapping J W X ! X such that Jg.x/ WD 2g

x 2

for all x 2 A. It follows from (2.61) that   x  x x x L    '.x; x; x/ f .x/  2f  ' ; ; 2 B 2 2 2 2

(2.67)

2.4 Stability of Cauchy–Jensen Functional Equations

for all x 2 A. Hence d.f ; Jf /  H W A ! B such that

L . 2

43

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(2.68)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (2.68) such that there exists C 2 .0; 1/ satisfying kH.x/  f .x/kB  C'.x; x; x/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality lim 2n f

n!1

x D H.x/ 2n

for all x 2 A; 1 (3) d.f ; H/  1L d.f ; Jf /, which implies the inequality d.f ; H/ 

L ; 2  2L

which implies that the inequality (2.67) holds. It follows from (2.58), (2.66) and (2.69) that  x C y   x  y    Cz CH C z  H.x/  2H.z/ H B 2 2  x C y    x  z  z z xy   D lim 2n f nC1 C n C f nC1 C n  f n  2f n  n!1 B 2 2 2 2 2 2 x y z  lim 2n ' n ; n ; n n!1 2 2 2 x y z  lim 4n ' n ; n ; n n!1 2 2 2 D0 for all x; y; z 2 A and so H

x C y 2

 x  y  Cz CH C z D H.x/ C 2H.z/ 2

(2.69)

44

2 Stability of Functional Equations in Banach Algebras

for all x; y; z 2 A. By Lemma 2.22, the mapping H W A ! B is Cauchy additive. By (2.58), the mapping H W A ! B is R-linear. It follows from (2.59) that   xy   x   y    kH.xy/  H.x/H.y/kB D lim 4n f n  f n f n  n!1 B 4 2 2 x y   lim 4n ' n ; n ; 0 n!1 2 2 D0 for all x; y 2 A and so H.xy/ D H.x/H.y/ for all x; y 2 A. Thus H W A ! B is a homomorphism satisfying (2.67). This completes the proof.  Corollary 2.26. Let r > 2,  be nonnegative real numbers and f W A ! B be a mapping satisfying (2.64) and (2.65). If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique homomorphism H W A ! B such that kf .x/  H.x/kB 

3 kxkrA 2

2r

for all x 2 A. Proof. The proof follows from Theorem 2.25 by taking '.x; y; z/ WD .kxkrA C kykrA C kzkrA / for all x; y; z 2 A and L D 21r .



2.4.2 Stability of Generalized Derivations in Real Banach Algebras Assume that A is a real Banach algebra with the norm k  kA . For any mapping f W A ! A, we define Df .x; y; z/ WD 2f

x C y 2

 C z  f .x/  f .y/  2f .z/

for all x; y; z 2 A. Now, we prove the Hyers-Ulam stability of generalized derivations on real Banach algebras for the functional equation Df .x; y; z/ D 0.

2.4 Stability of Cauchy–Jensen Functional Equations

45

Theorem 2.27. Let f W A ! A be a mapping for which there exists a function ' W A3 ! Œ0; 1/ satisfying (2.57) such that kDf .x; y; z/kA  '.x; y; z/

(2.70)

kf .xyz/  f .xy/z C xf .y/z  xf .yz/kA  '.x; y; z/

(2.71)

and

for all x; y; z 2 A. If there exists L < 1 such that '.x; x; x/  2L'

x x x ; ; 2 2 2

for all x 2 A and f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique generalized derivation ı W A ! A such that kf .x/  ı.x/kA 

1 '.x; x; x/ 4  4L

(2.72)

for all x 2 A. Proof. Consider the set X WD fg W A ! Ag and introduce the generalized metric on X defined by d.g; h/ D inffC 2 RC W kg.x/  h.x/kA  C'.x; x; x/; 8x 2 Ag; which .X; d/ is complete. Now, we consider the linear mapping J W X ! X such that Jg.x/ WD

1 g.2x/ 2

for all x 2 A. Now, we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting y D z D x in (2.70), we get k2f .2x/  4f .x/kA  '.x; x; x/ for all x 2 A and so

(2.73)

46

2 Stability of Functional Equations in Banach Algebras

  1 1   f .x/  f .2x/  '.x; x; x/ A 2 4 for all x 2 A. Hence d.f ; Jf /  14 . By Theorem 1.3, there exists a mapping ı W A ! A such that (1) ı is a fixed point of J, i.e., ı.2x/ D 2ı.x/

(2.74)

for all x 2 A. The mapping ı is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that ı is a unique mapping satisfying (2.74) such that there exists C 2 .0; 1/ satisfying kı.x/  f .x/kA  C'.x; x; x/ for all x 2 A; (2) d.J n f ; ı/ ! 0 as n ! 1. This implies the equality f .2n x/ D ı.x/ n!1 2n lim

for all x 2 A; 1 d.f ; Jf /, which implies the inequality (3) d.f ; ı/  1L d.f ; ı/ 

1 : 4  4L

This implies that the inequality (2.72) holds. It follows from (2.57), (2.70) and (2.75) that  x C y     C z  ı.x/  ı.y/  2ı.z/ 2ı A 2 1 D lim n k2f .2n1 .x C y/ C 2n z/  f .2n x/  f .2n y/  2f .2n z/kA n!1 2 1  lim n '.2n x; 2n y; 2n z/ n!1 2 D0 for all x; y; z 2 A and so

(2.75)

2.4 Stability of Cauchy–Jensen Functional Equations

2ı

x C y 2

47

 C z D ı.x/ C ı.y/ C 2ı.z/

for all x; y; z 2 A. By Lemma 2.22, the mapping ı W A ! A is Cauchy additive. By (2.71), the mapping ı W A ! A is R-linear. It follows from (2.71) that kı.xyz/  ı.xy/z C xı.y/z  xı.yz/kA 1 kf .8n xyz/  f .4n xy/  2n z C 2n xf .2n y/  2n z  2n xf .4n yz/kA 8n 1  lim n '.2n x; 2n y; 2n z/ n!1 8 1  lim n '.2n x; 2n y; 2n z/ n!1 2 D0

D lim

n!1

for all x; y; z 2 A and so ı.xyz/ D ı.xy/z  xı.y/z C xı.yz/ for all x; y; z 2 A. Thus ı W A ! A is a generalized derivation satisfying (2.72). This completes the proof.  Corollary 2.28. Let r < 1,  be nonnegative real numbers and f W A ! A be a mapping such that r

r

r

kDf .x; y; z/kA    kxkA3  kykA3  kzkA3

(2.76)

and kf .xyz/  f .xy/z C xf .y/z  xf .yz/kA r 3

r 3

(2.77)

r 3

   kxkA  kykA  kzkA for all x; y; z 2 A. If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique generalized derivation ı W A ! A such that kf .x/  ı.x/kA 

 kxkrA 4  2rC1

for all x 2 A. Proof. The proof follows from Theorem 2.27 by taking r

r

r

'.x; y; z/ WD   kxkA3  kykA3  kzkA3 for all x; y; z 2 A and L D 2r1 .



48

2 Stability of Functional Equations in Banach Algebras

Theorem 2.29. Let f W A ! A be a mapping for which there exists a function ' W A3 ! Œ0; 1/ satisfying (2.70) and (2.71) such that 1 X jD0

8j '

x y z ; ; <1 2j 2j 2j

(2.78)

for all x; y; z 2 A. If there exists L < 1 such that '.x; x; x/ 

1 L'.2x; 2x; 2x/ 2

for all x 2 A and f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique generalized derivation ı W A ! A such that kf .x/  ı.x/kA 

L '.x; x; x/ 4  4L

(2.79)

for all x 2 A. Proof. We consider the linear mapping J W X ! X such that x Jg.x/ WD 2g. / 2 for all x 2 A. It follows from (2.73) that   x  1 x x x L    '.x; x; x/ f .x/  2f   ' ; ; 2 A 2 2 2 2 4 for all x 2 A. Hence d.f ; Jf /  L4 . By Theorem 1.3, there exists a mapping ı W A ! A such that (1) ı is a fixed point of J, i.e., ı.2x/ D 2ı.x/

(2.80)

for all x 2 A. The mapping ı is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that ı is a unique mapping satisfying (2.80) such that there exists C 2 .0; 1/ satisfying kı.x/  f .x/kA  C'.x; x; x/ for all x 2 A;

2.4 Stability of Cauchy–Jensen Functional Equations

(2) d.J n f ; ı/ ! 0 as n ! 1. This implies the equality x lim 2n f n D ı.x/ n!1 2

49

(2.81)

for all x 2 A; 1 (3) d.f ; ı/  1L d.f ; Jf /, which implies the inequality d.f ; ı/ 

L ; 4  4L

which implies that the inequality (2.79) holds. It follows from (2.70), (2.78) and (2.81) that   x C y    C z  ı.x/  ı.y/  2ı.z/ 2ı A 2  x C y x y  z  z  n D lim 2 2f nC1 C n  f n  f n  2f n  n!1 A 2 2 2 2 2 x y z  lim 2n '. n ; n ; n / n!1 2 2 2 x y z n  lim 8 ' n ; n ; n n!1 2 2 2 D0 for all x; y; z 2 A and so 2ı

x C y 2

 C z D ı.x/ C ı.y/ C 2ı.z/

for all x; y; z 2 A. By Lemma 2.22, the mapping ı W A ! A is Cauchy additive. It is straight forward to show that the mapping ı W A ! A is R-linear. It follows from (2.71) that kı.xyz/  ı.xy/z C xı.y/z  xı.yz/kA   xyz   xy  z x y z x  yz     f   D lim 8n f C f  f n  n n n n n n n n!1 A 8 4 2 2 2 2 2 4 x y z n  lim 8 ' n ; n ; n n!1 2 2 2 D0 for all x; y; z 2 A and so ı.xyz/ D ı.xy/z  xı.y/z C xı.yz/ for all x; y; z 2 A. Thus ı W A ! A is a generalized derivation satisfying (2.80). This completes the proof. 

50

2 Stability of Functional Equations in Banach Algebras

Corollary 2.30. Let r > 3,  be nonnegative real numbers and f W A ! A be a mapping satisfying (2.76) and (2.77). If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique generalized derivation ı W A ! A such that kf .x/  ı.x/kA 

 2rC1

4

kxkrA

for all x 2 A. Proof. The proof follows from Theorem 2.29 by taking r

r

r

'.x; y; z/ WD   kxkA3  kykA3  kzkA3 for all x; y; z 2 A and L D 21r .



Chapter 3

Stability of Functional Equations in C -Algebras

In this chapter, we study the stability of some important functional equations in -algebras by using both the direct and fixed point methods. In Sect. 3.1, we consider the linear bijection h W A ! B of a unital C -algebra A onto a unital C -algebra B and show that it is a C -algebra isomorphism when h.3n uy/ D h.3n u/h.y/ for all unitaries u 2 A, y 2 A and n 2 Z by the Ulam method. In Sect. 3.2, we introduce a new functional equation, which is called the Apollonius type additive functional equation, and a solution of the functional equation is called the Apollonius type additive mapping:  x C y 1 : L.z  x/ C L.z  y/ D  L.x C y/ C 2L z  2 4 Also, we investigate homomorphisms and derivations in C -algebras associated with the Apollonius type additive functional equation, homomorphisms and derivations on Lie C -algebras associated with the Apollonius type additive functional equation. Finally, we study homomorphisms and derivations on JC -algebras associated with the Apollonius type additive functional equation. In Sect. 3.3, by using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms in C -algebras and Lie C -algebras and derivations on C -algebras and Lie C -algebras for the following Jensen type functional equation f

x C y 2

Cf

x  y 2

D f .x/:

In Sect. 3.4, we introduce the following additive functional equation : n 1 X f 2 jD1

X 1in;i¤j

1 X  1  X ri xi  rj xj C ri f .xi / D nf ri xi ; 2 2 iD1 iD1 n

© Springer International Publishing Switzerland 2015 Y.J. Cho et al., Stability of Functional Equations in Banach Algebras, DOI 10.1007/978-3-319-18708-2_3

n

51

3 Stability of Functional Equations in C -Algebras

52

where r1 ;    ; rn 2 R: Using the fixed point method, we investigate the Hyers-Ulam stability of the above functional equation in Banach modules over a C -algebra. These results are applied to investigate C -algebra homomorphisms in unital C -algebras. In Sect. 3.5, we show that, if an odd mapping f W X ! Y satisfies the functional equation:  Pd rf

jD1 xj

r



C

X

rf

 Pd .1/.j/ xj  jD1

.j/ D 0; 1 Pd .j/ Dl jD1

D .d1 Cl d1 Cl1 C 1/

d X

r

f .xj /;

jD1

then the odd mapping f W X ! Y is additive. Also, we prove the Hyers-Ulam stability of the above functional equation in Banach modules over a unital C algebra. As an application, we show that every almost linear bijection h W A ! B of a unital C -algebra A onto a unital C -algebra B is a C -algebra isomorphism when n n h. 2rn uy/ D h. 2rn u/h.y/ for all unitaries u 2 A, y 2 A and n 2 N. In Sect. 3.6, we investigate the Hyers-Ulam stability of Jordan -derivations and f quadratic Jordan -derivations on real C -algebras and real JC -algebras. Also, we prove the superstability of Jordan -derivations and quadratic Jordan -derivations on real C -algebras and real JC -algebras under some conditions. In Sect. 3.7, we investigate the Hyers-Ulam stability of .˛; ˇ;  /-derivations on Lie C -algebras associated with the following functional equation: f

    x  x  3x1 C 3x3  x2 x1  3x3 2 1 Cf D f .x1 /: Cf 3 3 3

In Sects. 3.8 and 3.9, we introduce a square root functional equation and a 3rd root functional equation. By using both the fixed point method and direct method, we prove the Hyers-Ulam stability of the square root functional equation and of the 3rd root functional equation in C -algebras. In Sect. 3.10, we introduce the following functional equation: T

     1 1 m 1 1 m D T.x/ m C T.y/ m xm C ym

for all x; y 2 AC and a fixed integer m greater than 1, which is called a positiveadditive functional equation. Using the fixed point and direct methods, we prove the stability of the positive-additive functional equation in C -algebras. Finally, in Sect. 3.11, we show that every almost unital almost linear mapping f W A ! B of JC -algebra A to a JC -algebra B is a homomorphism when f .2n u ı y/ D f .2n u/ ı f .y/ for all unitaries u 2 A, y 2 A and n  0 and every almost

3.1 Isomorphisms in Unital C -Algebras

53

unital almost linear continuous mapping f W A ! B of a JC -algebra A of real rank zero to a JC -algebra B is a homomorphism when f .2n u ı y/ D f .2n u/ ı f .y/ for all u 2 fv 2 A W v D v  ; kvk D 1; v is invertibleg, y 2 A and n  0. Furthermore, we prove the Hyers-Ulam stability of -homomorphisms in JC -algebras and C-linear -derivations on JC -algebras.

3.1 Isomorphisms in Unital C -Algebras It is shown that every almost linear bijection h W A ! B of a unital C -algebra A onto a unital C -algebra B is a C -algebra isomorphism when h.3n uy/ D h.3n u/h.y/ for all unitaries u 2 A, y 2 A, n 2 Z and an almost linear continuous bijection h W A ! B of a unital C -algebra A of real rank zero onto a unital C -algebra B is a C -algebra isomorphism when h.3n uy/ D h.3n u/h.y/ for all u 2 fv 2 A j v D v  , kvk D 1; v is invertibleg, y 2 A and n 2 Z. Assume that X and Y are left normed modules over a unital C -algebra A. It is shown that every surjective isometry T W X ! Y satisfying T.0/ D 0 and T.ux/ D uT.x/ for all x 2 X and unitaries u 2 A is an A-linear isomorphism. This is applied to investigate C -algebra isomorphisms in unital C -algebras. Let X and Y be Banach spaces with the norms k  k and k  k, respectively. Consider a mapping f W X ! Y such that f .tx/ is continuous in t 2 R for each fixed x 2 X. Th. M. Rassias [267] introduced the following inequality, which is called the Cauchy-Rassias inequality. Assume that there exist constants   0 and p 2 Œ0; 1/ such that kf .x C y/  f .x/  f .y/k  .jjxjjp C jjyjjp / for all x; y 2 X. Th. M. Rassias [267] showed that there exists a unique R-linear mapping T W X ! Y such that kf .x/  T.x/k 

2 jjxjjp 2  2p

for all x 2 X. The above inequality has provided a lot of influence in the development of what we now call the Hyers–Ulam stability of functional equations. Beginning around the year 1980 the topic of approximate homomorphisms or the stability of the equation of homomorphism was studied by a number of mathematicians. In [145], Jun and Lee proved the following: Denote by ' W X  X ! Œ0; 1/ a function such that '.x; Q y/ D

1 X 1 '.3j x; 3j y/ < 1 j 3 jD0

3 Stability of Functional Equations in C -Algebras

54

for all x; y 2 X. Suppose that f W X ! Y is a mapping satisfying f .0/ D 0 and   xCy   /  f .x/  f .y/  '.x; y/ 2f . 2 for all x; y 2 X. Then there exists a unique additive mapping T W X ! Y such that kf .x/  T.x/j 

1 .'.x; Q x/ C '.x; Q 3x// 3

for all x 2 X. In [242], Park and Park applied Jun and Lee’s result to Jensen’s equation in Banach modules over a C -algebra. Throughout this section, let A be a unital C -algebra with the norm k  k and the unit e and B be a unital C -algebra with the norm k  k. Let U.A/ be the set of unitary elements in A, Asa D fx 2 A j x D x g and I1 .Asa / D fv 2 Asa j kvk D 1; v is invertibleg. We prove that every almost linear bijection h W A ! B is a C -algebra isomorphism when h.3n uy/ D h.3n u/h.y/ for all u 2 U.A/, y 2 A, n 2 Z and, for a unital C -algebra A of real rank zero, every almost linear continuous bijection h W A ! B is a C -algebra isomorphism when h.3n uy/ D h.3n u/h.y/ for all u 2 I1 .Asa /, y 2 A and n 2 Z. Also, we prove that every surjective isometry satisfying some conditions is a C -algebra isomorphism.

3.1.1 C -Algebra Isomorphisms in Unital C -Algebras Now, we investigate C -algebra isomorphisms in unital C -algebras. Theorem 3.1. Let h W A ! B be a bijective mapping satisfying h.0/ D 0 and h.3n uy/ D h.3n u/h.y/ for all u 2 U.A/, y 2 A and n 2 Z, for which there exists a function ' W A  A ! Œ0; 1/ such that '.x; Q y/ WD

1 X 1 '.3j x; 3j y/ < 1; j 3 jD0

(3.1)

   x C y     h.x/  h.y/  '.x; y/; 2h 2

(3.2)

kh.3n u /  h.3n u/ k  '.3n u; 3n u/

(3.3)

for all 2 S1 WD f 2 C W jj D 1g, u 2 U.A/, n 2 Z and x; y 2 A. Assume that h.3n e/ n!1 3n lim

(3.4)

is invertible. Then the bijective mapping h W A ! B is a C -algebra isomorphism.

3.1 Isomorphisms in Unital C -Algebras

55

Proof. Put D 1 2 S1 . By direct method there exists a unique additive mapping H W A ! B such that kh.x/  H.x/k 

 1 '.x; Q x/ C '.x; Q 3x/ 3

(3.5)

for all x 2 A. The additive mapping H W A ! B is given by H.x/ D lim

n!1

1 h.3n x/ 3n

for all x 2 A. By the assumption, for each 2 S1 ,   3n x  1 1  n   h.3 x/   n '.3n x; 0/; 2h n 3 2 3 which tends to zero as n ! 1 for all x 2 A. Hence we have 2H

 x  2

2h. 3 2 x / h.3n x/ D lim D H.x/ n n!1 n!1 3 3n n

D lim

for all 2 S1 and x 2 A. Since H W A ! B is additive, H. x/ D 2H

 x  2

D H.x/

(3.6)

for all 2 S1 and x 2 A. Now, let  2 C . ¤ 0/ and M be an integer greater than 4jj. Then we have ˇˇ 1 2 1 ˇ ˇ ˇ ˇ< <1 D : M 4 3 3 By Kadison and Pedersen [167], there exist three elements 1 ; 2 ; 3 2 S1 such that 3 M D 1 C 2 C 3 . So, by (3.6), we have 1     M    x DMH 3 x D H 3 x 3 M 3 M 3 M M M D H. 1 x C 2 x C 3 x/ D .H. 1 x/ C H. 2 x/ C H. 3 x// 3 3  M M  3 H.x/ D . 1 C 2 C 3 /H.x/ D 3 3 M D H.x/

H.x/ D H

M

3

3 Stability of Functional Equations in C -Algebras

56

for all x 2 A. Hence we have H. x C y/ D H. x/ C H. y/ D H.x/ C H.y/ for all ; 2 C . ; ¤ 0/ and x; y 2 A and H.0x/ D 0 D 0H.x/ for all x 2 A. Thus the unique additive mapping H W A ! B is a C-linear mapping. By (3.1) and (3.3), we get h.3n u / h.3n u/ D lim n!1 n!1 3n 3n   h.3n u/  D lim D H.u/ n!1 3n

H.u / D lim

for all u 2 U.A/. Since H is C-linear and P each x 2 A is a finite linear combination of unitary elements (see [168]), i.e., x D m jD1 j uj for all j 2 C and uj 2 U.A/, we have H.x / D H

m X

m m  X X j uj D j H.uj / D j H.uj /

jD1

D

m X jD1

jD1

j H.uj /



D H.

jD1 m X

j uj / D H.x/

jD1

for all x 2 A. Since h.3n uy/ D h.3n u/h.y/ for all u 2 U.A/, y 2 A and n 2 Z, we have H.uy/ D lim

n!1

1 1 h.3n uy/ D lim n h.3n u/h.y/ D H.u/h.y/ n!1 3 3n

(3.7)

for all u 2 U.A/ and y 2 A. By the additivity of H and (3.7), we have 3n H.uy/ D H.3n uy/ D H.u.3n y// D H.u/h.3ny/ for all u 2 U.A/ and y 2 A. Hence it follows that H.uy/ D

1 1 H.u/h.3ny/ D H.u/ n h.3n y/ 3n 3

(3.8)

for all u 2 U.A/ and y 2 A. Taking n ! 1 in (3.8), we obtain H.uy/ D H.u/H.y/

(3.9)

for all u 2 U.A/ and y 2 A. Since H is C-linear and each x 2 A is a finite linear P  combination of unitary elements, i.e., x D m j jD1 uj for all j 2 C and uj 2 U.A/, it follows from (3.9) that

3.1 Isomorphisms in Unital C -Algebras

H.xy/ D H

m X

m m  X X j u j y D j H.uj y/ D j H.uj /H.y/

jD1

DH

m X

57

jD1

jD1

 j uj H.y/ D H.x/H.y/

jD1

for all x; y 2 A. By (3.7) and (3.9), we have H.e/H.y/ D H.ey/ D H.e/h.y/ for all y 2 A. Since limn!1

h.3n e/ 3n

D H.e/ is invertible, H.y/ D h.y/

for all y 2 A. Therefore, the bijective mapping h W A ! B is a C -algebra isomorphism. This completes the proof.  Corollary 3.2. Let h W A ! B be a bijective mapping satisfying h.0/ D 0 and h.3n uy/ D h.3n u/h.y/ for all u 2 U.A/, y 2 A and n 2 Z, for which there exist constants   0 and p 2 Œ0; 1/ such that   x C y      h.x/  h.y/  .kxkp C kykp / 2h 2 and kh.3n u /  h.3n u/ k  2  3np  n

for all 2 S1 , u 2 U.A/, n 2 Z and x; y 2 A. Assume that limn!1 h.33n e/ is invertible. Then the bijective mapping h W A ! B is a C -algebra isomorphism. Proof. Defining '.x; y/ D .kxkp C kykp / (Rassias upper bound in the Cauchy-Rassias inequality) and applying Theorem 3.1, we get the desired result.  Theorem 3.3. Let h W A ! B be a bijective mapping satisfying h.0/ D 0 and h.3n uy/ D h.3n u/h.y/ for all u 2 U.A/, y 2 A and n 2 Z, for which there exists a function ' W A  A ! Œ0; 1/ satisfying (3.1), (3.3) and (3.4) such that    x C y     h.x/  h.y/  '.x; y/ 2h 2

(3.10)

for D 1; i and x; y 2 A. If h.tx/ is continuous in t 2 R for each fixed x 2 A, then the bijective mapping h W A ! B is a C -algebra isomorphism.

3 Stability of Functional Equations in C -Algebras

58

Proof. Put D 1 in (3.10). Then, there exists a unique additive mapping H W A ! B satisfying (3.5). Also, by (3.10), the additive mapping H W A ! B is R-linear. Put D i and y D 0 in (3.10). By the same method as in the proof of Theorem 3.1, one can obtain that H.ix/ D 2H

 ix  2

2h. 3 2ix / ih.3n x/ D lim D lim D iH.x/ n!1 n!1 3n 3n n

for all x 2 A. For each  2 C, let  D s C it for all s; t 2 R. So, we have H.x/ D H.sx C itx/ D sH.x/ C tH.ix/ D sH.x/ C itH.x/ D .s C it/H.x/ D H.x/ for all  2 C and x 2 A. Thus we have H. x C y/ D H. x/ C H. y/ D H.x/ C H.y/ for all ; 2 C and x; y 2 A. Hence the additive mapping H W A ! B is C-linear. The rest of the proof is the same as in the proof of Theorem 3.1. This completes the proof.  From now on, assume that A is a unital C -algebra of real rank zero, where “real rank zero” means that the set of invertible self-adjoint elements is dense in the set of self-adjoint elements (see [54]). Now, we investigate continuous C -algebra isomorphisms in unital C -algebras. Theorem 3.4. Let h W A ! B be a continuous bijective mapping satisfying h.0/ D 0 and h.3n uy/ D h.3n u/h.y/ for all u 2 I1 .Asa /, y 2 A and n 2 Z, for which there exists a function ' W A  A ! Œ0; 1/ satisfying (3.1), (3.2), (3.3) and (3.4). Then the bijective mapping h W A ! B is a C -algebra isomorphism. Proof. It is straight forward to show that, there exists a unique C-linear involution H W A ! B satisfying (3.5). Since h.3n uy/ D h.3n u/h.y/ for all u 2 I1 .Asa /, y 2 A and n 2 Z, we have H.uy/ D lim

n!1

1 1 h.3n uy/ D lim n h.3n u/h.y/ D H.u/h.y/ n!1 3 3n

for all u 2 I1 .Asa / and y 2 A. By the additivity of H and (3.11), we have 3n H.uy/ D H.3n uy/ D H.u.3n y// D H.u/h.3ny/

(3.11)

3.1 Isomorphisms in Unital C -Algebras

59

for all u 2 I1 .Asa / and y 2 A. Hence we have H.uy/ D

1 1 H.u/h.3ny/ D H.u/ n h.3n y/ n 3 3

(3.12)

for all u 2 I1 .Asa / and y 2 A. Taking n ! 1 in (3.12), we obtain H.uy/ D H.u/H.y/

(3.13)

for all u 2 I1 .Asa / and y 2 A. By (3.11) and (3.13), we have H.e/H.y/ D H.ey/ D H.e/h.y/ for all y 2 A. Since limn!1

h.3n e/ 3n

D H.e/ is invertible, H.y/ D h.y/

for all y 2 A. So H W A ! B is continuous. But, by the assumption that A has real rank zero, it is easy to show that I1 .Asa / is dense in fx 2 Asa W kxk D 1g. Thus, for each w 2 fz 2 Asa W kzk D 1g, there exists a sequence fj g such that j ! w as j ! 1 and j 2 I1 .Asa /. Since H W A ! B is continuous, it follows from (3.13) that H.wy/ D H. lim j y/ D lim H.j y/ D lim H.j /H.y/ j!1

j!1

j!1

D H. lim j /H.y/ D H.w/H.y/

(3.14)

j!1





for all w 2 fz 2 Asa W kzk D 1g and y 2 A. For each x 2 A, x D xCx C i xx , where 2 2i   xCx xx x1 WD 2 and x2 WD 2i are self-adjoint. First, consider the case that x1 ¤ 0 and x2 ¤ 0. Since H W A ! B is C-linear, it follows from (3.14) that H.xy/ D H.x1 y C ix2 y/  x1 x2  D H kx1 k y C ikx2 k y kx1 k kx2 k  x   x  1 2 y C ikx2 kH y D kx1 kH kx1 k kx2 k  x   x  1 2 D kx1 kH H.y/ C ikx2 kH H.y/ kx1 k kx2 k n   x1  x2 o D H kx1 k C iH kx2 k H.y/ kx1 k kx2 k D H.x1 C ix2 /H.y/ D H.x/H.y/ for all y 2 A.

3 Stability of Functional Equations in C -Algebras

60

Next, consider the case that x1 ¤ 0; x2 D 0. Since H W A ! B is C-linear, it follows from (3.14) that   x  x1  1 y D kx1 kH y H.xy/ D H.x1 y/ D H kx1 k kx1 k kx1 k   x  x1  1 H.y/ D H kx1 k H.y/ D kx1 kH kx1 k kx1 k D H.x1 /H.y/ D H.x/H.y/ for all y 2 A. Finally, consider the case that x1 D 0; x2 ¤ 0. Since H W A ! B is C-linear, it follows from (3.14) that   x  x2  2 y D ikx2 kH y H.xy/ D H.ix2 y/ D H ikx2 k kx2 k kx2 k   x  x2  2 D ikx2 kH H.y/ D H ikx2 k H.y/ kx2 k kx2 k D H.ix2 /H.y/ D H.x/H.y/ for all y 2 A. Hence H.xy/ D H.x/H.y/ for all x; y 2 A. Therefore, the bijective mapping h W A ! B is a C -algebra isomorphism. This completes the proof.  Corollary 3.5. Let h W A ! B be a continuous bijective mapping satisfying h.0/ D 0 and h.3n uy/ D h.3n u/h.y/ for all u 2 I1 .Asa /, y 2 A and n 2 Z, for which there exist constants   0 and p 2 Œ0; 1/ such that   x C y      h.x/  h.y/  .kxkp C kykp / 2h 2 and kh.3n u /  h.3n u/ k  2  3np  n

for all 2 S1 , u 2 I1 .Asa /, n 2 Z and x; y 2 A. Assume that limn!1 h.33n e/ is invertible. Then the bijective mapping h W A ! B is a C -algebra isomorphism. Proof. Defining '.x; y/ D .kxkp C kykp / (Rassias upper bound in the Cauchy-Rassias inequality) and applying Theorem 3.4, we get the desired result. 

3.1 Isomorphisms in Unital C -Algebras

61

Remark 3.6. If h W A ! B is a continuous bijective mapping satisfying h.0/ D 0 and h.3n uy/ D h.3n u/h.y/ for all u 2 I1 .Asa /, y 2 A and n 2 Z, for which there exists a function ' W A  A ! Œ0; 1/ satisfying (3.1), (3.3), (3.4) and (3.10). Then the bijective mapping h W A ! B is a C -algebra isomorphism.

3.1.2 On the Mazur-Ulam Theorem in Modules over C -Algebras Now, we prove the Mazur-Ulam theorem in modules over C -algebras. Lemma 3.7 ([114]). If T is an isometry from a normed vector space X onto a normed vector space Y, then T.x C y/ D T.x/ C T.y/  T.0/ and T.rx/ D rT.x/ C .1  r/T.0/ for all r 2 R. Corollary 3.8 ([114]). If T is an isometry from a normed vector space X onto a normed vector space Y and T.0/ D 0, then T is R-linear. Theorem 3.9. Let X and Y be left normed modules over a unital C -algebra A. If T W X ! Y is a surjective isometry with T.0/ D 0 and T.ux/ D uT.x/ for all u 2 U.A/ and x 2 X, then T W X ! Y is an A-linear isomorphism. Proof. By Corollary 3.8, T W X ! Y is R-linear. Since i 2 U.A/, T.ix/ D iT.x/ for all x 2 X. For each  2 C,  D 1 C i2 for all 1 ; 2 2 R. Thus we have T.x/ D T.1 x C i2 x/ D T.1 x/ C T.i 2 x/ D 1 T.x/ C iT.2 x/ D .1 C i2 /T.x/ D T.x/ for all x 2 PX. Since each a 2 A is a finite linear combination of unitary elements, i.e., a D njD1 j uj for all j 2 C and uj 2 U.A/, T.ax/ D T

n X jD1

n n  X X j u j x D j T.uj x/ D j uj T.x/ D aT.x/ jD1

jD1

3 Stability of Functional Equations in C -Algebras

62

for all x 2 X and so T.ax C by/ D T.ax/ C T.by/ D aT.x/ C bT.y/ for all a; b 2 A and x; y 2 X. This completes the proof.







Now, we investigate C -algebra isomorphisms in unital C -algebras. Theorem 3.10. If T W A ! B is a surjective isometry with T.0/ D 0, T.iu/ D iT.u/, T.u / D T.u/ and T.uv/ D T.u/T.v/ for all u; v 2 U.A/, then T W A ! B is a C -algebra isomorphism. Proof. It is straight forward to show that, T W A ! B is R-linear and T.u/ D T.u/ for all  2 C and u 2 U.A/. P Since each a 2 A is a finite linear combination of unitary elements, i.e., a D njD1 j uj for all j 2 C and uj 2 U.A/, we have T.a/ D T

n X

n n  X X  j uj D j T.uj / D  j T.uj /

jD1

D T

jD1

n X

jD1

 j uj D T.a/

jD1

for all  2 C and a 2 A. Thus T W A ! B is C-linear. Furthermore, we have T.a / D T

n X

n n  X X j uj D j T.uj / D j T.uj /

jD1

DT

n X

jD1

j u j



jD1

D T.a/

jD1

for all a 2 A and T.av/ D T

n X jD1

DT

n X

n n  X X j u j v D j T.uj v/ D j T.uj /T.v/ jD1

jD1

 j uj T.v/ D T.a/T.v/

jD1

for all a 2 A and v 2 U.A/. P Since each b 2 A is a finite linear combination of unitary elements, i.e., b D m jD1 j vj for all j 2 C and vj 2 U.A/, we have

3.2 Apollonius Type Additive Functional Equations

T.ab/ D T

m X

m m  X X

j avj D

j T.avj / D

j T.a/T.vj /

jD1

D T.a/T

63

jD1

m X

jD1



j vj D T.a/T.b/

jD1

for all a; b 2 A and so T W A ! B is multiplicative. Therefore, T W A ! B is a C -algebra isomorphism. This completes the proof. 

3.2 Apollonius Type Additive Functional Equations In an inner product space, the following equality: kz  xk2 C kz  yk2 D

 x C y 1 2  kx  yk2 C 2z   2 2

holds, which is called the Apollonius’ identity. The following functional equation, which was motivated by this equation, Q.z  x/ C Q.z  y/ D

 x C y 1 Q.x  y/ C 2Q z  2 2

(3.15)

is quadratic. For this reason, the function equation (3.15) is called a quadratic functional equation of Apollonius type and each solution of the functional equation (3.15) is called a quadratic mapping of Apollonius type. In [144], Jun and Kim investigated the quadratic functional equation of Apollonius type. In this section, modifying the above equality (3.15), we consider a new functional equation, which is called the Apollonius type additive functional equation and whose solution of the functional equation is said to be the Apollonius type additive mapping [209]:  1 x C y L.z  x/ C L.z  y/ D  L.x C y/ C 2L z  : 2 4 In [126], Gilányi showed that, if f has it’s values in an inner product space and satisfies the functional inequality: k2f .x/ C 2f .y/  f .xy1 /k  kf .xy/k; then f satisfies the Jordan–von Neumann functional inequality 2f .x/ C 2f .y/ D f .xy/ C f .xy1 /:

(3.16)

3 Stability of Functional Equations in C -Algebras

64

In [113] and [127], Fechner and Gilányi proved the stability of the functional inequality (3.16), respectively. In [253], Park et al. proved the stability of functional inequalities associated with Jordan–von Neumann type additive functional equations. In 1932, Jordan observed that L.H/ is a (non-associative) algebra via the anticommutator product x ı y WD xyCyx 2 . A commutative algebra X with the product x ı y is called a Jordan algebra. A Jordan C -subalgebra of a C -algebra endowed with the anti-commutator product is called a JC -algebra. A C -algebra C endowed with the Lie product Œx; y D xyyx on C is called a Lie C -algebra. 2 In this section, we investigate homomorphisms and derivations in C -algebras associated with the Apollonius type additive functional equation. Also, we investigate homomorphisms and derivations on Lie C -algebras associated with the Apollonius type additive functional equation. Finally, we investigate homomorphisms and derivations on JC -algebras associated with the Apollonius type additive functional equation.

3.2.1 Homomorphisms and Derivations on C -Algebras Now, we study homomorphisms and derivations on C -algebras. Theorem 3.11. Let A be a uniquely 2-divisible Abelian group and B be a normed linear space. A mapping f W A ! B satisfies the following:     1 x C y      f .z  x/ C f .z  y/ C f .x C y/  2f z   B B 2 4

(3.17)

for all x; y; z 2 A if and only if f W A ! B is additive. Proof. Letting x D y D z D 0 in (3.17), we get 5 kf .0/kB  2kf .0/kB 2 and so f .0/ D 0. Letting z D 0 and y D x in (3.17), we get kf .x/ C f .x/kB  2kf .0/kB D 0 for all x 2 A and hence f .x/ D f .x/ for all x 2 A. Letting x D y D 2z in (3.17), we get   1   2f .z/ C f .4z/  k2f .0/kB D 0 B 2 for all z 2 A and hence f .4z/ D 4f .z/ D 4f .z/

3.2 Apollonius Type Additive Functional Equations

for all z 2 A. Letting z D

xCy 4

65

in (3.17), we get

   3x C y   x  3y  1   Cf C f .x C y/  k2f .0/kB D 0 f B 4 4 2 for all x; y 2 A and so  3x C y   x  3y  1 f Cf C f .x C y/ D 0 4 4 2 for all x; y 2 A. Let w1 D

3xCy 4

and w2 D

x3y 4

(3.18)

in (3.18). Then we have

w C w  1 1 1 2 f .w1 / C f .w2 / D  f .2w1  2w2 / D f .2w1 C 2w2 / D 2f 2 2 2 for all w1 ; w2 2 A and so f is additive. It is clear that each additive mapping satisfies the inequality (3.17). This completes the proof.  Now, we investigate C -algebra homomorphisms between C -algebras and linear derivations on C -algebras associated with the Apollonius type additive functional equation. From now on, assume that A is a C -algebra with the norm k  kA and B is a C -algebra with the norm k  kB . Lemma 3.12. Let f W A ! B be an additive mapping such that f . x/ D f .x/ for all x 2 A and 2 T1 D f 2 C W jj D 1g: Then the mapping f is C-linear. Proof. Let  2 C . ¤ 0/ and M an integer greater than 4jj. Then we have j M j < 14 < 1  23 D 13 . By Theorem 1 of [54], there exist three elements 1 ; 2 ; 3 2 T1 such that 3 M D 1 C 2 C 3 . And f .x/ D f .3  13 x/ D 3f . 31 x/ for all x 2 A and so f . 13 x/ D 13 f .x/ for all x 2 A. Thus 1     M    x DMf 3 x D f 3 x 3 M 3 M 3 M M M D f . 1 x C 2 x C 3 x/ D .f . 1 x/ C f . 2 x/ C f . 3 x// 3 3  M M  3 f .x/ D . 1 C 2 C 3 /f .x/ D 3 3 M D f .x/

f .x/ D f

M

3

for all x 2 A. Hence we have f . x C y/ D f . x/ C f . y/ D f .x/ C f .y/ for all ; 2 C . ; ¤ 0/ and all x; y 2 A. And f .0x/ D 0 D 0f .x/ for all x 2 A. So the additive mapping f W A ! B is a C-linear mapping. This completes the proof. 

3 Stability of Functional Equations in C -Algebras

66

Theorem 3.13. Let r > 1,  be nonnegative real numbers and f W A ! B be a mapping such that    1 xCy      / ; f .z  x/ C f .z  y/ C f .x C y/  2f .z  B B 2 4

(3.19)

kf .xy/  f .x/f .y/kB    kxkrA  kykrA ;

(3.20)

kf .x /  f .x/ kB  2kxkrA

(3.21)

for all 2 T1 WD f 2 C W jj D 1g and x; y; z 2 A. Then the mapping f W A ! B is a C -algebra homomorphism. Proof. Let D 1 in (3.19). By Theorem 3.11, the mapping f W A ! B is additive. Letting y D x and z D 0 in (3.19), we get kf . x/ C f .x/kB  k2f .0/kB D 0 for all x 2 A and 2 T1 and so f . x/ C f .x/ D f . x/ C f .x/ D 0 for all x 2 A and 2 T1 . Hence f . x/ D f .x/ for all x 2 A and 2 T1 . So, the mapping f W A ! B is C-linear. It follows from (3.20) that   xy   x   y    kf .xy/  f .x/f .y/kB D lim 4n f n n  f n f n  n!1 B 2 2 2 2 n 4   lim nr  kxkrA  kykrA n!1 4 D0 for all x; y 2 A and so f .xy/ D f .x/f .y/ for all x; y 2 A. It follows from (3.21) that   x   x     kf .x /  f .x/ kB D lim 2n f n  f n  n!1 B 2 2 2nC1  kxkrA n!1 2nr D0  lim

3.2 Apollonius Type Additive Functional Equations

67

for all x 2 A and so f .x / D f .x/ for all x 2 A. Therefore, the mapping f W A ! B is a C -algebra homomorphism. This completes the proof.  Remark 3.14. Let r < 1,  be positive real numbers and f W A ! B be a mapping satisfying (3.19), (3.20) and (3.21). Then the mapping f W A ! B is a C -algebra homomorphism. Theorem 3.15. Let r > 1,  be nonnegative real numbers and f W A ! A be a mapping satisfying (3.21) such that kf .xy/  f .x/y  xf .y/kA    kxkrA  kykrA

(3.22)

for all x; y 2 A. Then the mapping f W A ! A is a linear derivation. Proof. By applying Lemma 3.12, the mapping f W A ! A is C-linear. It follows from (3.22) that   xy  xy x  y    kf .xy/  f .x/y  xf .y/kA D lim 4n f n  f n n  n f n  n!1 A 4 2 2 2 2 4n   lim nr  kxkrA  kykrA n!1 4 D0 for all x; y 2 A and so f .xy/ D f .x/y C xf .y/ for all x; y 2 A. Thus the mapping f W A ! A is a linear derivation. This completes the proof.  Remark 3.16. Let r < 1,  be positive real numbers and f W A ! A be a mapping satisfying (3.19) and (3.22). Then the mapping f W A ! A is a linear derivation.

3.2.2 Homomorphisms and Derivations in Lie C -Algebras Assume that A is a Lie C -algebra with the norm k  kA and B is a Lie C -algebra with the norm k  kB . We recall that a C-linear mapping H W A ! B is called a Lie C -algebra homomorphism if H W A ! B satisfies the following: H.Œx; y/ D ŒH.x/; H.y/

3 Stability of Functional Equations in C -Algebras

68

for all x; y 2 A. A C-linear mapping D W A ! A is called a Lie derivation if D W A ! A satisfies the following: D.Œx; y/ D ŒD.x/; y C Œx; D.y/ for all x; y 2 A. Now, we investigate Lie C -algebra homomorphisms in Lie C -algebras and Lie derivations on Lie C -algebras associated with the Apollonius type additive functional equation. Theorem 3.17. Let r > 1,  be nonnegative real numbers and f W A ! B be a mapping satisfying (3.19) such that kf .Œx; y/  Œf .x/; f .y/kB    kxkrA  kykrA

(3.23)

for all x; y 2 A. Then the mapping f W A ! B is a Lie C -algebra homomorphism. Proof. It is straight forward to show that, the mapping f W A ! B is C-linear. It follows from (3.23) that    Œx; y  h  x   y i      f .Œx; y/  Œf .x/; f .y/ D lim 4n f n n  f n ; f n  n!1 B B 2 2 2 2 4n   lim nr  kxkrA  kykrA n!1 4 D0 for all x; y 2 A and so f .Œx; y/ D Œf .x/; f .y/ for all x; y 2 A. Hence the mapping f W A ! B is a Lie C -algebra homomorphism. This completes the proof.  Remark 3.18. If r < 1,  is positive real numbers and f W A ! B be a mapping satisfying (3.19) and (3.23). Then the mapping f W A ! B is a Lie C -algebra homomorphism. Theorem 3.19. Let r > 1,  be nonnegative real numbers and f W A ! A be a mapping satisfying (3.19) such that kf .Œx; y/  Œf .x/; y  Œx; f .y/kA    kxkrA  kykrA

(3.24)

for all x; y 2 A. Then the mapping f W A ! A is a Lie derivation. Proof. It is straight forward to show that, the mapping f W A ! A is C-linear. It follows from (3.24) that

3.2 Apollonius Type Additive Functional Equations

69

kf .Œx; y/  Œf .x/; y  Œx; f .y/kA   Œx; y  h  x  y i h x  y i    f n ; n  n;f n  D lim 4n f n!1 A 4n 2 2 2 2 n 4   lim nr  kxkrA  kykrA n!1 4 D0 for all x; y 2 A and so f .Œx; y/ D Œf .x/; y C Œx; f .y/ for all x; y 2 A. Thus the mapping f W A ! A is a Lie derivation. This completes the proof.  Remark 3.20. If r < 1,  is positive real numbers and f W A ! A be a mapping satisfying (3.19) and (3.24). Then the mapping f W A ! A is a Lie derivation.

3.2.3 Homomorphisms and Derivations in JC -Algebras Assume that A is a JC -algebra with the norm k  kA and B is a JC -algebra with the norm k  kB . A C-linear mapping H W A ! B is called a JC -algebra homomorphism if H W A ! B satisfies the following: H.x ı y/ D H.x/ ı H.y/ for all x; y 2 A. A C-linear mapping D W A ! A is called a Jordan derivation if D W A ! A satisfies the following: D.x ı y/ D D.x/ ı y C x ı D.y/ for all x; y 2 A. Now, we investigate JC -algebra homomorphisms between JC -algebras and Jordan derivations on JC -algebras associated with the Apollonius type additive functional equation. Remark 3.21. Let r > 1,  be nonnegative real numbers and f W A ! B be a mapping satisfying (3.19) such that kf .x ı y/  f .x/ ı f .y/kB    kxkrA  kykrA

(3.25)

for all x; y 2 A. Then the mapping f W A ! B is a JC -algebra homomorphism.

3 Stability of Functional Equations in C -Algebras

70

Remark 3.22. Let r < 1,  be positive real numbers and f W A ! B be a mapping satisfying (3.19) and (3.25). Then the mapping f W A ! B is a JC -algebra homomorphism. Remark 3.23. Let r > 1,  be nonnegative real numbers and f W A ! A be a mapping satisfying (3.19) such that kf .x ı y/  f .x/ ı y  x ı f .y/kA    kxkrA  kykrA

(3.26)

for all x; y 2 A. Then the mapping f W A ! A is a Jordan derivation. Remark 3.24. Let r < 1,  be positive real numbers and f W A ! A be a mapping satisfying (3.19) and (3.26). Then the mapping f W A ! A is a Jordan derivation.

3.3 Stability of Jensen Type Functional Equations in C -Algebras Using the fixed point method, we consider [250] the Hyers-Ulam stability of homomorphisms in C -algebras and Lie C -algebras and derivations on C algebras and Lie C -algebras for the following Jensen type functional equation: f

x C y 2

Cf

x  y 2

D f .x/:

3.3.1 Stability of Homomorphisms in C -Algebras Assume that A is a C -algebra with the norm k  kA and B is a C -algebra with the norm k  kB . For any mapping f W A ! B, we define D f .x; y/ WD f

x C y 2

C f

x  y 2

 f . x/

for all 2 T1 WD f 2 C W j j D 1g and x; y 2 A. Now, we prove the Hyers-Ulam stability of homomorphisms in C -algebras for the functional equation D f .x; y/ D 0. Theorem 3.25. Let f W A ! B be a mapping for which there exists a function ' W A2 ! Œ0; 1/ such that 1 X jD0

2j '.2j x; 2j y/ < 1;

(3.27)

3.3 Stability of Jensen Type Functional Equations in C -Algebras

71

kD f .x; y/kB  '.x; y/;

(3.28)

kf .xy/  f .x/f .y/kB  '.x; y/;

(3.29)

kf .x /  f .x/ kB  '.x; x/

(3.30)

for all 2 T1 and x; y 2 A. If there exists L < 1 such that x  '.x; 0/  2L' ; 0 2 for all x 2 A, then there exists a unique C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

L '.x; 0/ 1L

(3.31)

for all x 2 A. Proof. Consider the set X WD fg W A ! Bg and introduce the generalized metric on X defined by d.g; h/ D inffC 2 RC W kg.x/  h.x/kB  C'.x; 0/; 8x 2 Ag; which .X; d/ is complete. Now, we consider the linear mapping J W X ! X such that Jg.x/ WD

1 g.2x/ 2

for all x 2 A. Thus we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting D 1 and y D 0 in (3.28), we get  x     f .x/  '.x; 0/ 2f B 2 for all x 2 A and so   1 1   f .x/  f .2x/  '.2x; 0/  L'.x; 0/ B 2 2 for all x 2 A. Hence d.f ; Jf /  L.

(3.32)

3 Stability of Functional Equations in C -Algebras

72

By Theorem 1.3, there exists a mapping H W A ! B such that (1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(3.33)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (3.33) such that there exists C 2 .0; 1/ satisfying kH.x/  f .x/kB  C'.x; 0/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality f .2n x/ D H.x/ n!1 2n lim

(3.34)

for all x 2 A; 1 d.f ; Jf /, which implies the inequality (3) d.f ; H/  1L d.f ; H/ 

L : 1L

This implies that the inequality (3.31) holds. It follows from (3.27), (3.28) and (3.34) that  x C y  x  y   CH  H.x/ H B 2 2 1 D lim n kf .2n1 .x C y// C f .2n1 .x  y//  f .2n x/kB n!1 2 1  lim n '.2n x; 2n y/ n!1 2 D0 for all x; y 2 A and so H

x C y

for all x; y 2 A. Letting z D

2 xCy 2

CH

x  y

and w D

2 xy 2

D H.x/

in (3.35), we get

(3.35)

3.3 Stability of Jensen Type Functional Equations in C -Algebras

73

H.z/ C H.w/ D H.z C w/ for all z; w 2 A. So the mapping H W A ! B is Cauchy additive, i.e., H.z C w/ D H.z/ C H.w/ for all z; w 2 A. Letting y D x in (3.28), we get f .x/ D f . x/ for all 2 T1 and x 2 A. By a similar method to above, we get H.x/ D H. x/ for all 2 T1 and x 2 A. Thus one can show that the mapping H W A ! B is C-linear. It follows from (3.29) that 1 kf .4n xy/  f .2n x/f .2n y/kB 4n 1  lim n '.2n x; 2n y/ n!1 4 1  lim n '.2n x; 2n y/ n!1 2 D0

kH.xy/  H.x/H.y/kB D lim

n!1

for all x; y 2 A. Then we have H.xy/ D H.x/H.y/ for all x; y 2 A. It follows from (3.30) that 1 kf .2n x /  f .2n x/ kB n!1 2n 1  lim n '.2n x; 2n x/ n!1 2 D0

kH.x /  H.x/ kB D lim

for all x 2 A. Then we have H.x / D H.x/ for all x 2 A. Thus H W A ! B is a C -algebra homomorphism satisfying (3.31). This completes the proof. 

3 Stability of Functional Equations in C -Algebras

74

Corollary 3.26. Let r < 1,  be nonnegative real numbers and f W A ! B be a mapping such that kD f .x; y/kB  .kxkrA C kykrA /; kf .xy/  f .x/f .y/kB 

.kxkrA

C

kykrA /;

kf .x /  f .x/ kB  2kxkrA

(3.36) (3.37) (3.38)

for all 2 T1 and x; y 2 A. Then there exists a unique C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

2r  kxkrA 2  2r

(3.39)

for all x 2 A. Proof. The proof follows from Theorem 3.25 by taking '.x; y/ WD .kxkrA C kykrA / 

for all x; y 2 A and L D 2r1 .

Theorem 3.27. Let f W A ! B be a mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.28), (3.29) and (3.30) such that 1 X jD0

4j '

x y ; <1 2j 2j

(3.40)

for all x; y 2 A. If there exists L < 1 such that '.x; 0/ 

1 L'.2x; 0/ 2

for all x 2 A, then there exists a unique C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

L '.x; 0/ 2  2L

for all x 2 A. Proof. We consider the linear mapping J W X ! X such that Jg.x/ WD 2g

x 2

(3.41)

3.3 Stability of Jensen Type Functional Equations in C -Algebras

75

for all x 2 A. It follows from (3.32) that   x  x  L     ' ; 0  '.x; 0/ f .x/  2f 2 B 2 2 for all x 2 A and hence d.f ; Jf /  H W A ! B such that

L . 2

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(3.42)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (3.42) such that there exists C 2 .0; 1/ satisfying kH.x/  f .x/kB  C'.x; 0/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality lim 2n f .

n!1

x / D H.x/ 2n

for all x 2 A; 1 d.f ; Jf /, which implies the inequality (3) d.f ; H/  1L d.f ; H/ 

L ; 2  2L

which implies that the inequality (3.41) holds. The rest of the proof is similar to the proof of Theorem 3.25. This completes the proof.  Corollary 3.28. Let r > 2,  be nonnegative real numbers and f W A ! B be a mapping satisfying (3.36), (3.37) and (3.38). Then there exists a unique C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB  for all x 2 A.

 kxkrA 2r  2

(3.43)

3 Stability of Functional Equations in C -Algebras

76

Proof. The proof follows from Theorem 3.27 by taking '.x; y/ WD .kxkrA C kykrA / for all x; y 2 A and L D 21r .



Theorem 3.29. Let f W A ! B be an odd mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.27), (3.28), (3.29) and (3.30). If there exists L < 1 such that '.x; 3x/  2L'

 x 3x  ; 2 2

for all x 2 A, then there exists a unique C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

1 '.x; 3x/ 2  2L

(3.44)

for all x 2 A. Proof. Consider the set X WD fg W A ! Bg and introduce the generalized metric on X defined by d.g; h/ D inffC 2 RC W kg.x/  h.x/kB  C'.x; 3x/; 8x 2 Ag; which .X; d/ is complete. Now, we consider the linear mapping J W X ! X such that Jg.x/ WD

1 g.2x/ 2

for all x 2 A. Now, we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting D 1 and replacing y by 3x in (3.28), we get kf .2x/  2f .x/kB  '.x; 3x/ for all x 2 A and so 1 1 kf .x/  f .2x/kB  '.x; 3x/ 2 2

(3.45)

3.3 Stability of Jensen Type Functional Equations in C -Algebras

77

for all x 2 A. Hence d.f ; Jf /  12 . By Theorem 1.3, there exists a mapping H W A ! B such that (1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(3.46)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (3.46) such that there exists C 2 .0; 1/ satisfying kH.x/  f .x/kB  C'.x; 3x/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality f .2n x/ D H.x/ n!1 2n lim

for all x 2 A; 1 d.f ; Jf /, which implies the inequality (3) d.f ; H/  1L d.f ; H/ 

1 : 2  2L

This implies that the inequality (3.44) holds. The rest of the proof is similar to the proof of Theorem 3.25. This completes the proof.  Corollary 3.30. Let r < 12 ,  be nonnegative real numbers and f W A ! B be an odd mapping such that kD f .x; y/kB    kxkrA  kykrA ; kf .xy/  f .x/f .y/kB   

kxkrA



kykrA ;

kf .x /  f .x/ kB  kxk2r A

(3.47) (3.48) (3.49)

for all 2 T1 and x; y 2 A. Then there exists a unique C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB  for all x 2 A.

3r  kxk2r A 2  22r

(3.50)

3 Stability of Functional Equations in C -Algebras

78

Proof. The proof follows from Theorem 3.29 by taking '.x; y/ WD   kxkrA  kykrA for all x; y 2 A and L D 22r1 .



Theorem 3.31. Let f W A ! B be an odd mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.28), (3.29), (3.30) and (3.40). If there exists L < 1 such that '.x; 3x/ 

1 L'.2x; 6x/ 2

for all x 2 A, then there exists a unique C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

L '.x; 3x/ 2  2L

(3.51)

for all x 2 A. Proof. We consider the linear mapping J W X ! X such that Jg.x/ WD 2g

x 2

for all x 2 A. It follows from (3.45) that   x 3x  L  x     '.x; 3x/ f .x/  2f  ' ; 2 B 2 2 2 for all x 2 A and hence d.f ; Jf /  H W A ! B such that

L . 2

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(3.52)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (3.52) such that there exists C 2 .0; 1/ satisfying kH.x/  f .x/kB  C'.x; 3x/ for all x 2 A;

3.3 Stability of Jensen Type Functional Equations in C -Algebras

79

(2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality lim 2n f

n!1

x D H.x/ 2n

for all x 2 A; 1 (3) d.f ; H/  1L d.f ; Jf /, which implies the inequality d.f ; H/ 

L ; 2  2L

which implies that the inequality (3.51) holds. The rest of the proof is similar to the proof of Theorem 3.25. This completes the proof.  Corollary 3.32. Let r > 1,  be nonnegative real numbers and f W A ! B be an odd mapping satisfying (3.47), (3.48) and (3.49). Then there exists a unique C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

 kxk2r A 22r  2

(3.53)

for all x 2 A. Proof. The proof follows from Theorem 3.31 by taking '.x; y/ WD   kxkrA  kykrA for all x; y 2 A and L D 212r .



3.3.2 Stability of Derivations in C -Algebras Now, we prove the Hyers-Ulam stability of derivations on C -algebras for the functional equation D f .x; y/ D 0. Theorem 3.33. Let f W A ! A be a mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.27) such that kD f .x; y/kA  '.x; y/

(3.54)

kf .xy/  f .x/y  xf .y/kA  '.x; y/

(3.55)

and

for all 2 T1 and x; y 2 A. If there exists L < 1 such that

3 Stability of Functional Equations in C -Algebras

80

'.x; 0/  2L'

x 2

;0



for all x 2 A. Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

L '.x; 0/ 1L

(3.56)

for all x 2 A. Proof. It is straight forward to show that, there exists a unique involutive C-linear mapping ı W A ! A satisfying (3.56). The mapping ı W A ! A is given by f .2n x/ n!1 2n

ı.x/ D lim for all x 2 A. It follows from (3.55) that kı.xy/  ı.x/y  xı.y/kA

1 kf .4n xy/  f .2n x/  2n y  2n xf .2n y/kA 4n 1  lim n '.2n x; 2n y/ n!1 4 1  lim n '.2n x; 2n y/ n!1 2 D0

D lim

n!1

for all x; y 2 A and so ı.xy/ D ı.x/y C xı.y/ for all x; y 2 A. Thus ı W A ! A is a derivation satisfying (3.56). This completes the proof.  Corollary 3.34. Let r < 1,  be nonnegative real numbers and f W A ! A be a mapping such that kD f .x; y/kA  .kxkrA C kykrA /

(3.57)

kf .xy/  f .x/y  xf .y/kA  .kxkrA C kykrA /

(3.58)

and

for all 2 T1 and x; y 2 A. Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA  for all x 2 A.

2r  kxkrA 2  2r

(3.59)

3.3 Stability of Jensen Type Functional Equations in C -Algebras

81

Proof. The proof follows from Theorem 3.33 by taking '.x; y/ WD .kxkrA C kykrA / 

for all x; y 2 A and L D 2r1 .

Remark 3.35. If f W A ! A is a mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.40), (3.54) and (3.55). If there exists L < 1 such that '.x; 0/  12 L'.2x; 0/ for all x 2 A, then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

L '.x; 0/ 2  2L

(3.60)

for all x 2 A. Corollary 3.36. Let r > 2,  be nonnegative real numbers and f W A ! A be a mapping satisfying (3.57) and (3.58). Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

 kxkrA 2r  2

(3.61)

for all x 2 A. Proof. The proof follows from Theorem 3.35 by taking '.x; y/ WD .kxkrA C kykrA / for all x; y 2 A and L D 21r .



Remark 3.37. For the inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 3.29 and 3.31 and Corollaries 3.30 and 3.32.

3.3.3 Stability of Homomorphisms in Lie C -Algebras Assume that A is a Lie C -algebra with the norm k  kA and B is a Lie C -algebra with the norm k  kB . Now, we prove the Hyers-Ulam stability of homomorphisms in Lie C -algebras for the functional equation D f .x; y/ D 0. Theorem 3.38. Let f W A ! B be a mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.27) and (3.28) such that kf .Œx; y/  Œf .x/; f .y/kB  '.x; y/

(3.62)

3 Stability of Functional Equations in C -Algebras

82

for all x; y 2 A. If there exists L < 1 such that '.x; 0/  2L'

x 2

;0



for all x 2 A, then there exists a unique Lie C -algebra homomorphism H W A ! B satisfying (3.31). Proof. It is straight forward to show that, there exists a unique C-linear mapping H W A ! B satisfying (3.31). The mapping H W A ! B is given by f .2n x/ n!1 2n

H.x/ D lim for all x 2 A. It follows from (3.62) that kH.Œx; y/  ŒH.x/; H.y/kB

1 kf .4n Œx; y/  Œf .2n x/; f .2n y/kB 4n 1  lim n '.2n x; 2n y/ n!1 4 1  lim n '.2n x; 2n y/ n!1 2 D0

D lim

n!1

for all x; y 2 A and so H.Œx; y/ D ŒH.x/; H.y/ for all x; y 2 A. Thus H W A ! B is a Lie C -algebra homomorphism satisfying (3.31). This completes the proof.  Corollary 3.39. Let r < 1,  be nonnegative real numbers and f W A ! B be a mapping satisfying (3.36) such that kf .Œx; y/  Œf .x/; f .y/kB  .kxkrA C kykrA /

(3.63)

for all x; y 2 A. Then there exists a unique Lie C -algebra homomorphism H W A ! B satisfying (3.39). Proof. The proof follows from Theorem 3.38 by taking '.x; y/ WD .kxkrA C kykrA / for all x; y 2 A and L D 2r1 .



3.3 Stability of Jensen Type Functional Equations in C -Algebras

83

Remark 3.40. If f W A ! B is a mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.28), (3.40) and (3.62). If there exists L < 1 such that '.x; 0/  12 L'.2x; 0/ for all x 2 A, then there exists a unique Lie C -algebra homomorphism H W A ! B satisfying (3.41). Corollary 3.41. Let r > 2,  be nonnegative real numbers and f W A ! B be a mapping satisfying (3.36) and (3.63). Then there exists a unique Lie C -algebra homomorphism H W A ! B satisfying (3.43). Proof. The proof follows from Theorem 3.40 by taking '.x; y/ WD .kxkrA C kykrA / for all x; y 2 A and L D 21r .



Remark 3.42. For the inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 3.29 and 3.31 and their corollaries.

3.3.4 Stability of Lie Derivations in C -Algebras Assume that A is a Lie C -algebra with the norm k  kA . Now, we prove the Hyers-Ulam stability of derivations on Lie C -algebras for the functional equation D f .x; y/ D 0. Theorem 3.43. Let f W A ! A be a mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.27) and (3.54) such that kf .Œx; y/  Œf .x/; y  Œx; f .y/kA  '.x; y/

(3.64)

for all x; y 2 A. If there exists L < 1 such that x  '.x; 0/  2L' ; 0 2 for all x 2 A. Then there exists a unique Lie derivation ı W A ! A satisfying (3.56). Proof. It is easy to show that, there exists a unique involution C-linear mapping ı W A ! A satisfying (3.56). The mapping ı W A ! A is given by f .2n x/ n!1 2n

ı.x/ D lim for all x 2 A. It follows from (3.62) that

kı.Œx; y/  Œı.x/; y  Œx; ı.y/kA D lim

n!1

1 kf .4n Œx; y/  Œf .2n x/; 2n y  Œ2n x; f .2n y/kA 4n

3 Stability of Functional Equations in C -Algebras

84

1 '.2n x; 2n y/ 4n 1  lim n '.2n x; 2n y/ n!1 2 D0

 lim

n!1

for all x; y 2 A and so ı.Œx; y/ D Œı.x/; y C Œx; ı.y/ for all x; y 2 A. Thus ı W A ! A is a derivation satisfying (3.56). This completes the proof.  Corollary 3.44. Let r < 1,  be nonnegative real numbers and f W A ! A be a mapping satisfying (3.57) such that kf .Œx; y/  Œf .x/; y  Œx; f .y/kA  .kxkrA C kykrA /

(3.65)

for all x; y 2 A. Then there exists a unique Lie derivation ı W A ! A satisfying (3.59). Proof. The proof follows from Theorem 3.38 by taking '.x; y/ WD .kxkrA C kykrA / for all x; y 2 A and L D 2r1 .



Remark 3.45. If f W A ! A is a mapping for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.40), (3.54) and (3.64). Whenever there exists L < 1 such that '.x; 0/ 

1 L'.2x; 0/ 2

for all x 2 A, then there exists a unique Lie derivation ı W A ! A satisfying (3.60). Corollary 3.46. Let r > 2,  be nonnegative real numbers and f W A ! A be a mapping satisfying (3.57) and (3.65). Then there exists a unique Lie derivation ı W A ! A satisfying (3.61). Proof. The proof follows from Theorem 3.45 by taking '.x; y/ WD .kxkrA C kykrA / for all x; y 2 A and L D 21r .



Remark 3.47. For the inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 3.29 and 3.31 and their corollaries.

3.4 Generalized Additive Mapping

85

3.4 Generalized Additive Mapping Recently, Park and Park [243] introduced and investigated the following generalized additive functional equation n X

0 1 ! ! n n n X X X @ A ri L rj .xi  xj / C ri L ri xi

iD1

D

jD1 n X

! ri

iD1

iD1

n X

iD1

ri L.xi /

(3.66)

iD1

for all r1 ;    ; rn 2 .0; 1/ whose solution is called a generalized additive mapping. In this section, we consider [210] the following additive functional equation which is somewhat different from (3.66): n 1 X f 2 jD1

X 1in;i¤j

1 X  1  X ri xi  rj xj C ri f .xi / D nf ri xi 2 2 iD1 iD1 n

n

(3.67)

for all r1 ;    ; rn 2 R: Every solution of the functional equation (3.67) is said to be a generalized additive mapping. Using the fixed point method, we investigate the Hyers–Ulam stability of the functional equation (3.67) in Banach modules over a C -algebra. These results are applied to investigate C -algebra homomorphisms in unital C -algebras. Throughout this section, assume that A is a unital C -algebra with the norm k  kA and the unit e that B is a unital C -algebra with the norm k  kB and X and Y are left Banach modules over a unital C -algebra A with the norms k  kX and k  kY ; respectively. Let U.A/ be the group of unitary elements in A and let r1 ;    ; rn 2 R:

3.4.1 Hyers–Ulam Stability of Functional Equations in Banach Modules over a C -Algebra For any mapping f W X ! Y; u 2 U.A/ and 2 C; we define Du;r1 ; ;rn f and D ;r1 ; ;rn f W X n ! Y by Du;r1 ; ;rn f .x1 ;    ; xn / WD

n 1 X f 2 jD1

X 1in;i¤j

 X  1 X 1 ri uxi  rj uxj C ri uf .xi /  nf ri uxi 2 2 iD1 iD1 n

n

3 Stability of Functional Equations in C -Algebras

86

and D ;r1 ; ;rn f .x1 ;    ; xn / WD

n 1 X f 2 jD1

X 1in;i¤j

1 X  X  1 ri xi  rj xj C ri f .xi /  nf ri xi 2 2 iD1 iD1 n

n

for all x1 ;    ; xn 2 X, respectively. Lemma 3.48. Let X and Y be linear spaces and r1 ;    ; rn be real numbers with Pn kD1 rk ¤ 0 and ri ¤ 0, rj ¤ 0 for some 1  i < j  n: Assume that a mapping L W X ! Y satisfies the functional equation (3.67) for all x1 ;    ; xn 2 X: Then the mapping L is additive. Moreover, L.rk x/ D rk L.x/ for all x 2 X and 1  k  n: P Proof. Since nkD1 rk ¤ 0, putting x1 D    D xn D 0 in (3.67), we get L.0/ D 0. Without loss of generality, we assume that r1 ; r2 ¤ 0. Letting x3 D    D xn D 0 in (3.67), we get   r x  r x  r1 x1 C r2 x2 1 1 2 2 CL C r1 L.x1 / C r2 L.x2 / L 2 2   r1 x1 C r2 x2 (3.68) D 2L 2 for all x1 ; x2 2 X. Letting x2 D 0 in (3.68), we get  r x  r x  1 1 1 1 L r1 L.x1 / D L 2 2 for all x1 2 X. Similarly, by putting x1 D 0 in (3.68), we get  r x  r x  2 2 2 2 L r2 L.x2 / D L 2 2 for all x2 2 X. It follows from (3.68), (3.69) and (3.70) that   r x  r x  r1 x1 C r2 x2 1 1 2 2 CL L 2 2 r x  r x   r x   r x  1 1 2 2 1 1 2 2 CL CL L L 2 2 2 2   r1 x1 C r2 x2 D 2L 2 for all x1 ; x2 2 X. Replacing x1 and x2 by

2x r1

and

2x r2

(3.69)

(3.70)

(3.71)

in (3.71), we get

L.x C y/ C L.x  y/ C L.x/ C L.y/  L.x/  L.y/ D 2L.x C y/

(3.72)

3.4 Generalized Additive Mapping

87

for all x; y 2 X. Letting y D x in (3.72), we get that L.2x/ C L.2x/ D 0 for all x 2 X. Then the mapping L is odd. Therefore, it follows from (3.72) that the mapping L is additive. Moreover, let x 2 X and 1  k  n. Setting xk D x and xl D 0 for all 1  l  n with l ¤ k in (3.67) and, using the oddness of L, it follows that L.rk x/ D rk L.x/. This completes the proof.  Using the same method as Pin the proof of Lemma 3.48, we have an alternative result of Lemma 3.48 when nkD1 rk D 0. Lemma 3.49. Let X and Y be linear spaces and r1 ;    ; rn be real numbers with ri ¤ 0; rj ¤ 0 for some 1  i < j  n: Assume that a mapping L W X ! Y with L.0/ D 0 satisfies the functional equation (3.67) for all x1 ;    ; xn 2 X: Then the mapping L is additive. Moreover, L.rk x/ D rk L.x/ for all x 2 X and all 1  k  n: Now, we investigate the Hyers-Ulam stability of a generalized additive mapping in Banach modules over a unital C -algebra. Here r1 ;    ; rn are real numbers such that ri ¤ 0 and rj ¤ 0 for fixed 1  i < j  n: Theorem 3.50. Let f W X ! Y be a mapping satisfying f .0/ D 0 for which there exists a function ' W X n ! Œ0; 1/ such that kDe;r1 ; ;rn f .x1 ;    ; xn /kY  '.x1 ;    ; xn /

(3.73)

for each x1 ;    ; xn 2 X. Let  'ij .x; y/ WD ' 0;    ; 0; „ƒ‚… x ; 0;    ; 0; i th

y ; 0;    ; 0 „ƒ‚…



j th

for all x; y 2 X and 1  i < j  n: If there exists 0 < C < 1 such that '.2x1 ;    ; 2xn /  2C'.x1 ;    ; xn / for all x1 ;    ; xn 2 X, then there exists a unique generalized additive mapping L W X ! Y such that kf .x/  L.x/kY 

x 1 n  2x 2x  x C 2'ij ;  'ij ; 4  4C ri rj ri rj  2x  x  C 'ij ; 0 C 2'ij ; 0 ri ri  2x   x o C 2'ij 0;  C 'ij 0; rj rj

for all x 2 X: Moreover, L.rk x/ D rk L.x/ for all x 2 X and 1  k  n:

(3.74)

88

3 Stability of Functional Equations in C -Algebras

Proof. For each 1  k  n with k ¤ i; j; let xk D 0 in (3.73). Then we get the following inequality:   r x C r x  r x  r x  r x C r x  i i j j i i j j i i j j  Cf  2f f 2 2 2   C ri f .xi / C rj f .xj / Y    ' 0;    ; 0; xi ; 0;    ; 0; xj ; 0;    ; 0 „ƒ‚… „ƒ‚… i th

(3.75)

j th

for all xi ; xj 2 X: Letting xi D 0 in (3.75), we get   rx   r x  j j j j   f C rj f .xj /  'ij .0; xj / f  2 2 Y

(3.76)

for all xj 2 X: Similarly, letting xj D 0 in (3.75), we get   rx   r x    i i i i f C ri f .xi /  'ij .xi ; 0/ f  2 2 Y

(3.77)

for all xi 2 X: It follows from (3.75), (3.76) and (3.77) that   r x C r x  r x  r x  r x C r x  i i j j i i j j i i j j  Cf  2f f 2 2 2 r x   rx   r x  r x  i i j j i i j j  Cf f  f  Cf  Y 2 2 2 2  'ij .xi ; xj / C 'ij .xi ; 0/ C 'ij .0; xj / for all xi ; xj 2 X: Replacing xi and xj by

2x ri

and

2y rj

(3.78)

in (3.78), it follows that

kf .x C y/ C f .x  y/  2f .x C y/ C f .x/ C f .y/  f .x/  f .y/kY       2y 2x 2y 2x C 'ij  'ij ; ; 0 C 'ij 0; ri rj ri rj

(3.79)

for all x; y 2 X: Putting y D x in (3.79), we get k2f .x/  2f .x/  2f .2x/kY  2x 2x   2x   2x  C 'ij  'ij ; ; 0 C 'ij 0; ri rj ri rj

(3.80)

3.4 Generalized Additive Mapping

89

for all x 2 X: Replacing x and y by

x 2

and  2x in (3.79), respectively, we get

kf .x/ C f .x/kY x x   x x C 'ij ; 0 C 'ij 0;   'ij ;  ri rj ri rj

(3.81)

for all x 2 X: It follows from (3.80) and (3.81) that 1  1   .x/  f .2x/  f .x/  Y 2 4

(3.82)

for all x 2 X; where  2x 2x  x x C 2'ij ;  ; ri rj ri rj  2x  x   2x   x C 2'ij 0;  : ; 0 C 2'ij ; 0 C 'ij 0; C 'ij ri ri rj rj

.x/ WD 'ij

Consider the set W WD fg W X ! Yg and introduce the generalized metric on W defined by d.g; h/ D inffC 2 RC W kg.x/  h.x/kY  C .x/; 8x 2 Xg: It is easy to show that .W; d/ is complete. Now, we consider the linear mapping J W W ! W such that Jg.x/ WD

1 g.2x/ 2

(3.83)

for all x 2 X. It follows that d.Jg; Jh/  Cd.g; h/ for all g; h 2 W and so d.f ; Jf /  14 . By Theorem 1.3, there exists a mapping L W X ! Y such that (1) L is a fixed point of J, i.e., L.2x/ D 2L.x/

(3.84)

for all x 2 X. The mapping L is a unique fixed point of J in the set Z D fg 2 W W d.f ; g/ < 1g: This implies that L is a unique mapping satisfying (3.84) such that there exists C 2 .0; 1/ satisfying kL.x/  f .x/kY  C .x/ for all x 2 X;

3 Stability of Functional Equations in C -Algebras

90

(2) d.J n f ; L/ ! 0 as n ! 1. This implies the equality f .2n x/ D L.x/ n!1 2n lim

for all x 2 X; 1 d.f ; Jf /, which implies the inequality d.f ; L/  (3) d.f ; L/  1C implies that the inequality (3.74) holds.

1 44C :

This

Since '.2x1 ;    ; 2xn /  2C'.x1 ;    ; xn /, we have 1 kDe;r1 ; ;rn f .2k x1 ;    ; 2k xn /kY 2k 1  lim k '.2k x1 ;    ; 2k xn / k!1 2

kDe;r1 ; ;rn L.x1 ;    ; xn /kY D lim

k!1

 lim Ck '.x1 ;    ; xn / D 0 k!1

for all x1 ;    ; xn 2 X: Therefore, the mapping L W X ! Y satisfies the Eq. (3.67) and L.0/ D 0: Hence, by Lemma 3.49, L is a generalized additive mapping and L.rk x/ D rk L.x/ for all x 2 X and all 1  k  n: This completes the proof.  Theorem 3.51. Let f W X ! Y be a mapping satisfying f .0/ D 0 for which there exists a function ' W X n ! Œ0; 1/ satisfying kDu;r1 ; ;rn f .x1 ;    ; xn /k  '.x1 ;    ; xn /

(3.85)

for all x1 ;    ; xn 2 X and u 2 U.A/: If there exists 0 < C < 1 such that '.2x1 ;    ; 2xn /  2C'.x1 ;    ; xn / for all x1 ;    ; xn 2 X, then there exists a unique A-linear generalized additive mapping L W X ! Y satisfying (3.74) for all x 2 X: Moreover, L.rk x/ D rk L.x/ for all x 2 X and 1  k  n: Proof. By Theorem 3.50, there exists a unique generalized additive mapping L W X ! Y satisfying (3.74) and, moreover, L.rk x/ D rk L.x/ for all x 2 X and 1  k  n: By the assumption, for all u 2 U.A/, we get     x ; 0    ; 0/ Du;r1 ; ;rn L.0;    ; 0; „ƒ‚…

Y

i th

 1   k f .0;    ; 0; 2 x ; 0    ; 0/ D  u;r ; ;r 1 n „ƒ‚… k!1 2k Y

D lim

i th

3.4 Generalized Additive Mapping

91

 1  k ' 0;    ; 0; 2 x ; 0    ; 0 „ƒ‚… k!1 2k

 lim

i th

  x ;0 ;0  lim Ck ' 0;    ; 0; „ƒ‚… k!1

i th

D0 for all x 2 X and so ri uL.x/ D L.ri ux/ for all u 2 U.A/ and x 2 X: Since L.ri x/ D ri L.x/ for all x 2 X and ri ¤ 0; we have L.ux/ D uL.x/ for all u 2 U.A/ and x 2 X: Now, we have L.ax C by/ D L.ax/ C L.by/ D aL.x/ C bL.y/ for all a; b 2 A .a; b ¤ 0/ and x; y 2 X: Since L.0x/ D 0 D 0L.x/ for all x 2 X; the unique generalized additive mapping L W X ! Y is an A-linear mapping. This completes the proof.  Theorem 3.52. Let f W X ! Y be a mapping satisfying f .0/ D 0 for which there exists a function ' W X n ! Œ0; 1/ such that kDe;r1 ; ;rn f .x1 ;    ; xn /kY  '.x1 ;    ; xn /

(3.86)

for all x1 ;    ; xn 2 X. If there exists 0 < C < 1 such that '.x1 ;    ; 2n / 

C '.2x1 ;    ; 2xn / 2

for all x1 ;    ; xn 2 X, then there exists a unique generalized additive mapping L W X ! Y such that kf .x/  L.x/kY 

x C n  2x 2x  x C 2'ij ;  'ij ; 4  4C ri rj ri rj  2x  x   2x   x o C 2'ij 0;  C 'ij ; 0 C 2'ij ; 0 C 'ij 0; ri ri rj rj

(3.87)

for all x 2 X; where 'ij is defined in the statement of Theorem 3.50. Moreover, L.rk x/ D rk L.x/ for all x 2 X and 1  k  n:

3 Stability of Functional Equations in C -Algebras

92

Proof. It follows from (3.82) that   x  1   f .x/  f   2 Y 2

x 2



C 4

.x/

for all x 2 X; where is defined in the proof of Theorem 3.50. The rest of the proof is similar to the proof of Theorem 3.50. This completes the proof.  Remark 3.53. Let f W X ! Y be a mapping with f .0/ D 0 for which there exists a function ' W X n ! Œ0; 1/ satisfying kDu;r1 ; ;rn f .x1 ;    ; xn /k  '.x1 ;    ; xn /

(3.88)

for all x1 ;    ; xn 2 X and u 2 U.A/: If there exists 0 < C < 1 such that '.x1 ;    ; 2n / 

C '.2x1 ;    ; 2xn / 2

for all x1 ;    ; xn 2 X, then there exists a unique A-linear generalized additive mapping L W X ! Y satisfying (3.87) for all x 2 X: Moreover, L.rk x/ D rk L.x/ for all x 2 X and 1  k  n: P Remark 3.54. In Theorems 3.52 and 3.53, one can assume that nkD1 rk ¤ 0 instead of f .0/ D 0:

3.4.2 Homomorphisms in Unital C -Algebras Now, we investigate C -algebra homomorphisms in unital C -algebras. We use the following lemma in the proof of the following theorem. Lemma 3.55 ([229]). Let f W A ! B be an additive mapping such that f . x/ D f .x/ for all x 2 A and 2 S1 WD f 2 C W jj D 1g: Then the mapping f W A ! B is C-linear. Theorem 3.56. Let f W A ! B be a mapping with f .0/ D 0 for which there exists a function ' W An ! Œ0; 1/ satisfying   D ;r ; ;r f .x1 ;    ; xn /  '.x1 ;    ; xn /; 1 n B     k  f .2 u /  f .2k u/   ' 2k u;    ; 2k u ; B „ ƒ‚ …

(3.89) (3.90)

n times

   k  f .2 ux/  f .2k u/f .x/  ' 2k ux;    ; 2k ux B ƒ‚ … „ n times

(3.91)

3.4 Generalized Additive Mapping

93

for all x; x1 ;    ; xn 2 A; u 2 U.A/; k 2 N and 2 S1 . If there exists 0 < C < 1 such that '.2x1 ;    ; 2xn /  2C'.x1 ;    ; xn / for all x1 ;    ; xn 2 A, then the mapping f W A ! B is a C -algebra homomorphism. Proof. Since jJj  3; letting D 1 and xk D 0 for all 1  k  n with k ¤ i; j in (3.89), we get  f

 r x  r x  ri xi C rj xj i i j j Cf C ri f .xi / C rj f .xj / 2 2   ri xi C rj xj D 2f 2

for all xi ; xj 2 A: By Lemma 3.48, the mapping f is additive and f .rk x/ D rk f .x/ for all x 2 A and k D i; j: So, by letting xi D x and xk D 0 for all 1  k  n with k ¤ i in (3.89), it follows that f . x/ D f .x/ for all x 2 A and 2 S1 . Therefore, by Lemma 3.55, the mapping f is C-linear. Hence it follows from (3.90) and (3.91) that  1  f .2k u /  f .2k u/  B k 2   1  lim k ' 2k u;    ; 2k u „ ƒ‚ … k!1 2

kf .u /  f .u/ kB D lim

k!1



n times

 lim C ' u;    ; u „ ƒ‚ … k!1 k



n times

D0 and  1  f .2k ux/  f .2k u/f .x/ B k k!1 2   1  lim k ' 2k ux;    ; 2k ux „ ƒ‚ … k!1 2

kf .ux/  f .u/f .x/kB D lim



n times

 lim Ck ' ux;    ; ux „ ƒ‚ … k!1 n times

D0



3 Stability of Functional Equations in C -Algebras

94

for all x 2 A and u 2 U.A/ and so f .u / D f .u/ ;

f .ux/ D f .u/f .x/

for all x 2 A and u 2 U.A/: Since f is C-linear and each P x 2 A is a finite linear combination of unitary elements (see [168]), i.e., x D m kD1 k uk ; where k 2 C and uk 2 U.A/ for all 1  k  n; we have m X



f .x / D f

! k uk

kD1

D

m X

m X

D

kD1

!

k f .uk /

m   X k f uk D k f .uk /

Df

kD1

kD1

m X

! k u k

D f .x/

kD1

and f .xy/ D f

m X kD1

D

m X

! k u k y D

m X

k f .uk y/

kD1

k f .uk /f .y/ D f

kD1

m X

! k uk f .y/ D f .x/f .y/

kD1

for all x; y 2 A: Therefore, the mapping f W A ! B is a C -algebra homomorphism. This completes the proof.  Remark 3.57. Let f W A ! B be a mapping with f .0/ D 0 for which there exists a function ' W An ! Œ0; 1/ satisfying   kD ;r1 ; ;rn f .x1 ;    ; xn /kB  ' x1 ;    ; xn ;   u   u    u u    f k  f k    k ;    ; k ; B 2 2 2 ƒ‚ 2… „

(3.92)

n times

  ux   u  ux ux    f k  f k f .x/   k ;    ; k B 2 2 „2 ƒ‚ 2…

(3.93)

n times

for all x; x1 ;    ; xn 2 A; u 2 U.A/; k 2 N and 2 S1 . If there exists 0 < C < 1 such that '.x1 ;    ; 2n / 

C '.2x1 ;    ; 2xn / 2

for all x1 ;    ; xn 2 A, then the mapping f W A ! B is a C -algebra homomorphism.

3.4 Generalized Additive Mapping

95

Remark 3.58. In Theorem 3.56 and last remark, one can assume that instead of f .0/ D 0:

Pn

kD1 rk

¤0

Theorem 3.59. Let f W A ! B be a mapping with f .0/ D 0 for which there exists a function ' W An ! Œ0; 1/ satisfying (3.90), (3.91) and kD ;r1 ; ;rn f .x1 ;    ; xn /kB  '.x1 ;    ; xn /; for all x1 ;    ; xn 2 A and 2 S1 . Assume that limk!1 there exists 0 < C < 1 such that

1 f .2k e/ 2k

(3.94) is invertible. If

'.2x1 ;    ; 2xn /  2C'.x1 ;    ; xn / for all x1 ;    ; xn 2 A, then the mapping f W A ! B is a C -algebra homomorphism. Proof. Consider the C -algebras A and B as left Banach modules over the unital C -algebra C: By Theorem 3.51, there exists a unique C-linear generalized additive mapping H W A ! B defined by 1 f .2k x/ k!1 2k

H.x/ D lim for all x 2 A: By (3.90) and (3.91), we get

          H u  H.u/  D lim 1 f 2k u  f 2k u   B B k k!1 2  1   lim k ' 2k u;    ; 2k u „ ƒ‚ … k!1 2 n times

D0 and    1  f 2k ux  f .2k u/f .x/ B k 2   1  lim k ' 2k ux;    ; 2k ux ƒ‚ … „ k!1 2

kH.ux/  H.u/f .x/kB D lim

k!1

n times

D0 for all u 2 U.A/ and x 2 A and so   H u D H.u/ ;

H.ux/ D H.u/f .x/

3 Stability of Functional Equations in C -Algebras

96

for all u 2 U.A/ and x 2 A: Therefore, by the additivity of H, we have H.ux/ D lim

k!1

1  k  1   H 2 ux D H.u/ lim k f 2k x D H.u/H.x/ k!1 2 2k

(3.95)

for all u 2 U.A/ and x 2 A: Since H is C-linear and each x 2 A is a finite linear P combination of unitary elements, i.e., x D m  uk ; where k 2 C and uk 2 U.A/ k kD1 for all 1  k  n; it follows from (3.95) that ! m m X X H.xy/ D H k u k y D k H.uk y/ kD1

D

m X

kD1

k H.uk /H.y/ D H

kD1

m X

! k uk H.y/

kD1

D H.x/H.y/ and m X   H x D H k uk

! D

kD1

D

m X

k H.uk /

kD1 

k H.uk / D

kD1

DH

m X

m X

! k H.uk /

kD1

X m

k u k



D H.x/

kD1

for all x; y 2 A: Since H.e/ D limk!1

1 f .2k e/ 2k

is invertible and

H.e/H.y/ D H.ey/ D H.e/f .y/;

H.y/ D f .y/

for all y 2 A: Therefore, the mapping f W A ! B is a C -algebra homomorphism. This completes the proof.  Remark 3.60. Let f W A ! B be a mapping with f .0/ D 0 for which there exists a function ' W An ! Œ0; 1/ satisfying (3.92), (3.93) and kD ;r1 ; ;rn f .x1 ;    ; xn /kB  '.x1 ;    ; xn /; for all x1 ;    ; xn 2 A and 2 S1 . Assume that limk!1 2k f . 2ek / is invertible. If there exists 0 < C < 1 such that '.x1 ;    ; 2n / 

C '.2x1 ;    ; 2xn / 2

for all x1 ;    ; xn 2 A, then the mapping f W A ! B is a C -algebra homomorphism.

3.4 Generalized Additive Mapping

97

In the last Remark, one can assume that

Pn

kD1 rk

¤ 0 instead of f .0/ D 0:

Theorem 3.61. Let f W A ! B be a mapping with f .0/ D 0 for which there exists a function ' W An ! Œ0; 1/ satisfying (3.90), (3.91) and kD ;r1 ; ;rn f .x1 ;    ; xn /kB  '.x1 ;    ; xn /

(3.96)

for D i; 1 and x1 ;    ; xn 2 A. Assume that limk!1 21k f .2k e/ is invertible and, for each fixed x 2 A, the mapping t 7! f .tx/ is continuous in t 2 R. If there exists 0 < C < 1 such that '.2x1 ;    ; 2xn /  2C'.x1 ;    ; xn / for all x1 ;    ; xn 2 A, then the mapping f W A ! B is a C -algebra homomorphism. Proof. Put D 1 in (3.96). By the same reasoning as in the proof of Theorem 3.50, there exists a unique generalized additive mapping H W A ! B defined by f .2k x/ k!1 2k

H.x/ D lim

for all x 2 A: It is straight forward to show that, the generalized additive mapping H W A ! B is R-linear. By the same method as in the proof of Theorem 3.51, we have     x ; 0    ; 0/ D ;r1 ; ;rn H.0;    ; 0; „ƒ‚… Y

j th

 1   k f .0;    ; 0; 2 x ; 0    ; 0/  D ;r ; ;r 1 n „ƒ‚… k!1 2k Y

D lim

j th

 lim

k!1

1 '.0;    ; 0; „ƒ‚… 2k x ; 0    ; 0/ 2k j th

D0 for all x 2 A and so rj H.x/ D H.rj x/ for all x 2 A: Since H.rj x/ D rj H.x/ for all x 2 X and rj ¤ 0; we have H. x/ D H.x/

3 Stability of Functional Equations in C -Algebras

98

for all x 2 A and D i; 1. For each element  2 C, we have  D s C it; where s; t 2 R. Thus it follows that H.x/ D H.sx C itx/ D sH.x/ C tH.ix/ D sH.x/ C itH.x/ D .s C it/H.x/ D H.x/ for all  2 C and x 2 A: So, we have H. x C y/ D H. x/ C H. y/ D H.x/ C H.y/ for all ; 2 C and x; y 2 A: Hence the generalized additive mapping H W A ! B is C-linear. The rest of the proof is the same as in the proof of Theorem 3.59. This completes the proof.  The following is an alternative result of Theorem 3.61. Remark 3.62. Let f W A ! B be a mapping with f .0/ D 0 for which there exists a function ' W An ! Œ0; 1/ satisfying (3.92), (3.93) and kD ;r1 ; ;rn f .x1 ;    ; xn /kB  '.x1 ;    ; xn / for all x; x1 ;    ; xn 2 A and D i; 1: Assume that limk!1 2k f . 2ek / is invertible and, for each fixed x 2 A, the mapping t 7! f .tx/ is continuous in t 2 R. If there exists 0 < C < 1 such that '.x1 ;    ; 2n / 

C '.2x1 ;    ; 2xn / 2

for all x1 ;    ; xn 2 A, then the mapping f W A ! B is a C -algebra homomorphism. P Remark 3.63. Also, one can assume that nkD1 rk ¤ 0 instead of f .0/ D 0:

3.5 Generalized Additive Mappings in Banach Modules Let X; Y be vector spaces. It is shown that, if an odd mapping f W X ! Y satisfies the functional equation ! ! Pd Pd .j/ X jD1 xj jD1 .1/ xj rf rf C r r .j/ D 0; 1 Pd jD1 .j/ D l D .d1 Cl d1 Cl1 C 1/

d X jD1

f .xj /;

(3.97)

3.5 Generalized Additive Mappings in Banach Modules

99

then the odd mapping f W X ! Y is additive. Also, we consider the Hyers-Ulam stability of the functional equation (3.97) in Banach modules over a unital C algebra. As an application, we show that every almost linear bijection h W A ! B of a unital C -algebra A onto a unital C -algebra B is a C -algebra isomorphism when n n h. 2rn uy/ D h. 2rn u/h.y/ for all unitaries u 2 A, y 2 A and n  0 [20]. Throughout this section, assume that r is a positive rational number and d, l are integers with 1 < l < d2 .

3.5.1 Odd Functional Equations in d–Variables Now, assume that X and Y are real linear spaces. Lemma 3.64. An odd mapping f W X ! Y satisfies (3.97) for all x1 ; x2 ;    ; xd 2 X if and only if f is Cauchy additive. Proof. Assume that f W X ! Y satisfies (3.97) for all x1 ; x2 ;    ; xd 2 X. Note that f .0/ D 0 and f .x/ D f .x/ for all x 2 X since f is an odd mapping. Putting x1 D x; x2 D y and x3 D    D xd D 0 in (3.97), we get .d2 Cl d2 Cl2 C 1/rf

x C y

r D .d1 Cl d1 Cl1 C 1/.f .x/ C f .y//

(3.98)

for all x; y 2 X. Since d2 Cl d2 Cl2 C 1 Dd1 Cl d1 Cl1 C 1, we have rf

x C y r

D f .x/ C f .y/

for all x; y 2 X. Letting y D 0 in (3.98), we get rf . xr / D f .x/ for all x 2 X. Hence we have x C y D f .x/ C f .y/ f .x C y/ D rf r for all x; y 2 X. Thus f is Cauchy additive. The converse is obviously true. This completes the proof. In the proof of Lemma 3.64, we prove the following. Corollary 3.65. An odd mapping f W X ! Y satisfies rf

x C y r

D f .x/ C f .y/

for all x; y 2 X if and only if f is Cauchy additive.



3 Stability of Functional Equations in C -Algebras

100

3.5.2 Stability of Odd Functional Equations in Banach Modules over a C -Algebra Assume that A is a unital C -algebra with the norm j  j and a unitary group U.A/ and X, Y are left Banach modules over a unital C -algebra A with norms k  k and k  k, respectively. For any mapping f W X ! Y, we set Du f .x1 ;    ; xd /  Pd WD rf

jD1

uxj 

r

X

C

rf

 Pd .1/.j/ uxj  jD1

.j/ D 0; 1 Pd .j/ Dl jD1

.d1 Cl d1 Cl1 C 1/

d X

r

uf .xj /

jD1

for all u 2 U.A/ and x1 ;    ; xd 2 X. Theorem 3.66. Let r ¤ 2. Let f W X ! Y be an odd mapping for which there exists a function ' W X d ! Œ0; 1/ such that '.x Q 1 ;    ; xd / WD

1 j  j X r 2 2j  ' x ;    ; xd < 1 1 2j r j rj jD0

(3.99)

and kDu f .x1 ;    ; xd /k  '.x1 ;    ; xd /

(3.100)

for all u 2 U.A/ and x1 ;    ; xd 2 X. Then there exists a unique A-linear generalized additive mapping L W X ! Y such that kf .x/  L.x/k 

  'Q x; x; 0;    ; 0 „ ƒ‚ … 2.d2 Cl d2 Cl2 C 1/ 1

(3.101)

d  2 times

for all x 2 X. Proof. Note that f .0/ D 0 and f .x/ D f .x/ for all x 2 X since f is an odd mapping. Let u D 1 2 U.A/. Putting x1 D x2 D x and x3 D    D xd D 0 in (3.100), we have   2    rf x  2f .x/  r

  ' x; x; 0;    ; 0 „ ƒ‚ … d2 Cl d2 Cl2 C 1 1

d  2 times

3.5 Generalized Additive Mappings in Banach Modules

101

for all x 2 X. Letting t WDd2 Cl d2 Cl2 C 1, we get   r  2  1    f .x/  f x   ' x; x; 0;    ; 0 „ ƒ‚ … 2 r 2t d  2 times

for all x 2 X. Hence we have  rn  2n  rnC1  2nC1     n f n x  nC1 f nC1 x  2 r 2 r  n  r  2 2n  rn   2   x  D n f n x  f 2 r 2 r rn  r n  2n 2n  nC1 ' n x; n x; 0;    ; 0 2 t r r „ ƒ‚ …

(3.102)

d  2 times

for all x 2 X and n  1. By (3.102), we have  rm  2m  rn  2n     mf mx  nf nx  2 r 2 r n1   2k 2k X rk  ' x; x; 0;    ; 0 2kC1 t rk rk „ ƒ‚ … kDm

(3.103)

d  2 times

for all x 2 X and m; n  1 with m < n. This shows that the sequence f 2r n f . 2rn x/g n n is a Cauchy sequence for all x 2 X. Since Y is complete, the sequence f 2r n f . 2rn x/g converges for all x 2 X. So we can define a mapping L W X ! Y by n

n

r n  2n  f nx n!1 2n r

L.x/ WD lim

for all x 2 X. Since f .x/ D f .x/ for all x 2 X, we have L.x/ D L.x/ for all x 2 X. Also, we get  2n rn  2n    D1 f n x1 ;    ; n xd  n n!1 2 r r r n  2n 2n   lim n ' n x1 ;    ; n xd n!1 2 r r D0

kD1 L.x1 ;    ; xd /k D lim

for all x1 ;    ; xd 2 X. So L is a generalized additive mapping. Putting m D 0 and letting n ! 1 in (3.103), we get (3.101). Now, let L0 W X ! Y be another additive mapping satisfying (3.101). By Lemma 3.64, L and L0 are additive. So we have

3 Stability of Functional Equations in C -Algebras

102

 n   n  rn   2  0 2 x  L x  L 2n rn rn  n   2n    2n   2n  rn   2     n L n x  f n x  C L0 n x  f n x / 2 r r r r  2rn  2n 2n  nC1 'Q n x; n x; 0;    ; 0 ; 2 t r r „ ƒ‚ …

kL.x/  L0 .x/k D

d  2 times

which tends to zero as n ! 1 for all x 2 X. So we can conclude that L.x/ D L0 .x/ for all x 2 X. This proves the uniqueness of L. By the assumption, for each u 2 U.A/, we get   2n rn    kDu L.x; 0;    ; 0/k D lim n Du f n x; 0;    ; 0  „ ƒ‚ … „ ƒ‚ … n!1 2 r d  1 times

d  1 times

 r ' x; 0;    ; 0 n!1 2n rn „ ƒ‚ …

 lim

n

 2n

d  1 times

D0 for all x 2 X and so

 ux  .d1 Cl d1 Cl1 C 1/rL D .d1 Cl d1 Cl1 C 1/uL.x/ r

for all u 2 U.A/ and x 2 X. Since L is additive,  ux  L.ux/ D rL D uL.x/ r

(3.104)

for all u 2 U.A/ and x 2 X. Now, let a 2 A .a ¤ 0/ and M be an integer greater than 4jaj. Then we have ˇaˇ 1 2 1 ˇ ˇ ˇ ˇ< <1 D : M 4 3 3 By Kadison and Pederson [167], there exist three elements u1 ; u2 ; u3 2 U.A/ such that 3 Ma D u1 C u2 C u3 . So, by (3.104), we have M 1 a  a  M  a  L.ax/ D L 3 x DML 3 x D L 3 x 3 M 3 M 3 M M M D L.u1 x C u2 x C u3 x/ D .L.u1 x/ C L.u2 x/ C L.u3 x// 3 3 a M M  3 L.x/ D .u1 C u2 C u3 /L.x/ D 3 3 M D aL.x/

3.5 Generalized Additive Mappings in Banach Modules

103

for all a 2 A and x 2 X. Hence we have L.ax C by/ D L.ax/ C L.by/ D aL.x/ C bL.y/ for all a; b 2 A with a; b ¤ 0 and x; y 2 X and L.0x/ D 0 D 0L.x/ for all x 2 X. So the unique generalized additive mapping L W X ! Y is an A-linear mapping. This completes the proof.  Corollary 3.67. Let r > 2 and , p > 1 be positive real numbers or let r < 2 and , p < 1 be positive real numbers. Let f W X ! Y be an odd mapping such that kDu f .x1 ;    ; xd /k  

d X

jjxj jjp

jD1

for all u 2 U.A/ and x1 ;    ; xd 2 X. Then there exists a unique A-linear generalized additive mapping L W X ! Y such that kf .x/  L.x/k 

rp1  kxkp .rp1  2p1 /.d2 Cl d2 Cl2 C 1/

for all x 2 X. Proof. Defining '.x1 ;    ; xd / D  the desired result.

Pd jD1

jjxj jjp and applying Theorem 3.66, we get 

Theorem 3.68. Let r ¤ 2. Let f W X ! Y be an odd mapping for which there exists a function ' W X d ! Œ0; 1/ satisfying (3.100) such that '.x Q 1 ;    ; xd / WD

1 X 2j  r j rj  ' x ;    ; xd < 1 1 r j 2j 2j jD1

for all u 2 U.A/ and x1 ;    ; xd 2 X. Then there exists a unique A-linear generalized additive mapping L W X ! Y such that kf .x/  L.x/k 

  'Q x; x; 0;    ; 0 „ ƒ‚ … 2.d2 Cl d2 Cl2 C 1/ 1

d  2 times

for all x 2 X. Proof. Note that f .0/ D 0 and f .x/ D f .x/ for all x 2 X since f is an odd mapping. Let u D 1 2 U.A/. Putting x1 D x2 D x and x3 D    D xd D 0 in (3.100), we have  2     rf x  2f .x/  r

  ' x; x; 0;    ; 0 „ ƒ‚ … d2 Cl d2 Cl2 C 1 1

d  2 times

3 Stability of Functional Equations in C -Algebras

104

for all x 2 X. Letting t WDd2 Cl d2 Cl2 C 1, we get   2  r  1 r r   f .x/  f x   ' x; x; 0;    ; 0 r 2 rt 2 2 „ ƒ‚ … d  2 times

for all x 2 X. The rest of the proof is similar to the proof of Theorem 3.66. This completes the proof.  Corollary 3.69. Let r < 2 and , p > 1 be positive real numbers or let r > 2 and , p < 1 be positive real numbers. Let f W X ! Y be an odd mapping such that kDu f .x1 ;    ; xd /k  

d X

kxj kp

jD1

for all u 2 U.A/ and x1 ;    ; xd 2 X. Then there exists a unique A-linear generalized additive mapping L W X ! Y such that kf .x/  L.x/k 

rp1  kxkp .2p1  rp1 /.d2 Cl d2 Cl2 C 1/

for all x 2 X. Proof. Defining '.x1 ;    ; xd / D 

d X

jjxj jjp

jD1

and applying Theorem 3.68, we get the desired result.



Now, we investigate the Hyers–Ulam stability of linear mappings for the case d D 2. Theorem 3.70. Let r ¤ 2. Let f W X ! Y be an odd mapping for which there exists a function ' W X 2 ! Œ0; 1/ such that '.x; Q y/ WD

1 j  j X r 2 2j  ' x; y < 1 2j r j r j jD0

and   ux C uy      uf .x/  uf .y/  '.x; y/ rf r

(3.105)

3.5 Generalized Additive Mappings in Banach Modules

105

for all u 2 U.A/ and x; y 2 X. Then there exists a unique A-linear mapping L W X ! Y such that kf .x/  L.x/k 

1 '.x; Q x/ 2

for all x 2 X. Proof. Let u D 1 2 U.A/. Putting x D y in (3.105), we have  2     rf x  2f .x/  '.x; x/ r for all x 2 X and so  r  2    1 f .x/  f x   '.x; x/ 2 r 2 for all x 2 X. The rest of the proof is the same as in the proof of Theorem 3.66. This completes the proof.  Corollary 3.71. Let r > 2 and , p > 1 be positive real numbers or let r < 2 and , p < 1 be positive real numbers. Let f W X ! Y be an odd mapping such that   ux C uy      uf .x/  uf .y/  .kxkp C kykp / rf r for all u 2 U.A/ and x; y 2 X. Then there exists a unique A-linear mapping L W X ! Y such that kf .x/  L.x/k 

rp1  jjxjjp  2p1

rp1

for all x 2 X. Proof. Defining '.x; y/ D .jjxjjp C jjyjjp / and applying Theorem 3.70, we get the desired results.  Theorem 3.72. Let r ¤ 2. Let f W X ! Y be an odd mapping for which there exists a function ' W X 2 ! Œ0; 1/ satisfying (3.105) such that '.x; Q y/ WD

1 X 2j  r j r j  ' x; y < 1 r j 2j 2j jD1

for all u 2 U.A/ and all x; y 2 X. Then there exists a unique A-linear mapping L W X ! Y such that

3 Stability of Functional Equations in C -Algebras

106

kf .x/  L.x/k 

1 '.x; Q x/ 2

for all x 2 X. Proof. Let u D 1 2 U.A/. Putting x D y in (3.105), we have  2     rf x  2f .x/  '.x; x/ r for all x 2 X and so    2  r   1 r r  f .x/  f x   ' x; x r 2 r 2 2 for all x 2 X. The rest of the proof is similar to the proof of Theorem 3.66. This completes the proof.  Corollary 3.73. Let r < 2 and , p > 1 be positive real numbers or let r > 2 and , p < 1 be positive real numbers. Let f W X ! Y be an odd mapping such that   ux C uy      uf .x/  uf .y/  .kxkp C kykp / rf r for all u 2 U.A/ and x; y 2 X. Then there exists a unique A-linear mapping L W X ! Y such that kf .x/  L.x/k 

rp1  kxkp  rp1

2p1

for all x 2 X. Proof. Defining '.x; y/ D .kxkp C kykp / and applying Theorem 3.72, we get the desired results. 

3.5.3 Isomorphisms in Unital C -Algebras Assume that A is a unital C -algebra with the norm jj  jj and the unit e and B is a unital C -algebra with the norm k  k. Let U.A/ be the set of unitary elements in A. Now, we investigate C -algebra isomorphisms in unital C -algebras. Theorem 3.74. Let r ¤ 2. Let h W A ! B be an odd bijective mapping satisfying n n h. 2rn uy/ D h. 2rn u/h.y/ for all u 2 U.A/, y 2 A and n  0 for which there exists a function ' W Ad ! Œ0; 1/ such that

3.5 Generalized Additive Mappings in Banach Modules

107

1 X 2j  r j  2j ' j x1 ;    ; j xd < 1; j 2 r r jD0

(3.106)

kD h.x1 ;    ; xd /k  '.x1 ;    ; xd /;  2n   2n 2n 2n    h. n u /  h. n u/   ' n u;    ; n u r r ƒ‚ r … „r

(3.107)

d times

for all 2 S1 WD f 2 C W jj D 1g, u 2 U.A/, n  0 and x1 ;    ; xd 2 A. Assume n n that limn!1 2r n h. 2rn e/ is invertible. Then the odd bijective mapping h W A ! B is a C -algebra isomorphism. Proof. Consider the C -algebras A and B as left Banach modules over the unital C -algebra C. By Theorem 3.66, there exists a unique C-linear generalized additive mapping H W A ! B such that kh.x/  H.x/k 

  'Q x; x; 0;    ; 0 „ ƒ‚ … 2.d2 Cl d2 Cl2 C 1/ 1

(3.108)

d  2 times

for all x 2 A. The generalized additive mapping H W A ! B is given by r n  2n  h nx n!1 2n r

H.x/ D lim

for all x 2 A. By (3.106) and (3.107), we get r n  2n   rn  2n  h n u D lim n h n u n n!1 2 n!1 2 r r  n  n  r 2 D lim n h n u D H.u/ n!1 2 r

H.u / D lim

for all u 2 U.A/. Since H is C-linear and P each x 2 A is a finite linear combination of unitary elements (see [168]), i.e., x D m jD1 j uj for all j 2 C and uj 2 U.A/, we have 

H.x / D H

m X

j uj



D

jD1

D

m X

m X

j H.uj /

D

jD1

j H.uj /

jD1



DH

m X

j H.uj /

jD1 m X

j u j



D H.x/

jD1

for all x 2 A. Since h. 2rn uy/ D h. 2rn u/h.y/ for all u 2 U.A/, y 2 A and n  0, we have n

n

3 Stability of Functional Equations in C -Algebras

108

r n  2n  r n  2n  h uy D lim h n u h.y/ n!1 2n n!1 2n rn r D H.u/h.y/

H.uy/ D lim

(3.109)

for all u 2 U.A/ and y 2 A. By the additivity of H and (3.109), we have  2n    2n   2n  2n H.uy/ D H uy D H u y D H.u/h y rn rn rn rn for all u 2 U.A/ and y 2 A. Hence we have  2n  r n  2n  rn H.u/h y D H.u/ h y 2n rn 2n r n

H.uy/ D

(3.110)

for all u 2 U.A/ and y 2 A. Taking n ! 1 in (3.110), we obtain H.uy/ D H.u/H.y/

(3.111)

for all u 2 U.A/ and y 2 A. Since H is C-linear and each x 2 A is a finite linear P combination of unitary elements, i.e., x D m jD1 j uj for all j 2 C and uj 2 U.A/, it follows from (3.111) that H.xy/ D H

m X

m m  X X j u j y D j H.uj y/ D j H.uj /H.y/

jD1

D H.

m X

jD1

jD1

j uj /H.y/ D H.x/H.y/

jD1

for all x; y 2 A. By (3.109) and (3.111), we have H.e/H.y/ D H.ey/ D H.e/h.y/ for all y 2 A. Since limn!1

n rn h. 2rn e/ 2n

D H.e/ is invertible,

H.y/ D h.y/ for all y 2 A. Therefore, the odd bijective mapping h W A ! B is a C -algebra isomorphism. This completes the proof.  Corollary 3.75. Let r > 2 and , p > 1 be positive real numbers or let r < 2 and , p < 1 be positive real numbers. Let h W A ! B be an odd bijective mapping n n satisfying h. 2rn uy/ D h. 2rn u/h.y/ for all u 2 U.A/, y 2 A and n  0 such that

3.5 Generalized Additive Mappings in Banach Modules

kD h.x1 ;    ; xd /k  

109

d X

jjxj jjp

jD1

and   2n   2n   2pn   h n u  h n u   d pn  r r r for all 2 S1 , u 2 U.A/, n  0 and x1 ;    ; xd 2 A. Assume that limn!1 2r n h. 2rn e/ is invertible. Then the odd bijective mapping h W A ! B is a C -algebra isomorphism. n

n

Proof. Defining '.x1 ;    ; xd / D 

d X

jjxj jjp

jD1

and applying Theorem 3.74, we get the desired results.



Theorem 3.76. Let r ¤ 2. Let h W A ! B be an odd bijective mapping satisfying n n h. 2rn uy/ D h. 2rn u/h.y/ for all u 2 U.A/, y 2 A and n  0 for which there exists a n n function ' W Ad ! Œ0; 1/ satisfying (3.106), (3.107). Assume that limn!1 2r n h. 2rn e/ is invertible such that kD h.x1 ;    ; xd /k  '.x1 ;    ; xd /

(3.112)

for all x1 ;    ; xd 2 A and D 1; i. If h.tx/ is continuous in t 2 R for each fixed x 2 A, then the odd bijective mapping h W A ! B is a C -algebra isomorphism. Proof. Put D 1 in (3.112). By the same reasoning as in the proof of Theorem 3.74, there exists a unique generalized additive mapping H W A ! B satisfying (3.108). By (3.112), the generalized additive mapping H W A ! B is R-linear. Put D i in (3.112). By the same method as in the proof of Theorem 3.66, one can obtain that r n  2n  i r n  2n  h ix D lim h n x D iH.x/ n!1 2n n!1 2n rn r

H.ix/ D lim

for all x 2 A. For each element  2 C, let  D s C it, where s; t 2 R. Then we have H.x/ D H.sx C itx/ D sH.x/ C tH.ix/ D sH.x/ C itH.x/ D .s C it/H.x/ D H.x/ for all  2 C and x 2 A. Thus we have

110

3 Stability of Functional Equations in C -Algebras

H. x C y/ D H. x/ C H. y/ D H.x/ C H.y/ for all ; 2 C and x; y 2 A. Hence the generalized additive mapping H W A ! B is C-linear. The rest of the proof is the same as in the proof of Theorem 3.66. This completes the proof. 

3.6 Jordan -Derivations and Quadratic Jordan -Derivations Jordan -derivations were introduced in [307, 308] for the first time and the structure of such derivations has been investigated in [52]. The reason for introducing these mappings was the fact that the problem of representing quadratic forms by sesquilinear ones is closely connected with the structure of Jordan -derivations. In [13], An et al. investigated Jordan -derivations on C -algebras and Jordan -derivations on JC -algebras associated with a special functional inequality. In this section, we consider the Hyers-Ulam stability of Jordan -derivations and quadratic Jordan -derivations on real C -algebras and real JC -algebras. We also prove the superstability of Jordan -derivations and quadratic Jordan -derivations on real C -algebras and real JC -algebras under some conditions [50].

3.6.1 Stability of Jordan -Derivations Here we prove the Hyers–Ulam stability of Jordan -derivations on real C -algebras and real JC -algebras. Definition 3.77. Let A be a real C -algebra. An R-linear mapping D W A ! A is called a Jordan -derivation if D.a2 / D a D.a/ C D.a/a for all a 2 A. The mapping Dx W A ! A; a 7! a x  xa , where x is a fixed element in A, is a Jordan -derivation. A real C -algebra A endowed with the Jordan product a ı b WD abCba on A is called a real JC -algebra (see [13, 225]). 2 Definition 3.78. Let A be a real JC -algebra. An R-linear mapping ı W A ! A is called a Jordan -derivation if ı.a2 / D a ı D.a/ C D.a/ ı a for all a 2 A.

3.6 Jordan -Derivations and Quadratic Jordan -Derivations

111

Theorem 3.79. Let A be a real C -algebra. Suppose that f W A ! A is a mapping with f .0/ D 0 for which there exists a function ' W A3 ! Œ0; 1/ such that '.a; Q b; c/ WD

1 X 1 '.2n a2n b; 2n c/ < 1 nC1 2 nD0

(3.113)

and kf .a C b C c2 /  f .a/  f .b/  f .c/c  c f .c/k  '.a; b; c/

(3.114)

for all  2 R and a; b; c 2 A. Then there exists a unique Jordan -derivation ı on A satisfying kf .a/  ı.a/k  '.a; Q a; 0/

(3.115)

for all a 2 A. Proof. Setting a D b; c D 0 and  D 1 in (3.114), we have kf .2a/  2f .a/k  '.a; a; 0/ for all a 2 A. One can use induction to show that  n1  n  f .2 a/ f .2m a/  X 1   '.2k a; 2k a; 0/  2n  2m  D kC1 2 kDm

(3.116)

for all n > nm  0 and a 2 A. It follows from (3.113) and (3.116) that the sequence f f .22na/ g is a Cauchy sequence. Due to the completeness of A, this sequence is convergent. Define f .2n a/ n!1 2n

d.a/ WD lim

(3.117)

for all a 2 A. Then we have ı

1  1 f .2nk a/ 1 a D lim k D k d.a/ k nk n!1 2 2 2 2

for each k 2 N. Putting c D 0 and replacing a and b by 2n a and 2n b, respectively, in (3.114), we get 1  1 1 1    n f .2n .a C b//   n f .2n a/  n f .2n b/  n '.2n a; 2n b; 0/: 2 2 2 2

3 Stability of Functional Equations in C -Algebras

112

Taking n ! 1, we obtain ı.a C b/ D ı.a/ C ı.b/ for all a; b 2 A and  2 R. So ı is R-linear. Putting a D b D 0 and substituting c by 2n c in (3.114), we get  1  1 1    2n f .22n c2 /  2n f .2n c/.2n c /  2n .2n c /f .2n c/ 2 2 2 1  2n '.0; 0; 2n c/ 2 1  n '.0; 0; 2n c/: 2 Taking n ! 1, we obtain ı.c2 / D ı.c/c C c ı.c/ for all c 2 A. Moreover, it follows from (3.116) with m D 0 and (3.117) that kı.a/  f .a/k  '.a; Q a; 0/ for all a 2 A. For the uniqueness of ı, let ıQ W A ! B be another Jordan -derivation satisfying (3.115). Then we have 1 Q Q n a/k kı.a/  ı.a/k D n kı.2n a/  ı.2 2 1 Q n a/k/  n .kı.2n a/  f .2n a/k C kf .2n a/  ı.2 2 1 X 1 '.2nCj a; 2nCj a; 0/ 2 nCj 2 jD1 D2

1 X 1 '.2j a; 2j a; 0/; j 4 jDn

which tends to zero as n ! 1 for all a 2 A. So ı is unique. Therefore, ı is a Jordan -derivation on A. This completes the proof.  Remark 3.80. Let A be a real C -algebra. Suppose that f W A ! A is a mapping with f .0/ D 0 for which there exists a function ' W A3 ! Œ0; 1/ satisfying (3.114) and

3.6 Jordan -Derivations and Quadratic Jordan -Derivations

'.a; Q b; c/ WD

1 X

2n1 '

nD1

113

a b c <1 ; ; 2n 2n 2n

for all a; b; c 2 A. Then there exists a unique Jordan -derivation ı on A satisfying kf .a/  ı.a/k  '.a; Q a; 0/ for all a 2 A. Corollary 3.81. Let A be a real C -algebra and "; p be positive real numbers with p ¤ 1. Suppose that f W A ! A is a mapping satisfying kf .a C b C c2 /  f .a/  f .b/  cf .c/  f .c/c k  ".kakp C kbkp C kckp / for all  2 R and a; b; c 2 A. Then there exists a unique Jordan -derivation ı on A satisfying kf .a/  ı.a/k 

2" kakp j2  2p j

(3.118)

for all a 2 A. Proof. Putting '.a; b; c/ D ".kakp C kbkp C kckp / in Theorem 3.79, we get the desired result.



Now, we consider the Hyers-Ulam stability of Jordan -derivations on a real JC -algebra A. Since the proofs are similar to the above results, here we omit them. Remark 3.82. Let A be a real JC -algebra. Suppose that f W A ! A is a mapping with f .0/ D 0 for which there exists a function ' W A3 ! Œ0; 1/ such that '.a; Q b; c/ WD

1 X 1 '.2n a; 2n b; 2n c/ < 1 nC1 2 nD0

and kf .a C b C c2 /  f .a/  f .b/  f .c/ ı c  c ı f .c/k  '.a; b; c/

(3.119)

3 Stability of Functional Equations in C -Algebras

114

for all  2 R and a; b; c 2 A. Then there exists a unique Jordan -derivation ı on A satisfying kf .a/  ı.a/k  '.a; Q a; 0/ for all a 2 A. Remark 3.83. Let A be a real JC -algebra. Suppose that f W A ! A is a mapping with f .0/ D 0 for which there exists a function ' W A3 ! Œ0; 1/ satisfying (3.119) and '.a; Q b; c/ WD

1 X

2n1 '

nD1

a b c <1 ; ; 2n 2n 2n

for all a; b; c 2 A. Then there exists a unique Jordan -derivation ı on A satisfying kf .a/  ı.a/k  '.a; Q a; 0/ for all a 2 A. Corollary 3.84. Let A be a real JC -algebra and ", p be positive real numbers with p ¤ 1. Suppose that f W A ! A is a mapping satisfying kf .a C b C c2 /  f .a/  f .b/  c ı f .c/  f .c/ ı c k  ".kakp C kbkp C kckp / for all  2 R and a; b; c 2 A. Then there exists a unique Jordan -derivation ı on A satisfying kf .a/  ı.a/k 

2" kakp j2  2p j

for all a 2 A. Proof. The result follows from Remarks 3.82 and 3.83 by putting '.a; b; c/ D ".kakp C kbkp C kckp /:



3.6.2 Stability of Quadratic Jordan -Derivations Now, we prove the Hyers-Ulam stability of quadratic Jordan -derivations on real C -algebras and real JC -algebras.

3.6 Jordan -Derivations and Quadratic Jordan -Derivations

115

Definition 3.85. Let A be a real C -algebra. A mapping D W A ! A is called a quadratic Jordan -derivation if D is a quadratic R-homogeneous mapping, that is, D is quadratic, D.a/ D 2 D.a/ for all a 2 A and  2 R and D.a2 / D .a /2 D.a/ C D.a/.a /2 for all a 2 A. The mapping Dx W A ! A, a 7! .a /2 x  x.a /2 , where x is a fixed element in A, is a quadratic Jordan -derivation. Definition 3.86. Let A be a real JC -algebra. A mapping ı W A ! A is called a quadratic Jordan -derivation if ı is a quadratic R-homogeneous mapping and ı.a2 / D .a /2 ı D.a/ C D.a/ ı .a /2 for all a 2 A. Theorem 3.87. Let A be a real C -algebra. Suppose that f W A ! A is a mapping with f .0/ D 0 for which there exists a function ' W A2 ! Œ0; 1/ such that '.a; Q b/ WD

1 X 1 '.2k a; 2k b/ < 1; k 4 kD0

kf .a C b/ C f .a  b/  22 f .a/  22 f .b/k  '.a; b/;

(3.120)

kf .a2 /  f .a/.a /2  .a /2 f .a/k  '.a; a/

(3.121)

for all a; b 2 A and  2 R. Then there exists a unique quadratic Jordan -derivation ı on A satisfying kf .a/  ı.a/k 

1 '.a; Q a/ 4

for all a 2 A. Proof. Putting a D b and  D 1 in (3.120), we have kf .2a/  4f .a/k  '.a; a/ for all a 2 A. One can use induction to show that

(3.122)

3 Stability of Functional Equations in C -Algebras

116

n1  f .2n a/ f .2m a/  1 X '.2k a; 2k a/     n   4 4m 4 kDm 4k

(3.123)

f .2n a/ for all n > m  0 and a 2 A. It follows from (3.123) that the sequence 4n is a Cauchy sequence. Since A is complete, this sequence is convergent. Define



f .2n a/ n!1 4n

ı.a/ WD lim

for all a 2 A. Since f .0/ D 0, we have ı.0/ D 0. Replacing a and b by 2n a and 2n b, respectively, in (3.120), we get  f .2n .a C b// n n  f .2n .a  b//  2 f .2 a/ 2 f .2 b/  C  2  2   4n 4n 4n 4n '.2n a; 2n b/  : 4n Taking n ! 1, we obtain ı.a C b/ C ı.a  b/ D 22 ı.a/ C 22 ı.b/

(3.124)

for all a; b 2 A and  2 R. Putting  D 1 in (3.124), we obtain that ı is a quadratic mapping. It is easy to check that the quadratic mapping ı satisfying (3.122) is unique (see the proof of Theorem 3.79). Setting b WD a in (3.124), we get ı.2a/ D 42 ı.a/ for all a 2 A and  2 R. Hence ı.a/ D 2 ı.a/ for all a 2 A and  2 R. Replacing a by 2n a in (3.121), we get  f .2n a  2n a/ 22n .a /2 f .2n a/ f .2n a/22n .a /2        2n 2n 2n 4 4 4  f .22n a2 / 22n .a /2 f .2n a/ f .2n a/ 22n .a /2      D  42n 22n 4n 4n 22n '.2n a; 2n a/  42n '.2n a; 2n a/  4n for all a 2 A. Thus we have '.2n a; 2n a/ D 0: n!1 4n

kı.a2 /  .a /2 ı.a/  ı.a/.a/2 k  lim

Therefore, ı is a quadratic Jordan -derivation on A. This completes the proof. 

3.6 Jordan -Derivations and Quadratic Jordan -Derivations

117

Remark 3.88. Let A be a real C -algebra. Suppose that f W A ! A is a mapping with f .0/ D 0 for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.120), (3.121) and '.a; Q b/ WD

1 X

4k '

a 2

kD1

; k

b <1 2k

for all a; b 2 A. Then there exists a unique quadratic Jordan -derivation ı on A satisfying kf .a/  ı.a/k 

1 '.a; Q a/ 4

for all a 2 A. Corollary 3.89. Let A be a real C -algebra and ", p be positive real numbers with p ¤ 2. Suppose that f W A ! A is a mapping such that kf .a C b/ C f .a  b/  22 f .a/  22 f .b/k  ".kakp C kbkp / and kf .a2 /  a2 f .a/  f .a/.a /2 k  2"kakp for all a; b 2 A and  2 R. Then there exists a unique quadratic Jordan -derivation ı on A satisfying kf .a/  ı.a/k 

2" kakp j4  2p j

(3.125)

for all a 2 A. Proof. Putting '.a; b/ D ".kakp C kbkp / in Theorem 3.87, we get the desired result.



Here we assume that A is a real JC -algebra. Then we can get the Hyers-Ulam stability of quadratic Jordan -derivations on A. Remark 3.90. Suppose that f W A ! A is a mapping with f .0/ D 0 for which there exists a function ' W A2 ! Œ0; 1/ such that '.a; Q b/ WD

1 X 1 '.2k a; 2k b/ < 1; k 4 kD0

3 Stability of Functional Equations in C -Algebras

118

kf .a C b/ C f .a  b/  22 f .a/  22 f .b/k  '.a; b/;

(3.126)

kf .a2 /  .a /2 ı f .a/  f .a/ ı .a /2 k  '.a; a/

(3.127)

for all a; b 2 A and  2 R. Then there exists a unique quadratic Jordan -derivation ı on A satisfying kf .a/  ı.a/k 

1 '.a; Q a/ 4

for all a 2 A. Remark 3.91. Suppose that f W A ! A is a mapping with f .0/ D 0 for which there exists a function ' W A2 ! Œ0; 1/ satisfying (3.126), (3.127) and '.a; Q b/ WD

1 X

4k '

a 2

kD1

; k

b <1 2k

for all a; b 2 A. Then there exists a unique quadratic Jordan -derivation ı on A satisfying kf .a/  ı.a/k 

1 '.a; Q a/ 4

for all a 2 A. We can obtain the following Remark by letting '.a; b/ D ".kakp C kbkp / in Remarks 3.90 and 3.91. Remark 3.92. Let " and p be positive real numbers with p ¤ 2. Suppose that f W A ! A is a mapping such that kf .a C b/ C f .a  b/  22 f .a/  22 f .b/k  ".kakp C kbkp / and kf .a2 /  .a /2 ı f .a/  f .a/ ı .a /2 k  2"kakp for all a; b 2 A and  2 R. Then there exists a unique quadratic Jordan -derivation ı on A satisfying kf .a/  ı.a/k  for all a 2 A.

2" kakp j4  2p j

3.6 Jordan -Derivations and Quadratic Jordan -Derivations

119

3.6.3 Stability of Jordan -Derivations: The Fixed Point Method Now, we assume that A is a real C -algebra and prove the stability of Jordan derivations by the fixed point method. Theorem 3.93. Let f W A ! A be a mapping with f .0/ D 0 and ' W A3 ! Œ0; 1/ be a function such that kf .a C b C c2 /  f .a/  f .b/  f .c/c  c f .c/k  '.a; b; c/

(3.128)

for all  2 R and a; b; c 2 A: If there exists a constant k 2 .0; 1/ such that '.2a; 2b; 2c/  2k'.a; b; c/

(3.129)

for all a; b; c 2 A, then there exists a unique Jordan -derivation ı W A ! A satisfying kf .a/  ı.a/k 

1 '.a; a; 0/ 2.1  k/

(3.130)

for all a 2 A. Proof. It follows from (3.129) that '.2j a; 2j b; 2j c/ D0 j!1 2j lim

for all a; b; c 2 A: Putting  D 1; a D b and c D 0 in (3.128), we have kf .2a/  2f .a/k  '.a; a; 0/ for all a 2 A and so  1  1   f .a/  f .2a/  '.a; a; 0/ 2 2

(3.131)

for all a 2 A: We consider the set ˝ WD fh W A ! A W h.0/ D 0g and introduce the generalized metric on ˝ defined as follows: d.h1 ; h2 / WD inffC 2 .0; 1/ W kh1 .a/  h2 .a/k  C'.a; a; 0/; 8a 2 Ag if there exists such a constant C and, otherwise, d.h1 ; h2 / D 1. We know that d is a generalized metric on ˝ and the metric space .˝; d/ is complete. We now define the linear mapping T W ˝ ! ˝ by

3 Stability of Functional Equations in C -Algebras

120

Th.a/ D

1 h.2a/ 2

(3.132)

for all a 2 A. For any h1 ; h2 2 ˝, let C 2 RC be an arbitrary constant with d.h1 ; h2 /  C, that is, kh1 .a/  h2 .a/k  C'.a; a; 0/

(3.133)

for all a 2 A. Substituting a by 2a in the inequality (3.133) and using the equalities (3.129) and (3.132), we have 1 kh1 .2a/  h2 .2a/k 2 1  C'.2a; 2a; 0/ 2  Ck'.2a; 2a; 0/

kTh1 .a/  Th2 .a/k D

for all a 2 A and so d.Th1 ; Th2 /  Ck. Therefore, we conclude that d.Th1 ; Th2 /  kd.h1 ; h2 / for all h1 ; h2 2 ˝. It follows from (3.131) that d.Tf ; f / 

1 : 2

(3.134)

By Theorem 1.3, the sequence fT n f g converges to a unique fixed point ı W A ! A in the set ˝1 D fh 2 ˝ W d.f ; h/ < 1g, that is, f .2n a/ D ı.a/ n!1 2n lim

for all a 2 A. By Theorem 1.3 and (3.134), we have d.f ; ı/ 

1 d.Tf ; f /  : 1k 2.1  k/

The above inequalities show that (3.130) holds for all a 2 A. Thus, by the same proof of Theorem 3.79, we can deduce that ı is R-linear by letting c D 0 and replacing a and b by 2n a and 2n b, respectively, in (3.128). Also, we have ı.c2 / D ı.c/c C c ı.c/ for all c 2 A. This completes the proof.



The following shows that we can obtain a more accurate approximation of (3.118) in the case p < 1.

3.6 Jordan -Derivations and Quadratic Jordan -Derivations

121

Corollary 3.94. Let p,  be non-negative real numbers with p < 1 and f W A ! A be a mapping with f .0/ D 0 such that kf .a C b C c2 /  f .a/  f .b/  f .c/c  c f .c/k  .kakp C kbkp C kckp / for all  2 R and a; b; c 2 A: Then there exists a unique Jordan -derivation ı W A ! A satisfying kf .a/  ı.a/k 

 kakp 2  2p

for all a 2 A. Proof. The result follows from Theorem 3.93 by taking '.a; b; c/ D .kakp C kbkp C kckp /:  In the following, we show that, under some conditions, the superstability for Jordan -derivations on real C -algebras. Corollary 3.95. Let p, q, r,  be non-negative real numbers such that p C q C r 2 .1; 1/. Suppose that a mapping f W A ! A satisfies the following: kf .a C b C c2 /  f .a/  f .b/  f .c/c  c f .c/k  .kakp kbkq kckr /

(3.135)

for all a; b; c 2 A. Then f is a Jordan -derivation on A. Proof. Letting a D b D c D 0 in (3.135), we have f .0/ D 0. Once more, if we put  D 1; c D 0 and a D b in (3.135), then we get f .2a/ D 2f .a/n for all a 2 A. By induction, it is easy to see that f .2n a/ D 2n f .a/ and so f .a/ D f .22na/ for all a 2 A and n 2 N. Now, it follows from Theorem 3.93 that f is a Jordan -derivation.  Note that, in Corollary 3.95, if p C q C r 2 .0; 1/ and p > 0 such that the inequality (3.135) holds, then, by applying '.x; y/ D .kakp kbkq kckr / in Theorem 3.93, f is again a Jordan -derivation. Theorem 3.96. Let f W A ! A be a mapping with f .0/ D 0 and ' W A2 ! Œ0; 1/ be a function such that kf .a C b/ C f .a  b/  22 f .a/  22 f .b/k  '.a; b/

(3.136)

3 Stability of Functional Equations in C -Algebras

122

and kf .a2 /  f .a/.a /2  .a /2 f .a/k  '.a; a/ for all a; b 2 A and  2 R. If there exists a constant k 2 .0; 1/ such that '.2a; 2b/  4k'.a; b/

(3.137)

for all a; b 2 A, then there exists a unique quadratic Jordan -derivation ı W A ! A satisfying kf .a/  ı.a/k 

1 '.a; a/ 4.1  k/

(3.138)

for all a 2 A. Proof. By the same proof of Theorem 3.93, we consider the set ˝ D fg W A ! A W g.0/ D 0g and define the mapping d on ˝  ˝ as follows: d.g; h/ WD inffc 2 .0; 1/ W kg.a/  h.a/k  c.a; a/; 8a 2 Ag if there exists such a constant c and, otherwise, d.g; h/ D 1. One can easily show that .˝; d/ is complete. Now, we consider the mapping T W ˝ ! ˝ defined by Tg.a/ D

1 g.2a/ 4

for all a 2 A. For any g; h 2 ˝ with d.g; h/ < c, by the definition of d and T, we get 1  1 1    g.2a/  h.2a/  c'.2a; 2a/ 4 4 4 for all a 2 A. Using (3.137), we have 1  1    g.2a/  h.2a/  ck'.a; a/ 4 4 for all a 2 A. The above inequality shows that d.Tg; Th/  kd.g; h/ for all g; h 2 ˝. Hence T is a strictly contractive mapping on ˝ with the Lipschitz constant k. Now, we prove that d.Tf ; f / < 1. Putting a D b and  D 1 in (3.136), we obtain kf .2a/  4f .a/k  '.a; a/ for all a 2 A and so  1 1    f .2a/  f .a/  '.a; a/ 4 4

(3.139)

3.6 Jordan -Derivations and Quadratic Jordan -Derivations

123

for all a 2 A. We deduce from (3.139) that d.Tf ; f /  14 . It follows from Theorem 1.3 that d.T n g; T nC1 g/ < 1 for all n  0 and so, in Theorem 1.3, we have n0 D 0. Thus Theorem 1.3 hold on the whole ˝. Hence there exists a unique mapping ı W A ! A such that ı is a fixed point of T and T n f ! ı as n ! 1. Thus we have f .2n a/ D ı.a/ n!1 4n lim

for all a 2 A and so d.f ; ı/ 

1 1 d.Tf ; f /  : 1k 4.1  k/

The above equalities show that (3.138) is true for all a 2 A. Now, it follows from (3.137) that '.2n a; 2n b/ D 0: n!1 4n lim



The rest of the proof is easy.

In the following, we find a more accurate approximation relative to Corollary 3.89 with the same conditions on the mapping f when p < 2. In fact, we obtain a refinement of the inequality (3.125). Corollary 3.97. Let  and p be positive real numbers with p < 2. Suppose that f W A ! A is a mapping such that kf .a C b/ C f .a  b/  22 f .a/  22 f .b/k  .kakp C kbkp / and kf .a2 /  .a /2 f .a/  f .a/.a /2 k  2kakp for all a; b 2 A and  2 R. Then there exists a unique quadratic Jordan -derivation ı on A satisfying kf .a/  ı.a/k 

 kakp 4  2p

for all a 2 A. Proof. If we put '.a; b/ D .kakp C kbkp / in Theorem 3.96, then we obtain the desired result.



3 Stability of Functional Equations in C -Algebras

124

The following shows that, under which conditions, a quadratic Jordan derivation on a real C -algebra is superstable. Corollary 3.98. Let , p, q be positive real numbers with p C q ¤ 2. Suppose that f W A ! A is a mapping such that kf .a C b/ C f .a  b/  22 f .a/  22 f .b/k  .kakp kbkq /

(3.140)

and kf .a2 /  .a /2 f .a/  f .a/.a /2 k  kakpCq

(3.141)

for all a; b 2 A and  2 R. Then f is a quadratic Jordan -derivation on A. Proof. Putting a D b D 0 in (3.140), we get f .0/ D 0. Now, if we put a D b and  D 1 in (3.140), then we have f .2a/ D 4f .a/ for all a 2 A. It is easy to see, by n induction, that f .2n a/ D 4n f .a/ and so f .a/ D f .24na/ for all a 2 A and n 2 N. It follows from Theorem 3.96 that f is a quadratic homogeneous mapping. Letting '.a; b/ D .kakp kbkq / in Theorem 3.96, the we can obtain the desired result. 

3.7 .˛; ˇ; /-Derivations on Lie C -Algebras: The Direct Method Let A be a Lie C -algebra. A C-linear mapping D W A ! A is a called an .˛; ˇ;  /derivation of A if there exist ˛; ˇ;  2 C such that ˛D.Œx; y/ D ˇŒD.x/; y C  Œx; D.y/ for all x; y 2 A: In this section, we review some works of Eshaghi Gordji et al. [98, 100] on the Hyers-Ulam stability of .˛; ˇ;  /-derivations on Lie C -algebras associated with the following functional equation: f

x  x   x  3x   3x C 3x  x  2 1 1 3 1 3 2 Cf Cf D f .x1 /: 3 3 3

In fact, we investigate the superstability and the Hyers–Ulam stability of .˛; ˇ;  /-derivations on Lie C -algebras. Assume that A is a Lie C -algebra with the norm k  k: Now, we remember a lemma to be used in the last sections.

3.7 .˛; ˇ; /-Derivations on Lie C -Algebras: The Direct Method

125

Lemma 3.99. Let X, Y be linear spaces and f W X ! Y be an additive mapping such that f . x/ D f .x/ for all x 2 X and 2 T1 WD f 2 C W jj D 1g: Then the mapping f W X ! Y is C-linear. Lemma 3.100. Let f W A ! A be a mapping such that  x  x   x  3x   3x C 3x  x   2 1 1 3 1 3 2  Cf Cf  f A 3 3 3  kf .x1 /k

(3.142)

for all x1 ; x2 ; x3 2 A: Then the mapping f W A ! A is additive. Proof. Letting x1 D x2 D x3 D 0 in (3.142), we get k3f .0/kA  kf .0/kA D 0 and so f .0/ D 0. Letting x1 D x2 D 0 in (3.142), we get kf .x3 / C f .x3 /kA  kf .0/kA D 0 for all x3 2 A. Hence f .x3 / D f .x3 / for all x3 2 A. Letting x1 D 0 and x2 D 6x3 in (3.142), we get kf .2x3 /  2f .x3 /kA  kf .0/kA D 0 for all x3 2 A and so f .2x3 / D 2f .x3 / for all x3 2 A. Letting x1 D 0 and x2 D 9x3 in (3.142), we get kf .3x3 /  3f .x3 /kA  kf .0/kA D 0 for all x3 2 A and so f .3x3 / D 3f .x3 / for all x3 2 A. Letting x1 D 0 in (3.142), we get  x   x2  2   C f .x3 / C f x3    kf .0/kA D 0 f 3 3 A for all x2 ; x3 2 A. Then we have x   x2  2 C f .x3 / C f x3  D0 f 3 3 for all x2 ; x3 2 A. Letting t1 D x3 

x2 3

and t2 D

x2 3

(3.143)

in (3.143), we get

f .t2 /  f .t1 C t2 / C f .t1 / D 0 for all t1 ; t2 2 A and so f is additive. This completes the proof.

 

Now, we prove the superstability of .˛; ˇ;  /-derivations in Lie C -algebra A.

126

3 Stability of Functional Equations in C -Algebras

Theorem 3.101. Let p ¤ 1,  be nonnegative real numbers and f W A ! A be a mapping such that, for some ˛; ˇ;  2 C,   x  x   x  3 x   3x C 3x  x  2 1 1 3 1 3 2   Cf C f  f 3 3 3  kf .x1 /k

(3.144)

and k˛f .Œx1 ; x2 /  ˇŒf .x1 /; x2    Œx1 ; f .x2 /k  .kx1 k2p C kx2 k2p /

(3.145)

for all 2 T1 and x1 ; x2 ; x3 2 A: Then the mapping f W A ! A is an .˛; ˇ;  /derivation. Proof. Assume p > 1. Let D 1 in (3.144). By Lemma 3.100, the mapping f W A ! A is additive. Letting x1 D x2 D 0 in (3.144), we get kf . x3 / C f .x3 /kA  kf .0/kA D 0 for all x3 2 A and 2 T1 . Then we have f . x3 / C f .x3 / D f . x3 / C f .x3 / D 0 for all x3 2 A and 2 T1 . Hence f . x3 / D f .x3 / for all x3 2 A and 2 T1 . By Lemma 3.99, the mapping f W A ! A is C-linear. Since f is additive, it follows from (3.145) that k˛f .Œx1 ; x2 /  ˇŒf .x1 /; x2    Œx1 ; f .x2 /kA   Œx ; x   h x hx x2 i x2 i 1 2 1 1    ˇ f .   D lim 4n ˛f /; ; f . /  n!1 A 4n 2n 2n 2n 2n 4n  2p 2p  lim np .kx1 kA C kx2 kA / n!1 4 D0 for all x1 ; x2 2 A. Thus for some ˛; ˇ;  2 C ˛f .Œx1 ; x2 / D ˇŒf .x1 /; x2  C  Œx1 ; f .x2 / for all x1 ; x2 2 A. Hence the mapping f W A ! A is an .˛; ˇ;  /-derivation. Similarly, one obtains the results for the case p < 1. This completes the proof.  Now, we prove the Hyers-Ulam stability of .˛; ˇ;  /-derivations on Lie C -algebras.

3.7 .˛; ˇ; /-Derivations on Lie C -Algebras: The Direct Method

127

Theorem 3.102. Let p > 1,  be nonnegative real numbers and f W A ! A be a mapping with f .0/ D 0 such that, for some ˛; ˇ;  2 C,   x  x    x  3 x   3x C 3x  x  2 1 1 3 1 3 2   Cf C f  f .x1 / f 3 3 3  .kx1 kp C kx2 kp C kx3 kp / (3.146) and k˛f .Œx1 ; x2 /  ˇŒf .x1 /; x2    Œx1 ; f .x2 /k  .kx1 k2p C kx2 k2p /

(3.147)

for all 2 T1 and x1 ; x2 ; x3 2 A: Then there exists a unique .˛; ˇ;  /-derivation D W A ! A such that kD.x1 /  f .x1 /k 

3p .1 C 2p /kx1 kp 3p  3

(3.148)

for all x1 2 A: Proof. Letting D 1; x2 D 2x1 and x3 D 0 in (3.146), we get   x  1    f .x1 /  .1 C 2p /kx1 kp 3f 3

(3.149)

for all x1 2 A: By induction, we have n1  x   X 1  n  p p 3i.1p/ 3 f n  f .x1 /  .1 C 2 /kx1 k 3 iD0

for all x1 2 A: Hence   x   x  1 1   nCm f nCm  3m f m  3 3 3  .1 C 2p /kx1 kp

n1 X

3.iCm/.1p/

iD0

 .1 C 2p /kx1 kp

nCm1 X

3i.1p/

(3.150)

iDm

for all m; n  1 and x1 2 A: This implies that the sequence f3n f . 3x1n /g is a Cauchy sequence for all x1 2 A: Since A is complete, the sequence f3n f . 3x1n /g converges. Thus one can define the mapping D W A ! A by

3 Stability of Functional Equations in C -Algebras

128

D.x1 / WD lim 3n f n!1

x  1

3n

for all x1 2 A: Moreover, letting m D 0 and n ! 1 in (3.150), we get (3.148). It follows from (3.146) that    x  x   x  3 x   3x C 3x  x    2 1 1 3 1 3 2 CD C D  D.x1 / D 3 3 3   x  x   x  3 x  2 1 1 3  D lim 3n f Cf n!1 3nC1 3nC1  x   3x C 3x  x  1 3 2 1   f n  C f 3nC1 3 3n   lim np .kx1 kp C kx2 kp C kx3 kp / n!1 3 D0 for all 2 T1 and x1 ; x2 ; x3 2 A and so D

 x  3 x   3x C 3x  x   x  x  2 1 1 3 1 3 2 CD C D 3 3 3 D D.x1 /

(3.151)

for all 2 T1 and x1 ; x2 ; x3 2 A: Let D 1 in (3.151). Then the mapping D W A ! A satisfies the inequality (3.142). By Lemma 3.100, the mapping D  W A  ! A is additive. Letting x1 D x2 D 0 in (3.151), we get   3 C D 3x33 D 0 and so D. x3 / D D.x3 / for all 2 T1 and all D 3 x 3 x3 2 A: By Lemma 3.99, D is C-linear. It follows from (3.147) that k˛D.Œx1 ; x2 /  ˇŒD.x1 /; x2    Œx1 ; D.x2 /k   Œx ; x   h x  x i h x  x i 1 2 1 2 1 2    ˇ f n ; n   n;f n  D lim 9n ˛f n n!1 9 3 3 3 3 9n   lim np .kx1 k2p C kx2 k2p / D 0 n!1 9 for all x1 ; x2 2 A: So, for some ˛; ˇ;  2 A, we have ˛D.Œx1 ; x2 / D ˇŒD.x1 /; x2  C  Œx1 ; D.x2 / for all x1 ; x2 2 A: Thus D is an .˛; ˇ;  /-derivation. 0 Now, let D W A ! A be another .˛; ˇ;  /-derivation satisfying (3.148). Then we have

3.7 .˛; ˇ; /-Derivations on Lie C -Algebras: The Direct Method

129

0

kD.x1 /  D .x1 /k  x    0 x1  1  D 3n D n  D n  3 3   x   x   0  x   x    1 1  1 1   3n D n  f n  C D n  f n  3 3 3 3 2  3n .1 C 2p /3p  kxkp ; 3np .3p  3/ which tends to zero as n ! 1 for all x1 2 A: So we can conclude that 0 D.x1 / D D .x1 / for all x1 2 A: This proves the uniqueness of D: Therefore, the mapping D W A ! A is a unique .˛; ˇ;  /-derivation satisfying (3.148). This completes the proof.  Theorem 3.103. Let p < 1,  be nonnegative real numbers and f W A ! A with f .0/ D 0 be a mapping satisfying (3.146) and (3.147). Then there exists a unique .˛; ˇ;  /-derivation D W A ! A such that kD.x1 /  f .x1 /k 

3p .1 C 2p /kx1 kp 3  3p

(3.152)

for all x1 2 A: Proof. It follows from (3.149) that  3p  1   f .x1 /  f .3x1 /  .1 C 2p /kx1 kp 3 3 for all x1 2 A: By induction, we have n   X 1   3i.p1/ f .x1 /  n f .3n x1 /  .1 C 2p /kx1 kp 3 iD1

for all x1 2 A: Hence we have   1 1    m f .3m x1 /  nCm f .3nCm x1 / 3 3 n X p p 3.iCm/.p1/  .1 C 2 /kx1 k iD1

 .1 C 2p /kx1 kp

nCm X

3i.p1/

(3.153)

iDmC1

for all m  1 and x1 2 A: This implies that the sequence f 31n f .3n x1 /g is a Cauchy sequence for all x1 2 A: Since A is complete, the sequence f 31n f .3n x1 /g converges. Thus one can define the mapping D W A ! A by

3 Stability of Functional Equations in C -Algebras

130

D.x1 / WD lim

n!1

1 f .3n x1 / 3n

for all x1 2 A: Moreover, letting m D 0 and n ! 1 in (3.153), we get (3.152). The rest of the proof is similar to the proof of Theorem 3.102. This completes the proof.  Corollary 3.104. Let  be a nonnegative real number. Let f W A ! A with be a mapping f .0/ D 0 such that, for some ˛; ˇ;  2 C,    x  x   x  3 x   3x C 3x  x  2 1 1 3 1 3 2   Cf C f  f .x1 / f 3 3 3  and k˛f .Œx1 ; x2 /  ˇŒf .x1 /; x2    Œx1 ; f .x2 /k   for all 2 T1 and x1 ; x2 ; x3 2 A: Then there exists a unique .˛; ˇ;  /-derivation D W A ! A such that kD.x1 /  f .x1 /k 

 2

for all x1 2 A:

3.8 Square Roots and 3rd Root Functional Equations: The Direct Method In this section, we introduce a square root functional equation and a 3rd root functional equation. We prove the Hyers-Ulam stability of the square root functional equation and of the 3rd root functional equation in C -algebras. Definition 3.105 ([96]). Let A be a C -algebra and x 2 A be a self-adjoint element, i.e., x D x. Then x is said to be positive if it is of the form yy for some y 2 A. The set of positive elements of A is denoted by AC . Note that AC is a closed convex cone (see [96]). It is well-known that, for a positive element x and a positive integer n, there exists a unique positive element 1 y 2 AC such that x D yn . We denote y by x n (see [128]). In this section, we introduce a square root functional equation:   1 1 1 1 1 1 S x C y C x 4 y 2 x 4 C y 4 x 2 y 4 D S.x/ C S.y/

(3.154)

3.8 Square Roots and 3rd Root Functional Equations: The Direct Method

131

and a 3rd root functional equation:   1 1 1 1 1 1 T x C y C 3x 3 y 3 x 3 C 3y 3 x 3 y 3 D T.x/ C T.y/

(3.155)

for all x; y 2 AC . Each solution of the square root functional equation is called a square root mapping and each solution of the 3rd root functional equation is called a 3rd root mapping. p p 1 1 Note that the functions S.x/ D x D x 2 and T.x/ D 3 x D x 3 in the set of nonnegative real numbers are solutions of the functional equations (3.154) and (3.155), respectively. Throughout this section, let AC and BC be the sets of positive elements in C algebras A and B, respectively.

3.8.1 Stability of the Square Root Functional Equation Here we investigate the square root functional equation in C -algebras. Lemma 3.106. Let S W AC ! BC be a square root mapping satisfying (3.154). Then S satisfies S.4n x/ D 2n S.x/

(3.156)

for all x 2 AC and n 2 Z. Proof. Putting x D y D 0 in (3.154), we obtain S.0/ D 0. Letting y D 0 in (3.154), we obtain S.40x/ D S.x/ D 20 S.x/ for all x 2 AC . First of all, we use the induction on n to prove the equality (3.155) for all n  1. Replacing y by x in (3.154), we get S.4x/ D 2S.x/

(3.157)

for all x 2 AC . So the equality (3.156) holds for n D 1. Assume that S.4k x/ D 2k S.x/

(3.158)

holds for a positive integer k. Replacing x by 4x in (3.158) and using (3.157), we obtain S.4kC1x/ D S.4k  4x/ D 2k S.4x/ D 2kC1 S.x/

3 Stability of Functional Equations in C -Algebras

132

for all x 2 AC . So the equality (3.156) holds for n D k C 1. Thus, by induction, we have S.4n x/ D 2n S.x/

(3.159)

for all x 2 AC and n  1. Next, replacing x by 4n x in (3.159), we obtain S.x/ D S.4n  4n x/ D 2n S.4n x/ for all x 2 AC and n  1 and so S.4n x/ D 2n S.x/ for all x 2 AC and n  1. Therefore, we have S.4n x/ D 2n S.x/ for all x 2 AC and n 2 Z. This completes the proof.



Now, we prove the Hyers-Ulam stability of the square root functional equation in C -algebras. Theorem 3.107. Let f W AC ! BC be a mapping for which there exists a function ' W AC  AC ! Œ0; 1/ such that '.x; Q y/ WD

1 X jD1

2j '

x y <1 ; 4j 4j

(3.160)

and     1 1 1 1 1 1   f x C y C x 4 y 2 x 4 C y 4 x 2 y 4  f .x/  f .y/  '.x; y/

(3.161)

for all x; y 2 AC . Then there exists a unique square root mapping S W AC ! AC satisfying (3.154) and kf .x/  S.x/k 

1 '.x; Q y/ 2

(3.162)

for all x 2 AC . Proof. Letting y D x in (3.161), we get     f .4x/  2f .x/  '.x; x/

(3.163)

3.8 Square Roots and 3rd Root Functional Equations: The Direct Method

133

for all x 2 AC . It follows from (3.163) that   x  x x   ; f .x/  2f ' 4 4 4 for all x 2 AC . Hence m  x  x  1 X x x  l  2j ' j ; j 2 f l  2m f m   4 4 2 jDlC1 4 4

(3.164)

C for all m; l˚  1with (3.164) that the  m > l and x 2 A . It follows fromC(3.160) and x k sequence 2 f 4k is a Cauchy sequence for all x 2 A . Since BC is complete, the ˚   sequence 2k f 4xk converges and so one can define the mapping S W AC ! BC by x S.x/ WD lim 2k f k k!1 4

for all x 2 AC . By (3.163) and (3.164), we have     1 1 1 1 1 1   S x C y C x 4 y 2 x 4 C y 4 x 2 y 4  S.x/  S.y/   x C y C x 14 y 12 x 14 C y 14 x 12 y 41  x  y    D lim 2k f  f  f  k!1 4k 4k 4k x y  lim 2k ' k ; k k!1 4 4 D0 for all x; y 2 AC and so   1 1 1 1 1 1 S x C y C x 4 y 2 x 4 C y 4 x 2 y 4  S.x/  S.y/ D 0: Hence the mapping S W AC ! BC is a square root mapping. Moreover, letting l D 0 and m ! 1 in (3.164), we get (3.162). So, there exists a square root mapping S W AC ! BC satisfying (3.154) and (3.162). Now, let S0 W AC ! BC be another square root mapping satisfying (3.154) and (3.162). Then we have  x  x    kS.x/  S0 .x/k D 2q S q  S0 q  4 4  x  x  x   x       2q S q  f q  C 2q S0 q  f q  4 4 4 4 2  2q  x x  'Q q ; q ;  2 4 4 which tends to zero as q ! 1 for all x 2 AC . So we can conclude that S.x/ D S0 .x/ for all x 2 AC . This proves the uniqueness of S. This completes the proof. 

3 Stability of Functional Equations in C -Algebras

134

Corollary 3.108. Let p > 12 and 1 ; 2 be non-negative real numbers, and let f W AC ! BC be a mapping such that     1 1 1 1 1 1   f x C y C x 4 y 2 x 4 C y 4 x 2 y 4  f .x/  f .y/ p

p

 1 .kxkp C kykp / C 2  kxk 2  kyk 2

(3.165)

for all x; y 2 AC . Then there exists a unique square root mapping S W AC ! BC satisfying (3.154) and kf .x/  S.x/k 

21 C 2 kxkp 4p  2

for all x 2 AC . Proof. Define p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 and apply Theorem 3.107. Then we get the desired result. C



C

Theorem 3.109. Let f W A ! B be a mapping for which there exists a function ' W AC  AC ! Œ0; 1/ satisfying (3.161) such that '.x; Q y/ WD

1 X

2j '.4j x; 4j y/ < 1

jD0

for all x; y 2 AC . Then there exists a unique square root mapping S W AC ! BC satisfying (3.154) and kf .x/  S.x/k 

1 '.x; Q x/ 2

for all x 2 AC . Proof. It follows from (3.163) that     f .x/  1 f .4x/  1 '.x; x/   2 2 for all x 2 AC . The rest of the proof is similar to the proof of Theorem 3.107.



Corollary 3.110. Let 0 < p < 12 , 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping satisfying (3.165). Then there exists a unique square root mapping S W AC ! BC satisfying (3.154) and

3.8 Square Roots and 3rd Root Functional Equations: The Direct Method

kf .x/  S.x/k 

135

21 C 2 kxkp 2  4p

for all x 2 AC . Proof. Define p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 and apply Theorem 3.109. Then we get the desired result.



3.8.2 Stability of the 3rd Root Functional Equation Now, we investigate the 3rd root functional equation in C -algebras. Lemma 3.111. Let T W AC ! BC be a 3rd root mapping satisfying (3.155). Then T satisfies T.8n x/ D 2n T.x/ for all x 2 AC and n 2 Z. Proof. The proof is similar to the proof of Lemma 3.106.



Now, we prove the Hyers-Ulam stability of the 3rd root functional equation in C -algebras. Theorem 3.112. Let f W AC ! BC be a mapping for which there exists a function ' W AC  AC ! Œ0; 1/ such that '.x; Q y/ WD

1 X jD1

2j '

x y <1 ; 8j 8j

and     1 1 1 1 1 1   f x C y C 3x 3 y 3 x 3 C 3y 3 x 3 y 3  f .x/  f .y/  '.x; y/

(3.166)

for all x; y 2 AC . Then there exists a unique 3rd root mapping T W AC ! AC satisfying (3.155) and kf .x/  T.x/k  for all x 2 AC .

1 '.x; Q y/ 2

3 Stability of Functional Equations in C -Algebras

136

Proof. Letting y D x in (3.166), we get kf .8x/  2f .x/k  '.x; x/

(3.167)

for all x 2 AC . The rest of the proof is similar to the proof of Theorem 3.107. Corollary 3.113. Let p > be a mapping such that

1 , , 3 1 2



C

be non-negative real numbers and f W A ! BC

    1 1 1 1 1 1   f x C y C 3x 3 y 3 x 3 C 3y 3 x 3 y 3  f .x/  f .y/ p

p

 1 .kxkp C kykp / C 2  kxk 2  kyk 2

(3.168)

for all x; y 2 AC . Then there exists a unique 3rd root mapping T W AC ! BC satisfying (3.155) and kf .x/  T.x/k 

21 C 2 kxkp 8p  2

for all x 2 AC . Proof. Define p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 and apply Theorem 3.112. Then we get the desired result. C



C

Theorem 3.114. Let f W A ! B be a mapping for which there exists a function ' W AC  AC ! Œ0; 1/ satisfying (3.166) such that '.x; Q y/ WD

1 X

2j '.8j x; 8j y/ < 1

jD0

for all x; y 2 AC . Then there exists a unique 3rd root mapping T W AC ! BC satisfying (3.155) and kf .x/  T.x/k 

1 '.x; Q x/ 2

for all x 2 AC . Proof. It follows from (3.167) that   1 1   f .x/  f .8x/  '.x; x/ 2 2 for all x 2 AC . The rest of the proof is similar to the proof of Theorem 3.107.



3.9 Square Root and 3rd Root Functional Equations: The Fixed Point Method

137

Corollary 3.115. Let 0 < p < 13 , 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping satisfying (3.168). Then there exists a unique 3rd root mapping T W AC ! BC satisfying (3.155) and kf .x/  T.x/k 

21 C 2 kxkp 2  8p

for all x 2 AC . p

p

Proof. Define '.x; y/ D 1 .kxkp Ckykp /C2 kxk 2 kyk 2 and apply Theorem 3.114. Then we get the desired result. 

3.9 Square Root and 3rd Root Functional Equations: The Fixed Point Method In this section, we prove the Hyers-Ulam stability of the square root functional equation and the 3rd root functional equation in C -algebras via fixed point method [235].

3.9.1 Stability of the Square Root Functional Equation In this section, we investigate the square root functional equation in C -algebras. Now, we prove the Hyers-Ulam stability of the square root functional equation in C -algebras. Theorem 3.116. Let ' W AC  AC ! Œ0; 1/ be a function such that there exists L < 1 with '.x; y/ 

L '.4x; 4y/ 2

(3.169)

for all x; y 2 AC . Let f W AC ! BC be a mapping satisfying     1 1 1 1 1 1   f x C y C x 4 y 2 x 4 C y 4 x 2 y 4  f .x/  f .y/  '.x; y/

(3.170)

for all x; y 2 AC . Then there exists a unique square root mapping S W AC ! AC satisfying (3.154) and kf .x/  S.x/k  for all x 2 AC .

L '.x; x/ 2  2L

(3.171)

3 Stability of Functional Equations in C -Algebras

138

Proof. Letting y D x in (3.170), we get kf .4x/  2f .x/k  '.x; x/

(3.172)

for all x 2 AC . Consider the set X WD fg W AC ! BC g and introduce the generalized metric on X defined by d.g; h/ D inff 2 RC W kg.x/  h.x/k  '.x; x/; 8x 2 AC g; where, as usual, inf  D C1 which .X; d/ is complete. Now, we consider the linear mapping J W X ! X such that Jg.x/ WD 2g

x 4

for all x 2 AC . Let g; h 2 X be given such that d.g; h/ D ". Then kg.x/  h.x/k  '.x; x/ for all x 2 AC . Hence we have  x  x     2h kJg.x/  Jh.x/k D 2g   L'.x; x/ 4 4 for all x 2 AC . So, d.g; h/ D " implies that d.Jg; Jh/  L". This means that d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. It follows from (3.172) that   x  L     '.x; x/ f .x/  2f 4 2 for all x 2 AC and so d.f ; Jf /  L2 . By Theorem 1.3, there exists a mapping S W AC ! BC satisfying the following: (1) S is a fixed point of J, i.e., S

x 4

D

1 S.x/ 2

for all x 2 AC . The mapping S is a unique fixed point of J in the set M D fg 2 X W d.f ; g/ < 1g:

(3.173)

3.9 Square Root and 3rd Root Functional Equations: The Fixed Point Method

139

This implies that S is a unique mapping satisfying (3.173) such that there exists a 2 .0; 1/ satisfying kf .x/  S.x/k  '.x; x/ for all x 2 AC ; (2) d.J n f ; S/ ! 0 as n ! 1. This implies the equality lim 2n f

n!1

x D S.x/ 4n

for all x 2 AC ; 1 (3) d.f ; S/  1L d.f ; Jf /, which implies the inequality d.f ; S/ 

L : 2  2L

This implies that the inequality (3.171) holds. By (3.169) and (3.170), we have   x C y C x 14 y 12 x 14 C y 14 x 12 y 41  x  y    f n f n  2n f n 4 4 4 x y  2n ' n ; n 4 4 n  L '.x; y/ for all x; y 2 AC and n 2 N. So, we have     1 1 1 1 1 1   S x C y C x 4 y 2 x 4 C y 4 x 2 y 4  S.x/  S.y/ D 0 for all x; y 2 AC . Thus the mapping S W AC ! BC is a square root mapping. This completes the proof.  Corollary 3.117. Let p > 12 , 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping such that     1 1 1 1 1 1   f x C y C x 4 y 2 x 4 C y 4 x 2 y 4  f .x/  f .y/ p

p

 1 .kxkp C kykp / C 2  kxk 2  kyk 2

(3.174)

for all x; y 2 AC . Then there exists a unique square root mapping S W AC ! BC satisfying (3.154) and

140

3 Stability of Functional Equations in C -Algebras

21 C 2 jjxjjp 4p  2

kf .x/  S.x/k  for all x 2 AC .

Proof. The proof follows from Theorem 3.116 by taking p p '.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 for all x; y 2 AC and L D 212p .



Theorem 3.118. Let ' W AC  AC ! Œ0; 1/ be a function such that there exists L < 1 with x y ; '.x; y/  2L' 4 4 for all x; y 2 AC . Let f W AC ! BC be a mapping satisfying (3.170). Then there exists a unique square root mapping S W AC ! AC satisfying (3.154) and kf .x/  S.x/k 

1 '.x; x/ 2  2L

for all x 2 AC . Proof. Let .X; d/ be the generalized metric space defined in the proof of Theorem 3.116. Consider the linear mapping J W X ! X such that Jg.x/ WD

1 gt.4x/ 2

for all x 2 AC . It follows from (3.172) that   1 1   f .x/  f .4x/  '.x; x/ 2 2 for all x 2 AC . So d.f ; Jf /  12 . The rest of the proof is similar to the proof of Theorem 3.116. This completes the proof.  Corollary 3.119. Let 0 < p < 12 , 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping satisfying (3.174). Then there exists a unique square root mapping S W AC ! BC satisfying (3.154) and kf .x/  S.x/k  for all x 2 AC .

21 C 2 kxkp 2  4p

3.9 Square Root and 3rd Root Functional Equations: The Fixed Point Method

141

Proof. The proof follows from Theorem 3.118 by taking p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 for all x; y 2 AC and L D 22p1 .



3.9.2 Stability of the 3rd Root Functional Equation Now, we investigate the 3rd root functional equation in C -algebras. Now, we prove the Hyers-Ulam stability of the 3rd root functional equation in C -algebras. Theorem 3.120. Let ' W AC  AC ! Œ0; 1/ be a function such that there exists an L < 1 with '.x; y/ 

L '.8x; 8y/ 2

for all x; y 2 AC . Let f W AC ! BC be a mapping satisfying     1 1 1 1 1 1   f x C y C 3x 3 y 3 x 3 C 3y 3 x 3 y 3  f .x/  f .y/  '.x; y/

(3.175)

for all x; y 2 AC . Then there exists a unique 3rd root mapping T W AC ! AC satisfying (3.155) and kf .x/  T.x/k 

L '.x; x/ 2  2L

for all x 2 AC . Proof. Letting y D x in (3.175), we get kf .8x/  2f .x/k  '.x; x/

(3.176)

for all x 2 AC . Let .X; d/ be the generalized metric space defined in the proof of Theorem 3.116. Consider the linear mapping J W X ! X such that Jg.x/ WD 2g for all x 2 AC .

x 8

3 Stability of Functional Equations in C -Algebras

142

Now, we consider the linear mapping J W X ! X such that Jg.x/ WD 2g

x 8

for all x 2 AC . It follows from (3.176) that   x  L     '.x; x/ f .x/  2f 8 2 for all x 2 X and so d.f ; Jf /  L2 . The rest of the proof is similar to the proof of Theorem 3.116. This completes the proof.  Corollary 3.121. Let p > 13 , 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping such that     1 1 1 1 1 1   f x C y C 3x 3 y 3 x 3 C 3y 3 x 3 y 3  f .x/  f .y/ p

p

 1 .kxkp C kykp / C 2  kxk 2  kyk 2

(3.177)

for all x; y 2 AC . Then there exists a unique 3rd root mapping T W AC ! BC satisfying (3.155) and kf .x/  T.x/k 

21 C 2 kxkp 8p  2

for all x 2 AC . Proof. The proof follows from Theorem 3.120 by taking p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 for all x; y 2 AC and L D 213p .



Theorem 3.122. Let ' W AC  AC ! Œ0; 1/ be a function such that there exists L < 1 with x y ; '.x; y/  2L' 8 8 for all x; y 2 AC . Let f W AC ! BC be a mapping satisfying (3.175). Then there exists a unique 3rd root mapping T W AC ! AC satisfying (3.155) and kf .x/  T.x/k  for all x 2 AC .

1 '.x; x/ 2  2L

3.10 Positive-Additive Functional Equation

143

Proof. Let .X; d/ be the generalized metric space defined in the proof of Theorem 3.116. Consider the linear mapping J W X ! X such that Jg.x/ WD

1 g.8x/ 2

for all x 2 AC . It follows from (3.176) that   1 1   f .x/  f .8x/  '.x; x/ 2 2 for all x 2 AC and so d.f ; Jf /  12 . The rest of the proof is similar to the proof of Theorem 3.116. This completes the proof.  Corollary 3.123. Let 0 < p < 13 , 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping satisfying (3.177). Then there exists a unique 3rd root mapping T W AC ! BC satisfying (3.155) and kf .x/  T.x/k 

21 C 2 kxkp 2  8p

for all x 2 AC . Proof. The proof follows from Theorem 3.122 by taking p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 for all x; y 2 AC and L D 23p1 .



3.10 Positive-Additive Functional Equation In this section, we consider a positive-additive functional equation in C -algebras [258]. Using fixed point and direct methods, we prove the stability of the positiveadditive functional equation in C -algebras. Definition 3.124 ([96]). Let A be a C -algebra and x 2 A be a self-adjoint element, i.e., x D x. Then x is said to be positive if it is of the form yy for some y 2 A. The set of positive elements of A is denoted by AC . Note that AC is a closed convex cone (see [96]). It is well known that for a positive element x and a positive integer n there exists a unique positive element 1 y 2 AC such that x D yn . We denote y by x n (see [128]).

3 Stability of Functional Equations in C -Algebras

144

In this section, we introduce the following functional equation: T

     1 1 m 1 1 m D T.x/ m C T.y/ m xm C ym

(3.178)

for all x; y 2 AC and a fixed integer m greater than 1, which is called a positiveadditive functional equation. Each solution of the positive-additive functional equation is called a positive-additive mapping. Note that the function f .x/ D cx for any c  0 in the set of non-negative real numbers is a solution of the functional equation (3.178). Throughout this section, let AC and BC be the sets of positive elements in C -algebras A and B, respectively. Assume that m is a fixed integer greater than 1.

3.10.1 Stability of the Positive-Additive Functional Equations: The Fixed Point Method Here we investigate the positive-additive functional equation (3.178) in C -algebras. Lemma 3.125. Let T W AC ! BC be a positive-additive mapping satisfying (3.178). Then T satisfies T.2mn x/ D 2mn T.x/ for all x 2 AC and n 2 Z. Proof. Putting x D y in (3.178), we obtain T.2m x/ D 2m T.x/ for all x 2 AC . So, one can show that T.2mn x/ D 2mn T.x/ for all x 2 AC and n 2 Z.



Using the fixed point method, we prove the Hyers-Ulam stability of the positiveadditive functional equation (3.178) in C -algebras. Note that the fundamental ideas in the proofs of the main results in this section are contained in [62–64]. Theorem 3.126. Let ' W AC  AC ! Œ0; 1/ be a function such that there exists L < 1 with '.x; y/ 

L ' .2m x; 2m y/ 2m

for all x; y 2 AC . Let f W AC ! BC be a mapping satisfying

(3.179)

3.10 Positive-Additive Functional Equation

145

  1      1 m 1 1 m   f .x/ m C f .y/ m   '.x; y/ f x m C y m

(3.180)

for all x; y 2 AC . Then there exists a unique positive-additive mapping T W AC ! AC satisfying (3.178) and kf .x/  T.x/k 

2m

L '.x; x/  2m L

(3.181)

for all x 2 AC . Proof. Letting y D x in (3.180), we get kf .2m x/  2m f .x/k  '.x; x/ for all x 2 AC . Consider the set X WD fg W AC ! BC g and introduce the generalized metric on X defined by d.g; h/ D inff 2 RC W kg.x/  h.x/k  '.x; x/; 8x 2 AC g; where, as usual, inf  D C1 which .X; d/ is complete. Now, we consider the linear mapping J W X ! X such that Jg.x/ WD 2m g

 x  2m

for all x 2 AC . Let g; h 2 X be given such that d.g; h/ D ". Then we have kg.x/  h.x/k  '.x; x/ for all x 2 AC and so   x   x    kJg.x/  Jh.x/k D 2m g m  2m h m   L'.x; x/ 2 2 for all x 2 AC . So, d.g; h/ D " implies that d.Jg; Jh/  L". This means that d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. It follows from (3.182) that   x  L  f .x/  2m f m k  m '.x; x/ 2 2

(3.182)

3 Stability of Functional Equations in C -Algebras

146

for all x 2 AC and so d.f ; Jf /  2Lm . By Theorem 1.3, there exists a mapping T W AC ! BC satisfying the following: (1) T is a fixed point of J, i.e., T

 x  1 D m T.x/ m 2 2

(3.183)

for all x 2 AC . The mapping T is a unique fixed point of J in the set M D fg 2 X W d.f ; g/ < 1g: This implies that T is a unique mapping satisfying (3.183) such that there exists a 2 .0; 1/ satisfying kf .x/  T.x/k  '.x; x/ for all x 2 AC ; (2) d.J n f ; T/ ! 0 as n ! 1. This implies the equality lim 2mn f

n!1

 x  D T.x/ 2mn

for all x 2 AC ; 1 (3) d.f ; T/  1L d.f ; Jf /, which implies the inequality d.f ; T/ 

L : 2m  2m L

This implies that the inequality (3.181) holds. By (3.179) and (3.180), we have   x m1 C y m1 m    x  m1   y  m1 m   mn mn 2 f  2 f C 2 f  2mn 2mn 2mn  x  y  2mn ' mn ; mn 2 2  Lmn '.x; y/ mn 

for all x; y 2 AC and n 2 N and so   1      1 m 1 1 m   T.x/ m C T.y/ m  D 0 T x m C y m for all x; y 2 AC . Thus the mapping T W AC ! BC is positive-additive. This completes the proof. 

3.10 Positive-Additive Functional Equation

147

Corollary 3.127. Let p > 1, 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping such that   1      1 m 1 1 m   f .x/ m C f .y/ m  f x m C y m p

p

 1 .kxkp C kykp / C 2  kxk 2  kyk 2

(3.184)

for all x; y 2 AC . Then there exists a unique positive-additive mapping T W AC ! BC satisfying (3.178) and kf .x/  T.x/k 

21 C 2 kxkp 2mp  2m

for all x 2 AC . Proof. The proof follows from Theorem 3.126 by taking p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 for all x; y 2 AC and L D 2mmp .



Theorem 3.128. Let ' W AC  AC ! Œ0; 1/ be a function such that there exists an L < 1 with  x y  '.x; y/  2m L' m ; m 2 2 for all x; y 2 AC . Let f W AC ! BC be a mapping satisfying (3.180). Then there exists a unique positive-additive mapping T W AC ! AC satisfying (3.178) and kf .x/  T.x/k 

1 '.x; x/ 2m  2m L

for all x 2 AC . Proof. Let .X; d/ be the generalized metric space defined in the proof of Theorem 3.126. Consider the linear mapping J W X ! X such that Jg.x/ WD

1 g .2m x/ 2m

for all x 2 AC . It follows from (3.182) that     f .x/  1 f .2m x/  1 '.x; x/   2m m 2 for all x 2 AC and so d.f ; Jf /  21m . The rest of the proof is similar to the proof of Theorem 3.126. This completes the proof. 

3 Stability of Functional Equations in C -Algebras

148

Corollary 3.129. Let 0 < p < 1, 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping satisfying (3.184). Then there exists a unique positiveadditive mapping T W AC ! BC satisfying (3.178) and kf .x/  T.x/k 

21 C 2 kxkp 2m  2mp

for all x 2 AC . Proof. The proof follows from Theorem 3.128 by taking p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 for all x; y 2 AC and L D 2mpm .



3.10.2 Stability of the Positive-Additive Functional Equations: The Direct Method Now, using the direct method of Hyers, we prove the Hyers-Ulam stability of the positive-additive functional equation (3.178) in C -algebras. Theorem 3.130. Let f W AC ! BC be a mapping for which there exists a function ' W AC  AC ! Œ0; 1/ satisfying (3.180) and '.x; Q y/ WD

1 X jD1

2mj '

 x y  <1 ; 2mj 2mj

(3.185)

for all x; y 2 AC . Then there exists a unique positive-additive mapping T W AC ! AC satisfying (3.178) and kf .x/  T.x/k 

1 '.x; Q y/ 2m

(3.186)

for all x 2 AC . Proof. It follows from (3.182) that   x   x x    f .x/  2m f m   ' m ; m 2 2 2 for all x 2 AC and so k   x   x  1 X mj  x x   ml  2 ' mj ; mj 2 f ml  2mk f mk   m 2 2 2 jDlC1 2 2

(3.187)

3.10 Positive-Additive Functional Equation

149

for all k; l  1 with l and x 2 AC . It follows from (3.185) and (3.187) ˚ mj k  >x  that the sequence 2 f 2mj is a Cauchy sequence for all x 2 AC . Since BC is ˚   complete, the sequence 2mj f 2xmj converges and so one can define the mapping T W AC ! BC by T.x/ WD lim 2mj f j!1

 x  2mj

for all x 2 AC . By (3.180) and (3.185), we have   1      1 m 1 1 m   T.x/ m C T.y/ m  T x m C y m   x m1 C y m1 m   D lim 2mj f j!1 2mj   x  m1   y  m1 m    2mj f mj C 2mj f mj  2 2  x y   lim 2mj ' mj ; mj j!1 2 2 D0 for all x; y 2 AC and so  T

1

1

xm C ym

m 

  1 1 m  T.x/ m C T.y/ m D0

for all x; y 2 AC . Hence the mapping T W AC ! BC is positive-additive. Moreover, letting l D 0 and passing the limit k ! 1 in (3.187), we get (3.186). So there exists a positive-additive mapping T W AC ! BC satisfying (3.178) and (3.186). Now, let T 0 W AC ! BC be another positive-additive mapping satisfying (3.178) and (3.186). Then we have kT.x/  T 0 .x/k   x   x    D 2mq T mq  T 0 mq  2 2   x    x   x   x       2mq T mq  f mq  C 2mq T 0 mq  f mq  2 2 2 2 2  2mq  x x   'Q mq ; mq ; m 2 2 2 which tends to zero as q ! 1 for all x 2 AC . So, we can conclude that T.x/ D T 0 .x/ for all x 2 AC , which proves the uniqueness of T. This completes the proof. 

3 Stability of Functional Equations in C -Algebras

150

Corollary 3.131. Let p > 1, 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping satisfying (3.184). Then there exists a unique positive-additive mapping T W AC ! BC satisfying (3.178) and kf .x/  T.x/k 

21 C 2 kxkp 2mp  2m

for all x 2 AC . Proof. Define p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 and apply Theorem 3.130. Then we get the desired result.



Theorem 3.132. Let f W AC ! BC be a mapping for which there exists a function ' W AC  AC ! Œ0; 1/ satisfying (3.180) such that '.x; Q y/ WD

1 X

2mj '.2mj x; 2mj y/ < 1

jD0

for all x; y 2 AC . Then there exists a unique positive-additive mapping T W AC ! BC satisfying (3.178) and kf .x/  T.x/k 

1 '.x; Q x/ 2m

for all x 2 AC . Proof. It follows from (3.182) that   1 1   f .x/  m f .2m x/  m '.x; x/ 2 2 for all x 2 AC . The rest of the proof is similar to the proof of Theorem 3.130.



Corollary 3.133. Let 0 < p < 1, 1 , 2 be non-negative real numbers and f W AC ! BC be a mapping satisfying (3.184). Then there exists a unique positiveadditive mapping T W AC ! BC satisfying (3.178) and kf .x/  T.x/k  for all x 2 AC .

21 C 2 kxkp 2m  2mp

3.11 Stability of -Homomorphisms in JC -Algebras

151

Proof. Define p

p

'.x; y/ D 1 .kxkp C kykp / C 2  kxk 2  kyk 2 and apply Theorem 3.132. Then we get the desired result.



3.11 Stability of -Homomorphisms in JC -Algebras It is shown that every almost unital almost linear mapping f W A ! B of JC algebra A to a JC -algebra B is a homomorphism when f .2n u ı y/ D f .2n u/ ı f .y/ for all unitaries u 2 A, y 2 A and n  0 and every almost unital almost linear continuous mapping f W A ! B of a JC -algebra A of real rank zero to a JC algebra B is a homomorphism when f .2n u ı y/ D f .2n u/ ı f .y/ for all u 2 fv 2 A W v D v  ; kvk D 1; v is invertibleg, y 2 A and n  0. Furthermore, we prove the Hyers-Ulam stability of -homomorphisms in JC algebras and C-linear -derivations on JC -algebras. Our knowledge concerning the continuity properties of epimorphisms on Banach algebras, Jordan–Banach algebras, and, more generally, non-associative complete normed algebras, is now fairly complete and satisfactory (see [143] and [326]). A basic continuity problem consists in determining algebraic conditions on a Banach algebra A which ensure that derivations on A are continuous. In 1996, Villena [326] proved that derivations on semisimple Jordan–Banach algebras are continuous. Let E1 and E2 be Banach spaces with the norms k  k and k  k, respectively. Consider f W E1 ! E2 to be a mapping such that f .tx/ is continuous in t 2 R for each fixed x 2 E1 . Assume that there exist constants   0 and p 2 Œ0; 1/ such that kf .x C y/  f .x/  f .y/k  .kxkp C kykp / for all x; y 2 E1 . In [267], Th. M. Rassias showed that there exists a unique R-linear mapping T W E1 ! E2 such that kf .x/  T.x/k 

2 kxkp 2  2p

for all x 2 E1 . G˘avruta [123] generalized the Rassias’ result. In [146], Jun et al. proved the following: Let X and Y be Banach spaces. Denote by ' W X  X ! Œ0; 1/ a function such that ".x/ WD

1 X jD1

2j .'.2j1 x; 0/ C '.0; 2j1 x/ C '.2j1 x; 2j1 x// < 1

3 Stability of Functional Equations in C -Algebras

152

for all x 2 X. Suppose that f ; g; h W X ! Y are mappings satisfying k2f .

xCy /  g.x/  h.y/k  '.x; y/ 2

for all x; y 2 X. Then there exists a unique additive mapping T W X ! Y such that x k2f . /  T.x/k  kg.0/k C kh.0/k C ".x/; 2 kg.x/  T.x/k  kg.0/k C 2kh.0/k C '.x; 0/ C ".x/; kh.x/  T.x/k  2kg.0/k C kh.0/k C '.0; x/ C ".x/ for all x 2 X. In Theorem 7.2 of Johnson [143], Johnson also investigated almost algebra homomorphisms between Banach -algebras: Suppose that U and B are Banach -algebras which satisfy the conditions of Theorem 3.1 in [143]. Then, for each positive  and K, there exists a positive ı such that if T 2 L.U; B/ with kTk < K, kT _ k < ı and kT.x /  T.x/k  ıkxk for all x 2 U, then there exists a -homomorphism T 0 W U ! B with kT  T 0 k < . Here L.U; B/ is the space of bounded linear mappings from U into B and T _ .x; y/ D T.xy/  T.x/T.y/ for all x; y 2 U (see [143] for details). The original motivation to introduce the class of nonassociative algebras known as Jordan algebras came from quantum mechanics (see [323]). Let H be a complex Hilbert space, regarded as the “state space” of a quantum mechanical system. Let L.H/ be the real vector space of all bounded selfadjoint linear operators on H, interpreted as the (bounded) observables of the system. In 1932, Jordan observed that L.H/ is a (nonassociative) algebra via the anticommutator product x ı y WD xyCyx . A commutative algebra X with product x ı y 2 is called a Jordan algebra if x2 ı .x ı y/ D x ı .x2 ı y/: A complex Jordan algebra C with the product x ı y and involution x 7! x is called a JB -algebra if C carries a Banach space norm k  k satisfying kx ı yk  kxk  kyk;

kfxx xgk D kxk3 :

Here fxy zg WD x ı .y ı z/  y ı .z ı x/ C z ı .x ı y / denotes the Jordan triple product of x; y; z 2 C. A unital Jordan C -subalgebra of a C -algebra endowed with the anticommutator product is called a JC -algebra.

3.11 Stability of -Homomorphisms in JC -Algebras

153

Throughout this section, let A be a JC -algebra with the norm k  k and the unit e and B be a JC -algebra with the norm k  k and the unit e0 . Let U.A/ D fu 2 A W u u D uu D eg;

Asa D fx 2 A W x D x g;

I1 .Asa / D fv 2 Asa W kvk D 1; v is invertibleg: In this section, we prove that every almost unital almost linear mapping h W A ! B is a homomorphism when h.3n u ı y/ D h.3n u/ ı h.y/ for all u 2 U.A/, y 2 A and n  0 and, for a JC -algebra A of real rank zero, every almost unital almost linear continuous mapping h W A ! B is a homomorphism when h.3n u ı y/ D h.3n u/ ı h.y/ for all u 2 I1 .Asa /, y 2 A and n  0. Furthermore, we prove the Hyers-Ulam stability of -homomorphisms between JC -algebras and C-linear -derivations on JC -algebras.

3.11.1 -Homomorphisms in JC -Algebras Now, we investigate -homomorphisms in JC -algebras. Theorem 3.134. Let f ; g; h W A ! B be mappings satisfying f .0/ D 0, g.0/ D 0 and h.0/ D 0 and let f .2n u ı y/ D f .2n u/ ı f .y/, g.2n u ı y/ D g.2n u/ ı g.y/ and h.2n u ı y/ D h.2n u/ ı h.y/ for u 2 U.A/, y 2 A and n  0 for which there exists a function ' W A n f0g  A n f0g ! Œ0; 1/ such that '.x; Q y/ WD

1 X

2j '.2j1 x; 2j1 y/ < 1;

(3.188)

jD0

  x C y      g.x/  h.y/  '.x; y/; 2f 2

(3.189)

kf .2n u /  f .2n u/ k  '.2n u; 2n u/

(3.190)

for all 2 T1 WD f 2 C j jj D 1g, u 2 U.A/, x; y 2 A and n  0. Assume that f .2n e/ D e0 : n!1 2n lim

(3.191)

Then the mappings f ; g; h W A ! B are -homomorphisms. Proof. Put D 1 2 T1 . It follows from Corollary 2.5 of [146] that there exists a unique additive mapping H W A ! B such that   x    H.x/  ".x/; 2f 2

kg.x/  H.x/k  '.x; 0/ C ".x/;

3 Stability of Functional Equations in C -Algebras

154

kh.x/  H.x/k  '.0; x/ C ".x/

(3.192)

for all x 2 A n f0g, where ".x/ WD

1 X

2j .'.2j1 x; 0/ C '.0; 2j1 x/ C '.2j1 x; 2j1 x// < 1

jD1

for all x 2 A n f0g. The additive mapping H W A ! B is given by H.x/ D lim

n!1

1 f .2n x/ 2n

for all x 2 A and lim 2n f .2n x/ D lim 2n g.2n x/ D lim 2n h.2n x/

n!1

n!1

n!1

for all x 2 A. Let fQ .x/ D 2f . 2x / for all x 2 A. Then we have lim

n!1

1Q n 1 f .2 x/ D lim n f .2n x/ n!1 2 2n

for all x 2 A. By the assumption, we have kf .2n x/  f .2n x/k   1 1 1 1   D f .2n x/  g.2n x/  h.2n x/ C g.2n x/ C h.2n x/  f .2n x/ 2 2 2 2 1 1  '.2n x; 2n x/ C j j'.2n x; 2n x/ 2 2 n n D '.2 x; 2 x/ for all 2 T1 and x 2 A n f0g. Thus 2n kf .2n x/  f .2n x/k ! 0 as n ! 1 for all 2 T1 and x 2 A n f0g. Hence we have f .2n x/ f .2n x/ D lim D H.x/ n n!1 n!1 2 2n

H. x/ D lim

(3.193)

for all 2 T1 and x 2 A n f0g. Now, let  2 C . ¤ 0/ and M be an integer greater than 2jj. Then we have j M j < 12 D 1  24 . By Theorem 1 of Kadison and Pedersen [167], there exist four elements 1 ; 2 ; 3 ; 4 2 T1 such that 4 M D 1 C 2 C 3 C 4 . Note that H.x/ D H.2  12 x/ D 2H. 21 x/ for all x 2 A and so H. 21 x/ D 12 H.x/ for all x 2 A. Thus, by (3.193), we have

3.11 Stability of -Homomorphisms in JC -Algebras

1     x DMH 4 x 4 M 4 M    M M H 4 x D H. 1 x C 2 x C 3 x C 4 x/ 4 M 4 M .H. 1 x/ C H. 2 x/ C H. 3 x/ C H. 4 x// 4 M . 1 C 2 C 3 C 4 /H.x/ 4 M   4 H.x/ D H.x/ 4 M

H.x/ D H D D D D

155

M

4

for all x 2 A. Hence we have H. x C y/ D H. x/ C H. y/ D H.x/ C H.y/ for all ; 2 C n f0g and x; y 2 A and H.0x/ D 0 D 0H.x/ for all x 2 A. So the unique additive mapping H W A ! B is a C-linear mapping. By (3.188) and (3.190), we get  f .2n u / f .2n u/ f .2n u/  D lim D lim D H.u/ n!1 n!1 n!1 2n 2n 2n

H.u / D lim

for all u 2 U.A/. Since H is C-linear andP each x 2 A is a finite linear combination of unitary elements (see [168]), say, x D m jD1 j uj for all j 2 C and uj 2 U.A/, it follows that H.x / D H

m X

m m  X X j uj D j H.uj / D j H.uj /

jD1

D

m X

j H.uj /

jD1

jD1



DH

jD1 m X

j u j



D H.x/

jD1

for all x 2 A. Since f .2n u ı y/ D f .2n u/ ı f .y/ for all u 2 U.A/, y 2 A and n  0, we have 1 1 f .2n u ı y/ D lim n f .2n u/ ı f .y/ n n!1 2 n!1 2 D H.u/ ı f .y/

H.u ı y/ D lim

for all u 2 U.A/ and y 2 A. By the additivity of H and (3.194), we have 2n H.u ı y/ D H.2n u ı y/ D H.u ı .2n y// D H.u/ ı f .2n y/ for all u 2 U.A/ and y 2 A and so

(3.194)

3 Stability of Functional Equations in C -Algebras

156

H.u ı y/ D

1 1 H.u/ ı f .2n y/ D H.u/ ı n f .2n y/ 2n 2

(3.195)

for all u 2 U.A/ and y 2 A. Taking n ! 1 in (3.195), we obtain H.u ı y/ D H.u/ ı H.y/

(3.196)

for all u 2 U.A/ and y 2 A. Since H is C-linear and each x 2 A is a finite linear P combination of unitary elements, i.e., x D m jD1 j uj for all j 2 C and uj 2 U.A/, it follows from (3.196) that H.x ı y/ D H

m X

m  X j u j ı y D j H.uj ı y/

jD1

D

m X

jD1

j H.uj / ı H.y/ D H

jD1

m X

 j uj ı H.y/

jD1

D H.x/ ı H.y/ for all x; y 2 A. By (3.194) and (3.196), we have H.e/ ı H.y/ D H.e ı y/ D H.e/ ı f .y/ for all y 2 A. Since limn!1

f .2n e/ 2n

D H.e/ D e0 , it follows that H.y/ D f .y/

for all y 2 A. Similarly, H.y/ D g.y/ D h.y/ for all y 2 A. Therefore, the mapping f ; g; h W A ! B are -homomorphisms. This completes the proof.  Corollary 3.135. Let f ; g; h W A ! B be mappings satisfying f .0/ D 0, g.0/ D 0 and h.0/ D 0 and let f .2n u ı y/ D f .2n u/ ı f .y/, g.2n u ı y/ D g.2n u/ ı g.y/ and h.2n u ı y/ D h.2n u/ ı h.y/ for all u 2 U.A/, y 2 A and n  0 for which there exist constants   0 and p 2 Œ0; 1/ such that   x C y      g.x/  h.y/  .kxkp C kykp / 2f 2 and kf .2n u /  f .2n u/ k  2npC1  for all 2 T1 , u 2 U.A/, n  0 and x; y 2 Anf0g. Assume that limn!1 Then the mappings f ; g and h are -homomorphisms.

f .2n e/ 2n

D e0 .

Proof. Define '.x; y/ D .kxkp C kykp / for all x; y 2 A n f0g and apply Theorem 3.134. The we have the conclusion. 

3.11 Stability of -Homomorphisms in JC -Algebras

157

Theorem 3.136. Let f ; g; h W A ! B be mappings satisfying f .0/ D 0, g.0/ D 0 and h.0/ D 0 and let f .2n u ı y/ D f .2n u/ ı f .y/, g.2n u ı y/ D g.2n u/ ı g.y/ and h.2n u ı y/ D h.2n u/ ı h.y/ for all u 2 U.A/, y 2 A and n  0 for which there exists a function ' W A  A ! Œ0; 1/ satisfying (3.188), (3.190) and (3.191) such that    x C y     g.x/  h.y/  '.x; y/ 2f 2

(3.197)

for all x; y 2 A n f0g and D 1; i. If f .tx/ is continuous in t 2 R for each fixed x 2 A, then the mappings f ; g; h W A ! B are -homomorphisms. Proof. Put D 1 in (3.197). By the same reasoning as the proof of Theorem 3.134, there exists a unique additive mapping H W A ! B satisfying the inequality (3.192). It is easy to show that, the additive mapping H W A ! B is R-linear. Put D i in (3.197). By the same method as the proof of Theorem 3.134, one can obtain that f .2n ix/ if .2n x/ D lim D iH.x/ n!1 n!1 2n 2n

H.ix/ D lim

for all x 2 A. For each element  2 C,  D s C it for all s; t 2 R. So, we have H.x/ D H.sx C itx/ D sH.x/ C tH.ix/ D sH.x/ C itH.x/ D .s C it/H.x/ D H.x/ for all  2 C and x 2 A and H. x C y/ D H. x/ C H. y/ D H.x/ C H.y/ for all ; 2 C and x; y 2 A. Hence the additive mapping H W A ! B is C-linear. The rest of the proof is the same as the proof of Theorem 3.134. This completes the proof.  From now on, assume that A is a JC -algebra of real rank zero, where “real rank zero” means that the set of invertible self-adjoint elements is dense in the set of self-adjoint elements (see [54]). Now, we investigate continuous -homomorphisms between JC -algebras. Theorem 3.137. Let f ; g; h W A ! B be continuous mappings satisfying f .0/ D 0, g.0/ D 0 and h.0/ D 0 and let f .2n u ı y/ D f .2n u/ ı f .y/, g.2n u ı y/ D g.2n u/ ı g.y/ and h.2n u ı y/ D h.2n u/ ı h.y/ for all u 2 I1 .Asa /, y 2 A and all n  0 for which there exists a function ' W A  A ! Œ0; 1/ satisfying (3.188), (3.189), (3.190) and (3.191). Then the mappings f ; g; h W A ! B are -homomorphisms.

3 Stability of Functional Equations in C -Algebras

158

Proof. By the same reasoning as the proof of Theorem 3.134, there exists a unique C-linear involution mapping H W A ! B satisfying the inequality (3.192). Since f .2n u ı y/ D f .2n u/ ı f .y/ for all u 2 I1 .Asa /, y 2 A and n  0, we have 1 1 f .2n u ı y/ D lim n f .2n u/ ı f .y/ n!1 2 2n D H.u/ ı f .y/

H.u ı y/ D lim

n!1

(3.198)

for all u 2 I1 .Asa / and y 2 A. By the additivity of H and (3.198), we have 2n H.u ı y/ D H.2n u ı y/ D H.u ı .2n y// D H.u/ ı f .2n y/ for all u 2 I1 .Asa / and y 2 A. Hence it follows that H.u ı y/ D

1 1 H.u/ ı f .2n y/ D H.u/ ı n f .2n y/ n 2 2

(3.199)

for all u 2 I1 .Asa / and y 2 A. Taking n ! 1 in (3.199), we obtain H.u ı y/ D H.u/ ı H.y/

(3.200)

for all u 2 I1 .Asa / and y 2 A. By (3.198) and (3.200), we have H.e/ ı H.y/ D H.e ı y/ D H.e/ ı f .y/ for all y 2 A. Since limn!1

f .2n e/ 2n

D H.e/ D e0 , we have H.y/ D f .y/

for all y 2 A. Similarly, H.y/ D g.y/ D h.y/ for all y 2 A. So H W A ! B is continuous. But, by the assumption that A has real rank zero, it is easy to show that I1 .Asa / is dense in fx 2 Asa W kxk D 1g. So, for each w 2 fz 2 Asa W kzk D 1g, there exists a sequence fj g such that j ! w as j ! 1 and j 2 I1 .Asa /. Since H W A ! B is continuous, it follows from (3.200) that   H.w ı y/ D H lim j ı y D lim H.j ı y/ j!1

j!1

  D lim H.j / ı H.y/ D H lim j ı H.y/ j!1

j!1

D H.w/ ı H.y/ for all w 2 fz 2 Asa W kzk D 1g and y 2 A.   For each x 2 A, let x D xCx C i xx , where x1 WD 2 2i self-adjoint.

(3.201) xCx 2

and x2 WD

xx 2i

are

3.11 Stability of -Homomorphisms in JC -Algebras

159

First, consider the case that x1 ¤ 0; x2 ¤ 0. Since H W A ! B is C-linear, it follows from (3.201) that   x1 x2 ı y C ikx2 k ıy H.x ı y/ D H.x1 ı y C ix2 ı y/ D H kx1 k kx1 k kx2 k    x  x 1 2 D kx1 kH ı y C ikx2 kH ıy kx1 k kx2 k  x   x  1 2 ı H.y/ C ijjx2 jjH ı H.y/ D kx1 kH kx1 k kx2 k n   x1  x2 o D H kx1 k C iH kx2 k ı H.y/ kx1 k kx2 k D H.x1 C ix2 / ı H.y/ D H.x/ ı H.y/ for all y 2 A. Next, consider the case that x1 ¤ 0 and x2 D 0. Since H W A ! B is C-linear, it follows from (3.201) that     x x1 1 ı y D kx1 kH ıy H.x ı y/ D H.x1 ı y/ D H kx1 k kx1 k kx1 k    x  x1 1 ı H.y/ D H kx1 k ı H.y/ D kx1 kH kx1 k kx1 k D H.x1 / ı H.y/ D H.x/ ı H.y/ for all y 2 A. Finally, consider the case that x1 D 0; x2 ¤ 0. Since H W A ! B is C-linear, it follows from (3.201) that     x x2 2 ı y D ikx2 kH ıy H.x ı y/ D H.ix2 ı y/ D H ikx2 k kx2 k kx2 k   x  x2  2 ı H.y/ D H ikx2 k ı H.y/ D ikx2 kH kx2 k kx2 k D H.ix2 / ı H.y/ D H.x/ ı H.y/ for all y 2 A. Hence we have H.x ı y/ D H.x/ ı H.y/ for all x; y 2 A. Therefore, the mappings f ; g; h W A ! B are -homomorphisms. This completes the proof.  Corollary 3.138. Let f ; g; h W A ! B be continuous mappings satisfying f .0/ D 0, g.0/ D 0 and h.0/ D 0 and let f .2n u ı y/ D f .2n u/ ı f .y/, g.2n u ı y/ D g.2n u/ ı g.y/ and h.2n u ı y/ D h.2n u/ ı h.y/ for all u 2 I1 .Asa /, y 2 A and n  0 for which there

3 Stability of Functional Equations in C -Algebras

160

exist constants   0 and p 2 Œ0; 1/ such that    x C y     g.x/  h.y/  .kxkp C kykp / 2f 2 and kf .2n u /  f .2n u/ kł 2npC1  for all 2 T1 , u 2 I1 .Asa /, x; y 2 A n f0g and n  0. If limn!1 the mappings f ; g; h W A ! B are -homomorphisms.

f .2n e/ 2n

D e0 , then

Proof. Define '.x; y/ D .kxkp C kykp / for all x; y 2 A n f0g and then apply Theorem 3.137. Then we get the desired result.  Remark 3.139. Let f ; g; h W A ! B be continuous mappings satisfying f .0/ D 0, g.0/ D 0 and h.0/ D 0 and let f .2n u ı y/ D f .2n u/ ı f .y/, g.2n u ı y/ D g.2n u/ ı g.y/ and h.2n u ı y/ D h.2n u/ ı h.y/ for all u 2 I1 .Asa /, y 2 A and n  0 for which there exists a function ' W A  A ! Œ0; 1/ satisfying (3.188), (3.190), (3.191) and (3.197). Then the mappings f ; g; h W A ! B are -homomorphisms. Note that there exists a unique C-linear mapping H W A ! B satisfying the system of the inequalities (3.192).

3.11.2 Stability of -Homomorphisms in JC -Algebras Now, we prove the Hyers-Ulam stability of -homomorphisms in JC -algebras. Theorem 3.140. Let f ; g; h W A ! B be mappings with f .0/ D 0, g.0/ D 0 and h.0/ D 0 for which there exists a function ' W A4 ! Œ0; 1/ such that '.x; Q y; z; w/ D

1 X

2j '.2j x; 2j y; 2j z; 2j w/ < 1;

(3.202)

jD0

   x C y C z ı w     g.x/  h.y/  f .z/ ı f .w/ 2f 2  '.x; y; z; w/; kf .2n u /  f .2n u/ k  '.2n u; 2n u; 0; 0/

(3.203) (3.204)

3.11 Stability of -Homomorphisms in JC -Algebras

161

for all 2 T1 , u 2 U.A/, x; y; z; w 2 A n f0g and n  0. Then there exists a unique -homomorphism H W A ! B such that      x  2f 2  H.x/  ".x/; kg.x/  H.x/k  '.x; 0; 0; 0/ C ".x/; kh.x/  H.x/k  '.0; x; 0; 0/ C ".x/

(3.205)

for all x 2 A n f0g, where ".x/ WD

1 X

2j .'.2j1 x; 0; 0; 0/ C '.0; 2j1 x; 0; 0/ C '.2j1 x; 2j1 x; 0; 0//

jD1

< 1: Proof. Put z D w D 0 and D 1 2 T1 in (3.203). By the same reasoning as in the proof of Theorem 3.134, there exists a unique C-linear involutive mapping H W A ! B satisfying the inequality (3.205). The C-linear mapping H W A ! B is given by H.x/ D lim

n!1

1 f .2n x/ 2n

(3.206)

for all x 2 A. It follows from (3.206) that f .22n x/ n!1 22n

H.x/ D lim

(3.207)

for all x 2 A. Let x D y D 0 in (3.203). Then we get  z ı w     f .z/ ı f .w/  '.0; 0; z; w/ 2f 2 for all z; w 2 A. Since

1 '.0; 0; 2n z; 2n w/ 22n



1 n n 2n '.0; 0; 2 z; 2 w/,

  1 1    n n n n 2 z ı 2 w  f .2 z/ ı f .2 w/ 2f  22n 2 1  2n '.0; 0; 2n z; 2n w/ 2 1  n '.0; 0; 2n z; 2n w/ 2 for all z; w 2 A. By (3.206), (3.207) and (3.208), we have

it follows that (3.208) (3.209)

3 Stability of Functional Equations in C -Algebras

162

2H

z ı w 2

2f . 12 22n z ı w/ 2f . 21 2n z ı 2n w/ D lim n!1 n!1 22n 2n  2n  f .2n z/ f .2n w/  f .2n z/ f .2n w/ D lim D lim ı ı lim n n n n!1 n!1 n!1 2 2 2 2n D H.z/ ı H.w/ D lim

for all z; w 2 A. But, since H is C-linear, z ı w D H.z/ ı H.w/ H.z ı w/ D 2H 2 for all z; w 2 A. Hence the C-linear mapping H W A ! B is a -homomorphism satisfying the inequality (3.205). This completes the proof.  Corollary 3.141. Let f ; g; h W A ! B be a mapping with f .0/ D 0, g.0/ D 0 and h.0/ D 0 for which there exist constants   0 and p 2 Œ0; 1/ such that   x C y C z ı w      g.x/  h.y/  f .z/ ı f .w/ 2f 2  .kxkp C kykp C kzkp C kwkp / and kf .2n u /  f .2n u/ k  2npC1  for all 2 T1 , u 2 U.A/, x; y; z; w 2 A n f0g and n  0. Then there exists a unique -homomorphism H W A ! B such that  x  1    H.x/  kxkp ; 2f 2 2  2p 3  2p kg.x/  H.x/k  kxkp ; 2  2p 3  2p kh.x/  H.x/k  kxkp 2  2p for all x 2 A n f0g. Proof. Define '.x; y; z; w/ D .kxkp C kykp C kzkp C kwkp / and apply Theorem 3.140. Then we get the desired result.



3.11 Stability of -Homomorphisms in JC -Algebras

163

Remark 3.142. Let f ; g; h W A ! B be a mapping with f .0/ D 0, g.0/ D 0 and h.0/ D 0 for which there exists a function ' W A4 ! Œ0; 1/ satisfying (3.202) and (3.203) such that   x C y C z ı w      g.x/  h.y/  f .z/ ı f .w/  '.x; y; z; w/ 2f 2 for all x; y; z; w 2 A n f0g and D 1; i. If f .tx/ is continuous in t 2 R for each fixed x 2 A, then there exists a unique -homomorphism H W A ! B satisfying the inequality (3.205). Note that there exists a unique C-linear mapping H W A ! B satisfying the inequality (3.205).

Chapter 4

Stability of Functional Inequalities in Banach Algebras

In this chapter, we study functional inequalities in Banach algebras via the direct and fixed point methods. In Sect. 4.1, we consider the following additive functional inequality: kf .2x/ C f .2y/ C 2f .z/k  k2f .x C y C z/k: Next, we approximate the homomorphisms in proper CQ -algebras and derivations on proper CQ -algebras associated with the above additive functional inequality by direct method. In Sect. 4.2, we consider the functional inequality kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k: Next, we prove that, if f W A ! B is a multiplicative mapping such that kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k for all x; y; z; w 2 A and all 2 T WD f 2 C W jj D 1g. Then the mapping f W A ! B is a C -algebra homomorphism. Moreover, by using fixed point method, we prove the Hyers-Ulam stability of the functional following inequality: kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k p

p

p

p

C.kxkp C kykp C kzkp C kwkp C kxk 4  kyk 4  kzk 4  kwk 4 / in real Banach spaces.

© Springer International Publishing Switzerland 2015 Y.J. Cho et al., Stability of Functional Equations in Banach Algebras, DOI 10.1007/978-3-319-18708-2_4

165

166

4 Stability of Functional Inequalities in Banach Algebras

4.1 Stability of Additive Functional Inequalities in Banach Algebras In this section, by the direct method, we prove the Hyers-Ulam stability of the following additive functional inequality: kf .2x/ C f .2y/ C 2f .z/k  k2f .x C y C z/k:

(4.1)

Then we consider homomorphisms in proper CQ -algebras and derivations on proper CQ -algebras associated with the additive functional inequality (4.1) (see [181, 182, 186, 254]).

4.1.1 Stability of C-Linear Mappings in Banach Spaces Now, we investigate the Hyers-Ulam stability of C-linear mappings in Banach spaces associated with the additive functional inequality. Now, we assume that X; Y are Banach spaces. Lemma 4.1. Let f W X ! Y be a mapping satisfying the following: kf .2x/ C f .2y/ C 2f .z/kY  k2f .x C y C z/kY

(4.2)

for all x; y; z 2 X. Then f is additive. Proof. Letting x D y D z D 0 in (4.2), we get k4f .0/kY  k2f .0/kY and so f .0/ D 0. Letting z D 0 and replacing y by x in (4.2), we get kf .2x/ C f .2x/kY  k2f .0/kY D 0 for all x 2 X. Hence f .2x/ D f .2x/ and so f .x/ D f .x/ for all x 2 X. Letting y D 0 and replacing z by x in (4.2), we get kf .2x/ C 2f .x/kY  k2f .0/kY D 0 for all x 2 X. Thus we have f .2x/ D 2f .x/ for all x 2 X. Letting replacing z by x  y in (4.2), we get kf .2x/ C f .2y/  2f .x C y/kY  k2f .0/kY D 0 for all x; y 2 X. Thus we have f .2x C 2y/ D f .2x/ C f .2y/

4.1 Stability of Additive Functional Inequalities in Banach Algebras

167

for all x 2 X and so f .x C y/ D f .x/ C f .y/ for all x; y 2 X. This completes the proof.



Theorem 4.2. Let f W X ! Y be a mapping with f .0/ D 0. If there exists a function ' W X 3 ! Œ0; 1/ satisfying the following; kf .2x/ C f .2y/ C 2f .z/kY  k2f .x C y C z/kY C '.x; y; z/

(4.3)

1 X  1  '.x; Q y; z/ WD ' .2/j x; .2/j y; .2/j z < 1 j 2 jD0

(4.4)

and

for all x; y; z 2 X, then there exists a unique additive mapping L W X ! Y such that kf .x/  L.x/kY 

1 '.0; Q x; x/ 2

(4.5)

for all x 2 X. Proof. Replacing x; y; z by 0; .2/n x; .2/n x, respectively, and dividing by 2nC1 in (4.3). Since f .0/ D 0, we get    f ..2/nC1 x/ f ..2/n x/  1 n n    .2/nC1  .2/n   2nC1 '.0; .2/ x; .2/ x/ Y for all x 2 X. From the above inequality, we have    n1  X  f ..2/jC1 x/ f ..2/j x/   f ..2/n x/ f ..2/q x/       .2/jC1  .2/j   .2/n  .2/q   Y Y jDq 

n1 X 1 '.0; .2/j x; .2/j x/ jC1 2 jDq

x/ for all x 2 X and q; n  1 with q < n. From (4.4), the sequence f f ..2/ g is .2/n n

x/ a Cauchy sequence for all x 2 X. Since Y is complete, the sequence f f ..2/ .2/n g converges for all x 2 X and so we can define a mapping L W X ! Y by n

f ..2/n x/ n!1 .2/n

L.x/ WD lim for all x 2 X.

168

4 Stability of Functional Inequalities in Banach Algebras

In order to prove that L satisfies (4.5), if we put q D 0 and let n ! 1 in the above inequality, then we obtain kf .x/  L.x/kY 

1 X 1 1 '.0; .2/j x; .2/j x/ D '.0; Q x; x/ jC1 2 2 jD0

for all x 2 X. Replacing x; y; z by .2/n x; .2/n y; .2/n z, respectively, and dividing by 2n in (4.3), we get    f ..2/n 2x/ f ..2/n 2y/ 2f ..2/n z/     .2/n C .2/n C .2/n  Y   n  f ..2/ .x C y C z//   C 1 '..2/n x; .2/n y; .2/n z/   2 .2/n 2n Y for all x; y; z 2 X. Since (4.4) gives lim

n!1

1 '..2/n x; .2/n y; .2/n z/ D 0 2n

for all x; y; z 2 X, if we let n ! 1 in the above inequality, then we get kL.2x/ C L.2y/ C 2L.z/kY  k2L.x C y C z/kY ; and so L is additive by Lemma 4.1. Now, to prove the uniqueness of L, let L0 W X ! Y be another additive mapping satisfying (4.5). Since L and L0 are additive, we have  1  L.2n x/  L0 .2n x/ Y n 2   1  n .kL.2n x/  f .2n x/kY C L0 .2n x/  f .2n x/Y / 2 1  n  2'.0; Q 2n x; 2n x/ 2 1 X 1 D2 '.0; .2/jCn x; .2/jCn x/; nCj 2 jD0

kL.x/  L0 .x/kY D

which goes to zero as n ! 1 for all x 2 X by (4.4). Consequently, L is a unique additive mapping satisfying (4.5). This completes the proof. 

4.1 Stability of Additive Functional Inequalities in Banach Algebras

Corollary 4.3. Let f W X ! Y be a mapping. If there exists a function ' W X 3 ! Œ0; 1/ satisfying (4.3) and   1 X x y z i <1 2' ; ; '.x; Q y; z/ WD .2/i .2/i .2/i iD1

169

(4.6)

for all x; y; z 2 X, then there exists a unique additive mapping L W X ! Y such that kf .x/  L.x/kY 

1 '.0; Q x; x/ 2

(4.7)

for all x 2 X. Proof. Since '.0; Q 0; 0/ < 1 in (4.6), we have '.0; 0; 0/ D 0 and so f .0/ D 0. x x n1 Replacing x; y; z by 0;  .2/ in (4.3), n ; .2/n , respectively, and multiplying by 2 we get        x x n .2/n1 f   .2/ f  n1 n .2/ .2/ Y   x x n1  2 ' 0;  ; .2/n .2/n for all x 2 X. From the above inequality, we have        x x q .2/n f   .2/ f  n q .2/ .2/ Y      n X   x x i i1    .2/ f .2/i  .2/ f .2/i1  Y iDqC1   n X x x  2i1 ' 0;  ; i .2/i .2/ iDqC1  

x for all x 2 X and q; n  1 with q < n. From (4.6), the sequence .2/n f .2/ n is sequence for all x 2 X. Since Y is complete, the sequence a Cauchy 

 x converges for all x 2 X and so we can define a mapping .2/n f .2/ n L W X ! Y by

 L.x/ WD lim .2/n f n!1

x .2/n



for all x 2 X. In order to prove that L satisfies (4.7), if we put q D 0 and let n ! 1 in the above inequality, then we obtain   1 X 1 x x Q x; x/ D '.0; 2i1 ' 0;  ; kf .x/  L.x/kY  i i .2/ .2/ 2 iD1

170

4 Stability of Functional Inequalities in Banach Algebras

for all x 2 X. Replacing x; y; z by by 2n in (4.3), we get

y x z .2/n ; .2/n ; .2/n ,

respectively, and multiplying

         2x 2y z n n  .2/n f C .2/ C 2  .2/ f f  n n n .2/ .2/ .2/ Y       xCyCz  x y z n  C 2n ' 2  .2/  f ; ;   .2/n .2/n .2/n .2/n Y for all x; y; z 2 X. Since (4.6) gives   x y z D0 lim 2n ' ; ; n!1 .2/n .2/n .2/n for all x; y; z 2 X, if we let n ! 1 in the above inequality, then we get kL.2x/ C L.2y/ C 2L.z/kY  k2L.x C y C z/kY and so L is additive by Lemma 4.1. The rest of the proof is the same as in the corresponding part of the proof of Theorem 4.2. This completes the proof.  Lemma 4.4. Let f W X ! Y be a mapping satisfying kf .2x/ C f .2y/ C 2f .z/kY  k2f .x C y C z/kY

(4.8)

for all 2 T1 and all x; y; z 2 X. Then f is C-linear. Proof. If we put D 1 in (4.8), then f is additive by Lemma 4.1. Replacing x; y; z by x; x; 0 in (4.8), respectively, we get f .2 x/C f .2x/ D 0 and so f . x/ D f .x/ for all 2 T1 and x 2 X. Thus we have f . x C x/ N D f . x/ C f . x/ N D f .x/ C f N .x/ for all 2 T1 and x 2 X and so f .tx/ D tf .x/ for any real number t with jtj  1 and all x 2 X. On the other hand, since f .2x/ D 2f .x/, we get f .2n x/ D 2n f .x/ for all n  1. So, for any real number t, there exists an integer n  1 with jtj  2n . Thus we have  t   t  t f .tx/ D f 2n  n x D 2n f n x D 2n  n f .x/ D tf .x/: 2 2 2 Now, we consider any ˛ 2 C with ˛ D t C si for some real numbers t; s. Since f .ix/ D if .x/ all x 2 X, we have f .˛x/ D f .tx/ C f .six/ D tf .x/ C sf .ix/ D tf .x/ C sif .x/ D ˛f .x/ and so f is C-linear. This completes the proof.



4.1 Stability of Additive Functional Inequalities in Banach Algebras

171

Theorem 4.5. Let f W X ! Y be a mapping with f .0/ D 0. If there exists a function ' W X 3 ! Œ0; 1/ satisfying (4.4) and kf .2x/ C f .2y/ C 2f .z/kY  k2f .x C y C z/kY C '.x; y; z/

(4.9)

for all 2 T1 and x; y; z 2 X, then there exists a unique C-linear mapping L W X ! Y satisfying (4.5). Proof. If we put D 1 in (4.9), then, by Theorem 4.2, there exists a unique additive mapping L W X ! Y defined by f ..2/n x/ n!1 .2/n

L.x/ WD lim

for all x 2 X, which satisfies (4.5). By the similar method to the corresponding part in the proof of Theorem 4.2, L satisfies kL.2x/ C L.2y/ C 2L.z/kY  k2L.x C y C z/kY for all 2 T1 and x; y; z 2 X. Thus Lemma 4.4 gives that L is C-linear. This completes the proof.  Corollary 4.6. Let f W X ! Y be a mapping. If there exists a function ' W X 3 ! Œ0; 1/ satisfying (4.6) and (4.9), then there exists a unique C-linear mapping L W X ! Y satisfying (4.7). Proof. If we put D 1 in (4.9), then, by Corollary 4.3, there exists a unique additive mapping L W X ! Y defined by  L.x/ WD lim .2/n f n!1

x .2/n



for all x 2 X which satisfies (4.7). The rest of the proof is the same as in the corresponding part of the proof of Theorem 4.5. 

4.1.2 Stability of Homomorphisms in Proper CQ -Algebras Now, we investigate the Hyers-Ulam stability of isomorphisms in proper CQ algebras associated with the additive functional inequality. Theorem 4.7. Let f W A ! B be a mapping with f .0/ D 0. If there exists a function ' W A3 ! Œ0; 1/ satisfying kf .2x/ C f .2y/ C 2f .z/kB  k2f .x C y C z/kB C '.x; y; z/

(4.10)

172

4 Stability of Functional Inequalities in Banach Algebras

and '.x; Q y; z/ WD

1 X  1  ' .2/j x; .2/j y; .2/j z < 1; j 2 jD0

(4.11)

for all 2 T1 and x; y; z 2 A. If, in addition, there exists a function  W A2 ! Œ0; 1/ satisfying kf .xy/  f .x/f .y/kB  .x; y/

(4.12)

and lim

n!1

1 ..2/n x; .2/n y/ D 0 4n

(4.13)

for all x; y 2 A whenever the multiplication is defined, then there exists a unique proper CQ -algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

1 '.0; Q x; x/ 2

(4.14)

for all x 2 A. Proof. By Theorem 4.5, we have a unique C-linear mapping h W A ! B defined by f ..2/n x/ n!1 .2/n

h.x/ WD lim

for all x 2 A which satisfies (4.14). Now, we show that h.xy/ D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined. Replacing x; y by .2/n x; .2/n y, respectively, and dividing by 4n in (4.12), we get   1   Œf ..2/n x.2/n y/  f ..2/n x/f ..2/n y/  4n  

B

1 ..2/n x; .2/n y/ 4n

for all x; y 2 A whenever the multiplication is defined. Also, we have 1 f ..2/2n xy/ n n f ..2/ x.2/ y/ D lim D h.xy/ n!1 4n n!1 .2/2n lim

4.1 Stability of Additive Functional Inequalities in Banach Algebras

173

and 1 f ..2/n x/ f ..2/n x/ n n f ..2/ x/f ..2/ y/ D lim  lim n!1 4n n!1 n!1 .2/n .2/n lim

D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined. If we let n ! 1 in the above inequality, then (4.13) gives h.xy/ D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined. This completes the proof.  Corollary 4.8. Let ; p be nonnegative real numbers with p < 1 and f W A ! B be a mapping satisfying kf .2x/ C f .2y/ C 2f .z/kB p

p

p

 k2f .x C y C z/kB C .kxkA C kykA C kzkA / and 2p

2p

kf .xy/  f .x/f .y/kB  .kxkA C kykA / for all 2 T1 and x; y; z 2 A whenever the multiplication is defined. Then there exists a unique proper CQ -algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

2 p kxkA 2  2p

for all x 2 A Proof. Let ' W A3 ! Œ0; 1/ be a mapping defined by p

p

p

'.x; y; z/ D .kxkA C kykA C kzkA /: for all x; y; z 2 A. For any p < 1, we have '.x; Q y; z/ W D

D

1 X  1  ' .2/j x; .2/j y; .2/j z j 2 jD0 1 X 2pj jD0

D

2j

p

p

p

.kxkA C kykA C kzkA /

2 p p p .kxkA C kykA C kzkA /: 2  2p

174

4 Stability of Functional Inequalities in Banach Algebras 2p

2p

In addition, let  W A2 ! Œ0; 1/ be .x; y/ D .kxkA C kykA /. For any p < 1, we have 1 22pn 2p 2p n n ..2/ x; .2/ y/ D lim .kxkA C kykA / D 0 n!1 4n n!1 4n lim

for all x; y 2 A. By applying Theorem 4.7, there exists a unique proper CQ -algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

2 1 p '.0; Q x; x/ D kxkA p 2 22

for all x 2 A. This completes the proof.



Corollary 4.9. Let ; p be nonnegative real numbers with p < 1 and f W A ! B be a mapping satisfying kf .2x/ C f .2y/ C 2f .z/kB p

p

p

 k2f .x C y C z/kB C .kxkA C kykA C kzkA / and p

p

kf .xy/  f .x/f .y/kB    kxkA  kykA for all x; y 2 A whenever the multiplication is defined. Then there exists a unique proper CQ -algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

2 p kxkA 2  2p

for all x 2 A Proof. Let a mapping ' W A3 ! Œ0; 1/ be defined by p

p

p

'.x; y; z/ D .kxkA C kykA C kzkA / for all x; y; z 2 A and  W A2 ! Œ0; 1/ be a mapping defined by p

p

.x; y/ D   kxkA  kykA for all x; y 2 A. When p < 1, we have '.x; Q y; z/ < 1 and 1 22pn p p ..2/n x; .2/n y/ D lim n    kxkA  kykA D 0 n n!1 4 n!1 4 lim

4.1 Stability of Additive Functional Inequalities in Banach Algebras

175

for all x; y; z 2 A. By applying Theorem 4.7, there exists a unique proper CQ algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

2 1 p '.0; Q x; x/ D kxkA 2 2  2p

for all x 2 A. This completes the proof.



Theorem 4.10. Let f W A ! B be a mapping. Suppose that there exists a function ' W A3 ! Œ0; 1/ satisfying (4.10) and '.x; Q y; z/ WD

1 X

 2i '

iD1

x y z ; ; .2/i .2/i .2/i

 <1

(4.15)

for all x; y; z 2 A If, in addition, there exists a function  W A2 ! Œ0; 1/ satisfying (4.12) and  lim 4n 

n!1

x y ; .2/n .2/n

 D0

(4.16)

for all x; y 2 A whenever the multiplication is defined, then there exists a unique proper CQ -algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

1 '.0; Q x; x/ 2

(4.17)

for all x 2 A. Proof. By Corollary 4.12, we have a unique C-linear mapping h W A ! B defined by  h.x/ WD lim .2/n f n!1

x .2/n



for all x 2 A, which satisfies (4.17). Now, replacing x; y by and multiplying by 4n in (4.12), we get

x ; y , .2/n .2/n

respectively,

   n 4 f 

       y x x y    f f  n n n n .2/ .2/ .2/ .2/ B   x y  4n  ; .2/n .2/n

for all x; y 2 A whenever the multiplication is defined. Since  lim 4 f n

n!1

x y  .2/n .2/n



2n

D lim .2/ f n!1



xy .2/2n

 D h.xy/

176

4 Stability of Functional Inequalities in Banach Algebras

and  y lim 4 f n!1 .2/n     x y n D lim .2/n f  lim .2/ f n!1 n!1 .2/n .2/n 

n

x .2/n

  f

D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined. If we let n ! 1 in the above inequality then (4.16) gives h.xy/ D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined. This completes the proof.  Corollary 4.11. Let ; p be nonnegative real numbers with p > 1 and f W A ! B be a mapping satisfying kf .2x/ C f .2y/ C 2f .z/kB p

p

p

 k2f .x C y C z/kB C .kxkA C kykA C kzkA / and 2p

2p

kf .xy/  f .x/f .y/kB  .kxkA C kykA / for all 2 T1 and x; y; z 2 A whenever the multiplication is defined. Then there exists a unique proper CQ -algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

2 p kxkA 2p  2

for all x 2 A Proof. Let ' W A3 ! Œ0; 1/ be a mapping defined by p

p

p

'.x; y; z/ D .kxkA C kykA C kzkA / for all x; y; z 2 A. When p > 1, we get '.x; Q y; z/ W D

1 X iD1

 2' i

y z x ; ; .2/i .2/i .2/i



D

1 X 2i p p p .kxkA C kykA C kzkA / pi 2 iD1

D

2 p p p .kxkA C kykA C kzkA / 2p  2

4.1 Stability of Additive Functional Inequalities in Banach Algebras

177

for all x; y; z 2 A. In addition, let  W A2 ! Œ0; 1/ be a mapping defined by 2p

2p

.x; y/ D .kxkA C kykA / for all x; y 2 A. When p > 1, we get  lim 4n 

n!1

x y ; .2/n .2/n



4n 2p 2p .kxkA C kykA / D 0 n!1 22pn

D lim

for all x; y 2 A. By applying Theorem 4.10, there exists a unique proper CQ algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

2 1 p '.0; Q x; x/ D p kxkA 2 2 2

for all x 2 A. This completes the proof.



Corollary 4.12. Let ; p be nonnegative real numbers with p > 1 and f W A ! B be a mapping satisfying kf .2x/ C f .2y/ C 2f .z/kB p

p

p

 k2f .x C y C z/kB C .kxkA C kykA C kzkA / and p

p

kf .xy/  f .x/f .y/kB    kxkA  kykA for all 2 T1 and x; y; z 2 A whenever the multiplication is defined. Then there exists a unique proper CQ -algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

2 p kxkA 2p  2

for all x 2 A. Proof. Let ' W A3 ! Œ0; 1/ be a mapping defined by p

p

p

'.x; y; z/ D .kxkA C kykA C kzkA / for all x; y; z 2 A and  W A2 ! Œ0; 1/ be a mapping defined by p

p

.x; y/ D   kxkA  kykA for all x; y 2 A. For any p > 1, we have '.x; Q y; z/ < 1 and  lim 4  n

n!1

x y ; .2/n .2/n



4n p p    kxkA  kykA D 0 n!1 22pn

D lim

178

4 Stability of Functional Inequalities in Banach Algebras

for all x; y; z 2 A. By applying Theorem 4.10, there exists a unique proper CQ algebra homomorphism h W A ! B such that kf .x/  h.x/kB 

2 1 p '.0; Q x; x/ D p kxkA 2 2 2

for all x 2 A. This completes the proof.



4.1.3 Stability of Derivations in Proper CQ -Algebras Now, we consider the Hyers-Ulam stability of derivations on proper CQ -algebras associated with the additive functional inequality. Theorem 4.13. Let f W A ! A be a mapping with f .0/ D 0. Suppose that there exists a function ' W A3 ! Œ0; 1/ such that kf .2x/ C f .2y/ C 2f .z/kA  k2f .x C y C z/kA C '.x; y; z/

(4.18)

1 X  1  '.x; Q y; z/ WD ' .2/j x; .2/j y; .2/j z < 1 j 2 jD0

(4.19)

and

for all 2 T1 and x; y; z 2 A. If, in addition, there exists a function such that kf .xy/  f .x/y  xf .y/kA 

.x; y/

W A2 ! Œ0; 1/

(4.20)

and lim

n!1

1 ..2/n x; .2/n y/ D 0 4n

(4.21)

for all x; y 2 A whenever the multiplication is defined, then there exists a unique derivation ı on A such that kf .x/  ı.x/kA 

1 '.0; Q x; x/ 2

(4.22)

for all x 2 A. Proof. By Theorem 4.5, we have a unique C-linear mapping ı W A ! A defined by f ..2/n x/ n!1 .2/n

ı.x/ WD lim for all x 2 A which satisfies (4.22).

4.1 Stability of Additive Functional Inequalities in Banach Algebras

179

Now, we show that ı.xy/ D ı.x/ı.y/ for all x; y 2 A whenever the multiplication is defined. Replacing x; y by .2/n x; .2/n y, respectively, and dividing by 4n in (4.3), we have   1   Œf ..2/n x.2/n y/  f ..2/n x/.2/n y  .2/n xf ..2/n y/  4n  A 

1 ..2/n x; .2/n y/ 4n

for all x; y 2 A whenever the multiplication is defined. Also, we have 1 f ..2/2n xy/ n n f ..2/ x.2/ y/ D lim D ı.xy/; n!1 4n n!1 .2/2n lim

1 f ..2/n x/ .2/n y n n f ..2/ x/  .2/ y D lim  D ı.x/y n!1 4n n!1 .2/n .2/n lim

and 1 .2/n x .2/n y n n .2/ x  f ..2/ y/ D lim  D xı.y/ n!1 4n n!1 .2/n .2/n lim

for all x; y 2 A whenever the multiplication is defined. If we let n ! 1 in the above inequality, then (4.21) gives ı.xy/ D ı.x/y  xı.y/ for all x; y 2 A whenever the multiplication is defined. This completes the proof.  Corollary 4.14. Let ; p be nonnegative real numbers with p < 1 and f W A ! A be a mapping satisfying kf .2x/ C f .2y/ C 2f .z/kA p

p

p

 k2f .x C y C z/kA C .kxkA C kykA C kzkA / and 2p

2p

kf .xy/  f .x/y  xf .y/kA  .kxkA C kykA / for all 2 T1 and x; y; z 2 A whenever the multiplication is defined. Then there exists a unique derivation ı on A such that kf .x/  ı.x/kA  for all x 2 A.

2 p kxkA 2  2p

180

4 Stability of Functional Inequalities in Banach Algebras

Proof. Let ' W A3 ! Œ0; 1/ be a mapping defined by p

p

p

'.x; y; z/ D .kxkA C kykA C kzkA / for all x; y; z 2 A. For any p < 1, we have '.x; Q y; z/ W D

D

1 X  1  ' .2/j x; .2/j y; .2/j z j 2 jD0 1 X 2pj jD0

D In addition, let

2j

p

p

p

.kxkA C kykA C kzkA /

2 p p p .kxkA C kykA C kzkA /: 2  2p

W A2 ! Œ0; 1/ be a mapping defined by 2p

2p

.x; y/ D .kxkA C kykA / for all x; y 2 A. For any p < 1, we have 1 22pn 2p 2p n n ..2/ x; .2/ y/ D lim .kxkA C kykA / D 0 n!1 4n n!1 4n lim

for all x; y 2 A. By applying Theorem 4.13, there exists a unique proper CQ -algebra homomorphism h W A ! B such that kf .x/  ı.x/kA 

2 1 p '.0; Q x; x/ D kxkA 2 2  2p

for all x 2 A. This completes the proof.



Corollary 4.15. Let ; p be nonnegative real numbers with p < 1 and f W A ! B be a mapping satisfying p

p

p

kf .2x/ C f .2y/ C 2f .z/kA  k2f .x C y C z/kA C .kxkA C kykA C kzkA / and p

p

kf .xy/  f .x/y  xf .y/kA    kxkA  kykA for all x; y; z 2 A whenever the multiplication is defined. Then there exists a unique derivation ı on A such that kf .x/  ı.x/kA  for all x 2 A.

2 p kxkA 2  2p

4.1 Stability of Additive Functional Inequalities in Banach Algebras

181

Proof. Let ' W A3 ! Œ0; 1/ be a mapping defined by p

p

p

'.x; y; z/ D .kxkA C kykA C kzkA / for all x; y; z 2 A and

W A2 ! Œ0; 1/ be a mapping by p

p

.x; y/ D   kxkA  kykA for all x; y 2 A. For any p < 1, we have '.x; Q y; z/ < 1 and 1 22pn p p n n ..2/ x; .2/ y/ D lim    kxkA  kykA D 0 n!1 4n n!1 4n lim

for all x; y; z 2 A. By applying Theorem 4.13, there exists a unique proper CQ algebra homomorphism ı W A ! A such that kf .x/  ı.x/kA 

2 1 p '.0; Q x; x/ D kxkA 2 2  2p

for all x 2 A. This completes the proof.



Theorem 4.16. Let f W A ! A be a mapping. Suppose that there exists a function ' W A3 ! Œ0; 1/ satisfying (4.18) and '.x; Q y; z/ WD

1 X

 2i '

iD1

x y z ; ; j j .2/ .2/ .2/j

 <1

for all x; y; z 2 A. If, in addition, there exists a function satisfying (4.20) and  lim 4n

n!1

x y ; n .2/ .2/n

(4.23) W A2 ! Œ0; 1/

 D0

(4.24)

for all x; y 2 A, then there exists a unique derivation ı on A satisfying kf .x/  ı.x/kA 

1 '.0; Q x; x/ 2

(4.25)

for all x 2 A. Proof. By Corollary 4.18, we have a unique C-linear mapping ı W A ! A defined by  ı.x/ WD lim .2/n f n!1

x .2/n



182

4 Stability of Functional Inequalities in Banach Algebras

for all x 2 A, which satisfies (4.25). Now, replacing x; y by and multiplying by 4n in (4.20), we get

y x .2/n ; .2/n ,

respectively,

   n 4 f 

       x x y y   f f   n n n n .2/ .2/ .2/ .2/ A   x y  4n ; .2/n .2/n

for all x; y 2 A whenever the multiplication is defined. Also, we have  lim 4 f n

n!1

x y  n .2/ .2/n 



2n



D lim .2/ f n!1

xy .2/2n

 D ı.xy/;



y .2/n   .2/n y x n  lim D lim .2/ f D ı.x/y n!1 n!1 .2/n .2/n

lim 4n f

n!1

x .2/n



and x f lim 4 n!1 .2/n n



y .2/n



.2/n x  lim .2/n f n!1 .2/n n!1



D lim

y .2/n

 D xı.y/

for all x; y 2 A whenever the multiplication is defined. If we let n ! 1 in the above inequality, then (4.24) gives ı.xy/ D ı.x/y  xı.y/ for all x; y 2 A whenever the multiplication is defined. This completes the proof.  Corollary 4.17. Let ; p be nonnegative real numbers with p > 1 and f W A ! A be a mapping satisfying kf .2x/ C f .2y/ C 2f .z/kA p

p

p

 k2f .x C y C z/kA C .kxkA C kykA C kzkA / and 2p

2p

kf .xy/  f .x/y  xf .y/kA  .kxkA C kykA /

4.1 Stability of Additive Functional Inequalities in Banach Algebras

183

for all 2 T1 and x; y; z 2 A whenever the multiplication is defined. Then there exists a unique derivation ı on A such that kf .x/  ı.x/kA 

2 p kxkA 2p  2

for all x 2 A. Proof. Let ' W A3 ! Œ0; 1/ be a mapping by p

p

p

'.x; y; z/ D .kxkA C kykA C kzkA / for all x; y; z 2 A. For any p > 1, we have '.x; Q y; z/ W D

1 X

 2' i

iD1

y z x ; ; .2/i .2/i .2/i



D

1 X 2i p p p .kxkA C kykA C kzkA / pi 2 iD1

D

2 p p p .kxkA C kykA C kzkA /: 2p  2

W A2 ! Œ0; 1/ be a mapping defined by

In addition, let

2p

2p

.x; y/ D .kxkA C kykA / for all x; y 2 A. For any p > 1, we get  lim 4

n!1

n

x y ; n .2/ .2/n



4n 2p 2p .kxkA C kykA / D 0 n!1 22pn

D lim

for all x; y 2 A. By applying Theorem 4.16, there exists a unique proper CQ algebra homomorphism ı W A ! A such that kf .x/  ı.x/kA 

2 1 p '.0; Q x; x/ D p kxkA 2 2 2

for all x 2 A. This completes the proof.



Corollary 4.18. Let ; p be nonnegative real numbers with p > 1 and f W A ! B be a mapping satisfying kf .2x/ C f .2y/ C 2f .z/kA p

p

p

 k2f .x C y C z/kA C .kxkA C kykA C kzkA /

184

4 Stability of Functional Inequalities in Banach Algebras

and p

p

kf .xy/  f .x/y  xf .y/kA    kxkA  kykA for all 2 T1 and x; y; z 2 A whenever the multiplication is defined. Then there exists a unique derivation ı on A such that kf .x/  ı.x/kA 

2 p kxkA 2

2p

for all x 2 A. Proof. Let ' W A3 ! Œ0; 1/ be a mapping defined by p

p

p

'.x; y; z/ D .kxkA C kykA C kzkA / for all x; y; z 2 A and

W A2 ! Œ0; 1/ be a mapping defined by p

p

.x; y/ D   kxkA  kykA for all x; y 2 A. When p > 1, we have '.x; Q y; z/ < 1 and  lim 4  n

n!1

x y ; .2/n .2/n



4n p p    kxkA  kykA D 0 n!1 22pn

D lim

for all x; y; z 2 A. By applying Theorem 4.10, there exists a unique proper CQ algebra homomorphism ı W A ! A such that kf .x/  ı.x/kA 

2 1 p '.0; Q x; x/ D p kxkA 2 2 2

for all x 2 A. This completes the proof.



4.2 Stability of Functional Inequalities over C -Algebras In this section, we investigate a module linear mapping associated with the following functional inequality. kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k:

(4.26)

This is applied to understand homomorphisms in C -algebras. Moreover, we prove the Hyers-Ulam stability of the functional following inequality:

4.2 Stability of Functional Inequalities over C -Algebras

185

kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k p

p

p

p

C.kxkp C kykp C kzkp C kwkp C kxk 4  kyk 4  kzk 4  kwk 4 /

(4.27)

in real Banach spaces. Using the fixed point method, we prove the Hyers-Ulam stability of the functional inequality (4.27) in real Banach spaces.

4.2.1 Functional Inequalities in Normed Modules over C -Algebras Throughout this section, let A be a unital C -algebra with the unitary group U.A/ and the unit e and let B be a C -algebra. Assume that X is a normed A-module with the norm k  k and Y is a normed A-module with the norm k  k. Now, we investigate an A-linear mapping associated with the functional inequality (4.26). Theorem 4.19. Let f W X ! Y be a mapping such that kf .x/ C f .y/ C f .z/ C uf .w/k  kf .x/ C f .y C z C uw/k

(4.28)

for all x; y; z; w 2 X and u 2 U.A/. Then the mapping f W X ! Y is A-linear. Proof. Letting x D y D z D w D 0 and u D e 2 U.A/ in (4.28), we have k4f .0/k  k2f .0/k: So f .0/ D 0. Letting x D w D 0 in (4.28), we have kf .y/ C f .z/k  kf .y C z/k

(4.29)

for all y; z 2 X. Replacing y and z by x and y C z C w in (4.29), respectively, we get kf .x/ C f .y C z C w/k  kf .x C y C z C w/k for all x; y; z; w 2 X and so kf .x/ C f .y/ C f .z/ C f .w/k  kf .x C y C z C w/k for all x; y; z; w 2 X. Letting z D w D 0 and y D x in (4.30), we have kf .x/ C f .x/k  kf .0/k D 0

(4.30)

186

4 Stability of Functional Inequalities in Banach Algebras

for all x 2 X. Hence f .x/ D f .x/ for all x 2 X. Letting z D x  y and w D 0 in (4.30), we have kf .x/ C f .y/  f .x C y/k D kf .x/ C f .y/ C f .x  y/k  kf .0/k D 0 for all x; y 2 X. Thus f .x C y/ D f .x/ C f .y/ for all x; y 2 X. Letting z D uw and x D y D 0 in (4.28), we have k  f .uw/ C uf .w/k D kf .uw/ C uf .w/k  k2f .0/k D 0 for all w 2 X and all u 2 U.A/. Thus we have f .uw/ D uf .w/

(4.31)

for all u 2 U.A/ and all w 2 X. Now, let a 2 A .a ¤ 0/ and M be an integer greater than 4jaj. Then we have j

a 1 2 1 j< <1 D : M 4 3 3

By Theorem 1 of Kadison and Pedersen [167], there exist three elements u1 ; u2 ; u3 2 U.A/ such that 3 Ma D u1 C u2 C u3 . So, by (4.31), we have 1 a  a  M  a  x D Mf 3 x D f 3 x 3 M 3 M 3 M M M D h.u1 x C u2 x C u3 x/ D .f .u1 x/ C f .u2 x/ C f .u3 x// 3 3 a M M  3 f .x/ D af .x/ D .u1 C u2 C u3 /f .x/ D 3 3 M

f .ax/ D f

M

3

for all x 2 X. Therefore, f W X ! Y is A-linear. This completes the proof.



Corollary 4.20. Let f W A ! B be a multiplicative mapping such that kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k

(4.32)

for all x; y; z; w 2 A and 2 T WD f 2 C W jj D 1g. Then the mapping f W A ! B is a C -algebra homomorphism. Proof. By Theorem 4.19, the multiplicative mapping f W A ! B is C-linear since C -algebras are normed modules over C. So, the multiplicative mapping f W A ! B is a C -algebra homomorphism. 

4.2 Stability of Functional Inequalities over C -Algebras

187

Assume that X is a real normed linear space and Y is a real Banach space. Now, we prove the Hyers-Ulam stability of the functional inequality (4.27) in real Banach spaces. Theorem 4.21. Assume that f W X ! Y is an odd mapping for which there exist the constants   0 and p 2 R such that p ¤ 1 and f W X ! Y satisfies the following functional inequality: kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k p

p

p

p

C.kxkp C kykp C kzkp C kwkp C kxk 4  kyk 4  kzk 4  kwk 4 /

(4.33)

for all x; y; z; w 2 X. Then there exists a unique Cauchy additive mapping A W X ! Y such that kf .x/  A.x/k 

2p C 2 kxkp j2p  2j

(4.34)

for all x 2 X. If, in addition, f W X ! Y is a mapping such that the transformation t ! f .tx/ is continuous in t 2 R for each fixed x 2 X; then A is an R-linear mapping. Proof. Since f is odd, f .0/ D 0 and f .x/ D f .x/ for all x 2 X. Letting x D 0, z D y and w D 2y in (4.33), we have k2f .y/  f .2y/k  .2 C 2p /jjyjjp

(4.35)

for all y 2 X and so   x  2 C 2p   jjxjjp  f .x/  2f 2 2p for all x 2 X. Hence we have  x  x  2 C 2p m1 X 2j  l  m  2 f f kxkp 2  l m p pj 2 2 2 2 jDl

(4.36)

for all m; l  1 with m > l and x 2 X. Assume that p > 1. It follows from (4.36) that the sequence f2k f . 2xk /g is a Cauchy sequence for all x 2 X. Since Y is complete, the sequence f2k f . 2xk /g converges and so one can define the mapping A W X ! Y by x A.x/ WD lim 2k f k k!1 2 for all x 2 X. Letting l D 0 and m ! 1 in (4.36), we get kf .x/  A.x/k 

2p C 2 jjxjjp 2p  2

188

4 Stability of Functional Inequalities in Banach Algebras

for all x 2 X. It follows from (4.33) that  x y z  w   k  2 f k C 2k f k C 2k f k C 2k f k  2 2 2 2       k x yCzCw  k    2 f k C 2 f  2 2k C

p p p p 2k  .kxkp C kykp C kzkp C kwkp C kxk 4  kyk 4  kzk 4  kwk 4 / (4.37) pk 2

for all x; y; z; w 2 X. Letting k ! 1 in (4.37), we get kA.x/ C A.y/ C A.z/ C A.w/k  kA.x/ C A.y C z C w/k

(4.38)

for all x; y; z; w 2 X. It is easy to show that A W X ! Y is odd. Letting w D y  z and x D 0 in (4.38), we get A.y C z/ D A.y/ C A.z/ for all y; z 2 X. So, there exists a Cauchy additive mapping A W X ! Y satisfying (4.34) for the case p > 1. Now, let T W X ! Y be another Cauchy additive mapping satisfying (4.33). Then we have  x  x    kA.x/  T.x/k D 2q A q  T q  2 2   x   x    x   x       2q L q  f q  C T q  f q  2 2 2 2 2p C 2 2  2q  p  kxkp ; 2  2 2pq which tends to zero as q ! 1 for all x 2 X. So, we can conclude that A.x/ D T.x/ for all x 2 X. This proves the uniqueness of A. Assume that p < 1. It follows from (4.35) that   p   f .x/  1 f .2x/  2 C 2 jjxjjp   2 2 for all x 2 X. Hence we have   m1 p X 2pj 1   f .2l x/  1 f .2m x/  2 C 2 kxkp  2l  j 2m 2 2 jDl

(4.39)

for all m; l  1 with m > l and x 2 X. It follows from (4.39) that the sequence f 21k f .2k x/g is a Cauchy sequence for all x 2 X. Since Y is complete, the sequence f 21k f .2k x/g converges and so one can define the mapping A W X ! Y by A.x/ WD lim

k!1

1  k  f 2x 2k

for all x 2 X. Letting l D 0 and m ! 1 in (4.39), we get

4.2 Stability of Functional Inequalities over C -Algebras

kf .x/  A.x/k 

189

2p C 2 jjxjjp 2  2p

for all x 2 X. The rest of the proof is similar to the above proof. So, there exists a unique Cauchy additive mapping A W X ! Y such that kf .x/  A.x/k 

2p C 2 kxkp j2p  2j

(4.40)

for all x 2 X. Assume that f W X ! Y is a mapping such that the transformation t ! f .tx/ is continuous in t 2 R for each fixed x 2 X. By the same reasoning as in the proof of Theorem 4.19, one can prove that A is an R-linear mapping. This completes the proof.  Using the fixed point method, we prove the Hyers-Ulam stability of the functional inequality (4.27) in Banach spaces. Theorem 4.22. Let f W X ! Y be an odd mapping for which there exists a function ' W X 4 ! Œ0; 1/ such that there exists L < 1 such that '.x; y; z; w/ 

1 L'.2x; 2y; 2z; 2w/ 2

for all x; y; z; w 2 X and kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k C '.x; y; z; w/

(4.41)

for all x; y; z; w 2 X. Then there exists a unique Cauchy additive mapping A W X ! Y such that kf .x/  A.x/k 

L '.0; x; x; 2x/ 2  2L

(4.42)

for all x 2 X. Proof. Consider the set S WD fg W X ! Yg and introduce the generalized metric on S as follows: d.g; h/ D inffK 2 RC W kg.x/  h.x/k  K'.0; x; x; 2x/; 8x 2 Xg: which .S; d/ is complete. Now, we consider the linear mapping J W S ! S defined by Jg.x/ WD 2g

x 2

190

4 Stability of Functional Inequalities in Banach Algebras

for all x 2 X. Now, we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 S. Since f W X ! Y is odd, f .0/ D 0 and f .x/ D f .x/ for all x 2 X. Letting z D y D x and w D 2x in (4.41), we have k2f .x/  f .2x/k D k2f .x/ C f .2x/k  '.0; x; x; 2x/

(4.43)

for all x 2 X. It follows from (4.43) that   x x  x   L     ' 0; ; ; x  '.0; x; x; 2x/ f .x/  2f 2 2 2 2 for all x 2 X. Hence d.f ; Jf /  A W X ! Y satisfying the following:

L 2.

By Theorem 1.3, there exists a mapping

(1) A is a fixed point of J, i.e., A

x 2

D

1 A.x/ 2

(4.44)

for all x 2 X. Then A W X ! Y is an odd mapping. The mapping A is a unique fixed point of J in the set M D fg 2 S W d.f ; g/ < 1g: This implies that A is a unique mapping satisfying (4.44) such that there exists a K 2 .0; 1/ satisfying kf .x/  A.x/k  K'.0; x; x; 2x/ for all x 2 X; (2) d.J n f ; A/ ! 0 as n ! 1. This implies the equality lim 2n f

n!1

x D A.x/ 2n

for all x 2 X; 1 (3) d.f ; A/  1L d.f ; Jf /, which implies the inequality d.f ; A/ 

L : 2  2L

This implies that the inequality (4.42) holds.

(4.45)

4.2 Stability of Functional Inequalities over C -Algebras

191

It follows from (4.41) and (4.45) that kA.x/ C A.y/ C A.z/ C A.w/k  kA.x/ C A.y C z C w/k for all x; y; z; w 2 X. By Theorem 4.19, the mapping A W X ! Y is a Cauchy additive mapping. Therefore, there exists a unique Cauchy additive mapping A W X ! Y satisfying (4.43). This completes the proof.  Corollary 4.23. Let r > 1 and  be nonnegative real numbers and f W X ! Y be an odd mapping such that kf .x/ C f .y/ C f .z/ C f .w/k  kf .x/ C f .y C z C w/k r

r

r

r

C.kxkr C kykr C kzkr C kwkr C kxk 4  kyk 4  kzk 4  kwk 4 /

(4.46)

for all x; y; z; w 2 X. Then there exists a unique Cauchy additive mapping A W X ! Y such that kf .x/  A.x/k 

2r C 2 kxkr 2r  2

for all x 2 X. Proof. The proof follows from Theorem 4.22 by taking '.x; y; z; w/ r

r

r

r

WD .kxkr C kykr C kzkr C kwkr C kxk 4  kyk 4  kzk 4  kwk 4 / for all x; y; z; w 2 X and L D 21r and so we get the desired result.



Remark 4.24. Let f W X ! Y be an odd mapping for which there exists a function ' W X 4 ! Œ0; 1/ satisfying (4.41). By the similar method to the proof of Theorem 4.22, one can show that, if there exists L < 1 such that x y z w ; ; ; '.x; y; z; w/  2L' 2 2 2 2 for all x; y; z; w 2 X, then there exists a unique Cauchy additive mapping A W X ! Y such that kf .x/  A.x/k 

1 '.0; x; x; 2x/ 2  2L

for all x 2 X. For the case 0 < r < 1, one can obtain the similar result to Corollary 4.23: Let 0 < r < 1,   0 be real numbers and f W X ! Y be an odd mapping satisfying (4.46). Then there exists a unique Cauchy additive mapping A W X ! Y such that

192

4 Stability of Functional Inequalities in Banach Algebras

kf .x/  A.x/k 

2 C 2r kxkr 2  2r

for all x 2 X.

4.2.2 On Additive Functional Inequalities in Normed Modules over C -Algebras In [12], An investigated the following additive functional inequality: kf .x/ C f .y/ C f .z/ C f .w/k  kf .x C y/ C f .z C w/k

(4.47)

in normed modules over a C -algebra. This is applied to understand homomorphisms in C -algebras. Moreover, he proved the Hyers-Ulam stability of the functional following inequality: kf .x/ C f .y/ C f .z/ C f .w/k  kf .x C y C z C w/k C kxkp kykp kzkp kwkp

(4.48)

in real Banach spaces, where  and p are positive real numbers with 4p ¤ 1. Gilányi [126] showed that, if f satisfies the functional following inequality: k2f .x/ C 2f .y/  f .x  y/k  kf .x C y/k;

(4.49)

then f satisfies the Jordan-von Neumann functional equation: 2f .x/ C 2f .y/ D f .x C y/ C f .x  y/: Fechner [113] and Gilányi [127] proved the Hyers-Ulam stability of the functional inequality (4.49). Park et al. [253] investigated the functional following inequality: kf .x/ C f .y/ C f .z/k  kf .x C y C z/k

(4.50)

in Banach spaces and proved the Hyers-Ulam stability of the functional inequality (4.50) in Banach spaces. Now, let A be a unital C -algebra with the unitary group U.A/ and the unit e and let B be a C -algebra. Assume that X is a normed A-module with the norm k  kX and Y is a normed A-module with the norm k  kY . Theorem 4.25. Let f W X ! Y be a mapping such that kuf .x/ C f .y/ C f .z/ C f .w/kY  kf .ux C y/ C f .z C w/kY

(4.51)

4.2 Stability of Functional Inequalities over C -Algebras

193

for all x; y; z; w 2 X and u 2 U.A/. Then the mapping f W X ! Y is A-linear. Proof. Letting x D y D z D w D 0 and u D e 2 U.A/ in (4.51), we have k4f .0/kY  k2f .0/kY and so f .0/ D 0. Letting z D w D 0 in (4.51), we have kf .x/ C f .y/kY  kf .x C y/kY

(4.52)

for all x; y 2 X. Replacing x and y by x C y and z C w in (4.52), respectively, we have kf .x C y/ C f .z C w/kY  kf .x C y C z C w/kY for all x; y; z; w 2 X and so kf .x/ C f .y/ C f .z/ C f .w/kY  kf .x C y C z C w/kY

(4.53)

for all x; y; z; w 2 X. Letting z D w D 0 and y D x in (4.53), we have kf .x/ C f .x/kY  kf .0/kY D 0 for all x 2 X. Hence f .x/ D f .x/ for all x 2 X. Letting z D x  y and w D 0 in (4.53), we have kf .x/ C f .y/  f .x C y/kY D kf .x/ C f .y/ C f .x  y/kY  kf .0/kY D 0 for all x; y 2 X. Thus we have f .x C y/ D f .x/ C f .y/ for all x; y 2 X. Letting y D ux and y D w D 0 in (4.51), we have kf .ux/  f .ux/kY D kf .ux/ C f .ux/kY  k2f .0/kY D 0 for all x 2 X and u 2 U.A/. Thus we have f .ux/ D uf .x/

(4.54)

for all u 2 U.A/ and x 2 X. Now, let a 2 A with a ¤ 0 and M be an integer greater than 4jaj. Then we have ˇaˇ 1 2 1 ˇ ˇ ˇ ˇ< <1 D : M 4 3 3

194

4 Stability of Functional Inequalities in Banach Algebras

By Theorem 1 of Kadison and Pedersen [167], there exist three elements u1 ; u2 ; u3 2 U.A/ such that 3 Ma D u1 C u2 C u3 and so, by (4.54), we have 1 a  a  M  a  x D Mf 3 x D f 3 x 3 M 3 M 3 M M M D h.u1 x C u2 x C u3 x/ D .f .u1 x/ C f .u2 x/ C f .u3 x// 3 3 a M M  3 f .x/ D af .x/ D .u1 C u2 C u3 /f .x/ D 3 3 M

f .ax/ D f

M

3

for all x 2 X. Therefore, f W X ! Y is A-linear. This completes the proof.



Corollary 4.26. Let f W A ! B be a multiplicative mapping such that k f .x/ C f .y/ C f .z/ C f .w/k  kf . x C y/ C f .z C w/k

(4.55)

for all x; y; z; w 2 A and 2 T WD f 2 C W jj D 1g. Then the mapping f W A ! B is a C -algebra homomorphism. Proof. By Theorem 4.25, the multiplicative mapping f W A ! B is C-linear since C -algebras are normed modules over C. So, the multiplicative mapping f W A ! B is a C -algebra homomorphism.  Theorem 4.27. Let X be a real normed linear space and Y be a real Banach space. Assume that f W X ! Y is an approximately additive odd mapping for which there exist the constants   0 and p 2 R such that 4p ¤ 1 and f satisfies the general Cauchy-Rassias inequality: kf .x/ C f .y/ C f .z/ C f .w/k  kf .x C y C z C w/k C kxkp kykp kzkp kwkp

(4.56)

for all x; y; z; w 2 X. Then there exists a unique additive mapping L W X ! Y such that kf .x/  L.x/k 

3p  kxk4p j81p  3j

(4.57)

for all x 2 X. If, in addition, f W X ! Y is a mapping such that the transformation t ! f .tx/ is continuous in t 2 R for each fixed x 2 X; then L is an R-linear mapping. Proof. Since f is odd, f .0/ D 0 and f .x/ D f .x/ for all x 2 X. Letting y D z D x and w D 3x in (4.56), we have k3f .x/  f .3x/k  3p jjxjj4p for all x 2 X and so   x       p jjxjj4p f .x/  3f 3 27

(4.58)

4.2 Stability of Functional Inequalities over C -Algebras

195

for all x 2 X. Hence we have m1  x  x   X 3j  l  jjxjj4p 3 f l  3m f m   p 3 3 27 jDl 81pj

(4.59)

for all m; l  1 with m > l and x 2 X. Assume that p > 14 . It follows from (4.59) that the sequence f3k f . 3xk /g is a Cauchy sequence for all x 2 X. Since Y is complete, the sequence f3k f . 3xk /g converges and so one can define the mapping L W X ! Y by L.x/ WD lim 3k f k!1

x 3k

for all x 2 X. Letting l D 0 and m ! 1 in (4.59), we have kf .x/  L.x/k 

3p  jjxjj4p 81p  3

for all x 2 X. It follows from (4.56) that  x y z  w   k  3 f k C 3k f k C 3k f k C 3k f k  3 3 3 3    k  k xCyCzCw    C 3  kxkp kykp kzkp kwkp  3 f  81pk 3k

(4.60)

for all x; y; z; w 2 X. Letting k ! 1 in (4.60), we get kL.x/ C L.y/ C L.z/ C L.w/k  kL.x C y C z C w/k

(4.61)

for all x; y; z; w 2 X. It is easy to show that L W X ! Y is odd. Letting z D x  y and w D 0 in (4.61), we get L.x C y/ D L.x/ C L.y/ for all x; y 2 X. So, there exists an additive mapping L W X ! Y satisfying (4.57) for the case p > 14 . Now, let T W X ! Y be another additive mapping satisfying (4.57). Then we have  x  x    kL.x/  T.x/k D 3q L q  T q  3 3   x   x    x   x       3q L q  f q  C T q  f q  3 3 3 3 3p  2  3q  p jjxjj4p ;  81  3 81q which tends to zero as q ! 1 for all x 2 X. So we can conclude that L.x/ D T.x/ for all x 2 X. This proves the uniqueness of L.

196

4 Stability of Functional Inequalities in Banach Algebras

Assume that p < 14 . It follows from (4.58) that     f .x/  1 f .3x/  3p1 kxk4p   3 for all x 2 X. Hence we have   m1 X 81pj 1   f .3l x/  1 f .3m x/  3p1  jjxjj4p  3l  j 3m 3 jDl

(4.62)

for all m; l  1 with m > l and x 2 X. It follows from (4.62) that the sequence f 31k f .3k x/g is a Cauchy sequence for all x 2 X. Since Y is complete, the sequence f 31k f .3k x/g converges and so one can define the mapping L W X ! Y by 1  k  f 3x k!1 3k

L.x/ WD lim

for all x 2 X. Letting l D 0 and m ! 1 in (4.62), we have kf .x/  L.x/k 

3p  jjxjj4p 3  81p

for all x 2 X. The rest of the proof is similar to the above proof. So, there exists a unique additive mapping L W X ! Y satisfying kf .x/  L.x/k 

3p  kxk4p j81p  3j

(4.63)

for all x 2 X. Assume that f W X ! Y is a mapping such that the transformation t ! f .tx/ is continuous in t 2 R for each fixed x 2 X. By the same reasoning as in the proof of Theorem 4.25, one can prove that L is an R-linear mapping This completes the proof. 

4.2.3 Generalization of Cauchy-Rassias Inequalities via the Fixed Point Method Now, we improve the results of An’s results [12], which presented at last pages. In fact, we get a better error estimation of main result of An by applying the fixed point alternative theorem.

4.2 Stability of Functional Inequalities over C -Algebras

197

Theorem 4.28. Let X be a real normed linear space and Y be a real Banach space. Assume that f W X ! Y is an approximately additive odd mapping satisfying the general Cauchy-Rassias inequality: kf .x/ C f .y/ C f .z/ C f .w/k  kf .x C y C z C w/k C '.x; y; z; w/

(4.64)

for all x; y; z; w 2 X, where ' W X 4 ! Œ0; 1/ is a given function. If there exists 0 < L < 1 such that '.x; y; z; w/ 

1 L'.3x; 3y; 3z; 3w/ 3

(4.65)

for all x; y; z; w 2 X. Then there exists a unique additive mapping A W X ! Y such that kf .x/  A.x/k 

L '.x; x; x; 3x/ 3  3L

(4.66)

for all x 2 X. If, in addition, f W X ! Y is a mapping such that the transformation t ! f .tx/ is continuous in t 2 R for each fixed x 2 X; then A is an R-linear mapping. Proof. Since f is odd, f .0/ D 0 and f .x/ D f .x/ for all x 2 X. Consider the set S WD fg W X ! Yg and introduce the generalized metric on S as follows: d.g; h/ D inffC 2 RC W kg.x/  h.x/k  C'.x; x; x; 3x/; 8x 2 Xg: Now, we show that .S; d/ is complete. Let fhn g be a Cauchy sequence in .S; d/. Then, for any " > 0, there exists an integer N" > 0 such that d.hm ; hn / < " for all m; n  N" . Then there exists C 2 .0; "/ such that khm .x/  hn .x/k  C'.x; x; x; 3x/  "'.x; x; x; 3x/

(4.67)

for all m; n  N" and x 2 X. Since Y is complete, fhn .x/g converges for each x 2 X. Thus a mapping h W X ! Y can be defined by h.x/ WD lim hn .x/ n!1

(4.68)

for all x 2 X. Letting n ! 1 in (4.67), we have m  N" H) khm .x/  hn .x/k  "'.x; x; x; 3x/ H) " 2 fC 2 RC W kg.x/  h.x/k  C'.x; x; x; 3x/; 8x 2 Xg H) d.hm ; h/  " for all x 2 X. This means that the Cauchy sequence fhn g converges to h in .S; d/. Hence .S; d/ is complete.

198

4 Stability of Functional Inequalities in Banach Algebras

Now, we consider the linear mapping  W S ! S defined by g.x/ WD 3g

x

(4.69)

3

for all x 2 X. We show that  is a strictly contractive on S. For any g; h 2 S, let Cg;h  0 be an arbitrary constant with d.g; h/  Cg;h . Then we have d.g; h/  Cg;h H) kg.x/  h.x/k  Cg;h '.x; x; x; 3x/ x  x x x x  3h k  3Cg;h ' ; ; ; x H) k3g 3 3 3 3 3 x x  3h k  LCg;h '.x; x; x; 3x/ H) k3g 3 3 for all x 2 X. This means d.g; h/  LCg;h . Hence we see that d.g; h/  Ld.g; h/ for any g; h 2 S. Therefore,  is a strictly contractive on S with the Lipschitz constant 0 < L < 1. Letting y D z D x and w D 3x in (4.64), we have k3f .x/  f .3x/k  '.x; x; x; 3x/

(4.70)

for all x 2 X and so   x  x x x  1   ; ; ; x  L'.x; x; x; 3x/ f .x/  3f ' 3 3 3 3 3 for all x 2 X. Thus we have d.f ; f / 

L : 3

Therefore, it follows of Theorem 1.3 that the sequence fn f g converges to the unique fixed point A of , i.e., A.x/ D .f /.x/ DWD lim 3k f k!1

x 3k

and A.3x/ D 3A.x/ for all x 2 X. Also, we have d.A; f / 

L 1 d.f ; f /  ; 1L 3  3L

which means that (4.66) holds. Assume that f W X ! Y is a mapping such that the transformation t ! f .tx/ is continuous in t 2 R for each fixed x 2 X. By the same reasoning as in the proof of Theorem 4.25, one can prove that A is an R-linear mapping. This completes the proof. 

4.2 Stability of Functional Inequalities over C -Algebras

199

In the following, we get a better error estimation of the main result of An [12]. Corollary 4.29. Let X be a real normed linear space and Y be a real Banach space. Assume that f W X ! Y is an approximately additive odd mapping for which there exist the constants   0 and p 2 R such that 4p ¤ 1 and f satisfies the general Cauchy-Rassias inequality: kf .x/ C f .y/ C f .z/ C f .w/k  kf .x C y C z C w/k C kxkp kykp kzkp kwkp for all x; y; z; w 2 X. Then there exists a unique additive mapping A W X ! Y such that kf .x/  A.x/k 

3p1  kxk4p j81p  3j

for all x 2 X. If, in addition, f W X ! Y is a mapping such that the transformation t ! f .tx/ is continuous in t 2 R for each fixed x 2 X; then A is an R-linear mapping. Proof. In Theorem 4.28, take '.x; y; z; w/ WD kxkp kykp kzkp kwkp for all x; y; z; w 2 X and L D 811p . Then we have desired result.



Corollary 4.30. Let X be a real normed linear space and Y be a real Banach space. Assume that f W X ! Y is an approximately additive odd mapping for which there exist the constants   0 and p 2 R such that 4p ¤ 1 and f satisfies the general Cauchy-Rassias inequality: kf .x/ C f .y/ C f .z/ C f .w/k  kf .x C y C z C w/k C .kxkp C kykp C kzkp C kwkp / for all x; y; z; w 2 X. Then there exists a unique additive mapping A W X ! Y such that kf .x/  A.x/k 

.3p C 3/ kxkp j3p  3j

for all x 2 X. If, in addition, f W X ! Y is a mapping such that the transformation t ! f .tx/ is continuous in t 2 R for each fixed x 2 X; then A is an R-linear mapping. Proof. In Theorem 4.28, take '.x; y; z; w/ WD kxkp C kykp C kzkp C kwkp for all x; y; z; w 2 X and L D 31p . Then we have desired result.



Chapter 5

Stability of Functional Equations in C -Ternary Algebras

Ternary algebraic operations were considered in the nineteenth century by several mathematicians such as Cayley [68] who introduced the notion of cubic matrix which, in turn, was generalized by Kapranov et al. [169]. The simplest example of such non-trivial ternary operation is given by the following composition rule: fa; b; cgijk D

X

anil bljm cmkn

1l;m;nN

for each i; j; k D 1; 2;    ; N. Ternary structures and their generalization, the so-called n-ary structures, raise certain hopes in view of their applications in physics. Some significant applications are as follows (see [171] and [172]): (1) The algebra of nonions generated by two matrices 0

1 010 @0 0 1A; 100

1 0 1 0 @ 0 0 !A ! 2 0 0; 0

2i

where ! D e 3 , was introduced by Sylvester as a ternary analog of Hamiltons quaternions (see [2]). (2) The quark model inspired a particular brand of ternary algebraic systems. The so-called Nambu mechanics is based on such structures (see [93]). There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory and the Yang-Baxter equation (see [2, 171] and [325]). In Sect. 5.1, we prove the Hyers-Ulam stability of C -ternary 3-derivations and of C -ternary 3-homomorphisms for the following functional equation:

© Springer International Publishing Switzerland 2015 Y.J. Cho et al., Stability of Functional Equations in Banach Algebras, DOI 10.1007/978-3-319-18708-2_5

201

5 Stability of Functional Equations in C -Ternary Algebras

202

f .x1 C x2 ; y1 C y2 ; z1 C z2 / D

X

f .xi ; yj ; zk /

1i;j;k2

in C -ternary algebras. In Sect. 5.2, we consider the following Apollonius type additive functional equation:   xCy 1 f .z  x/ C f .z  y/ D  f .x C y/ C 2f z  2 4 and prove the superstability of C -ternary homomorphisms, C -ternary derivations, JB -triple homomorphisms and JB -triple derivations by using the fixed point method. In Sect. 5.3, we prove the Hyers-Ulam stability of bi--derivations on JB -triples.

5.1 C -Ternary 3-Homomorphism and C -Ternary 3-Derivations In this section, we prove the Hyers-Ulam stability of C -ternary 3-derivations and of C -ternary 3-homomorphisms for the following functional equation: f .x1 C x2 ; y1 C y2 ; z1 C z2 / D

X

f .xi ; yj ; zk /

1i;j;k2

in C -ternary algebras (see [94]). Let X and Y be complex vector spaces. A mapping f W XXX ! Y is called a 3additive mapping if f is additive for each variable and a mapping f W XXX ! Y is called a 3-C-linear mapping if f is C-linear for each variable. A 3-C-linear mapping H W A  A  A ! B is called a C -ternary 3-homomorphism if it satisfies H.Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / D ŒH.x1 ; x2 ; x3 /; H.y1 ; y2 ; y3 /; H.z1 ; z2 ; z3 / for all x1 ; y1 ; z1 ; x2 ; y2 ; z2 ; x3 ; y3 ; z3 2 A. For a given mapping f W A3 ! B, we define D; ; f .x1 ; x2 ; y1 ; y2 ; z1 ; z2 / WD f .x1 C x2 ; y1 C y2 ; z1 C z2 /  

X

f .xi ; yj ; zk /

1i;j;k2

for all ; ; 2 S1 WD f 2 C W jj D 1g and x1 ; x2 ; y1 ; y2 ; z1 ; z2 2 A.

5.1 C -Ternary 3-Homomorphism and C -Ternary 3-Derivations

203

Bae and Park [21] proved the Hyers–Ulam stability of 3-homomorphisms in C -ternary algebras for the following functional equation: D; ; f .x1 ; x2 ; y1 ; y2 ; z1 ; z2 / D 0: Lemma 5.1 ([21]). Let X and Y be complex vector spaces and f W X  X  X ! Y be a 3-additive mapping such that f .x; y; z/ D  f .x; y; z/ for all ; ; 2 S1 and x; y; z 2 X. Then f is 3-C-linear. Theorem 5.2 ([21]). Let p; q; r 2 .0; 1/ with p C q C r < 3 and  2 .0; 1/ and f W A3 ! B be a mapping such that kD; ; f .x1 ; x2 ; y1 ; y2 ; z1 ; z2 /k

(5.1)

   maxfkx1 k; kx2 kg  maxfky1 k; ky2 kg  maxfkz1 k; kz2 kg p

q

r

and kf .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œf .x1 ; x2 ; x3 /; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k 

3 X

(5.2)

kxi kp  kyi kq  kzi kr

iD1

for all ; ; 2 S1 and x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Then there exists a unique C -ternary 3-homomorphism H W A3 ! B such that kf .x; y; z/  H.x; y; z/k 

23

 kxkp  kykq  kzkr  2pCqCr

(5.3)

for all x; y; z 2 A.

5.1.1 C -Ternary 3-Homomorphisms in C -Ternary Algebras Now, we investigate C -ternary 3-homomorphisms in C -ternary algebras. Theorem 5.3. Let p; q; r 2 .0; 1/ with p C q C r < 3 and  2 .0; 1/ and f W A3 ! B be a mapping satisfying (5.1) and (5.2). If there exists .x0 ; y0 ; z0 / 2 A3 such that lim

n!1

1 f .2n x0 ; 2n y0 ; 2n z0 / D e0 ; 8n

then the mapping f is a C -ternary 3-homomorphism.

5 Stability of Functional Equations in C -Ternary Algebras

204

Proof. By Theorem 5.2, there exists a unique C -ternary 3-homomorphism H W A3 ! B satisfying (5.3). Note that 1 f .2n x; 2n y; 2n z/ n!1 8n

H.x; y; z/ WD lim

for all x; y; z 2 A. By the assumption, we have H.x0 ; y0 ; z0 / D lim

n!1

1 f .2n x0 ; 2n y0 ; 2n z0 / D e0 : 8n

It follows from (5.2) that kŒH.x1 ; x2 ; x3 /; H.y1 ; y2 ; y3 /; H.z1 ; z2 ; z3 / ŒH.x1 ; x2 ; x3 /; H.y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k D kH.Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / ŒH.x1 ; x2 ; x3 /; H.y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k D lim

n!1

1 kf .Œ2n x1 ; 2n y1 ; z1 ; Œ2n x2 ; 2n y2 ; z2 ; Œ2n x3 ; 2n y3 ; z3 / 82n Œf .2n x1 ; 2n x2 ; 2n x3 /; f .2n y1 ; 2n y2 ; 2n y3 /; f .z1 ; z2 ; z3 /k

3 2n.pCq/ X kxi kp  kyi kq  kzi kr n!1 82n iD1

 lim D0

for all x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A and so ŒH.x1 ; x2 ; x3 /; H.y1 ; y2 ; y3 /; H.z1 ; z2 ; z3 / D ŒH.x1 ; x2 ; x3 /; H.y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 / for all x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Letting x1 D y1 D x0 ; x2 D y2 D y0 and x3 D y3 D z0 in the last equality, we get f .z1 ; z2 ; z3 / D H.z1 ; z2 ; z3 / for all z1 ; z2 ; z3 2 A. Therefore, the mapping f is a C -ternary 3-homomorphism. This completes the proof.  Theorem 5.4. Let pi ; qi ; ri 2 .0; 1/ .i D 1; 2; 3/ such that pi ¤ 1 or qi ¤ 1 or ri ¤ 1 for some 1  i  3, ; 2 .0; 1/ and f W A3 ! B be a mapping such that kD; ; f .x1 ; x2 ; y1 ; y2 ; z1 ; z2 /k  .kx1 kp1  kx2 kp2  ky1 kq1  ky2 kq2 Cky1 kq1  ky2 kq2  kz1 kr1  kz2 kr2 C kx1 kp1  kx2 kp2  kz1 kr1  kz2 kr2 /

(5.4)

5.1 C -Ternary 3-Homomorphism and C -Ternary 3-Derivations

205

and kf .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œf .x1 ; x2 ; x3 /; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k  kx1 kp1  kx2 kp2  kx3 kp3

(5.5)

ky1 k  ky2 k  ky3 k  kz1 k  kz2 k  kz3 k q1

q2

q3

r1

r2

r3

for all ; ; 2 S1 and x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Then the mapping f W A3 ! B is a C -ternary 3-homomorphism. Proof. Letting xi D yj D zk D 0 .i; j; k D 1; 2/ in (5.4), we get f .0; 0; 0/ D 0. Putting  D D D 1; x2 D 0 and yj D zk D 0 .j; k D 1; 2/ in (5.4), we have f .x1 ; 0; 0/ D 0 for all x1 2 A. Similarly, we get f .0; y1 ; 0/ D f .0; 0; z1 / D 0 for all y1 ; z1 2 A. Setting  D D D 1; x2 D 0; y2 D 0 and z1 D z2 D 0, we have f .x1 ; y1 ; 0/ D 0 for all x1 ; y1 2 A. Similarly, we get f .x1 ; 0; z1 / D f .0; y1 ; z1 / D 0 for all x1 ; y1 ; z1 2 A. Now, letting  D D D 1 and y2 D z2 D 0 in (5.4), we have f .x1 C x2 ; y1 ; z1 / D f .x1 ; y1 ; z1 / C f .x2 ; y1 ; z1 / for all x1 ; x2 ; y1 ; z1 2 A. Similarly, one can show that the other equations hold and so f is 3-additive. Letting x2 D y2 D z2 D 0 in (5.4), we get f .x1 ; y1 ; z1 / D  f .x1 ; y1 ; z1 / for all ; ; 2 S1 and x1 ; y1 ; z1 2 A. So, by Lemma 5.1, the mapping f is 3-C-linear. Without any loss of generality, we may suppose that p1 ¤ 1. Let p1 < 1. It follows from (5.5) that kf .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œf .x1 ; x2 ; x3 /; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k D lim

n!1

1 kf .Œ3n x1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / 3n Œf .3n x1 ; x2 ; x3 /; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k

3np1 .kx1 kp1  kx2 kp2  kx3 kp3  ky1 kq1  ky2 kq2 n!1 3n ky3 kq3  kz1 kr1  kz2 kr2  kz3 kr3 /

 lim

D0

5 Stability of Functional Equations in C -Ternary Algebras

206

for all x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Let p1 > 1. It follows from (5.5) that kf .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œf .x1 ; x2 ; x3 /; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k  h 1 i   D lim 3n f x ; y ; z 1 1 1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3  n n!1 3  i h 1   f n x1 ; x2 ; x3 ; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /  3 3n  lim np .kx1 kp1  kx2 kp2  kx3 kp3  ky1 kq1  ky2 kq2 n!1 3 1 ky3 kq3  kz1 kr1  kz2 kr2  kz3 kr3 / D0 for all x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Therefore, we have f .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / D Œf .x1 ; x2 ; x3 /; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 / for all x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Therefore, the mapping f W A3 ! B is a C -ternary 3-homomorphism. This completes the proof.  Remark 5.5. Let ' W A6 ! Œ0; 1/ and

W A9 ! Œ0; 1/ be the functions such that

'.x1 ;   ; x6 / D 0 if xi D 0 for some 1  i  6 and 1 .x1 ;   ; 3n xi ;   ; x9 / D 0 3n or 3n



x1 ;   ;

 1 D 0: x ;   ; x i 9 3n

Suppose that f W A3 ! B is a mapping satisfying kD; ; f .x1 ; x2 ; y1 ; y2 ; z1 ; z2 /k  '.x1 ; x2 ; y1 ; y2 ; z1 ; z2 / and kf .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œf .x1 ; x2 ; x3 /; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k 

.x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 /

5.1 C -Ternary 3-Homomorphism and C -Ternary 3-Derivations

207

for all ; ; 2 S1 and x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Then the mapping f is a C -ternary 3-homomorphism. Corollary 5.6. Let pi ; qi ; ri 2 .0; 1/ .i D 1; 2; 3/ such that pi ¤ 1 or qi ¤ 1 or ri ¤ 1 for some 1  i  3, ; 2 .0; 1/ and f W A3 ! B be a mapping such that kD; ; f .x1 ; x2 ; y1 ; y2 ; z1 ; z2 /k  kx1 kp1  kx2 kp2  ky1 kq1  ky2 kq2  kz1 kr1  kz2 kr2 and kf .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œf .x1 ; x2 ; x3 /; f .y1 ; y2 ; y3 /; f .z1 ; z2 ; z3 /k  kx1 kp1  kx2 kp2  kx3 kp3  ky1 kq1  ky2 kq2  ky3 kq3  kz1 kr1  kz2 kr2  kz3 kr3 for all ; ; 2 S1 and x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Then the mapping f W A3 ! B is a C -ternary 3-homomorphism.

5.1.2 C -Ternary 3-Derivations in C -Ternary Algebras Now, we investigate C -ternary 3-derivations in C -ternary algebras. Definition 5.7. A 3-C-linear mapping D W A3 ! A is called a C -ternary 3-derivation if it satisfies the following: D.Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / D ŒD.x1 ; x2 ; x3 /; Œy1 ; y2 ; y3 ; Œz1 ; z2 ; z3  CŒŒx1 ; x2 ; x3 ; D.y1 ; y2 ; y3 /; Œz1 ; z2 ; z3  CŒŒx1 ; x2 ; x3 ; Œy1 ; y2 ; y3 ; D.z1 ; z2 ; z3 / for all x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Theorem 5.8. Let p; q; r 2 .0; 1/ with p C q C r < 3 and  2 .0; 1/, and let f W A3 ! A be a mapping such that kD; ; f .x1 ; x2 ; y1 ; y2 ; z1 ; z2 /k

(5.6)

   maxfkx1 k; kx2 kg  maxfky1 k; ky2 kg  maxfkz1 k; kz2 kg p

q

and kf .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œf .x1 ; x2 ; x3 /; Œy1 ; y2 ; y3 ; Œz1 ; z2 ; z3 

r

5 Stability of Functional Equations in C -Ternary Algebras

208

ŒŒx1 ; x2 ; x3 ; f .y1 ; y2 ; y3 /; Œz1 ; z2 ; z3  ŒŒx1 ; x2 ; x3 ; Œy1 ; y2 ; y3 ; f .z1 ; z2 ; z3 /k 

3 X

kxi kp  kyi kq  kzi kr

(5.7)

iD1

for all ; ; 2 S1 and x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Then there exists a unique C -ternary 3-derivation ı W A3 ! A such that kf .x; y; z/  ı.x; y; z/k 

 kxkp  kykq  kzkr 23  2pCqCr

(5.8)

for all x; y; z 2 A. Proof. By the same method as in the proof of [21, Theorem 1.2], we obtain a 3-Clinear mapping ı W A3 ! A satisfying (5.8) and the mapping ı.x; y; z/ WD lim

j!1

1 f .2j x; 2j y; 2j z/ 8j

for all x; y; z 2 A. It follows from (5.7) that kı.Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œı.x1 ; x2 ; x3 /; Œy1 ; y2 ; y3 ; Œz1 ; z2 ; z3  ŒŒx1 ; x2 ; x3 ; ı.y1 ; y2 ; y3 /; Œz1 ; z2 ; z3  ŒŒx1 ; x2 ; x3 ; Œy1 ; y2 ; y3 ; ı.z1 ; z2 ; z3 /k D lim

n!1

1 kf .23n Œx1 ; y1 ; z1 ; 23n Œx2 ; y2 ; z2 ; 23n Œx3 ; y3 ; z3 / 83n Œf .2n x1 ; 2n x2 ; 2n x3 /; Œ2n y1 ; 2n y2 ; 2n y3 ; Œ2n z1 ; 2n z2 ; 2n z3  ŒŒ2n x1 ; 2n x2 ; 2n x3 ; f .2n y1 ; 2n y2 ; 2n y3 /; Œ2n z1 ; 2n z2 ; 2n z3  ŒŒ2n x1 ; 2n x2 ; 2n x3 ; Œ2n y1 ; 2n y2 ; 2n y3 ; f .2n z1 ; 2n z2 ; 2n z3 /k

3 2n.pCqCr/ X kxi kp  kyi kq  kzi kr D 0 n!1 83n iD1

 lim

for all x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Now, let T W A3 ! A be another 3-derivation satisfying (5.8). Then we have 1 kı.2n x; 2n y; 2n z/  T.2n x; 2n y; 2n z/k 8n 1  n kı.2n x; 2n y; 2n z/  f .2n x; 2n y; 2n z/k 8

kı.x; y; z/  T.x; y; z/k D

5.2 Apollonius Type Additive Functional Equations

C 

209

1 kf .2n x; 2n y; 2n z/  T.2n x; 2n y; 2n z/k 8n

2.pCqCr3/nC1 kxkp  kykq  kzkr ; 23  2pCqCr

which tends to zero as n ! 1 for all x; y; z 2 A. So we can conclude that ı.x; y; z/ D T.x; y; z/ for all x; y; z 2 A. This proves the uniqueness of ı. Therefore, the mapping ı W A3 ! A is a unique C -ternary 3-derivation satisfying (5.8). This completes the proof.  Corollary 5.9. Let  2 .0; 1/ and f W A3 ! A be a mapping satisfying kD; ; f .x1 ; x2 ; y1 ; y2 ; z1 ; z2 /k   and kf .Œx1 ; y1 ; z1 ; Œx2 ; y2 ; z2 ; Œx3 ; y3 ; z3 / Œf .x1 ; x2 ; x3 /; Œy1 ; y2 ; y3 ; Œz1 ; z2 ; z3  ŒŒx1 ; x2 ; x3 ; f .y1 ; y2 ; y3 /; Œz1 ; z2 ; z3  ŒŒx1 ; x2 ; x3 ; Œy1 ; y2 ; y3 ; f .z1 ; z2 ; z3 /k  3 for all ; ; 2 S1 and x1 ; x2 ; x3 ; y1 ; y2 ; y3 ; z1 ; z2 ; z3 2 A. Then there exists a unique C -ternary 3-derivation ı W A3 ! A such that kf .x; y; z/  ı.x; y; z/k 

 7

for all x; y; z 2 A.

5.2 Apollonius Type Additive Functional Equations C. Park and Th. M. Rassias proved the superstability of C -ternary homomorphisms, C -ternary derivations, JB -triple homomorphisms and JB -triple derivations associated with the following Apollonius type additive functional equation:   xCy 1 f .z  x/ C f .z  y/ D  f .x C y/ C 2f z  2 4 by using the direct method (see [244–248]).

210

5 Stability of Functional Equations in C -Ternary Algebras

In this section, under the conditions of the theorems, we can show that the mappings f must be zero and we correct some conditions. Furthermore, we prove the superstability of C -ternary homomorphisms, C -ternary derivations, JB -triple homomorphisms and JB -triple derivations by using fixed point method. In an inner product space, the following equality holds: kz  xk2 C kz  yk2 D

 x C y 1  2 kx  yk2 C 2z   ; 2 2

which is called the Apollonius’ identity. The following functional equation, which was motivated by this equation: Q.z  x/ C Q.z  y/ D

 1 x C y Q.x  y/ C 2Q z  ; 2 2

(5.9)

is quadratic. For this reason, the function equation (5.9) is called a quadratic functional equation of Apollonius type and each solution of the functional equation (5.9) is said to be a quadratic mapping of Apollonius type. Jun and Kim [144] investigated the quadratic functional equation of Apollonius type. Now, employing the above equality (5.9), we introduce a new functional equation, which is called the Apollonius type additive functional equation and whose solution of the functional equation is said to be the Apollonius type additive mapping:  1 x C y L.z  x/ C L.z  y/ D  L.x C y/ C 2L z  : 2 4

5.2.1 Homomorphisms in C -Ternary Algebras Assume that A is a C -ternary algebra with the norm k  kA and that B is a C -ternary algebra with the norm k  kB . Now, we investigate homomorphisms in C -ternary algebras. Lemma 5.10. Let f W A ! B be a mapping such that  1 xCy    / kf .z  x/ C f .z  y/ C f .x C y/kB  2f .z  B 2 4 for all x; y; z 2 A. Then f is additive.

(5.10)

    Proof. Letting x D y D z D 0 in (5.10), we get  52 f .0/  k2f .0/kB and so B f .0/ D 0. Letting z D 0 and y D x in (5.10), we have kf .x/ C f .x/kB  k2f .0/kB D 0

5.2 Apollonius Type Additive Functional Equations

211

for all x 2 A. Hence f .x/ D f .x/ for all x 2 A. Letting x D y D 2z in (5.10), we have   1   2f .z/ C f .4z/  k2f .0/kB D 0 B 2 for all z 2 A and so f .4z/ D 4f .z/ D 4f .z/ for all z 2 A. Letting z D

xCy 4

in (5.10), we have

  3x C y    x  3y  1   Cf C f .x C y/  k2f .0/kB D 0 f B 4 4 2 for all x; y 2 A and so f

 3x C y  4

for all x; y 2 A. Let w1 D

3xCy 4

Cf

 x  3y  4

and w2 D

1 C f .x C y/ D 0 2

x3y 4

(5.11)

in (5.11). Then we have

w C w  1 1 1 2 f .w1 / C f .w2 / D  f .2w1  2w2 / D f .2w1 C 2w2 / D 2f 2 2 2 for all w1 ; w2 2 A and so f is additive. This completes the proof.



Theorem 5.11. Let r ¤ 1,  be a nonnegative real number and f W A ! B be a mapping such that     f . z  x/ C f .z  y/ C f .x C y/ 2 B     xCy    (5.12)  2f z  4 B and kf .Œx; y; z/  Œf .x/; f .y/; f .z/kB 3r 3r  .kxk3r A C kykA C kzkA /

(5.13)

for all 2 T1 WD f 2 C W jj D 1g and all x; y; z 2 A. Then the mapping f W A ! B is a C -ternary algebra homomorphism. Proof. Assume r > 1. Let D 1 in (5.12). By Lemma 5.10, the mapping f W A ! B is additive. Letting y D x and z D 0, we get kf . x/ C f .x/kB  k2f .0/kB D 0

5 Stability of Functional Equations in C -Ternary Algebras

212

for all x 2 A and 2 T1 and so f . x/ C f .x/ D f . x/ C f .x/ D 0 for all x 2 A and 2 T1 . Hence f . x/ D f .x/ for all x 2 A and 2 T1 . By the same reasoning as in the proof of Theorem 2.1 of Park [227], the mapping f W A ! B is C-linear. It follows from (5.13) that kf .Œx; y; z/  Œf .x/; f .y/; f .z/kB   Œx; y; z  h  x   y   z i   D lim 8n f n n n  f n ; f n ; f n  n!1 B 2 2 2 2 2 2 8n  3r 3r  lim nr .kxk3r A C kykA C kzkA / n!1 8 D0 for all x; y; z 2 A and so f .Œx; y; z/ D Œf .x/; f .y/; f .z/ for all x; y; z 2 A. Hence the mapping f W A ! B is a C -ternary algebra homomorphism. Similarly, one obtains the result for the case r < 1. This completes the proof. 

5.2.2 Derivations in C -Ternary Algebras Assume that A is a C -ternary algebra with the norm k  kA . Now, we investigate derivations on C -ternary algebras. Theorem 5.12. Let r ¤ 1 and  be nonnegative real numbers, and let f W A ! A be a mapping such that     f . z  x/ C f .z  y/ C f .x C y/ 2 A      x C y    2f z  4 A and kf .Œx; y; z/  Œf .x/; y; z  Œx; f .y/; z  Œx; y; f .z/kA 3r 3r  .kxk3r A C kykA C kzkA /

for all x; y; z 2 A. Then the mapping f W A ! A is a C -ternary derivation.

(5.14)

5.2 Apollonius Type Additive Functional Equations

213

Proof. Assume r > 1. By the same reasoning as in the proof of Theorem 5.11, the mapping f W A ! A is C-linear. It follows from (5.15) that kf .Œx; y; z/  Œf .x/; y; z  Œx; f .y/; z  Œx; y; f .z/kA   Œx; y; z  h  x  y z i   f n ; n; n D lim 8n f n!1 8n 2 2 2 h x  y  z i h x y  z i   n;f n ; n  n; n;f n  A 2 2 2 2 2 2 8n  3r 3r  lim nr .kxk3r A C kykA C kzkA / n!1 8 D0 for all x; y; z 2 A and so f .Œx; y; z/ D Œf .x/; y; z C Œx; f .y/; z C Œx; y; f .z/ for all x; y; z 2 A. Thus the mapping f W A ! A is a C -ternary derivation. Similarly, one obtains the result for the case r < 1. This completes the proof. 

5.2.3 Homomorphisms in JB -Triples Assume that J is a JB -triple with the norm k  kJ and that L is a JB -triple with the norm k  kL . Now, we investigate homomorphisms in JB -triples. Theorem 5.13. Let r ¤ 1,  be a nonnegative real number and f W J ! L be a mapping such that     f . z  x/ C f .z  y/ C f .x C y/ 2 L      xCy    2f z  4 L

(5.15)

and kf .fxyzg/  ff .x/f .y/f .z/gkL 3r 3r  .kxk3r J C kykJ C kzkJ /

(5.16)

for all 2 T1 and all x; y; z 2 J . Then the mapping f W J ! L is a JB -triple homomorphism.

5 Stability of Functional Equations in C -Ternary Algebras

214

Proof. Assume r > 1. By the same reasoning as in the proof of Theorem 5.11, the mapping f W J ! L is C-linear. It follows from (5.16) that kf .fxyg/  ff .x/f .y/f .z/gkL   fxyzg  n  x   y   z o  D lim 8n f n n n  f n f n f n kL n!1 2 2 2 2 2 2 n 8  3r 3r  lim nr .kxk3r J C kykJ C kzkJ / n!1 8 D0 for all x; y; z 2 J and so f .fxyzg/ D ff .x/f .y/f .z/g for all x; y; z 2 J . Hence the mapping f W J ! L is a JB -triple homomorphism. Similarly, one obtains the result for the case r < 1. This completes the proof. 

5.2.4 Derivations in JB -Triples Assume that J is a JB -triple with the norm k  kJ . Now, we investigate derivations on JB -triples. Theorem 5.14. Let r ¤ 1,  be a nonnegative real number and f W J ! J be a mapping such that     f . z  x/ C f .z  y/ C f .x C y/ 2 J      xCy    2f z  4 J

(5.17)

and kf .fxyzg/  ff .x/yzg  fxf .y/zg  fxyf .z/gkJ 3r 3r  .kxk3r J C kykJ C kzkJ /

(5.18)

for all x; y; z 2 J . Then the mapping f W J ! J is a JB -triple derivation. Proof. Assume r > 1. By the same reasoning as in the proof of Theorem 5.11, the mapping f W J ! J is C-linear. It follows from (5.18) that

5.2 Apollonius Type Additive Functional Equations

215

kf .fxyzg/  ff .x/yzg  fxf .y/zg  fxyf .z/gkJ   fxyzg  n  x  y z o   f n n n D lim 8n f n!1 8n 2 2 2 n x  y  z o n x y  z o   nf n n  n nf n  J 2 2 2 2 2 2 n 8  3r 3r  lim nr .kxk3r J C kykJ C kzkJ / n!1 8 D0 for all x; y; z 2 J and so f .fxyzg/ D ff .x/yzg C fxf .y/zg C fxyf .z/g for all x; y; z 2 J . Thus the mapping f W J ! J is a JB -triple derivation. Similarly, one obtains the result for the case r < 1. This completes the proof. 

5.2.5 C -Ternary Homomorphisms: Fixed Point Method Now, we prove the superstability of C -ternary homomorphisms by the using fixed point method. Theorem 5.15. Let ' W A3 ! Œ0; 1/ be a function such that there exists ˛ < 1 with x y z ; ; (5.19) '.x; y; z/  8˛' 2 2 2 for all x; y; z 2 A. Let f W A ! B be a mapping satisfying (5.12) and kf .Œx; y; z/  Œf .x/; f .y/; f .z/kB  '.x; y; z/

(5.20)

for all x; y; z 2 A: Then the mapping f W A ! B is a C -ternary homomorphism. Proof. By the same reasoning as in the proof of Theorem 5.11, one can show that the mapping f W A ! B is C-linear. It follows from (5.19) that lim

n!1

1 '.2n x; 2n y; 2n z/ D 0 8n

(5.21)

for all x; y; z 2 A. Since f W A ! B is additive, it follows from (5.20) and (5.21) that f .Œx; y; z/ D Œf .x/; f .y/; f .z/ for all x; y; z 2 A. Thus the mapping f W A ! B is a C -ternary homomorphism. This completes the proof. 

5 Stability of Functional Equations in C -Ternary Algebras

216

Theorem 5.16. Let ' W A3 ! Œ0; 1/ be a function such that there exists ˛ < 1 with '.x; y; z/ 

˛ ' .2x; 2y; 2z/ 2

(5.22)

for all x; y; z 2 A. Let f W A ! B be a mapping satisfying (5.12) and (5.20). Then the mapping f W A ! B is a C -ternary homomorphism. Proof. By the same reasoning as in the proof of Theorem 5.11, one can show that the mapping f W A ! B is C-linear. It follows from (5.22) that lim 2n '

n!1

x y z D0 ; ; 2n 2n 2n

(5.23)

for all x; y; z 2 A. Since f W A ! B is additive, it follows from (5.20) and (5.23) that f .Œx; y; z/ D Œf .x/; f .y/; f .z/ for all x; y; z 2 A. Thus the mapping f W A ! B is a C -ternary homomorphism. This completes the proof.  Remark 5.17. Theorem 5.11 follows from Theorems 5.15 and 5.16 by taking '.x; y; z/ D .kxk3r C kyk3r C kzk3r / for all x; y; z 2 A.

5.2.6 C -Ternary Derivations: The Fixed Point Method Now, we prove the superstability of C -ternary derivations by using the fixed point method. Theorem 5.18. Let ' W A3 ! Œ0; 1/ be a function satisfying (5.19). Let f W A ! A be a mapping satisfying (5.14) and kf .Œx; y; z/  Œf .x/; y; z  Œx; f .y/; z  Œx; y; f .z/kA  '.x; y; z/

(5.24)

for all x; y; z 2 A: Then the mapping f W A ! A is a C -ternary derivation. Proof. The proof is similar to the proof of Theorem 5.15. 3



Theorem 5.19. Let ' W A ! Œ0; 1/ be a function satisfying (5.22). Let f W A ! A be a mapping satisfying (5.14) and (5.24). Then the mapping f W A ! A is a C ternary derivation.

5.2 Apollonius Type Additive Functional Equations

217



Proof. The proof is similar to the proof of Theorem 5.16. Remark 5.20. Theorem 5.12 follows from Theorems 5.18 and 5.19 by taking '.x; y; z/ D .kxk3r C kyk3r C kzk3r / for all x; y; z 2 A.

5.2.7 JB -Triple Homomorphisms: The Fixed Point Method Now, we prove the superstability of JB -triple homomorphisms by using the fixed point method. Theorem 5.21. Let ' W J 3 ! Œ0; 1/ be a function such that there exists ˛ < 1 with x y z ; ; (5.25) '.x; y; z/  8˛' 2 2 2 for all x; y; z 2 J . Let f W J ! L be a mapping satisfying (5.15) and kf .fxyzg/  ff .x/f .y/f .z/gkL  '.x; y; z/

(5.26)

for all x; y; z 2 J : Then the mapping f W J ! L is a JB -triple homomorphism. Proof. By the same reasoning as in the proof of Theorem 5.13, one can show that the mapping f W J ! L is C-linear. It follows from (5.25) that 1 '.2n x; 2n y; 2n z/ D 0 n!1 8n lim

(5.27)

for all x; y; z 2 J . Since f W J ! L is additive, it follows from (5.26) and (5.27) that f .Œx; y; z/ D Œf .x/; f .y/; f .z/ for all x; y; z 2 J . Thus the mapping f W J ! L is a JB -triple homomorphism. This completes the proof.  Theorem 5.22. Let ' W J 3 ! Œ0; 1/ be a function such that there exists an ˛ < 1 with '.x; y; z/ 

˛ ' .2x; 2y; 2z/ 2

(5.28)

for all x; y; z 2 J . Let f W J ! L be a mapping satisfying (5.15) and (5.26). Then the mapping f W J ! L is a JB -triple homomorphism.

5 Stability of Functional Equations in C -Ternary Algebras

218

Proof. By the same reasoning as in the proof of Theorem 5.13, one can show that the mapping f W J ! L is C-linear. It follows from (5.28) that lim 2n '

n!1

x y z D0 ; ; 2n 2n 2n

(5.29)

for all x; y; z 2 J . Since f W J ! L is additive, it follows from (5.26) and (5.29) that f .Œx; y; z/ D Œf .x/; f .y/; f .z/ for all x; y; z 2 J . Thus the mapping f W J ! L is a C -ternary homomorphism. This completes the proof.  Remark 5.23. Theorem 5.13 follows from Theorems 5.21 and 5.22 by taking '.x; y; z/ D .kxk3r C kyk3r C kzk3r / for all x; y; z 2 J .

5.2.8 JB -Triple Derivations: Fixed Point Method Now, we prove the superstability of JB -triple derivations by using the fixed point method. Theorem 5.24. Let ' W J 3 ! Œ0; 1/ be a function satisfying (5.25). Let f W J ! J be a mapping satisfying (5.17) and kf .fxyzg/  ff .x/yzg  fxf .y/zg  fxyf .z/gkJ  '.x; y; z/

(5.30)

for all x; y; z 2 J : Then the mapping f W J ! J is a JB -triple derivation. Proof. The proof is similar to the proof of Theorem 5.21.



Theorem 5.25. Let ' W J 3 ! Œ0; 1/ be a function satisfying (5.28). Let f W J ! J be a mapping satisfying (5.17) and (5.30). Then the mapping f W J ! J is a JB -triple derivation. Proof. The proof is similar to the proof of Theorem 5.22. Remark 5.26. Theorem 5.14 follows from Theorems 5.24 and 5.25 by taking '.x; y; z/ D .kxk3r C kyk3r C kzk3r / for all x; y; z 2 J .



5.3 Bi- -Derivations in JB -Triples

219

5.3 Bi--Derivations in JB -Triples In this section, we prove the Hyers-Ulam stability of bi--derivations in JB triples (see [237]). Definition 5.27 ([97]). Let J be a complex JB -triple and  W J ! J be a C-linear mapping. A C-bilinear mapping D W J  J ! J is called a bi--derivation on J if D.fx; y; zg; w/ D fD.x; w/; .y/; .z/g C f.x/; D.y; w/; .z/g Cf.x/; .y/; D.z; w/g and D.x; fy; z; wg/ D fD.x; y/; .z/; .w/g C f.y/; D.x; z/; .w/g Cf.y/; .z/; D.x; w/g for all x; y; z; w 2 J. The w-variable of the left side in the first equality is C-linear and the x-variable of the left side in the second equality is C-linear. But the w-variable of the right side in the first equality is not C-linear and the x-variable of the right side in the second equality is not C-linear. Thus we correct the definition of bi--derivation as follows: Definition 5.28. Let J be a complex JB -triple and  W J ! J be a C-linear mapping. A C-bilinear mapping D W J  J ! J is called a bi--derivation on J if D.fx; y; zg; w/ D fD.x; w/; .y/; .z/g C f.x/; D.y; w /; .z/g Cf.x/; .y/; D.z; w/g and D.x; fy; z; wg/ D fD.x; y/; .z/; .w/g C f.y/; D.x ; z/; .w/g Cf.y/; .z/; D.x; w/g for all x; y; z; w 2 J. Under the conditions of [97, Theorem 2.5], we can easily show that the mapping D W J  J ! J must be zero. In particular, if f is bi-additive, then D must be zero. In this section, we correct the statements of the results, and prove the corrected theorems and corollaries. Throughout this section, assume that J is a JB -triple with the norm k  k: For any mapping f W J  J ! J, we define

5 Stability of Functional Equations in C -Ternary Algebras

220

E; f .x; y; z; w/ D f .x C y; z  w/ Cf .x  y; z C w/  2 f .x; z/ C 2 f .y; w/ for all x; y; z; w 2 J and ; 2 T1 WD f 2 C W jj D 1g. Lemma 5.29 ([21]). Let f W J  J ! J be a mapping satisfying E; f .x; y; z; w/ D 0 for all x; y; z; w 2 J and ; 2 T1 . Then the mapping f W J  J ! J is C-bilinear. Now, we prove the Hyers-Ulam stability of bi--derivations on JB -triples. Theorem 5.30. Let p; " be positive real numbers with p < 1 and f W J  J ! J with f .0; 0/ D 0, h W J ! J with h.0/ D 0 be the mappings such that kE; f .x; y; z; w/ C h. a C b/  h.a/  h.b/k  ".kxkp C kykp C kzkp C kwkp C kakp C kbkp /

(5.31)

and kf .fx; y; zg; w/  ff .x; w/; h.y/; h.z/g fh.x/; f .y; w /; h.z/g  fh.x/; h.y/; f .z; w/gk Ckf .x; fy; z; wg/  ff .x; y/; h.z/; h.w/g fh.y/; f .x ; z/; h.w/g  fh.y/; h.z/; f .x; w/gk  ".kxkp C kykp C kzkp C kwkp /

(5.32)

for all ; 2 T1 and x; y; z; w 2 J: If the mapping f W J  J ! J satisfies the following: lim

n!1

1 1 f .2n x; 2n y/ D lim f .2n x; 8n y/ n n!1 4 16n 1 D lim f .8n x; 2n y/; n!1 16n

(5.33)

then there exist a unique C-linear mapping  W J ! J and a unique bi--derivation D W J  J ! J such that kh.a/  .a/k 

2" kakp 2  2p

(5.34)

5.3 Bi- -Derivations in JB -Triples

221

and kf .x; z/  D.x; z/k 

5" .kxkp C kzkp / 4  2p

(5.35)

for all a; x; z 2 J: Proof. Letting x D y D z D w D 0 in (5.31), we have kh. a C b/  h.a/  h.b/k  ".kakp C kbkp / for all a; b 2 J. By the same reasoning as in the proof of Park [227, Theorem 2.1], one can show that there exists a unique C-linear mapping  W J ! J satisfying (5.34) and the mapping  W J ! J is given by .a/ WD lim

n!1

1 h.2n a/ 2n

for all a 2 J. Letting  D D 1, a D b D 0, y D x and w D z in (5.31), we have kf .2x; 2z/  2f .x; z/ C 2f .x; z/k  2".kxkp C kzkp /

(5.36)

for all x; z 2 J. Letting  D D 1, a D b D 0, y D x and w D z in (5.31), we have kf .2x; 2z/  2f .x; z/ C 2f .x; z/k  2".kxkp C kzkp /

(5.37)

for all x; z 2 J. Letting  D D 1, a D b D 0, x D z D 0 in (5.31), we have kf .y; w/ C f .y; w/ C 2f .y; w/k  ".kykp C kwkp /

(5.38)

for all y; w 2 J. Replacing y; w by x; z in (5.38), respectively, we have kf .x; z/ C f .x; z/ C 2f .x; z/k  ".kxkp C kzkp /

(5.39)

for all x; z 2 J. By (5.36) and (5.39), we obtain kf .2x; 2z/  4f .x; z/ C f .x; z/  f .x; z/k  3".kxkp C kzkp /

(5.40)

for all x; z 2 J. By (5.36) and (5.37), we obtain kf .x; z/  f .x; z/k  2".kxkp C kzkp /

(5.41)

222

5 Stability of Functional Equations in C -Ternary Algebras

for all x; z 2 J. By (5.40) and (5.41), we obtain kf .2x; 2z/  4f .x; z/k  5".kxkp C kzkp /

(5.42)

for all x; z 2 J. It follows from (5.42) that  m1  1  X 5" 2pj  f .2l x; 2l z/  1 f .2m ; 2m z/  .kxkp C kzkp /  4l  j 4m 4 4 jDl

(5.43)

˚for1 alln x; z n 2 J and m; l  1 with m > l. This implies that the sequence f .2 x; 2 z/ is a Cauchy sequence for all x; z 2 J: Since J is complete, the 4n  ˚ sequence 41n f .2n x; 2n z/ converges and so one can define the mapping D W J  J ! J by 1 f .2n x; 2n z/ n!1 4n

D.x; z/ WD lim

for all x; z 2 J: Moreover, letting l D 0 and passing the limit m ! 1 in (5.43), we have (5.35). Let a D b D 0 in (5.31). Then, by the definition of the mapping D, we have 1 kE; f .2n x; 2n y; 2n z; 2n w/k 4n 2pn  lim n ".kxkp C kykp C kzkp C kwkp / n!1 4 D0

kE; D.x; y; z; w/k D lim

n!1

for all ; 2 T1 and all x; y; z; w 2 J: By Lemma 5.29, the mapping D W J  J ! J is C-bilinear. It follows from (5.32) and (5.33) that kD.fx; y; zg; w/  fD.x; w/; .y/; .z/g  f.x/; D.y; w /; .z/g  f.x/; .y/; D.z; w/gk C kD.x; fy; z; wg/  fD.x; y/; .z/; .w/g  f.y/; D.x ; z/; .w/g  f.y/; .z/; D.x; w/gk  1 n1 o 1 1  D lim  4n f .8n fx; y; zg; 2n w/  n f .2n x; 2n w/; n h.2n y/; n h.2n z/ n!1 2 4 2 2 o n1 1 1  n h.2n x/; n f .2n y; 2n w /; n h.2n z/ 2 4 2 n1 o 1 1  n h.2n x/; n h.2n y/; n f .2n z; 2n w/ 2 2 4  1 n1 o 1 1  C  4n f .2n x; 8n fy; z; wg/  n f .2n x; 2n y/; n h.2n z/; n h.2n w/ 2 4 2 2

5.3 Bi- -Derivations in JB -Triples

223

n1 o 1 1 n n  n n h.2 y/; f .2 x ; 2 z/; h.2 w/ 2n 4n 2n n1 o 1 1   n h.2n y/; n h.2n z/; n f .2n x; 2n w/  2 2 4 2pn "  lim 4n .kxkp C kykp C kzkp C kwkp / n!1 2 

D0 for all x; y; z; w 2 J and so D.fx; y; zg; w/ D fD.x; w/; .y/; .z/g Cf.x/; D.y; w /; .z/g C f.x/; .y/; D.z; w/g and D.x; fy; z; wg/ D fD.x; y/; .z/; .w/g Cf.y/; D.x ; z/; .w/g C f.y/; .z/; D.x; w/g for all x; y; z; w 2 J: Let T W J J ! J be another C-bilinear mapping satisfying (5.35). Then we have kD.x; z/  T.x; z/k 1 kD.2n x; 2n z/  T.2n x; 2n z/k 4n 1 1  n kD.2n x; 2n z/  f .2n x; 2n z/k C n kf .2n x; 2n z/  T.2n x; 2n z/k 4 4 2pn 5" 2 n .kxkp C kzkp /; 4 4  2p

D

which tends to zero as n ! 1 for all x; z 2 J. This proves the uniqueness of D. Therefore, the mapping D W J  J ! J is a unique bi--derivation satisfying (5.35). This completes the proof.  Similarly, one can obtain the following theorem. Remark 5.31. Let p; " be positive real numbers with p > 4 and f W J  J ! J with f .0; 0/ D 0, h W J ! J with h.0/ D 0 be the mappings satisfying (5.31), (5.32) and (5.33). Then there exist a unique C-linear mapping  W J ! J and a unique bi--derivation D W J  J ! J such that kh.a/  .a/k 

2" kakp 2

2p

5 Stability of Functional Equations in C -Ternary Algebras

224

and kf .x; z/  D.x; z/k 

5" .kxkp C kzkp / 2p  4

for all a; x; z 2 J: Theorem 5.32. Let p; "; ı be nonnegative real numbers with 0 < p < 1 and f W J  J ! J with f .0; 0/ D 0, h W J ! J with h.0/ D 0 be the mappings satisfying (5.33) and kE; f .x; y; z; w/ C h. a C b/  h.a/  h.b/k  "kxkp kykp kzkp kwkp kakp kbkp C ı

(5.44)

and kf .fx; y; zg; w/  ff .x; w/; h.y/; h.z/g fh.x/; f .y; w /; h.z/g  fh.x/; h.y/; f .z; w/gk Ckf .x; fy; z; wg/  ff .x; y/; h.z/; h.w/g fh.y/; f .x ; z/; h.w/g  fh.y/; h.z/; f .x; w/gk  "kxkp kykp kzkp kwkp C ı

(5.45)

for all ; 2 T1 and x; y; z; w 2 J: Then there exist a unique C-linear mapping  W J ! J and a unique bi--derivation D W J  J ! J such that kh.a/  .a/k  ı

(5.46)

kf .x; z/  D.x; z/k  ı

(5.47)

and

for all a; x; z 2 J: Proof. Letting x D y D z D w D 0 in (5.44), we have kh. a C b/  h.a/  h.b/k  ı for all a; b 2 J. By the same reasoning as in the proof of Park [227, Theorem 2.1], one can show that there exists a unique C-linear mapping  W J ! J satisfying (5.46) and the mapping  W J ! J is given by .a/ WD lim

n!1

1 h.2n a/ 2n

5.3 Bi- -Derivations in JB -Triples

225

for all a 2 J. Letting  D D 1, a D b D 0, y D x and w D z in (5.44), we have kf .2x; 2z/  2f .x; z/ C 2f .x; z/k  ı

(5.48)

for all x; z 2 J. Letting  D D 1, a D b D 0, y D x and w D z in (5.44), we have kf .2x; 2z/  2f .x; z/ C 2f .x; z/k  ı

(5.49)

for all x; z 2 J. Letting  D D 1, a D b D 0, x D z D 0 in (5.44), we have kf .y; w/ C f .y; w/ C 2f .y; w/k  ı

(5.50)

for all y; w 2 J. Replacing y; w by x; z in (5.50), respectively, we have kf .x; z/ C f .x; z/ C 2f .x; z/k  ı

(5.51)

for all x; z 2 J. By (5.48) and (5.51), we obtain kf .2x; 2z/  4f .x; z/ C f .x; z/  f .x; z/k  2ı

(5.52)

for all x; z 2 J. By (5.48) and (5.49), we obtain kf .x; z/  f .x; z/k  ı

(5.53)

for all x; z 2 J. By (5.52) and (5.53), we obtain kf .2x; 2z/  4f .x; z/k  3ı

(5.54)

for all x; z 2 J. It follows from (5.54) that  m1   X 3ı 1 1  f .2l x; 2l z/  1 f .2m ; 2m z/    4l m 4 4 4j jDl

(5.55)

˚for1 alln x; z n 2 J and m; l  1 with m > l. This implies that the sequence f .2 x; 2 z/ is a Cauchy sequence for all x; z 2 J: Since J is complete, the 4n ˚  sequence 41n f .2n x; 2n z/ converges and so one can define the mapping D W JJ ! J by 1 f .2n x; 2n z/ n!1 4n

D.x; z/ WD lim

for all x; z 2 J: Moreover, letting l D 0 and passing the limit m ! 1 in (5.55), we have (5.47).

5 Stability of Functional Equations in C -Ternary Algebras

226

Let a D b D 0 in (5.44). Then by the definition of the mapping D, we have 1 kE; f .2n x; 2n y; 2n z; 2n w/k n!1 4n ı  lim n n!1 4 D0

kE; D.x; y; z; w/k D lim

for all ; 2 T1 and x; y; z; w 2 J: By Lemma 5.29, the mapping D W J  J ! J is C-bilinear. It follows from (5.33) and (5.45) that kD.fx; y; zg; w/  fD.x; w/; .y/; .z/g  f.x/; D.y; w /; .z/g  f.x/; .y/; D.z; w/gk C kD.x; fy; z; wg/  fD.x; y/; .z/; .w/g  f.y/; D.x ; z/; .w/g  f.y/; .z/; D.x; w/gk  1 n1 o 1 1  D lim  4n f .8n fx; y; zg; 2n w/  n f .2n x; 2n w/; n h.2n y/; n h.2n z/ n!1 2 4 2 2 o n1 1 1  n h.2n x/; n f .2n y; 2n w /; n h.2n z/ 2 4 2 n1 o 1 1  n h.2n x/; n h.2n y/; n f .2n z; 2n w/ 2 2 4  1 n1 o 1 1  C  4n f .2n x; 8n fy; z; wg/  n f .2n x; 2n y/; n h.2n z/; n h.2n w/ 2 4 2 2 n1 o 1 1  n h.2n y/; n f .2n x ; 2n z/; n h.2n w/ 2 4 2 n1 o 1 1   n h.2n y/; n h.2n z/; n f .2n x; 2n w/  2 2 4  24pn " ı  kxkp kykp kzkp kwkp C 4n  lim 4n n!1 2 2 D0 for all x; y; z; w 2 J and so D.fx; y; zg; w/ D fD.x; w/; .y/; .z/g C f.x/; D.y; w /; .z/g C f.x/; .y/; D.z; w/g and D.x; fy; z; wg/ D fD.x; y/; .z/; .w/g C f.y/; D.x ; z/; .w/g C f.y/; .z/; D.x; w/g for all x; y; z; w 2 J:

5.3 Bi- -Derivations in JB -Triples

227

Let T W J J ! J be another C-bilinear mapping satisfying (5.47). Then we have kD.x; z/  T.x; z/k 1 kD.2n x; 2n z/  T.2n x; 2n z/k 4n 1 1  n kD.2n x; 2n z/  f .2n x; 2n z/k C n kf .2n x; 2n z/  T.2n x; 2n z/k 4 4 2ı  n; 4

D

which tends to zero as n ! 1 for all x; z 2 J. This proves the uniqueness of D. Therefore, the mapping D W J  J ! J is a unique bi--derivation satisfying (5.47). This completes the proof.  Theorem 5.33. Let p; " be positive real numbers with p ¤ 1 and f W J  J ! J with f .0; 0/ D 0, h W J ! J with h.0/ D 0 be the mappings such that kE; f .x; y; z; w/ C h. a C b/  h.a/  h.b/k  "kxkp kykp kzkp kwkp kakp kbkp

(5.56)

and kf .fx; y; zg; w/  ff .x; w/; h.y/; h.z/g fh.x/; f .y; w /; h.z/g  fh.x/; h.y/; f .z; w/gk Ckf .x; fy; z; wg/  ff .x; y/; h.z/; h.w/g fh.y/; f .x ; z/; h.w/g  fh.y/; h.z/; f .x; w/gk  "kxkp kykp kzkp kwkp

(5.57)

for all ; 2 T1 and x; y; z; w 2 J: Then the mapping h W J ! J is a C-linear mapping and the mapping f W J  J ! J is a bi-h-derivation. Proof. Letting x D y D z D w D 0 in (5.56), we have kh. a C b/  h.a/  h.b/k  0 for all a; b 2 J. By the same reasoning as in the proof of [227, Theorem 2.1], one can show that the mapping h W J ! J is a C-linear mapping. Letting a D b D 0 in (5.56), we have kE; f .x; y; z; w/k D 0

5 Stability of Functional Equations in C -Ternary Algebras

228

for all ; 2 T1 and x; y; z; w 2 J: By Lemma 5.29, the mapping f W J  J ! J is C-bilinear. For the case p < 1, it follows from (5.57) that kf .fx; y; zg; w/  ff .x; w/; h.y/; h.z/g  fh.x/; f .y; w /; h.z/g  fh.x/; h.y/; f .z; w/gk C kf .x; fy; z; wg/  ff .x; y/; h.z/; h.w/g  fh.y/; f .x ; z/; h.w/g  fh.y/; h.z/; f .x; w/gk  1 n1 o 1 1  D lim  4n f .8n fx; y; zg; 2n w/  n f .2n x; 2n w/; n h.2n y/; n h.2n z/ n!1 2 4 2 2 o n1 1 1  n h.2n x/; n f .2n y; 2n w /; n h.2n z/ 2 4 2 n1 o 1 1  n h.2n x/; n h.2n y/; n f .2n z; 2n w/ 2 2 4  1 n1 o 1 1  C  4n f .2n x; 8n fy; z; wg/  n f .2n x; 2n y/; n h.2n z/; n h.2n w/ 2 4 2 2 n1 o 1 1  n h.2n y/; n f .2n x ; 2n z/; n h.2n w/ 2 4 2 n1 o 1 1   n h.2n y/; n h.2n z/; n f .2n x; 2n w/  2 2 4 24pn " kxkp kykp kzkp kwkp n!1 24n

 lim D0

for all x; y; z; w 2 J and so the mapping f W J  J ! J is a bi-h-derivation. Similarly, for the case p > 1, one can show that the mapping f W J  J ! J is a bi-h-derivation. Therefore, the mapping h W J ! J is a C-linear mapping and the mapping f W J  J ! J is a bi-h-derivation. This completes the proof. 

Chapter 6

Stability of Functional Equations in Multi-Banach Algebras

In this chapter, we extend some results from last chapters in multi-Banach algebras (see [7, 91, 218, 252]). In Sect. 6.1, we consider the stability of the m-variable additive functional equation: m  m m m  X  X  X X f mxi C xj C f xi D 2f mxi iD1

jD1; j¤i

iD1

iD1

for each m  2, which was presented at Sect. 2.2 of Chap. 2 and, by the fixed point method, we approximate homomorphisms and derivations in multi-Banach algebras. In Sect. 6.2, by using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms in multi-C -ternary algebras and derivations on multi-C ternary algebras for the additive functional equation: m  m m m  X  X  X X f mxi C xj C f xi D 2f mxi iD1

jD1; j¤i

iD1

iD1

for each m  2. In Sect. 6.3, we consider the functional equation (3.97) presented at Sect. 3.5 of Chap. 3 and we use a fixed point method to prove the Hyers-Ulam stability of the functional equation (3.97) in multi-Banach modules over a unital multi-C -algebra.

© Springer International Publishing Switzerland 2015 Y.J. Cho et al., Stability of Functional Equations in Banach Algebras, DOI 10.1007/978-3-319-18708-2_6

229

230

6 Stability of Functional Equations in Multi-Banach Algebras

As an application, we show that every almost linear bijection h W A ! B of a unital multi-C -algebra A onto a unital multi-C -algebra B is a C -algebra isomorphism when  2n   2n  h n uy D h n u h.y/ r r for all unitaries u 2 U.A/, y 2 A and n  0. In Sect. 6.4, we approximate the following additive functional inequality: dC1 dC1  X X   f .x1i /;    ; f .xki // . iD1

k

iD1

  PdC1 x   PdC1 x    iD1 1i iD1 ki ;    ; mf   mf  k m m for all x11 ;    ; xk dC1 2 X where d  2 is a fixed integer. Also, we investigate homomorphisms in proper multi-CQ -algebras and derivations on proper multiCQ -algebras associated with the above additive functional inequality. In Sect. 6.5, by using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms and derivations on multi-C –ternary algebras for the additive functional equation: 2f

 Pp

jD1 xj

2

C

p d d  X X X yj D f .xj / C 2 f .yj /: jD1

jD1

jD1

6.1 Stability of m-Variable Additive Mappings For any mapping f W A ! B, we define D f .x1 ;    ; xm / WD

m X iD1

m m m  X   X   X f mxi C xj C f xi  2f mxi jD1; j¤i

iD1

iD1

for all 2 T1 WD f 2 C W j j D 1g and x1 ;    ; xm 2 A.

6.1.1 Stability of Homomorphisms in Multi-Banach Algebras Now, we prove the Hyers-Ulam stability of homomorphisms in multi-Banach algebras for the functional equation D f .x1 ;    ; xm / D 0.

6.1 Stability of m-Variable Additive Mappings

231

Theorem 6.1. Let ..Bk ; k  kk / W k  1/ be a multi-Banach algebra. Let f W A ! B be a mapping for which there exists the functions ' W Amk ! Œ0; 1/ and W A2k ! Œ0; 1/ such that lim mj '.mj x11 ;    ; mj x1m ;    ; mj xk1 ;    ; mj xkm / D 0;

j!1

(6.1)

k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk  '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm /;

(6.2)

k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk 

.x1 ; y1 ;    ; xk ; yk /

(6.3)

and lim m2j .mj x1 ; mj y1 ;    ; mj xk ; mj yk / D 0

(6.4)

j!1

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A. If there exists L < 1 such that m m m ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ' mx11 ; 0;    ; 0; mx21 ; 0;    ; 0;    ; mxk1 ; 0;    ; 0 m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   mL' x11 ; 0;    ; 0; x21 ; 0;    ; 0;    ; xk1 ; 0;    ; 0

for all x11 ; x21 ;    ; xk1 2 A, then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  1 ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0  m  mL

(6.5)

for all x1 ;    ; xk 2 A. Proof. Consider the set X WD fg W A ! Bg and introduce the generalized metric on X as follows: d.g; h/ D inffC 2 RC W k.g.x1 /  h.x1 /;    ; g.xk /  h.xk //kk m

m

m

‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  C'.x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0/; 8x1 ;    ; xk 2 Ag; which .X; d/ is complete.

232

6 Stability of Functional Equations in Multi-Banach Algebras

Now, we consider the linear mapping J W X ! X such that Jg.x/ WD

1 g.mx/ m

for all x 2 A. Now, we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting D 1, xi1 D x1 and xi2 D    D xim D 0, 1  i  k, in (6.2), we have k.f .mx1 /  mf .x1 /;    ; f .mxk /  mf .xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0

(6.6)

for all x1 ;    ; xk 2 A and so k.f .x1 / 

1 1 f .mx1 /;    ; f .xk /  f .mxk //kk m m m

m

m

‚ …„ ƒ  1  ‚ …„ ƒ ‚ …„ ƒ  ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0 m for all x1 ;    ; xk 2 A. Hence d.f ; Jf /  H W A ! B such that

1 m.

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.mx/ D mH.x/

(6.7)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (6.7) such that there exists C 2 .0; 1/ satisfying k.H.x1 /  f .x1 /;    ; H.xk /  f .xk //kk m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   C' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0

for all x1 ;    ; xk 2 A;

6.1 Stability of m-Variable Additive Mappings

233

(2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality f .mn x/ D H.x/ n!1 mn

(6.8)

lim

for all x 2 A; 1 d.f ; Jf /, which implies the inequality (3) d.f ; H/  1L 1 : m  mL

d.f ; H/ 

This implies that the inequality (6.4) holds. It follows from (6.1), (6.2) and (6.8) that m m m m  X  X  X   X  H mx1i C x1j C H x1i  2H mx1i ;  iD1

iD1

jD1; j¤i

 ;

iD1

m m m m  X  X   X X  H mxki C xkj C H xki  2H mxki  iD1

iD1

jD1; j¤i

iD1

k

m  m   X 1   X nC1 f m x C mn x1j  1i n n!1 m iD1

D lim

jD1; j¤i

Cf

m X iD1

 ;

m X iD1

Cf

m X iD1

 lim

n!1



mn x1i  2f

m X

 mnC1 x1i ;

iD1 m   X f mnC1 xki C mn xkj jD1; j¤i m  X   mn xki  2f mnC1 xki 

k

iD1

1 '.mn x11 ;    ; mn x1m ;    ; mn xk1 ;    ; mn xkm / mn

D0 for all x11 ;    ; x1m ;    ; xk1 ;    ; xkm 2 A and so m m m m  X  X   X X H mxi C xj C H xi D 2H mxi iD1

jD1; j¤i

iD1

iD1

for all x1 ;    ; xm 2 A. So H is additive. By a similar method to above, we get H.mx/ D H.m x/

(6.9)

234

6 Stability of Functional Equations in Multi-Banach Algebras

for all 2 T1 and x 2 A. Thus one can show that the mapping H W A ! B is C-linear. It follows from (6.3), (6.4) and (6.8) that k.H.x1 y1 /  H.x1 /H.y1 /;    ; H.xk yk /  H.xk /H.yk //kk D lim

n!1

1 k.f .m2n x1 y1 /  f .mn x1 /f .mn y1 /; m2n

   ; f .m2n xk yk /  f .mn xk /f .mn yk //kk  lim

n!1

1 .mn x1 ; mn y1 ;    ; mn xk ; mn yk / m2n

D0 for all x1 ; y1 ;    ; xk ; yk 2 A and so H.xy/ D H.x/H.y/ for all x; y 2 A. Thus H W A ! B is a homomorphism satisfying (6.5). This completes the proof.  Corollary 6.2. Let ..Bk ; k  kk / W k  1/ be a multi-Banach algebra. Let r < 1 and  be nonnegative real numbers and f W A ! B be a mapping such that k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk 

m X jD1

kx1j krA C    C

m X

kxkm krA

 (6.10)

jD1

and k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk     kx1 krA  ky1 krA C    C kxk krA  kyk krA

(6.11)

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A. Then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk    r r r r kx  k C ky k C    C kx k C ky k 1 1 k k A A A A m  mr for all x1 ;    ; xk 2 A.

6.1 Stability of m-Variable Additive Mappings

235

Proof. The proof follows from Theorem 6.1 by taking '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm / WD 

m X

kx1j krA C    C

m X

jD1

kxkm krA



jD1

and   .x1 ; y1 ;    ; xk ; yk / WD  kx1 krA  ky1 krA C    C kxk krA  kyk krA for all x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A and L D mr1 .



Theorem 6.3. Let ..Bk ; kkk / W k  1/ be a multi-Banach algebra. Let f W A ! B be a mapping for which there are functions ' W Amk ! Œ0; 1/ and W A2k ! Œ0; 1/ such that lim mj '.mj x11 ;    ; mj x1m ;    ; mj xk1 ;    ; mj xkm / D 0;

j!1

(6.12)

k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk  '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm /;

(6.13)

k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk 

.x1 ; y1 ;    ; xk ; yk /

(6.14)

and lim m2j .mj x1 ; mj y1 ;    ; mj xk ; mj yk / D 0

j!1

(6.15)

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A. If there exists L < 1 such that m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  ' x11 ; 0;    ; 0; x21 ; 0;    ; 0;    ; xk1 ; 0;    ; 0 m

m

m

…„ ƒ ‚ …„ ƒ ‚ …„ ƒ L ‚  ' mx11 ; 0;    ; 0; mx21 ; 0;    ; 0;    ; mxk1 ; 0;    ; 0 m for all x11 ; x21 ;    ; xk1 2 A, then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  L  ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0 m  mL for all x1 ;    ; xk 2 A.

(6.16)

236

6 Stability of Functional Equations in Multi-Banach Algebras

Proof. We consider the linear mapping J W X ! X such that Jg.x/ WD mg

x m

for all x 2 A. It follows from (6.6) that  x   x    1 k ;    ; f .xk /  mf  f .x1 /  mf  k m m ‚ x

m

m

…„ ƒ ‚ …„ ƒ x2 1 ; 0;    ; 0; ; 0;    ; 0;    ' m m m

m

‚ …„ ƒ  xk ; ; 0;    ; 0 m

m

(6.17)

m

‚ …„ ƒ  L  ‚ …„ ƒ ‚ …„ ƒ  ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0 m for all x1 ;    ; xk 2 A. Hence we have d.f ; Jf / 

L : m

By Theorem 1.3, there exists a mapping H W A ! B such that (1) H is a fixed point of J, i.e., H.mx/ D mH.x/

(6.18)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (6.18) such that there exists C 2 .0; 1/ satisfying k.H.x1 /  f .x1 /;    ; H.xk /  f .xk //kk m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   C' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0

for all x1 ;    ; xk 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality lim mn f

n!1

for all x 2 A;

 x  D H.x/ mn

6.1 Stability of m-Variable Additive Mappings

(3) d.f ; H/ 

1 1L d.f ; Jf /,

237

which implies the inequality d.f ; H/ 

L ; m  mL

which implies that the inequality (6.16) holds. The rest of the proof is similar to the proof of Theorem 6.1. This completes the proof.  Corollary 6.4. Let ..Bk ; k  kk / W k  1/ be a multi-Banach algebra. Let r > 1,  be nonnegative real numbers and f W A ! B be a mapping such that k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk 

m X

kx1j krA C    C

jD1

m X

kxkm krA

 (6.19)

jD1

and k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk     kx1 krA  ky1 krA C    C kxk krA  kyk krA

(6.20)

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A. Then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk 

mr

   kx1 krA C ky1 krA C    C kxk krA C kyk krA m

for all x1 ;    ; xk 2 A. Proof. The proof follows from Theorem 6.3 by taking '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm / WD 

m X jD1

kx1j krA C    C

m X

 kxkm krA ;

jD1

  .x1 ; y1 ;    ; xk ; yk / WD  kx1 krA  ky1 krA C    C kxk krA  kyk krA for all x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A and L D m1r .



238

6 Stability of Functional Equations in Multi-Banach Algebras

6.1.2 Stability of Derivations in Multi-Banach Algebras Now, we prove the Hyers-Ulam stability of derivations in multi-Banach algebras for the following functional equation: D f .x1 ;    ; xm / D 0 for all 2 T1 WD f 2 C W j j D 1g and x1 ;    ; xm 2 A. Theorem 6.5. Let ..Ak ; k  kk / W k  1/ be a multi-Banach algebra. Let f W A ! A be a mapping for which there exist the functions ' W Amk ! Œ0; 1/ and W A2k ! Œ0; 1/ such that lim mj '.mj x11 ;    ; mj x1m ;    ; mj xk1 ;    ; mj xkm / D 0;

j!1

(6.21)

k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk  '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm /;

(6.22)

k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk 

.x1 ; y1 ;    ; xk ; yk /

(6.23)

and lim m2j .mj x1 ; mj y1 ;    ; mj xk ; mj yk / D 0

(6.24)

j!1

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A. If there exists L < 1 such that m m m ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ' mx11 ; 0;    ; 0; mx21 ; 0;    ; 0;    ; mxk1 ; 0;    ; 0 m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   mL' x11 ; 0;    ; 0; x21 ; 0;    ; 0;    ; xk1 ; 0;    ; 0

for all x11 ; x21 ;    ; xk1 2 A, then there exists a unique derivation ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  1 ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0  m  mL for all x1 ;    ; xk 2 A.

(6.25)

6.1 Stability of m-Variable Additive Mappings

239

Proof. By the same reasoning as in the proof of Theorem 6.1, there exists a unique C-linear mapping ı W A ! A satisfying (6.24). The mapping ı W A ! A is given by ı.x/ D lim

n!1

f .mn x/ mn

(6.26)

for all x 2 A. It follows from (6.21), (6.24) and (6.26) that k.ı.x1 y1 /  ı.x1 /y1  x1 ı.y1 /;    ; ı.xk yk /  ı.xk /yk  xk ı.yk //kk D lim

n!1

1 k.f .m2n x1 y1 /  f .mn x1 /  mn y1  mn x1 f .mn y1 /; m2n

   ; f .m2n xk yk /  f .mn xk /  mn yk  mn xk f .mn yk //kk  lim m2n .mn x1 ; mn y1 ;    ; mn xk ; mn yk / n!1

D0 for all x; y 2 A and so ı.xy/ D ı.x/y C xı.y/ for all x; y 2 A. Thus ı W A ! A is a derivation satisfying (6.23). This completes the proof.  Corollary 6.6. Let ..Ak ; k  kk / W k  1/ be a multi-Banach algebra. Let r < 1,  be nonnegative real numbers and f W A ! A be a mapping such that k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk 

m X jD1

kx1j krA C    C

m X

kxkj krA

 (6.27)

jD1

and k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk   (6.28)   kx1 krA  ky1 krA C    C kxk krA  kyk krA for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A. Then there exists a unique derivation ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk  for all x 2 A.

 kxk2r A m  mr

240

6 Stability of Functional Equations in Multi-Banach Algebras

Proof. The proof follows from Theorem 6.5 by taking '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm / WD 

m X

kx1j krA C    C

m X

jD1

 kxkj krA ;

jD1

  .x1 ; y1 ;    ; xk ; yk / WD  kx1 krA  ky1 krA C    C kxk krA  kyk krA for all x; y 2 A and L D mr .



Remark 6.7. Let ..A ; k  kk / W k  1/ be a multi-Banach algebra. Let f W A ! A be a mapping for which there exist the functions ' W Amk ! Œ0; 1/ and W A2k ! Œ0; 1/ such that k

lim mj '.mj x11 ;    ; mj x1m ;    ; mj xk1 ;    ; mj xkm / D 0;

j!1

(6.29)

k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk  '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm /;

(6.30)

k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk 

.x1 ; y1 ;    ; xk ; yk /

(6.31)

and lim m2j .mj x1 ; mj y1 ;    ; mj xk ; mj yk / D 0

j!1

(6.32)

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A. If there exists L < 1 such that m m m ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ' mx11 ; 0;    ; 0; mx21 ; 0;    ; 0;    ; mxk1 ; 0;    ; 0 m

m

m

‚ …„ ƒ  L  ‚ …„ ƒ ‚ …„ ƒ  ' x11 ; 0;    ; 0; x21 ; 0;    ; 0;    ; xk1 ; 0;    ; 0 m for all x11 ; x21 ;    ; xk1 2 A, then there exists a unique derivation ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  L ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0  m  mL for all x1 ;    ; xk 2 A.

(6.33)

6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C -Ternary. . .

241

Corollary 6.8. Let ..Ak ; k  kk / W k  1/ be a multi-Banach algebra. Let r > 1,  be nonnegative real numbers and f W A ! A be a mapping such that k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk 1 0 m m X X @ kx1j krA C    C kxkj krA A jD1

jD1

and k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk     kx1 krA  ky1 krA C    C kxk krA  kyk krA for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; x; y1 ;    ; yk 2 A. Then there exists a unique derivation ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk 

mr

 kxk2r A m

for all x 2 A. Proof. The proof follows from Remark 6.7 by taking '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm / WD 

m X

kx1j krA C    C

jD1

m X

 kxkj krA ;

jD1

  .x1 ; y1 ;    ; xk ; yk / WD  kx1 krA  ky1 krA C    C kxk krA  kyk krA for all x; y 2 A and L D m1r .



6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C -Ternary Algebras In this section, using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms in multi-C -ternary algebras and derivations on multi-C -ternary algebras for the following additive functional equation: m  m m m  X  X  X X f mxi C xj C f xi D 2f mxi iD1

jD1; j¤i

iD1

iD1

for each m  2. Throughout this section, assume that A, B are C -ternary algebras.

242

6 Stability of Functional Equations in Multi-Banach Algebras

6.2.1 Stability of Homomorphisms in Multi-C -Ternary Algebras For any mapping f W A ! B, we define D f .x1 ;    ; xm / WD

m X iD1

m m m  X   X   X f mxi C xj C f xi  2f mxi iD1

jD1; j¤i

iD1

for all 2 T1 WD f 2 C W j j D 1g and x1 ;    ; xm 2 A. Using Theorem 1.3, we prove the Hyers-Ulam stability of homomorphisms in multi-C ternary algebras for the functional equation D f .x1 ;    ; xm / D 0: Theorem 6.9. Let ..Bk ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! B be a mapping for which there exist the functions ' W Amk ! Œ0; 1/ and W A2k ! Œ0; 1/ such that lim mj '.mj x11 ;    ; mj x1m ;    ; mj xk1 ;    ; mj xkm / D 0;

j!1

(6.34)

k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk  '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm /;

(6.35)

k.f .Œx1 ; y1 ; z1 /  Œf .x1 /; f .y1 /; f .z1 /;    ; f .Œxk ; yk ; zk /  Œf .xk /; f .yk /; f .zk //kk 

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /

(6.36)

and lim m3j .mj x1 ; mj y1 ; mj z1 ;    ; mj xk ; mj yk ; mj zk / D 0

j!1

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A. If there exists L < 1 such that m m m ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ' mx11 ; 0;    ; 0; mx21 ; 0;    ; 0;    ; mxk1 ; 0;    ; 0 m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   mL' x11 ; 0;    ; 0; x21 ; 0;    ; 0;    ; xk1 ; 0;    ; 0

(6.37)

6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C -Ternary. . .

243

for all x11 ; x21 ;    ; xk1 2 A, then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  1 ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0  m  mL

(6.38)

for all x1 ;    ; xk 2 A. Proof. Consider the set X WD fg W A ! Bg and introduce the generalized metric on X as follows: d.g; h/ D inffC 2 RC W k.g.x1 /  h.x1 /;    ; g.xk /  h.xk //kk m

m

m

‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  C'.x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0/; 8x1 ;    ; xk 2 Ag; which .X; d/ is complete. Now, we consider the linear mapping J W X ! X such that Jg.x/ WD

1 g.mx/ m

for all x 2 A. Now, we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting D 1, xi1 D xi and xi2 D    D xim D 0 (1  i  k) in (6.35), we have k.f .mx1 /  mf .x1 /;    ; f .mxk /  mf .xk //kk m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0

(6.39)

for all x1 ;    ; xk 2 A. Thus we have k.f .x1 / 

1 1 f .mx1 /;    ; f .xk /  f .mxk //kk m m m

m

m

‚ …„ ƒ  1  ‚ …„ ƒ ‚ …„ ƒ  ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0 m for all x1 ;    ; xk 2 A. Hence d.f ; Jf /  H W A ! B such that

1 . m

By Theorem 1.3, there exists a mapping

244

6 Stability of Functional Equations in Multi-Banach Algebras

(1) H is a fixed point of J, i.e., H.mx/ D mH.x/

(6.40)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (6.40) such that there exists C 2 .0; 1/ satisfying the following: k.H.x1 /  f .x1 /;    ; H.xk /  f .xk //kk m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   C' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0

for all x1 ;    ; xk 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality f .mn x/ D H.x/ n!1 mn lim

(6.41)

for all x 2 A; 1 d.f ; Jf /, which implies the inequality (3) d.f ; H/  1L d.f ; H/ 

1 : m  mL

This implies that the inequality (6.38) holds. It follows from (6.34), (6.35) and (6.41) that m m m m  X  X  X  X  H mx1i C x1j C H x1i  2H mx1i /;  iD1

 ;

iD1

jD1; j¤i m X

 H mxki C

iD1

m X



xkj C H

iD1

m X



xki  2H

iD1

jD1; j¤i

m   m  X 1   X nC1 n f m x C m x  1i 1j n!1 mn iD1

D lim

jD1; j¤i

Cf

m X iD1

m  X  mn x1i  2f mnC1 x1i ; iD1

m X iD1

mxki

  

k

6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C -Ternary. . .

 ;

m  m  X X f mnC1 xki C mn xkj iD1

Cf

m X

jD1; j¤i

m  X   mn xki  2f mnC1 xki 

iD1

 lim

n!1

245

iD1

k

1 '.mn x11 ;    ; mn x1m ;    ; mn xk1 ;    ; mn xkm / D 0 mn

for all x11 ;    ; x1m ;    ; xk1 ;    ; xkm 2 A and so m m m m  X  X   X X H mxi C xj C H xi D 2H mxi iD1

jD1; j¤i

iD1

(6.42)

iD1

for all x1 ;    ; xm 2 A. Thus H is additive. By the similar method, we have H.mx/ D H.m x/ for all 2 T1 and x 2 A. Thus one can show that the mapping H W A ! B is C-linear. It follows from (6.36), (6.37) and (6.41) that k.H.Œx1 ; y1 ; z1 /  ŒH.x1 /; H.y1 /; H.z1 /;    ; H.Œxk ; yk ; zk /  ŒH.xk /; H.yk /; H.zk //kk 1 k.f .Œmn x1 ; mn y1 ; mn z1 /  Œf .mn x1 /; f .mn y1 /; f .mn z1 /; m3n    ; f .Œmn xk ; mn yk ; mn zk /  Œf .mn xk /; f .mn yk /; f .mn zk //kk

D lim

n!1

 lim

n!1

1 .mn x1 ; mn y1 ;    ; mn xk ; mn yk / m3n

D0 for all x1 ; y1 ;    ; xk ; yk 2 A and so H.Œx; y; z/ D ŒH.x/; H.y/; H.z/ for all x; y 2 A. Thus H W A ! B is a homomorphism satisfying (6.38). This completes the proof.  Corollary 6.10. Let ..Bk ; k  kk /W k  1/ be a multi-C -ternary algebra. Let r < 1,  be nonnegative real numbers and f W A ! B be a mapping such that

246

6 Stability of Functional Equations in Multi-Banach Algebras

k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk 0 1 m m X X r r @ kx1j kA C    C kxkj kA A jD1

(6.43)

jD1

and k.f .Œx1 ; y1 ; z1 /  Œf .x1 /; f .y1 /; f .z1 /;    ; f .Œxk ; yk ; zk /  Œf .xk /; f .yk /; f .zk //kk     kx1 krA  ky1 krA  kz1 krA C    C kxk krA  kyk krA  kzk krA

(6.44)

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A. Then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk 

   kx1 krA C    C kxk krA r mm

for all x1 ;    ; xk 2 A. Proof. The proof follows from Theorem 6.9 by taking '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm / WD 

m X jD1

kx1j krA C    C

m X

 kxkj krA ;

jD1

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /   WD  kx1 krA  ky1 krA  kz1 krA C    C kxk krA  kyk krA  kzk krA for all x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A and L D mr1 .  Theorem 6.11. Let ..Bk ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! B be a mapping for which there exist the functions ' W Amk ! Œ0; 1/ and W A2k ! Œ0; 1/ satisfying the inequalities (6.35) and (6.36) such that lim mj '.mj x11 ;    ; mj x1m ;    ; mj xk1 ;    ; mj xkm / D 0

(6.45)

lim m3j .mj x1 ; mj y1 ; mj z1 ;    ; mj xk ; mj yk ; mj zk / D 0

(6.46)

j!1

and j!1

6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C -Ternary. . .

247

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xm ; y1 ;    ; yk ; z1 ;    ; zk 2 A. If there exists L < 1 such that m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  ' x11 ; 0;    ; 0; x21 ; 0;    ; 0;    ; xk1 ; 0;    ; 0 m

m

m

…„ ƒ ‚ …„ ƒ ‚ …„ ƒ L ‚  ' mx11 ; 0;    ; 0; mx21 ; 0;    ; 0;    ; mxk1 ; 0;    ; 0 m for all x11 ; x21 ;    ; xk1 2 A, then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  L ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0  m  mL

(6.47)

for all x1 ;    ; xk 2 A. Proof. We consider the linear mapping J W X ! X defined by Jg.x/ WD mg

x m

for all x 2 A. It follows from (6.39) that  x   x   1 k  ;    ; f .xk /  mf  f .x1 /  mf  k m m ‚ x

m

m

…„ ƒ ‚ …„ ƒ x2 1 ; 0;    ; 0; ; 0;    ; 0;    ' m m m

m

m

‚ …„ ƒ  xk ; ; 0;    ; 0 m m

‚ …„ ƒ  L  ‚ …„ ƒ ‚ …„ ƒ  ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0 m

(6.48)

for all x1 ;    ; xk 2 A. Hence we have d.f ; Jf / 

L : m

By Theorem 1.3, there exists a mapping H W A ! B such that (1) H is a fixed point of J, i.e., H.mx/ D mH.x/ for all x 2 A. The mapping H is a unique fixed point of J in the set

(6.49)

248

6 Stability of Functional Equations in Multi-Banach Algebras

Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (6.49) such that there exists C 2 .0; 1/ satisfying the following: k.H.x1 /  f .x1 /;    ; H.xk /  f .xk //kk m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   C' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0

for all x1 ;    ; xk 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality  x  D H.x/ lim mn f n!1 mn for all x 2 A; 1 (3) d.f ; H/  1L d.f ; Jf /, which implies the inequality d.f ; H/ 

L ; m  mL

which implies that the inequality (6.47) holds. The rest of the proof is similar to the proof of Theorem 6.9. This completes the proof.  Corollary 6.12. Let ..Bk ; k  kk / W k  1/ be a multi-C -ternary algebra. Let r > 1,  be nonnegative real numbers and f W A ! B be a mapping such that satisfying (6.43) and (6.44). Then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk 

mr

   kx1 krA C    C kxk krA m

for all x1 ;    ; xk 2 A. Proof. The proof follows from Theorem 6.11 by taking '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm / WD 

m X jD1

kx1j krA

CC

m X

 kxkj krA ;

jD1

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /   WD  kx1 krA  ky1 krA  kz1 krA C    C kxk krA  kyk krA  kzk krA for all x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A and L D m1r . 

6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C -Ternary. . .

249

6.2.2 Stability of Derivations in Multi-C -Ternary Algebras Now, we prove the Hyers-Ulam stability of derivations on multi-C -ternary algebras for the following functional equation: D f .x1 ;    ; xm / D 0 for all 2 T1 WD f 2 C W j j D 1g and x1 ;    ; xm 2 A. Theorem 6.13. Let ..Ak ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! A be a mapping for which there exist the functions ' W Amk ! Œ0; 1/ and W A3k ! Œ0; 1/ satisfying (6.34), (6.35) and (6.37) such that k.f .Œx1 ; y1 ; z1 /  Œf .x1 /; y1 ; z1   Œx1 ; f .y1 /; z1   Œx1 ; y1 ; f .z1 /;    ; f .Œxk ; yk ; zk /  Œf .xk /; yk ; zk  Œxk ; f .yk /; zk   Œxk ; yk ; f .zk //kk 

(6.50)

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /

for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xk ; y1 ;    ; yk , z1 ;    ; zk 2 A. If there exists L < 1 such that m m m ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ' mx11 ; 0;    ; 0; mx21 ; 0;    ; 0;    ; mxk1 ; 0;    ; 0 m m m  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ   mL' x11 ; 0;    ; 0; x21 ; 0;    ; 0;    ; xk1 ; 0;    ; 0

for all x11 ; x21 ;    ; xk1 2 A, then there exists a unique derivation ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  1 ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0  m  mL

(6.51)

for all x1 ;    ; xk 2 A. Proof. By the same reasoning as in the proof of Theorem 6.9, there exists a unique C-linear mapping ı W A ! A satisfying (6.50) and the mapping ı W A ! A is given by ı.x/ D lim

n!1

f .mn x/ mn

(6.52)

250

6 Stability of Functional Equations in Multi-Banach Algebras

for all x 2 A. It follows from (6.34), (6.37) and (6.52) that k.ı.Œx1 ; y1 ; z1 /  Œı.x1 /; y1 ; z1   Œx1 ; ı.y1 /; z1   Œx1 ; y1 ; f .z1 /;    ; ı.Œxk ; yk ; zk /  Œı.xk /; yk ; zk  Œxk ; ı.yk /; zk   Œxk ; yk ; ı.zk //kk 1 D lim 3n k.f .Œmn x1 ; mn y1 ; mn z1 /  Œf .mn x1 /; mn y1 ; mn z1  n!1 m Œmn x1 ; f .mn y1 /; mn z1   Œmn x1 ; mn y1 ; f .mn z1 /;    ; f .Œmn xk ; mn yk ; mn zk /  Œf .mn xk /; mn yk ; mn zk  Œmn xk ; f .mn yk /; mn zk   Œmn xk ; mn yk ; f .mn zk //kk  lim m3n .mn x1 ; mn y1 ; mn z1 ;    ; mn xk ; mn yk ; mn zk / n!1

D0 for all x1 ;    ; xm ; y1 ;    ; yk ; z1 ;    ; zk 2 A and so ı.Œx; y; z/ D Œı.x/; y; z C Œx; ı.y/; z C Œx; y; ı.z/ for all x; y; z 2 A. Thus ı W A ! A is a derivation satisfying (6.50). This completes the proof.  Corollary 6.14. Let ..Ak ; k  kk / W k  1/ be a multi-C -ternary algebra. Let r<1,  be nonnegative real numbers and f W A ! A be a mapping such that k.D f .x11 ;    ; x1m /;    ; D f .xk1 ;    ; xkm //kk 

m X

kx1j krA C    C

m X

jD1

kxkj krA

 (6.53)

jD1

and k.f .Œx1 ; y1 ; z1 /  Œf .x1 /; y1 ; z1   Œx1 ; f .y1 /; z1   Œx1 ; y1 ; f .z1 /;    ; f .Œxk ; yk ; zk /  Œf .xk /; yk ; zk  Œxk ; f .yk /; zk   Œxk ; yk ; f .zk //kk     kx1 krA  ky1 krA  kz1 krA C    C kxk krA  kyk krA  kyk krA for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A. Then there exists a unique derivation ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk  for all x1 ;    ; xk 2 A.

mr

   kx1 krA C    C kxk krA m

(6.54)

6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C -Ternary. . .

251

Proof. The proof follows from Theorem 6.50 by taking '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm / WD 

m X

kx1j krA C    C

jD1

m X

 kxkj krA ;

jD1

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /   WD  kx1 krA  ky1 krA  kz1 krA C    C kxk krA  kyk krA  kyk krA for all x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A and L D mr . This completes the proof.  Remark 6.15. Let ..Ak ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! A be a mapping for which there exist the functions ' W Amk ! Œ0; 1/ and W A3k ! Œ0; 1/ satisfying (6.35), (6.45), (6.46) and (6.50) for all 2 T1 and x11 ;    ; x1m ;    ; xk1 ;    ; xkm ; x1 ;    ; xk , y1 ;    ; yk ; z1 ;    ; zk 2 A. If there exists L < 1 such that m m m ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ' mx11 ; 0;    ; 0; mx21 ; 0;    ; 0;    ; mxk1 ; 0;    ; 0 m

m

m

‚ …„ ƒ  L  ‚ …„ ƒ ‚ …„ ƒ  ' x11 ; 0;    ; 0; x21 ; 0;    ; 0;    ; xk1 ; 0;    ; 0 m for all x11 ; x21 ;    ; xk1 2 A, then there exists a unique derivation ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk m

m

m

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  L ' x1 ; 0;    ; 0; x2 ; 0;    ; 0;    ; xk ; 0;    ; 0  m  mL for all x1 ;    ; xk 2 A. Corollary 6.16. Let ..Ak ; k  kk / W k  1/ be a multi-C -ternary algebra. Let r > 1,  be nonnegative real numbers and f W A ! A be a mapping satisfying (6.53) and (6.54). Then there exists a unique derivation ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk  for all x1 ;    ; xk 2 A.

mr

   kx1 krA C    C kxk krA m

252

6 Stability of Functional Equations in Multi-Banach Algebras

Proof. The proof follows from Theorem 6.15 by taking '.x11 ;    ; x1m ;    ; xk1 ;    ; xkm / WD 

m X

kx1j krA C    C

jD1

m X

 kxkj krA ;

jD1

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /   WD  kx1 krA  ky1 krA  kz1 krA C    C kxk krA  kyk krA  kyk krA for all x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A and L D m1r . This completes the proof. 

6.3 Generalized Additive Mappings and Isomorphisms in Multi-C -Algebras Let X and Y be vector spaces. It is shown that, if an odd mapping f W X ! Y satisfies the functional equation (3.97), then the odd mapping f is additive. Also, we use the fixed point method to prove the Hyers-Ulam stability of the functional equation (3.97) in multi-Banach modules over a unital multi-C -algebra. As an application, we show that every almost linear bijection h W A ! B of a unital multiC -algebra A onto a unital multi-C -algebra B is a C -algebra isomorphism when h

 2n

  2n  uy D h u h.y/ rn rn

for all unitaries u 2 U.A/, y 2 A and n  0.

6.3.1 Stability of Odd Functional Equations in Multi-Banach Modules over a Multi-C -Algebra We assume that ..Ak ; k  kk / W k  1/ is a unital multi-C -algebra and ..X k ; k  kk / W k  1/, ..Y k ; kkk / W k  1/ are multi-Banach left modules over ..Ak ; kkk / W k  1/. Moreover, by U.A/, we denote the unitary group of A. For any mapping f W X ! Y, we set

6.3 Generalized Additive Mappings and Isomorphisms in Multi-C -Algebras

Du f .x1 ;    ; xd /  Pd

jD1

WD rf

uxj 

r

X

C .j/

Pd

jD1

rf

253

 Pd .1/.j/ uxj  jD1 r

D 0; 1

.j/ D l

.d1 Cl d1 Cl1 C 1/

d X

uf .xj /

jD1

for all u 2 U.A/ and x1 ;    ; xd 2 X. Theorem 6.17. Let r ¤ 2 and f W X ! Y be an odd mapping such that, for each k  1, there exists a function 'k W X kd ! Œ0; 1/ such that rj 'k j!1 2j lim



2j 2j 2j 2j x11 ;    ; j x1d ;    ; j xk1 ;    ; j xkd j r r r r

 D0

(6.55)

and k.Du f .x11 ;    ; x1d /;    ; Du f .xk1 ;    ; xkd //kk  'k .x11 ;    ; x1d ;    ; xk1 ;    ; xkd /

(6.56)

for all u 2 U.A/ and x11 ;    ; x1d ;    ; xk1 ;    ; xkd 2 X. If there exists L < 1 such that

'k

‚ 2

d

…„ ƒ 2 x11 ; x11 ;    ; 0;    r r d

d

‚ …„ ƒ  2 2 ; xk1 ; xk1 ;    ; 0 r r d

‚ …„ ƒ ‚ …„ ƒ 2  L'k x11 ; x11 ;    ; 0;    ; xk1 ; xk1 ;    ; 0 r for all k  1 and x11 ;    ; xk1 2 X, then there exist a unique A-linear generalized additive mapping  W X ! Y such that k..x1 /  f .x1 /;    ; .xk /  f .xk //kk

(6.57)   1 'k x1 ; x1 ; 0; : : : ; 0; : : : ; xk ; xk ; 0; : : : ; 0  „ ƒ‚ … „ ƒ‚ … 2.d2 Cl d2 Cl2 C 1/.1  L/ d  2 times

for all k  1 and x1 ;    ; xk 2 X.

d  2 times

254

6 Stability of Functional Equations in Multi-Banach Algebras

Proof. Put S WD f W X ! Yg and d.; h/ D inffC 2 RC W k..x1 /  h.x1 /;    ; .xk /  h.xk //kk d

d

‚ …„ ƒ ‚ …„ ƒ  C'k .x1 ; x1 ; 0;    ; 0;    ; xk ; xk ; 0;    ; 0/; k  1; x1 ;    ; xk 2 Xg for all ; h 2 S which .S; d/ is a complete generalized metric space. Define a mapping J W S ! S by   r 2 J.x/ WD  x 2 r for all  2 S and x 2 X. Now, we have d.J; Jh/  Ld.; h/ for all ; h 2 S. For a fixed k  1, putting u D 1 2 U.A/, xi1 D xi2 D x1 and xi3 D    D xid D 0 for each i 2 f1;    ; kg in (6.56), we have   2  2      rf x1  2f .x1 /;    ; rf xk  2f .xk /  k r r   1 'k x1 ; x1 ; 0;    ; 0;    ; xk ; xk ; 0;    ; 0  „ ƒ‚ … „ ƒ‚ … d2 Cl d2 Cl2 C 1 d  2 times

d  2 times

since f is odd and t WDd2 Cl d2 Cl2 C 1 Dd1 Cl d1 Cl1 C 1. Thus we have  r 2  r  2     f .x1 /  f x1 ;    ; f .xk /  f xk  k 2 r 2 r   1  'k x1 ; x1 ; 0;    ; 0;    ; xk ; xk ; 0;    ; 0 „ ƒ‚ … „ ƒ‚ … 2t d  2 times

d  2 times

for all x1 ;    ; xk 2 X and so d.f ; Jf / 

1 : 2t

(6.58)

Consequently, by Theorem 1.3, there exists a mapping  W X ! Y such that (1)  is a fixed point of J, i.e.,  

 2 2 x D .x/ r r

for all x 2 X and  is unique in the set Y D f 2 X W d.f ; / < 1g:

(6.59)

6.3 Generalized Additive Mappings and Isomorphisms in Multi-C -Algebras

255

This means that  is a unique mapping satisfying (6.59) such that there exists C 2 .0; 1/ with k..x1 /  f .x1 /;    ; .xk /  f .xk //kk    C'k x1 ; x1 ; 0;    ; 0;    ; xk ; xk ; 0;    ; 0 „ ƒ‚ … „ ƒ‚ … d  2 times

d  2 times

for all k  1 and x1 ;    ; xk 2 X; (2) d.J n f ; / ! 0 as n ! 1. This implies the equality  n  rn 2 x D .x/ lim n f n!1 2 rn

(6.60)

for all x 2 X; 1 (3) d.f ; /  1L d.f ; Jf / which together with (6.58) gives d.f ; / 

1 2t  2tL

and so the inequality (6.57) holds for all x1 ;    ; xk 2 X. Next, note that the fact that the mapping f is odd and (6.60) imply that  is odd. Moreover, by (6.55) and (6.56), we have k.D1 .x11 ;    ; x1d /;    ; D1 .xk1 ;    ; xkd //kk     n  n  rn  2 2n 2 2n  D lim n  D1 f x11 ;    ; n x1d ;    ; D1 f xk1 ;    ; n xkd   n!1 2 rn r rn r k   n n n n n 2 2 2 2 r  lim n 'k n x11 ;    ; n x1d ;    ; n xk1 ;    ; n xkd n!1 2 r r r r D0 for all k  1 and x11 ;    ; x1d ;    ; xk1 ;    ; xkd 2 X and so  is a generalized additive mapping. For any fixed u 2 U.A/ and x 2 X, using (6.55) and (6.56), we have     .Du .x; 0;    ; 0/;    ; Du .x; 0;    ; 0// „ ƒ‚ … „ ƒ‚ … k d  1 times

d  1 times

   2n  2n  r    D lim n  Du f n x; 0;    ; 0 ;    ; Du f n x; 0;    ; 0  „ „ ƒ‚ … ƒ‚ … n!1 2 k r r n

d  1 times

 r n  2n 2n  lim n 'k n x; 0;    ; 0;    ; n x; 0;    ; 0 n!1 2 r „ ƒ‚ … r „ ƒ‚ … d  1 times

D0

d  1 times

d  1 times

256

6 Stability of Functional Equations in Multi-Banach Algebras

and so  ux  D .d1 Cl d1 Cl1 C 1/u.x/: .d1 Cl d1 Cl1 C 1/r r Since  is a generalized additive mapping, from Lemma 3.64 it follows that  is additive and so  ux  D u.x/ .ux/ D r r for all u 2 U.A/ and x 2 X. It is straight forward to show that  is an A-linear mapping (see also Theorem 3.1 in [20]). This completes the proof.  Corollary 6.18. Let r ¤ 2 and ; p 2 .0; 1/. Assume also that p > 1 for r > 2 and p < 1 for r < 2. If f W X ! Y is an odd mapping such that k.Du f .x11 ;    ; x1d /;    ; Du f .xk1 ;    ; xkd //kk 

d X

kx1j kp C : : : C

jD1

d X

kxkj kp



jD1

for all u 2 U.A/, k  1 and x11 ;    ; x1d ;    ; xk1 ;    ; xkd 2 X, then there exists a unique A-linear generalized additive mapping  W X ! Y such that k..x1 /  f .x1 /;    ; .xk /  f .xk //kk 

rp1  .jjx1 jjp C    C jjxk jjp / .rp1  2p1 /.d2 Cl d2 Cl2 C 1/

for all k  1 and x1 ;    ; xk 2 X. Proof. Taking L D

2p1 rp1

and

'k .x11 ;    ; x1d ;    ; xk1 ;    ; xkd / D 

d X

kx1j kp C : : : C

jD1

d X

kxkj kp



jD1

for all k  1 and x11 ;    ; x1d ;    ; xk1 ;    ; xkd 2 X in Theorem 6.17, we get the desired assertion.  Theorem 6.19. Let r ¤ 2. Let f W X ! Y be an odd mapping for which there exists a function ' W X kd ! Œ0; 1/ such that 2j ' j!1 rj lim



rj rj rj rj x ;    ; x ;    ; x ;    ; xkd 11 1d k1 2j 2j 2j 2j

 D0

6.3 Generalized Additive Mappings and Isomorphisms in Multi-C -Algebras

257

and k.Du f .x11 ;    ; x1d /;    ; Du f .xk1 ;    ; xkd //kk  '.x11 ;    ; x1d ;    ; xk1 ;    ; xkd / for all u 2 U.A/ and x11 ;    ; x1d ;    ; xk1 ;    ; xkd 2 X. If there exists L < 1 such that d

 ‚r

d

…„ ƒ ‚ …„ ƒ r r r x11 ; x11 ;    ; 0; x21 ; x21 ;    ; 0;    ' 2 2 2 2 d

d

‚ …„ ƒ r r ; xk1 ; xk1 ;    ; 0 2 2

d

d

…„ ƒ ‚ …„ ƒ ‚ …„ ƒ r ‚  L' x11 ; x11 ;    ; 0; x21 ; x21 ;    ; 0;    ; xk1 ; xk1 ;    ; 0 2 for all x11 ; x21 ;    ; xk1 2 X, then there exists a unique A-linear generalized additive mapping  W X ! Y such that sup k..x1 /  f .x1 /;    ; .xk /  f .xk //kk k1

 sup k1

 L ' x1 ; x1 ; 0;    ; 0; „ ƒ‚ … 2.d2 Cl d2 Cl2 C 1/.1  L/    ; xk ; xk ; 0;    ; 0 „ ƒ‚ …



d  2 times

d  2 times

for all x1 ;    ; xk 2 X. Proof. Note that f .0/ D 0 and f .x/ D f .x/ for all x 2 X since f is an odd mapping. Let u D 1 2 U.A/. Putting xi1 D xi2 D x1 and xi3 D    D xim D 0 (1  i  k) in (6.56), we have          rf 2 x1  2f .x1 /;    ; rf 2 xk  2f .xk /    r r k   1 ' x1 ; x1 ; 0;    ; 0;    ; xk ; xk ; 0;    ; 0 :  „ ƒ‚ … „ ƒ‚ … d2 Cl d2 Cl2 C 1 d  2 times

Letting t WDd2 Cl d2 Cl2 C 1, we have         f .x1 /  2 f r x1 ;    ; f .xk /  2 f r xk    r 2 r 2 k

d  2 times

258

6 Stability of Functional Equations in Multi-Banach Algebras



 1 r r r r ' x1 ; x1 ; 0;    ; 0;    ; xk ; xk ; 0;    ; 0 rt 2 2 „ ƒ‚ … 2 2 „ ƒ‚ … d  2 times

d  2 times

 L   ' x1 ; x1 ; 0;    ; 0;    ; xk ; xk ; 0;    ; 0 „ ƒ‚ … „ ƒ‚ … 2t d  2 times

d  2 times

for all x1 ;    ; xk 2 X. The rest of the proof is similar to the proof of Theorem 6.17. This completes the proof.  Corollary 6.20. Let r < 2 and , p > 1 be positive real numbers or let r > 2 and , p < 1 be positive real numbers. Let f W X ! Y be an odd mapping such that k.Du f .x11 ;    ; x1d /;    ; Du f .xk1 ;    ; xkd //kk 

d X

kx1j kp C    C

jD1

d X

kxkj kp



jD1

for all u 2 U.A/ and x11 ;    ; x1d ;    ; xk1 ;    ; xkd 2 X. Then there exists a unique A-linear generalized additive mapping  W X ! Y such that sup k..x1 /  f .x1 /;    ; .xk /  f .xk //kk k1

 sup k1

rp1  .jjx1 jjp C    C kxk kp / .2p1  rp1 /.d2 Cl d2 Cl2 C 1/

for all x 2 X. Proof. Define '.x11 ;    ; x1d ;    ; xk1 ;    ; xkd / D 

d X

kx1j kp C    C

jD1

and put L D

rp1 2p1

d X

kxkj kp



jD1

in Theorem 6.19. Then we get the desired result.



Now, we investigate the Hyers–Ulam stability of linear mappings for the case d D 2. Theorem 6.21. Let r ¤ 2. Let f W X ! Y be an odd mapping for which there exists a function ' W X 2k ! Œ0; 1/ such that rj ' j!1 2j lim



2j 2j 2j 2j x1 ; j y1 ;    ; j xk ; j yk j r r r r

 D0

6.3 Generalized Additive Mappings and Isomorphisms in Multi-C -Algebras

259

and   ux C uy  1 1   uf .x1 /  uf .y1 /;  rf r  ux C uy   k k     ; rf  uf .xk /  uf .yk /  k r  '.x1 ; y1 ;    ; xk ; yk /

(6.61)

for all u 2 U.A/ and x1 ;    xk ; y1    ; yk 2 X. If there exists L < 1 such that  '

2 2 2 2 2 2 x1 ; x1 ; x2 ; x2 ;    ; xk ; xk r r r r r r

 

2 L' .x1 ; x1 ; x2 ; x2 ;    ; xk ; xk / r

for all x1 ;    xk 2 X. Then there exists a unique A-linear generalized additive mapping  W X ! Y such that sup k..x1 /  f .x1 /;    ; .xk /  f .xk //kk k1

 sup k1

L '.x1 ; x1 ;    ; xk ; xk / 2.1  L/

for all x1 ;    ; xk 2 X. Proof. Let u D 1 2 U.A/. Putting xi D yi (1  i  k) in (6.61), we have          rf 2 x1  2f .x1 /;    ; rf 2 xk  2f .xk /    r r k  '.x1 ; x1 ;    ; xk ; xk / for all x 2 X and so         f .x1 /  r f 2 x1 ;    ; f .xk /  r f 2 xk    2 r 2 r k 

1 '.x1 ; x1 ;    ; xk ; xk / 2

for all x 2 X. The rest of the proof is the same as in the proof of Theorem 6.17. This completes the proof.  Corollary 6.22. Let r > 2 and , p > 1 be positive real numbers or let r < 2 and , p < 1 be positive real numbers. Let f W X ! Y be an odd mapping such that

260

6 Stability of Functional Equations in Multi-Banach Algebras

  ux C uy  1 1   uf .x1 /  uf .y1 /;  rf r   ux C uy  k k   uf .xk /  uf .yk /     ; rf k r 

k X

.kxj kp C kyj kp /

jD1

for all u 2 U.A/ and x1 ;    xk 2 X. Then there exists a unique A-linear generalized additive mapping  W X ! Y such that sup k..x1 /  f .x1 /;    ; .xk /  f .xk //kk  sup k1

k1

k X rp1  jjxj jjp rp1  2p1 jD1

for all x1 ;    xk 2 X. Proof. Define '.x1 ; y1 ;    ; xk ; yk / D 

k X

.jjxj jjp C jjyj jjp /

jD1



and apply Theorem 6.21. Then we get the desired result.

Theorem 6.23. Let r ¤ 2. Let f W X ! Y be an odd mapping for which there exists a function ' W X 2k ! Œ0; 1/ such that 2j lim j ' j!1 r and



rj rj rj rj x ; y ;    ; x ; yk 1 1 k 2j 2j 2j 2j

 D0

  ux C uy  1 1   uf .x1 /  uf .y1 /;  rf r   ux C uy   k k  uf .xk /  uf .yk /     ; rf k r  '.x1 ; y1 ;    ; xk ; yk /

(6.62)

for all u 2 U.A/ and x1 ;    xk ; y1    ; yk 2 X. If there exists L < 1 such that '

r r r r r  r x1 ; x1 ; x2 ; x2 ;    ; xk ; xk  L' .x1 ; x1 ; x2 ; x2 ;    ; xk ; xk / 2 2 2 2 2 2 2

r

for all x1 ;    xk 2 X. Then there exists a unique A-linear generalized additive mapping  W X ! Y such that

6.3 Generalized Additive Mappings and Isomorphisms in Multi-C -Algebras

261

sup k..x1 /  f .x1 /;    ; .xk /  f .xk //kk k1

 sup k1

1 '.x1 ; x1 ;    ; xk ; xk / 2.1  L/

for all x1 ;    ; xk 2 X. Proof. Let u D 1 2 U.A/. Putting xi D yi (1  i  k) in (6.62), we have          rf 2 x1  2f .x1 /;    ; rf 2 xk  2f .xk /    r r k  '.x1 ; x1 ;    ; xk ; xk / for all x1 ;    ; xk 2 X and so         f .x1 /  2 f r x1 ;    ; f .xk /  2 f r xk    r 2 r 2 k 1 r r r r  x1 ; x1 ;    ; xk ; xk  ' r 2 2 2 2 1  L'.x1 ; x1 ;    ; xk ; xk / 2 for all x1 ;    ; xk 2 X. The rest of the proof is similar to the proof of Theorem 6.17. This completes the proof.  Corollary 6.24. Let r > 2 and , p > 1 be positive real numbers or let r < 2 and , p < 1 be positive real numbers. Let f W X ! Y be an odd mapping such that   ux C uy   1 1  uf .x1 /  uf .y1 /;  rf r  ux C uy   k k     ; rf  uf .xk /  uf .yk /  k r 

k X

.kxj kp C kyj kp /

jD1

for all u 2 U.A/ and x1 ;    xk 2 X. Then there exists a unique A-linear generalized additive mapping  W X ! Y such that sup k..x1 /  f .x1 /;    ; .xk /  f .xk //kk  sup k1

for all x1 ;    xk 2 X.

k1

k X rp1  kxj kp 2p1  rp1 jD1

262

6 Stability of Functional Equations in Multi-Banach Algebras

Proof. Define '.x1 ; y1 ;    ; xk ; yk / D 

k X

.kxj kp C kyj kp /

jD1

and apply Theorem 6.23. Then we get the desired result.



6.3.2 Isomorphisms in Unital Multi-C -Algebras Assume that A and B are unital multi-C -algebras with the unit e. Let U.A/ be the set of unitary elements in A. Now, we investigate C -algebra isomorphisms in unital multi-C -algebras. Theorem 6.25. Let r ¤ 2. Let h W A ! B be an odd bijective mapping satisfying h

  2n  uy D h u h.y/ rn rn

 2n

for all u 2 U.A/, y 2 A and n  0 for which there exists a function ' W Akd ! Œ0; 1/ such that  j  2 2j 2j 2j rj lim ' j x11 ;    ; j x1d ;    ; j xk1 ;    ; j xkd D 0; j!1 2j r r r r k.D h.x11 ;    ; x1d /;    ; D h.xk1 ;    ; xkd //kk  '.x11 ;    ; x1d ;    ; xk1 ;    ; xkd / and   n   n   n    n     h 2 u  h 2 u1 ;    ; h 2 u  h 2 uk  1 k   rn rn rn rn k  2n 2n 2n 2n   ' n u1 ;    ; n u1 ;    ; n uk ;    ; n uk „r ƒ‚ r … „r ƒ‚ r … d times

d times

for all 2 S1 WD f 2 C W jj D 1g, u1 ;    ; uk 2 U.A/, n  0 and n n x11 ;    ; xkd 2 A. Assume that limn!1 2r n h. 2rn e/ is invertible. Then the odd bijective mapping h W A ! B is a C -algebra isomorphism. Proof. Consider the multi-C -algebras A and B as left Banach modules over the unital multi-C -algebra C. By Theorem 6.17, there exists a unique C-linear generalized additive mapping H W A ! B such that

6.3 Generalized Additive Mappings and Isomorphisms in Multi-C -Algebras

263

sup k.h.x1 /  H.x1 /;    ; h.xk /  H.xk //kk k1

 sup k1

  ' x1 ; x1 ; 0;    ; 0;    ; xk ; xk ; 0;    ; 0 „ ƒ‚ … „ ƒ‚ … 2.d2 Cl d2 Cl2 C 1/ 1

d  2 times

d  2 times

for all x1 ;    ; xk 2 A in which H W A ! B is given by r n  2n  h nx n!1 2n r

H.x/ D lim

for all x 2 A. The rest of the proof is easy. This completes the proof.



Corollary 6.26. Let r > 2 and , p > 1 be positive real numbers or let r < 2 and , p < 1 be positive real numbers. Let h W A ! B be an odd bijective mapping satisfying h

 2n

  2n  uy D h u h.y/ rn rn

for all u 2 U.A/, y 2 A and n  0 such that k.D h.x11 ;    ; x1d /;    ; D h.xk1 ;    ; xkd //kk 

d X .kx1j kp C    C kxkj kp / jD1

and   n   n   n    n     h 2 u  h 2 u1 ;    ; h 2 u  h 2 uk  1 k   n n n n r r r r

k

2pn  kd pn  r for all 2 S1 , u 2 U.A/, n  0 and x11 ;    ; xkd 2 A. Assume that limn!1 2r n h. 2rn e/ is invertible. Then the odd bijective mapping h W A ! B is a C -algebra isomorphism. n

n

Proof. Define '.x11 ;    ; x1d ;    ; xk1 ;    ; xkd / D 

d X .jjx1j jjp C    C jjxkj jjp / jD1

and apply Theorem 6.25. Then we get the desired result.



264

6 Stability of Functional Equations in Multi-Banach Algebras

6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras In this section, we approximate the following additive functional inequality: dC1 dC1 X  X   f .x1i /;    ; f .xki // . iD1

k

iD1

  PdC1 x   PdC1 x    iD1 1i iD1 ki ;    ; mf   mf  k m m

(6.63)

for all x11 ;    ; xk dC1 2 X, where d  2 is a fixed integer. Also, we investigate homomorphisms in proper multi-CQ -algebras and derivations on proper multiCQ -algebras associated with the above additive functional inequality.

6.4.1 Stability of C-Linear Mappings in Multi-Banach Spaces Now, we investigate the Hyers-Ulam stability of C-linear mappings in multi-Banach spaces associated with the multi-additive functional inequality (6.63). In this section, we assume that .X; k  k/ and .Y; k  k/ are Banach spaces such that .X k ; k  kk / and .Y k ; k  kk / are multi-Banach spaces. Lemma 6.27. Let f W X ! Y be a mapping satisfying (6.63) in which f .0/ D 0. Then f is additive. Proof. Letting x3 D    D xdC1 D 0 and replacing x1 by x and x2 by x in (6.63), we have kf .x/ C f .x/k  kmf .0/k D 0 for all x 2 X. Hence f .x/ D f .x/ for all x 2 X. Replacing x1 by x, x2 by y and x3 by x  y and putting x4 D    D xdC1 D 0 in (6.63), we have kf .x/ C f .y/  f .x C y/k D kf .x/ C f .y/ C f .x  y/k  kmf .0/kY D0 for all x; y 2 X. Thus we have f .x C y/ D f .x/ C f .y/ for all x; y 2 X. This completes the proof.



6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

265

Theorem 6.28. Let f W X ! Y be a mapping. If there exists a function ' W X kdCk ! Œ0; 1/ satisfying the following: dC1 dC1 X  X   f .x1i /;    ; f .xki // . iD1

iD1

k

  PdC1 x   PdC1 x    iD1 1i iD1 ki ;    ; mf   mf  k m m C '.x11 ;    ; x1 dC1 ;    ; xk1 ;    ; xk dC1 /

(6.64)

and '.x Q 11 ;    ; x1 dC1 ;    ; xk1 ;    ; xk dC1 / WD

1 X

  sup dj ' dj1 x11 ;    ; dj1 x1 dC1 ;    ; dj1 xk1 ;    ; dj1 xk dC1

jD0 k1

<1

(6.65)

for all x11 ;    ; xk dC1 2 X, then there exists a unique additive mapping L W X ! Y such that sup k.f .x1 /  L.x1 /;    ; f .xk /  L.xk //kk k1

 sup '.x Q 1 ; x1 ;    ; dx1 ;    ; xk ; xk ;    ; dxk /

(6.66)

k1

for all x1 ;    ; xk 2 X. Proof. Since '.0; Q    ; 0/ < 1 in (6.65), we have '.0;    ; 0/ D 0 and so f .0/ D 0. Replacing xi1 ;    ; xid by xi and xidC1 by dxi (1  i  k), respectively, in (6.64), since f .0/ D 0, we have k.df .x1 /  f .dx1 /;    ; df .xk /  f .dxk //kk D k.df .x1 / C f .dx1 /;    ; df .xk / C f .dxk //kk  k.mf .0/;    ; mf .0//kk C '.x1 ; x1 ;    ; dx1 ;    ; xk ; xk ;    ; dxk / for all x1 ;    ; xk 2 X. From the above inequality, we have  x   x  1 k   ;    ; f .xk /  df   f .x1 /  df d d k  x x xk xk 1 1 ; ;    ; x1 ;    ; ; ;    ; xk ' d d d d

266

6 Stability of Functional Equations in Multi-Banach Algebras

for all x1 ;    ; xk 2 X. Replacing xi by dn xi (1  i  k) in the above inequality, we have     x  x   x  1 k k   n x1 nC1 n nC1  d ;    ; d  d f f f   d f n nC1 n nC1 d d d d k  x x1 x1 xk xk xk  1 n  d ' nC1 ; nC1 ;    ;  n ;    ; nC1 ; nC1 ;    ;  n : d d d d d d From the above inequality, we have  x  x  x   x  1 1 k k   q n q sup  dn f  d f ;    ; d f  d f  n q n q d d d d k k1 



  x  x   x   x    1 1 k k j jC1 j  d ;    ; d  d sup  djC1 f f f f  jC1 j jC1 j d d d d k jDq k1

n1 X

n1 X

 sup d ' j

jDq k1

x1 x1 dx1 xk xk dxk ; ;    ;  jC1 ;    ; jC1 ; jC1 ;    ;  jC1 djC1 djC1 d d d d



˚   for all x1 ;    ; xk 2 X and q; n  1 with q < n. From (6.64), the sequence d n f dxn is a Cauchy sequence for all x 2 X and so it is convergent in the complete multinorm Y. Thus we can define a mapping L W X ! Y by L.x/ WD lim dn f n!1

x dn

for all x 2 X. In order to prove that L satisfies (6.66), if we put q D 0 and let n ! 1 in the above inequality, then we obtain sup k.f .x1 /  L.x1 /;    ; f .xk /  L.xk //kk k1



n1 X

 sup dj '

jD0 k1

x1 x1 dx1 xk xk dxk ; jC1 ;    ;  jC1 ;    ; jC1 ; jC1 ;    ;  jC1 jC1 d d d d d d



D sup '.x Q 1 ; x1 ;    ; dx1 ;    ; xk ; xk ;    ; dxk / k1

x

for all x1 ;    ; xk 2 X. Replacing xij by d1jn (1  i  k and 1  j  d C 1), respectively, and multiplying by dnC1 in (6.64), we have

6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

267

dC1 X  x   1i  dn f nC1   d iD1   PdC1 x   x x1dC1 x11 x1dC1  11  n iD1 1i  C d  mdn f ' ;    ; ;    ;    ;  mdnC1 dnC1 dnC1 dnC1 dnC1

for all x1j 2 X (1  j  d C 1). Since (6.65) gives  x x1dC1 x11 x1dC1  11 lim dn sup ' nC1 ;    ; nC1 ;    nC1 ;    ; nC1 D 0 n!1 d d d d k1 for all x1j 2 X (1  j  d C 1), letting n ! 1 in the above inequality, we have dC1 X    PdC1 x     iD1 1i  L.x1i /  mL   m iD1

(6.67)

and so L is additive by Lemma 6.27. Now, to prove the uniqueness of L, let L0 W X ! Y be another additive mapping satisfying (6.66). Since L and L0 are additive, we have kL.x/  L0 .x/k  x  x    D dn L n  L0 n  d d   x   x    x   x     0  f  dn L n  f  C L  n n d d d dn   x dx x x dx x ;    ; ; ;    ; ;    ; ;  dn  2'Q dn dn dn dn dn dn D2

1 X

sup dnCj ' .A/ ;

jD0 k1

where '.A/ 1 0 k ‚ …„ ƒ C B dC1 dC1 B‚ …„ ƒ ‚ …„ ƒC B x x x dx x dx C C B D ' B nCjC1 ;    ; nCjC1 ; nCjC1 ;    ; nCjC1 ;    ; nCjC1 ; nCjC1 C ; C Bd d d d d d C B A @ which goes to zero as n ! 1 for all x 2 X by (6.65). Consequently, L is the unique additive mapping satisfying (6.66). This completes the proof. 

268

6 Stability of Functional Equations in Multi-Banach Algebras

Corollary 6.29. Let f W X ! Y be a mapping with f .0/ D 0. If there exists a function ' W X kdCk ! Œ0; 1/ satisfying (6.64) and '.x Q 11 ;    ; xkdCk / WD

1 X

sup

jD0 k2N

1 djC1

  ' dj x11 ;    ; dj xk dC1 < 1

(6.68)

for all x11 ;    ; xk dC1 2 X, then there exists a unique additive mapping L W X ! Y such that sup k.f .x1 /  L.x1 /;    ; f .xk /  L.xk //kk k1

 sup '.x Q 1 ; x1 ;    ; dx1 ;    ; xk ; xk ;    ; dxk /

(6.69)

k1

for all x1 ;    ; xk 2 X. Proof. The proof is same as in the corresponding part of the proof of Theorem 6.28.  Lemma 6.30. Let f W X ! Y be a mapping satisfying the following: d X    Pd x C x  dC1     iD1 i f .xi / C f .xdC1 /  mf   m iD1

(6.70)

for all 2 T1 and x1 ;    ; xdC1 2 X. Then f is C-linear. Proof. If we put D 1 in (6.70), then f is additive by Lemma 6.27. Putting x1 D x, xi D 0 (2  i  d) and xdC1 D x, respectively, we get f . x/ C f .x/ D 0 and so f . x/ D f .x/ for all 2 T1 and x 2 X. Thus we have f . x C x/ N D f . x/ C f . x/ N D f .x/ C f N .x/ for all 2 T1 and x 2 X and so f .tx/ D tf .x/ for any real number t with jtj  1 and x 2 X. On the other hand, since f .mx/ D mf .x/, we get f .mn x/ D mn f .x/ for all n 2 N. So, for any real number t, there exists a positive integer n with jtj  mn . Thus we have   t  t  t f .tx/ D f mn  n x D mn f x D mn  n f .x/ D tf .x/: n m m m Now, we consider any ˛ 2 C with ˛ D t C si for some real numbers t; s. Since f .ix/ D if .x/ holds, we have f .˛x/ D f .tx/ C f .six/ D tf .x/ C sf .ix/ D tf .x/ C sif .x/ D ˛f .x/ and so f is C-linear. This completes the proof.



6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

269

Theorem 6.31. Let f W X ! Y be a mapping. If there exists a function ' W X kdCk ! Œ0; 1/ satisfying (6.64) and d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

k

iD1

  Pd x C x   Pd x C x   1dC1 k dC1  iD1 1i iD1 ki ;    ; mf   mf  k m m C '.x11 ;    ; xk dC1 /

(6.71)

for all 2 T1 and x11 ;    ; xk dC1 2 X, then there exists a unique C-linear mapping L W X ! Y satisfying (6.66). Proof. If we put D 1 in (6.71), then, by Theorem 6.28, there exists a unique additive mapping L W X ! Y defined by L.x/ WD lim dn f n!1

x dn

for all x 2 X which satisfies (6.66). By the similar method to the corresponding part of the proof of Theorem 6.28, L satisfies the following: d X    Pd x C x  dC1     iD1 i L.xi / C L.xdC1 /  mL   m iD1

for all 2 T1 and x1 ;    ; xdC1 2 X. Thus Lemma 6.30 gives that L is C-linear. This completes the proof.  Corollary 6.32. Let f W X ! Y be a mapping with f .0/ D 0. If there exists a function ' W X kdCk ! Œ0; 1/ satisfying (6.68) and (6.70), then there exists a unique C-linear mapping L W X ! Y satisfying (6.71). Proof. The proof is same as in the corresponding part of the proof of Theorem 6.31. 

6.4.2 Stability of Homomorphisms in Proper Multi-CQ -Algebras Now, we investigate the Hyers-Ulam stability of isomorphisms in proper multiCQ -algebras associated with the additive functional inequality. We assume that .A; k  k/ and .B; k  k/ are Banach algebras such that .Ak ; k  kk / and .Bk ; k  kk / are multi-Banach algebras.

270

6 Stability of Functional Equations in Multi-Banach Algebras

Theorem 6.33. Let f W A ! B be a mapping. Suppose that there exists a function ' W AkdCk ! Œ0; 1/ satisfying (6.65) and d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

k

iD1

  Pd x C x   Pd x C x   1dC1 k dC1  iD1 1i iD1 ki ;    ; mf   mf  k m m C '.x11 ;    ; xk dC1 /

(6.72)

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, there exists a function  W A2k ! Œ0; 1/ satisfying the following: k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk  .x1 ; y1 ;    ; xk ; yk /

(6.73)

and lim sup d2n .dn x1 ; dn y1 ;    ; dn xk ; dn yk / D 0

n!1 k1

(6.74)

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique proper CQ -algebra homomorphism h W A ! B such that sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk k1

 sup '.x Q 1 ; x1 ;    ; dx1 ;    ; xk ; xk ;    ; dxk /

(6.75)

k1

for all x1 ;    ; xk 2 A. Proof. By Theorem 6.31, we have a unique C-linear mapping h W A ! B defined by h.x/ WD lim dn f n!1

x dn

for all x 2 A which satisfies (6.75). Now, we show that h.xy/ D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined. Replacing xi ; yi by dn xi ; dn yi (1  i  k), respectively, and multiplying by d2n in (6.73), we have k.d2n Œf .dn x1 dn y1 /  f .dn x1 /f .dn y1 /;    ; d2n Œf .dn xk dn yk /  f .dn xk /f .dn yk //kk  d2n .dn x1 ; dn y1 ;    ; dn xk ; dn yk /

(6.76)

6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

271

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined. Also, we have lim d2n f .dn xdn y/ D lim d2n f .d2n xy/ D h.xy/

n!1

n!1

and lim d2n f .dn x/f .dn y/ D lim dn f .dn x/  lim dn f .dn y/

n!1

n!1

n!1

D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined. If we let n ! 1 in the above inequality, then (6.74) gives h.xy/ D h.x/h.y/ for all x; y 2 A whenever the multiplication is defined. This completes the proof.  Corollary 6.34. Let ; p be nonnegative real numbers with p > 1 and f W A ! B be a mapping satisfying the following: d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

    mf C 

iD1

 Pd

iD1 x1i

dC1 k X X

C x1dC1  ;    ; mf m

k

 Pd

iD1 xki

C xk dC1    k m

kxli kp

(6.77)

lD1 iD1

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk 

dC1 X .kxi k2p C kyi k2p /

(6.78)

iD1

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined. Then there exists a unique proper CQ -algebra homomorphism h W A ! B such that dp1 C 1 X  kxl kp sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk  sup p1  1 lD1 k1 k1 d k

for all x1 ;    ; xk 2 A.

272

6 Stability of Functional Equations in Multi-Banach Algebras

Proof. Let ' W AkdCk ! Œ0; 1/ be a mapping defined by '.x11 ;    ; x1 dC1 ;    ; xk1 ;    ; xk dC1 / D  

dC1 k X X

kxli kp :

lD1 iD1

When p > 1, we have '.x Q 11 ;    ; x1 dC1 ;    ; xk1 ;    ; xk dC1 / WD

1 X

sup dj '.dj1 x11 ;    ; dj1 x1 dC1 ;

jD0 k1

   ; dj1 xk1 ;    ; dj1 xk dC1 / D

dC1 1 k X X 1 X djC1 sup   kxli kp d jD0 d.jC1/p k1 lD1 iD1

D

XX  sup kxli kp : p d  d k1 lD1 iD1 k

dC1

In addition, let  W A2k ! Œ0; 1/ be a mapping defined by .x1 ; y1 ;    ; xk ; yk / D  

dC1 X .kxi k2p C kyi k2p /: iD1

When p > 1, we have lim d2n .dn x1 ; dn y1 ;    ; dn xk ; dn yk /

n!1

dC1 X d2n   .kxi k2p C kyi k2p / n!1 d 2pn iD1

D lim D0

for all x1 ;    ; xk ; y1 ;    ; yk 2 A. By applying Theorem 6.33, there exists a unique proper CQ -algebra homomorphism h W A ! B such that dp C d X  kxl kp dp  d lD1 k

sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk  sup k1

for all x1 ;    ; xk 2 A. This completes the proof.

k1



6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

273

Corollary 6.35. Let ; p be nonnegative real numbers with p > 1 and f W A ! B be a mapping satisfying the following: d d  X  X   f .x / C f .x /;    ; f .xki / C f .xk dC1 /   1i 1dC1 iD1

k

iD1

  Pd x C x   Pd x C x  1dC1 k dC1   iD1 1i iD1 ki ;    ; mf   mf  k m m C 

dC1 k X X

kxli kp

(6.79)

lD1 iD1

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk 

dC1 X

.kxi kp  kyi kp /

(6.80)

iD1

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined. Then there exists a unique proper CQ -algebra homomorphism h W A ! B such that dp1 C 1 X kxl kp  dp1  1 lD1 k

sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A. Proof. Let ' W AkdCk ! Œ0; 1/ be a mapping defined by '.x11 ;    ; x1 dC1 ;    ; xk1 ;    ; xk dC1 / D  

dC1 k X X lD1 iD1

When p > 1, we have '.x Q 11 ;    ; x1 dC1 ;    ; xk1 ;    ; xk dC1 / WD

1 X

sup dj '.dj1 x11 ;    ; dj1 x1 dC1 ;

jD0 k1

   ; dj1 xk1 ;    ; dj1 xk dC1 / D

dC1 1 k X X 1 X djC1 sup   kxli kp d jD0 d.jC1/p k1 lD1 iD1

D

XX  sup kxli kp : p d  d k1 lD1 iD1 k

dC1

kxli kp :

274

6 Stability of Functional Equations in Multi-Banach Algebras

In addition, let  W A2k ! Œ0; 1/ be a mapping defined by dC1 X .kxi kp  kyi kp /: .x1 ; y1 ;    ; xk ; yk / D   iD1

When p > 1, we have lim d2n .dn x1 ; dn y1 ;    ; dn xk ; dn yk /

n!1

dC1 X d2n   .kxi kp  kyi kp / n!1 d 2pn iD1

D lim D0

for all x1 ;    ; xk ; y1 ;    ; yk 2 A. By applying Theorem 6.33, there exists a unique proper CQ -algebra homomorphism h W A ! B such that dp C d X  kxl kp dp  d lD1 k

sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A. This completes the proof.



Remark 6.36. Let f W A ! B be a mapping with f .0/ D 0. Suppose that there exists a function ' W AkdCk ! Œ0; 1/ satisfying (6.68) and d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

    mf

 Pd

iD1 x1i

iD1

C x1dC1  ;    ; mf m

k

 Pd

iD1 xki

C xk dC1    k m

C '.x11 ;    ; xk dC1 /

(6.81)

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, there exists a function  W A2k ! Œ0; 1/ satisfying the following: k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk  .x1 ; y1 ;    ; xk ; yk /

(6.82)

6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

275

and lim sup d2n .dn x1 ; dn y1 ;    ; dn xk ; dn yk / D 0

(6.83)

n!1 k1

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique proper CQ -algebra homomorphism h W A ! B satisfying the following: sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk k1

 sup '.x Q 1 ; x1 ;    ; dx1 ;    ; xk ; xk ;    ; dxk /

(6.84)

k1

for all x1 ;    ; xk 2 A. Corollary 6.37. Let ; p be nonnegative real numbers with p < 1 and f W A ! B be a mapping satisfying the following: d d  X  X   f .x / C f .x /;    ; f .xki / C f .xk dC1 /   1i 1dC1 iD1

k

iD1

  Pd x C x   Pd x C x   1dC1 k dC1  iD1 1i iD1 ki ;    ; mf   mf  k m m C 

dC1 k X X

kxli kp

(6.85)

lD1 iD1

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk 

dC1 X .kxi k2p C kyi k2p /

(6.86)

iD1

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique proper CQ -algebra homomorphism h W A ! B satisfying d C dp X  kxl kp d  dp lD1 k

sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk  sup k1

for all x1 ;    ; xk 2 A.

k1

276

6 Stability of Functional Equations in Multi-Banach Algebras

Proof. Let  W A2k ! Œ0; 1/ be a mapping defined by dC1 X .kxi k2p C kyi k2p /: .x1 ; y1 ;    ; xk ; yk / D   iD1

When p < 1, we have lim d2n .dn x1 ; dn y1 ;    ; dn xk ; dn yk /

n!1

dC1 d2np X   .kxi k2p C kyi k2p / n!1 d 2n iD1

D lim D0

for all x1 ;    ; xk ; y1 ;    ; yk 2 A. By Remark 6.36, there exists a unique proper CQ -algebra homomorphism h W A ! B such that d C dp X  kxl kp d  dp lD1 k

sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A. This completes the proof.



Corollary 6.38. Let ; p be nonnegative real numbers with p < 1 and f W A ! B be a mapping satisfying the following: d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

iD1

k

  Pd x C x   Pd x C x   1dC1 k dC1  iD1 1i iD1 ki ;    ; mf   mf  k m m C 

dC1 k X X

kxli kp

(6.87)

lD1 iD1

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk 

dC1 X .kxi kp  kyi kp /

(6.88)

iD1

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique proper CQ -algebra homomorphism h W A ! B such that

6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

277

d C dp X  kxl kp d  dp lD1 k

sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A. Proof. Let  W A2k ! Œ0; 1/ be a mapping defined by .x1 ; y1 ;    ; xk ; yk / D  

dC1 X .kxi kp  kyi kp /: iD1

When p < 1, we have lim d2n .dn x1 ; dn y1 ;    ; dn xk ; dn yk /

n!1

dC1 d2pn X D lim 2n   .kxi kp  kyi kp / n!1 d iD1

D0 for all x1 ;    ; xk ; y1 ;    ; yk 2 A. By Remark 6.36, there exists a unique proper CQ -algebra homomorphism h W A ! B such that d C dp X  kxl kp d  dp lD1 k

sup k.f .x1 /  h.x1 /;    ; f .xk /  h.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A. This completes the proof.



6.4.3 Stability of Derivations in Proper CQ -Algebras Now, we investigate the Hyers-Ulam stability of derivations on proper multi-CQ algebras associated with the additive functional inequality. In this section, we assume that .A; k  k/ is a Banach algebra such that .Ak ; k  kk / is a multi-Banach algebra. Theorem 6.39. Let f W A ! A be a mapping. Suppose that there exists a function ' W AkdCk ! Œ0; 1/ satisfying (6.65) and d d  X  X   f .x / C f .x /;    ; f .xki / C f .xk dC1 /   1i 1dC1 iD1

    mf

 Pd

iD1 x1i

iD1

C x1dC1  ;    ; mf m

C '.x11 ;    ; xk dC1 /

k

 Pd

iD1 xki

C xk dC1    k m (6.89)

278

6 Stability of Functional Equations in Multi-Banach Algebras

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, there exists a function W A2k ! Œ0; 1/ satisfying the following: k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk 

.x1 ; y1 ;    ; xk ; yk /

(6.90)

and lim sup d2n .d n x1 ; dn y1 ;    ; dn xk ; dn yk / D 0

n!1 k2N

(6.91)

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique derivation ı W A ! A satisfying the following: sup k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk k1

 sup '.x Q 1 ; x1 ;    ; dx1 ;    ; xk ; xk ;    ; dxk /

(6.92)

k1

for all x1 ;    ; xk 2 A. Proof. By Theorem 6.31, we have a unique C-linear mapping ı W A ! A defined by x ı.x/ WD lim dn f n!1 dn for all x 2 A which satisfies (6.92). Now, we show that ı.xy/ D ı.x/ı.y/ for all x; y 2 A whenever the multiplication is defined. Replacing xi ; yi by dn xi ; dn yi (1  i  k), respectively, and multiplying by d2n in (6.90), we have k.d2n Œf .dn x1 dn y1 /  dn f .dn x1 /y1  dn x1 f .dn y1 /;    ; d2n Œf .dn xk dn yk /  dn f .dn xk /yk  dn xk f .dn yk //kk  d2n .d n x1 ; dn y1 ;    ; dn xk ; dn yk /

(6.93)

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined. Also, we have lim d2n f .dn xdn y/ D lim d2n f .d2n xy/ D ı.xy/;

n!1

n!1

2n

lim d f .d x/d y D lim dn f .dn x/  y D ı.x/y

n!1

n

n

n!1

and lim d2n dn xf .d n y/ D lim x  dn f .dn y/ D xı.y/

n!1

n!1

6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

279

for all x; y 2 A whenever the multiplication is defined. If we let n ! 1 in the above inequality, then (6.93) gives ı.xy/ D ı.x/y  xı.y/ for all x; y 2 A whenever the multiplication is defined. This completes the proof.  Corollary 6.40. Let ; p be nonnegative real numbers with p > 1 and f W A ! A be a mapping such that d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

k

iD1

  Pd x C x   Pd x C x    1dC1 k dC1 iD1 1i iD1 ki ;    ; mf   mf  k m m C 

dC1 k X X

kxli kp

(6.94)

lD1 iD1

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk 

dC1 X .kxi k2p C kyi k2p /

(6.95)

iD1

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique derivation ı W A ! A satisfying the following: dp C d X  kxl kp dp  d lD1 k

sup k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A. 

Proof. The proof is same to the proof given in Corollary 6.34.

Corollary 6.41. Let ; p be nonnegative real numbers with p > 1 and f W A ! A be a mapping such that d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

iD1

k

  Pd x C x   Pd x C x  1dC1 k dC1   iD1 1i iD1 ki ;    ; mf   mf  k m m C 

dC1 k X X lD1 iD1

kxli kp

(6.96)

280

6 Stability of Functional Equations in Multi-Banach Algebras

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk 

dC1 X

.kxi kp  kyi kp /

(6.97)

iD1

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique derivation ı W A ! A satisfying the following: dp C d X  kxl kp dp  d lD1 k

sup k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A. Remark 6.42. Let f W A ! A be a mapping with f .0/ D 0. Suppose that there exists a function ' W AkdCk ! Œ0; 1/ satisfying (6.68) and (6.89). If, in addition, there exists a function W A2k ! Œ0; 1/ such that k.f .x1 y1 /  f .x1 /f .y1 /;    ; f .xk yk /  f .xk /f .yk //kk 

.x1 ; y1 ;    ; xk ; yk /

(6.98)

and lim sup d2n .d n x1 ; dn y1 ;    ; dn xk ; dn yk / D 0

n!1 k1

(6.99)

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique derivation ı W A ! A such that sup k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk k1

 sup '.x Q 1 ; x1 ;    ; dx1 ;    ; xk ; xk ;    ; dxk /

(6.100)

k1

for all x1 ;    ; xk 2 A. Corollary 6.43. Let ; p be nonnegative real numbers with p < 1 and f W A ! A be a mapping satisfying the following:

6.4 Additive Functional Inequalities in Proper Multi-CQ -Algebras

281

d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

k

iD1

  Pd x C x   Pd x C x  1dC1 k dC1   iD1 1i iD1 ki ;    ; mf   mf  k m m C 

dC1 k X X

kxli kp

(6.101)

lD1 iD1

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk 

dC1 X

.kxi k2p C kyi k2p /

(6.102)

iD1

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined. Then there exists a unique derivation ı W A ! A satisfying d C dp X  kxl kp d  dp lD1 k

sup k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A. Corollary 6.44. Let ; p be nonnegative real numbers with p < 1 and f W A ! A be a mapping satisfying the following: d d  X  X   f .x1i / C f .x1dC1 /;    ; f .xki / C f .xk dC1 /   iD1

    mf C 

 Pd

iD1

iD1 x1i

dC1 k X X

C x1dC1  ;    ; mf m

k

 Pd

iD1 xki

C xk dC1    k m

kxli kp

(6.103)

lD1 iD1

for all 2 T1 and x11 ;    ; xk dC1 2 A. If, in addition, k.f .x1 y1 /  f .x1 /y1  x1 f .y1 /;    ; f .xk yk /  f .xk /yk  xk f .yk //kk 

dC1 X iD1

.kxi kp  kyi kp /

(6.104)

282

6 Stability of Functional Equations in Multi-Banach Algebras

for all x1 ;    ; xk ; y1 ;    ; yk 2 A whenever the multiplication is defined, then there exists a unique derivation ı W A ! A such that d C dp X  kxl kp d  dp lD1 k

sup k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk  sup k1

k1

for all x1 ;    ; xk 2 A.

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras Using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms and derivations on multi-C -ternary algebras for the additive functional equation: 2f

 Pp

jD1 xj

2

C

p d d  X X X yj D f .xj / C 2 f .yj /: jD1

jD1

jD1

6.5.1 Stability of Homomorphisms Assume that A, B are C -ternary algebras. For any mapping f W A ! B; we define C f .x1 ;    ; xp ; y1 ;    ; yd / p d d  X  Pp xj X X jD1 C WD 2f yj  f .xj /  2 f .yj / 2 jD1 jD1 jD1 for all 2 T1 WD f  2 C W jj D 1g and x1 ;    ; xp ; y1 ;    ; yd 2 A: One can easily show that a mapping f W A ! B satisfies C f .x1 ;    ; xp ; y1 ;    ; yd / D 0 for all 2 T1 and x1 ;    ; xp ; y1 ;    ; yd 2 A if and only if f . x C y/ D f .x/ C f .y/ for all ;  2 T1 and x; y 2 A:

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

283

Now, we introduce the following lemma for the main results in this section: Lemma 6.45. Let fxn g, fyn g and fzn g be the convergent sequences in A. Then the sequence fŒxn ; yn ; zn g is convergent in A Proof. Let x; y; z 2 A be such that lim xn D x;

n!1

lim yn D y;

n!1

lim zn D z:

n!1

Since Œxn ; yn ; zn   Œx; y; z D Œxn  x; yn  y; zn ; z C Œxn ; yn ; z C Œx; yn  y; zn  C Œxn ; y; zn  z for all n  1, we get kŒxn ; yn ; zn   Œx; y; zk D kxn  xkkyn  ykkzn  zk C kxn  xkkyn kkzk Ckxkkyn  ykkzn k C kxn kkykkzn  zk for all n  1 and so lim Œxn ; yn ; zn  D Œx; y; z:

n!1



This completes the proof.

Using Theorem 1.3, we prove the Hyers-Ulam stability of homomorphisms in multi-C –ternary algebras for the following functional equation: C f .x1 ;    ; xm / D 0: Theorem 6.46. Let ..Bk ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! B be a mapping for which there exist the functions ' W A.pCd/k ! Œ0; 1/ and W A3k ! Œ0; 1/ such that lim  n '. n x11 ;    ;  n x1p ;  n y11 ;    ;  n y1p ;

n!1

   ;  n xk1 ;    ;  n xkp ;    ;  n yk1 ;    ;  n ykd / D 0;    C f .x11 ;    ; x1p ; y11 ;      ; y1d /;    ; C f .xk1 ;    ; xkp ; yk1 ;    ; ykd /  k

 '.x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd /;

(6.105)

(6.106)

284

6 Stability of Functional Equations in Multi-Banach Algebras

   f .Œx1 ; y1 ; z1 /  Œf .x1 /; f .y1 /; f .z1 /;      ; f .Œxk ; yk ; zk /  Œf .xk /; f .yk /; f .zk / 

k



.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /;

(6.107)

lim  3n . n x1 ;  n y1 ;  n z1 ;    ;  n xk ;  n yk ;  n zk / D 0

(6.108)

lim  2n . n x1 ;  n y1 ; z1 ;    ;  n xk ;  n yk ; zk / D 0

(6.109)

n!1

and n!1

for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A, where  D pC2d . If there exists L < 1 2 such that pCd pCd  ‚ pCd …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  '  x1 ;    ;  x1 ;  x2 ;    ;  x2 ;    ;  xk ;    ;  xk pCd pCd  ‚ pCd …„ ƒ ‚ …„ ƒ ‚ …„ ƒ    L' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk

(6.110)

for all x1 ; x2 ;    ; xk 2 A, then there exists a unique multi-C -ternary algebra homomorphism H W A ! B such that      f .x1 /  H.x1 /;    ; f .xk /  H.xk /  k

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  1 ' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk  .1  L/2 pCd

pCd

pCd

(6.111)

for all x1 ;    ; xk 2 A. Proof. Let D 1 and xij D yij D xi for 1  i  k in (6.106). Then we have      f . x1 /   f .x1 /;    ; f . xk /   f .xk /  

k

‚ …„ ƒ  1  ‚ …„ ƒ ‚ …„ ƒ ' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk 2 pCd

pCd

pCd

(6.112)

for all x1 ;    ; xk 2 A. Consider the set E WD fg W A ! Bg and introduce the generalized metric on E as follows:

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

285

 n    d.g; h/ D inf C 2 RC W  g.x1 /  h.x1 /;    ; g.xk /  h.xk / 

k

 ‚ …„ ƒ ‚ …„ ƒ o ‚ …„ ƒ   C' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk ; 8x1 ;    ; xk 2 A pCd

pCd

pCd

which .E; d/ is complete. Now, we consider the linear mapping  W E ! E such that g.x/ WD

1 g. x/ 

for all x 2 A. Now, we have d.g; h/  Ld.g; h/ for all g; h 2 E. Let g; h 2 E and C 2 Œ0; 1 be an arbitrary constant with d.g; h/  C. From the definition of d, we have pCd    ‚ pCd …„ ƒ ‚ …„ ƒ     g.x1 /  h.x1 /;    ; g.xk /  h.xk /   C' x1 ;    ; x1 ;    ; xk ;    ; xk k

for all x1 ;    ; xk 2 A. By the assumption and the last inequality, we have      g.x1 /  h.x1 /;    ; g.xk /  h.xk / 

k

  1   D  g. x1 /  h. x1 /;    ; g. xk /  h. xk /  k  ‚ …„ ƒ  C  ‚ …„ ƒ '  x1 ;    ;  x1 ;    ;  xk ;    ;  xk  pCd



pCd

pCd  ‚ pCd …„ ƒ ‚ …„ ƒ   CL' x1 ;    ; x1 ;    ; xk ;    ; xk

for all x1 ;    ; xk 2 A and so      f .x1 /  f .x1 /;    ; f .xk /  f .xk / 

k

 1  1   D  f . x1 /  f .x1 /;    ; f . xk /  f .xk /  k      1  D  f . x1 /   f .x1 /;    ; f . xk /   f .xk /  k  ‚ …„ ƒ  1  ‚ …„ ƒ ' x1 ;    ; x1 ;    ; xk ;    ; xk  2 pCd

pCd

286

6 Stability of Functional Equations in Multi-Banach Algebras

for all x1 ;    ; xk 2 A. Hence d.f ; f /  21 . By Theorem 1.3, the sequence fn f g converges to a fixed point H of , i.e., H W A ! B is a mapping defined by H.x/ D lim .n f /.x/ D lim n!1

n!1

1 f . n x/ n

(6.113)

and H. x/ D  H.x/ for all x 2 A. Also, H is the unique fixed point of  in the set E0 D fg 2 E W d.f ; g/ < 1g and d.H; f / 

1 1 d.f ; f /  1L .1  L/2

i.e., the inequality (6.111) holds for all x1 ;    ; xk 2 A. Thus it follows from the definition of H, (6.105) and (6.106) that p d d   Pp x1j X  X X  jD1  C y H.x /  2 H.y1j /; 2H  1j 1j 2 jD1 jD1 jD1

   ; 2H

 Pp

2

  1  D lim n  2f  n n!1  

xkj

jD1

p X

C

d X

p d  X  X  ykj  H.xkj /  2 H.ykj / 

jD1

Pp jD1

x1j

2

jD1

C n

f . n x1j /  2

jD1



d X

f . n y1j /;

jD1

    ; 2f  n p X

y1j

jD1

jD1



d X

jD1

Pp jD1

xkj

2

f . n xkj /  2

C n

d X

ykj



jD1 d X

  f . n ykj / 

jD1

k

 1  C f . n x11 ;    ;  n x1p ;  n y11 ;    ;  n y1d /;  n!1  n

 lim

     ; C f . n xk1 ;    ;  n xkp ;  n yk1 ;    ;  n ykd / 

k

1  lim n '. n x11 ;    ;  n x1p : n y11 ;    ;  n y1d ; n!1     ;  n xk1 ;    ;  n xkp ;  n yk1 ;    ;  n ykd / D 0

k

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

287

for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd 2 A. Hence we have 2H

 Pp

jD1

2

xij

C

d X jD1

p d  X X yij D H.xij / C 2 H.yij / jD1

jD1

for all 2 T1 , x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd 2 A and 1  i  k and so H.x C y/ D H.x/ C H.y/ for all ; 2 T1 and x; y 2 A. Therefore, by Lemma 3.12, the mapping H W A ! B is C-linear. On the other hand, it follows from (6.107) and (6.108) that    H.Œx1 ; y1 ; z1 /  ŒH.x1 /; H.y1 /; H.z1 /;      ; H.Œxk ; yk ; zk /  ŒH.xk /; H.yk /; H.zk /  k

   h i 1   D lim 3n  f Πn x1 ;  n y1 ;  n z1   f . n x1 /; f . n y1 /; f . n z1 / ; n!1   h i      ; f Πn xk ;  n yk ;  n zk   f . n xk /; f . n yk /; f . n zk /  k

 lim

n!1

1 . n x1 ;  n y1 ;  n z1 ;    ;  n xk ;  n yk ;  n zk /  3n

D0 for all x1 ; y1 ; z1 ;    ; xk ; yk ; zk 2 A. Thus we have H.Œx; y; z/ D ŒH.x/; H.y/; H.z/ for all x; y; z 2 A. Thus H W A ! B is a homomorphism satisfying (6.111). Now, let T W A ! B be another multi-C -ternary algebras homomorphism 1 satisfying (6.111). Since d.f ; T/  .1L/2 and T is C-linear, we get T 2 E0 and .T/.x/ D 1 .T x/ D T.x/ for all x 2 A, i.e., T is a fixed point of . Since H is the unique fixed point of  2 E0 , we have H D T. This completes the proof.  Theorem 6.47. Let ..Bk ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! B be a mapping for which there exist the functions ' W A.pCd/k ! Œ0; 1/ and W A3k ! Œ0; 1/ satisfying the inequalities (6.106) and (6.107) such that

288

6 Stability of Functional Equations in Multi-Banach Algebras

lim  n '

n!1

lim  3n

n!1

x

x1p y11 y1p xk1 ; n ; ; n ; ; n ; n     xkp yk1 ykd     ; n ;    ; n ;    ; n D 0;   

11 ; n

x

1 ; n

;

(6.114)

y1 z1 xk yk zk  D0 ; ;    ; ; ; n n n n n

(6.115)

y1 xk yk  ; z1 ;    ; n ; n ; zk D 0 n   

(6.116)

and lim  2n

n!1

x

1 ; n

for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A, where  D pC2d 2 . If there exists L < 1 such that pCd

pCd

…„ ƒ ‚ …„ ƒ ‚ x1 x1 x2 x2 ' ; ; ; ; ; ;    

‚ …„ ƒ  L  ‚ …„ ƒ ‚ …„ ƒ ' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk  pCd



pCd

‚ …„ ƒ  xk xk ; ; ;  

pCd

pCd

(6.117)

for all x1 ; x2 ;    ; xk 2 A, then there exists a unique multi-C -ternary algebra homomorphism H W A ! B such that      f .x1 /  H.x1 /;    ; f .xk /  H.xk / 

k

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  1 ' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk  .1  L/2 pCd

pCd

pCd

(6.118)

for all x1 ;    ; xk 2 A. Proof. If we replace xi in (6.112) by

xi for 

1  i  k, then we have

 1  1    ;    ; f .xk /   f   f .x1 /   f k x1 xk pCd

pCd

pCd

‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ 1 1  1 1 1 1 1   ' ; ; ; ; ; ; ; ; ; 2 x1 x1 x2 x2 xk xk

(6.119)

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

289

for all x1 ;    ; xk 2 A. Consider the set E WD fg W A ! Bg and introduce the generalized metric on E as follows:  n    d.g; h/ D inf C 2 RC W  g.x1 /  h.x1 /;    ; g.xk /  h.xk / 

k

 ‚ …„ ƒ ‚ …„ ƒ o ‚ …„ ƒ   C' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk ; 8x1 ;    ; xk 2 A ; pCd

pCd

pCd

which .E; d/ is complete. Now, we consider the linear mapping  W E ! E such that x g.x/ WD  g  for all x 2 A. Now, we have d.g; h/  Ld.g; h/ for all g; h 2 E. Let g; h 2 E and C 2 Œ0; 1 be an arbitrary constant with d.g; h/  C. From the definition of d, we have pCd    ‚ pCd …„ ƒ ‚ …„ ƒ     g.x1 /  h.x1 /;    ; g.xk /  h.xk /   C' x1 ;    ; x1 ;    ; xk ;    ; xk k

for all x1 ;    ; xk 2 A. By the assumption and the last inequality, we have     g.x1 /  h.x1 /;    ; g.xk /  h.xk / kk   x  x  x   x  1 1 k k   D  g h ; ;g h  k     pCd

…„ ƒ ‚ x1 x1 ; ; ;  C'  

pCd

‚ …„ ƒ  xk xk ; ; ;  

pCd  ‚ pCd …„ ƒ ‚ …„ ƒ   CL' x1 ;    ; x1 ;    ; xk ;    ; xk

for all x1 ;    ; xk 2 A and so d.g; h/  Ld.g; h/ for any g; h 2 E. It follows from (6.119) that d.f ; f /  21 . Therefore, according to Theorem 1.3, the sequence fn f g converges to a fixed point H of , i.e., H W A ! B is a mapping defined by

290

6 Stability of Functional Equations in Multi-Banach Algebras

H.x/ D lim .n f /.x/ D lim  n f n!1

n!1

 x n

(6.120)

for all x 2 A. The rest of the proof is similar to the proof of Theorem 6.46 and so we omit it. This completes the proof.  Theorem 6.48. Let r and  be non-negative real numbers such that r … Œ1; 3 and ..Bk ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! B be a mapping such that    C f .x11 ;    ; x1p ; y11 ;    ; y1d /;      ; C f .xk1 ;    ; xkp ; yk1 ;    ; ykd /  (6.121) k



p X

kx1j krA C

jD1

d X

ky1j krA C    C

jD1

p X

kxkj krA C

jD1

d X

kykj krA



jD1

and    f .Œx1 ; y1 ; z1 /  Œf .x1 /; f .y1 /; f .z1 /;      ; f .Œxk ; yk ; zk /  Œf .xk /; f .yk /; f .zk / 

(6.122)

k

 .kx1 krA  ky1 krA :kz1 krA C    C kxk krA  kyk krA  kzk krA / for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A: Then there exists a unique C -ternary algebra homomorphism H W A ! B such that      f .x1 /  H.x1 /;    ; f .xk /  H.xk /  B



2 .p C d/ .kx1 krA C    C kxk krA / j2.p C 2d/r  .p C 2d/2r j r

for all x1 ;    ; xk 2 A. Proof. The proof follows from Theorem 6.46 by taking '.x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd / WD 

p X jD1

kxij krA C

d X jD1

kyij krA C    C

p X jD1

kxkj krA C

d X jD1

 kykj krA ;

(6.123)

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

291

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk / WD .kx1 krA  ky1 krA  kz1 krA C    C kxk krA  kyk krA  kzk krA / for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A, L D 21r .p C 2d/r1 , when 0 < r < 1, and L D 2  21r .p C 2d/r1 , when r > 3. This completes the proof.  Theorem 6.49. Let ..Bk ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! B be a mapping for which there exist the functions ' W A.pCd/k ! Œ0; 1/ and W A3k ! Œ0; 1/ such that lim dn '.dn x11 ;    ; dn x1p ; dn y11 ;    ; dn y1p ;

n!1

   ; dn xk1 ;    ; dn xkp ;    ; dn yk1 ;    ; dn ykd / D 0;

(6.124)

k.C f .x11 ;    ; x1p ; y11 ;    ; y1d /;    ; C f .xk1 ;    ; xkp ; yk1 ;    ; ykd //kk

(6.125)

 '.x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd /; k.f .Œx1 ; y1 ; z1 /  Œf .x1 /; f .y1 /; f .z1 /;    ; f .Œxk ; yk ; zk /  Œf .xk /; f .yk /; f .zk //kk 

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /;

(6.126)

lim d3n .d n x1 ; dn y1 ; dn z1 ;    ; dn xk ; dn yk ; dn zk / D 0

(6.127)

lim d2n .d n x1 ; dn y1 ; z1 ;    ; dn xk ; dn yk ; zk / D 0

(6.128)

n!1

and n!1

for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A, where  D pC2d 2 . If there exists L < 1 such that pCd

pCd

pCd

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  ' dx1 ;    ; dx1 ; dx2 ;    ; dx2 ;    ; dxk ;    ; dxk p

p

(6.129) p

d d d  ‚ …„ ƒ ‚ …„ ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ƒ  dL' 0;    ; 0; x1 ;    ; x1 ; 0;    ; 0; x2 ;    ; x2 ;    ; 0;    ; 0; xk ;    ; xk

292

6 Stability of Functional Equations in Multi-Banach Algebras

for all x1 ; x2 ;    ; xk 2 A, then there exists a unique homomorphism H W A ! B such that k.f .x1 /  H.x1 /;    ; f .xk /  H.xk //kk p

p

d  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ 1  ' 0;    ; 0; x1 ;    ; x1 ; 0;    ; 0; .1  L/2d

(6.130)

p

d d ‚ …„ ƒ ‚ …„ ‚ …„ ƒ ƒ x2 ;    ; x2 ;    ; 0;    ; 0; xk ;    ; xk

for all x1 ;    ; xk 2 A. Proof. Let D 1 and xij D 0;

yij D xi for 1  i  k in (6.125). Then we get

k.f .dx1 /  df .x1 /;    ; f .dxk /  df .xk //kk p

(6.131)

p

p

d d d ‚ …„ ƒ ‚ …„ ƒ 1  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  ' 0;    ; 0; x1 ;    ; x1 ; 0;    ; 0; x2 ;    ; x2 ;    ; 0;    ; 0; xk ;    ; xk 2

for all x1 ;    ; xk 2 A. Consider the set E WD fg W A ! Bg and introduce the generalized metric on E as follows: d.g; h/

˚  D inf C 2 RC W k.g.x1 /  h.x1 /;    ; g.xk /  h.xk //kk ; 8x1 ;    ; xk 2 A p

p

p

d d d  ‚ …„ ƒ ‚ …„ ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ ƒ  C' 0;    ; 0; x1 ;    ; x1 ; 0;    ; 0; x2 ;    ; x2 ;    ; 0;    ; 0; xk ;    ; xk ;

which .E; d/ is complete. Now, we consider the linear mapping  W E ! E defined by g.x/ WD

1 g.dx/ d

for all x 2 A. Now, we have d.g; h/  Ld.g; h/ for all g; h 2 E. Let g; h 2 E and let C 2 Œ0; 1 be an arbitrary constant with d.g; h/  C. From the definition of d, we have k.g.x1 /  h.x1 /;    ; g.xk /  h.xk //kk p

p

d d  ‚ …„ ƒ ‚ …„ ‚ …„ ƒ ‚ …„ ƒ ƒ  C' 0;    ; 0; x1 ;    ; x1 ;    ; 0;    ; 0; xk ;    ; xk

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

293

for all x1 ;    ; xk 2 A. By the assumption and the last inequality, we have k.g.x1 /  h.x1 /;    ; g.xk /  h.xk //kk D

1 k.g.dx1 /  h.dx1 /;    ; .g.dxk /  h.dxk /kk d p

p

d

d

‚ …„ ƒ ‚ …„ ƒ  C  ‚ …„ ƒ ‚ …„ ƒ  ' 0;    ; 0; dx1 ;    ; dx1 ;    ; 0;    ; 0; dxk ;    ; dxk d p

p

d d  ‚ …„ ƒ ‚ …„ ‚ …„ ƒ ‚ …„ ƒ ƒ  CL' 0;    ; 0; x1 ;    ; x1 ;    ; 0;    ; 0; xk ;    ; xk

for all x1 ;    ; xk 2 A. Thus we have k.f .x1 /  f .x1 /;    ; f .xk /  f .xk //kk  1  1   D . f .dx1 /  f .x1 /;    ; f .dxk /  f .xk // k d d 1 D k.f .dx1 /  df .x1 /;    ; f .dxk /  df .xk //kk d p

p

d d ‚ …„ ƒ ‚ …„ ƒ 1  ‚ …„ ƒ ‚ …„ ƒ ' 0;    ; 0; x1 ;    ; x1 ;    ; 0;    ; 0; xk ;    ; xk  2d 1 for all x1 ;    ; xk 2 A. Hence d.f ; f /  2d . By Theorem 1.3, the sequence fn f g converges to a fixed point H of , i.e., H W A ! B is a mapping defined by

H.x/ D lim .n f /.x/ D lim n!1

n!1

1 f .dn x/ dn

(6.132)

and H.dx/ D dH.x/ for all x 2 A. Also, H is the unique fixed point of  in the set E0 D fg 2 E W d.f ; g/ < 1g and d.H; f / 

1 1 d.f ; f /  ; 1L .1  L/2d

i.e., the inequality (6.130) holds for all x1 ;    ; xk 2 A. It follows from the definition of H, (6.124) and (6.125) that p d d   Pp x1j X  X X jD1  y1j  H.x1j /  2 H.y1j /; C  2H 2 jD1 jD1 jD1

   ; 2H

 Pp

jD1

2

xkj

C

d X jD1

p d  X  X  ykj  H.xkj /  2 H.ykj /  jD1

jD1

k

294

6 Stability of Functional Equations in Multi-Banach Algebras

  1  D lim n  2f d n n!1 d X

Pp jD1

x1j

2

C dn

f .dn x1j /  2

d X

jD1

X



f .dn y1j /;

jD1

    ; 2f dn

Pp jD1

xkj

2

f .dn xkj /  2

jD1

Cd

n

d X

ykj



jD1

p



y1j

jD1

p



d X

d X

  f .dn ykj / 

k

jD1

 1   C f .dn x11 ;    ; dn x1p ; dn y11 ;    ; dn y1d /; n n!1 d

 lim

     ; C f .dn xk1 ;    ; dn xkp ; dn yk1 ;    ; dn ykd / 

k

C lim

n!1

1 '.dn x11 ;    ; dn x1p ; dn y11 ;    ; dn y1d ; dn    ; dn xk1 ;    ; dn xkp ; dn yk1 ;    ; dn ykd / D 0

for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd 2 A. Hence we have 2H

 Pp

jD1

2

xij

C

d X jD1



yij D

p X

H.xij / C 2

jD1

d X

H.yij /

jD1

for all 2 T1 , x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd 2 A and 1  i  k and so H.x C y/ D H.x/ C H.y/ for all ; 2 T1 and x; y 2 A. Therefore, by Lemma 3.12, the mapping H W A ! B is C-linear. On the other hand, it follows from (6.126) and (6.127) that   H.Œx1 ; y1 ; z1 /  ŒH.x1 /; H.y1 /; H.z1 /;      ; H.Œxk ; yk ; zk /  ŒH.xk /; H.yk /; H.zk / k

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

295

  1   n n n Œd x ; d y ; d z  f 1 1 1 n!1 d 3n i h  f .dn x1 /; f .dn y1 /; f .dn z1 / ;      ; f Œdn xk ; dn yk ; dn zk  i h   f .dn xk /; f .dn yk /; f .dn zk / 

D lim

k

1  lim 3n .d n x1 ; dn y1 ; dn z1 ;    ; dn xk ; dn yk ; dn zk / n!1 d D0 for all x1 ; y1 ; z1 ;    ; xk ; yk ; zk 2 A. Thus H.Œx; y; z/ D ŒH.x/; H.y/; H.z/ for all x; y; z 2 A. Thus H W A ! B is a multi-C -ternary algebra homomorphism satisfying (6.129). Now, let T W A ! B be another C -ternary algebras homomorphism satis1 fying (6.130). Since d.f ; T/  .1L/2d and T is C-linear, we have T 2 E0 and 1 .T/.x/ D d .T x/ D T.x/ for all x 2 A, i.e., T is a fixed point of . Since H is the unique fixed point of  2 E0 , we have H D T. This completes the proof.  Theorem 6.50. Let r; s;  be nonnegative real numbers such that 0 < r ¤ 1, 0 < s ¤ 3 and let d  2: Suppose that f W A ! B is a mapping with f .0/ D 0 satisfying (6.121) and    f .Œx1 ; y1 ; z1 /  Œf .x1 /; f .y1 /; f .z1 /;      ; f .Œxk ; yk ; zk /  Œf .xk /; f .yk /; f .zk /  k     kx1 ksA  ky1 ksA  kz1 ksA C    C kxk ksA  kyk ksA  kzk ksA

(6.133)

for all 2 T1 and x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A: Then there exists a unique C –ternary algebra homomorphism H W A ! B such that      f .x1 /  H.x1 /;    ; f .xk /  H.xk /   for all x1 ;    ; xk 2 A

d 2jd  dr j

  kx1 krA C    C kxK krA

K

(6.134)

296

6 Stability of Functional Equations in Multi-Banach Algebras

Proof. Let 0 < r < 1 and 0 < s < 3: Similarly, one can prove the theorem for other cases. The proof follows from Theorem 6.49 by taking '.x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd / WD 

p X

kx1j krA C

jD1

d X

ky1j krA C    C

jD1

p X

kxkj krA C

jD1

d X

 kykj krA ;

jD1

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk / WD .kx1 ksA  ky1 ksA  kz1 ksA C    C kxk ksA  kyk ksA  kzk ksA / for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A and L D dr1 , when 0 < r < 1 and 0 < s < 3 and L D 2  dr1 , when r > 1 and s > 3. This completes the proof.  Now, assume that A is a unital multi-C -ternary algebra with the norm k  k and the unit e and B is a unital C -ternary algebra with the norm k  k and the unit e0 . We investigate homomorphisms in multi-C -ternary algebras associated with the following functional equation: C f .x1 ;    ; xp ; y1 ;    ; yd / D 0: Theorem 6.51. Let r < 1,  be nonnegative real numbers and f W A ! B be a mapping satisfying (6.121) and (6.122). If there exist a real number  > 1 .resp., 0 <  < 1/ and an element x0 2 A such that limn!1 1n f .n x0 / D e0 .resp., limn!1 n f . x0n / D e0 /, then the mapping f W A ! B is a multi–C –ternary algebra homomorphism. Proof. By using the proof of Theorem 6.48, there exists a unique multi-C -ternary algebra homomorphism H W A ! B satisfying (6.123). It follows from (6.123) that 1 f .n x/ n!1 n

H.x/ D lim

  x  resp:; H.x/ D lim n f n n!1 

for all x 2 A and  > 1 .0 <  < 1/: Therefore, by the assumption, we get that H.x0 / D e0 : Let  > 1 and limn!1 1n f .n x0 / D e0 : It follows from (6.122) that    ŒH.x1 /; H.y1 /; H.z1 /  ŒH.x1 /; H.y1 /; f .z1 /;      ; ŒH.xk /; H.yk /; H.zk /  ŒH.xk /; H.yk /; f .zk /    D  HŒx1 ; y1 ; z1   ŒH.x1 /; H.y1 /; f .z1 /;      ; HŒxk ; yk ; zk   ŒH.xk /; H.yk /; f .zk / 

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

297

 1   f .Œn x1 ; n y1 ; z1 /  Œf .n x1 /; f .n y1 /; f .z1 /;  n!1 2n      ; f .Œn xk ; n yk ; zk /  Œf .n xk /; f .n yk /; f .zk / 

D lim

rn .kx1 krA  ky1 krA  kz1 krA C    C kxk krA  kyk krA  kzk krA / n!1 3n D0  lim

for all x1 ;    ; xk 2 A and so ŒH.x/; H.y/; H.z/ D ŒH.x/; H.y/; f .z/ for all x; y; z 2 A: Letting x D y D x0 in the last equality, we get f .z/ D H.z/ for all z 2 A: Similarly, one can show that H.x/ D f .x/ for all x 2 A when 0 <  < 1 and limn!1 n f . x0n / D e0 : Similarly, one can show the theorem for the case  > 1. Therefore, the mapping f W A ! B is a multi-C -ternary algebra homomorphism. This completes the proof.  Remark 6.52. Let r > 1,  be nonnegative real numbers and f W A ! B be a mapping satisfying (6.121) and (6.122). If there exist a real number  > 1 .resp., 0 <  < 1/ and an element x0 2 A such that limn!1 1n f .n x0 / D e0 .resp., limn!1 n f . x0n / D e0 /, then the mapping f W A ! B is a multi-C -ternary algebra homomorphism.

6.5.2 Stability of Derivations in Multi-C -Ternary Algebras Assume that A is a C -ternary algebra with the norm k  k. Park [231] proved the Hyers-Ulam stability of derivations on C -ternary algebras for the following functional equation: C f .x1 ;    ; xp ; y1 ;    ; yd / D 0: For any mapping f W A ! A; let Df .x; y; z/ D f .Œx; y; z/  Œf .x/; y; z  Œx; f .y/; z  Œx; y; f .z/ for all x; y; z 2 A. Theorem 6.53. Let ..Ak ; k  kk / W k 2 N/ be a multi-C -ternary algebra. Let f W A ! A be a mapping for which there exist the functions ' W A.pCd/k ! Œ0; 1/ and W A3k ! Œ0; 1/ satisfying the inequalities (6.105), (6.106) and (6.108) such that

298

6 Stability of Functional Equations in Multi-Banach Algebras

     Df .x1 ; y1 ; z1 /;    ; Df .xk ; yk ; zk /  

.x1 ; y1 ; z1 ;    ; xk ; yk ; zk /

(6.135)

for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd ; x1 ;    ; xk ; y1 ;    ; yk ; z1 ;    ; zk 2 A, where  D pC2d 2 . If there exists L < 1 such that pCd pCd  ‚ pCd …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  '  x1 ;    ;  x1 ;  x2 ;    ;  x2 ;    ;  xk ;    ;  xk pCd pCd  ‚ pCd …„ ƒ ‚ …„ ƒ ‚ …„ ƒ    L' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk

(6.136)

for all x1 ; x2 ;    ; xk 2 A, then there exists a unique multi-C -ternary derivation ı W A ! B such that      f .x1 /  ı.x1 /;    ; f .xk /  ı.xk /  k

 ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  1 ' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk .1  L/2 pCd



pCd

pCd

(6.137)

for all x1 ;    ; xk 2 A. Proof. By the same reasoning as in the proof of Theorem 6.46, there exists a unique C-linear mapping ı W A ! A satisfying (6.135). The mapping ı W A ! A is given by ı.x/ D lim .n f /.x/ D lim n!1

n!1

1 f . n x/ n

(6.138)

and ı. x/ D  ı.x/ for all x 2 A. Also, H is the unique fixed point of  in the set E0 D fg 2 E W d.f ; g/ < 1g and d.ı; f / 

1 1 d.f ; f /  ; 1L .1  L/2

i.e., the inequality (6.110) holds for all x1 ;    ; xk 2 A. It follows from the definition of ı, (6.105), (6.106) and (6.138) that      C ı.x11 ;    ; x1p y11 ;    y1d /;    ; C ı.xk1 ;    ; xkp yk1 ;    ykd /   1  D lim n  C f . n x11 ;    ;  n x1p ;  n y11 ;    ;  n y1d /; n!1 

k

6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

299

     ; C f . n xk1 ;    ;  n xkp ;  n yk1 ;    ;  n ykd / 

k

1  lim n '. n x11 ;    ;  n x1p : n y11 ;    ;  n y1d ; n!1     ;  n xk1 ;    ;  n xkp ;  n yk1 ;    ;  n ykd / D0 for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd 2 A. Hence we have 2ı

 Pp

jD1

2

xij

C

d X

p d  X X yij D ı.xij / C 2 ı.yij /

jD1

jD1

jD1

for all 2 T1 , x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd 2 A and 1  i  k and so ı.x C y/ D ı.x/ C ı.y/ for all ; 2 T1 and x; y 2 A. Therefore, by Lemma 3.12, the mapping ı W A ! B is C-linear. On the other hand, it follows from (6.108) and (6.135) that      Dı.x1 ; y1 ; z1 /;    ; Dı.xk ; yk ; zk /  k

  1    D lim 3n f Df . n x1 ;  n y1 ;  n z1 /;    ; f . n xk ;  n yk ;  n zk /  n!1   lim

n!1

1 . n x1 ;  n y1 ;  n z1 ;    ;  n xk ;  n yk ;  n zk /  3n

D0 for all x1 ; y1 ; z1 ;    ; xk ; yk ; zk 2 A and so .ı.Œx1 ; y1 ; z1 /;    ; ı.Œxk ; yk ; zk // D .Œı.x1 /; .y1 /; .z1 / C Œx1 ; ı.y1 /; z1  C Œx1 ; y1 ; ı.z1 /;

(6.139)

   ; Œı.xk /; .yk /; .zk / C Œxk ; ı.yk /; zk  C Œxk ; yk ; ı.zk // for all x; y; z 2 A and so the mapping ı W A ! A is a C -ternary derivation. It follows from (6.135) and (6.108) that

300

6 Stability of Functional Equations in Multi-Banach Algebras

   ıŒx1 ; y1 ; z1   Œı.x1 /; y1 ; z1   Œx1 ; ı.y1 /; z1   Œx; y; f .z1 /;      ; ıŒxk ; yk ; zk   Œı.xk /; yk ; zk   Œxk ; ı.yk /; zk   Œx; y; f .zk /   1   f Œ n x1 ;  n y1 ; z1   Œf . n x1 /;  n y1 ; z1   n!1  2n

D lim

 Œ n x1 ; f . n y1 /; z1   Œ n x1 ;  n y1 ; f .z1 /;    ; f Œ n xk ;  n yk ; zk   Œf . n xk /;  n yk ; zk     Œ n xk ; f . n yk /; zk   Œ n xk ;  n yk ; f .zk /   lim

n!1

1 . n x1 ;  n y1 ; z1 ;    ;  n xk ;  n yk ; zk /  2n

D0 for all x1 ; y1 ; z1 ;    ; xk ; yk ; zk 2 A and so .ıŒx; y; z/ D Œı.x/; y; z C Œx; ı.y/; z C Œx; y; f .z/

(6.140)

for all x; y; z 2 A: Hence it follows from (6.139) and (6.140) that Œx; y; ı.z/ D Œx; y; f .z/

(6.141)

for all x; y; z 2 A: Letting x D y D f .z/  ı.z/ in (6.141), we have h i   kf .z/  ı.z/k3 D  f .z/  ı.z/; f .z/  ı.z/; f .z/  ı.z/  D 0

(6.142)

for all z1 ;    ; zk 2 A and hence f .z/ D ı.z/ for all z 2 A. Therefore, the mapping f W A ! A is a multi-C –ternary derivation. This completes the proof.  Corollary 6.54. Let r < 1; s < 2,  be non-negative real numbers and f W A ! A be a mapping satisfying (6.121) and   k Df .x1 ; y1 ; z1 /;    ; Df .xk ; yk ; zk / k  .kx1 ksA  ky1 ksA  kz1 ksA C    C kxk ksA  kyk ksA  kzk ksA / for all x1 ; y1 ; z1 ;    ; xk ; yk ; zk 2 A: Then the mapping f W A ! A is a multi-C ternary derivation. Proof. Define '.x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd / D

p X jD1

kx1j krA C

d X jD1

ky1j krA ;    ;

p X jD1

kxkj krA C

d X jD1

kykj krA



6.5 Stability of Homomorphisms and Derivations in Multi-C –Ternary Algebras

301

and .x1 ; y1 ; z1 ;    ; xk ; yk ; zk /   D  kx1 ksA  ky1 ksA  kz1 ksA C    C kxk ksA  kyk ksA  kzk ksA for all x1 ; y1 ; z1 ;    ; xk ; yk ; zk ; x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd 2 A and applying Theorem 6.53. Then we get the desired result.  Theorem 6.55. Let ..Ak ; k  kk / W k  1/ be a multi-C -ternary algebra. Let f W A ! A be a mapping for which there exist the functions ' W A.pCd/k ! Œ0; 1/ and W A3k ! Œ0; 1/ satisfying (6.106), (6.114), (6.115) and (6.135) for all 2 T1 and x11 ;    ; x1p ; y11 ;    ; y1d ;    ; xk1 ;    ; xkp ; yk1 ;    ; ykd ; x1 ;    ; xk ; . If there exists L < 1 such that y1 ;    ; yk ; z1 ;    ; zk 2 A, where  D pC2d 2 pCd

pCd

…„ ƒ ‚ …„ ƒ ‚ x1 x1 x2 x2 ' ; ; ; ; ; ;    

‚ …„ ƒ  L  ‚ …„ ƒ ‚ …„ ƒ ' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk  pCd



pCd

‚ …„ ƒ  xk xk ; ; ;  

pCd

(6.143)

pCd

(6.144)

for all x1 ; x2 ;    ; xk 2 A, then there exists a unique multi-C -ternary algebra homomorphism ı W A ! A such that k.f .x1 /  ı.x1 /;    ; f .xk /  ı.xk //kk  ‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ  1 ' x1 ;    ; x1 ; x2 ;    ; x2 ;    ; xk ;    ; xk  .1  L/2 pCd

pCd

pCd

(6.145)

for all x1 ;    ; xk 2 A. Proof. By the same reasoning as in the proof of Theorem 6.47, there exists a unique C-linear mapping ı W A ! A satisfying (6.135). The rest of the proof is similar to the proof of Theorem 6.53 and so we omit it. 

Chapter 7

Stability of Functional Equations in Non-Archimedean Banach Algebras

In [203], Moslehian and Rassias proved the Hyers-Ulam stability of the Cauchy and quadratic functional equations in non-Archimedean normed spaces. After their results, some papers (see, for instance, [71, 83, 103]) on the stability of other equations in such spaces have been published. Next, Eshaghi-Gordji et al. and Cho et al. applied the direct method or the fixed point method to prove the stability of some functional equations in nonArchimedean Banach algebras. In this chapter, we study the directions mentioned above and apply the fixed point method to show the Hyers-Ulam stability of some wide classes of functional equations in non-Archimedean Banach algebras and nonArchimedean C -algebras. In Sect. 7.1, we extend the results presented in Chap. 3 to the setting of nonArchimedean C -algebras. In Sect. 7.2, we extend the results presented in Chap. 2 to the setting of nonArchimedean C -algebras. In fact, by using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean C -algebras and non-Archimedean Lie C -algebras for the following additive functional equation: m  m m m  X  X  X X f mxi C xj C f xi D 2f mxi iD1

jD1; j¤i

iD1

iD1

for each m  2.

© Springer International Publishing Switzerland 2015 Y.J. Cho et al., Stability of Functional Equations in Banach Algebras, DOI 10.1007/978-3-319-18708-2_7

303

304

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach Using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms in non-Archimedean C -algebras and non-Archimedean Lie C -algebras and derivations on non-Archimedean C -algebras and non-Archimedean Lie C algebras for the following Jensen type functional equation: f

x C y 2

Cf

x  y 2

D f .x/:

7.1.1 Stability of Homomorphisms in Non-Archimedean C -Algebras Throughout this section, assume that A is a non-Archimedean C -algebra with the norm k  kA and B is a non-Archimedean C -algebra with the norm k  kB . For any mapping f W A ! B, we define D f .x; y/ WD f

x C y 2

C f

x  y 2

 f . x/

(7.1)

for all 2 T1 WD f 2 C W j j D 1g and x; y 2 A. Now, we prove the Hyers-Ulam stability of homomorphisms in nonArchimedean C -algebras for the functional equation D f .x; y/ D 0. Theorem 7.1. Let f W A ! B be a mapping for which there exist the functions '; W A2 ! Œ0; 1/ and W A ! Œ0; 1/ such that kD f .x; y/kB  '.x; y/; kf .xy/  f .x/f .y/kB 

.x; y/

(7.2) (7.3)

and kf .x /  f .x/ kB  .x/

(7.4)

for all 2 T1 and x; y 2 A. If there exists L < 1 such that j2j < 1 and '.2x; 2y/  j2jL'.x; y/;

(7.5)

.2x; 2y/  j4jL .x; y/

(7.6)

and

.2x/  j2jL .x/

(7.7)

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach

305

for all x; y 2 A, then there exists a unique non-Archimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

L '.x; 0/ 1L

(7.8)

for all x 2 A. Proof. It follows from (7.5), (7.6), (7.7) and L < 1 that lim

1 '.2n x; 2n y/ D 0; j2jn

(7.9)

lim

1 .2n x; 2n y/ D 0 j2j2n

(7.10)

1

.2n x/ D 0 j2jn

(7.11)

n!1

n!1

and lim

n!1

for all x; y 2 A. Consider the set X WD fg W A ! Bg and introduce the generalized metric on X as follows: d.g; h/ D inffC 2 RC W kg.x/  h.x/kB  C'.x; 0/; 8x 2 Ag; which .X; d/ is complete. Now, we consider the linear mapping J W X ! X defined by Jg.x/ WD

1 g.2x/ 2

for all x 2 A. Now, we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting D 1 and y D 0 in (7.2), we have  x     f .x/  '.x; 0/ 2f B 2

(7.12)

for all x 2 A and so   1 1   '.2x; 0/  L'.x; 0/ f .x/  f .2x/  B 2 j2j for all x 2 A. Hence d.f ; Jf /  L. By Theorem 1.3, there exists a mapping H W A ! B such that

306

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

(1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(7.13)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (7.13) such that there exists C 2 .0; 1/ such that kH.x/  f .x/kB  C'.x; 0/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality f .2n x/ D H.x/ n!1 2n lim

(7.14)

for all x 2 A; 1 d.f ; Jf /, which implies the inequality (3) d.f ; H/  1L d.f ; H/ 

L : 1L

This implies that the inequality (7.8) holds. It follows from (7.5) and (7.14) that  x C y  x  y   CH  H.x/ H B 2 2 1 D lim kf .2n1 .x C y// C f .2n1 .x  y//  f .2n x/kB n!1 j2jn  lim

n!1

1 '.2n x; 2n y/ j2jn

D0 for all x; y 2 A. Then we have H

x C y

for all x; y 2 A. Letting z D

2 xCy 2

CH

x  y

and w D

2 xy 2

D H.x/

in (7.15), we have

H.z/ C H.w/ D H.z C w/

(7.15)

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach

307

for all z; w 2 A and so the mapping H W A ! B is Cauchy additive, i.e., H.z C w/ D H.z/ C H.w/ for all z; w 2 A. Letting y D x in (7.2), we get f .x/ D f . x/ for all 2 T1 and all x 2 A. By the similar method as in above, we have H.x/ D H. x/ f or all 2 T1 and x 2 A. Thus one can show that the mapping H W A ! B is C-linear. It follows from (7.6) that kH.xy/  H.x/H.y/kB D lim

n!1

 lim

n!1

1 kf .4n xy/  f .2n x/f .2n y/kB j4jn 1 .2n x; 2n y/ j4jn

D0 for all x; y 2 A and so H.xy/ D H.x/H.y/ for all x; y 2 A. It follows from (7.7) that 1 kf .2n x /  f .2n x/ kB n!1 j2jn

kH.x /  H.x/ kB D lim  lim

n!1

1

.2n x/ j2jn

D0 for all x 2 A and so H.x / D H.x/ for all x 2 A. Thus H W A ! B is a non-Archimedean C -algebra homomorphism satisfying (7.8). This completes the proof. 

308

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

Corollary 7.2. Let r < 1,  be nonnegative real numbers and f W A ! B be a mapping such that kD f .x; y/kB  .kxkrA C kykrA /; kf .xy/  f .x/f .y/kB 

.kxkrA

C

kykrA /

(7.16) (7.17)

and kf .x /  f .x/ kB  kxkrA

(7.18)

for all 2 T1 and x; y 2 A. Then there exists a unique non-Archimedean C algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

j2jr  kxkrA j2j  j2jr

(7.19)

for all x 2 A. Proof. The proof follows from Theorem 7.1 by taking '.x; y/ D

.x; y/ WD .kxkrA C kykrA /;

.x/ WD .kxkrA / 

for all x; y 2 A and L D j2jr1 .

Theorem 7.3. Let f W A ! B be a mapping for which there exist the functions '; W A2 ! Œ0; 1/ and W A ! Œ0; 1/ satisfying (7.2), (7.3) and (7.4). If there exists L < 1 such that j2j < 1 and x y ;  L'.x; y/; 2 2 x y ;  L .x; y/ j4j 2 2 j2j'

(7.20) (7.21)

and j2j

x 2

 L .x/

(7.22)

for all x; y 2 A, then there exists a unique non-Archimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB  for all x 2 A.

L '.x; 0/ j2j  j2jL

(7.23)

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach

309

Proof. It follows from (7.20), (7.21), (7.22) and L < 1 that lim j2jn '

n!1

lim j2j2n

n!1

x y D 0; ; 2n 2n x y D0 ; 2n 2n

and lim j2jn

n!1

x D0 2n

for all x; y 2 A. We consider the linear mapping J W X ! X defined by Jg.x/ WD 2g

x 2

for all x 2 A. It follows from (7.12) that   x  x  L   '.x; 0/   ' ;0  f .x/  2f 2 B 2 j2j for all x 2 A. Hence d.f ; Jf /  H W A ! B such that

L j2j .

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(7.24)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (7.24) such that there exists C 2 .0; 1/ such that kH.x/  f .x/kB  C'.x; 0/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality lim 2n f

n!1

x D H.x/ 2n

for all x 2 A; 1 (3) d.f ; H/  1L d.f ; Jf /, which implies the inequality

310

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

d.f ; H/ 

L ; j2j  j2jL

which implies that the inequality (7.23) holds. The rest of the proof is similar to the proof of Theorem 7.1. This completes the proof.  Corollary 7.4. Let r > 2,  be nonnegative real numbers and f W A ! B be a mapping satisfying (7.16), (7.17) and (7.18). Then there exists a unique nonArchimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

j2jr

 kxkrA  j2j

(7.25)

for all x 2 A. Proof. The proof follows from Theorem 7.3 by taking '.x; y/ D

.x; y/ WD .kxkrA C kykrA /;

.x/ WD kxkrA

for all x; y 2 A and L D j2j1r .



Theorem 7.5. Let f W A ! B be an odd mapping for which there exist the functions '; W A2 ! Œ0; 1/ and W A ! Œ0; 1/ satisfying (7.2), (7.3) and (7.4). If there exists L < 1 such that '.x; 3x/  j2jL'

 x 3x  ; 2 2

for all x 2 A and (7.5), (7.6) and (7.7) hold then there exists a unique nonArchimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

1 '.x; 3x/ j2j  j2jL

(7.26)

for all x 2 A. Proof. Consider the set X WD fg W A ! Bg and introduce the generalized metric on X as follows: d.g; h/ D inffC 2 RC W kg.x/  h.x/kB  C'.x; 3x/; 8x 2 Ag; which .X; d/ is complete. Now, we consider the linear mapping J W X ! X defined by Jg.x/ WD

1 g.2x/ 2

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach

311

for all x 2 A. Now, we have d.Jg; Jh/  Ld.g; h/ for all g; h 2 X. Letting D 1 and replacing y by 3x in (7.2), we have kf .2x/  2f .x/kB  '.x; 3x/

(7.27)

for all x 2 A and so   1 1   '.x; 3x/ f .x/  f .2x/  B 2 j2j for all x 2 A. Hence d.f ; Jf /  H W A ! B such that

1 . j2j

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(7.28)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (7.28) such that there exists C 2 .0; 1/ such that kH.x/  f .x/kB  C'.x; 3x/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality f .2n x/ D H.x/ n!1 2n lim

for all x 2 A; 1 (3) d.f ; H/  1L d.f ; Jf /, which implies the inequality d.f ; H/ 

1 : j2j  j2jL

This implies that the inequality (7.26) holds. The rest of the proof is similar to the proof of Theorem 7.1. This completes the proof. 

312

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

Corollary 7.6. Let r < 12 ,  be nonnegative real numbers and f W A ! B be an odd mapping such that kD f .x; y/kB    kxkrA  kykrA ; kf .xy/  f .x/f .y/kB   

kxkrA



kykrA

(7.29) (7.30)

and kf .x /  f .x/ kB  kxk2r A

(7.31)

for all 2 T1 and x; y 2 A. Then there exists a unique non-Archimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

3r  kxk2r A j2j  j2j2r

(7.32)

for all x 2 A. Proof. The proof follows from Theorem 7.5 by taking '.x; y/ D

.x; y/ WD   kxkrA  kykrA ;

.x/ WD   kxkrA

for all x; y 2 A and L D j2j2r1 .



Theorem 7.7. Let f W A ! B be an odd mapping for which there exist the functions '; W A2 ! Œ0; 1/ and W A ! Œ0; 1/ satisfying (7.2), (7.3), and (7.4). If there exists L < 1 such that '.x; 3x/ 

1 L'.2x; 6x/ j2j

for all x 2 A, and also (7.20), (7.21) and (7.22) hold, then there exists a unique non-Archimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

L '.x; 3x/ j2j  j2jL

for all x 2 A. Proof. We consider the linear mapping J W X ! X defined by x Jg.x/ WD 2g 2 for all x 2 A. It follows from (7.27) that   x 3x  L x     '.x; 3x/ f .x/  2f . /  ' ; 2 B 2 2 j2j

(7.33)

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach

for all x 2 A. Hence d.f ; Jf /  H W A ! B such that

L . 2

313

By Theorem 1.3, there exists a mapping

(1) H is a fixed point of J, i.e., H.2x/ D 2H.x/

(7.34)

for all x 2 A. The mapping H is a unique fixed point of J in the set Y D fg 2 X W d.f ; g/ < 1g: This implies that H is a unique mapping satisfying (7.34) such that there exists C 2 .0; 1/ such that kH.x/  f .x/kB  C'.x; 3x/ for all x 2 A; (2) d.J n f ; H/ ! 0 as n ! 1. This implies the equality x lim 2n f n D H.x/ n!1 2 for all x 2 A; 1 d.f ; Jf /, which implies the inequality (3) d.f ; H/  1L d.f ; H/ 

L ; 2  2L

which implies that the inequality (7.33) holds. The rest of the proof is similar to the proof of Theorem 7.1.  Corollary 7.8. Let r > 1,  be nonnegative real numbers and f W A ! B be an odd mapping satisfying (7.29), (7.30) and (7.31). Then there exists a unique nonArchimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

j2j2r

 kxk2r A  j2j

(7.35)

for all x 2 A. Proof. The proof follows from Theorem 7.7 by taking '.x; y/ D

.x; y/ WD   kxkrA  kykrA ;

for all x; y 2 A and L D j2j12r .

.x/ WD   kxkrA 

314

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

7.1.2 Stability of Derivations in Non-Archimedean C -Algebras Assume that A is a non-Archimedean C -algebra with the norm k  kA . Now, we prove the Hyers-Ulam stability of derivations in non-Archimedean C algebras for the functional equation D f .x; y/ D 0. Theorem 7.9. Let f W A ! A be a mapping for which there exist the functions '; W A2 ! Œ0; 1/ such that kD f .x; y/kA  '.x; y/

(7.36)

and kf .xy/  f .x/y  xf .y/kA 

.x; y/

(7.37)

for all 2 T1 and all x; y 2 A. If there exists L < 1 such that x  '.x; 0/  j2jL' ; 0 2 for all x 2 A, (7.5) and (7.6) hold. Then there exists a unique non-Archimedean derivation ı W A ! A such that kf .x/  ı.x/kA 

L '.x; 0/ 1L

(7.38)

for all x 2 A. Proof. By the same reasoning as in the proof of Theorem 7.1, there exists a unique involution C-linear mapping ı W A ! A satisfying (7.38). Also, the mapping ı W A ! A is given by f .2n x/ n!1 2n

ı.x/ D lim for all x 2 A. It follows from (7.37) that

kı.xy/  ı.x/y  xı.y/kA 1 D lim kf .4n xy/  f .2n x/  2n y  2n xf .2n y/kA n!1 j4jn 1  lim .2n x; 2n y/ n!1 j4jn D0 for all x; y 2 A. Then we have ı.xy/ D ı.x/y C xı.y/ for all x; y 2 A. Thus ı W A ! A is a derivation satisfying (7.38). This completes the proof. 

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach

315

Corollary 7.10. Let r < 1,  be nonnegative real numbers and f W A ! A be a mapping such that kD f .x; y/kA  .kxkrA C kykrA /

(7.39)

kf .xy/  f .x/y  xf .y/kA  .kxkrA C kykrA /

(7.40)

and

for all 2 T1 and x; y 2 A. Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

j2jr  kxkrA j2j  j2jr

(7.41)

for all x 2 A. Proof. The proof follows from Theorem 7.9 by taking '.x; y/ D

.x; y/ WD .kxkrA C kykrA /

for all x; y 2 A and L D j2jr1 .



Remark 7.11. Let f W A ! A be a mapping for which there exist the functions '; W A2 ! Œ0; 1/ satisfying (7.36) and (7.37). If there exists L < 1 such that '.x; 0/ 

1 L'.2x; 0/ j2j

for all x 2 A, (7.20) and (7.21) hold, then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA 

L '.x; 0/ j2j  j2jL

(7.42)

for all x 2 A. Corollary 7.12. Let r > 2,  be nonnegative real numbers and f W A ! A be a mapping satisfying (7.39) and (7.40). Then there exists a unique derivation ı W A ! A such that kf .x/  ı.x/kA  for all x 2 A.

 kxkrA j2jr  j2j

(7.43)

316

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

Proof. The proof follows from Remark 7.11 by taking '.x; y/ D

.x; y/ WD .kxkrA C kykrA /

for all x; y 2 A and L D j2j1r .



Remark 7.13. For the inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 7.5, 7.7 and Corollaries 7.6, 7.8.

7.1.3 Stability of Homomorphisms in Non-Archimedean Lie C -Algebras A non-Archimedean C -algebra C endowed with the Lie product Œx; y WD

xy  yx 2

on C is called a non-Archimedean Lie C -algebra (see [224, 225, 227]). Definition 7.14. Let A and B be non-Archimedean Lie C -algebras. A C-linear mapping H W A ! B is called a non-Archimedean Lie C -algebra homomorphism if H.Œx; y/ D ŒH.x/; H.y/ for all x; y 2 A. Throughout this section, assume that A is a non-Archimedean Lie C -algebra with the norm k  kA and B is a non-Archimedean Lie C -algebra with the norm k  kB . Now, we prove the Hyers-Ulam stability of homomorphisms in nonArchimedean Lie C -algebras for the functional equation D f .x; y/ D 0. Theorem 7.15. Let f W A ! B be a mapping for which there are functions '; W A2 ! Œ0; 1/ satisfying (7.2) such that kf .Œx; y/  Œf .x/; f .y/kB 

.x; y/

(7.44)

for all x; y 2 A. If there exists L < 1 such that '.x; 0/  j2jL'

x 2

;0



for all x 2 A, and also (7.5) and (7.6) hold, then there exists a unique nonArchimedean Lie C -algebra homomorphism H W A ! B satisfying (7.8).

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach

317

Proof. By the same reasoning as in the proof of Theorem 7.1, there exists a unique C-linear mapping H W A ! B satisfying (7.8). Also, the mapping H W A ! B is given by f .2n x/ n!1 2n

H.x/ D lim for all x 2 A. It follows from (7.44) that

kH.Œx; y/  ŒH.x/; H.y/kB D lim

1 kf .4n Œx; y/  Œf .2n x/; f .2n y/kB j4jn

 lim

1 .2n x; 2n y/ j4jn

n!1

n!1

D0 for all x; y 2 A. Then we have H.Œx; y/ D ŒH.x/; H.y/ for all x; y 2 A. Thus H W A ! B is a non-Archimedean Lie C -algebra homomorphism satisfying (7.8). This completes the proof.  Corollary 7.16. Let r < 1,  be nonnegative real numbers and f W A ! B be a mapping satisfying (7.16) such that kf .Œx; y/  Œf .x/; f .y/kB  .kxkrA C kykrA /

(7.45)

for all x; y 2 A. Then there exists a unique non-Archimedean Lie C -algebra homomorphism H W A ! B satisfying (7.19). Proof. The proof follows from Theorem 7.15 by taking '.x; y/ D

.x; y/ WD .kxkrA C kykrA /

for all x; y 2 A and L D j2jr1 .



Remark 7.17. Let f W A ! B be a mapping for which there exist the functions '; W A2 ! Œ0; 1/ and W A ! Œ0; 1/ satisfying (7.2), (7.5), (7.6) and (7.44). If there exists L < 1 such that '.x; 0/ 

1 L'.2x; 0/ j2j

for all x 2 A, then there exists a unique non-Archimedean Lie C -algebra homomorphism H W A ! B satisfying (7.23).

318

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

Corollary 7.18. Let r > 2,  be nonnegative real numbers and f W A ! B be a mapping satisfying (7.16) and (7.45). Then there exists a unique non-Archimedean Lie C -algebra homomorphism H W A ! B satisfying (7.25). Proof. The proof follows from Remark 7.17 by taking '.x; y/ D

.x; y/ WD .kxkrA C kykrA /

for all x; y 2 A and L D j2j1r .



Remark 7.19. For the inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 7.5, 7.7 and their corollaries.

7.1.4 Stability of Non-Archimedean Lie Derivations in C -Algebras First, we give the following definition on the non-Archimedean Lie derivation in a non-Archimedean Lie C -algebra. Definition 7.20. Let A be a non-Archimedean Lie C -algebra. A C-linear mapping ı W A ! A is called a non-Archimedean Lie derivation if ı.Œx; y/ D Œı.x/; y C Œx; ı.y/ for all x; y 2 A. Assume that A is a non-Archimedean Lie C -algebra with the norm k  kA . Now, we prove the Hyers-Ulam stability of non-Archimedean Lie derivations on non-Archimedean Lie C -algebras for the functional equation D f .x; y/ D 0: Theorem 7.21. Let f W A ! A be a mapping for which there exists a function '; W A2 ! Œ0; 1/ satisfying (7.5), (7.6) and (7.36) such that kf .Œx; y/  Œf .x/; y  Œx; f .y/kA 

.x; y/

(7.46)

for all x; y 2 A. If there exists L < 1 such that '.x; 0/  j2jL'

x 2

;0



for all x 2 A. Then there exists a unique non-Archimedean Lie derivation ı W A ! A satisfying (7.38).

7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach

319

Proof. By the same reasoning as in the proof of Theorem 7.1, there exists a unique involution C-linear mapping ı W A ! A satisfying (7.38). Also, the mapping ı W A ! A is given by f .2n x/ n!1 2n

ı.x/ D lim for all x 2 A. It follows from (7.44) that

kı.Œx; y/  Œı.x/; y  Œx; ı.y/kA D lim

1 kf .4n Œx; y/  Œf .2n x/; 2n y  Œ2n x; f .2n y/kA j4jn

 lim

1 .2n x; 2n y/ j4jn

n!1

n!1

D0 for all x; y 2 A. Then we have ı.Œx; y/ D Œı.x/; y C Œx; ı.y/ for all x; y 2 A. Thus ı W A ! A is a non-Archimedean Lie derivation satisfying (7.38). This completes the proof.  Corollary 7.22. Let r < 1,  be nonnegative real numbers and f W A ! A be a mapping satisfying (7.39) such that kf .Œx; y/  Œf .x/; y  Œx; f .y/kA  .kxkrA C kykrA /

(7.47)

for all x; y 2 A. Then there exists a unique non-Archimedean Lie derivation ı W A ! A satisfying (7.41). Proof. The proof follows from Theorem 7.15 by taking '.x; y/ D

.x; y/ WD .kxkrA C kykrA /;

for all x; y 2 A and L D j2jr1 .

.x/ WD kxkrA 

Remark 7.23. Let f W A ! A be a mapping for which there exist functions '; W A2 ! Œ0; 1/ and W A ! Œ0; 1/ satisfying (7.20), (7.21), (7.22), (7.36) and (7.46). If there exists L < 1 such that '.x; 0/ 

1 L'.2x; 0/ j2j

for all x 2 A, then there exists a unique non-Archimedean Lie derivation ı W A ! A satisfying (7.42).

320

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

Corollary 7.24. Let r > 2,  be nonnegative real numbers and f W A ! A be a mapping satisfying (7.39) and (7.47). Then there exists a unique non-Archimedean Lie derivation ı W A ! A satisfying (7.43). Proof. The proof follows from Remark 7.23 by taking '.x; y/ D

.x; y/ WD .kxkrA C kykrA /

for all x; y 2 A and L D j2j1r .



Remark 7.25. For the inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems 7.5, 7.7 and their corollaries.

7.2 Stability for m-Variable Additive Functional Equations Using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean C -algebras and non-Archimedean Lie C -algebras for the following additive functional equation: m  m m m  X  X  X X f mxi C xj C f xi D 2f mxi iD1

iD1

jD1; j¤i

(7.48)

iD1

for each m  2. For any mapping f W A ! B, we define D f .x1 ;    ; xm / WD

m X iD1



f mxi C

m X jD1; j¤i

 xj

m m  X   X  Cf xi  2f mxi iD1

iD1

for all 2 T1 WD f 2 C W j j D 1g and x1 ;    ; xm 2 A.

7.2.1 Stability of Homomorphisms and Derivations in Non-Archimedean C -Algebras Now, we prove the Hyers–Ulam stability of homomorphisms in non-Archimedean C -algebras for the functional equation D f .x1 ;    ; xm / D 0. Theorem 7.26. Let f W A ! B be a mapping for which there exist the functions ' W Am ! Œ0; 1/, W A2 ! Œ0; 1/ and W A ! Œ0; 1/ such that jmj < 1 is far from zero and kD f .x1 ;    ; xm /kB  '.x1 ;    ; xm /; kf .xy/  f .x/f .y/kB 

.x; y/

(7.49) (7.50)

7.2 Stability for m-Variable Additive Functional Equations

321

and kf .x /  f .x/ kB  .x/

(7.51)

for all 2 T1 and x1 ;    ; xm ; x; y 2 A. If there exists L < 1 such that '.mx1 ;    ; mxm /  jmjL'.x1 ;    ; xm /; .mx; my/  jmj2 L .x; y/

(7.52) (7.53)

and

.mx/  jmjL .x/

(7.54)

for all x; y; x1 ;    ; xm 2 A, then there exists a unique non-Archimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

1 '.x; 0;    ; 0/ jmj  jmjL

(7.55)

for all x 2 A. Proof. It follows from (7.52), (7.53), (7.54) and L < 1 that 1 '.mn x1 ;    ; mn xm / D 0; n!1 jmjn lim

lim

n!1

(7.56)

1 .mn x; mn y/ D 0 jmj2n

(7.57)

1

.mn x/ D 0 jmjn

(7.58)

and lim

n!1

for all x; y; x1 ;    ; xm 2 A. Let us define ˝ to be the set of all mappings g W A ! B and introduce a generalized metric on ˝ as follows: d.g; h/

(7.59)

D inffk 2 .0; 1/ W kg.x/  h.x/kB < k.x; 0;    ; 0/; 8x 2 Ag; which .˝; d/ is a generalized complete metric space. Now, we consider the function J W ˝ ! ˝ defined by Jg.x/ D m1 g.mx/ for all x 2 A and g 2 ˝. Note that, for all g; h 2 ˝, d.g; h/ < k H) kg.x/  h.x/kB < k.x; 0;    ; 0/

322

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

 1 1 k   .mx; 0;    ; 0/ H)  g.mx/  h.mx/ < B m m jmj 1  1   H)  g.mx/  h.mx/ < kL.mx; 0;    ; 0/ B m m H) d.Jg; Jh/ < kL:

(7.60)

From this, it is easy to see that d.Jg; Jk/  Ld.g; h/ for all g; h 2 ˝, that is, J is a self-function of ˝ with the Lipschitz constant L. Putting D 1, x D x1 and x2 D x3 D    D xm D 0 in (7.49) we have kf .mx/  mf .x/kB  .x; 0;    ; 0/

(7.61)

  1 1   .x; 0;    ; 0/ f .x/  f .mx/  B m jmj

(7.62)

for all x 2 A. Then

for all x 2 A, that is, d.Jf ; f / 

1 < 1: jmj

Now, from the fixed point method, it follows that there exists a fixed point H of J in ˝ such that 1 f .mn x/ n!1 mn

H.x/ D lim

(7.63)

for all x 2 A since limn!1 d.J n f ; H/ D 0. On the other hand, it follows from (7.49), (7.56) and (7.63) that   1   kD H.x1 ;    ; xm /kB D lim  n Df .mn x1 ;    ; mn xm / n!1 m B 1  lim .mn x1 ;    ; mn xm / n!1 jmjn D 0: By the similar method as in above, we have H.mx/ D H.m x/

(7.64)

7.2 Stability for m-Variable Additive Functional Equations

323

for all 2 T1 and x 2 A. Thus one can show that the mapping H W A ! B is C-linear. It follows from (7.50), (7.57) and (7.63) that kH.xy/  H.x/H.y/kB D lim

1 kf .m2n xy/  f .mn x/f .mn y/kB jmj2n

 lim

1 .mn x; mn y/ jmj2n

n!1

n!1

D0

(7.65)

for all x; y 2 A and so H.xy/ D H.x/H.y/ for all x; y 2 A. Thus H W A ! B is a homomorphism satisfying (7.55). Also, by (7.51), (7.58), (7.63) and the similar method, we have H.x / D H.x/ . This completes the proof.  Corollary 7.27. Let r > 1,  be nonnegative real numbers and f W A ! B be a mapping such that kD f .x1 ;    ; xm /kB    .kx1 krA C kx2 krA C    C kxm krA /; kf .xy/  f .x/f .y/kB    .kxkrA  kykrA /

(7.66)

and kf .x /  f .x/ kB    kxkrA for all 2 T1 and x1 ;    ; xm ; x; y 2 A. Then there exists a unique non-Archimedean C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

 kxkrA jmj  jmjr

(7.67)

for all x 2 A. Proof. The proof follows from Theorem 7.26 by taking '.x1 ;    ; xm / WD   .kx1 krA C kx2 krA C    C kxm krA /; .x; y/ WD   .kxkrA  kykrA /;

.x/ WD   kxkrA for all x1 ;    ; xm ; x; y 2 A and L D jmjr1 .

(7.68) 

324

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

Now, we prove the Hyers-Ulam stability of derivations on non-Archimedean C -algebras for the functional equation D f .x1 ;    ; xm / D 0: Remark 7.28. Let f W A ! A be a mapping for which there exist the functions ' W Am ! Œ0; 1/, W A2 ! Œ0; 1/ and W A ! Œ0; 1/ such that jmj < 1 is far from zero and kD f .x1 ;    ; xm /kA  '.x1 ;    ; xm /; kf .xy/  f .x/y  xf .y/kA 

.x; y/

(7.69)

and kf .x /  f .x/ kA  .x/ for all 2 T1 and x1 ;    ; xm ; x; y 2 A. If there exists L < 1 such that (7.52), (7.53) and (7.54) hold, then there exists a unique non-Archimedean C -algebra derivation ı W A ! A such that kf .x/  ı.x/kA 

1 '.x; 0;    ; 0/ jmj  jmjL

for all x 2 A.

7.2.2 Stability of Homomorphisms and Derivations in Non-Archimedean Lie C -Algebras Now, we prove the Hyers-Ulam stability of homomorphisms in non-Archimedean Lie C -algebras for the functional equation D f .x1 ;    ; xm / D 0. Theorem 7.29. Let f W A ! B be a mapping for which there exist the functions ' W Am ! Œ0; 1/ and W A2 ! Œ0; 1/ such that (7.49) and (7.51) hold and kf .Œx; y/  Œf .x/; f .y/kB 

.x; y/

(7.70)

for all 2 T1 and x; y 2 A. If there exists L < 1 and (7.52) and (7.53) hold, then there exists a unique non-Archimedean Lie C -algebra homomorphism H W A ! B such that (7.55) hold.

7.2 Stability for m-Variable Additive Functional Equations

325

Proof. By the same reasoning as in the proof of Theorem 7.26, we can find the mapping H W A ! B given by f .mn x/ n!1 mn

H.x/ D lim

(7.71)

for all x 2 A. It follows from (7.53) and (7.71) that kH.Œx; y/  ŒH.x/; H.y/kB D lim

1 kf .m2n Œx; y/  Œf .mn x/; f .mn y/kB jmj2n

 lim

1 .mn x; mn y/ jmj2n

(7.72)

H.Œx; y/ D ŒH.x/; H.y/

(7.73)

n!1

n!1

D0 for all x; y 2 A and so

for all x; y 2 A. Thus H W A ! B is a non-Archimedean Lie C -algebra homomorphism satisfying (7.55). This completes the proof.  Corollary 7.30. Let r > 1,  be nonnegative real numbers and f W A ! B be a mapping such that kD f .x1 ;    ; xm /kB  .kx1 krA C kx2 krA C    C kxm krA /; kf .Œx; y/  Œf .x/; f .y/kB    kxkrA  kykrA

(7.74)

and kf .x /  f .x/ kB    kxkrA for all 2 T1 and x1 ;    ; xm ; x; y 2 A. Then there exists a unique non-Archimedean Lie C -algebra homomorphism H W A ! B such that kf .x/  H.x/kB 

 kxkrA jmj  jmjr

(7.75)

for all x 2 A. Proof. The proof follows from Theorem 7.29 and the method similar to Corollary 7.27. 

326

7 Stability of Functional Equations in Non-Archimedean Banach Algebras

Definition 7.31. Let A be a non-Archimedean Lie C -algebra. A C-linear mapping ı W A ! A is called a non-Archimedean Lie derivation if ı.Œx; y/ D Œı.x/; y C Œx; ı.y/ for all x; y 2 A. Now, we prove the Hyers-Ulam stability of derivations on non-Archimedean Lie C -algebras for the functional equation D f .x1 ;    ; xm / D 0: Theorem 7.32. Let f W A ! A be a mapping for which there exist the functions ' W Am ! Œ0; 1/ and W A2 ! Œ0; 1/ such that (7.49) and (7.51) hold and kf .Œx; y/  Œf .x/; y  Œx; f .y/kA 

.x; y/

(7.76)

for all x; y 2 A. If there exists L < 1 and (7.52) and (7.53) hold, then there exists a unique non-Archimedean Lie derivation ı W A ! A such that such that (7.55) holds. Proof. It is straight forward to show, there exists a unique C-linear mapping ı W A ! A satisfying (7.55) and the mapping ı W A ! A is given by f .mn x/ n!1 mn

ı.x/ D lim

(7.77)

for all x 2 A. It follows from (7.53) and (7.75) that kı.Œx; y/  Œı.x/; y  Œx; ı.y/kA D lim

1 kf .m2n Œx; y/  Œf .mn x/; mn y  Œmn x; f .mn y/kA jmj2n

 lim

1 .mn x; mn y/ jmj2n

n!1

n!1

(7.78)

D0 for all x; y 2 A and so ı.Œx; y/ D Œı.x/; y C Œx; ı.y/ for all x; y 2 A. Thus ı W A ! A is a non-Archimedean Lie derivation satisfying (7.55). This completes the proof. 

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Index

Symbols .˛; ˇ; /-derivation, 124 .˛; ˇ; /-derivations, 52 -derivation, 11 -homomorphisms, 53 CQ -algebra, 16 C -algebra, 10 C -ternary 3-derivation, 207 C -ternary 3-homomorphism, 204 C -ternary algebra, 12 C -ternary algebra homomorphism, 12 C -ternary derivations, 216 C -ternary homomorphisms, 215 JB -algebra, 152 JB -triple, 13 JB -triple derivation, 13 JB -triple derivations, 218 JB -triple homomorphism, 13 JB -triple homomorphisms, 217 JB -triples, 220 JC -algebra, 64, 69, 152 JC -algebra homomorphism, 69 JC -algebras, 53, 151 J  -triple, 13 -additive, 5 p-Banach algebra, 10, 36 p-Banach space, 10 p-adic, 14 p-norm, 10

A additive function, 1 additive functional equation, 51 additive functional inequality, 171, 178

anticommutator product, 64, 152 Aoki–Rolewicz theorem, 10 Apollonius type additive functional equation, 51, 63, 202, 209 Apollonius type additive mapping, 51, 63, 210 Apollonius’ identity, 63

B Banach algebra, 9 Banach contraction principle, 7 Banach fixed point theorem, 7 bi- -derivation, 219 bi- -derivations, 202, 219 biadditive function, 1

C Cauchy difference, 3 Cauchy functional equation, 1 Cauchy-Jensen functional equations, 20, 38 Cauchy-Rassias inequality, 53, 57, 60 complete non-Archimedean normed space, 14 complex normed algebra, 9

D derivation, 9–11 derivations on Banach algebras, 23 direct method, 3, 20

F fixed point method, 303

© Springer International Publishing Switzerland 2015 Y.J. Cho et al., Stability of Functional Equations in Banach Algebras, DOI 10.1007/978-3-319-18708-2

341

342 G generalized additive mapping, 85, 87, 91 generalized complete metric space, 8 generalized derivation, 36, 44 generalized Leibniz rule, 36 generalized metric, 8, 25, 143

H homomorphism, 9, 10, 26 Hyers-Ulam stability, 19, 24, 29, 31, 32, 36, 38, 44, 51–53, 85, 104, 110, 124, 165, 203 Hyers-Ulam stability of homomorphisms, 24 Hyers-Ulam-Rassias stability, 6

I involution, 10 isomorphisms, 20

J Jensen type functional equation, 51, 70 Jensen’s equation, 54 Jordan -derivation, 52, 110, 112, 114 Jordan algebra, 64, 152 Jordan derivation, 69 Jordan triple product, 152 Jordan triple system, 12 Jordan–von Neumann, 64 Jordan–von Neumann functional inequality, 63 Jordan-von Neumann functional equation, 192

L Lie C -algebra, 11, 64, 67 Lie C -algebra homomorphism, 11, 67 Lie C -algebras, 51 Lie derivation, 11, 68

M maximum multi-norm, 15 minimum multi-norm, 15 modulus of concavity, 9 multi-C ternary algebras, 229, 241 Multi-C -algebra, 16 multi-C -ternary algebra, 17 multi-CQ -algebra, 17 multi-Banach algebra, 16 multi-Banach space, 15 multi-convergent, 15 multi-norm, 14

Index multi-normed algebra, 16 multi-normed space, 15

N non-Archimedean C -algebras, 303 non-Archimedean Banach algebra, 14 non-Archimedean field, 13 non-Archimedean Lie C -algebra, 316 non-Archimedean Lie C -algebra homomorphism, 316 non-Archimedean Lie derivation, 318, 326 non-Archimedean norm, 13 non-Archimedean normed space, 13 non-Archimedean normed spaces, 303 normed A-module, 185 normed modules, 186

O observables, 152

P positive, 143 positive-additive functional equation, 144 positive-additive mapping, 144 proper CQ -algebra homomorphism, 11 proper CQ -algebras, 11, 16, 166 proper multi-CQ -algebras, 230, 264

Q quadratic functional equation of Apollonius type, 63, 210 quadratic Jordan -derivations, 115 quadratic mapping of Apollonius type, 63 quasi--algebra, 16 quasi-Banach algebra, 10, 35 quasi-Banach algebras, 19, 31 quasi-Banach space, 9 quasi-norm, 9 quasi-normed algebra, 10 quasi-normed space, 9

S square root functional equation, 52, 130 square root mapping, 131 stable, 3, 7 superstability, 52, 110, 124, 202, 209 superstable, 7

Index T theorem of the alternative, 8 topological, 16 trivial valuation, 13

343 U ultrametric, 13 unital C -algebra, 12, 51, 53 unital Banach algebra, 9