Introduction to Functional Equations

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<ul><li><p>Introduction to </p><p>Functional Equations</p><p>K11911_FM.indd 1 12/21/10 11:21 AM</p></li><li><p>Introduction to </p><p>Functional Equations</p><p>Prasanna K. Sahoo </p><p>Palaniappan Kannappan</p><p>K11911_FM.indd 3 12/21/10 11:21 AM</p></li><li><p>Chapman &amp; Hall/CRCTaylor &amp; Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742</p><p> 2011 by Taylor and Francis Group, LLCChapman &amp; Hall/CRC is an imprint of Taylor &amp; Francis Group, an Informa business</p><p>No claim to original U.S. Government works</p><p>Printed in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1</p><p>International Standard Book Number: 978-1-4398-4111-2 (Hardback)</p><p>This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.</p><p>Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.</p><p>For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.</p><p>Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.</p><p>Library of Congress CataloginginPublication Data</p><p>Sahoo, Prasanna,Introduction to functional equations / Prasanna K. Sahoo, Palaniappan Kannappan.</p><p>p. cm.Includes bibliographical references and index.ISBN 978-1-4398-4111-2 (hardback)1. Functional equations. I. Kannappan, Pl. (Palaniappan) II. Title.</p><p>QA431.S15 2011515.75--dc22 2010045164</p><p>Visit the Taylor &amp; Francis Web site athttp://www.taylorandfrancis.com</p><p>and the CRC Press Web site athttp://www.crcpress.com </p><p>K11911_FM.indd 4 12/21/10 11:21 AM</p><p>www.copyright.comwww.copyright.com</p></li><li><p>Dedication</p><p>Dedicated by</p><p>Prasanna Sahooto his wife Sadhna and son Amit</p><p>and</p><p>Palaniappan Kannappanto his wife Renganayaki and his grandchildren</p></li><li><p>Contents</p><p>Preface xiii</p><p>1 Additive Cauchy Functional Equation 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Functional Equations . . . . . . . . . . . . . . . . . . . 21.3 Solution of Additive Cauchy Functional Equation . . . 31.4 Discontinuous Solution of Additive Cauchy Equation . 91.5 Other Criteria for Linearity . . . . . . . . . . . . . . . . 141.6 Additive Functions on the Complex Plane . . . . . . . 161.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . 191.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 21</p><p>2 Remaining Cauchy Functional Equations 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Solution of the Exponential Cauchy Equation . . . . . . 252.3 Solution of the Logarithmic Cauchy Equation . . . . . 282.4 Solution of the Multiplicative Cauchy Equation . . . . 302.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 342.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 36</p><p>3 Cauchy Equations in Several Variables 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Additive Cauchy Equations in Several Variables . . . . 393.3 Multiplicative Cauchy Equations in Several Variables . 433.4 Other Two Cauchy Equations in Several Variables . . . 443.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 453.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 46</p><p>4 Extension of the Cauchy Functional Equations 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Extension of Additive Functions . . . . . . . . . . . . . 494.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . 554.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 58</p><p>vii</p></li><li><p>viii Contents</p><p>5 Applications of Cauchy Functional Equations 615.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Area of Rectangles . . . . . . . . . . . . . . . . . . . . . 625.3 Definition of Logarithm . . . . . . . . . . . . . . . . . . 645.4 Simple and Compound Interest . . . . . . . . . . . . . . 655.5 Radioactive Disintegration . . . . . . . . . . . . . . . . 675.6 Characterization of Geometric Distribution . . . . . . . 685.7 Characterization of Discrete Normal Distribution . . . 715.8 Characterization of Normal Distribution . . . . . . . . 745.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . 76</p><p>6 More Applications of Functional Equations 796.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Sum of Powers of Integers . . . . . . . . . . . . . . . . . 79</p><p>6.2.1 Sum of the first n natural numbers . . . . . . . . 806.2.2 Sum of square of the first n natural numbers . . 816.2.3 Sum of kth power of the first n natural numbers 81</p><p>6.3 Sum of Powers of Numbers on Arithmetic Progression . 846.4 Number of Possible Pairs Among n Things . . . . . . . 866.5 Cardinality of a Power Set . . . . . . . . . . . . . . . . 886.6 Sum of Some Finite Series . . . . . . . . . . . . . . . . 886.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . 90</p><p>7 The Jensen Functional Equation 937.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 937.2 Convex Function . . . . . . . . . . . . . . . . . . . . . . 937.3 The Jensen Functional Equation . . . . . . . . . . . . . 957.4 A Related Functional Equation . . . . . . . . . . . . . . 997.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 1017.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 103</p><p>8 Pexiders Functional Equations 1078.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1078.2 Pexiders Equations . . . . . . . . . . . . . . . . . . . . 1078.3 Pexiderization of the Jensen Functional Equation . . . 1118.4 A Related Equation . . . . . . . . . . . . . . . . . . . . 1128.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 1158.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 117</p><p>9 Quadratic Functional Equation 1199.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1199.2 Biadditive Functions . . . . . . . . . . . . . . . . . . . . 1199.3 Continuous Solution of Quadratic Functional Equation 1239.4 A Representation of Quadratic Functions . . . . . . . . 126</p></li><li><p>Contents ix</p><p>9.5 Pexiderization of Quadratic Equation . . . . . . . . . . 1299.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . 1349.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 139</p><p>10 dAlembert Functional Equation 14310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 14310.2 Continuous Solution of dAlembert Equation . . . . . . 14310.3 General Solution of dAlembert Equation . . . . . . . . 14910.4 A Charcterization of Cosine Functions . . . . . . . . . . 15710.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 15910.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 163</p><p>11 Trigonometric Functional Equations 16511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 16511.2 Solution of a Cosine-Sine Functional Equation . . . . . 16611.3 Solution of a Sine-Cosine Functional Equation . . . . . 17011.4 Solution of a Sine Functional Equation . . . . . . . . . 17311.5 Solution of a Sine Functional Inequality . . . . . . . . . 18311.6 An Elementary Functional Equation . . . . . . . . . . . 18411.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . 18711.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 194</p><p>12 Pompeiu Functional Equation 19712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 19712.2 Solution of the Pompeiu Functional Equation . . . . . . 19712.3 A Generalized Pompeiu Functional Equation . . . . . . 19912.4 Pexiderized Pompeiu Functional Equation . . . . . . . 20212.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 20812.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 209</p><p>13 Hosszu Functional Equation 21113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 21113.2 Hosszu Functional Equation . . . . . . . . . . . . . . . 21113.3 A Generalization of Hosszu Equation . . . . . . . . . . 21413.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 22213.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 224</p><p>14 Davison Functional Equation 22714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 22714.2 Continuous Solution of Davison Functional Equation . 22714.3 General Solution of Davison Functional Equation . . . 23014.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 23114.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 232</p></li><li><p>x Contents</p><p>15 Abel Functional Equation 23515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 23515.2 General Solution of the Abel Functional Equation . . . 23615.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . 23915.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 240</p><p>16 Mean Value Type Functional Equations 24316.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 24316.2 The Mean Value Theorem . . . . . . . . . . . . . . . . 24316.3 A Mean Value Type Functional Equation . . . . . . . . 24516.4 Generalizations of Mean Value Type Equation . . . . . 24716.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 26116.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 266</p><p>17 Functional Equations for Distance Measures 26917.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 26917.2 Solution of Two Functional Equations . . . . . . . . . . 27317.3 Some Auxiliary Results . . . . . . . . . . . . . . . . . . 27817.4 Solution of a Generalized Functional Equation . . . . . 28617.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 28717.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 290</p><p>18 Stability of Additive Cauchy Equation 29318.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 29318.2 Cauchy Sequence and Geometric Series . . . . . . . . . 29418.3 Hyers Theorem . . . . . . . . . . . . . . . . . . . . . . 29518.4 Generalizations of Hyers Theorem . . . . . . . . . . . . 30018.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 30518.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 309</p><p>19 Stability of Exponential Cauchy Equations 31319.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 31319.2 Stability of Exponential Equation . . . . . . . . . . . . 31319.3 Ger Type Stability of Exponential Equation . . . . . . 32019.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 32219.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 326</p><p>20 Stability of dAlembert and Sine Equations 32920.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 32920.2 Stability of dAlembert Equation . . . . . . . . . . . . . 32920.3 Stability of Sine Equation . . . . . . . . . . . . . . . . . 33620.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 34020.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 345</p></li><li><p>Contents xi</p><p>21 Stability of Quadratic Functional Equations 34921.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 34921.2 Stability of the Quadratic Equation . . . . . . . . . . . 34921.3 Stability of Generalized Quadratic Equation . . . . . . 35321.4 Stability of a Functional Equation of Drygas . . . . . . 36021.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . 36921.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 376</p><p>22 Stability of Davison Functional Equation 38122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 38122.2 Stability of Davison Functional Equation . . . . . . . . 38122.3 Generalized Stability of Davison Equation . . . . . . . 38422.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 38722.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 390</p><p>23 Stability of Hosszu Functional Equation 39123.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 39123.2 Stability of Hosszu Functional Equation . . . . . . . . . 39223.3 Stability of Pexiderized Hosszu Functional Equation . . 39423.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . 40123.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 403</p><p>24 Stability of Abel Functional Equation 40524.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 40524.2 Stability Theorem . . . . . . . . . . . . . . . . . . . . . 40524.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . 40924.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 415</p><p>Bibliography 417</p><p>Index 441</p></li><li><p>Preface</p><p>The subject of functional equations forms a modern branch of mathe-matics. The origin of functional equations came about the same time asthe modern definition of function. From 1747 to 1750, J. dAlembertpublished three papers. These three papers were the first on func-tional equations. The first significant growth of the discipline of func-tional equations was stimulated by the problem of the parallelogramlaw of forces (for a history see Aczel (1966)). In 1769, dAlembertreduced this problem to finding solutions of the functional equationf(x + y) + f(x y) = 2f(x)f(y). Many celebrated mathematiciansincluding N.H. Abel, J. Bolyai, A.L. Cauchy, J. dAlembert, L. Eu-ler, M. Frechet, C.F. Gauss, J.L.W.V. Jensen, A.N. Kolmogorov, N.I.Lobacevskii, J.V. Pexider, and S.D. Poisson have studied functionalequations because of their apparent simplicity and harmonic nature.</p><p>Although the modern study of functional equations originated morethan 260 years ago, a significant growth of this discipline occurred dur-ing the last sixty years. In 1900, David Hilbert suggested in connectionwith his fifth problem that, while the theory of differential equations pro-vides elegant and powerful techniques for solving functional equations,the differentiability assumptions are not inherently required. Motivatedby Hilberts suggestion many researchers have treated various functionalequations without any (or with only mild) regularity assumption. Thiseffort has given rise to the modern theory of functional equations. Thecomprehensive books by S. Pincherle (...</p></li></ul>

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