Functional Equations and Inequalities

334 Pages·2000·14.818 MB·English

by  Themistocles M. Rassias (auth.)

Functional Equations and Inequalities
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 518
Functional Equations
and Inequalities
by
Themistocles M. Rassias
Department of Mathematics,
National Technical University of Athens,
Zografou Campus,
Athens, Greece
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-5869-8 ISBN 978-94-011-4341-7 (eBook)
DOI 10.1007/978-94-011-4341-7
Printed on acid-free paper
All Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
Softcover reprint of the hardcover 1st edition 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
dedicatedtothememoryof
HiroshiHaruki
and
DonaldH. Hyers
inadmiration
Contents
Preface Xl
On the Stability of a Functional Equation for Generalized Trigonometric
Functions
Roman Badora 1
SomeNotesonTwo-ScaleDifferenceEquations
LotharBergandGerlindPlonka .. . .. . .. .. .. .. .. . .. .. . .. .. . . .. .. 7
SomeDemandFunctions ina DuopolyMarketwith Advertising
E. Castillo,J Maria Sarabia andA. Mercedes Gonzalez. .. . ... . . .... 31
Solutionsofa Functional Inequality ina Special ClassofFunctions
MarekCzerni 55
OnDependenceofLipschitzianSolutionsofNonlinearFunctionalInequalityon
an ArbitraryFunction
MarekCzerni 65
TheProblemofExpressibilityin SomeExtensionsofFreeGroups
ValeriIFalziev 77
Ona PythagoreanFunctional EquationInvolving CertainNumberFields
J L. Garcia-RoigandJ Sali/las 91
Ona Conditional CauchyFunctional EquationInvolvingCubes
J L. Garcia-RoigandE. Martin-Gutierrez 97
vii
viii Contents
Hyers-Ulam StabilityofHossm's Equation
P. Gavruta 105
TheFunctional EquationoftheSquareRootSpiral
K. J Heuvers, D. S. MoakandB. Boursaw 111
Onthe SuperstabilityoftheFunctional Equationf(xY)=f(xY
Soon-MoJung 119
ReplicativityandFunction Spaces
Hans-Heinrich Kairies 125
Normal Distributionsandthe Functional Equationf(x+y) g(x-y)
= f(x)frY) g(x) g(-y)
PL. Kannappan 139
OnthePolynomial-LikeIterativeFunctional Equation
JanuszMatkowski and Weinian Zhang . . .. .. . .. .. . ... .. .. .. . ... 145
DistributionofZerosandInequalitiesforZerosofAlgebraicPolynomials
GradimirV Milovanovic and ThemistoclesM Rassias 171
AFunctional DefinitionofTrigonometricFunctions
NicolaeN Neam{U 205
A QualitativeStudyofLobachevksy's ComplexFunctionalEquation
NicolaeN Neam!u 215
SmoothSolutionsofan IterativeFunctionalEquation
Jian-Guo Si, Wei-Nian ZhangandSui-Sun Cheng 221
Set-ValuedQuasiconvexFunctions andtheirConstantSelections
Wilhelmina Smajdor 233
Entire SolutionsoftheHille-typeFunctionalEquation
AndrzejSmajdorand Wilhelmina Smajdor 249
Ulam's Problem,Hyers's Solution- andtoWheretheyLed
LaszlOSzekelyhidi 259
Contents ix
A SeparationLemmafortheConstructionofFinite Sums Decompositions
Wolfgang Tutschke 287
AleksandrovProblemandMappings whichPreserveDistances
Shuhuang)(iang 297
On Some Subclasses ofHarmonicFunctions
S. Yalfin,M OztiirkandM Yamankaradeniz ....................325
Index 333
Preface
Functional Equations andInequalities providesanextensive studyofsomeof
the most important topics of current interest in functional equations and
inequalities. Subjects dealt with include: a Pythagorean functional equation, a
functional definition oftrigonometric functions, the functional equation ofthe
square root spiral, a conditional Cauchy functional equation, an iterative
functional equation, the Hille-type functional equation, the polynomial-like
iterativefunctional equation, distribution ofzeros and inequalities for zerosof
algebraicpolynomials,aqualitativestudyofLobachevsky'scomplexfunctional
equation,functional inequalitiesinspecialclassesoffunctions, replicativityand
function spaces, normal distributions, some difference equations, finite sums
decompositions of functions, harmonic functions, set-valued quasiconvex
functions, the problem of expressibility in some extensions of free groups,
Aleksandrovproblemandmappingswhichpreservedistances, Ulam'sproblem,
stability ofsome functional equation for generalized trigonometric functions,
Hyers-Ulam stability of Hosszil's equation, superstability of a functional
equation, andsomedemandfunctions ina duopoly marketwithadvertising.
Itisapleasuretoexpressmydeepestappreciationtoallthemathematicianswho
contributed to this volume. Finally, we wish to acknowledge the superb
assistanceprovided bythe staffofKluwerAcademicPublishers.
June 2000 Themistocles M. Rassias
xi
ON THE STABILITY OF A FUNCTIONAL EQUATION FOR
GENERALIZED TRIGONOMETRIC FUNCTIONS
ROMAN BADORA
lnstytutMatematyki, Uniwersytet Sli;ski, ul. Bankowa 14, PL-40-007Katowice,
Poland, e-mail: [email protected]
Abstract. In the present paper the stability result concerning a functional equation
for generalized trigonometric functions is presented.
Z. Gajdain [3] presented the following result:
Let (X,+) be an abelian group, let G be a finite group of authomorphisms of
X and let IG I be the order ofG. Then a function f : X ~ ([; satisfies the
functional equation
L
(1) f(x +u(y)) =1 G 1f(x)f(y), x, y EX
<7EG
ifand only ifthere exists an exponential function m :X ~ ([; (i.e. m(x+ y) =
m(x)m(y), x, y EX) such that
TG1f L
(2) f(x) = m(u(x)), x E X.
<7EG
Classical examples ofEq.(l) are Cauchy's functional equation:
f(x +y) =f(x)f(y), x, y EX,
while G = {idx}, and the cosine (d'Alembert's) functional equation:
f(x +y) +f(x - y) = 2f(x)f(y), x, Y EX,
=
while G {idx, -idx}.
For the stability properties ofEq.(l) W. Forg-Rob and J. Schwaiger proved in
[2] the following generalization ofBaker's result on the stability ofthe cosine
equation:
T.M.Rassias(ed.),FunctionalEquationsandInequalities, 1-5.
©2000KluwerAcademicPublishers.