Functional Equations, Inequalities and Applications

219 Pages·2003·5.86 MB·English

by  Jae-Hyeong Bae

Functional Equations, Inequalities and Applications
Functional Equations,
Inequalities and
Applications
Edited 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 978-90-481-6406-6 ISBN 978-94-017-0225-6 (eBook)
DOI 10.1007/978-94-017-0225-6
Printed on acid-free paper
All Rights Reserved
© 2003 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2003
Softcover reprint ofthe hardcover Ist edition 2003
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilrning, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
Contents
Preface vii
1
Hyers-Ulam stability of a quadratic functional equation in Banach 1
modules
Jae-Hyeong Bae and Won-Gil Park
2
Cauchy and Pexider operators in some function spaces 11
Stefan Czerwik and K rzysztof Dlutek
3
The median principle for inequalities and applications 21
Sever S. Dragomir
4
On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic 39
equation II
Kil-Woung Jun and Yang-Hi Lee
5
On the Hyers-Ulam-Rassias stability of a functional equation 67
Soon-Mo lung
6
A pair of functional inequalities of iterative type related to a Cauchy 73
functional equation
Dorota Krassowska and Janusz Matkowski
7
On approximate algebra homomorphisms 91
Chun-Gil Park
8
Hadamard and Dragomir-Agarwal inequalities, the Euler formulae 105
and convex functions
Josip Pecaric and Ana Vukelic
v
vi Functional Equations, Inequalities and Applications
9
On Ulam stability in the geometry of PDE's 139
Agostino Prastaro and Themistocles M. Rassias
10
On certain functional equations and mean value theorems 149
Themistocles M. Rassias and Young-Ho Kim
11
Some general approximation error and convergence rate estimates 159
in statistical learning theory
Saburou Saitoh
12
Functional equations on hypergroups 167
Laszlo Szekelyhidi
13
The generalized Cauchy functional equation 183
Abraham A. Ungar
14
On the Aleksandrov-Rassias problem for isometric mappings 191
Shuhuang Xiang
Index 223
Preface
Functional equations, inequalities and applications provides an extensive
study of several important equations and inequalities useful in a number of
problems in mathematical analysis. Subjects dealt with include: The general
ized Cauchy functional equation, the Ulam stability theory in geometric partial
differential equations, stability of a quadratic functional equation in Banach
modules, functional equations and mean value theorems, isometric mappings,
functional inequalities of iterative type related to a Cauchy functional equa
tion, stability of a Pexiderized quadratic equation, functional equations on hy
pergroups, some general approximation error and convergence rate estimates
in statistical learning theory, the median principle for inequalities and applica
tions, Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and
convex functions, approximate algebra homomorphisms, stability of the ho
mogeneous functions and applications.
In addition to these, applications to certain problems in pure and applied
mathematics are included.
I wish to express my appreciation to the distinguished mathematicians who
contributed to this volume. We also wish to acknowledge the superb assistance
provided by the staff of Kluwer Academic Publishers.
I would also like to take this opportunity to thank my High School Professor
Elias Karakitsos, who inspired in me the love for mathematics early in life.
Themistocles M. Rassias
Professor of Mathematics
National Technical University ofA thens,
May2003
vii
Chapter 1
HYERS-ULAM STABILITY OF A
QUADRATIC FUNCTIONAL EQUATION
IN BANACH MODULES
Jae-Hyeong Bae
Department of Mathematics, Chungnam National University
Taejon 305-764, Korea
j hbae@math .en u.ac. kr
Won-Gil Park
Department of Mathematics, Chungnam National University
Taejon 305-764, Korea
[email protected]
Abstract We extend the Hyers-Ulam-Rassias stability of a quadratic functional equation
f(x + y + z) + f(x- y) + f(y- z) + f(x- z) = 3f(x) + 3f(y) + 3f(z)
to Banach modules over a Banach algebra.
2000 MSC: 39B72, 39B32 (Primary).
1. Introduction
In 1940, S.M. Ulam gave a wide ranging talk before the Mathematics Club
of the University of Wisconsin in which he discussed a number of important
unsolved problems ([16]). Among those was the question concerning the sta
bility of homomorphisms:
Let G 1 be a group and let G2 be a metric group with metric d( ·, · ).
Given c: > 0, does there exist a 8 > 0 such that if a function
h: G1 ---> G2 satisfies the inequality d(h(xy), h(x)h(y)) < 8 for
all x, y E G1 then there is a homomorphism H: G1 ---> G2 with
d(h(x),H(x)) < c:forall x E G1?
1
Th.M. Rassias (ed.), Functional Equations, Inequalities and Applications, 1-/0.
© 2003 Kluwer Academic Publishers.
2 Functional Equations, Inequalities and Applications
The case of approximately additive mappings was solved by D.H. Hyers
[7] under the assumption that G1 and G2 are Banach spaces. In 1978, Th.M.
Rassias [10] gave a generalization of the Hyers's result. Recently, Gavruta [6]
also obtained a further generalization of the Hyers-Ulam-Rassias theorem.
The quadratic functional equation
f(x + y) + f(x- y)- 2f(x)- 2f(y) = 0 (A)
clearly has f (x ) = cx2 as a solution with c an arbitrary constant when f is a
real function of a real variable. We define any solution of (A) to be a quadratic
function. A Hyers-Ulam stability theorem for the equation (A) was proved
by F. Skof for functions f : V --+ X where V is a normed space and X a
Banach space ([15]). In the paper [5], S. Czerwik proved the Hyers-Ulam
Rassias stability of the quadratic functional equation (A) and this result was
generalized by a number of mathematicians ([3], [4], [8], [9], [11], [12], [13],
[14]).
Consider the following functional equation:
f(x + y + z) + f(x- y) + f(y- z) + f(x- z)
= 3f(x) + 3f(y) + 3f(z). (B)
Recently, the first author investigated the generalized Hyers-Ulam stability
problem of the equation (B) and proved the Hyers-Ulam stability of the equa
tion on restricted domains ([1], [2]). In sections 2 and 3 of this paper, we extend
the generalized Hyers-Ulam stability of the quadratic functional equation (B)
to Banach modules over a Banach algebra, and prove the stability.
2. The Results on B-Quadratic Mappings
In this paper, let B be a unital Banach algebra with norm 1·1, and B1 = {a E
B : lal = 1}, let BB1 and BB2 be left Banach B-modules with norms II· II
and 11·11, respectively, and let <p: BB1 \ {0} x BB1 \ {0} x BB1 \ {0}--+ [0, oo)
be a function satisfying
f
(p(x, y, z) := g~+l <p(3mx, 3my, 3mz) < oo (i)
m=O
for all x, y, z E BB1 \ {0}, or
f,
gmi'(J:+l'
ij!(x,y,z) '= 3.,!'+1' 3,:+1) < oo (ii)
for all x, y, z E Bllll1 \ {0}.
Hyers-Ulam stability of a quadratic functional equation in Banach modules 3
Definition 2.1 A quadratic mapping Q: nlB1 ---t nlB2 is called B-quadratic if
Q(ax) = a2Q(x) for all a E Band all x E nlB1.
Theorem 2.1 Let f: nlB1 ---t nlB2 be a mapping such that
\\a2 f(x + y + z) + a2 f(x- y) + a2 f(y- z) + a2 f(x- z)
- 3f(ax)- 3f(ay)- 3f(az)\\::; cp(x,y,z) (2.1)
for all a E B1 and all x, y, z E nlB1 \ {0}, and let f(O) = Ofor the case (ii). If
f ( tx) is continuous in t E lRfor each fixed x E nlB1, then there exists a unique
B-quadratic mapping Q: nlB1 ---t nlB2 such that
IIQ(x)- f(x) + ~f(O)II::; $(x,x,x) (2.2)
for all x E nlB1 \ {0}. The mapping Q: nlB1 ---t nlB2 is given by
.
{1
f(3nx) if .. (; ·
Q(x) = ~IDn---+oo ngn x i cpsatis1.es(1)
hmn---+oo 9 f ( 3n) if 'P satisfies (ii)
for all x E nlB1.
PROOF: By [2], it follows from the inequality of the statement for a = 1 that
there exists a unique quadratic mapping Q: nlB1 ---t nlB2 satisfying (2.2). The
quadratic mapping Q is similar to the additive mapping T given in the proof
of [ 10 ]. Under the assumption that f (t x) is continuous in t E lR for each fixed
x E nlB1. by the same reasoning as the proof of [10], the quadratic mapping
Q: nlB1 ---t nllll2 satisfies Q(tx) = t2Q(x) for all x E nllll1 and all t E R
Suppose that cp satisfies (i). Replacing x, y and z by 3n-lx in (2.1),
for all a E B1 and all x E nllll1 \ {0}. Putting a= 1 and replacing x, y and z
by x in (2.1),
\\f(3x) + 3f(O)- 9f(x)\\ ::; cp(x, x, x) (2.4)
for all x E nllll1 \ {0}. Replacing x by 3n-1ax in (2.4),
\\f(3nax) + 3f(O)-9J(3n-lax)\\ :<; cp(3n-lax, 3n-lax, 3n-lax) (2.5)
for all a E B1 and all x, y, z E sllll1 \ {0}.
Note that for each a E B and each w E nllll2,
\law\1 ::; K\a! · \lw\1 (2.6)
4 Functional Equations, Inequalities and Applications
for some K > 0. Using (2.3), (2.5) and (2.6),
lla2 f(3nx)- f(3nax)!!
:::; lla2 f(3nx) + 3a2 f(O)- 9f(3n-1ax)!!
+ ll9f(3n-1ax)- f(3nax)- 3a2 f(O)II
:::; ~(3n-1x,3n-1x,3n-1x)
+ l!f(3nax) + 3f(O)- 9f(3n-1ax)!l +II- 3f(O) + 3a2 f(O)II
:::; ~(3n-1x,3n-1x,3n-1x)
+ ~(3n-1ax, 3n-1ax, 3n-1ax) + 3Kia2 - ll · llf(O)II
for all a E B1 and all x E slB\1 \ {0}.
So g-nlla2 f(3nx) - f(3nax)!l --+ 0 as n --+ oo for all a E B1 and all
x E slB\1 \ {0}. By the definition of Q, for each element a E B1,
a2Q(x) = lim a2 f(3nx) = lim f(3nax) = Q(ax)
n-->oo gn n-->oo gn
for all x E slB\1 \ {0}. For a E B1 \ {0},
2
Q(ax) = Q ( !a!a~ x ) = lal 2Q ( ~a x ) = lal 2 ~aQ(x) =a2 Q (x),
for all x E slB\1 \ {0}. So Q is B-quadratic.
By the same reasoning as above, for the case (ii), one can show that the
JR-quadratic mapping Q is B-quadratic, as desired. 0
Notice that, for the case (ii), Q(x) = limn-->oo gn(f(3-nx)- f(O)) in [9].
To get Q(ax) = a2Q(x) for all a E B1 and for all x E slB\1, we need the
assumption f(O) = 0. In this paper, assume that f(O) = 0 for the case (ii).
Corollary 2.1 Let p -=/=- 2, () > 0 be real numbers and f: slB\1 --+ slB\2 a
mapping such that
lla2 f(x + y + z) + a2 f(x- y) + a2 f(y- z) + a2 f(x- z)
- 3f(ax)- 3f(ay)- 3f(az)l! :::; ()(llxiiP + IIYIIP + llziiP)
for all a E B1 and all x, y, z E slB\1 \ {0}. If f(tx) is continuous in t E
lR for each fixed x E slB\1, then there exists a unique B-quadratic mapping
Q: slB\1 --+ slB\2 such that