Table Of ContentProgress in Nonlinear Differential Equations
and Their Applications
Volume 34
Editor
Haim Brezis
Universite Pierre et Marie Curie
Paris
and
Rutgers University
New Brunswick, N.J.
Editorial Board
Antonio Ambrosetti, Scuola Normale Superiore, Pisa
A. Bahri, Rutgers University, New Brunswick
F. Browder, Rutgers University, New Brunswick
Luis Cafarelli, Institute for Advanced Study, Princeton
L. C. Evans, University of California, Berkeley
Mariano Giaquinta, University of Pisa
David Kinderlehrer, Carnegie-Mellon University, Pittsburgh
S. Klainerman, Princeton University
Robert Kohn, New York University
P. L. Lions, University of Paris IX
Jean Mawhin, Universite Catholique de Louvain
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
John Toland, University of Bath
Donald H. Hyers
George Isac
Themistocles M. Rassias
Stability of Functional Equations
in Several Variables
Springer Science+Business Media, LLC
Donald H. Hyers (deceased)
George Isac
Themistoclcs M. Rassias Royal Militar)' College of Canada
National Technical University of Athens Department of Mathematics
Department of Mathematics and Computer Science
Zografou Campus Kingston,Ontano,
15780 Athens, Greece Canada K7K 5LO
Library of Congress Cataioging.in.Publication Data
Hyers, Donald H.
Stability of functional equations in several variables / Donald H.
Hyers, George Isac, Themistocles M. Rassias.
p. cm. -- (Progress in nonlinear differential equations and
their applications ; v. 34)
Includes bibliographical references and index.
ISBN 978-1-4612-7284-7 ISBN 978-1-4612-1790-9 (eBook)
DOI 10.1007/978-1-4612-1790-9
paper)
1. Functional equations--Numerical solutions. 2. Functions of
several real variables. 1. Isac, George. II. Rassias,
Themistocles M., 1951- . III. TiUe. IV. Series.
QA43l.H94 1998
515' .8--dc21 98-2593
CIP
AMS Subject Classification: 39-01. 39872
Printed on acid-frec paper
Birkhăuser ~
© 1998 Springer Science+Business Media New York
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Contents
Preface ................................................................ vii
Prologue ................................................................. 1
Chapter 1. Introduction ............................................... 11
Chapter 2. Approximately Additive and Approximately
Linear Mappings ........................................... 15
Chapter 3. Stability of the Quadratic Functional Equation ............. 45
Chapter 4. Generalizations. The Method of Invariant Means ........... 78
Chapter 5. Approximately Multiplicative Mappings. Superstability .... 102
Chapter 6. The Stability of Functional Equations for
Trigonometric and Similar Functions ...................... 132
Chapter 7. Functions with Bounded nth Differences ................... 155
Chapter 8. Approximately Convex Functions ......................... 166
Chapter 9. Stability of the Generalized Orthogonality
Functional Equation ...................................... 180
Chapter 10. Stability and Set-Valued Functions ........................ 204
Chapter 11. Stability of Stationary and Minimum Points .............. 232
Chapter 12. Functional Congruences .................................. 246
Chapter 13. Quasi-Additive Functions and Related Topics ............. 277
References ........................................................... 290
Index ................................................................. 307
Preface
The notion of stability of functional equations of several variables in
the sense used here had its origins more than half a century ago when
S. Ulam posed the fundamental problem and Donald H. Hyers gave the first
significant partial solution in 1941. The subject has been revised and de
veloped by an increasing number of mathematicians, particularly during the
last two decades. Three survey articles have been written on the subject by
D.H. Hyers (1983), D.H. Hyers and Th. M. Rassias (1992), and most recently
by G.L. Forti (1995). None of these works included proofs of the results which
were discussed. Furthermore, it should be mentioned that wider interest in
this subject area has increased substantially over the last years, yet the pre
sentation of research has been confined mainly to journal articles.
The time seems ripe for a comprehensive introduction to this subject,
which is the purpose of the present work. This book is the first to cover the
classical results along with current research in the subject. An attempt has
been made to present the material in an integrated and self-contained fashion.
In addition to the main topic of the stability of certain functional equa
tions, some other related problems are discussed, including the stability of the
convex functional inequality and the stability of minimum points.
A sad note. During the final stages of the manuscript our beloved co
author and friend Professor Donald H. Hyers passed away. We express here
our grief and our sorrow for the fact that he was not able to see this completed
work. This book is dedicated to his memory.
We wish to express our thanks to Stefan Czerwik, P. Gavruta, Soon-Mo
Jung, Jiirg Ratz, Wilhelmina Smajdor, J6zef Tabor and Jacek Tabor for their
useful suggestions and comments. Thanks also to Halm Brezis for his help
and encouragement.
Finally, it is a pleasure to acknowledge the superb assistance provided
by Ann Kostant of Birkhauser Boston in the publication of this book, as well
as the support by L.L.C.F. The author would also like to thank Elizabeth
Loew of 'IEXniques, Inc. and Tom Grasso of Birkhauser Boston for their aid
throughout the typesetting and production process.
The authors
October, 1997
Prologue
To quote S.M. Ulam (1960) (page 63)1 for very general functional equa
tions, one can ask the following question. When is it true that the solution of
an equation differing slightly from a given one, must of necessity be close to
the solution of the given equation? Similarly, if we replace a given functional
equation by a functional inequality, when can one assert that the solutions of
the inequality lie near the solutions of the strict equation? The present work
is designed to provide an introduction to the formulation and solution of such
problems by examination of various researches most of which has been carried
out during the last two decades.
The following problem (Ibid., p. 64) of this kind had been formulated by
Ulam during a talk before a Mathematical Colloquium at the University of
Wisconsin, Madison, in 1940. Given a metric group G(·,p), a number E: > 0
and a mapping f: G ---+ G which satisfies the inequality p(f(x·y), f(x)· f(y)) <
E: for all x, y in G, does there exist an automorphism a of G and a constant
:s:
k > 0, depending only on G, such that p(a(x), f(x)) kE: for all x in G? If
the answer is affirmative, we would call the equation a(x . y) = a(x) . a(y) of
automorphism stable.
In trying to solve problems of this kind, most authors have considered
homomorphisms between different groups or vector spaces, rather than auto
morphisms, for perturbations.
In particular, D.H. Hyers (1941) dealt with E:-additive mappings f:
El ---+ E2 between Banach spaces, i.e. f must satisfy the inequality
IIf(x + y) - f(x) - f(y)11 < E: for all x,y in E1·
It was shown in this case that an additive function a: El ---+ E2 exists
:s:
with Ila(x) - f(x)11 E: (see Chapter 1). Thus, in this special case, k = 1.
Here the Cauchy equation f (x + y) - f (x) - f (y) = 0 is stable. Moreover, the
relation between f and a is given by the formula
a(x) = lim f(2nx)/2n.
n->oo
A method leading to such a formula will be called a direct method.
1 Here and elsewhere in this book an author's name followed by a year number
in parentheses stands for a reference listed at the end of the book.
D. H. Hyers et al., Stability of Functional Equations in Several Variables and Isometries
© Birkhäuser Boston 1998
2 Stability of Functional Equations of Several Variables
The matter rested there until Th.M. Rassias (1978) considered a gener
alized version of the previous result which permitted the Cauchy difference to
become unbounded. That is, he assumed that
Ilf(x + y) - f(x) - f(y) II s (IIxliP + IIyllP)c for x, yin EI, where 0 S p < 1.
By using a direct method, he showed that, in this case too, there is an
additive function a from El into E2 given by the same formula as before such
s
that IIa(x) - f(x)II kcllxllP where k depends on p as well as c (see Chapter
2). This result was later extended to all pi-I and generalized by Z. Gajda,
Th.M. Rassias and P. Semrl, G. Isac and Th.M. Rassias, R. Ger, P. Gavruta,
S.-M. Jung.
Beginning around the year 1980 the topic of approximate homomor
phisms, or the stability of the equation of homomorphism, was taken up by a
number of mathematicians. Some of these studies are presented in Chapters
1, 2, 4 and 5 of this book. In general, these authors used a whole vector space,
group or semigroup for the domain of the approximately additive mapping.
However, F. Skof (1983) studied the stability problem for o-additive mappings
defined on restricted domains in the plane (see Theorem 2.5).
It was soon apparent (see Chapter 1) that there was no difficulty in re
placing the vector space El by a commutative semigroup in the theorem of
Hyers alluded to above. On the other hand, an example by G.L. Forti (1985),
included here in Chapter 4, shows that not all non-commutative groups or
semigroups can serve as domains of homomorphisms with stable equations.
If not commutativity, then some weaker restriction is necessary. In this re
gard, J. Ratz (1980) considered mappings f: X -+ Y, where X is a "power
associative groupoid" , not necessarily commutative, and Y is a sequentially
complete topological vector space (see Chapter 4). Once again a direct method
was used here for the proof of the stability theorem.
Another significant development was to replace commutativity of the
semigroup S by the left or right amenability of S in dealing with mappings
from S into the real or complex numbers. Thus, for the stability problems
for such mappings, we have a new method at our disposal, the method of
invariant means, and we are no longer confined to only commutative do
mains. This new method was first used for stability problems by L. Szekelyhidi
(1982a). Later, the last named author applied the method to the stability of
the Cauchy equation for functions defined on groups (1985) and G.L. Forti
did the same for semigroups (1987) (see Theorem 4.17). Moreover, by the
"bootstrap" theorems of L. Szekelyhidi, G.L. Forti and Z. Gajda, the results
Prologue 3
were extended to mappings from a left (or right) amenable semigroup into a
locally convex Hausdorff topological space which is sequentially complete (see
Theorem 4.19).
A critique concerning various definitions from the literature was made by
z. Moszner (1985), and this is presented at the end of Chapter 4.
There are cases in which each approximate homomorphism is actually a
true homomorphism. In such cases, we call the equation of homomorphism
"superstable". A study of approximately multiplicative functions from a
vector space V into the real numbers was made by J.A. Baker, J. Lawrence
and F. Zoritto (1979). They proved that for a given 8 > 0 and a function
f: V -- R such that If(x+y) - f(x)f(y)1 < 8 for all x, y in V, then either f(x)
+
remains bounded, with a bound depending on 8, or else f(x y) = f(x)f(y)
for all x, yin V. In this case, we might say that this equation is superstable
modulo the bounded functions on V. This result was generalized with a
simplified proof by J.A. Baker (1980), which is presented below in Chapter 5.
This is followed by another generalization of the same result by L. Szekelyhidi
(1982), who builds on the idea of superstability modulo the bounded functions.
Next we look at a paper of J. Lawrence (1985) in which he considered
mappings f from a semi group S into a real or complex normed algebra A that
satisfy the inequality Ilf(xy) - f(x)f(y)11 ~ 8 for a given 8 > o. Among other
results, he obtained a stability theorem for the equation of homomorphism
h(xy) = h(x)h(y) in the case when j, h: S -t M2(C), where the semigroup S
is commutative and M2 (C) denotes the normed algebra of 2 x 2 matrices with
complex entries.
Most of the rest of Chapter 5 is devoted to the theory of approximately
multiplicative linear functionals defined on a commutative complex Banach
algebra A as given by B.E. Johnson (1986). He defines a linear functional ep on
A to be a "8-multiplicative" when the norm of the bilinear functional ep( ab)
ep(a)ep(b) does not exceed 8 > o. Denote by M the class of all multiplicative
linear functionals on A. For each ep in A* put deep) = inf{llep - ~II: ~ EM}.
The stability question here is, given f > 0 does there exist a 8 > 0 such that
deep) < f, when ep is a 8-multiplicative element of A*? The answer turns out
to be in the affirmative for many of the standard examples of commutative
complex Banach algebras.
The definition of approximately multiplicative mappings as used by the
authors cited at the beginning of Chapter 5 has been criticized by R. Ger in a
talk at the 26th International Symposium on Functional Equations, entitled
4 Stability of Functional Equations of Several Variables
"Superstability is not Natural". This is explained, along with a modification
of Ger's definition by J. Tabor, at the end of Chapter 5 under "Comments".
In Chapter 3, we deal with stability of the quadratic functional equation
and related topics. Digressing from approximate homomorphisms, we first
consider stability questions for the quadratic functional equation
q(x + y) + q(x - y) - 2q(x) - 2q(y) = O.
It turns out that direct methods work well for all the stability results in this
chapter. F. Skof (1983b) proved a stability theorem for this equation for
functions f: X -> E, where X is a vector space and E is a Banach space.
Here f was assumed to be 8-quadratic, in the sense that
Ilf(x + y) + f(x - y) - 2f(x) - 2f(y) II < 8
for x, y in X and some fixed 8 > O. The direct method used here resulted in
the formula q(x) = limn-+co 4-n f(2nx) for the unique solution of the quadratic
functional equation which satisfies the inequality Ilf(x) - q(x) II :::; 8/2. Details
are given in the work of S. Czerwik (1992a), who, pursuing the idea of Th.M.
Rassias (1978) in the case of approximately additive mappings, replaced the
8 in the above definition by the quantity c(lIxllP + lIylip) , where p is real and
different from 2, and obtained stability theorems in this case as well.
The second topic in Chapter 3 is the work of F. Skof and S. Terracini
(1987) on the problem of stability of the quadratic functional equations for
functions defined on restricted real domains with values in a Banach space.
The third topic in Chapter 3 involves functionals which are approximately
additive on A-orthogonal pairs. Here A is a bounded selfadjoint linear oper
ator on a Hilbert space X whose range AX has dimension greater than 2.
The inner product is denoted by (" .). A functional 1jJ on X with complex
values is called additive on A-orthogonal pairs if for x, y in X such that
(Ax, y) = 0 we have 1jJ(x+y) = 1jJ(x)+1jJ(y). One might ask, what is this state
ment concerning additivity doing in this chapter on the quadratic functional
equation? An answer is given by a representation theorem of F. Vajzovic
(1966), which tells us that if 1jJ is continuous and additive on A-orthogonal
pairs, then there exists a unique scalar f3 and unique vectors u, v in X such
that 1jJ(x) = (x, u) + (v, x) + f3(Ax, x). Thus, we see that the odd part of 1jJ is
additive while the even part is quadratic.
H. Drljevic and Z. Mavar (1982) defined a functional cp on X to be ap
e
proximately additive on A-orthogonal pairs if there exist constants > 0 and
