Theory of Approximate Functional Equations: In Banach Algebras, Inner Product Spaces and Amenable Groups

By Madjid Eshaghi Gordji and Sadegh Abbaszadeh

Presently no other book deals with the stability problem of functional equations in Banach algebras, inner product spaces and amenable groups. Moreover, in most stability theorems for functional equations, the completeness of the target space of the unknown functions contained in the equation is assumed. Recently, the question, whether the stability of a functional equation implies this completeness, has been investigated by several authors.

In this book the authors investigate these developments in the theory of approximate functional equations.

• A useful text for graduate seminars and of interest to a wide audience including mathematicians and applied researchers
• Presents recent developments in the theory of approximate functional equations
• Discusses the stability problem of functional equations in Banach algebras, inner product spaces and amenable groups

• Language: English
• Publisher: Academic Press
• Release date: Mar 3, 2016
• ISBN: 9780128039717

Madjid Eshaghi Gordji

Full Professor of Mathematics, Editor in chief of “International Journal of Nonlinear Analysis and Applications” Editor in chief of “Asian Journal of Scientific Research”

Book preview

Theory of Approximate Functional Equations

in Banach algebras, inner product spaces and amenable groups

First Edition

Madjid Eshaghi Gordji

Institute for Cognitive Science Studies, Shahid Beheshti University, Tehran, Iran

Sadegh Abbaszadeh

Intelligent systems and perception recognition laboratory, CS group of Mathematics department, Shahid Beheshti University, Tehran, Iran

Table of Contents

Cover image

Title page

Copyright

1: Introduction

Abstract

2: Approximate Cauchy functional equations and completeness

Abstract

2.1 Theorem of Hyers

2.2 Theorem of Themistocles M. Rassias

2.3 Completeness of normed spaces

3: Stability of mixed type functional equations

Abstract

3.1 Binary mixtures of functional equations

3.2 Ternary mixtures of functional equations

3.3 Mixed foursome of functional equations

4: Stability of functional equations in Banach algebras

Abstract

4.1 Approximate homomorphisms and derivations in ordinary Banach algebras

4.2 Approximate homomorphisms and derivations in C*-algebras

4.3 Stability problem on C*-ternary algebras

4.4 General solutions of some functional equations

4.5 Some open problems

5: Stability of functional equations in inner product spaces

Abstract

5.1 Introduction

5.2 Orthogonal derivations in orthogonality Banach algebras

5.3 Some open problems

6: Amenability of groups (semigroups) and the stability of functional equations

Abstract

6.1 Introduction

6.2 The stability of homomorphisms and amenability

6.3 Some open problems

Bibliography

Index

Copyright

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ISBN: 978-0-12-803920-5

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1

Introduction

Abstract

In this chapter, the general idea of approximation theory is presented, and the scope of the book is defined. By giving some basic theorems and definitions, the conventional notation for the whole book is presented.

Keywords

Hyers-Ulam stability; Hyperstable equation; Completeness; Mixed type functional equations; Inner product spaces; Amenability of groups.

In 1940, an interesting talk presented by Stanislaw M. Ulam triggered the study of stability problems for various functional equations. In his talk, Ulam discussed the notion of the stability of mathematical theorems considered from a rather general point of view: When is it true that by changing a little the hypotheses of a theorem, one can still assert that the thesis of the theorem remains true or approximately true? Particularly, for every general functional equation 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? Indeed, Ulam proposed the following problem:

Given a group G1, a metric group G2 with the metric d(⋅,⋅) and a positive number ε, does there exist δ > 0 such that, if a mapping f : G1 → G2 satisfies

for all x,y ∈ G1, then a homomorphism h : G1 → G2 exists with

for all x ∈ G1? If the problem accepts a solution, the equation is said to be stable (see [90]).

In trying to solve problems of this kind, most authors have considered homomorphisms between different groups or vector spaces, rather than automorphisms, for perturbations.

In the following year, 1941, Donald H. Hyers [6] was able to give a partial solution to Ulam’s question that was the first significant breakthrough and step toward more solutions in this area. He considered the case of approximately additive mappings between Banach spaces and proved the following results.

Theorem 1.0.1

Let E1 and E2 be Banach spaces and let f : E1→E≥ 0 such that

for all x,y ∈ Eexists for all x ∈ Eis a unique linear transformation such that

Theorem 1.0.2

If under the hypotheses of Theorem 1.0.1f(txfor each x ∈ E1, then the mapping h -linear.

The term Hyers-Ulam stability originates from this historical background. The method that was provided by Hyers is called the direct method, and produces the additive mapping h. The direct method is the most important and powerful tool for studying the stability of various functional equations.

The theorem of Hyers was generalized by Aoki [1] for additive mappings by considering an unbounded Cauchy difference.

Theorem 1.0.3

(Aoki [1]) If f(x) is an approximately linear transformation from E into E′, ie,

where K0 ≥ 0 and 0 ≤ p < 1, then there is a unique linear transformation Q(x) near f(x), ie, there exists K ≥ 0 such that

In 1978, Themistocles M. Rassias [8] succeeded in extending the result of Hyers’s theorem by weakening the condition for the Cauchy difference.

Theorem 1.0.4

Let E1 and E2 be two Banach spaces and let f : E1→E2 be a mapping such that f(txfor each fixed x. Assume that there exists ε > 0 and 0 ≤ p < 1 such that

such that

The stability phenomenon that was presented by Th.M. Rassias is called the generalized Hyers-Ulam stability. This terminology may also be applied to the cases of other functional equations. Since then, a large number of papers have been published in connection with various generalizations of Ulam’s problem and Hyers’s theorem.

There are cases in which each approximate mapping is actually a true mapping. In such cases, we say that the functional equation is hyperstable. Indeed, a functional equation is hyperstable if every solution satisfying the equation approximately is an exact solution of it. For the history and various aspects of this theory, we refer the reader to [2–5].

The concept of Hyers-Ulam stability is quite significant in realistic problems in numerical analysis, biology, and economics. This concept actually means that if one is studying a Hyers-Ulam stable system, then one does not have to reach the exact solution (which usually is quite difficult or time consuming). Hyers-Ulam stability guarantees that there is a close exact solution. This is quite useful in many applications, eg, numerical analysis, optimization, biology, and economics, etc., where finding the exact solution is quite difficult. It also helps, if the stochastic effects are small, to use a deterministic model to approximate a stochastic one.

The notion of stability arose naturally in problems of mechanics. There it involves, mathematically speaking, the continuity of the solution of a problem in its dependence on initials parameters. This continuity may be defined in various ways. Often it is sufficient to prove the boundedness of the solutions for arbitrarily long times, eg, the boundedness of the distance between the point representing the system at any time for the initial point, etc. Needless to say, problems of stability occur in other branches of physics and, in a way, also even in pure mathematics.

Unfortunately, there are no books completely dealing with the stability problem of functional equations in Banach algebras, inner product spaces, and amenable groups. Moreover, in most stability theorems for functional equations, the completeness of the target space of the unknown functions contained in the equation is assumed. Recently, the question of whether the stability of a functional equation implies this completeness has been investigated by several authors. In this book, we are going to deal with the above-mentioned developments in the theory of approximate functional equations.

This book introduces the latest and new results on the following topics:

(1) approximate Cauchy functional equations and completeness;

(2) stability of binary, ternary, and foursome mixtures of functional equations;

(3) approximations of homomorphisms and derivations in different Banach algebras;

(4) stability of functional equations in inner product spaces; and

(5) amenability of groups (semigroups) and the stability of functional equations.

References

[1] Aoki T. On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950;2:64–66.

[2] Baker J. The stability of the cosine equation. Proc. Am. Math. Soc. 1980;80:411–416.

[3] Baker J., Lawrence J., Zorzitto