Ulam Stability of Operators

By Janusz Brzdek, Dorian Popa, Ioan Rasa and Bing Xu

Ulam Stability of Operators presents a modern, unified, and systematic approach to the field. Focusing on the stability of functional equations across single variable, difference equations, differential equations, and integral equations, the book collects, compares, unifies, complements, generalizes, and updates key results. Whenever suitable, open problems are stated in corresponding areas. The book is of interest to researchers in operator theory, difference and functional equations and inequalities, differential and integral equations.

• Allows readers to establish expert knowledge without extensive study of other books
• Presents complex math in simple and clear language
• Compares, generalizes and complements key findings
• Provides numerous open problems

• Language: English
• Publisher: Academic Press
• Release date: Jan 10, 2018
• ISBN: 9780128098301

Janusz Brzdek

Janusz Brzdek has published numerous papers on Ulam’s type stability (e.g., of functional, difference, differential and integral equations), its applications and connections to other areas of mathematics. He has been editor of several books and special volumes focused on such subjects. He was the chairman of the organizing and/or scientific committees of several conferences on Ulam’s type stability and on functional equations and inequalities.

Book preview

Ulam Stability of Operators

Series Editor

Themistocles M. Rassias

Authors

Janusz Brzdęk

Pedagogical University, Department of Mathematics, Podchorążych 2, 30-084 Kraków, Poland

Dorian Popa

Technical University of Cluj-Napoca, Department of Mathematics, 28 Memorandumului Street, 400114, Cluj-Napoca, Romania

Ioan Raşa

Technical University of Cluj-Napoca, Department of Mathematics, 28 Memorandumului Street, 400114, Cluj-Napoca, Romania

Bing Xu

Sichuan University, Department of Mathematics, No. 29 Wangjiang Road, 610064, Chengdu, China

Mathematical Analysis and its Applications Series

Table of Contents

Cover

Title page

Copyright

Dedication

Acknowledgment

Preface

About the Authors

Chapter 1: Introduction to Ulam stability theory

Abstract

1 Historical background

2 Stability of additive mapping

3 Approximate isometries

4 Other functional equations and inequalities in several variables

5 Stability of functional equations in a single variable

6 Iterative stability

7 Differential and integral equations

8 Superstability and hyperstability

9 Composite type equations

10 Nonstability

Chapter 2: Ulam stability of operators in normed spaces

Abstract

1 Introduction

2 Ulam stability with respect to gauges

3 Closed operators

4 Some differential operators on bounded intervals

5 Stability of the linear differential operator with respect to different norms

6 Some classical operators from the approximation theory

Chapter 3: Ulam stability of differential operators

Abstract

1 Introduction

2 Linear differential equation of the first order

3 Linear differential equation of a higher order with constant coefficients

4 First-order linear differential operator

5 Higher-order linear differential operator

6 Partial differential equations

7 Laplace operator

Chapter 4: Best constant in Ulam stability

Abstract

1 Introduction

2 Best constant for Cauchy, Jensen, and Quadratic functional equations

3 Best constant for linear operators

4 Ulam stability of operators with respect to different norms

Chapter 5: Ulam stability of operators of polynomial form

Abstract

1 Introduction

2 Auxiliary results

3 A general stability theorem

4 Complementary results for the second-order equations

5 Linear difference equation with constant coefficients

6 Difference equation with a matrix coefficient

7 Linear functional equations with constant coefficients

8 Linear differential equations

9 Integral equations

Chapter 6: Nonstability theory

Abstract

1 Preliminary information

2 Possible definitions of nonstability

3 Linear difference equation of the first order

4 Linear difference equation of a higher order

5 Linear functional equation of the first order

6 Linear functional equation of a higher order

Index

Copyright

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ISBN: 978-0-12-809829-5

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Dedication

Janusz Brzdęk

Dorian Popa

Ioan Raşa

Bing Xu

We dedicate this monograph to Professor Themistocles M. Rassias, the editor of this series of books, on the occasion of the 40th anniversary of the publication of his first paper on the stability of functional equations, which together with his numerous other papers strongly influenced the development of the theory of Ulam stability.

Acknowledgment

Janusz Brzdęk; Dorian Popa; Ioan Raşa; Bing Xu June 30, 2017

We thank the Elsevier staff for guidance throughout the publishing process; we especially thank Susan E. Ikeda and her team. We are also grateful to all the anonymous referees for carefully reviewing and improving our preliminary proposal of this monograph.

Preface

Janusz Brzdęk; Dorian Popa; Ioan Raşa; Bing Xu June 30, 2017

The aim of this book is not to present a survey of various papers dealing with the Ulam stability. We would not be able to do this in one book, because this area of research is too vast at the moment. Moreover, there are several such books published already and we do not want to copy their approach. Rather, we try to propose a somewhat new systematic approach to investigating Ulam stability. Therefore, after presenting some general results we show numerous examples of applications in various forms of difference, differential, functional and integral equations. Certainly, we use various previously published results, but in this book they are very often extended, generalized, and/or modified. So, it can be said that this book contains numerous outcomes that are new and unpublished, so far.

In this way we would like to show possible directions for future research and thus stimulate further investigations of Ulam stability as well as other related areas of mathematics. For this reason we do not tend to obtain the most general version of outcomes and further possible generalizations of them can be easily visible in many cases.

In the first chapter we present a brief introduction to the subject and cite several somewhat randomly selected results, providing references to sources with more detailed information on Ulam stability.

Our book presents, for the first time in unified and systematic way, some novel approaches to Ulam stability of numerous, mainly linear, operators. Moreover, it has a unique position of presenting up-to-date knowledge on subjects that have been treated only marginally in other similar books published. It includes, in particular, a lot of information on stability of several difference equations, functional equations in a single variable, various types of differential equations, and some integral equations.

It collects and compares suitable results from papers that have been published several years ago but also those published very recently; also, it unifies, complements, generalizes, and updates that information. Whenever it is suitable, open problems have been stated that suggest further possible exploration in the corresponding areas. The book is of interest to specialized researchers in the fields of various types of analysis, operator theory, difference and functional equations and inequalities, and differential and integral equations.

About the Authors

Janusz Brzdęk is Professor of Mathematics at Pedagogical University of Cracow (Poland). He has published numerous papers on Ulam’s type stability (e.g., of difference, differential, functional, and integral equations), its applications, and connections to other areas of mathematics; he has been editor of several books and special volumes focused on such subjects. He was also the chairman of the organizing and scientific committees of several international conferences on Ulam’s type stability and functional equations and inequalities.

Dorian Popa is Professor of Mathematics at Technical University of Cluj-Napoca (Romania). He is the author of numerous papers on Ulam’s type stability of functional equations, differential equations, linear differential operators, and positive linear operators in approximation theory. His other papers deal with the connections of Ulam’s type stability with some topics pertaining to multivalued analysis (e.g., the existence of a selection of a multivalued operator satisfying a functional inclusion associated with a functional equation).

Ioan Raşa is Professor of Mathematics at Technical University of Cluj-Napoca (Romania). He has published papers on Ulam’s type stability of differential operators and several types of positive linear operators arising in approximation theory. He is author/co-author of many papers connecting Ulam’s stability with other areas of mathematics (functional analysis, approximation theory, and differential equations). He is a co-author (with. F. Altomare et al.) of the book Markov Operators, Positive Semigroups and Approximation Processes, de Gruyter, 2014.

Bing Xu is Professor of Mathematics at Sichuan University (China). She has published many papers on Ulam’s type stability (e.g., of difference, differential, functional, and integral equations), its applications and connections to iterative equations, and multivalued analysis. Xu is co-author with W. Zhang et al. of the book Ordinary Differential Equations, Higher Education Press, 2014.

Chapter 1

Introduction to Ulam stability theory

Abstract

We describe the origin of Ulam stability theory, methods, and approaches, as well as some relevant results on this topic. In particular, we mention the preliminary result of G. Pólya and G. Szegö (published in 1925), describe the problem of S.M. Ulam (1909-1984), posed in 1940, and the partial solution to it that was published in 1941 by D.H. Hyers. Next, we present the further analogous outcomes of Ulam and Hyers (e.g., those published in 1945, 1947, 1952) and the results of T. Aoki (1950), D.G. Bourgin (1949, 1951), Th.M. Rassias (1978), J. Rätz (1980), P. Găvruţa (1994) and others. We then discuss the stability results for various equations (difference, differential, functional, and integral) providing suitable examples of them. We also depict the notions of superstability and hyperstability, and we present some remarks on the notion of nonstability.

Keywords

Additive mapping; Approximate isometry; Differential equation; Functional equation; Hyers-Ulam stability; Hyperstability; Integral equation; Iterative stability; Nonstability; Stability; Superstability; Ulam stability theory

Contents

1.Historical background

2.Stability of additive mapping

3.Approximate isometries

4.Other functional equations and inequalities in several variables

5.Stability of functional equations in a single variable

6.Iterative stability

7.Differential and integral equations

8.Superstability and hyperstability

9.Composite type equations

10.Nonstability

References

1 Historical background

The stability problem of functional equations was originally raised by Stanisław Marcin Ulam (cf. [73, 144]) in the fall term of the year 1940, when he gave a wide ranging talk before the Mathematics Club of the University of Wisconsin, discussing a number of unsolved problems. Among these was the following question concerning the approximate homomorphisms of groups:

We are given a group G1 and a metric group G2 with metric d. Given ε > 0, does there exist a δ > 0 such that if f : G1 → G2 satisfies

then a homomorphism g : G1 → G2 exists with

However, a somewhat similar problem was considered earlier by G. Pólya and G. Szegö (reals). They have obtained the following result.

Theorem 1

Suppose that a sequence of real numbers satisfies

Then the limit

exists and satisfies

The first partial answer to Ulam’s question came within a year, when D.H. Hyers [73] proved a result that can be stated as follows:

Theorem 2

Let E1 and E2 be Banach spaces and let f : E1 → E2 be a transformation such that, for some δ > 0,

Then the limit

exists for each x ∈ E1 and g : E1 → E2 is the unique additive transformation satisfying

Moreover, if f is continuous at least in one point x ∈ E1, then g is continuous everywhere in E1.

Furthermore, if the function is continuous for each x ∈ E1, then g is linear.

So if G1, G2 are the additive groups of Banach spaces, this theorem provides a positive answer to Ulam’s question with ε = δ. Shortly, we describe that result stating that the additive Cauchy equation,

is Hyers-Ulam stable (or has the Hyers-Ulam stability) in the class of functions f : E1 → E2.

Below, we describe the method of proof used in [73]; we call it the direct method; for information and further references concerning other methods see [32, 36, 74].

It is easy to prove, by induction, that

Write

Then

  

(1.1)

is a Cauchy sequence for each x ∈ E1, and since E2 is complete, there exists the limit function g : E1 → E2,

Clearly,

whence letting n → ∞ we obtain the additivity of g. Next, (1.1) with m = 0, yields

   (1.2)

, satisfying the inequality

we have

.

Clearly, if f is continuous at point y ∈ E1, then from (1.2) we deduce that g is bounded in a neighborhood of y, and consequently g is continuous.

Finally, if for a fixed xis additive and bounded on any finite interval, whence continuous and therefore linear. So, the assumption of continuity of fx for each x ∈ E1 implies the linearity of g.

2 Stability of additive mapping

T. Aoki [10] extended the result of Hyers by considering the case where the Cauchy difference f(x + y) − f(x) − f(y) is not necessarily bounded. He proved the following:

Theorem 3

Let E1 and E2 be Banach spaces, and f : E1 → E2 be such that

with some K ≥ 0 and p ∈ [0, 1). Then