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1998 | Buch

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Stability of Functional Equations in Several Variables

verfasst von: Donald H. Hyers, George Isac, Themistocles M. Rassias

Verlag: Birkhäuser Boston

Buchreihe : Progress in Nonlinear Differential Equations and Their Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Prologue

Abstract

To quote S.M. Ulam (1960) (page 63)¹ for very general functional equations, 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.

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 1. Introduction

Abstract

The most famous functional equation is the Cauchy equation

$$ f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)$$

(1.1)

any solution of which is called additive. It is known that, for real valued functions defined on the real line, every Lebesgue measurable solution of (1.1) is of the form f (x) =cx for some constant c. Nonmeasurable solutions also exist but they are “wild”, being discontinuous everywhere and unbounded on every interval. A discussion of these facts may be found in Chapter 2 of the book by J. Aczél and J. Dhombres (1989).

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 2. Approximately Additive and Approximately Linear Mappings

Abstract

We have mentioned the concept of stability and given an example for the Cauchy functional equation (1.1). A definition of stability in the case of homomorphisms between metric groups was suggested by a problem posed by S.M. Ulam in 1940 (see Ulam (1960), p. 64). Given a group (G₁, •), a metric group (G₂,*) with metric d and a positive number η, suppose that there exists a positive numberε= ε(η) such that, if d(f (x • y), f (x) * f (y)) <ε for some f : G ₁ → G₂ and all x and y in G₁, then a homomorphism h: G ₁ → G ₂ exists with d(f (x), h(x)) <η for all x in G ₁. In this case, the equation of homomorphism h(x • y) = h(x) * h(y) is called stable. Theorem 1.1 with G ₁ = E ₁, G ₂ = E ₂ and with addition as the group operation in each case shows that Cauchy’s equation is stable by this definition with η= ε.

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 3. Stability of the Quadratic Functional Equation

Abstract

The quadratic functional equation

$$ f\left( {x + y} \right) + f\left( {x - y} \right) - 2f\left( x \right) - 2f\left( y \right) = 0$$

(3.1)

clearly has f(x) = cx ² as a solution with c an arbitrary constant when f is a real function of a real variable. We define any solution of (3.1) to be a quadratic function, even in more general contexts. We shall be interested in functions f: E ₁ → E ₂ where both E ₁ and E ₂ are real vector spaces, and we need a few facts concerning the relation between a quadratic function and a biadditive function sometimes called its polar. This relation is explained in Proposition 1, p. 166, of the book by J. Aczél and J. Dhombres (1989) for the case where E ₂ = R, but the same proof holds for functions f: E ₁ → E ₂. It follows then that f: E ₁ → E ₂ is quadratic if and only if there exists a unique symmetric function B: E ₁ × E ₁ → E ₂, additive in x for fixed y, such that f (x) = B(x, x). The biadditive function B, the polar of f, is given by

$$B\left( {x,y} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ - \\ 4 \end{array}} \right)\left( {f\left( {x + y} \right) - f\left( {x - y} \right)} \right)$$

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 4. Generalizations. The Method of Invariant Means

Abstract

As was seen above in Corollary 1.2, there is no difficulty in generalizing Theorem 1.1 to mappings from a commutative semigroup S to a Banach space. We note here that the property of commutativity was used only in demonstrating the additivity of the mapping g.

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 5. Approximately Multiplicative Mappings. Superstability

Abstract

The stability of the functional equation f (x + y) = f (x) f (y) was studied by J. Baker, J. Lawrence and F. Zorzitto (1979). They proved that if f is a function from a real vector space W to the real numbers satisfying |f (x+y) -f (x) f (y) | < δ for some fixed δ> 0 and all x, y in W, then f is either bounded or else f (x + y) = f (x)f (y) for all x,y in W.

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 6. The Stability of Functional Equations for Trigonometric and Similar Functions

Abstract

In this section, we will be interested primarily in the stability of the following equations:

(6.1) $f\left( {x + y} \right) + f\left( {x - y} \right) = 2f\left( x \right)f\left( y \right)$ (d’Alembert)

(6.2) $f{\left( {\frac{{x + y}}{2}} \right)^2} = f\left( x \right)f\left( y \right) $ (Lobačevskiĭ)

(6.3a) $f\left( {x + y} \right)f\left( {x - y} \right) = f{\left( x \right)^2} - {\left( y \right)^2}$ (sine equation)

(6.3b) $f\left( {x + y} \right) = f\left( x \right)g\left( y \right) + g\left( x \right)f\left( y \right)$ (sine equation)

(6.4) $f\left( {x + y} \right) = f\left( x \right)f\left( y \right) - g\left( x \right)g\left( y \right)$ (cosine equation)

where f and g may be defined on a group or semigroup with values in a field K which usually is the field of real or complex numbers. Methods of solving such equations are described in the books by Aczél (1966) and Aczél and Dhombres (1989).

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 7. Functions with Bounded nth Differences

Abstract

Let S be a commutative semigroup with a zero element and Y a real Banach space. For functions f: S → Y, we define the difference operator △_h f (x)by △_h f (x)= f (x + h) - f (x). Similarly, we define$\Delta _{{h_1}{h_2}}^2f\left( x \right) = {\Delta _{{h_2}}}\left( {{\Delta _{{h_1}}}f\left( x \right)} \right)and\Delta _{{h_1} \cdots {h_{n + 1}}}^{n + 1}f\left( x \right) = {\Delta _{{h_{n + 1}}}}\left( {\Delta _{{h_1} \cdots {h_n}}^nf\left( x \right)} \right),n = 1,2,....$Note that the nth difference is symmetric in the increments h ₁,…, h _n. When all the increments are equal to h we write$\Delta _h^nf\left( x \right)$instead of$\Delta _{h \cdots h}^nf\left( x \right)$.

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 8. Approximately Convex Functions

Abstract

So far we have discussed the stability of various functional equations. In the present section, we consider the stability of a well-known functional inequality, namely the inequality defining convex functions:

$$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right),$$

(8.1)

where$0 \leqslant \lambda \leqslant 1$, with x and y in R ⁿ, A function f : S→R, where S is a convex subset of R ⁿ, will be called ε-convex (where ε > 0) if the inequality:

$$f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right) + \varepsilon $$

(8.2)

holds for all $\lambda$in [0, 1] and all x, y in S.

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 9. Stability of the Generalized Orthogonality Functional Equation

Abstract

Let (E,.) denote a real Hilbert space of dimension greater than one, and let T: E → E be a mapping which satisfies the functional equation

$$\left| {T\left( x \right) \cdot T\left( y \right)} \right| = \left| {x \cdot y} \right|$$

(9.1)

known as the generalized orthogonality equation. We will follow the work of C. Alsina and J.L. Garcia-Roig (1991) (see also Th.M. Rassias (1997)). It is easy to see that any solution of (9.1) will satisfy the following conditions for x, y in E and real µ:

(A)

||T(x)|| =||x||.

(B)

T(x) = 0 if and only if x = 0.

(C)

T(x) • T(y) = 0 if and only if x • y = 0.

(D)

I cos A(x, y) I = I cos A(T (x), T (y))1, where A(u, v) denotes the angle between the vectors u and v.

(E)

T(µx) = ±µT(x).

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 10. Stability and Set-Valued Functions

Abstract

Some interesting connections between the stability results of Chapter 1 as well as the theory of subadditive set-valued functions has been pointed out by several authors. We begin with a work by W. Smajdor (1986) which generalizes for set-valued functions some well-known theorems on linearity for ordinary functions, starting with an example (see also Th.M. Rassias (1998)).

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 11. Stability of Stationary and Minimum Points

Abstract

It is of course well known that two functions may differ uniformly by a small amount and yet their derivatives may differ widely. However, there are certain cases in which these derivatives may even be equal, provided they are evaluated at slightly different points, as in case n = 1 of Theorem 11.1 below. This was proved by S.M. Ulam and D.H. Hyers (1954) (see also Th.M. Rassias (m)).

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 12. Functional Congruences

Abstract

In the following we consider a mapping f:E → R, where E is a real topological vector space, such that

$$f\left( {x + y} \right) - f\left( x \right) - f\left( y \right) \in Z$$

(12.1)

for all x, y in E.

Donald H. Hyers, George Isac, Themistocles M. Rassias

Chapter 13. Quasi-Additive Functions and Related Topics

Abstract

J. Tabor (1988) introduced the class of functions f: R → R which satisfy the inequality

$$\left| {f\left( {x + y} \right) - f\left( x \right) - f\left( y \right)} \right| \leqslant \varepsilon \min \left\{ {\left| {f\left( {x + y} \right)} \right|,\left| {f\left( x \right) + f\left( y \right)} \right|} \right\}$$

for all real x and y,where εis a fixed number satisfying 0 ≤ ε < 1. Later the same author (see J. Tabor (1990)) generalized the concept by considering functions f: X — Y, where XandYare real normed spaces. He called the class of functions satisfying the inequality

$$\left\| {f\left( {x + y} \right) - f\left( x \right) - f\left( y \right)} \right\| \leqslant \varepsilon \min \left\{ {\left\| {f\left( {x + y} \right)} \right\|,\left\| {f\left( x \right) + f\left( y \right)} \right\|} \right\}$$

(13.1)

for x,y in X, and for some ε∈ [0,1), quasi-additive.

Donald H. Hyers, George Isac, Themistocles M. Rassias

Backmatter

Titel: Stability of Functional Equations in Several Variables
verfasst von: Donald H. Hyers
George Isac
Themistocles M. Rassias
Verlag: Birkhäuser Boston
Electronic ISBN: 978-1-4612-1790-9
Print ISBN: 978-1-4612-7284-7
DOI: https://doi.org/10.1007/978-1-4612-1790-9

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Inhaltsverzeichnis

Frontmatter

Prologue

Chapter 1. Introduction

Chapter 2. Approximately Additive and Approximately Linear Mappings

Chapter 3. Stability of the Quadratic Functional Equation

Chapter 4. Generalizations. The Method of Invariant Means

Chapter 5. Approximately Multiplicative Mappings. Superstability

Chapter 6. The Stability of Functional Equations for Trigonometric and Similar Functions

Chapter 7. Functions with Bounded nth Differences

Chapter 8. Approximately Convex Functions

Chapter 9. Stability of the Generalized Orthogonality Functional Equation

Chapter 10. Stability and Set-Valued Functions

Chapter 11. Stability of Stationary and Minimum Points

Chapter 12. Functional Congruences

Chapter 13. Quasi-Additive Functions and Related Topics

Backmatter