Functional Equations and Inequalities in Several Variables

The following sections are included:

• Preface

• Contents

The following sections are included:

• Basic definitions

• Semilinear topology

• Notes

The following sections are included:

• Properties of convex functions

• Continuous convex functions

• Notes

The idea of the lower hull of a convex function has been presented for the first time by F. Berstein and G. Doetsch [15] (for a functions of a single real variable)…

The theorems of Bernstein-Doetsch, Piccard and Mehdi are very important in the theory of convex functions. In this section we present them under some very general assumptions. The proofs are based on the paper [116]…

Let's look again into the Theorem 4.1.

If X is a linear space with a semilinear topology , then every J-convex function defined on an open and convex set A ⊂ X and bounded above on a non-empty open subset of A has to be continuous in A. This justifies the following definition…

The following sections are included:

• Cauchy's additive equation

• Cauchy's exponential equation

• Notes

The following functional equation is refered to as d'Alembert's equation

The following sections are included:

• Solution of equations (8.1) and (8.2)

• Solution of D'Alembert's equation

• Notes

The following sections are included:

• The diference operator

• Polynomial functions

• Extensions of polynomial functions

• Notes

The following sections are included:

• Continuous quadratic functionals

• Continuous quadratic functionals on Hilbert spaces

• Notes

In this chapter, we give the general solutions of the quadratic functional equation on an interval of real number. All the results we shall present here are due to K. Dłutek [47]…

Let X and Y be groups and let f : X → Y be a function. We define , the quadratic difference of f, by the formula

Very often instead of a functional equation, we consider a functional inequality and one can ask the following question: when can one assert that the solutions of the inequality lie near to the solutions of the equation?

The problem of this kind had been formulated by S. M. Ulam [212] during his talk before the Mathematics Club of the University of Wisconsis, in 1940…

Let (G, ·) be a semigroup and E a normed algebra with multiplicative norm, i.e. ‖a · b‖ = ‖a‖ · ‖b‖ for a, b ∈ E. A mapping f : G → E is said to be multiplicative iff f satisfies the equation (multiplicative Cauchy equation)

An interesting result for the stability of Jensen's functional equation was obtained by Z. Kominek [118]. His result reads as follows…

First we establish some notations. Let (S, +) be a semigroup and Y a topological vector space over the field ℚ of all rational numbers. A set-valued function F: S → 2^Y is said to be subadditive iff

where "+" on the right side means the algebraic sum of sets…

The functional equation

We will now consider the D'Alembert's functional equation (called also the cosine equation)

We shall use the following symbols: ℕ, ℝ, ℝ₀, ℝ₊ to denote the sets of all natural, real, non-zero real or nonnegative real numbers respectively. Moreover, denote

The equation

The following sections are included:

• Stability in L^p norms

The following sections are included:

• Stability of the Cauchy equation

• Cauchy difference operator

The following sections are included:

• Stability of the Pexider equation

• Pexider linear operator

The following sections are included:

• Cauchy operator

• Pexider operator

The following sections are included:

• Stability in the Lipschitz norms

• Notes

The following sections are included:

• Preliminaries

• Stability of iteration procedures

The following sections are included:

• Superstability of the quadratic functional equation

The following sections are included:

• Set-valued functions in I(ℝ)

• Additive set-valued functions with values in topological vector spaces

• Notes

The following sections are included:

• Continuous solutions of Jensen's functional equation

In this section we give a characterization of set-valued solutions of the Pexider functional equation

Consider a real normed space X. Let us remind that a set-valued function F: ℝ → 2^X is called quadratic provided that

The following sections are included:

• Subadditive s.v. functions in topological vector spaces

• Notes

Let X and Y be semigroups. A s.v.f. F: X → 2^Y is superadditive iff

The following sections are included:

• Hahn-Banach theorem

• Generalizations of the Hahn-Banach theorem

• Applications to stability of functional equations

It follows from Corollary 20.1, that if X is an abelian group and Y is a Banach space and f : X → Y satisfies the inequality

The following sections are included:

• Continuity properties for K-midconvex set-valued functions

• Midconvex set-valued functions with values in ℝ

• K-midconcave set-valued functions

Let X be a non-empty set. A family

The following sections are included:

• References

• Index