In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.

On the Stability of Functional Equations and a Problem of Ulam

On the Stability of Functional Equations in Banach Spaces

The paper is a survey about Hyers—Ulam stability of functional equations and systems in several variables.

Hyers—Ulam stability of functional equations in several variables

We present a survey of ideas and results stemming from the following stability problem of S. M. Ulam. Given a groupG1, a metric groupG2 and e > 0, find δ > 0 such that, iff: G1 →G2 satisfiesd(f(xy),f(x)f(y)) ⩽ δ for allx, y ∈G1, then there exists a homomorphismg: G1 →G2 such thatd(f(x),g(x))⩽e for allx ∈ Gl. For Banach spaces the problem was solved by D. Hyers (1941) with δ = e and

$$g(x) = \mathop {\lim }\limits_{n \to \infty } f(2^n x)/2^n .$$

Section 2 deals with the case whereG1 is replaced by an Abelian semigroupS andG2 by a sequentially complete locally convex topological vector spaceE. The necessity for the commutativity ofS and the sequential completeness ofE are also considered.

Approximate homomorphisms

In this paper, we show that the theorems of Hyers, Rassias and Gajda concerning the stability of the Cauchy's functional equation in Banach spaces, are direct consequences of the alternative of fixed point.

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The fixed point alternative and the stability of functional equations

In this paper we answer a question of Th. M. Rassias concerning an extension of validity of his result proved in [3].

https://downloads.hindawi.com/journals/ijmms/1991/817959.pdf

On stability of additive mappings

/pdf/a-characterization-of-exponential-polynomials-by-a-class-of-2mb8vir3.pdf

Zbigniew Gajda

Papers

On stability of additive mappings

A characterization of exponential polynomials by a class of functional equations