Stability of a functional equation of Deeba on semigroups

TL;DR

This paper investigates the stability of a specific functional equation on semigroups, showing it is generally unstable but stable on certain classes like periodic and abelian semigroups, and explores embedding into stable structures.

Contribution

It establishes conditions under which the functional equation is stable on semigroups and demonstrates embeddings into stable semigroups with reduction laws.

Findings

The equation is not stable on arbitrary semigroups.

The equation is stable on periodic and abelian semigroups.

Embedding semigroups with reduction laws into stable ones is possible.

Abstract

Let $S$ be a semigroup and $X$ a Banach space. The functional equation $\phi (xyz)+ \phi (x) + \phi (y) + \phi (z) = \phi (xy) + \phi (yz) + \phi (xz)$ is said to be stable for the pair $(X, S)$ if and only if $f: S\to X$ satisfying $\| f(xyz)+f(x) + f(y) + f(z) - f(xy)- f(yz)-f(xz)\| \leq \delta $ for some positive real number $\delta$ and all $x, y, z \in S$, there is a solution $\phi : S \to X$ such that $f-\phi$ is bounded. In this paper, among others, we prove the following results: 1) this functional equation, in general, is not stable on an arbitrary semigroup; 2) this equation is stable on periodic semigroups; 3) this equation is stable on abelian semigroups; 4) any semigroup with left (or right) law of reduction can be embedded into a semigroup with left (or right) law of reduction where this equation is stable.