Stability of a functional equation of Deeba on semigroups

Abstract

Let $S$ be a semigroup and $X$ a Banach space. The functional equation $\varphi (xyz)+\varphi (x)+\varphi (y)+\varphi (z)=\varphi (xy)+\varphi (yz)+\varphi (xz)$ is said to be stable for the pair $(X,S)$ if and only if $f:S\to X$ satisfying $\Vert f(xyz)+f(x)+f(y)+f(z)-f(xy)-f(yz)-f(xz)\Vert \le \delta$ for some positive real number $\delta$ and all $x,y,z\in S$, there is a solution $\varphi :S\to X$ such that $f-\varphi$ 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.

29 pages