Stability of a functional equation of Deeba on semigroups

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.