Hyers-Ulam stability of Jensen functional equation on amenable semigroups

TL;DR

This paper proves the Hyers-Ulam stability of the Jensen functional equation on amenable semigroups with an involution, extending stability results to a broader algebraic context.

Contribution

It establishes the stability of the Jensen functional equation on amenable semigroups with involution, a novel extension in the field of functional equations.

Findings

Hyers-Ulam stability holds for the Jensen equation on amenable semigroups.

The proof utilizes properties of amenable semigroups and involutions.

The results generalize previous stability results to a wider class of algebraic structures.

Abstract

In this paper, we give a proof of the Hyers-Ulam stability of the Jensen functional equation $$f(xy)+f(x\sigma(y))=2f(x),\phantom{+} x,y\in{G},$$ where $G$ is an amenable semigroup and $\sigma$ is an involution of $G.$