On a type of exponential functional equation and its superstability in the sense of Ger

TL;DR

This paper investigates a specific exponential functional equation on commutative semigroups and proves its superstability in the sense of Ger, contributing to the understanding of functional equation stability.

Contribution

It establishes the superstability of a particular exponential functional equation in the sense of Ger for functions on commutative semigroups.

Findings

The functional equation is superstable in the sense of Ger.

Superstability holds under certain conditions on the functions.

The result extends the theory of stability for exponential functional equations.

Abstract

In this paper, we deal with a type exponential functional equation as follows $$f(xy)=f(x)^{g(y)},$$ where $f$ and $g$ are two real valued functions on a commutative semigroup. Our aim of this paper is to proved that the above functional equation in the sense of Ger is superstable.