On the stability of the linear functional equation in a single variable on complete metric groups

TL;DR

This paper investigates the Hyers-Ulam stability of a specific linear functional equation within complete metric groups, providing conditions under which solutions are stable under perturbations.

Contribution

It extends the theory of functional equation stability to the setting of complete metric groups for a particular class of equations.

Findings

Established stability conditions for the functional equation

Extended stability results to metric group context

Provided new insights into functional equation behavior in abstract algebraic structures

Abstract

In this paper we obtain a result on Hyers-Ulam stability of the linear functional equation in a single variable $f(\varphi(x)) = g(x) \cdot f(x)$ on a complete metric group.