Solutions and stability of variant of Wilson's functional equation

TL;DR

This paper investigates solutions and stability of a generalized Wilson's functional equation on groups, providing explicit solutions, stability results, and demonstrating superstability under certain conditions.

Contribution

It offers new solutions for the functional equation when involving involutive automorphisms and anti-automorphisms, and establishes stability and superstability results.

Findings

Solutions characterized for involutive automorphisms.

Hyers-Ulam stability established for the equation.

Superstability demonstrated for a related functional equation.

Abstract

In this paper we will investigate the solutions and stability of the generalized variant of Wilson's functional equation $$ (E):\;\;\;\; f(xy)+\chi(y)f(\sigma(y)x)=2f(x)g(y),\; x,y\in G,$$ where $G$ is a group, $\sigma$ is an involutive morphism of $G$ and $\chi$ is a character of $G$. (a) We solve $(E)$ when $\sigma$ is an involutive automorphism, and we obtain some properties about solutions of $(E)$ when $\sigma$ is an involutive anti-automorphism. (b) We obtain the Hyers Ulam stability of equation $(E)$. As an application, we prove the superstability of the functional equation $f(xy)+\chi(y)f(\sigma(y)x)=2f(x)f(y),\; x,y\in G.$