The superstability of a variant of Wilson’s functional equation on an arbitrary group

Abstract

The aim of this paper is to investigate the superstability problem for the variant $$\begin{aligned} f(xy)+f(\sigma (y)x)=2f(x)g(y),\ \ \ \ x,y\in G, \end{aligned}$$ of Wilson’s functional equation where \(G\) is an arbitrary group, \(f,g\) are complex valued functions and \(\sigma \) is an involution of \(G\). As a consequence, we obtain the superstability of the Pexider type functional equation $$\begin{aligned} f(xy)+f(\sigma (y)x)=2g(x)h(y),\ \ x,y\in G, \end{aligned}$$ on any group.