Solutions and stability of variant of Wilson's 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 $f(xy)+\chi (y)f(\sigma (y)x)=2f(x)f(y),x,y\in G.$

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