Kannappan’s integral equation with an anti-endomorphism

Abstract

Let G be a locally compact Hausdorff group and $\psi :G\rightarrow G$ is a continuous anti-endomorphism that need not be involutive. Our main goal is to extend previous results about Kannappan’s functional equation $g(xyz_{0})+g(x\psi (y)z_{0})=g(x)g(y),$ where $z_{0}$ is an arbitrarily fixed element in G. We introduce and study the continuous solutions $h,f,g:G\rightarrow {\mathbb {C}}$ of the integral-functional equation

$$\begin{aligned} \int _{G}\{h(xyt)+\mu (y)h(x\psi (y)t)\}\,d\nu (t)=f(x)g(y),\,\,\,x,y\in G, \end{aligned}$$

where $\mu :G\rightarrow {\mathbb {C}}$ is a continuous multiplicative function satisfying $\mu (x\psi (x))=1$ for all $x\in G$, and $\nu$ is a regular, compactly supported, complex-valued Borel measure on G.