A Variant of d’Alembert Functional Equation on Monoids

Abstract

In this paper, we determine the complex-valued solutions of the functional equation $$\begin{aligned} f(x\sigma (y))+f(\tau (y)x)=2f(x)f(y) \end{aligned}$$for all \(x,y \in M\), where M is a monoid, \(\sigma \): \(M\longrightarrow M\) is an involutive automorphism and \(\tau \): \(M\longrightarrow M\) is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of 2-dimensional irreducible representations of M.