Variants of Wilson's functional equation on semigroups

Abstract

Given a semigroup $S$ generated by its squares equipped with an involutive automorphism $\sigma$ and a multiplicative function $\mu :S\to \mathbb{C}$ such that $\mu (x\sigma (x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equations \begin{equation*}f(xy)+μ(y)f(σ(y)x)=2f(x)g(y),\, x,y\in S\end{equation*} and \begin{equation*}f(xy)+μ(y)f(σ(y)x)=2f(y)g(x),\, x,y\in S\end{equation*}

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