Functional equations on semigroups with an endomorphism

Abstract

Let S be a semigroup, let H be an abelian group which is 2-torsion free, and let ${\varphi \colon S \to S}$ be an endomorphism. We determine the solutions ${ g \colon S \to \mathbb{C}}$ of the functional equation

$$g(xy)+g(\varphi(y)x)=2g(x)g(y), \quad x,y \in S,$$

in terms of multiplicative functions on S, and we show that any solution ${ {g \colon S\to H }}$, when ${\varphi}$ is surjective, of the functional equation

$$g(xy)+g(\varphi(y)x)=2g(x), \quad x,y \in S,$$

has the form ${g=A+c }$, where ${A \colon S\rightarrow H}$ is an additive map such that ${{ A\circ\varphi =-A }}$, and where ${c\in H}$ is a constant. The endomorphism ${ \varphi}$ need not be involutive.