On a Pexider-Drygas functional equation on semigroups with an endomorphism

Abstract

Let S be a semigroup that need not be abelian, let (H,+) be a uniquely
2-divisible abelian group, and let $\varphi $ be an endomorphism of S. We
characterize the solutions f,g:S\rightarrow H of the pexiderized version
of the variant of Drygas' equation, that is
\begin{equation*}
f(xy)+f(\varphi (y)x)=2f(x)+g(y),\ x,y\in S.
\end{equation*}
Interesting consequences of this result are presented.