The Drygas Functional Equation on Abelian Semigroups with Endomorphisms

Let $$(S,+)$$ be an abelian semigroup, let $$(H,+)$$ a uniquely 2-divisible, group and $$\sigma ,\tau $$ two endomorphisms of S. In this paper, we find the solutions $$f:S\rightarrow H$$ following Drygas type functional Eq. 1 $$\begin{aligned} f(x+\sigma (y))+f(x+\tau (y))=2f(x)+f(\sigma (y))+f(\tau (y)), \quad x,y\in S, \end{aligned}$$ in terms additive bi-additive maps. Further, solve partly Pexiderized version above Eq use its to determine some related Eqs. all results it is assumed that at least one $$\tau surjective.