The Drygas Functional Equation on Abelian Semigroups with Endomorphisms

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