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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) 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.

We investigate a generalization of many functional equations. Namely, we consider the following functional equation ?S f(x + y + t) d?(t) + ?S f(x + ?(y) + t) d?(t) = f(x) + h(y), x, y ? S, where (S,+) is an abelian semigroup, ? is a surjective endomorphism of S, E is a linear space over the field K ? {R, C} and ?,? are linear combinations of Dirac measures. Under appropriate conditions on ? and ? and based on Stetkar?s result [9], we find and characterize solutions of the previous functional equation.

Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation f ( x ϕ ( y ) ) + f ( ψ ( y ) x ) = 2 f ( x ) f ( y ) ,             x , y   ∈   S , f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,S, where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.

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