AN INTEGRAL FUNCTIONAL EQUATION ON ABELIAN SEMIGROUPS.

We investigate a generalization of many functional equations. Namely, we consider the following functional equation ∫S f(x + y + t) dμ(t) + ∫S f(x + φ(y) + t) dν(t) = f(x) + h(y), x, y ∈ S, where (S,+) is an abelian semigroup, φ is a surjective endomorphism of S, E is a linear space over the field K ∈ {R, C} and μ,ν are linear combinations of Dirac measures. Under appropriate conditions on μ and ν and based on Stetkar's result [9], we find and characterize solutions of the previous functional equation. [ABSTRACT FROM AUTHOR]