An integral functional equation on abelian semigroups

Abstract

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.