An integral functional equation on abelian semigroups

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