Some generalization of Cauchy's and the quadratic functional equations

Abstract

We find the solutions f, g, h: S → H of each of the functional equations

λ∈Λ f(x + λy) = |Λ|g(x) + h(y), x,y∈ S, where (S, +) is an abelian semigroup, Λ is a finite subgroup of the automorphism group of S, (H, +) is an abelian group.