A SPECIAL CLASS OF FUNCTIONAL EQUATIONS

We obtain in terms of additive and multi-additive functions the solutions f, h: S > H of the functional equation Sigma(lambda is an element of Phi)f(x + lambda y + a(lambda)) - N f (x) + h(y), x, y is an element of S, where (S,+) is an abelian monoid, Phi is a finite group of automorphisms of S, N = vertical bar Phi vertical bar designates the number of its elements, {a(lambda), lambda is an element of Phi} are arbitrary elements of S and (H, +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy's, Jensen's, Lukasik's, quadratic or Phi-quadratic equations. (C) 2018 Mathematical Institute Slovak Academy of Sciences