Solution of Generalized Jensen and Quadratic Functional Equations

We obtain in terms of additive and multi-additive functions the general solution f : S → H of each of the functional equations $$\displaystyle \sum _{\lambda \in \varPhi } f(x+\lambda y+a_{\lambda })=Nf(x),\ x,y\in S, $$ $$\displaystyle \sum _{\lambda \in \varPhi }f(x+\lambda y+a_{\lambda })=Nf(x)+Nf(y),\ x,y\in S, $$ where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, \(N=\left \vert \varPhi \right \vert \) designates the number of its elements, \( \left \{ a_{\lambda },\lambda \in \varPhi \right \} \) are arbitrary elements of S, and (H, +) is an abelian group. In addition, some applications are given. These equations provide a common generalization of many functional equations (Cauchy’s, Jensen’s, quadratic, Φ-quadratic equations, …).