On pexider additive functional equations

Abstract

Let Y(+) egy Abelian group, D in Z2 ( Z(+,≤) denotes the ordered group of the integers).Dx:={uϵZ | exists vϵ X; (u,v)ϵ D}, Dy:={vϵZ | exists uϵ X; (u,v)ϵ D}, Dx+y:={zϵZ | exists (u,v)ϵ D; z=u+v}.The aim of the article is to find sets D in Z2 that the general solution of the functional equation f(x+y) = g(x) + h(y) ((x,y ϵ D) with unknown functions f, g and h is in the form of f(u)=a(u)+C1+C2 (uϵDx+y), g(v)=a(v)+C1 (vϵDx); h(z)=a(z)+C2 (zϵDy) where a is an additive function, C1, C2 ϵ Y are constants).