On generalizations of the Pompeiu functional equation

Abstract

In this paper, we determine the general solution of the functional equations f(x+y+xy)=p(x)+q(y)+g(x)h(y),     (∀x,y∈ℜ*) and f(ax+by+cxy)=f(x)+f(y)+f(x)f(y),     (∀x,y∈ℜ) which are generalizations of a functional equation studied by Pompeiu. We present a method which is simple and direct to determine the general solutions of the above equations without any regularity assumptions.