Note on some functional equations of Gołąb-Schinzel type

Abstract

In the present paper, we consider the following functional equation of Gołąb-Schinzel type: f(f(x)[n]y + f(y)[k]x) = ф(f(х), f(y)) (1) where ф : R[2] → R is a given function, f : E → R is the unknown function, E is a real vector space, k and n are given positive integers. We prove that, under suitable hypotheses on ф, if k and n are distinct, the only solutions of (1) having some regularity properties are the constant functions.