On bounded solutions of a generalized Schilling's problem

Abstract

In this note we prove that, under some assumptions on a, b, c, h, g and Q, the zero function is the only solution φ : R → R of the functional equation φ(g(x)) = a(x)φ(h(x)) + b(x)φ(h[-1](x)) + c(x)φ(x), bounded in a neighbourhood of the origin and such that φ(x) = 0 for |x| > Q.