“Pexiderized” homogeneity almost everywhere

In the paper we examine Pexiderized φ-homogeneity equation almost everywhere. Assume that G and H are groups with zero, (X,G) and (Y,H) are a G- and an H-space, respectively. We prove, under some assumption on (Y,H), that if functions φ :G→H and F1,F2 :X→Y satisfy Pexiderized φ-homogeneity equation F1(αx) = φ(α)F2(x) almost everywhere in G×X then either φ = 0 almost everywhere in G or F2 = θ almost everywhere in X or there exists a homomorphism φ˜ :G→H such that φ = aφ˜ almost everywhere in G and there exists a function F :X→Y such that F(αx) = φ˜(α)F(x) for α ∈ G\ {0}, x ∈ X, and F1 = aF almost everywhere in X, F2 = F almost everywhere in X, where a ∈ H∗ is a constant. From this result we derive solution of the classical Pexider equation almost everywhere. © 2006 Elsevier Inc. All rights reserved