Abstract :
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
