Linear functional equations and their solutions in generalized Orlicz spaces

Assume that Ω ⊂ Rk is an open set, V is a real separable Banach space and f1, … , fN: Ω → Ω , g1, … , gN: Ω → R, h: Ω → V are given functions. We are interested in the existence and uniqueness of solutions φ: Ω → V of the linear equation φ=∑k=1Ngk·(φ∘fk)+h0 in generalized Orlicz spaces