Linear functional equations and their solutions in generalized Orlicz spaces

ume that Ω⊂ℝ^k is an open set, V is a real separable Banach space and f_1,… ,f_N :Ω→Ω , g_1,… , g_N:Ω→ℝ , h_0:Ω→ V are given functions. We are interested in the existence and uniqueness of solutions φ :Ω→ V of the linear equation φ =∑ _k=1^Ng_k· (φ∘ f_k)+h_0 in generalized Orlicz spaces.