Inhomogeneous poly-scale refinement type equations and Markov operators with perturbations

Abstract

Given measure spaces (Ω1,A1,μ1), . . . , (ΩN,AN,μN), functions φ1 : Rm×Ω1 Rm, . . . ,φN : Rm×ΩN Rm and g : Rm R, we present results on the existence of solutions f : Rm R of the inhomogeneous poly-scale refinement type equation of the form f(x) = ΣN n=1 ∫ Ωn det(φn)′x(x, ωn) f (φn(x, ωn)) dμn(ωn) + g(x) in some special classes of functions. The results are obtained by Banach fixed point theorem applied to a perturbed Markov operator connected with the considered inhomogeneous poly-scale refinement type equation. Mathematics Subject Classification. Primary 37H99, 37N99; Secondary 39B12.