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

Abstract

Given measure spaces \({(\Omega_{1}, \mathcal{A}_{1}, \mu_{1}),...,(\Omega_{N}, \mathcal{A}_{N}, \mu_{N}),}\) functions \({\varphi_{1}: \mathbb{R}^{m} \times \Omega_{1} \rightarrow \mathbb{R}^{m},...,\varphi_{N}: \mathbb{R}^{m} \times \Omega_{N} \rightarrow \mathbb{R}^{m}}\) and \({g: \mathbb{R}^{m} \rightarrow \mathbb{R}}\), we present results on the existence of solutions \({f: \mathbb{R}^{m} \rightarrow \mathbb{R}}\) of the inhomogeneous poly-scale refinement type equation of the form $$f(x) = \sum_{n=1}^{N} \int_{\Omega_{n}}|{\rm det}(\varphi_{n})^{\prime} _{x}(x, \omega_{n})|f(\varphi_{n}(x, \omega_{n}))d\mu_{n}(\omega_{n}+g(x)$$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.