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

Rafał Kapica,Janusz Morawiec

doi:10.1007/s11784-015-0226-3

Rafał Kapica, Janusz Morawiec

Open Access

https://doi.org/10.1007/s11784-015-0226-3

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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.

Highlights

Fix m, N ∈ N, measure spaces (Ω1, A1, μ1), . . . , (ΩN, AN, μN ) and functions φ1 : Rm ×Ω1 → Rm, . . . , φN : Rm ×ΩN → Rm, g : Rm → R
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
JM. oMroarwaiweciec the function φn : Rm × Ωn → Rm is of the form φn(x, ωn) = Kn(ωn)x − Mn(ωn) with given functions

Summary

Introduction

N ∈ N, measure spaces (Ω1, A1, μ1), . . . , (ΩN , AN , μN ) and functions φ1 : Rm ×Ω1 → Rm, . . . , φN : Rm ×ΩN → Rm, g : Rm → R. If homogeneous poly-scale refinement type equation (1.6) has no nontrivial Lebesgue integrable solution, we can ask if its inhomogeneous counterpart (1.1), obtained by adding to the right-hand side of (1.6) a perturbation function g, has such a solution. If F is a given class of functions, the existence of a solution f ∈ F of inhomogeneous poly-scale refinement type equation (1.1) is a consequence of the existence of a fixed point of the operator P : F → F given by. We are going to examine the Banach fixed point theorem to obtain results on the existence of a pth power Lebesgue integrable solution as well as a continuous and bounded solution of equation (1.1) and of its special case (1.3). We are looking for conditions, on the spaces (Ω1, A1, μ1), . . . , (ΩN , AN , μN ) and the given functions φ1, . . . , φN and g, guaranteeing that the operator P is well defined and satisfies assumptions of the Banach fixed point theorem

Notation

L1-solutions

Lp-solutions

Continuous and bounded solutions