Integrable solutions of inhomogeneous refinement type equations on intervals

TL;DR

This paper investigates the existence of solutions to inhomogeneous refinement equations involving probability measures and functions on intervals, contributing to the mathematical understanding of such equations.

Contribution

It provides new results on the existence of solutions for inhomogeneous refinement equations with probabilistic and functional conditions.

Findings

Established conditions for solution existence

Extended refinement equations to inhomogeneous cases

Analyzed solutions in the space L^1(I)

Abstract

Given a probability measure $P$ on a $\sigma$-algebra of subsets of a set $\Omega$, an interval $I\subset\mathbb R$, $g\in L^1(I)$, and a function $\varphi\colon I\times\Omega\to I$ fulfilling some conditions we obtain results on the existence of solutions $f\in L^1(I)$ of the inhomogeneous refinement type equation $$ f(x)=\int_{\Omega}\big|\varphi'_x(x,\omega)\big|f(\varphi(x,\omega))dP(\omega)+g(x). $$