Lipschitzian solutions to inhomogeneous linear iterative equations

TL;DR

This paper investigates the existence, uniqueness, and stability of Lipschitz solutions to a class of inhomogeneous linear iterative equations involving measure integration, expanding understanding of their mathematical properties.

Contribution

It provides new conditions for the existence and uniqueness of Lipschitz solutions to these integral equations, and analyzes their continuous dependence on parameters.

Findings

Established criteria for solution existence and uniqueness.

Proved continuous dependence of solutions on data.

Extended the theory of linear iterative equations with measure integration.

Abstract

We study the problems of the existence, uniqueness and continuous dependence of Lipschitzian solutions $\varphi$ of equations of the form $$ \varphi(x)=\int_{\Omega}g(\omega)\varphi\big(f(x,\omega)\big)\mu(d\omega)+F(x), $$ where $\mu$ is a measure on a $\sigma$-algebra of subsets of a set $\Omega$.