An application of medial limits to iterative functional equations

TL;DR

This paper uses medial limits to analyze bounded solutions of a class of iterative functional equations with boundary conditions, providing a new approach to existence and characterization of solutions.

Contribution

It introduces a novel application of medial limits to describe solutions of functional equations with probabilistic components and boundary constraints.

Findings

Characterization of solutions using medial limits

Existence results for bounded solutions

Application to boundary value problems in functional equations

Abstract

Assume that $(\Omega,\mathcal A,P)$ is a probability space, $f\colon[0,1] \times \Omega\to[0,1]$ is a function such that $f(0,\omega)=0$, $f(1,\omega)=1$ for every $\omega\in\Omega$, $g\colon[0,1]\to\mathbb R$ is a bounded function such that $g(0)=g(1)=0$, and $a,b\in\mathbb R$. Applying medial limits we describe bounded solutions $\varphi\colon[0,1] \to \mathbb R$ of the equation \begin{equation*} \varphi(x) = \int_\Omega \varphi(f(x,\omega)) dP(\omega)+g(x) \end{equation*} satisfying the boundary conditions $\varphi(0)=a$ and $\varphi(1)=b$.