An application of medial limits to iterative functional equations

Abstract

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