An Application of Medial Limits to Iterative Functional Equations

Abstract

Assume that (Ω,A, P) is a probability space, f : [0, 1] × Ω → [0, 1] is a function such that f(0, ω) = 0, f(1, ω) = 1 for every ω ∈ Ω, g: [0, 1] → R is a bounded function such that g(0) = g(1) = 0, and a, b ∈ R. Applying medial limits we describe bounded solutions ϕ: [0, 1] → R of the equation ϕ(x) =

Ω ϕ(f(x, ω))dP(ω) + g(x) satisfying the boundary conditions ϕ(0) = a and ϕ(1) = b.