An application of medial limits to iterative functional equations, II

Abstract

Applying medial limits we describe bounded solutions ϕ: S → R of the functional equation ϕ(x) = Ω g(ω)ϕ(f(x, ω))dμ(ω) + G(x), where (Ω,A, μ) is a measure space, S ⊂ R, f : S × Ω → S, g: Ω → R is integrable and G: S → R is bounded. The main purpose of this paper is to extend results obtained in Morawiec (Results Math 75(3):102, 2020) to the above general functional equation in wider classes of functions and under weaker assumptions.