An Application of Medial Limits to Iterative Functional Equations, II

Abstract

AbstractApplying medial limits we describe bounded solutions $$\varphi :S\rightarrow {\mathbb {R}}$$ φ : S → R of the functional equation $$\begin{aligned} \varphi (x)=\int _{\Omega }g(\omega )\varphi (f(x,\omega ))d\mu (\omega )+G(x), \end{aligned}$$ φ ( x ) = ∫ Ω g ( ω ) φ ( f ( x , ω ) ) d μ ( ω ) + G ( x ) , where $$(\Omega ,{\mathcal {A}},\mu )$$ ( Ω , A , μ ) is a measure space, $$S\subset \mathbb R$$ S ⊂ R , $$f:S\times \Omega \rightarrow S$$ f : S × Ω → S , $$g:\Omega \rightarrow {\mathbb {R}}$$ g : Ω → R is integrable and $$G:S\rightarrow {\mathbb {R}}$$ 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.