An Application of Medial Limits to Iterative Functional Equations, II

Abstract

Applying 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.