Continuous solutions to two iterative functional equations

Abstract

Based on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions φ of the equations φ(x)=F(x)−∫Ωφ(f(x,ω))P(dω),φ(x)=F(x)+∫Ωφ(f(x,ω))P(dω),

where P is a probability measure on a σ -algebra of subsets of Ω.