Continuous solutions to two iterative functional equations

Abstract Based on iteration of random-valued functions we study the problem solvability in class continuous and Hölder $$\varphi $$ φ equations $$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ (x)=F(x)+\int ), \end{aligned}$$ ( x ) = F - ∫ Ω f , ω P d + where P is a probability measure $$\sigma σ -algebra subsets $$\Omega .