Based on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions \(\varphi \) of the equations

$$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ), \end{aligned}$$

where P is a probability measure on a \(\sigma \)-algebra of subsets of \(\Omega \).

中文翻译：

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两个迭代函数方程的连续解

基于随机值函数的迭代，我们研究方程组的连续函数和Hölder连续函数\（\ varphi \）类中的可解性问题

$$ \ begin {aligned} \ varphi（x）= F（x）-\ int _ {\ Omega} \ varphi \ big（f（x，\ omega）\ big）P（d \ omega），\\ \ varphi（x）= F（x）+ \ int _ {\ Omega} \ varphi \ big（f（x，\ omega）\ big）P（d \ omega），\ end {aligned} $$

其中P是\（\ Omega \）的子集的\（\ sigma \）-代数上的概率测度。