Assume \( (\Omega , {\mathcal {A}}, {\mathbb {P}}) \) is a probability space, \((X,\rho )\) is a complete and separable metric space with the \( \sigma \) –algebra \( {\mathcal {B}} \) of all its Borel subsets and \( f: X \times \Omega \rightarrow X \) is measurable for \( {\mathcal {B}} \otimes {\mathcal {A}}\) and such that \(\begin{aligned} \int _{\Omega } \rho \big (f(x, \omega ), f(z, \omega )\big ) {\mathbb {P}}(d\omega ) \le \beta \big (\rho (x, z)\big ) \quad \text {for } x, z \in X \end{aligned}\) with a concave \(\beta : [0,\infty ) \rightarrow [0,\infty )\) satisfying \(\beta (t)<t\) for \(t \in (0,\infty )\) , and \(\int _{\Omega } \rho \big (f(x_0, \omega ), x_0\big ) {\mathbb {P}}(d \omega ) <\infty \) for an \(x_0 \in X.\) We consider the weak limit of the sequence of iterates of f and problems of the existence and uniqueness of solutions \(\varphi \) of the equations \(\begin{aligned} & \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big ){\mathbb {P}}(d\omega ), \\ & \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big ){\mathbb {P}}(d\omega ) \end{aligned}\) in some classes of continuous functions mapping X into a separable Banach space.
