Weakly contractive in mean random-valued functions and two linear functional equations

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.