Given a probability space (Ω,A,P)\documentclass[12pt]{minimal}
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\begin{document}$$ (\Omega , {\mathcal {A}}, P) $$\end{document}, a complete and separable metric space X with the σ\documentclass[12pt]{minimal}
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\begin{document}$$ \sigma $$\end{document}-algebra B\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {B}} $$\end{document} of all its Borel subsets, a B⊗A\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {B}} \otimes {\mathcal {A}} $$\end{document}-measurable and contractive in mean f:X×Ω→X\documentclass[12pt]{minimal}
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\begin{document}$$ f: X \times \Omega \rightarrow X $$\end{document}, and a Lipschitz F mapping X into a separable Banach space Y we characterize the solvability of the equation φ(x)=∫Ωφf(x,ω)P(dω)+F(x)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varphi (x)=\int _{\Omega }\varphi \left( f(x,\omega )\right) P(d\omega )+F(x) \end{aligned}$$\end{document}in the class of Lipschitz functions φ:X→Y\documentclass[12pt]{minimal}
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\begin{document}$$\varphi : X \rightarrow Y$$\end{document} with the aid of the weak limit πf\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^f$$\end{document} of the sequence of iterates fn(x,·)n∈N\documentclass[12pt]{minimal}
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\begin{document}$$\left( f^n(x,\cdot )\right) _{n \in {\mathbb {N}}}$$\end{document} of f, defined on X×ΩN\documentclass[12pt]{minimal}
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\begin{document}$$ X \times \Omega ^{{\mathbb {N}}}$$\end{document} by f0(x,ω)=x\documentclass[12pt]{minimal}
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\begin{document}$$f^0(x, \omega ) = x$$\end{document} and fn(x,ω)=ffn-1(x,ω),ωn\documentclass[12pt]{minimal}
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\begin{document}$$ f^n(x, \omega ) = f\left( f^{n-1}(x, \omega ), \omega _n\right) $$\end{document} for n∈N\documentclass[12pt]{minimal}
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\begin{document}$$n \in {\mathbb {N}}$$\end{document}, and propose a characterization of πf\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^f$$\end{document} for some special rv-functions in Hilbert spaces.
