LAUSR

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