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Weak limit of iterates of some random-valued functions and its application

  • Karol BaronEmail author
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Abstract

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

Keywords

Random-valued functions Iterates Weak limit Iterative equations Lipschitzian solutions Bochner integral Gaussian measures 

Mathematics Subject Classification

Primary 39B12 26A18 Secondary 60B12 58D20 

Notes

Acknowledgements

I thank Professor Gregory Derfel for calling my attention to the paper by O. K. Zakusilo.

This research was supported by the University of Silesia Mathematics Department (Iterative Functional Equations and Real Analysis program).

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Instytut MatematykiUniwersytet Śla̧skiKatowicePoland

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