Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations

Karol Baron

Published: 2020-05-08

Vol. 34 No. 1 (2020)

Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations

Karol Baron

Abstract

The paper consists of two parts. At first, assuming that (Ω,A,P) is a probability space and (X,ρ) is a complete and separable metric space with the σ-algebra B of all its Borel subsets we consider the set R_c of all B ⊗A measurable and contractive in mean functions f:X×Ω→X with finite integral ∫_Ωρ(f(x,ω),x)P(dω) for x∈X, the weak limit π^f of the sequence of iterates of f∈R_c, and investigate continuity-like property of the function f↦π^f, f∈?R_c, and Lipschitz solutions ϕ that take values in a separable Banach space of the equation
ϕ(x) = ∫_Ωϕ(f(x,ω))P(dω) + F(x).
Next, assuming that X is a real separable Hilbert space, Λ:X→X is linear and continuous with ‖Λ‖<1, and μ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions ϕ:X→ℂ of the equation
ϕ(x) = μ^{^}(x)ϕ(Λx)
which characterizes the limit distribution π^f for some special f∈R_c.

Keywords:

random-valued functions , iterates , convergence in law , continuous dependence on the given function , Fourier transform , iterative functional equations , continuous and Lipschitz solutions

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Baron, K. (2020). Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations. Annales Mathematicae Silesianae, 34(1), 36–44. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13629

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References

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Vol. 34 No. 1 (2020)
Published: 2020-07-20

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Licence CC

This work is licensed under a Creative Commons Attribution 4.0 International License.

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