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

Karol Baron

Published: 2020-05-08
Language: EN

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,????,P) is a probability space and (X,ρ) is a complete and separable metric space with the σ-algebra ???? of all its Borel subsets we consider the set ????_c of all ????⊗???? 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∈????_c, and investigate continuity-like property of the function f↦π^f, f∈????_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∈????_c.

(EN)

Keywords:

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

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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://trrest.vot.pl/ojsus/index.php/AMSIL/article/view/13629

References

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3. K. Baron, Weak law of large numbers for iterates of random-valued functions, Aequationes Math. 93 (2019), 415–423.
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5. R. Kapica, Sequences of iterates of random-valued vector functions and solutions of related equations, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 213 (2004), 113–118 (2005).
6. R. Kapica, The geometric rate of convergence of random iteration in the Hutchinson distance, Aequationes Math. 93 (2019), 149–160.
7. M. Kuczma, B. Choczewski and R. Ger, Iterative functional equations, Encyclopedia of Mathematics and its Applications, vol. 32, Cambridge University Press, Cambridge, 1990. Google Scholar

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

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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