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2020 | 34 (1) | 36--44
Tytuł artykułu

Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations

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Języki publikacji
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 πf 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)x)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)= μ̂(x)ϕ(Λx) which characterizes the limit distribution π^f for some special f∈R_c . (original abstract)
Opis fizyczny
  • University of Silesia in Katowice, Poland
  • K. Baron, On the convergence in law of iterates of random-valued functions, Aust. J. Math. Anal. Appl. 6 (2009), no. 1, Art. 3, 9 pp.
  • K. Baron, On the continuous dependence in a problem of convergence of iterates of random-valued functions, Grazer Math. Ber. 363 (2015), 1-6.
  • K. Baron, Weak law of large numbers for iterates of random-valued functions, Aequationes Math. 93 (2019), 415-423.
  • K. Baron, Weak limit of iterates of some random-valued functions and its application, Aequationes Math. DOI: 10.1007/s00010-019-00650-z.
  • 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).
  • R. Kapica, The geometric rate of convergence of random iteration in the Hutchinson distance, Aequationes Math. 93 (2019), 149-160.
  • M. Kuczma, B. Choczewski and R. Ger, Iterative functional equations, Encyclopedia of Mathematics and its Applications, vol. 32, Cambridge University Press, Cambridge, 1990.
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