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