Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations - Baron, Karol (Annales Mathematicae Silesianae, 2020)
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 Rc 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∈Rc, and investigate continuity-like property of the function f↦πf, f∈?Rc, 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 πf for some special f∈Rc.