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 𝒝 of all its Borel subsets we consider the set 𝒭_c of all 𝒝 ⊗ 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 ∈ 𝒭_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).\varphi \left( x \right) = \int_\Omega {\varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right)} + F\left( x \right).

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 → 𝔺 of the equation

φ(x)=μ⌢(x)φ(Λx)\varphi \left( x \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \mu } \left( x \right)\varphi \left( {\Lambda x} \right)

which characterizes the limit distribution π ^f for some special f ∈ 𝒭_c.