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

Abstract

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 ∫ Ω ϱ (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 ↦ πf, f ∈ 𝒭c, and Lipschitz solutions φ that take values in a separable Banach space of the equation

Keywords

• random-valued functions
• iterates
• convergence in law
• continuous dependence on the given function
• fourier transform
• iterative functional equations
• continuous and lipschitz solutions
• 26a18
• 39b12
• 39b32
• 39b52
• 60b12
• 60b15