Abstract
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 ↦ π f, f ∈ c , and Lipschitz solutions φ that take values in a separable Banach space of the equation φ ( x ) = ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) + 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 ( Λ x ) \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 .
Highlights
If f ∈ Rc, (3) holds and so every Lipschitz function mapping X into a separable Banach space is Bochner integrable with respect to πf
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
In [2] we considered continuity-like property of the function f → πf
Summary
If f ∈ Rc, (3) holds and so every Lipschitz function mapping X into a separable Banach space is Bochner integrable with respect to πf . Assume F is a Lipschitz mapping of X into a separable Banach space Y .
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.