<abstract xmlns="http://www.w3.org/1999/xhtml">The paper consists of two parts. At first, assuming that (Ω, <italic>A, P</italic>) is a probability space and (<italic>X, ϱ</italic>) is a complete and separable metric space with the σ-algebra 𝒝 of all its Borel subsets we consider the set 𝒭<italic>c</italic> of all 𝒝 ⊗ 𝒜<italic>-</italic>measurable and contractive in mean functions <italic>f</italic> : <italic>X ×</italic> Ω → <italic>X</italic> with finite integral ∫ Ω<italic>ϱ</italic> (<italic>f</italic>(<italic>x, ω</italic>), <italic>x</italic>) <italic>P</italic> (<italic>dω</italic>) for <italic>x</italic> ∈ <italic>X</italic>, the weak limit π <italic>f</italic> of the sequence of <italic>iterates</italic> of <italic>f</italic> ∈ 𝒭<italic>c</italic>, and investigate continuity-like property of the function <italic>f ↦</italic> π<italic>f, f</italic> ∈ 𝒭<italic>c</italic>, and Lipschitz solutions <italic>φ</italic> that take values in a separable Banach space of the equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0015_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><msub><mo>∫</mo><mi>Ω</mi></msub><mrow><mi>φ</mi><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>ω</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mi>P</mi><mrow><mo>(</mo><mrow><mi>d</mi><mi>ω</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>.</mo></mrow></math><tex-math>\varphi \left( x \right) = \int_\Omega {\varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right)} + F\left( x \right).</tex-math></alternatives></disp-formula>Next, assuming that <italic>X</italic> is a real separable Hilbert space, Λ: <italic>X</italic> → <italic>X</italic> is linear and continuous with ||Λ || <italic>&lt;</italic> 1, and <italic>µ</italic> is a probability Borel measure on <italic>X</italic> with finite first moment we examine continuous at zero solutions <italic>φ</italic> : <italic>X</italic> → 𝔺 of the equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0015_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mover><mi>μ</mi><mo>⌢</mo></mover><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>φ</mi><mrow><mo>(</mo><mrow><mi>Λ</mi><mi>x</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>\varphi \left( x \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \mu } \left( x \right)\varphi \left( {\Lambda x} \right)</tex-math></alternatives></disp-formula>which characterizes the limit distribution π <italic>f </italic>for some special <italic>f</italic> ∈ 𝒭<italic>c</italic>.</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 ∫ 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 ↦ πf, f ∈ 𝒭c, and Lipschitz solutions φ that take values in a separable Banach space of the equationφ(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 equationφ(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.

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

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

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...

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

Published Online: Jul 09, 2020

Page range: 36 - 44

Received: May 21, 2019

Accepted: Dec 19, 2019

DOI: https://doi.org/10.2478/amsil-2019-0015

Keywords
random-valued functions, iterates, convergence in law, continuous dependence on the given function, Fourier transform, iterative functional equations, continuous and Lipschitz solutions

© 2020 Karol Baron, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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

Published Online: Jul 09, 2020

Page range: 36 - 44

Received: May 21, 2019

Accepted: Dec 19, 2019

DOI: https://doi.org/10.2478/amsil-2019-0015

Keywordsrandom-valued functions, iterates, convergence in law, continuous dependence on the given function, Fourier transform, iterative functional equations, continuous and Lipschitz solutions

© 2020 Karol Baron, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
random-valued functions, iterates, convergence in law, continuous dependence on the given function, Fourier transform, iterative functional equations, continuous and Lipschitz solutions