<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>We study the problem of solvability of the equation
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0031_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><msub><mo>∫</mo><mi mathvariant="normal">Ω</mi></msub><mrow><mi>g</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow></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><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  {g\left( w \right)} \varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right) + F\left( x \right),</tex-math>
</alternatives>
</disp-formula>
where <italic>P</italic> is a probability measure on a <italic>σ</italic>-algebra of subsets of Ω, assuming Hölder continuity of <italic>F</italic> on the range of <italic>f</italic>.</p>
</abstract>

We study the problem of solvability of the equation



ϕ(x)=∫Ωg(w)ϕ(f(x,ω))P(dω)+F(x),

\varphi \left( x \right) = \int_\Omega  {g\left( w \right)} \varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right) + F\left( x \right),


where P is a probability measure on a σ-algebra of subsets of Ω, assuming Hölder continuity of F on the range of f.

Note on an Iterative Functional Equation

Annales Mathematicae Silesianae

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

{"article-title":"Note on an Iterative Functional Equation"}

We study the problem of solvability of the equation



ϕ(x)=∫Ωg(w)ϕ(f(x,ω))P(dω)+F(x),

\varphi \left( x \right) = \int_\Omega  {g\left( w \right)} \varphi...

Note on an Iterative Functional Equation

Published Online: Jan 10, 2024

Page range: 12 - 17

Received: Aug 31, 2023

Accepted: Dec 21, 2023

DOI: https://doi.org/10.2478/amsil-2023-0031

Keywords
iterative functional equations, Hölder continuous functions

© 2023 Karol Baron et al., published by Sciendo

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

Note on an Iterative Functional Equation

Published Online: Jan 10, 2024

Page range: 12 - 17

Received: Aug 31, 2023

Accepted: Dec 21, 2023

DOI: https://doi.org/10.2478/amsil-2023-0031

Keywordsiterative functional equations, Hölder continuous functions

© 2023 Karol Baron et al., published by Sciendo

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

Keywords
iterative functional equations, Hölder continuous functions