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

<p>We investigate the discrete equations of the form
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2023-0014_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi mathvariant="normal">Δ</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>r</mi></mrow><mi>n</mi></msub><mi mathvariant="normal">Δ</mi><msub><mrow><mi>x</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>a</mi></mrow><mi>n</mi></msub><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>σ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>b</mi></mrow><mi>n</mi></msub><mo>.</mo></mrow></math>
<tex-math>\Delta \left( {{r_n}\Delta {x_n}} \right) = {a_n}f\left( {{x_{\sigma \left( n \right)}}} \right) + {b_n}.</tex-math>
</alternatives>
</disp-formula>
Using the Knaster-Tarski fixed point theorem, we study solutions with prescribed asymptotic behaviour. Our technique allows us to control the degree of approximation. In particular, we present the results concerning harmonic and geometric approximations of solutions.</p>
</abstract>

We investigate the discrete equations of the form



Δ(rnΔxn)=anf(xσ(n))+bn.
\Delta \left( {{r_n}\Delta {x_n}} \right) = {a_n}f\left( {{x_{\sigma \left( n \right)}}} \right) + {b_n}.


Using the Knaster-Tarski fixed point theorem, we study solutions with prescribed asymptotic behaviour. Our technique allows us to control the degree of approximation. In particular, we present the results concerning harmonic and geometric approximations of solutions.

Asymptotic Properties of Solutions to Discrete Sturm-Liouville Monotone Type Equations

Tatra Mountains Mathematical Publications

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

We investigate the discrete equations of the form



Δ(rnΔxn)=anf(xσ(n))+bn.
\Delta \left( {{r_n}\Delta {x_n}} \right) = {a_n}f\left( {{x_{\sigma \left( n...

Asymptotic Properties of Solutions to Discrete Sturm-Liouville Monotone Type Equations

Published Online: Jun 28, 2023

Page range: 35 - 44

Received: Nov 14, 2022

DOI: https://doi.org/10.2478/tmmp-2023-0014

Keywords
Sturm-Liouville difference equation, prescribed asymptotic behaviour, convergent solution, asymptotically periodic solution, degree of approximation

© 2023 Janusz Migda et al., published by Sciendo

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

Asymptotic Properties of Solutions to Discrete Sturm-Liouville Monotone Type Equations

Published Online: Jun 28, 2023

Page range: 35 - 44

Received: Nov 14, 2022

DOI: https://doi.org/10.2478/tmmp-2023-0014

KeywordsSturm-Liouville difference equation, prescribed asymptotic behaviour, convergent solution, asymptotically periodic solution, degree of approximation

© 2023 Janusz Migda et al., published by Sciendo

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

Keywords
Sturm-Liouville difference equation, prescribed asymptotic behaviour, convergent solution, asymptotically periodic solution, degree of approximation