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

<p>We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2023-0016_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">Δ</mi></mrow><mn>2</mn></msup></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><msup><mrow><mi mathvariant="normal">Δ</mi></mrow><mn>2</mn></msup></mrow></msup><mo>=</mo><mi>b</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mrow></math>
<tex-math>{\left( {a\left( t \right){x^{{\Delta ^2}}}\left( t \right)} \right)^{{\Delta ^2}}} = b\left( t \right)f\left( {x\left( t \right)} \right) + c\left( t \right),</tex-math>
</alternatives>
</disp-formula>
such that for a given function <italic>y</italic> : 𝕋 → ℝ there exists a solution <italic>x</italic> : 𝕋 → ℝ to considered equation with asymptotic behaviour <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2023-0016_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>o</mi><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msup><mrow><mi>t</mi></mrow><mi>β</mi></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>x\left( t \right) = y\left( t \right) + o\left( {{1 \over {{t^\beta }}}} \right)</tex-math>
</alternatives>
</inline-formula>. The presented result is applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case 𝕋 = ℝ.</p>
</abstract>

We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales



(a(t)xΔ2(t))Δ2=b(t)f(x(t))+c(t),
{\left( {a\left( t \right){x^{{\Delta ^2}}}\left( t \right)} \right)^{{\Delta ^2}}} = b\left( t \right)f\left( {x\left( t \right)} \right) + c\left( t \right),


such that for a given function y : 𝕋 → ℝ there exists a solution x : 𝕋 → ℝ to considered equation with asymptotic behaviour 


x(t)=y(t)+o(1tβ)
x\left( t \right) = y\left( t \right) + o\left( {{1 \over {{t^\beta }}}} \right)

. The presented result is applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case 𝕋 = ℝ.

Asymptotic Properties of Solutions to Fourth-Order Difference Equations on Time Scales

Tatra Mountains Mathematical Publications

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

We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time...

Asymptotic Properties of Solutions to Fourth-Order Difference Equations on Time Scales

Pubblicato online: 28 giu 2023

Pagine: 61 - 76

Ricevuto: 05 gen 2023

DOI: https://doi.org/10.2478/tmmp-2023-0016

Parole chiave
prescribed asymptotic behaviour, time scales, fixed point theorem, fourth-order

© 2023 Urszula Ostaszewska 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 Fourth-Order Difference Equations on Time Scales

Pubblicato online: 28 giu 2023

Pagine: 61 - 76

Ricevuto: 05 gen 2023

DOI: https://doi.org/10.2478/tmmp-2023-0016

Parole chiaveprescribed asymptotic behaviour, time scales, fixed point theorem, fourth-order

© 2023 Urszula Ostaszewska et al., published by Sciendo

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

Parole chiave
prescribed asymptotic behaviour, time scales, fixed point theorem, fourth-order