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

Ostaszewska, Urszula; Schmeidel, Ewa; Zdanowicz, Małgorzata

doi:10.2478/tmmp-2023-0016

Abstract

We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales ${(a (t) x^{Δ^{2}} (t))}^{Δ^{2}} = b (t) f (x (t)) + c (t),$ {\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 $x (t) = y (t) + o (\frac{1}{t^{β}})$ 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.

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

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