Asymptotic properties of solutions to fourth-order difference equations on time scales

Urszula Ostaszewska, Ewa Schmeidel, Małgorzata Zdanowicz

Abstract


We provide  sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales
\[
\left(a(t)x^{\Delta ^2}(t)\right)^{\Delta ^2}=b(t)f(x(t))+c(t),
\]
such that for a given function $y \colon \mathbb{T} \to \mathbb{R}$  there exists a solution $x \colon \mathbb{T} \to \mathbb{R} $ to  considered equation with asymptotic behaviour $x(t)=y(t)+ \small{o}\left(\frac{1}{t^\beta}\right)$. The presented result is
applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case $\mathbb{T}=\mathbb{R}$.

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