Asymptotic properties of solutions to fourth-order difference equations on time scales
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 $\mathbb{T}=\mathbb{R}$.
\[
\left(a(t)x^{\Delta ^2}(t)\right)^{\Delta ^2}=b(t)f(x(t))+c(t),
\]
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 $\mathbb{T}=\mathbb{R}$.
Full Text:
Subscribers OnlyDOI: https://doi.org/10.2478/tmmp-2023-0016