Oscillation Results for Fourth-Order Nonlinear Neutral Dynamic Equations

Abstract

In this paper, the authors study the oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form \begin{equation} \tag{H} (r(t)(y(t)+p(t)y(\alpha(t)))^{{\Delta }^2})^{{\Delta }^2} + q(t)G(y(\beta(t)))h(t)H(y(\gamma(t)))=0 \end{equation} and \begin{equation} \tag{NH} (r(t)(y(t)+p(t)y(\alpha(t)))^{{\Delta }^2})^{{\Delta }^2} + q(t)G(y(\beta(t)))h(t)H(y(\gamma(t)))=f(t), \end{equation} where $\mathbb {T}$ is a time scale with $\sup \mathbb {T}=\infty$, $t \in [t_0,\infty)_\mathbb{T}$, and $t_0\geqslant 0$. They assume that $\int _{t_0}^\infty \frac{\sigma (t)}{r(t)}\Delta t \lt \infty$ and obtain results for various ranges of values of $p(t)$. Examples illustrating the results are included.