NEW OSCILLATION CRITERIA FOR A CLASS OF HIGHER-ORDER NEUTRAL FUNCTIONAL DYNAMIC EQUATIONS ON TIME SCALES

Abstract

In this paper, we study the higher-order neutral functional dynamic equations of the form \begin{equation*} L_ny(t)+q(t)f(|y(\theta(t))|^{\beta}sgn(y(\theta(t))))=0,\ \ t\in[t_0,\infty)_\mathbb{T}, \end{equation*} on an arbitrary time scale $\mathbb{T}$ with $\sup\mathbb{T}=\infty$, where \begin{equation*} L_1y(t)=[y(t)+r(t)y(\tau(t))]^{\Delta},\ L_{i+1}y(t)=[p_i(t)|L_iy(t)|^{\alpha_i}sgn(L_iy(t))]^{\Delta}, \end{equation*} $\alpha_i$, $1\leq i\leq n-1$ and $\beta$ are positive constants, $p_i$, $1\leq i\leq n-1$ and $q$ are rd-continuous functions from $[t_0,\infty)_{\mathbb{T}}$ to $[0,\infty)$ and $r\in \mathrm{C}_{\mathrm{rd}}(\mathbb{T},[0,1))$. The functions $\tau,\theta\in \mathrm{C}_{\mathrm{rd}}(\mathbb{T},\mathbb{T})$ satisfy $\tau(t)\leq t$ and $\lim_{t\rightarrow\infty}\tau(t)=\lim_{t\rightarrow\infty}\theta(t)=\infty$. Criteria are established for the oscillation of solutions for both even and odd order cases. The obtained results here generalize and improve some known results for oscillation of the corresponding higher-order ordinary differential equations [?], but the proof of these counterparts are quite different from the literature. Finally, some interesting examples are given to illustrate the versatility of our main results.