Oscillatory Behavior of Solutions of Third-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Abstract

The main objective of this article is to investigate oscillation criteria of solutions of the third order nonlinear neutral delay dynamic equations of the form

$$\begin{aligned} (r_{1}(t)((r_{2}(t)z^{\Delta }(t))^{\Delta })^{\gamma })^{\Delta }+P(t,x(\tau _{1}(t)))+F(t,x(\tau _{2}(t)))=0,\quad t\ge t_{0}\text {,} \end{aligned}$$

where $z(t)=x^{\xi }(t)+p(t)x(\tau _{0}(t))$, $\gamma , \xi$ are ratios of odd positive integers, $0<\xi \le 1$ and $0\le p(t)<\infty$ in both canonical and semi-canonical forms are considered. The obtained results improve and complement some of those recently published in the literature. An illustrative example is presented.