Oscillation Behavior Of Third-Order Quasilinear Neutral Delay Dynamic Equations On Time Scales

Abstract

The aim of this paper is to give oscillation criteria for the third-order quasilinear neutral delay dynamic equation \begin{equation*}\bigg[r(t)\big([x(t)+p(t)x(\tau_{0}(t))]^{\Delta\Delta}\big)^{\gamma}\bigg]^{\Delta}+q_{1}(t)x^{\alpha}(\tau_{1}(t))+q_{2}(t)x^{\beta}(\tau_{2}(t))=0,\end{equation*}on a time scale $\mathbb{T}$, where $0<\alpha<\gamma<\beta$. By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.