Oscillation Behavior Of Third-Order Nonlinear Neutral Dynamic Equations On Time Scales With Distributed Deviating Arguments

Abstract

The aim of this paper is to give oscillation criteria for the third-order neutral dynamic equations with distributed deviating arguments of the form\begin{equation*}\bigg[r(t)\big([x(t)+p(t)x(\tau(t))]^{\Delta\Delta}\big)^{\gamma}\bigg]^{\Delta}+\int_{c}^{d}f(t,x[\phi(t,\xi)])\Delta\xi=0,\end{equation*}where $\gamma>0$ is the quotient of odd positive integers with $r$ and $p$ real-valued rd-continuous positive functions defined on $\mathbb{T}$. $\phi(t,\xi)\in C_{rd}([t_{0},\infty)\times[c,d],\mathbb{T})$ is not decreasing function for $\xi$ and such that$\phi(t,\xi)\leq t$ and $\lim_{t\rightarrow\infty}\min_{\xi\in[c,d]}\phi(t,\xi)=\infty$. We establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.