# Oscillation criteria for third order neutral Emden–Fowler delay dynamic equations on time scales

Research paper by Yunlong Shi, Zhenlai Han; Chuanxia Hou

Indexed on: 13 Aug '16Published on: 29 Jun '16Published in: Journal of Applied Mathematics and Computing

#### Abstract

Abstract This paper is concern with a class of third-order neutral Emden–Fowler dynamic equation \begin{aligned} (a((rz^\Delta )^\Delta )^\alpha )^\Delta (t)+q(t)x^\alpha (\delta (t))=0, \end{aligned} where $$z(t):=x(t)+p(t)x(\tau (t)), \alpha$$ is a quotient of odd positive integers. By generalized Riccati transformation and comparison principles, some new criteria which ensure that every solution is oscillatory are established, which improve and supplement some known results in literatures.AbstractThis paper is concern with a class of third-order neutral Emden–Fowler dynamic equation \begin{aligned} (a((rz^\Delta )^\Delta )^\alpha )^\Delta (t)+q(t)x^\alpha (\delta (t))=0, \end{aligned} where $$z(t):=x(t)+p(t)x(\tau (t)), \alpha$$ is a quotient of odd positive integers. By generalized Riccati transformation and comparison principles, some new criteria which ensure that every solution is oscillatory are established, which improve and supplement some known results in literatures. \begin{aligned} (a((rz^\Delta )^\Delta )^\alpha )^\Delta (t)+q(t)x^\alpha (\delta (t))=0, \end{aligned} \begin{aligned} (a((rz^\Delta )^\Delta )^\alpha )^\Delta (t)+q(t)x^\alpha (\delta (t))=0, \end{aligned} $$z(t):=x(t)+p(t)x(\tau (t)), \alpha$$ $$z(t):=x(t)+p(t)x(\tau (t)), \alpha$$