New oscillation criteria for second-order neutral dynamic equations on time scales via Riccati substitution

Abstract

In this paper, we consider the second-order nonlinear neutral functional dynamic equation

$\left( p(t)\left( \left[ y(t)+r(t)y(\tau (t))\right] ^{\Delta }\right) ^{\gamma }\right) ^{\Delta }+f(t,y(\delta (t)))=0$

on a time scale $\mathbb{T}$ and establish some new sufficient conditions for oscillation. Our results improve oscillation results for neutral delay dynamic equations on time scales and are new when $\delta (t)>t$ and/or $% 0<\gamma <1.$ Furthermore our results can be applied on the time scales $% \mathbb{T=}h\mathbb{T}$, for $h>0$, $\mathbb{T=}q^{\mathbb{N}}=\{t:t=q^{k}$% \textbf{, }$k\in \mathbb{N}$, $q>1$, $\mathbb{T=N}^{2}=\{t^{2}:t\in \mathbb{N% }\},$ $\mathbb{T}_{2}\mathbb{=}\{\sqrt{n}:n\in \mathbb{N}_{0}\},$ $\mathbb{T}% _{3}\mathbb{=}\{\sqrt[3]{n}:n\in \mathbb{N}_{0}\},$ and when $\mathbb{T=T}% _{n}=\{t_{n}:n\in \mathbb{N}_{0}\}$ where $\{t_{n}\}$ is the set of harmonic numbers, etc.