Existence of the nonoscillatory solutions of higher order neutral dynamic equations on time scales

In this paper, we investigate the existence of the nonoscillatory solutions of the following higher order neutral dynamic equation: {rn(t)[(rn−1(t)(⋯(r1(t)(x(t)−q(t)x(τ(t)))Δ)Δ⋯)Δ)Δ]γ}Δ+f(t,x(δ(t)))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{r_{n}(t)[(r_{n-1}(t)(\cdots(r_{1}(t)(x(t)-q(t)x(\tau(t)))^{\Delta })^{\Delta}\cdots)^{\Delta})^{\Delta}]^{\gamma}\}^{\Delta} +f(t,x(\delta (t)))=0$\end{document} for t∈[t0,∞)T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\in \left .[t_{0},\infty) \right ._{\mathbb{T}}$\end{document}, and obtain some necessary and sufficient conditions for the existence of nonoscillatory bounded solutions for this equation.