Asymptotic behaviour and existence of nonoscillatory solutions of second-order neutral delay difference equations

In this paper, we give a classification of nonoscillatory solution of a second-order neutral delay difference equation of the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Delta ^2 (x_n - c_n x_{n - \tau } ) = f(n, x_{g_1 (n)} ,..., x_{g_m (n)} ).$$ \end{document} Some existence results for each kind of nonoscillatory solutions are also established.