Abstract
In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the second order neutral difference equation \t\t\tΔ(an(Δzn)β)+qnxn−ℓγ=0,n≥n0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\Delta \\bigl( a_{n} ( \\Delta z_{n} )^{\\beta} \\bigr) + q_{n}x_{n - \\ell}^{\\gamma} = 0,\\quad n \\ge n_{0}, $$\\end{document} where z_{n} = x_{n} + p_{n}x_{n - k}^{alpha} . The established results extend, unify and improve some of the results reported in the literature. Examples are provided to illustrate the importance of the main results.
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