Abstract
In this paper, we establish some sufficient conditions which ensure that the solutions of the third order delay difference equation with a negative middle term Δ(anΔ(Δwn)α)−pn(Δwn+1)α−qnh(wn−l)=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}$$ \\Delta \\bigl(a_{n}\\Delta (\\Delta w_{n})^{\\alpha } \\bigr)-p_{n}(\\Delta w_{n+1})^{ \\alpha }-q_{n}h(w_{n-l})=0,\\quad n\\geq n_{0}, $$\\end{document}are oscillatory. Moreover, we study the asymptotic behavior of the nonoscillatory solutions. Two illustrative examples are included for illustration.
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