Classification and existence of positive solutions of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations 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}$$(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad \mathit{\Delta}^2 \left( {p_n \left( {\mathit{\Delta}^2 y_n } \right)^{\alpha } } \right) + q_n y_{n + 3}^{\beta } = 0,\quad n \in \mathbb{N}, $$\end{document}where α and β are the ratios of odd positive integers, and {pn} and {qn} are positive real sequences defined for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n \in \mathbb{N}\left( {n_0 } \right) $$\end{document} satisfying the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum\limits_{{n = n_0 }}^{\infty } {n\left( {\frac{n}{{p_n }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. } \alpha }}} < \infty .} $$\end{document} We classify the nonoscillatory solutions of (Ω) and establish necessary and/or sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior.