On the structure of positive solutions of a class of fourth order nonlinear differential equations

A detailed analysis is made of the structure of positive solutions of fourth-order differential 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}$$\displaylines{ (p(t)|x^{\prime\prime}|^{\alpha- 1}x^{\prime\prime})^{\prime\prime} + q(t)|x|^{\beta - 1}x = 0,{\rm (A)}} $$\end{document} under the assumption that α, β are positive constants, p(t), q(t) are positive continuous functions on [a,∞), and p(t) satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaylines{ \int_{a}^{\infty}t^{1+({1}/{\alpha})}(p(t))^{-{1}/{\alpha}}\,{\rm d}t<\infty.} $$\end{document}