Asymptotics and oscillation of nth-order nonlinear differential equations with p-Laplacian like operators

This paper is concerned with nth-order nonlinear differential equations of the form (a(t)|x(n−1)(t)|p−2x(n−1)(t))′+r(t)|x(n−1)(t)|p−2x(n−1)(t)+q(t)|x(g(t))|p−2x(g(t))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(a(t)|x^{(n-1)}(t)|^{p-2}x^{(n-1)}(t) )^{\prime}+ r(t)|x^{(n-1)}(t)|^{p-2}x^{(n-1)}(t)+q(t)|x(g(t))|^{p-2}x(g(t))=0 $\end{document} with n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\ge2$\end{document}. By discussing the signs of ith-order derivatives of eventually positive solutions, for i=1,…,n−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i=1,\ldots,n-1$\end{document}, and using the generalized Riccati technique and integral averaging technique, we derive new criteria for oscillation and asymptotic behavior of the equation. Our results generalize and improve many existing results in the literature.