Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation

Sufficient conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the second-order nonlinear differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{\prime\prime}(t)+q(t)|x(t)|^{\gamma}\,{\rm sgn}\, x(t)=0, \quad \quad (A)$$\end{document}where γ is a positive constant different from 1 and q : [a, ∞) → (0, ∞) is a continuous integrable function. We show how an application of the theory of regular variation gives the possibility of determining the precise asymptotic behavior of solutions of both superlinear and sublinear equation (A).