Positive decreasing solutions of second order quasilinear ordinary differential equations in the framework of regular variation

Abstract

This paper is concerned with asymptotic analysis of positive decreasing solutions of the second-order
quasilinear ordinary differential equation
$$
\left(p(t)\f(|x'(t)|)\right)^\prime=q(t)\psi(x(t)),\leqno{\rm(E)}
$$
with the regularly varying coefficients $p,\;q,\;\f,\;\psi$. An application of the theory of regularly variation gives the possibility of determining the precise information about asymptotic behavior at infinity of solutions of equation (E) such that $\dlim_{\tti} x(t)=0,\; \dlim_{\tti} p(t)\f(-x'(t))=\infty$.