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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 $\dlim_{\tti} x(t)=0,\; \dlim_{\tti} p(t)\f(-x'(t))=\infty$.

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