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In this lecture we shall consider the second-order quasilinear differential equations$$\bigl(p(t)|x'|^{\alpha-1}x'\bigr)' +\omega\,q(t)|x|^{\beta-1}x = 0,\qquad \omega=\pm1,%\alpha>\beta>0\leqno{\textrm{(E)}}$$where $p,q:[a,\infty)\rightarrow (0,\infty)$ are continuous  functions. Asymptotic of nonoscilllatory solutions of (E) is essentially affected by the function $p(t)$, more precisely, by the integrals$$    P(t)= \int_a^t\frac{ds}{p(s)^{\frac{1}{\alpha}}} \qquad \textrm{or } \qquad    \pi(t)= \int_t^\infty\frac{ds}{p(s)^{\frac{1}{\alpha}}},$$in case $\int_a^\infty p(t)^{-{1}/{\alpha}}dt$ is either divergent or convergent. Therefore, to analyze  the asymptotic of positive solutions,equations (E)  is considered in the framework of the generalized Karamata functions with respect to $P(t)$ or $\pi(t)$.  The purpose of this lecture to fully describe the overall structureof generalized regularly varying solutions with respect to $P(t)$ or $\pi(t)$,on the basis of behavior of coefficients $p(t)$ and $q(t)$ which are assumed to be generalized regularly varying functions.  An application of the theory of regular variation gives the possibility of obtaining the necessary and sufficient conditions for the existence of three (four) possible types of positive solutions of eq. (E) with $\omega=1$ ($\omega=-1$), together with the precise information about asymptotic behavior at infinity of all types of solutions.

Emden-Fowler differential equations, regularly varying functions, asymptotic behavior, strongly increasing solutions, strongly decreasing solutions, intermediate solutions