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Positive solutions of the nonlinear second-order differential equation $\left(p\left(t\right)\right|x^{\text{'}}{|}^{\alpha -1}{x}^{\text{'}}{)}^{\text{'}}+q\left(t\right){\left|x\right|}^{\beta -1}x=0,\alpha >\beta >0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.

New oscillation criteria are given for the second order sublinear differential equation $${\left[a\left(t\right)\psi \left(x\left(t\right)\right)x^{\text{'}}\left(t\right)\right]}^{\text{'}}+q\left(t\right)f\left(x\left(t\right)\right)=0,t\ge t_0>0,$$ where $a\in C^1\left([t_0,\infty )\right)$ is a nonnegative function, $\psi ,f\in C\left(ℝ\right)$ with $\psi \left(x\right)\ne 0$, $xf\left(x\right)/\psi \left(x\right)>0$ for $x\ne 0$, $\psi$, $f$ have continuous derivative on $ℝ\setminus \left\{0\right\}$ with ${[f\left(x\right)/\psi \left(x\right)]}^{\text{'}}\ge 0$ for $x\ne 0$ and $q\in C\left([t_0,\infty )\right)$ has no restriction on its sign. This oscillation criteria involve integral averages of the coefficients $q$ and $a$ and extend known oscillation criteria for the equation $x^{\text{'}\text{'}}\left(t\right)+q\left(t\right)x\left(t\right)=0$.