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.

In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for $$\left\}\begin{array}{c}{(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0,\text{in}(\alpha ,\infty ),u^{\left(i\right)}\left(\xi \right)=0,i=0,1,\cdots ,n-1,\text{and}\xi \in (\alpha ,\infty ),\hfill \end{array}\right.$$ must be unbounded, provided $f(t,z)z\ge 0$, in $E\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E\times I$. (B) Every bounded solution for ${(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0$, in $ℝ$, must be constant, provided $f(t,z)z\ge 0$ in $ℝ\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $ℝ\times I$.

Sufficient conditions are given under which the sequence of the absolute values of all local extremes of $y^{\left[i\right]}$, $i\in \{0,1,\cdots ,n-2\}$ of solutions of a differential equation with quasiderivatives $y^{\left[n\right]}=f(t,y^{\left[0\right]},\cdots ,y^{[n-1]})$ is increasing and tends to $\infty$. The existence of proper, oscillatory and unbounded solutions is proved.

Inequalities for some positive solutions of the linear differential equation with delay ẋ(t) = -c(t)x(t-τ) are obtained. A connection with an auxiliary functional nondifferential equation is used.

In this paper new generalized notions are defined: $\Psi$-boundedness and $\Psi$-asymptotic equivalence, where $\Psi$ is a complex continuous nonsingular $n\times n$ matrix. The $\Psi$-asymptotic equivalence of linear differential systems $y^{\text{'}}=A\left(t\right)y$ and $x^{\text{'}}=A\left(t\right)x+B\left(t\right)x$ is proved when the fundamental matrix of $y^{\text{'}}=A\left(t\right)y$ is $\Psi$-bounded.

Asymptotic representations of some classes of solutions of nonautonomous ordinary differential $n$-th order equations which somewhat are close to linear equations are established.