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 $ℝ\times I$.

We give an equivalence criterion on property A and property B for delay third order linear differential equations. We also give comparison results on properties A and B between linear and nonlinear equations, whereby we only suppose that nonlinearity has superlinear growth near infinity.

In the paper the fourth order nonlinear differential equation $y^{\left(4\right)}+{\left(q\left(t\right)y^{\text{'}}\right)}^{\text{'}}+r\left(t\right)f\left(y\right)=0$, where $q\in C^1\left([0,\infty )\right)$, $r\in C^0\left([0,\infty )\right)$, $f\in C^0\left(R\right)$, $r\ge 0$ and $f\left(x\right)x>0$ for $x\ne 0$ is considered. We investigate the asymptotic behaviour of nonoscillatory solutions and give sufficient conditions under which all nonoscillatory solutions either are unbounded or tend to zero for $t\to \infty$.

Asymptotic behaviour of oscillatory solutions of the fourth-order nonlinear differential equation with quasiderivates $y^{\left[4\right]}+r\left(t\right)f\left(y\right)=0$ is studied.

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.

In the paper we study the existence of nonoscillatory solutions of the system $x_i^{\left(n\right)}\left(t\right)={\sum }_{j=1}^2{p}_{ij}\left(t\right)f_{ij}\left(x_j\left(h_{ij}\left(t\right)\right)\right),n\ge 2,i=1,2$, with the property $li{m}_{t\to \infty }{x}_i\left(t\right)/t^{k_i}=const\ne 0$ for some $k_i\in \{1,2,...,n-1\},i=1,2$. Sufficient conditions for the oscillation of solutions of the system are also proved.

The aim of this paper is to study asymptotic properties of the solutions of the third order delay differential equation $${\left(\frac{1}{r\left(t\right)}{y}^{\text{'}}\left(t\right)\right)}^{\text{'}\text{'}}-p\left(t\right)y^{\text{'}}\left(t\right)+g\left(t\right)y\left(\tau \left(t\right)\right)=0.*$$ Using suitable comparison theorem we study properties of Eq. () with help of the oscillation of the second order differential equation.