In this paper we consider the third-order nonlinear delay differential equation (*) $${\left(a\left(t\right){\left(x^{\text{'}\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+q\left(t\right)x^{\gamma }\left(\tau \left(t\right)\right)=0,t\ge t_0,$$ where $a\left(t\right)$, $q\left(t\right)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau \left(t\right)\le t$ satisfies ${lim}_{t\to infty}\tau \left(t\right)=infty$. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria....

The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation $${\left[a\left(t\right){\left({[x\left(t\right)+p\left(t\right)x\left(\delta \left(t\right)\right)]}^{\text{'}\text{'}}\right)}^{\alpha }\right]}^{\text{'}}+q\left(t\right)x^{\alpha }\left(\tau \left(t\right)\right)=0,E$$ where $\alpha >0$, $0\le p\left(t\right)\le p_0<\infty$ and $\delta \left(t\right)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.

In this paper we shall study some oscillatory and nonoscillatory properties of solutions of a nonlinear third order differential equation, using the results and methods of the linear differential equation of the third order.

Conditions are given for a class of nonlinear ordinary differential equations $x^{\text{'}\text{'}}+a\left(t\right)w\left(x\right)=0$, $t\ge t_0\ge 1$, which includes the linear equation to possess solutions $x\left(t\right)$ with prescribed oblique asymptote that have an oscillatory pseudo-wronskian $x^{\text{'}}\left(t\right)-\frac{x\left(t\right)}{t}$.