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.

The aim of the paper is to study the structure of oscillatory solutions of a nonlinear third order differential equation $y^{\text{'}\text{'}\text{'}}+p{y}^{\text{'}\text{'}}+q{y}^{\text{'}}+rf(y,y^{\text{'}},y^{\text{'}\text{'}})=0$.

The aim of this paper is to study the global structure of solutions of three differential inequalities with respect to their zeros. New information for the differential equation of the third order with quasiderivatives is obtained, too.

Oscillation and nonoscillation criteria are established for the equation $$u^{\text{'}\text{'}}+{p\left(t\right)\left|u\right|}^{\alpha }{\left|u^{\text{'}}\right|}^{1-\alpha }sgnu=0,$$ where $\alpha \in ]0,1]$, and $p:[0,+\infty [\to [0,+\infty [$ is a locally summable function.

We obtain sufficient conditions for every solution of the differential equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ to oscillate or to tend to zero as $t$ approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when $G$ has sub-linear growth at infinity. Our results also apply to the neutral equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)G\left(y\left(g\left(t\right)\right)\right)=f\left(t\right)$$ when $q\left(t\right)$ has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$x^{\text{'}\text{'}\text{'}}\left(t\right)+q\left(t\right)x^{\text{'}}\left(t\right)+r\left(t\right){\left|x\right|}^{\lambda }\left(t\right)\mathrm{sgn}x\left(t\right)=0,t\ge 0.$$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \le 1$ and if the corresponding second order differential equation $h^{\text{'}\text{'}}+q\left(t\right)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.