In the paper we offer criteria for oscillation of the third order Euler differential equation with delay $$y^{\text{'}\text{'}\text{'}}\left(t\right)+\frac{k^2}{t^3}y\left(ct\right)=0.$$ We provide detail analysis of the properties of this equation, we fill the gap in the oscillation theory and provide necessary and sufficient conditions for oscillation of equation considered.

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.

The aim of this paper is to present new oscillatory criteria for the second order neutral differential equation with mixed argument $${(x\left(t\right)-px(t-\tau ))}^{\text{'}\text{'}}-q\left(t\right)x\left(\sigma \left(t\right)\right)=0.$$ The results include also sufficient conditions for bounded and unbounded oscillation of the equations considered.

In this paper, the authors present some new results for the oscillation of the second order nonlinear neutral differential equations of the form $${\left(r\left(t\right)\psi \left(x\left(t\right)\right){\left[x\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)\right]}^{\text{'}}\right)}^{\text{'}}+q\left(t\right)f\left(x\left[\sigma \left(t\right)\right]\right)=0$$ . Easily verifiable criteria are obtained that are also new for differential equations without neutral term i.e. for p(t)≡0.

This paper is concerned with a class of even order nonlinear differential equations of the form $$\frac{d}{dt}\left(|{\left(x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)\right)}^{(n-1)}{|}^{\alpha -1}{(x\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right))}^{(n-1)}\right)+F(t,x\left(g\left(t\right)\right))=0,$$ where $n$ is even and $t\ge t_0$. By using the generalized Riccati transformation and the averaging technique, new oscillation criteria are obtained which are either extensions of or complementary to a number of existing results. Our results are more general and sharper than some previous results even for second order equations.