In this paper, the oscillation criteria for solutions of the neutral delay differential equation (NDDE) $${\left(y\left(t\right)-p\left(t\right)y\left(t-\tau \right)\right)}^{\left(n\right)}+\alpha Q\left(t\right)G\left(y\left(t-\sigma \right)\right)=f\left(t\right)$$ has been studied where $p\left(t\right)=1$ or $p\left(t\right)\le 0$, $\alpha =\pm 1$, $Q\in C\left([0,\infty ),R^{+}\right)$, $f\in C\left(\right[0,\infty ),R)$, $G\in C(R,R)$. This work improves and generalizes some recent results and answer some questions that are raised in [1].

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.

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 we present some new oscillatory criteria for the $n$-th order neutral differential equations of the form $${(x\left(t\right)\pm p\left(t\right)x\left[\tau \left(t\right)\right])}^{\left(n\right)}+q\left(t\right)x\left[\sigma \left(t\right)\right]=0.$$ The results obtained extend and improve a number of existing criteria.

Some oscillation criteria for solutions of a general perturbed second order ordinary differential equation with damping (r(t)x′ (t))′ + h(t)f (x)x′ (t) + ψ(t, x) = H(t, x(t), x′ (t)) with alternating coefficients are given. The results obtained improve and extend some existing results in the literature.