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.

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.

The authors study the n-th order nonlinear neutral differential equations with the quasi – derivatives $L_n[x\left(t\right)+{(-1)}^{r}P\left(t\right)x\left(g\left(t\right)\right)]+\delta Q\left(t\right)f\left(x\left(h\left(t\right)\right)\right)=0,$ where $n\ge 2,r\in \{1,2\},$ and $\delta =\pm 1.$ There are given sufficient conditions for solutions to be either oscillatory or they converge to zero.

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.