In this paper, we investigate a class of higher order neutral functional differential equations, and obtain some new oscillatory criteria of solutions.

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 oscillatory properties of solutions of systems $$\begin{array}{ccc}\hfill {[y_1\left(t\right)-a\left(t\right)y_1\left(g\left(t\right)\right)]}^{\text{'}}=& p_1\left(t\right)y_2\left(t\right),y_2^{\text{'}}\left(t\right)=\hfill & \hfill -p_2\left(t\right)f\left(y_1\left(h\left(t\right)\right)\right),t\ge t_0.\end{array}$$

In the paper ordinary neutral differential equations with ?maxima? are considered. Sufficient conditions for oscillation of all solutions are obtained.

In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form $$y_i^{\text{'}}\left(t\right)-p_i\left(t\right)y_{i+1}\left(t\right)=0,i=1,2,\cdots ,n-2,y_{n-1}^{\text{'}}\left(t\right)-p_{n-1}\left(t\right)|y_n\left(h_n\left(t\right)\right){|}^{\alpha }\mathrm{s}gn\left[y_n\left(h_n\left(t\right)\right)\right]=0,y_n^{\text{'}}\left(t\right)\mathrm{s}gn\left[y_1\left(h_1\left(t\right)\right)\right]+p_n\left(t\right){\left|y_1\left(h_1\left(t\right)\right)\right|}^{\beta }\le 0,$$ where $n\ge 3$ is odd, $\alpha >0$, $\beta >0$.

Oscillation criteria, extended Kamenev and Philos-type oscillation theorems for the nonlinear second order neutral delay differential equation with and without the forced term are given. These results extend and improve the well known results of Grammatikopoulos et. al., Graef et. al., Tanaka for the nonlinear neutral case and the recent results of Dzurina and Mihalikova for the neutral linear case. Some examples are considered to illustrate our main results.

We consider nonlinear neutral delay differential equations with variable coefficients. Finite and infinite integral conditions for oscillation are obtained. As an example, the neutral delay logistic differential equation is discussed.

2000 Mathematics Subject Classification: 34K15, 34C10.In this paper, we study the oscillatory behavior of first order nonlinear neutral delay differential equation (x(t) − q(t) x(t − σ(t))) ′ +f(t,x( t − τ(t))) = 0, where σ, τ ∈ C([t0,∞),(0,∞)), q О C([t0,∞), [0,∞)) and f ∈ C([t0,∞) ×R,R). The obtained results extended and improve several of the well known previously results in the literature. Our results are illustrated with an example.