New oscillation criteria are obtained for all solutions of a class of first order nonlinear delay differential equations. Our results extend and improve the results recently obtained by Li and Kuang [7]. Some examples are given to demonstrate the advantage of our results over those in [7].

We study oscillatory properties of solutions of the Emden-Fowler type differential equation $$u^{\left(n\right)}\left(t\right)+p\left(t\right){\left|u\left(\sigma \left(t\right)\right)\right|}^{\lambda }signu\left(\sigma \left(t\right)\right)=0,$$ where $0<\lambda <1$, $p\in L_{\mathrm{loc}}({ℝ}_{+};ℝ)$, $\sigma \in C({ℝ}_{+};{ℝ}_{+})$ and $\sigma \left(t\right)\ge t$ for $t\in {ℝ}_{+}$. Sufficient (necessary and sufficient) conditions of new type for oscillation of solutions of the above equation are established. Some results given in this paper generalize the results obtained in the paper by Kiguradze and Stavroulakis (1998).

Qualitative comparison of the nonoscillatory behavior of the equations $$L_{n}y\left(t\right)+H(t,y\left(t\right))=0$$ and $$L_{n}y\left(t\right)+H(t,y\left(g\left(t\right)\right))=0$$ is sought by way of finding different nonoscillation criteria for the above equations. $L_n$ is a disconjugate operator of the form $$L_n=\frac{1}{p_n\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{p_{n-1}\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}...\frac{\mathrm{d}}{\mathrm{d}t}\frac{·}{p_0\left(t\right)}.$$ Both canonical and noncanonical forms of $L_n$ have been studied.

We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations $$u^{\text{'}\text{'}}\left(x\right)+\sum _i{p}_i\left(x\right)u^{\text{'}}\left(h_i\left(x\right)\right)+\sum _i{q}_i\left(x\right)u\left(g_i\left(x\right)\right)=0$$ without the delay conditions $h_i\left(x\right),g_i\left(x\right)\le x$, $i=1,2,...$, and $$u^{\text{'}\text{'}}\left(x\right)+{\int }_0^{\infty }{u}^{\text{'}}\left(s\right){\mathrm{d}}_s{r}_1(x,s)+{\int }_0^{\infty }u\left(s\right){\mathrm{d}}_s{r}_0(x,s)=0.$$

In this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form $${\left(r_2\left(t\right){\left(r_1\left(t\right)x^{\text{'}}\right)}^{\text{'}}\right)}^{\text{'}}+p\left(t\right)x^{\text{'}}+q\left(t\right)f\left(x\left(g\left(t\right)\right)\right)=0.$$ We establish some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero.

We study oscillatory properties of solutions of the systems of differential equations of neutral type.