Several recent oscillation criteria are obtained for nonlinear delay impulsive differential equations by relating them to linear delay impulsive differential equations or inequalities, and then comparison and oscillation criteria for the latter are applied. However, not all nonlinear delay impulsive differential equations can be directly related to linear delay impulsive differential equations or inequalities. Moreover, standard oscillation criteria for linear equations cannot be applied directly...

Consider the forced higher-order nonlinear neutral functional differential equation $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}[x\left(t\right)+C\left(t\right)x(t-\tau )]+\sum _{i=1}^m{Q}_i\left(t\right)f_i\left(x(t-{\sigma }_i)\right)=g\left(t\right),t\ge t_0,$$ where $n,m\ge 1$ are integers, $\tau ,{\sigma }_i\in {ℝ}^{+}=[0,\infty )$, $C,Q_i,g\in C([t_0,\infty ),ℝ)$, $f_i\in C(ℝ,ℝ)$, $(i=1,2,\cdots ,m)$. Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general $Q_i\left(t\right)$$(i=1...$

In this paper, we study the existence of oscillatory and nonoscillatory solutions of neutral differential equations of the form $x\left(t\right)-cx(t-r)$’$P\left(t\right)x(t-\theta )-Q\left(t\right)x(t-\delta )$=0 where $c>0$, $r>0$, $\theta >\delta \ge 0$ are constants, and $P$, $Q\in C({ℝ}^{+},{ℝ}^{+})$. We obtain some sufficient and some necessary conditions for the existence of bounded and unbounded positive solutions, as well as some sufficient conditions for the existence of bounded and unbounded oscillatory solutions.

The higher order neutral functional differential equation $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}\left[x\left(t\right)+h\left(t\right)x\left(\tau \right(t\left)\right)\right]+\sigma f\left(t,x\left(g\right(t\left)\right)\right)=0\left(1\right)$$ is considered under the following conditions: $n\ge 2$, $\sigma =\pm 1$, $\tau \left(t\right)$ is strictly increasing in $t\in [t_0,\infty )$, $\tau \left(t\right)<t$ for $t\ge t_0$, ${lim}_{t\to \infty }\tau \left(t\right)=\infty$, ${lim}_{t\to \infty }g\left(t\right)=\infty$, and $f(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and nondecreasing in $u\in (0,\infty )$. A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).