Existence for nonoscillatory solutions of higher order nonlinear neutral differential equations

摘要

summary:Consider the forced higher-order nonlinear neutral functional differential equation \[ \frac{{\mathrm d}^n}{{\mathrm d}t^n}[x(t)+C(t) x(t-\tau )]+\sum ^m_{i=1} Q_i(t)f_i(x(t-\sigma _i))=g(t), \quad t\ge t_0, \] where $n, m \ge 1$ are integers, $\tau , \sigma _i\in {\mathbb{R}}^+ =[0, \infty )$, $C, Q_i, g\in C([t_0, \infty ), {\mathbb{R}})$, $f_i\in C(\mathbb{R}, \mathbb{R})$, $(i=1,2,\dots ,m)$. Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general $Q_i(t)$ $(i=1,2,\dots ,m)$ and $g(t)$ which means that we allow oscillatory $Q_i(t)$ $(i=1,2,\dots ,m)$ and $g(t)$. Our results improve essentially some known results in the references.