Existence of Nonoscillatory Solutions for a Second-Order Nonlinear Neutral Delay Differential Equation

DOI：10.3770/j.issn:1000-341X.2011.03.016

 作者 单位 郭振宇 辽宁石油化工大学理学院, 辽宁 抚顺 113001

本文研究了一个二阶非线性中立时滞微分方程, 给出对于此方程的非振荡解的一些充分条件,并根据函数$P(t)$值域的不同用五个定理加以阐述.本文的结果推广了文[3, 8, 14]中的相应结果,本文的优点在于省略了$Q_1(t)$ 和$Q_2(t)$ 的限制条件,而这个严格的限制条件在文[3, 8, 14]中是必不可少的.最后举出例子说明本文结果是文[3, 8, 14]的真推广.

A new second-order nonlinear neutral delay differential equation \align \Big(r(t)&\big(x(t) P(t)x(t-\tau)\big)' cr(t)\big(x(t)-x(t-\tau)\big)\Big)' \\&F\big(t,x(t-\sigma_1),x(t-\sigma_2),\ldots,x(t-\sigma_n)\big)=G(t),~~t\ge t_0,\endalign where $\tau>0,\sigma_1,\sigma_2,\ldots,\sigma_n\ge0,P,r\in C([t_0, \infty),\R),F\in C([t_0, \infty)\times \R^n, \R),G\in C([t_0, \infty), \R)$ and $c$ is a constant, is studied in this paper, and some sufficient conditions for existence of nonoscillatory solutions for this equation are established and expatiated through five theorems according to the range of value of function $P(t)$. Two examples are presented to illustrate that our works are proper generalizations of the other corresponding results. Furthermore, our results omit the restriction of $Q_1(t)$ dominating $Q_2(t)$ (See condition $C$ in the text).