一类非线性中立型脉冲时滞微分方程解的渐近性

数学物理学报 ›› 2010, Vol. 30 ›› Issue (3): 753-763.

一类非线性中立型脉冲时滞微分方程解的渐近性

魏耿平¹|申建华²

1. 怀化学院数学系湖南怀化 418008|2. 杭州师范大学数学系杭州 310028

收稿日期:2008-04-21 修回日期:2009-06-23 出版日期:2010-05-25 发布日期:2010-05-25
基金资助:
国家自然科学基金(10571050)和湖南省自然科学基金(07JJ6010)资助

Asymptotic Behavior for a Class of Nonlinear Impulsive Neutral Delay Differential Equations

WEI Geng-Ping¹, SHEN Jian-Hua²

1. Department of Mathematics, Huaihua College, Hunan Huaihua 418008|2. Department of Mathematics, Hangzhou Normal University, Hangzhou 310028

Received:2008-04-21 Revised:2009-06-23 Online:2010-05-25 Published:2010-05-25
Supported by:
国家自然科学基金(10571050)和湖南省自然科学基金(07JJ6010)资助

摘要/Abstract

摘要：

该文研究具有正负系数的非线性中立型脉冲时滞微分方程
$$
\left\{
\begin{array}{l}
[x(t)-c(t)x(t-\tau)]'+p(t)f(x(t-\delta))-q(t)f(x(t-\sigma))=0,
\quad t\geq t_0,~ t\not= t_k, \\
x(t_k)= b_kx(t_k^-)\\
\disp\qquad \quad\ \ +(1-b_k)\Big(\int^{t_k}_{t_k-\delta}p(s+\delta)f(x(s)){\rm d}s-
\int^{t_k}_{t_k-\sigma}q(s+\sigma)f(x(s)){\rm d}s\Big),\\
\hskip 9cm k=1,2,3,\cdots,
\end{array}
\right.
$$
获得了该方程的每一个解当$t\to \infty$时趋于一个常数的充分条件.

关键词: 渐近性, Liapunov泛函, 中立型时滞微分方程, 脉冲

Abstract:

This paper is concerned with the
nonlinear impulsive neutral delay differential equation with
positive and negative coefficients,
$$
\left\{
\begin{array}{ll}
[x(t)-c(t)x(t-\tau)]'+p(t)f(x(t-\delta))-q(t)f(x(t-\sigma))=0,\quad
t\geq t_0,~ t\not= t_k, \\
x(t_k)= b_kx(t_k^-)\\
\disp\qquad \quad\ \ +(1-b_k)\left(\int^{t_k}_{t_k-\delta}p(s+\delta)f(x(s)){\rm d}s-
\int^{t_k}_{t_k-\sigma}q(s+\sigma)f(x(s)){\rm d}s\right),\\
\hskip 9.7cm k=1,2,3,\cdots.
\end{array}
\right.
$$
Sufficient conditions are obtained for every solution of
the above equation tending to a constant as $t\to \infty$.

Key words: Asymptotic behavior, Liapunov functional, Neutral delay differential equation, Impulse

中图分类号:

34K20

魏耿平, 申建华. 一类非线性中立型脉冲时滞微分方程解的渐近性[J]. 数学物理学报, 2010, 30(3): 753-763.

WEI Geng-Ping, SHEN Jian-Hua. Asymptotic Behavior for a Class of Nonlinear Impulsive Neutral Delay Differential Equations[J]. Acta mathematica scientia,Series A, 2010, 30(3): 753-763.

参考文献

[1] Lakshimikantham V, Bainov D D, et al. Theory of Impulsive Differential Equations. Singapore: World Scientific, 1989

[2] Bainov D D, Simeonov P S. Systems with Impulse Effect: Stability Theory and Applications. New York: Ellis Horwood Ltd, 1989

[3] Bainov D D, Dimitrova M B. Oscillatory properties of the solutions of impulsive differential equations with a deviating argument and nonconstant coefficients. Rocky Mountain J Math, 1997, 27: 1027--1040

[4] Chen Y S, Feng W Z. Oscillations of second order nonlinear ODE with impulses. J Math Anal Appl, 1997, 210: 150--169

[5] Yu J S, Zhang B G. Stability theorem for delay differential equations with impulses. J Math Anal Appl, 1996, 199: 162--175

[6] Luo Z G, Shen J H. Stability and boundedness for impulsive differential equations with infinite delays. Nonlinear Anal, 2001, 46: 475--493

[7] Shen J H. Global existence and uniqueness, oscillation and nonoscillation of impulsive delay differential equations. Acta Math Sinica, 1997, 40: 53--59

[8] Graef J R, Shen J H, et al. Oscillation of impulsive neutral delay differential equations. J Math Anal Appl, 2002, 268: 310--333

[9] Zhao A M, Yan J R. Asymptotic behavior of solutions of impulsive delay differential equations. J Math Anal Appl, 1996, 201: 943--954

[10] Liu X Z, Shen J H. Asymptotic behavior of solutions of impulsive neutral differential equations. Appl Math Letter, 1999, 12: 51--58

[11] Liu X Z, Ballinger G. Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear Anal, 2002, 51: 633--647

[12] Shen J H, Lou Z G, et al. Impulsive stabilization for functional differential equations via Liapunov functionals. J Math Anal Appl, 1999, 240: 1--15

[13] Ladas G. Asymptotic behavior of solutions of retarded differential equations. Proc Amer Math Soc, 1983, 88: 247--253

[14] Ladas G, Sficas Y G. Asymptotic behavior of oscillatory solutions. Hiroshima Math J, 1998, 18: 351--359

[15] Shen J H, Yu J S. Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients. J Math Anal Appl, 1995, 195: 517--526