具有脉冲的Dirichlet边值问题的Lyapunov不等式及其应用

具有脉冲的Dirichlet边值问题的Lyapunov不等式及其应用

Generalization of Lyapunov Inequality for |Dirichlet BVPs with Impulses and its Applications

摘要/Abstract

参考文献

Metrics

数学物理学报 ›› 2011, Vol. 31 ›› Issue (1): 82-91.

翁爱治¹, 孙继涛²

1.上海政法学院经济管理系上海 201701|.2.同济大学数学系上海 200092

收稿日期:2008-10-08 修回日期:2009-11-06 出版日期:2011-02-25 发布日期:2011-02-25
基金资助:
国家自然科学基金(60874027)和上海高校选拔培养优秀青年教师科研专项基金(szf08004)资助

WENG Ai-Zhi¹, SUN Ji-Tao²

1.Department of Economics and Management, Shanghai University of Political Science and Law, Shanghai 201701; 2.Department of Mathematics, Tongji University, Shanghai 200092

Received:2008-10-08 Revised:2009-11-06 Online:2011-02-25 Published:2011-02-25
Supported by:
国家自然科学基金(60874027)和上海高校选拔培养优秀青年教师科研专项基金(szf08004)资助

摘要：

该文首先研究具有脉冲的线性Dirichlet边值问题

{\begin{cases} x^{″} (t) + a (t) x (t) = 0, t \neq τ_{k}, \\ Δ x (τ_{k}) = c_{k} x (τ_{k}), \\ Δ x^{'} (τ_{k}) = d_{k} x (τ_{k}), \\ x (0) = x (T) = 0, \end{cases} (k = 1, 2 \dots, m)

$\left\{ \begin{array}{ll} x''(t)+a(t)x(t)=0, t\neq \tau_{k}, \\ \Delta x(\tau_{k})=c_{k}x(\tau_{k}),\\ \Delta x'(\tau_{k})=d_{k}x(\tau_{k}), \\ x(0)=x(T)=0, \end{array} \right. (k=1,2\cdots,m)$
给出该Dirichlet边值问题仅有零解的两个充分条件, 其中

a : [0, T] \to R

$a:[0,T]\rightarrow R$ ,

c_{k}, d_{k}, k = 1, 2,

$c_{k}, d_{k}, k=1,2,$

\dots, m

$\cdots,m$ 是常数, $0<\tau_{1}<\tau_{2}\cdots<\tau_{m}

{\begin{cases} x^{″} (t) + f (t, x (t)) = 0, t \neq τ_{k}, \\ Δ x (τ_{k}) = I_{k} (x (τ_{k})), \\ Δ x^{'} (τ_{k}) = M_{k} (x (τ_{k})), \\ x (0) = x (T) = 0 \end{cases} (k = 1, 2 \dots, m)

$\left\{ \begin{array}{ll} x''(t)+f(t,x(t))=0, t\neq \tau_{k}, \\ \Delta x(\tau_{k})=I_{k}(x(\tau_{k})), \\ \Delta x'(\tau_{k})=M_{k}(x(\tau_{k})), \\ x(0)=x(T)=0 \end{array} \right. (k=1,2\cdots,m)$ 解的存在性和唯一性, 其中

f \in C ([0, T] \times R, R)

$f\in C([0,T]\times R,R)$ ,

I_{k}, M_{k} \in C (R, R), k = 1, 2, \dots, m

$I_{k},M_{k}\in C(R, R),k=1,2,\cdots,m$ .
该文主要定理的一个推论将经典的Lyaponov不等式比较完美地推广到脉冲系统.

关键词: 脉冲;边值问题, Lyapunov不等式, Leray-Schauder度, 存在唯一性

Abstract:

In this paper, first the authors obtain the nonexistence of nontrivial solutions for the linear Dirichlet boundary value problem with impulses

{\begin{cases} x^{″} (t) + a (t) x (t) = 0, t \neq τ_{k}, \\ Δ x (τ_{k}) = c_{k} x (τ_{k}), \\ Δ x^{'} (τ_{k}) = d_{k} x (τ_{k}), \\ x (0) = x (T) = 0, \end{cases} (k = 1, 2 \dots, m)

a : [0, T] \to R

$a:[0,T]\rightarrow R$ ,

c_{k}

$c_{k}$ and

d_{k}

$d_{k}$ are
constants,

k = 1, 2, \dots, m

$k=1,2,\cdots,m$ ,

Δ x (τ_{k}) = x (τ_{k}^{+}) - x (τ_{k}^{-})

$\Delta x(\tau_{k})=x(\tau_{k}^{+})-x(\tau_{k}^{-})$ ,

Δ x^{'} (τ_{k}) = x^{'} (τ_{k}^{+}) - x^{'} (τ_{k}^{-})

$\Delta x'(\tau_{k})=x'(\tau_{k}^{+})-x'(\tau_{k}^{-})$ ,
$0<\tau_{1}<\tau_{2}<\cdots<\tau_{m} Secondly, by applying Leray-Schauder degree, the authors obtain the existence and uniqueness of solutions for the nonlinear Dirichlet boundary value problem with impulses

{\begin{cases} x^{″} (t) + f (t, x (t)) = 0, t \neq τ_{k}, \\ Δ x (τ_{k}) = I_{k} (x (τ_{k})), \\ Δ x^{'} (τ_{k}) = M_{k} (x (τ_{k})), \\ x (0) = x (T) = 0, \end{cases} (k = 1, 2 \dots, m)

f \in C ([0, T] \times R, R)

$f\in C([0,T]\times R, R)$ ,

I_{k}, M_{k} \in C (R, R)

$I_{k},M_{k}\in C(R,R)$ ,

k = 1, 2, \dots, m

$k=1,2,\cdots,m$ .
As a corollary of the results, the Lyapunov inequality is extended to impulsive systems.

Key words: Impulses, Boundary value problem, Lyapunov inequality, Leray-Schauder degree, Existence and uniqueness

中图分类号:

34B37

翁爱治, 孙继涛. 具有脉冲的Dirichlet边值问题的Lyapunov不等式及其应用[J]. 数学物理学报, 2011, 31(1): 82-91.

WENG Ai-Zhi, SUN Ji-Tao. Generalization of Lyapunov Inequality for |Dirichlet BVPs with Impulses and its Applications[J]. Acta mathematica scientia,Series A, 2011, 31(1): 82-91.

[1] 康平, 刘立山. 二阶奇异微分方程边值问题正解的存在性. 数学物理学报,2008, 28A(1): 73--80

[2] 刘衍胜. Banach空间中一类带奇异性的脉冲微分方程边值问题的正解. 数学物理学报,2002, 22A(3): 391--398

[3] Torres P. Weak singularities may help periodic solutions to exist. J Differential Equations, 2007, 232: 277--284

[4] Torres P. Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed
point theorem. J Differential Equations, 2003, 190: 643--662

[5] Jiang D, Chu J, Zhang M. Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J Differential Equations, 2005, 211: 282--302

[6] Torres P. Non-trivial periodic solutions of a non-linear Hill's equation with positively homogeneous term. Nonlinear Anal, 2006: 841--844

[7] Chen H, Li Y. Rate of decay of stable periodic solutions of Duffing equations. J Differential Equations, 2007, 236: 493--503

[8] Wong F, Yu S, Ye C, Lian W. Lyapunov's inequality on timescales. Applied Mathematics Letters, 2006, 19: 1293--1299

[9] Torres P, Zhang M. A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum
principle. Math Nach, 2003, 251: 101--107

[10] Zhang M, Li W. A Lyapunov-type stability criterion using L^α norms. Proc Amer Math Soc, 2002, 130: 3325--3333

[11] Deimling K. Nonlinear Functional Analysis. Berlin: Springer, 1985

[12] Cañada A, Montero J, Villegas S. Lyapunov inequalities for partial differential equations. J Funct Anal, 2006, 237: 176--193

[13] Liapunov A. Probleme général de la stabilitédu mouvement. Ann of Math Stud, Vol 17. Princeton, NJ: Princeton Univ Press, 1949

\REF{
[14]} 郭大均, 孙经先, 刘兆理. 非线性常微分方程泛函方法. 山东: 山东科学技术出版社, 2005

Just accepted

Online first

Just accepted

Online first

Viewed

Full text

	From	local

	Times	19
	Rate	100%

Abstract

119

Just accepted	Online first	Issue

0	0	119

From	Others	local

Times	101	18
Rate	85%	15%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

[1]	赵伟霞. 剥脱现象中的某个自由边值问题的整体经典解[J]. 数学物理学报, 2013, 33(6): 1001-1012.
[2]	金山, 鲁世平. 一类多项式系统极限环的唯一性与分支[J]. 数学物理学报, 2011, 31(6): 1669-1673.
[3]	陈淑红, 郭柏灵. 费米子气体附近BCS-BEC跨越的Ginzburg-Landau方程组整体解的存在性[J]. 数学物理学报, 2011, 31(5): 1359-1368.
[4]	栾世霞, 万菲菲, 张新光. 具有变号非线性项的多点边值问题的非平凡解[J]. 数学物理学报, 2010, 30(6): 1503-1513.
[5]	张诚坚, 吕鹏. 空间中的时滞积分微分方程数值方法及其牛顿迭代解的存在唯一性[J]. 数学物理学报, 2010, 30(2): 456-464.
[6]	袁洪君, 佟丽宁. 带有非线性源的以有限Radon测度为初值的拟线性双曲方程的BV解[J]. 数学物理学报, 2010, 30(1): 54-70.
[7]	朱波; 韩宝燕. 非Lipschitz条件下的倒向重随机微分方程[J]. 数学物理学报, 2008, 28(5): 977-984.
[8]	张全举. 一类可扩充杆横截挠度方程的Cauchy问题[J]. 数学物理学报, 2003, 23(6): 704-710.
[9]	陈任昭, 李健全. 与年龄相关的非线性时变种群发展方程解的存在与唯一性[J]. 数学物理学报, 2003, 23(4): 385-400.
[10]	邢家省. 耗散Schrodinger-Poisson方程组的Cauchy问题[J]. 数学物理学报, 2003, 23(2): 215-223.
[11]	李功胜, 马逸尘. 半线性热方程的源项反问题[J]. 数学物理学报, 2001, 21(2): 230-234.
[12]	郝新生. 半线性波方程的一个反问题[J]. 数学物理学报, 2000, 20(zk): 602-609.
[13]	周海云. 带紧扰动的极大单调算子的零点定理[J]. 数学物理学报, 1998, 18(4): 378-383.
[14]	周海云, 陈东青. 带紧扰动的单调算子的特征值问题[J]. 数学物理学报, 1998, 18(3): 330-334.
[15]	保继光, 李美生. Burgers-KdV方程的初边值问题[J]. 数学物理学报, 1996, 16(3): 324-330.