王佳,盧金芳

(西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,甘肅 蘭州 730070)

一類滯后脈沖微分方程有界變差解的唯一性

王佳,盧金芳

(西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,甘肅 蘭州 730070)

借助Henstock-Kurzweil積分,在建立了一類滯后脈沖微分方程有界變差解存在性定理的基礎(chǔ)上,建立其解的唯一性定理并給出證明.這個(gè)結(jié)果將唯一性定理從Lebesgue積分意義下推廣到Henstock-Kurzweil積分意義下.

滯后脈沖微分方程;局部有界變差;有界變差解;唯一性

1 引言

脈沖微分方程性質(zhì)研究已有廣泛的結(jié)果,在滯后脈沖微分方程的研究也有許多結(jié)果.關(guān)于滯后脈沖微分方程與廣義常微分方程的關(guān)系最早由文獻(xiàn)[1-2]提出.此后,文獻(xiàn)[3]證明了它們的關(guān)系,同樣的結(jié)果在文獻(xiàn)[4]中進(jìn)一步得到證明.

設(shè)G([a,b],Rn)是正則函數(shù)x:[a,b]→Rn所構(gòu)成的空間,x(t)成為正則函數(shù)是指:x(t)的左右極限存在且有限,

顯然,x在區(qū)間[t0,t0+σ]上是絕對(duì)連續(xù)的.

在本文中,對(duì)于滯后脈沖微分方程的條件比文獻(xiàn) [6]中更廣泛.考慮的 f為 Henstock-Kurzweil可積且φ為正則函數(shù).

本文結(jié)構(gòu)分為四節(jié):第二節(jié)中給出了Henstock-Kurzweil積分的定義以及控制收斂定理.第三節(jié)中回顧滯后脈沖微分方程并給出一些新的關(guān)于其解的相關(guān)結(jié)論.第四節(jié)中建立了滯后脈沖微分方程解的唯一性定理.

2 Henstock-Kurzweil積分與控制收斂定理

3 滯后脈沖微分方程

4 滯后脈沖微分方程有界變差解的唯一性

[1]Imaz C,Vorel Z.Generalized ordinary diff erentialequations in Banach spaces and app lications to functional equations[J].Bol.Soc.M at.M exicana,1966,11:47-59.

[2]O liva F,Vorel Z.Functional equations and generalized ordinary diff erential equations[J].Bol.Soc.M at. M exicana,1966,11:36-40.

[3]Federson M,T′aboas P Z.Topological dynam ics of retarded functional diff erential equations[J].J.Diff. Equations,2003,195:313-331.

[4]Schwabik S.Generalized O rdinary D iff erential Equations[M].Singapore:World Scientific,1992.

[5]Fedrson M,Schwabik S.Generalized ODEs approach to im pulsive retarded diff erential equations[J].Diff erential and Integral Equations,2006,19:1201-1234.

[6]Federson M.Coverse Lyapunov theorem s for retarded functional diff erential equation[J].Cadernos De M atem′atica,2007,8:285-303.

[7]Chew T S.On retarded functionaldiff erentialequationsand Henstock-Kurzweil integrals[J].Diff erent Integral Equat.,1990,9:1-12.

Uniqueness of bounded variatonal solu tions for a class of im pu lsive retarded functional d iff erential equations

Wang Jia,Lu Jinfang
(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

In this paper,based on the existence theorem of bounded variation solution for im pu lsive retarded functional diff erential equations,using the Henstock-Kurzweil integral we estab lish the uniqueness theorem of bounded variation solution for these equations.This result generalizes theorem concerning uniqueness in Lebesgue integral setting to a Henstock-Kurzweil integral setting.

im pu lsive retarded diff erential equations,locally bounded variation solution, bounded variational solutions,uniqueness

O175.12

A

1008-5513(2012)06-0809-10

2012-09-05.

國(guó)家自然科學(xué)基金(11061031).

王佳(1987-),碩士生,研究方向：常微分方程拓?fù)鋭?dòng)力系統(tǒng).

2010 M SC:35A 02