衛(wèi)麗娟,王曉云,2

(1.中北大學(xué)數(shù)學(xué)系,山西 太原 030051;2.山西警官高等?？茖W(xué)校,山西 太原 030021)

一類非線性快慢系統(tǒng)非局部問(wèn)題的攝動(dòng)解

衛(wèi)麗娟1,王曉云1,2

(1.中北大學(xué)數(shù)學(xué)系,山西 太原 030051;2.山西警官高等?？茖W(xué)校,山西 太原 030021)

主要討論了一類非線性快慢系統(tǒng)非局部問(wèn)題的攝動(dòng)解,在適當(dāng)?shù)臈l件下,根據(jù)不同邊界層利用伸長(zhǎng)變量和冪級(jí)數(shù)展開(kāi)理論,構(gòu)造了問(wèn)題的形式漸近解,并利用微分不等式理論在整個(gè)區(qū)間上證明了形式漸近解的一致有效性,把奇攝動(dòng)問(wèn)題的攝動(dòng)解推廣到快慢系統(tǒng)非局部問(wèn)題的攝動(dòng)解.

快慢系統(tǒng);非局部問(wèn)題;漸近展開(kāi)式;微分不等式

1 引言

奇異攝動(dòng)理論自應(yīng)用于控制理論的研究以來(lái),一直伴隨著控制理論的發(fā)展而壯大.其中,奇攝動(dòng)快慢動(dòng)力系統(tǒng)出現(xiàn)在很多領(lǐng)域中[16],也是國(guó)際學(xué)術(shù)界研究中的一個(gè)熱門話題[610].近年來(lái),許多學(xué)者做了大量的工作,得到一系列的成果,如Tikhonov定理,O'Malley-Vasil'eva展開(kāi),慢流形的逼近等.本文利用邊界層理論討論了一類非線性快慢系統(tǒng)非局部問(wèn)題的攝動(dòng)解,并給出解的一致有效性證明.

考慮如下快慢系統(tǒng)的非局部問(wèn)題:

2 形式漸近解的構(gòu)造

系統(tǒng)(N)的漸近解可分為外部解和邊界層的校正項(xiàng)兩部分.首先構(gòu)造外部解,然后在此基礎(chǔ)上構(gòu)造校正項(xiàng).

假設(shè)系統(tǒng)(N)有外部漸近展開(kāi)解:

由(19)式可以求得X0,Y0,X1,Y1,依次可以確定Xi,Yi序列,其中i=0,1,2,…,于是得到系統(tǒng)的外部展開(kāi)式,但是外部解未必滿足(3),(4),(7),(8)式,所以需要構(gòu)建系統(tǒng)在t=0和t=1處的校正項(xiàng).

3 邊界層的校正

4 解的一致有效性證明

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Perturbation solution of a class of nonlinear slow-fast system nonlocal problem

Wei Lijuan1,Wang Xiaoyun1,2
(1.Department of Mathematics,North University of China,Taiyuan 030051,China;
2.Shanxi Police Academy,Taiyuan 030021,China)

A class of nonlinear speed system perturbed solution nonlocal problem is discussed in this paper. Under suitable conditions,according to di ff erent boundary layer and using stretchy variable and power series launched theory,the asymptotic expansions of solution of this problem is shown and proved to be uniformly e ff ective using the theory of di ff erential inequality in the whole interval.This paper extends the perturbed solution of singularly perturbed problems to nonlinear slow-fast system nonlocal problem.

speed system,nonlocal problem,asymptotic expansions,di ff erential inequality

O175.14

A

1008-5513(2012)01-0129-08

2011-03-19.

山西省自然科學(xué)基金(2011011002-1);中國(guó)博士后特別資助基金(201104653).

衛(wèi)麗娟(1985-),碩士生,研究方向：應(yīng)用數(shù)學(xué).

王曉云(1972-),副教授,研究方向：應(yīng)用數(shù)學(xué).

2010 MSC:35B25