劉柏楓,韓玉良,孫喜東

(山東工商學(xué)院 數(shù)學(xué)與信息科學(xué)學(xué)院,山東 煙臺(tái) 264005)

0 引 言

概周期函數(shù)是比周期函數(shù)更廣的一類(lèi)函數(shù),在物理、 生物、 數(shù)學(xué)、 控制論及神經(jīng)網(wǎng)絡(luò)等領(lǐng)域應(yīng)用廣泛,目前已有許多研究結(jié)果[1-3]. 文獻(xiàn)[4-11]研究了It隨機(jī)微分方程的概周期解、 偽概周期解及概自守解等. 積分-微分方程在力學(xué)、 電磁學(xué)、 原子反應(yīng)動(dòng)力系統(tǒng)及人口動(dòng)力系統(tǒng)中應(yīng)用廣泛. 文獻(xiàn)[8]研究了如下隨機(jī)積分-微分方程:

文獻(xiàn)[8]在假設(shè)隨機(jī)微分方程(1)對(duì)應(yīng)線(xiàn)性部分是指數(shù)穩(wěn)定的條件下,給出了方程(1)均方概周期解的存在唯一性定理. 但由于指數(shù)型穩(wěn)定性是一個(gè)非常強(qiáng)的條件,因此即使形如A=diag{1,-1}的系統(tǒng)指數(shù)穩(wěn)定也很難做到,而這樣的系統(tǒng)顯然滿(mǎn)足指數(shù)型二分性. 本文在指數(shù)型二分性的條件下研究形如隨機(jī)積分-微分方程(1)的均方概周期解的存在唯一性.

1 線(xiàn)性隨機(jī)積分-微分方程

對(duì)x∈L2(P,Rn),定義

易驗(yàn)證當(dāng)賦予范數(shù)‖·‖∞時(shí),L2(P,Rn)是一個(gè)Banach空間. 用AP(R,L2(P,Rn))表示所有x: R →L2(P,Rn)均方概周期隨機(jī)過(guò)程的集合,用AP(R×L2(P,Rn),L2(P,Rn))表示所有f: R×L2(P,Rn) →L2(P,Rn)一致概周期隨機(jī)過(guò)程的集合.

類(lèi)似于文獻(xiàn)[2]的證明,有:

考慮如下線(xiàn)性隨機(jī)積分-微分方程:

(2)

定義1對(duì)于一個(gè) Rn值的{Ft,t∈R}相適的隨機(jī)過(guò)程{x(t)}t∈R,如果對(duì)任意的t≥s,有

則稱(chēng){x(t)}t∈R為方程(2)的一個(gè)解.

定義2如果存在投影算子P(P是線(xiàn)性的且滿(mǎn)足P2=P)及常數(shù)α1>0,α2>0和β1>1,β2>1,使得:

‖X(t)PX-1(s)‖2≤β1e-α1(t-s),t≥s,

(3)

‖X(t)(I-P)X-1(s)‖2≤β2eα2(s-t),t≤s,

(4)

則系統(tǒng)(2)所對(duì)應(yīng)的齊次線(xiàn)性微分方程稱(chēng)為滿(mǎn)足指數(shù)型二分性. 其中X(t)是方程(2)所對(duì)應(yīng)齊次線(xiàn)性微分方程的基本解矩陣.

定理1假設(shè)方程(2)所對(duì)應(yīng)的齊次線(xiàn)性微分方程滿(mǎn)足指數(shù)型二分性,A(t)∈AP(R,Rn×Rn). 則對(duì)給定的f1,f2,gj∈AP(R,L2(P,Rn))(j=1,2,…,m),方程(2)存在唯一均方概周期解.

證明：設(shè)隨機(jī)過(guò)程{x(t)}t∈R由下式定義:

由式(3)～(5)、 Cauchy-Schuwarz不等式和隨機(jī)積分的It等距公式[12]易證

所以由式(5)定義的隨機(jī)過(guò)程{x(t)}t∈R是有意義的,并且容易驗(yàn)證對(duì)t∈R,由式(5)定義的x(t)是方程(2)的一個(gè)解.

為了證明x(t)是均方概周期的,分別定義

(6)

(7)

設(shè)

2 非線(xiàn)性隨機(jī)積分-微分方程

考慮如下非線(xiàn)性隨機(jī)積分-微分方程:

其中:A(t)是一個(gè)n×n連續(xù)矩陣;F1：R×L2(P,Rn) →L2(P,Rn),F2：R×L2(P,Rn) →L2(P,Rn),Gj：R×L2(P,Rn) →L2(P,Rn)(j=1,2,…,m)是連續(xù)隨機(jī)過(guò)程;B,C,W(t)如前所述.

假設(shè)：

(H1) 方程(10)所對(duì)應(yīng)的齊次線(xiàn)性微分方程滿(mǎn)足指數(shù)型二分性,A(t)∈AP(R,Rn×Rn);

(H2)F1∈AP(R×L2(P,Rn),L2(P,Rn)),F2∈AP(R×L2(P,Rn),L2(P,Rn)),Gj∈AP(R×L2(P,Rn),L2(P,Rn))(j=1,2,…,m),并且分別滿(mǎn)足Lipschitz條件,即存在L1>0,L2>0,L3>0,使得對(duì)任意的x,y∈L2(P,Rn),t∈R,

(11)

又由定理1知,Λ把AP(R,L2(P,Rn))映到自身. 分別考慮作用在Banach空間AP(R,L2(P,Rn))上的非線(xiàn)性算子:

對(duì)x,y∈AP(R,L2(P,Rn)),t∈R,由Cauchy-Schwarz不等式及隨機(jī)積分的It等距公式,有

再由Cauchy-Schwarz不等式,有

(14)

(15)

對(duì)t∈R,由式(13)～(15),有

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