分?jǐn)?shù)階微分方程耦合系統(tǒng)多點積分邊值問題的解

張　寧,　張　娣,　史小藝

(1.中國礦業(yè)大學(xué) 理學(xué)院, 江蘇 徐州 221116; 2.中國礦業(yè)大學(xué) 管理學(xué)院, 江蘇 徐州 221116)

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分?jǐn)?shù)階微分方程耦合系統(tǒng)多點積分邊值問題的解

張寧1,張娣2,史小藝1

(1.中國礦業(yè)大學(xué) 理學(xué)院, 江蘇 徐州 221116; 2.中國礦業(yè)大學(xué) 管理學(xué)院, 江蘇 徐州 221116)

文中討論了一類非線性Caputo型分?jǐn)?shù)階微分方程耦合系統(tǒng)多點積分邊值問題解的存在性。在一定條件下，給出格林函數(shù)，用Schauder不動點定理得到了解存在的充分條件。數(shù)值算例說明了所得定理的適用性。

分?jǐn)?shù)階微分方程； 多點邊值問題； Green函數(shù)； 不動點定理

0　引　言

近些年來,分?jǐn)?shù)階導(dǎo)數(shù)及分?jǐn)?shù)階微分方程在科學(xué)、工程和數(shù)學(xué)等領(lǐng)域得到了重要應(yīng)用,分?jǐn)?shù)階微分方程邊值問題的理論研究,獲得了不少研究成果[1-8]。值得注意的是,分?jǐn)?shù)階耦合系統(tǒng)多點邊值問題作為分?jǐn)?shù)階邊值問題的一種情況,近年來得到研究者們的重視[5-8],如文獻(xiàn)[5]研究了分?jǐn)?shù)階耦合系統(tǒng)反周期邊值問題

受上述文獻(xiàn)啟發(fā),筆者利用不動點定理,研究一類非線性的Caputo型分?jǐn)?shù)階微分方程耦合系統(tǒng)多點積分邊值問題(BVP)

(1)

1　預(yù)備知識

定義1[3]函數(shù)u:(0,+∞)→R的α階Riemann-Liouville分?jǐn)?shù)階積分為

其中α>0,Γ(·)為Gamma函數(shù)。

定義2[3]函數(shù)u:(0,+∞)→R的α階Caputo分?jǐn)?shù)階導(dǎo)數(shù)為

其中α>0,Γ(·)為Gamma函數(shù),n為大于或等于α的最小整數(shù)。

引理1[5]令u∈Cm[0,T]且α∈(m-1,m],m∈N。那么對于t∈[0,T],

引理2假設(shè)y(t)∈C[0,1],那么分?jǐn)?shù)階微分方程邊值問題(BVP)

(2)

等價于積分方程

其中,

證明應(yīng)用引理1,有

u′(0)=

因此,

(3)

反過來,也很容易證明滿足式(3)的解也是BVP(2)的解,得證。

2　主要結(jié)果

假設(shè)(u,v)∈X×X是BVP(1)的一個解,則

定義映射T:X×X→X×X,

(4)

(5)

其中,

為方便,引入記號：

定理1假設(shè)下列條件成立,

(C1)f1,f2:[0,1]×R4→R是連續(xù)的;

其中,常數(shù)ci,di>0(i=1,2,3,4),那么BVP(1)有一個解。

證明(1)先證明映射T:X×X→X×X是連續(xù)的。

對于0

因為

(2)定義Ω={(u,v)∈X×X:‖(u,v)‖≤r},取c1g1(r)+c2g2(r)+c3g3(r)+c4g4(r)≤(r/2-l-l′)/M,d1h1(r)+d2h2(r)+d3h3(r)+d4h4(r)≤(r/2-n-n′)/M′,證明T(Ω)是相對緊的。

?(u,v)∈Ω,由條件(C2),有

l+(c1g1(r)+c2g2(r)+c3g3(r)+c4g4(r))·

l+(c1g1(r)+c2g2(r)+c3g3(r)+c4g4(r))·

同時有

l′+(c1g1(r)+c2g2(r)+c3g3(r)+c4g4(r))·

l′+(c1g1(r)+c2g2(r)+c3g3(r)+c4g4(r))·

故‖T(u,v)‖≤r,即T(Ω)是相對緊的。

0≤t≤1,‖(u,v)≤r‖),

對于任意0≤t1≤t2≤1,有

同時有

同理可得

由于t1→t2,那么

都趨于0,所以T等度連續(xù)。由Ascoli-Arzela引理,知映射T:X×X→X×X是全連續(xù)的。再由Schauder不動點定理,知T有一個不動點,從而BVP(1)有一個解。

3　算　例

例1考慮邊值問題

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對于

[1]LIU XIPING, JIA MEI, XIANG XIUFEN. On the solvability of a fractional differential equation model involving thep-Laplacian operator[J]. Computers & Mathematics with Applications, 2012, 64(10): 3267-3275.

[2]GUEZANE-LAKOUD A, KHALDI R. Solvability of a fractional boundary value problem with fractional integral condition[J]. Nonlinear Analysis: Theory,Methods & Applications, 2012, 75(4): 2692-2700.

[3]CHEN TAIYONG, LIU WENBIN, HU ZHIGANG. A boundary value problem for fractional differential equation withp-Laplacian operator at resonance[J]. Nonlinear Analysis:Theory,Methods & Applications, 2012, 75(2): 3210-3217.

[4]朱彥, 李鑫. 一類非線性分?jǐn)?shù)階微分方程三點邊值問題的解[J]. 黑龍江科技學(xué)院學(xué)報, 2012, 22(1): 93-97.

[5]SUN JIHUA, LIU YILIANG, LIU GUIFANG. Existence of solutions for fractional differential systems with antiperiodic boundary conditions[J]. Computers & Mathematics with Applications, 2012, 64(6): 1557-1566.

[6]WANG GANG, LIU WENBIN, ZHU SINIAN, et al. Existence results for a coupled system of nonlinear fractional 2m-point boundary value problems at resonance[J]. Advances in Difference Equations, Springer, 2011, 2011(1): 44.

[7]ZHANG YINGHAN, BAI ZHANBING, FENG TINGTING. Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance[J]. Computers & Mathematics with Applications, 2011, 61(4): 1032-1047.

[8]SU XINWEI. Boundary value problem for a coupled system of nonlinear fractional differential equations[J]. Applied Mathematics Letters, 2009, 22(1): 64-69.

(編輯王冬)

Solutions for multi-point boundary value problem of coupled system of fractional differential equations with integral boundary conditions

ZHANGNing1，ZHANGDi2，SHIXiaoyi1

(1.College of Sciences, China University of Mining & Technology, Xuzhou 221116, China; 2.School of Management, China University of Mining & Technology, Xuzhou 221116, China)

This paper discusses the existence of solutions to multi-point boundary value problem of a coupled system of nonlinear fractional differential equations with integral boundary conditions. In certain conditions, the study starts with Green’s function, followed by the sufficient conditions for the existence of solutions obtained by using Schauder fixed point theorem. The study ends with the illustration of the applicability of the theorem by a numerical example.

fractional differential equation; muti-point boundary value problem; Green’s function; fixed point theorem

1671-0118(2012)06-0635-05

2012-05-17

國家自然科學(xué)基金項目(10771212)

張寧(1985-),女,山西省晉城人,碩士,研究方向:微分方程邊值問題,E-mail:ninging_love@163.com。

O175.8

A