王翠菁，張金陵

（1.中國(guó)礦業(yè)大學(xué)理學(xué)院，江蘇徐州 221008；

2.徐州工業(yè)職業(yè)技術(shù)學(xué)院信息管理技術(shù)學(xué)院，江蘇徐 州221140；3.徐州高等師范學(xué)校數(shù)理系，江蘇徐州 221116）

分?jǐn)?shù)階微分方程耦合奇異系統(tǒng)解的存在性

王翠菁1,2，張金陵1,3

（1.中國(guó)礦業(yè)大學(xué)理學(xué)院，江蘇徐州 221008；

2.徐州工業(yè)職業(yè)技術(shù)學(xué)院信息管理技術(shù)學(xué)院，江蘇徐 州221140；3.徐州高等師范學(xué)校數(shù)理系，江蘇徐州 221116）

討論非線性分?jǐn)?shù)階微分方程耦合系統(tǒng)三點(diǎn)奇異邊值問(wèn)題，應(yīng)用Green函數(shù)，將其轉(zhuǎn)化為等價(jià)的積分方程耦合系統(tǒng)，利用Schauder不動(dòng)點(diǎn)定理，考察了解的存在性.

三點(diǎn)邊值；Schauder不動(dòng)點(diǎn)定理；耦合系統(tǒng)

0 引言

分?jǐn)?shù)階微分方程在工程、生物、經(jīng)濟(jì)各個(gè)領(lǐng)域都起著重要作用.所以分?jǐn)?shù)階微分方程受到越來(lái)越多學(xué)者的重視，他們?cè)谖⒎址匠滔到y(tǒng)方面得到了不少的研究成果[1-8].

蘇新衛(wèi)[1]研究分?jǐn)?shù)階微分方程耦合系統(tǒng)兩點(diǎn)邊值問(wèn)題解的存在性

其中1＜α,β≤2,u,v＞0,α-v≥1,β-u≥1,f,g:[] 0,1×R×R→R是連續(xù)的，D表示標(biāo)準(zhǔn)的Riemann-Liouville型分?jǐn)?shù)階導(dǎo)數(shù).

Bashir Ahmad[2]研究分?jǐn)?shù)階微分方程耦合系統(tǒng)三點(diǎn)邊值問(wèn)題解的存在性

其中1＜α,β＜2,p,q,γ＞0,0＜η＜1,α-q≥1,β-p≥1,γηα-1＜1,γηβ-1＜1，f,g:[] 0,1×R×R→R是連續(xù)的，D表示標(biāo)準(zhǔn)的Riemann-Liouville型分?jǐn)?shù)階導(dǎo)數(shù).

Feng[3]研究分?jǐn)?shù)階微分方程耦合系統(tǒng)奇異問(wèn)題解的存在性

1 預(yù)備知識(shí)

定義1.1［2］函數(shù)y:() 0,∞→R的α階Riemann-Liouville分?jǐn)?shù)階積分為

定義1.2［2］連續(xù)函數(shù)y:() 0,∞→R的α階Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù)為

其中α＞0，Γ(·)為gamma函數(shù)，n=[α]+1.

引理1.1［2］若α＞0,u∈C(0,1)∩L1(0,1),則分?jǐn)?shù)階微分方程

有唯一解

其中N為大于或等于α的最小整數(shù).

其中G1(t,s)表示分?jǐn)?shù)階邊值問(wèn)題（2）的Green函數(shù),且G1(t,s)＞0,具體形式如下:

證明由引理2.2知

定理1.1[9]（Schauder不動(dòng)點(diǎn)定理）若U是Banach空間X的一個(gè)有界閉凸子集,且T:U→U是全連續(xù)的，則在U中至少有一個(gè)不動(dòng)點(diǎn).

2 主要結(jié)果

證明由于在[0,1]tεF(t)是連續(xù)函數(shù)，故存在M＞0,使得|tεF(t)|≤M,t∈[0,1]且

易知H(0)=0，我們分三種情況證明:

（i）t0=0,?t∈(0,1].

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[8]Wang Jinhua,Xiang Hongjun,Liu Zhigang.Positive solution to Nonzero boundary values problem for a coupled system of nonlinear fractional differential equations[J].International Journal of Differential Equations,2010,Article ID 186928,12 pages,doi:10.1155/ 2010/186928.

Existence of Solutions for a Singular Coupled System of Fractional Differential Equations

WANG Cui-jing1,2,ZHANG Jin-ling1,3
(1.College of Science,China University of Mining&Technology,Xuzhou 221008,China; 2.College of Information Management&Technology,Xuzhou College of Industrial Technology,Xuzhou 221140,China; 3.Department of Math and Physics,Xuzhou Normal School,Xuzhou 221116,China)

This paper discussed the existence of positive solution for a singular coupled system of nonlinear fractional differential equations with three-point boundary conditions.The existence of solutions relies on the Schauder fixed point theorem and the reduction of the considered problem to the equivalent coupled system of integral equations.

three-point boundary;Schauder fixed point theorem;coupled system

O175.8

A

1008-2794（2011）10-0028-07

2011-09-02

王翠菁（1982—），女，山東淄博人，徐州工業(yè)職業(yè)技術(shù)學(xué)院講師，在讀碩士，研究方向：微分方程邊值問(wèn)題.