一類分?jǐn)?shù)階奇異微分方程邊值問題正解的存在性

張金陵，張福珍

（1.中國礦業(yè)大學(xué)理學(xué)院，江蘇徐州 221008；2.徐州高等師范學(xué)校數(shù)理系，江蘇徐州 221116；3.九州職業(yè)技術(shù)學(xué)院高等數(shù)學(xué)教研室，江蘇徐州 221116）

一類分?jǐn)?shù)階奇異微分方程邊值問題正解的存在性

張金陵1，2，張福珍1，3

（1.中國礦業(yè)大學(xué)理學(xué)院，江蘇徐州 221008；2.徐州高等師范學(xué)校數(shù)理系，江蘇徐州 221116；3.九州職業(yè)技術(shù)學(xué)院高等數(shù)學(xué)教研室，江蘇徐州 221116）

利用Leray-Schauder型非線性抉擇和Krasnoselskii錐壓縮拉伸不動(dòng)點(diǎn)定理，給出了一類非線性分?jǐn)?shù)階奇異微分方程邊值問題正解的存在性的充分條件.

非線性抉擇；不動(dòng)點(diǎn)定理；奇異微分方程；正解

0 引言

分?jǐn)?shù)算子在各個(gè)學(xué)科領(lǐng)域中得到了廣泛的應(yīng)用，如物理、機(jī)械、化學(xué)、工程等.值得注意的是，分?jǐn)?shù)階微分方程的理論研究剛起步，分?jǐn)?shù)階微分方程邊值問題作為分?jǐn)?shù)階微分方程理論研究的重要分支之一，近年來得到了研究者們的重視，也獲得了不少研究成果，如文獻(xiàn)[1-15]，本文受文獻(xiàn)[16]啟發(fā)，研究分?jǐn)?shù)階奇異邊值問題.

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

其中α＞0，Γ(?)為gamma函數(shù).

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),則存在分?jǐn)?shù)階微分方程

引理1.6[17]（Leray-Schauder非線性抉擇）假設(shè)Ω是Banach空間U上凸集K的一個(gè)相對(duì)子集．令F→K是緊的且0∈Ω，則

（2）存在一個(gè)點(diǎn)u∈?Ω和λ∈(0,1)，使得u=λTu.

2 主要結(jié)果

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Problem for Fractional Differential Equation

ZHANG Jin-ling1,2,ZHANG Fu-zhen1,3
(1.College of Science,China University of Mining and Technology,Xuzhou 221008,China; 2.Department of Math and Physics,Xuzhou Normal School,Xuzhou 221116,China; 3.Advanced Math Teaching and Research Office,Jiuzhou College of Vocational&Technology,Xuzhou 221116,China)

This paper discussed the existence of positive solution for a singular boundary value problem for frac?tional differential equation by using nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem in a cone.

nonlinear alternative;fixed point theorem;singular boundary value problem;positive solution

O175.8

A

1008-2794（2011）08-0015-07

2011-5-19

張金陵（1974—），女，江蘇徐州人，徐州高等師范學(xué)校數(shù)理系講師，研究方向：微分方程．