含积分边界条件的分数阶微分方程边值问题的正解的存在性

张立新,王海菊

(北京联合大学基础部,北京 100101)

含积分边界条件的分数阶微分方程边值问题的正解的存在性

张立新,王海菊

(北京联合大学基础部,北京 100101)

研究了含积分边界条件的分数阶微分方程的边值问题,首先给出格林函数及性质,其次将问题转化为一个等价的积分方程,最后应用Krasnoselkii及Leggett-W illiam s不动点定理得到了一个及多个正解的存在性,推广了以往的结果.

积分边界条件;分数阶微分方程;不动点定理;正解

DO I：10.3969/j.issn.1008-5513.2013.05.002

1 引言

近年来,分数阶微分方程问题得到了广泛的研究.除了在数学各方面的应用外,分数阶微分方程还在流体力学、分数控制系统与分数控制器、各种电子回路、电分析化学、生物系统的电传导等方面有广泛的应用[1].关于分数阶微分方程的边值问题的研究取得了很多成果[28],但关于分数阶微分方程积分边值问题的研究相对较少[910].

本文考虑下面的分数阶边值问题:

2 预备知识和引理

3 主要结论

4 例子

参考文献

[1]K illbas A A,Srivastava H M,Trujillo J J.Theory and App lication of Fractional D if erential Equation[M]. Am sterdam:Elsevier B V,2006.

[2]BaiZhanbing,L¨u Haisheng.Positive solutions for boundary value p roblem of non linear fractionaldif erential equation[J].J.M ath.Anal.A pp l.,2005,311:495-505.

[3]Bai Zhanbing.On positive solu tions of a non local fractional boundary value p rob lem[J].Non linear Anal., 2010,72:916-924.

[4]Wang Yongqing,Liu Lishan,Wu Yonghong.Positive solutions for a class of fractional boundary value p rob lem w ith changing sign non linearity[J].Nonlinear Anal.,2011,74:6434-6441.

[5]Liang Sihua,Zhang Jihui.Positive solutions for boundary value p rob lem s of non linear fractional dif erential equation[J].Non linear Anal.,2009,71:5545-5550.

[6]Zhao Yige,Sun Shurong,Han Zhen lai,et al.The existence of mu ltiple positive solutions for boundary value prob lem s of nonlinear fractional d if erential equations[J].Comm un.Nonlinear Sci.Num er.Simu lat., 2011,16:2086-2097.

[7]许晓婕,孙新国,吕炜.非线性分数阶微分方程边值问题正解的存在性[J].数学物理学报:A辑,2011,31(2):401-409.

[8]芦芳,周宗福.一类分数阶微分方程三点边值问题正解的存在性[J].纯粹数学与应用数学,2011,27(5):672-678.

[9]Cabada A lberto,Wang Guotao.Positive solutions of nonlinear fractional dif erential equationswith integral boundary conditions[J].J.M ath.Anal.App l.,2012,389:403-411.

[10]金京福,刘锡平,窦丽霞,等.分数阶微分方程积分边值问题正解的存在性[J].吉林大学学报:理学版, 2011,49(5):823-828.

[11]K rasnoselskiiM A.Positive Solutions of Operator Equations[M].Groningen:Noordhof,1964.

[12]Leggett RW,W illiam s L R.M u ltiple positive fxed points of non linear operators on ordered Banach spaces [J].Ind iana Univ.M ath.J.,1979,28:673-688.

The ex istence o f p ositive so lu tions for boundary value p rob lem s of fractional d if eren tial equations w ith in tegral boundary cond itions

Zhang Lixin,Wang Haiju

(Department of Basic Courses,Beijing Union University,Beijing 100101,China)

In this paper,we consider the existence of positive solu tions for fractional boundary value p rob lem s w ith integral boundary conditions.First,w e give the properties of G reen′s function.Second,the p rob lem has been reduced to the equivalent Fredholm integral equation.Finally,using K rasnoselkii f xed point theorem and Leggett-W illiam s f xed point theorem,some resu ltson the existenceof positive solutions are obtained.Thework is an extension of the p revious resu lts.

integral boundary conditions,fractional dif erential equation,f xed point theorem, positive solu tion

O175.8

A

1008-5513(2013)05-0450-08

2013-05-02.

北京市自然科学基金(1122016);北京市教委科技计划面上项目(KM 201311417006);北京联合大学中自然科学类新起点计划项目(zk201203).

张立新(1971-),硕士,副教授,研究方向：微分方程边值问题.

2010 MSC:34A 08,34B18