含積分邊界條件的分?jǐn)?shù)階微分方程邊值問題的正解的存在性

張立新,王海菊

(北京聯(lián)合大學(xué)基礎(chǔ)部,北京 100101)

含積分邊界條件的分?jǐn)?shù)階微分方程邊值問題的正解的存在性

張立新,王海菊

(北京聯(lián)合大學(xué)基礎(chǔ)部,北京 100101)

研究了含積分邊界條件的分?jǐn)?shù)階微分方程的邊值問題,首先給出格林函數(shù)及性質(zhì),其次將問題轉(zhuǎn)化為一個(gè)等價(jià)的積分方程,最后應(yīng)用Krasnoselkii及Leggett-W illiam s不動(dòng)點(diǎn)定理得到了一個(gè)及多個(gè)正解的存在性,推廣了以往的結(jié)果.

積分邊界條件;分?jǐn)?shù)階微分方程;不動(dòng)點(diǎn)定理;正解

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

1 引言

近年來,分?jǐn)?shù)階微分方程問題得到了廣泛的研究.除了在數(shù)學(xué)各方面的應(yīng)用外,分?jǐn)?shù)階微分方程還在流體力學(xué)、分?jǐn)?shù)控制系統(tǒng)與分?jǐn)?shù)控制器、各種電子回路、電分析化學(xué)、生物系統(tǒng)的電傳導(dǎo)等方面有廣泛的應(yīng)用[1].關(guān)于分?jǐn)?shù)階微分方程的邊值問題的研究取得了很多成果[28],但關(guān)于分?jǐn)?shù)階微分方程積分邊值問題的研究相對(duì)較少[910].

本文考慮下面的分?jǐn)?shù)階邊值問題:

2 預(yù)備知識(shí)和引理

3 主要結(jié)論

4 例子

參考文獻(xiàn)

[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]許曉婕,孫新國(guó),呂煒.非線性分?jǐn)?shù)階微分方程邊值問題正解的存在性[J].數(shù)學(xué)物理學(xué)報(bào):A輯,2011,31(2):401-409.

[8]蘆芳,周宗福.一類分?jǐn)?shù)階微分方程三點(diǎn)邊值問題正解的存在性[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),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,劉錫平,竇麗霞,等.分?jǐn)?shù)階微分方程積分邊值問題正解的存在性[J].吉林大學(xué)學(xué)報(bào):理學(xué)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.

北京市自然科學(xué)基金(1122016);北京市教委科技計(jì)劃面上項(xiàng)目(KM 201311417006);北京聯(lián)合大學(xué)中自然科學(xué)類新起點(diǎn)計(jì)劃項(xiàng)目(zk201203).

張立新(1971-),碩士,副教授,研究方向：微分方程邊值問題.

2010 MSC:34A 08,34B18