含積分邊界條件的分數階微分方程邊值問題的正解的存在性

張立新,王海菊

(北京聯合大學基礎部,北京 100101)

含積分邊界條件的分數階微分方程邊值問題的正解的存在性

張立新,王海菊

(北京聯合大學基礎部,北京 100101)

研究了含積分邊界條件的分數階微分方程的邊值問題,首先給出格林函數及性質,其次將問題轉化為一個等價的積分方程,最后應用Krasnoselkii及Leggett-W illiam s不動點定理得到了一個及多個正解的存在性,推廣了以往的結果.

積分邊界條件;分數階微分方程;不動點定理;正解

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

1 引言

近年來,分數階微分方程問題得到了廣泛的研究.除了在數學各方面的應用外,分數階微分方程還在流體力學、分數控制系統與分數控制器、各種電子回路、電分析化學、生物系統的電傳導等方面有廣泛的應用[1].關于分數階微分方程的邊值問題的研究取得了很多成果[28],但關于分數階微分方程積分邊值問題的研究相對較少[910].

本文考慮下面的分數階邊值問題:

2 預備知識和引理

3 主要結論

4 例子

參考文獻

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[7]許曉婕,孫新國,呂煒.非線性分數階微分方程邊值問題正解的存在性[J].數學物理學報:A輯,2011,31(2):401-409.

[8]蘆芳,周宗福.一類分數階微分方程三點邊值問題正解的存在性[J].純粹數學與應用數學,2011,27(5):672-678.

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

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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.

北京市自然科學基金(1122016);北京市教委科技計劃面上項目(KM 201311417006);北京聯合大學中自然科學類新起點計劃項目(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