一类分数阶微分方程三点边值问题正解的存在性

汤小松

一类分数阶微分方程三点边值问题正解的存在性

汤小松

(井冈山大学数理学院，江西，吉安 343009)

利用锥不动点指数理论，研究一类非线性分数阶微分方程的三点边值问题，获得至少存在一个正解的充分条件。由此推广了整数阶微分方程的相应结果。

分数阶微分方程；三点边值问题；锥不动点指数；正解；存在性

0 引言

Zhang[8]利用锥上不动点定理和Leggett- Williams不动点定理讨论了如下两点边值问题

受以上文献的启发，本文研究下面一类非线性分数阶微分方程的三点边值问题

为方便起见，本文总假定：

本文的主要结果如下：

1 准备知识和引理

则为Banach空间。

有唯一解

证：利用引理２可知分数阶方程等价于积分方程

于是，由(5)可得

引理3得证，证毕。

证：由引理3可得

又由引理3，可得

因此，有

因此，有

由(6)和(7)，有

引理5得证，证毕。

另一方面，有

为方便起见，我们引入下面记号。

2 主要结果的证明

下面分有界，无界两种情形考虑。

[1] Miller K S, Ross B. An introduction to the fractional calculus and fractional differential equation[M]. New York: Wiley,1993.

[2] Kibas A A, Anatoly A Srivasfava. Theory and Applications of Fractional Differential Equati-ons[C]. in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006.

[3] Delbosco D, Rodino L. Existence and uniqueness for a nonlinear fractional differential equation[J]. J. Math. Anal. Appl., 1996, 204: 609-625.

[4] Salem H A H. On the existence of continuous solutions for a singular system of nonlinear fractional differential equations[J]. Appl. Math. Comput., 2008, 198: 445-452.

[5] Su X. Boundary value problem for a coupled system of nonlinear fractional differential equations[J]. Appl. Math. Lett., 2009, 22: 64-69.

[6] Ahmad B, Nieto J J. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions[J]. Comput. Math. Appl., 2009, 58: 1838-1843.

[7] BaiZ, Lü H. Positive solutions for boundary value problem of nonlinear fractional differential equation[J]. J. Math. Anal. Appl., 2005, 311: 495-505.

[8] Zhang S. Positive solution for boundary value problem of nonlinear fractional differential equations[J]. Elect. J. Diff. Eqns., 2006(5): 1-12.

[9] Liang Sihua, Zhang Jihui. Positive solution for boundary value problem of nonlinear fractional differential equation[J]. Nonlinear Anal., 2009, 71: 5545-5550.

[10] Goodrich C S. Existence of a positive solution to a class of fractional differential equations[J]. Appl. Math. Lett., 2010, 23: 1050-1055.

[11] Zhao Y, Sun S, Han Z, et al. The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations[J]. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(4) : 2086-2097.

[12] Li X, Liu S, Jiang W. Positive solutions for boundary value problem of nonlinear fractional functional differential equations[J]. Appl. Math. Comput., 2011, 217: 9278-9285.

[13] Guo D, Lakshmikantham V. Nonlinear Problems in Abstract Cones[M]. New York: Academic press, 1988.

Existence of Positive Solutions for a Class of Fractional Differential Equations of Three-point Boundary Value Problem

TANG Xiao-song

(School of Mathematics and Physics, Jinggangshan Universtiy, ji’an, Jiangxi 343009, China)

We study the three-point boundary value problems for nonlinear fractional differential equations by the fixed-point index theory in cone. Furthermore, we obtain the sufficient conditions for the existence of at least one positive solution for this problem, which promotes the corresponding ones of ordinary differential equations of integer order.

fractional differential equations; three-point boundary value problems; cone’s fixed point index; positive solution; existence

O175.8

A

10.3969/j.issn.1674-8085.2012.02.004

1674-8085(2012)02-0014-05

2011-11-16；

2012-01-12

安徽省教育厅自然科学基金(KJ2010B138)，井冈山大学2009年科研课题项目

汤小松(1975-)，男，江西永新人，讲师，硕士，主要从事微分方程和动力系统研究(E-mail: tangxs@jgus.edu.cn).