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International Journal of Differential Equations
Volume 2011 (2011), Article ID 328394, 10 pages
http://dx.doi.org/10.1155/2011/328394
Research Article

Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 29 May 2011; Revised 24 August 2011; Accepted 28 September 2011

Academic Editor: Om Agrawal

Copyright © 2011 Hongliang Gao and Xiaoling Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary condition , , , is considered, where is a real number, is the standard Riemann-Liouville differentiation, and with , .

1. Introduction

Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For details, see [16] and references therein.

It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations in terms of special functions [68]. Recently, there are some papers that deal with the existence and multiplicity of solution (or positive solution) of nonlinear initial fractional differential equation by the use of techniques of nonlinear analysis (fixed-point theorems, Leray-Shauder theory, etc.); see [917].

Recently, Bai and Lü [15] studied the existence of positive solutions of nonlinear fractional differential equation where is a real number, is the standard Riemann-Liouville differentiation, and is continuous.

In this paper, we study the existence of positive solutions for fractional differential equation with nonlocal boundary condition where is a real number, is the standard Riemann-Liouville differentiation, and with , .

We assume the following conditions hold throughout the paper:(H1) is both constants with ,(H2), on ,(H3).

Remark 1.1. To our knowledge, there are no results about the existence of positive solutions for problem (1.2).

2. The Preliminary Lemmas

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can be found in the recent literature.

Definition 2.1. The fractional integral of order of a function is given by provided the right side is pointwise defined on .

Definition 2.2. The fractional derivative of order of a function is given by where , provided the right side is pointwise defined on .

Definition 2.3. The map is said to be a nonnegative continuous concave functional on a cone of a real Banach space , provided that is continuous and for all and .

Remark 2.4. As a basic example, we quote for , giving in particular , where is the smallest integer greater than or equal to .
From Definition 2.2 and Remark 2.4, we then obtain the following.

Lemma 2.5. Let . If one assumes , then the fractional differential equation has , where is the smallest integer greater than or equal to , as unique solutions.

Lemma 2.6. Assume that with a fractional derivative of order that belongs to . Then, for some .

Lemma 2.7 (see [15]). Given and , the unique solution of is where

Lemma 2.8. Suppose holds. Given and , the unique solution of is where

Proof. By applying Lemmas 2.6 and 2.7, we have Because by (H1), is converge. Therefore, is converge. is continuous function on , so is converge.
By , , there are . Therefore,

Lemma 2.9 (see [15]). The function defined by (2.9) satisfies the following conditions:(1), for ,(2)there exists a positive function such that

Lemma 2.10 (see [18]). Let E be a Banach space, a cone and two bounded open sets of with . Suppose that is a completely continuous operator such that either(i) and , or(ii) and holds. Then, has a fixed point in .

Lemma 2.11 (see [19]). Let be a cone in real Banach space a nonnegative continuous concave functional on such that , for all , and . Suppose that is completely continuous, and there exist constants such that(C1), and ,(C2), for ,(C3) for with .Then, has at least three fixed points with

Remark 2.12. If there holds , then condition (C1) of Lemma 2.11 implies condition (C3) of Lemma 2.11.

3. The Main Results

Let be endowed with the ordering if for all , and the maximum norm, . Define the cone by .

Let the nonnegative continuous concave functional on the cone be defined by .

Lemma 3.1 (see [15]). Let be the operator defined by , then is completely continuous.

Lemma 3.2. Let be the operator defined by then is completely continuous.

Proof. The proof is similar to Lemma 3.1, so we omit.
Denote

Theorem 3.3. Assume (H1)–(H3) hold, and there exist two positive constants such that(1), (2), where is defined in (*),then problem (1.2) has at least one positive solution such that .

Proof. By Lemmas 2.8 and 3.2, we know is completely continuous, and problem (1.2) has a solution if and only if solves the operator equation . In order to apply Lemma 2.10, we separate the proof into the following two steps.
Step 1. Let . For , we have for all . It follows from (1) that for , Therefore, Step 2. Let . For , we have for all . By assumption (2), for , there is So, Therefore, by (ii) of Lemma 2.10, we complete the proof.

Example 3.4. Consider the problem where .
A simple computation showed . Choosing , we have
With the use of Theorem 3.3, problem (3.6) has at least one positive solutions such that .

Theorem 3.5. Assume (H1)–(H3) hold, and there exist constants such that the following assumptions hold:(A1) for ,(A2) for ,(A3) for , where is defined in (*).Then, the boundary value problem (1.2) has at least three positive solutions with

Proof. We show that all the conditions of Lemma 2.9 are satisfied.
If , then . Assumption (A3) implies for . Consequently, Hence, . In the same way, if , then assumption (A1) yields . Therefore, condition (C2) of Lemma 2.11 is satisfied.
To check condition (C1) of Lemma 2.11, we choose . It is easy to see that , and consequently, Hence, if , then for . From assumption (A2), we have for . So, , for all .
This shows that condition (C1) of Lemma 2.11 is also satisfied.
By Lemma 2.11 and Remark 2.12, the boundary value problem (1.2) has at least three positive solutions with The proof is complete.

Example 3.6. Consider the problem where ,
We have . Choosing , there hold With the use of Theorem 3.5, problem (3.12) has at least three positive solutions , and with

Acknowledgment

This work is sponsored by the NSFC (11101335, 11026060, Gansu Provincial Department of Education Fund (1101-02), NWNU-KJCXGC (03-69).

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