Abstract and Applied Analysis

Volume 2014 (2014), Article ID 602604, 8 pages

http://dx.doi.org/10.1155/2014/602604

Research Article

Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

Hongjie Liu,¹ Xiao Fu,² and Liangping Qi²

¹School of Land Science and Technology, China University of Geosciences, Beijing 10083, China

²School of Science, China University of Geosciences, Beijing 10083, China

Received 18 March 2014; Accepted 26 April 2014; Published 20 May 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Hongjie Liu et al. 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.

We are concerned with the following nonlinear three-point fractional boundary value problem: , , , and , where , , , is the standard Riemann-Liouville fractional derivative, is continuous for , and is continuous on . By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

1. Introduction

Fractional differential equations have been of great interest recently. With the development of nonlinear science, the researchers found that nonlinear fractional differential equations could describe something's changing rules more accurately. Therefore, it is significant to study nonlinear fractional differential equations to solve the nonlinear problems. Recently, many researchers paid attention to existence and multiplicity of solution of the boundary value problem for fractional differential equations with different boundary conditions, such as [1–10]. Bai and Lü [1] investigated the existence and multiplicity of positive solutions for nonlinear fractional boundary value problem:

where is a real number, is the standard Riemann-Liouville fractional derivative, and is continuous. By means of some fixed-point theorems on cone, some existence and multiplicity results of positive solutions are obtained. Agarwal et al. [2] investigated the existence of positive solutions for the singular fractional boundary value problem:

where are real numbers, is positive, and is the standard Riemann-Liouville fractional derivative. By means of a fixed point theorem on cone, the existence of positive solutions is obtained. Delbosco and Rodino [3] investigated the nonlinear Dirichlet-type problem:

They proved that if is a Lipschitzian function, then the problem has at least one solution in a certain subspace of . In this paper, we study the following three-point fractional boundary value problem. Consider

where is the standard Riemann-Liouville fractional derivative, is continuous, and is continuous on . By using Krasnoesel'skii's fixed-point theorem, we get the existence of at least one positive solution.

2. Background Materials and Preliminaries

For convenience of the readers, we present here the necessary definitions from fractional calculus theory. These definitions can be found in the recent literature [1–5].

Definition 1. The fractional integral of order of a function is given by

provided that the right side is pointwise defined on .

Definition 2. The fractional derivative of order of a function is given by

where , provided that the right side is pointwise defined on .

Lemma 3. Let , if we assume , then the fractional differential equation has

as unique solutions.

Lemma 4. Assume that with a fractional derivative of order that belongs to . Then

for some .

Definition 5. Let be a real Banach space. A nonempty closed convex set is called cone of if it satisfies the following conditions:

(1)  implies  ;(2)  implies  .

Lemma 6. Let be a Banach space and let be a cone in . Assume that and are open subsets of with . Let be a completely continuous operator. In addition, suppose that either

(H1) and or(H2) and

holds. Then has a fixed point in .

In the following, we present the Green function of fractional differential equation boundary value problem.

Lemma 7 (see [4]). Let , then the boundary value problem

has a unique solution

where

Lemma 8. The function defined by (11) satisfies   for  .

Proof. (1) For ,

Let . There is

With , there is . Hence, function is monotonically decreasing in . With , . Let , there is

With , we have . With , there is , then , there is .

(2) For ,

Let ; it is obvious that function is monotonically increasing in , when ,

With , there is ; we have , then .

(3) For ,

Let , then is monotonically decreasing in ,

With , there is , then , hence .

(4) For ,

Therefore, for . The proof is complete.

Lemma 9. for .

Proof. To get , firstly, we prove the following inequality:

where , . Let , then

With , there is ; with , there is ; then . It means that is monotonically increasing in , then ; we have

Here we prove .

(1) For ,

With ,   is equivalent to

or

(a)For ;(b)for , ;(c)for ,

 if , ; if , from (22), we have ;(d)for , , if ; it is obvious that . If , by (22), there is ; then

Consider

It means that holds.

(2) For ,

With ,    is equivalent to

or

From (22), we have ; with , we have ; then ; with , there is

It means that holds.

(3) For ,

With , is equivalent to

or

With , , there is ; with , there is

It means that holds.

(4) For ,

It is obvious that . The proof is complete.

Lemma 10. for , where is an integer and

Proof. From (11), we have

It is obvious that , for ,

(1) For , let

By Lemma 8, for ,

With , then ; with . It means that is monotonically increasing in . Similarly,

It means that is monotonically decreasing in . Therefore

(2) For , let

By Lemma 8, for ,

It means that is monotonically increasing in

It means that is monotonically decreasing in

(3) For , let

By Lemma 8, for ,

then is monotonically decreasing in . Consider

then is monotonically increasing in . Therefore,

(4) For , let

It is obvious that is monotonically increasing in and monotonically decreasing in ; then

we have

there is

Let

Therefore, .

The proof is complete.

We consider the Banach space equipped with standard norm . We define a cone by

Define an integral operator by

Lemma 11. It holds the following.

(1) is completely continuous.(2) is a positive solution of the fractional boundary value problem (4) if and only if is a fixed point of the operator in cone .

Proof. For , by Lemma 10, ; for ,

It is obvious that . Thus . In addition, standard arguments show that is completely continuous.

(2) It is obvious that is the positive solution of BVP (4) if and only if

It can be proved by the definition of integral operator .

3. Main Results

We denote some important constants as follows:

Here we assume that if , if , if , and if .

Theorem 12. Suppose that , then for each , BVP (4) has at least one positive solution.

Proof. We choose sufficiently small such that . By the definition of , we can see that there exists , such that for . For with , we have

Then we have . Thus if we let , then   for  . We choose and such that . There exists , such that for . Therefore, for each with , we have

Thus if we let , then and for . Condition (H1) of Krasnoesel'skii's fixed point theorem is satisfied. So there exists a fixed point of in . This completes the proof.

Theorem 13. Suppose that , then for each , BVP (4) has at least one positive solution.

Proof. Choose sufficiently small such that . From the definition of , we see that there exists , such that for . If with , we have

Let ; then we have for . Choose ; then we have . There exists , such that for . For each , we have

Let ; then , and we have for . Condition (H2) of Krasnoesel'skii's fixed-point theorem is satisfied. So there exists a fixed point of in . This completes the proof.

4. An Example

Example 1. Consider the following three-point fractional boundary value problem:

where .

By calculations,

Let , then

Consider

The condition is obtained. From Theorem 12, we see that if , the problem (67) has a positive solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the reviewers for the valuable suggestions. This paper was supported by the Fundamental Research Funds for the Central Universities (no. 2652012141) and Beijing Higher Education Young Elite Teacher Project.

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