Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 736108, 6 pages
http://dx.doi.org/10.1155/2015/736108

Research Article

Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

Jun-Rui Yue,¹ Jian-Ping Sun,¹ and Shuqin Zhang²

¹Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
²Department of Mathematics, China University of Mining and Technology, Beijing 100083, China

Received 16 July 2015; Accepted 14 September 2015

Academic Editor: Chris Goodrich

Copyright © 2015 Jun-Rui Yue 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 consider the following boundary value problem of nonlinear fractional differential equation: ,  ,  ,  , where is a real number, denotes the standard Caputo fractional derivative, and is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.

1. Introduction

Fractional calculus has demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering, for example, fluid flow, rheology, electrical networks, chemical physics, control theory of dynamical systems, optics, and signal processing [1].

Since the discussion of many problems can be summed up in the study of boundary value problems (BVPs for short) for nonlinear fractional differential equations, recently, the existence or uniqueness of solutions or positive solutions of BVPs for nonlinear fractional differential equations has received much attention from many authors; see [2–13] and the references therein.

In particular, Zhang [6] studied the existence and multiplicity of positive solutions to the following BVP of nonlinear fractional differential equation:

where is a real number and denotes the standard Caputo fractional derivative. The main tools used were the Guo-Krasnoselskii and Leggett-Williams fixed point theorems.

In 2009, by using the nonlinear alternative of Leray-Schauder type and Guo-Krasnoselskii fixed point theorem, Bai and Qiu [12] discussed the existence of a positive solution to the following singular BVP of nonlinear fractional differential equation:

where is a real number, denotes the standard Caputo fractional derivative, and is singular at .

Motivated greatly by the above-mentioned works, in this paper, we consider the following BVP of nonlinear fractional differential equation:

where is a real number, denotes the standard Caputo fractional derivative, and is continuous. First, Green’s function for the associated linear BVP is constructed. It is necessary to point out that is singular at . Next, some useful properties of are studied. Finally, existence results of at least one positive solution for BVP (3) are obtained.

In order to obtain our main results, we need the following Guo-Krasnoselskii fixed point theorem [14, 15].

Theorem 1. Let be a Banach space and let be a cone in . Assume that and are bounded open subsets of such that , and let be a completely continuous operator such that either

(1) or(2)

Then has a fixed point in .

2. Preliminaries

For the convenience of the reader, we present here some necessary definitions and results from fractional calculus theory. These definitions and results can be found in the recent literature [1].

In this section, we always assume that and denotes the integer part of .

Definition 2. The Riemann-Liouville fractional integral of order is defined by

Definition 3. The Riemann-Liouville fractional derivative of order is defined by

Definition 4. The Caputo fractional derivative of order is defined via the above Riemann-Liouville fractional derivative by

where

Lemma 5. Let be given by (7) and or . Then

where .

3. Main Results

In the remainder of this paper, we always assume that is a real number and is continuous.

Lemma 6. Let be a given function. Then the BVP

has a unique solution:

where

Proof. In view of the equation in (9) and Lemma 5, we know that

and so,

which together with the boundary conditions in (9) implies that

Therefore, BVP (9) has a unique solution:

Now, for , we denote

Remark 7. From the definition of function , it is not difficult to get that for and .

Lemma 8. Green’s function defined by (11) has the following properties. Consider

(1) and for ,(2)    

Proof. Since (1) is obvious, we only need to prove (2).
First, for any , if , then

and if , then

Next, for any , if , then

and if , then

Let be equipped with norm and

Then it is easy to check that is a Banach space and is a cone in .

Now, we define an operator on by

Since is singular at , we need to prove that operator is well defined. In fact, for any fixed , we have If we let

then

which together with Lemma 8 implies that

Since is convergent, we obtain that is uniformly convergent on . This shows that operator is well defined. Furthermore, we know that if , then and if is a fixed point of in , then is a nonnegative solution of BVP (3).

Lemma 9. is completely continuous.

Proof. First, for any , in view of Lemma 8, we have

So,

which together with Lemma 8 implies that

which shows that

Therefore, .
Next, we prove that is continuous. Suppose that and . Then there exists such that for any . If we let

then for any and , in view of Lemma 8, we have

By applying Lebesgue dominated convergence theorem, we get

which indicates that is continuous.
Finally, we show that is compact. Assume that is a bounded set. Then there exists a constant such that for any . In what follows, we will prove that is relatively compact. Let

On the one hand, for any , there exists such that , and so, it follows from (34) and Lemma 8 that

which together with the fact that is convergent implies that is uniformly bounded. On the other hand, for any , since is convergent, we may choose such that

since is uniformly continuous on , there exists such that for any with ,

For any , there exists such that , and so, for any with , it follows from (36), (37), and Lemma 8 that

which indicates that is equicontinuous. By Arzela-Ascoli theorem, we know that is relatively compact. Therefore, is completely continuous.

For convenience, we denote

Theorem 10. Assume that there exist two different positive constants and such that the following conditions are fulfilled:

Then BVP (3) has a positive solution with .

Proof. Without loss of generality, we assume that . Let

Then for any , we get , which together with (40) and Lemma 8 implies that

This shows that

For any , we get , which together with (41) and Lemma 8 implies that

This indicates that

Therefore, it follows from Theorem 1, Lemma 9, (44), and (46) that the operator has a fixed point , which is a desired positive solution of BVP (3).

Example 11. We consider the BVP

Since , a direct calculation shows that

So,

Let . If we choose and , then it is easy to verify that

So, all the hypotheses of Theorem 10 are fulfilled. Therefore, it follows from Theorem 10 that BVP (47) has a positive solution satisfying

Conflict of Interests

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

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