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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 301560, 20 pages
http://dx.doi.org/10.1155/2013/301560
Research Article

The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm-Liouville Functional Differential Equation

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

Received 25 April 2013; Accepted 20 June 2013

Academic Editor: Bashir Ahmad

Copyright © 2013 Yanan Li 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 study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative: + , , , , , , , where , denote the Caputo fractional derivatives, is a nonnegative continuous functional defined on , , , , are suitably small, , and , . By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results.

1. Introduction

Fractional calculus is a branch of mathematics, it is an emerging field in the area of the applied mathematics that deals with derivatives and integrals of arbitrary orders as well as with their applications. The origins can be traced back to the end of the seventeenth century. During the history of fractional calculus, it was reported that the pure mathematical formulations of the investigated problems started to be addressed with more applications in various fields. With the help of fractional calculus, we can describe natural phenomena and mathematical models more accurately. Therefore, fractional differential equations have received much attention, and the theory and its applications have been greatly developed; see [16].

Recently, there have been many papers focused on boundary value problems of fractional ordinary differential equations [719] and initial value problems of fractional functional differential equations [10, 2027]. But the results dealing with the boundary value problems of fractional functional differential equations with delay are relatively scarce [2832]. It is well known that in practical problems, the behavior of systems not only depends on the status just at the present but also on the status in the past [33]. Thus, in many cases, we must consider fractional functional differential equations with delay in order to solve practical problems. Consequently, our aim in this paper is to study the existence of solutions for boundary value problems of fractional functional differential equations.

In 2005, by means of the fixed point index theorem, Bai and Ma [34] established some criteria for the existence of solutions for the boundary value problem expressed by second order differential equations with delay: where , are suitably small and , , .

In 2011, Li et al. [21] investigated the existence of positive solutions for the nonlinear Caputo fractional functional differential equation: where is the Caputo fractional order derivative, subject to the following boundary conditions: They obtained the existence results of positive solutions by using some fixed point theorems.

In 2012, Zhao et al. [22] studied the existence of positive solutions for the nonlinear Caputo fractional functional differential equation: By constructing a special cone and using the Guo-Krasnoselskii fixed point theorem, they obtained the existing results.

Motivated by the works above, in this paper, we study the existence of positive solutions of boundary value problems for nonlinear fractional functional differential equation: where denote the Caputo fractional derivatives, is a nonnegative continuous functional defined on , , , , are suitably small, with and , with , and is a positive measurable continuous function defined on and satisfies the following condition: where , , and is a constant.

When , problem (5) is reduced to the problem of second order differential equations with delay and has been studied by Bai and Ma [34]. To the best of our knowledge, no one has studied the existence of positive solutions for boundary value problem (5). Key tools in finding our main results are fixed point index theorem and the Guo-Krasnoselskii fixed point theorem, and our main results of this paper are to extend and supplement some results in [21, 22, 34].

The paper is organized as follows. In Section 2, we will introduce some definitions and lemmas to prove our main results. In Section 3, we investigate the existence of positive solution for boundary value problems (5) by the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem. As an application, an example is presented to illustrate our main results.

2. Preliminaries

In the following section, we introduce the definitions and lemmas which are used throughout the paper.

Definition 1 (see [4]). The fractional integral of order ( of a function is given by where is the gamma function, provided that the right side is pointwise defined on .

Definition 2 (see [4]). The Caputo fractional derivative of order () of a function is given by where is the gamma function, provided that the right side is pointwise defined on .
Obviously, the Caputo derivative for every constant function is equal to zero.

From the definition of the Caputo derivative, we can acquire the following statements.

Lemma 3 (see [5]). Let . Then,

Lemma 4 (see [5]). Let . Then, for some , , where is the smallest integer greater than or equal to .

Assume that is the solution of (5) with ; then it can be expressed as where

Next, we introduce the Green function of boundary value problems for fractional functional differential equations.

Lemma 5. Let and be continuous. Then, the boundary value problem for fractional functional differential equation has a unique solution: where

Proof. It is easy to know that for and for . From (13), we know that From Lemma 4, we have Then, we get that According to (14) and conditions , we imply that Then, we obtain Therefore, where is defined by (17). The proof is completed.

Lemma 6. Let , and be continuous. Then, the boundary value problem for fractional functional differential equation has a unique solution: where is defined by (17), and

Proof. From (25), we know that From Lemma 4, we have According to the conditions and , we derive that Therefore, Thus, Then boundary value problem (25) is equivalent to the following problem: Lemma 5 implies that boundary value problem (33) has a unique solution: where and are defined as (17) and (27), respectively. The proof is completed.

The following properties of the Green function play important roles in this paper.

Lemma 7 (see [21]). The function defined by (17) satisfies the following conditions: (1) is continuous on ;(2)for , we have for ; (3) for ; (4)there exist positive numbers such that where .

Remark 8. From the definition of and , we know that . Furthermore, its easy to get from the proof of Lemma 3.3 in [21] that there exists a positive number such that where .

Lemma 9 (see [18]). The function defined by (27) satisfies the following conditions: (1) for ; (2)there exists a positive function for such that where

The following lemmas are fundamental in proof of our main results.

Lemma 10 (see [35]). Let be a Banach space, and let be a cone. Assume that , are open and bounded subsets of with , , and let be a completely continuous operator such that(i), , and , ; (ii), , and , .
Then, has a fixed point in .

Lemma 11 (see [36]). Assume that is a Banach space and is a cone. Let . Furthermore, assume that is compact and for . Thus, one has the following conclusions: (i)if for , then ; (ii)if for , then .

3. Main Results

In this section, we discuss the existence of positive solutions for boundary value problem (5).

Let be a Banach space with the maximum norm for . Let be a cone in defined by

Let . Noting that for , we have where Define an operator on as follows:

Lemma 12. The operator is completely continuous.

Proof. For , we have , from properties of and . It follows from (55) that we have for , and for , Hence, we get that By Lemma 7 and Remark 8, we have and Thus, .
Let be a bounded subset in , that is, there exists a positive constant such that , for all . Let . Then, for , in view of Lemma 9, we have Hence, is bounded in .
Now, we divide three cases to prove that is equicontinuous.
Case  1. If , .
Then,
Case  2. If , .
Then,
Case  3. If , .
Then, Let . Then, for , , we obtain Thus, is equicontinuous.
Next, we show that is continuous. For any , , with as . Then, for , we have This implies that as . Hence, is continuous. According to the Ascoli-Arzelà Theorem, is completely continuous. The proof is completed.

For convenience, we give some conditions, which will play roles in this paper for as follows...

Lemma 13. Let and hold. Then, there exist positive numbers such that

Proof. Choose such that By , there is an such that implies that Thus, for , we have In view of (55) and (56), we get that Hence, . It is obvious that for . Therefore, by Lemma 11, we conclude that .
In the same way, for the same satisfying (55), implies that there is such that Choose Thus, for , , we have In view of (55) and (59), we get that Therefore, . By Lemma 11, we conclude that . The proof is completed.

Lemma 14. Let hold. Then, there exists a such that

Proof. By , for any there exists such that Setting we get that Choose By (67), for , , we have Hence, . By Lemma 11, . The proof is completed.

Now, we prove the existence of solutions for boundary value problem (5) by using the Guo-Krasnoselskii fixed point theorem.

Theorem 15. Let , and let and hold. Then, boundary value problem (5) has at least a positive solution.

Proof. By , there exists a such that where
Taking we get that Assume that By (73), for any satisfied and , we have Now, if we let then (76) shows that
On the other hand, by , there is an such that implies where satisfies Thus for and , , we have In view of (78) and (79), we get that Now, if we let then (81) shows that
Thus, by the first part of Lemma 10, has a fixed point with , and accordingly, is a positive solution of problem (5). The proof is completed.

Theorem 16. Let , and let and hold. Then, boundary value problem (5) has at least a positive solution.

Proof. By for any satisfying (126), there is a such that Choose Thus for , and , we have In view of (55) and (59), we get that
Now, if we let then (144) shows that
On the other hand, from Theorem 15, by , there exists a such that where Now, if we let