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

Yanan Li, Shurong Sun, Zhenlai Han, and Hongling Lu

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 [1–6].

Recently, there have been many papers focused on boundary value problems of fractional ordinary differential equations [7–19] and initial value problems of fractional functional differential equations [10, 20–27]. But the results dealing with the boundary value problems of fractional functional differential equations with delay are relatively scarce [28–32]. 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

then (119) shows that

Thus, by the second part of Lemma 10, has a fixed point with , and accordingly, is a positive solution of problem (5). The proof is completed.

Next, we study the existence of solutions for boundary value problem (5) by the fixed point index theorem. For this purpose, we first give the following lemma.

Lemma 17. Let . Assume that

If there exists a function such that

Then boundary value problem for the following fractional functional differential equation:

has at least a positive solution for .

Proof. Let . We define the operator on by

In view of Lemma 12, it can be verified that is completely continuous. Then, we prove that has a fixed point in .

Set . From the assumption, for any given , there exists such that

For and , we have

Now, if we let , then it shows that

Denote

For and , we have

Now, if we let , then it shows that

Thus, by Lemma 10, has a fixed point between and . Accordingly, is a positive solution to (96). The proof is completed.

From Lemma 17, we assume that is an eigenvalue of

subject to the following conditions:

and the corresponding eigenfunction for such that

For convenience, we give some conditions as follows.

, where

, where

, where

There is a such that , , and imply that

where . There is a such that , , and imply that

where .

Lemma 18. Let hold. Then, .

Proof. Let . Then,

Thus, for , we have by that

Hence, . By Lemma 11, we get . The proof is complete.

Lemma 19. Let hold. Then, .

Proof. Let . Then,

Thus, for , we have by that

Hence, . By Lemma 11, we get . The proof is complete.

Theorem 20. Let , , and hold. Then, boundary value problem (5) has at least two positive solutions and such that

Proof. According to Lemma 18, we get that . Let and , for . Then, satisfy and .

Define by

It is easy to know that is a completely continuous operator.

Also, according to Lemma 13, we have that there exist , such that

Define by . For any , , we have

Note that is uniformly bounded in . Thus, is continuous on uniformly for . Then, we conclude that is a completely continuous operator on .

By and definition of , there exists a and such that

Next, we prove that for all and .

In fact, if there exist and such that , then satisfies the following:

and the following boundary conditions:

Multiplying both sides of (122) by , then integrating from to and using Green’s formula, we obtain

Thus, we know that

Then, we have

By the definition of , we have

Combining with (126), we get that

which is a contradiction for definition of in . Thus, for and . In view of (118) and homotopy invariance of the fixed point index, we obtain

On the other hand, by and definition of , there are and such that

Setting

and it follows that

Then, we prove that there exists a such that for all and , . In fact, if there exist and such that , we get that

Then, we have

By the definition of , we get that

Combining with (134), we conclude that

that is,

Let . Then, we obtain for and . In view of (118) and homotopy invariance of the fixed point index, we obtain

By using additivity, we get that

Since

it follows that and satisfy

The proof is completed.

Theorem 21. Let , , and hold. Assume that ; then boundary value problem (5) has at least two positive solutions and such that

Proof. According to Lemmas 18 and 19, we obtain

By and definition of , there are and such that

Let

and it follows that

Define by . For any , , we have

Thus, is continuous on uniformly for . Then, we conclude that is also a completely continuous operator on .

Next, we prove that there exists a such that for all and , .

In fact, if there exist and such that , then satisfies the following:

Multiplying both sides of (149) by then integrating from to and using Green’s formula, we get that

Thus, we get that

Then, we have

By the definition of , we get that

This combined with (126) gives,

that is,

Let . Then, we obtain for and . In view of (143), (144), and homotopy invariance of the fixed point index, we obtain

By using additivity, we get that

Thus has fixed points and in and , which means that and are positive solutions of boundary value problem (5) and

The proof is completed.

Corollary 22. Let , , and hold. Assume that . Then, boundary value problem (5) has at least two positive solutions and such that

Corollary 23. Let , and hold. Assume . Then boundary value problem (5) has at least two positive solutions and such that

4. Example

In this section, we will present an example to illustrate our main results.

Consider the following fractional functional differential equations with delay:

By a simple computation, we can get that and

as .

Similarly, we can obtain as .

Then, conditions and are satisfied. Then, by Theorem 15, boundary value problem (161) has at least a positive solution.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This research is supported by the Natural Science Foundation of China (11071143), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119), Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2011AL007), Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

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