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Abstract and Applied Analysis
Volume 2010, Article ID 601492, 15 pages
http://dx.doi.org/10.1155/2010/601492
Research Article

Positive Solutions for Multipoint Boundary Value Problem of Fractional Differential Equations

School of Mathematics and Computer Sciences, Jishou University, 120 Renmin South Road, Jishou, Hunan 416000, China

Received 28 August 2010; Revised 10 December 2010; Accepted 16 December 2010

Academic Editor: Ferhan M. Atici

Copyright © 2010 Wenyong Zhong. 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 the existence and multiplicity of positive solutions for the fractional -point boundary value problem , , , , where , is the standard Riemann-Liouville fractional derivative, and is continuous. Here, for , , and with . In light of some fixed point theorems, some existence and multiplicity results of positive solutions are obtained.

1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order. It has been applied to almost every field of science, engineering, and mathematics in the last three decades [15]. But the investigation of the theory of fractional differential equations has only been started quite recently.

Among all the researches on the theory of the fractional differential equations, the study of the boundary value problems for fractional differential equations recently has attracted a great deal of attention from many researchers. And some results have been obtained on the existence of solutions (or positive solutions) of the boundary value problems for some specific fractional differential equations [611].

More specifically, Bai [12] discussed the existence of positive solutions for the boundary value problem (BVP for short) where is the standard Riemann-Liouville fractional derivative. Some existence results of at least one positive solution for the above-mentioned BVP are obtained by the use of fixed point index theory.

In [13], Salem considered the existence of Pseudosolutions for the nonlinear -point BVP, where takes values in a reflexive Banach space . Here, with , and denotes the th Pseudoderivative of while denotes the Pseudofractional differential operator of order . In light of the fixed point theorem given by O'Regan, the criteria for the existence of at least one Pseudo solution for the -point BVP are established.

Very recently, M. El-Shahed [14] considered the existence and nonexistence of positive solutions to the fractional differential equation subject to the boundary conditions Their analysis relies on Krasnoselskii's fixed point theorem.

Goodrich [15] then considered the BVP for the higher-dimensional fractional differential equation as follows: and a Harnack-like inequality associated with the Green's function related to the above problem is obtained improving the results in [16].

Motivated by the aforementioned results and techniques in coping with those boundary value problems of the fractional differential equations, we then turn to investigate the existence and multiplicity of positive solutions for the following BVP: where is the standard Riemann-Liouville fractional derivative of order . Here, by a positive solution of BVP (1.6), we mean a function which is positive on and satisfies the equation and boundary conditions in (1.6).

Throughout the paper, we will assume that the following conditions hold.(H1) is continuous.(H2) for , , and with .

There is a vast literature concerning the multipoint BVPs for the integer-order differential equations. An important recent paper was given by Webb and Infante [17]; and they established a new unified method for the existence of multiple positive solutions to a large number of nonlinear nonlocal BVPs for the integer-order differential equations. While in the setting of the fractional-order derivatives, as far as we know, the existence of positive solutions for the multipoint BVP (1.6) has not been discussed in the literature.

The rest of the paper is organized as follows. Section 2 preliminarily provides some definitions and lemmas which are crucial to the following discussion. In Section 3, we obtain the existence and multiplicity results of positive solutions for the BVP (1.6) by means of some fixed point theorems. Finally, we give a concrete example to illustrate the possible application of our analytical results.

2. Preliminaries

In this section, we preliminarily provide some definitions and lemmas which are useful in the following discussion.

Definition 2.1 (see [3]). The fractional integral of order of a function is given by provided the right side is pointwise defined on .

Definition 2.2 (see [3]). The standard Riemann-Liouville fractional derivative of order of a continuous function is given by where provided the right side is pointwise defined on .

Lemma 2.3 (see [6]). Assume that with a fractional derivative of order , then for some , where is the smallest integer greater than or equal to .

By Lemma 2.3, we next present an integral presentation of the solution for the BVP of the linearized equation associated with the BVP (1.6).

Lemma 2.4. Let , then the BVP has a unique solution where the Green function is given by , where Here, , and denotes the characteristic function of the set for .

Proof. Lemma 2.3 yields This, with the condition that , gives . Furthermore, differentiating both sides of the expression of with respect to , we obtain Now, by the conditions that and , we get that and Hence, The proof is complete.

The functions have important properties as follows.

Lemma 2.5. Assume (H2) holds, then for all and , where .

Proof. The asserted relation for is a direct result of Lemma 2.8 in [14]. In addition, the case for follows from a direct application of the definition of and the fact that holds for . The proof is complete.

Remark 2.6. The definition of and Lemma 2.5 yield

Remark 2.7. It is necessary to mention that Bai and Lü [6] showed that their Green's function did not satisfy a classical Harnack-like inequality (HLI) for the homogenous two-point BVP of fractional differential equation with order in . They proved that , where as , which is a challenge for our seeking positive solutions.
On the other hand, Goodrich [15], for the homogenous BVP of fractional differential equation with order in , established a HLI: , where is a positive constant. Using this inequality, the author obtained sufficient conditions on the existence of positive solutions. In Remark 2.6, we get a generalized HLI contrasting to the result in [15]. In fact, for a constant , it follows from Remark 2.6 that .
From the above discussion, we may infer that it is closely related to both the order of fractional differential equations and boundary conditions that whether or not the Green's function satisfies a traditional HLI, which needs more detailed and rigorous investigations.

Next, we introduce some fixed point theorems which will be adopted to prove the main results in the following section.

Lemma 2.8 (see [18]). Let be a Banach space, a cone, and two bounded open balls of centered at the origin with . Suppose that is a completely continuous operator such that either(B1) and or(B2) and Holds, then has a fixed point in .

Definition 2.9. 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 .

Lemma 2.10 (see [19]). Let be a cone in a 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 for ,(C2) for ,(C3) for with ,then has at least three fixed points , , and such that

Remark 2.11. If there holds , then condition (C1) of Lemma 2.10 implies condition (C3) of Lemma 2.10.

3. Main Results

In order to apply the fixed point theorems to the BVP (1.6), we first import some notations and operator.

Let be the classical Banach space with the norm . Furthermore, define the cone by Notice that for each . For a positive number , define the function space by Also define the operator by

Next, we show some properties of the operator .

Lemma 3.1. If (H1)-(H2) hold, then .

Proof. From the definition of the operator and Remark 2.6, it follows that, for , Therefore, . This completes the proof.

Lemma 3.2. Assume that (H1)-(H2) hold, then the operator is completely continuous.

Proof. Lemma 3.1 implies that . Moreover, the uniform continuity of the function on the compact set yields that the operator is continuous.
We now show that is bounded. To this end, let , then, for , it follows from Remark 2.6 that which implies that is bounded.
In addition, for each , writing , we have the following estimation: Now, using the fact that the functions and are uniformly continuous on , we conclude that is an equicontinuous set on . It follows from the Arzelà-Ascoli Theorem that is a relatively compact set. As a consequence, we complete the whole proof.

Lemma 3.3. Assume that (H1)-(H2) hold, then is a solution of the BVP (1.6) if and only if it is a fixed point of in .

Proof. If and , then Thus, and . Furthermore, differentiation of (3.7) with respect to produces This yields and Therefore, is a positive solution of the BVP (1.6).
On the other hand, if is a positive solution of the BVP (1.6), then Lemma 2.4 implies . Moreover, in view of the proof of Lemma 3.1, we also get for . Hence, is a fixed point of in . We consequently complete the proof.

In the following, fix in (0,1) and set We now present two main results on the existence of positive solutions for the BVP (1.6).

Theorem 3.4. Assume that (H1)-(H2) hold. In addition, and suppose that one of the following two conditions holds: (H3), (H4), then the BVP (1.6) has at least one positive solution.

Proof. Notice that Lemma 3.2 guarantees that the operator is completely continuous.
Now, assume that condition (H3) holds. Since , there exists an such that where the constant is chosen, so that Thus, This, together with the definitions of and Remark 2.6, implies that for any , That is, for .
On the other hand, from , it follows that there exists a such that where the constant satisfies Now, write . Then the relation (3.16) yields Set and let , then Remark 2.6 and (3.18) imply Thus, the operator satisfies condition (B2) of Lemma 2.8. Consequently, the operator has at least one fixed point , which is one positive solution of the BVP (1.6).
Next, we suppose that (H4) holds. The proof is similar to that of the case in which assumption (H3) holds and will only be sketched here. Select two positive constants and with and , respectively. Then, there exist two positive numbers and , such that It follows from Remark 2.6 and (3.20) that for ,
In addition, let . If , then and by (3.21), This yields that, for , Thus, for . Hence, the operator satisfies condition (B1) in Lemma 2.8. As a consequence, the operator has at least one fixed point . This means that the BVP (1.6) has at least one positive solution . We complete the whole proof.

Theorem 3.5. Assume (H1)-(H2) hold. In addition, suppose that there exist constants with , such that the following assumptions hold:(H5), for ,(H6), for ,(H7), for ,then the BVP (1.6) has at least three positive solutions.

Proof. Define the nonnegative continuous concave functional on the cone by Next, we intend to verify that all the conditions in Lemma 2.10 hold with respect to the operator . Lemma 2.10 involves parameters , and with . Now, let , then by Remark 2.11, it is sufficient to verify that the conditions (C1) and (C2) in Lemma 2.10 hold. To this end, let , then . This together with assumption (H7) implies that for . This relation and Remark 2.6 yield This, with Lemma 3.2, clearly manifests that the operator is completely continuous. In a similar argument, if , then assumption (H5) yields . Therefore, condition (C2) of Lemma 2.10 is satisfied.
Moreover, the set is not empty, since the constant function is contained in the set . Now, let , then for and . Thus, we obtain Assumption (H6) and (3.28) imply Hence, it follows from (3.29) and Lemma 2.5 that Accordingly, the validity of condition (C1) in Lemma 2.10 is verified. Consequently, by virtue of Lemma 2.10 and Remark 2.11, the operator has at least three fixed points , and satisfying These fixed points are positive solutions for the BVP (1.6). The proof is complete.

4. Illustrative Example

Consider the BVP where

Let . Clearly, the parameters ,. Choosing , then . Now, we can verify the validity of conditions (H5)–(H7) in Theorem 3.5. Indeed, indirect computations yield Hence, conditions (H5)–(H7) in Theorem 3.5 are satisfied for the above-specified functions and parameters. Therefore, in light of Theorem 3.5, we conclude that the above BVP has at least three positive solutions , , and defined on satisfying , and with .

Acknowledgment

This work is supported by the Scientific Research Foundation of the Higher Education Institutions of Hunan Province, China (Grant no. 10C1125).

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