Abstract and Applied Analysis

Volume 2012 (2012), Article ID 214042, 17 pages

doi:10.1155/2012/214042

Research Article

Positive Solutions for a Fractional Boundary Value Problem with Changing Sign Nonlinearity

Yongqing Wang,¹ Lishan Liu,1,2 and Yonghong Wu²

¹School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China

²Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 14 April 2012; Accepted 25 June 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 Yongqing Wang 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 discuss the existence of positive solutions to the following fractional m-point boundary value problem with changing sign nonlinearity , where is a positive parameter, , , with , is the standard Riemann-Liouville derivative, and may be singular at and/or and also may change sign. The work improves and generalizes some previous results.

1. Introduction

In this paper, we consider the following fractional differential equation with -point boundary conditions:

where , is a parameter, , with , is the standard Riemann-Liouville derivative, and may be singular at and/or and also may change sign. In this paper, by a positive solution to (1.1), we mean a function which is positive on and satisfies (1.1).

In recent years, great efforts have been made worldwide to study the existence of solutions for nonlinear fractional differential equations by using nonlinear analysis methods [1–24]. Fractional-order multipoint boundary value problems (BVP) have particularly attracted a great deal of attention (see, e.g., [13–19]). In [10], the authors discussed some properties of the Green function for the Direchlet-type BVP of nonlinear fractional differential equations

where , is the standard Riemann-Liouville derivative and is continuous. By using the Krasnosel'skii fixed-point theorem, the existence of positive solutions was obtained under some suitable conditions on .

In [14], the authors investigated the existence and multiplicity of positive solutions by using some fixed-point theorems for the fractional differential equation given by

where , , , with , , is nonnegative.

It should be noted that in most of the works in literature the nonlinearity needs to be nonnegative in order to establish positive solutions. As far as we know, semipositone fractional nonlocal boundary value problems with have been seldom studied due to the difficulties in finding and analyzing the corresponding Green function.

In [23], the authors investigated the following fractional differential equation with three-point boundary conditions:

where , , , ,, , and satisfies the Caratheodory conditions. The authors obtained the properties of the Green function for (1.4) as follows:

By using the Schauder fixed-point theorem, the authors obtained the existence of positive solution of (1.4) with the following assumptions:

for each , there exists a function such that for a.e. , for all ; there exist , and , such that

here is continuous and nonincreasing, is continuous, and is nondecreasing;

There exist two positive constants such that

Here

The assumptions on nonlinearity are not suitable for frequently used conditions, such as superlinear or some sublinear. For instance, , , obviously, does not satisfy .

Inspired by the previous work, the aim of this paper is to establish conditions for the existence of positive solutions of the more general BVP (1.1). Our work presented in this paper has the following new features. Firstly, we consider few cases of which has been studied before, and in dealing with the difficulties related to the Green function for this case, some new properties of the Green function have been discovered. Secondly, the BVP (1.1) possesses singularity; that is, may be singular at and/or . Thirdly, the nonlinearity may change sign and may be unbounded from below. Finally, we impose weaker positivity conditions on the nonlocal boundary term; that is, some of the coefficients may be negative.

The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that are to be used to prove our main results. We also discover some new positive properties of the corresponding Green function. In Section 3, we discuss the existence of positive solutions of the semipositone BVP (1.1). In Section 4, we give an example to demonstrate the application of our theoretical results.

2. Basic Definitions and Preliminaries

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can also be found in the recent literature.

Definition 2.1. The fractional integral of order of a function is given by

provided that the right-hand side is pointwisely defined on .

Definition 2.2. The Riemann-Liouville fractional derivative of order of a function is given by

where and denotes the integer part of the number , provided that the right-hand side is pointwisely defined on .

Lemma 2.3 (see [3]). Let . Then the following equality holds for , ;

where , , .

Set

where

For convenience in presentation, we here list the assumption to be used throughout the paper.

, on .

Remark 2.4. If , we have and . If and , we have on .

Lemma 2.5 (see [14]). Assume that and . Then

Lemma 2.6. Assume holds, and . Then the unique solution of the problem

is

where is the Green function of the boundary value problem (2.9).

Proof. From Lemma 2.3, the solution of (2.9) is

Consequently,

From , we have .

By Lemma 2.5, we have

Therefore,

By , we have

Therefore, the solution of (2.9) is

Lemma 2.7. The function has the following properties:

(1), for ;(2), for ;(3),  for  ,

where

Proof. (1) When , it is clear that

When , we have

(2) By (2.4), for any , we have

In the following, we will prove (3).

(i) When , noticing that , we have

Therefore,

which implies

On the other hand, we have

Therefore, , which implies

Then

(ii) When , we have

On the other hand, clearly we have

The inequalities (2.23)–(2.28) imply that (3) holds.

By Lemma 2.7, we have the following results.

Lemma 2.8. Assume holds, then the Green function defined by (2.6) satisfies

(1), for all ;(2), for all ;(3), for all ,

where

Lemma 2.9. Assume holds, then the function satisfies

(1), for all ;(2), for all ;(3), for all .

For convenience, we list here four more assumptions to be used later:

satisfies

where , .

(), . There exists such that

There exists such that

Remark 2.10. The second limit of implies

where

Proof. By , for any , there exists , such that for any we have

Let , for any we have

Therefore, .

Lemma 2.11. Assume holds and is nonnegative, then the BVP

has a unique solution with , where

Proof. By Lemma 2.6, is the unique solution of (2.37). By (2) of Lemma 2.8, we have

Let be endowed with the maximum norm and define a cone by

and then set , , .

Next we consider the following singular nonlinear BVP:

where , , is defined in Lemma 2.11.

Let

Clearly, if is a fixed point of , then is a positive solution of (2.41).

Lemma 2.12. Suppose that ()–() hold. Then is a completely continuous operator.

Proof. It is clear that is well defined on . For any , Lemma 2.9 implies

On the other hand,

Therefore, . Noticing that

we have .

Using the Ascoli-Arzela theorem, we can then get that is a completely continuous operator.     

Lemma 2.13 (see [25]). Let be a real Banach space and let be a cone. Assume that and are two-bounded open subsets of with , a completely continuous operator such that either

(1), and , , or(2), and , .

Then has a fixed point in .

3. Existence of Positive Solutions

Theorem 3.1. Suppose that ()–() hold. Then there exists such that the BVP (1.1) has at least one positive solution for any .

Proof . Choose . Let

where

In the following of the proof, we suppose .

For any , noticing and Lemma 2.11, we have

Therefore,

Thus,

Now choose a real number

By , there exists a constant such that

Let

Then for any , we have

Thus, for any , we have . Hence, we get

Therefore,

By Lemma 2.13, has a fixed point such that . Let . Since , by (3.3) we have on and . Notice that is the solution of (2.37) and is the solution of (2.41). Thus, is a positive solution of the BVP (1.1).

Theorem 3.2. Suppose that – and hold. Then there exists such that the BVP (1.1) has at least one positive solution for any .

Proof. By the first limit of , there exists such that

Let

In the following part of the proof, we suppose .

Let

Then for any , we have

Therefore, , for any and . Then

This implies

On the other hand, is continuous on , and thus from the second limit of , we have

where is defined by (3.2). For

there exists such that for any and .

Let

For any , by (3.16) we can get , for all . Therefore,

Thus,

By Lemma 2.13, has a fixed point such that . Let . Since , by (3.16) we have on and . Notice that is a solution of (2.37) and is a solution of (2.41). Thus, is a positive solution of the BVP (1.1).

By the proof of Theorem 3.2, we have the following corollary.

Corollary 3.3. The conclusion of Theorem 3.2 is valid if is replaced by . There exist and such that for any and ,

4. Example

Example 4.1 (a 4-point BVP with coefficients of both signs). Consider the following problem:

where

We have

By direct calculations, we have and , which implies that holds.

Let , , . It is easy to see that and hold. Moreover,

Therefore, the assumptions of Theorem 3.1 are satisfied. Thus, Theorem 3.1 ensures that there exists such that the BVP (4.1) has at least one positive solution for any .

Remark 4.2. Noticing that does not satisfy , therefore, the work in the present paper improves and generalizes the main results of [23].

Acknowledgments

The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11101237) and the Natural Science Foundation of Shandong Province of China (ZR2011AQ008, ZR2011AL018). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

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