Abstract and Applied AnalysisVolume 2012 (2012), Article ID 149849, 12 pagesdoi:10.1155/2012/149849

Research Article

Positive Solutions of 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, China²Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 12 December 2011; Revised 8 February 2012; Accepted 22 February 2012

Academic Editor: Benchawan Wiwatanapataphee

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 of a boundary value problem of nonlinear fractional differential equation with changing sign nonlinearity. We first derive some properties of the associated Green function and then obtain some results on the existence of positive solutions by means of the Krasnoselskii's fixed point theorem in a cone.

1. Introduction

Recently, much attention has been paid to the existence of solutions for fractional differential equations due to its wide range of applications in engineering, economics, and many other fields, and for more details see, for instance, [1–17] and the references therein. In most of the works in literature, the nonlinearity needs to be nonnegative to get positive solutions [10–17]. In particular, by using the Krasnosel’skii fixed-point theorem and the Leray-Schauder nonlinear alternative, Bai and Qiu [14] consider the positive solution for the following boundary value problem: where is a real number, is the Caputo fractional derivative, is continuous and singular at .

To the best of our knowledge, there are only very few papers dealing with the existence of positive solutions of semipositone fractional boundary value problems due to the difficulties in finding and analyzing the corresponding Green function. The purpose of this paper is to establish the existence of positive solutions to the following nonlinear fractional differential equation boundary value problem: where is a real number, is the Caputo fractional derivative, is a positive parameter, and may change sign and may be singular at . In this paper, by a positive solution to (1.1), we mean a function , which is positive on and satisfies (1.1).

The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. We also develop some properties of the associated Green function. In Section 3, we discuss the existence of positive solutions of the semipositone BVP (1.1). In Section 4, we give two examples to illustrate the application of our main results.

2. Basic Definitions and Preliminaries

In this section, we present some preliminaries and lemmas that are useful to the proof of our main results. For the convenience of the reader, we also present here some necessary definitions from fractional calculus theory. These definitions can be found in the recent literature.

Definition 2.1. The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on .

Definition 2.2. The Caputo’s fractional derivative of order of a function is given by where , provided that the right-hand side is pointwise defined on .

Lemma 2.3 (see [14]). Given , the unique solution of the problem is where

Lemma 2.4. The function has the following properties: (1) for,(2) for ,(3) for .

Proof. It is obvious that (1) holds. In the following, we will prove (2) and (3).(i) When , as , we have therefore On the other hand, since , we have (ii) When , we have On the other hand, as for , we have The proof is completed.

Remark 2.5. By Lemma 2.4, there exists such that the positive solution in [14] satisfies where .

Proof. In [14], the positive solution of (P) is equivalent to the fixed point of in , where and For any , by (1) of Lemma 2.4, we have On the other hand, by of Lemma 2.4, we get which implies .If is a positive solution of (P), then is a fixed point of in , therefore where . The proof is completed.

For the convenience of presentation, we list here the hypotheses to be used later. and satisfies where , . , . There exists such that There exists such that

Lemma 2.6. Assume that and hold, then the boundary value problem has a unique solution with

Proof. By Lemma 2.3, we have that is the unique solution of (2.19). By (1) of Lemma 2.4, we have The proof is completed.

Let be endowed with the maximum norm . Define a cone by Set , , .

Next we consider the following boundary value problem: where , is defined in Lemma 2.6, .

Let It is easy to check that is a solution of (2.23) if and only if is a fixed point of .

Lemma 2.7. is a completely continuous operator.

Proof. For any , Lemma 2.4 implies that On the other hand Then , which implies .According to the Ascoli-Arzela theorem, we can easily get that is a completely continuous operator. The proof is completed.

Lemma 2.8 (see [18]). Let be a real Banach space, and let be a cone. Assume that and are two bounded open subsets of with , and is a completely continuous operator such that either (1) and , or(2) and .Then T has a fixed point in .

3. Existence of Positive Solutions

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

Proof. Choose . Let where In the rest of the proof, we suppose .For any , noting that and using (2.20), we have Therefore, Thus, Now choose a real number By , there exists a constant such that for any , , we have Select Then for any , we have . Moreover, by the selection of we have Thus for any , as , we get Noting that , we have Hence we get Thus, By Lemma 2.8, has a fixed point such that . Since , by (3.4) we have . Let . As is the solution of (2.19) and is the solution of (2.23), is a positive solution of the singular semipositone boundary value problem (1.1). The proof is completed.

Theorem 3.2. Suppose that , , and hold. Then there exists such that the boundary value problem (1.1) has at least one positive solution for any .

Proof. By the first limit of , we have that there exists such that, for any and , we have Select In the rest of the proof, we suppose .Let Then, for any , we have and therefore on , . Then, which implies On the other hand, as is continuous on , from the second limit of , we have where is defined by (3.2). In fact, by for any , there exists such that for any we have . Let , for any we have . Therefore, For there exists such that when , for any , we have Select Then, for any , we get Thus, By Lemma 2.8, has a fixed point such that . Since , by (3.18), we have . Let . As is a solution of (2.19) and is a solution of (2.23), is a positive solution of the singular semipositone boundary value problem (1.1). The proof is completed.

Corollary 3.3. The conclusion of Theorem 3.2 is valid if is replaced by there exists such that

4. Examples

Example 4.1. Consider the following problem where . Let , . By direct calculation, we have , and So all conditions of Theorem 3.1 are satisfied. By Theorem 3.1, BVP (4.1) has at least one positive solution provided is sufficiently small.

Example 4.2. Consider the following problem where . Let . By direct calculation, we have , , and So all conditions of Theorem 3.2 are satisfied. By Theorem 3.2, BVP (4.3) has at least one positive solution provided is sufficiently large.

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.

References

O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002.

V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 8, pp. 828–834, 2008.

M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 7-8, pp. 2391–2396, 2009.

T. G. Bhaskar, V. Lakshmikantham, and S. Leela, “Fractional differential equations with a Krasnoselskii-Krein type condition,” Nonlinear Analysis, vol. 3, no. 4, pp. 734–737, 2009.

B. Ahmad and A. Alsaedi, “Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type,” Nonlinear Analysis, vol. 3, no. 4, pp. 501–509, 2009.

Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009.

M. El-Shahed and J. J. Nieto, “Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3438–3443, 2010.

A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 72, no. 2, pp. 580–586, 2010.

M. ur Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1038–1044, 2010.

Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.

H. Jafari and V. Daftardar-Gejji, “Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 700–706, 2006.

M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 7-8, pp. 2391–2396, 2009.

S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 11, pp. 5545–5550, 2009.

Z. Bai and T. Qiu, “Existence of positive solution for singular fractional differential equation,” Applied Mathematics and Computation, vol. 215, no. 7, pp. 2761–2767, 2009.

D. Jiang and C. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 72, no. 2, pp. 710–719, 2010.

C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010.

C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050–1055, 2010.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.