Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 850871, 10 pagesdoi:10.1155/2012/850871

Research Article

Multiple Positive Solutions for Nonlinear Semipositone Fractional Differential Equations

Wen-Xue Zhou,1,2 Ji-Gen Peng,¹ and Yan-Dong Chu²

¹Department of Mathematics, Xi'an Jiaotong University, Shaanxi, Xi'an 710049, China²Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China

Received 22 September 2011; Accepted 13 December 2011

Academic Editor: Chuanxi Qian

Copyright © 2012 Wen-Xue Zhou 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 present some new multiplicity of positive solutions results for nonlinear semipositone fractional boundary value problem , where is a real number and is the standard Riemann-Liouville differentiation. One example is also given to illustrate the main result.

1. Introduction

This paper is mainly concerned with the multiplicity of positive solutions of nonlinear fractional differential equation boundary value problem (BVP for short) where is a real number and is the standard Riemann-Liouville differentiation, and is a given function satisfying some assumptions that will be specified later.

In the last few years, fractional differential equations (in short FDEs) have been studied extensively the motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For an extensive collection of such results, we refer the readers to the monographs by Kilbas et al. [1], Miller and Ross [2], Oldham and Spanier [3], Podlubny [4], and Samko et al. [5].

Some basic theory for the initial value problems of FDE involving the Riemann-Liouville differential operator has been discussed by Lakshmikantham and Vatsala [6–8], Babakhani and Daftardar-Gejji [9–11], and Bai [12], and others. Also, there are some papers that deal with the existence and multiplicity of solutions (or positive solution) for nonlinear FDE of BVPs by using techniques of nonlinear analysis (fixed point theorems, Leray-Schauders theory, topological degree theory, etc.), see [13–22] and the references therein.

Bai and Lü [15] studied the following two-point boundary value problem of FDEs where is the standard Riemann-Liouville fractional derivative. They obtained the existence of positive solutions by means of the Guo-Krasnosel’skii fixed point theorem and Leggett-Williams fixed point theorem.

Zhang [22] considered the existence and multiplicity of positive solutions for the nonlinear fractional boundary value problem where is a real number, , and is the standard Caputo’s fractional derivative. The author obtained the existence and multiplicity results of positive solutions by means of the Guo-Krasnosel’skii fixed point theorem.

From the above works, we can see the fact that although the fractional boundary value problems have been investigated by some authors to the best of our knowledge, there have been few papers that deal with the boundary value problem (1.1) for nonlinear fractional differential equation. Motivated by all the works above, in this paper we discuss the boundary value problem (1.1), using the Guo-Krasnosel’skii fixed point theorem, and we give some new existence of multiple positive solutions criteria for boundary value problem (1.1).

The paper is organized as follows. In Section 2, we give some preliminary results that will be used in the proof of the main results. In Section 3, we establish the existence of multiple positive solutions for boundary value problem (1.1) by the Guo-Krasnosel’skii fixed point theorem. In the end, we illustrate a simple use of the main result.

2. Preliminaries and Lemmas

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can be found in the recent literature such as [1, 4, 15].

Definition 2.1 (see [1, 4]). The Riemann-Liouville fractional integral of order of a function is given by provided that the right side is pointwise defined on , where is the gamma function.

Definition 2.2 (see [1, 4]). The Riemann-Liouville fractional derivative of order of a continuous function is given by provided that the right side is pointwise defined on , where and denotes the integer part of .

Lemma 2.3 (see [15]). Let . If one assumes , then fractional differential equation has as unique solutions, where is the smallest integer greater than or equal to .

Lemma 2.4 (see [15]). Assume that with a fractional derivative of order that belongs to . Then for some , , where is the smallest integer greater than or equal to .

In the following, we present Green’s function of the fractional differential equation boundary value problem.

Lemma 2.5. Let and , then the unique solution of is given by where is Green’s function given by

The following properties of Green’s function form the basis of our main work in this paper.

Lemma 2.6. The function defined by (2.8) possesses the following properties: (i) for ;(ii) for , ;(iii) for , ;(iv) for .

The following Krasnosel’skii’s fixed point theorem will play a major role in our next analysis.

Lemma 2.7 (see [23]). Let be a Banach space, and let be a cone in . Assume are open subsets of with , and let be a completely continuous operator such that either (i), , or(ii), .Then has a fixed point in .

3. Main Results

In this section, we establish some new existence results for the fractional differential equation (1.1). Given , we write a , if for , and it is positive in a set of positive measure.

Let us list the following assumptions:

(H1) is continuous, ;

(H2) there exists , such that

In view of Lemmas 2.5 and 2.6, we obtain the following.

Lemma 3.1. Let with on (0,1), and is the unique solution of Then where  , .

Next, we consider where Then (3.4) is equivalent to the following integral equation:

Lemma 3.2. Let for , and is positive solution of the problem (3.4). Then is positive solution of the problem (1.1).

Proof. In fact, let . Then and . Since is positive solution of the problem (3.4), we have So For our constructions, we will consider the Banach space equipped with standard norm .Define a cone by Let the operator be defined by the formula

Lemma 3.3. Assume that (H1) holds. Then .

Proof. Notice from (3.10) and Lemma 2.6 that, for , on and On the other hand, we have Thus we have . The proof is finished.

It is standard that is continuous and completely continuous.

For convenience, we introduce the following notations: , , .

Theorem 3.4. Assume that (H1) and (H2) are satisfied. Also suppose the following conditions are satisfied: (A1) there exists a constant such that for all ;(A2) there exists a constant such that for all ;(A3). Then the problem (1.1) has at least two positive solutions.

Proof. To show that (1.1) has at least two positive solutions, we will assume the problem (3.4) has at least two positive solutions and with .We now show To see this, let , then for , , by Lemma 3.1 and (A1), we have Thus, we see, from Lemma 2.6 and (A1), that from which we see that , for .Next we now show To see this, let ; then, for , , by , we have For ; , then, it follows from (3.17) that In view of (A2), (3.17) and Lemma 2.6, we have that for all , from which we see that , for .On the other hand, let , where Supposing that (A3) holds, one can find , so that Setting then , and so from which we see that , for .In view of Lemma 2.7, the problem (3.4) has at least two positive solutions and with . Since , we have Therefore are solutions of the problem (1.1). This completes the proof.

Theorem 3.5. Suppose that (H1), (H2) are satisfied. Furthermore assume that (A4) there exists a constant such that for all ;(A5) there exists a constant such that for all ;(A6). Then the problem (1.1) has at least two positive solutions.

4. An Example

As an application of the main results, we consider Set Then we have ,, letting, , then , choosing , , then , ; therefore, we have , , , and .

It is clear that is continuous. Since all the conditions of Theorem 3.5 are satisfied, the problem (4.1) has at least two positive solutions.

Acknowledgments

This work supported by the Nature Science Foundation of China under the Contact no. 10901075 and the Key Project of Chinese Ministry of Education (210226).

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