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Abstract and Applied Analysis
Volume 2010, Article ID 971824, 12 pages
http://dx.doi.org/10.1155/2010/971824
Research Article

Upper and Lower Solutions Method for a Class of Singular Fractional Boundary Value Problems with p-Laplacian Operator

Department of Mathematics, Xiangnan University, Chenzhou 423000, China

Received 24 May 2010; Accepted 25 August 2010

Academic Editor: Paul Eloe

Copyright © 2010 Jinhua Wang and Hongjun Xiang. 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

The upper and lower solutions method is used to study the -Laplacian fractional boundary value problem , , , , , and , where . Some new results on the existence of at least one positive solution are obtained. It is valuable to point out that the nonlinearity can be singular at or

1. Introduction

It is well know that the upper and lower solutions method is a powerful tool for proving the existence results for boundary value problem. It has been used to deal with many multipoint boundary value problem of integer ordinary differential equations (see, e.g., [13] and references therein).

Recently, boundary value problems of nonlinear fractional differential equations have aroused considerable attention. Many people pay attention to the existence and multiplicity of solutions or positive solutions for boundary value problems of nonlinear fractional differential equations by means of some fixed point theorems, such as the Krasnosel'skii fixed-point theorem, the Leggett-Williams fixed-point theorem, and the Schauder fixed-point theorem (see [48]). To the best of our knowledge, the upper and lower solutions method is seldom considered in the literatures, and there are few papers devoted to investigate -Laplacian fractional boundary value problems.

In this paper, we deal with the following -Laplacian fractional boundary value problem: where , and is the standard Riemann-Liouville fractional differential operator of order , , . By using upper and lower solutions method, the existence results of at least one positive solution for the above fractional boundary value problem are established, and an example is given to show the effectiveness of our results. It is valuable to point out that the nonlinearity can be singular at or .

The remaining part of the paper is organized as follows. In Section 2, we will present some definitions and lemmas. In Section 3, some results are given. In Section 4, we present an example to demonstrate our results.

2. Basic Definitions and Preliminaries

In this section, we present some necessary definitions and lemmas.

Definition 2.1 (see [9]). The integral where , is called the Riemann-Liouville fractional integral of order

Definition 2.2 (see [10]). For a function given in the interval , the expression where , and denotes the integer part of number , is called the Riemann-Liouville fractional derivative of order

Remark 2.3. From the definition of the Riemann-Liouville fractional derivative, we quote for , then
In particular, where is the smallest integer greater than or equal to

Lemma 2.4 (see [4]). Assume that with a fractional derivative of order that belongs to . Then for some where is the smallest integer greater than or equal to .

Lemma 2.5 (see [6]). Let and , the unique solution of is where

Lemma 2.6. Let , , , . The fractional boundary value problem has a unique solution where and is defined by (2.7).

Proof. At first, by Lemma 2.4, the (2.8) is equivalent to the integral equation that is, By the boundary conditions we have Therefore, the solution of boundary value problem (2.8) satisfies Consequently, So, fractional boundary value problem (2.8) is equivalent to the following problem: Lemma 2.5 implies that fractional boundary value problem (2.8) has a unique solution The proof is completed.

Lemma 2.7. Let , , . The functions and defined by (2.7) and (2.10), respectively, are continuous on and satisfy(1) (2) (3)

Proof. The proofs of part (1) and part (2) can be obtained in [6]. Here, we only prove part (3).
If then
If , then
If , then
If , then The above inequalities imply that part (3) of Lemma 2.7 holds.

From Lemmas 2.5 and 2.7, it is easy to obtain the following lemma.

Lemma 2.8. Let , , . If and , then fractional boundary value problem (2.5) has a unique solution
Let Now one introduces the following definitions about the upper and lower solutions of fractional boundary value problem (1.1).

Definition 2.9. A function is called a lower solution of fractional boundary value problem (1.1), if and satisfies

Definition 2.10. A function is called an upper solution of fractional boundary value problem (1.1), if and satisfies

3. Main Result

In this section, our objective is to give and prove our main results.

For the sake of simplicity, we assume that( ) and is nonincreasing relative to ;( )For any constant , ( )There exist a continuous function and a positive number such that , and

We define And define an operator by

Theorem 3.1. Suppose that conditions are satisfied, then the boundary value problem (1.1) has at least one positive solution which satisfies for some positive number .

Proof. We will divide our proof into four steps.
Step 1. We prove that .
Firstly, from Lemma 2.7 and conditions ( ) ( ), for any , there exists such that Thus,
Secondly, it follows from Lemma 2.7 that Consequently, is well defined and
In the meanwhile, by direct computations, we can obtain
Step 2. We will prove that the functions , are lower and upper solutions of fractional boundary value problem (1.1), respectively.
From , we know that is nonincreasing relative to . Combining , we have Therefore, .
By Step 1, we know And by (3.6) (3.8), we obtain that is, and are lower and upper solutions of fractional boundary value problem (1.1), respectively.
Step 3. We will show that the fractional boundary value problem has a positive solution, where Thus, we consider the operator defined as follows: where is defined as (2.7), is defined as (2.10). It is clear that , and a fixed point of the operator is a solution of the fractional boundary value problem (3.10).
Since , there exists a positive number such that It follows from that Consequently, for all we have which implies that the operator is uniformly bounded.
On the other hand, since is continuous on , it is uniformly continuous on . So, for fixed and for any , there exists a constant , such that any and Then, for all , that is to say, is equicontinuous. Thus, from the Arzela-Ascoli Theorem, we know that is a compact operator. By the Schauder's fixed-point theorem, the operator has a fixed point; that is, the fractional boundary value problem (3.10) has a positive solution.
Step 4. We will prove that fractional boundary value problem (1.1) has at least one positive solution.
Suppose that is a solution of (3.10), we only need to prove that

Let be a solution of (3.10). We have In addition, the function is nonincreasing in , we know that It follows from (3.8) and that By (3.6) and we obtain Together with (3.7), (3.17)–(3.20), we obtain Let we obtain and

By Lemma 2.8, we know , which implies that Since is monotone increasing, we have that is, By Lemma 2.8 and (3.21), we have Therefore,

In the similar way, we can prove that Consequently, is a positive solution of fractional boundary value problem (1.1). This completes the proof.

Theorem 3.2. If is nonincreasing relative to and does not vanish identically for any , then the fractional boundary value problem (1.1) has at least one positive solution , which satisfies for some positive number

The proof is similar to Theorem 3.1, we omit it here.

4. Example

Example 4.1. As an example, we consider the fractional boundary value problem

Proof. It is clear that holds. For any , which implies that holds.
For , Let , by (3.5), we have and that is, there exist positive numbers , , such that , .
Choose a positive number and combining the monotonicity of , we have Taking , then, that is to say, the condition holds. Theorem 3.1 implies that the fractional boundary value problem (4.1) has at least one positive solution.

Acknowledgments

This work was jointly supported by Science Foundation of Hunan Provincial under Grants 2009JT3042, 2010GK3008 and 10JJ6007, the Construct Program of the Key Discipline in Hunan Province, Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

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