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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 137890, 8 pages
http://dx.doi.org/10.1155/2013/137890
Research Article

Eigenvalue of Fractional Differential Equations with -Laplacian Operator

Department of Mathematics, Aba Teachers College, Wenchuan, Sichuan 623002, China

Received 27 December 2012; Accepted 19 February 2013

Academic Editor: Fuyi Xu

Copyright © 2013 Wenquan Wu and Xiangbing Zhou. 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 investigate the existence of positive solutions for the fractional order eigenvalue problem with -Laplacian operator , where are the standard Riemann-Liouville derivatives and -Laplacian operator is defined as is continuous and can be singular at and By constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of fractional differential equation is established.

1. Introduction

Differential equations of fractional order have been recently proved to be valuable tools in the modeling of many phenomena arising from science and engineering, such as viscoelasticity, electrochemistry, control, porous media, and electromagnetism. For detail, see the monographs of Kilbas et al. [1], Miller and Ross [2], and Podlubny [3] and the papers [423] and the references therein.

In [16], the authors investigated the nonlinear nonlocal boundary value problem: where . By using Krasnoselskii's fixed point theorem and the Leggett-Williams theorem, some sufficient conditions for the existence of positive solutions to the above BVP are obtained. In [17], by using the upper and lower solutions method, under suitable monotone conditions, the authors investigated the existence of positive solutions to the following nonlocal problem: where . Recently, by means of the fixed point theorem on cones, Chai [18] investigated two-point boundary value problem of fractional differential equation with -Laplacian operator: Some existence and multiplicity results of positive solutions are obtained.

As far as we know, no result has been obtained for the existence of positive solution for the fractional order eigenvalue problem with -Laplacian operator: where , , are the standard Riemann-Liouville derivatives with with , -Laplacian operator is defined as can be singular at , and . In order to obtain the existence of positive solutions of the fractional order eigenvalue problem (4), we will apply the upper and lower solutions method associated with the Schauder's fixed point theorem. It is worth emphasizing that the problem (4) not only includes the well-known Sturm-Liouville boundary value problems and the nonlocal boundary value problems as special case, but also can be singular at and .

The organization of this paper is as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. In Section 3, we put forward and prove our main results. Finally, we will give an example to demonstrate our main results.

2. Preliminaries and Lemmas

In this section, we introduce some preliminary facts which are used throughout this paper.

Definition 1 (see [13]). 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 (see [13]). The Riemann-Liouville fractional derivative of order of a function is given by where and denotes the integer part of number , provided that the right-hand side is pointwise defined on .

Proposition 3 (see [13]). (1) If , , then
(2) If , , then

Proposition 4 (see [13]). Let , and is integrable, then where and is the smallest integer greater than or equal to .

Definition 5. A continuous function is called a lower solution of the BVP (4), if it satisfies

Definition 6. A continuous function is called an upper solution of the BVP (4), if it satisfies

For forthcoming analysis, we first consider the following linear fractional differential equation:

Lemma 7 (see [15]). If and , then the boundary value problem (12) has the unique solution where is the Green function of the boundary value problem (12) and

Lemma 8. The Green function in Lemma 7 has the following properties:(i) is continuous on ;(ii) for any ;(iii), for , where

Let satisfy the relation , where is given by (4). To study BVP (4), we first consider the associated linear BVP: for and . For convenience, let then we have the following lemma.

Lemma 9. The associated linear BVP (17) has the unique positive solution

Proof. In fact, let . By Proposition 4, the solution of initial value problem is given by . From the relations , it follows that , and so Noting that , it follows from (21) that the solution of (17) satisfies By Lemma 7, the solution of (22) can be written as Since , , we have , , which implies that the solution of (22) is given by

The following lemma is a straightforward conclusion of Lemma 9.

Lemma 10. If satisfies and for any , then , for .

3. Main Results

Set

We present the following two assumptions.

is continuous and decreasing in .

For any , and

Let , and Clearly, , so is nonempty. For any , define an operator by

Theorem 11. Suppose conditions and hold. In addition, if the following condition holds:
for uniformly holds. Then there exists a constant such that the BVP (4) has at least one positive solution for any , and there exists one positive constant 1 such that

Proof. The proof is divided into four steps.
Step 1. We show that is well defined on and , and is decreasing in .
In fact, for any , by the definition of , there exists two positive numbers such that for any . It follows from Lemma 8 and (H1)-(H2) that
Now take , by , for any , . Thus by the continuity of and Lemma 8 and (32), we have
Take then by (32) and (33), which implies that is well defined and . And the operator is decreasing in from . Moreover, by direct computations, we also have
Step 2. In this step, we will focus on lower and upper solutions of the fractional boundary value problem (4).
By Lemma 8, we have Let it follows from (37) that
On the other hand, take then by monotonicity of in and (37)–(40), for any , we have From , we have uniformly on . Thus there exists large enough , such that, for any , which yields Letting and by Lemma 9, (39), (44), and (45), one has
By Step  1 and (46), (47), we know . And it follows from (45)–(47) that Consequently, it follows from (44)–(48) that that is, and are a couple of lower and upper solutions of fractional boundary value problem (4) by (46)–(49), respectively.
Step 3. Let It follows from and (46) that is continuous.
We will show that the fractional boundary value problem has a positive solution.
To see this, we consider the operator defined as follows: Obviously, a fixed point of the operator is a solution of the BVP (51). Noting that , then there exists a constant such that . Thus for all , it follows from Lemma 8, (50), and that which implies that the operator is uniformly bounded.
From the uniform continuity of and the Lebesgue dominated convergence theorem, we easily obtain that is equicontinuous. Thus by the means of the Arzela-Ascoli theorem, we have that is completely continuous. The Schauder fixed point theorem implies that has at least a fixed point such that .
Step 4. We will prove that the boundary value problem (4) has at least one positive solution.
In fact, we only need to prove that By (46), (47) and noticing that is fixed point of , we know that Notice that the definition of and the function is nonincreasing in , we obtain So by (48) and (56), Thus one has by (57) Let ; then and (55) implies that . It follows from (21) that and then Notice that is monotone increasing; we have It follows from Lemma 10 and (55) that Thus we have on . By the same way, we also have on . So Consequently, . Then is a positive solution of the problem (4).
Finally, by (48) and (64) and , we have where

In the end of this work we also remark the above results to the problem (4) with which is nonsingular at and ; that is, we have the following result.

Theorem 12. If is continuous, decreasing in and , for any , then the boundary value problem (4) has at least one positive solution for any , and there exists a constant such that

Proof. The proof is similar to Theorem 11; we omit it here.

Example 13. Consider the following boundary value problem:

Let , , , , and Firstly, And, it is easy to check that holds. For any , and which implies that holds.

On the other hand, Thus also holds.

By Theorem 11, the boundary value problem (68) has at least one positive solution.

Acknowledgments

This work was supported by the Natural Sciences of Education and the Science Office Bureau of Sichuan Province of China, under Grants nos. 10ZC060, 2010JY0J41.

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