Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 725494, 11 pagesdoi:10.1155/2011/725494

Research Article

Positive Solutions of Nonlinear Eigenvalue Problems for a Nonlocal Fractional Differential Equation

Xiaoling Han and Hongliang Gao

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 13 September 2011; Revised 21 November 2011; Accepted 25 November 2011

Academic Editor: Gerhard-Wilhelm Weber

Copyright © 2011 Xiaoling Han and Hongliang Gao. 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.

By using the fixed point theorem, positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation are considered, where is a real number, is a positive parameter, is the standard Riemann-Liouville differentiation, and with , .

1. Introduction

Fractional differential equations have been of great interest recently. This is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For details, see [1–6] and references therein.

Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by the use of techniques of nonlinear analysis, see [7–21] and the reference therein. Bai and Lü [7] studied the existence of positive solutions of nonlinear fractional differential equation where is a real number, is the standard Riemann-Liouville differentiation, and is continuous. They derived the corresponding Green function and obtained some properties as follows.

Proposition 1.1. Green’s function satisfies the following conditions: (R1), and for ;(R2) there exists a positive function such that where

It is well known that the cone plays a very important role in applying Green’s function in research area. In [7], the authors cannot acquire a positive constant taken instead of the role of positive function with in (1.2). In [9], Jiang and Yuan obtained some new properties of the Green function and established a new cone. The results can be stated as follows.

Proposition 1.2. Green’s function defined by (1.3) has the following properties: and

Proposition 1.3. The function has the following properties: where .

In this paper, we study the existence of positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation where is a real number, is a positive parameter, is the standard Riemann-Liouville differentiation, and with , .

We assume the following conditions hold throughout the paper:(H1) are both constants with ;(H2), ;(H3), and there exist such that where .

2. The Preliminary 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.

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

Definition 2.2. The fractional derivative of order of a function is given by where , provided the right side is pointwise defined on .

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

Lemma 2.4. Assume that with a fractional derivative of order that belongs to . Then for some .

Lemma 2.5 (see [7]). Given and , the unique solution of is where

Lemma 2.6. Suppose (H1) holds. Given and , the unique solution of is where

Proof. By applying Lemmas 2.4 and 2.5, we have Because by (H1), is convergent; therefore, is convergent. Note that is continuous function on , so is convergent.From , we have . Therefore,

Lemma 2.7 (see [7]). Let be a Banach space, a cone, and two bounded open sets of with . Suppose that is a completely continuous operator such that either (i), and , or(ii), and ,holds. Then has a fixed point in .

3. The Main Results

Let Then is the solution of BVP (1.6) if and only if , where is the operator defined by By similar arguments to Proposition 1.3, we obtain the following result.

Lemma 3.1. Suppose (H1) holds. The function has the following properties: where .

Let be endowed with the ordering if for all , and the maximum norm . Define the cone by , and where is defined by (3.3).

It is easy to see that and are cones in . For any , let , and .

For convenience, we introduce the following notations: By similar arguments to Lemma 4.1 of [9], we obtain the following result.

Lemma 3.2. Assume that (H1)–(H3) hold. Let be the operator defined by Then is completely continuous.

Theorem 3.3. Assuming (H1)–(H3) hold, exist. Then, for each satisfying there exists at least one positive solution of BVP (1.6) in .

Theorem 3.4. Assuming (H1)–(H3) hold, exist. Then, for each satisfying there exists at least one positive solution of BVP (1.6) in .

Proof of Theorem 3.3. Let be given as in (3.7), and choose such that Beginning with , there exists an such that , for . So and . For , we have Thus, . So, if we let then It remains to consider . There exists an such that , for all . There are the two cases, (a), where is bounded, and (b), where is unbounded.Case a. Suppose is such that , for all .Let . Then, for with , we have So, if we let then Case b. Let be such that . Choosing with , and so . For this case, if we let then Therefore, by (ii) of Lemma 2.7, has a fixed point such that and satisfies It is obvious that is solution of (1.6) for , and Next, we will prove . From and (H1)–(H3), we have Thus, . then is solution of (1.6) for .

Proof of Theorem 3.4. Let be given as in (3.8), and choose such that Beginning with , there exists an such that , for . So, for and , we have Thus, . So, if we let then Next, considering , there exists an such that , for all . Let . Then, . For , we have and so . For this case, if we let then Therefore, by (i) of Lemma 2.7, has a fixed point such that and satisfies By similar method to Theorem 3.3, we can get , then is solution of (1.6) for . We complete the proof.

Acknowledgments

This work is supported by the NSFC (11061030, 11101335, 11026060), Gansu Provincial Department of Education Fund (No. 1101-02), and Science and Technology Bureau of Lanzhou City (No. 2011-2-72). The authors are very grateful to the anonymous referees for their valuable suggestions.

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