Abstract and Applied Analysis

Volume 2012, Article ID 512127, 16 pages

http://dx.doi.org/10.1155/2012/512127

## Positive Solutions of Eigenvalue Problems for a Class of Fractional Differential Equations with Derivatives

^{1}School of Mathematical and Informational Sciences, Yantai University, Yantai, Shandong 264005, China^{2}School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China^{3}Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand^{4}Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 2 January 2012; Accepted 15 March 2012

Academic Editor: Shaoyong Lai

Copyright © 2012 Xinguang Zhang 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

By establishing a maximal principle and constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of a class of fractional differential equations is discussed. Some sufficient conditions for the existence of positive solutions are established.

#### 1. Introduction

In this paper, we discuss the existence of positive solutions for the following eigenvalue problem of a class fractional differential equation with derivatives where is a parameter, , , , , with , and is the standard Riemann-Liouville derivative. is continuous, and may be singular at , and .

As fractional order derivatives and integrals have been widely used in mathematics, analytical chemistry, neuron modeling, and biological sciences [1–6], fractional differential equations have attracted great research interest in recent years [7–17]. Recently, ur Rehman and Khan [8] investigated the fractional order multipoint boundary value problem: where , , , with . The Schauder fixed point theorem and the contraction mapping principle are used to establish the existence and uniqueness of nontrivial solutions for the BVP (1.2) provided that the nonlinear function is continuous and satisfies certain growth conditions. But up to now, multipoint boundary value problems for fractional differential equations like the BVP (1.1) have seldom been considered when has singularity at and (or) 1 and also at . We will discuss the problem in this paper.

The rest of the paper is organized as follows. In Section 2, we give some definitions and several lemmas. Suitable upper and lower solutions of the modified problems for the BVP (1.1) and some sufficient conditions for the existence of positive solutions are established in Section 3.

#### 2. Preliminaries and Lemmas

For the convenience of the reader, we present here some definitions about fractional calculus.

*Definition 2.1 (See [1, 6]). *Let with . Suppose that . Then the th Riemann-Liouville fractional integral is defined by
whenever the right-hand side is defined. Similarly, for with , we define the th Riemann-Liouville fractional derivative by
where is the unique positive integer satisfying and .

*Remark 2.2. *If with order , then

Lemma 2.3 (See [6]). *One has the following.*

(1)* If *,* then *

(2) *If *,* then *

Lemma 2.4 (See [6]). *Let . Assume that . Then**
where , and is the smallest integer greater than or equal to . *

Let

and for , we have

Lemma 2.5. *Let ; If , then the unique solution of the linear problem
**
is given by
**
where
**
is the Green function of the boundary value problem (2.9).*

*Proof. *Applying Lemma 2.4, we reduce (2.9) to an equivalent equation:
From (2.12) and noting that , we have . Consequently the general solution of (2.9) is
Using (2.13) and Lemma 2.3, we have
Thus,
and for ,
Using , (2.15), and (2.16), we obtain
So the unique solution of the problem (2.9) is
The proof is completed.

Lemma 2.6. *
The function has the following properties.*

where

*Proof. *It is obvious that holds.

From (2.11), we obtainFrom (2.8), we have
The proof is completed.

Consider the modified problem of the BVP (1.1):

Lemma 2.7. *Let ; then problem (1.1) is turned into (2.22). Moreover, if is a solution of problem (2.22), then the function is a positive solution of the problem (1.1).*

*Proof. *Substituting into (1.1) and using Definition 2.1 and Lemmas 2.3 and 2.4, we obtain
Consequently, . It follows from that . Using , , we transform (1.1) into (2.22).

Now, let be a solution for problem (2.22). Using Lemma 2.3, (2.22), and (2.23), one has
Noting
we have
It follows from the monotonicity and property of that
Consequently, is a positive solution of the problem (1.1).

*Definition 2.8. *A continuous function is called a lower solution of the BVP (2.22), if it satisfies

*Definition 2.9. *A continuous function is called an upper solution of the BVP (2.22), if it satisfies

By Lemmas 2.5 and 2.6, we have the maximal principle.

Lemma 2.10. *If and satisfies
**and for any , then , for .*

Set

To end this section, we present here two assumptions to be used throughout the rest of the paper. is decreasing in and , and for any ,

uniformly on . For any , , and

#### 3. Main Results

The main result is summarized in the following theorem.

Theorem 3.1. *Provided that and hold, then there is a constant such that for any , the problem (1.1) has at least one positive solution , which satisfies , .*

*Proof. *Let ; we denote a set and an operator in as follows:
Clearly, is a nonempty set since . We claim that is well defined and .

In fact, for any , by the definition of , there exists one positive number such that for any . It follows from Lemma 2.6 and that

Setting , from , we have . By the continuity of on , we have . On the other hand,From (3.3), one has
It follows from Lemma 2.6 and (3.3) that
where
Using (3.3) and (3.6), we know that is well defined and .

Next we will focus on the upper and lower solutions of problem (2.22). From and (3.2), we know that the operator is decreasing in . Usingand letting
we have

On the other hand, letting , since is decreasing with respect to and , for any , we haveFrom (3.2), (3.3), and , for all , we have
uniformly on . Thus there exists large enough , such that, for any ,
From Lemma 2.6, one has
Letting
and using Lemmas 2.3 and 2.7, we obtain
Obviously, . By (3.16), we have
which implies that
Consequently, it follows from (3.17)-(3.18) that
From (3.16) and (3.18)–(3.20), we know that are upper and lower solutions of the problem (2.22), and .

Define the function and the operator in by
It follows from and (3.21) that is continuous. Consider the following boundary value problem:
Obviously, a fixed point of the operator is a solution of the BVP (3.22). For all , it follows from Lemma 2.6, (3.21), and that
So is bounded. From the continuity of and , it is obviously that is continuous.

From the uniform continuity of and the Lebesgue dominated convergence theorem, we easily get that is equicontinuous. Thus from the Arzela-Ascoli theorem, is completely continuous. The Schauder fixed point theorem implies that has at least one fixed point such that .

Now we prove
Let . Since is the upper solution of problem (2.22) and is a fixed point of , we have

From (3.17), (3.18), and the definition of , we obtainSo
From (3.18) and (3.20), one has
By (3.27), (3.28), and Lemma 2.10, we get which implies that on . In the same way, we have on . Thus we obtain
Consequently, . Then is a positive solution of the problem (2.22). It thus follows from Lemma 2.7 that is a positive solution of the problem (1.1).

Finally, by (3.29), we haveThus,

Corollary 3.2. *Suppose that condition holds, and that for any , , and
**Then there exists a constant such that for any , the problem (1.1) has at least one positive solution , which satisfies , .*

We consider some special cases in which has no singularity at or .

We give the following assumption.

is decreasing in .

Then, is nonsingular at and for all , , which implies that . Thus

naturally holds; we then have the following corollary.

Corollary 3.3. *If holds and**
then there exists a constant such that for any , the problem (1.1) has at least one positive solution , which satisfies , .**Proof. *In the proof of Theorem 3.1, we replace the set by
and the inequalities (3.18)–(3.20) by
Since , we have
The rest of the proof is similar to that of Theorem 3.1.

If is nonsingular at and , we have the conclusion.

Corollary 3.4. *If is continuous and decreasing in , the problem (1.1) has at least one positive solution , which satisfies , .*

*Example 3.5. *Consider the existence of positive solutions for the following eigenvalue problem of fractional differential equation:

Let Then is decreasing in , and for any , uniformly on . Thus holds.

On the other hand, for any and ,

thus we have

which implies that holds. From Theorem 3.1, there is a constant such that for any the problem (3.38) has at least one positive solution and

#### Acknowledgments

This work is supported financially by the National Natural Science Foundation of China (11071141, 11126231) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017).

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