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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 512127, 16 pages
Positive Solutions of Eigenvalue Problems for a Class of Fractional Differential Equations with Derivatives
1School of Mathematical and Informational Sciences, Yantai University, Yantai, Shandong 264005, China
2School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
3Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
4Department 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.
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.
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  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 ). One has the following.
(1) If , then
(2) If , then
Lemma 2.4 (See ). Let . Assume that . Then where , and is the smallest integer greater than or equal to .
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.(1)(2),
Consider the modified problem of the BVP (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
Lemma 2.10. If and satisfies and for any , then , for .
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
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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