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
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####^~^~^~^~^~^####x2013;6], fractional differential equations have attracted great research interest in recent years [7####^~^~^~^~^~^####x2013;17]. Recently, ur Rehman and Khan  investigated the fractional order multipoint boundary value problem:
where ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;, , ####^~^~^~^~^~^####x2009; 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 .
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
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
and for ,
Using , (2.15), and (2.16), we obtain
So the unique solution of the problem (2.9) is
The proof is completed.
The function has the following properties.
Proof. It is obvious that holds. From (2.11), we obtainFrom (2.8), we have
The proof is completed.
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
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####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;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 .
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 ####^~^~^~^~^~^####x2009; ####^~^~^~^~^~^####x2009;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
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
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
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)####^~^~^~^~^~^####x2013;(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)####^~^~^~^~^~^####x2013;(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:
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
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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