Existence and uniqueness of positive solution to singular fractional differential equations
© Wang et al.; licensee Springer 2012
Received: 6 April 2012
Accepted: 10 July 2012
Published: 28 July 2012
In this paper, we discuss the existence and uniqueness of a positive solution to the following singular fractional differential equation with nonlocal boundary value conditions:
where , , with , is the standard Riemann-Liouville derivative, f may be singular at , , and .
Keywordsfractional differential equation positive solution iterative scheme singular boundary value problem
where , , with , is the standard Riemann-Liouville derivative, may be singular at , , and . In this paper, by a positive solution to (1.1), we mean a function which satisfies , positive on and satisfies (1.1).
where , is the standard Riemann-Liouville derivative, . By using the Krasnosel’skii fixed point theorem, the existence of positive solutions were obtained under suitable conditions on f.
where , , , with , , satisfied Carathéodory type conditions.
In , , and g, h have different monotone properties. By using the fixed point theorem for the mixed monotone operator, Zhang obtained (1.4) and had a unique positive solution with . But the results are not true since is a positive solution of (1.5), and . What causes it lies in the unsuitable using of properties of the Green function.
In , , is increasing for , . By using the positive properties of the Green function obtained in and fixed point theory for the concave operator, the authors obtained the uniqueness of a positive solution for the BVP (1.4).
Motivated by the works mentioned above, in this paper we aim to establish the existence and uniqueness of a positive solution to the BVP (1.1). Our work presented in this paper has the following features. Firstly, the BVP (1.1) possesses singularity, that is, f may be singular at , , and . Secondly, we impose weaker positivity conditions on the nonlocal boundary term, that is, some of the coefficients can be negative. Thirdly, the unique positive solution can be approximated by an iterative scheme.
The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. We also develop some new positive properties of the Green function. In Section 3, we discuss the existence and uniqueness of a positive solution of the BVP (1.1), we also give an example to demonstrate the application of our theoretical results.
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can be found in recent literature.
provided the right-hand side is defined pointwise on .
where , denotes the integral part of the number α, provided the right-hand side is pointwisely defined on .
Definition 2.3 By , we mean .
Lemma 2.1 ()
where , , .
For the convenience in presentation, we here list the assumption to be used throughout the paper.
() , on .
Remark 2.1 If (), we have and . If () and , we have and on .
Lemma 2.2 ()
where is called the Green function of BVP (2.5).
From , we have .
, for ;
, for ;
- (3), for , where(2.6)
- (i)When , noticing , we have(2.7)
- (ii)When , we have(2.10)
(2.8)-(2.11) implies (3) holds. □
By Lemma 2.4 we have the following results.
- (3), , where
For convenience, we list here two more assumptions to be used later:
Set , where θ is the zero element of E. We have the following lemma.
Lemma 2.7 Suppose that ()-() hold. Then .
where . This implies that A is well defined in .
Therefore, . Combining with (2.16), we have . □
Remark 2.3 By () and (2.15), A is a mixed monotone operator.
3 Main results
Theorem 3.1 Suppose that ()-() hold. Then the BVP (1.1) has a unique positive solution.
This implies that .
Since , we have is a positive fixed point of A.
In the following, we will prove the positive fixed point of A is unique.
Thus, , which contradicts the definition of . Consequently, the positive fixed point of A is unique.
Lemma 2.3 implies is a positive solution of (1.1).
which implies u is a positive fixed point of A.
Then is the unique positive solution of the BVP (1.1). □
Remark 3.1 The unique positive solution y of (1.1) can be approximated by the iterative schemes: for any , let , be defined as (3.2) and , , , then .
Example 3.1 (A 4-point BVP with coefficients of both signs)
By direct calculations, we have and , which implies () holds.
Therefore () holds. It is easy to get that () holds. Therefore, the assumptions of Theorem 3.1 are satisfied. Thus Theorem 3.1 ensures that the BVP (3.8) has a unique positive solution.
The authors are grateful to the anonymous referee for his/her valuable suggestions. The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11126231) and Project of Shandong Province Higher Educational Science and Technology Program (J11LA06). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
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