Keywords:fractional differential equation; positive solution; iterative scheme; singular boundary value problem
In this paper, we consider the following fractional differential equation:
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).
Recently, many results were obtained dealing with the existence of solutions for nonlinear fractional differential equations by using the techniques of nonlinear analysis; see [1-23] and references therein. The multi-point boundary value problems (BVP for short) have provoked a great deal of attention, for example [13-19]. In , the authors discussed some positive properties of the Green function for Direchlet-type BVP of nonlinear fractional differential equation
In , the authors investigated the existence and multiplicity of positive solutions by using some fixed point theorems for the fractional differential equation
In order to obtain the existence of positive solutions of (1.4), they considered the following fractional differential equation:
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.
Lemma 2.1 ()
For the convenience in presentation, we here list the assumption to be used throughout the paper.
Lemma 2.2 ()
Proof From Lemma 2.1, we have the solution of (2.5) given by
By Lemma 2.2, we have
Therefore, the solution of (2.5) is
Proof It is obvious that (1), (2) hold. In the following, we will prove (3).
On the other hand, we have
On the other hand, clearly we have
(2.8)-(2.11) implies (3) holds. □
By Lemma 2.4 we have the following results.
For convenience, we list here two more assumptions to be used later:
Remark 2.2 Inequality (2.12) is equivalent to
On the other hand, by (3) of Lemma 2.6, we have
3 Main results
It is easy to see that
By induction, we can get
By (3.4), (3.5), we have
(3.5) and (3.6) imply that
By the mixed monotone property of A and (3.6), we have
In the following, we will prove the positive fixed point of A is unique.
which implies u is a positive fixed point of A.
Example 3.1 (A 4-point BVP with coefficients of both signs)
Consider the following problem:
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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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