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.
- Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivatives (Theory and Applications). Gordon & Breach, Switzerland; 1993.Google Scholar
- Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Agrawal OP: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 2002, 272: 368-379. 10.1016/S0022-247X(02)00180-4MathSciNetView ArticleGoogle Scholar
- Lakshmikantham V, Vatsala AS: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 2008, 21: 828-834. 10.1016/j.aml.2007.09.006MathSciNetView ArticleGoogle Scholar
- Kosmatov N: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal. 2009, 70: 2521-2529. 10.1016/j.na.2008.03.037MathSciNetView ArticleGoogle Scholar
- Agarwal RP, O’Regan D, Stanek S: Positive solutions for Dirichlet problem of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 2010, 371: 57-68. 10.1016/j.jmaa.2010.04.034MathSciNetView ArticleGoogle Scholar
- Ahmad B, Alsaedi A: Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type. Nonlinear Anal. 2009, 3: 501-509.MathSciNetGoogle Scholar
- Liang S, Zhang J: Positive solutions for boundary value problems of fractional differential equation. Nonlinear Anal. 2009, 71: 5545-5550. 10.1016/j.na.2009.04.045MathSciNetView ArticleGoogle Scholar
- Jiang D, Yuan C: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. 2010, 72: 710-719. 10.1016/j.na.2009.07.012MathSciNetView ArticleGoogle Scholar
- Goodrich CS: Existence of positive solution to a class of fractional differential equations. Appl. Math. Lett. 2010, 23: 1050-1055. 10.1016/j.aml.2010.04.035MathSciNetView ArticleGoogle Scholar
- Arara A, Benchohra M, Hamidi N, Nieto JJ: Fractional order differential equations on an unbounded domain. Nonlinear Anal. 2010, 72: 580-586. 10.1016/j.na.2009.06.106MathSciNetView ArticleGoogle Scholar
- Salem HAH: On the fractional order m -point boundary value problem in reflexive Banach spaces and weak topologies. Comput. Math. Appl. 2009, 224: 565-572. 10.1016/j.cam.2008.05.033MathSciNetView ArticleGoogle Scholar
- Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 2010, 59: 1363-1375. 10.1016/j.camwa.2009.06.029MathSciNetView ArticleGoogle Scholar
- Benchohraa M, Hamania S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 2009, 71: 2391-2396. 10.1016/j.na.2009.01.073MathSciNetView ArticleGoogle Scholar
- El-Shahed M, Nieto JJ: Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order. Comput. Math. Appl. 2010, 59: 3438-3443. 10.1016/j.camwa.2010.03.031MathSciNetView ArticleGoogle Scholar
- Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033MathSciNetView ArticleGoogle Scholar
- Rehman M, Khan RA: Existence and uniqueness of solution for multi-point boundary value problems for fractional differential equations. Appl. Math. Lett. 2010, 23: 1038-1044. 10.1016/j.aml.2010.04.033MathSciNetView ArticleGoogle Scholar
- Allison J, Kosmatov N: Multi-point boundary value problems of fractional order. Commun. Appl. Anal. 2008, 12(4):451-458.MathSciNetGoogle Scholar
- Zhang S: Positive solution to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 2010, 59: 1300-1309. 10.1016/j.camwa.2009.06.034MathSciNetView ArticleGoogle Scholar
- Yang L, Chen H: Unique positive solutions for fractional differential equation boundary value problems. Appl. Math. Lett. 2010, 23: 1095-1098. 10.1016/j.aml.2010.04.042MathSciNetView ArticleGoogle Scholar
- Chang Y, Kavitha V, Mallika Arjunan M: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. 2009, 71: 5551-5559. 10.1016/j.na.2009.04.058MathSciNetView ArticleGoogle Scholar
- Shua X, Lai Y, Chen Y: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. 2011, 74: 2003-2011. 10.1016/j.na.2010.11.007MathSciNetView ArticleGoogle Scholar