- Research Article
- Open Access
Positive Solution of Singular Boundary Value Problem for a Nonlinear Fractional Differential Equation
© Changyou Wang et al. 2011
- Received: 16 August 2010
- Accepted: 9 January 2011
- Published: 17 January 2011
The method of upper and lower solutions and the Schauder fixed point theorem are used to investigate the existence and uniqueness of a positive solution to a singular boundary value problem for a class of nonlinear fractional differential equations with non-monotone term. Moreover, the existence of maximal and minimal solutions for the problem is also given.
- Fractional Order
- Fractional Derivative
- Fractional Calculus
- Fractional Differential Equation
- Minimal Solution
where , denotes Caputo derivative, and with (i.e., is singular at ). Their analysis relies on Krasnoselskii's fixed-point theorem and nonlinear alternative of Leray-Schauder type in a cone. More recently, Caballero Mena et al.  have proved the existence and uniqueness of a positive and non-decreasing solution to this problem by a fixed-point theorem in partially ordered sets. Other related results on the boundary value problem of the fractional differential equations can be found in the papers [11–23]. A study of a coupled differential system of fractional order is also very significant because this kind of system can often occur in applications [24–26].
However, in the previous works [9, 10], the nonlinear term has to satisfy the monotone or other control conditions. In fact, the nonlinear fractional differential equation with non-monotone term can respond better to impersonal law, so it is very important to weaken control conditions of the nonlinear term. In this paper, we mainly investigate the fractional differential (1.2) without any monotone requirement on nonlinear term by constructing upper and lower control function and exploiting the method of upper and lower solutions and Schauder fixed-point theorem. The existence and uniqueness of positive solution for (1.2) is obtained. Some properties concerning the maximal and minimal solutions are also given. This work is motivated by the above references and my previous work . This paper is organized as follows. In Section 2, we recall briefly some notions of the fractional calculus and the theory of the operators for integration and differentiation of fractional order. Section 3 is devoted to the study of the existence and uniqueness of positive solution for (1.2) utilizing the method of upper and lower solutions and Schauder fixed-point theorem. The existence of maximal and minimal solutions for (1.2) is given in Section 4.
For the convenience of the reader, we present here the necessary definitions and properties from fractional calculus theory, which are used throughout this paper.
provided that the right-hand side is pointwise defined on .
where , , provided that the right-hand side is pointwise defined on .
Lemma 2.3 (see ).
Lemma 2.4 (see ).
is valid when , , .
Lemma 2.5 (see ).
where denotes the beta function. Thus, as . In the similar way, we can prove that as .
Namely, (2.7) follows.
Namely, (1.2) holds. The proof is therefore completed.
Hence, it is follow from (2.6) that , for and .
Let is the Banach space endowed with the infinity norm, is a nonempty closed subset of defined as . The positive solution which we consider in this paper is a function such that .
then we have the following lemma.
Lemma 2.8 (see ).
Let , , is a continuous function and . If is continuous function on , then the operator is completely continuous.
Let , , is a continuous function, , and is continuous function on . Take , and . For any , , we define the upper-control function , and lower-control function , it is obvious that are monotonous non-decreasing on and .
then the function are called a pair of order upper and lower solutions for (1.2).
Now, we give and prove the main results of this paper.
endowed with the norm , then we have . Hence is a convex, bounded, and closed subset of the Banach space . According to Lemma 2.8, the operator is completely continuous. Then we need only to prove .
Hence , , that is, . According to Schauder fixed-point theorem, the operator has at least a fixed-point , . Therefore the boundary value problem (1.2) has at least one solution , and , .
then when , the boundary value problem (1.2) has a unique positive solution .
Note that, from Lemma 2.5, is a continuous function on . Thus, when , the operator is the contraction mapping. Then by Banach contraction fixed-point theorem, the boundary value problem (1.2) has a unique positive solution .
In this section, we consider the existence of maximal and minimal solutions for (1.2).
Let be a solution of (1.2) in , then is said to be a maximal solution of (1.2), if for every solution of (1.2) existing on , the inequality , , holds. A minimal solution may be defined similarly by reversing the last inequality.
respectively, furthermore, if then is the unique solution in , and hence the proof is completed.
Finally, we give an example to illuminate our results.
The authors are grateful to the referee for the comments. This work is supported by Natural Science Foundation Project of CQ CSTC (Grants nos. 2008BB7415, 2010BB9401) of China, Ministry of Education Project (Grant no. 708047) of China, Science and Technology Project of Chongqing municipal education committee (Grant no. KJ100513) of China, the NSFC (Grant no. 51005264) of China.
- Mainardi F: The fundamental solutions for the fractional diffusion-wave equation. Applied Mathematics Letters 1996, 9(6):23-28. 10.1016/0893-9659(96)00089-4View ArticleMathSciNetGoogle Scholar
- Buckwar E, Luchko Y: Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. Journal of Mathematical Analysis and Applications 1998, 227(1):81-97. 10.1006/jmaa.1998.6078View ArticleMathSciNetGoogle Scholar
- Zhu ZY, Li GG, Cheng CJ: Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation. Applied Mathematics and Mechanics 2002, 23(1):1-12. 10.1007/BF02437724View ArticleGoogle Scholar
- Nahušev AM: The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms. Doklady Akademii Nauk SSSR 1977, 234(2):308-311.MathSciNetGoogle Scholar
- Aleroev TS: The Sturm-Liouville problem for a second-order differential equation with fractional derivatives in the lower terms. Differentsial'nye Uravneniya 1982, 18(2):341-342.MathSciNetGoogle Scholar
- Zhang S: Existence of solution for a boundary value problem of fractional order. Acta Mathematica Scientia B 2006, 26(2):220-228. 10.1016/S0252-9602(06)60044-1View ArticleGoogle Scholar
- Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, 36: 1-12.Google Scholar
- Bai ZB, Lü HS: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005, 311(2):495-505. 10.1016/j.jmaa.2005.02.052View ArticleMathSciNetGoogle Scholar
- Qiu T, Bai Z: Existence of positive solutions for singular fractional differential equations. Electronic Journal of Differential Equations 2008, 149: 1-9.MathSciNetGoogle Scholar
- Caballero Mena J, Harjani J, Sadarangani K: Existence and unqiueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems. Boundary Value Problems 2009, 2009:-10.Google Scholar
- Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
- Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009, 49(3-4):605-609. 10.1016/j.mcm.2008.03.014View ArticleMathSciNetGoogle Scholar
- Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Computers & Mathematics with Applications 2009, 58(9):1838-1843. 10.1016/j.camwa.2009.07.091View ArticleMathSciNetGoogle Scholar
- Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Applied Mathematics Letters 2009, 22(1):64-69. 10.1016/j.aml.2008.03.001View ArticleMathSciNetGoogle Scholar
- Zhang SQ: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Computers & Mathematics with Applications 2010, 59(3):1300-1309.View ArticleMathSciNetGoogle Scholar
- Ahmad B: Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations. Applied Mathematics Letters 2010, 23(4):390-394. 10.1016/j.aml.2009.11.004View ArticleMathSciNetGoogle Scholar
- Ahmad B, Sivasundaram S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis. Hybrid Systems 2010, 4(1):134-141. 10.1016/j.nahs.2009.09.002View ArticleMathSciNetGoogle Scholar
- Daftardar-Gejji V, Jafari H: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. Journal of Mathematical Analysis and Applications 2007, 328(2):1026-1033. 10.1016/j.jmaa.2006.06.007View ArticleMathSciNetGoogle Scholar
- Momani S, Qaralleh R: An efficient method for solving systems of fractional integro-differential equations. Computers & Mathematics with Applications 2006, 52(3-4):459-470. 10.1016/j.camwa.2006.02.011View ArticleMathSciNetGoogle Scholar
- Hosseinnia SH, Ranjbar A, Momani S: Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Computers & Mathematics with Applications 2008, 56(12):3138-3149. 10.1016/j.camwa.2008.07.002View ArticleMathSciNetGoogle Scholar
- Abdulaziz O, Hashim I, Momani S: Solving systems of fractional differential equations by homotopy-perturbation method. Physics Letters. A 2008, 372(4):451-459. 10.1016/j.physleta.2007.07.059View ArticleMathSciNetGoogle Scholar
- Abdulaziz O, Hashim I, Momani S: Application of homotopy-perturbation method to fractional IVPs. Journal of Computational and Applied Mathematics 2008, 216(2):574-584. 10.1016/j.cam.2007.06.010View ArticleMathSciNetGoogle Scholar
- Hashim I, Abdulaziz O, Momani S: Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation 2009, 14(3):674-684. 10.1016/j.cnsns.2007.09.014View ArticleMathSciNetGoogle Scholar
- Chen Y, An H-L: Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives. Applied Mathematics and Computation 2008, 200(1):87-95. 10.1016/j.amc.2007.10.050View ArticleMathSciNetGoogle Scholar
- Gafiychuk V, Datsko B, Meleshko V: Mathematical modeling of time fractional reaction-diffusion systems. Journal of Computational and Applied Mathematics 2008, 220(1-2):215-225. 10.1016/j.cam.2007.08.011View ArticleMathSciNetGoogle Scholar
- Deng WH, Li CP: Chaos synchronization of the fractional Lü system. Physica A 2005, 353(1–4):61-72.View ArticleGoogle Scholar
- Wang C: Existence and stability of periodic solutions for parabolic systems with time delays. Journal of Mathematical Analysis and Applications 2008, 339(2):1354-1361. 10.1016/j.jmaa.2007.07.082View ArticleMathSciNetGoogle Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.Google Scholar
This article is published under license to BioMed Central Ltd. 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.