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.
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.
2. Preliminaries and Notations
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).
3. Existence and Uniqueness of Positive Solution
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 .
4. Maximal and Minimal Solutions Theorem
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.
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