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 differential equation can be extensively applied to various disciplines such as physics, mechanics, chemistry, and engineering, see [1–3]. Hence, in recent years, fractional differential equations have been of great interest, and there have been many results on existence and uniqueness of the solution of boundary value problems for fractional differential equations, see [4–7]. Especially, in  the authors have studied the following type of fractional differential equations:
where is a real number, is continuous and is the fractional derivative in the sense of Riemann-Liouville. Recently, Qiu and Bai  have proved the existence of a positive solution to boundary value problems of the nonlinear fractional differential equations
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.
The Riemann-Liouville fractional integral of order of a function is given by
provided that the right-hand side is pointwise defined on .
The Caputo fractional derivative of order of a continuous function is given by
where , , provided that the right-hand side is pointwise defined on .
Lemma 2.3 (see ).
Let , , , then
Lemma 2.4 (see ).
is valid when , , .
Lemma 2.5 (see ).
Let , ; is a continuous function and . If is continuous function on , then the function
is continuous on , where
Let , ; is a continuous function and . If is continuous function on , then the boundary value problems (1.2) are equivalent to the Volterra integral equations
From Lemma 2.5, the Volterra integral equation (2.7) is well defined. If satisfies the boundary value problems (1.2), then applying to both sides of (1.2) and using Lemma 2.3,one has
where , . Since is continuous in , there exists a constant , such that , for . Hence
where denotes the beta function. Thus, as . In the similar way, we can prove that as .
By Lemma 2.4 we have
From the boundary conditions , one has
Therefore, it follows from (2.8) that
Namely, (2.7) follows.
Conversely, suppose that satisfies (2.7), then we have
From Lemmas 2.3 and 2.4 and Definition 2.2, one has
as well as
Thus, from (2.12), (2.14), and (2.15), it is follows that
Namely, (1.2) holds. The proof is therefore completed.
For , since , we can obtain
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 .
According to Lemma 2.6, (1.2) is equivalent to the fractional integral equation (2.7). The integral equation (2.7) is also equivalent to fixed-point equation , , where operator is defined as
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 .
Let , , and satisfy, respectively
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.
Let , ; is a continuous function with , and is a continuous function on . Assume that are a pair of order upper and lower solutions of (1.2), then the boundary value problem (1.2) has at least one solution , moreover,
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 .
For any , we have . In view of Remark 2.7, Definition 2.9, and the definition of control function, one has
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 , .
Let , ; is a continuous function with , and is a continuous function on . Assume that there exist two distinct positive constant , such that
then the boundary value problem (1.2) has at least a positive solution , moreover
By assumption (3.4) and the definition of control function, we have
Now, we consider the equation
From Lemmas 2.5 and 2.6, (3.7) has a positive continuous solution on
Namely, is a upper solution of (1.2). In the similar way, we obtain is the lower solution of (1.2). An application of Theorem 3.1 now yields that the boundary value problem (1.2) has at least a positive solution , moreover
If the conditions in Theorem 3.1 hold. Moreover for any , , there exists , such that
then when , the boundary value problem (1.2) has a unique positive solution .
According to Theorem 3.1, if the conditions in Theorem 3.1 hold, then the boundary value problems (1.2) have at least a positive solution in . Hence we need only to prove that the operator defined in (2.18) is the contraction mapping in . In fact, for any , by assumption (3.10), we have
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.
Let , , is a continuous function with , and is a continuous function on . Assume that is monotone non-decreasing with respect to the second variable, and there exist two positive constants such that
Then there exist maximal solution and minimal solution of (1.2) on , moreover
It is easy to know from Corollary 3.2 that and are the upper and lower solutions of (1.2), respectively. Then by using , as a pair of coupled initial iterations we construct two sequences , from the following linear iteration process:
It is easy to show from the monotone property of and the condition (4.1) that the sequences , possess the following monotone property:
The above property implies that
exist and satisfy the relation
Letting in (4.3) shows that and satisfy the equations
It is easy to verify that the limits and are maximal and minimal solutions of (1.2) in
respectively, furthermore, if then is the unique solution in , and hence the proof is completed.
Finally, we give an example to illuminate our results.
We consider the fractional order differential equation
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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