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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 425408, 16 pages
Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative
1College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
3College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
Received 13 August 2012; Accepted 5 September 2012
Academic Editor: Seenith Sivasundaram
Copyright © 2012 Changyou Wang et al. 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.
This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained.
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 FDE, see [4–8]. Especially, Diethelm and Ford  have gained existence, uniqueness, and structural stability of solution of the type of fractional differential equation
where is a real number, denotes the Riemann-Liouville differential operator of order , and is the Taylor polynomial of order for the function at . Recently, Daftardar-Gejji and Jafari  have discussed the existence, uniqueness, and stability of solution of the system of nonlinear fractional differential equation where and denotes Caputo fractional derivative (see Definition 2.3). Delbosco and Rodino  have proved existence and uniqueness theorems for the nonlinear fractional equation where , is the Riemann-Liouville fractional derivative. Zhang  used the method of the upper and lower solution and cone fixed point theorem to obtain the existence and uniqueness of positive solution to (1.3). Yao  considered the existence of positive solution to (1.3) controlled by the power function employing Krasnosel’skii fixed point theorem of cone expansion-compression type. The existence of the local and global solution for (1.3) was obtained by Lakshmikantham and Vatsala  utilizing classical differential equation theorem.
More recently, Zhang  shows the existence of positive solutions to the singular boundary value problem for fractional differential equation where is the Riemann-Liouville fractional derivative of order , .
However, in the previous works, the nonlinear term has to satisfy the monotone or others control conditions. In fact, the fractional differential equations with nonmonotone function can respond better to impersonal law, so it is very important to weaken monotone condition. Considering this, in this paper, we mainly investigate the fractional differential Equation (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 the Schauder fixed point theorem. The existence and uniqueness of positive solution for (1.2) are obtained. Some properties concerning the maximal and minimal solutions are also given. This work is motivated by the above references and my previous work [16, 17]. Other related results on the fractional differential equations can be found in [18–24].
This paper is organized as follow. 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 upper and lower solution method and the Schauder fixed point theorem. The existence of maximal and minimal solutions for (1.2) is given in Section 4.
2. Preliminaries and Notations
First, we give some basic definitions and theorems which are basically used throughout this paper. denotes the space of continuous functions defined on and denotes the class of all real valued functions defined on which have continuous th order derivative.
Definition 2.1. Let and , then the expression is called the (left-sided) Riemann-Liouville integral of order .
Definition 2.2. Let , then the expression is called the (left-sided) Riemann-Liouville derivative of of order whenever the expression on the right-hand side is defined.
Definition 2.3. Let and , then the expression is called the (left-sided) Caputo derivative of of order .
In further discussion we will denote , , and as , , and , respectively.
Proof. Suppose satisfies the initial value problem (1.2), then applying to both sides of (1.2) and using Lemma 2.4 (2.7) follows. Conversely, suppose satisfies (2.7). Then observe that exists and is integrable, because
which exists and is integrable as is . Thus exists.
Applying on both sides of (2.7), one has as is continuous and . Hence satisfies (1.2). Moreover, from (2.4), hold.
Let be the Banach space endowed with the infinity norm and a nonempty closed subset of defined as . The positive solution which we consider in this paper is a function such that .
According to Lemma 2.5, (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 2.6. Let a given continuous function. Then the operator is completely continuous.
Proof. Let be bounded, that is, there exists a positive constant such that for any . Since is a given continuous function, we have
Let , then for any , we have Thus, Hence is uniformly bounded.
Now, we prove that is continuous. Since is continuous function in a compact set , then it is uniformly continuous there. Thus given , we can find such that whenever , where . Then proving the continuity of the operators .
Now, we will prove that the operator is equicontinuous. For each , any and . Let , then when , we have The Arzela-Ascoli Theorem implies that is completely continuous. The proof is therefore completed.
Lemma 2.7. If the operator is the contraction mapping, where is the Banach space, then has a unique fixed point in .
Let be a given function. Take , and . For any one defines the upper-control function , and lower-control function , obviously is monotonous nondecreasing on and .
Definition 2.8. Let , and satisfy then the functions 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 2.6, the operator is completely continuous. Then we need only to prove .
For any , we have , then Hence , that is, . According to Schauder fixed point theorem, the operator exists at least one fixed point . Therefore the boundary value problem (1.2) exists at least one solution , and .
Corollary 3.2. Assume is continuous, and there exist , such that then the boundary value problem (1.2) exists at least one positive solution , moreover
Proof. By assumption (3.4) and the definition of control function, we have Now, we consider the equation Obviously, (3.7) has a positive solution 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) exists at least one positive solution , moreover
Corollary 3.3. Assume is continuous, where , moreover then the boundary value problem (1.2) has at least one positive solution .
Proof. By assumption (3.9), there are positive constants , such that whenever . Let , then , . By the definition of control function, one has .
Now, we consider the equation Obviously, (3.11) has a positive solution namely, is the upper solution of (1.2). In the similar way, we obtain is the lower solution of (1.2). Therefore, the boundary value problem of (1.2) has at least one positive solution , what is more, we have
Corollary 3.4. Assume is continuous, where , moreover then the boundary value problem (1.2) exists at least at one positive solution .
Proof. According to , there exists , such that for any , we have
By the definition of control function, we have
We now consider the equation
According to Lemma 2.5, (3.17) is equivalent to the integral equation
Let be an operator as follows:
by Lemma 2.6, the operator is completely continuous.
Let where and satisfies that , then is convex, bounded, and closed subset of the Banach space . For any , we have then thus Hence, the Schauder fixed theorem assures that the operator has at least one fixed point and then (3.17) has at least one positive solution , therefore we have Combining condition (3.16), we have Obviously, is the upper solution of initial value problem (1.2), and is the lower solution. By Theorem 3.1, system (1.2) has at least one positive solution .
Corollary 3.5. Assume is continuous and there exists , such that then the boundary value problem (1.2) has at least one positive solution , moreover
Proof. According to Theorem 3.1, if the conditions in Theorem 3.1 hold, then the boundary value problem (1.2) has at least one positive solution in . Hence we need only to prove that the operator defined in (2.10) is the contraction mapping in . In fact, for any , by assumption (3.30), we have Thus, when , the operator is the contraction mapping. Then by Lemma 2.7, 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).
Definition 4.1. 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.
Theorem 4.2. Let be a given continuous and monotone nondecreasing with respect to the second variable. Assume that there exist two positive constants such that Then there exist maximal solution and minimal solution of (1.2) on , moreover
Proof. It is easy to know 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 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.
The authors are grateful to the referee for the comments. This work is supported by the Science and Technology Project of Chongqing Municipal Education Committee (Grants nos. KJ110501 and KJ120520) of China, Natural Science Foundation Project of CQ CSTC (Grant no. cstc2012jjA20016) of China, and the NSFC (Grant no. 11101298) of China.
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