#### Abstract

We establish some sufficient conditions for the existence and uniqueness of positive solutions to a class of initial value problem for impulsive fractional differential equations involving the Caputo fractional derivative. Our analysis relies on a fixed point theorem for mixed monotone operators. Our result can not only guarantee the existence of a unique positive solution but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate our main result.

#### 1. Introduction

The purpose of this paper is to investigate the existence and uniqueness of positive solutions to the following impulsive initial value problem (IVP for short) for Caputo fractional-order differential equations: where , , is the Caputo fractional derivative, and are given functions, , , , , and and represent the right and left limits of at .

Fractional differential equations arise in many fields, such as physics, mechanics, chemistry, economics, engineering, and biological sciences; see [1–6], for example. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Miller and Ross [3], Podlubny [5], Kilbas et al. [6], and the papers [7–21] and the references therein. In these papers, many authors have investigated the existence of positive solutions for nonlinear fractional differential equation boundary value problems. On the other hand, the uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems has been studied by some authors; see [10, 15, 19, 22, 23], for example.

In [7], Ahmad and Sivasundaram considered the following impulsive hybrid boundary value problem for nonlinear fractional differential equations: where is the Caputo fractional derivative, is a continuous function, , with , and , for . Based on contraction mapping principle and the Krasnoselskii fixed point theorem, they discussed some existence results for (2).

In [24], Benchohra and Slimani concerned the existence and uniqueness of solutions for the following initial value problem for Caputo fractional-order differential equations: where , , is a given function, , , , , and and represent the right and left limits of at . They gave an existence and uniqueness result for the IVP (3) which was based on the Banach fixed point theorem and also obtained two existence and uniqueness results; the first one was based on the Schaefer fixed point theorem and the second one was based on the nonlinear alternative of the Leray-Schauder type.

Different from the above works [7, 24], in this paper, we will use a fixed point theorem for mixed monotone operators to study the existence and uniqueness of positive solutions for the IVP (1). Our result can not only guarantee the existence of unique positive solution but also be applied to construct iterative scheme for approximating it.

With this context in mind, the outline of this paper is as follows. In Section 2, we will recall certain results from the theory of fractional calculus and some definitions, notations, and results of mixed monotone operators. In Section 3, we will provide some conditions under which the IVP (1) will have a unique positive solution. Finally, in Section 4, we will provide one example, which explicates the applicability of our main result.

#### 2. Preliminaries

For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proof of our main theorem.

*Definition 1 (see [9]). *Let with . Suppose that . Then the th Riemann-Liouville fractional integral is defined to be
whenever the right-hand side is defined. Similarly, with and , one defines the th Riemann-Liouville fractional derivative to be
where is the unique positive integer satisfying and .

*Definition 2 (see [11]). *For a function given on the interval , the Caputo fractional-order derivative of order of is defined by
where .

In the sequel, we present some basic concepts in the ordered Banach spaces for completeness and one fixed point theorem which will be used later. For convenience of readers, we suggest that one refers to [23, 25, 26] for details.

Suppose that is a real Banach space which is partially ordered by a cone . That is, if and only if . If and , then we denote that or . By we denote the zero element of . Recall that a nonempty closed convex set is a cone if it satisfies , ; , .

is called normal if there exists a constant such that, for all , , implies that ; in this case is called the normality constant of . If , , the set is called the order interval between and .

*Definition 3 (see [23, 25, 26]). *Consider that is said to be a mixed monotone operator if is increasing in and decreasing in ; that is, , , , and imply that . Element is called a fixed point of if .

Lemma 4 (see [25]). *Let be normal and let be a mixed monotone operator. Suppose that** there exist and , such that and ,** for any , there exists , such that
**Then has a unique fixed point in with . Moreover, for any initial values , , constructing successively the sequences
**
one has and as .*

#### 3. Main Result

In this section, we apply Lemma 4 to study the IVP (1) and then we obtain a new result on the existence and uniqueness of positive solutions. The existence and uniqueness result is relatively new to the fractional differential equations in this literature.

In our considerations we will work in the Banach space with the standard norm . Notice that this space can be equipped with a partial order given by Set , the standard cone. It is clear that is a normal cone in and the normality constant is 1. Consider the following set of functions: Then is a Banach space with the norm Denote by ; then is a standard and normal cone in .

Lemma 5 (see [24]). *Let and let be continuous. A function is a solution of the fractional integral equation:
**
where , if and only if is a solution of the fractional IVP
*

*Remark 6. *The concept of solutions for fractional differential equations has been argued extensively; see [7, 18, 21, 24] and the references therein. In [18], the authors gave a counterexample to show that the result in Lemma 5 is not reasonable. However, the counterexample is inappropriate. From the very recent paper [21], the approach for finding the solution of impulsive fractional differential equations in [18] is inappropriate and some arguments like Lemma 5 are plausible.

For the convenience, we list the following conditions: × × is continuous and × is continuous; is increasing in for fixed and and decreasing in for fixed and , and is decreasing in for fixed ; for any , there exist , , and which depend on , such that
the functions are continuous and nondecreasing and there exists a constant such that for each , ;.

Theorem 7. *Assume that are satisfied and there exists such that
**
where , .**Then the IVP (1) has a unique positive solution in with , . Moreover, for any initial values , , constructing successively the sequences
**
where , then one has and as .*

*Proof. *To begin with, from Lemma 5, the IVP (1) has an integral formulation given by
Define an operator by
It is easy to prove that is the solution of the IVP (1) if and only if is the fixed point of . From , , and , we know that
for any , . So, . In the sequel we check that satisfies all assumptions of Lemma 4.

Firstly, we prove that is a mixed monotone operator. In fact, for , with , , we know that , , and , and by and Definition 3, we have that
That is, .

Next, we show that satisfies the condition of Lemma 4. Now, we set . Then, .

On the one hand, from and , we have
That is to say, . From (15), for any , we obtain
That is, .

On the other hand, we take such that
where , , and . For any ,
That is to say, . Therefore, condition of Lemma 4 holds.

Finally, we show that satisfies condition of Lemma 4. Let and
For any , , and , by we have
That is to say, , for all , . Hence, condition of Lemma 4 is satisfied. So, the conclusion of Theorem 7 follows from Lemma 4.

#### 4. An Example

We present one example to illustrate Theorem 7.

*Example 1. *Consider the following fractional IVP:
where , .

In this example,
Then, , is continuous and increasing in for fixed and and decreasing in for fixed and . And , is continuous and decreasing in for fixed . Moreover, for any , we take
For any , , and , we have
We show that
if and only if we can prove that
That is to say,
From (29), we get
and then
Hence, (33) is satisfied. So, (31) holds.

Similarly, we obtain that
Further, is continuous, increasing in , and
And, we also have .

Finally, owing to , we have that
We take such that
And thus,
Then,
Hence, (15) holds. Therefore, all the conditions of Theorem 7 are satisfied. An application of Theorem 7 implies that problem (27) has a unique positive solution.

#### Acknowledgments

This research was supported by the Youth Science Foundation of China (11201272), the Science Foundation of Shanxi Province (2013011003-3), and the Science Foundation of Business College of Shanxi University (2012050).