Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 672543, 8 pages

doi:10.1155/2012/672543

Research Article

Uniqueness of Positive Solutions for a Perturbed Fractional Differential Equation

Chen Yang and Jieming Zhang

Department of Mathematics, Business College of Shanxi University, Shanxi, Taiyuan 030031, China

Received 29 September 2012; Accepted 1 November 2012

Academic Editor: To Ma

Copyright © 2012 Chen Yang and Jieming Zhang. 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.

Abstract.

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem: , , , , , where is the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem of generalized concave operators. An example is given to illustrate the main result.

1. Introduction

In this paper, we are interested in the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem:

where is the standard Riemann-Liouville fractional derivative and is continuous.

Fractional differential equations arise in many fields, such as physics, mechanics, chemistry, economics, engineering, and biological sciences; see [1–15], for example. In recent years, the study of positive solutions for fractional differential equation boundary value problems has attracted considerable attention, and fruits from research into it emerge continuously. For a small sample of such work, we refer the reader to [16–26] and the references therein. On the other hand, the uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems has been studied by some authors; see [19–22, 25], for example.

When , Yang and Chen [22] investigated the existence and uniqueness of positive solutions for the problem (1.1) by means of a fixed-point theorem for concave operators. They present the following result.

Theorem 1.1 (see [22]). Assume that

is increasing for , , and is not identically vanishing; for any , , , there exist constants , , such that

.

Then the problem (1.1) with has a unique positive solution.

In this paper, we will remove the condition and improve on . And we will use a fixed-point theorem of generalized concave operators to show the existence and uniqueness of positive solutions for the problem (1.1). Our main result is summarized in the following theorem.

Theorem 1.2. Assume that holds and

for any and , , there exists a number such that

Then the problem (1.1) has a unique positive solution which satisfies , , where .

Remark 1.3. Some examples of which satisfy the condition are

(1), where ,(2) with , for all .

Remark 1.4. It is easy to see that the condition is weaker than the condition . Moreover, we do not need the condition in our main result.

2. Preliminaries and Previous Results

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

Definition 2.1 (see [4, Definition 2.1]). The integral

is called the Riemann-Liouville fractional integral of order , where and denotes the gamma function.

Definition 2.2 (see [4, page 36-37]). For a function given in the interval , the expression

where , denotes the integer part of number , is called the Riemann-Liouville fractional derivative of order .

By using the same method in [21], the problem (1.1) can be transformed into the following boundary value problem:

Moreover, from Lemma 2.5 and Lemma 2.7 in [21], we can easily obtain the following result.

Lemma 2.3. If is a positive solution of the problem (2.3), then is a positive solution of the problem (1.1). On the other hand, if is a positive solution of the problem (2.3), then the solution is

where

Here is called the Green function of the problem (2.3). Evidently, for .

The following property of the Green function plays important roles in this paper.

Lemma 2.4 (see [22]). The Green function in Lemma 2.3 has the following property:

In the sequel, we present some basic concepts in ordered Banach spaces for completeness and a fixed-point theorem which we will be used later. For convenience of readers, we suggest that one refers to [27, 28] 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 or . By we denote the zero element of . Recall that a nonempty closed convex set is a cone if it satisfies (i) , ; (ii) , .

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

For all , the notation means that there exist and such that . Clearly, is an equivalence relation. Given  (i.e., and ), we denote by the set . It is easy to see that is convex and for all .

In a recent paper [28], Zhai et al. considered the following operator equation:

They established the existence and uniqueness of positive solutions for the above equation, and they present the following interesting result.

Theorem 2.5 (see Theorem 2.1 in [28]). Let and be a normal cone. Assume that

is increasing; satisfies ; for any and , there exists such that .

Then the operator equation (2.7) has a unique solution in .

Remark 2.6. An operator is said to be generalized concave if satisfies condition .

3. Proof of Theorem 1.2

In this section, we apply Theorem 2.5 to study the problem (1.1), and we obtain a new result on the existence and uniqueness of positive solutions. The method used here is new to the literature and so is the existence and uniqueness result to the fractional differential equations.

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.

Proof of Theorem 1.2. Let , . Then

For any , we define

where is given as in Lemma 2.3. Noting that and , it follows from that . In the sequel we check that and satisfy all assumptions of Theorem 2.5.

Firstly, we prove that is an increasing operator. In fact, for , with , we know that , , by the monotonicity of Riemann-Liouville fractional integral and ,

That is . Hence, the condition in Theorem 2.5 is satisfied.

Next we show that the condition holds. From , for any and , we obtain

That is , for all , . So the condition in Theorem 2.5 is satisfied. Now we show that the condition is also satisfied. On one hand, it follows from and Lemma 2.4 that

On the other hand, also from and Lemma 2.4, we obtain

Let

Since is continuous and , we can get . Consequently,

Next we consider . If , then ; if , let , then . It is easy to prove that

Hence,

Further,

Hence . Finally, using Theorem 2.5, has a unique solution in . That is, is a unique positive solution of the problem (2.3) in . So there are , with such that , . From Lemma 2.3, is the solution of the problem (1.1). Evidently, is a unique positive solution of the problem (1.1).

Remark 3.1. Let . Then the conditions , are satisfied and the problem (2.3) has a unique solution , . From Lemma 2.4, the unique solution is a positive solution and satisfies . So is a unique positive solution of the problem (1.1).

To illustrate how our main result can be used in practice we present an example.

Example 3.2. Consider the following problem:

where are continuous with .

In this example, we have . Let . Evidently, is increasing in for , , and increasing in for , . Moreover, . Set . Then

Hence, all the conditions of Theorem 1.2 are satisfied. An application of Theorem 1.2 implies that the problem (3.13) has a unique positive solution.

Acknowledgments

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

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