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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 856302, 11 pages
http://dx.doi.org/10.1155/2012/856302
Research Article

A Contraction Fixed Point Theorem in Partially Ordered Metric Spaces and Application to Fractional Differential Equations

Department of Computer Science, Aba Teachers College, Sichuan, Wenchuan 623002, China

Received 23 September 2012; Accepted 31 October 2012

Academic Editor: Xinguang Zhang

Copyright © 2012 Xiangbing Zhou 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.

Abstract

We generalize a fixed point theorem in partially ordered complete metric spaces in the study of A. Amini-Harandi and H. Emami (2010). We also give an application on the existence and uniqueness of the positive solution of a multipoint boundary value problem with fractional derivatives.

1. Introduction

Let denote the class of those functions which satisfies the condition The following generalization of Banach's contraction principle is due to Geraghty [1].

Theorem 1.1. Let be a complete metric space and let be a map. Suppose there exists such that for each , Then has a unique fixed point , and converges to , for each .

And then, Amini-Harandi and Emami [2] proved a version of Theorem 1.1 in the context of partially ordered complete metric spaces.

Theorem 1.2. Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping, and there exists an element with . Suppose that there exists such that Assume that either is continuous or is such that Besides, if Then has a unique fixed point.

Theorem 2.2 was also applied to obtain the existence and uniqueness of the solution of a periodic boundary value problem by Amini-Harandi and Emami [2] and a singular fractional three-point boundary value problem by Cabrera et al. [3].

2. Main Results

In this section, we firstly define a class of functions by , and there exists a constant such that

Remark 2.1. Function classes include all bounded functions on with upper bound which are more extensive than those of . For example, but not in .

Now, we give an extended version of Theorem 1.2 in the context of partially ordered complete metric spaces.

Theorem 2.2. Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping, and there exists s an element with . Suppose that there exists a constant and such that Assume that either is continuous or Besides, if then has a unique fixed point.

Proof. We first show that has a fixed point. Since and is an increasing function, we obtain by induction that Put , For each integer , from (2.5), we have , then by (2.2) If there exists such that , then and is a fixed point; in this case, the proof is finished. Otherwise, for any , . Then by (2.6), we have that is,
Now, we show that is a Cauchy sequence. By the triangle inequality and (2.2), we have and then which implies that is a Cauchy sequence in . Since is a complete metric space, then there exists a such that . To prove that is a fixed point of , if is continuous, then hence . If case (2.3) holds, then we claim that still holds. In fact, since , taking limit as , then , this proves that ; consequently, .
Let be another fixed point of . From (2.4) there exists which is comparable to and . Monotonicity implies that is comparable to and for Moreover, Taking limit, and then . Similar to [2], we have . The proof of the uniqueness of the fixed point is completed.

3. Application to Fractional Differential Equations

In this section, we consider the unique positive solution for a general higher order fractional differential equation by using the generalized fixed point theorem where ,   and with and , , satisfying and is the standard Riemann-Liouville derivative, . Recently, there has been a significant development in the study of fractional differential equations; for more details we refer the reader to [412] and the references cited therein.

For the convenience of the reader, we present some notations and lemmas which will be used in the proof of our results.

Definition 3.1 (see [13, 14]). The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on .

Definition 3.2 (see [13, 14]). The Riemann-Liouville fractional derivative of order of a function is given by where , denotes the integer part of number , provided that the right-hand side is pointwise defined on .

Proposition 3.3 (see [13, 14]). Consider the following.(1) If , , then (2) If ,  , then

Proposition 3.4 (see [13, 14]). Let , and is integrable, then where   , is the smallest integer greater than or equal to .

Lemma 3.5 (see [15]). Let , then BVP (3.1) is equivalent to the following BVP: Moreover, if is a solution of problem (3.7), then the function is a positive solution of problem (3.1).

Let from [15], the Green function of (3.7) is and with property

In our considerations, we will work in the Banach space with the classical metric given by . Notice that this space can be equipped with a partial order given by In [2], it is proved that satisfies condition (2.3) of Theorem 2.2. Moreover, for , as the function ,   satisfies condition [15]. Consider the cone Note that as is a closed set of , is a complete metric space.

It is well known that the BVP (3.7) is equivalent to the integral equation Now, for we define the operator by Then from the assumption on and (3.10), we have

We also introduce the following class of nondecreasing functions by satisfying the following: Clearly, if , then . The standard functions , for example, , , and .

Theorem 3.6. Suppose is nondecreasing in on ; moreover, there exist positive constants , that satisfy and there exist a function and constants , , , such that for , with and . Then problem (3.1) has a unique nonnegative solution.

Proof. We check that the hypotheses in Theorem 2.2 are satisfied.
Firstly, the operator is nondecreasing by the hypothesis. Then for any , we have Noticing that and take
Thus, for any , we have For , we have and this inequality is obviously satisfied for . Thus, we have
Finally, since the zero function satisfies , Theorem 2.2 tells us that the operator has a unique fixed point in , or, equivalently, the BVP (3.1) has a unique nonnegative solution in .

Theorem 3.7. If the assumptions of Theorem 3.6 are satisfied, and there exists such that , then the unique solution of (3.1) is positive (a positive solution means a solution satisfying for ).

Proof. By Theorem 3.6, the problem (3.1) has a unique nonnegative solution. We prove the nonnegative solution is also positive.
Otherwise, there exists such that , and Then Consequently, this yields Note that , then we have
But on the other hand, since , , we have , by the continuity of , we can find a set satisfying and the Lebesgue measure such that for any . This contradicts to (3.29). Therefore, , that is, is positive solution of (3.1).

Example 3.8. Consider the following fractional boundary value problem: Then the BVP (3.30) has a unique positive solution.

Proof. Since Thus,
Let and , and take Then, for any , , , where Thus, and all of the conditions of Theorem 3.6 are satisfied.
On the other hand, , by Theorems 3.6 and 3.7, the BVP (19) has a unique positive solution.

Remark 3.9. In Example 3.8, which does not possess property and Thus, the unique positive solution of BVP (3.30) cannot be obtained via Theorem 1.2, but we obtain the unique positive solution of BVP (3.30) by using the generalized fixed point Theorem 2.2, which implies that Theorem 2.2 is an essential promotion of Theorem 1.2.

Acknowledgment

This work is supported by the Natural Sciences of Education and Science Office Bureau of Sichuan Province of China under Grant no. 2010JY0J41,11ZB152.

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