Abstract and Applied Analysis

Volume 2012 (2012), Article ID 856302, 11 pages

doi:10.1155/2012/856302

Research Article

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

Xiangbing Zhou, Wenquan Wu, and Hongjiang Ma

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 [4–12] 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.

References

M. A. Geraghty, “On contractive mappings,” Proceedings of the American Mathematical Society, vol. 40, pp. 604–608, 1973.

A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis, vol. 72, no. 5, pp. 2238–2242, 2010.

I. J. Cabrera, J. Harjani, and K. B. Sadarangani, “Existence and uniqueness of positive solutions for a singular fractional three-point boundary value problem,” Abstract and Applied Analysis, vol. 2012, Article ID 803417, 18 pages, 2012.

B. Ahmad, A. Alsaedi, and B. S. Alghamdi, “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1727–1740, 2008.

J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 3403–3410, 2009.

Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity,” Nonlinear Analysis, vol. 74, no. 17, pp. 6434–6441, 2011.

X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555–560, 2012.

X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012.

X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012.

X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012.

Y. Wang, L. Liu, and Y. Wu, “Positive solutions of a fractional boundary value problem with changing sign nonlinearity,” Abstract and Applied Analysis, vol. 2012, Article ID 149849, 12 pages, 2012.

M. Jia, X. Liu, and X. Gu, “Uniqueness and asymptotic behavior of positive solutions for a fractional-order integral boundary value problem,” Abstract and Applied Analysis, vol. 2012, 21 pages, 2012.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.

X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012.