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International Journal of Analysis
Volume 2013 (2013), Article ID 438029, 7 pages
http://dx.doi.org/10.1155/2013/438029
Research Article

A Simple Method for Obtaining Coupled Fixed Points of --Contractive Type Mappings

1Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
2Department of Mathematics, Azarbaijan Shahid Madani University, Azarshahr, Tabriz, Iran

Received 17 August 2012; Revised 11 October 2012; Accepted 18 October 2012

Academic Editor: Ahmed Zayed

Copyright © 2013 Sh. Rezapour and J. Hasanzade Asl. 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

In 2012, the notion of --contractive type mappings was introduced by Samet, C. Vetro, and P. Vetro. By using a simple method, we give some coupled fixed point results for --contractive type mappings.

1. Introduction

In 1987, Guo and Lakshmikantham introduced the notion of coupled fixed points [1]. Then some authors proved some coupled fixed point results via some applications in the last decade of the previous century [26]. Later, this field was completed by some researchers by using different sights (see for example [718]). In 2012, Samet et al. introduced the notion of --contractive type mappings [19]. Also, Amini-Harandi has provided a method for obtaining coupled fixed point results [20]. The aim of this paper is to provide a simple method for obtaining some coupled fixed point results for --contractive type mappings.

Denote by the family of nondecreasing functions such that for all , where is the th iterate of . It is known that for all and [19]. Let be a metric space and a self-map on . Then is called a --contraction mapping whenever there exist and such that for all [19]. Also, we say that is -admissible whenever implies [19]. Also, we say that has the property () if is a sequence in such that for all and , then for all [19]. Let be a complete metric space and a -admissible --contractive mapping on . Suppose that there exists such that . If is continuous or has the property (), then has a fixed point (see [19]; Theorems 2.1 and 2.2). Also, we say that has the property () whenever for each there exists such that and . If has the property () in the Theorems 2.1 and 2.2, then has a unique fixed point ([19]; Theorem 2.3). It is considerable that the results of Samet et al. generalize similar ordered results in the literature (see the results of third section in [19]). Let be a mapping, where is a metric space. We say that is a coupled fixed point of whenever and . Define by for all . Then, it is easy to check that is a coupled fixed point of if and only if is a fixed point of .

2. Main Results

In this section, we define and for all , where is a metric space and is a mapping. Now, we are ready to state and prove our main results.

Lemma 1. Let be a complete metric space, a function, and a self-map on such that for all . Suppose that is -admissible and there exists such that . If has the property (), then has a fixed point.

Proof. Take such that and define the sequence in by for all . If for some , then is a fixed point of . Assume that for all . Since is -admissible, it is easy to check that for all . Thus, for each natural number one has If , then which is contradiction. Thus, ,  for all . Hence, and so for all . It is easy to check that is a Cauchy sequence. Thus, there exists such that . By using the assumption, we have for all . Thus, for all . Hence, and so .

By using a similar argument we can prove the following results.

Lemma 2. Let be a complete metric space, a function, and a continuous self-map on such that for all . Suppose that is -admissible and there exists such that . Then has a fixed point.

Lemma 3. Let be a complete metric space, a function, and a self-map on such that for all . Suppose that is -admissible and there exists such that . If has the property (), then has a fixed point.

Lemma 4. Let be a complete metric space, a function, and a continuous self-map on such that for all . Suppose that is -admissible and there exists such that . Then has a fixed point.

Theorem 5. Let be a complete metric space and a mapping. Suppose that there exist a mapping and such that for all , implies for all and there exists such that and . Also, suppose that for each convergent sequences and with , , and for all , we have and for all . Then has a coupled fixed point.

Proof. It is easy to check that for all in , the metric space is complete and for all , , where Thus, is a --contractive mapping. It is easy to check that is -admissible and . Since has the property (), has a fixed point by using Lemma 1 and so has a coupled fixed point.

Theorem 6. Let be a complete metric space and a continuous mapping. Suppose that there exist a mapping and such that for all , implies for all and there exists such that and . Then has a coupled fixed point.

Proof. The metric space is complete and for all , where Thus, is a continuous --contractive mapping. Also, it is easy to check that is -admissible and . Now by using Lemma 2, has a fixed point and so has a coupled fixed point.

Example 7. Let and for all . Define the mapping by whenever and whenever . Then is continuous mapping. Also, define by Thus, for each we have where for all . Also, it is easy to check that implies for all . Also, and , where . Thus by using Theorem 6, has a coupled fixed point.

Corollary 8. Let be a complete metric space, a mapping and an order on . Suppose that there exist such that and are comparable and also and are comparable, and a mapping such that for all comparable elements in . Suppose that is comparable with whenever is comparable with . Also, suppose that for each convergent sequences and with , , is comparable with and is comparable with for all , one gets that is comparable with and is comparable with for all . Then has a coupled fixed point.

Proof. Define the mapping by whenever and are comparable and otherwise. Then by using Theorem 5, has a coupled fixed point.

Corollary 9. Let be a complete metric space, a continuous mapping, a fixed element and an order on . Suppose that there exist such that , , and are comparable with , and a mapping such that for all and in which are comparable with . Assume that and are comparable with whenever and so are. Then has a coupled fixed point.

Proof. Define the mapping by whenever and are comparable with and otherwise. Then by using Theorem 6, has a coupled fixed point.

Theorem 10. Let be a complete metric space and a mapping. Suppose that there exist a mapping and such that for all , implies for all and there exists such that and . Also, suppose that for each convergent sequences and with , , and for all , one has and for all . Then has a coupled fixed point.

Proof. The metric space is complete and for all , where Thus, for all . Also, it is easy to check that is -admissible and . Since has the property (), has a fixed point by using Lemma 3, and so has a coupled fixed point.

Theorem 11. Let be a complete metric space and a continuous mapping. Suppose that there exist a mapping and such that for all , implies for all and there exists such that and . Then has a coupled fixed point.

Proof. The metric space is complete and for all , where Thus, for all . Also, it is easy to check that is a continuous -admissible mapping and . Now by using Lemma 4, has a fixed point and so has a coupled fixed point.

Example 12. Let and for all . Define the mapping by whenever and whenever . Then is a discontinuous mapping. Define by Thus, for each we have where for all . Also, implies for all . Finally, and , where . Thus by using Theorem 10, has a coupled fixed point.

Corollary 13. Let be a complete metric space, a mapping and an order on . Suppose that there exists a mapping such that for all comparable elements and in . Suppose that there exists such that and are comparable and also and are comparable, is comparable with whenever is comparable with and for each sequences and in such that , , and are comparable elements of for all , and are comparable elements for all . Then has a coupled fixed point.

Proof. Define the mapping by whenever and are comparable and otherwise. Then by using Theorem 10, has a coupled fixed point.

Corollary 14. Let be a complete metric space, a continuous mapping, a fixed element and an order on . Suppose that there exist such that , , and are comparable with , a mapping such that for all and in which are comparable with . Assume that and are comparable with whenever and so are. Then has a coupled fixed point.

Proof. Define the mapping by whenever and are comparable with and otherwise. Then by using Theorem 11, has a coupled fixed point.

Now, we give an application of Lemma 1. In this way, we study the nonlinear fractional differential equation for via the two-point boundary value condition , where is a continuous function and . Recall that the Green function associated to the equation is given by , where is the two-parametric Mittag-Leffler function defined by for and (see [21, 22]). Let and Now by considering some conditions, we give the following result about existence of solution of the nonlinear fractional differential equation.

Theorem 15. Suppose that (i)there exist a function and such that for all and with , (ii)there exist such that for all , for each and , implies (iii) if is a sequence in with and for all , then for all .Then the nonlinear fractional differential equation has at least one solution.

Proof. It is well known that is a solution of the nonlinear fractional differential equation if and only if is a solution of the integral equation for all . Define the operator by for all . Thus, for finding a solution of the the nonlinear fractional differential equation it is sufficient we find a fixed point of the continuous operator . Let be such that for all . By using (i), we get Note that, for all . Thus, . Now, define by whenever for all and otherwise. Hence, and so for all . Thus, is an --contractive mapping. By using (iii), implies . Therefore, is -admissible. From (ii), there exists such that . Now by using (iv) and Lemma 1, there exists such that .

Similar to the work of Samet et al. [19], we say that the space has the property whenever for each there exists such that , , and . It is easy to check that if has the property in before results, then the mapping has a unique coupled fixed point.

If there is an order on , then one can construct the order on by whenever and . By using this idea, it has been provided some coupled fixed point results for mixed monotone mappings on ordered metric spaces [7]. Let be a partially ordered set and a mapping. We say that has the mixed monotone property whenever implies and implies for all [7]. Thus by considering the provided corollaries and the explanations in the text-body, we can obtain some similar coupled fixed point results for mixed monotone mappings as special case of above results. Finally, it has been published interesting fixed point results on metric spaces with a graph (see e.g., [2326]). There is a connection between fixed point results on ordered metric spaces and fixed point results on metric spaces with a graph [23, 27]. It is notable that one can get some similar coupled fixed point results on metric spaces with a graph.

Acknowledgments

The authors express their gratitude to the referees for their helpful suggestions which improved final version of this paper.

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