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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 352927, 10 pages
http://dx.doi.org/10.1155/2013/352927
Research Article

Some Common Coupled Fixed Point Results for Generalized Contraction in Complex-Valued Metric Spaces

1Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
2Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad, Pakistan
3Università Degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34,90123 Palermo, Italy

Received 11 April 2013; Revised 22 May 2013; Accepted 22 May 2013

Academic Editor: Erdal Karapinar

Copyright © 2013 Marwan Amin Kutbi 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 introduce and study the notion of common coupled fixed points for a pair of mappings in complex valued metric space and demonstrate the existence and uniqueness of the common coupled fixed points in a complete complex-valued metric space in view of diverse contractive conditions. In addition, our investigations are well supported by nontrivial examples.

1. Introduction

Azam et al. [1] introduced the concept of complex-valued metric spaces and obtained sufficient conditions for the existence of common fixed points of a pair of contractive type mappings involving rational expressions. Subsequently, several authors have studied the existence and uniqueness of the fixed points and common fixed points of self-mappings in view of contrasting contractive conditions. Some of these investigations are noted in [226].

In [27], Bhaskar and Lakshmikantham introduced the concept of coupled fixed points for a given partially ordered set . Recently Samet et al. [28, 29] proved that most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of well-known fixed point theorems in the literature. In this paper, we deal with the corresponding definition of coupled fixed point for mappings on a complex-valued metric space along with generalized contraction involving rational expressions. Our results extend and improve several fixed point theorems in the literature.

2. Preliminaries

Let be the set of complex numbers and . Define a partial order   on as follows: Note that and ,  implies .

Definition 1. Let be a nonempty set. Suppose that the self-mapping satisfies the following:(1)for all  and if and only if ;(2) for all ;(3), for all .
Then is called a complex valued metric on , and  is known as a complex valued metric space. A point is called interior point of a set whenever, there exists such that

A point is a limit point of whenever, for every ,

is called open whenever each element of is an interior point of . Moreover, a subset is called closed whenever each limit point of belongs to . The family is a subbasis for a Hausdorff topology on .

Let be a sequence in  and . If for every with  there is such that, for all , then is said to be convergent,  converges to , and  is the limit point of . We denote this by , or , as . If for every with there is such that, for all , then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete complex valued metric space. We require the following lemmas.

Lemma 2 (see [1]). Let  be a complex valued metric space, and let be a sequence in . Then converges to if and only if .

Lemma 3 (see [1]). Let  be a complex valued metric space, and let be a sequence in . Then is a Cauchy sequence if and only if.

Definition 4 (see [27]). An element is called a coupled fixed point of if

Definition 5. An element is called a coupled coincidence point of if

Example 6. Let and defined as and for all . Then , , and are coupled coincidence points of  and .

Example 7. Let and defined as and for all . Then and are coupled coincidence points of  and .

Definition 8. An element is called a common coupled fixed point of if

Example 9. Let and defined as and for all . Then , , and are common coupled fixed points of  and .

In the following, we provide common coupled fixed point theorem for a pair of mappings satisfying a rational inequality in complex valued metric spaces.

Theorem 10. Let  be a complete complex-valued metric space, and let the mappings satisfy for all and , and are nonnegative reals with . Then and have a unique common coupled fixed point.

Proof. Let and be arbitrary points in . Define , and , , for
Then, which implies that Since , so we get and hence Similarly, one can show that Also, so that As , therefore Similarly, one can show that Adding (12)–(17), we get If , then from (18), we get Now if , then Without loss of generality, we take . Since , so we get This implies that and are Cauchy sequences in . Since is complete, there exists such that and as . We now show that and . We suppose on the contrary that and so that and ; we would then have so that By taking , we get which is a contradiction so that . Similarly, one can prove that . It follows similarly that and . So we have proved that is a common coupled fixed point of and . We now show that and have a unique common coupled fixed point. For this, assume that is a second common coupled fixed point of and . Then so that Since , so we get Similarly, one can easily prove that If we add (26) and (27), we get which is a contradiction because . Thus, we get and , which proves the uniqueness of common coupled fixed point of and .

By setting in Theorem 10, one deduces the following.

Corollary 11. Let  be a complete complex-valued metric space, and let the mapping  satisfy for all , where , and are nonnegative reals with . Then has a unique coupled fixed point.

Corollary 12. Let  be a complete complex-valued metric space, and let the mapping  satisfy for all , where , and are nonnegative reals with . Then, has a unique coupled fixed point.

Theorem 13. Let  be a complete complex-valued metric space, and let the mappings  satisfy for all , where and are nonnegative reals with . Then and have a unique common coupled fixed point.

Proof. Let and be arbitrary points in . Define , and , , for .
Now, we assume that
Then, which implies that as Therefore, Similarly, one can easily prove that
Now, if we get which implies that as Therefore,
Similarly, if , one can easily prove that Adding the inequalities (36)–(43), we get If , then, from (44), we get Now if , then Without loss of generality, we take . Since , so we get This implies that and are Cauchy sequences in . Since is complete, so there exists such that and as . We now show that and . We suppose on the contrary that and so that and ; we would then have so that By taking , we get which is a contradiction so that . Now which implies that Which, on making , gives us which is a contradiction so that . It follows similarly that and . So we have proved that is a common coupled fixed point of and . As in Theorem 10, the uniqueness of common coupled fixed point remains a consequence of contraction condition (31).
We have obtained the existence and uniqueness of a unique common coupled fixed point if for all . Now, assume that for some . From we obtain that and . If , using (8), we deduce That is, (this equality holds also if ). The equalities ensure that is a unique common coupled fixed point of and . The same holds if either , , or .

From Theorem 13, if we assume , we obtain the following corollary.

Corollary 14. Let   be a complete complex-valued metric space, and let the self-mappings  satisfy for all , where and is a nonnegative real such that . Then and have a unique common coupled fixed point.

Corollary 15. Let  be a complete complex-valued metric space, and let the mapping  satisfy for all , where and are nonnegative reals with . Then has a unique coupled fixed point.

Corollary 16. Let  be a complete complex-valued metric space, and let the mapping  satisfy for all , where and are nonnegative reals with . Then has a unique coupled fixed point.

Now, we furnished a nontrivial example to support our main result (Theorem 10).

Example 17. Let and let . Consider a complex valued metric  as follows: with and . Then is a complex valued metric space. Define as follows: By a routine calculation, one can easily verify that the maps and satisfy the contraction condition (8) with , , and . Notice that the point remains fixed under and and is indeed unique common coupled fixed point.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper. Marwan Amin Kutbi gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. Cristina Di Bari is supported by Università degli Studi di Palermo (Local University Project R.S. ex 60%).

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