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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 604215, 6 pages

http://dx.doi.org/10.1155/2013/604215

## Some Common Fixed-Point Theorems for Generalized-Contractive-Type Mappings on Complex-Valued Metric Spaces

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 14 December 2012; Accepted 9 March 2013

Academic Editor: Somyot Plubtieng

Copyright © 2013 Chakkrid Klin-eam and Cholatis Suanoom. 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

Fixed-point theory in complex valued metric spaces has greatly developed in recent times. In this paper, we prove certain common fixed-point theorems for two single-valued mappings in such spaces. The mappings we consider here are assumed to satisfy certain metric inequalities with generalized fixed-point theorems due to Rouzkard and Imdad (2012). This extends and subsumes many results of other authors which were obtained for mappings on complex-valued metric spaces.

#### 1. Introduction

The existence and uniqueness of fixed-point theorems of operators or mappings has been a subject of great interest since the work of Banach in 1992 [1]. The Banach contraction mapping principle is widely recognized as the source of metric fixed-point theory. A mapping , where is a metric space, is said to be a contraction mapping if for all , According to the Banach contraction mapping principle, any mapping satisfying (1) in a complete metric space will have a unique fixed point. This principle includes different directions in different spaces adopted by mathematicians; for example, metric spaces, -metric spaces, partial metric spaces, cone metric spaces, quasimetric spaces have already been obtained.

A new space called the complex-valued metric space which is more general than well-known metric spaces has been introduced by Azam et al. [2]. Azam proved some fixed-point theorems for mappings satisfying a rational inequality. Naturally, this new idea can be utilized to define complex-valued normed spaces and complex-valued inner product spaces which, in turn, offer a wide scope for further investigation. Several authors studied many common fixed point results on complex-valued metric spaces (see [3–5]).

In 2012, Rouzkard and Imdad [6] extended and improved the common fixed-point theorems which are more general than the result of Azam et al. [2].

Theorem 1 (see [6, Theorem 1]). *If and are self-mappings defined on a complete complex-valued metric space satisfying the condition
**
for all where , , and are nonnegative with , then and have a unique common fixed point. *

Though complex-valued metric spaces from a spacial class of cone metric spaces, yet this idea is intended to define rational expressions which are not meaningful in cone metric spaces, and thus many results of analysis cannot be generalized to cone metric spaces. The aim of this paper is to establish some common fixed-point theorems for two nonlinear general contraction mappings in complex-valued metric spaces. Our results generalized Theorem 1.

#### 2. Preliminaries

Let be the set of complex numbers and , we define a partial order and on as follows:(i) if and only if and ;(ii) if and only if and .

Now, we briefly review the notation about complex valued metric space and some lemma for prove our main results.

*Definition 2. *Let be a nonempty set. Suppose that the mapping satisfies the following conditions:(d_{1}) for all ;(d_{2}) if and only if for all ;(d_{3}) for all ;(d_{4}) for all .

Then, is called a *complex-valued metric* on , and is called a *complex valued metric space*.

*Definition 3. *Let be a complex-valued metric space.(i)A point is called *interior point* of a set whenever there exists such that .(ii)A point is called *limit point* of a set whenever for every such that .(iii)A subset is called *open* whenever each element of is an interior point of .(iv)A subset is called *closed* whenever each limit point of belongs to . (v)The family is a subbasis for a topology on . We denote this complex topology by . Indeed, the topology is Hausdorff.

*Definition 4 (see [2]). *Let be a complex-valued metric space, and let be a sequence in and .(i)If for every , with there is such that for all , , then is said to be *convergent*, converges to and is limit point of . We denote this by as or .(ii)If for every , with there is such that for all , , where , then is said to be *Cauchy sequence. *(iii)If every Cauchy sequence in is convergent, then is said to be a *complete complex-valued metric space. *

Lemma 5 (see [2]). *Let be a complex-valued metric space, and let be a sequence in . Then, converges to if and only if as .*

Lemma 6 (see [2]). *Let be a complex-valued metric space, and let be a sequence in . Then, is a Cauchy sequence if and only if as , where .*

*Definition 7. *Two families of self-mappings and are said to be pairwise commuting if:(i), .(ii), .(iii), , .

*Definition 8. *Let and be self-mappings of a nonempty set .(i)A point is said to be a *fixed point* of if .(ii)A point is said to be a *common fixed point* of and if .

*Remark 9. *We obtain that the following statements hold:(i)If and , then .(ii)If , , and , then .(iii)If , then .

#### 3. Main Results

In this section, we will prove some common fixed-point theorems for the generalized contractive mappings in complex-valued metric space.

Theorem 10. *If and are self-mappings defined on a complete complex valued metric space satisfying the condition
**
for all , where , , , , and are nonnegative with , then and have a unique common fixed point. *

*Proof. *Let be an arbitrary in . Since and , we construct the sequence in such that and for all . From the definition of and (3), we obtain that
Since implies that ; therefore,
by Remark 9 and , we have
From (6) and Definition 2, we have
it follows that .

Similarly, we get
Since implies that ; therefore,
by Remark 9 and , we have
From (10) and Definition 2, we have
it follows that .

Putting , we obtain that
Thus, for any , we have
it follows that as .

By Lemma 6, the sequence is a Cauchy. Since is compete, there exists a point such that as .

Next, we will show that . By the notion of a complete complex-valued metric , we have
which implies that
Taking , we have ; it is obtained that . Thus, . It follows that similarly . Therefore, is common fixed point of and .

Finally, to prove the uniqueness of common fixed point, let be another common fixed point of and such that . Consider
so that
Since , therefore .

This is contradiction to . Hence, . Therefore, is a unique common fixed point of and .

Corollary 11. *If is a self-mapping defined on a complete complex-valued metric space satisfying the condition
**
for all , where , , , , and are nonnegative with , then has a unique fixed point.*

*Proof. *We can prove this result by applying Theorem 10 by setting .

Corollary 12. *If and are self-mappings defined on a complete complex valued metric space satisfying the condition
**
for all , where , , , and are nonnegative with , then and have a unique common fixed point.*

*Proof. *We can prove this result by applying Theorem 10 by setting .

Corollary 13. *If is a self-mapping defined on a complete complex valued metric space satisfying the condition
**
for all , where , , , and are nonnegative with , then has a unique fixed point.*

*Proof. *We can prove this result by applying Corollary 12 by setting and .

Corollary 14. *If and are self-mappings defined on a complete complex valued metric space satisfying the condition
**
for all where , , , and are nonnegative with , then and have a unique common fixed point.*

*Proof. *We can prove this result by applying Theorem 10 by setting .

Corollary 15. *If is a self-mapping defined on a complete complex valued metric space satisfying the condition
**
for all , where , , , and are nonnegative with , then has a unique fixed point.*

*Proof. *We can prove this result by applying Corollary 14 by setting .

*Remark 16. *(i) By choosing in Theorem 10, we get Theorem of [6].

(ii) By choosing and in Theorem 10, we get Corollary of [6].

(iii) By choosing in Theorem 10, we get Theorem of Azam et al. [2].

(iv) By choosing and in Theorem 10, we get Corollary of Azam et al. [2].

Theorem 17. *If and are two finite pairwise commuting finite families of self-mapping defined on complete complex-valued metric space such that the mappings and with and satisfy condition (3), then the component maps of the two families and have a unique common fixed point.*

*Proof. *By Theorem 10, one can infer that and have a unique common fixed point . Now, we will show that is a common fixed point of all the component maps of both families. In view of pairwise commutativity of the families and , for every , we can write
It implies that is also a common fixed point of and . By using the uniqueness of common fixed point, we have . Hence, is a common fixed point of the family . Similarly, we can show that is a common fixed point of the family . This completes the proof of the theorem.

Corollary 18. *If and are self-mappings defined on a complete complex-valued metric space satisfying the condition
**
for all , where , , , , and are nonnegative with , then and have a unique common fixed point. *

*Proof. *We can prove this result by applying Theorem 17 by setting and .

Corollary 19. *If is a self-mapping defined on a complete complex valued metric space satisfying the condition
**
for all , where , , , , and are nonnegative with , then has a unique fixed point.*

*Proof. *We can prove this result by applying Corollary 18 by setting .

*Remark 20. *(i) By choosing in Theorem 17, we get Theorem of [6].

(ii) By choosing in Corollary 18, we get Corollary of [6].

(iii) By choosing in Corollary 19, we get Corollary of [6].

(iv) By choosing in Corollary 19, we get Corollary of Azam et al. [2].

Corollary 21 (see [5]). *If is a mapping defined on a complete complex-valued metric space satisfying the condition
**
for all , where is nonnegative reals , then has a unique fixed point.*

The following example demonstrates the superiority of Bryant theorem [5] over Banach contraction theorem.

*Example 22. *Let be the set of complex numbers. Define as
where and . Then, is a complete complex-valued metric space. Define as
Now, for and , we get
Thus, , which is a contradiction as . However, notice that , so that , which shows that satisfies the requirement of Bryant theorem and is the unique fixed point of .

#### Acknowledgment

The authors would like to thank the Faculty of Science, Naresuan University, Phitsanulok, Thailand, for the financial support.

#### References

- S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,”
*Fundamenta Mathematicae*, vol. 3, pp. 133–181, 1992. - A. Azam, B. Fisher, and M. Khan, “Common fixed point theorems in complex valued metric spaces,”
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