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The Scientific World Journal

Volume 2014, Article ID 587825, 6 pages

http://dx.doi.org/10.1155/2014/587825
Research Article

Some Common Fixed Point Theorems in Complex Valued -Metric Spaces

Department of Mathematical Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

Received 6 January 2014; Accepted 12 March 2014; Published 25 May 2014

Academic Editors: A. Alotaibi and S. A. Mohiuddine

Copyright © 2014 Aiman A. Mukheimer. 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

Azam et al. (2011), introduce the notion of complex valued metric spaces and obtained common fixed point result for mappings in the context of complex valued metric spaces. Rao et al. (2013) introduce the notion of complex valued -metric spaces. In this paper, we generalize the results of Azam et al. (2011), and Bhatt et al. (2011), by improving the conditions of contraction to establish the existence and uniqueness of common fixed point for two self-mappings on complex valued -metric spaces. Some examples are given to illustrate the main results.

1. Introduction

Banach contraction principle in [1] was the starting point for many researchers during last decades in the field of nonlinear analysis. In 1989, Bakhtin [2] introduced the concept of -metric space as a generalization of metric spaces. The concept of complex valued -metric spaces was introduced in 2013 by Rao et al. [3], which was more general than the well-known complex valued metric spaces that were introduced in 2011 by Azam et al. [4]. The main purpose of this paper is to present common fixed point results of two self-mappings satisfying a rational inequality on complex valued -metric spaces. The results presented in this paper are generalization of work done by Azam et al. in [4] and Bhatt et al in [5].

Definition 1 (see [6]). Let be a nonempty set and let be a given real number. A function is called a -metric if for all the following conditions are satisfied: (i) if and only if ;(ii) ;(iii) .The pair is called a -metric space. The number is called the coefficient of .

Example 2 (see [7]). Let be a metric space and , where is a real number. Then is a -metric space with .

Let be the set of complex numbers and . Define a partial order on as follows: Thus if one of the following holds:(1) and ,(2) and ,(3) and ,(4) and .We will write if and one of (2), (3), and (4) is satisfied; also we will write if only (4) is satisfied.

Remark 3. We can easily check that the following statements are held:(i)if and , then for all ;(ii)if , then ;(iii)if and , then .

Definition 4 (see [4]). Let be a nonempty set. A function is called a complex valued metric on if for all the following conditions are satisfied:(i) and if and only if ;(ii) ;(iii) .The pair is called a complex valued metric space.

Example 5 (see [8]). Let . Define the mapping by Then is a complex valued metric space.

Example 6 (see [9]). Let . Define the mapping by Then is a complex valued metric space.

Definition 7 (see [3]). Let be a nonempty set and let be a given real number. A function is called a complex valued -metric on if for all the following conditions are satisfied:(i) and if and only if ;(ii) ;(iii) .The pair is called a complex valued -metric space.

Example 8 (see [3]). Let . Define the mapping by Then is a complex valued -metric space with .

Definition 9 (see [3]). Let be a complex valued -metric space. Consider the following.(i)A point is called interior point of a set whenever there exists such that .(ii)A point is called a limit point of a set whenever, for every ,   .(iii)A subset is called open whenever each element of is an interior point of .(iv)A subset is called closed whenever each element of belongs to .(v)A subbasis for a Hausdorff topology on is a family and .

Definition 10 (see [3]). Let be a complex valued -metric space and a sequence in and . Consider the following.(i)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   .(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 11 (see [3]). Let be a complex valued -metric space and let be a sequence in . Then converges to if and only if   as   .

Lemma 12 (see [3]). Let be a complex valued -metric space and let be a sequence in . Then is a Cauchy sequence if and only if   as   , where .

Theorem 13 (see [4]). Let be a complete complex valued metric space and let be nonnegative real numbers such that . Suppose that are mappings satisfying for all . Then have a unique common fixed point in .

Theorem 14 (see [5]). Let be a complete complex valued metric space and let be mappings satisfying for all , where . Then have a unique common fixed point in .

2. Main Result

Our next theorem is a generalization of Theorem 13 in complex valued -metric spaces.

Theorem 15. Let be a complete complex valued -metric space with the coefficient and let be mappings satisfying for all , where are nonnegative reals with . Then have a unique common fixed point in .

Proof. For any arbitrary point, . Define sequence in such that Now, we show that the sequence is Cauchy. Let and in (7); we have which implies that Since , we get and hence Similarly, we obtain Since and , we get .

Therefore, with , and for all , consequently, we have Thus for any ,   ,   , and since , we get By using (14) we get Therefore, and hence Thus, is a Cauchy sequence in .

Since is complete, there exists some such that as . Assuming not, then there exist such that So by using the triangular inequality and (7), we get which implies that Taking the limit of (21) as , we obtain that , a contradiction with (19). So . Hence . Similarly, we obtain .

Now we show that and have unique common fixed point of and . To show this, assume that is another common fixed point of and . Then This implies that , a contradiction. So which proves the uniqueness of common fixed point in . This completes the proof.

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

Proof. We can prove this result by applying Theorem 15 with .

Corollary 17. Let be a complete complex valued -metric space with the coefficient and let be a mapping satisfying for all , where are nonnegative reals with . Then has a unique fixed point in .

Proof. From Corollary 20, we obtain such that The uniqueness follows from By taking modulus of (26) and since , we obtain , a contradiction. So, . Hence Therefore, the fixed point of is unique. This completes the proof.

Example 18. Let . Define a function such that where and .

To verify that is a complete complex valued -metric space with , it is enough to verify the triangular inequality condition.

Let , and ; then, Therefore, .

Now, define two self-mappings as follows: such that and . Let and , and since , we have Note that for , so for all and with . So all conditions of Corollary 21 are satisfied to get a unique fixed point of .

Our next theorem is a generalization of Theorem 14 in complex valued -metric spaces.

Theorem 19. Let be a complete complex valued -metric space with the coefficient and let be mappings satisfying for all , where . Then have a unique common fixed point in .

Proof. For any arbitrary point, . Define sequence in such that Now, we show that the sequence is Cauchy. Let and in (33); we have which implies that and hence Similarly, we can see that Since and , we get .

Therefore, for all , consequently, we have Thus for any ,   ,   , we have By using (39), we get Therefore, Now, since , we deduce Thus, is a Cauchy sequence in .

Since is complete, there exists some such that as . Assuming not, then there exist such that So by using the triangular inequality and (33), we get which implies that Taking the limit of (48) as , we obtain that , a contradiction with (44). So . Hence . Similarly, we obtain .

Now we show that and have unique common fixed point of and . To show this, assume that is another common fixed point of and . Then This implies that , and then which proves the uniqueness of common fixed point in . This completes the proof.

Corollary 20. Let be a complete complex valued -metric space with the coefficient and let be a mapping satisfying for all , where . Then has a unique fixed point in .

Corollary 21. Let be a complete complex valued -metric space with the coefficient and let be a mapping satisfying for all , where and . Then has a unique fixed point in .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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