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Journal of Complex Analysis
Volume 2016, Article ID 9786063, 7 pages
http://dx.doi.org/10.1155/2016/9786063
Research Article

Complex Valued -Metric Spaces and Common Fixed Point Theorems under Rational Contractions

Department of Mathematics, Bhilai Institute of Technology, Bhilai House, Durg, Chhattisgarh 491001, India

Received 16 November 2015; Revised 2 March 2016; Accepted 10 May 2016

Academic Editor: Arcadii Z. Grinshpan

Copyright © 2016 Anil Kumar Dubey. 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

The aim of this paper is to prove the existence and uniqueness of a common fixed point for a pair of mappings satisfying certain rational contraction conditions in complex valued -metric space. The obtained results generalize and extend some of the well-known results in the literature.

1. Introduction

Banach contraction principle in [1] gives appropriate and simple conditions to establish the existence and uniqueness of a solution of an operator equation . Later, a number of papers were devoted to the improvement and generalization of that result. Most of these results deal with the generalizations of the different contractive conditions in metric spaces. There have been a number of generalizations of metric spaces such as vector valued metric spaces, -metric spaces, pseudometric spaces, fuzzy metric spaces, -metric spaces, cone metric spaces, and modular metric spaces. Bakhtin [2] introduced the notion of -metric space which is a generalized form of metric spaces.

In [3], Czerwik proved the contraction mapping principle in -metric spaces. Subsequently, many authors obtained fixed point results for single valued and multivalued operators in -metric spaces.

A new space called the complex valued metric space which is more general than the well-known metric space has been introduced by Azam et al. [4]. They proved some fixed point results for a pair of mappings for contraction condition satisfying a rational expression. Azam et al. [4] improved the Banach contraction principle in the context of complex valued metric space involving rational inequality which could not be meaningful in cone metric spaces. Several authors studied many common fixed point theorems on complex valued metric spaces (see [59]).

The concept of complex valued -metric spaces was introduced in 2013 by Rao et al. [10]. In sequel, Mukheimer [11] proved some common fixed point theorems in complex valued -metric spaces.

In this paper, we continue the study of fixed point theorems in complex valued -metric spaces. The obtained results are generalizations of recent results proved by Dubey et al. [12, 13], Nashine et al. [5, 6], and Rao et al. [10].

2. Preliminaries

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

if and only if , .

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.

It follows that(i) implies ;(ii) and imply ;(iii) implies ;(iv)if , and , then for all .The following definition is recently introduced by Rao et al. [10].

Definition 1. 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 2 (see [10]). If , define the mapping by , for all .
Then is complex valued -metric space with .

Definition 3 (see [10]). 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 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 .

Definition 4 (see [10]). 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 the limit point of . We denote this by 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 [10]). Let be a complex valued -metric space and let be a sequence in . Then converges to if and only if as .

Lemma 6 (see [10]). Let be a complex valued -metric space and let be a sequence in . Then is a Cauchy sequence if and only if as , where .

3. Main Results

Theorem 7. Let be a complete complex valued -metric space with the coefficient and let be mappings satisfyingfor all , such that , , where , are nonnegative reals with or if .
Then and have a unique common fixed point.

Proof. For any arbitrary point , define sequence in such thatNow, we show that the sequence is Cauchy.
Let and in (1); we haveso thatAs (owing to triangular inequality)thereforeSimilarly, we obtainSince and , we get . Therefore, with , and for all , and consequently, we haveThat is,Thus, for any , , we haveBy using (9), we getTherefore,and henceThus, is a Cauchy sequence in . Since is complete, there exists some such that as . Assume not, then there exists such thatSo by using the triangular inequality and (1), we getwhich implies thatTaking the limit of (16) as , we obtain that , a contradiction with (14). So Hence . Similarly, we obtain .
Now we show that and have unique common fixed point of and . To prove this, assume that is another common fixed point of and . Thenso thatso that which proves the uniqueness of common fixed point.
Now, we consider the second case: . Put and in this expression; we get (for any ) which implies so that . Thus, we have , so there exist and such that , where and . Using foregoing arguments, one can also show that there exist and such that , where and . As (due to definition) implies , therfore . Thus we obtain that . Similarly, one can also have . As implies , therefore is common fixed point of and . For uniqueness of common fixed point, assume that in is another common fixed point of and . Then we have .
As , therefore .
This implies that . This completes the proof of the theorem.

Corollary 8 (see [12]). Let be a complete complex valued -metric space with the coefficient . Let be a mapping satisfyingfor all such that , , where , are nonnegative reals with or if . Then has a unique fixed point in .

Proof. We can prove this result by applying Theorem 7 with .

Corollary 9 (see [12]). Let be a complete complex valued -metric space with the coefficient and let be a mapping satisfying (for some fixed )for all such that , , where , are nonnegative reals with or if . Then has a unique fixed point in .

Theorem 10. Let be a complete complex valued -metric space with the coefficient and let be mappings satisfyingfor all such that , where , , and are nonnegative reals with or if . Then and have a unique common fixed point in .

Proof. For any arbitrary point , define sequence in such thatNow, we show that the sequence is a Cauchy sequence. Let and in (21); we haveso thatAsthereforeSimilarly, we obtainSince and we get .
Therefore, with , and for all , and consequently, we haveThat is,Thus, for any , , we getBy using (29) we getThereforeand henceThus, is a Cauchy sequence in . Since is complete, there exists some such that as . Assume not, then there exists such thatSo by using the triangular inequality and (21), we getwhich implies thatTaking the limit of (36) as , we obtain that , a contradiction with (34). So ; hence .
Similarly, we obtain that . Now, we show that and have a unique common fixed point of and . To show this, assume that is another common fixed point of and . Then, a contradiction. So , which proves the uniqueness of common fixed point in For the second case if , the proof of unique common fixed point can be completed in the line of Theorem 7. This completes the proof of the theorem.

Corollary 11. Let be a complete complex valued -metric space with the coefficient and let be a mapping satisfyingfor all such that , where , , and are nonnegative reals with or if . Then has a unique fixed point.

Proof. We can prove this result by applying Theorem 10 with .

Corollary 12. Let be a complete complex valued -metric space with the coefficient and let be a mapping satisfying (for some fixed )for all , such that , where , , and are nonnegative reals with or if . Then has a unique fixed point in .

Proof. From Corollary 11, we obtain that such that . The uniqueness follows fromBy Taking modulus of (40) and since , we obtain , a contradiction. So . Hence . Therefore, the fixed point of is unique. This completes the proof.

Competing Interests

The author declares that they have no competing interests.

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