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Journal of Operators
Volume 2013 (2013), Article ID 813707, 6 pages
http://dx.doi.org/10.1155/2013/813707
Research Article

Some Common Fixed Point Results for Rational Type Contraction Mappings in Complex Valued Metric Spaces

1Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar 143001, India
2Department of Mathematics, Lovely Professional University, Phagwara 144411, India

Received 21 March 2013; Accepted 8 May 2013

Academic Editor: Ram U. Verma

Copyright © 2013 Sumit Chandok and Deepak Kumar. 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 prove some common fixed point theorems for two pairs of weakly compatible mappings satisfying a rational type contractive condition in the framework of complex valued metric spaces. The proved results generalize and extend some of the known results in the literature.

1. Introduction and Preliminaries

The famous Banach contraction principle states that if is a complete metric space and is a contraction mapping (i.e., for all , where is a nonnegative number such that ), then has a unique fixed point. This principle is one of the cornerstones in the development of nonlinear analysis. Fixed point theorems have applications not only in the different branches of mathematics, but also in economics, chemistry, biology, computer science, engineering, and others. Due to its importance, generalizations of Banach’s contraction principle have been investigated heavily by several authors. Fixed point and common fixed point theorems for different types of nonlinear contractive mappings have been investigated extensively by various researchers (see [135] and references cited therein).

Recently, Azam et al. [1] introduced the complex valued metric space, which is more general than the well-known metric spaces. Many researchers have obtained fixed point, common fixed point, coupled fixed point, and coupled common fixed point results in partially ordered metric spaces, complex valued metric spaces, and other spaces. In this paper, we prove some common fixed point theorems for two pairs of weakly mappings satisfying a contractive condition of rational type in the framework of complex valued metric spaces. The proved results generalize and extend some of the results in the literature.

To begin with, we recall some basic definitions, notations, and results.

The following definitions of Azam et al. [1] are needed in the sequel.

Let be the set of complex numbers, and let . Define a partial order on as follows: It follows that if one of the following conditions is satisfied:(1), and ;(2), and ;(3), and ;(4), and .In particular, we will write if and one of (1), (2), and (3) is satisfied, and we will write if only (3) is satisfied.

Note. We obtained that the following statements hold:(i) and ,  for all ;(ii);(iii) and imply .

Definition 1. Let be a nonempty set. Suppose that the mapping satisfies the following conditions:(i) for all and if and only if ;(ii) for all ;(iii) for all .Then, is called a complex valued metric on , and is called a complex valued metric space.

Example 2. Let . Define a mapping by where . Then, is a complex valued metric space.

A point is called an interior point of a set whenever there exists such that . A subset in is called open, whenever each point of is an interior point of . The family is a subbasis for a Hausdorff topology on .

A point is called a limit point of , whenever for every , . A subset is called closed, whenever each limit point of belongs to .

Let be a sequence in and . If for every , with , there is such that for all , , then is called the limit of , and we write .

If for every , with , there is an 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.

Lemma 3 (see [1]). Let be a complex valued metric space and a sequence in . Then, converges to if and only if as .

Lemma 4 (see [1]). Let be a complex valued metric space and a sequence in . Then, is a Cauchy sequence if and only if as .

Let be a nonempty subset of a metric space . Let and be mappings from a metric space into itself. A point is a common fixed (resp., coincidence) point of and if (resp., ). The set of fixed points (resp., coincidence points) of and is denoted by (resp., ).

In 1986, Jungck [22] introduced the more generalized commuting mappings in metric spaces, called compatible mappings, which also are more general than the concept of weakly commuting mappings (i.e., the mappings are said to be weakly commuting if for all ) introduced by Sessa [28].

Definition 5. Let and be mappings from a metric space into itself. The mappings and are said to be compatible if whenever is a sequence in such that for some .

In general, commuting mappings are weakly commuting, and weakly commuting mappings are compatible, but the converses are not necessarily true, and some examples can be found in [2224].

Definition 6. The mappings and are said to be weakly compatible if they commute at coincidence points of and .

Definition 7. Let be two self-mappings of a complex valued metric space . The pair is said to satisfy (E.A.)  property (see [35]) if there exists a sequence in such that , for some .

Pathak et al. [27] showed that weakly compatibility and (E.A.) property are independent of each other.

Definition 8. The self mappings and from to are said to satisfy the common limit in the range of property (  property) (see [31]) if , for some .

Some recent papers related to (CLR)  property and the complex valued metric spaces can be found in [1, 3, 27, 3135] and references cited therein.

2. Main Results

2.1. Common Fixed Point Theorem Using (E.A.) Property

In this section, we prove some common fixed point theorems using (E.A.) property in the complex valued metric spaces.

Theorem 9. Let be a complex valued metric space and , , , four self-mappings satisfying the following conditions: (i), ; (ii) for all and ,  (iii) the pairs and are weakly compatible; (iv) one of the pairs or satisfies (E.A.)-property.If the range of one of the mappings or is a closed subspace of , then the mappings , , , and have a unique common fixed point in .

Proof. First, we suppose that the pair satisfies (E.A.) property. Then, by Definition 7 there exists a sequence in such that for some .
Further, since , there exists a sequence in such that . Hence, . We claim that . Let , then putting , in condition (ii), we have Letting , we have Then, ; hence, and that is, .
Now suppose that is a closed subspace of , then for some . Subsequently, we have We claim that . Put and in contractive condition (ii), and we have Letting and using (7), we have Then, , which is contradiction. Hence, is a coincidence point of .
Now the weak compatibility of pair implies that or .
On the other hand, since , there exists in such that . Thus, . Now, we show that is a coincidence point of ; that is, . Put , in contractive condition (ii), and we have or whence , which is a contradiction. Thus, . Hence, , and is the coincidence point of and .
Further, the weak compatibility of pair implies that , or . Therefore, is a common coincidence point of , , , and .
Now, we show that is a common fixed point. Put and in contractive condition (ii), and we have or , which is a contradiction. Thus, . Hence, .
Similar argument arises if we assume that is closed subspace of . Similarly, the (E.A.)-property of the pair will give a similar result.
For uniqueness of the common fixed point, let us assume that is another common fixed point of , , , and . Then, put , in contractive condition (ii), and we have or , which is a contradiction. Thus, . Hence, , and is the unique common fixed point of , , , and .

Remark 10. (a) Continuity of mappings , , , and   is relaxed in Theorem 9.
(b) Completeness of space is relaxed in Theorem 9.
If and in Theorem 9, we have the following result.

Corollary 11. Let be a complex valued metric space and self-mappings satisfying the following conditions: (i); (ii) for all and ,  (iii) the pair is weakly compatible; (iv) the pair satisfies (E.A.)-property.If the range of the mapping is a closed subspace of , then and have a unique common fixed point in .

Theorem 12. Let be a complex valued metric space and , , , four self-mappings satisfying the following conditions: (i), ; (ii) for all and , where ; (iii) the pairs and are weakly compatible; (iv) one of the pairs or satisfies (E.A.)-property.If the range of one of the mappings or is a closed subspace of , then the mappings , , , and have a unique common fixed point in .

Proof. Using the same arguments as in Theorem 9, we have the following result.

2.2. Fixed Point Theorem Using (CLR)-Property

In this section, we prove some common fixed point theorems using (CLR) property in the complex valued metric spaces.

Theorem 13. Let be a complex valued metric space and , , , and four self-mappings satisfying the following conditions: (i), ; (ii) for all and ,  (iii) the pairs and are weakly compatible.If the pair satisfies property or satisfies property, then , , and have a unique common fixed point in .

Proof. First, we suppose that the pair satisfies property. Then, by Definition 8, there exists a sequence in such that for some .
Further, since , we have , for some . We claim that . Put and in contractive condition (ii), and we have letting and using (17), we have Then, , which is contradiction. Thus, . Hence, .
Now, the weak compatibility of pair implies that, or .
Further, since , there exists in such that . Thus, .
Now, we show that is a coincidence point of that is, . Put , in contractive condition (ii), and we have or whence , which is a contradiction. Thus, . Hence, , and is coincidence point of and .
Further, the weak compatibility of pair implies that , or . Therefore, is a common coincidence point of , , , and .
Now, we show that is a common fixed point. Put and in contractive condition (ii), and we have or , which is a contradiction. Thus, . Hence, . The uniqueness of the common fixed point follows easily.
In a similar way, the argument that the pair satisfies property will also give the unique common fixed point of , , and . Hence the result follows.

Following the similar steps as in Theorem 13, we have the following result.

Theorem 14. Let be a complex valued metric space and , , , and four self-mappings satisfying the following conditions: (i), ; (ii) for all and , where ; (iii) the pairs and are weakly compatible.If the pair satisfy property or satisfies property, then , , , and have a unique common fixed point in .

Remark 15. In this paper, we used the (E.A.) property and CLR property to claim the existence of common fixed point of some rational type contraction mappings. (E.A.) property requires the condition of closedness of subspace. However, property never requires any condition on closedness of subspace, continuity of one or more mappings and containment of ranges of involved mappings. So, property is an interesting auxiliary tool to claim the existence of a common fixed point.

Acknowledgment

The authors are thankful to the learned referees for the very careful reading and valuable suggestions.

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