`Abstract and Applied AnalysisVolume 2012, Article ID 394830, 15 pageshttp://dx.doi.org/10.1155/2012/394830`
Research Article

## Common Fixed Point for Two Pairs of Mappings Satisfying (E.A) Property in Generalized Metric Spaces

1College of Mathematics and Information Science, Jiangxi Normal University, Jiangxi, Nanchang 330022, China
2Department of Mathematics, Lahore University of Management Science, Lahore 54792, Pakistan
3Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

Received 3 July 2012; Accepted 18 August 2012

Copyright © 2012 Wei Long 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

Recently, Abbas et al. (2012) obtained some unique common fixed-point results for a pair of mappings satisfying (E.A) property under certain generalized strict contractive conditions in the framework of a generalized metric space. In this paper, we present common coincidence and common fixed points of two pairs of mappings when only one pair satisfies (E.A) property in the setup of generalized metric spaces. We present some examples to support our results. We also study well-posedness of common fixed-point problem.

#### 1. Introduction and Preliminaries

Mustafa and Sims [1] generalized the concept of a metric in which the real number is assigned to every triplet of an arbitrary set. Based on the notion of generalized metric spaces, Mustafa et al. [26] obtained some fixed-point theorems for mappings satisfying different contractive conditions. Chugh et al. [7] obtained some fixed-point results for maps satisfying property in metric spaces. Saadati et al. [8] studied fixed-point of contractive mappings in partially ordered metric spaces. Shatanawi [9] obtained fixed-points of maps in metric spaces. Study of common fixed-point theorems in generalized metric spaces was initiated by Abbas and Rhoades [10] (see also, [1114]). Recently, Abbas et al. [15] obtained some unique common fixed-point results for a pair of mappings satisfying (E.A) property under certain generalized strict contractive conditions in the framework of a generalized metric space.

The aim of this paper is to study common fixed-point of two pairs of mappings for which only one pair needs to satisfy (E.A) property in the framework of metric spaces. Our results do not rely on any commuting or continuity condition of mappings.

Consistent with Mustafa and Sims [2], the following definitions and results will be needed in the sequel.

Definition 1.1. Let be a nonempty set. Suppose that a mapping satisfies the following:: if ;: for all , with ;: for all , with ;: (symmetry in all three variables);: for all .
Then is called a metric on and is called a metric.

Definition 1.2. A sequence in a metric space is(i)a Cauchy sequence if, for any , there is an (the set of natural numbers) such that for all ,(ii)a convergent sequence if, for any , there is an and an , such that for all .
A metric space on is said to be complete if every Cauchy sequence in is convergent in . It is known that   converges to if and only if as .

Proposition 1.3. Let be a metric space. Then the following are equivalent:(1) is convergent to .(2) as .(3) as .(4) as .

Definition 1.4. A metric on is said to be symmetric if for all .

Proposition 1.5. Every metric on will define a metric on by For a symmetric metric However, if is nonsymmetric, then the following inequality holds: It is also obvious that Now, we give an example of a nonsymmetric metric.

Example 1.6. Let and a mapping be defined as shown in Table 1.

Table 1

Note that satisfies all the axioms of a generalized metric but for distinct in . Therefore, is a nonsymmetric metric on .

Sessa [16] introduced the notion of the weak commutativity of mappings in metric spaces.

Definition 1.7 (see [13]). Let be a metric space. Mappings , are called (i) compatible if, whenever a sequence in is such that and are convergent to some , then , (ii) noncompatible if there exists at least one sequence in such that and are convergent to some , but is either nonzero or does not exist.
Jungck [17] defined and to be weakly compatible if implies .
In 2002, Aamri and Moutaawakil [18] introduced (E.A) property to obtain common fixed-point of two mappings. Recently, Babu and Negash [19] employed this concept to obtain some new common fixed-point results (see also [2022]).
Recently, Abbas et al. [15] studied (E.A) property in the frame work of metric space.

Definition 1.8 (see [15]). Let be a metric space. Self-maps and on are said to satisfy the (E.A) property if there exists a sequence in such that and are convergent to some .

Example 1.9 (see [15]). Let be a metric space with Let be defined by For a decreasing sequence in such that and . So, and are noncompatible. Note that, there exists a sequence in such that , take for each . Hence and satisfy (E.A) property.

Definition 1.10 (see [23]). The control functions and are defined as follows:(a) is a continuous nondecreasing function with if and only if ,(b) is a lower semicontinuous function with if and only if .

#### 2. Common Fixed-Point Theorems

In this section, we obtain some common fixed-point results for two pairs of mappings satisfying certain contractive conditions in the frame work of a generalized metric space. It is worth mentioning to note that, one needs (E.A) property of only one pair to prove the existence of coincidence point of mappings involved therein. We start with the following result.

Theorem 2.1. Let be a metric space and be mappings with and such that or hold for all , where and are control functions. Suppose that one of the pairs and satisfies (E.A) property and one of the subspace ,,, is closed in . If for every sequence in , one of the following conditions hold:(a) is bounded in case satisfies (E.A) property, (b) is bounded in case satisfies (E.A) property.
Then, the pairs and have a common point of coincidence in . Moreover, if the pairs and are weakly compatible, then , and have a unique common fixed-point.

Proof. Suppose that the pair satisfies (E.A) property, there exists a sequence in satisfying for some . As , there exists a sequence in such that . As is bounded, and are finite numbers. Note that Since as , therefore . Indeed, using that , we obtain subsequences and such that and are convergent to . Replacing by and by in (2.1), we have which on taking limit as implies that Now which on taking upper limit gives and so . Hence, , and so, .
If is a closed subspace of . Then, there exist a in such that . From (2.1), we have where Hence, we have and . Hence , is the coincidence point of pair . As , there exist a point in such that . We claim that . From (2.1), we get where Hence, we have which implies . Hence , so is the coincidence point of pair . Thus . Now, weakly compatibility of pairs and give that and . From (2.1), we have where From (2.14), we obtain and so . Therefore . Similarly, it can be shown that . Therefore . To prove uniqueness of , suppose that . From (2.1) we have the following: where Thus from (2.17), we obtain which implies that , and so . The proof using (2.2) is similar.

Example 2.2. Let be a set with metric defined by Table 2.

Table 2

Note that is a nonsymmetric as . Let be defined by Table 3.

Table 3

Clearly, and with the pairs and being weakly compatible. Also a pair satisfy (E.A) property, indeed, for each is the required sequence. Note that pair is not commuting at . The control functions are defined by To check contractive conditions (2.1) and (2.2) for all , we consider the following cases:

Note that for cases (I) , (II) , (III) , (IV) , (V) , and (VI) ,

We have , and hence (2.1) and (2.2) are obviously satisfied now.

(VII) If , then , . Also

(VIII) For , then .

(IX) Now, when , then . Also

Hence, all of the conditions of Theorem 2.1 are satisfied. Moreover, is the unique common fixed-point of , , , and .

As two noncompatible selfmappings on metric space satisfy the (E.A) property, so above result remains true if any one of the pair of mapping is noncompatible.

Above theorem is true for any choice of control functions, for example if we take and for in Theorem 2.1, we have the following corollary.

Corollary 2.3. Let be a metric space and be mappings with and such that or hold for all , where hold for all , where and are control functions. Suppose that one of the pairs and satisfies (E.A) property and one of the subspace , and is closed in . If for every sequence in , one of the following conditions hold:(a) is bounded in case satisfies (E.A) property (b) is bounded in case satisfies (E.A) property.
Then, the pairs and have a point of coincidence in . Moreover, if the pairs and are weakly compatible, then , and have a unique common fixed-point.
If we take and with for all in Theorem 2.1, we obtain the following corollary which extends Theorem 3.1 of [19] to generalized metric space.

Corollary 2.4. Let be a metric space and be mappings with such that or hold for all , where are control functions. Suppose that the pair satisfy (E.A) property and one of the subspaces , is closed in . Then, the pair has a common point of coincidence in . Moreover, if the pair is weakly compatible, then and have a unique common fixed-point.

#### 3. Well-Posedness

The notion of well-posedness of a fixed-point problem has evoked much interest of several mathematicians, (see [2427]).

Definition 3.1. Let be a metric space and be a mapping. The fixed-point problem of is said to be well-posed if:(a) has a unique fixed-point in ;(b)for any sequence of points in such that , we have .

Definition 3.2. Let be a metric space and be a set of mappings on . Common fixed-point problem is said to be well-posed if:(a) is a unique common fixed-point of all mappings in ;(b)for any sequence of points in such that for each , we have .

Theorem 3.3. Let be a metric space and be mappings such that or hold for all , where is a control function. Suppose that one of the pairs and satisfies (E.A) property and one of the subspace is closed in . If for every sequence in , one of the following conditions hold:(a) is bounded in case satisfies (E.A) property; (b) is bounded in case satisfies (E.A) property.
If pairs and are weakly compatible, then is well-posed.

Proof. From Theorem 2.1, the mappings have a unique common fixed-point (say) in . Let be a sequence in such that Now by using (3.1), we have which further implies On taking limit as implies that and by the property of , we have Hence the result follows.

Remark 3.4. A metric naturally induces a metric given by . If metric is not symmetric, either of the inequalities (2.1) or (2.2) does not reduce to any metric inequality with the metric . Hence our theorems do not reduce to fixed-point problems in the corresponding metric space .

#### Acknowledgments

Wei Long acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province, and the Foundation of Jiangxi Provincial Education Department (GJJ12205).

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