Research Article | Open Access

# Weakly Compatible Maps Using E.A. and (CLR) Properties in Complex Valued -Metric Spaces

**Academic Editor:**Jianxin Chen

#### Abstract

We introduce the notion of complex valued -metric spaces and prove common fixed point theorems for weakly compatible maps along with E.A. and (CLR) properties in complex valued -metric spaces.

#### 1. Introduction

The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Recently, Mustafa and Sims [1, 2] have shown that most of the results concerning Dhage’s -metric spaces are invalid; therefore, they introduced an improved version of the generalized metric space structure and called it -metric spaces.

In 2006, Mustafa and Sims [2] introduced the concept of -metric spaces as follows.

*Definition 1. *Let be a nonempty set, and let be a function satisfying the following properties:(G1) if ;(G2) for all , with ;(G3)for all with ;(G4) = = (symmetry in all three variables);(G5) for all (rectangle inequality);

Then the function is called a generalized metric or, more specially a -metric on , and the pair () is called a -metric space.

The idea of complex metric space was initiated by Azam et al. [3] to exploite the idea of complex valued normed spaces and complex valued Hilbert spaces.

*Definition 2. *Let be the set of complex numbers and , . Define a partial order on as follows:

That is, if one of the following holds:(C1) Re() Re() and Im() Im();(C2) Re() Re() and Im() Im();(C3) Re() Re() and Im() Im();(C4) Re() Re() and Im() Im().

In particular, we will write that if and one of (C2), (C3), and (C4) is satisfied, and we will write if only (C4) is satisfied.

*Remark 3. *We noted that the following statements hold:(i) and for all ;(ii);(iii) and .

Now we introduce the notion of complex valued -metric space akin to the notion of complex valued metric spaces [3] as follows.

*Definition 4. *Let be a non-empty set. Let be a function satisfying the following properties:(CG1) if ;(CG2) for all with ;(CG3)for all with ;(CG4) (symmetry in all three variables);(CG5) for all .

Then the function is called a complex valued generalized metric or more specially, a complex valued -metric on , and the pair is called a complex valued -metric space.

#### 2. The Complex Valued -Metric Topology

A point is called *interior point* of a set , whenever there exists such that

A point is called *limit point* of a set , whenever there exists :

The set is called *open,* whenever each element of is an interior point of . A subset is called *closed,* whenever each limit point of belongs to .

Proposition 5. *Let () be complex valued -metric space, then for any and 0, one has the following:*(1)*if *,
* then *;
(2)*if **, then there exists ** such that **.*

Proposition 6. *Let () be complex valued -metric space; then for all and , one has
**
where .*

Proposition 7. *Let () be a complex valued -metric space. Then for any , and in it follows that: *(i)*if ** and *;
(ii);
(iii);(iv);(v);(vi).

Proposition 8. *Let () be a complex valued -metric space. Then the following are equivalent:*(i)* is symmetric;*(ii)*, for all **;*(iii)* for all **.*

#### 3. Convergence, Continuity, and Completeness in Complex Valued -Metric Spaces

Now we discuss some definition regarding convergence, continuity, and completeness in complex valued -metric spaces.

*Definition 9. *Let be a complex valued -metric space. Let be a sequence of points of ; we say that is complex valued -convergent to if, for any , there exists such that for all . We refer to as the limit of the sequence , and we write .

Proposition 10. *Let be complex valued -metric spaces. For a sequence and point , the following are equivalent:*(1)* is complex valued **convergent to*(2)* as **;*(3)* as **;*(4)* as *.

*Definition 11. *Let and be two complex valued -metric spaces. Then a function is complex valued -continuous at a point if for all . We say is complex valued -continuous if it complex valued -continuous at all points of ; that is, continuous as a function from with the -topology to with -topology.

Since complex valued -metric topologies are metric topologies, therefore we have some proposition in this regard.

Proposition 12. *Let and be two complex valued -metric spaces. Then a function is complex valued -continuous at a point if and only if it is complex valued -sequentially continuous at ; that is, whenever is complex valued -convergent to , one has ( is complex valued -convergent to .*

Proposition 13. *Let () be a complex valued -metric spaces. The function is jointly continuous in all three of its variables.*

*Proof. *Suppose that , , and are complex valued -convergent to , and , respectively. Then by (CG5), we have
so
Similarly,
Combining the above inequality and using (iii) of Proposition 7, we have
→ 0, as , and the result follows by Proposition 12.

*Definition 14. *Let () be a complex valued -metric space. A sequence is complex valued -Cauchy if, given , there exists such that for all .

*Definition 15. *A complex valued -metric space () is said to be complex valued -complete if every complex valued -Cauchy sequence is complex valued -convergent in ().

Proposition 16. *Let () be a complex valued -metric space. Then the following are equivalent:*(1)*the sequence ** is a complex valued **-Cauchy in X;*(2)*for every **, there exists ** such that ** for all **;*(3)* is a Cauchy sequence in the complex valued metric space *,
.

Proposition 17. *Let () be a complex valued -metric space, and let be a sequence in . Then is complex valued -convergent to if and only if as .*

*Proof. *Suppose that is complex valued -convergent to . For a given real number , let
Then , and there is a natural number , such that for all .

Therefore,
It follows that as .

Conversely, suppose that as . Then given with , there exists a real number , such that, for ,
For this , there is a natural number such that
This means that for all . Hence is complex valued -convergent to .

Proposition 18. *Let () be a complex valued -metric space, and let be a sequence in . Then is complex valued -Cauchy sequence if and only if as .*

*Proof. *Suppose that is complex valued -Cauchy sequence. For a given real number , let
Then , and there is a natural number , such that for all .

Therefore,
It follows that as .

Conversely, suppose that as . Then given with , there exists a real number , such that, for ,
For this , there is a natural number such that
This means that for all ≥ . Hence is complex valued -Cauchy sequence.

#### 4. Weakly Compatible Maps

In 1996, Jungck [4] introduced the concept of weakly compatible maps as follows.

*Definition 19. *Two self-maps and are said to be weakly compatible if they commute at coincidence points.

Now we prove our main result for a pair of self-mappings.

Theorem 20. *Let () be a complete complex valued -metric space. Let be self-mappings satisfying the following conditions:*(2.1)*; *(2.2)* any one of the subspace or is complete;*(2.3)* for all , where ;*(2.4)* and are weakly compatible self-maps.**
Then and have a unique common fixed point in X.*

*Proof. *Let be an arbitrary point in . By (2.1), one can choose a point in such that . In general, choose such that
Now, we prove that is a complex valued -Cauchy sequence in X.

Putting , , and in (2.3), we have
Continuing in the same way, we have
This implies that .

Then, for all , , we have by (CG5)
Therefore,
Proceeding limit as and since 0 , then ; that is, .

For , (CG5) implies that
Therefore,
Taking limit as , we get 0; that is, . So is complex valued -Cauchy sequence. Since either or is complete. Without loss of generality, we assume that is complete subspace of , and then the subsequence of must get a limit in , say . Then for some , as is a complex valued -Cauchy sequence containing a convergent subsequence.

Next we show that . On setting , and in (2.3), we have
Taking limit as , we have .

Therefore, implies that .

Therefore, . That is, is coincidence point of and . Since and are weakly compatible, it follows that ; that is, .

We now show that . Suppose that ; therefore implies that .

Putting , , and in (2.3), we have
That is, , which is a contradiction; therefore . Thus ; that is, is a common fixed point of and .*Uniqueness*. To prove uniqueness, suppose that be another common fixed point of and . Then implies that .

Putting , and in (2.3), we have
that is, , which is a contradiction; therefore . Thus ; that is, is a unique common fixed point of and .

*Example 21. *Let , and let be complex valued -metric space defined as follows:
Then () is complex valued -metric space. Define as and . Here we note that (2.1) , (2.3) ) holds for all , , and (2.4) and are weakly compatible because and commute at their coincidence point, that is, at , and is the unique common fixed point of and and and also satisfy the condition (2.2).

#### 5. E.A. Property and Weakly Compatible Maps

In 2002, Aamri and Moutawakil [5] introduced the notion of E.A. property as follows.

*Definition 22. *Two self-mappings and of a metric space () are said to satisfy E.A. property if there exists a sequence in such that for some in .

In a similar mode, we use these notions in complex valued -metric spaces.

*Example 23. *Let . Let be complex valued -metric space defined as follows:
Then () is complex valued -metric space. Define , : as and ; for all .

Consider a sequence , , in ; then
Thus and satisfy E.A. property.

Now we prove a common fixed point theorem for weakly compatible maps along with E.A. property.

Theorem 24. *Let and be self-mappings of a complex valued -metric space () satisfying (2.3), (2.4), and the following:*(3.1)* and satisfy E.A. property;*(3.2)* is a closed subspace of .**Then and have a unique common fixed point.*

*Proof. *Since and satisfy the E.A. property, therefore, there exists a sequence in such that . Since is a closed subspace of , therefore every convergent sequence of points of has a limit point in .

Then for some in . This implies that .

On setting , and , in (2.3), we have
Taking limit as , we have .

Therefore, implies that .

Therefore, . That is, is coincidence point of and . Since and are weakly compatible, it follows that .

Next we show that . On setting , , and in (2.3), we have
Therefore, implies that . Hence is a common fixed point of and .

Uniqueness easily follows from Theorem 20. Hence is a unique common fixed point of and .

#### 6. (CLR) Property and Weakly Compatible Maps

In 2011, Kumam and Sintunavarat [6] introduced the notion of () property as follows.

*Definition 25. *Two self-mappings and of a metric space () are said to satisfy () property if there exists a sequence in such that for some in .

In a similar mode, we use these notions in complex valued -metric spaces.

*Example 26. *Let . Let be complex valued -metric space defined as follows:
Then () is complex valued -metric space. Define as and for all .

Consider a sequence , , in ; then
Also, we have
Thus and satisfy () property

Now we prove a common fixed point theorem for weakly compatible maps along with () property.

Theorem 27. *Let and be self-mappings of a complex valued -metric space () satisfying (2.1), (2.3), (2.4), and the following:*(4.1)* and satisfy () property.**
Then and have a unique common fixed point.*

*Proof. *Since and satisfy the () property, therefore, there exists a sequence in such that . Then for some in . This implies that .

On setting , , and in (2.3), we have
Taking limit as *, *we have ().

Therefore, implies that .

Therefore, . That is, is coincidence point of and . Since and are weakly compatible, it follows that ; that is, .

Next we show that . On setting , and in (2.3), we have
Therefore, implies that . Hence is a common fixed point of and .

Uniqueness easily follows from Theorem 20. Hence is a unique common fixed point of and .

#### Acknowledgment

S. Kumar would like to acknowledge UGC for providing Major Research Project Financial Grant under Reference (39-41/2010(SR)).

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#### Copyright

Copyright © 2013 Balbir Singh 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.