#### 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)).