Abstract

We prove a common fixed point theorem for a pair of mappings. Also, we prove a common fixed point theorem for pairs of self-mappings along with weakly commuting property.

1. Introduction

Azam et al. [1] introduced the notion of complex valued metric spaces and established some fixed point theorems for the mappings satisfying a rational inequality. The definition of a cone metric space banks on the underlying Banach space which is not a division Ring. The idea of rational expressions is not meaningful in cone metric spaces, and therefore many results of analysis cannot be generalized to cone metric spaces. The complex valued metric spaces form a special class of cone metric space, and we can study improvements of host results of analysis involving divisions.

A complex number is an ordered pair of real numbers, whose first coordinate is called and second coordinate is called .

Let be the set of complex numbers and , . Define a partial order on as follows: if and only if and ; that is, , if one of the following holds:(C1) and ;(C2) and ;(C3) and ;(C4) and .In particular, we will write if and one of (C2), (C3), and (C4) is satisfied, and we will write if only (C4) is satisfied.

Remark 1. We note that the following statements hold:(i), and   ,(ii),(iii) and .Azam et al. [1] defined the complex valued metric space (, ) as follows.

Definition 2. 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 3. Let . Define the mapping by Then, is a complex valued metric space.

Definition 4. Let be a complex valued metric space. A sequence in is said to be (i)convergent to , if for every with there is such that, for all , . We denote this by as or ;(ii)Cauchy, if for every with there is such that for all , , where ;(iii)complete, if every Cauchy sequence in converges in .

Lemma 5. Let be a complex valued metric space, and let be a sequence in . Then, converges to if and only if as .

Lemma 6. Let be a complex valued metric space, and let be a sequence in . Then, is a Cauchy sequence if and only if as , where .

In 1982, Sessa [2] introduced the notion of weak commutativity as follows.

Definition 7. Two self-maps and of a metric space are said to be weakly commuting if , for all in .

In a similar mode, we introduce the notion of weak commutativity in complex valued metric spaces as follows.

Definition 8. Two self maps and of a complex valued metric space are said to be weakly commuting if ,  for all in .

Example 9. Let , and define by , for all , .
Then, is a complex valued metric space.
Define and .
Then, clearly , for all .
Thus, and are weakly commuting.

2. Main Theorem

Theorem 10. Let and be self mappings of a complex valued metric space satisfying the following: Then, and have a common fixed point.
Further, if implies that , then and have a unique common fixed point.

Proof. Let . Define a sequence in by , , .
Let .
From (2), we have
Similarly, we have Consequently, it can be concluded that Now, for all , Therefore, we have Hence,
Hence, is a Cauchy sequence in . But is complete metric space, so is convergent to some point, say , in , that is, as .
Now, we will prove that .
Let, if possible, .
Now, using the triangular inequality and (2), we have Thus, we have
Letting , we have a contradiction.
Hence, we get ; that is, is the fixed point of .
Again assume that .
From (2) and using the triangular inequality, we have Thus, we have
Letting , we have , a contradiction.
Hence, we get ; that is, is a fixed point of .
Therefore, we find that is a common fixed point of and .

Uniqueness. Let () be another fixed point of . Suppose that implies .

Now, Therefore, we get Hence, and have a unique common fixed point.

Corollary 11. Let be a self-map of a complex valued metric space satisfying the following: Then, has a fixed point.
Further, if implies that then has a unique common fixed point.

Proof. By putting in Theorem 10, we get Corollary 11.

3. Weakly Commuting Property

Theorem 12. Let , , , and be self mappings of a complex valued metric space satisfying the following:(3.1) SX BX, TX AX,(3.2) the pairs and are weakly commuting,(3.3) for all , in , either if , where < 1 and ;  , if .
If any of , , , or is continuous, then , , , and have a unique common fixed point .

Proof. Let . Since , so there exists a point in such that . Also, since , we can choose a point in such that .
Continuing this process, we have and , for .
Define and .
Suppose and for .
From (3.3), we have Thus, we have
In general, we have Letting , we have Therefore, as .
We get the following sequence:(3.4), which is a Cauchy sequence in the complete complex valued metric space , therefore converges to a limit point in .
Therefore, the sequences and , which are the subsequences of (3.4) hence also converge to the same point in .
Now, suppose that is continuous so that the sequences and converge to the same point . Since and are weakly commuting, we have
Letting , we have
Now, we will show that . Let, if possible, .
Now, using the triangle inequality and (3.3), we get Thus, we have
Letting , we have Hence, .
Now, we will prove that .
Again, using the triangle inequality and (3.3), we have Thus, we have Letting , we have Now, since , there exists a point in such that .
Thus, we have
Since and are weakly commuting, so we have Now, we will prove that . Let, if possible, .
From (3.3), we have
Thus, we have Hence, and .
So, is the common fixed point of , , , and .
Now, if one of the mappings , , or is continuous instead of , then one can show that , , , and have a common fixed point.
To show that is unique, let be another common fixed point of and .
From (3.3), we have Thus, we have, that is, and have a unique common fixed point.
In the same way, it can be shown that is the unique common fixed point of and .