## New Contribution to the Advancement of Fixed Point Theory, Equilibrium Problems, and Optimization Problems

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# Some Common Coupled Fixed Point Results for Generalized Contraction in Complex-Valued Metric Spaces

**Academic Editor:**Erdal Karapinar

#### Abstract

We introduce and study the notion of common coupled fixed points for a pair of mappings in complex valued metric space and demonstrate the existence and uniqueness of the common coupled fixed points in a complete complex-valued metric space in view of diverse contractive conditions. In addition, our investigations are well supported by nontrivial examples.

#### 1. Introduction

Azam et al. [1] introduced the concept of complex-valued metric spaces and obtained sufficient conditions for the existence of common fixed points of a pair of contractive type mappings involving rational expressions. Subsequently, several authors have studied the existence and uniqueness of the fixed points and common fixed points of self-mappings in view of contrasting contractive conditions. Some of these investigations are noted in [2–26].

In [27], Bhaskar and Lakshmikantham introduced the concept of coupled fixed points for a given partially ordered set . Recently Samet et al. [28, 29] proved that most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of well-known fixed point theorems in the literature. In this paper, we deal with the corresponding definition of coupled fixed point for mappings on a complex-valued metric space along with generalized contraction involving rational expressions. Our results extend and improve several fixed point theorems in the literature.

#### 2. Preliminaries

Let be the set of complex numbers and . Define a partial order on as follows: Note that and , implies .

*Definition 1. *Let be a nonempty set. Suppose that the self-mapping satisfies the following:(1)for all and if and only if ;(2) for all ;(3), for all .

Then is called a complex valued metric on , and is known as a complex valued metric space. A point is called interior point of a set whenever, there exists such that

A point is a limit point of whenever, for every ,

is called open whenever each element of is an interior point of . Moreover, a subset is called closed whenever each limit point of belongs to . The family is a subbasis for a Hausdorff topology on .

Let be a sequence in and . If for every with there is such that, for all , then is said to be convergent, converges to , and is the limit point of . We denote this by , or , as . If for every with there is 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. We require the following lemmas.

Lemma 2 (see [1]). *Let be a complex valued metric space, and let be a sequence in . Then converges to if and only if .*

Lemma 3 (see [1]). *Let be a complex valued metric space, and let be a sequence in . Then is a Cauchy sequence if and only if.*

*Definition 4 (see [27]). *An element is called a coupled fixed point of if

*Definition 5. *An element is called a coupled coincidence point of if

*Example 6. *Let and defined as and for all . Then , , and are coupled coincidence points of and .

*Example 7. *Let and defined as and for all . Then and are coupled coincidence points of and .

*Definition 8. *An element is called a common coupled fixed point of if

*Example 9. *Let and defined as and for all . Then , , and are common coupled fixed points of and .

In the following, we provide common coupled fixed point theorem for a pair of mappings satisfying a rational inequality in complex valued metric spaces.

Theorem 10. *Let be a complete complex-valued metric space, and let the mappings satisfy
**
for all and , and are nonnegative reals with . Then and have a unique common coupled fixed point. *

*Proof. *Let and be arbitrary points in . Define , and , , for

Then,
which implies that
Since , so we get
and hence
Similarly, one can show that
Also,
so that
As , therefore
Similarly, one can show that
Adding (12)–(17), we get
If , then from (18), we get
Now if , then
Without loss of generality, we take . Since , so we get
This implies that and are Cauchy sequences in . Since is complete, there exists such that and as . We now show that and . We suppose on the contrary that and so that and ; we would then have
so that
By taking , we get which is a contradiction so that . Similarly, one can prove that . It follows similarly that and . So we have proved that is a common coupled fixed point of and . We now show that and have a unique common coupled fixed point. For this, assume that is a second common coupled fixed point of and . Then
so that
Since , so we get
Similarly, one can easily prove that
If we add (26) and (27), we get
which is a contradiction because . Thus, we get and , which proves the uniqueness of common coupled fixed point of and .

By setting in Theorem 10, one deduces the following.

Corollary 11. *Let be a complete complex-valued metric space, and let the mapping satisfy
**
for all , where , and are nonnegative reals with . Then has a unique coupled fixed point. *

Corollary 12. *Let be a complete complex-valued metric space, and let the mapping satisfy
**
for all , where , and are nonnegative reals with . Then, has a unique coupled fixed point. *

Theorem 13. *Let be a complete complex-valued metric space, and let the mappings satisfy
**
for all , where and are nonnegative reals with . Then and have a unique common coupled fixed point. *

*Proof. *Let and be arbitrary points in . Define , and , , for .

Now, we assume that

Then,
which implies that
as
Therefore,
Similarly, one can easily prove that

Now, if
we get
which implies that
as
Therefore,

Similarly, if , one can easily prove that
Adding the inequalities (36)–(43), we get
If , then, from (44), we get
Now if , then
Without loss of generality, we take . Since , so we get
This implies that and are Cauchy sequences in . Since is complete, so there exists such that and as . We now show that and . We suppose on the contrary that and so that and ; we would then have
so that
By taking , we get which is a contradiction so that . Now
which implies that
Which, on making , gives us which is a contradiction so that . It follows similarly that and . So we have proved that is a common coupled fixed point of and . As in Theorem 10, the uniqueness of common coupled fixed point remains a consequence of contraction condition (31).

We have obtained the existence and uniqueness of a unique common coupled fixed point if
for all . Now, assume that for some . From
we obtain that and . If , using (8), we deduce
That is, (this equality holds also if ). The equalities
ensure that is a unique common coupled fixed point of and . The same holds if either , , or .

From Theorem 13, if we assume , we obtain the following corollary.

Corollary 14. *Let be a complete complex-valued metric space, and let the self-mappings satisfy
**
for all , where and is a nonnegative real such that . Then and have a unique common coupled fixed point.*

Corollary 15. *Let be a complete complex-valued metric space, and let the mapping satisfy
*