# Coupled Coincidence Point and Coupled Common Fixed Point Theorems in Partially Ordered Metric Spaces with -Distance

- Mujahid Abbas
^{1}, - Dejan Ilić
^{2}Email author and - MuhammadAli Khan
^{1}

**2010**:134897

https://doi.org/10.1155/2010/134897

© Mujahid Abbas et al. 2010

**Received: **7 April 2010

**Accepted: **18 October 2010

**Published: **24 October 2010

## Abstract

We introduce the concept of a -compatible mapping to obtain a coupled coincidence point and a coupled point of coincidence for nonlinear contractive mappings in partially ordered metric spaces equipped with -distances. Related coupled common fixed point theorems for such mappings are also proved. Our results generalize, extend, and unify several well-known comparable results in the literature.

## 1. Introduction and Preliminaries

In 1996, Kada et al. [1] introduced the notion of -distance. They elaborated, with the help of examples, that the concept of -distance is general than that of metric on a nonempty set. They also proved a generalization of Caristi fixed point theorem employing the definition of -distance on a complete metric space. Recently, Ilić and Rakočević [2] obtained fixed point and common fixed point theorems in terms of -distance on complete metric spaces (see also [3–9]).

Definition 1.1.

Let be a metric space. A mapping is called a -distance on if the following are satisfied:

(w_{2}) for any
,
is lower semicontinuous,

(w_{3}) for any
there exists
such that
and
imply
, for any
.

The metric is a -distance on . For more examples of -distances, we refer to [10].

Definition 1.2.

The next Lemma is crucial in the proof of our results.

Lemma 1.3 (see [1]).

Let be a metric space, and let be a -distance on . Let and be sequences in , let and be sequences in converging to 0, and let . Then the following hold.

(1)If and for any , then . In particular, if , then .

(2)If and for any , then converges to .

(3)If for any with , then is a Cauchy sequence.

(4)If for any , then is a Cauchy sequence.

Bhaskar and Lakshmikantham in [11] introduced the concept of coupled fixed point of a mapping and investigated some coupled fixed point theorems in partially ordered sets. They also discussed an application of their result by investigating the existence and uniqueness of solution for a periodic boundary value problem. Sabetghadam et al. in [12] introduced this concept in cone metric spaces. They investigated some coupled fixed point theorems in cone metric spaces. Recently, Lakshmikantham and Ćirić [13] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces which extend the coupled fixed point theorem given in [11]. The following are some other definitions needed in the sequel.

Definition 1.4 . (see [12]).

Let be any nonempty set. Let and be two mappings. An ordered pair is called

(1)a coupled fixed point of a mapping if and ,

(2)a coupled coincidence point of hybrid pair if and and is called coupled point of coincidence,

(3)a common coupled fixed point of hybrid pair if and .

Note that if is a coupled fixed point of , then is also a coupled fixed point of the mapping .

Definition 1.5.

Let be any nonempty set. Mappings and are called -compatible if whenever and .

Definition 1.6.

Let be a metric space with -distance . A mapping is said to be -continuous at a point with respect to mapping if for every there exists a such that implies that for all .

Definition 1.7.

Definition 1.8.

Kada et al. [1] gave an example to show that is not symmetric in general. We denote by and , respectively, the class of all -distances on and the class of all -distances on which are symmetric for comparable elements in . Also in the sequel, we will consider that and are comparable with respect to ordering in if and .

## 2. Coupled Coincidence Point

In this section, we prove coincidence point results in the frame work of partially ordered metric spaces in terms of a -distance.

Theorem 2.1.

for all with or and . Let and whenever , for some . If is complete and there exist such that and , then and have a coupled coincidence point.

Proof.

using Lemma 1.3(1), we obtain . Similarly, we can prove that . Hence is coupled coincidence point of and .

Theorem 2.2.

Let be a partially ordered metric space with a -distance having the following properties.

(1)If is in with for all and for some , then for all .

(2)If is in with for all and for some , then for all .

for all with or and . Let and whenever , for some . If is complete and there exist such that and , then and have a coupled coincidence point.

Proof.

implies that . Similarly, we can prove that . Hence is coupled coincidence point of and .

## 3. Coupled Common Fixed Point

In this section, using the concept of -compatible maps, we obtain a unique coupled common fixed point of two mappings.

Theorem 3.1.

Let all the hypotheses of Theorem 2.1 (resp., Theorem 2.2) hold with . If for every there exists that is comparable to and with respect to ordering in , then there exists a unique coupled point of coincidence of and . Moreover if and are -compatible, then and have a unique coupled common fixed point.

Proof.

Let be another coupled coincidence point of and . We will discuss the following two cases.

Case 1.

gives that . The result follows using Lemma 1.3(1).

Case 2.

*w*-compatible, we obtain

Consequently . Therefore . Hence is a coupled common fixed point of and .

Remark 3.2.

If in addition to the hypothesis of Theorem 2.1 (resp., Theorem 2.2) we suppose that , and are comparable, then .

Proof.

implies that . Since , therefore . Similarly we can prove that . Hence by Lemma 1.3(1), we have . Similarly, if , we can show that for each and .

## Declarations

### Acknowledgment

The present version of the paper owes much to the precise and kind remarks of the learned referees.

## Authors’ Affiliations

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