# Coupled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Quasi-Metric Spaces with a *Q*-Function

- N Hussain
^{1}Email author, - MH Shah
^{2}and - MA Kutbi
^{1}

**2011**:703938

https://doi.org/10.1155/2011/703938

© N. Hussain et al. 2011

**Received: **20 August 2010

**Accepted: **16 September 2010

**Published: **28 September 2010

## Abstract

Using the concept of a mixed *g*-monotone mapping, we prove some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete quasi-metric spaces with a *Q*-function *q*. The presented theorems are generalizations of the recent coupled fixed point theorems due to Bhaskar and Lakshmikantham (2006), Lakshmikantham and Ćirić (2009) and many others.

## 1. Introduction

The Banach contraction principle is the most celebrated fixed point theorem and has been generalized in various directions (cf. [1–31]). Recently, Bhaskar and Lakshmikantham [8], Nieto and Rodríguez-López [28, 29], Ran and Reurings [30], and Agarwal et al. [1] presented some new results for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham [8] noted that their theorem can be used to investigate a large class of problems and discussed the existence and uniqueness of solution for a periodic boundary value problem. For more on metric fixed point theory, the reader may consult the book [22].

Recently, Al-Homidan et al. [2] introduced the concept of a -function defined on a quasi-metric space which generalizes the notions of a -function and a -distance and establishes the existence of the solution of equilibrium problem (see also [3–7]). The aim of this paper is to extend the results of Lakshmikantham and Ćirić [24] for a mixed monotone nonlinear contractive mapping in the setting of partially ordered quasi-metric spaces with a -function . We prove some coupled coincidence and coupled common fixed point theorems for a pair of mappings. Our results extend the recent coupled fixed point theorems due to Lakshmikantham and Ćirić [24] and many others.

Recall that if is a partially ordered set and such that for implies , then a mapping is said to be nondecreasing. Similarly, a nonincreasing mapping is defined. Bhaskar and Lakshmikantham [8] introduced the following notions of a mixed monotone mapping and a coupled fixed point.

Definition 1.1 (Bhaskar and Lakshmikantham [8]).

Definition 1.2 (Bhaskar and Lakshmikantham [8]).

The main theoretical result of Lakshmikantham and Ćirić in [24] is the following coupled fixed point theorem.

Theorem 1.3 (Lakshmikantham and Ćirić [24, Theorem ]).

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(b) has the following property:

(i) if a nondecreasing sequence ,then for all

(ii) if a nonincreasing sequence ,then for all

that is, and have a coupled coincidence.

Definition 1.4.

Let be a nonempty set. A real-valued function is said to be quasi-metric on if

The pair is called a quasi-metric space.

Definition 1.5.

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

if and is a sequence in such that it converges to a point (with respect to the quasi-metric) and for some then ;

for any , there exists such that , and implies that

Remark 1.6 (see [2]).

If is a metric space, and in addition to the following condition is also satisfied:

for any sequence in with and if there exists a sequence in such that then

then a -function is called a -function, introduced by Lin and Du [27]. It has been shown in [27]that every -distance or -function, introduced and studied by Kada et al. [21], is a -function. In fact, if we consider as a metric space and replace by the following condition:

for any , the function is lower semicontinuous,

then a -function is called a -distance on . Several examples of -distance are given in [21]. It is easy to see that if is lower semicontinuous, then holds. Hence, it is obvious that every -function is a -function and every -function is a -function, but the converse assertions do not hold.

- (a)

- (b)

Then is a -function on However, is neither a -function nor a -function, because is not a metric space.

The following lemma lists some properties of a
-function on
which are similar to that of a
-function (see [21])**.**

Lemma 1.8 (see [2]).

Let be a -function on Let and be sequences in , and let and be such that they converge to and Then, the following hold:

(1) if and for all , then . In particular, if and , then ;

(2) if and for all , then converges to ;

(3) if for all with , then is a Cauchy sequence;

## 2. Main Results

Analogous with Definition 1.1, Lakshmikantham and Ćirić [24] introduced the following concept of a mixed -monotone mapping.

Definition 2.1 (Lakshmikantham and Ćirić [24]).

Note that if is the identity mapping, then Definition 2.1 reduces to Definition 1.1.

Definition 2.2 (see [24]).

Definition 2.3 (see [24]).

Following theorem is the main result of this paper.

Theorem 2.4.

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(b) has the following property:

(i) if a nondecreasing sequence , then for all

(ii) if a nonincreasing sequence , then for all

that is, and have a coupled coincidence.

Proof.

Case 1.

*Suppose that the assumption (a) holds*. Taking the limit as in (2.26), and using the continuity of , we get

Case 2.

*Suppose that the assumption (b) holds*. Let . Now, since is continuous, is nondecreasing with for all , and is nonincreasing with for all , so is nondecreasing, that is,

Therefore, by the triangle inequality and ( ), we have for

Case 3.

Case 4.

Hence, by Lemma 1.8, and Thus, and have a coupled coincidence point.

The following example illustrates Theorem 2.4.

Example 2.5.

and , are both continuous on their domains and . Let be such that and There are four possibilities for (2.5) to hold. We first compute expression on the left of (2.5) for these cases:

Thus, satisfies the contraction condition (2.5) of Theorem 2.4. Now, suppose that be, respectively, nondecreasing and nonincreasing sequences such that and , then by Theorem 2.4, and for all

Therefore, by Theorem 2.4, there exists such that and

Corollary 2.6.

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(b) has the following properties:

(i) if a nondecreasing sequence , then for all

(ii) if a nonincreasing sequence , then for all .

that is, and have a coupled coincidence.

Proof.

Taking in Theorem 2.4, we obtain Corollary 2.6.

From Theorem 2.4, it follows that the set of coupled coincidences is nonempty.

Theorem 2.7.

Proof.

Therefore, is a coupled common fixed point of and To prove the uniqueness, assume that is another coupled common fixed point. Then, by (2.55), we have and

Corollary 2.8.

Also suppose that either

(b) has the following properties:

(i) if a nondecreasing sequence , then for all

(ii) if a non-increasing sequence , then for all

Furthermore, if are comparable, then that is,

Proof.

where . Hence, by Lemma 1.8, that is,

Corollary 2.9.

Also, suppose that either

(b) has the following properties:

(i) if a nondecreasing sequence , then for all

(ii) if a nonincreasing sequence , then for all

Furthermore, if are comparable, then that is,

Proof.

Taking in Corollary 2.8, we obtain Corollary 2.9.

Remark 2.10.

As an application of fixed point results, the existence of a solution to the equilibrium problem was considered in [2–7]. It would be interesting to solve Ekeland-type variational principle, Ky Fan type best approximation problem and equilibrium problem utilizing recent results on coupled fixed points and coupled coincidence points.

## Declarations

### Acknowledgment

The first and third author are grateful to DSR, King Abdulaziz University for supporting research project no. (3-74/430).

## Authors’ Affiliations

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