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

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## 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.

### Keywords

Equilibrium Problem Fixed Point Theorem Monotone Property Couple Fixed Point Couple Coincidence Point## 1. Introduction

The Banach contraction principle is the most celebrated fixed point theorem and has been generalized in various directions (cf. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 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 Open image in new window -function defined on a quasi-metric space which generalizes the notions of a Open image in new window -function and a Open image in new window -distance and establishes the existence of the solution of equilibrium problem (see also [3, 4, 5, 6, 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 Open image in new window -function Open image in new window . 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 Open image in new window is a partially ordered set and Open image in new window such that for Open image in new window Open image in new window implies Open image in new window , then a mapping Open image in new window 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 Open image in new window ]).

for all Open image in new window for which Open image in new window and Open image in new window Suppose that Open image in new window and Open image in new window is continuous and commutes with Open image in new window , and also suppose that either

(a) Open image in new window is continuous or

(b) Open image in new window has the following property:

(i) if a nondecreasing sequence Open image in new window ,then Open image in new window for all Open image in new window

(ii) if a nonincreasing sequence Open image in new window ,then Open image in new window for all Open image in new window

that is, Open image in new window and Open image in new window have a coupled coincidence.

Definition 1.4.

Let Open image in new window be a nonempty set. A real-valued function Open image in new window is said to be quasi-metric on Open image in new window if

Open image in new window for all Open image in new window

Open image in new window if and only if Open image in new window

Open image in new window for all Open image in new window .

The pair Open image in new window is called a quasi-metric space.

Definition 1.5.

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

for all Open image in new window

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

for any Open image in new window , there exists Open image in new window such that Open image in new window , and Open image in new window implies that Open image in new window

Remark 1.6 (see [2]).

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

for any sequence Open image in new window in Open image in new window with Open image in new window and if there exists a sequence Open image in new window in Open image in new window such that Open image in new window then Open image in new window

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

for any Open image in new window , the function Open image in new window is lower semicontinuous,

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

- (a)

- (b)

Then Open image in new window is a Open image in new window -function on Open image in new window However, Open image in new window is neither a Open image in new window -function nor a Open image in new window -function, because Open image in new window is not a metric space.

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

Lemma 1.8 (see [2]).

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

(1) if Open image in new window and Open image in new window for all Open image in new window , then Open image in new window . In particular, if Open image in new window and Open image in new window , then Open image in new window ;

(2) if Open image in new window and Open image in new window Open image in new window for all Open image in new window , then Open image in new window converges to Open image in new window ;

(3) if Open image in new window for all Open image in new window Open image in new window with Open image in new window , then Open image in new window is a Cauchy sequence;

(4) if Open image in new window for all Open image in new window , then Open image in new window is a Cauchy sequence;

(5) if Open image in new window are Open image in new window -functions on Open image in new window , then Open image in new window Open image in new window is also a Open image in new window -function on Open image in new window .

## 2. Main Results

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

Definition 2.1 (Lakshmikantham and Ćirić [24]).

Note that if Open image in new window is the identity mapping, then Definition 2.1 reduces to Definition 1.1.

Definition 2.2 (see [24]).

Definition 2.3 (see [24]).

for all Open image in new window

Following theorem is the main result of this paper.

Theorem 2.4.

for all Open image in new window for which Open image in new window and Open image in new window Suppose that Open image in new window and Open image in new window is continuous and commutes with Open image in new window , and also suppose that either

(a) Open image in new window is continuous or

(b) Open image in new window has the following property:

(i) if a nondecreasing sequence Open image in new window , then Open image in new window for all Open image in new window

(ii) if a nonincreasing sequence Open image in new window , then Open image in new window for all Open image in new window

that is, Open image in new window and Open image in new window have a coupled coincidence.

Proof.

We now show that Open image in new window and Open image in new window

Case 1.

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

Case 2.

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

with Open image in new window , Open image in new window for all Open image in new window .

Therefore, by the triangle inequality and ( ), we have Open image in new window for Open image in new window

Case 3.

Case 4.

Hence, by Lemma 1.8, Open image in new window and Open image in new window Thus, Open image in new window and Open image in new window have a coupled coincidence point.

The following example illustrates Theorem 2.4.

Example 2.5.

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

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

Therefore, by Theorem 2.4, there exists Open image in new window such that Open image in new window and Open image in new window

Corollary 2.6.

for all Open image in new window for which Open image in new window and Open image in new window Suppose that Open image in new window and Open image in new window is continuous and commutes with Open image in new window , and also suppose that either

(a) Open image in new window is continuous or

(b) Open image in new window has the following properties:

(i) if a nondecreasing sequence Open image in new window , then Open image in new window for all Open image in new window

(ii) if a nonincreasing sequence Open image in new window , then Open image in new window for all Open image in new window .

that is, Open image in new window and Open image in new window have a coupled coincidence.

Proof.

Taking Open image in new window in Theorem 2.4, we obtain Corollary 2.6.

From Theorem 2.4, it follows that the set Open image in new window of coupled coincidences is nonempty.

Theorem 2.7.

Proof.

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

Corollary 2.8.

Also suppose that either

(a) Open image in new window is continuous or

(b) Open image in new window has the following properties:

(i) if a nondecreasing sequence Open image in new window , then Open image in new window for all Open image in new window

(ii) if a non-increasing sequence Open image in new window , then Open image in new window for all Open image in new window

Furthermore, if Open image in new window are comparable, then Open image in new window that is, Open image in new window

Proof.

where Open image in new window . Hence, by Lemma 1.8, Open image in new window that is, Open image in new window

Corollary 2.9.

Also, suppose that either

(a) Open image in new window is continuous or

(b) Open image in new window has the following properties:

(ii) if a nonincreasing sequence Open image in new window , then Open image in new window for all Open image in new window

Furthermore, if Open image in new window are comparable, then Open image in new window that is, Open image in new window

Proof.

Taking Open image in new window 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, 3, 4, 5, 6, 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.

## Notes

### Acknowledgment

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

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