Open Access

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

Fixed Point Theory and Applications20102010:134897

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

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 [39]).

Definition 1.1.

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

(w1) for all ,

(w2) for any , is lower semicontinuous,

(w3) 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.

Let be a nonempty set with a -distance on . Ones denotes the -closure of a subset of by which is defined as
(1.1)

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.

Let be a partially ordered set. Mapping is called strictly monotone increasing mapping if
(1.2)

Definition 1.8.

Let be a partially ordered set. A mapping is said to be a mixed monotone if is monotone nondecreasing in and monotone nonincreasing in , that is, for any ,
(1.3)

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.

Let be a partially ordered metric space with a -distance and a strictly monotone increasing mapping. Suppose that a mixed monotone mapping is -continuous with respect to such that
(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.

Let and for some ; this can be done since . Following the same arguments, we obtain and . Put
(2.2)
Similarly for all ,
(2.3)
Since is strictly monotone increasing and has the mixed monotone property, we have
(2.4)
Similarly
(2.5)
Now for all , using (2.1), we get
(2.6)
From (2.6),
(2.7)
where . Continuing, we conclude that
(2.8)
if is odd, where . Also,
(2.9)
if is even, where
(2.10)
Let ; then for every in we have
(2.11)
where
(2.12)
Hence,
(2.13)
For , we get
(2.14)
which further implies that
(2.15)
Lemma 1.3(3) implies that and are Cauchy sequences in . Since is complete, there exist such that and . Since is lower semicontinuous, we have
(2.16)
which implies that
(2.17)
Similarly
(2.18)
Let be given. Since is -continuous at with respect to , there exists such that for each
(2.19)
Since and , for , there exists such that, for all ,
(2.20)
Now,
(2.21)
implies that . Since
(2.22)

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 .

Let be a mixed monotone and a strict monotone increasing mapping such that
(2.23)

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.

Construct two sequences and such that and for all and and for some , as given in the proof of Theorem 2.1. Now, we need to show that and . Let . Since and , there exists such that, for all , we have
(2.24)
Consider
(2.25)
which implies that . Also, from Theorem 2.1, we have
(2.26)
Therefore,
(2.27)

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.

If is comparable to with respect to ordering in , then
(3.1)
implies that . Hence . Also,
(3.2)

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

Case 2.

If is not comparable to , then there exists an upper bound or lower bound of . Again since is strictly monotone increasing mapping and satisfies mixed monotone property, therefore, for all , is comparable to and . Following similar arguments to those given in the proof of Theorem 2.1, we obtain
(3.3)
where and . On taking limit as on both sides of (3.3), we have
(3.4)
and . By the same lines as in Case 1, we prove that . Again Lemma 1.3(1) implies that and . Hence is unique coupled point of coincidence of and . Note that if is a coupled point of coincidence of and , then are also a coupled points of coincidence of and . Then and therefore is unique coupled point of coincidence of and . Let . Since and are w-compatible, we obtain
(3.5)

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.

Recall that . Now, if , then . We claim that, for all , . Since is strictly monotone increasing and satisfies mixed monotone property, we have
(3.6)
Assuming that , since is strictly monotone increasing, so . By the mixed monotone property of , we have
(3.7)
Therefore,
(3.8)
Letting , there exists an such that and for all . Now,
(3.9)

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

(1)
Department of Mathematics, Lahore University of Management Sciences
(2)
Department of Mathematics, Faculty of Sciences and Mathematics, University of Niŝ

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© Mujahid Abbas et al. 2010

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