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

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:

(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 .