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Some Common Fixed Point Theorems for Four Mappings in Dislocated Metric Space ()

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*Advances in Pure Mathematics*,

**6**, 695-712. doi: 10.4236/apm.2016.610057.

1. Introduction

In 1986, S. G. Matthews [1] introduced some concepts of metric domains in the context of domain theory. In 2000, P. Hitzler and A. K. Seda [2] introduced the concept of dislocated topology where the initiation of dislocated metric space is appeared. Since then, many authors have established fixed point theorems in dislocated metric space. In the literature, one can find many interesting recent articles in the field of dislocated metric space (see examples [3] - [12] ). Dislocated metric space plays very important role in topology, semantics of logical programming and in electronics engineering.

The purpose of this article is to establish some common fixed point theorems for two pairs of weakly compatible mappings with (E. A.) and (CLR) property in dislocated metric space.

2. Preliminaries

We start with the following definitions, lemmas and theorems.

Definition 1. [2] Let X be a non empty set and let be a function satisfying the following conditions:

1)

2) implies

3) for all.

Then, d is called dislocated metric (or d-metric) on X and the pair (X, d) is called the dislocated metric space (or d-metric space).

Definition 2. [2] A sequence in a d-metric space is called a Cauchy sequence if for given, there corresponds such that for all, we have.

Definition 3. [2] A sequence in d-metric space converges with respect to d (or in d) if there exists such that as

Definition 4. [2] A d-metric space is called complete if every Cauchy se- quence in it is convergent with respect to d.

Lemma 1. [2] Limits in a d-metric space are unique.

Definition 5. Let A and S be two self mappings on a set X. If for some, then x is called coincidence point of A and S.

Definition 6. [13] Let A and S be mappings from a metric space into itself. Then, A and S are said to be weakly compatible if they commute at their coincident point; that is, for some implies

Definition 7. [14] Let A and S be two self mappings defined on a metric space. We say that the mappings A and S satisfy (E. A.) property if there exists a sequence such that

for some.

Definition 8. [15] Let A and S be two self mappings defined on a metric space. We say that the mappings A and S satisfy property if there exists a sequence such that

3. Main Results

Now, we establish a common fixed point theorem for two pairs of weakly compatible mappings using E. A. property.

Theorem 1. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

(1)

(2)

.

1) The pairs or satisfy E. A. property.

2) The pairs and are weakly compatible.

If T(X) is closed then

1) The maps A and T have a coincidence point.

2) The maps B and S have a coincidence point.

3) The maps A, B, S and T have an unique common fixed point.

Proof. Assume that the pair satisfy E. A. property, so there exists a sequence such that

(3)

for some. Since, so there exists a sequence such that. Hence,

(4)

From condition (2), we have

Taking limit as, we get

Since

Therefore we have,

which is a contradiction, since. Hence,. Now, we have

Assume is closed, then there exits such that. We claim that. Now, from condition (2)

(5)

Since

So, taking limit as in (5), We conclude that

(6)

which is a contradiction. Hence,. Now, we have

(7)

This proves that v is the coincidence point of.

Again, since so there exists such that

Now, we claim that. From condition (2)

which is a contradiction.

Hence,

Therefore,.

This represents that w is the coincidence point of the maps B and S.

Hence,

Since the pairs and are weakly compatible so,

We claim. From condition (2)

which is a contradiction.

Hence,

Therefore,. Similary,. Hence,. This represents that u is the common fixed point of the mappings and.

Uniqueness:

If possible, let be other common fixed point of the mappings, then by the condition (2)

which is a contradiction.

Hence, This establishes the uniqueness of the common fixed point of four mappings.

From the above theorem, one can obtain the following corollaries easily.

Corollary 1. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

.

1) The pairs or satisfy E. A. property.

2) The pairs and are weakly compatible.

If T(X) is closed then,

1) The maps A and T have a coincidence point.

2) The maps A and S have a coincidence point.

3) The maps A, S and T have an unique common fixed point.

Corollary 2. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

.

1) The pairs or satisfy E. A. property.

2) The pairs and are weakly compatible.

If T(X) is closed then,

1) The maps A and S have a coincidence point.

2) The maps B and S have a coincidence point.

3) The maps A, B and S have an unique common fixed point.

Corollary 3. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

.

1) The pair satisfy E. A. property.

2) The pair is weakly compatible.

If S(X) is closed, then the mappings A and S have an unique common fixed point.

Now, we establish the following theorem.

Theorem 2. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

(8)

(9)

.

1) The pairs or satisfy E. A. property.

2) The pairs and are weakly compatible.

If T(X) is closed then,

1) The maps A and T have a coincidence point.

2) The maps B and S have a coincidence point.

3) The maps A, B, S and T have an unique common fixed point.

Proof. Assume that the pair satisfy E. A. property, so there exists a sequence such that

(10)

for some. Since, so there exists a sequence such that. Hence,

(11)

From condition (9), we have

Taking limit as we get

Since

Therefore we have,

which is a contradiction, since. Hence,. Now, we have

Assume is closed, then there exits such that. We claim that. Now from condition (9)

(12)

Since

So, taking limit as in (12), We conclude that

(13)

which is a contradiction. Hence,. Now, we have

(14)

This proves that v is the coincidence point of.

Again, since so there exists such that

Now we claim that. From condition (9)

Since

So if or we get the contradiction, since

or

Hence,

Therefore,.

This represents that w is the coincidence point of the maps B and S.

Hence,

Since the pairs and are weakly compatible so,

We claim. From condition (9)

Since

So if or or we get the contradiction. Since,

or

Hence,

Therefore,. Similary,. Hence,. This represents that u is the common fixed point of the mappings and.

Uniqueness:

If possible, let be other common fixed point of the mappings, then by the condition (9)

Since

So if or or we get the contradiction, since

or

Hence, This establishes the uniqueness of the common fixed point of four mappings.

From the above theorem, we can establish the following corollaries:

Corollary 4. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

.

1) The pairs or satisfy E. A. property.

2) The pairs and are weakly compatible.

If T(X) is closed then

1) The maps A and T have a coincidence point.

2) The maps A and S have a coincidence point.

3) The maps A, S and T have an unique common fixed point.

Corollary 5. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

.

1) The pairs or satisfy E. A. property.

2) The pairs and are weakly compatible.

if T(X) is closed then

1) The maps A and S have a coincidence point.

2) The maps B and S have a coincidence point.

3) The maps A, B and S have an unique common fixed point.

Corollary 6. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

.

1) The pair satisfy E. A. property.

2) The pair is weakly compatible.

If S(X) is closed, then the mappings A and S have an unique common fixed point.

Now, we establish a common fixed point theorem for weakly compatible mappings using (CLR)-property.

Theorem 3. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

(15)

(16)

where,

(17)

1) The pairs or satisfy CLR-property.

2) The pairs and are weakly compatible.

Then

1) The maps A and T have a coincidence point.

2) The maps B and S have a coincidence point.

3) The maps A, B, S and T have an unique common fixed point.

Proof. Assume that the pair satisfy property, so there exists a se- quence such that

(18)

for some. Since, so there exists a sequence such that. We show that

(19)

From condition (16), we have

(20)

where

Taking limit as in (20), we get

(21)

Since

Hence, we have

which is a contradiction, since.

Therefore,

Now we have

Assume, then there exits such that.

We claim that.

Now from condition (16)

(22)

where

Since

So, taking limit as in (22), we conclude that

(23)

which is a contradiction.

Hence,.

This proves that v is the coincidence point of the maps B and S.

Therefore,.

Since the pair (B, S) is weakly compatible, so

Since, there exists a point such that We show that

From condition (16),

where,

Therefore,.

This proves that u is the coincidence point of the maps A and T.

Since the pair is weakly compatible so,

We show that.

From condition (16)

where

which is a contradiction.

Hence,. Similarly, we obtain.

. Hence, w is the common fixed point of four mappings and.

Uniqueness:

Let be other common fixed point of the mappings and, then by the condition (16)

(24)

where

which is a contradiction.

Hence, This establishes the uniqueness of the common fixed point.

Now we have the following corollaries:

Corollary 7. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

where

1) The pairs or satisfy CLR-property.

2) The pairs and are weakly compatible.

Then

1) The maps A and S have a coincidence point.

2) The maps B and S have a coincidence point.

3) The maps A, B and S have an unique common fixed point.

Corollary 8. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

where

1) The pair and satisfy CLR-property.

2) The pairs and are weakly compatible.

Then

1) The maps A and T have a coincidence point.

2) The maps A and S have a coincidence point.

3) The maps A, S and T have an unique common fixed point.

Corollary 9. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

where

1) The pair satisfy CLR-property.

2) The pair is weakly compatible.

Then

1) The maps A and S have a coincidence point.

2) The maps A and S have an unique common fixed point.

Now, we establish the following theorem.

Theorem 4. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

(25)

(26)

where

(27)

1) The pairs or satisfy CLR-property.

2) The pairs and are weakly compatible.

then

1) The maps A and T have a coincidence point.

2) The maps B and S have a coincidence point.

3) The maps A, B, S and T have an unique common fixed point.

Proof. Assume that the pair satisfy property, so there exists a se- quence such that

(28)

for some. Since, so there exists a sequence such that. We show that

(29)

From condition (26), we have

(30)

where

Taking limit as in (30), we get

(31)

Since

Hence, we have

which is a contradiction, since.

Hence,

Now, we have

Assume, then there exits such that.

We claim that.

Now from condition (26)

(32)

where

Since

So, taking limit as in (32) We conclude that

(33)

which is a contradiction. Hence,. This proves that v is the coincidence point of of the maps B and S.

Hence,.

Since the pair (B, S) is weakly compatible, so

Since there exists a point such that We show that

From condition (26)

where

Hence

Since

So if or, we get the contradic- tion for both cases.

Therefore,.

This proves that u is the coincidence point of the maps A and T.

Since the pair is weakly compatible so,

We show that.

From condition (26)

where

Since

So if or or

we have

which give contradictions for all three cases.

Hence,. Similarly, we obtain.

. Hence, w is the common fixed point of four mappings and T.

Uniqueness:

Let be other common fixed point of the mappings and T, then by the condition (26)

(34)

where

Since

So if or or we have

or

which give contradictions for all three cases.

Hence, This establishes the uniqueness of the common fixed point.

Now, we have the following corollaries:

Corollary 10. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

where

1) The pairs or satisfy CLR-property.

2) The pairs and are weakly compatible.

Then

1) The maps A and S have a coincidence point.

2) The maps B and S have a coincidence point.

3) The maps A, B and S have an unique common fixed point.

Corollary 11. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

where

1) The pair and satisfy CLR-property.

2) The pairs and are weakly compatible.

Then

1) The maps A and T have a coincidence point.

2) The maps A and S have a coincidence point.

3) The maps A, S and T have an unique common fixed point.

Corollary 12. Let (X, d) be a dislocated metric space. Let satisfying the following conditions

where

1) The pair satisfy CLR-property.

2) The pair is weakly compatible.

Then

1) The maps A and S have a coincidence point.

2) The maps A and S have an unique common fixed point.

Remarks: Our results generalize and extend the results of A. Amri and D. Moutawakil [14] , W. Sintunavarat and P. Kumam [15] in dislocated metric space.

Conflicts of Interest

The authors declare no conflicts of interest.

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