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

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http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368

Some Common Fixed Point Theorems for Four

Mappings in Dislocated Metric Space

Dinesh Panthi1, Kumar Subedi2

1_{Department of Mathematics, Valmeeki Campus, Nepal Sanskrit University, Kathmandu, Nepal}

2_{Department of Mathematics Education, Mahendra Ratna Multiple Campus, Tribhuvan University, Ilam, Nepal}

Abstract

In this article, we establish some common fixed point theorems for two pairs of weakly compatible mappings with (E. A.) and (CLR) property in dislocated metric space which generalize and extend some similar results in the literature.

Keywords

Dislocated Metric, Weakly Compatible Maps, Common Fixed Point

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 dis-located topology where the initiation of disdis-located 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 to-pology, 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 d X: ×X →

[

0,∞

)

be a function

satisfying the following conditions: 1) d x y

(

,

)

=d y x

(

,

)

.

How to cite this paper: Panthi, D. and Subedi, K. (2016) Some Common Fixed Point Theorems for Four Mappings in Dis-located Metric Space. Advances in Pure Mathematics, 6, 695-712.

http://dx.doi.org/10.4236/apm.2016.610057

Received: August 9, 2016 Accepted: September 19, 2016 Published: September 22, 2016

http://creativecommons.org/licenses/by/4.0/

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2) d x y

(

,

)

=d y x

(

,

)

=0 implies x=y.

3) d x y

(

,

)

≤d x z

( )

, +d z y

(

,

)

for all x y z, , ∈X .

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

{ }

xn in a d-metric space

(

X d,

)

is called a Cauchy sequence if for given >0, there corresponds n₀∈N such that for all m n, ≥n0, we

have d x

(

_m,x_n

)

<.

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

(

_n,

)

→0 as n→ ∞.

Definition 4. [2] A d-metric space

(

X d,

)

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 Ax=Sx for some x∈X , then x is called coincidence point of A and S.

Definition 6. [13] Let A and S be mappings from a metric space

(

X d,

)

into itself.

Then, A and S are said to be weakly compatible if they commute at their coincident point; that is, Ax=Sx for some x∈X implies ASx=SAx.

Definition 7. [14] Let A and S be two self mappings defined on a metric space

(

X d,

)

. We say that the mappings A and S satisfy (E. A.) property if there exists a

sequence

{ }

xn ∈X such that

lim n lim n n→∞Ax =n→∞Sx =u for some x∈X .

Definition 8. [15] Let A and S be two self mappings defined on a metric space

(

X d,

)

. We say that the mappings A and S satisfy

(

CLR_A

)

property if there exists a

sequence

{ }

xn ∈X such that

lim _n lim _n

n→∞Ax =n→∞Sx =Ax

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 A B S T X, , , : →X satisfying the following conditions

( )

and

( )

A X ⊆S X B X ⊆T X (1)

(

,

)

{

(

,

)

(

,

)

(

,

)

(

,

)

(

,

)

}

d Ax By ≤k d Sy Ax +d Tx Sy +d Tx Ax +d By Sy +d Tx By (2)

1 0,

8

k∈_{ }_.

1) The pairs

(

A T,

)

or

(

B S,

)

satisfy E. A. property.

2) The pairs

(

A T,

)

and

(

B S,

)

are weakly compatible.

If T(X) is closed then

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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

(

A T,

)

satisfy E. A. property, so there exists a sequence

{ }

xn ∈X such that

lim _n lim _n

n→∞Ax =n→∞Tx =u (3) for some u∈X . Since A X

( )

⊆S X

( )

, so there exists a sequence

{ }

y_n ∈X such that

n n

Ax =Sy . Hence,

lim n lim n

n→∞Ax =n→∞Sy =u (4) From condition (2), we have

(

n, n

)

{

(

n, n

)

(

n, n

)

(

n, n

)

(

n, n

)

(

n, n

)

}

d Ax By ≤k d Sy Ax +d Tx Sy +d Tx Ax +d By Sy +d Tx By

Taking limit as n→ ∞, we get

(

)

{

(

)

(

)

(

)

(

)

(

)

}

lim , lim , , ,

, ,

n n n n n n n n

n n

n n n n

d Ax By k d Sy Ax d Tx Sy d Tx Ax

d By Sy d Tx By

→∞ ≤ →∞ + +

+ +

Since

(

)

(

)

(

)

lim n, n lim n, n lim n, n 0

n→∞d Tx Sy =n→∞d Tx Ax =n→∞d Sy Ax =

(

)

(

)

(

)

lim n, n lim n, n n,

n→∞d By Sy =n→∞d Tx By =d By u

Therefore we have,

(

)

(

)

lim , n 2 lim , n

n→∞d u By ≤ kn→∞d u By

which is a contradiction, since 0,1 8

k∈_{ }_. Hence, limn→∞Byn =u. Now, we have

lim _n lim _n lim _n lim _n

n→∞Ax =n→∞Tx =n→∞By =n→∞Sy =u

Assume T X

( )

is closed, then there exits v∈X such that Tv=u. We claim that Av=u. Now, from condition (2)

(

, n

)

{

(

n,

)

(

, n

)

(

,

)

(

n, n

)

(

, n

)

}

d Av By ≤k d Sy Av +d Tv Sy +d Tv Av +d By Sy +d Tv By (5)

Since

(

)

(

)

lim n, ,

n→∞d Sy Av =d u Av

(

)

(

)

(

)

lim , n lim n, n lim , n 0

n→∞d Tv Sy =n→∞d By Sy =n→∞d Tv By =

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

(

,

)

2

(

,

)

d Av u ≤ kd u Av (6)

which is a contradiction. Hence, d Av u

(

,

)

= ⇒0 Av=u. Now, we have

.

Av= =u Tv (7)

This proves that v is the coincidence point of

(

A T,

)

.

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Av=Sw=u

Now, we claim that Bw=u. From condition (2)

(

)

(

)

(

)

(

)

(

) (

)

(

)

{

}

( )

(

)

(

)

{

}

( )

(

)

{

}

(

)

, , , , , , , , , , , , ,

3 , 2 ,

8 ,

d u Bw d Av Bw

k d Sw Av d Tv Sw d Tv Av d Bw Sw d Tv Bw k d u u d u u d u u d Bw u d u Bw

k d u u d Bw u kd u Bw

=

≤ + + +

= + + + +

= +

≤

which is a contradiction.

Hence, d u Bw

(

,

)

= ⇒0 Bw=u.

Therefore, Bw= =u Sw.

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

u=Bw=Sw=Tv= Av

Since the pairs

(

B S,

)

and

(

A T,

)

are weakly compatible so,

,

BSw=SBw TAv=ATv

and

Tu=TAv= ATv=Au Su=SBw=BSw=Bu

We claim Bu=u. From condition (2)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

{

}

(

)

(

)

( )

(

)

(

)

{

}

(

)

( )

(

)

{

}

(

)

, , , , , , , , , , , ,

3 , , ,

7 ,

d u Bu d Av Bu

k d Su Av d Tv Su d Tv Av d Bu Su d Tv Bu

k d Bu u d u Bu d u u d Bu Bu d u Bu k d u Bu d u u d Bu Bu

kd u Bu

=

≤ + + + +

= + + + +

= + +

≤

Hence, d u Bu

(

,

)

= ⇒0 Bu=u.

Therefore, u=Bu=Su. Similary, Au= =u Tu . Hence, u=Au=Bu=Su=Tu .

This represents that u is the common fixed point of the mappings A B S, , and T.

Uniqueness:

If possible, let z

( )

≠u be other common fixed point of the mappings, then by the

condition (2)

( )

(

)

(

)

(

)

(

)

(

)

(

)

{

}

( )

{

}

( )

{

}

( )

, , , , , , , , , , , ,

3 , , ,

7 ,

d u z d Au Bz

k d Sz Au d Tu Sz d Tu Au d Bz Sz d Tu Bz k d z u d u z d u u d z z d u z

k d u z d u u d z z kd u z

=

≤ + + + +

= + + + +

= + +

≤ which is a contradiction.

Hence, d u z

( )

, = ⇒ =0 u z. This establishes the uniqueness of the common fixed

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

Corollary 1. Let (X, d) be a dislocated metric space. Let A S T X, , : →X satisfying the following conditions

( )

and

( )

A X ⊆S X A X ⊆T X

(

,

)

{

(

,

)

(

,

)

(

,

)

(

,

)

(

,

)

}

d Ax Ay ≤k d Sy Ax +d Tx Sy +d Tx Ax +d Ay Sy +d Tx Ay

1 0,

8

k∈_{ }_.

1) The pairs

(

A T,

)

or

(

A S,

)

2) The pairs

(

A T,

)

and

(

A S,

)

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 A B S X, , : →X satisfying the following conditions

( )

and

( )

A X B X ⊆S X

(

,

)

{

(

,

)

(

,

)

(

,

)

(

,

)

(

,

)

}

d Ax By ≤k d Sy Ax +d Sx Sy +d Sx Ax +d By Sy +d Sx By

1 0,

8

k∈_{ }_.

1) The pairs

(

A S,

)

or

(

B S,

)

2) The pairs

(

A S,

)

and

(

B S,

)

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 A S X, : →X satisfying the following conditions

( )

A X ⊆S X

(

,

)

{

(

,

)

(

,

)

(

,

)

(

,

)

(

,

)

}

d Ax Ay ≤k d Sy Ax +d Sx Sy +d Sx Ax +d Ay Sy +d Sx Ay

1 0,

8

k∈_{ }_.

1) The pair

(

A S,

)

2) The pair

(

A S,

)

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.

( )

and

( )

(6)

(

,

)

max

{

(

,

) (

, ,

) (

, ,

) (

, ,

) (

, ,

)

}

d Ax By ≤k d Sy Ax d Tx Sy d Tx Ax d By Sy d Tx By (9)

1 0,

2

k∈_{ }_.

1) The pairs

(

A T,

)

or

(

B S,

)

2) The pairs

(

A T,

)

and

(

B S,

)

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

(

A T,

)

satisfy E. A. property, so there exists a sequence

{ }

xn ∈X such that

lim _n lim _n

n→∞Ax =n→∞Tx =u (10) for some u∈X . Since A X

( )

⊆S X

( )

{ }

n n

Ax =Sy . Hence,

lim _n lim _n

n→∞Ax =n→∞Sy =u (11) From condition (9), we have

(

_n, _n

)

max

{

(

_n, _n

) (

, _n, _n

) (

, _n, _n

) (

, _n, _n

) (

, _n, _n

)

}

d Ax By ≤k d Sy Ax d Tx Sy d Tx Ax d By Sy d Tx By

Taking limit as n→ ∞ we get

(

)

{

(

) (

)

}

lim n, n lim max n, n , n, n , n, n , n, n , n, n n→∞d Ax By ≤n→∞k d Sy Ax d Tx Sy d Tx Ax d By Sy d Tx By

Since

(

)

(

)

(

)

(

)

(

)

(

)

lim n, n lim n, n n,

n→∞d By Sy =n→∞d Tx By =d By u

Therefore we have,

(

)

(

)

lim , n lim , n

n→∞d u By ≤kn→∞d u By

k∈_{ }_. Hence, limn→∞Byn =u. Now, we have

n→∞Ax =n→∞Tx =n→∞By =n→∞Sy =u

Assume T X

( )

is closed, then there exits v∈X such that Tv=u. We claim that Av=u. Now from condition (9)

(

, _n

)

max

{

(

_n,

) (

, , _n

) (

, ,

) (

, _n, _n

) (

, , _n

)

}

d Av By ≤k d Sy Av d Tv Sy d Tv Av d By Sy d Tv By (12)

Since

(

)

(

)

lim n, ,

n→∞d Sy Av =d u Av

(

)

(

)

(

)

lim , n lim n, n lim , n 0

n→∞d Tv Sy =n→∞d By Sy =n→∞d Tv By =

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(

,

)

(

,

)

d Av u ≤kd u Av (13)

which is a contradiction. Hence, d Av u

(

,

)

= ⇒0 Av=u. Now, we have

.

Av= =u Tv (14)

This proves that v is the coincidence point of

(

A T,

)

.

Again, since A X

( )

⊆S X

( )

so there exists w∈X such that Av=Sw=u

Now we claim that Bw=u. From condition (9)

(

)

(

)

(

) (

)

{

}

( ) ( ) ( ) (

) (

)

{

}

( ) (

)

{

}

, ,

max , , , , , , , , ,

max , , ,

d u Bw d Av Bw

k d Sw Av d Tv Sw d Tv Av d Bw Sw d Tv Bw k d u u d u u d u u d Bw u d u Bw

k d u u d Bw u

= ≤ = =

Since

( )

, 2

(

,

)

d u u ≤ d u Bw

So if max

{

d u u

( ) (

, ,d Bw u,

)

}

=d u u

( )

, or d Bw u

(

,

)

we get the contradiction, since

(

,

)

2

(

,

)

d u Bw ≤ kd u Bw

or

(

,

)

(

,

)

d u Bw ≤kd u Bw

Hence, d u Bw

(

,

)

= ⇒0 Bw=u.

Therefore, Bw= =u Sw.

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

u=Bw=Sw=Tv= Av

Since the pairs

(

B S,

)

and

(

A T,

)

are weakly compatible so,

,

BSw=SBw TAv=ATv

and

Tu=TAv= ATv=Au Su=SBw=BSw=Bu

We claim Bu=u. From condition (9)

(

)

(

)

(

) (

)

{

}

(

) (

) ( ) (

) (

)

{

}

(

) ( ) (

)

{

}

, ,

max , , , , , , , , ,

max , , , , ,

d u Bu d Av Bu

k d Su Av d Tv Su d Tv Av d Bu Su d Tv Bu k d Bu u d u Bu d u u d Bu Bu d u Bu k d u Bu d u u d Bu Bu

= ≤

= =

Since

( )

, 2

(

,

)

and

(

,

)

2

(

,

)

d u u ≤ d u Bu d Bu Bu ≤ d u Bu

So if max

{

d u Bu

(

,

) ( ) (

,d u u, ,d Bu Bu,

)

}

=d u Bu

(

,

)

or d u u

( )

, or d Bu Bu

(

,

)

we

get the contradiction. Since,

(

,

)

(

,

)

(

,

)

(8)

or

(

,

)

2

(

,

)

d u Bu ≤ kd u Bu

Hence, d u Bu

(

,

)

= ⇒0 Bu=u.

Therefore, u=Bu=Su. Similary, Au= =u Tu . Hence, u=Au=Bu=Su=Tu .

This represents that u is the common fixed point of the mappings A B S, , and T.

Uniqueness:

If possible, let z

( )

≠u be other common fixed point of the mappings, then by the

condition (9)

( )

(

)

(

) (

)

{

}

( ) ( ) ( ) ( ) ( )

{

}

( ) ( ) ( )

{

}

, ,

max , , , , , , , ,

max , , , , , , , , ,

max , , , , ,

d u z d Au Bz

k d Sz Au d Tu Sz d Tu Au d Bz Sz d Tu Bz

k d z u d u z d u u d z z d u z

k d u z d u u d z z

= ≤ = =

Since

( )

, 2

( )

, and

( )

, 2

( )

,

d u u ≤ d u z d z z ≤ d z u

So if max

{

d u z

( ) ( ) ( )

, ,d u u, ,d z z,

}

=d u z

( )

, or d u u

( )

, or d z z

( )

, we get the

contradiction, since

( )

,

(

,

)

( )

,

d u z =d Au Bz ≤kd u z

or

( )

, 2

( )

,

d u z ≤ kd u z

Hence, d u z

( )

, = ⇒ =0 u z. This establishes the uniqueness of the common fixed

point of four mappings.

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

( )

and

( )

A X ⊆S X A X ⊆T X

(

,

)

max

{

(

,

) (

, ,

) (

, ,

) (

, ,

) (

, ,

)

}

d Ax Ay ≤k d Sy Ax d Tx Sy d Tx Ax d Ay Sy d Tx Ay

1 0,

2

k∈_{ }_.

1) The pairs

(

A T,

)

or

(

A S,

)

2) The pairs

(

A T,

)

and

(

A S,

)

If T(X) is closed then

( )

and

( )

(9)

(

,

)

max

{

(

,

) (

, ,

) (

, ,

) (

, ,

) (

, ,

)

}

d Ax By ≤k d Sy Ax d Sx Sy d Sx Ax d By Sy d Sx By

1 0,

2

k∈_{ }_.

1) The pairs

(

A S,

)

or

(

B S,

)

2) The pairs

(

A S,

)

and

(

B S,

)

if T(X) is closed then

( )

A X ⊆S X

(

,

)

max

{

(

,

) (

, ,

) (

, ,

) (

, ,

) (

, ,

)

}

d Ax Ay ≤k d Sy Ax d Sx Sy d Sx Ax d Ay Sy d Sx Ay

1 0,

2

k∈_{ }_.

1) The pair

(

A S,

)

2) The pair

(

A S,

)

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.

( )

and

( )

A X ⊆S X B X ⊆T X (15)

(

)

( )

1

, , , 0,

8

d Ax By ≤kM x y k∈_{ }_ (16)

where,

(

,

)

{

(

,

)

(

,

)

(

,

)

(

,

)

(

,

)

}

M x y = d Tx Sy +d Tx Ax +d By Sy +d Tx By +d Sy Ax (17)

1) The pairs

(

A T,

)

or

(

B S,

)

satisfy CLR-property.

2) The pairs

(

A T,

)

and

(

B S,

)

Then

(

A T,

)

satisfy

(

CLR_A

)

property, so there exists a se-

quence

{ }

xn ∈X such that

lim n lim n

n→∞Ax =n→∞Tx =Ax (18) for some x∈X . Since A X

( )

⊆S X

( )

{ }

limn→∞ Axn =limn→∞Syn=Ax. We show that

lim _n

(10)

From condition (16), we have

(

_n, _n

)

(

_n, _n

)

,

d Ax By ≤kM x y (20)

where

(

_n, _n

)

{

(

_n, _n

)

(

_n, _n

)

(

_n, _n

)

(

_n, _n

)

(

_n, _n

)

}

M x y = d Tx Sy +d Tx Ax +d By Sy +d Tx By +d Sy Ax

Taking limit as n→ ∞ in (20), we get

(

)

(

)

lim n, n lim n, n ,

n→∞d Ax By ≤kn→∞M x y (21)

Since

(

)

(

)

(

)

(

)

(

)

(

)

lim n, n lim , n lim n, n

n→∞d Ax By =n→∞d Ax By =n→∞d By Sy

Hence, we have

(

)

(

)

lim , n 2 lim , n ,

n→∞d Ax By ≤ kn→∞d Ax By

k∈_{ }_.

Therefore,

(

)

lim , n 0 lim n .

n→∞d Ax By = ⇒n→∞By =Ax

Now we have

n→∞Ax =n→∞Tx =n→∞By =n→∞Sy = Ax

Assume A X

( )

⊆S X

( )

, then there exits v∈X such that Ax=Sv.

We claim that Bv=Sv.

Now from condition (16)

(

_n,

)

(

_n,

)

d Ax Bv ≤k M x v (22)

where

(

_n,

)

{

(

_n,

)

(

_n, _n

)

(

,

)

(

_n,

)

(

, _n

)

}

M x v = d Tx Sv +d Tx Ax +d Bv Sv +d Tx Bv +d Sv Ax

Since

(

)

(

)

(

)

lim n, , ,

n→∞d Tx Bv =d Ax Bv =d Sv Bv

(

)

(

)

(

)

lim n, lim n, n lim , n 0

n→∞d Tx Sv =n→∞d Tx Ax =n→∞d Sv Ax =

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

(

,

)

2

(

,

)

d Sv Bv ≤ k d Sv Bv (23)

Hence, d Sv Bv

(

,

)

= ⇒0 Sv=Bv.

This proves that v is the coincidence point of the maps B and S. Therefore, Sv=Bv=Ax=w Say

(

)

.

(11)

BSv=SBv⇒Bw=Sw

Since B X

( )

⊆T X

( )

, there exists a point u∈X such that Bv=Tu. We show that

Tu=Au=w

From condition (16),

(

,

)

( )

, ,

d Au Bv ≤k M u v

where,

( )

{

(

)

(

)

(

)

(

)

(

)

}

(

)

(

)

(

)

(

)

(

)

{

}

(

)

(

)

{

}

(

)

, , , , , , , , , , ,

3 , 2 ,

8 ,

M u v d Tu Sv d Tu Au d Bv Sv d Tu Bv d Sv Au

d Bv Bv d Bv Au d Bv Bv d Bv Bv d Bv Au

d Bv Bv d Bv Au

d Bv Au

= + + + +

= +

≤

Therefore, d Au Bv

(

,

)

= ⇒0 Au=Bv.

Au Bv Tu w

∴ = = =

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

(

A T,

)

is weakly compatible so,

ATu=TAu⇒Aw=Tw

We show that Aw=w.

From condition (16)

(

,

)

(

,

)

(

,

)

,

d Aw w =d Aw Bv ≤k M w v

where

(

)

{

(

)

(

)

(

)

(

)

(

)

}

(

)

(

)

(

)

(

)

(

)

{

}

(

)

(

)

(

)

{

}

(

)

, , , , , , , , , , ,

3 , , ,

7 ,

M w v d Tw Sv d Tw Aw d Bv Sv d Tw Bv d Sv Aw

d Aw w d Aw Aw d w w d Aw w d w Aw

d Aw w d Aw Aw d w w

kd Aw w

= + + + +

= + +

≤

Hence, d Aw w

(

,

)

= ⇒0 Aw=w. Similarly, we obtain Bw=w.

Aw Bw Sw Tw w

∴ = = = = _{. Hence, w is the common fixed point of four mappings}

, ,

A B S and T.

Uniqueness:

Let z

(

≠w

)

be other common fixed point of the mappings A B S, , and T , then

by the condition (16)

(

,

)

(

,

)

(

,

)

d w z =d Aw Bz ≤k M w z (24)

where

(

)

{

(

)

(

)

(

)

(

)

(

)

}

(

)

(

)

( )

(

)

(

)

{

}

(

)

(

)

( )

{

}

(

)

, , , , , , , , , , ,

3 , , ,

7 ,

M w z d Tw Sz d Tw Aw d Bz Sz d Tw Bz d Sz Aw

d w z d w w d z z d w z d z w

d w z d w w d z z

d w z

= + + + +

= + +

≤

(12)

Hence, d w z

(

,

)

= ⇒ =0 w z. This establishes the uniqueness of the common fixed

point.

Now we have the following corollaries:

( ) ( )

,

( )

A X B X ⊆S X

(

)

( )

1

, , , 0,

8

d Ax By ≤k M x y k∈_{ }_

where

(

,

)

{

(

,

)

(

,

)

(

,

)

(

,

)

(

,

)

}

M x y = d Sx Sy +d Sx Ax +d By Sy +d Sx By +d Sy Ax

1) The pairs

(

A S,

)

or

(

B S,

)

2) The pairs

(

A S,

)

and

(

B S,

)

Then

( )

and

( )

A X ⊆S X A X ⊆T X

(

)

( )

1

, , , 0,

8

d Ax Ay ≤k M x y k∈_{ }_

where

(

,

)

{

(

,

)

(

,

)

(

,

)

(

,

)

(

,

)

}

M x y = d Tx Sy +d Tx Ax +d Ay Sy +d Tx Ay +d Sy Ax

1) The pair

(

A T,

)

and

(

A S,

)

2) The pairs

(

A T,

)

and

(

A S,

)

Then

( )

A X ⊆S X

(

)

( )

1

, , , 0,

8

d Ax Ay ≤k M x y k∈ _

  where

(

,

)

{

(

,

)

(

,

)

(

,

)

(

,

)

(

,

)

}

M x y = d Sx Sy +d Sx Ax +d Ay Sy +d Sx Ay +d Sy Ax

1) The pair

(

A S,

)

(13)

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.

( )

and

( )

A X ⊆S X B X ⊆T X (25)

(

)

( )

1

, , , 0,

2

d Ax By ≤kM x y k∈_{ }_ (26)

where

(

,

)

max

{

(

,

) (

, ,

) (

, ,

) (

, ,

) (

, ,

)

}

M x y = d Tx Sy d Tx Ax d By Sy d Tx By d Sy Ax (27)

1) The pairs

(

A T,

)

or

(

B S,

)

2) The pairs

(

A T,

)

and

(

B S,

)

then

(

A T,

)

satisfy

(

CLR_A

)

property, so there exists a se-

quence

{ }

xn ∈X such that

lim n lim n

n→∞Ax =n→∞Tx =Ax (28)

for some x∈X . Since A X

( )

⊆S X

( )

{ }

lim_n_→∞Ax_n =lim_n_→∞Sy_n =Ax. We show that

lim n

n→∞Bx = Ax (29)

From condition (26), we have

(

_n, _n

)

(

_n, _n

)

,

d Ax By ≤k M x y (30)

where

(

n, n

)

max

{

(

n, n

) (

, n, n

) (

, n, n

) (

, n, n

) (

, n, n

)

}

M x y = d Tx Sy d Tx Ax d By Sy d Tx By d Sy Ax

Taking limit as n→ ∞ in (30), we get

(

)

(

)

lim n, n lim n, n ,

n→∞d Ax By ≤kn→∞M x y (31)

Since

(

)

(

)

(

)

(

)

(

)

(

)

lim n, n lim n, n lim , n

n→∞d Tx By =n→∞d By Sy =n→∞d Ax By

Hence, we have

(

)

(

)

lim , n lim , n ,

(14)

k∈_{ }_.

Hence,

(

)

lim , n 0 lim n .

n→∞d Ax By = ⇒n→∞By =Ax

Now, we have

lim n lim n lim n lim n

n→∞Ax =n→∞Tx =n→∞By =n→∞Sy = Ax

Assume A X

( )

⊆S X

( )

, then there exits v∈X such that Ax=Sv.

We claim that Bv=Sv.

Now from condition (26)

(

_n,

)

(

_n,

)

d Ax Bv ≤kM x v (32)

where

(

n,

)

max

{

(

n,

) (

, n, n

) (

, ,

) (

, n,

) (

, , n

)

}

M x v = d Tx Sv d Tx Ax d Bv Sv d Tx Bv d Sv Ax

Since

(

)

(

)

(

)

lim n, , ,

n→∞d Tx Bv =d Ax Bv =d Sv Bv

(

)

(

)

(

)

lim n, lim n, n lim , n 0

n→∞d Tx Sv =n→∞d Tx Ax =n→∞d Sv Ax =

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

(

,

)

(

,

)

,

d Sv Bv ≤k d Sv Bv (33)

which is a contradiction. Hence, d Sv Bv

(

,

)

= ⇒0 Sv=Bv. This proves that v is the

coincidence point of of the maps B and S. Hence, Sv=Bv=Ax=w Say

(

)

.

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

BSv=SBv⇒Bw=Sw

Since B X

( )

⊆T X

( )

there exists a point u∈X such that Bv=Tu. We show that

Tu=Au=w

From condition (26)

(

,

)

( )

, ,

d Au Bv ≤k M u v

where

( )

{

(

) (

)

}

(

) (

)

{

}

(

) (

)

{

}

, max , , , , , , , , ,

max , , , , , , , , ,

max , , ,

M u v d Tu Sv d Tu Au d Bv Sv d Tu Bv d Sv Au d Bv Bv d Bv Au d Bv Bv d Bv Bv d Bv Au d Bv Bv d Bv Au

= = =

Hence

(

,

)

max

{

(

,

) (

, ,

)

}

d Au Bv ≤k d Bv Bv d Bv Au

Since

(

,

)

2

(

,

)

(15)

So if max

{

d Bv Bv d Bv Au

(

,

) (

, ,

)

}

=d Bv Bv

(

,

)

or d Bv Au

(

,

)

, we get the contradic-

tion for both cases.

Therefore, d Au Bv

(

,

)

= ⇒0 Au=Bv.

Au Bv Tu w

∴ = = =

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

(

A T,

)

is weakly compatible so,

ATu=TAu⇒Aw=Tw

We show that Aw=w.

From condition (26)

(

,

)

=

(

,

)

(

,

)

,

d Aw w d Aw Bv ≤kM w v

where

(

) (

)

{

}

(

) (

)

{

}

(

) (

)

{

}

( , ) max , , , , , , , , ,

max , , , , , , , , ,

max , , , , ,

M w v d Tw Sv d Tw Aw d Bv Sv d Tw Bv d Sv Aw

d Aw w d Aw Aw d w w d Aw w d w Aw

d Aw w d Aw Aw d w w

= = =

Since

(

,

)

2

(

,

)

and

(

,

)

2

(

,

)

d Aw Aw ≤ d Aw w d w w ≤ d Aw w

So if max

{

d Aw w d Aw Aw d w w

(

,

) (

, ,

) (

, ,

)

}

=d Aw w

(

,

)

or d Aw Aw

(

,

)

or

(

,

)

d w w we have

(

,

)

(

,

)

or 2

(

,

)

d Aw w ≤k d Aw w ≤ kd Aw w

which give contradictions for all three cases.

Hence, d Aw w

(

,

)

= ⇒0 Aw=w. Similarly, we obtain Bw=w.

Aw Bw Sw Tw w

∴ = = = = _{. Hence, w is the common fixed point of four mappings}

, ,

A B S and T.

Uniqueness:

Let z

(

≠w

)

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

(

,

)

(

,

)

(

,

)

d w z =d Aw Bz ≤kM w z (34)

where

(

)

{

(

) (

)

}

(

) (

) ( ) (

) (

)

{

}

(

) (

) ( )

{

}

, max , , , , , , , , ,

max , , , , , , , , ,

max , , , , ,

M w z d Tw Sz d Tw Aw d Bz Sz d Tw Bz d Sz Aw

d w z d w w d z z d w z d z w

d w z d w w d z z

= = =

Since

(

,

)

2

(

,

)

and

( )

, 2

(

,

)

d w w ≤ d w z d z z ≤ d z w

So if max

{

d w z

(

,

) (

,d w w d z z,

) ( )

, ,

}

=d w z

(

,

)

or d w w

(

,

)

or d z z

( )

, we have

(

,

)

(

,

)

(

,

)

(

,

)

d w z =d Aw Bz ≤kM w z ≤kd w z

or

(

,

)

2

(

,

)

(16)

which give contradictions for all three cases.

Hence, d w z

(

,

)

= ⇒ =0 w z. This establishes the uniqueness of the common fixed

point.

Now, we have the following corollaries:

( ) ( )

,

( )

A X B X ⊆S X

(

)

( )

1

, , , 0,

2

d Ax By ≤k M x y k∈_{ }_

where

( )

, max

{

(

,

) (

, ,

) (

, ,

) (

, ,

) (

, ,

)

}

M x y = d Sx Sy d Sx Ax d By Sy d Sx By d Sy Ax

1) The pairs

(

A S,

)

or

(

B S,

)

2) The pairs

(

A S,

)

and

(

B S,

)

Then

( )

and

( )

A X ⊆S X A X ⊆T X

(

)

( )

1

, , , 0,

2

d Ax Ay ≤k M x y k∈_{ }_

where

( )

, max

{

(

,

) (

, ,

) (

, ,

) (

, ,

) (

, ,

)

}

M x y = d Tx Sy d Tx Ax d Ay Sy d Tx Ay d Sy Ax

1) The pair

(

A T,

)

and

(

A S,

)

2) The pairs

(

A T,

)

and

(

A S,

)

Then

( )

A X ⊆S X

(

)

( )

1

, , , 0,

2

d Ax Ay ≤k M x y k∈_{ }_

where

( )

, max

{

(

,

) (

, ,

) (

, ,

) (

, ,

) (

, ,

)

}

M x y = d Sx Sy d Sx Ax d Ay Sy d Sx Ay d Sy Ax

(17)

2) The pair

(

A S,

)

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.

References

[1] Matthews, S.G. (1986) Metric Domains for Completeness. Technical Report 76, Ph.D. The-sis, Department of Computer Science, University of Warwick, Coventry.

[2] Hitzler, P. and Seda, A.K. (2000) Dislocated Topologies. Journal of Electrical Engineering, 51, 3-7.

[3] Kumari, P.S., Zoto, K. and Panthi, D. (2015) d-Neighborhood System and Generalized F- Contraction in Dislocated Metric Space. SpringerPlus, 4, 1-10.

http://dx.doi.org/10.1186/s40064-015-1095-3

[4] Kumari, P.S. (2012) Common Fixed Point Theorems on Weakly Compatible Maps on Dis-located Metric Spaces. Mathematical Sciences, 6, 71.

http://dx.doi.org/10.1186/2251-7456-6-71

[5] Kumari, P.S. and Panthi, D. (2015) Cyclic Contractions and Fixed Point Theorems on Var-ious Generating Spaces. Fixed Point Theory and Applications, 2016, 28.

http://dx.doi.org/10.1186/s13663-016-0521-8

[6] Kumari, P.S., Ramana, C.V., Zoto, K. and Panthi, D. (2015) Fixed Point Theorems and Ge-neralizations of Dislocated Metric Spaces. Indian Journal of Science and Technology, 8, 154-158. http://dx.doi.org/10.17485/ijst/2015/v8iS3/62247

[7] Panthi, D. (2015) Common Fixed Point Theorems for Compatible Mappings in Dislocated Metric Space. International Journal of Mathematical Analysis, 9, 2235-2242.

http://dx.doi.org/10.12988/ijma.2015.57177

[8] Panthi, D. (2014) Fixed Point Results in Cyclic Contractions of Generalized Dislocated Me-tric Spaces. Annals of Pure and Applied Mathematics, 5, 192-197.

[9] Panthi, D. and Kumari, P.S. (2016) Some Integral Type Fixed Point Theorems in Dislocated Metric Space. American Journal of Computational Mathematics, 6, 88-97.

http://dx.doi.org/10.4236/ajcm.2016.62010

[10] Panthi, D. and Kumari, P.S. (2015) Common Fixed Point Theorems for Mappings of Com-patible Type(A) in Dislocated Metric Space. Nepal Journal of Science and Technology, 16, 79-86. http://dx.doi.org/10.3126/njst.v16i1.14360

[11] Panthi, D. and Jha, K. (2012) A Common Fixed Point of Weakly Compatible Mappings in Dislocated Metric Space. Kathmandu University Journal of Science, Engineering and Tech- nology, 8, 25-30.

[12] Sarma, I.R., Rao, J.M., Kumari, P.S. and Panthi, D. (2014) Convergence Axioms on Dislo-cated Symmetric Spaces. Abstract and Applied Analysis, 2014, Article ID: 745031. [13] Jungck, G. and Rhoades, B.E. (1998) Fixed Points for Set Valued Functions without

Conti-nuity. Indian Journal of Pure and Applied Mathematics, 29, 227-238.

[14] Amri, M. and El Moutawakil, D. (2002) Some New Common Fixed Point Theorems under Strict Contractive Conditions. Journal of Mathematical Analysis and Applications, 270, 181-188. http://dx.doi.org/10.1016/S0022-247X(02)00059-8

(18)

Com-patible Maps in Fuzzy Metric Spaces.Journal of Applied Mathematics, 2011, 1-14. http://dx.doi.org/10.1155/2011/637958

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