Vol 8, No 2 (2017)

(1)

Available online throug

ISSN 2229 – 5046

COMMON FIXED POINT THEOREMS

FOR FOUR SELF MAPS ON A MULTIPLICATIVE METRIC SPACE

K. P. R. SASTRY

1

_{, K. K. M. SARMA}

2

_{, S. LAKSHMANA RAO}

3*

1

_{8-28-8/1, Tamil Street, Chinna Waltair Visakhapatnam - 530 017, India.}

2

_{Department of Mathematics, Andhra University, Visakhapatnam - 30 003, India.}

3

_{Department of Mathematics, Andhra University, Visakhapatnam - 530 003, India.}

(Received On: 26-10-16; Revised & Accepted On: 27-02-17)

ABSTRACT

I

n this paper we improve and extend theorem 3.1 of Khan and Imdad[9] replacing

λ

by

ϕ

,

where

+

1 : R

0,

2 ϕ

→



_{ }



_

is an increasing and continuous function. We also show that the example 4.1 mentioned in Khan

and Imdad does not illustrate theorem 3.1 of Khan and Imdad. However an example to illustrate theorem 3.1 of Khan and Imdad is given.

Key words: Multiplicative metric space, multiplicative contraction, Coincidence point, Coincidentally Commuting

mappings, Common fixed point.

Mathematics Subject Classification (2010): 54H25, 47H10.

1. INTRODUCTION AND PRELIMINARIES

The study of fixed points of mappings satisfying certain contractions has many applications and has been at the center of various research activities. In the past years many authors generalized Banach's fixed point theorem in various spaces such as Quasi metric space, Fuzzy metric space, Z-metric space, Partial metric space, probabilistic metric space and generalized metric spaces. On the other hand, in 2008 Bashirov et al. [1] defined a new distance so called a multiplicative distance by using the concept of multiplicative absolute value. After than, in 2012, by using the same idea of multiplicative distance, M.Ozavsar and A.C.Civikel [7] investigate multiplicative metric space and introduced the concept of a multiplicative contraction mappings and proved some fixed point theorems for multiplicative contraction, such mappings on a complete multiplicative metric spaces. In 2014, He et al. [12] proved a Common fixed point theorem for four self mappings in complete multiplicative metric space.

Recently motivated by the concept of compatible mappings, compatible mapping of type(A), and R-Weak commutativity given by G.Jungck et al. [4,5,6], B.C.Dhage [2] termed a pair of self mappings to be Coincidentally Commuting(Weakly commuting), and proved some fixed point theorems for these mappings.

In this paper we improve the result 3.1 of Q.H.Khan and M.Imdad in [9]. Also the example 4.1 mentioned in [9] does not illustrate the result 3.1 of Q.H.Khan and M.Imdad [9], however, we gave an example to illustrates the result 3.1 of Q.H.Khan and M.Imdad [9].

The following one way implications are obviously true but their converse are not true.

Commuting maps

⇒

Weakly Commuting maps

⇒

Compatible maps

⇒

Coincidentally Commuting maps.

Corresponding Author: S. Lakshmana Rao

3*

(2)

Definition 1.1 (A.E.Bashirov, E.M.Kurplnara, A.Ozyapici [1]): Let

X

be a nonempty set. A multiplicative metric

is a mapping

d X

:

×

X

→



+ satisfying the following conditions: (i)

d x y

( , )

≥

1

for all

x y

,

∈

X

and

d x y

( , )

=

1

, if and only if

x

=

y

.

(ii)

d x y

( , )

=

d y x

( , )

for all

x y

,

∈

X

.

(iii)

d x y

( , )

≤

d x z d z y

( , ). ( , )

for all

x y z

, ,

∈

X

. (Multiplicative triangle inequality)

Definition 1.2 (M.Ozavsar, A.C.Civikel [7]): Let

( , )

X d

be a multiplicative metric space, { }

x

_n be a sequence in

X

and

x

∈

X

. the sequence { }

x

_n is called a multiplicative Cauchy sequence if, for each

ε

>

1

, there exists

N

∈

N

such that

d x x

(

_n

,

_m

)

<

ε

, for all

m n

,

≥

N

.

Definition 1.3 (M.Ozavsar, A.C.Civikel [7]): A multiplicative metric space

X

is complete, if every multiplicative Cauchy sequence in it is multiplicative convergent to

x

∈

X

.

Definition 1.4 (F.Gu, L.M.Cui, Y.H.Wu [3]): Suppose that

S

and

T

are two self mappings of a multiplicative

metric space

( , )

X d

;

S

,

T

are called commutative mappings if,

STx

=

TSx

for all

x

∈

X

.

S

and

T

metric space

( , )

X d

; The pair (

S

,

T

) are called weak commutative mappings if

d STx TSx

(

,

)

≤

d Tx Sx

(

,

)

for all

x

∈

X

.

S

and

T

metric space

( , )

X d

; A point

x

∈

X

is said to be a coincident point of

S

and

T

if

d Sx Tx

(

,

)

=

1

.

Definition 1.7 (M.Ozavsar, A.C.Civikel [7]): Let

( , )

X d

be a multiplicative metric space. A mapping

f

:

X

→

X

is called a multiplicative contraction if there exists a real constant

λ

∈

[0,1)

such that

d fx fy

(

,

)

≤

d x y

( , )

λ for all

,

x y

∈

X

.

Definition 1.8 (M.Ozavsar, A.C.Civikel [7])(Multiplicative continuity): Let

( ,

X d

_X

)

and

( ,

Y d

_Y

)

be two multiplicative metric spaces and

f

:

X

→

Y

be a function. If for every

ε

>

1

, there exists

δ

>

1

such that

(

( ))

( ( ))

f B x

_δ

⊂

B

_ε

f x

, then we call

f

multiplicative continuous at

x

∈

X

.

Definition 1.9 (M.Ozavsar, A.C.Civikel [7])(Semi-Multiplicative continuity): Let

( , )

X d

be a multiplicative

metric space, and

( , )

Y d

be a metric space and

f

:

X

→

Y

be a function. If for all

ε

>

0

, there exists

δ

>

1

such

that

f B x

(

_δ

( ))

⊂

B

_ε

( ( ))

f x

, then we call

f

Semi multiplicative continuous at

x

∈

X

. Similarly a function

:

g Y

→

X

is also said to be semi multiplicative continuous at

y

∈

Y

if it satisfies a similar requirement.

Definition 1.10 (M.Ozavsar, A.C.Civikel [7])(Multiplicative convergence): Let

( , )

X d

be a multiplicative metric

space,

{ }

x

_n be a sequence in

X

and

x

∈

X

. If for every multiplicative open ball

( )

{ / ( , )

},

1 B x

_ε

=

y d x y

<

ε ε

>

there exists a natural number

N

such that

n

≥

N

,

x

_n

∈

B x

_ε

( )

. The sequence

{ }

x

_n is said to be multiplicative converging to

x

, denoted by

x

_n

→

x

(

n

→ ∞

)

.

Lemma 1.11 (M.Ozavsar, A.C.Civikel [7]): Let

( , )

X d

be a multiplicative metric space, { }

x

_n be a sequence in

X

and

x

∈

X

then

x

_n

→

x

(

n

→ ∞

)

if and only if

d x x

( , )

_n

→

1 (

n

→ ∞

)

(3)

( ,

X d

_X

)

and

( ,

Y d

_Y

)

be two multiplicative metric spaces and

:

f

X

→

Y

be a mapping and

{ }

x

_n be any sequence in

X

. Then

f

is multiplicative continuous at the point

x

∈

X

if and only if

f x

(

_n

)

→

f x

( )

for every sequence

{ }

x

_n with

x

_n

→

x

(

n

→ ∞

)

.

Theorem 1.14 (M.Ozavsar, A.C.Civikel [7]): Let

( , )

X d

be a multiplicative metric space and

{ }

x

_n be a sequence in

X

. If the sequence is multiplicative convergent, then it is multiplicative Cauchy sequence.

( , )

X d

be a multiplicative metric space and { }

x

_n be a sequence in

X

. Then { }

x

_n is multiplicative Cauchy sequence if and only if

d x

(

_m

,

x

_n

)

→

1 ( ,

m n

→ ∞

)

Theorem 1.16 (M.Ozavsar, A.C.Civikel [7]): Let

( , )

X d

be a multiplicative metric space and let

f

:

X

→

X

be a

multiplicative contraction. If

( , )

X d

is complete, then

f

has a unique fixed point.

In 2014, X.He, M.Song and D.Chen [12] proved the fixed point theorem for two pairs of of weak commutative mappings on a multiplicative metric space.

Theorem 1.17 (X.He, M.Song, D.Chen [12]): Let

S T A

, ,

and

B

be self mappings of a complete multiplicative

metric space

X

, satisfying the following conditions: (i)

S X

(

)

⊂

B X T X

(

),

(

)

⊂

A X

(

),

(ii)

A

and

S

are weak commutative,

B

and

T

also are weak commutative, (iii) One of

S T A

, ,

and

B

is continuous,

(iv)

d Sx Ty

(

,

) {

≤

max d Ax By d Ax Sx d Ty By d Sx By d Ax Ty

{ (

,

), (

,

), (

,

), (

,

), (

,

)}}

λ

(0, )

1

2 λ

∈

,

∀

x y

∈

X

.

Then

S T A

, ,

and

B

have a unique common fixed point.

Now we given below the examples of multiplicative metrics.

Example 1.18 (M.Ozavsar, A.C.Civikel [7]): Let

X

= ∞

[1, )

, define

d X

:

×

X

→



+ by _{d x y}_{( , )} _max x y_,

y x

 

= _ _

 

then

d

is a multiplicative metric on

X

.

Example 1.19 (M.Sarwar and R.Badsahah-e [8]): Let

( , )

X d

be a metric space, let a > 1, define

:

d

′

X

× →

X



+ by

( , )

1 if

.

0 if

y.

d x y

x

y

d x y

a

x

=



′

=

_{= }

≠



for all

x y

,

∈

X

. Then d' is a multiplicative metric

on

X

and

( ,

X d

′

)

is known as the discrete multiplicative metric space .

Example 1.20: (M.Sarwar and R.Badsahah-e [8]): Let

d

:

 

× → ∞

[1, )

be defined as

d x y

( , )

=

a

x y−

,

where

,

x y

∈



and a>1. Then

d

is a multiplicative metric and

( , )



d

is a multiplicative metric space. We may call it usual multiplicative metric space.

Recently, Q.H.Khan and M.Imdad improved theorem 1.17 by relaxing the continuity requirement of the maps completely and to reduce the commutativity requirement of the maps to coincidence point only.

Theorem 1.21: (Q.H.Khan, M.Imdad [9]). Let

A B I

, ,

and

J

be self mappings of a multiplicative metric space

( , )

X d

, satisfying

A X

(

)

⊂

J X

(

)

,

B X

(

)

⊂

I X

(

)

and

(

,

)

{ ( ,

), ( ,

), (

,

), (

,

), ( ,

)}

d Ax By

≤

max d Ix Jy d Ix Ax d By Jy d Ax Jy d Ix By

λ

(0, ) , ,

1 .

2 x y z

X

λ

∈

∀

∈

If one of

A X

(

), (

B X

), (

I X

), (

J X

)

is a complete subspace of

X

, then the following conclusions hold.

(i)

( , )

A I

has a coincidence point;

(4)

Further if the pairs

( , )

A I

and

( , )

B J

are coincidentally commuting, then

A I

,

B

and

J

2. MAIN RESULT

In this section we mainly improve and extend the theorem 1.21, in [9], replacing

λ

by

ϕ

, where

: R

₊

0,

1

2 ϕ

→





_

 

is an increasing and continuous function.

Theorem 2.1: Let

A B I

, ,

and

J

be self mappings of a multiplicative metric space

( , )

X d

, satisfying

(

)

(

)

A X

⊂

J X

,

B X

(

)

⊂

I X

(

)

and define

: R

₊

0,

1

2 ϕ

→



_{ }



_

such that

ϕ

is continuous and increasing and

( ( , ))

(

,

)

[

( , )]

M x y

d Ax By

≤

M x y

ϕ (2.1.1) where

M x y

( , )

=

max d Ix Jy d Ix Ax d By Jy d Ax Jy d Ix By

{ ( ,

), ( ,

), (

,

), (

,

), ( ,

)}

∀

x y z

, ,

∈

X

.

If one of

A X

(

), (

B X

), (

I X

), (

J X

)

X

, then the following conclusions hold.

(i)

( , )

A I

(ii)

( , )

B J

Further if the pairs

( , )

A I

and

( , )

B J

are coincidentally commuting, then

A B I

, ,

and

J

Proof: Let

x

₀

∈

X

, since

A X

(

)

⊂

J X

(

)

, then

∃

a point

x

₁

∈

X

such that

Ax

₀

=

Jx

₁.

Also since

B X

(

)

⊂

I X

(

)

,

∃

x

₂

∈

X

such that

Bx

₁

=

Ix

₂.

Using this argument repeatedly, we can construct a sequence

{ }

z

_n in

X

such that

2n 2n 2n 1

z

=

Ax

=

Jx

₊ and

z

₂_n₊₁

=

Bx

₂_n₊₁

=

Ix

₂_n₊₂

Suppose

z

₂_n₊₁

=

z

₂_n for some

n

.

2 1 2 2

(

_n

,

_n

)

d z

₊

z

₊ =

d Bx

(

₂_n₊₁

,

Ax

₂_n₊₂

)

=

d Ax

(

₂_n₊₂

,

Bx

₂_n₊₁

)

2 1 2 2

(

_n

,

_n

)

d z

₊

z

₊ =

d Ax

(

₂_n₊₂

,

Bx

₂_n₊₁

)

≤

( ( 2 2,2 1))

2 2 2 1

{

(

,

)}

M xn xn

n n

M x

x

ϕ + +

+ + (2.1.2)

where

2 2, 2 1

(

n n

)

M x

₊

x

₊

=

m x

a

{ (

d Ix

₂_n₊₂

,

Jx

₂_n₊₁

), (

d Ix

₂_n₊₂

,

Ax

₂_n₊₂

), (

d Bx

₂_n₊₁

,

Jx

₂_n₊₁

), (

d Ax

₂_n₊₂

,

Jx

₂_n₊₁

), (

d Ix

₂_n₊₂

,

Bx

₂_n₊₁

)}

=

max

{ (

d z

2n+1

,

z

2n

), (

d z

2n+1

,

z

2n+2

), (

d z

2n+2

,

z

2n

), (

d z

2n+2

,

z

2n

), (

d z

2n+1

,

z

2n+1

)}

=

max

{1, (

d z

2n+1

,

z

2n+2

)}

=

d z

(

2n+1

,

z

2n+2

)

∴

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

≤

( (2 1,2 2))

2 1 2 2

{ (

,

)}

d zn zn

n n

d z

z

ϕ + +

+ +

<

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

, if

z

₂_n₊₁

≠

z

₂_n₊₂. a contradiction.

2n 1 2n 2

z

₊

z

₊

∴

=

.

Again

d z

(

₂_n₊₂

,

z

₂_n₊₃

)

=

d Ax

(

₂_n₊₂

,

Bx

₂_n₊₃

)

≤

( (2 2, 2 3))

2 2 2 3

{

(

,

)}

M xn xn

n n

M x

x

ϕ + +

+ + _(2.1.3)

where

2 2, 2 3

(

_n _n

)

M x

₊

x

₊

=

m

ax d Ix

{ (

₂_n₊₂

,

Jx

₂_n₊₃

), (

d Ix

₂_n₊₂

,

Ax

₂_n₊₂

), (

d Bx

₂_n₊₃

,

Jx

₂_n₊₃

), (

d Ax

₂_n₊₂

,

Jx

₂_n₊₃

), (

d Ix

₂_n₊₂

,

Bx

₂_n₊₃

)}

=

max

{ (

d z

2n+1

,

z

2n

), (

d z

2n+1

,

z

2n+2

), (

d z

2n+3

,

z

2n+2

), (

d z

2n+2

,

z

2n+2

), (

d z

2n+1

,

z

2n+3

)}

=

max

{1, (

d z

2n+2

,

z

2n+3

)}

=

d z

(

2n+2

,

z

2n+3

)

∴

d z

(

₂_n₊₂

,

z

₂_n₊₃

)

≤

( (2 2,2 3))

2 2 2 3

{ (

,

)}

d zn zn

n n

d z

z

ϕ + +

+ +

(5)

2n 2 2n 3

z

₊

z

₊

∴

=

.

Similarly we get

z

₂_n₊₃

=

z

₂_n₊₄ = ....

2 2 1 2 2 2 3

...

z

_n

z

_n₊

z

_n₊

z

_n₊

∴ =

=

(2.1.4)

Since

z

₂_n

=

Ax

₂_n

=

Jx

₂_n₊₁

2n 1 2n 1 2n 2

z

₊

=

Bx

₊

=

Ix

₊

Also

z

₂_n₊₂

=

Ax

₂_n₊₂

=

Jx

₂_n₊₃

...

∴

Bx

₂_n₊₁

=

Jx

₂_n₊₁

⇒

BJx

₂_n₊₁

=

JBx

₂_n₊₁

( ( , )



B J is coincidently commuting

)

⇒

Bz

₂_n

=

Jz

₂_n i.e.,

Bz

=

Jz

and

Ax

₂_n₊₂

=

Ix

₂_n₊₂

⇒

AIx

₂_n₊₂

=

IAx

₂_n₊₂

( ( , )



A I is coincidently commuting

)

⇒

Az

2n

=

Iz

2n i.e.,

Az

=

Iz

Az

Iz

Bz

Jz

∴

= =

=

Now

d Az B AZ

(

, (

))

≤

{

M z Az

( ,

}

ϕ(M z Az( , ))

( ,

)

M z Az

=

max d Iz J Az

{ ( , (

)), ( ,

d Iz Az d B Az J Az

), ( (

), (

)), (

d Az J Az

, (

)), ( , (

d Iz B Az

))}

=

max d Az B Az

{ (

, (

)), (

d Az Az d B Az B Az

,

), ( (

), (

)), (

d Az B Az

, (

)), (

d Az B Az

, (

))}

=

d Az B AZ

(

, (

))

(

, (

))

d Az B AZ

∴

≤

( ( , ( )))

{ (

d Az B Az

, (

))}

ϕ d Az B Az

<

d Az B Az

(

, (

))

, if

Az

≠

BAz

.a contradiction.

∴

Az

=

B Az

(

)

.

∴

Az

is a fixed point of

B

.

Also

d A Bz BZ

( (

),

)

≤

{

M Bz z

(

, }

ϕ(M Bz z( , ))

(

, )

M Bz z

=

max d I Bz Jz d I Bz A Bz

{ ( (

),

), ( (

), (

)), (

d Bz Jz d A Bz Jz d I Bz Bz

,

), ( (

),

), ( (

),

)}

=

max d A Bz Bz d A Bz A Bz

{ ( (

),

), ( (

), (

)), (

d Bz Bz d A Bz Bz d A Bz Bz

,

), ( (

),

), ( (

),

)}

=

d A Bz Bz

( (

),

)

∴

d A Bz Bz

( (

),

)

≤

{ ( (

d A Bz Bz

),

)}

ϕ( ( (d A Bz Bz), ))

<

d A Bz Bz

( (

),

)

, if

ABz

≠

Bz

.a contradiction.

∴

A Bz

(

)

=

Bz

.

∴

Bz

is a fixed point of

A

.

Similarly we can proved

I

and

J

.

∴

A B I

, ,

and

J

have unique common fixed point.

Suppose

z

₂_n₊₁

≠

z

₂_n for every

n

.

2 1 2 2

(

_n

,

_n

)

d z

₊

z

₊ =

d Bx

(

₂_n₊₁

,

Ax

₂_n₊₂

)

=

d Ax

(

₂_n₊₂

,

Bx

₂_n₊₁

)

2 1 2 2

(

_n

,

_n

)

d z

₊

z

₊ =

d Ax

(

₂_n₊₂

,

Bx

₂_n₊₁

)

≤

( ( 2 2,2 1))

2 2 2 1

{

(

,

)}

M xn xn

n n

M x

x

ϕ + +

+ + (2.1.5)

Where

2 2 2 1

(

_n

,

_n

)

max

M x

₊

x

₊

=

{ (

d Ix

₂_n₊₂

,

Jx

₂_n₊₁

), (

d Ix

₂_n₊₂

,

Ax

₂_n₊₂

), (

d Bx

₂_n₊₁

,

Jx

₂_n₊₁

), (

d Ax

₂_n₊₂

,

Jx

₂_n₊₁

),

d Ix

(

2n+2

,

Bx

2n+1

)}

=

max

{ (

d z

₂_n₊₁

,

z

₂_n

), (

d z

₂_n₊₁

,

z

₂_n₊₂

), (

d z

₂_n₊₁

,

z

₂_n

), (

d z

₂_n₊₂

,

z

₂_n

), (

d z

₂_n₊₁

,

z

₂_n₊₁

)}

(6)

From (2.1.5)

2 1 2 2

(

n

,

n

)

{

d z

₊

z

₊

≤

max

{ { (2 ,2 1), ( 2 1,2 2), ( 2 ,2 2)}}

2 2 1 2 1 2 2 2 2 2

{ (

,

), (

,

), (

,

)}}

ma xd znzn d zn zn d znzn

n n n n n n

d z

z

d z

z

d z

z

ϕ + + + +

+ + + +

Claim:

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

<

( (

d z

₂_n

,

z

₂_n₊₁

)

( )

i

If

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

is maximum of

d z

(

₂_n

,

z

₂_n₊₁

)

,

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

and

d z

(

₂_n

,

z

₂_n₊₂

)

.

then

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

≤

( (2 1,2 2))

2 1 2 2

[

,

]

d zn zn

n n

d z

z

φ + +

+ +

<

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

. a contradiction.

Hence

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

is not a maximum.

( )

ii

If

d z

(

₂_n

,

z

₂_n₊₁

)

is maximum of

( (

d z

₂_n

,

z

₂_n₊₁

)

,

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

and

d z

(

₂_n

,

z

₂_n₊₂

)

.then

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

≤

( ( 2 ,2 1))

2 2 1

[ (

,

)]

d znzn

n n

d z

z

ϕ +

+ <

( (

d z

2n

,

z

2n+1

)

. Hence the claim established.

( )

iii

if

d z

(

₂_n

,

z

₂_n₊₂

)

is maximum of

( (

d z

₂_n

,

z

₂_n₊₁

)

,

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

and

d z

(

₂_n

,

z

₂_n₊₂

)

.then

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

≤

( ( 2 ,2 2))

2 2 2

[ (

,

)]

d zn zn

n n

d z

z

ϕ +

+ .

We have

( (

d z

₂_n

,

z

₂_n₊₂

)

≤

d z

(

₂_n

,

z

₂_n₊₁

). (

d z

₂_n₊₁

,

z

₂_n₊₂

)

Now

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

≤

( (2 ,2 2))

2 2 2

[ (

,

)]

d znzn

n n

d z

z

ϕ +

+

≤

( ( 2 ,2 2))

2 2 1 2 1 2 2

[ (

,

). (

,

)]

d znzn

n n n n

d z

z

d z

z

ϕ +

+ + +

≤

( 2 ,2 2)

2 1 2 2 2 1 2 2

[ (

,

). (

,

)]

d znzn

n n n n

d z

z

d z

z

ϕ +

+ + + +

(

Suppose d z

(

2n

,

z

2n+1

)

≤

(

d z

2n+1

,

z

2n+2

))

∴

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

≤

2. ( 2 ,2 2)

2 1 2 2

[ (

,

)]

d znzn

n n

d z

z

ϕ +

+ +

<

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

. a contradiction.

∴

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

<

d z

(

₂_n

,

z

₂_n₊₁

)

(2.1.7)

Again

d z

(

₂_n₊₂

,

z

₂_n₊₁

)

=

d Ax

(

₂_n₊₂

,

Bx

₂_n₊₁

)

2 2 2 1

(

_n

,

_n

)

d z

₊

z

₊

≤

( ( 2 2,2 1))

2 2 2 1

{

(

,

)}

M xn xn

n n

M x

x

ϕ + +

+ + (2.1.8)

Where

2 2 2 1

(

_n

,

_n

)

M x

₊

x

₊ =

max

{ (

d Ix

₂_n₊₂

,

Jx

₂_n₊₁

), (

d Ix

₂_n₊₂

,

Ax

₂_n₊₂

), (

d Bx

₂_n₊₁

,

Jx

₂_n₊₁

), (

d Ax

₂_n₊₂

,

Jx

₂_n₊₁

), (

d Ix

₂_n₊₂

,

Bx

₂_n₊₁

)}

=

max

{ (

d z

₂_n₊₁

,

z

₂_n

), (

d z

₂_n₊₁

,

z

₂_n₊₂

), (

d z

₂_n₊₁

,

z

₂_n

), (

d z

₂_n₊₂

,

z

₂_n

), (

d z

₂_n₊₁

,

z

₂_n₊₁

)}

=

max

{ (

d z

2n+1

,

z

2n

), (

d z

2n+1

,

z

2n+2

), (

d z

2n

,

z

2n+2

)}

From (2.1.8)

d z

(

₂_n₊₂

,

z

₂_n₊₁

)

≤

{

max

{ { (2 1,2 ), (2 1,2 2), (2 ,2 2)}}

2 1 2 2 1 2 2 2 2 2

{ (

,

), (

,

), (

,

)}}

ma xd zn zn d zn zn d znzn

n n n n n n

d z

z

d z

z

d z

z

ϕ + + + +

+ + + +

(2.1.9)

Claim:

d z

(

₂_n₊₂

,

z

₂_n₊₁

)

<

( (

d z

₂_n₊₁

,

z

₂_n

)

( )

i

If

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

is maximum of

d z

(

₂_n₊₁

,

z

₂_n

)

,

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

and

d z

(

₂_n

,

z

₂_n₊₂

)

. then

2 2 2 1

(

_n

,

_n

)

d z

₊

z

₊

≤

( ( 2 2,2 1))

2 2 2 1

[

,

]

d zn zn

n n

d z

z

ϕ + +

+ +

<

d z

(

₂_n₊₂

,

z

₂_n₊₁

)

. a contradiction.

Hence

d z

(

₂_n₊₂

,

z

₂_n₊₁

)

is not a maximum.

( )

ii

If

d z

(

₂_n₊₁

,

z

₂_n

)

is maximum of

( (

d z

₂_n₊₁

,

z

₂_n

)

,

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

and

d z

(

₂_n

,

z

₂_n₊₂

)

. then

2 2 2 1

(

_n

,

_n

)

d z

₊

z

₊

≤

( ( 2 1,2 ))

2 1 2

[ (

,

)]

d zn zn

n n

d z

z

ϕ +

+ <

( (

d z

2n+1

,

z

2n

)

.

(7)

( )

iii

if

d z

(

₂_n

,

z

₂_n₊₂

)

is maximum of

( (

d z

₂_n₊₁

,

z

₂_n

)

,

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

and

d z

(

₂_n

,

z

₂_n₊₂

)

. Then

d z

(

2n+2

,

z

2n+1

)

≤

2 2 2

( ( , ))

2 2 2

[ (

,

)]

d zn zn

n n

d z

z

ϕ +

+ .

We have

( (

d z

₂_n

,

z

₂_n₊₂

)

≤

d z

(

₂_n

,

z

₂_n₊₁

). (

d z

₂_n₊₁

,

z

₂_n₊₂

)

Now

d z

(

₂_n₊₂

,

z

₂_n₊₁

)

≤

( (2 ,2 2))

2 2 2

[ (

,

)]

d znzn

n n

d z

z

ϕ +

+

≤

( (2 ,2 2))

2 2 1 2 1 2 2

[ (

,

). (

,

)]

d znzn

n n n n

d z

z

d z

z

ϕ +

+ + +

≤

( 2 ,2 2)

2 1 2 2 2 1 2 2

[ (

,

). (

,

)]

d zn zn

n n n n

d z

z

d z

z

ϕ +

+ + + +

(

Suppose d z

(

2n

,

z

2n+1

)

≤

(

d z

2n+1

,

z

2n+2

))

∴

d z

(

₂_n₊₂

,

z

₂_n₊₁

)

≤

2. ( 2 ,2 2)

2 1 2 2

[ (

,

)]

d znzn

n n

d z

z

ϕ +

+ +

<

d z

(

2n+1

,

z

2n+2

)

. a contradiction.

∴

d z

(

₂_n₊₂

,

z

₂_n₊₁

)

<

d z

(

₂_n₊₁

,

z

₂_n

)

(2.1.10)

From (2.1.7) and (2.1.10)

d z

(

_n₊₁

,

z

_n

)

<

d z z

( ,

_n _n₋₁

)

for

n

=

1, 2, 3...

∴

d z

(

_n₊₁

,

z

_n₊₂

)

<

d z z

( ,

_n _n₊₁

)

for

n

=

1,2,3... (2.1.11)

∴

{ ( ,

d z z

_n _n₊₁

)}

is a strictly decreasing sequence of positive numbers.

Suppose

d z

(

_n₊₁

,

z

_n₊₂

)

→ ≥

r

1

We show that

r

=

1

.

From (2.1.6) and (2.1.9)

2 2 2 1

(

_n

,

_n

)

d z

₊

z

₊

≤

{

max

{ { (2 ,2 1), (2 1,2 2), (2 ,2 2)}}

2 2 1 2 1 2 2 2 2 2

{ (

,

), (

,

), (

,

)}}

n n n n n n

d z

z

d z

z

d z

z

ϕ + + + +

+ + + +

As

n

→ ∞

,

r

≤

{

max

{ , , }}

r r r

2 ϕ{ma xr r r{ , , }}2

r

≤

{ }

r

2 ϕ(r2) =

2

2. (r )

r

φ <

r

, if

r

>

1 ,

a contradiction

.

∴

r

=

1. ∴

lim (

₂_n ₂

,

₂_n ₁

)

n→∞

d z

+

z

+

= 1. i.e.,

lim (

₂_n ₁

,

₂_n ₂

)

n→∞

d z

+

z

+

= 1.

Now we show that { }

z

_n is a Cauchy sequence.

Given

ε

>

1,

∃

a natural number

N

such that

n

≥

N

( ,

_n _{n k}

)

d z z

₊

≤

d z z

( ,

_n _n₊₁

). (

d z

_n₊₁

,

z

_n₊₂

)... (

d z

_{n k}_{+ −}₁

,

z

_{n k}₊

)

(2.1.12)

We have

2 1 2 2

(

_n

,

_n

)

d z

₊

z

₊

≤

{

max

{ { (2 ,2 1), ( 2 1,2 2), ( 2 ,2 2)}}

2 2 1 2 1 2 2 2 2 2

{ (

,

), (

,

), (

,

)}}

n n n n n n

d z

z

d z

z

d z

z

ϕ + + + +

+ + + +

=

{

max

{ { ( 2 ,2 1), ( 2 ,2 2)}}

2 2 1 2 2 2

{ (

,

), (

,

)}}

ma xd znzn d zn zn

n n n n

d z

z

d z

z

ϕ + +

+ +

(



d is decreasing sequence

)

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

≤

2 [ (2 ,2 1) ]2

2 2 1

[ (

,

) ]

d znzn

n n

d z

z

ϕ +

+

∴

d z

(

₂_n₊₁

,

z

₂_n₊₂

)

≤

2. [ ( 2 ,2 1) )]2

2 2 1

[ (

,

)]

d znzn

n n

d z

z

ϕ +

+

Similarly

d z

(

₂_n

,

z

₂_n₊₁

)

≤

2. [ ( 2 1,2 ) ]2

2 1 2

[ (

,

)]

d zn zn

n n

d z

z

ϕ −

− and

1 2

(

_n

,

_n

)

d z

₊

z

₊

≤

2. [ ( , 1) ]2

1

[ ( ,

)]

d z zn n

n n

d z z

ϕ +

(8)

Put

λ

=

2. [ ( ,

ϕ

d z z

_n _n₊₁

) ].

2

2 3

(

_n

,

_n

)

d z

₊

z

₊

≤

2. [ ( 1, 2) ]2

1 2

[ (

,

)]

d zn zn

n n

d z

z

ϕ + +

+ +

≤

2. [ ( 1, 2) ]2

1

[[ ( ,

)] ]

d zn zn

n n

d z z

λ ϕ + +

+

(



d is decreasing and

ϕ

is increasing

)

≤

[[ ( ,

d z z

n n 1

)] ]

λ λ +

≤

[ ( ,

d z z

_n _n₊₁

)]

λ2

Similarly

d z

(

_n₊₃

,

z

_n₊₄

)

≤

2. [ ( 2, 3) ]2

2 3

[ (

,

)]

d zn zn

n n

d z

z

ϕ + +

+ +

≤

2 2. [ ( 2, 3) ]2

1

[[ ( ,

)] ]

d zn zn

n n

d z z

λ ϕ + +

+

≤

[[ ( ,

d z z

_n _n₊₁

)] ] .

λ λ2

≤

[ ( ,

d z z

_n _n₊₁

)] .

λ3 ... ...

(2.1.12)

From

d z z

( ,

_n _{n k}₊

)

≤

[ ( ,

d z z

_n _n₊₁

)].[ ( ,

d z z

_n _n₊₁

)] .[ ( ,

λ

d z z

_n _n₊₁

)] .[ ( ,

λ2

d z z

_n _n₊₁

)] ...[ ( ,

λ3

d z z

_n _n₊₁

)]

λn k+ −1

2 3 1

1 ... 1

[ ( ,

=

d z z

_n _n₊

)]

+ + +λ λ λ +λn k+ −

1 1

1

[ ( ,

=

d z z

_n _n₊

)]

−λ

<

( )

δ

−λ

<

ε

if

n

≥

N

Given

ε

>

1

, choose

δ

such that

1 1 λ

δ

−

_<

ε

. .,

i e

δ ε

<

1−λ

∴

1

1 < <

δ ε

−λ

∴

1 1 λ

δ

−

_<

ε

∴

d z z

( ,

_n _{n k}₊

)

→

1

, (

n k

,

→ ∞

).

∴

{ }

z

_n is a multiplicative Cauchy sequence.

Now suppose that

I X

( )

X

, then we observe that the subsequence

{

z

₂_n₊₁

}

which is contained in

I X

( )

has a limit point

z

in

I X

( )

.

∴

∃ ∈

u

X

such that

Iu

=

z

.

Now we prove that

Au

=

z

.

Take

x

=

u and y

,

=

x

₂_n₊₁

2 1

(

,

_n

)

d Au Bx

₊

≤

( ( ,2 1))

2 1

{

( ,

)}

M u xn

n

M u x

ϕ +

+

where

M u x

( ,

₂_n₊₁

)

=

max

{ (

d Iu Jx

,

₂_n₊₁

), (

d Iu Au d Bx

,

), (

₂_n₊₁

,

Jx

₂_n₊₁

), (

d Au Jx

,

₂_n₊₁

), (

d Iu Bx

,

₂_n₊₁

)}

=

max

{ ( ,

d z z

2n

), ( ,

d z Au d z

), (

2n+1

,

z

2n

), (

d Au z

,

2n

), ( ,

d z z

2n+1

)}

{ ( ,

d z z

₂_n

), ( ,

d z Au d z

), (

₂_n₊₁

,

z

₂_n

),

2 1

(

,

_n

) {

d Au z

₊

max

∴

≤

( { ( ,2 ), ( , ), (2 1,2 ), ( ,2 ), ( ,2 1)})

2 2 1

(

,

), ( ,

)}}

ma xd z zn d z Au d zn zn d Au zn d z z n

n n

d Au z

d z z

ϕ + +

+

On letting

n

→ ∞

.

(

, ) {

d Au z

≤

max

( { ( , ), ( , ), ( , ), ( , ), ( , )}

{ ( , ), ( ,

d z z d z Au d z z d Au z d z z

), ( , ), (

, ), ( , )}}

ϕ ma xd z z d z Au d z z d Au z d z z

( ( , ))

(

, ) { ( ,

)}

d z Au

d Au z

≤

d z Au

φ

<

d z Au

( ,

)

, if

z

≠

Au

, a contradiction.

∴

Au

=

z

∴

Au

= =

z

Iu

∴

u

is a coincident point of

A

and

I

.

(9)

Now we prove that

Bv

=

z

.

Take

x

=

x

₂_n₊₂ and

y

=

v

2 2

(

_n

,

)

d Ax

₊

Bv

≤

( ( 2 2, ))

2 2

{

, )}

M xn v

n

x

v

ϕ +

+

where

M x

(

₂_n₊₂

, )

v

=

max

{ (

d Ix

₂_n₊₂

,

Jv d Ix

), (

₂_n₊₂

,

Ax

₂_n₊₂

), (

d Bv Jv d Ax

,

), (

₂_n₊₂

,

Jv d Ix

), (

₂_n₊₂

,

Bv

)}

=

max

{ (

d z

2n+1

, ), (

z d z

2n+1

,

z

2n+2

), (

d Bv z d z

, ), (

2n+2

, ), (

z d z

2n+1

,

Bv

)}

2 2

(

_n

,

) {

d z

₊

Bv

max

∴

≤

{ (

d z

₂_n₊₁

, ), (

z d z

₂_n₊₁

,

z

₂_n₊₂

), (

d Bv z

, ),

2 1 2 1 2 2 2 2 2 1

( { ( , ), ( , ), ( , ), ( , ), ( , )}

2 2 2 1

(

, ), (

,

)}}

ma xd zn z d zn zn d Bv z d zn z d zn Bv

n n

d z

z d z

Bv

ϕ + + + + +

+ +

On letting

n

→ ∞

.

( ,

) {

d z Bv

≤

max

( { ( , ), ( , ), ( , ), ( , ), ( , )}

{ ( , ), ( , ), (

d z z d z z d Bv z d z z d z Bv

, ), ( , ), ( ,

)}}

ϕ ma xd z z d z z d Bv z d z z d z Bv

( ( , ))

( ,

) { ( ,

)}

d z Bv

d z Bv

≤

d z Bv

ϕ

<

d z Bv

( ,

)

, if

z

≠

Bv

, a contradiction.

∴

Bv

=

z

∴

Bv

= =

z

Jv

.

∴

v

is a coincident point of

B

and

J

.

∴

Au

=

Iu

=

Bv

=

Jv

=

z

.

Suppose

( , )

A I

is coincidentally commuting. Then

AIu

=

IAu

⇒

Az

=

Iz

Suppose

( , )

B J

is coincidentally commuting. Then

BJv

=

JBv

⇒

Bz

=

Jz

∴

d Au Bz

(

,

)

≤

{

M u z

( , )}

ϕ(M u z( , ))

where

M u z

( , )

=

max

{ (

d Iu Jz d Iu Au d Bz Jz d Au Jz d Iu Bz

,

), (

,

), (

,

), (

,

), (

,

)}

=

max d z Bz d z z d Bz Bz d z Bz d z Bz

{ ( ,

), ( , ), (

,

), ( ,

)}

=

max

{1, ( ,

d z Bz

)}

=

d z Bz

( ,

)

_.

( ,

)

d z Bz

≤

{ ( ,

_{d z Bz}

)}

ϕ( ( ,d z Bz)

<

d z Bz

( ,

)

, if

z

≠

Bz

, a contradiction.

∴

Bz

= =

z

Jz

.

∴

z

is a fixed point of

B

and

J

.

Now

d Az Bz

(

,

)

≤

{

M z z

( , )}

ϕ(M z z( , ))

where

M z z

( , )

=

max d Iz Jz d Iz Az d Bz Jz d Az Jz d Iz Bz

{ ( ,

), ( ,

), (

,

), (

,

), ( ,

)}

=

max d Az Bz d Az Az d Bz Bz d Az Bz d Az Bz

{ (

,

), (

,

), (

,

), (

,

), (

,

)}

=

max

{1, (

d Az Bz

,

)}

=

d Az Bz

(

,

)

_.

∴

d Az Bz

(

,

)

≤

{ (

d Az Bz

,

)}

ϕ( (d Az Bz, ) <

d Az Bz

(

,

)

, if

Az

≠

Bz

, a contradiction.

∴

Az

=

Bz

=

z

.

∴

Az

=

Bz

=

Jz

= =

Iz

z

.

∴

z

is a fixed point of A,B,I and

J

.

Suppose

w

is another fixed point of

A B I

, ,

and

J

.

( , )

d z w

=

d Az Bw

(

,

)

≤

{

M z w

( , )}

ϕ(M z w( , ))

where

M z w

( , )

=

max d Iz Jw d Iz Az d Bw Jw d Az Jw d Iz Bw

{ ( ,

), ( ,

), (

,

), (

,

), ( ,

)}

=

max

{ ( , ), ( , ), ( , ), ( , ), ( , )}

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

=

max

{1, ( , )}

d z w

=

d z w

( , )

.

( , )

d z w

≤

{ ( , )}

d z w

ϕ( ( , )d z w <

d z w

( , )

, if

z

≠

w

, a contradiction.

∴

z

=

w

.