Coupled Common Fixed Point Theorem for Two Maps in Modular Spaces

(1)

Coupled Common Fixed Point

Theorem for Two Maps in Modular Spaces

K.P.R.Rao1, Sk.Sadik2, K.V. Sivaparvati3

Professor, Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Andhra Pradesh, India1 Assistant Professor, Department of Mathematics, Sir CRR College of Engineering, Eluru, Andhra Pradesh, India2 Academic Consultant, Department of Applied Mathematics, Dr.MRAR PG Centre, Nuzvid, Andhra Pradesh, India3

ABSTRACT: In this paper, we obtain a coupled common fixed point theorem in modular spaces. We also give an example to illustrate our main theorem.

KEYWORDS: Modular spaces, Coupled fixed points,  - compatible mappings.

I. INTRODUCTION

The concept of a modular space was intiated by Nakano [5] and was redefined and generalized by Musielak and Orlicz [7]. Since then, several fixed point and common fixed point theorems in the frame work of modular spaces have been investigated. For more details see ([1] – [4], [6], [8], [10] – [20], [22] ). Bhaskar and Lakshmikantham [21] introduced the concept of coupled fixed points and later several authors obtained coupled fixed point and coupled common fixed point theorems in various spaces. Recently Abbas et al. [9] introduced

W

-compatible mappings in cone metric spaces. In this paper, combining these concepts, we obtain a coupled common fixed point theorem for Jungck type mappings in modular spaces. We also give an example to illustrate our main theorem.

Now we give some basic definitions on modular spaces.

Definition 1.1 Let

X

be an arbitrary real or complex vector space. A functional



:

X



[0,



)

is called modular if for any

x

,

y



X

, the following conditions hold:

)

(



₁ (x)=0 if and only if

x

=

0

, )

(



2



(



x

)

=|



|



(

x

)

for every scalar



with ||=1,

)

(



₃



(



x





y

)





(

x

)





(

y

)

whenever







=

1

and ,0. Note that

X

_

=

{

x



X

:



(



x

)



0 as





0}

is called a modular space.

Remark 1.2 Let

X

_ be a modular space. Then

)

(

i

₍ ₎

2 x

x



 

   

 _{for all}



X

x



Proof. ₀ ₍ ₎ ₍₀₎₌ ₍ ₎

2 1 2 1 =

2 x x x

x

   

   

  

 

 

   

 _{, from}

₍

₎

3



and (



1).

)

(

ii

(xy)(2x)(2y) for all

x

,

y



X

_

Proof. ₍₂ ₎ ₍₂ ₎ ₍₂ ₎

2 1 ) (2 2 1 = )

(x y  x y  x  y

  

  

 



(2)

X

_ be a modular space.

)

(

i

The sequence

{

x

_n

}

in

X

_ is called



-convergent to

x



X

_ if and only if

_lim

(

x

n

x

)

=

0 .

n



 



)

(

ii



-Cauchy if and only if

_lim

(

)

=

0

,m n m

n

x



 



.

)

(

iii

A subset

C

of

X

_ is called



-closed if the



-limit of a



-convergent sequence of

C

is still in

C

.

)

(

iv

A subset

C

of

X

_ is called



-complete if any



-Cauchy sequence in

C

is



-convergent and its



-limit belongs to

C

.

)

(

v



is said to satisfy the ₂-condition if



(2

x

_n

)



0

whenever



(

x

_n

)



0

as

n





.

)

(

vi

we say that



has the Fatou property if

(

x

y

)

_lim

inf

(

x

_n

y

)

n







 



for all

y



X

_whenever

0 )

(

x

n



x





as

n





.

X

 be a modular space. We say that

T

:

X





X

 is



-continuous if



(

Tx

n



Tx

)



0

whenever



(

x

_n



x

)



0

as

n





.

X

_ be a modular space and

F

:

X

_



X

_



X

_ and

g

:

X

_



X

_. Then the pair

(

F

,

g

)

is said to be



-compatible if



(

F

(

gx

_n

,

gy

_n

)



g

(

F

(

x

_n

,

y

_n

)))



0

and



(

F

(

gy

_n

,

gx

_n

)



g

(

F

(

y

_n

,

x

_n

,

)))



0

as

n





whenever there exist sequences

{

x

_n

}

and

{

y

_n

}

in

X

_ such that

F

x

y

gx

_n

t

n n n n

=

lim

=

)

,

(

lim

  



and

t

gy

x

y

F

n

n n n n



  



=

lim

=

)

,

(

lim

for some

t

and

t



in

X

_.

X

F

:

X

_



X

_



X

_ and

g

:

X

_



X

_. Then the pair

(

F

,

g

)

is said to be



- weakly compatible if

F

(

gx

,

gy

)

=

g

(

F

(

x

,

y

))

and

F

(

gy

,

gx

)

=

g

(

F

(

y

,

x

))

whenever there exist

x

and

y

in

X

_ such that

F

(

x

,

y

)

=

gx

and

F

(

y

,

x

)

=

gy

.

X

F

:

X

_



X

_



X

_. Then

F

is said to be



-continuous if

))

,

(

)

,

(

lim

=

0 =

))

,

(

)

,

(

lim

F

x

y

F

x

y

F

y

_n

x

_n

F

y

x

n n



  





whenever

lim

(

x

)

=

0 =

lim

(

y

_n

y

)

n n

n



  





.

Definition 1.8 ([9]) Let

X

be a nonempty set. An element

(

x

,

y

)



X



X

is called )

(i a coupled coincidence point of

F

:

X



X



X

and

g

:

X



X

if

gx

=

F

(

x

,

y

)

and

gy

=

F

(

y

,

x

)

.

)

(ii a common coupled fixed point of

F

:

X



X



X

and

g

:

X



X

if

x

=

gx

=

F

(

x

,

y

)

and

) , ( = =gy F y x

y .

(3)

Definition 1.9 ([19])Let

X

be a nonempty set and



:

X

2



X

2



R

be a function.Let

F

:

X



X



X

,

X

g

:



be mappings. Then

F

and

g

are said to be



-admissible if

1 )))

,

(

),

,

(

)),

,

(

),

,

(

((

1 ))

,

(

),

,

((

gx

gy

gu

gv







F

x

y

F

y

x

F

u

v

F

v

u





for all

x

,

y

,

u

,

v



X

.

If

g

=

I

(

I

dentity

map

)

,then the above definition is the concept of Mursaleen et al.[15].

We say that the pair (F,g) is triangular



-admissible if

F

and

g

are



-admissible and if ,

1 )) , ( ), ,

((x₁ y₁ x₂ y₂ 



((x₂,y₂),(x₃,y₃))1((x₁,y₁),(x₃,y₃))1,

x

₁

,

x

₂

,

x

₃

,

y

₁

,

y

₂

,

y

₃



X

.

Now we prove our main result.

II. MAINRESULT

Theorem 2.1 . Let

X

_be a



-complete modular space, where



satisfies the ₂-condition. Let 



X

F

:





and

g

:

X

_



X

_ be mappings satisfying (2.1.1).

F

(

X





X



)



g

(

X



)

,

(2.1.2).



((

gx

,

gy

),

(

gu

,

gv

))



(

F

(

x

,

y

)



F

(

u

,

v

))





M

_xu,,_yv for all

x

,

y

,

u

,

v



X

, where





(0,1)

and

   

    

 

   

    

 

   

 

   

       

  

   

 

   

   

   

  

 



 



2 ) , ( 2

) , ( 2

1

, 2

) , ( 2

1

)), , ( ( )), , ( ( )), , ( (

)), , ( ( ), (

), (

max = , ,

x y F gv u

v F gy

y x F gu v

u F gx

u v F gv v u F gu x y F gy

y x F gx gv gy gu gx

Mxuvy

 



 



(2.1.3).

F

and

g

are



-continuous, (2.1.4). the pair

(

F

,

g

)

is



-compatible,

(2.1.5) (a) ((gx₀,gy₀),(F(x₀,y₀),F(y₀,x₀)))1 and

(2.1.5) (b) ((gy₀,gx₀),(F(y₀,x₀),F(x₀,y₀)))1 for some

x

₀

,

y

₀



X

,

(2.1.6) the pair

(

F

,

g

)

is triangular



-admissible.

Then

F

and

g

have a coupled coincidence point in

X



X

. Further if we assume that

(2.1.7)



((

gx

,

gy

),

(

gu

,

gv

))



1,



((

gy

,

gx

),

(

gv

,

gu

))



1

whenever

(

x

,

y

)

and

(

u

,

v

)

are coupled coincidence points of

F

and

g

,

then

F

and

g

have a unique coupled common fixed point.

Proof. Let

x

₀

,

y

₀



X

_satisfying (2.1.5)(a) and (2.1.5)(b).From

(2.1.1)

, there exist sequences

{

x

_n

}

and

{

y

_n

}

in 

X

such that

)

,

(

=

1 n n

n

F

x

y

gx

_ ,

gy

_n_₁

=

F

(

y

_n

,

x

_n

)

,

n

=

0,1,2,



(1) From (2.1.5)(a), we have

(4)

. (1) from , 1 )) , ( ), , ((

(2.1.6) from 1, ))) , ( ), , ( ( )), , ( ), , ( ((

(1) from , 1 )) , ( ), , ((

(2.1.6) from 1, ))) , ( ), , ( ( )), , ( ), , ( ((

(1) from 1, )) , ( ), , ((

1 ))) , ( ), , ( ( ), , ((

3 3 2 2

2 2 2 2 1

1 1 1

2 2 1 1

1 1 1 1 0

0 0 0

1 1 0 0

0 0 0 0 0

0

 



gy gx gy gx

x y F y x F x y F y x F

gy gx gy gx

x y F y x F x y F y x F

gy gx gy gx

x y F y x F gy gx



Continuing in this way, we have

.

1 ))

,

(

),

,

((

gx

_n

gy

_n

gx

_n_₁

gy

_n_₁





n



(2)

Similarly from (2.1.5)(b), we can obtain

.

1 ))

,

(

),

,

((

gy

_n

gx

_n

gy

_n_₁

gx

_n_₁





n



(3)

Let

R

_n

=

max

{



(

gx

_n



gx

_n_₁

),



(

gy

_n



gy

_n_₁

)}

. Using (2), consider

1 , 1 ,

1 1 1

1 1 1 2

1

)) , ( ) , ( ( )) , ( ), , ((

)) , ( ) , ( ( = ) (

 

  

   





 

 

n y n x

n n n

n n

n n n

n n

n

M

y x F y x F gy

gx gy gx

y x F y x F gx

gx









(4)

where

.

) 2 ( ) 2 ( 2 1

, ) 2 ( ) 2 ( 2 1

), (

max =

1 1 2

2 1 2

1 1

1

1 , 1 ,

  

   

 

  

   

 

   



 

 

   



 

 

 



 



  

 



 

n n n

n

n n n

n

n n n

n n

n

n n n

n n

n

n y n x

gy gy gy

gy

gx gx gx

gx

gy gy gx

gx gy

gy

gx gx gy

gy gx

gx

M

 



 



Consider



( ), ( )



max 2

) ( from ), (

) (

2 =

2

2 1 1

3 2

1 1

2 1 1 2

  

   

 



 

 

   



   

   



 

n n n

n

n n n

n

n n n n n

n

gx gx gx

gx

gx gx gx

gx

gx gx gx gx gx

gx

 

  

Similary



( ), ( )



max 2

2 1 1 2

2

  

 _ _ _

   

  

n n n

n n

n gy _gy _gy _gy _gy

gy



.

Thus 1, 1 =max{ , ₁}.

,  



n n n

y n x

n y n

x R R

M

(4)becomes



(

gx

_n_₁



gx

_n_₂

)





max

{

R

_n

,

R

_n_₁

}

.

Similarly using(2.1.2) and (3),we have(gyn1gyn2)max{Rn,Rn1}. Thus

}

,

{

max

₁

1 





n n

n

R



(5)

=

}

,

{

max

R

(5)

1

1 





n

R



which in turn yields that

R

_n_₁

=

0

.

By proceeding as in the above we can show that

R

_n_₂

=

0,

R

_n_₃

=

0,



.

Thus

{

gx

_n

}

and

{

gy

_n

}

are constant Cauchy sequences.

Suppose

max

{

R

_n

,

R

_n_₁

}

=

R

_n for all

n

.

Then from (5), we have

R

_n_₁





R

_n for all

n

.

Thus

R

_n_₁





n

R

₁ for all

n

0 

as

n





. Thus

. 0

) (

0 )

(gxngxn1  and  gyngyn1  asn

 (6)

Suppose either

{

gx

_n

}

or

{

gy

_n

}

is not Cauchy.

Then there exists



>

0

for which we can find sub sequences

{

gx

_m₍_k₎

}

,

{

gx

_n₍_k₎

}

,

{

gy

_m₍_k₎

}

and

{

gy

_n₍_k₎

}

with

k

n

k

m

(

)

>

(

)

>

such that for every

k

.

)}

(

),

(

{

max



gx

_m₍_k₎



gx

_n₍_k₎



gy

_m₍_k₎



gy

_n₍_k₎





(7) Moreover, corresponding to

n

(

k

)

, we can choose

m

(

k

)

in such a way that it is the smallest integer with

)

(

>

)

(

k

n

k

m

and satisfying (7). Then

. < ))} (2(

)), (2(

{

max  gxm(k)1gxn(k)  gym(k)1gyn(k)  (8)

From (7), we have











(2(

)),

(2(

))



,

from

(8).

max

<

))

(2(

)),

(2(

max

))

(2(

)),

(2(

max

)

1.2(

Remark

from

,

))

(2(

))

(2(

)),

(2(

))

(2(

max

)

(

),

(

max

=

)}

(

),

(

{

max

1 ) ( ) ( 1

) ( ) (

) ( 1 ) ( )

( 1 ) (

1 ) ( ) ( 1

) ( ) (

) ( 1 ) ( 1

) ( ) (

) ( 1 ) ( 1

) ( ) (

) ( 1 ) ( 1 ) ( ) (

) ( ) ( )

( ) (

































































 

k m k m k

m k m

k n k

m k

n k

m

k m k m k

m k m

k n k

m k

m k m

k n k

m k

m k m

k n k

m k

m k m

k n k

m k

m k m

k n k m k

n k m

gy

gx

gy

gx

gy

gx

ii

gy

gx

gy

gx

gy

gx

From (6) and since



satisfies the ₂-condition, we have





(

),

(

)}

=

{

max

lim

m(k) n(k) m(k) n(k) k

gy

gx



 

(9)

Using (2) and triangular property of



, we have

1 ))

,

(

),

,

((

gx

_m₍_k₎_₁

gy

_m₍_k₎_₁

gx

_n₍_k₎_₁

gy

_n₍_k₎_₁





.

(6)

1 ) ( , 1 ) ( 1 ) ( , 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( ) ( ) (

))

,

(

)

,

(

))

,

(

),

,

((

))

,

(

)

,

(

=

)

(

               









k n y k n x k m y k m x k n k n k m k m k n k n k m k m k n k n k m k m k n k m

M

y

x

F

y

x

F

gy

gx

gy

gx

y

x

F

y

x

F

gx









(10) Where . 2 2 2 1 , 2 2 2 1 ), ( ), ( ), ( ), ( ), ( ), ( max = ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( , 1 ) ( 1 ) ( , 1 ) (                                                                                                       k m k n k n k m k m k n k n k m k n k n k n k n k m k m k m k m k n k m k n k m k n y k n x k m y k m x gy gy gy gy gx gx gx gx gy gy gx gx gy gy gx gx gy gy gx gx M           Consider (8) from )), (2( < ) 1.2( Remark from , )) (2( )) (2( ) ( = ) ( 1 ) ( ) ( 1 ) ( ) ( ) ( 1 ) ( 1 ) ( ) ( ) ( 1 ) ( 1 ) ( 1 ) (                  k n k n k n k n k n k m k n k n k n k m k n k m gx gx ii gx gx gx gx gx gx gx gx gx gx      

Similarly



(

gy

_m₍_k₎_₁



gy

_n₍_k₎_₁

)

<







(2(

gy

_n₍_k₎



gy

_n₍_k₎_₁

)).

Consider ) ( from ), ( ) ( 2 = 2 3 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( 1 ) (      k n k m k m k m k n k m k m k m k n k m gx gx gx gx gx gx gx gx gx gx                        Similarly



()1 () () ( )



) ( 1 ) ( ( )

2 nk nk nk mk

k m k n gx gx gx gx gx gx                 , ) ( ) (

2 ( )1 ( ) () ()

) ( 1 ) ( k n k m k m k m k n k m gy gy gy gy gy gy                 , ) ( ) (

2 ()1 () ( ) ( )

) ( 1 ) ( k m k n k n k n k m k n gy gy gy gy gy gy                 .

Thus (10) becomes

(7)

Similarly we have                                                                                 ) ( ) ( ) ( ) ( 2 1 , ) ( ) ( ) ( ) ( 2 1 ), ( ), ( ), ( ), ( )), (2( )), (2( max ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( 1 ) ( ) ( 1 ) ( ) ( ) ( ) ( k m k n k n k n k n k m k m k m k m k n k n k n k n k m k m k m k n k n k n k n k m k m k m k m k n k n k n k n k n k m gy gy gy gy gy gy gy gy gx gx gx gx gx gx gx gx gy gy gx gx gy gy gx gx gy gy gx gx gy gy                   Thus                                                                                            ) ( ) ( ) ( ) ( 2 1 , ) ( ) ( ) ( ) ( 2 1 ), ( ), ( ), ( ), ( )), (2( )), (2( max ) ( , ) ( max ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 ) ( 1 ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( k m k n k n k n k n k m k m k m k m k n k n k n k n k m k m k m k n k n k n k n k m k m k m k m k n k n k n k n k n k m k n k m gy gy gy gy gy gy gy gy gx gx gx gx gx gx gx gx gy gy gx gx gy gy gx gx gy gy gx gx gy gy gx gx                   

Letting

k





and using (6), (9) and ₂-condition, we get







. It is a contradiction.

Hence

{

gx

_n

}

and

{

gy

_n

}

are Cauchy sequences. Since

X

_ is



-complete space, there exist

x

,

y



X

_ such that

x

gx

_n



and

gy

_n



y

.

Since the pair

(

F

,

g

)

is



-compatible, we have



(

g

(

F

(

x

_n

,

y

_n

))



F

(

gx

_n

,

gy

_n

))



0

and

0 ))

,

(

))

,

(

g

F

y

_n

x

_n



F

gy

_n

gx

_n





.

Since

F

is



-continuous, we have



(

F

(

gx

_n

,

gy

_n

)



F

(

x

,

y

))



0

as

n





. Now                     2 2 ) , ( ) , ( 2 ) , ( = 4 ) , ( y x F gy gx F gy gx F gx y x F

gx n n n n

(8)

compatible g)

(F, and continuous

are F and g since , 0

1.2(i) Remark and

) ( from

)), , ( ) , ( ( )) , ( )) , ( ( ( )) , ( ( (

2

) , ( ) , ( 2

) , (

3

 

  

 

 

   



 

    

   

 



 



 

is n

as

y x F gy gx F gy

gx F y x F g y

x F g gx

y x F gy gx F gy

gx F gx

n n n

n n

n

n n n

n

Thus

gx

=

F

(

x

,

y

).

Similarly we can show that

gy

=

F

(

y

,

x

)

.

Thus

(

x

,

y

)

is a coupled coincidence point of

F

and

g

.

Claim: If (u,v) is another coupled coincidence point of

F

and

g

, then

gx

=

gu

and

gy

=

gv

. By (2.1.7), we have



((

gx

,

gy

),

(

gu

,

gv

))



1

.

Consider

1.2(i) Remark from )}, (

), ( { max =

2 2

2 1

, 2 2

2 1

),0,0,0,0, (

), (

max

2 ) , ( 2

) , ( 2

1

, 2

, ( 2

) , ( 2

1

)), , ( ( )), , ( ( )), , ( (

)), , ( ( ), ( ), (

max

)) , ( ) , ( ( )) , ( ), , ((

)) , ( ) , ( ( = ) (

gv gy gu gx

gy gv gv gy

gx gu gu

gx

gv gy gu gx

x y F gv u

v F gy

y x F gu v

u F gx

u v F gv v u F gu x y F gy

y x F gx gv gy gu gx

v u F y x F gv gu gy gx

v u F y x F gu gx

 

   

    

 

   

    

 

   

 

              

   

 

              

 



   

    

 

   

    

 

   

 

   

       

  

   

 

   

       

  

 



 





 

 

 



 



 



 





 

 

Similarly

)}. (

), (

{ max )

(gygv   gxgu  gygv



Thus

)} (

), ( { max )

( ), (

max gx gu gy gv

gv gy

gu gx

 

   

  

 

 

 



which in turn yields that

gx

=

gu

and

gy

=

gv

. Hence the claim. Denote

gx

=

p

and

gy

=

q

.

Then

gp

=

g

(

gx

)

=

g

(

F

(

x

,

y

))

=

F

(

gx

,

gy

)

=

F

(

p

,

q

)

, since



-compatibility of

(

F

,

g

)

implies



-weakly compatibility of

(

F

,

g

)

.

Similarly

gq

=

F

(

q

,

p

)

. Thus

(

p

,

q

)

is a coupled coincidence point of

F

and

g

. By the claim we have

gx

=

gp

and

gy

=

gq

.

Thus

p

=

gp

and

q

=

gq

.

(9)

Suppose

(

p



,

q



)

is another coupled common fixed point of

F

and

g

. Then from (2.1.7), we have



((

gp

,

gq

),

(

g

p



,

g

q



))



1 .

) 1.2( Remark from )}, ( ), ( { max =

2 2

2 1

, 2 2

2 1

), ( ), ( ), (

max

)) , ( ) , ( ( )) , ( ), , ((

)) , ( ) , ( ( = ) (

i q

q p p

q q q q

p p p

p

q q p p q q

p p q q p p

q p F q p F q g p g gq gp

q p F q p F p p

   

   

    

 

   

    

 

   

 

               

   

 

               

      

    



   

 

   



 



 



 





 

 

Similarly

)}

(

),

(

{

max

)

(

q



q









p



p





q



q





.

Thus

max {(pp),(qq)}max{(pp),(qq)} which in turn yields that

p

=

p



,

q

=

q



.

Thus

(

p

,

q

)

is the unique coupled common fixed point of

F

and

g

. Now, we give an example to illustrate Theorem

2.1

.

Example 2.2 Let

X

_

=

l

1, where



(

x

)

=||

x

||=

_

x

_i for

x



l

1. Define

F

:

l

1



l

1



l

1 and

g

:

l

1



l

1by 

  



  



,0,0, 12 , 12 = ) ,

(_x _y x1 y1 x2 y2

F , _

  

 



,0,0, 3 , 3 = x1 x2

gx for all

x

=

(

x

1

,

x

2

,



)



l

1 and

1 2

1

,

)

(

=

y

l

y





.

Define



:

X

_2



X

_2



R

 as

2

3 =

))

,

(

),

,

((

x

y

u

v



if

x

,

y

,

u

,

v



l

1 with

||

x

||=||

y

||=||

u

||=||

v

||



1

and 0 , otherwise.

Let

x

=

(

x

₁

,

x

₂

,



),

y

=

(

y

₁

,

y

₂

,



),

u

=

(

u

₁

,

u

₂

,



),

v

=

(

v

₁

,

v

₂

,



)



l

1. Case(a):Suppose

||

x

||=||

y

||=||

u

||=||

v

||



1

.

Consider

















_

  

 

      

   

 

   

   

   

 

      

   



      

   

 

   



  

    



  



2 2 1 1 2

2 1 1

2 2 2 2 1

1 1 1

2 2 2 2 1 1 1 1

2 2 1 1 2

2 1 1

3 1 3

1 8 3 =

12 1 12

1 2 3

12 1 12

1 2 3 =

,0,0, 12

, 12 2

3 =

,0,0, 12 , 12 ,0,0,

12 , 12 2

3 =

)) , ( ) , ( ( )) , ( ), , ((

v y v y u

x u x

v y u x v

y u x

v u y x v u y x

v u v u y

x y x

v u F y x F gv gu gy gx



 

 

 

x, y, u and v are not zero

x, y, u and v are not zero.

atat least one of first two values of all x,y, u and v is not zero and 0, otherwise

(10)







( ), ( )



max

4 3

) ( ) ( 8 3 =

gv gy gu gx

 



  

 

Case(b): suppose at least one of ||x||,||y||,||u|| and ||v|| is greater than 1. Then



((

x

,

y

),

(

u

,

v

))

=

0

.

Thus in both cases, (2.1.2) is satisfied with 4 3 =

 .One can easily verify the remaining conditions.Clearly

(0,0)

is the

unique coupled common fixed point of F and g.

III. CONCLUSION

By choosing , F and g in Theorem 2.1, one can obtain several fixed point results in modular spaces.

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x, y, u and v is zero