1. A common fixed point theorem for six mappings with a new binary operator

(1)

A COMMON FIXED POINT THEOREM FOR SIX

MAPPINGS WITH A NEW BINARY OPERATOR

Rekha Jain1 and Saurabh jain2

I. INTRODUCTION AND PRELIMINARIES

The concept of common fixed point theorem for commuting mapping was given by Jungck [4]. The notion of weak commutativity was introduced by Sessa [7]. Imdad and Khan [5] has proved a common fixed point theorem for six mappings which was extension of Fisher [1] [2] and Jeong- Rhoades [3]. A new binary operation was introduced by Sedghi and Shobe [6].

Definition 1.1[7]: A pair of self-mapping (A, B) on a metric space (X, d) is said to be weakly commuting if d(ABx, BAx) ≤ d(Bx, Ax) for all x in X. Obviously, commuting mappings are weakly commuting but the converse is not necessarily true.

Definition 1.2[5]: A pair of self mappings (A, B) of a metric space (X, d) is said to be compatible if

lim

n Axn =

lim

n Bxn =

t X



. Obviously, weakly commuting mappings are compatible but the converse is not necessarily true.

In what follows, N is the set of all natural numbers and R+ is the set of all positive real numbers. Let ◊

:R × R

+ +





R

+be a binary operator satisfying the following conditions:

(1) ◊ is associative and commutative ; (2) ◊ is continuous ;

Some typical examples of ◊ are: Examples:





max

,

;

/ (

0)

a b

 

a b

 

a b a b ab



 

 

a b a b

 

a b b



;

and





max

, ,1

ab

a b

 

for each

a b

, ,



R



Definition 1.3. [6]: The binary operation is said to satisfy



-property if there exists a positive real number



such that

1

Medi-Caps University,Indore 2

Indore Institute of Science and technology

DOI: http://dx.doi.org/10.21172/1.81.079

e-ISSN:2278-621X

Abstract: Using notion of compatibility, weak compatibility, commutativity and a new binary operation we have proved a fixed point theorem for six mappings satisfying a rational inequality.

2000 AMS Subject Classification: Primary: 54H25, Secondary: 47H10

(2)





max

,

a b

 



a b

Motivated by Sedghi and Shobe [6]. and Imdad and khan[5], in the present paper, using a new binary operator we prove a fixed point theorem by improving the contraction condition and choosing suitable weak commutativity conditions .

II. RESULTS

We prove the following.

Theorem 2.1. Let A, B, S, T, I and J be self mappings of a complete metric space (X,d)

Satisfying AB(X)



J(X), ST(X)



I(X) and for each x, y Є X,













































3 3

1 2 2

2 2

2

3

,

d ABx Jy

d STy Ix

d ABx STy

k

d ABx Jy

d STy Ix

d ABx Jy

d STy Ix

k

d ABx Jy

d STy Ix

k

d ABx Jy

d STy Ix



_

_

_

_

_





















_

_

_

_

_

_

_

_

_









_

_

_

_

_



















 





_

_{ }

_











_

 

_{ }





_

(2.1)



,



2



,



2

0 ,



,





,



0 if d ABx Jy



d STy Ix



d ABx Jy



d STy Ix



,

k

_i



0

(i = 1, 2, 3) with atleast one

k

_i non zero and

2 k

₁



2 k

₂



2 k

₃





1

and

(i) {AB, I} are compatible, I or AB is continuous and (ST, J) are weakly compatible or

(ii) {ST, J} are compatible, J or ST is continuous and (AB, I) are weakly compatible then AB, ST, I and J have a unique common fixed point. Furthermore if the pairs (A, B), (A, I), (B, I), (S, T), (S, J) and (T, J) are commuting mappings then A, B, S, T, I and J have a unique common fixed point.

Proof. Let x0 be an arbitrary point in X. Since AB(X)



J(X) we can find a point x1 in X such that AB x0 = Jx1. Also since ST(X) subset of I(X) we can choose a point x2 with STx1 = I x2. Using this argument repeatedly one can construct a sequence {zn} such that z2n = ABx2n = J x2n+1, z2n+1 = STx2n+1 = I x2n+2 for n = 0, 1, 2…. For brevity let us put

u2n = d (ABx2n , STx2n+1) and u2n+1 = d (STx2n+1 ,ABx2n+2) for n = 0, 1, 2…… Now we distinguish two cases (i) Suppose that u2n + u2n+1 ≠ 0 for n = 0, 1, 2….

Then using the inequality (2.1), we have









2n 1 2 1 2 2 2 1 2 2

u





d z

n

,

z

n



d STx

n

,

ABx

n

































3 3

2 2 2 1 2 1 2 1

1 2 ₂

2 2 2 1 2 1 2 2

2 2

2 2 2 1 2 1 2 1

2

2 2 2 1 2 1 2 2

3

,

n n n n

d ABx

Jx

d STx

Ix

k

d ABx

Jx

d STx

Ix

d ABx

Jx

d STx

Ix

k

d ABx

Jx

d STx

Ix

k

d

   





























_

_





_{  }

_



_

_

_

_

































_







_{  }

_



_

_













ABx

2n2

,

Jx

2n1



d STx



2n1

,

Ix

2n2





_

 

_{ }





_







or













1 2 3

2 1 2 2 2 2 1

1 2 3

,

1

n n n n

k

d z

z

d z

z

k



  





(3)

Similarly we can show that













1 2 3

2 2 1 2 1 2

1 2 3

,

1

n n n n

k

d z

z

d z

z

k



 









thus for every n we have

d z z



_n

,

_n_₁





kd z



_n_₁

,

z

_n



(2.2) which shows that

 

z

_n is a Cauchy sequence in the complete metric space (X, d) and so has a limit point z in X. hence the sequence ABx2n = Jx2n+1 and STx2n+1 = Ix2n+2 which are subsequences also converge to the point z.

Let us now assume that I is continuous so that the sequence



I x

2 ₂_n



and



IABx

₂_n



converges to

Iz

.

Also in view of compatibility of



I AB

,



,



ABIx

₂_n



converges to

Iz

.

Now,

















_

_





_

_





3 3 ₂

2 2 1 2 1 2

2 2 1 1 2 ₂ 2

2 2 1 2 1 2

2

2 ₂

2 2 1 2 1 2

2

2 2 1

,

n n n n

n n

n n n n

n n

d ABIx

Jx

d STx

I x

d ABIx

STx

k

d ABIx

Jx

d STx

I x

d ABIx

Jx

d STx

I x

k

d ABIx

Jx

       



_

_

_

_

_







_

_











_

_





























































_

_

2

2 1 2

2

3 2 2 1 2 1 2

,

n n

n n n n

d STx

I x

k

d ABIx

Jx

d STx

I x

  







_{ }



_

_



























_{ }



Which on letting

n

 

reduces to



1 

k

₁



k

₂



k

₃



 

d Iz z

,





0

yielding thereby

Iz



z

Now,





































3 3

2 1 2 1

2 1 1 2 ₂

2 1 2 1

2 2

2 1 2 1

2

2 1 2 1

, , , , , , , , , n n n n n n n n n

d ABz Jx d STx Iz

d ABz STx k

k

                       _ _  _{  } _ _ _      _ _ __ _               _ _{ } _   









3 2 1 2 1

k d ABz Jx, n    d STxn,Iz 

On letting

n

 

and using Iz = z we get



,

 

1 2 3

 

,



d ABz z



k



k



k



d ABz z

This implies AB z = z

Since

AB X

 



J X

 

then there always exists a point

z

'

such that

Jz

'



z

so that

STz



ST Jz



'



.

Now,

(4)









































3 3

1 2 2

2 2 2 3

,

'

',

,

'

',

,

'

',

,

'

',

,

'

',

d ABz Jz

d STz Iz

k

d ABz Jz

d STz Iz

d ABz Jz

d STz Iz

k

d ABz Jz

d STz Iz

k

d ABz Jz

d STz Iz



_

_

_

_

_





















_

_

_

_

_

_

_

_

_









_

_

_

_

_













 





 





_

_{ }

_





























k

1



k

2



k

3









d STz z



',







.

Hence, STz’ =z= Jz’ which shows that z’ is the coincidence point of ST and J. Now using the weak compatibility of



ST J

,



, now using the weak compatibility of (ST, J), we have

STz



ST Jz

(

')



J STz

(

')



Jz

which shows that z is also a coincidence point of the pair (ST, J). Now



,





,



d z STz



d ABz STz









































3 3

1 2 2

2 2 2 3

,

+

,

d ABz Jz

d STz Iz

k

d ABz Jz

d STz Iz

d ABz Jz

d STz Iz

k

d ABz Jz

d STz Iz

k

d ABz Jz

d STz Iz



_

_

_

_

_





















_

_

_

_

_

















_

_

_

_

_



















 





_

_{ }

_











 





 







k

₁



k

₂



k

₃



 

d z STz

,



Hence,

z



STz



Jz

which shows that

z

is a common fined point of

AB I ST

, ,

and

J

.

Now suppose that

AB

is continuous so that the sequences



AB x

2 ₂_n



and



ABIx

₂_n



converge to

ABz

since



AB I

,



are compatible it follows that



IABx

₂_n



also converge to

ABz

, thus

































3 3 2

2 2 1 2 1 2

2

2 2 1 1 ₂ 2 2

2 2 1 2 1 2

2 2

2

2 2 1 2 1 2

2

2 2 1

,

n n n n

n n

n n n n

n n

d AB x

Jx

d STx

IABx

d AB x

STx

k

d AB x

Jx

d STx

IABx

d AB x

Jx

d STx

IABx

k

d AB x

Jx

       



_

_{  }

_

























_{  }

_



_















_{  }

_



























2 1 2

2

3 2 2 1 2 1 2

,

n n

n n n n

d STx

IABx

k

d AB x

Jx

d STx

IABx

  







_{ }





 

_



_













_{  }

_









which on letting

n

 

reduces to

d ABz z



,

 



k

₁



k

₂



k

₃



 

d ABz z

,



which implies

ABz



z

(5)





































3 ₃

2

2 2

2

2 1 ₂ 2 2

2 2

2 ₂

2

2 2

2

2 2

,

'

',

,

'

+

,

'

',

,

'

',

,

'

',

n n

n

n n

d AB x

Jz

d STz IABx

d AB x

STz

k

d AB x

Jz

d STz IABx

d AB x

Jz

d STz IABx

k

d AB x

Jz

d STz IABx



_

_{  }

_



















_



  

_



_















_

_{  }

_

















_



  

_



_

 









2







3 2 2

k

d AB x

_n

,

Jz

'

d STz IABx

',

_n







_{  }

_



_





which on letting

n

 

reduces to

d z STz



,

'

 



k

₁



k

₂



k

₃



 

d z STz

,

'



This gives STz' = z =J

z

'

thus z' is a coincidence point of (ST, J). Since, the pair (ST, J) is weakly compatible hence

(

')

(

')

STz



ST Jz



J STz



Jz

which shows that STz = Jz . Further





































3 3

2 2

2 1 2 2

2 2

2

2 2

,

n n

n

n n

d ABx

Jz

d STz Ix

d ABx

STz

k

d ABx

Jz

d STz Ix

d ABx

Jz

d STz Ix

k

d ABx

Jz

d STz Ix



_

_

_

_

_





















_

_

_

_

_

_

_

_

_









_

_

_

_

_













 





 





_

_{ }

_















3 2n

,

2n

k



_

d ABx

Jz

 

_{ }



d STz Ix



_

which on letting

n

 

reduces to



,

 

1 2 3

 

,



d z STz



k



k



k



d z STz

’

It follows that

STz



z



Jz

The point

z

therefore is in the range of

ST

and since

ST X

 



I X

 

there exist a point

z

"

in

X

such that

"

Iz



z

.Thus









































3 3

1 2 2

2 2

2

",

,

"

",

,

"

",

,

"

",

,

"

d ABz

Jz

d STz Iz

d ABz

z

d ABz STz

k

d ABz

Jz

d STz Iz

d ABz

Jz

d STz Iz

k

d ABz

Jz

d STz Iz



_

_

_

_

_























_

_

_

_

_

















_

_

_

_

_













_{ }





_

 

_{ }





_















3

k



_

d ABz

",

Jz

 

_{ }



d STz Iz

,

"



_

Letting

n

 



"

 

1 2 3

 

",



d ABz z



k



k



k



d ABz

z

which shows that

ABz

"



z

Also since



AB I

,



are compatible and hence using weakly commuting we obtain



,





" ,

 

"



d ABz Iz



d AB Iz

I ABz

(6)

Therefore

ABz



Iz



z

Thus we have proved that

z

is common fixed point of

AB ST I

,

and

J

.

If the mappings

ST

or

J

is continuous instead of

AB

or

I

then proof of

z

is a common fixed point of

AB ST I

,

and

J

is similar.

Let

v

be another fixed point of

I J AB

, ,

and

ST

then



,





,



d z v



d ABz STv









































3 3

1 2 2

2 2

2

3

,

d ABz Jv

d STv Iz

k

d ABz Jv

d STv Iz

d ABz Jv

d STv Iz

k

d ABz Jv

d STv Iz

k

d ABz Jv

d STv Iz



_

_

_

_

_























_



_





_



_









_

_

_

_

_













 





 





_

_{ }

_











 





 





,

 

1 2 3

 

,



d z v



k



k



k



d z v

yielding thereby

z



v

Finally we need to show that

z

is also a common fixed point of

A B S T I

, , , ,

and

J

. For this let

z

be the unique common fixed point of both the pairs



AB I

,



and



ST J

,



. Then













,

 





Az



A ABz



A BAz



AB Az

Az



A lz



I Az

















.

 





Bz



B ABz



B A Bz



BA Bz



AB Bz

Bz



B lz



I Bz

which shows that

Az

and

Bz

is a common fixed point of



AB I

,



yielding thereby

Az



z



Bz



lz



ABz

in the view of uniqueness of the common fixed point of the pair



AB l

,



.

Similarly using the commutatively of



S T

,

 

,

S J



and



T J

,



it can be shown that

Sz



z



Tz



Jz



STz

.

Now we need to show that

Az



Sz Bz





Tz



also remains a common fixed point of both the pairs



AB I

,



and



ST J

,



. For this



,





 

,



d Az Sz



d A BAz

S TSz





 



,



d AB Az

ST Sz

(7)





 







  







 







  







 







  







 







  







 







  



3 3

1 2 2

2 2

2

3

,

d AB Az

J Sz

d ST Sz

I Az

k

d AB Az

J Sz

d ST Sz

I Az

d AB Az

J Sz

d ST Sz

I Az

k

d AB Az

J Sz

d ST Sz

I Az

k

d AB Az

J Sz

d ST Sz

I Az



_

_

_

_

_





















_

_

_

_

_

















_

_

_

_

_













_{ }



_

_

_

_

_

















 







 



Which implies that

d Az Sz



,





0



as

d AB Az



_

_

, (

J Sz





d ST Sz



_{  }

,

I Az

_





0



yielding thereby

Az



Sz

. Similarly it can be shown that

Bz



Tz

. Thus

z

is the unique common fixed of

A B S T l

, , , ,

and

J

. Corollary 2.1. Theorem 2.1 remains true if contraction conditions (2.1) and (2.2) are replaced by any of the following conditions:

(i)





















3 3

1 2 2

,

d ABx Jy

d STy Ix

d ABz STy

k

d ABx Jy

d STy Ix



_

_

_

_

_



















_

_

_

_

_























3

,

k

d ABx Jy

d STy Ix





_

 

_{ }





_



,



2



,



2

0,

if d ABx Jy

_



_





_



d STy lx



_



k k

₁

,

₃



0, 2

k

₁



2 k

₃



₃



1

or (A)



,



0 

,



2



,



2

0 d ABx STy



if



_

d ABx Jy



_





_

d STy lx



_



  



































 

3 3 2 2

1 2 2 2

, , , ,

ii , B

, , , ,

d ABx Jy d STy Ix d ABx Jy d STy Ix

d ABx STy k k

d ABx Jy d STy Ix d ABx Jy d STy Ix

_ _ __ _  _ _ __ _ 

       

   

 

_ _ __ _   _ _{ }_ _ 

       

   



,



2



,



2

0,



,





,



0, ,

1 2

0, 2

1

2

1 if d ABx Jy

_





_



_



d STy Ix

_







_

d ABx Jy

_{ }

 



d STy Ix

_





k k



k



k



(iii)





















3 3

1 2 2

,

d ABx Jy

d STy Ix

Either d ABx STy

k

d ABx Jy

d STy Ix



_

_

_

_

_



















_

_

_

_

_

















,



2



,



2

0 d ABx Jy

d STy Ix



_

_

_

_

_



_













,

k

1



0,

k

1



½

(C)

Proof. Corollaries corresponding to the contraction conditions (A), (B) and (C) can be deduced directly from. Theorem 2.1 by choosing

k

₂



0,

k

₃



0

and

k

₂



k

₃



0

, respectively.

REFERENCES

[1]. Fisher B. Mapping satisfying rational inequality, Nanta. Math. 12 (1979) 195 -199.

[2]. Fisher B. Common fixed point and constant mapping satisfying a rational inequality, Math. Sem. Notes 6 (1978) 29 -35.

(8)

[4]. Jeong G. S. and B. E. Rhoades. Some remark for improving fixed point theorem for more than two maps, Indain J. Pure Appl. Math. 28 (1997) 1177 - 1196.

[5]. Jungck. G. Compatible mappings and Common fixed points, Internat. J. Math. and Math. Sci. 9 (1986) 771 - 779.

[6]. Sedghi S. and Shobe N. Common fixed point theorems for four mappings in complete metric spaces, Bulletin of the Iranian Math. Soc. Vol. 33 (2) (2007), 37 – 47.