Fixed Point Theorem on Complete N - Metric Spaces

Fixed Point Theorem on Complete N - Metric

Spaces

S.C. Ghosh ; M. Srivastava Department of Mathematics D.A-V.College, Kanpur, U.P., India

ABSTRACT: In this paper we have proved the generalized fixed point theorem on complete N-Metric Space with the help of N-mappings. This theorem generalizes and extends the result of R.K. Jain, H.K. Sahu and B. Fisher [3], Kikina [2] and chaube, Gupta and Shau [1] from three, four, five metric space to N-Metric Spaces.

KEY WORDS: Complete metric spaces, Cauchy sequences, fixed point. 2010 AMS subjectclassification 47 H10; 54H25

INTRODUCTION: In this paper

X

₁

,

X

₂

,





X

_N are taken as complete N-Metric Spaces and

1 1

3 2

1 2

1

X

,

f

:

X

X

f

:

X

X

X

:

f

_N



_N







_N



as N-mappings which are all the

self mappings. The elements of the metric spaces are taken as

N N

n n

n

X

,

X

X

x

X

x

₁



_{1 2}



₂







where the first suffix indicating the respective metric

space and the second one are the number of elements where

n



N

(Set of natural number).

MAIN RESULT:

Theorem [1.1] Let



X

₁

,

d

₁

 

,

X

₂

,

d

₂

 

,

X

₃

,

d

₃









X

_N

,

d

_N



are N-complete metric

spaces and

f

₁

f

₂

f

₃









f

_N are self mappings such that

1 1

1 2

1 2

3

4 3

2 3 2

1 2

1

X

X

:

f

,

X

X

:

f

,

X

X

:

f

X

X

:

f

,

X

X

:

f

,

X

X

:

f

N N

N N

N

N N

N













 















Satisfying following conditions





_



_





1

1

1



1 2 1 1 1

1 2 1 1 1 1

1 3

2 1 1 2 1 3

2 1

1

.

.

x

,

x

x

,

x

x

f

f

f

f

,

x

f

f

f

f

d

_{N N}























_



_





1

1

2



1 3 1 2 2

1 3 1 2 2 1

2 1 3

2 1 1

3 1 3

2 1

2

.

.

x

,

x

x

,

x

x

f

f

f

f

f

,

x

f

f

f

f

f

d

_{N N N N}

























_



_





1

1

3



1 4 1 3 3

1 4 1 3 3 1

3 2 3

2 1 1 1 1 3 3

2 1 1

3

.

.

x

,

x

x

,

x

x

f

f

f

f

f

f

,

x

f

f

f

f

f

f

d

_{N N N N N N}







 

















_



_



 

1

1

4

1 5 1 4 4

1 5 1 4 4 1

4 3

2 1 1 2 1 5 4 3

2 1 1 2

4

.

.

x

,

x

x

,

x

x

f

f

f

f

f

f

f

,

x

f

f

f

f

f

f

f

d

_{N N N N N N N N}







  







_



_





1

1

5



1

1 1

1 1 1 1

1 2 3 1

1 1 1

4 3

2

.

.

x

,

x

x

,

x

x

f

f

f

f

f

,

x

f

f

f

f

f

d

N N

N N N

N N N

N N



















N

N

X

x

X

x

,

X

x

,

X

x











_{11 1 21 2 31 3}





₁

and





 

0

,

1

, Also



₁



₂







_N and



₁

,



₂

,



₃







_N indicated in above inequality are defined as follows,



_{11 21}



0

₂



_{21 31}



0

₃



_{31 41}



0



_{1 11}



0

1

















x

,

x

,

x

,

x





x

,

x





_N

x

_N

,

x













 





















_{21 11}





1

1

6



2

1 2 1 3

2 1 1 1 1 1

1 5

4

1 2 1 6

5 4 2 1

1 3

2 1 1 1 1

1 1 1 4

3 1 3 2 5

4 3 1

1 3

2 1 1 1 1 1 1 1 3

2

1 2 1 2 3

2 1

3 2 1 1 1 1 1

2 1 1 1

.

.

x

f

,

x

d

,

x

f

f

f

f

,

x

d

x

f

f

f

x

f

f

f

f

d

,

x

f

f

f

f

,

x

d

,

x

f

f

f

f

,

x

f

f

f

f

d

,

x

f

f

f

f

,

x

d

,

x

f

f

f

f

,

x

f

f

f

f

d

,

x

f

f

f

f

,

x

d

max

x

,

x

N

N N

N N

N

N N N

N

N N N

N

N N N

N N

  

 





 





























































 





















 





11 1 11





1

1

7



1 1 1 3 2 1 1 1 3 2 1 1 1 1 1 1 1 4 3 2 1 1 1 2 3 1 1 1 1 7 6 5 3 1 1 4 3 2 1 1 1 5 4 3 1 1 6 5 4 2 1 1 4 3 2 1 1 1 5 4 3 1 1 5 4 3 1 1 1 5 4 3 2 1 1 1 1

.

.

x

f

,

x

d

,

x

f

f

f

x

d

,

x

f

f

f

f

,

x

d

,

x

f

f

f

f

,

x

d

,

x

f

f

f

f

f

,

x

f

f

f

f

d

,

x

f

f

f

f

f

,

x

d

,

x

f

f

f

f

f

,

x

f

f

f

f

d

,

x

,

f

f

f

f

f

,

x

d

,

x

f

f

f

f

f

,

x

f

f

f

f

d

,

x

f

f

f

f

f

f

,

x

d

max

x

,

x

N N , N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N          





















































Again



₁



₂



₃







_N are defined as -









 





11 1 2 3 1 11



2 1 1 3 2 1 1 1 1 1 1 1 3 2 1 1 1 1 1 2 1 1 1    





N N N N N , N N ,

x

f

f

f

f

f

,

x

f

d

,

x

f

f

f

f

x

d

,

x

f

f

f

f

,

x

d

max

x

x





















 





_{1 21 1 1 2 3 2 21}





1

1

8



3 1 1 3 2 1 1 2 2 1 2 2 3 2 1 1 2 2 1 3 1 2 2

.

.

x

f

f

f

f

f

f

,

x

f

d

,

x

f

f

f

f

x

d

,

x

f

f

f

f

f

,

x

d

max

x

x

r

N N N N N N , N N N ,      



























 

,

,





1

.

1

.

9



,

,

max

1 1 4 3 2 1 1 1 1 1 2 1 5 4 3 2 , 1 11 4 3 2 1 11 , 1

x

f

f

f

f

f

x

f

d

x

f

f

f

f

f

f

x

d

x

f

f

f

f

x

d

x

x

r

N N N N N N N N N N N

















i)

iv)

f

_N2,

f

_N1

f

_N

f

₁

f

₂





f

_N3 has a unique fixed point



₄



X

₄ and so on for

,

X

₅

5







₆



X

₆ and finally we obtain

f

₂

f

₃

f

₄

f

₅





f

_N1

f

_N

f

₁ has a unique fixed point



_N



X

_N

.

Further

f

_N



₁





₂

,

f

_N1



₂





₃

,

f

_N2



₃





₄

,

f

_N3



₄





₅

,





f

₁



_N





₁

Proof : Let us take x10 an element in

X

₁ Now let us define the sequences

     

x

1n

,

x

2n

,

x

3n





 

x

Nn on the complete metric spaces

X

1

,

X

2

,

X

3

,

X

4





X

N

respectively and the sequences defined on

X

₁

,

X

₂

,

X

₃





X

_N as follows



1 2 3



10

1

f

f

f

f

x

x

n







N n - - - -(1.1.10)

n N

n n

N

n

f

x

,

x

f

,

x

x

₂



_{1 1 3}



_{1 3} - - - -(1.1.11)

n N

n

f

x

x

₄



_{2 4} - - - -(1.1.12)

n N n

N

f

x

x



₁ for all positive integer n First we take

1 3 3 1 2 2

1 1

1n



x

n

,

x

n



x

n

,

x

n



x

n

x

and

x

_Nn



x

_Nn1 and therefore we take -

2 1

2 3 1

3

3 3

2 2

1 1

 

























n N n

N n

n

N n

N n

n n

x

x

x

x

x

,

x

,

x

,

x









Again we can put

N n

N n

n

n

,

x

,

x

,

x

x

₁





_{1 2}





_{2 3}





₃









if

x

_2n



x

_2n1

,

x

_3n



x

_3n1

,

x

_Nn



x

_Nn1





















 

 





 

 















n n

 

n n





n n



n N n N N n n n n n n n n n n n n n n n n n n n N N n N N n n

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

,

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

x

,

x

x

f

f

f

f

f

,

x

f

f

f

f

f

d

x

,

x

d

3 3 3 1 2 2 2 2 2 2 1 1 2 2 2 4 1 4 4 1 2 2 2 3 1 3 3 1 2 2 2 1 1 1 1 1 2 2 2 1 3 2 2 1 3 2 2 1 1 3 2 1 1 3 2 3 2 1 2 1 2 2 2               





























 



















0

₂



_{2 2 1}



0



1 4 1 4 4 3 1 3 3 1 1 1 1 1 2 2 2

,

x

,

x

d

,

max

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

n n n N n N N n n n n n n n n      













 









1





1

1

13



4 1 4 4 3 1 3 3 1 1 1 1

.

.

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

n N n N N n n n n n n    









Again substituting

x

₃₁



x

_3n and

x

₄₁



x

_4n1 we obtain from the inequality (1.1.3)















3 3 1



3 1 3 3 3 2 2 3 3 2 1 1 1 3 3 3 2 1 1 3 1 3 3 3         











n n n n n N N N N n N N N n n

x

,

x

x

,

x

x

f

f

f

f

f

f

f

,

x

f

f

f

f

f

f

d

x

,

x

d













 

 

 

 









 















 

 











0

₃



_{3 3 1}



0



1

1 4 4 4 1 3 3 1 1 2 2 2 1 3 3 3

,

x

,

x

d

,

max

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

x

,

x

d

max

n n n N n N N

n n n

n n

n n

n

 

 

 



















 

 











3 3 1



3 1

1 4 4 4 1 3 3 1 1 2 2 2 1 3 3 3

 

 

 





n n n

N n N N

n n

n n n

n n

n

x

,

x

d

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max







₂



_{2 2} ₁

 

_{1 1 1} ₁

 

_{4 4 4} ₁





₁







max

d

x

_n

,

x

_n

,

d

x

_n

,

x

_n

,

d

x

_n

,

x

_n





d

_N

x

_Nn

,

x

_{N n}

Again for

d

₄



x

_4n

,

x

_4n1



we have,









 

 





_{1 1 1}





₁





1

1

14



1

1 1 1 1 1 2 2 2 1 3 3 3 1

4 4 4

.

.

x

,

x

d

,

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

n N n N N n

N n N N

n n n

n n

n n

n

 

 



 





















 

 





1 4 1



1



1 1 1





1





1

1

15



1

1 2 2 2 1 3 3 3 1 4 4 4 1

5 5 5

.

.

x

,

x

d

,

x

,

x

d

,

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

n N n N N n

N n N N n n

n n n

n n

n n

n

 

 

 

 













continuing this process we obtain,

















2 2 1

 

1 1 1 1





1

1

16



2

1 3 3

3 1

2 2

2 1

1 1

1

.

.

x

,

x

d

x

,

x

d

x

,

x

d

,

x

,

x

d

max

x

,

x

d

n n n

n

n N n N N n

N n N N n

N n N N

 

  

 

 

 

 



























_{3 3 1}



₂



_{2 2 1}

 

_{1 1 1 1}





1

1

17



3

1 2 2

2 1

1 1

1 1

.

.

x

,

x

d

,

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

n n n

n n

N n N N

n N n N N n

N n N N n

N n N N

 

  



  

 

 

 























11 21



1

1 2 1 1 1

1 1 3

2 1 21 1 3

2 1 1 1

1 1 1

x

,

x

x

,

x

x

f

f

f

f

,

x

f

f

f

f

d

x

,

x

d

n n N n











_















 

 

 





 





 





 

 





n n n n n n



n N n N N n n n

n n

n

n n n

n n

n n

n

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

,

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

2 2 2 1 1 1 1 1 1 1

1 1

1 1 1 1

4 4 4 1 1 1 1

1 3 3 3 1 1 1 1 1 2 2 2 1 1 1 1



 

 

 

 



















 

 









0

1



1 1 1



0



1

1 4 4 4 1 3 3 3 1 2 2 2 1 1 1 1

,

x

,

x

d

,

max

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

x

,

x

d

max

n n n N n N N

n n n

n n

n n

n

 

 

 













 

 





₁





1

1

18



1 4 4 4 1 3 3 3 1 2 2 2

.

.

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

n N n N N

n n n

n n

n



 











Now using all inequality (1.1.1) we obtain









 

 





_{1 1 1}

 

₁





1

1

19



1

1 3 3 4 1 3 3 3 1 2 2 2 2

1 1 1 1

.

.

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

n N n N N n

N n N N

n n n

n n

n n

n

 

 



 













Similarly for the other inequality we have,









 

 





_{5 5 1}





₁





1

1

20



5

1 3 4 4 1 3 3 3 1 1 1

1 2

1 2 2 2

.

.

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

n N n N N n

n

n n n

n n

n n

n

 

 

 













 

 





_{5 5 1}





₁





1

1

21



5

1 4 4 4 1 2 2 2 1 1 1 1 2

1 3 3 3

.

.

x

,

x

d

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

n N n N N n

n

n n n

n n

n n

n

 

 

 

















 

 





_{4 4 1}



₁



_{1 1 1}





1

1

22



4

1 3 3 3 1 2 2 2 1 1 1 1 2

1

.

.

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

n N n N N n

n

n n n

n n

n n

N n N N

  

 

 

 









Now by induction, we obtain from the above inequality













 





_{1 2}





1

1

23



2 3 1 3 3 2 2 1 2 2 12 1 1 1 1

1 1 1 1

.

.

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

N N N

n n

n





 













 

 





_{1 2}





1

1

24



2 3 1 3 3 2 2 1 2 2 12 1 1 1 1

1 2 2 2

.

.

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

N N N

n n

n





 











1



11 12

 

2 21 22





1 2





1

1

25



1 1

3 3

3

x

,

x

max

d

x

,

x

,

d

x

,

x

d

x

,

x

.

.

d

N N N

n n

n





 





...

...









 

 





_{1 2}





1

1

26



2 3 1 3 3 2 2 1 2 2 2 1 1 1 1 1

12 1 1 1

.

.

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

x

d

N N N n N

N N





 

















 

 





_{1 12}

 

_{1 2}





1

1

27



1

2 3 1 3 3 2 2 1 2 2 2 1 1 1 1 1

2 1

.

.

x

,

x

d

x

,

x

d

,

x

,

x

d

,

x

,

x

d

,

x

,

x

d

max

x

,

x

d

N N N N

N N n

N N N

   





















0

0

0

1 3 3

3

1 2 2

2

1 1 1

1







  

n n

n n

n n

x

,

x

d

x

,

x

d

x

,

x

d

... ... ...









0

0

1 1 1 1

1





   



n N n

N N

n N n

N N

x

,

x

d

x

,

x

d

i.e. The Sequences

       

x

_1n

x

_2n

x

_3n





x

_Nn are cauchy sequence. Since



X

1

d

1

 

,

X

2

d

2

 

,

X

3

d

3









X

N

d

N



are complete metric space. So we have

where



₁



₂







_N



X

₁

,

X

₂

,

X

₃

,

X

₄





X

_N respectively. Now substituting

n

x

x

₁₁



₁ and

x

₂₁





₂ we obtain,















1 2



1

2 1 1

1 3

2 1 2 1 3

2 1 1 1 1 2 1 3

2 1 1



















_{ }



,

x

,

x

x

f

f

f

f

,

f

f

f

f

d

x

,

f

f

f

f

d

n n

N N N

n

N

















 

 









 







3 1 2 3

 

1 1 1 1





1 2 1 2



1

1 1 1 2 1 3

2 1 2

2 1 1 1 1



















  

 



N n

N N n n n

N

n n N

N n

n

f

f

f

,

x

d

,

x

,

x

d

x

,

f

d

,

x

,

x

d

,

f

f

f

f

f

,

d

,

x

,

x

d

max











 

 

 

 





 

N n

N n

n n

N N

n N n

n n

n

n n

n

x

lim

x

;

x

lim

x

;

x

lim

x

;

x

lim





























 

 

 

 

 

1 1

4 4

3 3

2 2

For

n





we get

d

₁



f

₁

f

₂

f

₃

f

₄





f

_N1



₂

,



₁





0

which is a contradiction and as



1 2 3 1 2 1



0

1

f

f

f

f





,





d





N Therefore

d

1



f

1

f

2

f

3





f

N1



2

,



1





0

1 2

1 3

2

1









f

f

f





f

_N

Similarly we obtain that

2 3

2 3

2

1 N







N

f

f

f

f

f





3 4

3 2

1

1 







 N N

N

f

f

f

f

f





4 5

4 2

1 1

2  







 N N N

N

f

f

f

f

f

f





...

...

...

f

₂

f

₃

f

₄





f

_N1

f

_N



₁





_N

Now we claim that,



₁ is the fixed point of

f

₁

f

₂

f

₃





f

_N in

X

₁

2



is the fixed point of

f

₂

f

₃

f

₄





f

_N

f

₁ in

X

₂

3



is the fixed point of

f

₃

f

₄

f

₅





f

_N

f

₁

f

₂ in

X

₃

... ... ...



_N is the fixed point of

f

_N

f

_N1

f

_N2





f

₂

f

₁ in

X

_N

Now let us take

x

₂₁



x

_2n and

x

₃₁



f

_N1



₂ and substituting in (1.1.2) we obtain







N N N N n



n N

N

x

f

f

f

f

,

f

f

f

f

f

d

x

,

f

f

f

f

f

d

2 1 2

1 2

1 3 2 1 2

2 2 1 3

2 1 2

 





























 























 





3 2 1 2 3 1 2



3

1 2 2 2 2 1 3

2 1 2

2

2 2 1 4

3 2

1 2 2 2 2

2 1 2

1 3

1 2 2 2 2

2 1 3

2 1 1 1 2 2

2













 

 



 

 



N N

N n

n n N

N n

n N

N

n n n

N N

n n n

N n

n

f

f

f

f

f

f

,

x

d

x

,

x

d

,

f

f

f

f

f

,

x

d

max

x

f

f

f

f

d

,

x

,

x

d

,

x

,

f

f

f

f

d

,

x

,

x

d

,

x

,

f

f

f

f

d

,

x

,

x

d

max

























Now let

n





and put

f

₁

f

₂

f

₃





f

_N1



₂





₁ we obtain





_{ }



_



_





_

1 1 3 3 2 1 2

1 2

3 2 1 3 2 1 2

1 2

2 2

2 1 2

1 2























 

 



N N N

N

N N

N N

N

f

f

,

d

,

f

f

f

f

d

max

,

f

d

,

f

f

f

f

,

d

,

f

f

f

f

d













Now

if,

max

 

d

₂



₂

,

f

_N

f

₁

f

₂





f

_N1



₂



,

d

₃





₃

,

f

_N1

f

_N



₁





2 1 2 1 2



2







d

,

f

_N

f

f





f

_N

Then we have



d

₂



f

_N

f

₁

f

₂





f

_N1



₂

,



₂





d

₂



f

_N



₁

,



₂







_{ }

 

_{ }

 



_

1 1 3 3 2 1 2

1 2

2

3 2 1 3 2 1 2

1 2

2 2

2 1 2

1 2

























 

 



N N N

N

N N

N N

N

f

f

,

d

,

f

f

f

f

,

d

max

,

f

d

,

f

f

f

f

,

d

max

,

f

f

f

f

d

















d

₃



f

_N1



₂

,



₃



Again if

max

 

d

₂



₂

f

_N

f

₁

f

₂





f

_N1



₂



,

d

₃





₃

,

f

_N1

f

_N



₁













d

_{3 3}

,

f

_N1

f

_N

Then similarly we have,



1 2 1 2 2



2



1 2



3



1 2 3



2

f

f

f

f





,





d

f



,







d

f





,



d

N





N N N

Again put

x

₃₁



x

_3n and

x

₄₁



f

_N3



₃ we obtain,



1 1 2 3 2 3 3



4



2 3 3



3

f



f

f

f

f

f





,







d

f





,



d

N N





N N