Some Fixed Point and Common Fixed Theorems in Banach Space for Rational Expressions

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Some Fixed Point and Common Fixed Theorems in

Banach Space for Rational Expressions

NEHA JAIN Barkatullah University Govt. Sci. & Comm. College Benazir Bhopal (M.P.) India

Abstract:

In present paper we prove some fixed point and common fixed point theorems for non contractive mapping in rational expression in Banach space, which generalize the well known results.

.

Key words: Banach space, Fixed point, Common fixed point, non contractive mapping

INTRODUCTION:

The study of non-contraction mapping concerning the existence of fixed points draws attention of various authors in non-linear analysis .It is well known that the differential and integral equations that arise in physical problems are generally non-linear, therefore the fixed point methods specially Banach contraction principle provides a powerful tool for obtaining the solutions of these equations which were very difficult to solve by any other methods.

continuous . A normed linear space is a linear space N in which to each vector z , there corresponds a real number denoted by

x

and called the norm of x in such a manner that

 

 

 

0 0 0

i x and x x

ii x y x y

iii ax a x

   

  



The non-negative real number

x

is to be thought of as the

length of vector x. If we regard

x

as a real function defined

on N. It is easy to verify that the normed function is called norm on N . It is easy to verify that the normed linear space N is a metric space w.r.to the metric d defined by

d x y

 

,

 

x

y

.

A Banach space is a complete normed linear space.

1. DEFINITION

Banach Space :-

A Banach space is a vector space X over the field R of real numbers, or over the field C of complex numbers, which is equipped with a norm and which is complete with respect to that norm, that is to say, for every Cauchy sequence {xn} in X, there exists an element x in X such that

lim_nxn x or equivalently:

lim

_n

x

n



x

X



0

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space X is a Banach space if and only if each absolutely convergent series in X converges,

1 n X

n v

 

   Implies that

1 n

n v

 

Completeness of a normed space is preserved if the given norm is replaced by an equivalent one.

In this chapter, we prove the following theorems:-

2 MAIN RESULTS

Theorem 8.2.1

Let T be a continuous self mapping defined on a Banach space

X , further T satisfies the following condition

3

2

3

2

...(8.2.1 )

,

x Tx y Ty x Ty

x

y

Tx Ty

x

y

y Ty y Tx x Ty

x

y

x

y

x Tx

a

with x

y with x

y Ty

x Ty

y Tx

x

y

x y

X

















 













 





  

_

 



_

  

_

 

 

_









 

0

2

2

1

, , , ,

0,1 .

y

and

where

 



 

    





 



 



Then T has unique fixed point.

Proof : Let

x

₀ be an arbitrary point in X and we define a

sequence

 

x

_n by means of iterates of T by setting

T

n

 

x

₀



x

₁

where n is a positive integer.

If

x

_n



x

_n1 for some n, then

x

_n is a fixed point of T.

1 1

3

1 1 1 1

2 1

3

1 1 1 1 1

2 1

1 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

x x Tx Tx

x Tx x Tx x Tx x x

x x

x Tx x Tx x Tx x x

x x

x Tx x Tx







 

   



    



 

  

    





    





 _    _

1 1 1

n n n n

n n

x Tx x Tx

x Tx





  

 _    _

 

3

1 1 1

2 1

3

1 1 1 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 n n n

n n n n

x x x x x x x x

x x

x x x x x x x x

x x

x x x x

x x x x









  



   



 

 

    



    





 _    _

   



 



1



xn xn



 













1 1 1

1 1 1

1 1

n n n n n n

n n n n n n

n n n n

x x x x x x

x x x x x x

x x x x

 







    

 

  

  

 

    _    _

 _     _ 

        

















1 1 1

1 1

1 1

1

1

1, 2 2 1

1 ...

...

....

.

n n n n n n

n n n n

n n n n

n

n n

x x x x x x

x x x x

x x s x x

where

s Since

x x s

    

 

    

 

    

 



 

 

  

 

 



         

   

  

 

  

   

      

 

  x₀x₁

By the triangle inequality , we have for m>n

1 1 2 1

1

0 0

...

....

1

0 1

,

n m n n n n m m

n n n

m n

n m

x x x x x x x x

s s s upto m

so

s

x x x Tx

s as m n

   



       

 

_    _ 

   

  



  

So

 

x

_n is a Cauchy sequence in X so by the completeness of

X, there is a point

v



x

such that

x

_n



v

or

n

 

.

But by continuity of T in X implies

1

lim

_n

lim

_n

lim

_n

n n n

Tv

T

x

Tx

x

_

v

  







_

_







V is fixed point of T in X.

Uniqueness: Suppose there is any other point w in X , where

3

2

3

2

v w Tv Tw

v Tv w Tw v Tw v w

v w

w Tw w Tv v Tw v w

v w

v Tv w Tw

v Tw w Tw

v w











  

    





    





  _   _

  _   _

 









2

2

.

v w

i e

v w v w

 

 







 

 

Which is a contradiction because 2

 

 1 .

So

v



w

. Hence fixed point is unique.

Theorem 8.2.2

Let T be self mapping defined on a Banach space such that

(8.2.1a) holds. If for some positive integerp,

T

p is continuous,

then T has a unique fixed point.

Proof:

We define a sequence as in Theorem (8.2.1 a). Since it converges to some point

v

in X . Therefore its subsequence

 

x

nk where



nk



np



also converges to

v

. Also

 

 

 

1

li

m

lim

lim

p p p

nk nk

k k

nk k p

T

v

T

x

T

x

x

v

T

v

v

 

 







_

_











So

v

is fixed point of p

Now we show that

Tv



v

. Let m be the smallest positive

integer such that

 

 



1



1

1, 2, 3,...,

1

because T v

v

,

m n

m m

m

for n

m

Tv v

Tv T T

v

put

x

T

v

v

T

v

v

v

v

y

T

 





 

































3

1 1 1 1

2 1

3

1 1 1 1 1

2 1

1 1

m m m m

m

m m m m m

m

m m

v Tv T v T T v v T T v v T v

v T v

T v T T v T v Tv v T T v v T v

v T v

v Tv T v T T v







   



    



 

 

   _  _



  

 

 

   _  _



  

 

 

 _    _

















1 1

1

1 1

1 1

1

1 1

.

m m

m

m m

m m

m

m

v T T v T v Tv

v T v

T v v T v Tv

T v v T v v v Tv

T v v

i e

Tv v s T v v

wher





  



  



   

 



 

 





 

 _  _ 

 

     

 

     _    _

    

  

1

1

1

e s s

   

    

On continuous this process we get

 

1

m

Tv v  s v Tv

So

v

is fixed point of T .

We can prove uniqueness as in Theorem 8.2.1

Now we further generalize the results of Theorem 8.2.1 in which T is neither continuous nor satisfies (8.2.1 a) In what

condition m

T

satisfies the same critical rational expression

and continuous , where m is a positive integer , still T has

unique fixed point.

Theorem 8.2.3.

Let T be a self map , defined on a Banach space X , such that

for some positive integer m satisfies the conditions :

 

 

3

2

3

2

m m m

m m

m m m

m m

m m

x T x y T y x T y

x

y

T

x

T

y

x

y

y T y y T x x T y

x

y

x

y

x T x

y T y

x T y

y T x















   

_

_





  











   

_

_



  











_



 

_







_



 

_

 

with

0

, , , ,

0,1

2

2

,

1

,

x

y

x y

x

x

y

x

y

with

   





 



 



























Then T has Unique fixed point in X .

Proof :

From the given condition of Theorem we assume that

T

m has

 

m

T

v



v

 

 

 

1

m m T v T T v

T v

v



 



We conclude that T is also a fixed point of m

T

and m

T

has unique fixed point

v

_.

So

v

_{is unique fixed point of T .}

Next we generalize Theorem (8.2.1) for three mapping F G, and T .

Theorem 8.2.4. : Let T and F be two self maps , defined

on a Banach space X satisfying the conditions

 

 

3

2

3

2

x Tx y

Fy x

Fy y Tx

x

y

T x

F y

x

y

y

Fy y Tx x

Fy

x

y

x

y

x Tx

y

Fy

x

Fy

y Tx

















   

_

_





  











   

_

_



  





  

_

 



_

  

_

 



_





x



y

...(8.2.4 )

a

 

with x

y with

0

, , ,

,

,

x

y

with

,

0,1 .

x y

X

    













(8.2.4b) T and F are continuous on X .

(8.2.4c) There exist an

x

0



X

, such that in the sequence

 

x

n , where

1

1

n n

n

Tx where n is even x

Fx where n is odd

 

  



Proof : we have

2 2 1 2 1 2

3

2 1 2 1 2 2 2 1 2 2 2 1 2 1 2

2

2 1 2

3

2 2 2 2 1 2 1 2 2 1 2

2

2 1 2

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 Tx Fx

x Tx x Fx x Fx x Tx x x

x x

x Fx x Tx x Fx x x

x x                       _  _           _  _      

2 1 2 2 2

2 1 2 2 2 1

2 1 2

2 1 2 2 2 1 2 1 2 1 2 2 2 1 2

n n n n

n n n n

n n

n n n n n n n n n n

x Fx x Fx

x Fx x Tx

x x

x x x x x x x x x x

              _    _  _    _   

     _  _3

2

2 1 2

3

2 2 1 2 2 2 1 2 1 2 1 2

2

2 1 2

2 1 2 1 2 2

2 1 2 2 2 1

n n

n n n n n n n n

n n

n n n n

n n n n

x x

x x x x x x x x

x x

x Tx x Fx

x Fx x Tx

                      _  _        _    _  _    _

2 1 2

 x_n x_n









2 2 1 2 1 2

2 2 1 2 1 2

2

2 2 1 0 1

2 1

2 1 2 2 0 1

1

1 1

...

...

n n n n

n n n n

n

n n

n

n n

x x x x

x x l x x

where

l

x x l x x

x x l x x

                                               

By the completeness of X ,

 

x

n converges to a pointX . Suppose

 

x

_n



p

,

And then the subsequence

 

x

_nk also converges to P .

Now

 



lim

nk



lim

nk 1

k k

TF P

TF

x

x

_

P

 







Now we will prove that

 

F P



P

Suppose

F P

 



P

then

 

 

 

 

 

   

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

3

2

3

2

P F P TF P F P

F P TF P P F P F P F P P TF P F P P F P P

P F P P TF P F P F P F P P F P P

F P F P P F P F P F P P TF P F P P





  

  

 

    _  _



  

 

 

   _  _



  

 

 

 _    _

 

 _    _

 





 

 

 

 





 

 

 





 

 

( )

( ) ( )

2

So

.

. 2 1

F P P F P TF P P F P P TF P

F P P F P P TF P P P F P P TF P

P F P

But

F P P

   



   



   

   

 

     _    _

 

 

    _      _

 

    

   

 

Which is a contradiction. So

 

F P



P

Now TF P

 

T F P_

 

_T P

 

P

That is a common fixed point of T and F . Uniqueness :

If possible let q (where

q



p

) be another common fixed point

of F and T . i.e .

F q

 



T q

 



q

Now

3

2

3

2

P q TP Fq

P TP q FP P Fq q TP P q

P q

q FP q TP P Fq P q

P q

P TP q Fq







  

       _{ }



  

 

      _{ }



  

 

  _   _





2

sin 2 1

( )

, ce

P F q q TP

P q

P q

so p q





 

 

 

 

  _   

   

 

 





 



Hence fixed point is unique.

 

 

 

 

 

 

 

 

 

 

 

 

3

2

3

2

x GF x y TF y x TF y y GF x x y GF x TF y

x y

y TF y y GF x x TF y x y x y

x GF x y TF y x TF y y





 

       _{ }

 

  

 

      _{ }



  

 

 

 _    _

  

 

3 2 1.

G, F, T has int . So common fixed po

GF x x y

If    

 

    

  

 

 

Proof : Let

x

0 be an arbitrary point and we defined a sequence

 





 





 









 





 

2 1 2 2 2 2 1

2 1 2 2 2 2 1

2 2 2 1 2 1 2 2 1

3

2 1 2 2 2 1

3

2 2 1

2 1 2 1 2 1 2

,

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 GF x x TF x

Now

x x GF x TF x

x GF x x TF x x TF x

x GF x x x

x x

x TF x x GF x

x 



  

  

  

 



  

 

  

  

  _  _



  

 

 



















 

3

2 2 1 2 2 1

2

2 2 1

2 2 1 2 1 2 1

2 2 1 2 1 2

2 2 1

n n n n

n n

n n n n

n n n n

n n

TF x x x

x x

x GF x x TF x

x TF x x GF x

x x







 



  

 



  _  _

  

 

 

 _    _

 

 _    _

 

2 2 1 2 1 2 2 2 2 2

3

2 1 2 1 2 2 1

2

2 2 1

2 1 2 2 2 1 2 1

3

2 2 2 2 2 1

2

2 2 1

2 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 n n

x x x x x x

x x x x

x x

x x x x

x x x x

x x

x x x







   

  



   

 





  

  _  _

  

 

 

  _  _

 



 

 







1 2 2

2 2 2 2 1 2 1

2 2 2

n

n n n n

n n

x

x x x x

x x





 

  





  



 

 

 _ _



2 2 1 2 2 1

2 2 1 2 1 2 2 2 1 2 2

2 2 1 2 1 2 2 2 2 1

n n n n

n n n n n n

n n n n n n

x x x x

x x x x x x

x x x x x x











   

 

    

   

   

 _      _

 _     _ 





   



2 2 1





2 1 2 2

2 1 2 2 2 2 1

2 1 2 2 2 2 1

2

1 2

1 1 2

3 2 1

n n n n

n n n n

n n n n

x x x x

x x x x

x x l x x

where

bec use l

a



 

    

 

    

 

 



 

  

  

  

   

     

 _{ } 

 

 

  

     _

 _{ } 

 

    



Proceeding in the same manner, we get

2 2 1

...

n

So

 

x

n is a Cauchy sequence in _X . By the completeness of X , there is a point pX such that

x

n



p

as

n

 

. Now we assume that

                       

2 1 2 1

2 1 2 2 3 2 2 2 2 2 2 2 , 0 n n n n n n

n n n

n

n

p TF p then p TF p

P TF P P x x TF P

x

x GF x

x

GF x TF P P

P TF P

P GF

P TF

TF P x x P

x P

x

P F x

P G                         _ _                       2 2 2 3 2 2 2 2

2 2 1

2 2 1 2

2 n n n n n n n n n

n n n

P TF P

P

TF P x P

x P

x GF x

x TF P x

x P P x

GF

P TF P x F

x x T P

           _ _        _   _    _   _                     

2 1 2

2

2 1 2

2

2

2 2 1

2 2 1

3 2 3 2 n n n n n n n n n n n

x x P

x P

x x TF P

x P

x P

x

P

P TF P P

P TF P x

x TF P P x

         _ _                           _   _    _      

2 2 1

xn P P xn

P TF P P TF P

Which is a contradiction because

 

 3



2

 

 1 .

So

TF P

 



P

.

Similarly we assume

P



GF

we get a contradiction.

Hence

TF P

 



GF P

 



P

So P is common fixed point of

TF and GF

.

So P is common fixed point of G, F and T as we proved

before Uniqueness can be proved easily .