Some Fixed Point and Common Fixed Point Theorems in 2-Banach Spaces

(1)

w w w . a j e r . u s Page 122

American Journal of Engineering Research (AJER) e-ISSN : 2320-0847 p-ISSN : 2320-0936 Volume-02, Issue-05, pp-122-127 www.ajer.us Research Paper Open Access

Some Fixed Point and Common Fixed Point Theorems in 2-Banach Spaces

A.S.Saluja, Alkesh Kumar Dhakde

Govt. J.H. P.G. College Betul (M.P.), India IES College of Technology Bhopal (M.P.), India

Abstract: In the present paper we prove some fixed point and common fixed point theorems in 2-Banach spaces for new rational expression. Which generalize the well known results.

AMS: 47H10, 54H25.

Keywords: Banach Space, 2-Banach Spaces, Fixed point, Common Fixed point.

I. 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’s contraction principle provides a powerful tool for obtaining the solutions of these equations which were very difficult to solve by any other methods. Recently Verma [9] described about the application of Banach’s contraction principle [2]. Ghalar [5] introduced the concept of 2-Banach spaces. Recently Badshah and Gupta [3], Yadava, Rajput and Bhardwaj [10], Yadava, Rajput, Choudhary and Bhardwaj [11] also worked for Banach and 2-Banch spaces for non contraction mappings. In present paper we prove some fixed point and common fixed point theorems for non-contraction mappings, in 2-Banach spaces motivated by above, before starting the main result first we write some definitions.

Definition (1.2A), 2-Banach Spaces:

In a paper Gahler [5] define a linear 2-normed space to be pair

  L , .,

where

L

is a linear space and

., .

is non negative , real valued function defined on

L

such that

a , b , c  L

(i)

a , b  0

if and only if

a

and

b

are linearly dependent (ii)

a , b  b , a

(iii)

a ,  b   a , b , 

is real (iv)

a , b  c  a , b  a , c

Hence

., .

is called a 2-norm.

Definition (1.2B):

A sequence

  x

n in a linear 2-normed space

L

, is called a convergent sequence if there is,

x  L

, such that

L y y

x x

_n

n

  





, 0 for all

lim

.

Definition (1.2C):

A sequence

  x

n in a linear 2-normed space

L

, is called a Cauchy sequence if there exists

y , z  L

, such that

y

and

z

are linearly independent and

(2)

lim , 0

,

 





x

_m

x

_n

y

n m

Definition (1.2D):

A linear 2-normed space in which every Cauchy sequence is convergent is called 2-Banach spaces.

II. MAIN RESULTS Theorem 1.1:

Let

T

be a mapping of a 2-Banach spaces into itself. If

T

satisfies the following conditions:

T

²

 I

, where

I

is identity mapping (1.1)

a y a x

Tx y a Ty x a

Ty y a Tx x

a y x

a y x a Ty x a Tx y a Ty y a

y x

a Ty y a Tx a x

Ty Tx

2 ,

, ,

2 , ,

,

, ,

₂

3



 



 



   

 



 



   









 



 













(1.2)

Where

x  y

,

a  0

is real with

8   10   4   2   3   4

.Then

T

has unique fixed point.

Proof: Suppose

x

is any point in 2-Banach space

X

. Taking

y   T  I  x , z  T   y

2 1

  Tx a T Ty a x T Ty a x

z  ,  

²

,   ,

     

   

a Tx a y

Ty Tx a Tx T y a

Tx T Tx a Ty y

a Tx y

a Tx y a Tx T y a Ty Tx a Tx T Tx a

Tx y

a Tx T Tx a Ty y

2 ,

, ,

2 , ,

,

, ,

, , ,

2

3



 



 



   

 



 



   









 



 











 

a Tx a y

Ty y a y Tx a x y a

x Tx a Ty y

a Tx x

a Tx y a x y a Ty y a y Tx a x Tx a

Tx x

a x Tx a Ty y

2 ,

, ,

, 2

, ,

4 , 1

, ,

, 2 ,

1 , ,

2

3



 



 



     

 



 



   













 



 











a Tx x a

Ty y a Tx x a Tx a x

x Tx a Ty y

a Tx x

a Tx x a Tx x a Ty y a Tx x a x Tx a

Ty y

2 , 2

, 2 ,

, 1 2

1

2 , ,

4 , 1

8 , , 1 2

, 1 2 ,

, 1

, 2

2

3



 

 





 





     

 



 



   







 

 

   









 







a Tx a x

Ty y a Tx x a

x Tx a Ty y

a Tx x

a Tx Ty x

y a Tx x a

Ty y

2 , 2

, ,

2 , ,

, , ,

2 4 1 , 2

2

₂

3



 



 



   

 



 



   





 





 





 

 

    







 



 

(3)

 

 ^y ^Ty â ^Tx ^x â   ^x ^Tx â ^y ^Ty â  ^x ^Tx â

a Tx x a Ty y a Tx x a

Ty y

2 , , 2 ,

, 2 ,

, ,

4 , 2 2

, 2











































 

           



 

 

















 



 



   

 



 



   





4 4 2 ,

3 1 2 ,

1 2 , 2 2

2 2 2 2 2 , 3

a Ty y a

Tx x

a Ty y a

Tx x

Now for

a Ty Tx a x z y a x

u  ,  2   ,   ,

a y a x

Tx y a Ty x a

Ty y a Tx x

a y x

a y x a Ty x a Tx y a Ty y a

y x

a Ty y a Tx x

2 ,

, ,

2 , ,

,

, ,

, , ,

2

3



 



 



   

 



 



   









 



 











a Tx x a

Tx x a Tx a x

Ty y a Tx x

a Tx x

a Tx x a

Tx x a Tx x a Ty y a

Tx x

a Ty y a Tx x

2 , 2

2 , , 1 2

1 2

, ,

4 , 1

8 , , 1 2

, 1 2

, 1

2 , 1

, ,

2

3



 

 





 





   

 



 



   





 

 

 







 

 







a Tx x a Tx x

a Ty y a Tx a x

Tx x a

Ty y a Ty y

2 , 2 ,

2 , , ,

, 2 ,

2 







 

 



   



















 





 

 



  



 



 



   



 , 2 2

2 2 2

,  2      

a Ty y a

Tx x

                 



 , 4 2

2 , 1

2 1 x Tx a y Ty a

Now

a u x a x z a u

z  ,   ,   ,

     



  













































2 4 2 ,

1

2 , 4 1

4 2 ,

3 1 2 ,

1

a Ty y

a Tx x a

Ty y a

Tx x

                                 



 , 4 4 4 2

2 3 1

2 ,

1 x Tx a y Ty a

                   



 , 8 6 2

2 2 1 2 2 4 2 ,

1 x Tx a y Ty a

(1.3) On the other hand

(4)

   

a y Ty

a y T y y T

a z y y T a u z

, 2

, 2 ,

















(1.4)

So

              ,  8   6   2    

2 2 1 2 2 4 2 ,

, 1

2 Ty y a x Tx a y Ty a

 

 4  8  6  2   y  Ty , a   4  2  2  2  x  Tx , a

        

 



8 10 4 2 3 4



as ,

,

2 , 2 2 4

2 6 8 , 4













 



 































a Ty y k a Tx x

a Ty y a

Tx x

Where

 

2 1 2 2 4

2 6 8

4 



 













 k 

Let

R   T  I 

2

1

, then

         

 

a Tx k x

a Ty y a

y y R

a x R x R R a x R x R

2 ,

2 , , 1

,

2

,



















By the definition of

R

we claim that

 ^R

ⁿ

  ^x 

is a Cauchy sequence in

X

,

 ^R

ⁿ

  ^x 

is converges to so element

x

0 in

X

. So

lim   R

ⁿ

  x x

0

n





 . So

 R   x

₀

  x

₀ . Hence

T   x

₀

 x

₀ So

x

₀ is a fixed point of

T

.

Uniqueness:

If possible let

y

₀

 x

₀ is another fixed point of

T

.Then

a Ty Tx a y

x

₀



₀

, 

₀



₀

,

a y a x

Tx y a Ty x a

Ty y a Tx x

a y x

a y x a Ty x a Tx y a Ty y a

y x

a Ty y a Tx x

2 ,

, ,

2 , ,

,

, ,

, , ,

0 0 0

0 0

0 0 0

2 0 0

3 0 0 0

0 0 0 0 0 0

0

0 0 0 0



 



 



   

 



 



  









 



 











a y x a y x a y

x

₀



₀

, 

₀



₀

, 

₀



₀

,

   

    ^x

0

 ^y

0

^, ^a

   

Which is contradiction so

x

₀

 y

₀.Hence fixed point in unique.

Theorem 2.1:

Let

T

and

G

be two expansion mappings of a 2-Banach space

X

into itself.

T

and

G

satisfy the following conditions:

(2.1)

T

and

G

commute

(2.1)

T

²

 I

and

G

²

 I

, where

I

is identity mapping.

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w w w . a j e r . u s Page 126 (2.3)

a Gy a Gx

Tx Gy a Ty Gx a

Ty Gy a Tx Gx

a Gy Gx

a Gy Gx a Ty Gx a Tx Gy a Ty Gy a

Gy Gx

a Ty Gy a Tx a Gx

Ty Tx

2 ,

, ,

, 2

, ,

,

, ,

, , ₂

3







 



  





 



   









 



 













For every

^x ^, ^y ^ ^X ^, ^ ^, ^ ^, ^ ^, ^ ^, ^ ^   ⁰ ^. ¹

^with

x  y

and

Gx Gy  0

and

      1

. Then there exists a unique common fixed of

T

and

G

such that

T   x

0

 x

0 and

G   x

₀

 x

₀

.

Proof: -

Suppose

x

is point in 2-Banach space

X

it is clear that

 

TG ² I

           

   

               

   

               

   

G x GG y a G

a x G T y G G a y G T x G G a

y G T y G G a x G T x G G

a y G G x G G

a y G G x G G a y G T x G G a x G T y G G a y G T y G G

a y G G x G G

a y G T y G G a x G T x G a G

y G TG x G TG

,

2

, ,

2

, ,

,

, ,

,

, , ,

. .

2 2

2 2 2

2 3 2

2 2















   













   









 



 













         

a Gy Gx

a Gx TG y G a Gx TG Gx a

Gy TG Gy a Gx TG Gx

a Gy Gx

a Gy Gx Gy

TG Gx a Gx TG Gy a Gy TG Gy a

Gy Gx

a Gy TG Gy a Gx TG Gx

,

2

, ,

2

, ,

,

, ,

2

3 2







 



   





 



   









 



 











Taking

^G   ^x  ^p

,

^G   ^y  ^q

, where

p  q

         

       

a q a p

p TG q a q TG p a

q TG q a p TG p

a q p

a q p q TG p a p TG q a q TG q a

q p

a q TG q a p TG p

2 ,

, ,

2

, ,

,

, ,

2

3







 



   





 



   









 



 











Taking

TG  R

we get

             

       

a q a p

p R q a q R p a

p R q a p R p

a q p

a q p q R p a p R q a q R q a

q p

a q R q a p R a p

q R p R

2 ,

, ,

2

, ,

,

, ,

,

, , ₂

3







 



   





 



   









 



 













It is clear by theorem (1.1); that

R  TG

has at least one fixed point say

x

₀ in

K

that is

  x

0

TG   x

0

x

0

R  

And so

T .      TG x

0

 T x

0

Or T²



Gx₀



T

 

x₀

G     x

0

 T x

0

Now

(6)

  x a Tx T T   x a T

Tx a x

Tx

₀



₀

, 

₀



² ₀

, 

₀

 .

₀

,

       

   

               

   

               

 

x G



Tx



a G

a x T Tx G a Tx T x G a

Tx T Tx G a x T x G

a Tx G x G

a Tx G x G a Tx T x G a x T Tx G a Tx T Tx G

a Tx G x G

a Tx T x GT a x T x G

,

2

, ,

2

, ,

,

, ,

,

, ,

0 0

2 0 0

3 0 0

0 0







 



   





 



   









 



 











    Tx

₀

 x

₀

, a

   

So

T   x

₀

 x

₀

       1 

That is

x

₀ is the fixed point of

T

. But

T   x

0

 G   x

0 so

G   x

0

 x

0. Hence

x

₀ is the fixed point of

T

and

G

.

Uniqueness:

If possible let

y

₀

 x

₀ is another common fixed point of

T

and

G

. Then

x

₀

 y

₀

, a  T

²

  x

₀

 T

²

  y

₀

, a  T  T   x

₀

  T  T   y

₀

 , a

       

   

               

   

               



^Tx



^G



^Ty



^a

G

a Tx T Ty G a Ty T Tx G a

Ty T Ty G a Tx T Tx G

a Ty G Tx G

a Ty G Tx G a Ty T Tx G a Tx T Ty G a Ty T Ty G

a Ty G Tx G

a Ty T Ty G a Tx T Tx G

,

2

, ,

2

, ,

,

, ,

,

, ,

0 0

2 0 0

3 0 0

0 0







 



   





 



   









 



 











a y x a y x a y

x

₀



₀

, 

₀



₀

, 

₀



₀

,

   

    x

₀

 y

₀

, a

   

But

      1

So

x

₀

 y

₀. So common fixed point in unique.

REFERENCES:

[1]. Ahmad, and Shakil, M. “Some fixed point theorems in Banach spaces” Nonlinear Funct.Anal. & Appl. 11 (2006) 343-349.

[2]. Banach, S. “Surles operation dans les ensembles abstraits et leur application aux equations integrals” Fund. Math.3 (1922) 133-181.

[3]. Badshah, V.H. and Gupta, O.P. “Fixed point theorems in Banach and 2-Banach spaces” Jnanabha 35(2005) 73-78.

[4]. Goebel, K. and Zlotkiewics, E. “Some fixed point theorems in Banach spaces” Colloq Math 23(1971) 103-106.

[5]. Gahlar, S. “2-metrche raume and ihre topologiscche structure” Math.Nadh.26 (1963-64) 115-148.

[6]. Isekey, K. “fixed point theorem in Banach space” Math.Sem.Notes, Kobe University 2 (1974) 111-115.

[7]. Jong, S.J.Viscosity approximation methods for a family of finite non expansive in Banach spaces” nonlinear Analysis 64 (2006) 2536-2552.

[8]. Sharma, P.L. and Rajput S.S. “Fixed point theorem in Banach space” Vikram Mathematical Journal 4 (1983) 35-38.

[9]. Verma, B.P. “Application of Banach fixed point theorem to solve non linear equations and its generalization” Jnanabha 36 (2006) 21-23.

[10]. Yadava, R.N., Rajput, S.S. and Bhardwaj, R.K. “Some fixed point and common fixed point theorems in Banach spaces” Acta Ciencia Indica 33 No 2 (2007) 453-460.

[11]. Yadava, R.N., Rajput, S.S. , Choudhary , S. and Bhardwaj , R.K. “Some fixed point and common fixed point theorems for non- contraction mapping on 2-Banach spaces” Acta Ciencia Indica 33 No 3 (2007) 737-744.