Vol 3, No 3 (2012)

Available online through

ISSN 2229 – 5046

RELATED FIXED POINT THEOREM ON N METRIC SPACES

Vishal Gupta

1

*, Ashima Kanwar

2

and R. K. Saini

3

1

Department of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana (India)

E-mail:

2

Department of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Hayana (India)

E-mail

3

Department of Mathematics, Bundelkhand University, Jhansi (U.P), (India)

E-mail:

(Received on: 22-02-12; Accepted on: 15-03-12)

________________________________________________________________________________________________ ABSTRACT

T

he aim of this paper to discuss about related fixed point theorem on n metric spaces. Generally, Fixed point theorems on two,three, four and five complete metric spaces have been proved by many authors. In this paper, we prove a new fixed point theorem on metric space using the concept of continous mapping and cauchy sequence on complete metric spaces. Our theorem generalize and extends many result.

________________________________________________________________________________________________

1. INTRODUCTION

In this paper, we are dealing with

n

metric space and we prove a fixed point theorem on

n

metric space..

In 2009, Faycal Merghadi, Abdelkrim Aliouche, Brain Fisher proved. a fixed point theorem in n complete metric spaces using the concept of implicit relation.

In 2010, Brat Ojha also proved common fixed point theorem for n metric spaces.

Now, in this paper we proved a new fixed point theorem on n metric spaces using continous mapping.

2. PRELIMINARIES

Definition 2.1: Let

( , )

X d

be a metric space. A sequence in

X

n is a function

x

:





X

.

The image

x n

( )

of

n

∈



is usually dentoed by

x

n, called the nth term of the sequence

{ }

x

n .

Definition 2.2: Let

( ,

X d

₁

)

and

( ,

Y d

₂

)

be two metric spaces and let

f

:

X



Y

be a mapping. Then

f

is said to be continuous at

x

∈

X

if and only if the following criterion is satisfied:

∀

ε

> 0

∃

a positive number

δ ε

( , )

x

such that

d

₂

(

f x

( ), ( ) <

f y

)

ε

for all points

y

∈

X

satisfying

d x y

₁

( , ) <

δ

. If

f

is continuous at every point of

X

, then

f

is called continuous function.

Definition 2.3: A sequence

{ }

x

_n in a metric space

( , )

X d

is said to be Cauchy sequence if and only iff

∀

ε

> 0

∃

a postive integer

n

₀

( )

ε

such that

d x

(

_m

,

x

_n

)

< ,

ε

∀

n m n

,



₀

Definition 2.4: A metric space

( , )

X d

is said to be complete if and only if every Cauchy sequence in

X

converges to a point of

X

.

________________________________________________________________________________________________

Page: 902-908

3. MAIN RESULT

Theorem 3.1: Let

(

X d

_i

,

_i

)

be

n

complete metric space and

T

_i be

n

mapping and

T T T

₁

,

₂

,

₃

,...,

T

_n−1 are continuous mapping such that

T X

_i

:

_i



X

_i+1 for

i

= 1, 2,



,

n

−

1

and

T

_n

:

X

_n



X

₁ and satisfying the following inequalities:

( )

( )

(

)

₍

(

1 1

₎

)

1 2

1 1

1 1 2 1 2 1 2 1 1 1

1 2

,

,

,

n n n n

cF x x

d T T

T T x

T T

T T x

G x x

−



−





(3.1)

( )

( )

(

)

₍

(

2 2

₎

)

2 2

2 2

2 1 3 2 2 1 3 2 2 2

2 2

,

,

,

n n

cF x x

d T T

T T x

T T

T T x

G

x x







(3.2)

( )

( )

(

)

₍

(

3 3

₎

)

3 2

3 3

3 2 1 4 3 2 2 1 5 4 3 3 3

3 2

,

,

,

n n

cF x x

d T T T

T T x

T T T

T T T x

G

x x







(3.3)

( )

(

( )

)

₍

(

2

₎

)

1 2 1 2 1 2 1

2

,

,

,

n n n

n n

n n n n n n n n n

n

cF x x

d T T

T T

x

T T

T T

x

G

x x

− −



− −





(3.4)

1 1 2 2 3 3

2 1 2 2 2 3 2

,

,

,

,

,

,

,

n

,

n _n

x x

X x x

X

x x

X

x x

X

∀

∈

∈

∈

_

∈

for which

G x x

₁

(

1

,

1₂

)

≠

0,

G

₂

(

x x

2

,

₂2

)

≠

0,

G

₃

(

x x

3

,

₂3

)

≠

0,



,

G

_n

(

x x

n

,

₂n

)

≠

0

where

0



c

< 1

.

(

1 1

)

1

,

2

F x x

= max

{

d

₁

(

x T T

1

,

_{n n−1}



T T x d

_{2 1} 1

)

_n

(

T T

_{n−1 n−2}



T T x T T

_{2 1 2}1

,

_{n−1 n−2}



T T x

_{2 1} 1

)

,

d x T T

₁

(

1

,

n n−₁



T T x d

_{2 1} 1

) (

n−₁

T

n−₂

T

n−₃



T T x T

_{2 1 2}1

,

n−₂

T

n−₃



T T x

_{2 1} 1

)

,

(

1 1

) (

2 1

) (

1 1

)

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

,

d x T T

,

_{n n−}

T T x d T T x T T x

,

,

d x T T

,

_{n n−}

T T x







d T x T T

₂

(

_{1 2}1

,

_{1 n}



T T T x

_{3 2 1 2}1

) (

,

d x T T

₁ 1

,

_{n n−1}



T T x

_{2 1 2}1

) (

d T x T x

_{2 1 2}1

,

₁ 1

)

}

(

2 2

)

{

(

2 2

) (

2 2

) (

2 2

)

2

,

2

= max

2

,

1 n 3 2 1 n n 1 3 2 2

,

n n 1 3 2

,

2

,

1 n 3 2

F x x

d

x T T



T T x

d T T

₋



T T x T T

₋



T T x

d

x T T



T T x

d

_n

(

T T

_{n−1 n−2}



T T x T T

_{4 3} 3

,

_{n−1 n−2}



T T x

_{3 2} 2

) (

,

d

₂

x T T

2

,

_{1 n}



T T x

_{3 2} 2

)

(

2 2

)

(

2 2

)

1 2 3 3 2 2

,

2 3 3 2

,

,

2

,

1 3 2

n n n n n n

d

₋

T

₋

T

₋



T T x T

₋

T

₋



T T x



d

x T T



T T x

d T x T T T

₃

(

_{2 2}2

,

_{2 1 n}



T T x

_{3 2 2}2

) (

,

d

₂

x T T

2

,

_{1 n}



T T x

_{3 2 2}2

) (

d T x T x

_{3 2 2}2

,

₂ 2

)

}

(

3 3

)

{

(

3 3

) (

3 3

) (

3 3

)

3

,

2

= max

3

,

2 1 n 4 3 2 1 n 4 3 2

,

1 n 4 3

,

3

,

2 1 n 4 3

F x x

d

x T T T



T T x

d T T



T T x T T



T T x

d

x T T T



T T x

d T T

₁

(

_{n n−1}



T T x T T

_{4 3 2}3

,

_{n n−1}



T T x

_{4 3} 3

) (

,

d

₃

x T T T

3

,

_{2 1 n}



T T x

_{4 3} 3

)

d

_n

(

T T

_{n−1 n−2}



T T x T T

_{4 3 2}3

,

_{n−1 n−2}



T T x

_{4 3} 3

) (

,

d

₃

x T T T

3

,

_{2 1 n}



T T x

_{4 3} 3

)

(

)

(

)

3 3 3 3

1 2 3 4 3 2

,

2 3 4 3

,

,

3

,

2 1 4 3

n n n n n n

d

₋

T

₋

T

₋



T T x T

₋

T

₋



T T x



d

x T T T



T T x

d T x T T T T

₄

(

_{3 2}3

,

_{3 2 1 n}



T T x

_{4 3 2}3

) (

,

d

₃

x T T T

3

,

_{2 1 n}



T T x

_{4 3 2}3

) (

d T x T x

_{4 3 2}3

,

₃ 3

)

}

  

(

,

2

)

= max

{

(

,

1 2 1

) (

1 2 3 1 2

,

2 3 1

)

,

n n n n n n

n n n n n n n n n n n n

F x x

d

x T T

_{− −}



T T x

d

₋

T

₋

T

₋



T T x T

₋

T

₋



T T x

(

n

,

_{1 2 1} n

) (

_{2 3 4 1 2}n

,

_{3 4 1} n

)

,

,

n n n n n n n n n n n

d

x T T

_{− −}



T T x

d

₋

T T

_{− −}



T T x T T

_{− −}



T T x



(

) (

1

) (

) (

)

}

1 2 1 1 2 1 2 1 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

© 2012, IJMA. All Rights Reserved 904

And

(

1 1

)

{

(

1 1

) (

1 1

)

(

1 1

)

}

1

,

2

= max

1

,

n n 1 2 1 2

,

1

,

n n 1 2 1

,

2 1

,

1 n 2 1 2

G x x

d x T T

₋



T T x

d x T T

₋



T T x

d T x T T



T T x

(

2 2

)

{

(

2 2

) (

2 2

)

(

2 2

)

}

2

,

2

= max

2

,

1 n 3 2 2

,

2

,

1 n 3 2

,

3 2

,

2 1 n 3 2 2

G

x x

d

x T T



T T x

d

x T T



T T x

d T x T T T



T T x

  

(

1

)

{

(

) (

)

(

)

}

1 2 1 1 1 2 1 1 1 1 1

,

= max

,

,

,

,

,

n n n n n n n

n n n n n n n n n n n n n

G

x x

d

x T T

_{− −}



T T x

d

x T T

_{− −}



T T x

d T x T T

₋



T T x

Then

T T

_{n n−1}



T T

_{2 1} has a unique fixed point

α

₁

∈

X

₁,

T T

_{1 n}



T T

_{3 2} has a unique fixed point

α

₂

∈

X

₂, and so on,

1 2 1

n n n

T T

_{− −}



T T

has a unique fixed point

α

n

∈

X

n.

Further,

T

₁

( )

α

₁

=

α

₂

,

T

₂

( )

α

₂

=

α

₃

,

T

₃

( )

α

₃

=

α

₄

,



,

T

_n−1

(

α

_n−1

)

=

α

_n

,

T

( )

α

_n

=

α

₁

Proof: Let

x

₀

′ ∈

X

, be any arbitrary point. We define sequences

{

x

1m

},

2

{

x

m

}

,

3

{

x

m

},



,{

x

mn

}

with

1

,

2

,

3

,

,

n

X X

X



X

respectively as follows:

(

)

1 1 2 1

1 2 1 0 1 1

=

m

,

=

,

m n n m m

x

T T

₋



T T

x x

T x

₋

x

_m3

=

T

₂

( )

x

_m2

,

x

_m4

=

T x

₃

( )

_m3

∀ ∈

m



We assume that

x

m1

≠

x

1m+₁

,

x

m2

≠

x

m2+₁

x

m3

≠

x

m3+₁

,

1

n n

m m

x

≠

x

₊

∀ ∈

m





Taking

x

2

=

x

_m2 and

x

₂2

=

x

_m2₋₁ in inequality (3.2), we get,

(

2 2

)

2 m

,

m 1

d

x

x

₊

(

2

( )

2

)

2 1 3 2 1 1 1 3 2

=

d T T

n



T T x

m−

,

T T T

n n−



T T

x

m

(

)

(

)

2 2

2 1

2 2

2 1

,

,

m m

m m

cF x

x

G

x

x

−

−



(

) (

{

) (

)

(

) (

) (

)

(

)

2 2 2 2 2 2

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

2 2 2 2 2 2

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

2 2 2

2 1 3 2 3 2 1 2 1 2

max

,

,

,

,

,

,

,

,

,

,

,

,

m n m n n m n n m m n m

n n n m n m m n m n n n m n m

m n m m n

c

d

x T T

T T x

d T T

T T x

T T

T x

d

x T T

T T x

d T T

T T x

T

T T x

d

x T T

T T x

d

T T

T T x

T

T T x

d

x T T

T T x

d T x

T T T

T

− − −

− − − − − − − − −

−

=























(



) (

) (

)

}

(

) (

) (

)

{

}

2 2 2 2 2

1 2 1 3 2 1 3 1 1 2

2 2 2 2 3 3

2 2 1 3

,

,

,

max

,

,

,

,

,

m m n m m m

m m m m m m

x

d

x T T

T T x

d T x

T x

d

x

x

d

x

x

d

x

x

− − −

+



(

2 2

)

{

(

1 1

) (

)

(

1 1

)

(

3 3

)

}

2

,

1

max

1 1

,

,

1

,

,

1 1

,

,

,

3 1

,

n n n n

m m m m n m m n m m m m

d

x

x

₊



c

d x

₋

x

d

x

₋

x

d

₋

x

−₋

x

−



d

x

₋

x

(3.5)

Now taking

x

3

=

x

_m3 and

x

₂3

=

x

_m3₋₁ in inequality (3.3), we obtain,

(

3 3

)

(

3 3

)

3 m

,

m 1

=

3 2 1 n 4 3 m 1

,

2 1 n 4 3 m

d

x

x

₊

d T T T



T T x

₋

T T T



T T x

(

)

(

)

3 3

3 1

3 3

3 1

,

,

m m

m m

cF x

x

G

x

x

−

−

Page: 902-908

(

) (

{

) (

)

(

) (

) (

)

(

) (

)

3 3 3 3 3 3

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

3 3 3 3 3 3

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

3 3 3 3

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

max

,

,

,

,

,

,

,

,

,

,

,

,

,

m n m n m n m m n m

n m n m m n m n n m n m

m n m m n m

c

d

x T T T

T x

d T T

T T x

T T

T T x

d

x T T T

T x

d T

T T x

T

T T x

d

x T T T

T T x

d

T

T T x

T

T T x

d

x T T T

T T x

d T x

T T T T

T T x

−

− − − − −

− −

=

























(

) (

)

}

(

) (

) (

)

{

}

(

)

3 3 3 3

3 2 1 3 4 3 1 4 3 1 3

3 3 3 3 4 4

3 3 1 4

,

,

max

,

,

,

,

,

m m m m

m m m m m m

d

x T T T

T T x

d T x

T x

d

x

x

d

x

x

d

x

x

− −

+



(

) (

) (

)

{

2 2 1 1

(

1 1

)

(

4 4

)

}

2 1 1 1 1 1 1 4 1

max

m

,

m

,

m

,

m

,

n mn

,

mn

,

n mn

,

mn

,

,

m

,

m

c

d

x

x

₊

d x

₋

x

d

x

₋

x

d

₋

x

−₋

x

−

d

x

₋

x

=

_

From inequality (3.5), we get,

(

3 3

)

3 m

,

m 1

d

x

x

₊



{

cd x

₁

(

_m1₋₁

,

x

_m1

)

,

cd

_n

(

x

_mn₋₁

,

x

_mn

)

,

cd

_n−1

(

x

_mn−₋1₁

,

x

_mn−1

)

,



,

cd

₃

(

x

_m3₋₁

,

x

_m3

) (

,

d x

₁ 1_m−1

,

x

_m1

)

,

d

_n

(

x

_mn₋₁

,

x

_mn

)

,

d

_n−1

(

x

_mn−₋1₁

,

x

_mn−1

)

,



,

d

₄

(

x

_m4₋₁

,

x

_m4

)

}

Since

0



c

< 1

, so, we get,

(

3 3

)

{

(

1

) (

)

(

1 1

)

(

4 4

) (

3 3

)

}

3

,

1

max

1 1

,

,

1

,

,

1 1

,

,

,

4 1

,

,

3 1

,

n n n n n

m m m m n m m n m m m m m m

d

x

x

₊



c

d x

₋

x

d

x

₋

x

d

₋

x

−₋

x

−



d

x

₋

x

d

x

₋

x

(3.6)

Similarly, we can get,

(

1 1

)

1

,

1

n n

n m m

d

₋

x

−

x

−₊



c

max

{

d x

₁

(

1_m−1

,

x

1_m

) (

,

d

₃

x

_m3₋₁

,

x

3_m

) (

,

d

₄

x

_m4₋₁

,

x

_m4

)

,



,

d

_n−1

(

x

_mn−₋1₁

,

x

_mn−1

) (

,

d

_n

x

_mn₋₁

,

x

_mn

)

}

(3.7)

Taking

x

₂n

=

x

_mn₋₁ and

x

n

=

x

_mn in inequality (3.4)

(

,

1

)

n n

n m m

d

x

x

₊

=

(

1 2 1 1

,

1 2 1

)

n n

n n n n m n n n m

d

T T

_{− −}



T T x

₋

T T

_{− −}



T T x

(

)

(

11

)

,

,

n n

n m m

n n

n m m

cF x

x

G

x

x

−

−



(

) (

)

{

(

)

(

)

(

)

(

) (

) (

)

}

1 2 2 1 1 2 1 1 2 1

1 2 1 2 3 1 1 3 1 1

1 1 1 1 1 1 1 1 1 1

max

,

,

,

,

,

,

,

,

,

,

,

,

max

n n n n

n m n n n m n n n m n n m

n n n n n n

n m n n n m n n n m n n m n m n n m

n n n n n n

n m n n n m n m n n m n m n m

n

c

d

x T T

T T T x

d

T

T T x

T

T T x

d

x T T

T T x

d

T

T T x

T

T x

d

x T

T T x

d T x

T T

T T x

d

x T

T T x

d T x

T x

d

− − − − − −

− − − − − − −

− − − − − −

=





















(

)

(

) (

)

}

{

,

1 2 1 1

,

,

1 2 1

,

1

,

1 2 1 1

n n n n n n

m n n n m n m n n n m n m n n n m

x T T

_{− −}



T T x

₋

d

x T T

_{− −}



T T x

d T x T T

₋



T T T x

₋

(

)

(

)

(

)

{

1 1 2 2 1 1

}

1 1 2 1 1 1

max

_{n m}n

,

_mn

,

_{n m}n

,

_mn

,

,

_m

,

_m

c

d

₋

x

−

x

−₊

d

₋

x

−

x

−₊

d x

₋

x

=



From (3.5), (3.6) and (3.7), we get,

(

)

{

(

1 1

) (

3 3

) (

4 4

)

(

1

) (

)

}

1 1 1 3 1 4 1 1 1 1

,

max

,

,

,

,

,

,

,

,

,

,

(3

)

.

8

n n n n n n

n m m m m m m m m n m m n m m

d

x

x

₊



c

d x

₋

x

d

x

₋

x

d

x

₋

x



d

₋

x

₋

x

−

d

x

₋

x

It now follows by induction on above inequalities, we get,

(

1 1

)

1

{

(

1 1

) (

3 3

)

(

1 1

) (

)

}

1

,

1

max

1 1

,

2

,

3 1

,

2

,

,

1 1

,

2

,

1

,

2

m n n n n

m m n n

d x

x

₊



c

−

d x x

d

x x



d

₋

x

−

x

−

d

x x

(

2 2

)

1

{

(

1 1

) (

3 3

)

(

1 1

) (

)

}

2

,

1

max

1 1

,

2

,

3 1

,

2

,

,

1 1

,

2

,

1

,

1

m n n n n

m m n n

© 2012, IJMA. All Rights Reserved 906



(

)

1

{

(

1 1

) (

3 3

)

(

1 1

) (

)

}

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

,

max

,

,

,

,

,

,

,

,

n n m n n n n

n m m n n

d

x

x

₊



c

−

d x x

d

x x



d

₋

x

−

x

−

d

x x

Since

0



c

< 1

, the sequence

{ } { } { } { } { }

x

m1

,

x

m2

,

x

m3

,

,

x

mn 1

,

x

mn −



are Cauchy sequence in

X X

₁

,

₂

,

X

₃

,



,

X

_n. So we have,

=

when = 1, 2, 3,

, .

lim

mi i

m

x

α

i

n

→∞



Since

T T T

₁

,

₂

,

₃

,



,

T

_n−1are continuous mapping.

T

₁

α α

₁

=

₂

,

T

₂

α

₂

=

α

₃

,...,

T

_n−1

α

_n−1

=

α

_n

Now taking

x

1

=

x

1m−₁ and

1 2

=

1

x

α

in inequality (3.1),

(

1

)

1

₍

(

1 1 1

₎

)

1 1 2 1 1 1 2 1 1

1 1 1

,

,

,

m

n n n n m

m

cF x

d T T

T T

T T

T T x

G x

α

α

α

−

− −

−







(

)

(

{

) (

)

(

)

(

) (

) (

)

(

)

1 1 1 1 1

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

1 1 1 1 1 1

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

1

2 1 1 1 2 1 1 1 1 1 1

max

,

,

,

,

,

,

,

,

,

,

,

,

,

m n m n n n m m n m

n n n m m n m m m n m

n m n n

c

d x

T

T T x

d

T

T T

T

T T x

d x

T

T T x

d

T

T T

T

T T x

d x

T

T T x

d T T

T T x

d x

T

T T x

d T

T T

T T

d x

T T T

α

α

α

α

α

− − − − − − −

− − − − − − − − −

− −

=





















(



) (

)

}

(

) (

) (

)

}

{

1 2 1 1 2 1 1 1 1

1 1 1 1

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

,

max

,

,

,

,

,

,

m

m n n m n n m m n

T T

d T

T x

d x

T T

T T

d x

T T

T T x

d T x

T T

T T

α

α

α

α

−

− −



− −



− −



As

m

→ ∞

, we get,

(

)

₍

1

(

1 1 1 2 1 1

_{) (}

) (

2 1 1 2

)

₎

_}

1 1 2 1 1 1

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

,

,

,

max{

,

,

,

n n n n

n n n

cd

T T T

T T

d T

d T T

T T

d

T T

T T

d

T T

T T

α

α

α α

α α

α

α

α

α

− −

−

≤









Since

T

₁ is continuous mapping.

(

)

1 n n 1 2 1 1

,

1

0

d T T

₋

T T

α α

⇒

_

_

Since, which is not possible

1 2 1 1

=

1

n n

T T

₋

T T

α

α

⇒

_

(3.9)

Now,

T T T

_{1 n n−1}



T T

_{3 2}

α

₂

=

T T T

_{1 n n−1}



T T T

_{3 2}

(

₁

α

₁

)

=

T T T

₁

(

_{n n−1}



T T T

_{3 2 1}

α

₁

)

=

T

1

α

1=

α

2

⇒

T T T

_{1 n n−1}



T T

_{3 2}

α

₂

=

α

₂

Since

T T T

1

,

2

,

3

,



,

T

n−1are continuous mapping, So in same above way,so we obtain

T T T

_{1 n n−1}



T

₂

α

₂

=

α

₂

2 1 n 3 3

=

3

T T T



T

α

α

3 2 1 4 4

=

4

T T T



T

α

α



1 2 1

=

n n n n

T

₋



T T T

α

α

Page: 902-908

Uniqueness

Let us assume that

α

₁

′

is another fixed point of

T T

_{n n−1}



T T

_{2 1} different from

α

₁ such that

α

_1′

≠

α

₁ and

1 2 1 1

=

1

n n

T T

₋



T T

α

′

α

′

,

T T

n n−1



T T

2 1

α

1

=

α

1.

Now taking

x

1

=

α

₁

′

and

x

1₂

=

α

₁ in inequality (3.1), we get,

(

)

1

,

1

d

α α

′

=

d T T

₁

(

_{n n−1}



T T

_{2 1}

α

₁

,

T T

_{n n−1}



T

₁

α

₁

′

)

(

)

(

)

1 1 1

1 1 1

,

,

cF

G

α α

α α

′

′



(

)

(

) (

)

{

(

)

(

) (

)

(

) (

) (

) (

)

}

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

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

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

max

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

=

n n n n n

n n n n

n n n n

c

d

T

T T

d

T

T T T T

T T

d

T

T T

d

T

T T T

T

T T

d

T

T T

d T T

T T

d

T

T T

d T

T T

T T T

d

T T

T T T

d T

T

α

α

α

α

α

α

α

α

α

α

α

α

α

α

α

α

α

α

α

− −

− − −

−

′

′

′

′

′

′

′

′

′

′

′

′

′























(

)

{

{

}

(

) (

)

(

max

d

1

α

1

,

T T

n n−1

T T T

3 2 1

α

1

,

d

1

α

1

,

T T

n n−1

T T

2 1

α

1

,

d T

2 1

α

1

,

T T

1 n

T T

2 1

α

1

}

)

























′

_

′

_

′

′

_

(

) (

)

(

) (

)

{

11 11 11 2 21 1 11 11 1

}

,

,

=

max

,

,

,

cd

d T

T

d

d T

T

α α

α α

α α

α

α

′

′

′

′

⇒

d

1

(

α α

1

,

1

′

)



cd T

2

(

1

α

1

,

T

1

α

1

′

)

(3.10)

In the same way, taking

x

₂2

=

T

₁

α

₁ and

x

2

=

T

₁

α

₁₁

′

in inequality (3.2), we get

(

)

2 1 n n 1 3 2 1 1

,

1 n n 1 2 1 1

d T T T

₋



T T T

α

T T T

₋



T T

α

′

(

)

(

)

2 1 1 1 1

2 1 1 1 1

,

,

cF T

T

G T

T

α

α

α

α

′

′



(

) (

{

)

(

) (

)

}

(

) (

) (

)

{

}

(

)

2 1 1 1 1 1 1 2 1 1

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

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

max

,

,

,

,

,

,

=

max

,

,

,

,

,

n n

n

c

d T

T

d T T

T T

T

T T

d T

T

d T T

T T

d T

T

d T

T

d T T

T T

α

α

α

α

α

α

α

α

α

α

α

α

α

α

−



′

′







′

′

′

















′

′

′

′







(

)

(

)

2 1 1

,

1 1 3 2 1 1

,

2 1 1

d T

α

T

α

′

cd T T

α

T T

α

′

⇒

_

(3.11)

Similarly, by taking

x

₂3

=

T T

_{2 1}

α

₁ and

x

3

=

T T

_{2 1}

α

₁

′

in inequality (3.3), we get,

(

)

(

)

3 2 1 1

,

2 1 1 4 3 2 1 1

,

3 2 1 1

d T T

α

T T

α

′



cd T T T

α

T T T

α

′

(3.12)

Continue like this, taking

x

₂n

=

T T

_{n−1 n−2}



T T

_{2 1}

α

₁ and

x

n

=

T

_n−1



T T

_{2 1}

α

₁

′

in inequality (3.4), we get,

(

1 2 2 1 1

,

1 2 1 1

)

n n n n

d

T T

_{− −}



T T

α

T

₋



T T

α

′



cd T T

₁

(

_{n n−1}



T T

_{2 1}

α

₁

,

T T

_{n n−1}



T T

_{2 1}

α

₁

′

)

© 2012, IJMA. All Rights Reserved 908

From inequalities (3.10),(3.11), (3.12)and(3.13), we get,

d

1

(

α α

1

,

1

′

)



cd T

2

(

1

α

1

,

T

1

α

1

′

)

(

)

2

3 2 1 1

,

2 1 1

c d T T

α

T T

α

′



(

)

3

4 3 2 1 1

,

3 2 1

c d T T T

α

T T

α′





(

)

1

1 2 1 1

,

1 2 1 1

n

n n n

c

−

d T

₋



T T

α

T

₋



T T

α

′



(

)

1 1

,

1

n

c d

α α

′



1

(

1

,

1

)

1

(

1

,

1

)

n

d

α α

′

c d

α α

′

⇒

_

Since

0



c

< 1

, so we get,

⇒

d

₁

(

α α

₁

,

₁

′

)

= 0

⇔

α

₁

=

α

₁

′

So,

α

₁ is unique fixed point of

T T

_{n n−1}



T T

_{2 1}.

In the similar way, the unicity of other fixed point can be proved.

We have to finally proved that

T

( )

α

_n

=

α

₁, To do this, note that

T

_n

( )

α

_n

=

T T

_n

(

_n−1



T T T

_{2 1 n}

α

_n

)

=

T T

_{n n−1}



T T T

_{2 1}

(

_n

α

_n

)

And so

T

_n

α

_nis fixed point of

T T

_{n n−1}



T T

_{2 1}. Since

α

₁ is the unique fixed point of

T T

_{n n−1}



T T

_{2 1}. It follows that

( )

n

=

1

T

α

α

So, we obtain

T T

_{n n−1}



T T

_{2 1} has a unique fixed point

α

₁

∈

X

₁,

T T

_{1 n}



T T

_{3 2} has a unique fixed point

α

₂

∈

X

₂, and so on,

T T

_{n−1 n−2}



T T

_{1 n} has a unique fixed point

α

_n

∈

X

_nAlso

T

₁

( )

α

₁

=

α

₂

,

T

₂

( )

α

₂

=

α

₃

,

T

₃

( )

α

₃

=

α

₄

,



,

(

)

( )

1 1

=

,

=

1

n n n n

T

₋

α

₋

α

T

α

α

REFERNCES

[1] F.Merghadi, A.Aliouche and B.Fisher, A related fixed point theorem in n complete metric spaces, Acta Universitatis Apulensis, no.19(2009), 91-99

[2] D.B.Ojha , Commom fixed point theorem for n metric spaces, International Journal of Applied Engineering Research,Dindigul,Vol.1(2010) no. 3, 324-329

[2] B. Fisher, Fixed points theorem on two metric spaces, Glasnik Mat., Vol. 16(1981) no.36, 333-337.

[2] L. Kikina and K. Kikina, A fixed point theorem on four spaces, Int. Math. J. Analysis,Vol. 3 (2009)no.32 , 1559-1568.

[3] N.P. Nung, A fixed point theorem for three metric spaces, Math. Seminar Notes Kobe University, Vol.1( 1983), 77-79.

[4] R.K. Jain, H.K. Sahu, and B. Fisher, A fixed point theorem on three metric spaces, Kyungpook Math. J.,Vol. 36 (1996), 151-154.

[5] S. Jain and B. Fisher, A related fixed point theorems on three complete metric spaces, Novisad J. Math. ,Vol.27 (1997), 27-35.