Vol 9, No 8 (2018)

(1)

Available online throug

ISSN 2229 – 5046

FIXED POINT THEOREMS THROUGH CONTRACTIVETYPE MAPPINGS

1

_{Dr. V. SRINIVASAKUMAR,}

2

_{K. KUMARA SWAMY AND}

3

**_{*K. SUJATHA}**

1

_{Assistant Professor, Department of Mathematics, JNTU College of Engineering,}

JNTU Hyderabad, Kukatpally -500044, Telangana State, India.

2

_{Assistant Professor, Department of Mathematics,}

GMR Institute of Technology, Rajam-532127, Srikakulam, (A.P.), India.

3

_{Research Scholar in JNTUH, Hyderabad-500044, Telangana State, India.}

(Received On: 02-07-18; Revised & Accepted On: 09-08-18)

ABSTRACT

I

n this research article, some fixed point theorems in 2- metric spaces are established. It also introduces contraction type mappings in 2-metric spaces. The theorems are generalizations of some fixed point theorems of Pal and Maiti [2].

AMS Subject Classification: 47H10, 54H25.

Key Words: Fixed point, 2-metric space, Contractive mapping, Generated Contractive mapping.

INTRODUCTION

The notion of 2-metric space was introduced by Gahler [1] in 1963 as a generalizationof area function for Euclidean

triangles. Fixed point theory was first studied by Poincare and developed by many mathematicians Brouwer, Banach,

Schauder, Rhoades [4], etc. Let X be a non-empty set. A function

f

:

X

→

X

from X into itself is called a self-map

on X. A point

z

∈

X

is called a fixed point of self-map

f

:

X

→

X

if

f z

( )

=

z

, Theorems concerning fixed points

of self-maps are known as fixed point theorems. Most of the fixed point theorems were proved for contraction mappings. It is well known that every contraction on a metric space is continuous. The converse is not necessarily true. The identity mapping on [0,1] simply serves the counter example.

In this present work, some of the fixed point theoremsof Pal and Maiti are extended to a more generalized 2-metric

space setting.

In what follows

X

stands for a 2-metric space.

1. PRELIMINARIES

This section is devoted to some basic definitions which are needed for the further study of this Article.

Definition 1.1: Let

X

be a non-empty set and

λ

:

X

× × →

X

R

. For all

x y z

, ,

and

u

in

X

, if

λ

satisfies the following conditions

(a)

λ

(

x y z

, ,

)

=

0

if atleast two of

x y z

, ,

are equal (b)

λ

(

x y z

, ,

)

=

λ

(

x z y

, ,

)

=

λ

(

y z x

, ,

)

=

...

(c)

λ

(

x y z

, ,

)

≤

λ

(

x y u

, ,

) (

+

λ

x u z

, ,

) (

+

λ

u y z

, ,

)

Then

d

is called a 2-metricon

X

and the pair

(

X

,

λ

)

is called a 2-metric space.

Corresponding Author:

3*

_{K. Sujatha}

(2)

(

X

,

λ

)

be a 2-metric space. A mapping

T X

:

→

X

is said to be a Contractive if

(

Tx Ty a

,

)

(

x y a

, ,

)

λ

<

λ

for all

x y a

, ,

in

X

.

Remark 1.3: It is well known that a contractive mapping on a complete metric space has a unique fixed point in

X

.

But a contractive mapping on a complete 2-metric space

(

X

,

λ

)

neednot have a fixed point. It can be seen from the

following example.

Example- 1.4:Let

X

=

{

x

∈

R x

/

≥

2 }

=

[

2,

∞

)

with 2-metric defined by

(

x y z

, ,

)

min

{

x

y

,

y

z z

,

x

}

λ

=

−

.

Define

f X

:

→

X

by

f x

( )

1 x

X

x

= ∀ ∈

.

(

)

1 1

,

fx fy a

a

x y

λ

_{= }

λ





_





1

1 min

,

a

,

a

x

y

x





=

_

−

_





{

}

min

,

<

x

−

y

y a a

−

x

(

x y a

, ,

)

λ

=

⇒

f

is a contractive mapping on

X

,but it has no fixed point in

X

.

(

X

,

λ

)

be a 2-metric space. A mapping

T X

:

→

X

is said to be a Generated Contractive if

(

)

{

(

) (

)

1 [

(

) (

)

]

}

,

max

, ,

,

2 Tx Ty a

x y a

x Tx a

y Ty a

x Ty a

y Tx a

λ

<

λ

+

λ

for all

x y a

, ,

in

X

.

Remark 1.6: A Generated contractive mapping is need not be contractive mapping. Itcan be seen from the following example.

Example 1.7: Let

X

=

[ ]

0, 6

with 2-metric defined as

λ

(

x y z

, ,

)

=

min

{

x

−

y

,

y

−

z

,

z

−

x

}

for all

x y z

, ,

in

X

.

We define a map

T

: 0, 6

[ ] [ ]

→

0, 6

by

( )

[ ]

(

]

3 0, 3

3

1 3, 6



∈



= 

−

∈



x

if

x

T x

x

if

x

Clearly

T

is generated contractive mapping.

But

λ

(

T

( ) ( )

3 ,

T

5 , 0

)

=

λ

(9,1 4, 0 ) 5

= >

λ

(

3, 5, 0

)

⇒

T

is not a contractive mapping.

Definition 1.8: A 2-metric space

(

X

,

λ

)

is said to be 2-compactif every sequence in

X

has a convergent subsequence.

2. FIXED POINT THEOREMS FOR GENERATED CONTRACTIVE MAPPINGS

In this section we proved fixed point theorems for Generated Contractive type mappings.

The following Theorem-2.1 is a generalization of Pal and Maiti [2] in 2-metric space setting.

Theorem 2.1: Let

T

be a continuous generated contractive self-mapping on a compact 2-metric space

X

such that

(

x Ty a

,

) (

y Tx a

,

)

2 (

x y a

, ,

)

(3)

Proof: Let

s

₀

∈

X

.

_.

Define a sequence

{ }

s

n in

X

such that 0

n n

s

=

T s

when

n

=

1, 2, 3,...

Consider

λ

(

x Ty a

,

) (

+

λ

y Tx a

,

)

<

2 λ

(

x y a

, ,

)

Put

x

=

s

_n₊₁_,

y

=

s

_n

(

s

n 1

,

Ts a

n

,

) (

s Ts

n

,

n 1

,

a

)

2 (

s

n 1

,

s a

n

,

)

λ

₊

+

λ

₊

<

λ

₊

(

s

n 1

,

s

n 1

,

a

) (

s s

n

,

n 2

,

a

)

2 (

s

n 1

,

s a

n

,

)

λ

₊ ₊

λ

₊

λ

₊

⇒

+

<

(

s s

n

,

n 2

,

a

)

2 (

s

n 1

,

s a

n

,

)

λ

₊

λ

₊

⇒

<

(

2

) (

1

)

1 ,

,

...

2 λ

s s

n n+

a

λ

s

n+

s a

n

⇒

<

( )

1

.Put

c

_n

=

d s s

(

_n

,

_n₊₁

,

a

)

Since

T

is generated contractive,

(

)

(

) (

)

1 (

) (

)

,

max

, ,

,

2 Tx Ty a

x y a

x Tx a

y Ty a

x Ty a

y Tx a

λ

<



_

λ



_

λ

+

λ



_



_





Put

x

=

s

_n₊₁_,

y

=

s

_n

(

Ts

n 1

,

Ts a

n

,

)

λ

+

(

1

) (

1 1

) (

)

(

1

) (

1

)

1 max

,

2

n n n n n n n n n n

s

s a

s

Ts

a

s Ts a

s

Ts a

s Ts

a

λ

₊

λ

₊ ₊

λ

₊

λ

₊



_

_



<

_

_

+

_

_





(

1

) (

1 2

) (

1

)

(

1 1

) (

2

)

1 max

,

2

n n n n n n n n n n

s

s a

s

a

s s

a

s

a

s s

a

λ

₊

λ

₊ ₊

λ

₊

λ

₊ ₊

λ

₊



_

_



=

_

_

+

_

_





(

2 1

)

(

1

) (

1 2

) (

2

)

1 ,

,

max

,

2

n n n n n n n n

s

a

s

s a

s

a

s s

a

λ

₊ ₊



λ

₊

λ

₊ ₊

λ

₊



⇒

<

_

_





Let

M

=

λ

(

s

_n₊₁

,

s

_n₊₂

,

a

)

Then

λ

(

s

_n₊₁

,

s

_n₊₂

,

a

) (

<

λ

s

_n₊₁

,

s

_n₊₂

,

a

)

This is a contradiction.

So

M

≠

λ

(

s

n+₁

,

s

n+₂

,

a

)

By

( )

1

,

M

=

λ

(

s s

_n

,

_n₊₁

,

a

)

So we get

λ

(

s

_n+₁

,

s

_n+₂

,

a

) (

<

λ

s s

_n

,

_n+₁

,

a

)

1

n n

c

+

c

⇒

<

Similarly

c

_n₊₁

< <

c

_n

c

_n₋₁

< <

...

c

₀

{ }

c

n

⇒

is a monotonically decreasing and bounded sequence of nonnegative real numbers.

{ }

c

n

⇒

Converges.

Suppose that

lim

_n

n→∞

c

=

l

(4)

Let

lim

_m

,

m→∞

p

=

u

where

u

∈

X

.

Since

T

is continuous and

lim

_m

,

m→∞

p

=

u

we have m

lim

→∞

Tp

m

=

Tu

1

lim

_m

m→∞

Tp

+

Tu

⇒

=

Now we show that

u

is a fixed point of

T

_{Assume that}

u

≠

Tu

Then

λ

(

u Tu a

,

)

≤

λ

(

u Tu p

,

_m

) (

+

λ

u p a

,

_m

,

) (

+

λ

p Tu a

_m

,

)

(

u Tu p

,

m

) (

u p a

,

m

,

) (

Tp

m1

,

Tu a

,

)

λ

₋

=

+

(

u Tu p

,

_m

) (

u p a

,

_m

,

)

λ

<

+

(

1

)

(

1 1

)

(

)

(

1

) (

1

)

1 max

, ,

,

2

m m m m m

p

u a

p

Tp

a

u Tu a

p

Tu a

u Tp

a

λ

₋

λ

₋ ₋

λ

₋

λ

₋





+

_



_

+



_

_





(

u Tu p

,

m

) (

u p a

,

m

,

)

λ

=

+

(

1

)

(

1

)

(

)

(

1

) (

)

1 max

, ,

,

2

m m m m m

p

u a

p

a

u Tu a

p

Tu a

u p a

λ

₋

λ

₋

λ

₋

λ





+

_



_

+



_

_





Letting

m

→ ∞

(

)

(

)

1 (

)

,

max

,

2 u Tu a

u Tu a

λ

<



_

λ



_





(

u Tu a

,

)

(

u Tu a

,

)

λ

<

λ

Therefore our assumption is false.

Hence

Tu

=

u

.

u

⇒

is a fixed point of

T

in

X

.

Now we show that it is unique.

Let

v

be another fixed point of

T

i.e,

Tv

=

v

Assume that

u

≠

v

Then

λ

(

u v a

, ,

)

=

λ

(

Tu Tv a

,

)

(

) (

)

1 (

) (

)

max

, ,

,

2 u v a

u Tu a

v Tv a

u Tv a

v Tu a

λ





<

_



_

+



_

_





(

) (

)

1 (

) (

)

max

, ,

,

, ,

,

, ,

,

, ,

2 u v a

u u a

v v a

u v a

v u a

λ





=

_



_

+



_

_





(

u v a

, ,

)

λ

=

(

u v a

, ,

)

(

u v a

, ,

)

λ

⇒

<

Hence

u

=

v

.

(5)

Theorem- 2.2: Let

(

X

,

λ

)

be a compact 2-metric space. Suppose that

S

and

T

are two continuous self-maps on a

X

such that

( ) (

1 λ

x Ty a

,

) (

+

λ

y S xa

,

)

<

2 λ

(

x y a

, ,

)

( ) (

)

(

) (

)

1 (

) (

)

2 ,

,

max

, ,

,

2 S xTy a

x y a

x S xa

y Ty a

x Ty a

y S xa

λ

<



_

λ



_

λ

+

λ



_



_





for all

x y a

, ,

in

X

.

Then

S

and

T

have a unique common fixed point in

X

_.

Proof: Let

p

0

∈

X

.

Define a sequence

{ }

p

n in

X

such that

p

2n+1

=

Sp

2n, where

n

=

0,1, 2, 3,...

And

p

₂_m

=

Tp

₂_m₋₁, where

m

=

1, 2, 3,...

Suppose that

λ

₂_n

=

λ

(

p

₂_n

,

p

₂_n+₁

,

a

)

, where

n

=

0,1, 2, 3,...

Consider

λ

(

x Ty a

,

) (

+

λ

y S xa

,

)

<

2 λ

(

x y a

, ,

)

Put

x

=

p

₂_nand

y

=

p

₂_n₋₁

(

p

2n

,

Tp

2n 1

,

a

) (

p

2n 1

,

S

2n

,

a

p

)

2 (

p

2n

,

p

2n 1

,

a

)

λ

₋

+

λ

₋

<

λ

₋

(

p

2n

,

p

2n

,

a

) (

p

2n 1

,

p

2n 1

,

a

)

2 (

p

2n

,

p

2n 1

,

a

)

λ

₋ ₊

λ

₋

⇒

+

<

(

p

2n 1

,

p

2n 1

,

a

)

2 (

p

2n

,

p

2n 1

,

a

)

λ

₋ ₊

λ

₋

⇒

<

(

2 1 2 1

) (

2 1 2

)

1 ,

,

2 λ

p

n−

p

n+

a

λ

p

n−

p

n

a

⇒

<

( )

1

Now, consider

(

,

)

max

(

, ,

) (

,

) (

,

)

,

1 (

,

) (

,

)

2 S xTy a

x y a

x S xa

y Ty a

x Ty a

y S xa

λ

<



_

λ



_

λ

+

λ



_



_





Put

x

=

p

₂_nand

y

=

p

₂_n₋₁

(

Sp

2n

,

Tp

2n 1

,

a

)

λ

−

(

2 2 1

) (

2 2

) (

2 1 2 1

)

(

2 2 1

) (

2 1 2

)

1 max

,

2

n n n n n n n n n n

p

a

p

S

p

a

p

Tp

a

p

Tp

a

p

S

p

a

λ

₋

λ

₋ ₋

λ

₋

λ

₋



_

_



<



_

+

_







(

2 2 1

) (

2 2 1

) (

2 1 2

)

(

2 2

) (

2 1 2 1

)

1 max

,

2

n n n n n n n n n n

p

a

p

a

p

a

p

a

p

a

λ

−

λ

+

λ

−

λ

− +



_

_



=

_

_

+

_

_





(

2 1 2

)

(

2 2 1

) (

2 2 1

) (

2 1 2 1

)

1 ,

,

max

,

2

n n n n n n n n

p

a

p

a

p

a

p

a

λ

₊



λ

₋

λ

₊

λ

₋ ₊



⇒

<

_

_





If

max

(

₂

,

₂ ₁

,

) (

,

₂

,

₂ ₁

,

)

,

1 (

₂ ₁

,

₂ ₁

,

)

(

₂

,

₂ ₁

,

)

2

n n n n n n n n

p

a

p

a

p

a

p

a

λ

₋

λ

₊

λ

₋ ₊

λ

₊



_{ =}









Then

λ

(

p

₂_n

,

p

₂_n+₁

,

a

) (

<

λ

p

₂_n

,

p

₂_n+₁

,

a

)

So

max

(

₂

,

₂ ₁

,

) (

,

₂

,

₂ ₁

,

)

,

1 (

₂ ₁

,

₂ ₁

,

)

(

₂

,

₂ ₁

,

)

2

n n n n n n n n

p

a

p

a

p

a

p

a

λ

₋

λ

₊

λ

₋ ₊

λ

₊



_{ ≠}





(6)

By

( )

1

, we get

(

2 2 1

) (

2 2 1

)

(

2 1 2 1

)

(

2 1 2

)

1 max

,

2

n n n n n n n n

p

a

p

a

p

a

p

a

λ

−

λ

+

λ

− +

λ

−



_{ =}









Similarly

d

₂_n

<

d

₂_n₋₁

<

d

₂_n₋₂

< <

...

d

₀

{ }

d

2n

⇒

is a monotonically decreasing and bounded sequence of nonnegative real numbers.

{ }

d

2n

⇒

Converges.

Suppose that

lim

₂_n

n→∞

d

=

l

Since

X

is compact, every sequence in

X

has a convergent subsequence

So

{ }

p

_n has a convergent subsequencein

X

.

Suppose that

{ }

p

₂_n is a convergent subsequence of

{ }

p

_n in

X

and let

lim

₂_n

,

n→∞

p

=

u

where

u

∈

X

.

Now we prove that

u

is acommon fixed point of

S

and

T

_.

Assume that

u

≠

Su

Since

S

is continuous on

X

and

lim

₂_n

,

n→∞

p

=

u

we have

lim

n→∞

Sp

2n

=

Su

2 1

lim

_n

n→∞

p

+

Su

⇒

=

Since

T

is continuous on

X

and

lim

₂_n ₁

,

n→∞

p

+

=

Su

we have

lim

₂_n ₁

n→∞

Tp

+

=

TSu

2 1

lim

_n

n→∞

Tp

+

TSu

⇒

=

2 2

lim

_n

n→∞

p

+

TSu

⇒

=

because

Tp

₂_n₊₁

=

p

₂_n₊₂

Then

λ

(

u Su a

,

)

≤

λ

(

u Su p

,

₂n

) (

+

λ

u p

,

₂n

,

a

) (

+

λ

p

₂n

,

Su a

,

)

(

u Su p

,

2n

) (

u p

,

2n

,

a

) (

Tp

2n 1

,

Su a

,

)

λ

₋

=

+

(

u S up

,

2n

) (

u p

,

2n

,

a

)

λ

<

+

(

2 1

) (

)

(

2 1 2 1

)

(

2 1

) (

2 1

)

1 max

,

2

n n n n n

u p

a

u Su a

p

Tp

a

u Tp

a

p

Su a

λ

−

λ

− −

λ

−

λ

−





+

_



_

+



_

_





(

u S up

,

2n

) (

u p

,

2n

,

a

)

λ

=

+

(

2 1

) (

)

(

2 1 2

)

(

2

) (

2 1

)

1 max

,

2

n n n n n

u p

a

u Su a

p

a

u p

a

p

Su a

λ

₋

λ

₋

λ

₋





+

_



_

+



_

_





Letting

n

→ ∞

(

)

(

)

1 (

)

,

max

,

2 u Su a

u Su a

λ

<



_

λ



_





(

u Su a

,

)

(

u Su a

,

)

λ

⇒

<

This is a contradiction.Soour assumption is false.

Hence

Su

=

u

.

u

⇒

is a fixed point of

S

in

X

.

2n 2n 1

d

₋

(7)

Consider

λ

(

x Ty a

,

) (

+

λ

y S xa

,

)

<

2 λ

(

x y a

, ,

)

Put

x

= =

y

u

(

u Tu a

,

) (

u S ua

,

)

2 (

u u a

, ,

)

λ

+

λ

<

λ

Since

Su

=

u

_,

λ

(

u Tu a

,

)

<

0

.

But

λ

(

u Tu a

,

)

≥

0

.

(

u Tu a

,

)

0 λ

⇒

=

Tu

u

⇒

=

Hence

u

is a common fixed point of

S

and

T

in

X

_.

Now we show that it is unique.

Let

v

be another fixed point of

T

i.e,

Sv

=

Tv

=

v

Assume that

u

≠

v

Then

λ

(

u v a

, ,

)

=

λ

(

S uTv a

,

)

(

) (

)

1 (

) (

)

max

, ,

,

2 u v a

u S ua

v Tv a

u Tv a

v S ua

λ





<

_



_

+



_

_





(

) (

)

1 (

) (

)

max

, ,

,

, ,

,

, ,

,

, ,

2 u v a

u u a

v v a

u v a

v u a

λ





=

_



_

+



_

_





(

u v a

, ,

)

λ

=

(

u v a

, ,

)

(

u v a

, ,

)

λ

⇒

<

Hence

u

=

v

.

Thus

S

and

T

have a unique common fixed point in

X

_.

Remark-2.3: If we take

S

=

T

in Theorem-2.2 then we obtain Theorem-2.1. So 2.2 is a further generalization of

Theorem-2.1. A point is a unique fixed point of

T X

:

→

X

iff it is unique fixed point of any positive power of

T

.

This fact leads us to the following theorem and proof of the following is similar to the proof of previous theorem.

Theorem- 2.4: Let

(

X

,

λ

)

be a compact 2-metric space. Suppose that

S

and

T

are two continuous self-maps on

X

such that

( )

1 λ

(

x T y a

,

q

,

) (

+

λ

y S x a

,

p

,

)

<

2 λ

(

x y a

, ,

)

( )

(

)

(

)

(

) (

)

1 (

) (

)

2 ,

,

max

, ,

,

2

p q p q q p

S x T y a

x y a

x S x a

y T y a

x T y a

y S x a

λ

_<



_

λ



_

λ

₊

λ



_



_





For all

x y a

, ,

in

X

and for all

p q

,

∈



.

(8)

REFERENCES

1. Gahler, S., 2-metrische Raumeandihre Topologische structure, MathNatch, Vol - 26, pp.115 –148,1963.

2. Pal, T.K and Maiti,M.: Extensions of fixed point theorems of Rhodes and Ciric, Proc.Amer.Math.

Soc.Vol-64(2), pp.283 –286, 1977.

3. Petryshyn,W.V.: Strjucture of the fixed points sets of the sets of K set- contractions Arch.Rational.mech.Anal.

Vol-40, pp. 312-328, (1970/71)

4. Rhoades, B. E., Contraction Type Mappings on a 2-Metric Space, Math. Nachr. Vol-91, pp. 151-155, 1979.

Source of support: Nil, Conflict of interest: None Declared.