Vol 2, No 10 (2011)

(1)

FIXED POINTS FOR NON-SELF MAPPINGS ON CONVEX VECTOR METRIC SPACES

Sushanta Kumar Mohanta*

Department of Mathematics, West Bengal State University, Barasat, 24 Parganas(North), Kolkata 700126, West Bengal, India

E-mail: [email protected]

(Received on: 13-09-11; Accepted on: 05-10-11)

________________________________________________________________________________________________ ABSTRACT

W

e introduce the concept of convex structure on a vector metric space and obtain some fixed point theorems for a class of non-self mappings satisfying certain contractive conditions in the setting of convex vector metric spaces.

Keywords and phrases: Convex vector metric space, non-self mapping, fixed point.

2000 Mathematics Subject Classification: 54H25, 54C60.

________________________________________________________________________________________________

1. INTRODUCTION:

Deterministic fixed point theorems are generally proved for self mappings only. In1970, W. Takahashi [5] had introduced the concept of convexity in a metric space and obtained some important fixed point theorems previously proved for Banach spaces. Afterwards, Gaji [2] obtained an important fixed point theorem for a class of non-self mappings in Takahashi convex metric spaces. In this paper, our attempt is to introduce the notion of convex structure on a vector metric space with relevant definitions with their properties. Finally, we prove some fixed point theorems for a class of non-self mappings over a subset of a convex vector metric space.

Let

S

be the vector space of all real sequences

α

=

{ }

a

_n and

θ

stands for the zero vector

{ }

0

. We can define a partial ordering

≤

on

S

by

α

≤

β

(

equivalent

ly

,

β

≥

α

)

if and only if

a

_n

≤

b

_n for all

n

, where

{ }

a

_n

=

{ }

b

_n

∈

S

=

β

α

,

. We write

α <

β

(equivalently,

β >

α

)

if

and

only

if

a

_n

<

b

_n

for

all

n

.

An element

α

∈

S

is called non-negative if

α

≥

θ

. An element

α

=

{ }

a

_n

∈

S

is called positive if

a

_n

>

0 for

all

n

. Also for any

α

=

{ }

a

_n

∈

S

and any real number

t

we define

t

α

=

{ }

ta

_n . If

α

=

{ }

a

_n

,

β

=

{ }

b

_n

∈

S

then

{

a

_n

+

b

_n

}

and

=

if

a

_n

=

b

_n

for

all

n

=

+

β

α

β

α

.

For

α

i

=

{ }

a

_ni _n

∈

S

;

i

=

1 ,

2 ,

,

k

, we define

{

:

1 ,

2 ,

,

}

{

max

(

),

max

(

),

,

max

(

),

}

max

1 2

1 1 1

i n k i i

k i i k i i

a

k

i

≤ ≤ ≤

≤ ≤

≤

=

α

.

Clearly,

max

{

α

i

:

i

=

1 ,

2 ,

,

k

}

∈

S

.

2. DEFINITIONS AND BASIC FACTS:

In this section, we recall some basic definitions and important results for vector metric spaces that will be needed in the sequel.

Definition: 2.1 [4] Let

X

be a non empty set. Then a function

V

:

X

×

X

→

S

is called a vector metric on

X

if the following conditions are satisfied:

(i)

V

(

x

,

y

)

≥

θ

for

all

x

,

y

∈

X

and

V

(

x

,

y

)

=

θ

if

and

only

if

x

=

y

,

(ii)

V

(

x

,

y

)

=

V

(

y

,

x

)

for

all

x

,

y

∈

X

,

(iii)

V

(

x

,

y

)

≤

V

(

x

,

z

)

+

V

(

z

,

y

)

for

all

x

,

y

,

z

∈

X

.

---

(2)

Page: 1800-1808

The pair

(

X

,

V

)

is called a vector metric space. We may verify that

V

(

x

,

y

)

is a continuous function of its arguments.

Theorem: 2.2[4] If

V

(

x

,

y

)

=

{

d

_n

(

x

,

y

)

}

be a vector metric then each

d

_n

(

x

,

y

)

is a quasi metric function; conversely if each

d

_n

(

x

,

y

)

is a quasi metric and the relations

d

_n

(

x

,

y

)

=

0

for all

n

imply

x

=

y

, then

{

(

,

)

}

)

,

(

x

y

d

x

y

V

=

_n is a vector metric.

Remark: 2.3 If

(

X

,

d

)

is a metric space and if

V

(

x

,

y

)

=

{

d

(

x

,

y

),

d

(

x

,

y

),

}

, then

(

X

,

V

)

is a vector metric space. So any metric space is a vector metric space and for the converse we can say from Theorem 2.2 that a vector metric space is a quasi metric space.

Example: 2.4 Let

X

=

R

n. If we define

V

on

X

×

X

by

V

(

x

,

y

)

=

{

x

₁

−

y

₁

,

x

₂

−

y

₂

,

x

_n

−

y

_n

,

0 ,

}

where

x

=

(

x

₁

,

x

₂

,

x

_n

),

y

=

(

y

₁

,

y

₂

,

y

_n

)

∈

X

, then

(

X

,

V

)

is a vector metric space.

Example: 2.5[3] Let

X

=

[

0 ,

1 ]

. If we define

V

on

X

×

X

by

−

+

−

+

−

=

,

0 ,

2

1

2 ,

0 ,

2 ,

0 ,

1 ,

0 ,

)

,

(

y

x

y

x

y

x

y

x

y

x

y

x

y

x

V

, then

(

X

,

V

)

is a vector metric

space.

Definition: 2.6[4] Let

(

X

,

V

)

be a vector metric space. A sequence

{ }

x

_k in

X

is said to converge to an element

X

x

∈

i.e.

x

_k

=

x

if

V

x

_k

x

→

as

k

→

∞

k

(

,

)

θ

lim

which is interpreted as

d

_n

(

x

_k

,

x

)

→

0 as

k

→

∞

for

all

n

where

V

(

x

_k

,

x

)

=

{

d

_n

(

x

_k

,

x

)

}

.

Definition: 2.7[3] A sequence

{ }

x

_k in

(

X

,

V

)

is said to be a vector Cauchy sequence if

V x x

( ,

_k _{k p}₊

)

→

θ

as k

→∞

,

for each p

i

.

e

.

d

_n

(

x

_k

,

x

_k₊_p

)

→

0 as

k

→

∞

for

each

p

and

for

each

n

.

In other words, a sequence

{ }

x

_k in

(

X

,

V

)

is said to be a vector Cauchy sequence if

V

(

x

_k

,

x

_j

)

→

θ

as

k

,

j

→

∞

,

i

.

e

.

d

_n

(

x

_k

,

x

_j

)

→

0 as

k

,

j

→

∞

and

for

each

n

.

Definition: 2.8[3] A vector metric space

(

X

,

V

)

is said to be complete if every vector Cauchy sequence in

X

converges to an element in

X

. Otherwise

X

is called incomplete.

It is to be noted that every complete metric space may be considered as a complete vector metric space.

Definition: 2.9 Let

(

X

,

V

)

be a vector metric space and

A

⊂

X

. Then

A

is called closed if and only if every sequence

{ }

x

_n in

A

with

x

_n

x

implies

that

x

A

n→∞

=

∈

lim

.

It is easy to verify that every closed subset of a complete vector metric space is complete.

Definition: 2.10 Let

(

X

,

V

)

be a vector metric space,

x

₀

∈

X

and

r

be a positive member of

S

. Then the set denoted by

B

(

x

₀

,

r

)

=

{

x

∈

X

:

V

(

x

₀

,

x

)

<

r

}

is called an open ball centered at

x

₀ and radius

r

in

X

.

(3)

Page: 1800-1808

Definition: 2.12 A subset

A

of a vector metric space

(

X

,

V

)

is called bounded if there exists a positive member

K

in

S

such that

V

(

x

,

y

)

≤

K

for

all

x

,

y

∈

A

.

Definition: 2.13 Diameter of a bounded set

A

in a vector metric space

(

X

,

V

)

, denoted by

Diam

(

A

)

or

δ

(

A

)

, is defined as

=

∈ ∈

),

,

(

sup

,

),

,

(

sup

),

,

(

sup

)

,

(

sup

)

(

, 2

, 1

, ,

y

x

a

y

x

a

y

x

a

y

x

V

A

Diam

_n

A y x A

y x A

y x

{

a

x

y

}

and

for

each

i

a

x

y

as

A

is

bounded

y

x

V

where

_i

A y x

i

<

+∞

=

∈

)

,

(

sup

,

)

,

(

)

,

(

,

.

In this case we write

δ

(

A

)

<

+∞

.

3. MAIN RESULTS:

Let

(

X

,

V

)

be a vector metric space, and

I

denote the closed unit interval

[0,1]

of reals.

Definition: 3.1 A mapping

W

:

X

×

X

×

I

→

X

is said to be a convex structure on

X

if for all

I

X

u

y

x

,

∈

,

λ

∈

;

V

(

u

,

W

(

x

,

y

,

λ

))

≤

λ

V

(

u

,

x

)

+

(

1 −

λ

)

V

(

u

,

y

)

.

Then

X

together with a convex structure

W

is called a convex vector metric space.

Remark: 3.2 It is observed that

W

(

x

,

y

,

1 )

=

x

and

W

(

x

,

y

,

0 )

=

y

.

Remark: 3.3 Any convex subset of a normed linear space is a convex vector metric space with convex structure

( , , )

(1

)

W x y

λ

=

λ

x

+

−

λ

y

.

Definition: 3.4 Let

(

X

,

V

)

be a convex vector metric space with a convex structure

W

.

For

x

,

y

∈

X

,

we

define

Seg

[

x

,

y

]

=

{

W

(

x

,

y

,

λ

)

:

λ

∈

[

0 ,

1 ]

}

.

[

x

y

]

Seg

y

x

Clearly

,

∈

,

, since

x

=

W

(

x

,

y

,

1 ),

y

=

W

(

x

,

y

,

0 ).

Proposition: 3.5 If

(

X

,

V

)

is a convex vector metric space with convex structure

W

, then for every

],

1 ,

0 [

,

y

∈

X

and

every

λ

∈

x

V

(

x

,

y

)

=

V

(

x

,

W

(

x

,

y

,

λ

))

+

V

(

W

(

x

,

y

,

λ

),

y

).

Proof:

For

every

x

,

y

∈

X

and

every

λ

∈

[

0 ,

1 ],

we have

V

(

x

,

y

)

≤

V

(

x

,

W

(

x

,

y

,

λ

))

+

V

(

W

(

x

,

y

,

λ

),

y

)

,

(

)

,

(

)

1 (

)

,

(

)

,

(

)

1 (

)

,

(

y

x

V

y

V

x

y

V

y

x

V

x

V

=

−

+

−

+

≤

λ

This implies that

V

(

x

,

y

)

=

V

(

x

,

W

(

x

,

y

,

λ

))

+

V

(

W

(

x

,

y

,

λ

),

y

).

Proposition: 3.6 Let

K

be a non-empty closed subset of the convex vector metric space

(

X

,

V

)

with convex structure

W

continuous on third variable. Let

x

∈

K

and

y

∉

K

. Then there exists a

λ

*

∈

[

0 ,

1 ]

such that

W

(

x

,

y

,

_λ

*

)

∈

Seg

[

x

,

y

]

∩

∂

K

(4)

Page: 1800-1808

Proof: Let us consider the set

A

=

{

q

:

q

≥

0 ,

W

(

x

,

y

,

η

)

∈

K

for

all

q

≤

η

≤

1 }

. Then

A

is non-empty, since

K

x

y

x

W

(

,

1 )

=

∈

. We put

q

A q∈

=

inf

λ

andlet

{ }

q

_n _n_∈_N

⊂

A

be a sequence satisfying

=

λ

∞ → n

n

q

lim

. Then, for

K

q

y

x

W

N

n

∈

,

(

,

_n

)

∈

. By usingthe continuity of

W

on the third variable andsince

K

is closed, we have

W

x

y

W

x

y

q

_n

K

n

∈

=

∞

→

(

,

)

lim

)

,

(

λ

.

Clearly

λ

>

0

since

W

(

x

,

y

,

0 )

=

y

∉

K

.

Now we prove that

W

(

x

,

y

,

λ

)

∈

∂

K

. For any positive member

ε

∈

S

,

B

(

W

(

x

,

y

,

λ

),

ε

)

∩

K

≠

Φ

. So we have to show that for any positive member

ε

∈

S

,

B

(

W

(

x

,

y

,

λ),

ε

) (

∩

X

−

K

)

≠

Φ

.

Since

λ

>

0

, we can find

λ

₁

∈

(

0 ,

1 )

such that

λ

₁

<

λ

and by definition of

λ

,

W

(

x

,

y

,

λ

₁

)

∉

K

. Using continuity of

W

on third variable, we obtain

V

(

W

(

x

,

y

,

λ

₁

),

W

(

x

,

y

,

λ

)

<

ε

.

Thus

W

(

x

,

y

,

λ

₁

)

∈

B

(

W

(

x

,

y

,

λ

),

ε

) (

∩

X

−

K

)

.

This completes the proof.

Theorem: 3.7 Let

(

X

,

V

)

be a complete convex vector metric space with convex structure

W

which is continuous on third variable,

C

be a non-empty closed subset of

X

and

T

:

C

→

X

be a non-self mapping satisfying the contractive type condition

V

(

T

(

x

),

T

(

y

)

≤

q

max

{

V

(

x

,

y

),

V

(

x

,

T

(

x

)),

V

(

y

,

T

(

y

)),

V

(

x

,

T

(

y

)),

V

(

y

,

T

(

x

))

}

for all

x

,

y

∈

C

and

0 <

q

<

1

. If

T

has the additional property

T

(

∂

C

)

⊂

C

where

∂

C

is the boundary of

C

, then

T

has a unique fixed point in

C

.

Proof:

For

x

∈

∂

C

,

we

put

x

0

=

x

. Then

T

(

∂

C

)

⊂

C

implies that

T

(

x

0

)

∈

C

. Let us put

x

1

=

T

(

x

0

)

. We

define

y

₂

=

T

(

x

₁

).

If

y

₂

∈

C

, let us take

x

₂

=

y

₂. If

y

₂

∉

C

, then by Proposition 3.6, there exists

].

,

[

₁ ₂ 2

C

Seg

x

y

x

∈

∂

∩

Continuing in this way we can obtain a sequence

{ }

x

_n such that

.

)

(

],

,

[

,

)

(

),

(

1 1

C

x

T

y

if

y

x

Seg

C

x

C

x

T

if

x

T

x

n n

n n n

n n

n

∉

=

∩

∂

∈

=

− −

We show that the sequences { }

x

_n

and

{

T

(

x

_n

)

}

are bounded.

{ }

{

(

)

}

.

,

1 0 1

0 n n

n i i n

i i

n

x

T

x

and

diam

A

define

we

N

n

For

=

∪

=

∈

− = −

=

α

We shall show that

α

_n

=

max

{

V

(

x

₀

,

T

(

x

_i

))

:

0 ≤

i

≤

n

−

1 }

.

We now discuss the following possible cases.

Case I: Suppose

α

_n

=

max

{

V

(

x

_i

,

T

(

x

_j

))

:

0 ≤

i

,

j

≤

n

−

1 }

. The case

i

=

0

is trivial. So we suppose that

i

≥

1

.

(1)

If

x

_i

=

T

(

x

_i₋₁

)

∈

C

, then

(

)

≤

− −

− − −

−

))

(

,

(

)),

(

,

(

)),

(

,

(

)),

(

,

(

),

,

(

max

)

(

),

(

1 1

1 1 1

1

i j j

i

j j i

i j i j

i

x

T

x

V

x

T

x

V

x

T

x

V

x

T

x

V

x

V

q

x

T

x

T

(5)

Page: 1800-1808

≤

q

α

_n

{

i j

}

n

V

x

T

x

i

j

n

and

i

q

So

,

α

=

max

(

,

(

))

:

0 ≤

,

≤

−

1 ≥

1 ≤

α

<

α

_n

,

a

contradict

ion

.

(2)

If

x

_i

≠

T

(

x

_i₋₁

)

i

.

e

.,

T

(

x

_i₋₁

)

∉

C

,

then

x

_i

∈

∂

C

∩

Seg

[

x

_i₋₁

,

T

(

x

_i₋₁

)].

(

₋₁

,

(

₋₁

),

)

∈

[

0 ,

1 ]

₋₁

=

(

₋₂

).

=

_i _i _i _i

i

W

x

T

x

for

some

and

x

T

x

So

λ

Now

V

(

x

_i

,

T

(

x

_j

))

=

V

(

W

(

x

_i₋₁

,

T

(

x

_i₋₁

),

λ

),

T

(

x

_j

))

.

)

1 (

))

(

),

(

)

1 (

))

(

),

(

))

(

),

(

)

1 (

))

(

,

(

1 2

1 1

n

n n

j i

q

x

T

x

T

V

x

T

x

T

V

x

T

x

T

V

x

T

x

V

α

λ

α

λ

=

−

+

≤

−

+

≤

−

+

≤

− −

{

V

x

T

x

i

j

n

and

i

}

q

a

contradict

ion

So

,

α

_n

=

max

(

_i

,

(

_j

))

:

0 ≤

,

≤

−

1 ≥

1 ≤

α

_n

<

α

_n

,

.

Case II:

Suppose

α

_n

=

max

{

V

(

x

_i

,

x

_j

)

:

0 ≤

i

,

j

≤

n

−

1 }

.

),

(

₋₁

=

j

T

x

If

we have Case I again.

1 1 1 1 2

(

),

[

, (

)],

2 (

)

j j j j j j j

If x

≠

T x

₋

then x

∈ ∂ ∩

C

Seg x

₋

T x

₋

j

≥

and x

₋

=

T x

₋ , so this is similar to Case I(2).

By an argument similar to that used above, the case

}

1 ,

0 :

))

(

),

(

{

max

≤

−

=

V

T

x

_i

T

x

_j

i

j

n

α

is also impossible. Thus, it must be the case that

{

(

,

(

))

:

0

1 }

max

₀

≤

−

=

V

x

T

x

_i

i

n

α

.

If

α

_n

=

θ

, then

x

₀ is the fixed point of

T

, so we can suppose that

α

_n

≠

θ

for

any

n

∈

N

.

))

(

),

(

))

(

,

(

))

(

,

(

,

1

0 ,

0 0

0

T

x

i

V

x

T

x

V

T

x

T

x

i

x

V

n

i

for

Further

+

≤

−

≤

V

(

x

₀

,

T

(

x

₀

))

+

q

α

_n.

{

i

}

n

V

x

T

x

i

n

V

x

T

x

q

So

,

α

=

max

(

₀

,

(

))

:

0 ≤

≤

−

1 ≤

(

₀

,

(

₀

))

+

α

i.e.,

(

,

(

))

1

0 0

T

x

V

q

n

−

≤

α

(3.1)

{ }

and

V

x

T

x

{

d

x

T

x

}

then

it

follows

from

that

If

α

_n

=

α

_nk _k

(

₀

,

(

₀

))

=

_k

(

₀

,

(

₀

))

_k

,

(

3 .

1 )

d

x

T

x

for

all

k

q

k

n

(

,

(

))

1

0 0

−

≤

α

.

{ }

n k n

k

fixed

For

,

α

is non-decreasing, so there exists

c

_k

∈

R

such that

(

,

(

))

1

1 lim

d

x

₀

T

x

₀

q

c

and

c

_nk _k _k

n k

−

≤

=

α

.

So there exists

c

=

(

c

₁

,

c

₂

,

c

_k

,

)

∈

S

such

that

(

,

(

))

1

0 0

T

x

V

q

c

−

(6)

Page: 1800-1808 We define

B

_n

=

{ }

x

_i _i_≥_n

∪

{

T

(

x

_i

)

}

_i_≥_n

,

and

β

_n

=

diam

B

_n for integer

n

≥

2

. Then by the same technique as given above, one can show that

{

V

x

_n

T

x

_j

j

n

}

n

=

sup

(

,

(

))

:

≥

β

and that if

β

_n

=

{ }

β

_nk _k, then for fixed

k

,

{ }

β

_nk _nis non-increasing and bounded. So there exists a limit and it must be zero(see [2] ). Therefore,

β =

_n

θ

n

lim

. So

{ }

x

_n

and

{

T

(

x

_n

)

}

are vector Cauchy sequences. Since

(

X

,

V

)

is a complete vector metric space and

C

is closed, there exists

z

∈

C

such

that _n

n

x

z

∞ →

=

lim

.

Again,

V

(

x

_n

,

T

(

x

_n

))

≤

β

_n

,

β

_n

→

θ

,

n

→

∞

implies that

.

))

(

,

(

x

T

x

→

as

n

→

∞

V

_n _n

θ

Now

∞

→

+

≤

V

T

x

V

x

z

as

n

z

x

T

V

(

_n

),

)

(

_n

),

_n

)

(

_n

,

)

θ

, gives that

lim

T

(

x

_n

)

z

.

n→∞

=

Assume that

z

≠

T

(

z

)

. Then

V

(

T

(

x

_n

),

T

(

z

))

≤

q

max

{

V

(

x

_n

,

z

),

V

(

x

_n

,

T

(

x

_n

)),

V

(

z

,

T

(

z

)),

V

(

x

_n

,

T

(

z

)),

V

(

z

,

T

(

x

_n

)

}

.

For

n

→

∞

, we have

,

))

(

,

(

))

(

,

(

))

(

,

(

z

T

z

q

V

z

T

z

V

z

T

z

V

≤

<

which is a contradiction. Therefore

z

=

T

(

z

)

. The uniqueness follows from contractive condition.

The following example shows that the contractive type condition in Theorem 3.7 can not be relaxed in order that Theorem 3.7 is true.

Example: 3.8 Let

X

=

[

0 ,

∞

)

with usual metric

d

. Then

(

X

,

d

)

is a complete metric space. So

(

X

,

V

)

is a complete vector metric space where

V

(

x

,

y

)

=

{

d

(

x

,

y

),

d

(

x

,

y

),

}

. We define

W

:

X

×

X

×

I

→

X

by

W

(

x

,

y

,

λ

)

=

λ

x

+

(

1 −

λ

)

y

for all

x

,

y

∈

X

and

λ

∈

I

=

[

0 ,

1 ]

.

For

x

,

y

,

u

∈

X

and

λ

∈

I

,

it

is

easy

to

check

that

V

(

u

,

W

(

x

,

y

,

λ

))

≤

λ

V

(

u

,

x

)

+

(

1 −

λ

)

V

(

u

,

y

)

.

Thus

(

X

,

V

)

is a complete convex vector metric space with convex structure

W

which is continuous on the third variable.

Let

C

=

[0,1]

be the closed unit interval of reals. Then

C

⊂

X

and let

T C

:

→

X

be a non-self mapping defined by

2

1 )

(

x

=

T

for

0 ≤ ≤

x

1

except

x

=

(

=

1 ,

2 ,

3 ,

⋅

)

2

1 ,

70

1 i

i

=

0

for

x

=

70

1

,

and

2

1

2

1

₊

=

i i

T

for

i

≥

1

.

Then

T

(

∂

C

)

⊂

C

, but

T

has no fixed point in

C

.

We now verify that for

x

=

,

3

1 y

=

70

1

,

T

does not satisfy the contractive condition

(7)

Page: 1800-1808

Now,

=

{

}

=

,

0 ,

2

1 ,

0 ,

2

1 )),

(

),

(

)),

(

),

(

))

(

),

(

T

x

T

y

d

T

x

T

y

d

T

x

T

y

d

V

=

,

2

1 ,

2

1

.

Similarly,

=

,

6

1 ,

6

1 ))

(

,

(

x

T

x

V

;

=

,

70

1 ,

70

1 ))

(

,

(

y

T

y

V

;

=

,

210

67 ,

210

67 )

,

(

x

y

V

;

=

,

3

1 ,

3

1 ))

(

,

(

x

T

y

V

;

=

,

35

17 ,

35

17 ))

(

,

(

y

T

x

V

.

{

(

,

),

(

,

(

)),

(

,

(

)),

(

,

(

)),

(

,

(

))

}

max

,

q

V

x

y

V

x

T

x

V

y

T

y

V

x

T

y

V

y

T

x

Therefore

<

=

<

=

,

2

1 ,

2

1 ,

35

34 .

2

1 ,

35

34 .

2

1 ,

35

17 ,

35

17 ,

35

17 ,

35

17 q

=

V

(

T

(

x

),

T

(

y

)),

for

0 <

q

<

1 .

Theorem: 3.9 Let

(

X

,

V

)

W

C

be a non- empty closed subset of

X

and

T

_i

:

C

→

X

be a non-self mapping satisfying the condition

V

(

T

_i

(

x

),

T

_i

(

y

)

≤

q

max

{

V

(

x

,

y

),

V

(

x

,

T

_i

(

x

)),

V

(

y

,

T

_i

(

y

)),

V

(

x

,

T

_i

(

y

)),

V

(

y

,

T

_i

(

x

))

}

for all

x

,

y

∈

C

and

0 <

q

<

1

with the additional property

T

_i

(

∂

C

)

⊂

C

. If

u

_i is the fixed point of

T

_i for

C

T

with

C

x

all

for

x

T

x

T

and

i

_i

i

∈

∂

⊂

=

∞

→

(

)

(

)

lim

)

(

,

3 ,

2 ,

1

, then

T

has a unique fixed point

u

in

i i

u

iff

C

∞ →

=

lim

.

Proof: We have

( )

lim ( )

_i

i

T x

→∞

=

for

x

∈

C

. For

x y

,

∈

C

, we have

(

T

(

x

),

T

(

y

)

q

max

{

V

(

x

,

y

),

V

(

x

,

T

(

x

)),

V

(

y

,

T

(

y

)),

V

(

x

,

T

(

y

)),

V

(

y

,

T

(

x

))

}

V

_i _i

≤

_i _i _i _i where

0 < <

q

1

.

As

i

→ ∞

,

(

T

(

x

),

T

(

y

)

q

max

{

V

(

x

,

y

),

V

(

x

,

T

(

x

)),

V

(

y

,

T

(

y

)),

V

(

x

,

T

(

y

)),

V

(

y

,

T

(

x

))

}

V

≤

where

0 < <

q

1

.

(8)

Page: 1800-1808 Now for

0 <

q

<

1

, we have

V

(

u

,

u

_i

)

=

V

(

T

(

u

),

T

_i

(

u

_i

))

+

≤

+

≤

))

(

,

(

)),

(

,

(

)),

(

,

(

)),

(

,

(

),

,

(

max

))

(

),

(

))

(

),

(

))

(

),

(

u

T

u

V

u

T

u

V

u

T

u

V

u

T

u

V

u

V

q

u

T

u

T

V

u

T

u

T

V

u

T

u

T

V

i i i i i i i i i i i i i i

≤

V

(

T

(

u

),

T

_i

(

u

))

+

q

max

{

V

(

u

,

u

_i

),

V

(

u

,

T

_i

(

u

)),

V

(

u

,

u

_i

)

+

V

(

u

,

T

_i

(

u

))

}

≤

V

(

T

(

u

),

T

_i

(

u

))

+

q

{

V

(

u

,

u

_i

)

+

V

(

T

(

u

),

T

_i

(

u

))

}

,

which implies that,

∞

→

−

+

≤

V

T

u

T

u

as

i

q

u

V

_i

(

),

_i

(

))

0

1

1 )

,

(

. i i

u

Therefore

∞ →

=

lim

,

.

Conversely, let

lim

_i

i

u

→∞

=

, then

≤

))

(

,

(

)),

(

,

(

)),

(

,

(

)),

(

,

(

),

,

(

max

))

(

),

(

u

T

u

V

u

T

u

V

u

T

u

V

u

T

u

V

u

V

q

u

T

u

T

V

i i i i i i i i i i i i

which implies that,

}

{

(

,

),

(

,

(

)),

(

,

(

))

max

)

),

(

T

u

q

V

u

V

u

T

u

V

u

T

u

V

_i _i

≤

_i _i _i _i .

Taking limit as

i

→ ∞

, we have

))

(

,

(

)

),

(

T

u

q

V

u

T

u

V

≤

.

So,

(

1 −

q

)

V

(

u

,

T

(

u

))

≤

θ

which

gives

that

T

(

u

)

=

u

.

We now prove a fixed point theorem for multivalued mappings. For this purpose we need the following notations.

Notations: Let

(

X

,

V

)

be a vector metric space and let

A B

,

be any two subsets of

X

. We denote

=

∈ ∈ ∈ ∈ ∈ ∈ ∈

∈

inf

(

,

)

inf

(

,

),

inf

(

,

),

,

inf

(

,

),

)

,

(

, 2 , 1 ,

,

V

x

y

a

x

y

a

x

y

a

x

y

B

A

D

_n B y A x B y A x B y A x B y A x ,

=

∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

),

,

(

sup

,

),

,

(

sup

),

,

(

sup

)

,

(

sup

)

,

(

, 2 , 1 , ,

y

x

a

y

x

a

y

x

a

y

x

V

B

A

_n B y A x B y A x B y A x B y A x

ρ

where

V

(

x

,

y

)

=

{

a

_i

(

x

,

y

)

}

.

{

Φ

≠

⊂

<

+∞

}

=

:

(

)

(

X

A

X

and

A

BN

δ

.

Theorem: 3.10 Let

(

X

,

V

)

W

C

be a non-empty closed subset of

X

and

F C

:

→

BN X

( )

be a multivalued mapping satisfying the condition

{

(

,

),

(

,

(

)),

(

,

(

)),

(

,

(

)),

(

,

(

))

}

max

))

(

),

(

F

x

F

y

q

V

x

y

ρ

x

F

x

ρ

y

F

y

D

x

F

y

D

y

F

x

ρ

≤

(3.2)

for all

x y

,

∈

C

and

0 < <

q

1

. If

F

has the additional property

F

(

∂

C

)

⊂

C

where

{

}

(

)

( ) :

F

∂

C

= ∪

F x

x

∈ ∂

C

, then

F

u

in

C

and

F u

( )

=

{ }

u

.

Proof: Take

0 <

a

<

1

and define a single valued mapping

T C

:

→

X

as follows: For each

x

∈

C

, let

T x

( )

be a point of

F x

( )

, which satisfies the condition

))

(

,

(

))

(

,

(

x

T

x

q

x

F

x

(9)

Page: 1800-1808

Then

T

has the property

T

(

∂

C

)

⊂

C

since

F

(

∂

C

)

⊂

C

.

Now for every

x y

,

∈

C

, we have

V

(

T

(

x

),

T

(

y

))

≤

ρ

(

F

(

x

),

F

(

y

))

≤

q

max

{

V

(

x

,

y

),

ρ

(

x

,

F

(

x

)),

ρ

(

y

,

F

(

y

)),

D

(

x

,

F

(

y

)),

D

(

y

,

F

(

x

))

}

=

−

))

(

,

(

)),

(

,

(

)),

(

,

(

)),

(

,

(

),

,

(

max

x

F

y

D

q

y

F

x

D

q

y

F

y

q

x

F

x

q

y

x

V

q

a a

a

ρ

≤

−

))

(

,

(

)),

(

,

(

)),

(

,

(

)),

(

,

(

),

,

(

max

1

x

T

y

V

y

T

x

V

y

T

y

V

x

T

x

V

y

x

V

q

a

for

0 <

q

<

1

.

Hence by Theorem 3.7,

T

has a unique fixed point in

C

. Let

u

∈

C

be such that

u

=

T u

( )

.

Clearly

u

=

T u

( )

implies

u

∈

F u

( )

. Since

F

satisfies (3.2),

u

∈

F u

( )

implies

))

(

,

(

))

(

),

(

F

u

F

u

q

ρ

u

F

u

ρ

≤

.

This may happen only if

F u

( )

=

{ }

u

. Therefore,

u

∈

C

is a fixed point of

T

iff

u

is a fixed point of

F

. Thus

F

u

in

C

and

F u

( )

=

{ }

u

.

REFERENCES:

[1] A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31-37.

[2] Ljiljana Gaji , Quasi-contractive non-self mappings on Takahashi convex metric spaces, Novi Sad J. Math., 30

(2000), 41-46.

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