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ISSN 2229 – 5046

International Journal of Mathematical Archive- 3 (6), June – 2012 2321

SOME COMMON FIXED POINT THEOREMS WITH COMPATIBLE MAPPINGS

AND CONVEXITY IN METRIC SPACE

Deepak Singh Kaushal* & S. S. Pagey

Sagar Institute of Science, Technology & Research, Ratibad, Bhopal (M.P.), India Institue for Excellence in Higher Education, Bhopal (M.P.), India

(Received on: 03-06-12; Accepted on: 20-06-12)

ABSTRACT

I

n this note we establish some common fixed point theorems of compatible mappings in the setting of convex metric spaces. Several results are also derived as special cases.

Keywords: Fixed point, Compatible mappings, Convex metric spaces.

1. INTRODUCTION

The concept of the commutativity has generalized in several ways.For this Sessa S. [13] has introduced the concept of weakly commuting and Gerald Jungck [5] initiated the concept of compatibility. It can be easily verified that when the two mappings are commuting then they are compatible but not conversely.

The study of common fixed point of mappings satisfying contractive type conditions has been a useful tool for obtaining more comprehensive fixed point theorems[3,6,7,8,9,10] during the last three decades.

Jungck [5] introduced more generalized commuting mappings called compatible mappings. Which are more general then commuting and weakly commuting mappings.

On the other hand, Takahanshi [14] introduced a notion of convex metric spaces, many authers have discussed the existence of fixed point and the convergence of iterative processes for non-expansive mappings in convex metric spaces.

The main purpose of this paper is to present fixed point results for a pair of maps satisfying a new contractive condition by using the concept of compatible maps in a convex metric space. Our main results are generalizations of some known results in [1, 2, 4, 6, 11, 12].

2. PRELIMINARIES

Definition 2.1: The pair (A, B) of self-mappings of a metric space (X, d) is said to be compatible on X if whenever

{ }

x

n is a sequence in

X

such that

Ax Bx

n

,

n

→ ∈

t

X

then

d BAx ABx

(

,

)

0

.

Definition 2.2: Let

(

X d

,

)

be a metric space and

J

=

[ ]

0,1

.A mapping

W X

:

× × →

X

J

X

is called a convex structure on

X

if for each

(

x y

, ,

λ

)

∈ × ×

X

X

J

and

u

X

,

d u W x y

(

,

(

, ,

λ

)

)

λ

d u x

( ) (

,

+ −

1

λ

) ( )

d u y

,

A metric space

X

together with a convex structure

W

is called a convex metric space.

Definition 2.3: A nonempty subset

K

of a convex metric space

(

X d

,

)

with a convex structure

W

is said to be convex if for all

(

x y

, ,

λ

)

∈ × ×

K

K

J

,

W x y

(

, ,

λ

)

K

.

*Corresponding author:Deepak Singh Kaushal*, Sagar Institute of Science, Technology & Research,

(2)

© 2012, IJMA. All Rights Reserved 2322

Definition 2.4: Let

K

be a convex subset of a convex metric space

(

X d

,

)

with a convex structure

W

. A mapping

:

I K

K

is said to be

W

-affine if for all

(

x y

, ,

λ

)

∈ × ×

K

K

J

,

IW x y

(

, ,

λ

)

=

W Ix Iy

(

,

,

λ

)

Definition 2.5: A sequence

{ }

x

n in a metric space

(

X d

,

)

is said to be convergent to a point

x

X

denoted by

lim

n

n→∞

x

=

x

if

lim (

n→∞

d x x

n

, )

=

0

.

3. MAIN RESULTS

Theorem 3.1: Let

(

X d

,

)

be a complete convex metric space with a convex structure

W

and

K

is a nonempty closed convex subset of

X

.

Suppose that the mappings

T

and

I

are compatible self –maps on

K

satisfying the following condition

(

)

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

,

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

,

2

d Tx Ty

ad Ix Iy

b

d Ix Tx d Iy Ty

c

d Iy Tx d Ix Ty

d

d Ix Iy d Ix Tx d Iy Ty

d Iy Tx

d Ix Ty

x y

K

+

+

+

+

Where a, b, c, d are non-negative real numbers such that

a b

+ +

2

c d

+ =

1

and

(

)(

)

(

)

2

.

2

a c b c

d

b c

+

+

>

− +

If

( )

( )

and is

T K

I K

I

W

-affine and continuous then there exists a unique common fixed point

of

and , and

z

T

I

T

is continuous at

z

.

Proof: Let

x

=

x

0be an arbitrary point in

K

and choose some points

x x

1

,

2

and

x

3in

K

such that

1

,

1 2

,

2 3

Tx

=

Ix Tx

=

Ix Tx

=

Ix

It is possible since

T K

( )

I K

( )

.

For

k

=

1, 2, 3

we have

(

k

,

k

)

(

k

,

k 1

)

d Tx Ix

=

d Tx Tx

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

1 1 1 1 1

1 1 1 1 1

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

2

k k k k k k k k k k

k k k k k k k k k k

ad Ix Ix

b

d Ix Tx

d Ix

Tx

c

d Ix

Tx

d Ix Tx

d

d Ix Ix

d Ix Tx

d Ix

Tx

d Ix

Tx

d Ix Tx

− − − − −

− − − − −

+

+

+

+

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

)

(

(

) (

)

)

1 1 1 1 1 1

1 1 1 1

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

2

k k k k k k k k k k

k k k k k k k k

ad Ix

Tx

b

d Ix Tx

d Ix

Tx

c

d Ix

Tx

d Ix Tx

d

d Ix

Tx

d Ix Tx

d Ix

Tx

d Ix Tx

− − − − − −

− − − −

+

+

+

+

If

(

k

,

k

)

(

k 1

,

k 1

)

d Tx Ix

>

d Tx

Ix

then we have

(

) (

) (

)

(

) (

)

,

2

,

,

,

Since

2

1

k k k k

k k k k

d Tx Ix

a b

c

d d Tx Ix

d Tx Ix

d Tx Ix

a b

c

d

< + +

+

<

+ +

+ =

Which is a contradiction and therefore

(

k

,

k

)

(

k 1

,

k 1

)

Where

1, 2, 3

(3)

© 2012, IJMA. All Rights Reserved 2323 Now

(

1

,

2

)

(

,

2

)

d Ix Tx

=

d Tx Tx

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

0 2 2 2 2 2

2 2 2 2 2

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

2

ad Ix Ix

b

d Ix Tx d Ix Tx

c

d Ix Tx d Ix Tx

d

d Ix Ix

d Ix Tx d Ix Tx

d Ix Tx

d Ix Tx

+

+

+

+

(

) (

)

{

}

(

)

(

) (

) (

) (

)

{

}

(

) (

) (

) (

)

(

) (

) (

) (

)

(

)

1 1

1 1 1 1 2 2

1 1

1 1 2 2 1 1

,

,

,

max

,

,

,

,

,

,

,

,

,

,

,

,

max 1

,

,

,

,

2

a d Ix Tx

d Ix Tx

b d Ix Tx

c

d Ix Tx

d Ix Tx

d Ix Tx

d Ix Tx

d Ix Tx

d Ix Tx

d Ix Tx d Ix Tx

d

d Ix Tx

d Ix Tx

d Ix Tx

d Ix Tx

+

+

+

+

+

+

+

+

+

+

(

) (

)

(

)

(

)

2

3

2

,

2

,

a b

c

d d Ix Tx

b c d Ix Tx

=

+ +

+

=

− +

2, 3 2 3 1 2

1

1

1

Let

,

then we know that

and ,

,

,

,

2

2

2

z

=

W x x

z

K

Iz

=

W Ix Ix

=

W Tx Tx

Now we obtain

(

1

)

1 1 2

1

,

,

,

,

2

d Iz Ix

=

d Ix W Tx Tx

(

)

(

)

(

)

(

) (

)

(

) (

)

(

)

1 1 1 2

1

1

,

,

2

2

1

1

,

2

3

2

,

2

2

1

1 2

3

2

,

2

3

,

2

d Ix Tx

d Ix Tx

d Ix Tx

a b

c

d d Ix Tx

a b

c

d d Ix Tx

b c

d Ix Tx

+

+

+ +

+

=

+

+ +

+

− +

=

(

2

)

2 1 2

1

,

,

,

,

2

d Iz Ix

=

d Ix W Tx Tx

(

)

(

)

(

)

(

)

(

)

2 1 2 2

2 2

1

1

,

,

2

2

1

1

,

,

2

2

1

,

2

d Ix Tx

d Ix Tx

d Ix Ix

d Ix Tx

d Ix Tx

+

+

=

(

3

)

3 1 2

1

,

,

,

,

2

d Iz Ix

=

d Ix W Tx Tx

(

)

(

)

(

)

(

)

3 1 3 2

2 2 3 3

1

1

,

,

2

2

1

1

,

,

2

2

d Ix Tx

d Ix Tx

d Ix Tx

d Ix Ix

+

+

(

)

1

,

2

d Ix Tx

(4)

© 2012, IJMA. All Rights Reserved 2324 Again we obtain

(

)

(

)

(

)

1 2 1 2

1

,

,

,

,

2

1

1

,

,

2

2

d Iz Tz

d Tz W Tx Tx

d Tz Tx

d Tz Tx

=

+

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

)

1 1 1 1 1

1 1 1 1 1

2 2 2 2 2

2 2 2 2

,

max

,

,

,

max

,

,

,

2

2

2

1

max

,

,

,

,

,

,

,

,

2

2

,

max

,

,

,

max

,

,

,

2

2

2

1

max

,

,

,

,

,

,

,

2

2

a

b

c

d Iz Ix

d Iz Tz d Ix Tx

d Ix Tz d Iz Tx

d

d Iz Ix

d Iz Tz d Ix Tx

d Ix Tz

d Iz Tx

a

b

c

d Iz Ix

d Iz Tz d Ix Tx

d Ix Tz d Iz Tx

d

d Iz Ix

d Iz Tz d Ix Tx

d Ix Tz

d Iz

+

+

+

+

+

+

+

+

(

+

(

,

Tx

2

)

)

(

)

(

)

{

(

) (

)

}

(

) (

)

(

)

(

) (

) (

)

(

)

(

)

3

,

max

,

,

,

4

2

1

1

max

(3

)

,

,

,

,

2

2

2

1

4

1

max

(3

)

,

,

,

,

,

,

,

,

2

2

4

2

a

b

b c d Ix Tx

d Iz Tz d Ix Tx

c

b c d Ix Tx

d Iz Tz

d Ix Tx

d

b c

b c d Ix Tx d Iz Tz d Ix Tx

d Ix Tx

d Iz Tz

− +

+

+

− +

+

− +

+

− +

+

(

)

{

(

) (

)

}

(

) (

)

(

)

(

) (

) (

)

(

(

) (

)

)

1

1

,

max

,

,

,

max

,

,

,

,

4

2

2

2

2

1

1

max

,

,

,

,

,

,

,

,

2

2

2

a

b

c

d Ix Tx

d Iz Tz

d Ix Tx

d Ix Tx

d Iz Tz

d Ix Tx

d

d Ix Tx d Iz Tz

d Ix Tx

d Ix Tx

d Iz Tz

+

+

+

+

+

+

If

d Iz Tz

(

,

)

>

d Ix Tx

(

,

)

then we have

(

)

1

(

2

)

(

)

,

1

2

2

2

2

2

,

8

d Iz Tz

≤ −

c

d

+

ab

+

ac

+

bc bd

+

+

cd

d Iz Tz

This is a contradiction since

(

)(

)

(

)

2

2

a

c b c

d

b c

+

+

>

− +

and

a b

+ +

2

c d

+ =

1

implies that 2

2

c

2

d

+

2

ab

+

2

ac

+

2

bc

+

bd

+

cd

>

0.

Hence we have

(

)

1

(

2

)

(

)

,

1

2

2

2

2

2

,

8

d Iz Tz

≤ −

c

d

+

ab

+

ac

+

bc bd

+

+

cd

d Ix Tx

Letting

1

1

(

2

2

2

2

2

2

)

8

c

d

ab

ac

bc bd

cd

λ

= −

+

+

+

+

+

We know that

0

< <

λ

1.

Then we have

d Iz Tz

(

,

)

λ

d Ix Tx

(

,

)

Since

x

0is an arbitrary point in

K

then there exists a sequence

{ }

z

n in

K

such that

(

)

(

)

(

)

(

)

(

)

(

)

0 0 0 0 1 1 0 0

1 1

,

,

,

,

...

...

,

,

n n n n

d Iz Tz

d Ix Tx

d Iz Tz

d Iz Tz

d Iz Tz

d Iz

Tz

λ

λ

λ

− −

(5)

© 2012, IJMA. All Rights Reserved 2325 Which yield that

(

,

)

1

(

0

,

0

)

n n n

d Iz Tz

λ

+

d Iz Tz

and so we have

(

n

,

n

)

0

d Iz Tz

Setting

(

)

1

:

,

,

1, 2, 3...

n

K

x

K d Ix Tx

n

n

=

=

Then

K

n

φ

,

n

=

1, 2, 3,...and

K

1

K

2

...

Obviously

1

,

1, 2, 3...

n n n

TK

φ

and TK

TK

+

n

=

For any

x y

,

K

n we have

(

)

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

,

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

2

d Tx Ty

ad Ix Iy

b

d Ix Tx d Iy Ty

c

d Iy Tx d Ix Ty

d

d Ix Iy d Ix Tx d Iy Ty

d Iy Tx

d Ix Ty

+

+

+

+

(

)

(

)

(

)

(

)

(

)

2

1

2

,

,

,

1

2

2

(

)

,

b

a

d Tx Ty

c

d Tx Ty

d

d Tx Ty

n

n

n

n

a b c

d

a

c

d d Tx Ty

n

+

+ +

+

+

+

=

+ + +

+ + +

Which yields that

(

)

(

1

) (

)

,

2

2

d Tx Ty

a b c

d

n b c

+ + +

+

Therefore we have

( )

(

)

lim

n

lim

n

0

n→∞

diam TK

=

n→∞

diam TK

=

Where

diam TK

( )

n denotes the diameter of

( )

TK

n .By the Cantor’s theorem there exists a point

u

K

such that

{ }

1

.

n

n n

u

TK

=∞

=

=

Since

u

K

for each

n

=

1, 2, 3,...,

there exists a point

x

n

K

nsuch that

d u Tx

(

,

n

)

1

n

<

, and so

.

n

Tx

u

Further, since

x

n

K

n, we have

d Tx Ix

(

n

,

n

)

1

n

<

and

Ix

n

u

.

Since

I

is continuous and the pair

( )

T I

,

is compatible, we can induce easily that

,

,

.

n n n

ITx IIx TIx

Iu

Now

(

,

)

(

,

n

) (

n

,

)

d Tu Iu

d Tu TIx

+

d TIx Iu

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

2

n n n n n

n n n n n

ad Iu IIx

b

d Iu Tu

d IIx TIx

c

d IIx Tu

d Iu TIx

d

d Iu IIx

d Iu Tu

d IIx TIx

d IIx Tu

d Iu TIx

+

+

+

+

(6)

© 2012, IJMA. All Rights Reserved 2326 Now letting

n

→ ∞

we have

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

2

ad Iu Iu

b

d Iu Tu

d Iu Iu

c

d Iu Tu

d Iu Iu

d

d Iu Iu

d Iu Tu

d Iu Iu

d Iu Tu

d Iu Iu

+

+

+

+

(

,

) (

) (

,

)

d Tu Iu

b c

d d Tu Iu

+ +

Which implies that

(

,

)

0

d Tu Iu

=

Tu

Iu

=

Hence

IIu

=

ITu

=

TIu

=

TTu

since the pair

( )

T I

,

is compatible.

Now

(

)

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

,

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

2

d TTu Tu

ad ITu Iu

b

d ITu TTu

d Iu Tu

c

d Iu TTu

d ITu Tu

d

d ITu Iu

d Iu Tu

d ITu TTu

d Iu TTu

d ITu Tu

+

+

+

+

(

,

) (

) (

,

)

Since

1

d TTu Tu

a c

d d TTu Tu

a c

d

+ +

+ + <

Which implies that

(

,

)

0

d TTu Tu

=

TTu

Tu

=

Now let

z

=

Tu

,

then

z

is a common fixed point of

T

& .

I

If

z

*is another common fixed point of

T

& .

I

then

( )

*

,

0.

d z z

>

Now

( ) (

)

(

)

{

(

)

(

)

}

{

(

) (

)

}

(

)

(

)

(

)

(

(

) (

)

)

* *

* * * * *

* * * * *

,

,

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

2

d z z

d Tz Tz

ad Iz Iz

b

d Iz Tz

d Iz Tz

c

d Iz Tz

d Iz Tz

d

d Iz Iz

d Iz Tz

d Iz Tz

d Iz Tz

d Iz Tz

=

+

+

+

+

( )

*

(

)

( )

*

,

,

d z z

a c d d z z

+ +

Which is a contradiction. Hence

z

is the unique common fixed point of

T

& .

I

To show that

T

is continuous at

z

.

Let

{ }

z

n

K

be a sequence converges to z. Since

I

is continuous then

.

n

(7)

© 2012, IJMA. All Rights Reserved 2327

(

)

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

)

(

(

) (

)

)

,

,

max

,

,

,

max

,

,

,

1

max

,

,

,

,

,

,

,

,

2

n n n n n n

n n n n n

d Tz Tz

ad Iz Iz

b

d Iz Tz

d Iz Tz

c

d Iz Tz

d Iz Tz

d

d Iz Iz

d Iz Tz

d Iz Tz

d Iz Tz

d Iz Tz

+

+

+

+

(

)

{

(

) ( )

}

{

(

) (

) (

)

}

(

) (

) ( )

(

(

) (

) (

)

)

,

max

,

,

,

max

,

,

,

,

1

max

,

,

,

,

,

,

,

,

,

2

n n

n n

ad Iz Iz

b

d Iz Tz

d z z

c

d Iz Tz

d Tz Tz

d Iz Tz

d

d Iz Iz

d Tz Tz

d z z

d Iz Tz

d Tz Tz

d Iz Tz

+

+

+

+

+

+

(

n

,

) (

) (

n

,

)

d Tz Tz

b c

d d Tz Tz

+ +

Taking the limit as

n

→ ∞

from which we see that

(

) (

)

(

)

lim sup

n

,

lim sup

n

,

n→∞

d Tz Tz

b c

d

n→∞

d Tz Tz

+ +

(

)

lim

n

,

0

Since

1

n→∞

d Tz Tz

=

b c

+ + <

d

Which implies that

T

is continuous at

z

.

This completes the proof of theorem.

Remark: If we put

c

=

0

in main inequality then we will obtain the result of Huang and Li [4].

Corollary 3.2: Let

(

X d

,

)

be a complete convex metric space with a convex structure

W

and

K

is a nonempty closed convex subset of

X

.

Suppose that the mappings

T

and

I

are compatible self –maps on

K

satisfying the following condition

(

)

(

)

(

) (

)

(

(

) (

)

)

(

) (

)

(

(

) (

)

)

1

,

,

max

,

,

,

,

,

,

2

1

max

,

,

,

,

,

,

2

d Tx Ty

ad Ix Iy

b

d Ix Tx d Iy Ty

d Ix Tx

d Iy Ty

c

d Iy Tx d Ix Ty

d Iy Tx

d Ix Ty

+

+

+

+

Where a,b,c are non-negative real numbers such that

a b

+ +

2

c

=

1

and

(

a c b c

+

)(

+ >

)

0

.If

( )

( )

and is

T K

I K

I

W

-affine and continuous then there exists a unique common fixed point

z

of

and , and

T

I

T

is continuous at

z

.

Corollary 3.3: Let

(

X d

,

)

be a complete convex metric space with a convex structure

W

and

K

is a nonempty closed convex subset of

X

.

Suppose that the mappings

T

and

I

are compatible self –maps on

K

satisfying the following condition

(

)

(

)

{

(

) (

)

}

{

(

) (

)

}

(

) (

) (

) (

) (

)

{

}

,

,

max

,

,

,

max

,

,

,

max

,

,

,

,

,

,

,

,

,

d Tx Ty

ad Ix Iy

b

d Ix Tx d Iy Ty

c

d Iy Tx d Ix Ty

d

d Ix Iy d Ix Tx d Iy Ty d Iy Tx d Ix Ty

+

+

+

Where a, b, c, d are non-negative real numbers such that

a b

+ +

2

c

+

2

d

=

1

and

(

a c

+ +

d

)(

b c

+ +

d

)

>

0.

If

( )

( )

and is

T K

I K

I

W

-affine and continuous then there exists a unique common fixed point

of

and , and

(8)

© 2012, IJMA. All Rights Reserved 2328

REFERENCES

[1] Fisher,B. and Sessa S., “On a fixed point theorem of Gregustype”, Publ. Math. Debrecen, 34 (1987), 83-89.

[2] Gregus Jr.M., “A fixed point theorem in Banach space”, Boll. Un. Mat. Ital., 517(1980), 193-198.

[3] Huang,N.J., “Stability for fixed point iterative procedures of nonlinear mappings in convex metric spaces”, J. Gannan Teacher’s College, 3(1991), 22-28.

[4] Huang,N.J. and Li,H.X., “Fixed point theorems of compatible mappings in convex metric spaces”, Soochow Journal of Mathematics, 22(1996),439-447.

[5] Jungck,G., “Compatible mapping and common fixed points” International J. Math. Math. Sci., 9 (1986), 771-779.

[6] Jungck,G., “On a fixed point theorem of Fisher and Sessa” International J. Math. Math. Sci., 13 (1990), 497-500.

[7] Jungck,G., “Compatible mappings and common fixed points ” International J. Math. Math. Sci., 17(1994), 37-40.

[8] Jungck,G., “Coincidence and fixed points for compatible and relatively nonexpansive maps” International J. Math. Math. Sci., 16 (1993),95-100.

[9] Jungck,G., “Compatible mapping and common fixed points(2)” International J. Math. Math. Sci., 11(1988), 265-288.

[10] Jungck,G. and Rhodes B.E., “Some fixed point theorems for compatible maps” International J. Math. Math. Sci., 16 (1993), 417-428.

[11] Li,B.Y. ,“Fixed point theorems for non expansive mappings in convex metric spaces”, Appl . Math. Mech., 10 (1989), 173-178.

[12] Mukherjee, R. N. and Verma , V., “A note on fixed point theorem of Gregus”, Math. Japonica, 33 (1988), 745-749.

[13] Sessa,S., “On a weak commutativity condition of mapping in a fixed point considerations”, Publ. Inst. Math. Debre. , 32(1982), 149-153.

[14] Takahashi,W. “A convexity in metric spaces and nonexpansive mappings” Kodai Math. Sem. Rep. 229(1970) 142-149.

References

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