FIXED POINTS OF MULTI-VALUED A -MAPPINGS IN ...

Pakistan Journal of Science (Vol. 66 No. 1 March, 2014)

* FIXED POINTS OF MULTI-VALUED A -MAPPINGS IN CONE METRIC SPACE.

M. Akram and Z. Afzal Department of Mathematics GC University, Katchery Road Lahore 54000, Pakistan. Corresponding Author Email: [email protected], [email protected], * ABSTRACT:In this paper, we extend the idea of A -mappings from metric spaces to cone metric * spaces. Further, we prove some fixed point theorems for multi-valued mapping using A -contractions

in cone metric space setup. Our results generalizes andimproves some results in the existing literature from metric spacessetup to cone metric spaces setup. Furthermore, some fixed point theorems in cone metics spaces are also improved. Key words:Cone metric, Fixed points,self-maps, Multi-valued mappings,

The least positive number satisfying the above inequality is called the normal constant of P , while

INTRODUCTION

x  y stands for y  x  intP (interior of P ).

The concept of cone metric spacewas introduced by (Huang and Zhang 2007) replacing the set of positive real numbers by an ordered Banach space. They obtained some fixed point theorems in cone metric spaces using the normality of the cone. The results of (Huang and Zhang 2007) were generalized by (Rezapour and Haghi 2008). Several other authers like (Aage and Salunke 2009), (Ilic and Rakocevic 2008) and (Abbas and Rhoades2009) were also investigating some common fixed point theorems for different types of contractive mappings in cone metric

(Rezapour and Hamlbarani Haghi 2008) proved that there is no normal cone with normal constant K < 1 and for each .

d : X  X  E satisfies: 1. 0  d ( x, y) for all x, y  X and d ( x, y) = 0 if the mapping

*

and only if x = y ;

2. d ( x, y) = d ( y, x) for all x, y  X ;

3. d ( x, y)  d ( x, z )  d ( z, y) for all x, y, z  X .

A* -maps in cone metric spaces.

Then

The concept of cone metric space more general than metric space.

Preliminaries: Let E be a topological vector space. A subset P of E is called a cone if and only if:

2.

ax  by  P ;

3.

imply

{xn } a sequence in X and x  X . For every c  E {xn } is with 0  c , we say that (i) a Cauchy sequence if there is a natural number N d ( xn , xm )  c . such that for all n, m > N ,

that

P( P) = 0

. Given a cone P  E , we define a partial

(ii) a convergent sequence if there is a natural number

x  y if and only if ordering  with respect to P by y  x  P . A cone P is said to be normal space E if there ia a number K > 0 such that for all x, y  E , 0  x  y implies

Let ( X , d ) be a cone metric space,

Definition 2.2

P is closed, nonempty and P  0 ;

a, b  R, a, b  0, x, y  P

d is called a cone metric on X and

( X , d ) is called a cone metric space.

AMS (2000) Mathematics Subject Classification: 47H10, 54H25

1.

k > 1 there is a cone with normal constant K > k

Definition 2.1 Let X be a nonempty set. Suppose that

space. The idea of general multi-valued A -maps was introduced by (Akram et al. 2003) and they proved some fixed point theorems for these maps in complete metric spaces. The purpose of this paper is to proved the existence of fixed point of a general class of multi-valued maps called

A* -mappings.

such that for all

x X

It is known that

x K y.

if

45

n > N , d ( xn , xm )  c for some

{ xn }

d ( xn , x)  0

N

as

converges to

n  .

x  X if and only

Pakistan Journal of Science (Vol. 66 No. 1 March, 2014)

T1 , T2 : Pcl ( X ) be a multi-valued i  j , for mappings suchthat for i, j {1,2} with u y  T j ( y) x, y  X u x  Ti (x)

Definition 2.3 A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X .

space

each that

Definition 2.4 A set A in a cone metric space X is

{xn } in a A which converges to some x  X implies that x  A . Let X be a cone metric space. We denote by

provided

such that

d ( x1 , x2 )   (d ( x0 , x1 ), d ( x0 , x1 ), d ( x1 , x2 ))  k d ( x0 , x1 ), where k  [0,1) . Similarly,

A* * -contraction as follows: Let a nonempty set A  : R3  R satisfying consisting of all functions

d ( x2 , x3 )  k d ( x1 , x2 )  k 2 d ( x0 , x1 ). Continuing in a similar way, we get

(i)  is continuous on the set of all triplets of nonnegative reals(with respects to the Euclidean metric on 3 

d ( xn , xn1 )  k d ( xn1 , xn )  ....  k n d ( x0 , x1 ), n  1.

R

m > n ; we get d ( xn , xm )  d ( xn , xn1 )  d ( xn1 , xn2 )  .....  d ( xm1 , xm )

R 3 ).

Then for

a  kb for some k [0,1) , whenever a   (a, b, b) or a   (b, a, b) or a   (b, b, a) for all a, b .

 (k n  k n1  k m1 ) d ( x0 , x1 )

(ii)  is non-decreasing in each coordinate variable; (iii)

A

multiple-valued

= k n d ( x0 , x1 )/(1  k ). Let c  0 be given, choose a symmetric open neighborhood V of 0 such that c  V  P . Also,

mapping

T : X  P( X ) is said to be A* -contractions. If  (Tx, Ty)   ( ( x, y),  ( x, Tx),  ( y, Ty)) (1) * x , y  X for some   A and for all . We also call

choose a natural number

N1

such that

k d ( x0 , x1 )/(1  k ) V , n  N1. n

these mappings as A*-mappings.

(3)

k d ( x0 , x1 )/(1  k )  c for all d ( xn , xm )  c m, n > N1 n

Which implies that

Throughout the sequel, CB(X ) would denote the set of all closed and bounded subsets of X , where X is a Complete cone metric space. For sets A and B in a cone metric space X , we use the symbols,

 ( A, B) = sup{d (a, b) : a  A, b  B} d ( A, B) = inf {d (a, b) : a  A, b  B}

and

x0 is an arbitrary point of X . For i, j 1,2 with i  j , Then x1  Ti ( x0 ) . There exist x2  T j ( x1 )

On the other hand, Akram et al. (2003) defined

2.5

(2)

Proof. Suppose that

x  Tx . The collection of all fixed points of T is denoted by F (T ) .

Definition

such

F (T1 ) = F (T2 )  Pcl ( X ) .

the family of all nonempty closed subsets of . A point x in X is called a fixed point of a

T : X  Pcl ( X )

and

F (T1 ) = F (T2 )  

Then

P(X ) the family of all nonempty subsets of X , and by Pcl (X ) X mapping

,

d (u x , u y )   (d ( x, y), d ( x, u x ), d ( y, u y )).

closed if for every sequence

multi-valued

and

n > N1

for all

{x }

n . Hence be a Cauchy sequence in X , since X is complete cone metric space, there exist an element

x  X s.t. xn  x as n   . Let 0  c be x2 n  T j ( x2 n1 ) un  Ti (x)

, and

H ( A, B) = max{supaA d (a, B), supbB d ( A, b)}

and hence

given and for such that

.

, there exist

,

d ( x2n , un )   (d ( x2n1 , x), d ( x2n1 , x2n ), d ( x, un )) . Taking limits n   , we get

Main Result for multi-valued maps Theorem 3.1 Let ( X , d ) be a Complete cone metric

lim d ( x2n , un )  lim [ (d ( x2 n1 , x), d ( x2 n1 , x2 n ), d ( x, un ))] n

46

n

Pakistan Journal of Science (Vol. 66 No. 1 March, 2014)

d ( x, lim un )   (d ( x, x), d ( x, x), d ( x, lim un ) n

a  b  c < 1 . Then F (T1 ) = F (T2 )   F (T1 ) = F (T2 )  Pcl ( X ) .

n

=  (0,0, d ( x, lim un )) n

 k (0) = 0. This

implies

that

lim n un = x

The following corollary extends Theorem 4.1 of (Latif and Beg 1997) to the case of two mappings on cone metric spaces.

Hence

un  x as n   . T (x) is closed, x  F (Ti ) and F (Ti )   . Since i T Let x  X be a fixed point of 1 Then by hypothesis, w T2 ( x) s.t. there exists

Corollary 3.3 Let ( X , d ) be a Complete cone metric P is a non-normal cone. If space and

T1 , T2 : X  Pcl ( X ) are two multi-valued mappings i  j , for each such that for i, j {1,2} with x, y  X , u x  Ti (x) and u y  T j ( y) , d (u x , u y )  hd ( x, u x )  d ( y, u y ).

d (w, x)   (d ( x, x), d ( x, x), d ( x, w)   (0,0, d ( x, w))  k (0) = 0.

(5)

F (T1 )  F (T2 ) , and so, x = w , Thus F (T2 )  F (T1 ) . Now we prove that F (Ti )

F (T1 ) = F (T2 )  Pcl ( X ) .

is closed.

The following corollary extends Theorem 4.1 of (Latif and Beg 1997) to cone metric spaces. Corollary 3.4 Let ( X , d ) be a Complete cone metric P is a non-normal cone. If space and

T : X  Pcl ( X ) is a multi-valued mapping such that u  T ( y) u  T (x) for each x, y  X , x and y such

such that

d ( xn , vn )   (d ( xn1 , x), d ( xn1 , xn ), d ( x, vn ))   (d ( xn1 , x), d ( xn1 , xn ), d ( x, vn )) . Taking limit n   , we get d ( x, limnvn )   (d ( x, x), d ( x, x), d ( x, limnvn )   (0,0, d ( x, limnvn ))  k (0) = 0. vn  x n  This implies that

vn  T j (x) x  T j (x)

as

that

d (u x , u y )  hd ( x, u x )  d ( y, u y ),

0  h < 1/2 . F (T )  Pcl ( X )

where

n  N and T j (x) is closed x  F (T j ) = F (Ti )

T : X  Pcl ( X )

is a multi-valued mapping such that

for each x, y  X , that

Corollary 3.2 Let ( X , d ) be a Complete cone metric

where

0  a < 1/2 . F (T )  Pcl ( X )

where

a, b, c  0

are

fixed

i  j , for u y  T j ( y)

constants

(7) Then

and

u y  T ( y)

F (T )  

such

and

.

be a multi-valued

x, y  X , and d (u x , u y )  a d ( x, y)  b d ( x, u x )  c d ( y, u y ),

each

u x  T (x)

d (u x , u y )  a d ( x, y),

mappings such that for i, j {1,2} with

u x  Ti (x)

and

Corollary 3.5 Let ( X , d ) be a Complete cone metric P is a non-normal cone. If space and

. Since

for each

T1 , T2 : X  Pcl ( X )

Then

(6)

F (T )  

.

. Hence, as required. Theorem 2 of (Abbas, Rhoades and Nazir 2009) on cone metric space become the corollary of Theorem 3.1 as follows.

space and

and

similarly,

{xn } be a Cauchy sequence in F (T j ) = F (Ti ) x  x as n   . Since xn  Ti ( xn1 ) , such that n vn  T j (x)

F (T1 ) = F (T2 )  

Then

Let

there exits

and

,

Corollary 3.6 Let ( X , d ) be a Complete cone metric P is a non-normal cone. If space and

(4)

is a multi-valued mapping such that

T : X  Pcl ( X )

for each x, y  X , that

with

47

u x  T (x)

and

u y  T ( y)

such

Pakistan Journal of Science (Vol. 66 No. 1 March, 2014)

d (u x , u y )  a d ( x, y)  b d ( x, u x )  cd ( y, u y ), a, b, c  0

where

are

fixed

constants

a  b  c < 1 . Then T has a fixed point.

sequence in ( X , d ) is a complete cone metric space,

(8)

there exist p  X s.t.

with

d (T1 ( p), p)  d ( p, xn )  d ( xn , T1 ( p))

 d ( p, xn )  H (T2 ( xn1 ), T1 ( p))  d ( p, xn )  H (T1 ( p), T2 ( xn1 ))  d ( p, xn )   (d ( p, xn1 ), d ( p, T1 ( p)), d ( xn1 , xn )).

Theorem 3.7 Let ( X , d ) be a complete cone metric

T1 , T2 : X  CB( X )

space and mappings the

following

conditions,

T1 ( x), T2 ( x)  CB( X ) ,

for

each

satisfying

x X

,

Taking

H (T1 ( x), T2 ( y))   (d ( x, y), d ( x, T1 x), d ( y, T2 y)) * where   A , then there exists p  X such that p  T1 ( x)T1 ( y)

 0   (0, d (T1 ( p)),0)  k (0) = 0.

x0  X

T (x ) Proof. Let , 1 0 is non-empty closed x  T1 ( x0 ) , for this x1 bounded subset of X . Choose 1 T (x ) by the same reason 2 1 is non-empty closed bounded

This implies that

Theorem 3.8

d ( x1 , x2 )  H (T1 ( x0 ), T2 ( x1 ))   (d ( x0 , x1 ), d ( x0 , T1 ( x0 )), d ( x1 ), T2 ( x1 ))   (d ( x0 , x1 ), d ( x0 , x1 ), d ( x1 , x2 ))  k d ( x0 , x1 ) x T (x ) where k  [0,1) . For this 2 , 1 0 is a non-empty

T2 ( x1 )

and

T1 ( x2 )

of X , there exist

and

, n=0,1,2,….

.

This implies that

H (Txn1 , Txn )  k d ( xn1 , xn )

and

.(9)

Similarly,

d ( xn1 , xn )  k d ( x0 , x1 ). n

d ( xn3 , xn 2 )  k d ( xn2 , xn1 )  k 2 d ( xn1 , xn ) for m > n , we get d ( xm , xn )  d ( xm , xm1 )  d ( xm1 , xm2 )  ...  d ( xn1 , xn )

0  c be given, choose a natural number N1 such k n d ( x0 , x1 )  c, n  N1 that . This implies that d ( xn , xn1 )  c. { xn } Let

Therefore,

xn1  Txn

d ( xn2 , xn1 )  H (Txn1 , Txn )  k d ( xn1 , xn )

On continuing this process, we get a sequence

xn1  T1 ( xn )

, consider

d ( xn2 , xn1 )  H (Txn1 , Txn )   (d ( xn1 , xn ), d ( xn1 , Txn1 ), d ( xn , Txn ))   (d ( xn1 , xn ), d ( xn1 , xn2 ), d ( xn , xn1 ))  k d ( xn1 , xn ) where k  [0,1) . But

such that

or

Let X be a cone complete metric space

Proof. Let Now, consider

 k d ( x1 , x2 )  k 2 d ( x0 , x1 ).

xn1  T2 ( xn )

p  T1 ( p)T2 ( p)

{xn } in X . Then T has a fixed point x* in X . Txn  Tx* . Moreover,

  (d ( x2 , x1 ), d ( x2 , T1 ( x2 )), d ( x1 , T2 ( x1 )))   (d ( x2 , x1 ), d ( x2 , x3 ), d ( x1 , x2 )) =  (d ( x1 , x2 ), d ( x2 , x3 ), d ( x1 , x2 ))

such that

p  T1 ( p) .

if there exists an asymptotically T -regular sequence

d ( x2 , x3 )  H (T2 ( x1 ), T1 ( x2 )) = H (T1 ( x2 ), T2 ( x1 ))

{ xn }

. Hence

or

and T : X  CB( X ) a mapping satisfying H (Tx, Ty)   (d ( x, y), d ( x, Tx), d ( y, Ty))

both are closed and bounded subset

x3  T1 ( x2 )

d (T1 ( p), p) = 0 ,

p  T ( p)

2 Similarly, Which is required.

subset of X .

x2  T2 ( x1 )

lim n , we get

d (T1 ( p), p)  d ( p, p)   (d ( p, p), d ( p, T1 ( p)), d ( p, p))

.

closed bounded subset of X . Since

xn  p.

 (k m  k m1  ...  k n1 )d ( x0 , x1 ).

is a Cauchy

48

Pakistan Journal of Science (Vol. 66 No. 1 March, 2014)

extended and improved severalresults of ( Abbas,Rhoads and Nazir 2009) for A*-mappings. Further, we generalised and improved fiew results of (Latif and Beg 1997) for A*-mappings in cone metric spaces setup.

{xn } be a Cauchy sequence in X . By {Txn } is a Cauchy sequence. Since equation(9), we get Hence

(CB( X ), H ) is a complete cone metric space. There * K *  CB( X ) , s.t. H (Txn , K )  0. Let exists a

REFERENCES

* x*  K * s.t. xn  x . Then d ( x* , Tx* )  H ( K * , Tx* ) = lim H (Txn , Tx* )

Aage, C. T. and J. N. Salunke. On common fixed points for contractive type mappings in cone metric spaces, Bull. Math. Anal. and Appl. Vol. 1(3): 10-15 (2009). Abbas, M. and B. E. Rhoades. Fixed and periodic point results in cone metric spaces. Appl. Math. Let., Vol. 22: 511-515 (2009). Abbas, M, B. E. Rhoades and T. Nazir. Common fixed points of generalized contractive multi-valued mappings in cone metric spaces. Math. Commun., Vol. 14(2): 365-378 (2009). Akram, M., A. A. Siddique and A. A. Zafar. Some fixed

n

 lim ( (d ( xn , x ), d ( xn , Txn ), d ( x* , Tx* ))) *

n

 lim ( (d ( xn , x* ), d ( xn , xn1 ), d ( x* , Tx* ))) n

  (d ( x* , x* ), d ( x* , x* ), d ( x* , Tx* ))  k (0) = 0. d ( x* , Tx* ) = 0 or x*  Tx* . Now This implies that

*

H ( K * , Tx* ) = lim H (Txn , Tx* ) = 0. n

This implies that

Tx* = K * = lim Txn . n

Corollary 3.9 Let X be complete metric space and

T : X  CB( X ) a mapping satisfying

H (Tx, Ty)  1 (d ( x, Tx))d ( x, Tx)   2 (d ( y, Ty)) d ( y, Ty)  : R  [0,1) , i = 1,2 . If for all x, y  X , where i there exists an asymptotically T -regular sequence

{ xn }

*

in X . Then T has fixed point x in X . Moreover,

Txn  Tx* . *

Conclusion: In this paper,we extend the idea of A -mappings from metric spaces to cone metric spaces. We,

49

point theorems for Multi-valued A - mappings. Korean J. Math. Science, Vol. 10:7-12 (2003). Beg, I. and A. Azam. Fixed point of asymptotically regular multivalued mappings. J. Austral. Math.Soc.(Series A) Vol. 53: 313-326 (1992). Huang, L. G. and X. Zhang. Cone metric spaces and fixed point theorems of contractive mappings. J.Math.Anal.Appl.,Vol. 332: 1467-1475 (2007). Ilic, D. and V. Rakocevic. Common fixed points for maps on cone metric space. J.Math.Anal.Appl.,Vol. 341: 876-882 (2008). Latif, A. and I. Beg. Geometric fixed points for single and multi-valued mappings. Demonstratio Math., Vol. 30: 791-800 (1997). Rezapour, Sh. and R. H. Haghi. Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings". J. Math. Anal.Appl., Vol. 345: 719-724 (2008).