A Unique Common Fixed Point Theorems in generalized D*- Metric Spaces

(1)

**A Unique Common Fixed Point Theorems in generalized D*-**

Metric Spaces

Salih Yousuf Mohammed Salih

Mathematics Department, University of Bakht Er-Ruda, Sudan. Nuraddeen Gafai Sayyadi

Mathematics Department, Umaru Musa Yar’adua University, Nigeria. Abstract

In this paper we establish some common fixed point theorem for contractive type mapping in cone metric spaces and prove some generalized complete D*- metric spaces.

Keywords: D*- metric space, common fixed points, normal cones. 1 Introduction and Preliminaries

Fixed point theorems play a major role in mathematics such as optimization, mathematical models, economy, military and medicine. So, the metric fixed point theory has been investigated extensively in the past two decades by numerous mathematicians. Some generalizations of a metric space concept have been studied by several authors. These different generalizations have been improved by Gahler [7, 10], by introducing 2-metric spaces, and Dhage [1] by studying the theory of D- metric spaces.

In 2005, Mustafa and Sims [11] introduced a new structure of generalized metric spaces which are called G-metric spaces as a generalization of metric spaces. Later, Mustafa et al. [12–14] obtained several fixed point theorems for mappings satisfying different contractive conditions in G-metric spaces. Later in 2007 Shaban Sedghi et.al [8] modified the D-metric space and defined D*-metric spaces and then C.T.Aage and J.N.Salunke [3] generalized the

D*-metric spaces by replacing the real numbers by an ordered Banach space and defined D*-cone metric spaces and prove the topological properties.

Further, Huang and Zhang [6] generalized the notion of metric spaces by replacing the real numbers by ordered Banach space and defined the cone metric spaces. They have investigated the convergence in cone metric spaces, introduced the completeness of cone metric spaces and have proved Banach contraction mapping theorem, some other fixed point theorems of contractive type mappings in cone metric spaces using the normality condition. Afterwards, Rezapour and Hamlbarani [9], Ilic and Rakocevic [5], contributed some fixed point theorems for contractive type mappings in cone metric spaces.

In this paper, we obtain a unique common fixed point theorem for generalized D*-metric spaces.

First, we present some known definitions and propositions in D*-metric spaces. Let E be a real Banach space and P a subset ofE. P is called a cone if and only if:

 

i P is closed, non-empty and P

 

0 ,

 

ii ax by P for all x y, P and non-negative real numbers a b, ,

 

iii P 

   

P  0 .

Given a conePE, we define a partial ordering  on E with respect to P by x y if and only if y x P . We shall write x y if x y and x y; we shall write

if int ,

x y y x P where int P denotes the interior ofP. The cone P is called normal if there is a number K0 such that for all x y, E

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The least positive number K satisfying the above is called the normal constant of P [12]. The cone P is called regular if every increasing sequence which is bounded from above is convergent. That is, if

 

1 n n

x _ is a sequence such that x₁x₂   y for someyE, then there is xE such that lim_n_ x_n x 0.

The cone P is regular if and only if every decreasing sequence which is bounded from below is convergent. Rezapour and Hamlbarani [9] proved every regular cone is normal and there are normal cone with normal constant M 1.

Definition (1.1) [4]. Let X be a non empty set. A generalized D*-metric on X is a function,

3

*:

D X E that satisfies the following conditions for allx y z a, , , X : (1) D*



x y z, ,



0,

(2) D*



x y z, ,



0 if and only if x y z,

(3) D*



x y z, ,



D*



p x y z



, ,





, (Symmetry) where p is a permutation function, (4) D*



x y z, ,



D*



x y a, ,



D*



a z z, ,



,

Then the function D* is called a generalized D*-metric and the pair



X D, *



is called a generalized D*-metric space

Example (1.2) [4]. Let ER P2, 



 

x y, E x y: , 0 ,



X R and D*:X  X X E

defined by









* , , , ,

D x y z  x    y y z x z  x    y y z x z where  0 is a constant. Then



X D, *



is a generalized D*- metric space.

Proposition (1.3) [4]. If



X D, *



be generalized D*- metric space, then for allx y z, , X, we have D*



x x y, ,



D*



x y y, ,



.

Proof. Let D*



x x y, ,



D*



x x x, ,



D*



x y y, ,



D*



x y y, ,



and similarly

















* , , * , , * , , * , , .

D y y x D y y y D y x x D y x x Hence we have

D*



x x y, ,



D*



x y y, ,



.

Definition (1.4) [4]. Let



X D, *



be a generalized D*- metric space. Let

 

x_n be a sequence in X and xX . If for every cE with 0 c there is N such that for all





, , * _m, _n, ,

m nN D x x x c then

 

x_n is said to be convergent and

 

x_n converges to x , and x is the limit of

 

x_n . We denote this byx_n x n



 



.

Lemma (1.5) [4]. Let



X D, *



be a generalized D*-metric space, P be a normal cone with normal constant K. Let

 

x_n be a sequence in X . Then

 

x_n converges to x if and only if







* _m, _n, , .

D x x x m n 

Proof. Let

 

x_n be a sequence in generalized D*-metric space X converge to xX and 0

  be any number. Then for any cE, with 0 c there is a positive integer N such that

,

m nN implies

D*



x_m,x x_n,



c D*



x_m,x x_n,



K c ,

since 0D*



x_m,x x_n,



c and K is normal constant. Choose c such that K c .Then





* _m, _n,

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Conversely let D*



x_m,x x_n,



0 as ,m n (in PE). So for any cE with 0 c

(i.e.cintP), we have c 0, let rdist c P



,



inf



c t :tP



,where P denotes the boundary of P , for this given , 0r  r c , there exist a positive integer N such that

,

m nN implies that



, ,



2

m

r

G x x x  c and for any tP,



*



, ,



*



, ,



2 2

m n m n

r r

cD x x x    t c t D x x x   r

which proves that c D *



x x x_m, _n,



int i.e. P G x x x



_m, _n,



c.

Remark. If

 

u_n is a sequence in PE and u_n u in E the uP as P is a closed subset of E. From this u_n   0 u 0. Thus if u_n v_n in P then limu_n limv_n, provided limit exist.

Lemma (1.6) [4]. Let



X D, *



be a generalized D*-metric space then the following are equivalent.

   

i x_n is *D convergent to .x

 

ii D *



x x x_n, _n,



0 as n 

 

iii D *



x x x_n, ,



0 as n . Proof.

   

i  ii by Lemma (1.5).

   

ii  i . Assume

 

ii , i.e. D*



x x x_n, ,



0 as n  i.e. for every cE with 0 c

there is N such that for all nN D, *



x x x_n, _n,



c/ 2,

D*



x x x_m, _n,



D*



x x x_m, ,



D*



x x x, _n, _n



D*



x x, _m,x_m



D*



x x x, _n, _n



c for all ,m nN.

Hence

 

x_n is D*-convergent to x .

   

ii  iii . Assume

 

ii , i.e. D*



x x x_n, _n,



0 as n  i.e. for every cE with 0 c

there is N such that for all nN, D*



x x x_n, _n,



c ,

D*



x x x_n, ,



D*



x x x, _n, _n



D*



x x x_n, _n,



c. Hence D*



x x x_n, ,



0 as n , since c is arbitrary.

   

iii  ii . Assume that

 

iii i.e. D*



x x x_n, ,



0 as n . Then for any cE with 0 c , there is an N such that nN implies *



, ,



.

2 n c D x x x Hence m n, N gives













* _m, _n, * _m, , * , _n, _n D x x x D x x x D x x x c. Thus

 

x_n is D*- convergent to x .

Lemma (1.7) [4]. Let



X D, *



be a generalized D*- metric space, P be a normal cone with normal constant K . Let

 

x_n be a sequence in X . If

 

x_n converges to x and

 

x_n

converges to y, then x y. That is the limit of

 

x_n , if exists, is unique.

Proof. For any cE with 0 c, there is N such that for all m n, N, D*



x_m,x x_n,



c . We have

(4)

0D*



x x y, ,



D*



x x x, , _n



D*



x y y_n, ,



D*



x x x_n, _n,



D*



x x y_n, _n,



c. for all nN.

Hence D*



x x y, ,



2K c .Since c is arbitrary, D*



x x y, ,



0, therefore x y.



X D, *



be a generalized D*- metric space,

 

x_n be a sequence in X . If for any cE with 0 c, there is N such that for all m n l, , N, D*



x_m,x x_n, _l



c , then

 

x_n is called a Cauchy sequence in X .



X D, *



be a generalized D*-metric space. If every Cauchy sequence in X is convergent in X , then X is called a complete generalized D*-metric space.

Lemma (1.10) [4]. Let



X D, *



be a generalized D*-metric space,

 

x_n be a sequence in X . If

 

x_n converges to x , then

 

x_n is a Cauchy sequence.

Proof. For any cE with 0 c , there is N such that for all m n l, , N ,





* _m, _n, / 2

D x x x c and D*



x x x_l, ,_l



c/ 2. Hence

D*



x x x_m, _n, _l



D*



x x x_m, _n,



D*



x x x, ,_l _l



c. Therefore

 

Lemma (1.11) [4]. Let



X D, *



 

x_n be a sequence in X . Then

 

x_n is a Cauchy sequence if and only if D*



x_m,x x_n, _l



0



m n l, ,  



.

Proof. Let

 

x_n be a Cauchy sequence in generalized D*-metric space



X D, *



and  0 be any real number. Then for any cE with 0 c, there exist a positive integer N such that m n l, , N implies D*



x_m,x x_n, _l



c 0 D*



x_m,x x_n, _l



c i.e.





* _m, _n, _l ,

D x x x K c where K is a normal constant of P in E . Choose c such that .

K c  Then D*



x_m,x x_n, _l



 for all m n l, , N , showing that





* _m, _n, _l 0 as , , .

D x x x  m n l 

Conversely let D*



x_m,x x_n, _l



0 as , ,m n l . For any cE with 0 c we have





0 0, as 0 and 1 .

K c  c  c  c IntP K  For given K c there is a positive integer N

such that m n l, , N D*



xm,x xn, l



K c . This proves that D*



xm,x xn, l



c for all

, ,

m n l N and hence

 



X D, *



,



X D', '*



be generalized D*-metric spaces, then a function f X: X' is said to be D*-continuous at a point xX if and only if it is

*

D -sequentially continuous at x , that is, whenever

 

x_n is D*-convergent to x we have

 

fxn is D*-convergent to fx.

Lemma (1.13) [4]. Let



X D, *



 

x_n ,

 

y_n , and

 

z_n be three sequences in X and x_n x y, _n y,

n

(5)

Proof. For every  0, choose cE with 0 c and . 6 3 c K    From xn x y, n y

and z_n z , there is N such that for all nN D, *



x x x_n, _n,



c D, *



y y y_n, _n,



c and





* _n, _n, . D z z z c We have D*



x y z_n, _n, _n



D*



x y z_n, _n,



D*



z z z, _n, _n



D*



z x y, _n, _n



D*



z z z_n, _n,



D*



z x y, _n,



D*



y y y, _n, _n



D*



z z z_n, _n,



D*



y z x, , _n



D*



y y y_n, _n,



D*



z z z_n, _n,



D*



y z x, ,



D*



x x x, _n, _n



D*



y y y_n, _n,



D*



z z z_n, _n,



3cD*



x y z, ,



Similarly, we infer D*



x y z, ,



D*



x y z_n, _n, _n



3 .c Hence 0D*



x y z, ,



 3c D*



x y z_n, _n, _n



6c

and

D*



x y z_n, _n, _n



D*



x y z, ,



 D*



x y z, ,



3cD*



x y z_n, _n, _n



 3c 



6K3



c , for all nN.

Therefore D*



x y z_n, _n, _n



D*



x y z, ,



n 



.

Remark. If xn x in generalized D*-metric space X , then every subsequence of

 

xn

converges to x in X . Let

 

n

k

x be any subsequence of

 

x_n and xn x in X then





* _m, _n, 0 as ,

D x x x  m n 

and also *



, ,



0 as , since for all . m n

k k n

D x x x  m n  k n n

Definition (1.14) [4]. Let f and g be self maps of a set X . If w fxgx for some x in X , then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.

Proposition (1.15) [4]. Let f and g be weakly compatible self maps of a setX . If f and g have a unique point of coincidencew fxgx, then w is the unique common fixed point of

f andg.

2. Main Results The first main result is

Theorem (2.1). Let



X D, *



be complete generalized D*-metric spaces, P be a normal cone with normal constant K and let T X: X , be a mapping satisfies the following conditions

D*



Tx Ty Tz, ,



aD*



x y z, ,



bD*



x Tx Tx, ,



cD*



y Ty Ty, ,



dD*



z Tz Tz, ,



(2)

for allx y z, , X, where a b c d, , , 0,a b c   d 1. Then T have a unique fixed point inX .

Proof. Let x0Xbe arbitrary, there exist x1X such thatTx0 x1, in this way we have a

sequence

 

x_n withTxn1xn. Then from the above inequality we have

(6)

aD*



x_n_₁,x x_n, _n



bD*



x_n_₁,Tx_n_₁,Tx_n_₁



cD*



x Tx Tx_n, _n, _n



dD*



x Tx Tx_n, _n, _n



aD*



x_n_₁,x x_n, _n



bD*



x_n_₁,x x_n, _n



cD*



x x_n, _n_₁,x_n_₁



dD*



x x_n, _n_₁,x_n_₁







a b D

 

* x_n_₁,x x_n, _n

 

 c d D

 

* x x_n, _n_₁,x_n_₁



This implies



1 



c d



D*



x x_n, _n_₁,x_n_₁

 

 a b D

 

* x_n_₁,x x_n, _n















1 1 1 * , , * , , 1 n n n n n n a b D x x x D x x x c d        D*



x x_n, _n_₁,x_n_₁



qD*



x_n_₁,x x_n, _n



where





1 a b q c d  

  , then0 q 1. By repeated the application of the above inequality we have

D*



x x_n, _n_₁,x_n_₁



q Dn *



x x x₀, ,₁ ₁



(3) Then for all n m,  ,nm we have by repeated use the triangle inequality and equality (3) that D*



x xn, m,xm



D*



x x xn, n, n1



D*



xn1,xn1,xn2



D*



x_n_₂,x_n_₂,x_n_₃



D*



x_m_₁,x_m_₁,x_m



D*



x x_n, _n_₁,x_n_₁



D*



x_n_₁,x_n_₂,x_n_₂



D*



x_n_₂,x_n_₃,x_n_₃



D*



x_m_₁,x_m,x_m







qn qn1qm1



D*



x x x₀, ,₁ ₁



*



0, ,1 1



1 n q D x x x q   . From (1) we infer *



, ,



*



₀, ,₁ ₁



1 n n m m q D x x x K D x x x q  

which implies that D*



x x_n, _m,x_m



0 as n m,  , since *



₀, ,₁ ₁



1 n q K D x x x q    as , n m . For n m l, ,  , and D*



Tx x_n, _m,x_l



D*



x x_n, _m,x_m



D*



x_m, ,x x_l _l



, from (1) D*



Tx xn, m,xl



K D*



x xn, m,xm



 D*



xm, ,x xl l





taking limit asn m l, ,  , we get D*



x x_n, _m,x_l



0. So

 

x_n is D*-Cauchy sequence, since X is D*-complete, there exists uX such that

 

x_n u as n , there exist pX such that pu. If T X is complete, then there exist

 

uT X

 

such that x_n u,

(7)

as T X

 

X , we have uX . Then there exist pX such that pu. We claim that Tpu, D*



Tp u u, ,



D*



Tp Tp u, ,



D*



Tp Tp Tx, , _n



D*



Tx u u_n, ,



aD*



p p x, , _n



bD*



p Tp Tp, ,



cD*



p Tp Tp, ,



dD*



x Tx Tx_n, _n, _n



D*



x_n_₁, ,u u



aD*



u u x, , _n



bD*



u Tp Tp, ,



cD*



u Tp Tp, ,



dD*



x x_n, _n_₁,x_n_₁



D*



x_n_₁, ,u u



This implies that

















1 1





1





1 * , , * , , * , , * , , 1 n n n n n D Tp Tp u aD u u x dD x x x D x u u b c         from (1)





_

_









1 1





1





1 * , , * , , * , , * , , 1 n n n n n D Tp Tp u K a D u u x d D x x x D x u u b c        

as n , right hand side approaches to zero. Hence D*



Tp Tp u, ,



0 and Tpu. i.e. Tp p. Now we show Thas a unique fixed point. For this, assume that there exists a point q in X such that qTq. Now

D*



Tp Tp Tq, ,



aD*



p p q, ,



bD*



p Tp Tp, ,



cD*



p Tp Tp, ,



dD*



q Tq Tq, ,



aD*



Tp Tp Tq, ,



bD*



Tp Tp Tp, ,



cD*



Tp Tp Tp, ,



dD*



Tq Tq Tq, ,



aD*



Tp Tp Tq, ,



we have D*



Tp Tp Tq, ,



aD*



Tp Tp Tq, ,



, i.e.



a1

 

D* Tp Tp Tq, ,



P, but



a1

 

D* Tp Tp Tq, ,



P, since k 1 0. Thus pis a unique common fixed point of T. Corollary (2.2). Let



X D, *



be a complete generalized D*-metric space, P be a normal

cone with normal constant K and let T X: X be a mappings satisfy the condition D*



Tx Ty Tzq, ,



a D_ *



x Ty Ty, ,



D*



y Tx Tx, ,



_

b D_ *



y Tz Tz, ,



D*



z Ty Ty, ,



_ c D_ *



x Tz Tz, ,



D*



z Tx Tx, ,



_

(4) for all x y z, , X, where a b c, , 0, 2a2b2c1. Then T has a unique fixed point in X . Proof. Let x0X be arbitrary, there exist x1X such that Tx0 x1, in this way we have

sequence

 

Tx_n with Txn xn1. Then from inequality (4), we have

D*



x x_n, _n_₁,x_n_₁



D*



Tx_n_₁,Tx Tx_n, _n



a D_ *



x_n_₁,Tx Tx_n, _n



D*



x Tx_n, _n_₁,Tx_n_₁



_ b D_ *



x Tx Tx_n, _n, _n



D*



x Tx Tx_n, _n, _n



_ c D_ *



x_n_₁,Tx Tx_n, _n



D*



x Tx_n, _n_₁,Tx_n_₁



_

(8)

a D *



xn₁,xn₁,xn₁



D*



x x xn, n, n





b D_ *



x x_n, _n_₁,x_n_₁



D*



x x_n, _n_₁,x_n_₁



_ c D_ *



x_n_₁,x_n_₁,x_n_₁



D*



x x x_n, _n, _n



_ 



a c





D*



x_n_₁,x x_n, _n



D*



x x_n, _n_₁,x_n_₁



2b D*



x x_n, _n_₁,x_n_₁





. This implies that

D*



x x_n, _n_₁,x_n_₁



q Dn *



x_n_₁,x x_n, _n



where









, 1 2 a c q a b c  

   then 0 q 1 and by repeated application of above inequality,

we have,

D*



x x_n, _n_₁,x_n_₁



q Dn *



x x x₀, ,₁ ₁



(5) Then, for all n m,  ,mn, we have, by repeated use of the rectangle inequality,

D*



x_m,x x_n, _n



D*



x_m,x_m_₁,x_m_₁



D*



x_m_₁,x x_n, _n



D*



x_m,x_m_₁,x_m_₁



D*



x_m_₁,x_m_₂,x_m_₂



D*



x_m_₂,x x_n, _n



D*



x_m,x_m_₁,x_m_₁



D*



x_m_₁,x_m_₂,x_m_₂



D*



x_n_₁,x x_n, _n





1 1







0 1 1 * , , m m n q q  q  D x x x    *



₀, ,₁ ₁



. 1 m q D x x x q   from (1) *



, ,



*



₀, ,₁ ₁



0 as , . 1 m m n n q D x x x K D x x x m n q     

since 0 q 1. So

 

x_n is D*-Cauchy sequence. By the completeness of X , there exists

uX such that

 

x_n is D*-convergent to u . Then there is pX , such that pu. If

 

T X is complete, then there exist uT X

 

such that x_n u, as T X

 

X, we have

uX .Then there exist pX such that pu. We claim that Tpu, D*



Tp Tp u, ,



D*



Tp Tp Tx, , _n



D*



Tx u u_n, ,



a D_ *



p Tp Tp, ,



D*



p Tp Tp, ,



_ b D_ *



p Tx Tx, _n, _n



D*



x Tp Tp_n, ,



_ c D *



p Tx Tx, n, n



D*



x Tp Tpn, ,



D*



Tx u un, ,



a D_ *



u Tp Tp, ,



D*



u Tp Tu, ,



_ b D *



u x, n₁,xn₁



D*



Tp Tp u, ,



D*



u x x, n, n



 c D *



u x, n₁,xn₁



D*



Tp Tp u, ,



D*



u x x, n, n





(9)

D*



x u u_n, ,



This implies that







 







1 1







1 * , , * , , * , , 1 2 n n n n D Tp Tp u b c D u x x D u x x a b c     _  _    D*



x_n_₁, ,u u





from (1)







 







1 1







1 * , , * , , * , , 1 2 n n n n D Tp Tp u b c D u x x D u x x a b c       _  _     D*



x_n_₁, ,u u





the right hand side approaches to zero asn . Hence D*



Tp Tp u, ,



0 and Tpu. Hence Tp p. Now we show that T has a unique fixed point. For this, assume that there exists a point q in X such that qTq. Now

D*



Tp Tp Tq, ,



a D_ *



p Tp Tp, ,



D*



p Tp Tp, ,



_ b D_ *



p Tq Tq, ,



D*



q Tp Tp, ,



_ c D_ *



p Tq Tq, ,



D*



q Tp Tp, ,



_ a D_ *



Tp Tp Tp, ,



D*



Tp Tp Tp, ,



_ b D_ *



Tp Tq Tq, ,



D*



Tq Tp Tp, ,



_ c D_ *



Tp Tq Tq, ,



D*



Tq Tp Tp, ,



_ b D_ *



Tp Tp Tq, ,



D*



Tp Tp Tq, ,



_ c D_ *



Tp Tp Tq, ,



D*



Tp Tp Tq, ,



_ 



2b2c D

 

* Tp Tp Tq, ,



D*



Tp Tp Tq, ,

 

 2b2c D

 

* Tp Tp Tq, ,



.

This implies



2b2c



1



D*



Tp Tp Tq, ,



P and



2b2c



1



D*



Tp Tp Tq, ,



P, since



2b2c



 1 0 . As P  P

 

0 , we have



2b2c



1



D*



Tp Tp Tq, ,



0, i.e.





* , , 0

D Tp Tp Tq  . Hence TpTq. Also pq, since Tp p. Hence p is a unique fixed point of T in X .

Corollary (2.3). Let



X D, *



be a complete generalized D*-metric space, P be a normal cone with normal constant K and let T X: X be a mapping which satisfies the following condition,

D*



Tx Ty Ty, ,



a D_ *



y Ty Ty, ,



D*



x Ty Ty, ,



_

bD*



y Tx Tx, ,



(6) for all x y, X , where a b, 0,3a b 1, Then T has a unique fixed point in X .

Proof. Let x0X be arbitrary, there exist x1X such that Tx0 x1, in this way we have

sequence

 

Tx_n with Txn xn1. Then from inequality (6), we have

D*



x x_n, _n_₁,x_n_₁



D*



Tx_n_₁,Tx Tx_n, _n



(10)

bD*



x Tx_n, _n_₁,Tx_n_₁



a D *



x xn, n₁,xn₁



D*



xn₁,xn₁,xn₁





bD*



x x x_n, _n, _n



a D *



x xn, n₁,xn₁



D*



xn₁,x xn, n



bD*



x x x_n, _n, _n



 This implies that

D*



x x_n, _n_₁,x_n_₁



rD*



x_n_₁,x x_n, _n



(7) where , 1 2 a r a 

 then 0 r 1. Then repeating application of (7), we get

D*



x x_n, _n_₁,x_n_₁



r Dn *



x x x₀, ,₁ ₁



Then, for all n m,  ,nm we have, by repeated use of the rectangle inequality,

D*



x x x_m, _n, _n



D*



x x_m, _m_₁,x_m_₁



D*



x_m_₁,x_m_₂,x_m_₂



D*



x_n_₁,x x_n, _n





1 1







0 1 1 * , , m m n r r  r  D x x x    *



₀, ,₁ ₁



. 1 m r D x x x r   From (1) *



, ,



*



₀, ,₁ ₁



0 as , . 1 m m n n r D x x x K D x x x m n r     

since 0 r 1. So

 

x_n is D*-Cauchy sequence. By the completeness of X , there exists

uX such that

 

x_n is D*-convergent to u . Then there is pX , such that pu. If

 

T X is complete, then there exist uT X

 

such that x_n u, as T X

 

X , we have

uX .Then there exists pX such that pu. We claim that Tpu, D*



Tp Tu u, ,



D*



Tp Tp x, , _n_₁



D*



x_n_₁, ,u u



a D_ *



p Tp Tp, ,



D*



p Tp Tp, ,



_ bD*



p Tp Tp, ,



D*



x_n_₁, ,u u



a D_ *



u Tp Tp, ,



D*



u Tp Tp, ,



_ bD*



u Tp Tp, ,



D*



x_n_₁, ,u u



a D_ *



Tp Tp u, ,



D*



Tp Tp u, ,



_ bD*



Tp Tp u, ,



D*



x_n_₁, ,u u



This implies that











1



1 * , , * , , 1 2 n D Tp u u D x u u a b     from (1)











1



1 * , , * , , 1 2 n D Tp u u K D x u u a b    

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right hand side approaches to zero as n . Hence D*



Tp u u, ,



0 and Tpu and pTp. Now we show that T has a unique fixed point. For this, assume that there exists a point q in X such that qTq. Now

D*



Tp Tq Tq, ,



a D_ *



q Tq Tq, ,



D*



p Tq Tq, ,



_bD*



q Tp Tp, ,



a D_ *



Tq Tq Tq, ,



D*



Tp Tq Tq, ,



_bD*



Tq Tp Tp, ,



aD*



Tp Tq Tq, ,



bD*



Tp Tq Tq, ,

 

 a b D

 

* Tp Tq Tq, ,



. This implies



a b 



1



D*



Tp Tq Tq, ,



P and



a b 



1



D*



Tp Tq Tq, ,



P, since





* , ,

D Tp Tq Tq P and



a b  



1 0. As P  P

 

0 , we have







a b 1



D*



Tp Tq Tq, ,



0, i.e. D*



Tp Tq Tq, ,



0. Hence TpTq. Also pq, since pTp. Hence p is a unique fixed point of T in X .

Theorem (2.4). Let



X D, *



be a generalized D*-metric space, P be a normal cone with normal constant K and let T X: X, be a mappings which satisfies the following condition D*



Tx Ty Tz, ,



a D_ *



z Tx Tx, ,



D*



y Tx Tx, ,



_

b D_ *



y Tz Tz, ,



D*



x Tz Tz, ,



_

c D_ *



x Ty Ty, ,



D*



z Ty Ty, ,



_ (8) for all x y z, , X, where a b c, , 0,3a2b3c1.Then T has a unique common fixed point in X .

Proof. Setting z y in condition (8), reduces it to condition (6), and the proof follows from Corollary (2.3).

References

[1] B.C. Dhage “Generalized metric spaces and mappings with fixed point,” Bull. Calcutta

Math. Soc., 84(1992), 329-336.

[2] B.C. Dhage, ‘‘Generalized metric spaces and topological structure I,’’ An. Stiint. Univ. Al.

I. Cuza Iasi.Mat. (N.S.) 46(2000), 3-24.

[3] C.T. Aage, J.N. Salunke, ‘‘On common fixed points for contractive type mapping in cone metric spaces, ‘Bulletin of Mathematical Analysis and Applications, 3(2009), 10-15. [4] C. T. Aage, j. N. Salunke, ‘‘some fixed point theorems in generalized D*-metric

spaces,’’ J. of Applied Sciences Vol. 12(2010), 1-13.

[5] D. Ilic, V. Rakocevic, ‘‘Common fixed points for maps on cone metric space,’’ J. Math.

Anal.App. XI (2008), 664-669.

[6] H. Long-Guang, Z. Xian, ‘‘Cone metric spaces and fixed point theorem of contractive mappings, ’’J. Math. Anal. Appl., 332(2007), 1468-1476.

[7] S. Gahler, ‘‘Zur geometric 2-metriche raume,’’ Revue Roumaine de Math. Pures et pl. XI (1966), 664-669.

[8] S. Shaban , Nabi S, Haiyun Z., ‘‘A common Fixed Point Theorem in D*- Metric Spaces.” Hindawi Publishing Corporation Fixed Point Theory and Applications. vol. 2007 (2007).

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[9] Sh. Rezapour, R. Hamlbarani, ‘‘Some notes on the paper’ Cone metric spaces and fixed point theorems of contractive mappings,’’ J. Math. Anal. Appl. 345 (2008), 719-724. [10] S. Nadler, ‘‘Multivalued contraction mappings,’’ Pac. J. of Math., vol. 20, No. 27(1)

(1968), 192-194.

[11] Z. Mustafa, ‘‘A New Structure for Generalized Metric Spaces with Applications to Fixed Point Theorem,’’ PhD Thesis, The University of Newcastle, Australia, 2005. [12] Z. Mustafa, B. Sims, ‘‘some remarks concerning D-metric spaces,’’ Proceedings of the

Int. Conference on Fixed Point Theory and Applications, Valencia, Spain, July,

(2003), 189-198.

[13] Z. Mustafa, B. Sims, ’’A new approach to generalized metric spaces,’’ J. Non-linear and

Convex Anal. 7 (2) (2006) 289–297.

[14] Z. Mustafa, O. Hamed, F. Awawdeh, ‘‘some fixed point theorem for mappings on complete G -metric spaces,’’ Fixed Point Theory and Applications, Volume 2008,

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