Some fixed point theorems in Gb-cone metric space

Komal Goyal and Bhagwati Prasad

Citation: 1802, 020004 (2017); doi: 10.1063/1.4973254View online: http://dx.doi.org/10.1063/1.4973254View Table of Contents: http://aip.scitation.org/toc/apc/1802/1Published by the American Institute of Physics

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Some Fixed Point Theorems in bG -cone Metric Space

Komal Goyal1 and Bhagwati Prasad1,a)

1Jaypee Institute of Information Technology, Department of Mathematics, Noida, India

a) Corresponding author: b_prasad10@yahoo.com

Abstract.The intent of the paper is to introduce a b

G -cone metric space and study its properties. Some fixed point theorems for the maps satisfying a general contractive condition are established in this setting. Some of the well known existing results are obtained as special cases.

Keywords- Fixed point; G -metric space; bG -cone metric space; Cone.

INTRODUCTION

The celebrated Banach contraction theorem (1922) is a classical and most powerful tool of nonlinear analysis which provides a constructive approach for solving various functional equations arising out of a number of physical problems related to diverse discipline of science and engineering. It has been extensively studied, generalized, enriched and extended in the literature by a number of authors in various setting for a variety of single valued and multi valued maps (see for instance [1-17], [20-32], [34] and several references thereof). For an excellent comparison of the various contractive conditions, one may refer to Rhoades [33]. Gahlar [12] introduced 2-metric space as a generalization of the usual notion of a metric space. However, Ha et al [14] observed that a 2-metric need not be a continuous function and there is no relationship between them. Dhage [11] introduced a new concept of metric space called D-metric space and subsequently developed the topological structures in this space through a series of papers. Mustafa and Sims [25] (also see Mustafa [22] and Mustafa et al [23-24]) in their seminal papers established that the claims made by Dhage for D-metric spaces were not correct. To overcome this, they introduced the notion of G-metric space. On the other hand, another generalization of a metric space was introduced by Bakhtin [5] which was studied by many authors such as Czerwik [10], Pacurar [26], Prasad et al [28-29] and Singh and Prasad [34] for the existence and uniqueness of the fixed point of single valued and multi-valued maps. Aghajani [3] generalized the concept of G -metric space to

bG -metric space through b-metric. Mustafa et al. [24]

obtained coupled coincidence point results for nonlinear ( , ) weakly contractive mappings in the setting of partially ordered

bG -metric spaces. Beg et al [6-7] generalized the cone metric spaces in the form of G-cone metric

spaces and studied topological properties such as convergence and completeness of these spaces and obtained fixed point theorems for the maps satisfying some general conditions. Huang and Zhang [17] generalized the notion of metric space and established fixed point results for maps under various contraction conditions in an ordered Banach space. Using these concepts, Hussain and Shah [18] obtained some results in cone b-metric space and established some topological properties. Later on, Huang and Xu [16] obtained some interesting results for contractive maps without the assumption of normality in cone b-metric spaces. In the present paper our intention is to introduce

bG

-cone metric spaces and study some basic properties of them. Some fixed point results in b

G -cone metric spaces are also established. Our results extend and generalize some of the well known previous results in G -metric space.

Mathematical Sciences and its ApplicationsAIP Conf. Proc. 1802, 020004-1–020004-11; doi: 10.1063/1.4973254

Published by AIP Publishing. 978-0-7354-1470-9/$30.00

020004-1

PRELIMINARIES

The basic concepts and relevant results required in the sequel are given below. Definition 1 [18, 19, 21]. Let E be a real Banach space and P a subset of E. By we denote the zero elements of E and by int P, the interior of P. The subset P is called a cone if and only if: ( 1)C P is closed nonempty and { };P

( 2) , , , , ;

( 3) ( ) { }.

C a b R a b x y P ax by P

C P P

A partial ordering with respect to P is defined by x y iff y x P while will represent inty x P . Definition 2 [22]. Let X be a nonempty set and :G X X X R satisfies the following properties: ( 1) ( , , ) 0G G x y z iff ;x y z ( 2) 0 ( , , )G G x y z for all , ,x y z X with ;x y ( 3) ( , , ) ( , , )G G x x y G x y z for all , ,x y z X with ;y z

( 4) ( , , ) ( , , ) ( , , ) ...G G x y z G x z y G y z x (symmetry in all variables); ( 5) ( , , ) ( , , ) ( , , )G G x y z G x a a G a y z for all , , , .x y z a X

Then, G is a generalized or G-metric and the pair (X, G) is a generalized or a G-metric space. Definition 3 [5, 34]. Let X be a non empty set and 1s be a given real number. A function :d X X R is said

to be a b-metric iff for all , ,x y z X , the following conditions are satisfied:

( 1) ( , ) 0 iff ,

( 2) ( , ) ( , ),

( 3) ( , ) ( ( , ) ( , )).

B d x y x y

B d x y d y x

B d x z s d x y d y z

The pair ( , )X d is called a b-metric space.

Definition 4 [3]. Let X be a nonempty set and :G X X X R with the constant 1s satisfies: ( 1) ( , , ) 0bG G x y z iff ;x y z ( 2) 0 ( , , )bG G x y z for all , ,x y z X with ;x y ( 3) ( , , ) ( , , )bG G x x y G x y z for all , ,x y z X with ;y z

( 4) ( , , ) ( , , ) ( , , ) ...bG G x y z G x z y G y z x (symmetry in all variables); ( 5) ( , , ) ( ( , , ) ( , , ))bG G x y z s G x a a G a y z for all , , , .x y z a X

Then, G is called a generalized b-metric and the pair (X, G) is a generalized b-metric space or a b

G -metric space. We extend the concept of

bG -metric space to

bG -cone metric space in the following manner.

Definition 5. Let X be a nonempty set and E a real Banach space with cone P. A vector-valued function : G X X X E is said to be a

bG -cone metric on X if it satisfies:

( 1) ( , , ) 0bG C G x y z iff ;x y z ( 2) 0 ( , , )bG C G x y z for all , ,x y z X with ;x y ( 3) ( , , ) ( , , )bG C G x x y G x y z for all , ,x y z X with ;y z

020004-2

( 4) ( , , ) ( , , ) ( , , ) ...bG C G x y z G x z y G y z x (symmetry in all variables); ( 5) ( , , ) ( ( , , ) ( , , ))bG C G x y z s G x a a G a y z for all , , ,x y z a X and 1.s

Then, the pair (X, G) is a b

G -cone metric space. It is to be noticed that for 1,s the ordinary triangle inequality of

cone metric space holds whereas it is not true for 1.s Thus the class of b

G -cone metric spaces are effectively

larger than that of the ordinary cone metric spaces. It is remarked that every cone metric space is a b

G -cone metric space, but the converse may not be true (see example 1). Example 1 [3]. Let (X, G) be a G-metric space, and * , , , ,pG x y z G x y z , where 1p is a real number. Note

that *G is a b

G -cone metric with 12 ps .

If 1 ,p 1( ) 2p p p pa b a b Thus, for each , , , ,x y z a X we obtain,

*

1

1 * *

, , , ,

( , , ( , , ))

2 (( ( , , ) ( , , ) )

2 ( , , , ,

)

p

p

p p p

p

G x y z G x y z

G x a a G a y z

G x a a G a y z

G x a a G a y z Thus is a bG -cone metric with 12 ps .

Let X = R and 1

, , for all , ,3

G x y z x y y z x z x y z R .Then,

* 2 21, , , , ( )

9G x y z G x y z x y y z x z

is a bG -cone metric on R with 2s , but it is not a G-metric on R.

Some well known concepts of b

G -metric may be easily extended in the setting of b

G -cone metric space in the following manner. Definition 6. Let X be a

bG -cone metric space and { }nx a sequence in X. Then,

(i) The sequence { }nx is a b

G -Cauchy sequence if, for every with c E c , there is a natural number 0n such

that for all 0

, , , , , ,b n m l

n m l n G x x x c

(ii) The sequence { }nx is a b

G -convergent sequence if, for every with c E c , there is an x X and an

0n N , such that for all 0, , ,

b nn n G x x x c for some fixed point x in X. We can say,

, , as b n

G x x x n .

Here, x is called the limit of sequence { }nx and is denoted by lim nn

x x .

A b

G -cone metric space on X is said to be complete if every Cauchy sequence in X is convergent in X. Sequence

{ }nx in b

G -converges to x X if and only if , ,b n mG x x x as , .n m Definition 7. A bG -cone metric space is called symmetric if for all ,x y X ,

, , , , .b b

G x y y G y x x

020004-3

Proposition 1. Let (X, G) be a b

G -cone metric space, P a normal cone with normal constant K, x X and { }nx a sequence in X. Then, (i) Every sequence has a unique limit point. (ii) Every convergent sequence is Cauchy.

Proof: (i) Suppose that the limit point of any sequence { }nx is not unique. Therefore, we have , ,b nG x x x as

n and , ,b n

G x y y as .n Now, from triangle inequality,

b b n n b n G x, y, y s G x, x , x G x , y, y as n

or , ,b

G x y y as n or , which is a contradiction. Hence proved. (ii) Since{ }nx is a

bG -convergent sequence then for every with c E c , there is an x X and an

0n N ,

such that for all 0 , , ,b nn n G x x x c we have for all 0, ,n m l n and some fixed x X . From triangle

inequality, , , , , , ,

b n m m b n b m mG x x x s G x x x G x x x c

or , ,b n m mG x x x c Therefore, every convergent sequence is Cauchy. Proposition 2. Let X be a

bG -cone metric space. Then, the following are equivalent.

(i) The sequence { }nx is convergent to x X . (ii) , ,

b n nG x x x as n (iii) , ,

b nG x x x as n

Proof: From Definitions 6 and 7, we have for some fixed point x in X, , , as b nG x x x n and

, , , , , for all , .b n b n nG x x x G x x x x y X

This shows ( ) ( )i ii . The implications ( ) ( )ii iii and ( ) ( )iii i are obvious.

The result of Remark 2.6 in Hussain and Shah [18] is obviously true for b

G -cone metric space. It can be presented

for b

G -cone metric space as follows:

Lemma 1. Let (X, G) be a

bG -cone metric space over the ordered real Banach space E with a cone P. Then the

following properties are often used: (i) If a b and b c, then a c. (ii) If and , then . (iii) If θ u c for each intc P , then u . (iv) If intc P , θ and ,na then there exists 0n such that for all 0n n we have .

(v) If θ and ,n na a b b , then a b, for each cone P. (vi) If E is a real Banach space with cone P and if a λa where a P and 0 1, then a .

020004-4

MAIN RESULT

Theorem 1. Let (X, G) be a complete symmetric b

G -cone metric space with 1. Let :s G X X X E satisfy the following condition

, , ( , , )G Tx Ty Ty G x y y (1) for all , ,x y X where [0,1) is a constant. Then, T has a unique fixed point in X. Furthermore, { }nT x converges to the fixed point of T in X. Proof: Choose

0x X and construct the sequence { }nx such that

1

1 0 , 0n

n nx Tx T x n .

Then, we have,

1 1 1 1 1 1 0 0, , , , , , , ,n

n n n n n n n n nG x x x G Tx Tx Tx G x x x G x x x . For any 1, 1m p , it follows that,

1 1 1

2 2

1 1 1, 2 2 2

2

1 1 1,

, , [ , , , , ]

, , , ( , , )

, ,

m p m m m p m p m p m p m m

m p m p m p m p m p m p m p m m

m p m p m p m p

G x x x s G x x x G x x x

s G x x x s G x x x s G x x x

s G x x x s G x 3

2 2 2 3 3

1 1

2 1 1 1

, , , ..

, , , ,

m p m p m p m p m p

p p

m m m m m m

x x s G x x x

s G x x x s G x x x

1 2 2 3 3

1 0 0 1 0 0 1 0 0

1 1 1

1 0 0 1 0 0

1 2 2 3 3

, , , , , , ..

+ , , , ,

[

m p m p m p

p m p m

m p m p m p

s G x x x s G x x x s G x x x

s G x x x s G x x x

s s s 1 1 1

1 0 0 1 0 0

1 2 2 3 2 1 1

1 0 0 1 0 0

.. ] , , , ,

.. , , , ,

p m p m

m p p p p m

s G x x x s G x x x

s s s s G x x x s G x x x11 1

1

1 0 0 1 0 01

11

1

1 0 0 1 0 0

11

1 0 0 1 0 0

{ 1} [ ] , , , ,

1

{ 1} [ ] , , , ,

, , , , .

p

m p p m

p

m p p m

p mp m

ss G x x x s G x x x

s

ss G x x x s G x x x

s

sG x x x s G x x x

s

Let c be given. Notice that 1

1

1 0 0

1 0 0

( , , ) as ( ) ( , , )

p mp ms

s G x x x ms G x x x for any k.

From Lemma 1 (iv), we find 0 m N for each 0m m such that 1

1

1 0 0 1 0 0

, , , ,

p mp ms

G x x x s G x x x cs

Then, for all 0m m and any p, we have 1

1

( ) 1 0 0

1 0 0

( , , ) ( , , )( ) ( , , )

p mp m

m p m m

sG x x x s G x x x c

s G x x x.

So, by definition 6 (i), { }nx is a Cauchy sequence in (X, G). Since (X, G) is a complete symmetric bG -cone metric

space, there exists *x X such that *

nx x . Take 0 n N such that for each 0n n , we have

020004-5

* * * * * *

* * *

( , , ) [ , , , , ]

[ { , , , , }]

n n n

n n n

G Tx x x s G Tx Tx Tx G Tx x x

s G x x x G x x x c

Then, by Lemma 1 (iii), we obtain * * *, ,G Tx x x , that is, * *Tx x .

For uniqueness, consider * y to be the other fixed point. Then, * * * * * * * * *, , ( , , ) , , ,G x y y G Tx Ty Ty G x y y

by Lemma 1 (vi) we have, * *x y . This completes the proof. Example 2. Let E=R, { , 0}P x R x be a cone and [0,1)X . Let :G X X X E be such that

, , , , ( , ),G x y z d x y d y z d z x

where ,d x y x y . Let :T X X be defined by, for all 4

xTx x X . Then,

, , , , ,

=

4 4 4 4 4 4

1 [ ]

4

, , where [0,1

G Tx Ty Tz d Tx Ty d Ty Tz d Tz Tx

Tx Ty Ty Tz Tz Tx

x y y z z x

x y y z z x

G x y z )

Hence conditions of Theorem 1 are satisfied and the point 0x is the unique fixed point of the map T. When 1s in above, we obtain following result of Mustafa [5] in G- metric space. Corollary 1 [22]. Let (X, G) be a complete metricG space and :T X X be a mapping satisfying the following condition for all , ;x y X

( , , ) ( , , )G Tx Ty Ty k G x y y where [0,1)k . Then, T has a unique fixed point in X. Now we extend Theorem 3.1 to a more general condition. Theorem 2. Let be a complete symmetric

bG -cone metric space with 1s and :T X X satisfies the

following condition for all ,x y X :

1 2 3 4 5 , , , , , , , , ( , , ) ( , , )G Tx Ty Ty G x Tx Tx G y Ty Ty G x Ty Ty G y Tx Tx G x y y (2)

where the constant 3 51 2 4[0,1) and for 1,2,3, 4,5, 1i i s .Then has a unique fixed point in

X. Moreover, the iterative sequence { }nT x converges to the fixed point of T.

Proof: Fix 0x X and set 1

1 0 for 0,1, 2...n

n nx Tx T x n Firstly, we see

020004-6

1 1 1

1 2 1 1 1 3 1 1 4 1

5 1 1

1 1 1

, , ( , , )

, , , , , , , ,

( , , )

, ,

n n n n n n

n n n n n n n n n n n n

n n n

n n n

G x x x G Tx Tx Tx

G x Tx Tx G x Tx Tx G x Tx Tx G x Tx Tx

G x x x

G x x x 2 1 3 4 1 1 1 5 1 1

1 1 1 2 1 4 1 4 1 1

1 4 1 1

5

2

, , , , , , ( , , )

, , ( ) , , [ , , , , ]

( ) , , (

n n n n n n n n n n n n

n n n n n n n n n n n n

n n n

G x x x G x x x G x x x G x x x

G x x x G x x x s G x x x G x x x

s G x x x 15 4 ) , ,n n ns G x x x

1 4 1 1 2 5 4 1 (1 ) , , ( ) , ,n n n n n ns G x x x s G x x x (3) Secondly,

1 1 1

1 1 1 1 2 3 1 4 1 1

5 1

1 1

, , ( , , )

, , , , , , , ,

( , , )

, ,

n n n n n n

n n n n n n n n n n n n

n n n

n n n

G x x x G Tx Tx Tx

G x Tx Tx G x Tx Tx G x Tx Tx G x Tx Tx

G x x x

G x x x 2 1 1 3 1 1 1 4 5 1

1 1 2 1 1 3 1 3 1 1

2 3 1 1 1

5

, , , , , , ( , , )

( , , , , [ , , , , ]

( ) , (

)

,

n n n n n n n n n n n n

n n n n n n n n n n n n

n n n

G x x x G x x x G x x x G x x x

G x x x G x x x s G x x x G x x x

s G x x x 3 15 ) , ,n n ns G x x x

2 3 1 1 1 5 3 1 (1 ) , , ( ) , ,n n n n n ns G x x x s G x x x (4)

On adding (3) and (4), we get, 1 2 3 41 1 1

1 2 3 4

5 ( ), , , ,

2

2

( )n n n n n n

sG x x x G x x x

s

Put 1 2 5 3 4

1 2 3 4

2 ( )

2 ( )

s

s, it is easy to see that 0 1.Thus,

1 1 1 1 0 0, , , , , ,n

n n n n n nG x x x G x x x G x x x .

Following similar argument as given in Theorem 1, there exists * x X such that *

nx x .

Let c be arbitrary. Since *

nx x , there exists N such that 2

* * 1 2 3 4

2

2 ( ), , for all .

2 2n

s s sG x x x c n N

s s

Next, we claim that *x is a fixed point of T. To prove * *Tx x . Then, * * * * * *

* * *

1

* * * * * * *

1 2 3 4 5

( , , ) [ , , , , ]

= , , , ,

[ , , , , , , , , ( , , )]

n n n

n n n

n n n n n n n n

G Tx x x s G Tx Tx Tx G Tx x x

sG Tx Tx Tx sG x x x

s G x Tx Tx G x Tx Tx G x Tx Tx G x Tx Tx G x x x* *

1

* * * * * * *

1 2 1 1 3 1 1 4 5

* *

1

* * * * *

1 2

, ,

[ , , , , , , , , ( , , )]

, ,

[ , , , ,{

n

n n n n n n n n

n

n

sG x x x

s G x Tx Tx G x x x G x x x G x Tx Tx G x x x

sG x x x

s G x Tx Tx s G x x x * * * *

1 1 3 1 1 4

* * * * * *

5 1

, , } , , , ,

{

, , } ( , , )] , ,

n n n n n

n n n

G x x x G x x x s G x x x

G x Tx Tx G x x x sG x x x

020004-7

* * * 2 * * 2 * *

1 2 2 1 1 3 1 1

2 * * 2 * * * * * *

4 4 5 1

2 *

1 4

)

, , , , ( , , ) ( , ,

, , , , ( , , ) , ,

,

n n n n n

n n n n

s G x Tx Tx s G x x x s G x x x s G x x x

s G x x x s G x Tx Tx s G x x x sG x x x

s s G x T * * 2 2 * * 2 *

2 4 5 2 3 1 1, , , ( ) , ,n n nx Tx s s s G x x x s s s G x x xwhich implies that

2 * * * 2 2 * * 2 *

1 4 2 4 2 3 1 15 (1 ) ( , , ) ( ) ( , , ) ( ) ( , , ).n n nss s G x Tx Tx s s G x x x s s s G x x x (5) On the other hand,

* * * * * *

* * *

* * * * * *

1 2 3

( , , ) [ , , , , ]

( , , ) ( , , )

, , [ , , , , , ,

n n n

n n n

n n n n n n

G x Tx Tx s G x Tx Tx G Tx Tx Tx

sG x Tx Tx sG Tx Tx Tx

sG x Tx Tx s G x Tx Tx G x Tx Tx G x Tx Tx* * *

4 5

* * * * * *

1 1 1 1 1 2 3

* * *

4 1 1 5

, , ( , , )]

, , [ , , , , , ,

, , ( , , )]

n n n

n n n n n n

n n n

G x Tx Tx G x x x

sG x x x s G x x x G x Tx Tx G x Tx Tx

G x x x G x x x* * * * * * *

1 1 1 1 1 2

* * * * * * * *

3 4 1 1 5

*

1 1

, , [ , , , , } , ,

, , , , } , , ( , , )]

, ,

{

{

n n n n n

n n n n

n n

sG x x x s s G x x x G x x x G x Tx Tx

s G x x x G x Tx Tx G x x x G x x x

sG x x x 2 * * 2 * * * *

1 1 1 1 2

2 * * 2 * * * * * *

3 3 4 1 1 5

2 * 2 * * *

1 4 1 1 2 3

, , , , , ,

, , , , , , ( , , )

, , ( ) , , (

n n n

n n n n

n n

s G x x x s G x x x s G x Tx Tx

s G x x x s G x Tx Tx s G x x x s G x x x

s s s G x x x s s G x Tx Tx s 5

2 2 * *

1 3 ) , ,nG x x xsswhich implies that

2 * * * 2 * 2 2 * *

2 3 1 4 1 1 51 3 (1 ) ( , , ) ( ) ( , , ) ( ) ( , , ).n n ns s G x Tx Tx s s s G x x x s s G x x xs (6) On adding (5) and (6),

2 2 * * * 2 2 *

1 2 3 4 1 2 3 4 1 1

2 2 2 2 * *

1 52 3 4

(2 ) , , 2 , ,

( ) , ,

2

n n

n

s s s s G x Tx Tx s s s s s G x x x

s s s s G x x xs2 * * 2 *

1 1 ( , , ( 2 ) , , .2 ) n n ns G x x x s s G x x xs

Simple calculation ensure that

2 * * 2 *

1 1* * *

2 2

1 2 3 4

2( , , ( 2 ) , ,, , .

(2 )

) n n ns G x x x s s G x x xG x Tx Tx c

s s s s

s

It is easy to see from Lemma 1 (iii), that * * *, , .G x Tx Tx Hence *x is a fixed point of T.

Finally, we show the uniqueness of fixed point. Indeed, if there is another fixed point *y , then * * * * * *

* * * * * * * * * * * * * * *

1 2 3 4 5

* * * * * * * * * * * * * * *

3 4 5

, , , ,

, , , , , , , , ( , , )

[ , , , , ] [ , , , , ] ( , , )

G x y y G Tx Ty Ty

G x Tx Tx G y Ty Ty G x Ty Ty G y Tx Tx G x y y

s G x y y G y Ty Ty s G y x x G x Tx Tx G x y y* * *

3 4 5 ( ) , , .s s G x y y

020004-8

Owing to 3 540 ( ) 1s s , we deduce from Lemma 1 (vi) that * *x y . This completes the proof.

Example 3. Let E=R, { , 0}P x R x be a cone. Let [0,1)X and :G X X X E be such that

, , max{ , , , , , } where ,G x y z d x y d y z d z x d x y x y . Let :T X X be defined

for all 9

xTx x X .

, , max , , , , , ,

1, , max , , , , , ,

98

, , max , , , , , ,9 9

8, , max , , , , , ,

9

9

, , max

G x y y d x y d y y d y x d x y x y

G Tx Ty Ty d Tx Ty d Ty Ty d Ty Tx d Tx Ty Tx Ty x y

xG x Tx Tx d x Tx d Tx Tx d Tx x d x Tx x Tx x x

yG y Ty Ty d y Ty d Ty Ty d Ty y d y Ty y Ty y y

G x Ty Ty d x, , , , , ,9

, , max , , , , , ,9

yTy d Ty Ty d Ty x d x Ty x Ty x

xG y Tx Tx d y Tx d Tx Tx d Tx y d y Tx y Tx y

So, we get, 51 2 3 4, , , , , , , , ( , , ) , , ,G Tx Ty Ty G x Tx Tx G y Ty Ty G x Ty Ty G y Tx Tx G x y y

for ,x y X , where the constant 1 52 3 4[0,1) and , 1,2,3,4,5, 1.i i s Hence Theorem 2 is verified and the unique fixed point of T is ‘0’. On putting 1s in Theorem 2, we get the following result [13]. Corollary 2 [13]. Let X be a complete symmetric G-cone metric space and :T X X satisfies the following conditions:

1 2 3 4 5

1 2 3 4 5

( ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , )

( ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , )

i G Tx Ty Ty a G x y y a G x Tx Tx a G y Tx Tx a G x Ty Ty a G y Ty Ty

ii G Ty Tx Tx a G y x x a G y Ty Ty a G y Tx Tx a G x Ty Ty a G x Tx Tx for all 1 2 3 4 5, and 1x y X a a a a a . Then T has unique fixed point. Further, when we put 51 2 3 40 nd a, 1 sa in Theorem 2, we get following result of Mustafa et al. [23]. Corollary 3 [23]. Let (X, G) be a complete G-metric space, and :T X X ,

, , , , , ,

or , , , , , ,

G Tx Ty Ty a G x Ty Ty G y Tx Tx

G Tx Ty Ty a G x x Ty G y y Tx

for all , x y X , with 1

0,2

a . Then T has a unique fixed point.

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