Some common Fixed Point Theorems for compatible  - contractions in G-metric Spaces

B. Ramu Naidu et al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 3, ( Part -3) March 2017, pp.21-32
www.ijera.com DOI: 10.9790/9622- 0703032132 21 | P a g e
Some common Fixed Point Theorems for compatible -
contractions in G-metric Spaces
1
B. Ramu Naidu, 2.
K.P.R. Sastry , 3
G. Appala Naidu , 4.
Ch. Srinivasa Rao,
1,2,3,4
Faculty in Mathematics, AU campus, Vizianagaram -535003 INDIA.
ABSTRACT
We prove some common fixed point theorems for compatible self mappings satisfying some kind of contractive
type conditions on complete G-metric spaces and obtain results of Kumara Swamy and Phaneendra[6] and
Sushanta Kumar Mohanta[16] as corollaries.
2010 Mathematics Subject Classification.47H10, 54H25
Keywords:- G-metric space, Compatible self-maps, G-Cauchy sequence, Common fixed point,
-contractions.
I. INTRODUCTION
The study of metric fixed point theory
plays an important role because the study finds
applications in many important areas as diverse as
differential equations, operation research,
mathematical economics and the like. Different
generalizations of the usual notion of a metric
space were proposed by several mathematicians
such as Gahler [3,4] (called 2-metric spaces) and
Dhage [1,2] (called D-metric spaces). K.S.Ha et al.
[5] have pointed out that the results cited by Gahler
are independent, rather than generalizations, of the
corresponding results in metric spaces. Moreover, it
was shown that Dhage’s notion of D-metric space
is flawed by errors and most of the results
established by him and others are invalid. These
facts determined Mustafa and Sims [11] to
introduce a new concept in the area, called G-
metric space. Recently, Mustafa et al. studied many
fixed point theorems for mappings satisfying
various contractive conditions on complete G-
metric spaces; see [8-13]. Subsequently, some
authors like Renu Chugh et al.[14], W.Shatanawi
[15] have generalized some results of Mustafa et al.
[7-8] and studied some fixed point results for self-
mappings in a complete G-metric space under some
contractive conditions related to a non-decreasing
       : 0, 0, w ith lim 0 for all 0, .
n
n
t t 
 
        
Kumara Swamy and Phaneendra[6] and Sushanta Kumar Mohanta[16] proved some fixed point theorems for
self-mappings on complete G-metric spaces.
In this paper we introduce  -contractions in G-metric spaces, prove fixed point results for such maps and
obtain results of Kumara Swamy and Phaneendra [6].
II. PRELIMINARIES
We begin by briefly recalling some basic definitions and results for G-metric spaces that will be needed in the
sequel.
Definition 2.1 :-(Mustafa and Sims [7]) Let X be a non–empty set, and let :G X X X R

   be a
function satisfying the following axioms:
   
   
     
        
       
1 , , 0 if ,
2 0 < , , , fo r all , , w ith ,
3 , , , , , fo r all , , , w ith ,
4 , , , , w h ere is a p erm u tatio n in , , ,
5 , , , , , , , fo r all , , , , r
G G x y z x y z
G G x x y x y X x y
G G x x y G x y z x y z X z y
G G x y z G x z y x y z
G G x y z G x a a G a y z x y z a X
 
  
 
  

    ectan g le in eq u ality .
RESEARCH ARTICLE OPEN ACCESS

Then the function G is called a generalized metric, or, more specifically a G- metric on X , and the pair
 ,X G is called a G-metric space.
Example 2.2:- (Mustafa and Sims [7]) Let R be the set of all real numbers define.
:G R R R R

   by
 , , , for all , , .G x y z x y y z z x x y z X      
Then it is clear that  ,R G is a G-metric space.
We use the following proposition in the Sequel without explicit mention.
Proposition 2.3:- (Mustafa and Sims [7]) Let  ,X G be a G-metric space. Then for any
, , , an d ,x y z a X it follows that
   
       
       
       
          
         
1 if , , 0 th en ,
2 , , , , , , ,
3 , , 2 , , , 2 .3 .1
4 , , , , , , ,
2
5 , , , , , , , , ,
3
6 , , , , , , , , .
G x y z x y z
G x y z G x x y G x x z
G x y y G y x x
G x y z G x a z G a y z
G x y z G x y a G x a z G a y z
G x y z G x a a G y a a G z a a
  
 
  
 
  
  
Definition 2.4: (Mustafa and Sims [7]) Let  ,X G be a G-metric space, let  n
x be a sequence of points
of X , we say that  n
x is G-convergent to x
   
 
0
,
0
if lim , , 0; th at is, fo r an y 0, th ere ex ists su ch th at , , ,
fo r all , , . W e refer to as th e lim it o f th e seq u en ce an d w rite .
n m n m
n m
n n
G x x x n N G x x x
n m n x x x x
 
     
  
The following proposition is used in the section 3.
Proposition 2.5:- (Mustafa and Sims [7]) Let  ,X G be a G-metric space. Then, the following are
equivalent:
   
   
   
   
1 is G -co n verg en t to .
2 , , 0, as .
3 , , 0, as .
4 , , 0, as , .
n
n n
n
m n
x x
G x x x n
G x x x n
G x x x m n
  
  
  
Definition 2.6:- (Mustafa and Sims [7]) Let  ,X G be a G-metric space, sequence n
x is called G-
Cauchy if given 0
0, th ere is n N   such that   0
, , , for all , ,n m l
G x x x n m l n 
that is if  , , 0 as , , .n m l
G x x x n m l  

Proposition 2.7:- (Mustafa and Sims [7]) In a G-metric space  ,X G , the following are equivalent:
(1) The sequence  n
x is G-Cauchy.
(2) For every  0 0
0, there exists such that , , for all , .n m m
n N G x x x n m n    
Definition 2.8:- (Mustafa and Sims [7]) Let  ,X G and  ,X G  be G-metric spaces and let
   : , ,f X G X G  be a function, then f is said to be G-continuous at a point
if given 0, there exists > 0a X   
        such that , ; , , im plies , , < .x y X G a x y G f a f x f y    A function f is
G-continuous on X if and only if it is G-continuous at all .a X
Proposition 2.9:- (Mustafa and Sims [7]) Let  ,X G and  ,X G  be G-metric spaces, then a function
:f X X  is G-continuous at a point x X if and only if it is G-sequentially continuous at x ; that is,
whenever n
x is G-convergent to   , n
x f x is G-convergent to  f x .
Proposition 2.10:- (Mustafa and Sims [7]) Let  ,X G be a G-metric space. Then, the function
 , ,G x y z is continuous in all variables.
Definition 2.11:- (Mustafa and Sims [7]) A G-metric space  ,X G is said to be G-complete (or a complete
G-metric space) if every G-Cauchy sequence in  ,X G is G-convergent in  ,X G .
Definition 2.12:- Two self-maps f and g on a G-metric space  ,X G are said to be compatible if
 lim , , 0n n n
G fgx gfx gfx
n

 
whenever  n
x is a sequence in X such that
lim lim fo r so m e .n n
fx g x p p X
n n
  
   
3. Main Result:-
Notation:-
     
1
: 0, 0, / is increasing, continuous and for 0 .
n
n
t t  


 
        
 

We observe that  0 0  and   .t t  (see Sastry,KPR et al.[15]) We also observe that, if we define
  for 0, w here 0 1 then .t k t t k      
Lemma 3.1:- Suppose 1,  and
       : 0, 0, is in creasin g an d 3 .1 .1
t
t 

      . Suppose  n

is a sequence of non-negative real numbers such that    1 1
a b 3.1.2n n n
    
   
Where a,b are positive real numbers such that a+b=  .

Then
1
n
n


 

 and 0 as , .k
m
m n
k n
   


Proof:- First we observe that   fo r 1, 2, 3, ....
n
n
t
t n n

 
Now
       1 1 1 1
1 1 1 1
a + b a + b a + b a+ b
.H en ce ,
n n n n n n n
n n n n n
      
     
   
   
   
      
a contradiction.
 
    
 
   
      
 
1
1
1
1 0
1 1 0 1
0
.
T h u s is a d ecreasin g seq u en ce
a+ ba b
sin ce is in creasin g
........ .
sin ce 1
1 1
H en ce 0 as , .
n n
n
nn n
n
n n
n
n n n n
n n
k
n
n n
m n
 

 
   
 
    

       


 






  
  

  
 
     
 
    
 
  
 
1
m
k 

Notation:- Let 1. 
Write      : 0, 0, / is in creasin g , co n tin u o u u s an d .
t
t   
  

 
      
 
We observe that
 
         1 0
sin ce
T h u s fo r > 1
H ere , 1, 2, ... w ith
n
n
n n
t
t
t t n t t
 

 


    

    
  
  
 
We prove the following common fixed point theorem.
Theorem 3.2:- Suppose that f and g are self-maps on a complete G-metric space  ,X G such that
     
 
 
          
     
a
b
c o r is co n tin u o u s,
d , , m ax , , , , , , ,
, , , , , , ,
f X g X
f g
G fx fy fz G g x fx fx G g y fy fy G g z fz fz
G g x fy fy G g y fx fx G g z fy fy
G



 
  
 
     
 
 
, , , , , ,
fo r all , , ............................ 3 .2 .1
e an d are co m p atib le.
T h en an d h ave a u n iq u e co m m o n fix ed p o in t.
g x fz fz G g y fz fz G g z fx fx
x y z X
f g
f g
 


Proof:- Let 0
x X . In view of (a), we can choose points
 
   
        
1 2 1
1
1 1 1 1 1 1 1
, , ..., ... su ch th at fo r 1, 2, ..... 3 .2 .2 .
W ritin g an d in 3 .2 .1 an d th en u sin g 3 .2 .2 , w e h ave
, , m ax , , , , , , ,
n n n
n n
n n n n n n n n n n n n
x x x fx g x n
x x y z x
G fx fx fx G fx fx fx G fx fx fx G fx fx fx


      
 
  
  
     1 1 1 1 1
, , , , , , ,n n n n n n n n n
G fx fx fx G fx fx fx G fx fx fx    
 
     
    
1 1 1 1 1
1 1 1
, , , , , ,
m ax , , 2 , , ,
n n n n n n n n n
n n n n n n
G fx fx fx G fx fx fx G fx fx fx
G fx fx fx G fx fx fx
    
  
 
 
     1 1 1 1 1
, , , , 3 .2 .3n n n n n n
G fx fx fx G fx fx fx    
 
 
     
         
5
1 1 1 1 1 1
1 1 1 1 1 1 1 1
F ro m G , w e h ave
, , , , + , ,
F ro m 2 .3 .1 w e g et , , + , , , , + , ,
n n n n n n n n n
G fx fx fx G fx fx fx G fx fx fx G fx fx fx
     
       


 
 
1 1
1 1
+ , ,
= , , + 2 , ,
n n n
n n n n n n
G fx fx fx
G fx fx fx G fx fx fx
 
    1
3 .2 .4

   
      
   
1 1 1 1 1
1 1 1
F ro m 3 .2 .3 an d 3 .2 .4 w e h ave
, , m ax , , 2 , , ,
, , 2 , ,
=
n n n n n n n n n
n n n n n n
n
G fx


    
  

 

      
   
   
1 1 1
1
1 1
, , 2 , , 3 .2 .5
W rite , , . T h en fro m 3 .2 .5 , w e h ave
2 3 .2 .6
n n n n n
n n n n
n n n
fx fx G fx fx fx
G fx fx fx
n

   
 

 
 

   
By taking 3, a 1 and b 2    in lemma 2.1 it follows that
1
n
n


 

 and hence
0 as , .k
m
m n
k n
   


 
     
     
     
5
1
1 1 1
1 1 1 2 2 2
N o w , b y ,
, , , , , ,
P u t
, , , , , ,
, , , , , ,
n m m n m m
n
n m m n n n n m m
n n n n n n n m m
G
G fx fx fx G fx w w G w fx fx
w fx

  
     
 

  
  

   
 
 
1 1
1
1
B y in d u ctio n ,
1
, , , ,
0
1
0 as , b y lem m a 3 .1
0
, , 0 as ,
an d h en ce ,
n m m n i n i n i
n i
n m m
n
m n
i
m n
m n
i
G fx fx fx m n
G g x g

    
 


 


 
   

   


 1 1
, 0 as ,m m
x g x m n 
  
 
 1
1
T h u s is C au ch y an d is C au ch y.
S in ce is G -co m p lete, w e can fin d a p o in t su ch th at
lim lim . 3 .2 .7
S u p p o se th at is co n tin u o u s.
T h en lim lim
n n
n n
n n
g x fx
X p X
fx g x p
n n
g
g fx g g x g p
n n



 
   
 
   
 
 
 
. 3 .2 .8
S in ce an d are co m p atib le,
lim , , 0 w h ich im p lies th at,
lim lim . 3 .2 .9
n n n
n n
f g
G fg x g fx g fx
n
g fx fg x g p
n n

 
 
   
 
         
 
1
B u t , fro m 3 .2 .1 , w e see th at
, , , , m ax , , , , , , ,
, ,
n n n n n n n n n n n n n n n
n n n n
G fg x fx fx G ffx fx fx G g fx ffx ffx G g x fx fx G g x fx fx
G g fx fx fx G g x

   
    
     
, , , ,
, , , , , ,
n n n n n
n n n n n n n n n
ffx ffx G g x fx fx
G g fx fx fx G g fx fx fx G g x ffx ffx

 
On taking the limit as n   , in view of (3.2.7),(3.2.8) and (3.2.9), it follow that
       
     
     
, , m ax , , , , , , ,
, , , , , , ,
, , , , , ,
G g p p p G g p g p g p G p p p G p p p
G g p p p G p g p g p G p p p
G g p p p G p p p G p g p g p
  
 
 
    
      
  
 
  2
, , , ,
, , 2 , , fro m 2 .3 .1
3 , ,
, , 0
fro m G
is a fix ed p o in t o f g .
G g p p p G p g p g p
G g p p p G g p p p
G g p p p
G g p p p
g p p
p



 
 

 
 


       
     
     
N o w , , , m ax , , , , , , ,
, , , , , , ,
, , , , , ,
P
n n n n
n n n
n n n
G fx fp fp G g x fx fx G g p fp fp G g p fp fp
G g x fp fp G g p fx fx G g p fp fp
G g x fp fp G g p fp fp G g p fx fx
  
 
 

       
     
   
ro ceed in g to th e lim it as , w e g et
, , m ax , , , , , , ,
, , , , , , ,
, , , ,
n
G p fp fp G p p p G g p fp fp G g p fp fp
G p fp fp G g p p p G g p fp fp
G p fp fp G g p fp fp G g

 
  
 
   
     
  
   
 
 2
, ,
m ax 0, 2 , , , 2 , , , 2 , ,
2 , ,
2 2
3 . , , , ,
3 3
, , = 0 .
B y G ,
p p p
G p fp fp G p fp fp G p fp fp
G p fp fp
G p fp fp G p fp fp
G p fp fp
fp p





 
  
 

 
is a co m m o n fix ed p o in t o f a n d .p f g
Suppose p and q are common fixed points of f and g .
so that and .fp qp p fq gq q   
Now from (3.2.1), we have
         
     
, , , , m ax , , , , , , ,
, , , , , , ,
,
G p q q G fp fq fq G g p fp fp G g q fq fq G g q fq fq
G g p fq fq G g q fp fp G g q fq fq
G g p fq
   
 
     
     
     
, , , , ,
m ax , , , , , , ,
, , , , , , ,
fq G g q fq fq G g q fp fp
G p p p G q q q G q q q
G p q q G q p p G q q q

 
  
 
     
   
   
, , , , , ,
m ax 0 0 0, , , , , 0,
, , 0 , ,
G p q q G q q q G q p p
G p q q G q p p
G p q q G q p p

 
    
 
    
       
   
, , , ,
, , 2 , , fro m 2 .3 .1
3 , ,
G p q q G q p p
G p q q G q q p
G p q q



 
 

     
 
, , 3 , ,
, , 0 .
.
T h u s an d h a ve u n iq u e co m m o n
G p q q G p q q
G p q q
p q
f g
 
 
 
fix ed p o in t.

From theorem (3.2), we obtain the following result of K.Kumara Swamy and T.Phaneendra [6] as corollary.
Theorem 3.3:- (K.Kumara swamy and T.Phaneendra [6]) Suppose that f and g are self-maps on a complete G-
metric space  ,X G such that
     
 
 
        
     
a
1
b o r is co n tin u o u s, an d 0 .
3
c an d are co m p atib le.
S u p p o se , , m ax , , , , , , ,
, , , , , , ,
f X g X
f g k
f g
G fx fy fz k G g x fx fx G g y fy fy G g z fz fz
G g x fy fy G g y fx fx G g z fy fy

 
  
 
     
 
, , , , , ,
fo r all , , ............................ 3 .3 .1
G g x fz fz G g y fz fz G g z fx fx
x y z X
 

Proof:- Take  
1
w h ere 0 .
3
t kt k   
The following result of Sushanta Kumar Mohanta([16], Theorem 3.9)follows as a corollary of theorem 3.2, by
taking   .t kt 
Theorem 3.4:- (Sushanta Kumar Mohanta([16], Theorem 3.9)) Let  ,X G be a complete G-metric space,
and let :T X X be a mapping which satisfies the following condition
                     
              
              
, , m ax , , , , , , ,
, , , , , , ,
, , , , , ,
G T x T y T z k G x T y T y G y T x T x G z T z T z
G y T z T z G z T y T y G x T x T x
G z T x T x G x T z T z G y T y T y
  
 
 
 
1
, , , an d 0 .
3
T h en h as a u n iq u e fix ed p o in t in .
P ro o f:- T ak e an d in th eo rem 3 .2 .
x y z X k
T X
f g T t kt
   
  
Theorem 3.5:- Suppose that f and g are self-maps on a complete G-metric space  ,X G such that
     
 
 
          
 
a
b
d , , m ax , , , , , , , , ,
, ,
fo r all , , ..
f X g X
f g
G fx fy fz G g x g y g z G g x fx fx G g y fy fy
G g z fz fz
x y z X



 

  
 
.......................... 3 .5 .1
T h en an d h ave a u n iq u e co m m o n fi x ed p o in t.
f g
f g

Proof:- Let 0
x X . In view of (a), we can choose points
 
 
        
1 2 1
1
1 1 1 1 1 1 1
, , ..., ... su ch th at , 0,1, 2, ...
W rite an d in 3 .5 .1 w e g et
, , m ax , , , , , , , , ,
n n n
n n
n
x x x f x g x n
x x y z x
G fx fx fx G g x g x g x G g x fx fx G g x fx fx
G g x



      

 
  

 1 1 1
, ,n n
fx fx 
      
 
    
     
   
1 1 1 1
1 1
1 1 1
1 1 1
1
m ax , , , , , , , , ,
, ,
m ax , , , , ,
m ax , , w h ere , ,
m ax , 3
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 fx fx fx
G fx fx fx


   
   
   
 
  
  



 
     
 
      
   
1
1 1
.5 .2
N o w a co n trad ictio n .
. T h u s is a d ecreasin g seq u en ce an d fro m 3 .5 .2
S u p p o se . T h u s
n n n n
n n n n n
n n
r r
    
     
   

 
  
  
 
   
 
 
 
 
 
1
1
0
0
1
1
0
0 .
S in ce , b y in d u ctio n w e h ave
H en ce
S o th at fo r ,
0 as , 3 .5 .3
S u p p o se . P u t , in 3 .5 .1
n n
n n
n
n
n
n
m n
n i
i
n m
n r r r
n m
n m
n m x x y z x
   
  
  
  



 
 

      


  

    
   
 

We get, by(G5)
     
     
     
1
1 1 1
1 1 1 2 2 2
, , , , , ,
P u t
, , , , , ,
, , , , , ,
n m m n m m
n
n m m n n n n m m
n n n n n n n m m
G fx fx fx G fx w w G w fx fx
w fx

  
     
 

  
  
   
  
 
1 1
1
1
B y in d u ctio n ,
1
, , , ,
0
1
0 as , fro m 3 .5 .3
0
, , 0 as ,
an d h en ce ,
n m m n i n i n i
n i
n m m
n m
m n
i
m n
m n
i
G fx fx fx m n
G g x g x

    
 
 

 


 
   

   


 1 1
, 0 as ,m
g x m n
  

 
   
     
 
1
B u t , fro m 3 .5 .1 , w e see th at
, , , ,
m ax , , , , , , , , ,
, ,
m ax
n n n n n n
n n n n n n n n n
n n n
G fg x fx fx G ffx fx fx
G g fx g x g x G g fx fx fx G g x fx fx
G g x fx fx
G g





      
 
1 1 1
1
, , , , , , , , ,
, , .
n n n n n n n n n
n n n
fx fx fx G g fx fx fx G fx fx fx
G fx fx fx
  

           
    
 
O n lettin g ,
, , m ax , , , , , , , , , , , .
, , , ,
, , 0 .
is a fix ed p o in t o f .
n
G g p p p G g p p p G g p g p g p G p p p G g p g p g p
G g p p p G g p p p
G g p p p
g p p
p g


 

 
 
 

           
           
         
 
N o w ,
, , m ax , , , , , , , , , , , .
O n lettin g , w e g et
, , m ax , , , , , , , , , , ,
m ax , , , , , , , , , , ,
, ,
n n n n n n
G fx fp fp G g x fp fp G g x fx fx G g p fp fp G fx g p g p
n
G p fp fp G p fp fp G p p p G g p fp fp G p g p g p
G p fp fp G p p p G p fp fp G p p p
G p fp fp




 


   
 
, ,
, , 0
is a co m m o n fix ed p o in t o f an d .
G p fp fp
G p fp fp
fp p
p f g

 
 

 
           
S u p p o se an d are co m m o n fix ed p o in t o f an d
so th at an d
N o w fro m 3 .5 .1 , w e h ave
, , , , , , m ax , , , , , , , , ,
p q f g
fp g p p fq g q q
G p p p p G p fq fq G fp fq fq G g p g q g q G g p fp fp G g q fq fq
   
  
 
         
, ,
m ax , , , , , , , , , , ,
G g q fq fq
G p q q G p p p G q q q G q q q
   
  
 
m ax , ,
, ,
, , 0
.
T h u s an d h ave u n i
G p q q
G p q q
G p q q
p q
f g




 
 
q u e co m m o n fix ed p o in t.

Corollary 3.6:- Suppose that f and g are self-maps on a complete G-metric space  ,X G such that
     
 
 
         
 
a
b 0 1
d , , m ax , , , , , , , , ,
, ,
fo r all , ,
f X g X
k
f g
G fx fy fz k G g x g y g z G g x fx fx G g y fy fy
G g z fz fz
x y z X

 

  
 
............................ 3 .5 .1
f g
f g
Proof:- Take   in T heorem 3.5t kt 
Theorem 3.7:- Suppose that f and g are self-maps on a complete G-metric space  ,X G
such that
     
 
 
         
 
a
b
d , , m ax , , , , , , , , ,
, ,
fo r all , , ..
f X g X
f g
G fx fy fz G g x g y g z G g x fx fx G g y fy fy
G g z fz fz
x y z X



 

  
 
.......................... 3 .7 .1
f g
f g
Proof:- The proof of the theorem is similar to that of 3.5
Corollary 3.8:- Suppose that f and g are self-maps on a complete G-metric space  ,X G
such that
     
 
 
         
 
a
b 0 1 .
d , , m ax , , , , , , , , ,
, ,
fo r all , ,
f X g X
k
f g
G fx fy fz k G g x g y g z G g x fx fx G g y fy fy
G g z fz fz
x y z

 

  
 
............................ 3 .8 .1
X
f g
f g

Proof:- Take   in T heorem 3.7t kt 
ACKNOWLEDGEMENT
Fourth Author B.Ramu Naidu is grateful to Special
officer, AU.P.G.Centre, Vizianagaram for
providing necessary permissions and facilities to
carry on this research.
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