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1. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 02 Issue: 01 | Jan-2013, Available @ http://www.ijret.org 1
FIXED POINT AND WEAK COMMUTING MAPPING
Yogita R Sharma
Abstract
In this paper we prove some new fixed point theorems for weak commuting mapping on complete metric space. Our results generalize
several corresponding relations in metric space of weak commuting mapping.
Keywords: Fixed point, weak commuting mapping, and metric space.
AMS Classification No (2000): 47H10, 54H25
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1. INTRODUCTION
In 1922, Banach [1] introduced very important concept of
contraction mapping, which is known as a Banach contraction
principle which states that “every contraction mapping of a
complete metric space into itself has a unique fixed point”.
Ciric [2] gave a generalization of Banach contraction principle
as:
Let S be a mapping of complete metric space  dX, into
itself, there exists an  1,0 such that
(1).  SySxd ,
     
    






SxydSyxd
SyydSxxdyxd
,,,
,,,,,,
max
For all Xyx , , then S has a unique fixed point.
In 1976, Jungck [4] investigated an interdependence between
commuting mappings and fixed points, proved the followings:
Let T be a continuous mapping of a complete metric space
 dX, into itself, then T has a fixed point in X , if and only if
there exists an  1,0 and a mapping XXS : which
commutes with T and satisfies
(2).    XTXS  and    TyTxdSySxd ,, 
For all Xyx , . Indeed S and T have a unique common
fixed point if and only if (.2) holds for some  1,0 .
Singh [9] further generalized the above result and proved the
followings:
Let two continuous and commuting mappings S and T from a
complete metric space  dX, into itself satisfying the
following conditions:
(3).    XTXS 
      
      TyTxdSxTydSyTxdb
SyTydSxTxdaSySxd
,,,
,,,


For all Xyx , , where cba ,, are non-negative real
numbers satisfying 1220  cba , then S and T have
a unique common fixed point.
Ranganathan [7] has further generalized the result of Jungck
[4] which gives criteria for the existence of a fixed point as
follows:
Let T be a continuous mapping of a complete metric space
 dX, into itself. Then T has a fixed point in X, if and only
if there exists a real number  1,0 and a mapping
XXS : which commutes with satisfying:
(4).    XTXS 
and
 SySxd ,
          TyTxdSxTydSyTxdSyTydSxTxd ,,,,,,,,,max
For all Xyx , . Indeed commuting mapping S and T have a
unique common fixed point.
Fisher [3] proved following theorem for two commuting
mappings S and T.

2. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 02 Issue: 01 | Jan-2013, Available @ http://www.ijret.org 2
Theorem 1 If S is a mapping and T is a continuous mapping of
the complete metric space X into itself and satisfying the
inequality.
(5).       STxSydTSyTxdkTSySTd ,,,  ,
For all Xyx , , where 210  k , then S and T have a
unique common fixed point.
Fisher further extended his theorem and proved a common
fixed point of commuting mappings S and T.
Theorem 2 If S is a mapping and T is a continuous mapping of
the complete metric space X into itself and satisfying the
inequality.
(6).       TSySydSTxTxdkTSySTd ,,,  ,
for all Xyx , ,where 210  k , then S and T have a
unique common fixed point.
2. WEAK** COMMUTING MAPPINGS AND
COMMON FIXED POINT
In this section Weak**
commuting mappings and unique
common fixed point theorem in metric space is established.
First we give the following definitions:
Definition 1 According to Sessa [8] two self maps S and T
defined on metric space  dX, are said to be weakly
commuting maps if and only if
   TxSxdTSxSTxd ,,  ,
For all Xx .
Definition 2 wo self mappings S and T of metric space
 dX, are said to be weak** commuted, if    XTXS 
and for any Xx ,
   
     xTxSdTSxSTxdSxTxSTd
xTSTxSdxSTxTSd
2222
222222
,,,
,,


Definition3 A map XXXS ,:  being a metric space,
is called an idempotent, if SS 2
.
Example1. Let ]1,0[X with Euclidean metric space
and define S and T by
3
,
3
x
Tx
x
x
Sx 

 for all Xx ,
then    1090,0,1  where  0,1Sx and
 1090,Tx
 2 2 2 2
,
4 81 36 81
x x
d S T x T S x
x x
 
 
  
2
32
4 81 36 81
x
x x

 
  
2
8
4 27 12 27
x
x x

 
4 27 12 27
x x
x x
 
 
 xTSTxSd 22
, .
Implies that    xTSTxSdxSTxTSd 222222
,, 
 xTSTxSd 22
, 4 27 12 27
x x
x x
 
 
  
2
8
4 27 12 27
x
x x

 
  27927
8 2


xx
x
27 9 27
x x
x x
 
 
 SxTxSTd 22
,
   SxTxSTdxTSTxSd 2222
,, 
 SxTxSTd 22
, 27 9 27
x x
x x
 
 
  
2
8
27 9 27
x
x x

 
  
2
2
9 3 9
x
x x

 
9 3 9
x x
x x
 
 
 TSxSTxd ,
   TSxSTxdSxTxSTd ,, 22

 TSxSTxd , 9 3 9
x x
x x
 
 
  
2
2
9 3 9
x
x x

 
 
2
4
9 4 9
x
x


9 4 9
x x
x
 

 xSxTd 22
,

3. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 02 Issue: 01 | Jan-2013, Available @ http://www.ijret.org 3
   xSxTdTSxSTxd 22
,,  .
Using  1,0 for Xx , we conclude that definition (2) as
follows:
   
     xTxSdTSxSTxdSxTxSTd
xTSTxSdxSTxTSd
2222
222222
,,,
,,


for any Xx .
We further generalized the result of Fisher [3], Pathak [6],
Lohani and Badshah [5] by using another type of rational
expression.
Theorem1. If S is a mapping and T is a continuous
mapping of complete metric space  TS, is weak commuting
pair and the following condition:
(7).
       
     22222222
32223222
2222
,,
,,
,
ySTySdxTSxTd
ySTySdxTSxTd
ySTxTSd


 
 ySxTd 22
, ,
for all x, y in X , where 0,  with 12   , then S
and T have a unique common fixed point.
Proof Let x be an arbitrary point X. Define
  n
n
xxTS 2
22
 or   12
222
 n
n
xxTST ,
where ,...,3,2,1,0n by contractive condition (7),
      xTSTxTSdxxd
nn
nn
22222
122 ,, 
     xTSTSTxTSTSd
nn 12222212222
,


     
      
     
       






















212222212222
2122221222
312222212222
3122221222
,
,
,
,
xTSTSTxTSTSd
xTSTSxTSTd
xTSTSTxTSTSd
xTSTSxTSTd
nn
nn
nn
nn

    xTSTSxTSTd
nn 122221222
,

 
     
     
 nn
nnnn
nnnn
xxd
xxdxxd
xxdxxd
2122
122
2
212
3
122
3
212
,
,,
,,






 
      nnnnnn xxdxxdxxd 212122212 ,,,   
     122212 ,,   nnnn xxdxxd 
       nnnn xxdxxd 212122 ,,1   
   
 
 nnnn xxdxxd 212122 ,
1
, 





 nn xxkd 212 , ,
where
 
 




1
k .
Proceeding in the same manner
   21
12
122 ,, xxdkxxd n
nn

  .
Also    

m
ni
iimn xxdxxd 1,, for nm  .
Since 1k , it follows that the sequence  nx is Cauchy
sequence in the complete metric space X and so it has a limit
in X, that is
122 limlim 

 n
n
n
n
xux
and since T is continuous, we have
  uTxTxu n
n
n
n
2
2
2
12 limlim 



.
Further,
    uSxTSTduSxd
n
n
212222
12 ,,

 
    uTSxTSTd
n 221222
,


for uTu 2

        
        212221222222
312221223222
,,
,,
xTSTxTSduTSuTd
xTSTxTSduTSuTd
nn
nn





  xTSuTd
n 1222
,

 
      uTxdxxduTSuTd nnn
2
223222
222
,,,   
      ,,, 223222
2
uxdxxduSud nnn   

4. IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 02 Issue: 01 | Jan-2013, Available @ http://www.ijret.org 4
Taking limit as n , it follows that
  ,0, 2
uSud
which implies
  0, 2
uSud and so uTuSu 22
 .
Now consider weak** commutativity of pair  TS, implies
that SuTuSTuTSTuSuSTuTS 22222222
,,  and
so TuTuS 2
and SuSuT 2
.
Now,
    SuSTuTSdSuud 2222
,, 
        
        
  SuSuTd
SuSTSuSduTSuTd
SuSTSuSduTSuTd
22
22222222
32223222
,
,,
,,






     
     
 Suud
SuSuduud
SuSuduud
,
,,
,,
 



    0,1  Suud .
This implies that   .01   Hence   0, Suud or
uSu  .
Similarly, we can show that uTu  . Hence, u is a common
fixed point of S and T.
Now suppose that v is another common fixed point of S and T.
Then
   vSTuTSdvud 2222
,, 
     
     
 vSuTd
vSTvSduTSuTd
vSTvSduTSuTd
22
22222222
32223222
,
,,
,,






      vudvvduud ,,,  
    0,1  vud .
Since,   01   , then   0, vud .
Thus, it follows that vu  .Hence S and T have a unique
common fixed point.
REFERENCES
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[3]. Fisher, B.Common fixed point and constant mappings on
metric spaces, Math. Sem. Notes, Kobe University 5(1977),
319-326.
[4]. Jungck, G.Commuting mappings and fixed points, Amer.
Math. Monthly 83(1976), 261-263.
[5]. Lohani, P. C. And Badshah, V. H. Common fixed point
and weak**commuting mappings. Bull. Cal. Math. Soc.
87(1995), 289-294.
[6]. Pathak, H. K.Weak* commuting mappings and fixed
points. Indian J. Pure Appl. Math. 17(1986), 201-211.
[7]. Ranganathan, S.A fixed point theorem for commuting
mapping. Math. Sem. Notes. Kobe. Univ. 6(1978), 351-357.
[8]. Sessa, S.On a weak commutativity condition in fixed
point considerations, Publ. Inst. Math. 32(1982), 149-153.
[9]. Singh, S. L.On common fixed point of commuting
mappings. Math. Sem. Notes. Kobe. Univ. 5(1977), 131-134.