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193
Fixed Point and Common Fixed Point Theorems in Complete
Metric Spaces
Pushpraj Singh Choudhary*
, Renu Praveen Pathak** and M.K. Sahu***,Ramakant Bhardwaj****
*Dept. of Mathematics & Computer Science Govt. Model Science College (Auto.) Jabalpur.
**Department of Mathematics, Late G.N. Sapkal College of Engineering,Anjaneri Hills, Nasik (M.H.), India
***Department of Mathematics,Govt PG College Gadarwara.
****Truba Institute of Engineering and Information Technology,Bhopal.
pr_singh_choudhary@yahoo.co.in, pathak_renu@yahoo.co.in
Abstract
In this paper we established a fixed point and a unique common fixed point theorems in four pair of weakly
compatible self-mappings in complete metric spaces satisfy weakly compatibility of contractive modulus.
Keywords : Fixed point, Common Fixed point, Complete metric space, Contractive modulus, Weakly
compatible maps.
Mathematical Subject Classification : 2000 MSC No. 47H10, 54H25
Introduction
The concept of the commutativity has generalised in several ways .In 1998, Jungck Rhoades [3] introduced the
notion of weakly compatible and showed that compatible maps are weakly compatible but conversely. Brian
Fisher [1] proved an important common fixed point theorem. Seesa S [9] has introduced the concept of weakly
commuting and Gerald Jungck [2] initiated the concept of compatibility. It can be easily verified that when the
two mappings are commuting then they are compatible but not conversely. Thus the study of common fixed
point of mappings satisfying contractive type condition have been a very active field of research activity during
the last three decades.
In 1922 the polished mathematician, Banach, proved a theorem ensures, under appropriate conditions, the
existence and uniqueness of a fixed point. His result is called Banach fixed point theorem or the Banach
contraction principle. This theorem provides a technique for solving a variety of a applied problems in a
mathematical science and engineering. Many authors have extended, generalized and improved Banach fixed
point theorem in a different ways. Jungck [2] introduced more generalised commuting mappings, called
compatible mappings, which are more general than commuting and weakly commuting mappings.
The main purpose of this paper is to present fixed point results for two pair of four self maps satisfying a new
contractive modulus condition by using the concept of weakly compatible maps in complete metric spaces.
Preliminaries
The definition of complete metric spaces and other results that will be needed are:
Definition 1: Let f and g two self-maps on a set X. Maps f and g are said to be commuting if
= .
Definition 2: Let f and g two self-maps on a set X. If = , ℎ called coincidence
point of f and g.
Definition 3: Let f and g two self-maps defined on a set X, then f and g are said to be compatible if they
commute at coincidence points. That is, if = , ℎ = .
Definition 4: Let f and g be two weakly compatible self-maps defined on a set X, If f and g have a unique point
of coincidence, that is, = = , ℎ .
Definition 5: A sequence in a metric space , ! ,
denoted by lim →& = , lim →& , = 0.
Definition 6: A sequence in a metric space , ( ℎ)
lim
*→&
, + = 0 , > .
Definition 7: A metric space , ( every Cauchy sequence in X is convergent.
Definition 8: A function -: /0, ∞ → /0, ∞ !
-: /0, ∞ → /0, ∞ - < > 0 .
Definition 9: A real valued function φ defined on RX ⊆ is said to be upper semi continuous
lim
→&
- ≤ - , ! ) ℎ → → ∞ .
Hence it is clear that every continuous function is upper semi continuous but converse may not true.

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194
MAIN RESULT
In this section we established a common fixed point theorem for two pairs of weakly compatible mappings in
complete metric spaces using a contractive modulus.
Theorem 1: Let (X,d) be a complete metric space. Suppose that the mapping E, F, G and H are four self maps
of X satisfying the following condition:
(i) 3 ⊆ 5 6 ⊆ 7
(ii) 869, 3:; ≤ - < , )
Where - is a upper semi continuous, contractive modulous and
< , ) = max
?
@
A
@
B 859, 7:;, 59, 69 , 87:, 3:;,
C
D
E 859, 3:; + 87:, 69;G ,
C
H
I
J8KL,MN;OJ KL,PL OJ8MN,PL;
COJ8KL,MN;J KL,PL J8MN,PL;
Q ,
R
D
I
J8MN,SN;OJ8MN,PL;OJ8MN,KL;
COJ8MN,SN;J8MN,PL;J8MN,KL;
Q
T
@
U
@
V
(iii) The pair (G, E) and (H, F) are weakly compatible then E, F, G& H have a unique common fixed point.
Proof : Suppose ( ) ℎ ) ℎ ℎ
) = 6 = 7 OC
) OC = 3 OC = 5 OD
By (ii) we have
) , ) OC = 6 , 3 OC
( )( )1,n nx x +≤ φ λ
Where < , OC =
?
@
A
@
B
5 , 7 OC , 5 , 6 , 7 OC, 3 OC ,
C
D
/ 5 , 3 OC + 7 OC, 6 W ,
C
H
X
J K9Y,M9YZ[ OJ K9Y,P9Y OJ M9YZ[,P9Y
COJ K9Y,M9YZ[ J K9Y,P9Y J M9YZ[,P9Y
,
R
D
X
J M9YZ[,S9YZ[ OJ M9YZ[,P9Y OJ M9YZ[,K9Y
COJ M9YZ[,S9YZ[ J M9YZ[,P9Y J M9YZ[,K9Y
T
@
U
@
V
=
?
@
@
A
@
@
B
3 ]C, 6 , 3 ]C, 6 , 6 , 3 OC ,
1
2
/ 3 ]C, 3 OC + 6 , 6 W ,
1
4
a
3 ]C, 6 + 3 ]C, 6 + 6 , 6
1 + 3 ]C, 6 3 ]C, 6 6 , 6
b ,
3
2
a
6 , 3 OC + 6 , 6 + 6 , 3 ]C
1 + 6 , 3 OC 6 , 6 6 , 3 ]C
b
T
@
@
U
@
@
V
≤ max d 3 ]C, 6 , 6 , 3 OC ,
1
2
/ 3 ]C, 3 OC W
1
4
e
3 ]C, 6
1
f
3
2
e
3 ]C, 3 OC
1
fg
≤ max I ) ]C, ) , ) , ) OC ,
1
2
/ ) ]C, ) OC W,
1
4
) ]C, ) ,
3
2
/ ) ]C, ) OC WQ
≤ max ) ]C, ) , ) , ) OC
Since - is a contractive modulus; < , OC = ) , ) OC , hℎ
) , ) OC ≤ - ) ]C, ) 1
Since - is an upper semi continuous contractive modulus, Equation (1) implies that the sequence ) , ) OC
is monotonic decreasing and continuous.
Hence there exists a real number, say ≥ 0 ℎ ℎ lim →& ) , ) OC =
∴ → ∞ , 5 1 ℎ
≤ -
Which is possible only If r = 0 because - is a contractive modulus, Thus
lim
→&
) , ) OC = 0
Now we show that ) is a Cauchy sequence.
Let If possible we assume that ) is not a cauchy sequence , Then there exist an ∈ > 0 and subsequence
l l ℎ ℎ l < l < lOC

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195
( ) ( )1
, and , im n m nd y y d y y −
≥ ∈ <∈ (2)
So that ( ) ( ) ( ) ( )1 1 1
, , , ,i i i i i i im n m n n n n nd y y d y y d y y d y y− − −
∈≤ ≤ + <∈ +
Therefore ( ) ( ) ( )( )lim , , ,i i i i i im n m n m n
n
d y y d Gx Hx x x
→∞
= ≤ φ λ
i.e. ( )( ),i im nx x∈≤ φ λ (3)
Where,
( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
, , , , , ,
1
, ,
2
, , ,1, max
4 1 , , ,
, , ,3
2 1 , ,
i i i i i i
i i i i
i i i i i i
i i
i i i i i i
i i i i i i
i i i
m n m m n n
m n n m
m n m m n m
m n
m n m m n m
n n n m n m
n n n
d Ex Fx d Ex Gx d Fx Hx
d Ex Hx d Fx Gx
d Ex Fx d Ex Gx d Fx Gxx x
d Ex Fx d Ex Gx d Fx Gx
d Fx Hx d Fx Gx d Fx Ex
d Fx Hx d Fx G
 + 
 + +λ =  
+ + 
 
+ +
+ ( ) ( ),i i im n mx d Fx Ex
 
 
 
 
 
  
 
 
 
  
  
     
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
1 1 1 1
1 1
1 1 1 1
1 1 1 1
1 1
, , , , , ,
1
, , ,
2
, , ,1max
4 1 , , ,
, ,3
2
i i i i i i
i i i i
i i i i i i
i i i i i i
i i i i i
m n m m n n
m n n m
m n m m n m
m n m m n m
n n n m n
d Hx Gx d Hx Gx d Gx Hx
d Hx Hx d Gx Gx
d Hx Gx d Hx Gx d Gx Gx
d Hx Gx d Hx Gx d Gx Gx
d Gx Hx d Gx Gx d Gx
− − − −
− −
− − − −
− − − −
− −
 + 
 + +=  
+ 
 
+ + ( )
( ) ( ) ( )
1 1
1 1 1 1
,
1 , , ,
i
i i i i i i
m
n n n m n m
Hx
d Gx Hx d Gx Gx d Gx Hx
− −
− − − −
 
 
 
 
 
  
 
 
 
  
  
 +    
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
1 1 1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1
, , , , , ,
1
, , ,
2
, , ,1max
4 1 , , ,
, , ,3
2 1 , ,
i i i i i i
i i i i
i i i i i i
i i i i i i
i i i i i i
i i i
m n m m n n
m n n m
m n m m n m
m n m m n m
n n n m n m
n n n
d y y d y y d y y
d y y d y y
d y y d y y d G y
d y y d y y d y y
d y y d y y d y y
d y y d y
− − − −
− −
− − − −
− − − −
− − − −
− −
 + 
 + +=  
+ 
 
+ +
+ ( ) ( )1 1
,i i im n my d y y− −
 
 
 
 
 
  
 
 
 
  
  
     
By taking limit as → ∞ ,
lim
l→&
< +l, l = I∈ ,0,0,
1
2
∈ +∈ ,
1
4
m
∈ +0 + 0
1 + 0
n ,
3
2
m
0 + 0+∈
1 + 0
nQ
( ) ( )1
, and , im n m nd y y d y y −
≥ ∈ <∈

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Thus we have ,lim*→& < +, =∈
Therefore from (3) ∈ ≤ - 5
This is a contraction because 0 <∈ - ! .
Thus ) ℎ) ,
, ℎ o ℎ ℎ lim →& ) = o.
Thus lim
→&
6 = lim
→&
7 OC = o lim
→&
3 OC = lim
→&
5 OD = o
i. e. lim
→&
6 = lim
→&
7 OC = lim
→&
3 OC = lim
→&
5 OD = o
since 6 ⊆ 7 , ℎ ∈ ℎ ℎ o = 5
Then by (ii), we have
6 , o ≤ 6 , 3 OC + 3 OC, o
≤ -8< , OC ; + 3 OC, o
Where, < , OC =
?
@
A
@
B
5 , 7 OC , 5 , 6 , 7 OC, 3 OC ,
C
D
/ 5 , 3 OC + 7 OC, 6 W,
C
H
X
J Kp,M9YZ[ OJ Kp,Pp OJ M9YZ[,Pp
COJ Kp,M9YZ[ J Kp,Pp J M9YZ[,Pp
,
R
D
X
J M9YZ[,S9YZ[ OJ M9YZ[,Pp OJ M9YZ[,Kp
COJ M9YZ[,S9YZ[ J M9YZ[,Pp J M9YZ[,Kp
T
@
U
@
V
=
?
@
@
A
@
@
B
o, 6 , o, 6 , 6 , 3 OC ,
1
2
/ o, 3 OC + 6 , 6 W,
1
4
a
o, 6 + o, 6 + 6 , 6
1 + o, 6 o, 6 6 , 6
b ,
3
2
a
6 , 3 OC + 6 , 6 + 6 , o
1 + 6 , 3 OC 6 , 6 6 , o
b
T
@
@
U
@
@
V
Taking the limit as → ∞ )
< , OC =
?
@
@
A
@
@
B o, o , o, 6 , o, o ,
1
2
/ o, o + o, 6 W,
,
1
4
a
o, o + o, 6 + o, 6
1 + o, o o, 6 o, 6
b ,
3
2
a
o, o + o, 6 + o, o
1 + o, o o, 6 o, o
b
T
@
@
U
@
@
V
= 6 , o
Thus as → ∞, 6 , o ≤ -8 6 , o ; + o, o = -8 6 , o ;
q 6 ≠ o ℎ 6 , o > 0 ℎ - ! -8 6 , o ; < 6 , o .
hℎ 6 , o < 6 , o ℎ ℎ .
hℎ 6 = o , 5 = 6 = o.
s 5 6 .
s ℎ 6 5 t ) ,
65 = 56 , . 6o = 5o .
u 6 ⊆ 7 , ℎ ! ∈ ℎ ℎ o = 7!.
hℎ ) , ℎ !
o, 3! = 6!, 3!
≤ - 8< , ! ;
ℎ , < , ! =
?
@@
A
@@
B 5 , 7! , 5 , 6 , 7!, 3! ,
1
2
/ 5 , 3! + 7!, 6 W,
1
4
a
5 , 7! + 5 , 6 + 7!, 6
1 + 5 , 7! 5 , 6 7!, 6
b ,
3
2
a
7!, 3! + 7!, 6 + 7!, 5
1 + 7!, 3! 7!, 6 7!, 5
b
T
@@
U
@@
V

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197
=
?
@
A
@
B o, o , o, o , o, 3! ,
C
D
/ o, 3! + o, o W,
C
H
X
J v,v OJ v,v OJ v,v
COJ v,v J v,v J v,v
,
R
D
X
J v,Sw OJ v,v OJ v,v
COJ v,Sw J v,v J v,v
T
@
U
@
V
= o, 3!
hℎ o, 3! ≤ - o, 3!
q 3! ≠ o ℎ o, 3! > 0 ℎ - ! ,
- o, 3! < o, 3! , ℎ ℎ .
Therefore 3! = 7! = o.
So ! 7 3.
Since the pair of maps F and H are weakly compatible,
73! = 37! , . 7o = 3o.
Now we show that z is a fixed point of G.
hℎ ) , ℎ ! 6o, o = 6o, 3!
≤ - 8< o, ! ;
Where,
< o, ! =
?
@@
A
@@
B 5o, 7! , 5o, 6o , 7!, 3! ,
1
2
/ 5o, 3! + 7!, 6o W,
1
4
a
5o, 7! + 5o, 6o + 7!, 6o
1 + 5o, 7! 5o, 6o 7!, 6o
b ,
3
2
a
7!, 3! + 7!, 6o + 7!, 5o
1 + 7!, 3! 7!, 6o 7!, 5o
b
T
@@
U
@@
V
=
?
@
A
@
B 6o, o , 6o, 6o , o, o ,
C
D
/ 6o, o + o, 6o W,
C
H
X
J Pv,v OJ Pv,Pv OJ v,Pv
COJ Pv,v J Pv,Pv J v,Pv
,
R
D
X
J v,v OJ v,Pv OJ v,Pv
COJ v,v J v,Pv J v,Pv
T
@
U
@
V
= 6o, o
hℎ 6o, o ≤ - 8 6o, o ;
q 6o ≠ o ℎ 6o, o > 0 ℎ - !
- 8 6o, o ; < 6o, o .
Therefore 6o = o
Hence 6o = 5o = o
By (ii), we have,
o, 3o = 5o, 3o
( )( ),z z≤ φ λ
Where,
< o, o =
?
@@
A
@@
B 5o, 7o , 5o, 6o , 7o, 3o ,
1
2
/ 5o, 3o + 7o, 6o W,
1
4
a
5o, 7o + 5o, 6o + 7o, 6o
1 + 5o, 7o 5o, 6o 7o, 6o
b ,
3
2
a
7o, 3o + 7o, 6o + 7o, 5o
1 + 7o, 3o 7o, 6o 7o, 5o
b
T
@@
U
@@
V
=
?
@
A
@
B o, 3o , o, o , 3o, 3o ,
C
D
/ o, 3o + 3o, o W,
C
H
X
J v,Sv OJ v,v OJ Sv,v
COJ v,Sv J v,v J Sv,v
,
R
D
X
J Sv,Sv OJ Sv,v OJ Sv,v
COJ Sv,Sv J Sv,v J Sv,v
T
@
U
@
V
= o, 3o
hℎ o, 3o ≤ - 8 o, 3o ;
If o ≠ 3o ℎ o, 3o > 0 ℎ - ! ,
- 8 o, 3o ; < o, 3o
Therefore o, 3o < o, 3o , ℎ ℎ .
Henceo = 3o

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198
Therefore 3o = 7o = o.
Therefore 6o = 5o = 3o = 7o = o , . o 5, 7, 6 3.
Uniqueness : For uniqueness, Let we assume that o , o ≠
5, 7, 6 3.
By (ii), we have,
o, = 6o, 3
( )( ),z w≤ φ λ
Where,
o, =
?
@
A
@
B 5o, 7 , 5o, 6o , 7 , 3 ,
C
D
/ 5o, 3 + 7 , 6o W,
C
H
X
J Kv,Mx OJ Kv,Pv OJ Mx,Pv
COJ Kv,Mx J Kv,Pv J Mx,Pv
,
R
D
X
J Mx,Sx OJ Mx,Pv OJ Mx,Kv
COJ Mx,Sx J Mx,Pv J Mx,Kv
T
@
U
@
V
=
?
@
A
@
B o, , o, o , , ,
C
D
/ o, + , o W,
C
H
X
J v,x OJ v,v OJ x,v
COJ v,x J v,v J x,v
,
R
D
X
J x,x OJ x,v OJ x,v
COJ x,x J x,v J x,v
T
@
U
@
V
= o,
Thus o, ≤ - 8 o, ;
Since o ≠ ℎ o, > 0 ℎ - ! ,
- 8 o, ; < o,
o, < o, ℎ ℎ .
Therefore o =
Thus z is the unique common fixed point of E, F, G & H .
Hence the theorem.
Corollary 1 : Let , ℎ ℎ 5, 6 3
, s ) ℎ
3 ⊆ 5 6 ⊆ 5
6 , 3) ≤ -8< , ) ;
Where - , !
< , ) =
?
@
A
@
B 859, 5:;, 59, 69 , 85:, 3:;,
C
D
E 859, 3:; + 85:, 69;G ,
C
H
I
J8KL,KN;OJ KL,PL OJ8KN,PL;
COJ8KL,KN;J KL,PL J8KN,PL;
Q ,
R
D
I
J8KN,SN;OJ8KN,PL;OJ8KN,KL;
COJ8KN,SN;J8KN,PL;J8KN,KL;
Q
T
@
U
@
V
hℎ 6, 5 3, 5 t )
Then E, G and H have a unique common fixed point.
Proof : By taking E = F in main theorem we get the proof.
Corollary 2 : Let (X,d) be a complete metric space, suppose that the mappings E and G are self maps of X,
satisfying the following conditions :
6 ⊆ 5
6 , 6) ≤ -8< , ) ;
Where - , !
< , ) =
?
@
@
A
@
@
B 859, 5:;, 59, 69 , 85:, 6:;,
1
2
E 859, 6:; + 85:, 69;G ,
1
4
e
859, 5:; + 59, 69 + 85:, 69;
1 + 859, 5:; 59, 69 85:, 69;
f ,
3
2
e
85:, 6:; + 85:, 69; + 85:, 59;
1 + 85:, 6:; 85:, 69; 85:, 59;
f
T
@
@
U
@
@
V
hℎ 6, 5 t )
Then E and G have a unique common fixed point.
Proof : By taking E = F and G = H in theorem 3.1 we get the proof.

7. Mathematical Theory and Modeling www.iiste.org
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199
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