Available online through
www.ijma.info
FIXED POINT THEOREM FOR FIVE MAPPINGS IN METRIC SPACE
Rajesh Shrivastava
*, Sarvesh Agrawal, Ramakant Bhardwaj and Dr. R. N. Yadava
**Department of Mathematics, Truba Group of Institutions Bhopal, INDIA
*Prof. & Head, Department of Mathematics, Govt. Science & Commerce College Benajeer, Bhopal, INDIA **
Director, Patel Group of Engineering Institutions, ratibad, Bhopal INDIA *E-mail: [email protected]
(Received on: 23-07-11; Accepted on: 07-08-11)
--- ABSTRACT
I
n this paper we prove a common fixed point theorem taking five mappings satisfying a weakly commuting condition. Our result generalize the results of Fisher [2], Pathak [9], Rao and Rao [10], Saluja and Dehariya [12].Key Words: Fixed point, Metric Space, Weakly commuting Mappings.
Subject Classification: 47H10, 54H25.
--- 2. INTRODUCTION & PRELIMINARIES:
There are many generalization of Banach Contraction Principle. Some of them are obtained by adding the terms Kannan[8], Fisher[2], Rhoades[11] generalized this principle in many ways. Jaggi [5], Jaggi and Das [6] extended this principle for rational expression. On the same way Bhardwaj et. al., Yadava et.al. also worked for rational expression. Initially Jungck [7] proved a common fixed point theorem for commuting mappings. This result was extended and generalized in various ways by many authors. On the way Sessa [13] gave the concept of weak commutativity.
In the present paper we prove a fixed point theorem for five mappings taking weakly commuting condition for certain mappings. Our result is motivated by Pathak[9], Rao and Rao[10], Saluja and Dehariya [12].
Definition 2.1[3, 4]: Let K be a non empty subset of a metric space (X, d),
T K
:
→
X and F K
:
→
CB X
( )
. The pair (F,T) is said to be weakly commuting if for everyx y
,
∈
K with x
∈
Fy and Ty
∈
K
,
we have(
,
)
(
,
)
d Tx FTy
=
d Ty Fy
Definition 2.2[4]: Let K be a non empty subset of a metric space (X, d),
T K
:
→
X and F K
:
→
CB X
( )
. The pair (F,T) is said to be point-wise R-weakly commuting on K if for givenx
∈
K and Tx
∈
K
,
there exists some R=R(x)>0 such that(
,
)
(
,
)
.
d Ty FTx
≤
R d Tx Fx
for each y
∈
K
∩
Fx
Definition 2.3: Let R, S, T, U and V are five mappings of a metric space (X, d). Then the pair (R S, T U) is called weakly commuting pair of two mappings with respect to V if
(
)
(
( )
( )
)
(
( )
( )
)
(
)
(
( )
( )
)
(
( )
( )
)
(
)
(
( )
( )
)
(
( )
( )
)
(
)
(
( )
( )
)
(
( )
( )
)
2.3.1 , ,
2.3.2 , ,
2.3.3 , ,
2.3.4 , ,
d SVRSV x TUV x d RSVSV x TUV x
d RSV x UVTUV x d RSV x TUVUV x
d VRSV x TUV x d RSVV x TUVV x
d RSV x VTUV x d RSV x TUVV x
≤ ≤ ≤
≤
3. MAIN RESULT:
Theorem 3.1: Let R, S, T, U and V are five self mappings of complete metric space (X, d) satisfying the conditions: (A) The pair (R, S, T, U) is weakly commuting pair of two mappings with respect to V.
---
Page: 1279-1285
© 2011, IJMA. All Rights Reserved 1280
( )
(
( )
( )
)
(
( )
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
(
)
( )
(
)
(
( )
)
(
( )
)
(
( )
)
( )
( )
(
)
(
)
(
)
2 2 , , ,1 , , , ,
, , , ,
, , ,
, , , , , , , 0 2 1 .
d x T U V y d y R S V x B d R S V x T U V y
d x T U V y d y R S V x d x T U V y d x y d y T U V y d x R S V x d y T U V y d y R S V x d R S V x T U V y d x y d x y
fo r a ll x y X x y w h e r e a n d
α
β γ
δ η
α β γ δ η α β γ δ η
≤
+ +
+ +
+ +
∈ ≠ ≥ + + + + <
Then R, S, T, U and V have a unique common fixed point in X. Proof: Let
x
0be an arbitrary point of X. We define a sequence{ }
x
nby x
=
x
2 ,ny
=
x
2n−1and RSV x
(
2 ,n)
=
x
2n+1,
TUV x
(
2n−1,)
=
x
2n(
)
(
(
)
(
)
)
(
)
(
)
(
(
)
)
(
)
(
)
(
(
)
)
(
(
)
)
(
)
(
)
(
)
(
(
)
)
(
(
)
)
(
(
)
)
(
)
(
)
(
)
(
)
2 22 2 1 2 2 1
2 2 1 2 1 2
2 2 1 2 1 2 2 2 1 2 2 1
2 2 2 1 2 1 2 1 2 1 2 1 2
2 2 1 2 1 2 2
, ,
, ,
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
n n n n
d x x d RSV x TUV x
d x TUV x d x RSV x
d x TUV x d x RSV x d x TUV x d x x
d x RSV x d x TUV x d x TUV x d x RSV x
d RSV x TUV x d x x d x
α
β
γ
δ
η
+ − − − − − − − − − − − − − − = ≤ + + + ++ +
(
)
21, 2
n− xn
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
)
2 2 2 1 2 1
2 2 2 1 2 1 2 2 2 2 1
2 2 1 2 1 2 2 1 2 2 1 2 1
2
2 1 2 2 1 2 2 1 2
,
,
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 n n
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
α
β
γ
δ
η
− + − + − + − − − + + − −≤
+
+
+
+
+
+
By using inequalities, we obtain
(
)
(
) (
)
(
)
(
)
(
)
(
) (
)
(
)
(
) (
)
(
)
(
) (
)
(
) (
)
(
)
2
2 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2 2 1
2
2 1 2 2 1 2 2 1 2
2
2 2 1 2 1 2 2 1 2 2 1 2 2 2 1
2
2 1 2 2 1 2 2 1 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 n
n n n n n n
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
β
γ
δ
η
β
γ
γ
δ
η
+ + − − − + + − − + − − − + + − −≤
+
+
+
+
≤
+
+
+
+
(
) (
) (
) (
) (
)
(
)
(
)
(
)
(
)
(
) (
)
22 2 1 2 1 2 2 1 2
2 2 2 2
2 2 1 2 2 1 2 1 2 2 1 2
,
,
,
.
.
,
,
,
,
,
2
n n n n n n
n n n n n n n n
d x
x
d x
x
d x
x
Since A M
G M then
d x
x
d x
x
d x
x
d x
x
β
γ
δ
γ
η
β
γ
δ
γ η
+ − − + + − −≤
+ +
+
+
≥
+ +
≤
+
+
+
(
)
(
)
(
)
(
)
2 22 2 1 2 1 2
2 2
2 2 1 2 1 2
3
2
,
,
1
2
,
,
n n n n
n n n n
d x
x
d x
x
d x
x
k d x
x
β
γ
δ
η
β
γ
δ
+ − + −
+
+
+
≤
+ +
−
≤
(
)
(
)
(
)
1 22 2 1 2 1 2
3
2
1, sin
2
1.
1
2
3.1.1
n,
n n,
nwhere k
ce
d x
x
k
d x
x
β
γ
δ
η
α
β
γ
δ η
β
γ
δ
+ −
+
+
+
=
<
+
+
+
+
<
+ +
−
Page: 1279-1285
Similarly on taking
x
=
x
2n+1,y
=
x
2nand RSV x
(
2n+1,)
=
x
2n+2,
TUV x
(
2 ,n)
=
x
2n+1in condition (3.1) (B), we obtain
(
)
(
(
)
(
)
)
(
)
(
)
(
(
)
)
(
)
(
)
(
(
)
)
(
(
)
)
(
)
(
)
(
)
(
(
)
)
(
(
)
)
(
(
)
)
(
)
(
)
(
)
(
)
2 2
2 1 2 2 2 1 2
2 1 2 2 2 1
2 1 2 2 2 1 2 1 2 2 2 1
2 1 2 1 2 2 2 2 2 2 1
2 1 2 2 1 2 2
,
,
,
,
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
n n n n n
d x
x
d RSV x
TUV x
d x
TUV x
d x
RSV x
d x
TUV x
d x
RSV x
d x
TUV x
d x
x
d x
RSV x
d x
TUV x
d x
TUV x
d x
RSV x
d RSV x
TUV x
d x
x
d x
α
β
γ
δ
η
+ + +
+ +
+ + + +
+ + +
+ + +
=
≤
+
+
+
+
+
+
(
1,
x
2n)
2(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
)
2 1 2 1 2 2 2
2 1 2 1 2 2 2 2 1 2 1 2 2 1
2 1 2 2 2 2 1 2 2 1 2 2 2
2
2 2 2 1 2 2 1 2 2 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 n n
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
α
β
γ
δ
η
+ + +
+ + + + + +
+ + + + +
+ + + +
≤
+
+
+
+
+
+
(
) (
)
(
)
(
)
(
)
(
) (
)
(
)
2 1 2 2 2 2 1 2 2 1 2 2 1 2 1 2 2
2
2 2 2 1 2 2 1 2 2 1
, , , , ,
, , ,
n n n n n n n n n n
n n n n n n
d x x d x x d x x d x x d x x
d x x d x x d x x
β
γ
δ
η
+ + + + + + +
+ + + +
≤ + +
+ +
(
) (
) (
) (
) (
)
(
)
(
)
(
) (
)
(
)
(
)
2
2 2 2 1 2 1 2 2 1 2
2 2 2
2 2 2 1 2 1 2 2 1 2
2 2
2 2 2 1 2 2 1
,
,
,
,
,
,
2
3
2
,
,
1
2
n n n n n n
n n n n n n
n n n n
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
d x
x
β
γ
δ
γ η
β
γ
δ
γ η
β
γ
δ
η
β
γ
δ
+ + + +
+ + + +
+ + +
≤
+ +
+
+
+ +
≤
+
+
+
+
+
+
≤
+ +
−
(
)
(
)
(
)
(
)
2 2
2 2 2 1 2 2 1
1 2
2 2 2 1 2 2 1
,
,
(3.1.2)
,
,
3
2
1, sin
2
1.
1
2
n n n n
n n n n
d x
x
k d x
x
d x
x
k d x
x
where
k
ce
β
γ
δ
η
α
β
γ
δ η
β
γ
δ
+ + +
+ + +
≤
≤
+
+
+
=
<
+
+
+
+
<
+ +
−
Hence by (3.1.1) and (3.1.2), we have
(
)
(
)
(
)
(
)
(
)
2
1 1
2 2
2 2 2 1 2 2 1 2 1 2
2 1 1 2
2 2 2 1 0 1
,
,
,
,
,
,
0
.
n n n n n n
n
n n
d x
x
k d x
x
k
d x
x
In general
d x
x
k
d x x
as n
+ + + −
+
+ +
≤
≤
≤
→
→ ∞
Thus
{ }
x
n is a D-Cauchy sequence in X. Since X is complete, there exists a point,
lim
n.
n
p
X such that
x
p
→∞
∈
=
Page: 1279-1285
© 2011, IJMA. All Rights Reserved 1282
( )
(
)
(
( )
(
)
)
(
)
(
)
(
( )
)
(
)
(
)
(
( )
)
(
(
)
)
(
)
( )
(
)
(
(
)
)
(
(
)
)
(
( )
)
( )
(
)
(
)
(
)
(
)
2 2
2 2 2 1
2 1 2 1
2 1 2 1 2 1 2 1
2 1 2 1 2 1 2 1 2 1
2
2 1 2 1 2 1
, ,
, ,
1 , , , ,
, , , ,
, , ,
n n
n n
n n n n
n n n n n
n n n
d RSV p x d RSV p TUV x
d p TUV x d x RSV p
d p TUV x d x RSV p d p TUV x d p x
d p RSV p d x TUV x d x TUV x d x RSV p
d RSV p TUV x d p x d p x
α
β
γ
δ
η
+ +
+ +
+ + + +
+ + + + +
+ + +
=
≤
+ +
+ +
+ +
(
)
(
( )
)
(
)
(
( )
)
(
) (
)
( )
(
)
(
)
(
)
(
( )
)
( )
(
)
(
)
(
)
2 2 2 1
2 2 2 1 2 2 2 1
2 1 2 2 2 1 2 2 2 1
2
2 2 2 1 2 1
,
,
1
,
,
,
,
,
,
,
,
,
,
,
n n
n n n n
n n n n n
n n n
d p x
d x
RSV p
d p x
d x
RSV p
d p x
d p x
d p RSV p d x
x
d x
x
d x
RSV p
d RSV p x
d x
p
d x
p
α
β
γ
δ
η
+ +
+ + + +
+ + + + +
+ + +
≤
+
+
+
+
+
+
( )
(
)
(
)
(
( )
)
(
)
(
( )
)
(
) (
)
( )
(
)
(
)
(
)
(
( )
)
( )
(
)
(
)
(
)
2
2
lim
,
,
,
,
1
,
,
,
,
,
,
,
,
,
,
,
making
it n
we have
d p p d p RSV p
d RSV p
p
d p p d p RSV p
d p p d p p
d p RSV p d p p
d p p d p RSV p
d RSV p
p d p p
d p p
α
β
γ
δ
η
→ ∞
≤
+
+
+
+
+
+
( )
(
)
( )
(
)
( )
2, 0
, 0
.
d RSV p p
d RSV p p
which is possible only when RSV p p
≤
≤
=
( )
.Similarly TUV p = p
Now, we shall prove that p is the common fixed point of RSV and TUV. Let q be another common fixed Point of RSV and TUV, then by condition (3.1)(B), we have
(
)
(
( )
( )
)
( )
(
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
(
)
( )
(
)
(
( )
)
(
( )
)
(
( )
)
( )
( )
(
)
(
)
(
)
2 2
2
, ,
, ,
1 , , , ,
, , , ,
, , ,
d p q d RSV p TUV q
d p TUV q d q RSV p
d p TUV q d q RSV p d p TUV q d p q
d p RSV p d q TUV q d q TUV q d q RSV p
d RSV p TUV q d p q d p q
α
β γ
δ η
=
≤
+ +
+ +
+ +
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
)
2,
,
1
,
,
,
,
,
,
,
,
,
,
,
d p q d q p
d p q d q p
d p q d q p
d p p d q q
d q q d q p
d p q d p q
d p q
α
β
γ
δ
η
≤
+
+
+
+
+
+
(
)
2(
) (
)
2,
,
, sin
2
1.
d p q
d p q
which is possible only when p
q
ce
α δ η
α
β
γ
δ η
≤
+
+
=
+
+
+
+
<
Now, we shall show that SV and UV have a common fixed point.
Page: 1279-1285
( )
(
)
(
( )
( )
)
(
( )
( )
)
( )
( )
(
)
(
( )
)
( )
( )
(
)
(
( )
)
(
( )
( )
)
(
( )
)
( )
( )
(
)
(
( )
)
(
( )
)
(
( )
)
( )
( )
(
)
(
( )
)
(
( )
)
2 2 2
2
, , ,
, ,
1 , , , ,
, , , ,
, , ,
d SV p p d SV R SV p T U V p d R SV SV p T U V p
d SV p T U V p d p R SV SV p
d SV p T U V p d p R SV SV p d SV p T U V p d SV p p
d SV p R SV SV p d p T U V p d p T U V p d p R SV SV p
d R SV SV p T U V p d SV p p d SV p p
α
β γ
δ η
= ≤
≤
+ +
+ +
+ +
( )
(
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
(
( )
)
( )
( )
(
)
(
)
(
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
2,
,
1
,
,
,
,
,
,
,
,
,
,
,
d SV p
p d p SV p
d SV p
p d p SV p
d SV p
p d SV p
p
d SV p
SV p
d p p
d p p d p SV p
d SV p
p d SV p
p
d SV p
p
α
β
γ
δ
η
≤
+
+
+
+
+
+
( )
(
)
( )
(
)
(
( )
)
(
( )
)
(
)
(
( )
)
( )
2
2 2
2
2
,
, ,
1 2 ,
,
, sin 2 1.
d SV p p
d SV p p d SV p p
d SV p p
d SV p p
which is possible only when p SV p ce
α
δ
η
α
δ
η
α
β
γ
δ
η
≤ + +
+
≤ + +
= + + + + <
Similarly, on taking x=p, y=UV (p), and applying definition (2.3.2), we have
( )
(
)
(
( )
( )
)
(
( )
( )
)
( )
(
)
(
( )
( )
)
( )
(
)
(
( )
( )
)
(
( )
)
(
( )
)
( )
(
)
(
( )
( )
)
( )
( )
(
)
(
( )
( )
)
( )
( )
(
)
(
( )
)
(
( )
)
2 2 2
2
, , ,
, ,
1 , , , ,
, ,
, ,
, , ,
d p UV p d RSV p U VTU V p d RSV p TUVUV p
d p TU VUV p d UV p RSV p
d p TUVU V p d UV p RSV p d p TUVUV p d p U V p
d p RSV p d UV p TUVUV p
d U V p TUVUV p d U V p RSV p
d RSV p TUVUV p d p UV p d p UV p
α
β
γ
δ η
= ≤
≤
+ +
+
+
+ +
( )
(
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
(
( )
)
(
)
(
( )
( )
)
(
( )
( )
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
2,
,
1
,
,
,
,
,
,
,
,
,
,
,
d UV p
p d p UV p
d UV p
p d p UV p
d UV p
p d UV p
p
d p p d UV p UV p
d UV p UV p
d UV p
p
d UV p
p d UV p
p
d UV p
p
α
β
γ
δ
η
≤
+
+
+
+
+
+
( )
(
)
( )
(
)
(
( )
)
(
( )
)
(
)
(
( )
)
2
2 2
2
2
,
, ,
1 2 ,
,
d U V p p
d U V p p d U V p p
d U V p p
d U V p p
α δ η
α δ η
≤ + +
+
≤ + +
( )
, sin
2
1.
which is possible only when p
=
UV p
ce
α
+
β
+
γ
+
δ
+
η
<
Thus
(3.1.3) SV (p) =UV (p) =p.
Page: 1279-1285
© 2011, IJMA. All Rights Reserved 1284
( )
(
)
(
( )
( )
)
(
( )
( )
)
( )
( )
(
)
(
( )
)
( )
( )
(
)
(
( )
)
(
( )
( )
)
(
( )
)
( )
( )
(
)
(
( )
)
(
( )
)
(
( )
)
( )
( )
(
)
(
( )
)
(
( )
)
2 2 2
2
, , ,
, ,
1 , , , ,
, , , ,
, , ,
d V p p d VRSV p TU V p d RSVV p TU VV p
d V p TU VV p d p RSVV p
d V p TU VV p d p RSVV p d V p TU VV p d V p p
d V p RSVV p d p TU VV p d p TU VV p d p RSVV p
d RSVV p TU VV p d V p p d V p p
α
β γ
δ η
= ≤
≤
+ +
+ +
+ +
( )
(
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
(
( )
)
(
)
(
( )
( )
)
(
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
2, ,
1 , , , ,
, , , ,
, , ,
d V p p d p V p
d V p p d p V p d V p p d V p p
d p p d V p V p d p p d V p p
d V p p d V p p d V p p
α
β γ
δ η
≤
+ +
+ +
+ +
( )
(
)
( )
(
)
(
( )
)
(
( )
)
(
)
(
( )
)
( )
2
2 2
2
2
,
, ,
1 2 ,
,
, sin 2 1.
d V p p
d V p p d V p p
d V p p
d V p p
which is possible only when p V p ce
α δ η
α δ η
α β γ δ η
≤ + +
+
≤ + +
= + + + + <
Similarly, on taking x=p, y=U(p) and applying definition (2.3.4), we have p = U(p). Thus
(3.1.4) U (p) = V (p) = p. Hence by (3.1.3) and (3.1.4), we have SV (p) = UV (p) =U (p) =V (p) = S (p) =p. Also by the uniqueness of p,
RSV (p) =TUV (p) =R(p) =T(p) =p.
Therefore, R(p) =S(p) =T(p) =U(p) =V(p) =p. Hence p is a common fixed point of R, S, T, Uand V.
Uniqueness: Let w be another common fixed point of R, S, T, U and V. Then by condition (3.1) (B), we have
(
)
(
( )
( )
)
( )
(
)
(
( )
)
( )
(
)
(
( )
)
(
( )
)
(
)
( )
(
)
(
( )
)
(
( )
)
(
( )
)
( )
( )
(
)
(
)
(
)
2 2
2
, ,
, ,
1 , , , ,
, , , ,
, , ,
d p w d RSV p TU V w
d p TU V w d w RSV p
d p TU V w d w RSV p d p TU V w d p w
d p RSV p d w TU V w d w TU V w d w RSV p
d RSV p TU V w d p w d p w
α
β γ
δ η
=
≤
+ +
+ +
+ +
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
) (
)
(
)
2, ,
1 , , , ,
, , , ,
, , ,
d p w d w p
d p w d w p d p w d w p
d p p d w w d w w d w p
d p w d p w d p w
α
β γ
δ η
≤
+ +
+ +
Page: 1279-1285
(
)
(
)
(
)
(
)
(
) (
)
2
2 2
2
2
,
,
,
1 2
,
,
d p w
d p w
d p w
d p w
d p w
α
δ
η
α
δ η
≤
+
+
+
≤
+
+
,sin
2
1.
which is possible only when p
=
w
ce
α
+
β
+
γ
+
δ η
+
<
This completes the proof of the theorem. 4. REFERENCE:
[1] Bhardwaj, R. K., Rajput,S. S., Yadava, R. N., “Application of fixed point theory in metric spaces,” Thai Journal of Mathematics, Vol.5, No. 2 (2007), 253-259.
[2] Fisher, B., “Common fixed point and constant mapping on metric space,” Math. Sem. Notes, Kobe Univ., 5 (1979), 319.
[3] Hadzic, O., “On coincidence points in Convex metric spaces,” Univ. u Novom Sadu Zb. Rad. Prirod-Mat. Fak. Ser. Mat., 19 (2) (1986), 233-240.
[4] Hadzic, O.and Gajic, Lj., “ Coincidence points for set-valued mappings in Convex metric spaces,” Univ. u Novom Sadu Zb. Rad. Prirod-Mat. Fak. Ser. Mat., 16 (1) (1986), 13-25
[5] Jaggi, D. S., “Some fixed point theorems”, I.J.P. Appl., 8(1977), 223-230.
[6] Jaggi, D. S. and Das, B. K., “An extension of Banach’s fixed point theorem through rational expression,” Bull. Cal. Math. Soc., 72(1980), 261-264.
[7] Jungck, G., “Commuting mappings and fixed points,” Amer. Math. Monthly, 83 (1976), 261-263. [8] Kannan, R., “Some generalization of the Banach contraction theorem,” Kobe, VIII, Vol.3, (1975), 1-11.
[9] Pathak, H. K., “A note on fixed point theorems of Rao and Rao,” Bull. of Cal. Math. Soc., 79 (1987), 267-273. [10] Rao,I.N.N. and Rao, K. P. R., “ Common fixed point theorem for three mappings,” Bull. of Cal. Math. Soc., 76 (1984), 228.
[11] Rhoades, B. E., “A comparison of various definitions of contractive mappings,” Trans. Amer. Math. Soc., 226(1976), 257-290
[12] Saluja, A. S. and Dehariya, R. D., “Fixed point theorem for five weakly commuting mappings,” J. Nanabha, Vol. 36 (2006), 119-124.
[13] Sessa, S., “On a weak commutativity condition of mappings in fixed point consideration,” Publ. Inst. Math., 32 (46) (1982), 149-153.
[14] Yadava, R. N., Rajput, S. S., Bhardwaj, R. K., “Some fixed point theorems in complete metric spaces,” Acta Ciencia India, Vol. XXXIIIM, NO. 2 (2007), 461-466.
[15] Yadava, R.N., Rajput, S. S., Bhardwaj, R. K., “Some fixed point and common fixed point theorems in Banach spaces,” Acta Ciencia India, Vol. XXXIIIM, NO. 2 (2007), 453-460.
