Applied Mathematics, 2011, 2, 595-599
doi:10.4236/am.2011.25079 Published Online May 2011 (http://www.SciRP.org/journal/am)
Common Fixed Points of Single and Multivalued Maps in
Fuzzy Metric Spaces
Palaniappan Vijayaraju1, Zahurullah Mohamed Ibrahim Sajath2
1Department of Mathematics, Anna University, Chennai, India
2Department of Information Techn ology, Higher Coll ege of Tech nology, Musc at , Sul t a n ate of Om a n
Received February 24, 20 1 1; revised March 25 , 20 11; accepted March 29, 2011
Abstract
In this paper we introduce the notion of common property (EA) in fuzzy metric spaces. Further we prove
some common fixed points theorems for hybrid pair of single and multivalued maps under hybrid contractive
conditions. Our results extend previous ones in fuzzy metric spaces.
Keywords: Fuzzy Metric Space, Common Fixed Point, Coincidence Point
1. Introduction
In 1965 Zadeh [1] introduced the theory of fuzzy sets.
Many authors introduced the notion of fuzzy metric
space in different ways. George and Veeramani [2] modi-
fied the concept of fuzzy metr ic space introduced by K ra-
mosil and Michalek [3] and defined Haussdorf topology
in fuzzy metric space. Several authors [4-11] studied and
developed the concept in different directions and proved
fixed point theorems in fuzzy metric spaces.
In 1986 Jungck [12] introduced the concept of compa-
tible mappings and utilized it to improve and generalize
the commutativity cond itions emplo yed in common fixed
point theorems. This induced interest in non-compatible
mappings initiated by Pant [13]. Recently Aamri and
Moutawakil [14] and Liu et al. [15] respectively defined
the property (E.A) and the common property (E.A) as a
generalization of non-compatibility and proved some
common fixed point theorems in metric spaces. The aim
of this paper is to define the common property (E.A) in
the settings of fuzzy metric space and utilize the same to
obtain some common fixed point theorems in fuzzy me-
tric spaces.
We begin with some definitions and preliminary con-
cepts.
2. Preliminaries
Definition 2.1. [16] A binary operation




*: 0,10,1


0,1 is called a continuous -norm if ([0,1],*) is an
abelian toplological monoid with unit 1 such that
t
**ab cd

whenever ac

and for all bd


,,,abcd
*=ab
0,1.
Examples of -norm areand t* =ab ab


min ,ab .
Definition 2.2. [3]. A triplet is sa id to b e a
fuzzy metric space if

,,*XM

X
is an arbitrary set, * is a
continuous -norm, and t
M
is a fuzzy set on


20,X


,, satisfying the following conditions: for all
x
yz X

and all , ,>0st
1)


,,0y=0Mx ;
2)


,,yt=1Mx for all if and only if >0t=
x
y;
3)




,,,, =
M
xytM yxt;
4)




*

,, ,,,,
M
xytM yMszsxzt;
5)






,,.y:0, 10,Mx is left continuous;
6)


,,yt=1
limtMx.
M
is called fuzzy metric on
X
. The functions


,,
M
xyt denote the degree of nearness between
x
and with respect to respectively.
yt
Definition 2.3. Let


*,,XM be a f u z z y met r i c space.
A sequence


n
x
in
X
is called Cauchy sequence
if and only if


,,
n
xt=1
limn np
Mx
 for each ,
. >0p >0tA sequence


n
x
in
X
is converging to
x
in
X
if and only if


=
nt  ,,
n
Mxx 1
lim .
A fuzzy metric space


*,,XM is said to be com-
plete if and only if every Cauchy sequence in
X
ison-
vergent in c
X
.
Definition 2.4. [8] Let

X
 denote the set of all
nonempty closed bounded subsets of
X
. Then for ev ery


,,
A
BC X and , >0t

P. VIJAYARAJU ET AL. 596


 

,, =
min min,,,min,,
aA bB
MABt

M
aBtM Abt



where


 
,,=max,, :
M
CytM zytzC
.
Remark 2.5. Obviously



,, ,,
M
ABtM aBt



,, =1Bt whenever and if and only if aAMA

=
A
B.
Definition 2.6. [9] Two mappings
f
and
g
are
compatible if and only if

g
 
,,t=1
limnnn
Mfgx fx


n
for all , whenever

>0t
x
is a sequence in
X
such that
 
==
llim im
nn
nno
f
xg
 xx  
,X. Lemma 2.7. Let
f
g be two compatible mappings
on
X
. If

=

f
xgx for some
x
in
X
, then

=

f
gx gfx.
Definition 2.8. [10,11] Maps ,:
f
gX X are said
to satisfy the property


EA if there exists a sequence

n
x
X such that ==
lim nn
n
limn
f
xgxtX
 

.
Definition 2.9. [11] A point in
X
is a coincidence
point (fixe d point) of
f
and T if
  
===
f
xTxTx fxx
.
Definition 2.10. [11] A point
x
in
X
is a coin-
cidence point of :
f
XX and


:TX X if
 
f
xTx. We denote the set of all coincidence
points of
f
and T by .

,T
:
Cf
Definition 2.11. [16] Maps
f
XX and
are weakly compatible if they com-
mute at their coincidence points,that is, if

:TX X
=
f
TxTfx
whenever
f
xTx

.
Definition 2.12. [18] Maps :
f
XX and


:TX X are said to be commuting at


x
X

if
f
Tx Tfx whenever
f
xTx
:. Definition 2.13. [19] Maps
f
XX and


:TX X are said to satisfy the property


EA
if there exists a sequence

n
x
X,some tX

and


A
X such that ==
lim lim
nn
nn
f
xtA
:TX
Tx
 

.
Definition 2.14. [19] Let . The map

X
:
f
XX is said to be weakly commuting at T
x
X

if
f
fx Tfx

.
3. Main Results
We begin with the following definition.
Definition 3.1. [11] Let be a fuzzy metric
space and

,,*XM

,., :
f
gFGXX. The maps pair


,
f
F
and


,
g
G are said to satisfy the common property


EA if there exist two sequences

,
nn
x
yX

and
some in t
X
such that
=
lim
===
limlim lim
nn
nnn
nnn
Gy
F
xgyfxt

X

.
Definition 3.2. Let


,,*XM be a fuzzy metric
space, ,:
f
gX X and


,:
F
GX X . The
maps pair


,
f
F and


,
g
G are said to satisfy the
common property


EA if there exist two sequences




nn
,
x
yX

,some tX

and


,
A
BX
such
that
=,=, ==
limlimlim lim
nnnn
nnnn
F
xA GyBfxgytAB
 

.
Theorem 3.3. Let ,
f
g

,,XM
be two self maps of the
fuzzy metric space and let 1)


,
f
F and


,
g
G satisfy the common property


EA ;

*,
F
G be two
maps from
X
into

X
 such that 2) for all
x
y

in
X
,
 
 
,,,, ,,,,
,,>min,,,
22
nn
M
fxFxtMgy Gy tMfx Gy tMgyFx t
MFxGwtMfxgwt ,












If and
fX
g
X are closed in
X
, then
1) and
f
F
have a coincidence point;
2)
g
and have a coincidence point; G
3) and
f
F
have a common fixed point provided
that is
f
F
weakly commuting at v and
for ; =ffvfv

F,fvC
4)
g
and have a common fixed point provided
that G
g
is G weakly commuting at v and =
g
gvgv
for ;

G,g
g
vC
5) and G have a common fixed point pro-
vided that both and

are true.
,,fF

c

d
Proof. Since


,
f
F and

,

g
G satisfy the com-
mon property


EA , there exist two sequences


n
x
and


n
y
in
X
, uX

and ,

A
BXh that
limn
 suc
=
n
F
xA
 and limn=
n
GyB ,

==
lim lim nn
nn
fxgyu A B
 
By virtue of


f
x and

g
x being closed, we have
and for some . Now we shall
show that
=ufv =ugw
Fv ,vw X fv

and
g
wGw. The condition (2)
implies that
 

 
,, ,,,, ,,
,,>min ,,,,
22
nnn n
nn
M
fxFxtMgwGw tMfxGwtMgwFxt
MFx GwtMfxgwt













P. VIJAYARAJU ET AL. 597
Taking the limit as , we get n
 
  
,,, ,, ,,,
,, min,,,,
22
M
fvAtM gwGwtMfvGwtM gwAt
MA Gw tMfvgw t
  












Since , we obtain
==ufvgw
  
,, ,,,,,,
,, min1,,
22
M
uAtMuGwtMuGwtMuAt
MAGwt
 







That is,
 

1,
,, 2
,
M
uGwt
MAGwt



, since uA

.
 

2,,1 ,,

M
AGwtM uGwt


 (3)
But from (2.5), we have




,, ,,
M
AGwtMuGwt


That is

2,,2,,

M
AGwtM uGwt

 (4)
Combining the inequalities (3) and (4) we get




1,,2,,2,

,
M
uGwtM AGwtM uGwt



This implies


1,,2,

,
M
uGwtM uGwt




,,MuGwt


=ugwGw
1

,,=MuGwt

 1
Hence

On the other hand by condition (2), we have
 
 
,,,,,, ,,
,,>min ,,,,
22
nnn n
nn
M
fvFvtMgy Gy tMfvGy tMgyFvt
MFvGyt Mfvgyt
 

 

 
 
 

Taking limit as , we get n
 

, ,,,,,, ,
,,min(, ,),
22
M
fvFvtMgwBtMfvBtMgwFvt
MFvBt Mfvgwt,













That is,

 
,,
,, 11,,
min 1,,
22
MFvBt
M
fvFv tMgw Fv t









Similarly, we obtain

,,=1MfvFvt
,which implies that . Thus
and fv Fvf
F
have a coincidence point v.
g
and G have
a coincidence point . This ends the proof of (a) and
w
(b).
By virtue of condition (c), we get . Thus ffv Ffv
=uf uFu

. This proves (c). Similarly (d) can be
proved. Then (e) follows immediately.
Corollary 3.4. Let be a self-map of fuzzy metric
space f


,,*XM and let
F
be a map from
X
into


X
 such that
1)


,
f
F satisfies the property ;

EA
2) for all
x
y

in
X
,
 
 
,, ,,,, ,,
,,>min,,,,
22
M
fxFxtMfyFy tMfxFy tMfyFxt
MFxFytMfx fyt












If is closed subset of
fX
X
, then
1) and
f
F
have a coincidence point;
2) and f
F
have a common fixed point provided
that is f
F
weakly commuting at v and ,
for . =ffv fv

F
=
,fvC
Proof. Let
F
Ga
Corollary 3.5. Let be a self-map of the fuzzy
metric space f


*,,XM and let
F
and be two
maps from G
X
into


X
 such that
1)


,
f
F and


,
f
G satisfy the common property


EA ;
n thed fn the result follows. =g,2) for all
x
y

in
X
.
 
 
,, ,,,, ,,
,,>min ,,,,
22
M
fxFxtMfy Gy tMfx GytMfyFx t
MFxGyt Mfxfyt













P. VIJAYARAJU ET AL. 598
If is closed subset of fX
X
, then
1) and ,fG
F
have a coicidence point;
2) and n ,fG
F
have a common fixed point provided
that f is both
F
weakly commuting and
weakly commuting at v =ffvfv
If both
Gand , for

.

,fF
of. L
vC
Pro et =fg, then the result follows.
F
and are single valued maps in theorem
3.3, then we have the following corollary.
Corollary 3.6. Let and be four self-
maps of the fuzzy
G
,,fgFG

,,*
metric space XM such that
1)


,
f
F and


,
f
G satisfy the common property


EA ;
2) for all
x
y

in
X
,
 
 
,,,, (,,)(, ,) gyGt
,,>min ,,,,
22
fxFxtMyM
M
fx Gy
MFxGyt MfxgyttMgyFx t















If and

fX
g
X are closed subsets of
X
, then
1) and f
F
haThe next theorem involves a continuousnction
ve a coincidence point;
2)
g
and have a coincidence point;
and G 3) f
F
hapoint d
th ve a common fixed provide at is
f
F
kly commuting at v d =ffv fvwea an
r

,vCfF; fo 
4)
g
and have a common fixed point pGrovided
that
g
is G weakly commuting at v and =
g
gvgv
for

vC ;
5) nd
,gG
g a

,, G have a common fixed point
provid that both (c) and (d) are true.
fF
ed
fu




:0,1 0,1   satisfying the following conditions:


A1

is nonincreasing on


0,1 ,




2>
A
tt 
for each

0,1t.
. Let ,fg be two self maps of Theorem 3.7the
fuzzy metric space


,,*XM ,
F
G and let be two
maps from
X
into


X
 th suchat
1)


,
f
F and


,
g
G satisfy the common property


EA ;
2) for all
x
y

in
X
,
 



,min,,,, ,,,,,,,,, ,



,
M
FytMfxgy tMfxFxtMGytMfxGytMgyFxt 

 x Ggy
If fX and
g
X are closed in
X
, then
1) and f
F
have a coincidee point;
2) nc
g
and have a coincidence point;
and G 3) f
F
haiprovided
th ve a common fxed point at is
f
F
kly commuting at v andwea
r

,;vCfF fo =ffvfv 
4)
g
and have a common fixed point pGrovided
that
g
is G weakly commuting at v and =
g
gvgv
for

;vCg
5) d
,G
g an

,,G have a common fixed point
provid that both

c and
fF
ed


d are true.
Proof. Since


,
f
F and

,
g
G satisfy the com-
mon property


EA , there exist two sequences


n
x
and


n
y
in
X
, uX

and

,
A
BX such that
=
nn
lim
F
xA
 and n
==
lim lim
nn
n
gyu A B
 .
Bye of
=
n
GyB ,
lim
n
fx



f
x and

g
virtu x e have
=u for sobeing closed, w and . we claim that fv
fv Fv=ugwme ,vw X

and
g
wGw

. Indeed condition (2) implies
that
 

 

,, ,
nn



, ,, ,,,,
nn
,min, ,,,
nn
,
M
Fx GwtMgwMfx

tMfxgw tMfxFx
 Gw tGw tMgwFxt

Taking limit as , we obtain

n
 



 
,,min,,,,,,,,,,,,,,
M
AGwtMfvgwt MfvAtMgwGwt MfvGwt MgwAt 
 

Since =fvgw, we get A B

,,= ,,=,,=1Mfvgwt MfvAt MgwAt

.
Therefore
 


,,,,>,,
M
AGwtM fvGwtMfvGwt 

.


,,> ,,

.
M
AGwtM fvGwt

 That is
This contradicts 2.5 and hence 1 . Thi s
implies that

,,=MAGwt

=
A
Gw . Therefore
On the other hand by condition (2) again, we have

n
=.fvgw Gw




 
,, min(,,),(,,),,,,,,,,
nn nnn
M
Fv
Gy tMfvgy t MfvFvt,Mgy Gy tMfvGy tMgyFvt

 


P. VIJAYARAJU ET AL. 599
, we obtain Similarly, taking limit as n


,,FvBt
=1M
=.
F
vB Thus, we get
=fvgwB,
f and fv Fv.
F
have a coinci
Since
Thus v,
g
dence point and
t Tof
part (a) and part (b). The rest of prf is similar to the
arrem 2.3
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