Description
In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is a point[1] that is mapped to itself by the function.
Fixed Point Result in Probabilistic Metric Space
ABSTRACT
In this paper we prove common fixed point theorem for four mapping with weak compatibility in probabilistic
metric space.
Keywords: Menger space, Weak compatible mapping, Semi-compatible mapping, Weakly commuting mapping,
common fixed point.
AMS Subject Classification: 47H10, 54H25.
1. INTRODUCTION:
Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis,
which is a very dynamic area of mathematical research. The notion of probabilistic metric space is introduced by
Menger in 1942 [9] and the first result about the existence of a fixed point of a mapping which is defined on a Menger
space is obtained by Sehgel and Barucha-Reid.
Recently, a number of fixed point theorems for single valued and multivalued mappings in menger
probabilistic metric space have been considered by many authors [1],[2],[3],[4],[5],[6]. In 1998, Jungck [7] introduced
the concept weakly compatible maps and proved many theorems in metric space. In this paper we prove common fixed
point theorem for four mapping with weak compatibility and rational contraction without appeal to continuity in
probabilistic metric space. Also we illustrate example in support of our theorem.
2. PRELIMINARIES:
Now we begin with some definition
Definition 2.1: Let R denote the set of reals and the non-negative reals. A mapping : ÷ is called a
distribution function if it is non decreasing left continuous with
i
t
nf
R
F (t ) = 0 and sup F (t ) =1 e
Definition 2.2: A probabilistic metric space is an ordered pair ( , ) where X is a nonempty set, L be set of all te R
distribution function and : × ÷ . We shall denote the distribution function by ( , ) or
,
; , e
and
,
( ) will represents the value of ( , ) ate . The function ( , ) is assumed to satisfy the following conditions:
1.
,
( ) = 1 >0 ! =
2.
,
(0) = 0 #$# ! , e
3.
,
= , #$# ! , e
4.
,
( ) = 1 ,'(!) = 1 (?# ,' ( + ! ) = 1 #$# ! , , e .
In metric space ( , ) , the metric d induces a mapping :× ÷
,
( )=
,
= + ( - ( , )) for every , e
ande , where H is the distribution function defined as such that
+( ) = -0,,iiff x s 02 1 x>0
Definition 2.3: A mapping- : [0, 1] [0, 1] ÷ [0, 1] is called t-norm if
1.
2.
( - 1) = ¬e [0,1]
(0 - 0) = 0, ¬ , 7 e [0,1]
3. ( - 7) = (7 - ),
4. (8 - ) > ( - 7 ) 8 > , > 7, and
5. ( ( - 7) - 8 ) = ( - (7 - 8 ))
Example: (i) ( - 7) = 7,
(ii) ( - 7) = : ( , 7)
(iii) ( - 7) = : ( + 7 ÷ 1; 0)
following condition Definition
2.4: A Menger space is a
triplet ( , ,-) where ( , )a PM-
space and ? is is a t-norm with the
The above inequality is called Menger's triangle inequality. ( ) ->,=(!)
EXAMPLE: Let = , ( - 7) = : ( , 7) , 7 e (0,1) and
259
(
) = - +( )
? = $2
1 0 ?=$
where +( ) = A 1
0 ss 12 s0
>1
Then ( , ,- ) is a Menger space.
Definition 2.5: Let ( , ,-) be a Menger space. If ? e , B > 0, C e (0, 1), then an (B, C) neighbourhood of u,
denoted by D
<
(B, C) is defined as
D
<
(B, C) = E$ e ;(B) > 1 ÷ CF.
If ( , ,-) be a Menger space with the continuous t-norm t, then the familyD
<
(B, C); ? e ; B > 0, C e (0,1) of
neighbourhood induces a hausdorff topology on X and if supJKL(a - a) = 1, it is metrizable.
Definition 2.6: A sequence N
O
P in ( , ,-) is said to be convergent to a pointe if for every B > 0 and
C > 0, there exists an integer Q = Q(B, C) such that
O
e D (B, C) for all> Q or equivalently
R
S,R(T) >
1 ÷ C for all> Q.
Definition 2.7: A sequence N
O
P in ( , ,-) is said to be Cauchy sequence if for every B > 0 and C > 0, there
exists an integer Q = Q(B, C) such that
S
, U(T) > 1 ÷ C for all , : > Q.
sequence in X converges to a point in X.
Definition 2.8: A Menger space ( , ,-) with the
continuous t-norm ? is said to be complete if every Cauchy
Definition 2.9: A coincidence point (or simply coincidence) of two mappings is a point in their domain having
the same image point under both mappings.
Formally, given two mappings , V ?÷ X we say that a point x in X is a coincidence point of f and g if
( ) = V ( ).
Definition 2.10: Let ( , ,-) be a Menger space. Two mappings , V ?÷ are said to be weakly compatible
if they commute at the coincidence point, i.e., the pair N , VP is weakly compatible pair if and only if =V
implies that V = V .
Example: Define the pair Y, Z: [0, 3] ÷ [0, 3] by
Y( ) = - 3,,
e [0, 1)2
e [1, 3] ,
Z( ) = -33÷ , ,
e [0, 1)2
e [1, 3].
Then for anye [1, 3], YZ = ZY , showing that A, S are weakly compatible maps on [0, 3].
Definition 2.11: Let ( , ,-) be a Menger space. Two mappings Y, Z ?÷ are said to be semi compatible if
[\RS,\R(() ÷ 1for all ( > 0 whenever N
O
P is a sequence in such that Y
O
, Z
O
÷ for some p in as÷·.
It follows that (Y, Z) is semi compatible and Y! = Z! imply YZ! = ZY! by taking N
O
P = ! = Y ! = Z! .
Lemma 2.12[15]: Let N
O
P be a sequence in Menger space ( , ,-) where- is continuous and ( - ) > for
alle [0, 1]. If there exists a constant ] e (0, 1) such that> 0 ande Q
S
, S^_ (] ) > S`_, S ( ), then
N
O
P is a Cauchy sequence.
Lemma 2.13[13]: If ( , ) is a metric space, then the metric d induces a mapping : × ÷ , defined by
( , ) = + ( - ( , )) , , e e . Further more if-: [0,1] × [0,1] ÷ [0,1] is defined by
( - 7) = : ( , 7), then ( , ,-) is a Menger space. It is complete if ( , ) is complete. The space ( , ,-)
so obtained is called the induced Menger space.
Lemma 2.14[10]: Let ( , ,-) be a Menger space. If there exists a constant ] e (0, 1) such that
R,a
(]() >
R,a((), for all , ! e and ( > 0 then = ! .
3.
Let A, B, T and S be mappings from X into itself such that Theorem
3.1: Let ( , ,-) be a complete Menger space where- is continuous and (( - () > (
for all ( e [0,1]. MAIN RESULT:
3 .1 .1 . Y ( ) c Z ( ) c( ) c d( )
3 .1 .2 . Z d are continuous
3.1.3. The pair (Z, Y) and (d, c) are Semi compatible
3.1.4. There exists a number ] e (0,1) such that
[R,ea (]() > \R,fa(() - \R,[R(() - [R,fa(() - fa,ea(() - \R,ea g(2 ÷ h)(i
, ! e , h e (0,2) ( > 0.
Then, Y, c, Z and d have a unique common fixed point in X.
Proof: Since Y( ) c Z( ) for any
j
e there exists a point
L
e such thatY
j
= Z L . Since c( ) c
d( ) for this point
L
we can choose a point
k
e such that d
L
= c
k
.
Inductively we can find a sequence N!
O
P as follows
!kO = Y kO = Z kO L
!kO
L
= c kO
L
= d kO k
260
For = 0, 1, 2, 3 .... by (3.1.4.), for all ( > 0 h = 1 ÷ withe (0,1), we have
amS,amS^_ (]() = [RmS ,eRmS^_ (]()
> \RmS,fRmS^_(() - \RmS^_,[RmS^_(() - [RmS,fRmS^_(() - fRmS^_,eRmS^_(() - \RmS,eRmS^_g(1 + )(i
= amS`_,amS(() - amS,amS^_(() - amS,amS^_(() -
amS,amS^_ (() - amS`_,amS^_g(1 + )(i
> amS`_,amS(() - amS,amS^_(() - amS`_,amS(() -amS,amS^_( ()
= amS`_,amS(() -amS,amS^_(() -amS,amS^_( ()
Since t-norm is continuous, letting÷ 1, we have
amS,amS^_(]() > amS`_,amS(() - amS,amS^_ (()
Similarly
amS^_,amS^m(]() > amS,amS^_(() - amS^_,amS^m(()
Similarly
amS^m,amS^n(]() > amS^_,amS^m (() - amS^m,amS^n(()
Therefore
aS,aS^_ (]() > aS`_,aS(() - aS,aS^_ (()
eo
Consequently aS,aS^_ (() > aS`_,aS(] () -, aS,aS^_ (] ()
pL pL
eo
Repeated application of this inequality will imply that
aS,aS^_ (() > aS`_,aS (] () - aS,aS^_ (] () >?... . > aS`_,aS (] () - aS,aS^_ (] (), e o
pL pL pL pr
Since
a
S,aS^_(] pr ( ) ÷ 1 s ÷·, it follows that
aS,aS^_ (() > aS`_,aS(] pL ( ) eo
Consequently
aS,aS^_ (]() > aS`_,aS(() for all
eo
Therefore by Lemma [2.12], N!
O
P is a Cauchy sequence in X. Since X is complete,N!
O
Pconverges to a point t e
z,
. Since NY kOP, Nc kO
L
P, NZ kO
L
P Nd kO
k
P are subsequences of N!
O
P , they also converge to the point
Case I: Since S is continuous. In this case we have . # .
as÷· , Y k
O
, c k O
L
, Z k O
L
d k O
k
÷ t .
ZY
O
÷ Z t , ZZ
O
÷ Zt
Also (Y, Z)is semi-compatible, we have YZ
O
÷ Zt
Step I: Let = Z
O
, ! =
O
u (? h = 1 in (3.1.4) we get
[\RS,eRS (]() > \\RS,fRS (() - \\RS,[\RS(() - [\RS,fRS (() - fRS,eRS (() - \\RS,eRS (()
\v,v(]() > \v,v(() - \v,\v(() - \v,v(() - v,v(() - \v,v(()
\v,v(]() > \v,v(()
So we get Zt = t.
Step II: By putting = t, ! =
O
u (? h = 1 in (3.1.4) we get
[v,eRS(]() > \v,fRS(() - \v,[v(() - [v,fRS (() - fRS,eRS(() - \v,eRS (()
[v,v(]() > v,v(() - v,[v(() - [v,v(() - v,v(()- v,v(()
[v,v(]() > [v,v(()
Case II: Since T is continuous. In this case we have So
we get Yt = t.
dc
O
÷ dt,
dd
O
÷ dt
Also (c, d)is semi-compatible, we have cd
O
÷ dt
Step I: Let =
O
, ! = d
O
u (? h = 1 in (3.1.4) we get
[RS,efRS (]() > \RS,ffRS (() - \RS,[RS (() - [RS,ffRS (() - ffRS,efRS (() - \RS,efRS (()
v,fv(]() > v,fv(() - v,v(() - v,fv(() - fv,fv(() - v,fv(()
v,fv(]() > v,fv(()
So we get dt = t.
Step II: By putting =
O
, ! = t u (? h = 1 in (3.1.4) we get
[RS,ev(]() > \RS,fv(() - \RS,[RS (() - [RS,fv(() -fv,ev(() - \RS,ev(()
v,ev(]() > v,v(() - v,v(() - v,v(() - v,ev(() - v,ev (()
ev,v(]() > ev,v(()
So we get ct = t.
Thus, we have Yt = Zt = dt = ct = t.
261
That is z is a common fixed point of Z, d, Y and c.
For uniqueness, let u (u = t) be another common fixed point of Z, d, Y and c .Then Yu = Zu == cu =
du = u .
Put = t, ! = u and w = 1, in (3.1.4.), we get
[v,e=(]() > \v,f=(() - \v,[=(() - [v,f=(() - f=,e=(() - \v,e= (()
v,=(]() > v,=(() - v,=(() - v,=(() - =,=(() - v,=(()
v,=(]() > v,=(() - v,=(() - v,=(() - 1 - v,=(()
v,=(]() > v,=(()
Thus we havet = u. Therefore z is a unique fixed point of A,S, B and T. This
completes the proof of the theorem.
COROLLARY 3.2: Let ( , ,-) be a complete Menger space where- is continuous and (( - () > ( for
all ( e [0,1]. Let A, and S be mappings from X into itself such that
3 .2 .1 . Y ( ) c Z ( )
3.2.2. Z is continuous
3.2.3. The pair (Z, Y) is semi compatible
3.2.4. There exists a number ] e (0,1) such that
[R,\a(]() > \R,\a(() - \R,[R(() - [R,\a(() - \a,[a (() - \R,[ag(2 ÷ h)(i
, ! e , h e (0,2)
Then, Y, and Z have a unique common fixed point in X.
doc_936901989.docx
In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is a point[1] that is mapped to itself by the function.
Fixed Point Result in Probabilistic Metric Space
ABSTRACT
In this paper we prove common fixed point theorem for four mapping with weak compatibility in probabilistic
metric space.
Keywords: Menger space, Weak compatible mapping, Semi-compatible mapping, Weakly commuting mapping,
common fixed point.
AMS Subject Classification: 47H10, 54H25.
1. INTRODUCTION:
Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis,
which is a very dynamic area of mathematical research. The notion of probabilistic metric space is introduced by
Menger in 1942 [9] and the first result about the existence of a fixed point of a mapping which is defined on a Menger
space is obtained by Sehgel and Barucha-Reid.
Recently, a number of fixed point theorems for single valued and multivalued mappings in menger
probabilistic metric space have been considered by many authors [1],[2],[3],[4],[5],[6]. In 1998, Jungck [7] introduced
the concept weakly compatible maps and proved many theorems in metric space. In this paper we prove common fixed
point theorem for four mapping with weak compatibility and rational contraction without appeal to continuity in
probabilistic metric space. Also we illustrate example in support of our theorem.
2. PRELIMINARIES:
Now we begin with some definition
Definition 2.1: Let R denote the set of reals and the non-negative reals. A mapping : ÷ is called a
distribution function if it is non decreasing left continuous with
i
t
nf
R
F (t ) = 0 and sup F (t ) =1 e
Definition 2.2: A probabilistic metric space is an ordered pair ( , ) where X is a nonempty set, L be set of all te R
distribution function and : × ÷ . We shall denote the distribution function by ( , ) or
,
; , e
and
,
( ) will represents the value of ( , ) ate . The function ( , ) is assumed to satisfy the following conditions:
1.
,
( ) = 1 >0 ! =
2.
,
(0) = 0 #$# ! , e
3.
,
= , #$# ! , e
4.
,
( ) = 1 ,'(!) = 1 (?# ,' ( + ! ) = 1 #$# ! , , e .
In metric space ( , ) , the metric d induces a mapping :× ÷
,
( )=
,
= + ( - ( , )) for every , e
ande , where H is the distribution function defined as such that
+( ) = -0,,iiff x s 02 1 x>0
Definition 2.3: A mapping- : [0, 1] [0, 1] ÷ [0, 1] is called t-norm if
1.
2.
( - 1) = ¬e [0,1]
(0 - 0) = 0, ¬ , 7 e [0,1]
3. ( - 7) = (7 - ),
4. (8 - ) > ( - 7 ) 8 > , > 7, and
5. ( ( - 7) - 8 ) = ( - (7 - 8 ))
Example: (i) ( - 7) = 7,
(ii) ( - 7) = : ( , 7)
(iii) ( - 7) = : ( + 7 ÷ 1; 0)
following condition Definition
2.4: A Menger space is a
triplet ( , ,-) where ( , )a PM-
space and ? is is a t-norm with the
The above inequality is called Menger's triangle inequality. ( ) ->,=(!)
EXAMPLE: Let = , ( - 7) = : ( , 7) , 7 e (0,1) and
259
(
) = - +( )
? = $2
1 0 ?=$
where +( ) = A 1
0 ss 12 s0
>1
Then ( , ,- ) is a Menger space.
Definition 2.5: Let ( , ,-) be a Menger space. If ? e , B > 0, C e (0, 1), then an (B, C) neighbourhood of u,
denoted by D
<
(B, C) is defined as
D
<
(B, C) = E$ e ;(B) > 1 ÷ CF.
If ( , ,-) be a Menger space with the continuous t-norm t, then the familyD
<
(B, C); ? e ; B > 0, C e (0,1) of
neighbourhood induces a hausdorff topology on X and if supJKL(a - a) = 1, it is metrizable.
Definition 2.6: A sequence N
O
P in ( , ,-) is said to be convergent to a pointe if for every B > 0 and
C > 0, there exists an integer Q = Q(B, C) such that
O
e D (B, C) for all> Q or equivalently
R
S,R(T) >
1 ÷ C for all> Q.
Definition 2.7: A sequence N
O
P in ( , ,-) is said to be Cauchy sequence if for every B > 0 and C > 0, there
exists an integer Q = Q(B, C) such that
S
, U(T) > 1 ÷ C for all , : > Q.
sequence in X converges to a point in X.
Definition 2.8: A Menger space ( , ,-) with the
continuous t-norm ? is said to be complete if every Cauchy
Definition 2.9: A coincidence point (or simply coincidence) of two mappings is a point in their domain having
the same image point under both mappings.
Formally, given two mappings , V ?÷ X we say that a point x in X is a coincidence point of f and g if
( ) = V ( ).
Definition 2.10: Let ( , ,-) be a Menger space. Two mappings , V ?÷ are said to be weakly compatible
if they commute at the coincidence point, i.e., the pair N , VP is weakly compatible pair if and only if =V
implies that V = V .
Example: Define the pair Y, Z: [0, 3] ÷ [0, 3] by
Y( ) = - 3,,
e [0, 1)2
e [1, 3] ,
Z( ) = -33÷ , ,
e [0, 1)2
e [1, 3].
Then for anye [1, 3], YZ = ZY , showing that A, S are weakly compatible maps on [0, 3].
Definition 2.11: Let ( , ,-) be a Menger space. Two mappings Y, Z ?÷ are said to be semi compatible if
[\RS,\R(() ÷ 1for all ( > 0 whenever N
O
P is a sequence in such that Y
O
, Z
O
÷ for some p in as÷·.
It follows that (Y, Z) is semi compatible and Y! = Z! imply YZ! = ZY! by taking N
O
P = ! = Y ! = Z! .
Lemma 2.12[15]: Let N
O
P be a sequence in Menger space ( , ,-) where- is continuous and ( - ) > for
alle [0, 1]. If there exists a constant ] e (0, 1) such that> 0 ande Q
S
, S^_ (] ) > S`_, S ( ), then
N
O
P is a Cauchy sequence.
Lemma 2.13[13]: If ( , ) is a metric space, then the metric d induces a mapping : × ÷ , defined by
( , ) = + ( - ( , )) , , e e . Further more if-: [0,1] × [0,1] ÷ [0,1] is defined by
( - 7) = : ( , 7), then ( , ,-) is a Menger space. It is complete if ( , ) is complete. The space ( , ,-)
so obtained is called the induced Menger space.
Lemma 2.14[10]: Let ( , ,-) be a Menger space. If there exists a constant ] e (0, 1) such that
R,a
(]() >
R,a((), for all , ! e and ( > 0 then = ! .
3.
Let A, B, T and S be mappings from X into itself such that Theorem
3.1: Let ( , ,-) be a complete Menger space where- is continuous and (( - () > (
for all ( e [0,1]. MAIN RESULT:
3 .1 .1 . Y ( ) c Z ( ) c( ) c d( )
3 .1 .2 . Z d are continuous
3.1.3. The pair (Z, Y) and (d, c) are Semi compatible
3.1.4. There exists a number ] e (0,1) such that
[R,ea (]() > \R,fa(() - \R,[R(() - [R,fa(() - fa,ea(() - \R,ea g(2 ÷ h)(i
, ! e , h e (0,2) ( > 0.
Then, Y, c, Z and d have a unique common fixed point in X.
Proof: Since Y( ) c Z( ) for any
j
e there exists a point
L
e such thatY
j
= Z L . Since c( ) c
d( ) for this point
L
we can choose a point
k
e such that d
L
= c
k
.
Inductively we can find a sequence N!
O
P as follows
!kO = Y kO = Z kO L
!kO
L
= c kO
L
= d kO k
260
For = 0, 1, 2, 3 .... by (3.1.4.), for all ( > 0 h = 1 ÷ withe (0,1), we have
amS,amS^_ (]() = [RmS ,eRmS^_ (]()
> \RmS,fRmS^_(() - \RmS^_,[RmS^_(() - [RmS,fRmS^_(() - fRmS^_,eRmS^_(() - \RmS,eRmS^_g(1 + )(i
= amS`_,amS(() - amS,amS^_(() - amS,amS^_(() -
amS,amS^_ (() - amS`_,amS^_g(1 + )(i
> amS`_,amS(() - amS,amS^_(() - amS`_,amS(() -amS,amS^_( ()
= amS`_,amS(() -amS,amS^_(() -amS,amS^_( ()
Since t-norm is continuous, letting÷ 1, we have
amS,amS^_(]() > amS`_,amS(() - amS,amS^_ (()
Similarly
amS^_,amS^m(]() > amS,amS^_(() - amS^_,amS^m(()
Similarly
amS^m,amS^n(]() > amS^_,amS^m (() - amS^m,amS^n(()
Therefore
aS,aS^_ (]() > aS`_,aS(() - aS,aS^_ (()
eo
Consequently aS,aS^_ (() > aS`_,aS(] () -, aS,aS^_ (] ()
pL pL
eo
Repeated application of this inequality will imply that
aS,aS^_ (() > aS`_,aS (] () - aS,aS^_ (] () >?... . > aS`_,aS (] () - aS,aS^_ (] (), e o
pL pL pL pr
Since
a
S,aS^_(] pr ( ) ÷ 1 s ÷·, it follows that
aS,aS^_ (() > aS`_,aS(] pL ( ) eo
Consequently
aS,aS^_ (]() > aS`_,aS(() for all
eo
Therefore by Lemma [2.12], N!
O
P is a Cauchy sequence in X. Since X is complete,N!
O
Pconverges to a point t e
z,
. Since NY kOP, Nc kO
L
P, NZ kO
L
P Nd kO
k
P are subsequences of N!
O
P , they also converge to the point
Case I: Since S is continuous. In this case we have . # .
as÷· , Y k
O
, c k O
L
, Z k O
L
d k O
k
÷ t .
ZY
O
÷ Z t , ZZ
O
÷ Zt
Also (Y, Z)is semi-compatible, we have YZ
O
÷ Zt
Step I: Let = Z
O
, ! =
O
u (? h = 1 in (3.1.4) we get
[\RS,eRS (]() > \\RS,fRS (() - \\RS,[\RS(() - [\RS,fRS (() - fRS,eRS (() - \\RS,eRS (()
\v,v(]() > \v,v(() - \v,\v(() - \v,v(() - v,v(() - \v,v(()
\v,v(]() > \v,v(()
So we get Zt = t.
Step II: By putting = t, ! =
O
u (? h = 1 in (3.1.4) we get
[v,eRS(]() > \v,fRS(() - \v,[v(() - [v,fRS (() - fRS,eRS(() - \v,eRS (()
[v,v(]() > v,v(() - v,[v(() - [v,v(() - v,v(()- v,v(()
[v,v(]() > [v,v(()
Case II: Since T is continuous. In this case we have So
we get Yt = t.
dc
O
÷ dt,
dd
O
÷ dt
Also (c, d)is semi-compatible, we have cd
O
÷ dt
Step I: Let =
O
, ! = d
O
u (? h = 1 in (3.1.4) we get
[RS,efRS (]() > \RS,ffRS (() - \RS,[RS (() - [RS,ffRS (() - ffRS,efRS (() - \RS,efRS (()
v,fv(]() > v,fv(() - v,v(() - v,fv(() - fv,fv(() - v,fv(()
v,fv(]() > v,fv(()
So we get dt = t.
Step II: By putting =
O
, ! = t u (? h = 1 in (3.1.4) we get
[RS,ev(]() > \RS,fv(() - \RS,[RS (() - [RS,fv(() -fv,ev(() - \RS,ev(()
v,ev(]() > v,v(() - v,v(() - v,v(() - v,ev(() - v,ev (()
ev,v(]() > ev,v(()
So we get ct = t.
Thus, we have Yt = Zt = dt = ct = t.
261
That is z is a common fixed point of Z, d, Y and c.
For uniqueness, let u (u = t) be another common fixed point of Z, d, Y and c .Then Yu = Zu == cu =
du = u .
Put = t, ! = u and w = 1, in (3.1.4.), we get
[v,e=(]() > \v,f=(() - \v,[=(() - [v,f=(() - f=,e=(() - \v,e= (()
v,=(]() > v,=(() - v,=(() - v,=(() - =,=(() - v,=(()
v,=(]() > v,=(() - v,=(() - v,=(() - 1 - v,=(()
v,=(]() > v,=(()
Thus we havet = u. Therefore z is a unique fixed point of A,S, B and T. This
completes the proof of the theorem.
COROLLARY 3.2: Let ( , ,-) be a complete Menger space where- is continuous and (( - () > ( for
all ( e [0,1]. Let A, and S be mappings from X into itself such that
3 .2 .1 . Y ( ) c Z ( )
3.2.2. Z is continuous
3.2.3. The pair (Z, Y) is semi compatible
3.2.4. There exists a number ] e (0,1) such that
[R,\a(]() > \R,\a(() - \R,[R(() - [R,\a(() - \a,[a (() - \R,[ag(2 ÷ h)(i
, ! e , h e (0,2)
Then, Y, and Z have a unique common fixed point in X.
doc_936901989.docx