Asian Journal of Current Engineering and Maths 1: 1 (2012) 5 – 8.
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ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS
Journal homepage: http://www.innovativejournal.in/index.php/ajcem
COMMON FIXED POINT THEOREM FOR WEAKLY COMPATIBLE MAPS SATISFYING
E.A. PROPERTY IN INTUITIONISTIC MENGER SPACES
Saurabh Manro1*, S. S. Bhatia1, Sanjay Kumar2
1School
of Mathematics and Computer Applications, Thapar University, Patiala (Punjab) India.
2Deenbandhu
ARTICLE INFO
Corresponding Author:
Saurabh Manro
School of Mathematics and
Computer Applications,
Thapar University, Patiala
(Punjab), India.
[email protected]
Chhotu Ram University of Science and Technology, Murthal (Sonepat), India.
ABSTRACT
In this paper, we use the notion of E.A. property in intuitionistic Menger spaces and
prove a common fixed point theorem for weakly compatible mappings using this
property.
KeyWords: Intuitionistic Menger spaces, E.A property, weakly compatible maps.
©2012, AJCEM, All Right Reserved.
INTRODUCTION
There have been a number of generalizations of metric
spaces. One such generalization is Menger space
introduced in 1942 by Menger[9] who used distribution
functions instead of nonnegative real numbers as values of
the metric. This space was expanded rapidly with the
pioneering works of Schweizer and Sklar [11, 12].
Modifying the idea of Kramosil and Michalek [7], George
and Veeramani [5] introduced fuzzy metric spaces which
are very similar that of Menger space. Atanassove [3]
introduced and studied the concept of intuitionistic fuzzy
sets as a generalization of fuzzy sets. In 2004, Park[10]
defined the notion of intuitionistic fuzzy metric space with
the help of continuous t-norms and continuous t-conorms.
Recently, in 2006, Alaca et al.[2] using the idea of
Intuitionistic fuzzy sets, defined the notion of intuitionistic
fuzzy metric space with the help of continuous t-norm and
continuous t-conorms as a generalization of fuzzy metric
space due to Kramosil and Michalek[8] .
Kutukcu et. al [8] introduced the notion of
intuitionistic Menger spaces with the help of t-norms and tconorms as a generalization of Menger space due to
Menger [9]. Further they introduced the notion of Cauchy
sequences and found a necessary and sufficient condition
for an intuitionistic Menger space to be complete.
On the other hand, Jungck [6] introduced the
notion of compatible mappings in metric spaces. The
concept of weakly compatible mappings is most general as
each pair of compatible mappings is weakly compatible but
the converse is not true. Recently, Amari and
Moutawakil[1] introduced a generalization of non
compatible maps as E.A. property. These observations
motivated us to prove a common fixed point theorem for
six weakly compatible maps in intuitionistic Menger
spaces.
In this paper, we use the notion of E.A. property
in intuitionistic Menger space and prove a common fixed
point theorem for weakly compatible mappings using this
property.
2. Preliminaries:
The concepts of triangular norms ( t- norm ) and triangular
conorms ( t- conorm ) are known as the axiomatic skelton
that we use are characterization fuzzy intersections and
union respectively. These concepts were originally
introduced by Menger [8] in study of statistical metric
spaces.
Definition 2.1[11]: A binary operation * : [0,1]×[0,1] 
[0,1] is continuous t-norm if * is satisfies the following
conditions:
(i) * is commutative and associative;
(ii) * is continuous;
(iii) a * 1 = a for all
a  0,1 ;
(iv) a * b  c * d whenever a  c and b  d for all
a, b, c, d  0,1 .
Definition 2.2[11]: A binary operation ◊ : [0,1]×[0,1] 
[0,1] is continuous t-conorm if ◊ is satisfies the following
conditions:
(i) ◊ is commutative and associative;
(ii) ◊ is continuous;
(iii) a ◊ 0 = a for all
a  0,1 ;
(iv) a ◊ b  c ◊ d whenever a 
a, b, c, d  0,1 .
c and b  d for all
Definition 2.3[12]: A distance distribution function is a
F : R  R which is left continuous on R, nondecreasing and inf F (t )  0;sup F (t )  1 . We will denote
function
tR
tR
by D the family of all distance distribution functions and by
H a special of D defined by
0, t  0,
H (t )  
1, t  0.
5
Manro et al/Common Fixed Point Theorem For Weakly Compatible Maps Satisfying E.A. Property In Intuitionistic
Menger Spaces
If X is a non-empty set, F : X  X  D is called a
probabilistic distance on X and F(x, y) is usually denoted by
Fx, y (t ) .
Definition 2.8: A pair of self mappings (A, S) of a
intuitionistic Menger space
( X , F , L,*, )is said to be
commuting if
FASx,SAx (t )  1 and LASx,SAx (t )  0 for all
Definition 2.4[12]: A non-distance distribution function is
L : R  R which is right continuous on R,
non-increasing and inf L(t )  1;sup L(t )  0 . We will
x X.
a function
tR
Definition 2.9: The self-maps A and S of an intuitionistic
Menger space ( X , F , L,*, ) are said to be compatible if
for all t > 0,
lim FASxn ,SAxn (t )  1 and lim LASxn , SAxn (t )  0.
tR
denote by E the family of all distance distribution functions
and by G a special of E defined by
1, t  0,
G (t )  
0, t  0.
If X is a non-empty set, L : X  X  E is called a
n 
whenever
n 
t, s  0 ,
(1) Fx , y (t )  Lx , y (t )  1 ,
and
(3)
Fx, y (t )  H (t ) if and only if x = y,
(4)
Fx, y (t )  Fy , x (t ) ,
(5) if
n 
Fx, z (t  s)  Fx, y (t )* Fy , z (s) ,
(7)
Lx, y (0)  1 ,
(8)
Lx, y (t )  G(t ) if and only if x = y,
(9)
Lx, y (t )  Ly , x (t ) ,
(10) if
(11)
and Sx =
Fx, y (t ) and Lx , y (t ) denote the degree of
n 
nearness and degree of non-nearness between x and y with
respect to t, respectively.
Remark 2.1. Every Menger space (X, F, * ) is intuitionistic
Menger space of the form ( X , F ,1  F ,*, ) such that tnorm * and t-conorm ◊ are as
x ◊ y = 1-((1-x) * (1-y)) for all x, y  X .
An
lim Lxm , xn (t )  0 .
x
4
3x
, for all x  X . Consider the sequence
4
Menger
n
Then, x = y.
Remark
2.3:
( X , F , L,*, ) ,
m, n 
intuitionistic
n
( X , F , L,*, ) be intuitionistic Menger
space and for all x, y  X , t  0 and if for a number
k  (0,1) ,
Fx, y (kt )  Fx, y (t ) and Lx, y (kt )  Lx, y (t ) .
In
intuitionistic
Menger
space
Fx, y (t ) is non-decreasing and Lx , y (t ) is
non-increasing for all x, y  X .
n 
2.7[8]:
n
Lemma 2.1. Let
lim Fxn , x (t )  1 and lim Lxn , x (t )  0 .
Definition
that
a contradiction, since 1 is not contained in X. Hence, S and A
do not satisfy E.A. property.
(b) a sequence {xn} in X is said to be convergent to a point
x  X if, for all t  0 ,
n 
such
 z 1 
lim xn  z  1 and lim xn  
 . Thus, z = 1, which is
n 
n 
 2 
Definition 2.6[8]: Let ( X , F , L,*, ) be an intuitionistic
Menger space. Then
(a) a sequence {xn} in X is said to be Cauchy sequence if, for
all t  0
lim Fxm , xn (t )  1 and
X
satisfy E.A. property.
Example 2.2. Let X = [2, +∞). Define S, A : X→X by Ax= x+1
and Sx= 2x+1, for all x  X . Suppose that the E.A. property
holds. Then, there exists in X, a sequence {xn} satisfying
lim Sxn  lim Axn  z for some z  X . Therefore,
Lx, z (t  s)  Lx, y (t )Ly , z (s) .
m, n
in
1

Sxn  lim Axn  0 . Then S and A
 xn   .Clearly, lim
n 
n
n

Lx, y (t )  0 and Ly , z (s)  0 , then Lx, z (t  s)  0 ,
The function
sequence
Example 2.1. Let X = [0, +∞). Define S, A : X→X by Ax =
Fx, y (t )  1 and Fy , z ( s)  1 , then Fx, z (t  s)  1 ,
(6)
a
Definition 2.10: A pair of self mappings (A, S) of a
intuitionistic Menger space ( X , F , L,*, ) is said to be
weakly compatible if they commute at the coincidence
points i.e. Au = Su for some u  X , then ASu = SAu.
Remark 2.2: If self-maps A and S of an intuitionistic Menger
space ( X , F , L,*, ) are compatible then they are weakly
compatible.
Recently, Amari and Moutawakil[1] introduced a
generalization of non compatible maps as E.A. property.
Definition 2.11[1]: Let A and S be two self-maps of a
metric space (X, d) .The pair (A,S) is said to satisfy E.A.
property, if there exists a sequence {xn} in X such that
lim Sxn  lim Axn  z for some z  X .
Definition 2.5[8]: A 5-tuple ( X , F , L,*, ) is said to be
an intuitionistic Menger space if X is an arbitrary set, * is a
continuous t-norm, ◊ is continuous t-conorm, F is a
probabilistic distance and L is a probabilistic non-distance
on X satisfying the following conditions: for all x, y, z  X
Fx, y (0)  0 ,
is
lim Sxn  lim Axn  z for some z  X .
probabilistic non-distance on X and L(x, y) is usually
denoted by Lx , y (t ) .
(2)
 xn 
n 
space
Lemma 2.2. Let
space and
( X , F , L,*, ) is said to be complete if and only if every
Cauchy sequence in X is convergent.
6
( X , F , L,*, ) be intuitionistic Menger
Manro et al/Common Fixed Point Theorem For Weakly Compatible Maps Satisfying E.A. Property In Intuitionistic
Menger Spaces
 xn  be
a sequence
Suppose that S(X) is a complete subspace of X. Then p = Su
for some u  X . Subsequently, we have
in X. If there exists a number
k  (0,1) such that:
Fxn2 , xn1 (kt )  Fxn1 , xn (t ) and Lxn2 , xn1 (kt )  Lxn1 , xn (t )
for all
lim Ayn  lim Syn  lim Bxn  lim Txn  p  Su.
n
t  0 and n = 1, 2,3, . . . . Then,  xn  is a Cauchy
and
 LSu ,Txn (t )LAu , Su (t )LBxn ,Txn (t ) 
LAu , Bxn (kt )  

 LBx , Su (2t )LAu ,Tx (t )
n
 n

Taking limit as n   we get
 FSu ,Su (t )* FAu ,Su (t )* FSu ,Su (t ) 
FAu ,Su (kt )  

 *F (2t )* F
Au , Su (t )
 Su ,Su

 LSu ,Su (t )LAu ,Su (t )LSu ,Su (t ) 
and LAu , Su (kt )  

 L (2t )L
Au , Su (t )
 Su ,Su

satisfying:
(3.1) A( X )  T ( X ) ,
B( X )  S ( X )
(3.2) for all x, y  X , k  (0,1) , t  0
 FSx ,Ty (t )* FAx , Sx (t )* FBy ,Ty (t ) 
FAx , By (kt )  

 *FBy , Sx (2t )* FAx ,Ty (t )


 LSx ,Ty (t )LAx ,Sx (t )LBy ,Ty (t ) 
and
LAx , By (kt )  

 LBy , Sx (2t )LAx ,Ty (t )


this gives,
FAu ,Su (kt )  FAu ,Su (t ) and LAu ,Su (kt )  LAu ,Su (t ) .
Now by using lemma 2.1, we have Au = Su. Therefore (A, S)
have coincidence point.
The weak compatibility of A and S implies that ASu = SAu
and thus AAu = ASu = SAu = SSu.
As A( X )  T ( X ) , there exists v in X such that Au = Tv.
We claim that Tv = Bv.
From (3.2), we have
(3.3) The pairs (A, S) or (B, T) satisfies E.A. property,
If one of A(X), B(X), S(X) or T(X) is complete subsets of X.
Then, pairs (A, S) and (B, T) have coincidence point.
Further, if (A, S) and (B, T) are weakly compatible then A, B,
S and T have unique common fixed point in X.
Proof: Suppose the pair (B, T) satisfies the E.A. property.
Then, there exists a sequence {xn} in X such that
lim Bxn  lim Txn  p for some p  X . Since
 FSu ,Tv (t ) * FAu , Su (t ) * FBv ,Tv (t ) 
FAu , Bv (kt )  

 *F (2t ) * F (t )
Au ,Tv
 Bv , Su

 FTv ,Tv (t ) * FAu , Au (t ) * FBv ,Tv (t ) 
FTv , Bv (kt )  

 *F (2t ) * F (t )
Tv ,Tv
 Bv ,Tv

FTv , Bv (kt )  FBv ,Tv (t )
n
B( X )  S ( X ) , there exists a sequence {yn} in X such that
Bxn = Syn . Hence lim Syn  p .
We shall show that
n 
lim Ayn  p .
n 
From (3.2), we have
 FSyn ,Txn (t )* FAyn , Syn (t )* FBxn ,Txn (t ) 
FAyn , Bxn (kt )  

 *FBx , Sy (2t )* FAy ,Tx (t )
n
n
n
n


L
(
t
)

L
(
t
)

L
 Syn ,Txn
Ayn , Syn
Bxn ,Txn (t ) 
and
LAyn , Bxn (kt )  

 LBx , Sy (2t )LAy ,Tx (t )
n
n
n
n


Taking limit as n   , we get
and
 LSu ,Tv (t )LAu , Su (t )LBv ,Tv (t ) 
LAu , Bv (kt )  

 L
 Bv , Su (2t )LAu ,Tv (t )

 LTv ,Tv (t )LAu ,Su (t )LBv ,Tv (t ) 
LTv , Bv (kt )  
 .
 L (2t )L (t )
Tv ,Tv
 Bv ,Tv

LTv , Bv (kt )  LBv ,Tv (t )
Now, by using lemma 2.1, we have Tv = Bv =Au. Thus we
have Au = Su = Tv = Bv.
The weak compatibility of B and T implies that BTv = TBv =
TTv = BBv.
Finally, we show that Au is the common fixed point of A, B, S
and T.
From (3.2), we have
 Fp , p (t )* FAyn , p (t )* Fp , p (t )* 
FAyn , p (kt )  
 and
 Fp , p (2t )* FAy , p (t )
n


 Lp , p (t )LAyn , p (t )Lp , p (t ) 
LAyn , p (kt )  

 Lp , p (2t )LAy , p (t )
n


Therefore, as n   ,
FAyn , p (kt )  FAyn , p (t ) and LAyn , p (kt )  LAyn , p (t ) .
 FSAu ,Tv (t )* FAAu , SAu (t )* 


FAu , AAu (kt )  FAAu , Bv (kt )   FBv ,Tv (t )*

F

 Bv , SAu (2t )* FAAu ,Tv (t ) 
FAu , AAu (kt )  FAAu , Bv (kt )  FAAu , Bv (t )
and
lim Ayn  lim Syn  p .
n 
n 
 FSu ,Txn (t )* FAu , Su (t )* FBxn ,Txn (t ) 
FAu , Bxn (kt )  

 *FBx , Su (2t )* FAu ,Tx (t )
n
n


a  b  min a, b and ab  max a, b , a, b  0,1 ,
Using lemma 2.1, we have
n 
Now, we shall show that Au = Su.
From (3.2), we have
sequence in X.
3. Weakly compatible maps and E.A. property:
Now, we prove our main result using using E.A property.
Our theorem generalise many known results in fixed point
theory in the following way:
(i) relaxing the continuity requirement of maps and,
(ii) relaxing the completeness of the space X .
Theorem 3.1: Let A, B, S and T be self maps of intuitionistic
menger metric space ( X , F , L,*, ) with continuous tnorm and continuous t- conorm defined by
n 
n 
n
7
Manro et al/Common Fixed Point Theorem For Weakly Compatible Maps Satisfying E.A. Property In Intuitionistic
Menger Spaces
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system, 20, 87-96(1986).
[4] D. Coker, An introduction to Intuitionistic Fuzzy
topological spaces, Fuzzy Sets and System, 88, 81- 89
(1997).
[5] A. Grorge and P. Veeramani, On some results in fuzzy
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contraction mapping principle in intuitionistic menger
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& Mech., 28:799:809, 2007.
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(USA),28 (1942).
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 LSAu ,Tv (t )LAAu , SAu (t ) 


LAu , AAu (kt )  LAAu , Bv (kt )   LBv ,Tv (t )

 L

(2
t
)

L
(
t
)
AAu ,Tv
 Bv , SAu

LAu , AAu (kt )  LAAu , Bv (kt )  LAAu , Bv (t )
Now, the use of lemma 2.1 gives AAu = Bv =Au and thus AAu
= Au.
Therefore, Au = AAu = SAu is the common fixed point of A
and S.
Similarly, we prove that Bv is the common fixed point of B
and T. Since Au = Bv, Au is common fixed point of A, B, S,
and T. The proof is similar when T(X) is assumed to be a
complete subspace of X. The cases in which A(X) or B(X) is a
complete subspace of X are similar to the cases in which
T(X) or S(X), respectively is complete subspace of X as
A( X )  T ( X ) and B( X )  S ( X ) . Uniqueness directly
follows from (3.2). Therefore, the mappings A, B, S, and T
have a unique common fixed point.
REFERENCES
[1] M. Aamri, D. El Moutawakil: some new common fixed
point theorems under strict contractive conditions. J. Math.
Anal. Appl., 270, 181-188 (2002).
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