Bd32360363

Deepak Singh, M.S. Rathore, Krishnapal Singh Sisodia / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 2, March -April 2013, pp.360-363
A Common Fixed Point Theorem through Weak and Semi-
Compatibility in Fuzzy Metric Space
Deepak Singh, M.S. Rathore, Krishnapal Singh Sisodia
*NITTTR, Bhopal (M.P.), India
**Department of Mathematics, Govt. P. G. College Sehore (M.P.), India
***Department of Mathematics, SISTec-E, Ratibad, Bhopal (M.P.), India

Abstract
In this paper, a common fixed point (fm5) M(x, y, .): (0, ) [0,1] is continuous.
theorem for six self mappings has been (fm6) for all x, y X.
established using the concept of semi- Then M is called a Fuzzy metric on X. The function
compatibility and weak compatibility in Fuzzy M (x, y, t) denote the degree of nearness between x
metric space, which generalizes the result of and y with respect to t.
Singh B.S., Jain A. and Masoodi A.A. [6]. Example 1: Let (X, d) be a metric space. Denote
a b=ab for a, b [0, 1] and let Md be Fuzzy set on
Keywords: Fuzzy metric space, common fixed
point, t-norm, compatible map, semi-compatible, X2 (0, ) defined as follows:
weak compatible map. Md (x, y, t) =
2000 Mathematics Subject Classification: 47H10,
54H25. Then (X, Md, ) is a Fuzzy metric space, we call this
Fuzzy metric induced by a metric d the standard
Introduction: The concept of Fuzzy sets was intuitionistic Fuzzy metric.
introduced by Zadeh [9]. Following the concept of Definition 3 [2]: Let (X, M, ) be a Fuzzy metric
Fuzzy sets, Fuzzy metric spaces have been space, then
introduced by Kramosil and Michalek [4] and (a) A sequence {xn} in X is said to be
George and Veeramani [2] modified the notion of
convergent to x in X if for each >0 and
Fuzzy metric spaces with the help of continuous t-
norms. Vasuki [8] investigated some fixed point each t>0, there exists n 0 N such that M
theorems in Fuzzy metric spaces for R-weakly (xn, x, t)>1- for all n n0.
commuting mappings. Inspired by the results of B. (b) A sequence {xn} in X is said to be Cauchy
Singh, A. Jain and A.A. Masoodi [6], in this paper, if for each >0 and each t>0, there exists
we prove a common fixed point theorem for six self
n0 N such that M (xn, xm, t)>1- for all n,
maps under the condition of weak compatibility and
semi-compatibility in Fuzzy metric spaces. m n0.
(c) A Fuzzy metric space in which every
Preliminaries: Cauchy sequence is convergent is said to
Definition 1: A binary operation : be complete.
[0,1] [0,1] [0,1] is a continuous t-norm if is Proposition 1: In a Fuzzy metric space (X, M, ), if
satisfying the following conditions: a a a for a [0, 1] then a b=min {a, b} for all a, b
(a) is commutative and associative; [0, 1].
(b) is continuous; Definition 4 [7]: Two self mappings A and S of a
(c) a 1=a for all a [0,1]; Fuzzy metric space (X, M, ) are called compatible
(d) a b c d whenever a c and b d and if whenever
a,b,c,d [0,1]. { } is a sequence in X such that
Definition 2 [2]: A 3-tuple (X, M, ) is said to be a for some in X.
Fuzzy metric space if X is an arbitrary set, is a
Definition 5 [7]: Two self mappings A and S of a
continuous t-norm and M is a Fuzzy set on
X2 (0, ) satisfying the following conditions: for Fuzzy metric space (X, M, ) are called semi-
all x,y,z X, s,t>0, compatible if
(fm1) M (x, y, t)>0; whenever { } is a sequence in X such that
(fm2) M (x, y, t) =1 iff x=y; for some in X.
(fm3) M (x, y, t) = M (y, x, t);
(fm4) M (x, y, t) M(y, z, s) M(x, z, t+s);

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Definition 6 [7]: Two self maps A and S of a Fuzzy
metric space (X, M, ) are called weakly compatible
(or coincidentally commuting) if they commute at
their coincidence points, i.e. if Ax=Sx for some
X then ASx=SAx.
Proposition 2 [7]: In a Fuzzy metric space (X,
By lemma 1 and lemma 2; we get
M, ), limit of a sequence is unique.
Proposition 3 [7]: In a Fuzzy metric space (X,
Similarly; we have
M, ), If (A, S) is a semi-compatible pair of self
maps and S is continuous, then (A, S) is compatible Thus we have
Remark 1: If self maps A and S of a Fuzzy metric
space (X, M, ) are compatible then they are weakly Using lemma 3; we have
compatible. { } is a Cauchy sequence in X.
Lemma 1 [3]: Let (X, M, ) be a Fuzzy metric Since (X, M, ) is complete, { } converges to
space. Then for all x, y єX, some point X. Also its subsequences converges
is a non decreasing function. to the same point i.e. X.
Lemma 2 [5]: Let (X, M, ) be a Fuzzy metric i.e. { } →z and { } →z ----------
space. If there exists k [0, 1] such that (1)
{ } →z and { } →z --------------- (2)
then x=y.
Case I: suppose P is continuous.
Lemma 3 [1]: let {yn} be a sequence in a Fuzzy
Step-2: since P is continuous and (P, AB) is semi-
metric space (X, M, ) with the condition (fm6). If compatible pair, we have
there exists k [0, 1] such that →Pz and
for all t>0 →ABz.
and n N, then {yn} is a Cauchy sequence in X. Since the limit in Fuzzy metric space is unique, we
Main Results: get
In this result, we are proving a theorem to obtain a . ---------------- (3)
unique common fixed point for six self mappings Step-3: putting x=Px2n and y=x2n+1 in condition (v);
using weak and semi-compatibility in a Fuzzy we have
metric space.
Theorem 1: Let (X, M, ) be a complete Fuzzy
metric space and let A, B, S, T, P and Q be Taking n→∞ and using (1), (2) and (3); we get
mappings from X into itself such that the following
conditions are satisfied: }
(i) P(X) ST(X), Q(X) AB(X);
(ii) AB=BA, ST=TS, PB=BP, QT=TQ }
(iii) Either AB or P is continuous; i.e.
(iv) The pair (P, AB) is semi-compatible and Therefore by using lemma 2; we have
(Q, ST) is weakly compatible. .
(v) There exists such that for every Step-4: putting x=Bz and y= x2n+1 in condition (v);
and t>0 we get

.
Then A, B, S, T, P and Q have a unique
common fixed point. BP=PB, AB=BA; so we have
Proof: Let x0 . By (i), there exists x1, x2 X such P(Bz)=B(Pz)=Bz and AB(Bz)=B(AB)z=Bz
that Px0=STx1= y0 and Qx1=ABx2=y1. Inductively, Taking n→∞ and using (1), we get
we can construct sequences {xn} and {yn} in X such
that Px2n=STx2n+1=y2n and Qx2n+1=ABx2n+2=y2n+1 for
n=0, 1, 2, ----. Therefore, by using lemma 2, we get
Step-1: putting x=x2n and y= x2n+1 in (v), we have
.
And also we have

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. Thus,
. ------------ (4) Now, we prove .
Step-5: since P(X) ST(X), there exists X such Step-8: putting and in
that (v); we get
--------------- (5)
Putting x=x2n and y= u in (v), we have

Taking n→∞ and using (2) and (5), we get
Taking n→∞ and using (1), (9) and (10); we get

Therefore by using lemma 2; we get
Therefore, by using lemma 2, we get
.
.
Hence Step-9: putting x=z and y= x2n+1 in condition (v); we
Since (Q, ST) is weakly compatible, therefore we get
have

Thus ----------- (6) Taking n→∞ and using (1); we get
Step-6: Putting x=x2n and y= z in (v), we have

Taking n→∞ and using (2) and (6); we get
Therefore, by using lemma 2, we get
.
.
Therefore by using lemma 2, we get Further, using step (4), we get
.
Step-7: putting x=x2n and y= Tz in (v), we have Thus, gives A
And so .
Also it follows from steps (5), (6) and (7) that
As QT=TQ and ST=TS, we have .
QTz=TQz=Tz and STTz=TSTz=T(STz)=T(Qz)=Tz
Hence we
Taking n→∞ and using (2); we get
get .
i.e. z is a common fixed point of A, B, P, Q, S and T
in this case also.
Uniqueness : Let w be another common fixed point
Therefore by using lemma 2; we get of A, B, P, Q, S and T,
Then
Now implies .
Hence -------------- (7) Putting x=z and y= w in condition (v); we get
Combining (4) and (7); we get

Hence z is the common fixed point of A, B, S, T, P
and Q.
Case II: suppose AB is continuous. Therefore, by using lemma 2, we get
Since AB is continuous and (P, AB) is semi- .
compatible, we have
Therefore z is the unique common fixed point of self
→ ------------- (8) maps A, B, P, Q, S and T.
----------- (9) Remark 2: If we take B=T=IX (the identity map on
----------- (10) X) in the theorem 1, then condition (ii) is satisfied
trivially and we get

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Corollary 1: Let (X, M, ) be a complete Fuzzy
metric space and let A, S, P and Q be mappings
from X into itself such that the following conditions
are satisfied:
(i) P(X) S(X), Q(X) A(X);
(ii) Either A or P is continuous;
(iii) The pair (P, A) is semi-compatible and (Q,
S) is weakly compatible.
(iv) There exists such that for every
and t>0

.
Then A, S, P and Q have a unique common
fixed point.

References:
[1] Y.J. Cho, H.K. Pathak, S.M. Kang, J.S.
Jung, Common Fixed points of compatible
maps of type (β) on Fuzzy metric spaces,
Fuzzy sets and systems, 93(1998), 99-111.
[2] A. George, P. Veeramani, On some results
in Fuzzy metric spaces, Fuzzy sets and
systems, 64(1994), 395-399.
[3] M. Grebiec, Fixed points in Fuzzy metric
space, Fuzzy sets and systems, 27(1998),
385-389.
[4] O. Kramosil and J. Michalek, Fuzzy metric
and statistical metric spaces, Kybernetika
11(1975), 326-334.
[5] S. Sharma, Common Fixed point theorems
in Fuzzy metric spaces, Fuzzy sets and
systems, 127(2002), 345-352.
[6] B. Singh, A. Jain and A.A. Masoodi, Semi-
compatibility, weak compatibility and fixed
point theorem in Fuzzy metric space,
International Mathematical Forum, 5, 2010,
no. 61, 3041-3051
[7] B. Singh and S. Jain, Semi-compatibility,
compatibility and fixed point theorems in
Fuzzy metric space, Journal of
Chungecheong Math. Soc. 18(1), (2005), 1-
22.
[8] R. Vasuki, Common Fixed points for R-
weakly commuting maps in Fuzzy metric
spaces, Indian J. Pure Appl. Math.,
30(1999), 419-423.
[9] L.A. Zadeh, Fuzzy sets, Inform and
Control, 8(1965), 338-353.

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