Common fixed point theorem for occasionally weakly compatible mapping in q fuzzy metric spaces

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications
1
Common Fixed Point Theorem for Occasionally Weakly
Compatible Mapping in Q-Fuzzy Metric Spaces
Kamal Wadhwa1
, Farhan Beg2
1. Govt. Narmada Mahavidhyalaya, Hoshangabad (M.P.)
2. Truba College Of Science & Technology Bhopal,(M.P.) India
E-mail: beg_farhan26@yahoo.com (corresponding author)
Abstract
This Paper present some common fixed point theorem for Occasionally Weakly Compatible mapping in Q-
fuzzy metric spaces under various conditions.
Keywords: Fixed point , Occasionally Weakly Compatible mapping, Q-fuzzy metric spaces , t-norm
1. Introduction:
The concept of fuzzy sets introduced by Zadeh [12] in 1965 plays an important role in topology and analysis.
Since then, there are many author to study the fuzzy set with application. Especially Kromosil and Michalek [10]
put forward a new concept of fuzzy metric spaces. George and Veermani [6] revised the notion of fuzzy metric
spaces with the help of continuous t-norm. As a result of many fixed point theorem for various forms of
mapping are obtained in fuzzy metric spaces. Dhage [5] introduced the definition of D-metric spaces and proved
many new fixed point theorem in D-metric spaces. Recently, Mustafa and Sims[13] presented a new definition
of G-metric space and made great contribution to the development of Dhage theory.
On the other hand ,Lopez-Rodrigues and Romaguera [11] introduced the concept of Hausdorff fuzzy metric
in a more general space .
The Q-fuzzy metrics spaces is introduced by Guangpeng Sun and kai Yang[7] which can be cosider as a
Generalization of fuzzy metric spaces. Sessa [18] improved commutativity condition in fixed point theorem by
introducing the notion of weakly commuting maps in metric space. R.Vasuki[14] proved fixed point theorems
for R-weakly commuting mapping Pant [14,15,16] introduced the new concept of reciprocally continuous
mappings and established some common fixed point theorems. The concept of compatible maps by [10] and
weakly compatible maps by [8] in fuzzy metric space is generalized by A.Al Thagafi and Naseer Shahzad [1] by
introducing the concept of occasionally weakly compatible mappings. Recent results on fixed point in Q-fuzzy
metric space can be viewed in[7]. In this paper we prove some fixed point theorems for four occasionally weakly
compatible owc mappings which improve the result of Ganpeng Sun and Kai Yang [7] in Q-fuzzy metric spaces.
2. Preliminary Notes:
Definition:2.1[2] A binary operation
∗:[0,1]×[0,1]→[0,1] is a continuous t-norm if it satisfy the following condition:
(i) ∗ is associative and commutative .
(ii) ∗ is continous function.
(iii) a∗1=a for all a∈ [0,1]
(iv) a∗b≤c∗d whenever a≤c and b≤d and a,b,c,d ∈ [0,1]
Definition 2.2[7] : A 3-tuple (X,Q, ∗ ) is
called a Q-fuzzy metric space if X is an arbitrary (non-empty) set ,* is a continuous t -norm, and Q is a fuzzy
set on
X3
×(0,∞),satisfying the following conditions for each x,y,z,a ∈ X and t ,s> 0 :
(i) Q( x,x,y,t)>0 and Q(x,x,y,t)≤ Q(x,y,z,t) for all x,y,z ∈ X with z≠y
(ii) Q ( x,y,z,t)=1 if and only if x =y = z
(iii) Q(x,y,z,t) = Q(p(x,y,z),t),(symmetry) where p is a permutation function ,
(iv) Q(x,a,a,t) ∗Q(a,y,z,s)≤Q(x,y,z,t+s),
(v) Q(x,y,z,. ):( 0 ,∞)→[0,1] is continuous
A Q-fuzzy metric space is said to be symmetric if Q( x,y,y,t)= Q(x,x,y,t) for all x,y ∈ X .
Example : Let X is a non empty set and G i s t h e G-metric on X. Denote a*b = a.b for all a, b ∈[0,1].For
each t > 0 :
Q(x,y,z,t) =
( , , )
Then (X,Q,∗) is a Q-fuzzy metric .
Definition 2.3[6] Let (X,Q,∗) be a Q-fuzzy metric space.For t>0, the open ballBQ(x,r,t) with center x∈X and
radius 0 <r<1 is defined by BQ( x,r,t)={y∈X:Q(x,y,y,t)>1-r}

2
A subset A of X is called open set if for each x∈A there exist t>0 and 0<r<1 such that BQ(x,r,t) ⊆ A .
A sequence { xn} in X converges to x if and only if Q (xm,xn,x ,t) →1 as n→∞,for each t>0. It is called a
Cauchy sequence if for each 0 <ε<1 and t>0,there existn0∈N such that Q(xm,xn,x1)>1- ε for each l, n ,m ≥
n0.The Q-fuzzy metric space is called to be complete if every Cauchy sequence is convergent. Following
similar argument in G-metric space, the sequence {xn} in X also converges to x if and only if Q(xn,xn,x,t)
→1 as n→∞, for each t >0 and it is a Cauchy sequence if for each 0 <g<1 and t>0, there exist n0∈N
suc h that Q(xm,xn,xn) > 1-ε for each n,m ≥ n0.
Lemma2.4[7] : If (X,Q, ∗) be a Q-fuzzy metric space,then Q(x,y,z,t) is non-decreasing with respect to t for all
x,y,z in X .
Proof: Proof is this is implicated in [7]
Lemma2.5[7] : Let (X,Q, ∗) be a Q-fuzzy metric space.(a) If there exists a positive number k<1 such that :
Q(yn + 2,yn + 1,yn + 1,kt)≥Q(yn+1,yn,yn, t ),t >0, n ∈N then{yn} is a Cauchy sequence in X.
(b) if there exists k ∈ ( 0,1) such that Q(x,y,y,kt) ≥ Q(x,y,y,t) for all x, y ∈X and t > 0 then x= y .
Proof: By the assume lim →∞ Q(x, y, z, t ) =1 and the property of non-decreasing, it is easy to get the results .
Definition 2.6[3]: Let X be a set, f and g Self maps of X. A point x ∈ is called a coincidence point of f and g
iff fx=gx .We shall call w=fx=gx a point of coincidence of f and g.
Definition 2.7[7]: Let f and g be self maps on a Q-fuzzy metric space (X ,Q,*) .Then the mappings
are said to be weakly compatible if they commute at their coincidence point, that is, f x =gx implies
that fgx = gfx .
Definition 2.8 [7]: Let f and g be self maps on a Q-fuzzy metric space (X,Q, *) . The pair (f ,g) is
said to be compatible if
lim →∞ ( ,gf , gf , )=1 whenever {xn} is a sequence in X such that
lim →∞ =lim →∞ =z for some z∈
Definition 2.9[3]: Two self maps f and g of a set X are occasionally weakly compatible (owc) iff there is a point
x in X which is coincidence point of f and g at which f and g commute.
Lemma 2.10 [9]: Let X be a set, f , g owc self maps of X. If f and g have unique point of coincidence, w = f x
= g x , then w is the unique common fixed point of f and g .
3. Main Result
Theorem 3.1 : Let (X,Q,∗) be complete symmetric Q-fuzzy metric space and f ,g ,S and T be a self
mapping of X. Let the pair {f, T} and {g, S} be owc. If there exist k ∈ (0,1) such that
Q(fx,gy,gy,kt) ≥ Q(Tx,Sy,Sy,t)∗Q(Tx,gy,gy,t) ∗Q(fx.Sy,Sy,t) ∗Q(fx,Tx,Tx,t)…(1)
For all x ,y ∈ "#$ %& "'' > 0,then there exist a unique point w ∈ X such that f w =Tw=w and a unique
point z ∈ X such that g z = S z =z, Moreover, z = w so that there is a unique common fixed point of f , g ,S and
T.
Proof:Let the pair {f,T} and {g ,S} be owc, so there are point x ,y ∈ such that fx =Tx and gy = Sy. We claim
that f x= gy. If not by inequality (1)
Q(fx,gy,gy,kt)≥Q(fx,gy,gy,t)∗Q(fx,gy,gy, t)∗Q(fx,gy,gy,t)∗Q(fx,fx,fx,t)
≥Q(fx, gy, gy ,t )∗1
≥Q(fx, gy, gy ,t )
Therefore f x = g y i.e. fx=Tx=gy=Sy.
Suppose that there is another point z such that f z=T z then by (1) we have f z = Tz =g y = S y , So f x = f
z and w = f x = T x is the unique point of coincidence of f and g by Lemma 2.10 w is the only common fixed
point of f and g. Similarly there is a unique point z∈ . such that z= gz =Sz.
Assume that w≠ * . We have
Q(w,z,z,kt) = Q(fw,gz,gz,kt)
≥Q(Tw,Sz,Sz,t)∗Q(Tw,gz,gz,t) ∗Q(fw.Sz,Sz,t) ∗ Q(fw,Tw,Tw,t)
≥Q(w,z ,z, t)∗Q(w ,z ,z ,t) ∗Q(w, z, z, t) ∗Q(w, w, w, t)
≥Q(w,z ,z, t)∗1
≥Q(w,z ,z, t)
Therefore we have z = w by Lemma 2.10 and z is unique common fixed point of f ,g, S and T. The uniqueness
of the fixed point holds from (1).
Theorem 3.2 : Let (X,Q,∗) be complete symmetric Q-fuzzy metric space and f ,g ,S and T be a self
mapping of X. Let the pair {f, T} and {g ,S} be owc. If there exist k ∈ (0,1) such that
Q(fx, gy, gy, kt) ≥ ∅ /01# 2
Q(Tx, Sy, Sy, t), Q(Tx, gy, gy, t)
, Q(fx. Sy, Sy, t), Q(fx, Tx, Tx, t)
56 ..................(2)
for all x,y ∈ and ∅: 80,19 → 80,19 such that ∅( ) > for all 0< t< 1,then there exist a unique common fixed
point of f ,g ,S and T.

3
Proof: The proof follows from Theorem 3.1 .
Theorem 3.3 Let (X,Q,∗) be complete symmetric Q-fuzzy metric space and f ,g ,S and T be a self mapping of
X. Let the pair {f, T} and {g, S} be owc. If there exist k ∈ (0,1) such that
Q(fx, gy, gy, kt) ≥ ∅ 2
Q(Tx, Sy, Sy, t), Q(Tx, gy, gy, t),
Q(fx. Sy, Sy, t), Q(fx, Tx, Tx, t)
5 .....(3)
For all x,y ∈ , t>0 and ∅[0,1]4
→[0,1] such that ∅(t,t,t,1) >t for all 0< t< 1 then there exist a unique common
fixed point of f ,g ,S and T.
Proof: Let the pair {f,T} and {g ,S} be owc, so there are point x ,y ∈ such that f x =Tx and gy = Sy. We
claim that f x= gy .If not by inequality (3)
Q(fx, gy, gy, kt) ≥ ∅;Q(fx, gy, gy, t), Q(fx, gy, gy, t), Q(fx, gy, gy, t), Q(fx, fx, fx, t)<
≥ ∅ {Q(fx,gy,gy,t), Q(fx,gy,gy,t), Q(fx,gy,gy,t),1}
> Q(fx,gy,gy,t)
Therefore f x = g y i.e. fx=Tx=gy=Sy.
Suppose that there is another point z such that f z=Tz then by (3) we have f z =T z =g y = S y , So f x = f z
and w = f x = T x is the unique point of coincidence of f and g by Lemma 2.10 w is the only common fixed
point of f and g. Similarly there is a unique point z∈ . such that z=gz=Sz.
Q(fx, gy, gy, kt) ≥ ∅ 2
Q(Tx, Sy, Sy, t), Q(Tx, gy, gy, t),
Q(fx. Sy, Sy, t), Q(fx, Tx, Tx, t)
5
Assume that w≠ * . We have
Q(w,z,z,t) = Q(fw,gz,gz,kt)
≥ ∅;Q(Tw, Sz, Sz, t), Q(Tw, gz, gz, t), Q(fw. Sz, Sz, t), Q(fw, Tw, Tw, t)<
≥ ∅;Q(w, z, z, t), Q(w, z, z, t), Q(w, z, z, t), Q(w, w, w, t)<
≥ ∅;Q(w, z, z, t), Q(w, z, z, t), Q(w, z, z, t), 1<
> Q(w,z,z,t)
Therefore we have z = w by Lemma 2.10 and z is unique common fixed point of f ,g, S and T. The uniqueness
of the fixed point holds from (3).
References
[1] A.Al -Thagafi and Naseer Shahzad, “Generalized I-NonexpansiveSelfmaps and Invariant Approximation”,
Acta Mathematica Sinica, English Series May, 2008, Vol.24, No.5, pp.867-876.
[2] B.Schweizer ans A.Sklar, “Statistical metric spaces”, Pacific J. Math. 10(1960), 313-334
[3] C.T. Aage, J.N.Salunke,”On fixed point theorem in Fuzzy Metric Spaces” Int. J.Open Problem Compt. Math.,
Vol. 3, No. 2, June 2010 ,pp 123-131.
[4] C.T. Aage, J.N.Salunke, ”On fixed point theorem in Fuzzy Metric Spaces Using A Control
Function”Submitted.
[5] Dhage, B.C., Generalised metric spaces and mappings with fixed point. Bull. Calcutta M ath. Soc.,84(4)
1992.:329-336
[6] George, A. and P. Veermani, On some results in fuzzy metric spaces. Fuzzy Sets Sys., 64,1994. 395-399.
[7] Guangpeng Sun and Kai Yang,“Generalized Fuzzy Metric Spaces with Properties” Research Journal of
Applied Sciences, Engineering and Technology 2(7) 2010,: 673-678,
[8] G. Jungck and B.E. Rhoades, ” Fixed Point for Occasionally Weakly Compatible Mappings” , Erratum, Fixed
Point Theory, Volume 9, No. 1,2008,383-384.
[ 9] G. Jungck and B. E. Rhoades, “Fixed Point Theorems for Occasionally Weakly compatible Mappings”,
Fixed Point Theory, Volume 7, No. 2, 2006, 287-296.
[10] Kramosil. J and J. Michalek, Fuzzy metric and statistical metric spaces. Kybernetica, 11,1975. : 326-334
[11] Lopez-Rodrigues,J.andS.Romaguera, The Hausdorff fuzzy metric on compact sets. Fuzzy Sets Sys., 147,
2004.: 273-283
[12] L.A. Zadeh,” Fuzzy sets”, Inform and Control 8 (1965), 338-353.
[13] Mustafa, Z. and B. Sims,. A new approach to generalized metric spaces. J. Nonlinear Convex Anal., 7,
2006: 289-297.
[14] R .Vasuki, “Common fixed points for R-weakly commuting maps in fuzzy metric spaces”, Indian J. Pure
Appl. Math. 30(1999), 419-423.
[15] R.P. Pant ,”Common fixed point of Four Mappings”, Bull. Cal. Math. Soc.90 (1998), 281-286.
[16] R.P. Pant ,”Common fixed point Theorems for contractive Mappings”, J. Math. Anal. Appl. 226 (1998),
251-258.
[17] R.P. Pant ,”A remark on Common fixed point of Four Mappings in a fuzzy metric space”,J. Fuzzy. Math.
12(2) (2004), 433-437.

4
[18] S.Sessa, “on a weak commutative condition in fixed point consideration”, Publ.Inst. Math(Beograd),
23(46)(1982), 146-153.

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