On fixed point theorems in fuzzy metric spaces

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
274
On Fixed Point theorems in Fuzzy Metric Spaces
Shailesh T.Patel ,Ramakant Bhardwaj*,Rakesh Shrivastava**,Shyam Patkar*,Sanjay Choudhary***
The Research Scholar of Singhania University, Pacheri Bari (Jhunjhunu)
*Truba Institutions of Engineering & I.T. Bhopal, (M.P.)
**JNCT, Bhopal.
***Prof.&Head Deptt.of Mathematics Govt.NMV Hoshangabad.
Abstract: This paper presents some common fixed point theorems for occasionally weakly compatible mappings
in fuzzy metric spaces.
Keywords: Occasionally weakly compatible mappings,fuzzy metric space.
1. Introduction
Fuzzy set was defined by Zadeh [7]. Kramosil and Michalek [5] introduced fuzzy metric space, George and
Veermani [2] modified the notion of fuzzy metric spaces with the help of continuous t-norms. Many researchers
have obtained common fixed point theorems for mappings satisfying different types.introduced the new concept
continuous mappings and established some common fixed point theorems.open problem on the existence of
contractive definition which generates a fixed point but does not force the mappings to be continuous at the fixed
point.this paper presents some common fixed point theorems for more general .
2 Preliminary Notes
Definition 2.1 [7] A fuzzy set A in X is a function with domain X and values in [0,1].
Definition 2.2 [6] A binary operation * : [0,1]× [0,1]→[0,1] is a continuous t-norms if *is satisfying conditions:
(1) *is an commutative and associative;
(2) * is continuous;
(3) a *1 = a forall a ϵ [0,1];
(4) a * b ≤ c * d whenever a ≤ c and b ≤ d, and a,b,c,d є [0,1].
Definition 2.3 [2] A 3-tuple (X,M,*) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-
norm and M is a fuzzy set on X2
× (0,∞) satisfying the following conditions, for all x,y,z є X, s,t>0,
(f1)M(x,y,t) > 0;
(f2)M(x,y,t) = 1 if and only if x = y;
(f3) M(x,y,t) = M(y,x,t);
(f4)M(x,y,t)* M(y,z,s) ≤ M(x,z,t+s) ;
(f5)M(x,y,.): (0,∞)→(0,1] is continuous.
Then M is called a fuzzy metric on X.Then M(x,y,t) denotes the degree of nearness between x and y with respect
to t.
Definition 2.4[2]Let (X,d) be a metric space.Denotea * b = ab for all a,bє [0,1] and Md be fuzzy sets onX2
× (0,∞)
defined as follows:
Md(x,y,t)= ),( yxdt
t
+
.
Then (X, Md, *) is a fuzzy metric space.Wecall this fuzzy metric induced by a metric d as the standard
intuitionistic fuzzy metric.
Definition 2.5[2]Let (X, M, *) is a fuzzy metric space.Then
(a) a sequence {xn} in X is said to convers to x in X if for each є>o and each t>o, Nno ∈∃ such
That M(xn,x,t)>1-є for all n≥no.
(b)a sequence {xn} in X is said to cauchy to if for each ϵ>o and each t>o, Nno ∈∃ such
That M(xn,xm,t)>1-є for all n,m≥no.
(c) A fuzzy metric space in which euery Cauchy sequence is convergent is said to be complete.Definition 2.6[3]
Two self mappings f and g of a fuzzy metric space (X,M,*) are called compatible if
1),,(lim =
∞→
tgfxfgxM nn
n
whenever {xn} is a sequencein X such that xgxfx n
n
n
n
==
∞→∞→
limlim
For some x in X.
Definition 2.7[1]Twoself mappings f and g of a fuzzy metric space (X,M,*) are called reciprocally continuous on
X if fxfgxn
n
=
∞→
lim and gxgfxn
n
=
∞→
lim whenever {xn} is a sequence in X such that
xgxfx n
n
n
n
==
∞→∞→
limlim for some x in X.

275
Lemma 2.8[4] Let X be a set, f,gowcself maps of X. If f and g have a unique point of coincidence, w = fx = gx,
then w is the unique common fixed point of f and g.
3 Main Results
Theorem 3.1Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) such that
∫
).,(
0
)(
qtRyPxM
dttξ
M(Px,Ry,qt)≥ min{ M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)} ……………(1)
For all x,yєX and for all t>o, then there exists a unique point wєX such that Pw = Sw = w and a unique point zєX
such that Rz = Tz = z. Moreover z = w so that there is a unique common fixed point of P,R,S and T.
Proof :Let the pairs {P,S} and {R,T} be owc, so there are points x,yϵX such that Px=Sx andRy=Ty. We claim
thatPx=Ry. If not, by inequality (1)
M(Px,Ry,qt) ≥ min{ M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)}
M(Px,Ry,qt) ≥ min{ M(Px,Ry,t), M(Px,Px,t), M(Ty,Ty,t), M(Px,Ry,t), M(Ry,Px,t),
M(Px,Ry,t), M(Px,Ry,t)* M(Px,Px,t)}
≥ min{ M(Px,Ry,t), M(Px,Px,t), M(Ty,Ty,t), M(Px,Ry,t),
M(Px,Ry,t),M(Px,Ry,t), M(Px,Ry,t)*1}
=M(Px,Ry,t).
Therefore Px = Ry, i.e. Px = Sx = Ry = Ty. Suppose that there is a another point z such that Pz = Sz then by
(1) we have Pz = Sz = Ry = Ty, so Px=Pz and w = Px = Sx is the unique point of coincidence of P and S.By
Lemma 2.8 w is the only common fixed point of P and S.Similarly there is a unique point zєX such that z = Rz =
Tz.
Assume that w ≠ z. we have
M(w,z,qt) = M(Pw,Rz,qt)
≥min{ M(Sw,Tz,t), M(Sw,Pw,t), M(Rz,Tz,t), M(Pw,Tz,t), M(Rz,Sw,t),
M(Pw,Rz,t), M(Sw,Tz,t)* M(Pw,Pw,t)}
≥ min{ M(w,z,t), M(w,w,t), M(z,z,t), M(w,z,t), M(z,w,t),
M(w,z,t), M(w,z,t)* M(w,w,t)}
=M(w,z,t).
Therefore we have z = w and z is a common fixed point of P,R,S and T. The uniqueness of the fixed point holds.
Theorem 3.2 Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
M(Px,Ry,qt) ≥ Ø( min{ M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)}) ……………(2)
For all x,yєXand Ø: [0,1]→[0,1] such that Ø(t) > t for all 0<t<1, then there existsa unique common fixed point of
P,R,S and T.
Proof :Let the pairs {P,S} and {R,T} be owc, so there are points x,yєX such that Px = Sx and Ry = Ty. We
claim that Px = Ry. If not, by inequality (2)
M(Px,Ry,qt) ≥ Ø( min{ M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)})
>Ø(M(Px,Ry,t)). From Theorem 3.1
=M(Px,Ry,t).
Assume that w ≠ z. we have
M(w,z,qt) = M(Pw,Rz,qt)
≥min{ M(Sw,Tz,t), M(Sw,Pw,t), M(Rz,Tz,t), M(Pw,Tz,t), M(Rz,Sw,t),

276
M(Pw,Rz,t), M(Sw,Tz,t)* M(Pw,Pw,t)}
=M(w,z,t). From Theorem 3.1
Therefore we have z = w and z is a common fixed point of P,R,S and T. The uniqueness of the fixed point holds.
M(Px,Ry,qt) ≥ Ø(M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t)) ……………(3)
For all x,yєXand Ø: [0,1]7
→[0,1] such that Ø(t,1,1,t,t,1,t) > t for all 0<t <1, then there exists a unique common
fixed point of P,R,S and T.
Proof: Let the pairs {P,S} and {R,T} be owc, so there are points x,yϵX such that Px = Sx and Ry = Ty. We
M(Px,Ry,qt) ≥ Ø(M(Sx,Ty,t), M(Sx,Px,t), M(Ry,Ty,t), M(Px,Ty,t), M(Ry,Sx,t),
M(Px,Ry,t), M(Sx,Ty,t)* M(Px,Px,t))
M(Px,Ry,qt) ≥ Ø(M(Px,Ry,t), M(Px,Px,t), M(Ty,Ty,t), M(Px,Ry,t), M(Ry,Px,t),
M(Px,Ry,t), M(Px,Ry,t)* M(Px,Px,t))
= Ø(M(Px,Ry,t), M(Px,Px,t), M(Ty,Ty,t), M(Px,Ry,t),
M(Px,Ry,t),M(Px,Ry,t), M(Px,Ry,t)*1)
= Ø(M(Px,Ry,t), 1, 1, M(Px,Ry,t), M(Px,Ry,t),M(Px,Ry,t), M(Px,Ry,t))
>M(Px,Ry,t).
A contradiction, therefore Px = Ry, i.e. Px = Sx = Ry = Ty. Suppose that there is a another point z such that Pz =
Sz then by (3) we have Pz = Sz = Ry = Ty, so Px=Pz and w = Px = Sx is the unique point of coincidence of P
and S.By Lemma 2.8 w is the only common fixed point of P and S.Similarly there is a unique point zϵX such
that z = Rz = Tz.Thus z is a common fixed point of P,R,S and T. The uniqueness of the fixed point holds from
(3).
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) for all x,yєX and t > 0
M(Px,Ry,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Ry,Sx,t)*
M(Px,Ry,t)* M(Sx,Ty,t) ………………… (4)
Then there existsa unique common fixed point of P,R,S and T.
Proof: Let the pairs {P,S} and {R,T} be owc, so there are points x,yєX such that Px = Sx and Ry = Ty. We
We have
M(Px,Ry,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Ry,Sx,t)*
M(Px,Ry,t)* M(Sx,Ty,t)
= M(Px,Ry,t)* M(Px,Px,t)* M(Ty,Ty,t)* M(Px,Ry,t)* M(Ry,Px,t)*
M(Px,Ry,t)* M(Px,Ry,t)
= M(Px,Ry,t)* 1* 1* M(Px,Ry,t)* M(Ry,Px,t)*
>M(Px,Ry,t).
Thus we have Px = Ry, i.e. Px = Sx = Ry = Ty. Suppose that there is a another point z such that Pz = Sz then by
(4) we have Pz = Sz = Ry = Ty, so Px=Pz and w = Px = Sx is the unique point of coincidence of P and
S.Similarly there is a unique point zϵX such that z = Rz = Tz.Thus w is a common fixed point of P,R,S and T.
Corollary 3.5 Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) for all x,yϵX and t > 0
M(Px,Ry,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Ry,Sx,2t)*
M(Px,Ry,t)* M(Sx,Ty,t) …………………(5)
Proof: We have
M(Px,Ry,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Ry,Sx,2t)*
≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t)* M(Sx,Ty,t)* M(Ty,Ry,t)*
≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Ry,Ty,t)* M(Px,Ty,t) * M(Px,Ry,t)*

277
M(Sx,Ty,t)
= M(Px,Ry,t)* M(Px,Px,t)* M(Ty,Ty,t)* M(Px,Ry,t)* M(Ry,Px,t)*
= M(Px,Ry,t)* 1* 1* M(Px,Ry,t)* M(Ry,Px,t)*
>M(Px,Ry,t).
And therefore from theorem 3.4, P,R,S and T have a common fixed point.
Corollary 3.6 Let (X, M, *) be a complete fuzzy metric space and let P,R,S and T be self-mappings of X. Let the
pairs {P,S} and {R,T} be owc.If there exists qє(0,1) for all x,yєX and t > 0
M(Px,Ry,qt) ≥ M(Sx,Ty,t) …………………(6)
Proof: The Proof follows from Corollary 3.5
Theorem 3.7 Let (X, M, *) be a complete fuzzy metric space.Then continuous self-mappings S and T of X have
a common fixed point in X if and only if there exites a self mapping P of X such that the following conditions
are satisfied
(i) PX ⊂ TX I SX
(ii) The pairs {P,S} and {P,T} are weakly compatible,
(iii) There exists a point qє(0,1) such that for all x,yєX and t > 0
M(Px,Py,qt) ≥ M(Sx,Ty,t)* M(Sx,Px,t)* M(Py,Ty,t)* M(Px,Ty,t)* M(Py,Sx,t)
…………………(7)
Then P,S and T havea unique common fixed point.
Proof: Since compatible implies ows, the result follows from Theorem 3.4
Theorem 3.8 Let (X, M, *) be a complete fuzzy metric space and let P and R be self-mappings of X. Let the P
and R areowc.If there exists qє(0,1) for all x,yєX and t > 0
M(Sx,Sy,qt) ≥αM(Px,Py,t)+β min{M(Px,Py,t), M(Sx,Px,t), M(Sy,Py,t), M(Sx,Py,t)}
…………………(8)
For all x,yϵ X where α,β> 0, α+β> 1. Then P and S have a unique common fixed point.
Proof: Let the pairs {P,S} be owc, so there are points x єX such that Px = Sx. Suppose that exist another point y
єX for whichPy = Sy. We claim that Sx = Sy. By inequality (8)
We have
M(Sx,Sy,qt) ≥αM(Px,Py,t)+ β min{M(Px,Py,t) , M(Sx,Px,t), M(Sy,Py,t), M(Sx,Py,t)}
=αM(Sx,Sy,t)+ β min{M(Sx,Sy,t) , M(Sx,Sx,t), M(Sy,Sy,t), M(Sx,Sy,t)}
=(α+β)M(Sx,Sy,t)
A contradiction, since (α+β)> 1.Therefore Sx = Sy. Therefore Px = Py and Px is unique.
From lemma2.8 , P and S have a unique fixed point.
Acknowledgement: One of the author (Dr. R.K. B.) is thankful to MPCOST Bhopal for the project No 2556
References
[1]P.Balasubramaniam,S.Murlisankar,R.P.Pant,”Common fixed points of four mappings in a fuzzy metric
spaces”,J.Fuzzy Math. 10(2) (2002), 379-384.
[2]A.George, P.Veeramani,”On some results in fuzzy metric spaces”,Fuzzy Sets and Systems, 64 (1994), 395-
399.
[3]G.Jungck,”Compatible mappings and common fixed points (2)”,Internat.J.Math.Sci. (1988), 285-288.
[4]G.Jungck and B.E.Rhoades,”Fixed Point Theorems for Occasionally Weakly compatible Mappings”,Fixed
Point Theory, Volume 7, No. 2, 2006, 287-296.
[5]O.Kramosil and J.Michalek,”Fuzzy metric and statistical metric spaces”,Kybernetika, 11 (1975), 326-334.
[6]B.Schweizer and A.Sklar,”Statistical metric spaces”,Pacific J. Math.10 (1960),313-334
[7]L.A.Zadeh, Fuzzy sets, Inform and Control 8 (1965), 338-353.

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On fixed point theorems in fuzzy metric spaces

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