On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-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
289
On Fixed Point theorems in Fuzzy 2-Metric Spaces and Fuzzy 3-
Metric Spaces
Shailesh T.Patel ,Ramakant Bhardwaj,Neena Vijayvargiya,Shyam Patkar
The Research Scholar of Singhania University, Pacheri Bari (Jhunjhunu)
Truba Institutions of Engineering & I.T. Bhopal, (M.P.)
Abstract
In this paper, we give some new definitions of compatible mappings of types (I) and (II) in fuzzy-2 metric space
and fuzzy-3 metric space prove some common fixed point theorems for mappings under the condition of
compatible mappings of types (I) and (II) in complete fuzzy-2 metric space and fuzzy-3 metric space. Our results
extend, generalize and improve the corresponding results given by many authors.
Keywords: Fuzzy metric space, Fuzzy 2-metric space, fuzzy-3 metric space ,Compatible mappings, Common
fixed point.
1. Introduction
Impact of fixed point theory in different branches of mathematics and its applications is immense. The first result
on fixed points for contractive type mapping was the much celebrated Banach’s contraction principle by S.
Banach [10] in 1922. In the general setting of complete metric space, this theorem runs as the follows, Theorem
1.1(Banach’s contraction principle) Let (X, d) be a complete metric space, c∈ (0, 1) and f: X→X be a mapping
such that for each x, y∈X, d ( , ) ≤ c d(x, y) Then f has a unique fixed point a∈X, such that for each
x∈X,lim →∞ = . After the classical result, R.Kannan [11] gave a subsequently new contractive mapping to
prove the fixed point theorem. Since then a number of mathematicians have been worked on fixed point theory
dealing with mappings satisfying various type of contractive conditions. In 2002, A. Branciari [1] analyzed the
existence of fixed point for mapping f defined on a complete metric space (X,d) satisfying a general contractive
condition of integral type.
.
2 Preliminary Notes
Definition 2.1:(a) A binary operation * : [0,1]× [0,1]× [0,1] → [0,1] is a continuous t-norms if ([0,1] ,*) is an
abeelian topological nonoid with unit 1 such that
a1* b1*c1≤ a2 * b2* c2 whenever a1 ≤ a2, b1 ≤ b2 and c1≤ c2
for all a1, a2, b1,b2 ,c1, c2 are in [0,1].
Definition 2.2:(a) A 3-tuple (X,M,*) is said to be a fuzzy 2- metric space if X is an arbitrary set, * is a
continuous t-norm and M is a fuzzy set on X3
× (0,∞) satisfying the following conditions:
for all x,y,z,t є X and t1,t2,t3 > 0,
(1)M(x,y,z,t) > 0;
(2)M(x,y,z,t) = 1, t > 0 when at least two of the three points are equal
(3) M(x,y,z,t) = M(x,z,y,t) = M(y,z,x,t)
(4)M(x,y,z,t1)* M(x,u,z,t2) * M(u,y,z,t3) ≤ M(x,y,z,t1+t2+t3)
The function value M(x,y,z,t) may be interpreted as the probability that the area of triangle is less than t.
(5)M(x,y,z,.): [0,1)→[0,1] is left continuous.
Definition2.3: (a)[08] Let (X, M,*) be a fuzzy- 2 metric space.
(1) A sequence { } in fuzzy -2 metric space X is said to be convergent to a point x ∈ X (denoted by
lim
→∞
= →
if for any λ ∈ (0,1) and t > 0, there exists ∈ N such that for all n ≥ and a ∈ X, M ( , , a, t) > 1 − λ .
That is
lim
→∞
M ( , , a, t) = 1 for all a ∈ X and t > 0.
(2) A sequence { } in fuzzy- 2 metric space X is called a Cauchy sequence, if for any λ ∈ (0,1) and t > 0, there
exists ∈ N such that for all m, n ≥ and a ∈ X, M ( , %, a, t) > 1 − λ .
(3) A fuzzy- 2 metric space in which every Cauchy sequence is convergent is said to be complete.
Definition2.4: (a)[08] Self mappings A and B of a fuzzy- 2 metric space (X, M,*) is said to be compatible, if
lim
→∞
M (AB , BA , a, t) = 1 (( ∈ ) * + > 0,

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Whenever { } is a sequence in X such that
lim
→∞
A = lim
→∞
B = , - ./ , ∈ ). 0ℎ/ lim
→∞
AB = 2,.
Definition 2.5:(a) Let (X, M, *) is a fuzzy-2 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:(a)[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:(a)[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.
Lemma2.2.1: [08] Let (X, M,*) be a fuzzy- 2 metric space. If there exists q ∈ (0, 1) such that
M(x, y, z, qt + 0) ≥ M(x, y, z, t) for all x, y, z ∈ X with z ≠ x, z ≠ y and t > 0, then x = y,
Lemma 2.2.2:[4] Let X be a set, f,g owc self 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 3.1:(a)Let (X, M, *) be a complete fuzzy 2-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 q є (0,1) such that
M(Px,Ry,a,qt)≥ min{ M(Sx,Ty,a,t), M(Sx,Px,a,t), M(Ry,Ty,a,t), M(Px,Ty,a,t), M(Ry,Sx,a,t),
M(Px,Ry,a,t), M(Sx,Ty,a,t)* M(Px,Px,a,t)} ……………(1)
For all x,y є X and for all t > o, then there exists a unique point w є X such that Pw = Sw = w and a unique point
z є X 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,y є X such that Px=Sx andRy=Ty. We claim
thatPx=Ry. If not, by inequality (1)
M(Px,Ry,a,qt)≥ min{ M(Sx,Ty,a,t), M(Sx,Px,a,t), M(Ry,Ty,a,t), M(Px,Ty,a,t), M(Ry,Sx,a,t),
M(Px,Ry,a,t), M(Sx,Ty,a,t)* M(Px,Px,a,t)}
≥ min{ M(Px,Ry,a,t), M(Px,Px,a,t), M(Ty,Ty,a,t), M(Px,Ry,a,t), M(Ry,Px,a,t),
M(Px,Ry,a,t), M(Px,Ry,a,t)* M(Px,Px,a,t)}
≥ min{ M(Px,Ry,a,t), M(Px,Px,a,t), M(Ty,Ty,a,t), M(Px,Ry,a,t),
M(Px,Ry,a,t),M(Px,Ry,a,t), M(Px,Ry,a,t)*1}
=M(Px,Ry,a,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,a,qt) = M(Pw,Rz,a,qt)
≥min{ M(Sw,Tz,a,t), M(Sw,Pw,a,t), M(Rz,Tz,a,t), M(Pw,Tz,a,t), M(Rz,Sw,a,t),
M(Pw,Rz,a,t), M(Sw,Tz,a,t)* M(Pw,Pw,a,t)}
≥ min{ M(w,z,a,t), M(w,w,a,t), M(z,z,a,t), M(w,z,a,t), M(z,w,a,t),
M(w,z,a,t), M(w,z,a,t)* M(w,w,a,t)}

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=M(w,z,a,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:(a) Let (X, M, *) be a complete fuzzy 2- metric space and let P,R,S and T be self-mappings of X.
M(Px,Ry,a,qt) ≥ Ø (min{ M(Sx,Ty,a,t), M(Sx,Px,a,t), M(Ry,Ty,a,t), M(Px,Ty,a,t), M(Ry,Sx,a,t),
M(Px,Ry,a,t), M(Sx,Ty,a,t)* M(Px,Px,a,t) )……………(2)
For all x,y є Xand 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,y є X such that Px = Sx and Ry = Ty. We
claim that Px = Ry. If not, by inequality (2)
M(Px,Ry,a,qt)≥ Ø (min{ M(Sx,Ty,a,t), M(Sx,Px,a,t), M(Ry,Ty,a,t), M(Px,Ty,a,t), M(Ry,Sx,a,t),
M(Px,Ry,a,t), M(Sx,Ty,a,t)* M(Px,Px,a,t)})
>Ø (M(Px,Ry,a,t)). From Theorem 3.1
=M(Px,Ry,a,t).
M(w,z,a,qt) = M(Pw,Rz,a,qt)
≥ Ø (min{ M(Sw,Tz,a,t), M(Sw,Pw,a,t), M(Rz,Tz,a,t), M(Pw,Tz,a,t), M(Rz,Sw,a,t),
M(Pw,Rz,a,t), M(Sw,Tz,a,t)* M(Pw,Pw,a,t)})
=M(w,z,a,t). From Theorem 3.1
M(Px,Ry,a,qt) ≥ Ø ( M(Sx,Ty,a,t), M(Sx,Px,a,t), M(Ry,Ty,a,t), M(Px,Ty,a,t), M(Ry,Sx,a,t),
M(Px,Ry,a,t), M(Sx,Ty,a,t)* M(Px,Px,a,t) ) ……………(3)
For all x,y є X and 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,y є X such that Px = Sx and Ry = Ty. We
M(Px,Ry,a,qt) ≥ Ø ( M(Sx,Ty,a,t), M(Sx,Px,a,t), M(Ry,Ty,a,t), M(Px,Ty,a,t), M(Ry,Sx,a,t),
M(Px,Ry,a,t), M(Sx,Ty,a,t)* M(Px,Px,a,t) )
≥ Ø(M(Px,Ry,a,t), M(Px,Px,a,t), M(Ty,Ty,a,t), M(Px,Ry,a,t), M(Ry,Px,a,t),
M(Px,Ry,a,t), M(Px,Ry,a,t)* M(Px,Px,a,t))
≥ Ø( M(Px,Ry,a,t), M(Px,Px,a,t), M(Ty,Ty,a,t), M(Px,Ry,a,t),
M(Px,Ry,a,t),M(Px,Ry,a,t), M(Px,Ry,a,t)*1)
= Ø(M(Px,Ry,a,t), 1, 1, M(Px,Ry,a,t), M(Px,Ry,a,t),M(Px,Ry,a,t), M(Px,Ry,a,t))
=M(Px,Ry,a,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 z є X 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).
Let the pairs {P,S} and {R,T} be owc.If there exists qє(0,1) for all x,y є X and t > 0

292
M(Px,Ry,a,qt) ≥ M(Sx,Ty,a,t)* M(Sx,Px,a,t)* M(Ry,Ty,a,t)* M(Px,Ty,a,t)* M(Ry,Sx,a,t)*
M(Px,Ry,a,t)* M(Sx,Ty,a,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,a,qt) ≥ M(Sx,Ty,a,t)* M(Sx,Px,a,t)* M(Ry,Ty,a,t)* M(Px,Ty,a,t)* M(Ry,Sx,a,t)*
M(Px,Ry,a,t)* M(Sx,Ty,a,t)
= M(Px,Ry,a,t)* M(Px,Px,a,t)* M(Ty,Ty,a,t)* M(Px,Ry,a,t)* M(Ry,Px,a,t)*
M(Px,Ry,a,t)* M(Px,Ry,a,t)
= M(Px,Ry,a,t)* 1* 1* M(Px,Ry,a,t)* M(Ry,Px,a,t)*M(Px,Ry,a,t)* M(Px,Ry,a,t)
> M(Px,Ry,a,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:(a) Let (X, M, *) be a complete fuzzy 2- 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 q є (0,1) for all x,y є X and t > 0
M(Px,Ry,a,qt) ≥ M(Sx,Ty,a,t)* M(Sx,Px,a,t)* M(Ry,Ty,a,t)* M(Px,Ty,a,t)* M(Ry,Sx,a,2t)*
M(Px,Ry,a,t)* M(Sx,Ty,a,t) …………………(5)
Proof: We have
M(Px,Ry,a,qt) ≥ M(Sx,Ty,a,t)* M(Sx,Px,a,t)* M(Ry,Ty,a,t)* M(Px,Ty,a,t)* M(Ry,Sx,a,2t)*
M(Px,Ry,a,t)* M(Sx,Ty,a,t)
≥ M(Sx,Ty,a,t)* M(Sx,Px,a,t)* M(Ry,Ty,a,t)* M(Px,Ty,a,t)* M(Sx,Ty,a,t)*
M(Ty,Ry,a,t)* M(Px,Ry,a,t)* M(Sx,Ty,a,t)
≥ M(Sx,Ty,a,t)* M(Sx,Px,a,t)* M(Ry,Ty,a,t)* M(Px,Ty,a,t) * M(Px,Ry,a,t)*
M(Sx,Ty,a,t)
= M(Px,Ry,a,t)* M(Px,Px,a,t)* M(Ty,Ty,a,t)* M(Px,Ry,a,t)* M(Ry,Px,a,t)*
= M(Px,Ry,a,t)* 1* 1* M(Px,Ry,a,t)* M(Ry,Px,a,t)*
>M(Px,Ry,a,t).
And therefore from theorem 3.4, P,R,S and T have a common fixed point.
Corollary 3.6:(a)Let (X, M, *) be a complete fuzzy 2-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,yє X and t > 0
M(Px,Ry,a,qt) ≥ M(Sx,Ty,a,t) …………………(6)
Proof: The Proof follows from Corollary 3.5
Theorem 3.7:(a) Let (X, M, *) be a complete fuzzy 2- 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,a,qt) ≥ M(Sx,Ty,a,t)* M(Sx,Px,a,t)* M(Py,Ty,a,t)* M(Px,Ty,a,t)* M(Py,Sx,a,t)
…………………(7)
Then P,S and T havea unique common fixed point.
Proof: Since compatible implies ows, the result follows from Theorem 3.4

293
Theorem 3.8:(a) Let (X, M, *) be a complete fuzzy 2- metric space and let P and R be self-mappings of X. Let
the P and R areowc.If there exists q є (0,1) for all x,y є X and t > 0
M(Sx,Sy,a,qt) ≥αM(Px,Py,a,t)+β min{M(Px,Py,a,t), M(Sx,Px,a,t), M(Sy,Py,a,t), M(Sx,Py,a,t)}
…………………(8)
For all x,y є 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,a,qt) ≥ α M(Px,Py,a,t) + β min{M(Px,Py,a,t) , M(Sx,Px,a,t), M(Sy,Py,a,
M(Sx,Py,a,t)}
=α M(Sx,Sy,a,t) + β min{M(Sx,Sy,a,t) , M(Sx,Sx,a,t), M(Sy,Sy,a,t), M(Sx,Sy,a,t)}
=(α+β)M(Sx,Sy,a,t)
A contradiction, since (α+β) > 1.Therefore Sx = Sy. Therefore Px = Py and Px is unique.
From lemma 2.2.2 , P and S have a unique fixed point.
Definition 3.1:(b)[09]: The 3- tuple (X, M,*) is called a fuzzy-3 metric space if X is an arbitrary set, * is a
continuous
t-norm and M is a fuzzy set in )4
× [0, ∞) satisfying the following conditions, for all x, y, z, w, u ∈ X and
+7, +8, +9, +4 > 0.
(1) M(x, y, z, w, 0) = 0,
(2) M(x, y, z, w, t) =1 for all t > 0, [only when the three simplex (x, y, z, w) degenerate]
(3) M(x, y, z, t) = M(x,w, z, y, t) = M( y, z,w, x, t) = M(z,w, x, y, t) = ......
(4) M(x, y, z, w,+7 + +8 + +9 + +4) ≥ M(x, y, z, w, +7 ) * M(x, y, z, w,+8) * M(x, y, z, w,+9) * M(x, y, z, w,+4)
(5) M(x, y, z, w): [0, 1) → [0, 1] is left continuous.
Definition 3.2:(b)[09]: Let (X, M,*) be a fuzzy 3-metric space, then
(1) A sequence {x<} in fuzzy 3-metric space X is said to be convergent to a point x∈ X if
lim
→∞
M ( , , a, b, t) = 1 (( , > ∈ ) * + > 0,
(2) A sequence {x<} in fuzzy 3-metric space X is called a Cauchy sequence, if
lim
→∞
M ( ?@, , a, b, t) = 1 (( , > ∈ ) * + > 0, A > 0.
(3) A fuzzy 3-metric space in which every Cauchy sequence is convergent is said to be complete.
Definition 3.3:(b)[09]: A function M is continuous in fuzzy 3-metric space iff whenever x< → x , y< → y, then
lim
→∞
M ( , y<, a, b, t) = M ( , , a, b, t) (( , > ∈ ) * + > 0.
Definition 3.4:(b)[09]: Two mappings A and S on fuzzy 3-metric space are weakly commuting iff M(ASu,
SAu,a,b,t) ≥ M(Au, Su,a,b,t) for all u, a, b∈ X and t > 0 .
3 Main Results
Theorem 3.1(b)Let (X, M, *) be a complete fuzzy 3-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 q є (0,1) such that
M(Px,Ry,a,b,qt)≥ min{ M(Sx,Ty,a,b,t), M(Sx,Px,a,b,t), M(Ry,Ty,a,b,t), M(Px,Ty,a,b,t), M(Ry,Sx,a,b,t),
M(Px,Ry,a,b,t), M(Sx,Ty,a,b,t)* M(Px,Px,a,b,t)} ……………(1)
For all x,y є X and for all t > o, then there exists a unique point w є X such that Pw = Sw = w and a unique point
z є X 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,y є X such that Px = Sx andRy = Ty. We
M(Px,Ry,a,b,qt)≥ min{ M(Sx,Ty,a,b,t), M(Sx,Px,a,b,t), M(Ry,Ty,a,b,t), M(Px,Ty,a,b,t),
M(Ry,Sx,a,b,t),M(Px,Ry,a,b,t), M(Sx,Ty,a,b,t)*M(Px,Px,a,b,t)}

294
= min{ M(Px,Ry,a,b,t), M(Px,Px,a,b,t), M(Ty,Ty,a,b,t), M(Px,Ry,a,b,t), M(Ry,Px,a,b,t),
M(Px,Ry,a,b,t), M(Px,Ry,a,b,t)* M(Px,Px,a,b,t)}
= min{ M(Px,Ry,a,b,t), M(Px,Px,a,b,t), M(Ty,Ty,a,b,t), M(Px,Ry,a,b,t), M(Ry,Px,a,b,t),
M(Px,Ry,a,b,t), M(Px,Ry,a,b,t)*1}
=M(Px,Ry,a,b,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 2.2.2 w is the only common fixed point of P and S.Similarly there is a unique point z є X such that z =
Rz = Tz.
M(w,z,a,b,qt) = M(Pw,Rz,a,b,qt)
≥min{ M(Sw,Tz,a,b,t), M(Sw,Pw,a,b,t), M(Rz,Tz,a,b,t), M(Pw,Tz,a,b,t), M(Rz,Sw,a,b,t),
M(Pw,Rz,a,b,t), M(Sw,Tz,a,b,t)* M(Pw,Pw,a,b,t)}
= min{ M(w,z,a,b,t), M(w,w,a,b,t), M(z,z,a,b,t), M(w,z,a,b,t), M(z,w,a,b,t),
M(w,z,a,b,t), M(w,z,a,b,t)* M(w,w,a,b,t)}
=M(w,z,a,b,t).
Theorem 3.2(b) Let (X, M, *) be a complete fuzzy 3- metric space and let P,R,S and T be self-mappings of X.
M(Px,Ry,a,b,qt) ≥ Ø (min{ M(Sx,Ty,a,b,t), M(Sx,Px,a,b,t), M(Ry,Ty,a,b,t), M(Px,Ty,a,b,t), M(Ry,Sx,a,b,t),
M(Px,Ry,a,b,t), M(Sx,Ty,a,b,t)* M(Px,Px,a,b,t)})……………(2)
For all x,y є X and 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,y є X such that Px = Sx and Ry = Ty. We
M(Px,Ry,a,b,qt) ≥ Ø (min{ M(Sx,Ty,a,b,t), M(Sx,Px,a,b,t), M(Ry,Ty,a,b,t), M(Px,Ty,a,b,t), M(Ry,Sx,a,b,t),
M(Px,Ry,a,b,t), M(Sx,Ty,a,b,t)* M(Px,Px,a,b,t)})
>Ø (M(Px,Ry,a,b,t)). From Theorem 3.1
=M(Px,Ry,a,b,t).
M(w,z,a,b,qt) = M(Pw,Rz,a,b,qt)
≥ Ø (min{ M(Sw,Tz,a,b,t), M(Sw,Pw,a,b,t), M(Rz,Tz,a,b,t), M(Pw,Tz,a,b,t), M(Rz,Sw,a,b,t),
M(Pw,Rz,a,b,t), M(Sw,Tz,a,b,t)* M(Pw,Pw,a,b,t)})
=M(w,z,a,b,t). From Theorem 3.1
M(Px,Ry,a,b,qt) ≥ Ø ({ M(Sx,Ty,a,b,t), M(Sx,Px,a,b,t), M(Ry,Ty,a,b,t), M(Px,Ty,a,b,t), M(Ry,Sx,a,b,t),
M(Px,Ry,a,b,t), M(Sx,Ty,a,b,t)* M(Px,Px,a,b,t)}) ……………(3)
For all x,y є X and 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.

295
M(Px,Ry,a,b,qt) ≥ Ø ({ M(Sx,Ty,a,b,t), M(Sx,Px,a,b,t), M(Ry,Ty,a,b,t), M(Px,Ty,a,b,t), M(Ry,Sx,a,b,t),
M(Px,Ry,a,b,t), M(Sx,Ty,a,b,t)* M(Px,Px,a,b,t)})
= Ø ({M(Px,Ry,a,b,t), M(Px,Px,a,b,t), M(Ty,Ty,a,b,t), M(Px,Ry,a,b,t), M(Ry,Px,a,b,t),
M(Px,Ry,a,b,t), M(Px,Ry,a,b,t)* M(Px,Px,a,b,t)})
= Ø ({M(Px,Ry,a,b,t), M(Px,Px,a,b,t), M(Ty,Ty,a,b,t), M(Px,Ry,a,b,t), M(Ry,Px,a,b,t),
M(Px,Ry,a,b,t), M(Px,Ry,a,b,t)*1})
=M(Px,Ry,a,b,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 z є X 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).
M(Px,Ry,a,b,qt) ≥ M(Sx,Ty,a,b,t)*M(Sx,Px,a,b,t)*M(Ry,Ty,a,b,t)*M(Px,Ty,a,b,t)*
M(Ry,Sx,a,b,t)*M(Px,Ry,a,b,t)* M(Sx,Ty,a,b,t) ………………… (4)
We have
M(Ry,Sx,a,b,t)*M(Px,Ry,a,b,t)* M(Sx,Ty,a,b,t)
= M(Px,Ry,a,b,t)* M(Px,Px,a,b,t)* M(Ty,Ty,a,b,t)* M(Px,Ry,a,b,t)*
M(Ry,Px,a,b,t)*M(Px,Ry,a,b,t)* M(Px,Ry,a,b,t)
= M(Px,Ry,a,b,t)* 1* 1* M(Px,Ry,a,b,t)* M(Ry,Px,a,b,t)*M(Px,Ry,a,b,t)*
M(Px,Ry,a,b,t)
> M(Px,Ry,a,b,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 z є X such that z = Rz = Tz.Thus w is a common fixed point of P,R,S and T.
Corollary 3.5(b) Let (X, M, *) be a complete fuzzy 3- 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 q є (0,1) for all x,y є X and t > 0
M(Ry,Sx,a,b,t)*M(Px,Ry,a,b,t)* M(Sx,Ty,a,b,t)…………………(5)
Then there exists a unique common fixed point of P,R,S and T.
Proof: We have
M(Ry,Sx,a,b,t)*M(Px,Ry,a,b,t)* M(Sx,Ty,a,b,t)
= M(Px,Ry,a,b,t)* M(Px,Px,a,b,t)* M(Ty,Ty,a,b,t)* M(Px,Ry,a,b,t)*
M(Ry,Px,a,b,t)*M(Px,Ry,a,b,t)* M(Px,Ry,a,b,t)
= M(Px,Ry,a,b,t)* 1* 1* M(Px,Ry,a,b,t)* M(Ry,Px,a,b,t)*M(Px,Ry,a,b,t)*
M(Px,Ry,a,b,t)
> M(Px,Ry,a,b,t).
And therefore from theorem 3.4, P,R,S and T have a common fixed point.
Corollary 3.6(b) Let (X, M, *) be a complete fuzzy 3-metric space and let P,R,S and T be self-mappings of X.
M(Px,Ry,a,qt) ≥ M(Sx,Ty,a,t) …………………(6)

296
Proof: The Proof follows from Corollary 3.5
Theorem 3.7(b) Let (X, M, *) be a complete fuzzy 3- 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 q є (0,1) such that for all x,y є X and t > 0
M(Px,Py,a,b,qt) ≥ M(Sx,Ty,a,b,t)* M(Sx,Px,a,b,t)* M(Py,Ty,a,b,t)* M(Px,Ty,a,b,t)*
M(Py,Sx,a,b,t)…………………(7)
Then P,S and T have a unique common fixed point.
Proof: Since compatible implies ows, the result follows from Theorem 3.4
Theorem 3.8(b) Let (X, M, *) be a complete fuzzy 3- metric space and let P and R be self-mappings of X. Let
the P and R are owc. If there exists q є (0,1) for all x,y є X and t > 0
M(Sx,Sy,a,b,qt) ≥α M(Px,Py,a,b,t)+β min{M(Px,Py,a,b,t), M(Sx,Px,a,b,t), M(Sy,Py,a,b,t),
M(Sx,Py,a,b,t)} …………………(8)
For all x,y є 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,a,b,qt) ≥α M(Px,Py,a,b,t)+β min{M(Px,Py,a,b,t), M(Sx,Px,a,b,t), M(Sy,Py,a,b,t),
M(Sx,Py,a,b,t)}
=α M(Sx,Sy,a,b,t) + β min{M(Sx,Sy,a,b,t) , M(Sx,Sx,a,b,t), M(Sy,Sy,a,b,t),
M(Sx,Sy,a,b,t)}
=(α+β)M(Sx,Sy,a,b,t)
A contradiction, since (α+β) > 1.Therefore Sx = Sy. Therefore Px = Py and Px is unique.
From lemma 2.2.2 , 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
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On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spaces

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