On fixed point theorem 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
148
On Fixed Point Theorem in Fuzzy Metric Spaces
Pankaj Tiwari1
, Rajesh Shrivastava1
, Rajendra Kumar Dubey2
1
Govt. Science & Comm. College Benjeer, Bhopal M.P., India
2
Govt. Science P.G. College, Rewa M.P., India
Abstract: - The Purpose of this paper, we prove common fixed point theorem using new continuity condition in
fuzzy metric spaces.
Keywords: - Compatible maps, R-weakly commuting maps, reciprocal continuity.
Mathematics subject classification: - 47H10, 54H25.
1 INTRODUCTION
The concept of fuzzy sets first introduced by Prof. Lofty Zadeh [20] in 1965 at University of California and
developed a basic frame work to treat mathematically the fuzzy phenomena or systems which due to in trinsic
indefiniteness, cannot themselves be characterized precisely. Fuzzy metric spaces have been introduced by
Kramosil and Michalek [7] and George and Veersamani [3] modified the notion of fuzzy metric with help of
continuous t-norms. Recently many have proved fixed point theorems involving fuzzy sets [1, 2, 4-6, 8-10, 14,
16-19]. Vasuki [19] investigated same fixed point theorems in fuzzy metric spaces for R-weakly commuting
mappings and part [12] introduced the notion of reciprocal continuity of mappings in metric spaces.
Balasubramaniam et al. and S. Muralishankar, R.P. Pant [1] proved the open problem of Rhoades [15] on the
existence of a contractive definition which generals a fixed point but does not force the mapping to be
continuous at the fixed point possesses an affirmative answer.
The purpose of this paper is to prove fixed point theorem in fuzzy metric spaces for using new continuity
condition in fuzzy metric spaces.
2 PRELIMINARIES
Before starting the main result we need some basic definitions and basic results
Definition 2.1: A fuzzy set A in x is a function with domain X and values in [0, 1]
Definition 2.2: A binary operation *: [0.1] → [0.1] is called a continuous t-norm of ([0.1], *) is an abelian
topological monoid with the unit 1 such that a * b ≤ C * d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1].
Example: Two typical examples of continuous t-norm are
(a) ∗ = ,
(b) ∗ = ( , )
Definition 2.3: A 3-tuple (x, M, *) is called a fuzzy metric space if x is an arbitrary (non-empty) set, * is a
continuous t-norm and M is a fuzzy set on
x2
×[0,∞) satisfying the following conditions for each x, y, z ∈ x and t, s > 0
( 1) ( , , 0) > 0;
( 2) ( , , ) = 1, ∀ > 0, = ;
( 3) ( , , ) = ( , , );
( 4) ( , , ) ∗ ( , , ) ≤ ( , , + );
( 5) ( , , . ): (0, ∞) → %0,1& ' ( ) )
( 6)
( ) xyxtyxM
t
∈∀=
∞→
,1,,lim
Then M is called a fuzzy metric on X. A function M(x, y, t) denote the degree of nearness between x and y with
respect to t.
Example: (Induced Fuzzy metric) [3] every metric space indices a fuzzy metric space. Let (x, d) be a metric
space
Define a * b = ab
And
( )
( )yxmdkt
kt
tyxM n
n
,
,,
+
=
K, m, n, t∈R+
. Then (x, M, *) is a fuzzy metric space if we put k = m – n = 1.
We get
( )
( )yxdt
t
tyxM
,
,,
+
=
The fuzzy metric induced by a metric d is referred to as a standard fuzzy metric.
Proposition 2.4 [21] in a fuzzy metric space (X, m, *), if a * a ≥ for all

149
a ∈[0, 1]. Then a * b = min {a, b} for all a, b ∈[0, 1].
Definition 2.5 ([2]): Two self mappings F and S of a fuzzy metric space (x, M, *) are called compatible if ∞→t
lim
M (FSxn, SFxn, t) = 1 when ever { *} is a sequence in x such that
xSxFx n
t
n
t
==
∞→∞→
limlim
for some x in X.
Definition 2.6 ([19]): Two self mappings A and S of a fuzzy metric space (x, M, *) are called weakly
commuting if M (Fsx, SFx, t) ≥ M (Fx, Sx, t) ∀ x in x and t > 0.
Definition 2.7 ([19]): Two self mappings A and S of a fuzzy metric space (x, M, *) are called point wise R-
weakly commuting if there exist R > 0 such that
M (Fsx, SFx, t) ≥ M (Fx, sx, t/R) for all x in X and t > 0.
Remark 1: Clearly, point R-weakly commutativity implies weak commutativity only when R ≤ 1.
Definition 2.8 ([1]): Two self maps F and S of a fuzzy metric space (x, M, *) are called reciprocally continuous
on x if
FxFsxn
t
=
∞→
lim
and
SxFxn
t
=
∞→
lim
when ever { *} is a sequences in x such that
xSxFx n
t
n
t
==
∞→∞→
limlim
for some x in X.
Lemma 2.9 ([16]): Let (X, M, *) be a fuzzy metric space. If there exists k∈(0, 1) such that M(x, y, kt) > M(x, y,
t) Then x = y.
Lemma-2.10 ([2]): Let {yn} be a sequence in a fuzzy metric space (X, M, *) with the condition (6). If there
exists, k∈ (0, 1) such that
M (Yn, yn+1, kt) ≥ M (yn-1, yn, t)
For all t > 0 and n∈N, Then {yn} is a Cauchy sequence in X.
The following theorems are basic theorems for our result
Theorem 2.11[1]: Let (A, S) and (B, T) be point wise R-weakly commuting pairs of self mappings of complete
fuzzy metric space (X, M, *) such that
1. AX ⊂TX, BX ⊂SX
2. M (Ax, By, t) ≥ M(x, y, ht), 0 < h < 1, x, y ∈x and t > 0.
Suppose that (A, S) and (B, T) is compatible pair of reciprocally continuous mappings X. Then A, B, S and T
have a unique common fixed point.
Theorem 2.12[14]: Let (A, S) and (B, T) be point wise R-weakly commuting pairs of self mappings of complete
fuzzy metric space (X, M, *) such that
1. AX ⊂TX, BX ⊂SX
2. M (Ax, By, t) ≥ M(x, y, ht), 0 < h < 1, x, y, ∈x and t > 0.
Let (A, S) and (B, T) is compatible mappings. If any of the mappings in compatible pairs (A, S) and (B, T) is
continuous then A, B, S and T have a unique common fixed point.
Remark 2: In [14], Pant and Jha proved that the theorem 2.12 is an analogue of the theorem 2.11 by obtaining
connection between continuity and reciprocal continuity in fuzzy metric space.
Lemma 2.13 [21]: Let (X, M,*) be a complete fuzzy metric space with a*a ≥ a for all aε[0,1] and the
condition(fm6). Let (A, S) and (B, T) be point wise R-weakly commuting pairs of self mappings of X such that
(a) AX ⊂TX, BX ⊂SX
There exists k ε (0, 1) such M (Ax, By, kt) ≥ N (x, y, t) for all x, y ε X, a ε (0,2) and t >0
Then the continuity of one of the mappings in compatible pair (A, S) or (B, T) on (X, M,*) implies their
reciprocal continuity.
3 MAIN RESULTS
Theorem 3.1: Let (x, M,*) be a complete fuzzy metric space a*a ≥ a, for all a ϵ [0, 1] and the condition
definition (3). Let (L, ST) and f (M, AB) be point wise R-weakly commuting pairs of self mappings of
x such that
3.1(a). L(x) ⊆ ST(x), M(x) ⊆ AB(x)
3.2(b). There exists k∈(0,1) such that
( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMyABxFtSTyABxqFtLxABxpF
ktMySTyFktABx,LxFktMyLxF
2,,,,
,,*,2
+≥
For all x, y∈X and t > 0 where 0 <q, q < 1 such that p + q = 1.
Then A, B, S, T, L and M have a unique common fixed point in X.
Proof. Suppose x0 ∈ X. From condition (3.1.1) ∃ x1, x2 ∈ X such that
Lx0 = STx1 and Mx1 = ABx2.
Inductively, we can construct sequences {xn} and {yn} in X such that

150
y2n = Lx2n = STx2n+1 and y2n+1 = Mx2n+1 = ABx2n+2 for n = 0, 1, 2, ……
Step 1. Taking x = x2n and y = x2n+1 in (3.1.5), we have
( )( ) ( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMxABxFtSTxABxqFtLxABxpF
ktMxSTxFktMxSTxFktLxABxFkt,MxLxF
nnnnnn
nnnnnnnn
2,.,,
,.,,,*
12212222
1212121222122
2
++
+++++
+≥
( )( ) ( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktyyFtyyqFtyypF
ktyyFktyyFktyyFktyyF
nnnnnn
nnnnnnnn
2,.,,
,.,.,*,
122212122
122212212122
2
+−−
+−−+
+≥
( )( ) ( )( ) ( )( )[ ] ( ) ( )( ) ( )( )ktyyFtyyFqpktyyFktyyFktyyF nnnnnnnnnn 2,.,.,*,, 1212122122212122 +−−+−+ +≥
( )( ) ( )( ) ( )( ) ( )( )ktynFytynFyktyyFktyyF nnnnnn 2,12.,122,, 1221212122 ++−+ −−≥
Hence, we have
( )( ) ( )( )tyyFktyyF nnnn 212122 ,, −+ ≥
Similarly, we also have
( )( ) ( )( )tyyFktyyF nnnn 1222212 ,, +++ ≥
In general, for all n even or odd, we have
( )( ) ( )( )tyyFktyyF nnnn ,, 11 −+ ≥
for all x, y ∈X and t > 0. Thus by lemma 2.1 {yn} is a Cauchy sequence in X. Since (X, F, ) is complete, it
converges to a point z in X. Also its subsequences converge as follows :
{Lx2n} → z, {ABx2n} → z, {Mx2n+1} → z and {STx2n+1} → z.
Step 1. Suppose AB is continuous.
As AB is continuous and (L, AB) is semi-compatible, we get
LABx2n+2 → LZ and LABx2n+2 → ABz.
Since the limit in Memnger space is unique, we get
Lz = ABz.
Step 2. By taking x = ABx2n and y = x2n+1 in (3.1.5), we have
( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMxABABxFtSTxABABxqFtLABxABABxpF
ktMxSTxFktLABxABABxFktMxLABxF
nnnnnn
nnnnnn
2,.,,
,.,*,
12212222
121222122
2
++
+++
+≥
Taking limit n → ∞
( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] ( )( )
( )( )[ ] ( )( )ktABzzFtABzzqFp
ktABzzFtABzzqFtABzABzpFktzzFktABzABzFktABzzF
,,
2,,,,.,*,2
+≥
+≥
( )( ) ( )( ) ( )( )ktABzzqFpktABZzqFpktABzzF ,,, +≥+≥
( )( ) 1
1
, =
−
≥
q
p
ktABzzF
For k∈(0,1) and all t > 0. Thus we have
Z = ABz.
Step 3. By taking x = z and y = x2n+1 in (3.1.5), we have
( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMxABzFtSTxABzqFtLzABzpF
ktMxSTxFktLzABzFktMxLzF
nn
nnn
2,.,,
,.,*,
1212
121212
2
++
+++
+≥
Taking limit n → ∞
( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] ( )( )ktzzFtzzqFtLzzpFktzzFktLzzFktLzzF 2,.,,,.,*,2
+≥
( )( ) ( )( ) ( )( ) qtLzzpFktLzzFktLzzF +≥ ,,*,2
Noting that ( )( ) 1,2
<ktLzzF and using (c) in definition 2.2, we have
( )( ) ( )( ) qtLzzpFktLzzF +≥ ,,
( )( ) qtLzzpF +≥ ,
( )( ) 1
1
, =
−
≥
p
q
ktLzzF

151
For k ∈(0,10 and all t > 0. Thus, we have z = Lz = ABz.
Step 4. By taking z = Bz and y = x2n+1 in (3.1.5), we have
( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMxABBzFtSTxABBzqFtLBzABBzpF
ktMxSTxFktLBzABBzFktMxLBzF
nn
nnn
2,.,,
,.,*,
1212
121212
2
++
+++
+≥
Since AB = BA and BL = LB, we have
L(Bz) = B(Lz) = Bz and
AB(Bz) = B(ABz) = Bz.
Taking limit n → ∞, we have
( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] ( )( )ktBzzFtBzzqFtBzBzpFktzzFktBzBzFktBzzF 2,.,,,.,*,2
+≥
( )( ) ( )( )[ ] ( )( )ktBzzFtBzzqFpktBzzF 2,,,2
+≥
( )( )[ ] ( )( )ktBzzFtBzzqFp ,,+≥
( )( ) ( )( )tBzzqFpktBzzF ,, +≥
( )( )ktBzzqFp ,+≥
( )( ) 1
1
, =
−
≥
q
p
ktBzzF
For k∈(0,1) and all t > 0.
Thus, we have z = Bz.
Since z = ABz, we also have
Z = Az.
Therefore, z = Az = Bz = Lz.
Step 5. Since L(X) ⊆ST(X) there exists v ∈X such that z = Lz = STv.
By taking x = x2n and y = v in (3.1.5), we get
( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMvABxFtSTvABxqFtLxABxpF
ktMvSTvFktLxABxFktMvLxF
nnnn
nnn
2,.,,
,.,*,
2222
222
2
+≥
Taking limit as n → ∞, we have
( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] ( )( )ktMvzFtzzqFtzzpFktMvzFktzzFktMvzF 2,.,,,.,*,2
+≥
( )( ) ( )( ) ( ) ( )( )ktMvzFqpktMvzFktMvzF 2,,*,2
+≥
Noting that ( )( ) 1,2
≤ktMvzF and using (c) in Definition 2.2, we have
( )( ) ( )( )
( )( )tMvzF
ktMvzFktMvzF
,
2,,
≥
≥
Thus we have
Z = Mv and so z = Mv = sTv.
Since (M, ST) is weakly compatible, we have
STMv = MSTv
Thus, STz = Mz.
Step 6. By taking x = x2n, y = z in (3.1.5) and using step 5, we have
( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMvABxFtSTzABxqFtLxABxpF
ktMzSTzFktLxABxFktMzLxF
nnnn
nnn
2,.,,
,.,*,
2222
222
2
+≥
Which implies that, as n → ∞
( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] ( )( )ktMzzFtMzzqFtzzpFktMzMzFktzzFktMzzF 2,.,,,.,*,2
+≥
( )( ) ( )( )[ ] ( )( )ktMzzFtMzzqFpktMzzF 2,,,2
+≥
( )( )[ ] ( )( )ktMzzFtMzzqFp ,,+≥
( )( ) ( )( )tMzzqFpktMzzF ,, +≥
( )( )ktMzzqFP ,+≥

152
( )( ) 1
1
, =
−
≥
q
p
ktMzzF
Thus, we have z = Mz and therefore z = Az = Bz = Lz = Mz = STz.
Step 7. By taking x = x2n, y = Tz in (3.1.5), we have
( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMTzABxFtSTTzABxqFtLxABxpF
ktMTzSTTzFktLxABxFktMTzLxF
nnnn
nnn
2,.,,
,.,*,
2222
222
2
+≥
Since MT = TM and ST = TS, we have
MTz = TMz = Tz and ST(Tz) = T(STz) = Tz.
Letting n → ∞, we have
( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] ( )( )ktTzzFtTzzqFtzzpFktTzTzFktzzFktTzzF 2,.,,,.,*,2
+≥
( )( ) ( )( )tTzzqFpktTzzF ,, +≥
( )( )ktTzzqFp ,+≥
( )( ) 1
1
, =
−
≥
q
p
ktTzzF
Thus, we have z = Tz. Since Tz = STz, we also have z = Sz.
Therefore, z = Az = Bz = Lz = Mz = Sz = Tz, that is, z is the common fixed point of the six maps.
Case II. L is continuous.
Since L is continuous, LLx2n → Lz and L(AB)x2n → Lz.
Step 8. By taking x = LLx2n, y = x2n+1 in (3.1.5), we have
( )( ) ( )( ) ( )( )[ ]
( )( ) ( )( )[ ] ( )( )ktMxABLxFtSTxABLXqFtLLxABLxpF
ktMxSTxFktLLxABLxFktMxLLxF
nnnnnn
nnnnnn
2,.,,
,.,*,
12212222
121222122
2
++
+++
+≥
Letting n → ∞, we have
( )( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] ( )( )ktLzzFtLzzqFtLzLzpFktzzFktLzLzFktLzzF 2,.,,,.,*,2
+≥
( )( ) ( )( )[ ] ( )( )ktLzzFtLzzqFpktLzzF 2,,,2
+≥
( )( )[ ] ( )( )ktLzzFtLzzqFp ,,+≥
( )( ) ( )( )tLzzqFpktLzzF ,, +≥
( )( )ktLzzqFp ,+≥ ,
( )( ) 1
1
, =
−
≥
q
p
ktLzzF
Thus, we have z = Lz and using steps 5-7, we have
Z = Lz = Mz = Sz = Tz.
Step 9. Since L is continuous,
LLx2n →Lz and LABx2n →Lz
Since(L, AB) is semi-compatible,
L(AB)x2n → ABz.
Since limit in Menger space is unique, so
Lz = ABz and using Step 4, we also have z = Bz.
Therefore, z = Az = Bz = Sz = Tz = Lz = Mz, that is, z is the common fixed point of the six maps in this case
also.
Step 10. For uniqueness, let w(w ≠ z)be another common fixed point of A, B, S, T, L and M.
Taking x = z, y = w in (3.1.5), we have
F2
(Lz,Mw)(kt)*
[F(ABz,Lz)(kt).F(STw,Mw)(kt)]
≥ [Pf(ABz,Lz)(t)+qF(ABz,STw)(t)].F(ABz,Mw)(2kt)
Which implies that
F2
(z,w)(kt) ≥ [p +qF(z,w)(t)]F(z,w)(2kt)
≥ [p + qF(z,w)(t)]F(z,w)(kt),
F(z,w)(kt) ≥ p + qF(z,w)(t)

153
( )( ) 1
1
, =
−
≥
q
p
ktwzF
Thus, we have z = w.
This completes the proof of the theorem.
If we take B = T = IX (the identity map on X) in theorem 3.1, we have the following:
References:
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395-399.
[4] M. Grabiec, "Fixed points in fuzzy metric specs, Fuzzy Sets and Systems" 27 (1988), 385-389.
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