Abstract:
In this paper we prove some common fixed point theorems for sequence of
mappings in complete M
� fuzzy metric space.
Keywords:
Complete M
� Fuzzy metric space, Common fixed point, Sequence of maps.
Mathematics Subject Classification:
47H10, 54H25.
1. Introduction and Preliminaries
Zadeh [16] introduced the concept of fuzzy
sets in 1965. George and Veeramani [2] modified the concept of fuzzy
metric space introduced by Kramosil and Michalek [6] and defined the
Hausdorff topology of fuzzy metric spaces. Many authors [4, 7] have
proved fixed point theorems in fuzzy metric space. Recently Sedghi and
Shobe [13] introduced D*-metric space as a probable modification
of the definition of D-metric introduced by Dhage [1], and prove
some basic properties in D*-metric spaces. Using D*-metric
concepts, Sedghi and Shobe define M � fuzzy metric space and proved a common
fixed point theorem in it. In this paper we prove some common fixed
point theorems for sequence of mappings in complete M
� fuzzy metric space.
Definition 1.1:
Let X be a nonempty set. A generalized metric (or D* -
metric) on X is a function: D* : X3�
[0, �),
that satisfies the following conditions for each x, y, z, a�X
(i) D* (x, y, z) �
0,
(ii) D* (x, y, z) = 0 iff
x = y = z,
(iii) D* (x, y, z) = D*
(p{x, y, z}), (symmetry) where p is a permutation
function,
(iv) D* (x, y, z) �D* (x, y, a) + D* (a, z, z).
The pair (X, D*), is called a
generalized metric (or D* - metric) space.
Example 1.2:
Examples of D* - metric are
(a) D* (x, y, z) = max { d(x,
y), d(y, z), d(z, x) },
(b) D* (x, y, z) = d(x,
y) + d(y, z) + d(z, x).
Here, d is the ordinary metric on X.
Definition 1.3:
A fuzzy set M in an arbitrary set X is a function
with domain X and values in [0, 1].
Definition 1.4:
A binary operation *: [0, 1] � [0, 1] �
[0, 1] is a continuous t-norm if it satisfies the following
conditions
(i) * is associative and commutative,
(ii) * is continuous,
(iii) a * 1 = a for all a
�
[0, 1],
(iv) a*b�c*d whenever a�c and b�d, for each a, b, c, d�
[0, 1].
Examples for continuous t-norm are
a*b = ab and a*b = min {a, b}.
Definition 1.5:
A 3-tuple (X,
M, *) is called M
� fuzzy metric space if X is an arbitrary non-empty set, * is a
continuous t-norm, and M
is a fuzzy set on X3 � (0, �),
satisfying the following conditions for each x, y, z, a�X and t, s > 0
(FM � 1) M
(x, y, z, t) > 0
(FM � 2) M
(x, y, z, t) = 1 iff x = y = z
(FM � 3) M
(x, y, z, t) = M
(p {x, y, z}, t), where p is a permutation
function
(FM � 4) M
(x, y, a, t) *
M (a, z, z, s) �M
(x, y, z, t+s)
(FM � 5) M
(x, y, z,
�) : (0, �)
�
[0, 1] is continuous
(FM � 6) limt ��M
(x, y, z, t) = 1.
Example 1.6:
Let X be a nonempty set and D* is the D* - metric
on X. Denote a*b = a.b for all a,
b�
[0, 1]. For each t�
(0, �),
define
M
(x, y, z, t) =
for all x, y, z �X, then (X, M,
*) is a M
� fuzzy metric space.
Example 1.7: Let (X, M, *) be a fuzzy metric
space. If we define M
: X3 � (0, �)
�
[0, 1] by
M
(x, y, z, t) = M (x, y, t)*M (y,
z, t)*M (z, x, t)
for all x, y, z �X, then (X, M,
*) is a M
� fuzzy metric space.
Lemma 1.8:([13])
Let (X, M,
*) be a M
� fuzzy metric space. Then for every t > 0 and for every x, y�X we have M(x,
x, y, t) = M(x,
y, y, t).
Lemma 1.9:([13])
Let (X, M,
*) be a M
� fuzzy metric space. Then M
(x, y, z, t) is non-decreasing with respect to t, for all
x, y, z in X.
Definition 1.10:
Let (X, M,
*) be a M
� fuzzy metric space. For t > 0, the open ball BM
(x, r, t) with center x �X and radius 0 < r < 1 is defined by
BM
(x, r, t) = {y �X: M
(x, y, y, t) > 1 � r }.
A subset A of X is called open
set if for each x �A there exist t > 0 and 0 < r < 1 such that BM
(x, r, t)
�A.
Definition 1.11:
Let (X, M,
*) be a M
� fuzzy metric space and {xn}
be a sequence in X
(a) {xn}
is said to be converges to a point x �X if limn ��M
(x, x, xn, t) =1 for all t > 0
(b) {xn}
is called Cauchy sequence if limn �� M
(xn+p, xn+p, xn, t) = 1 for all
t > 0 and p > 0
(c) A M � fuzzy metric space in which every Cauchy
sequence is convergent is said to be complete.
Remark 1.12:
Since * is continuous, it follows from (FM-4) that the limit of
the sequence is uniquely determined.
Definition 1.13: Let (X, M,
*) be a M
� fuzzy metric space, then M
is called of first type if for every x, y�X we have
M
(x, x, y, t) �M
(x, y, z, t)
for every z�X. Also it is called of second type if for every x, y, z �X we have
M
(x, y, z, t) = M (x, y, t)*M (y,
z, t)*M (z, x, t).
Example 1.14: Let X be a nonempty set and D*
is the D* - metric on X. If we define
M
(x, y, z, t) = ,
where D*(x, y, z)
= d(x, y) + d(y, z) + d(x,
z), then M
is first type.
Example 1.15: If(X, M, *) is a fuzzy
metric and M (x, y, t) =
, then
M
(x, y, z, t) = * *
is second type.
Definition 1.16:
A point x
�X is said to be a fixed point
of the map T: X�X if Tx = x.
Definition 1.17:
A point x
�X is said to be a common
fixed point of sequence of maps Tn: X�X if Tn(x) = x for all n.
2. Main Results
Theorem 2.1:
Let (X, M,
*) be a complete M � fuzzy metric space and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
3M
(Tix, Tjy, Tjy,
t) ≥ {M
(x, y, y, t/k) + M
(x, x, Tix, t/k) + M
(y, y, Tjy, t/k)}
for all i ≠ j and for all x,
y�X. Then {Tn} have a unique common fixed point.
Proof:
Let x0�X be any arbitrary element.
Define a sequence {xn} in
X as xn+1 = Tn+1xn
for n = 0, 1, 2,�
Uniqueness: Suppose x ≠ y such that Tny
= y for all n.
Then
3M
(x, y, y, t) = 3M
(Tix, Tjy, Tjy,
t)
≥ {M
(x, y, y, t/k) + M
(x, x, Tix, t/k)
+ M
(y, y, Tjy, t/k)}
= {M
(x, y, y, t/k) + M
(x, x, x, t/k) + M
(y, y, y, t/k)}
= M
(x, y, y, t/k) + 2
�M
(x, y, y, t) + 2
Therefore, 2M
(x, y, y, t) �
2
That is, M
(x, y, y, t) �
1
Hence M
(x, y, y, t) = 1, for all t>
0.
Therefore, x = y.
which is contradiction to x ≠ y.
Hence {Tn} have a unique
common fixed point.
This completes the proof.
Remark 2.2:
From the above theorem we have,
M
(Tix, Tjy, Tjy,
t)
≥
{M
(x, y, y, t/k) + M
(x, x, Tix, t/k)
+ M
(y, y, Tjy, t/k)}
�
min {M
(x, y, y, t/k), M
(x, x, Tix, t/k),
M
(y, y, Tjy, t/k)}
Therefore, M
(Tix, Tjy, Tjy,
t) �
min {M
(x, y, y, t/k), M
(x, x, Tix, t/k), M
(y, y, Tjy, t/k)}.
Hence we get the following corollary.
Corollary 2.3:
Let (X, M,
*) be a complete M � fuzzy metric space and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
M
(Tix, Tjy, Tjy,
t) �
min {M
(x, y, y, t/k), M
(x, x, Tix, t/k), M
(y, y, Tjy, t/k)}
for all i ≠ j and for all x,
y�X. Then {Tn} have a unique common fixed point.
Remark 2.4: By taking Ti = Tj
= T in the above corollary, we get the following corollary 2.5.
Corollary 2.5:
Let (X, M,
*) be a complete M � fuzzy metric space and let T: X�X be a mapping such that for all t > 0 and 0 < k <
1 satisfying the condition
M
(Tx, Ty, Ty, t) �
min {M
(x, y, y, t/k), M
(x, x, Tx, t/k), M (y, y, Ty, t/k)}
for all x, y�X. Then T has a unique fixed point.
Theorem 2.6:
Let (X, M,
*) be a complete M � fuzzy metric space with continuous t-norm
* is defined by a*b = min {a, b} and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
M
(Tix, Tjy, Tjy,
t) ≥ {M
(x, y, y, t/k)*M
(x, x, Tix, t/k)*M
(y, y, Tjy, t/k)}
for all i ≠ j and for all x,
y�X. Then {Tn} have a unique common fixed point.
Proof:
Let x0�X be any arbitrary element.
Define a sequence {xn} in
X as xn+1 = Tn+1xn
for n = 0, 1, 2,�
Now we prove that {xn} is a
Cauchy sequence in X.
For n�
0, we have
M
(xn+1,xn+2,xn+2, t) = M
(Tn+1xn, Tn+2xn+1,
Tn+2xn+1, t)
Uniqueness: Suppose x ≠ y such that Tny
= y for all n.
Then
M
(x, y, y, t) = M
(Tix, Tjy, Tjy,
t)
≥ {M
(x, y, y, t/k)*M
(x, x, Tix, t/k)
*M
(y, y, Tjy, t/k)}
= {M
(x, y, y, t/k)*M
(x, x, x, t/k)*M
(y, y, y, t/k)}
= {M
(x, y, y, t/k)*1*1}
= M
(x, y, y, t/k)
.
.
.
�M
(x, y, y, t/kn)
�
1 as n�
∞
Hence M
(x, y, y, t) = 1, for all t>
0.
Therefore, x = y.
which is contradiction to x ≠ y.
Hence {Tn} have a unique
common fixed point.
This completes the proof.
Remark 2.7: By taking Ti = Tj
= T in the above theorem, we get the following corollary 2.8
Corollary 2.8:
Let (X, M,
*) be a complete M � fuzzy metric space with continuous t-norm
* is defined by a*b = min {a, b} and let T: X�X be a mapping such that for all t > 0 and 0 < k <
1 satisfying the condition
M
(Tx, Ty, Ty, t) �
{M
(x, y, y, t/k) * M
(x, x, Tx, t/k)
* M
(y, y, Ty, t/k)}
for all x, y�X. Then T has a unique fixed point.
Theorem 2.9:
Let (X, M,
*) be a complete M � fuzzy metric space with continuous t-norm
* is defined by a*b = min {a, b} and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
M
(Tix, Tjy, Tjy,
t) ≥ min {M
(x, y, y, t/k), M
(x, x, Tix, t/k), M
(y, y, Tjy, t/k), M
(x, x, Tjy, 2t/k)}
for all i ≠ j and for all x,
y�X . Then {Tn} have a unique common fixed point.
Proof:
Let x0�X be any arbitrary element.
Define a sequence {xn} in
X as xn+1 = Tn+1xn
for n = 0, 1, 2,�
Now we prove that {xn} is a
Cauchy sequence in X.
For n�
0, we have
M
(xn+1,xn+2,xn+2, t) = M
(Tn+1xn, Tn+2xn+1,
Tn+2xn+1, t)
≥ min {M
(xn, xn+1, xn+1,
t/k), M
(xn, xn, Tn+1xn,
t/k),
M
(xn+1, xn+1,
Tn+2xn+1, t/k),
M
(xn, xn, Tn+2xn+1,
2t/k)}
= min {M
(xn, xn+1, xn+1,
t/k), M
(xn, xn, xn+1,
t/k),
M
(xn+1, xn+1,
xn+2, t/k), M
(xn, xn, xn+2,
2t/k)}
= min {M
(xn, xn+1, xn+1,
t/k), M
(xn, xn+1, xn+1,
t/k),
M
(xn+1, xn+2, xn+2,
t/k), M
(xn, xn, xn+2,
2t/k)}
= min {M
(xn, xn+1, xn+1,
t/k), M
(xn+1, xn+2, xn+2,
t/k),
M
(xn, xn, xn+2,
2t/k)}
�
min {M
(xn, xn+1, xn+1,
t/k), M
(xn+1, xn+2, xn+2,
t/k),
Therefore, M
(xn+1,xn+2,xn+2, t) �
{M
(xn, xn+1, xn+1,
t/k)*M
(xn+1, xn+2, xn+2,
t/k)}, which implies that
M
(xn+1,xn+2,xn+2, t) ≥ M
(xn, xn+1, xn+1,
t/k).
Continuing this way we get
M
(xn+1,xn+2,xn+2, t) ≥ M
(xn, xn+1, xn+1,
t/k)
≥ M
(x n-1, xn, xn,
t/k2)
.
.
.
≥ M
(x0, x1, x1, t/kn+1).
Now for any positive integer p and
t > 0, we have
M (xn, xn+p,
xn+p, t)
�M
(xn, xn+1,
xn+1,)
M
(xn+p-1,
xn+p, xn+p,)
�M
(x0, x1,
x1,)
M
(x0, x1,
x1,)
Therefore, by (FM-6), we have
limn ��M
(xn, xn+p,
xn+p, t) �
1 1 = 1
which implies that {xn}
is a Cauchy sequence in M � fuzzy metric space X. Since X
is M
� fuzzy complete, sequence {xn}
converges to a point x�X.
Now we prove that x is a fixed point
of {Tn} for all n.
Now we have
M
(Tmx, x, x, t) = limn
��M
(Tmx, xn+2,xn+2,
t)
= limn ��M
(Tmx, Tn+2xn+1,
Tn+2xn+1, t)
≥ limn ��
min {M
(x, xn+1, xn+1,
t/k), M
(x, x, Tmx, t/k), M
(xn+1, xn+1,
Tn+2xn+1, t/k),
M
(x, x, Tn+2xn+1,
2t/k)}
= limn ��
min {M
(x, xn+1, xn+1,
t/k), M
(x, x, Tmx, t/k), M
(xn+1, xn+1,
xn+2, t/k), M
(x, x, xn+2, 2t/k)}
= min {M
(x, x, x, t/k), M
(x, x, Tmx, t/k), M
(x, x, x, t/k), M
(x, x, x, 2t/k)}
= min {1, M
(x, x, Tmx, t/k), 1, 1}
= M
(x, x, Tmx, t/k)
= M
(Tmx, x, x, t/k)
.
.
.
≥
M (Tmx, x,
x, t/kn)
�
1 as n�
∞
Hence M
(Tmx, x, x, t) = 1, for all t
>
0.
Therefore, Tmx = x.
Hence Tnx = x for
all n.
Therefore x is a common fixed point of
{Tn}.
Uniqueness: Suppose x ≠ y such that Tny
= y for all n.
Then
M
(x, y, y, t) = M
(Tix, Tjy, Tjy,
t)
≥ min {M
(x, y, y, t/k), M
(x, x, Tix, t/k),
M
(y, y, Tjy, t/k), M
(x, x, Tjy, 2t/k)}
= min {M
(x, y, y, t/k), M
(x, x, x, t/k),
M
(y, y, y, t/k), M
(x, x, y, 2t/k)}
= min {M
(x, y, y, t/k), 1, 1,
M (x, y,
y, 2t/k)}
= M
(x, y, y, t/k)
.
.
.
�M
(x, y, y, t/kn)
�
1 as n�
∞
Hence M
(x, y, y, t) = 1, for all t>
0.
Therefore, x = y.
which is contradiction to x ≠ y.
Hence {Tn} have a unique
common fixed point.
This completes the proof.
Remark 2.10: By taking Ti = Tj
= T in the above theorem, we get the following corollary 2.11
Corollary 2.11:
Let (X, M,
*) be a complete M � fuzzy metric space with continuous t-norm
* is defined by a*b = min {a, b} and let T: X�X be a mapping such that for all t > 0 and 0 < k <
1 satisfying the condition
M
(Tx, Ty, Ty, t) �
min {M
(x, y, y, t/k), M
(x, x, Tx, t/k), M
(y, y, Ty, t/k), M
(x, x, Ty, 2t/k)}
for all x, y�X. Then T has a unique fixed point.
Theorem 2.12:
Let (X, M,
*) be a complete M � fuzzy metric space with continuous t-norm
* is defined by a*b = min {a, b} and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
M
(Tix, Tjy, Tjy,
t) ≥ {M
(x, y, y, t/k)*M
(x, x, Tix, t/k)*M
(y, y, Tjy, t/k)*M
(x, x, Tjy, 2t/k)}
for all i ≠ j and for all x,
y�X . Then {Tn} have a unique common fixed point.
Proof:
Let x0�X be any arbitrary element.
Define a sequence {xn} in
X as xn+1 = Tn+1xn
for n = 0, 1, 2,�
Now we prove that {xn} is a
Cauchy sequence in X.
For n�
0, we have
M
(xn+1,xn+2,xn+2, t) = M
(Tn+1xn, Tn+2xn+1,
Tn+2xn+1, t)
Uniqueness: Suppose x ≠ y such that Tny
= y for all n.
Then
M
(x, y, y, t) = M
(Tix, Tjy, Tjy,
t)
≥ {M
(x, y, y, t/k)*M
(x, x, Tix, t/k)*M
(y, y, Tjy, t/k)
*M
(x, x, Tjy, 2t/k)}
= {M
(x, y, y, t/k)*M
(x, x, x, t/k)*M
(y, y, y, t/k)
*M
(x, x, y, 2t/k)}
= {M
(x, y, y, t/k)*1*1*M
(x, y, y, 2t/k)}
= {M
(x, y, y, t/k)*
M (x, y, y, 2t/k)}
�
{M
(x, y, y, t/k)*M
(x, y, y, t/k)}
�M
(x, y, y, t/k)
.
.
.
�M
(x, y, y, t/kn)
�
1 as n�
∞
Hence M
(x, y, y, t) = 1, for all t>
0.
Therefore, x = y.
which is contradiction to x ≠ y.
Hence {Tn} have a unique
common fixed point.
This completes the proof.
Remark 2.13: By taking Ti = Tj
= T in the above theorem, we get the following corollary 2.14
Corollary 2.14:
Let (X, M,
*) be a complete M � fuzzy metric space with continuous t-norm
* is defined by a*b = min {a, b} and let T: X�X be a mapping such that for all t > 0 and 0 < k <
1 satisfying the condition
M
(Tx, Ty, Ty, t) �
{M
(x, y, y, t/k) * M
(x, x, Tx, t/k)
* M
(y, y, Ty, t/k) * M
(x, x, Ty, 2t/k)}
for all x, y�X. Then T has a unique fixed point.
Theorem 2.15:
Let (X, M,
*) be a complete M � fuzzy metric space with continuous t-norm
* is defined by a*b = min {a, b} and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
M
(Tix, Tjy, Tjy,
t) ≥ min {M
(x, y, y, t/k), M
(x, Tix, Tjy, 2t/k)}
for all i ≠ j and for all x,
y�X . Then {Tn} have a unique common fixed point.
Proof:
Let x0�X be any arbitrary element.
Define a sequence {xn} in
X as xn+1 = Tn+1xn
for n = 0, 1, 2,�
Now we prove that {xn} is a
Cauchy sequence in X.
For n�
0, we have
M
(xn+1,xn+2,xn+2, t) = M
(Tn+1xn, Tn+2xn+1,
Tn+2xn+1, t)
= limn ��
min {M
(x, xn+1, xn+1,
t/k), M
(x, Tmx, xn+2, 2t/k)}
= min {M
(x, x, x, t/k), M
(x, Tmx, x, 2t/k)}
= min {1, M
(Tmx, x, x, 2t/k)}
= M
(Tmx, x, x, 2t/k)
�M
(Tmx, x, x, t/k)
.
.
.
≥
M (Tmx, x,
x, t/kn)
�
1 as n�
∞
Hence M
(Tmx, x, x, t) = 1, for all t
>
0.
Therefore, Tmx = x.
Hence Tnx = x for
all n.
Therefore x is a common fixed point of
{Tn}.
Uniqueness: Suppose x ≠ y such that Tny
= y for all n.
Then
M
(x, y, y, t) = M
(Tix, Tjy, Tjy,
t)
≥ min {M
(x, y, y, t/k), M
(x, Tix, Tjy, 2t/k)}
= min {M
(x, y, y, t/k), M
(x, x, y, 2t/k)}
= min {
M (x, y, y, t/k),
M
(x, y, y, 2t/k)}
= M
(x, y, y, t/k)
.
.
.
�M
(x, y, y, t/kn)
�
1 as n�
∞
Hence M
(x, y, y, t) = 1, for all t>
0.
Therefore, x = y.
which is contradiction to x ≠ y.
Hence {Tn} have a unique
common fixed point.
This completes the proof.
Remark 2.16: By taking Ti = Tj
= T in the above theorem, we get the following corollary 2.17
Corollary 2.17:
Let (X, M,
*) be a complete M � fuzzy metric space with continuous t-norm
* is defined by a*b = min {a, b} and let T: X�X be a mapping such that for all t > 0 and 0 < k <
1 satisfying the condition
M
(Tx, Ty, Ty, t) �
min {M
(x, y, y, t/k), M
(x, Tx, Ty, 2t/k)}
for all x, y�X. Then T has a unique fixed point.
Theorem 2.18:
Let (X, M,
*) be a complete first type M � fuzzy metric space and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
Uniqueness: Suppose x ≠ y such that Tny
= y for all n.
Then
3M
(x, y, y, t) = 3M
(Tix, Tjy, Tky,
t)
�
{M
(x, y, y, t/k) + M
(x, Tix, Tjy, t/k)
+ �[M
(y, Tjy, Tky, t/k)+M
(y, Tky, Tix, t/k)]}
= {M
(x, y, y, t/k) + M
(x, x, y, t/k)
+ �[M
(y, y, y, t/k)+M
(y, y, x, t/k)]}
= {M
(x, y, y, t/k) + M
(x, y, y, t/k)
+ �[1+M
(x, y, y, t/k)]}
= �[5M
(x, y, y, t/k) + 1]
6M
(x, y, y, t) �
5M
(x, y, y, t/k) + 1
�
5M
(x, y, y, t) + 1
Therefore, M
(x, y, y, t) �
1
Hence M
(x, y, y, t) = 1, for all t>
0.
Therefore, x = y.
which is contradiction to x ≠ y.
Hence {Tn} have a unique
common fixed point. This completes the proof.
Remark 2.19:
From the above theorem we have,
M
(Tix, Tjy, Tkz,
t) ≥ {M
(x, y, z, t/k) + M
(x, Tix, Tjy, t/k)
+ �[M
(y, Tjy, Tkz, t/k)+M
(z, Tkz, Tix, t/k)]}
�
min {M
(x, y, z, t/k), M
(x, Tix, Tjy, t/k),
�[M
(y, Tjy, Tkz, t/k)+M
(z, Tkz, Tix, t/k)]}
Therefore,
M
(Tix, Tjy, Tkz,
t) ≥ min {M
(x, y, z, t/k), M
(x, Tix, Tjy, t/k),
�[M
(y, Tjy, Tkz, t/k)+M
(z, Tkz, Tix, t/k)]}
Hence we get the following corollary.
Corollary 2.20:
Let (X, M,
*) be a complete first type M � fuzzy metric space and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
M
(Tix, Tjy, Tkz,
t) ≥ min {M
(x, y, z, t/k), M
(x, Tix, Tjy, t/k),
�[M
(y, Tjy, Tkz, t/k)+M
(z, Tkz, Tix, t/k)]}
for all i ≠ j ≠ k and
for all x, y, z�X . Then {Tn} have a unique common fixed point.
Remark 2.21: By taking Ti = Tj
= Tk = T in the above corollary, we get the following
corollary 2.22
Corollary 2.22:
Let (X, M,
*) be a complete first type M � fuzzy metric space and let T: X�X be a mapping such that for all t > 0 and 0 < k <
1 satisfying the condition
M
(Tx, Ty, Tz, t) ≥ min {M
(x, y, z, t/k), M
(x, Tx, Ty, t/k), �[M
(y, Ty, Tz, t/k)+M
(z, Tz, Tx, t/k)]}
for all x, y, z�X. Then T has a unique fixed point.
Theorem 2.23:
Let (X, M,
*) be a complete first type M � fuzzy metric space and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
3M
(Tix, Tjy, Tkz,
t) ≥ {M
(x, y, z, t/k) + M
(y, z, Tkz, t/k) + �[M
(x, Tix, z, t/k)+M
(x, y, Tjy, t/k)]}
for all i ≠ j ≠ k and
for all x, y, z�X . Then {Tn} have a unique common fixed point.
Proof:
Let x0�X be any arbitrary element.
Define a sequence {xn} in
X as xn+1 = Tn+1xn
for n = 0, 1, 2,�
Uniqueness: Suppose x ≠ y such that Tny
= y for all n.
Then
3M
(x, y, y, t) = 3M
(Tix, Tjy, Tky,
t)
�
{M
(x, y, y, t/k) + M
(y, y, Tky, t/k)
+ �[M
(x, Tix, y, t/k)+M
(x, y, Tjy, t/k)]}
= {M
(x, y, y, t/k) + M
(y, y, y, t/k)
+ �[M
(x, x, y, t/k)+M
(x, y, y, t/k)]}
= {M
(x, y, y, t/k) + 1
+ �[M
(x, y, y, t/k)+M
(x, y, y, t/k)]}
3M
(x, y, y, t) �
2M
(x, y, y, t/k) + 1
�
2M
(x, y, y, t) + 1
Therefore, M
(x, y, y, t) �
1
Hence M
(x, y, y, t) = 1, for all t>
0.
Therefore, x = y. which is
contradiction to x ≠ y.
Hence {Tn} have a unique
common fixed point.
This completes the proof.
Remark 2.24:
From the above theorem we have,
M
(Tix, Tjy, Tkz,
t) ≥ {M
(x, y, z, t/k) + M
(y, z, Tkz, t/k) +�[M
(x, Tix, z, t/k)+M
(x, y, Tjy, t/k)]}
�
min {M
(x, y, z, t/k), M
(y, z, Tkz, t/k),
�[M
(x, Tix, z, t/k)+M
(x, y, Tjy, t/k)]}
Therefore,
M
(Tix, Tjy, Tkz,
t) �
min {M
(x, y, z, t/k), M
(y, z, Tkz, t/k), �[M
(x, Tix, z, t/k)+M
(x, y, Tjy, t/k)]}
Hence we get the following corollary.
Corollary 2.25:
Let (X, M,
*) be a complete first type M � fuzzy metric space and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
M
(Tix, Tjy, Tkz,
t) �
min {M
(x, y, z, t/k), M
(y, z, Tkz, t/k), �[M
(x, Tix, z, t/k)+M
(x, y, Tjy, t/k)]}
for all i ≠ j ≠ k and
for all x, y, z�X . Then {Tn} have a unique common fixed point.
Remark 2.26: By taking Ti = Tj
= Tk = T in the above corollary, we get the following
corollary 2.27
Corollary 2.27:
Let (X, M,
*) be a complete first type M � fuzzy metric space and let T: X�X be a mapping such that for all t > 0 and 0 < k <
1 satisfying the condition
M
(Tx, Ty, Tz, t) �
min {M
(x, y, z, t/k), M
(y, z, Tz, t/k), �[M
(x, Tx, z, t/k)+M
(x, y, Ty, t/k)]}
for all x, y, z�X . Then T has a unique fixed point.
Theorem 2.28:
Let (X, M,
*) be a complete first type M � fuzzy metric space and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
Uniqueness: Suppose x ≠ y such that Tny
= y for all n.
Then
5M
(x, y, y, t) = 5M
(Tix, Tjy, Tky,
t)
�
{M
(x, y, y, t/k) + M
(x, Tix, Tjy, t/k)
+ M
(y, y, Tky, t/k)
+ �[M
(y, Tjy, Tky, t/k)+M
(y, Tky, Tix, t/k)]
+ �[M
(x, Tix, y, t/k)+M
(x, y, Tjy, t/k)]}
= {M
(x, y, y, t/k) + M
(x, x, y, t/k) + M
(y, y, y, t/k)
+ �[M
(y, y, y, t/k)+M
(y, y, x, t/k)]
+ �[M
(x, x, y, t/k)+M
(x, y, y, t/k)]}
= {M
(x, y, y, t/k) + M
(x, y, y, t/k) + 1
+ �[1+M
(x, y, y, t/k)] + M
(x, y, y, t/k)}
= �[7M
(x, y, y, t/k) + 3]
10M
(x, y, y, t) �
7M
(x, y, y, t/k) + 3
�
7M
(x, y, y, t) + 3
Therefore, 3M
(x, y, y, t) �
3
That is, M
(x, y, y, t) �
1
Hence M
(x, y, y, t) = 1, for all t>
0.
Therefore, x = y.
which is contradiction to x ≠ y.
Hence {Tn} have a unique
common fixed point.
This completes the proof.
Corollary 2.29: Let (X, M,
*) be a complete first type M � fuzzy metric space and Tn:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition
M
(Tix, Tjy, Tkz,
t) ≥ min {M
(x, y, z, t/k), M
(x, Tix, Tjy, t/k),
M
(y, z, Tkz, t/k),�[M
(y, Tjy, Tkz, t/k)+M
(z, Tkz, Tix, t/k)],
�[M
(x, Tix, z, t/k)+M
(x, y, Tjy, t/k)]}
for all i ≠ j ≠ k and
for all x, y, z�X . Then {Tn} have a unique common fixed point.
Remark 2.30: By taking Ti = Tj
= Tk = T in the above corollary, we get the following
corollary 2.31
Corollary 2.31: Let (X, M,
*) be a complete first type M � fuzzy metric space and let T: X�X be a mapping such that for all t > 0 and 0 < k <
1 satisfying the condition
M
(Tx, Ty, Tz, t) ≥ min {M
(x, y, z, t/k), M
(x, Tx, Ty, t/k), M
(y, z, Tz, t/k), �[M
(y, Ty, Tz, t/k)+M
(z, Tz, Tx, t/k)], �[M
(x, Tx, z, t/k)+M
(x, y, Ty, t/k)]}
for all x, y, z�X . Then T has a unique fixed point.
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