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* : X^{3}�
[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 X^{3} � (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) lim_{t }_{�
�}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
: X^{3} � (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 B_{M}
(x, r, t) with center x �X and radius 0 < r < 1 is defined by

B_{M}
(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 B_{M}
(x, r, t)
�A.

Definition 1.11:
Let (X, M,
*) be a M
� fuzzy metric space and {x_{n}}
be a sequence in X

(a) {x_{n}}
is said to be converges to a point x �X if lim_{n }_{
�
�}M
(x, x, x_{n}, t) =1 for all t > 0

(b) {x_{n}}
is called Cauchy sequence if lim_{n }_{
�
�} M
(x_{n+p}, x_{n+p}, x_{n}, 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 T_{n}: X�X if T_{n}(x) = x for all n.

2. Main Results

Theorem 2.1:
Let (X, M,
*) be a complete M � fuzzy metric space and T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

3M
(T_{i}x, T_{j}y, T_{j}y,
t) ≥ {M
(x, y, y, t/k) + M
(x, x, T_{i}x, t/k) + M
(y, y, T_{j}y, t/k)}

for all i ≠ j and for all x,
y�X. Then {T_{n}} have a unique common fixed point.

Proof:
Let x_{0}�X be any arbitrary element.

Define a sequence {x_{n}} in
X as x_{n+}_{1 }= T_{n+}_{1}x_{n}
for n = 0, 1, 2,�

Now we prove that {x_{n}} is a
Cauchy sequence in X.

Uniqueness: Suppose x ≠ y such that T_{n}y
= y for all n.

Then

3M
(x, y, y, t) = 3M
(T_{i}x, T_{j}y, T_{j}y,
t)

≥ {M
(x, y, y, t/k) + M
(x, x, T_{i}x, t/k)

+ M
(y, y, T_{j}y, 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 {T_{n}} have a unique
common fixed point.

This completes the proof.

Remark 2.2:
From the above theorem we have,

M
(T_{i}x, T_{j}y, T_{j}y,
t)

≥
{M
(x, y, y, t/k) + M
(x, x, T_{i}x, t/k)

+ M
(y, y, T_{j}y, t/k)}

�
min {M
(x, y, y, t/k), M
(x, x, T_{i}x, t/k),

M
(y, y, T_{j}y, t/k)}

Therefore, M
(T_{i}x, T_{j}y, T_{j}y,
t) �
min {M
(x, y, y, t/k), M
(x, x, T_{i}x, t/k), M
(y, y, T_{j}y, t/k)}.

Hence we get the following corollary.

Corollary 2.3:
Let (X, M,
*) be a complete M � fuzzy metric space and T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{j}y,
t) �
min {M
(x, y, y, t/k), M
(x, x, T_{i}x, t/k), M
(y, y, T_{j}y, t/k)}

for all i ≠ j and for all x,
y�X. Then {T_{n}} have a unique common fixed point.

Remark 2.4: By taking T_{i} = T_{j}
= 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 T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{j}y,
t) ≥ {M
(x, y, y, t/k)*M
(x, x, T_{i}x, t/k)*M
(y, y, T_{j}y, t/k)}

for all i ≠ j and for all x,
y�X. Then {T_{n}} have a unique common fixed point.

Proof:
Let x_{0}�X be any arbitrary element.

Define a sequence {x_{n}} in
X as x_{n+}_{1 }= T_{n+}_{1}x_{n}
for n = 0, 1, 2,�

Now we prove that {x_{n}} is a
Cauchy sequence in X.

For n�
0, we have

M
(x_{n+}_{1},_{ }x_{n+2},_{
}x_{n+2}, t) = M
(T_{n+}_{1}x_{n}, T_{n+2}x_{n+}_{1},
T_{n+2}x_{n+}_{1}, t)

Uniqueness: Suppose x ≠ y such that T_{n}y
= y for all n.

Then

M
(x, y, y, t) = M
(T_{i}x, T_{j}y, T_{j}y,
t)

≥ {M
(x, y, y, t/k)*M
(x, x, T_{i}x, t/k)

*M
(y, y, T_{j}y, 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/k^{n})

�
1 as n�
∞

Hence M
(x, y, y, t) = 1, for all t>
0.

Therefore, x = y.

which is contradiction to x ≠ y.

Hence {T_{n}} have a unique
common fixed point.

This completes the proof.

Remark 2.7: By taking T_{i} = T_{j}
= 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 T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{j}y,
t) ≥ min {M
(x, y, y, t/k), M
(x, x, T_{i}x, t/k), M
(y, y, T_{j}y, t/k), M
(x, x, T_{j}y, 2t/k)}

for all i ≠ j and for all x,
y�X . Then {T_{n}} have a unique common fixed point.

Proof:
Let x_{0}�X be any arbitrary element.

Define a sequence {x_{n}} in
X as x_{n+}_{1 }= T_{n+}_{1}x_{n}
for n = 0, 1, 2,�

Now we prove that {x_{n}} is a
Cauchy sequence in X.

For n�
0, we have

M
(x_{n+}_{1},_{ }x_{n+2},_{
}x_{n+2}, t) = M
(T_{n+}_{1}x_{n}, T_{n+2}x_{n+}_{1},
T_{n+2}x_{n+}_{1}, t)

≥ min {M
(x_{n}, x_{n+}_{1}, x_{n+}_{1},
t/k), M
(x_{n}, x_{n}, T_{n+}_{1}x_{n},
t/k),

M
(x_{n+}_{1}, x_{n+}_{1},
T_{n+2}x_{n+}_{1}, t/k),
M
(x_{n}, x_{n}, T_{n+2}x_{n+}_{1},
2t/k)}

= min {M
(x_{n}, x_{n+}_{1}, x_{n+}_{1},
t/k), M
(x_{n}, x_{n}, x_{n+}_{1},
t/k),

M
(x_{n+}_{1}, x_{n+}_{1},
x_{n+2}, t/k), M
(x_{n}, x_{n}, x_{n+2},
2t/k)}

= min {M
(x_{n}, x_{n+}_{1}, x_{n+}_{1},
t/k), M
(x_{n}, x_{n+}_{1}, x_{n+}_{1},
t/k),

M
(x_{n+}_{1}, x_{n+2}, x_{n+2},
t/k), M
(x_{n}, x_{n}, x_{n+2},
2t/k)}

= min {M
(x_{n}, x_{n+}_{1}, x_{n+}_{1},
t/k), M
(x_{n+}_{1}, x_{n+2}, x_{n+2},
t/k),

M
(x_{n}, x_{n}, x_{n+2},
2t/k)}

�
min {M
(x_{n}, x_{n+}_{1}, x_{n+}_{1},
t/k), M
(x_{n+}_{1}, x_{n+2}, x_{n+2},
t/k),

M
(x_{n}, x_{n}, x_{n+}_{1},
t/k)*M
(x_{n+}_{1}, x_{n+2}, x_{n+2},
t/k)}

≥ lim_{n }_{
�
�}
min {M
(x, x_{n+}_{1}, x_{n+}_{1},
t/k), M
(x, x, T_{m}x, t/k), M
(x_{n+}_{1}, x_{n+}_{1},
T_{n+2}x_{n+}_{1}, t/k),
M
(x, x, T_{n+2}x_{n+}_{1},
2t/k)}

= lim_{n }_{
�
�}
min {M
(x, x_{n+}_{1}, x_{n+}_{1},
t/k), M
(x, x, T_{m}x, t/k), M
(x_{n+}_{1}, x_{n+}_{1},
x_{n+2}, t/k), M
(x, x, x_{n+2}, 2t/k)}

= min {M
(x, x, x, t/k), M
(x, x, T_{m}x, t/k), M
(x, x, x, t/k), M
(x, x, x, 2t/k)}

= min {1, M
(x, x, T_{m}x, t/k), 1, 1}

= M
(x, x, T_{m}x, t/k)

= M
(T_{m}x, x, x, t/k)

.

.

.

≥
M (T_{m}x, x,
x, t/k^{n})

�
1 as n�
∞

Hence M
(T_{m}x, x, x, t) = 1, for all t
>
0.

Therefore, T_{m}x = x.

Hence T_{n}x = x for
all n.

Therefore x is a common fixed point of
{T_{n}}.

Uniqueness: Suppose x ≠ y such that T_{n}y
= y for all n.

Then

M
(x, y, y, t) = M
(T_{i}x, T_{j}y, T_{j}y,
t)

≥ min {M
(x, y, y, t/k), M
(x, x, T_{i}x, t/k),

M
(y, y, T_{j}y, t/k), M
(x, x, T_{j}y, 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/k^{n})

�
1 as n�
∞

Hence M
(x, y, y, t) = 1, for all t>
0.

Therefore, x = y.

which is contradiction to x ≠ y.

Hence {T_{n}} have a unique
common fixed point.

This completes the proof.

Remark 2.10: By taking T_{i} = T_{j}
= 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 T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{j}y,
t) ≥ {M
(x, y, y, t/k)*M
(x, x, T_{i}x, t/k)*M
(y, y, T_{j}y, t/k)*M
(x, x, T_{j}y, 2t/k)}

for all i ≠ j and for all x,
y�X . Then {T_{n}} have a unique common fixed point.

Proof:
Let x_{0}�X be any arbitrary element.

Define a sequence {x_{n}} in
X as x_{n+}_{1 }= T_{n+}_{1}x_{n}
for n = 0, 1, 2,�

Now we prove that {x_{n}} is a
Cauchy sequence in X.

For n�
0, we have

M
(x_{n+}_{1},_{ }x_{n+2},_{
}x_{n+2}, t) = M
(T_{n+}_{1}x_{n}, T_{n+2}x_{n+}_{1},
T_{n+2}x_{n+}_{1}, t)

Uniqueness: Suppose x ≠ y such that T_{n}y
= y for all n.

Then

M
(x, y, y, t) = M
(T_{i}x, T_{j}y, T_{j}y,
t)

≥ {M
(x, y, y, t/k)*M
(x, x, T_{i}x, t/k)*M
(y, y, T_{j}y, t/k)

*M
(x, x, T_{j}y, 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/k^{n})

�
1 as n�
∞

Hence M
(x, y, y, t) = 1, for all t>
0.

Therefore, x = y.

which is contradiction to x ≠ y.

Hence {T_{n}} have a unique
common fixed point.

This completes the proof.

Remark 2.13: By taking T_{i} = T_{j}
= 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 T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{j}y,
t) ≥ min {M
(x, y, y, t/k), M
(x, T_{i}x, T_{j}y, 2t/k)}

for all i ≠ j and for all x,
y�X . Then {T_{n}} have a unique common fixed point.

Proof:
Let x_{0}�X be any arbitrary element.

Define a sequence {x_{n}} in
X as x_{n+}_{1 }= T_{n+}_{1}x_{n}
for n = 0, 1, 2,�

Now we prove that {x_{n}} is a
Cauchy sequence in X.

For n�
0, we have

M
(x_{n+}_{1},_{ }x_{n+2},_{
}x_{n+2}, t) = M
(T_{n+}_{1}x_{n}, T_{n+2}x_{n+}_{1},
T_{n+2}x_{n+}_{1}, t)

= lim_{n }_{
�
�}
min {M
(x, x_{n+}_{1}, x_{n+}_{1},
t/k), M
(x, T_{m}x, x_{n+2}, 2t/k)}

= min {M
(x, x, x, t/k), M
(x, T_{m}x, x, 2t/k)}

= min {1, M
(T_{m}x, x, x, 2t/k)}

= M
(T_{m}x, x, x, 2t/k)

�M
(T_{m}x, x, x, t/k)

.

.

.

≥
M (T_{m}x, x,
x, t/k^{n})

�
1 as n�
∞

Hence M
(T_{m}x, x, x, t) = 1, for all t
>
0.

Therefore, T_{m}x = x.

Hence T_{n}x = x for
all n.

Therefore x is a common fixed point of
{T_{n}}.

Uniqueness: Suppose x ≠ y such that T_{n}y
= y for all n.

Then

M
(x, y, y, t) = M
(T_{i}x, T_{j}y, T_{j}y,
t)

≥ min {M
(x, y, y, t/k), M
(x, T_{i}x, T_{j}y, 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/k^{n})

�
1 as n�
∞

Hence M
(x, y, y, t) = 1, for all t>
0.

Therefore, x = y.

which is contradiction to x ≠ y.

Hence {T_{n}} have a unique
common fixed point.

This completes the proof.

Remark 2.16: By taking T_{i} = T_{j}
= 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 T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{k}z,
t) ≥ min {M
(x, y, z, t/k), M
(x, T_{i}x, T_{j}y, t/k),
�[M
(y, T_{j}y, T_{k}z, t/k)+M
(z, T_{k}z, T_{i}x, t/k)]}

Hence we get the following corollary.

Corollary 2.20:
Let (X, M,
*) be a complete first type M � fuzzy metric space and T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{k}z,
t) ≥ min {M
(x, y, z, t/k), M
(x, T_{i}x, T_{j}y, t/k),
�[M
(y, T_{j}y, T_{k}z, t/k)+M
(z, T_{k}z, T_{i}x, t/k)]}

for all i ≠ j ≠ k and
for all x, y, z�X . Then {T_{n}} have a unique common fixed point.

Remark 2.21: By taking T_{i} = T_{j}
= T_{k }= 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 T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

3M
(T_{i}x, T_{j}y, T_{k}z,
t) ≥ {M
(x, y, z, t/k) + M
(y, z, T_{k}z, t/k) + �[M
(x, T_{i}x, z, t/k)+M
(x, y, T_{j}y, t/k)]}

for all i ≠ j ≠ k and
for all x, y, z�X . Then {T_{n}} have a unique common fixed point.

Proof:
Let x_{0}�X be any arbitrary element.

Define a sequence {x_{n}} in
X as x_{n+}_{1 }= T_{n+}_{1}x_{n}
for n = 0, 1, 2,�

Now we prove that {x_{n}} is a
Cauchy sequence in X.

Uniqueness: Suppose x ≠ y such that T_{n}y
= y for all n.

Then

3M
(x, y, y, t) = 3M
(T_{i}x, T_{j}y, T_{k}y,
t)

�
{M
(x, y, y, t/k) + M
(y, y, T_{k}y, t/k)

+ �[M
(x, T_{i}x, y, t/k)+M
(x, y, T_{j}y, 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 {T_{n}} have a unique
common fixed point.

This completes the proof.

Remark 2.24:
From the above theorem we have,

M
(T_{i}x, T_{j}y, T_{k}z,
t) ≥ {M
(x, y, z, t/k) + M
(y, z, T_{k}z, t/k) +�[M
(x, T_{i}x, z, t/k)+M
(x, y, T_{j}y, t/k)]}

�
min {M
(x, y, z, t/k), M
(y, z, T_{k}z, t/k),

�[M
(x, T_{i}x, z, t/k)+M
(x, y, T_{j}y, t/k)]}

Therefore,

M
(T_{i}x, T_{j}y, T_{k}z,
t) �
min {M
(x, y, z, t/k), M
(y, z, T_{k}z, t/k), �[M
(x, T_{i}x, z, t/k)+M
(x, y, T_{j}y, t/k)]}

Hence we get the following corollary.

Corollary 2.25:
Let (X, M,
*) be a complete first type M � fuzzy metric space and T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{k}z,
t) �
min {M
(x, y, z, t/k), M
(y, z, T_{k}z, t/k), �[M
(x, T_{i}x, z, t/k)+M
(x, y, T_{j}y, t/k)]}

for all i ≠ j ≠ k and
for all x, y, z�X . Then {T_{n}} have a unique common fixed point.

Remark 2.26: By taking T_{i} = T_{j}
= T_{k }= 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 T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

= {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 {T_{n}} 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 T_{n}:
X�X be a sequence of maps such that for all t > 0 and 0 <
k < 1 satisfying the condition

M
(T_{i}x, T_{j}y, T_{k}z,
t) ≥ min {M
(x, y, z, t/k), M
(x, T_{i}x, T_{j}y, t/k),
M
(y, z, T_{k}z, t/k),�[M
(y, T_{j}y, T_{k}z, t/k)+M
(z, T_{k}z, T_{i}x, t/k)],
�[M
(x, T_{i}x, z, t/k)+M
(x, y, T_{j}y, t/k)]}

for all i ≠ j ≠ k and
for all x, y, z�X . Then {T_{n}} have a unique common fixed point.

Remark 2.30: By taking T_{i} = T_{j}
= T_{k }= 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.

3. References

[1] Dhage .B.C, Generalised metric
spaces and mappings with fixed point, Bull. Calcutta Math. Soc., 84(4),
(1992), 329-336.

[2] George .A and Veeramani .P, On
some results in fuzzy metric space, Fuzzy Sets and Systems, 64 (1994),
395-399.

[3] Grabiec .M, Fixed points in fuzzy
metric spaces, Fuzzy Sets and Systems, 27 (1988), 385-389.

[4] Gregori V and Sapena A, On fixed
point theorem in fuzzy metric spaces, Fuzzy Sets and Systems, 125
(2002), 245-252.

[5] Klement .E.P, Mesiar .R and Pap
.E, Triangular Norms, Kluwer Academic Publishers.

[6] Kramosil .I and Michalek .J, Fuzzy
metric and statistical metric spaces, Kybernetica, 11 (1975), 326-334.

[7] Mihet D, A Banach contraction
theorem in fuzzy metric spaces, Fuzzy Sets and Systems, 144 (2004),
431-439.

[8] Naidu .S.V.R, Rao .K.P.R and
Srinivasa .R.N, On the topology of D-metric spaces and the generation of
D-metric spaces from metric space, Internet. J. Math. Math. Sci., 51
(2004), 2719-2740.

[9] Naidu .S.V.R, Rao .K.P.R and
Srinivasa .R.N, On the convergent sequences and fixed point theorems in
D-metric spaces, Internet. J. Math. Math. Sci., 12 (2005), 1969-1988.

[10] Park .J.H, Park .J.S and Kwun .Y.C,
Fixed point in M � fuzzy metric spaces, Optimization and
Decision Making, 7 (2008), No.4, 305-315.

[11] Schweizer .B and Sklar .A,
Statistical metric spaces, Pacific. J. Math., 10 (1960), 314-334.

[12] Sedghi .S and Shobe .N, A Common
fixed point theorems in two M � fuzzy metric spaces, Commun. Korean Math.
Soc. 22(2007), No.4, 513-526.

[13] Sedghi .S and Shobe .N, Fixed point
theorem in M � fuzzy metric spaces with property (E),
Advances in Fuzzy Mathematics, Vol.1, No.1 (2006), 55-65.

[14] Veerapandi .T, Jeyaraman .M and
Paul Raj Josph .J, Some fixed point and coincident point theorem in
generalized M
� fuzzy metric space, Int. Journal of Math. Analysis, Vol.3, (2009),
627-635.

[15] Veeapandi .T, Uthaya Sankar .G and
Subramanian .A, Some Common fixed point theorems in M
� fuzzy metric space, Int. Journal of Mathematical Archive-3(3), (2012),
1005-1016.

[16] Zadeh .L. A, Fuzzy sets,
Information and Control, 8 (1965), 338-353.

Copyrights statperson consultancy www

Copyrights
�
statperson consultancy www.statperson.com
2013. All Rights Reserved.