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Common Fixed Point Theorems for Sequence of Mappings in M - Fuzzy Metric Space

T. Veerapandi1, G. Uthaya Sankar2 and A. Subramanian3

1Associate Professor of Mathematics, P.M.T. College, Melaneelithanallur � 627953 INDIA

2Department of Mathematics, Mano College (An Institution of M.S. University, Tirunelveli), Sankarankovil � 627756 INDIA

3Department of Mathematics, The M.D.T. Hindu College, Tirunelveli � 627010 INDIA

1[email protected], 2[email protected], 3[email protected]

Research Article

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 � 6limt 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 ij 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

3M (xn+1, xn+2, xn+2, t)

= 3M (Tn+1xn, Tn+2xn+1, Tn+2xn+1, t)

≥ {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+1, xn+1, t/k) + M (xn, xn, xn+1, t/k)

+ M (xn+1, xn+1, xn+2, t/k)}

= 2M (xn, xn+1, xn+1, t/k) + M (xn+1, xn+2, xn+2, t/k)

2M (xn, xn+1, xn+1, t/k) + M (xn+1, xn+2, xn+2, t)

Therefore,

2M (xn+1, xn+2, xn+2, t) 2M (xn, xn+1, xn+1, t/k)

That is, 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

3M (Tmx, x, x, t) = limn   3M (Tmx, xn+2, xn+2, t)

= limn   3M (Tmx, Tn+2xn+1, Tn+2xn+1, t)

≥ limn   {M (x, xn+1, xn+1, t/k) + M (x, x, Tmx, t/k)

+ M (xn+1, xn+1, Tn+2xn+1, t/k)}

= limn   {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, x, t/k) + M (x, x, Tmx, t/k)

+ M (x, x, x, t/k)}

= {1 + M (x, x, Tmx, t/k) + 1}

= 2 + M (x, x, Tmx, t/k)

2 + M (x, x, Tmx, t)

Therefore, 2M (Tmx, x, x, t) 2

That is, M (Tmx, x, x, t) 1

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 xy 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 xy.

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 ij 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 ij 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)

≥ {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+1, xn+1, t/k)*M (xn, xn, xn+1, t/k)

*M (xn+1, xn+1, xn+2, t/k)}

= {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+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   {M (x, xn+1, xn+1, t/k)*M (x, x, Tmx, t/k)

*M (xn+1, xn+1, Tn+2xn+1, t/k)}

= limn   {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, x, t/k)*M (x, x, Tmx, t/k)*M (x, x, x, t/k)}

= {1*M (x, x, Tmx, t/k)*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 xy 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 xy.

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 ij 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),

M (xn, xn, xn+1, t/k)*M (xn+1, xn+2, xn+2, t/k)}

= {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 xy 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 xy.

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 ij 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)

≥ {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)}

= {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)}

= {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)}

{M (xn, xn+1, xn+1, t/k)*M (xn+1, xn+2, xn+2, t/k)

*M (xn, xn, xn+2, 2t/k)}

{M (xn, xn+1, xn+1, t/k)*M (xn+1, xn+2, xn+2, t/k)

*M (xn, xn, xn+1, t/k)*M (xn+1, xn+2, xn+2, t/k)}

{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   {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   {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)}

= {M (x, x, x, t/k)*M (x, x, Tmx, t/k)*M (x, x, x, t/k)

*M (x, x, x, 2t/k)}

= {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 xy 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 xy.

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 ij 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, Tn+1xn, Tn+2xn+1, 2t/k)}

= min {M (xn, xn+1, xn+1, t/k), M (xn, xn+1, 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+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) 11 = 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, Tmx, Tn+2xn+1, 2t/k)}

= 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 xy 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 xy.

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

3M (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)]}

for all ij 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,�

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

For n 0, we have

3M (xn+1, xn+2, xn+3, t)

= 3M (Tn+1xn, Tn+2xn+1, Tn+3xn+2, t)

{M (xn, xn+1, xn+2, t/k) + M (xn, Tn+1xn, Tn+2xn+1, t/k) + �[M(xn+1,Tn+2xn+1,Tn+3xn+2,t/k)+M(xn+2,Tn+3xn+2,Tn+1xn,t/k)]}

= {M (xn, xn+1, xn+2, t/k) + M (xn, xn+1, xn+2, t/k)

+ �[M (xn+1, xn+2, xn+3, t/k)+M (xn+2, xn+3, xn+1, t/k)]}

= {M (xn, xn+1, xn+2, t/k) + M (xn, xn+1, xn+2, t/k)

+ M (xn+1, xn+2, xn+3, t/k)}

= 2M (xn, xn+1, xn+2, t/k) + M (xn+1, xn+2, xn+3, t/k)

2M (xn, xn+1, xn+2, t/k) + M (xn+1, xn+2, xn+3, t)

Therefore,

2M (xn+1, xn+2, xn+3, t) 2M (xn, xn+1, xn+2, t/k)

That is, M (xn+1, xn+2, xn+3, t) M (xn, xn+1, xn+2, t/k).

Continuing this way we get

M (xn+1, xn+2, xn+3, t) M (xn, xn+1, xn+2, t/k)

M (xn-1, xn, xn+1, t/k2)

.

.

.

M (x0, x1, x2, t/kn+1)

1 as n

Since M is first type, we have

M (xn+1, xn+1, xn+2, t) M (xn+1, xn+2, xn+3, t)

Therefore, M (xn+1, xn+1, xn+2, t) 1 as n

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,)

Taking limit as n ∞ we get

limn M (xn, xn+p, xn+p, t) 1 1 = 1

Therefore, limn M (xn, xn+p, xn+p, t) = 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

3M (Tmx, x, x, t) = limn   3M (Tmx, xn+2, xn+3, t)

= limn   3M (Tmx, Tn+2xn+1, Tn+3xn+2, t)

limn   {M (x, xn+1, xn+2, t/k) + M (x, Tmx, Tn+2xn+1, t/k) +�[M(xn+1,Tn+2xn+1,Tn+3xn+2, t/k)+M(xn+2,Tn+3xn+2,Tmx,t/k)]}

= limn {M (x, xn+1, xn+2, t/k) + M (x, Tmx, xn+2, t/k)

+ �[M (xn+1, xn+2, xn+3, t/k)+M (xn+2, xn+3, Tmx, t/k)]}

= {M (x, x, x, t/k) + M (x, Tmx, x, t/k)

+ � [M (x, x, x, t/k)+M (x, x, Tmx, t/k)]}

= {1 + M (Tmx, x, x, t/k) + �[1+M (Tmx, x, x, t/k)]}

= �[3M (Tmx, x, x, t/k) + 3]

6M (Tmx, x, x, t) 3M (Tmx, x, x, t/k) + 3

3M (Tmx, x, x, t) + 3

Therefore, 3M (Tmx, x, x, t) 3

That is, M (Tmx, x, x, t) 1

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 xy 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 xy.

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 ij 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 ij 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,�

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

For n 0, we have

3M (xn+1, xn+2, xn+3, t)

= 3M (Tn+1xn, Tn+2xn+1, Tn+3xn+2, t)

{M (xn, xn+1, xn+2, t/k) + M (xn+1, xn+2, Tn+3xn+2, t/k)

+ �[M (xn, Tn+1xn, xn+2, t/k)+M (xn, xn+1, Tn+2xn+1, t/k)]}

= {M (xn, xn+1, xn+2, t/k) + M (xn+1, xn+2, xn+3, t/k)

+ �[M (xn, xn+1, xn+2, t/k)+M (xn, xn+1, xn+2, t/k)]}

= {M (xn, xn+1, xn+2, t/k) + M (xn+1, xn+2, xn+3, t/k)

+ M (xn, xn+1, xn+2, t/k)}

= 2M (xn, xn+1, xn+2, t/k) + M (xn+1, xn+2, xn+3, t/k)

2M (xn, xn+1, xn+2, t/k) + M (xn+1, xn+2, xn+3, t)

Therefore,

2M (xn+1, xn+2, xn+3, t) 2M (xn, xn+1, xn+2, t/k)

That is, M (xn+1, xn+2, xn+3, t) M (xn, xn+1, xn+2, t/k).

Continuing this way we get

M (xn+1, xn+2, xn+3, t) M (xn, xn+1, xn+2, t/k)

M (xn-1, xn, xn+1, t/k2)

.

.

.

M (x0, x1, x2, t/kn+1)

1 as n

Since M is first type, we have

M (xn+1, xn+1, xn+2, t) M (xn+1, xn+2, xn+3, t)

Therefore, M (xn+1, xn+1, xn+2, t) 1 as n

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,)

Taking limit as n ∞ we get

limn M (xn, xn+p, xn+p, t) 1 1 = 1

Therefore, limn M (xn, xn+p, xn+p, t) = 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

3M (Tmx, x, x, t) = limn   3M (Tmx, xn+2, xn+3, t)

= limn   3M (Tmx, Tn+2xn+1, Tn+3xn+2, t)

limn {M (x, xn+1, xn+2, t/k) + M (xn+1, xn+2, Tn+3xn+2, t/k) + �[M (x, Tmx, xn+2, t/k)+M (x, xn+1, Tn+2xn+1, t/k)]}

= limn {M (x, xn+1, xn+2, t/k) + M (xn+1, xn+2, xn+3, t/k) + �[M (x, Tmx, xn+2, t/k)+M (x, xn+1, xn+2, t/k)]}

= {M (x, x, x, t/k) + M (x, x, x, t/k)

+ �[M (x, Tmx, x, t/k)+M (x, x, x, t/k)]}

= {1 + 1 + �[M (Tmx, x, x, t/k)+1]}

= �[M (Tmx, x, x, t/k) + 5]

6M (Tmx, x, x, t) M (Tmx, x, x, t/k) + 5

M (Tmx, x, x, t) + 5

Therefore, 5M (Tmx, x, x, t) 5

That is, M (Tmx, x, x, t) 1

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 xy 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 xy.

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 ij 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

5M (Tix, Tjy, Tkz, t) ≥ {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 ij 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,�

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

For n 0, we have

5M (xn+1, xn+2, xn+3, t)

= 5M (Tn+1xn, Tn+2xn+1, Tn+3xn+2, t)

{M (xn, xn+1, xn+2, t/k) + M (xn, Tn+1xn, Tn+2xn+1, t/k) + M (xn+1, xn+2, Tn+3xn+2, t/k) + �[M (xn+1,Tn+2xn+1,Tn+3xn+2,t/k)+M (xn+2,Tn+3xn+2,Tn+1xn,t/k) + �[M (xn, Tn+1xn, xn+2, t/k)+M (xn, xn+1, Tn+2xn+1, t/k)]}

={M(xn,xn+1,xn+2,t/k)+M(xn,xn+1,xn+2,t/k)

+M(xn+1,xn+2,xn+3, t/k)

+ �[M (xn+1, xn+2, xn+3, t/k)+M (xn+2, xn+3, xn+1, t/k)]

+ �[M (xn, xn+1, xn+2, t/k)+M (xn, xn+1, xn+2, t/k)]}

= {M (xn, xn+1, xn+2, t/k) + M (xn, xn+1, xn+2, t/k)

+ M (xn+1, xn+2, xn+3, t/k) + M (xn+1, xn+2, xn+3, t/k)

+ M (xn, xn+1, xn+2, t/k)}

= 3M (xn, xn+1, xn+2, t/k) + 2 M (xn+1, xn+2, xn+3, t/k)

3M (xn, xn+1, xn+2, t/k) + 2 M (xn+1, xn+2, xn+3, t)

Therefore,

3M (xn+1, xn+2, xn+3, t) 3M (xn, xn+1, xn+2, t/k)

That is, M (xn+1, xn+2, xn+3, t) M (xn, xn+1, xn+2, t/k).

Continuing this way we get

M (xn+1, xn+2, xn+3, t) M (xn, xn+1, xn+2, t/k)

M (xn-1, xn, xn+1, t/k2)

.

.

.

M (x0, x1, x2, t/kn+1)

1 as n

Since M is first type, we have

M (xn+1, xn+1, xn+2, t) M (xn+1, xn+2, xn+3, t)

Therefore, M (xn+1, xn+1, xn+2, t) 1 as n

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,)

Taking limit as n ∞ we get

limn M (xn, xn+p, xn+p, t) 1 1 = 1

Therefore, limn M (xn, xn+p, xn+p, t) = 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

5M (Tmx, x, x, t) = limn   5M (Tmx, xn+2, xn+3, t)

= limn   5M (Tmx, Tn+2xn+1, Tn+3xn+2, t)

limn {M (x, xn+1, xn+2, t/k) + M (x, Tmx, Tn+2xn+1, t/k) + M (xn+1, xn+2, Tn+3xn+2, t/k) + �[M (xn+1,Tn+2xn+1,Tn+3xn+2,t/k) + M (xn+2,Tn+3xn+2,Tmx,t/k)] + �[M (x, Tmx, xn+2, t/k)+M (x, xn+1, Tn+2xn+1, t/k)]}

= limn {M (x, xn+1, xn+2, t/k)

+ M (x, Tmx, xn+2, t/k) + M (xn+1, xn+2, xn+3, t/k)

+ �[M (xn+1, xn+2, xn+3, t/k)+M (xn+2, xn+3, Tmx, t/k)]

+ �[M (x, Tmx, xn+2, t/k)+M (x, xn+1, xn+2, t/k)]}

= {M (x, x, x, t/k) + M (x, Tmx, x, t/k) + M (x, x, x, t/k)

+ �[M (x, x, x, t/k)+M (x, x, Tmx, t/k)]

+ �[M (x, Tmx, x, t/k)+M (x, x, x, t/k)]}

= {1 + M (Tmx, x, x, t/k) + 1 + �[1+M (Tmx, x, x, t/k)]

+ �[M (Tmx, x, x, t/k)+1]}

= 3 + 2M (Tmx, x, x, t/k)

3 + 2M (Tmx, x, x, t)

Therefore, 3M (Tmx, x, x, t) 3

That is, M (Tmx, x, x, t) 1

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 xy 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 xy.

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 ij 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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