Home| Journals | Statistical Calculator | About Us | Contact Us
    About this Journal  | Table of Contents
Untitled Document

 [Abstract] [PDF] [HTML] [Linked References]

 

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.

 

    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.

Developer Details