A Fixed Point Theorem in Modified Intuitionistic Fuzzy Metric Spaces

International Journal of Scientific & Engineering Research Volume 4, Issue 1, January-2013 1

ISSN 2229-5518

A Fixed Point Theorem in Modified Intuitionistic Fuzzy Metric Spaces Syed Shahnawaz Ali 1, Dr. Jainendra Jain 2, Dr. Anil Rajput 3

Abstract— Fuzzy Mathematics has seen an enormous growth since the introduction of notion of fuzzy sets by Zadeh in 1965. Kramosil and Michalek introduced the notion of fuzzy metric spaces which was later modified by George and Veeramani and others. The notion of intuitionistic fuzzy metric spaces was introduced by Park in 2004. Many authors have studied fixed point and common fixed theorems for mappings on fuzzy metric spaces and intuitionistic fuzzy metric spaces. In this paper we prove a common fixed point theorem for a sequence of mappings in an intuitionistic fuzzy metric space.

Index Terms— Fixed Points, Fuzzy sets, Fuzzy Metric Spaces, Intuitionistic Fuzzy Sets, Intuitionistic Fuzzy Metric Spaces, Triangular

Norm, Triangular Co norm.

1 INTRODUCTION

——————————  ——————————

he concept of fuzzy sets was introduced by Zadeh [22] in
1965. Since then, with a view to utilize this concept in topology and analysis, many authors have extensively developed the theory of fuzzy sets. In 1975, Kramosil and Michalek [7] introduced the concept of a fuzzy metric space by generalizing the concept of a probabilistic metric space to the fuzzy situation. The concept of Kramosil and Michalek [7] of a fuzzy metric space was later modified by George and Veeramani [1] in 1994. In 1988, Grabeic [11], following the concept of Kramosil and Michalek [7], obtained the fuzzy version of Banach’s fixed point theorem. Jungck [5] introduced the notion of compatible mappings in metric spaces and utilized the same as a tool to improve commutativity conditions in common fixed point theorems. This concept has frequently been employed to prove existence theorems on common fixed points. In recent past, several authors proved various fixed point theorems employing relatively more general contractive conditions. However, the study of common fixed points of non-compatible mappings is also equally interesting which was initiated by Pant [18]. The notion of Intuitionistic Fuzzy Sets was put forward by Atanassov [10] in 1986 and notion of Intuitionistic Fuzzy Metric Spaces was given by Park [9] in 2004 employing the
notions of continuous t − norm and continuous t − conorm.
Fixed point theory is one of the most fruitful and effective
tools in mathematics which has enormous applications in several branches of science especially in chaos theory, game theory, theory of differential equation, etc. Intuitionistic fuzzy metric notion is also useful in modeling some physical problems wherein it is necessary to study the relationship between two probability functions as noticed by Gregori et al. [21]. For instance, it has a concrete physical visualization in the
————————————————
1. Syed Shahnawaz Ali is working as Assisstant Professor in Mathematics at Sagar Institute of Research, Technology & Science, Bhopal, India. E-mail: shahnawaz_hunk@yahoo.com
2. Dr. Jainendra Jain is working as Professor in Mathematics at Sagar Institute of Research, Technology & Science, Bhopal, India. E-mail: jj.28481@gmail.com

3. Dr. Anil Rajput is working as Professor in Mathematics at Chandra Shekhar

context of two slit experiment as the foundation of E −infinity
theory of high energy physics whose details are available in El
Naschie in [12], [13], [14]. The topology induced by intuitionistic fuzzy metric coincides with the topology induced by fuzzy metric as noticed by Gregori et al. [21]. Following this, Saadati et al. [17] reframed the idea of intuitionistic fuzzy metric spaces and proposed a new notion under the name of modified intuitionistic fuzzy metric spaces
by introducing the notion of continuous t − representable
norm.

Fixed point and common fixed point properties for mappings defined on fuzzy metric spaces intuitionistic fuzzy metric

spaces and ℒ − fuzzy metric spaces have been studied by
many authors like H. Adibi et al. [6], S. Sharma [19], J. Goguen
[8], V. Gregori and A. Sapena [20], C. Alaca et al. [2], Saadati et al. [15], [16]. Most of the properties which provide the existence of fixed points and common fixed points are of linear contractive type conditions.
In this paper we prove a common fixed point theorem for a sequence of mappings in intuitionistic fuzzy metric spaces introduced by Park [9] and modified by Saadati et al. [17]. For the sake of completeness we recall some definitions and results in the next section.

2 PRELIMINARIES

Definition 1: Let ℒ = (L∗ , ≤L∗ ) be a complete lattice, and U a non empty set called a universe. An ℒ −fuzzy set Jl on U is defined as amapping Jl: U → L∗ . For each u in U, Jl(u) represents the degree (in L∗ ) to which u satisfies Jl.

Lemma 2: Consider the set L∗ and operation ≤L∗ defined by ∶

L∗ = {(x1 , x2 ) ∶ (x1, x2 ) ∈ [0, 1]2 & x1 + x2 ≤ 1},

(x1, x2 ) ≤ L∗ (y1 , y2 ) ⟺ x1 ≤ y1

and x2 ≥ y2 , for every (x1 , x2 ), (y1 , y2 ) ∈ L∗ .

Then, L∗ , ≤ L∗ ) is a complete lattice.

Definition 3: An intuitionistic fuzzy set JlJ,< in a universe U

is an object JlJ,< = {((Jl (u), 1Jl (u))|u ∈ U}, where, for all

u ∈ U, (Jl (u) ∈ [0, 1] and 1Jl (u) ∈ [0, 1] are called the

Azad P. G. College, Sehore, India. E-mail: dranilrajput@hotmail.com

http://www.ijser.org

International Journal of Scientific & Engineering Research Volume 4, Issue 1, January-2013 2

ISSN 2229-5518

membership degree and the non-membership degree,

≥L∗ T ℳM,N (x, z, t), ℳM,N (z, y, s) ;

∗

respectively of u in JlJ,< and furthermore they satisfy

5) ℳM,N (x, y,·): (0, ∞) → L

is continuous.

(Jl (u) + 1Jl (u) ≤ 1.

Definition 4: For every za = (xa , ya ) ∈ L∗ we define

∨ (za ) = (sup(xa ), inf(ya )).

Since z a ∈ L∗ then x a + y a ≤ 1 so sup(xa ) + inf (ya ) ≤

In this case ℳM,N is called an intuitionistic fuzzy metric space.

Here, ℳM,N (x, y, t) = (M(x, y, t ), N(x, y, t )).

Example 9: Let (X, d) be a metric space. Define T(a, b) =

{a1 b1, min(a2 + b2, 1)} for all a = (a1 , a2 ) and b = (b1, b2) ∈ L∗

sup(xa + ya ) ≤ 1, that is ∨ (za ) ∈ L∗ . We denote its units by

0L∗ = (0, 1) and 1L∗ = (1, 0).

and let M and N be fuzzy sets on X
follows:

× (0, ∞) defined as

Classically, a triangular norm ∗ = T on ([0,1], ≤) is defined as

an increasing, commutative, associative mapping T: [0,1]2 →

ℳM,N (x, y, t) = (M(x, y, t ), N(x, y, t))

[0,1] satisfying T(1, x) = 1 ∗ x = x, for all x ∈ [0,1]. A

triangular conorm S = ◊ is defined as an increasing,

ℎt n

= ℎt n + md(x, y) ,

md(x, y)

ℎt n + md(x, y) ,

commutative, associative mapping S: [0,1]2 → [0,1] satisfying

for all ℎ m n t ∈ ℝn
Then X ℳ

T is an intuitionistic

S(0, x) = 0 ◊ x = x, for all x ∈ [0,1]. Using the lattice (L∗ , ≤L∗ )

, , , .

( , M,N , )

these definitions can be straightforwardly extended.

Definition 5: [3, 4] A triangular norm (t–norm) on L∗ is a mapping T ∶ (L∗)2 → L∗ satisfying the following conditions:

1) (∀x ∈ L∗ )(T (x, 1L∗ ) = x), (boundary condition)

fuzzy metric space.

Example 10: Let X = N. Define T(a, b) = {(max(0, a1 + b1 −

1), a2 + b2 − a2 b2} for all a = (a1 , a2 ) and b = (b1 , b2 ) ∈ L

and let M and N be fuzzy sets on X 2 × (0, ∞) defined as

follows

2) (∀(x, y) ∈ (L∗ ) )(T(x, y) = T(y, x)), (commutativity) :

x y − x

if x ≤ y

3) (∀(x, y, z) ∈ (L∗ 3 ) T (x, T(y, z)) = T (T (x, y), z) ,

(associativity),

4) (∀ (x, x’, y, y’) ∈ (L∗ )(x ≤L∗ x’) and (y ≤L y’ ⟹

ℳM,N (x, y, t) = (M(x, y, t ), N(x, y, t)) =

y y

y x − y

x , x

if y ≤ x

T(x, y) ≤L∗ T (x’, y’)). (monotonicity).

for all x, y E X and t > 0. Then (X, ℳM,N , T ) is an intuitionistic

Definition 6: [3, 4] A continuous t–norm T on L∗ is called

continuous t– representable if and only if there exist a
continuous t– norm ∗ and a continuous t– conorm ◊ on [0, 1]

such that, for all x = (x1 , x2 ), y = (y1, y2 ) ∈ L∗ ,

T (x, y) = (x 1 ∗ y1 , x2 ◊ y2 ).

fuzzy metric space.

Definition 11: Let (X, ℳM,N , T ) be an intuitionistic fuzzy metric space and {xn } be a sequence in X.

1) A sequence {xn } is said to be convergent to x ∈ X in

the intuitionistic fuzzy metric space (X, ℳM,N , T ) and

Now define a sequence T n recursively by T 1 = T and

denoted by x

ℳrv,N

T n (x(1) , ⋯ , x (nn1) ) = T(T nn1(x(1) , ⋯ , x(n) ), x(n n1) )

for n ≥ 2 and x(i) ∈ L∗ .

We say the continuous t–representable norm is natural and

write Tn whenever Tn (a, b) = Tn (c, d) and a ≤L∗ c implies

n �⎯⎯ x if ℳM,N (xn , x, t) → 1L∗ as n → ∞

for every t > 0.

2) A sequence {xn } in an intuitionistic fuzzy metric space

(X, ℳM,N , T ) is called a Cauchy sequence if for each

0 < c < 1 and t > 0, there exists no ∈ ℕ such that

b ≤L∗ d.

ℳM N (x , x

, t) > ∗ (N (c), c), and for each n, m ≥ n ;

, n m L s o

Definition 7: A negator on L∗ is any decreasing mapping N ∶ L∗ → L∗ satisfying N(0L∗ ) = 1L∗ and N(1L∗ ) = 0L∗ . If N(N(x)) = x, for all x ∈ L∗ , then N is called an involutive negator. A negator on [0, 1] is a decreasing mapping N: [0, 1] → [0, 1] satisfying N(0) = 1 and N(1) = 0. NS denotes the standard negator on [0, 1] defined as NS (x) = 1 −

x for all x E [0, 1].

here Ns is the standard negator.
3) An intuitionistic fuzzy metric space is said to be
complete if and only if every Cauchy sequence in this
space is convergent. Henceforth, we assume that T is
a continuous t −norm on the lattice ℒ such that for

every 1 ∈ L∗ \ {0L∗ , 1L∗ }, there exists A ∈ L∗ \ {0L∗ , 1L∗ }

such that T nn1 (N(A), ⋯ , N(A)) ≥L∗ N(1).

Lemma 12: Let ℳM,N be an intuitionistic fuzzy metric. Then for

Definition 8: Let M, N are fuzzy sets from X 2 × (0, +∞) to

any t > 0 , ℳM,N

(x, y, t) is nondecreasing with respect to t in

[0, 1] such that M(x, y, t) + N(x, y, t) ≤ 1 for all x, y ∈ X and

t > 0. The 3 −tuple (X, ℳM,N , T ) is said to be an intuitionistic

fuzzy metric space if X is an arbitrary (non-empty) set, T is a
continuous t − representable norm and ℳM,N is a mapping

X 2 × (0, +∞) → L∗ (an intuitionistic fuzzy set, see Definition 3)

satisfying the following conditions for every x, y ∈ X and

t, s > 0:

( L∗ , ≤ L∗ ) for all x, y ∈ X.

Definition 13: Let (X, ℳM,N , T ) be an intuitionistic fuzzy

metric space. For t > 0, we define the open ball B(x, r, t ) with
center x ∈ X and 0 < r < 1 by

B(x, r, t) = {y ∈ X ∶ ℳM,N (x, y, t) >L∗ (Ns (r), r)}.

A subset A ⊂ X is called open if for each x ∈ A, there exist

1) ℳM,N (x, y, t) >L∗ 0L∗ ;

t > 0 and 0 < r < 1 such that B(x, r, t) ⊂ A. Let Tℳrv,N

denote

2) ℳM,N (x, y, t) = 1L∗ if and only if x = y;

the family of all open subset of X. Tℳrv,N
is called the topology

3) ℳM,N (x, y, t) = ℳM,N (y, x, t);

4) ℳM,N (x, y, t + s )

induced by the intuitionistic fuzzy metric space.

International Journal of Scientific & Engineering Research Volume 4, Issue 1, January-2013 3

ISSN 2229-5518

Definition 14: Let (X, ℳM,N , T ) be an intuitionistic fuzzy metric space. A subset A of X is said to be IF −bounded if there exist t > 0 and 0 < r < 1 such that ℳM,N (x, y, t) >L∗ (Ns (r), r) for each x, y ∈ A.

Definition 15: Let (X, ℳM,N , T ) be an intuitionistic fuzzy

Example 19: Let (X, d) be a metric space. Define T(a, b) = (a1 b1, a2 b2, min{a3 + b3, 1}) for all a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3) ∈ L∗ and let M and N be fuzzy sets on X 3 × (0, ∞) defined as follows:

ℳM,N (x, y, z, t) = (M(x, y, z t ), N(x, y, z , t))

metric space. ℳ is said to be continuous on X × X × ]0, ∞[ if

ℎt n

md x y z

nli→m ℳM,N (xn , yn , tn ) = ℳM,N (x, y, t)

( , , )

= , ,

whenever {(x , y , t )} is a sequence in X × X × ]0, ∞[ which

ℎt n + md(x, y, z)

n n n

converges to a point

(x, y, t) ∈ X × X × ]0, ∞[ i. e. , nli→m ℳM,N (xn , x, t)

= nli→m ℳM,N (yn , y, t) = 1L∗

for all ℎ, m, n, t ∈ ℝn . Then (X, ℳM N

fuzzy metric space.

, T ) is an intuitionistic

and lim

ℳ (x, y, t ) = ℳ

(x, y, t).

n→∞

M,N n

M,N

Definition 20: Let (X, ℳM,N , T ) be an intuitionistic fuzzy

metric space and {xn } be a sequence in X.

Lemma 16: Let (X, ℳM,N , T ) be an intuitionistic fuzzy metric

1) A sequence {x } is said to be convergent to x ∈ X in

space and define EA,ℳrv ,N ∶ X

→ ℝn

∪ {0} by

the intuitionistic fuzzy metric space

ℳrv,N

(X, ℳM,N , T ) and

EA,ℳrv,N (x, y) = inf{t > 0 ∶ ℳM,N (x, y, t) >L∗ N(A)}

denoted by

xn �⎯⎯ x if ℳM,N (xn , x, x, t) → 1L∗ as

for each A ∈ L∗ \ {0L∗ , 1L∗ } and x, y ∈ X here, N is an involutive

negator. Then we have

n → ∞ for every t > 0.

2) A sequence {xn } in an intuitionistic fuzzy metric space

(X, ℳM,N , T ) is called a Cauchy sequence if for each

0 < c < 1 and t > 0, there exists no ∈ ℕ such that

(i) For any 1 ∈ L∗ \ {0L∗ , 1L∗ }, there exists A ∈ L∗ \ {0L∗ , 1L∗ } such

ℳ (x , x , x

, t) > ∗ (N (c), c), and for each

M,N

n n m L s

that EJ1,ℳrv ,N (x1, xn ) ≤ EA,ℳrv,N (x1 , x2 ) + EA,ℳrv ,N (x2, x3 ) + ⋯ +

EA,ℳrv,N (xnn1 , xn ) for any x1 , x2 , x3 , ⋯ xn ∈ X.

(ii) The sequence {xn }n∈ℕ is convergent to x with respect to

intuitionistic fuzzy metric ℳM,N if and only if EA,ℳrv ,N (xn , x) → 0.

Also, the sequence {xn } is a Cauchy sequence with respect to

intuitionistic fuzzy metric ℳM,N if and only if it is a Cauchy

sequence with EA,ℳrv,N .

Lemma 17: Let (X, ℳM,N , T ) be an intuitionistic fuzzy metric

space. If

ℳM,N (xn , xnn1 , t) ≥L∗ ℳM,N xo , x1 , kn

for some k < 1 and n E ℕ then {xn } is a Cauchy sequence.

We now extend the above definitions and results.

n, m ≥ no ; here Ns is the standard negator.

3) An intuitionistic fuzzy metric space is said to be

complete if and only if every Cauchy sequence in this space is convergent.

Lemma 21: Let ℳM,N be an intuitionistic fuzzy metric. Then for any t > 0 , ℳM,N (x, y, z, t) is nondecreasing with respect to

t in ( L∗ , ≤ L∗ ) for all x, y, z ∈ X.

Definition 22: Let (X, ℳM,N , T ) be an intuitionistic fuzzy

metric space. For t > 0, we define the open ball B(x, r, t ) with
center x ∈ X and 0 < r < 1 by

B(x, r, t) = {y ∈ X ∶ ℳM,N (x, y, y, t) >L∗ (Ns (r), r)}

A subset A ⊂ X is called open if for each x ∈ A, there exist

t > 0 and 0 < r < 1 such that B(x, r, t) ⊂ A. Let Tℳrv,N denote

the family of all open subset of X. Tℳrv,N is called the topology
induced by the intuitionistic fuzzy metric space.

Definition 18: Let M, N are fuzzy sets from X 3 × (0, +∞) to

Definition 23: Let (X, ℳM,N

, T ) be an intuitionistic fuzzy

[0, 1]such that M(x, y , z, t) + N(x, y , z, t) ≤ 1 for all x, y, z ∈

X and t > 0. The 3 − tuple (X, ℳM,N , T ) is said to be an

metric space. ℳ is said to be continuous on X 3 × (0, ∞) if

lim ℳM,N (xn , yn , zn , tn ) = ℳM,N (x, y, z, t)

intuitionistic fuzzy metric space if X is an arbitrary (non-
Whenever

n→x∞ , y , z , t )} is a sequence in X 3 × (0, ∞) which

{( n n n n

empty) set, T is a continuous t −representable and ℳM,N is a

converges to a point (x, y, z, t) ∈ X 3 × (0, ∞), i. e.,

mapping X 3 × (0, +∞) → L∗ (an intuitionistic fuzzy set, see

limn→∞ xn

= x, limn→∞ yn

= y, limn→∞ zn

= z, and

Definition 3) satisfying the following conditions for every

x, y, z , w ∈ X and t, s > 0:

nli→m

∞

ℳM,N (x, y, z, tn ) = ℳM,N

(x, y, z, t).

1) ℳM,N (x, y, z, t) >L∗ 0L∗ ;

Lemma 24: Let (X, ℳM,N , T ) be an intuitionistic fuzzy metric

2) ℳM,N (x, y, z, t) = 1L∗ if and only if x = y = z;

space and define EA ℳ 3 n

3) ℳM,N (x, y, z, t) = ℳM,N (x, z, y, t) = ℳM,N (y, z, x, t);

∶ X → ℝ

, rv ,N

∪ {0} by

4) ℳM,N (x, y, z, t + s )

≥L∗ T ℳM,N (x, y, w, t), ℳM,N (w, z, z , s) ;

∗

EA,ℳrv,N

(x, y, z) = inf{t > 0 ∶ ℳM,N

(x, y, z, t) >L∗ N(A)}

5) ℳM,N (x, y, z,·) ∶ (0, ∞) → L

is continuous.

for each A ∈ L∗

∗ , 1 ∗ } and x, y, z ∈ X here, N is an involutive

In this case ℳM,N is called an intuitionistic fuzzy metric. Here,

ℳM,N (x, y, z, t) = (M(x, y, z, t ), N(x, y, z, t )).

\ {0L L

negator. Then we have

http://www.ijser.org

International Journal of Scientific & Engineering Research Volume 4, Issue 1, January-2013 4

ISSN 2229-5518

(i) For any 1 ∈ L∗ {0L∗ , 1L∗ } there exists A ∈ L∗ {0L∗ , 1L∗ } such that

= inf{t > 0 ∶ ℳM N (x , x

, x , t) > ∗ N(A)}

EJ1,ℳrv,N (x1, x2 , xn ) ≤ EA,ℳrv,N (x1 , x2 , x3 ) + EA,ℳrv,N (x2 , x3 , x ) +

, n nn1

nn2 L

⋯ + EA,ℳrv,N (xnn2 , xnn1, xn ) for any x1 , x2 , x3 , ⋯ xn ∈ X.

≤ inf{t > 0 ∶ ℳM,N xo , x1 , x2 , kn >L ∗ N(A)}

= inf{ kn t ∶ ℳM N (x , x

, x , t) > ∗ N(A)}

, o 1 2 L

(ii) The sequence {xn }n∈ℕ is convergent with respect to

= kn inf{t > 0 ∶ ℳM N (x , x

, x , t) > ∗ N(A)}

, o 1 2 L

intuitionistic fuzzy metric ℳM,N if and only if EA,ℳrv ,N (xn , x, x) →

0. Also the sequence {xn } is a Cauchy sequence with respect to

intuitionistic fuzzy metric ℳM,N if and only if it is a Cauchy

= k EA,ℳrv,N (xn , xnn1 , xnn2, t)

From lemma (24), for every 1 E L∗ \{0L∗ , 1L∗ } there exists

sequence with EA,ℳrv,N .

A ∈ L∗

\ {0L∗ , 1L∗ }, such that

Proof: For (i), by the continuity of t − norms, for every

EJ1,ℳrv,N (

n xnn1 ,

xm )

1 ∈ L∗ \ {0L∗ , 1L∗ }, we can find a A ∈ L∗ \{0L∗ , 1L∗ } such that ≤ E

(x , x

, x ) + E

(x , x

, x ) + ⋯

T(N(A), N(A)) ≥L∗ N(1). By definition 18 , we have

ℳM,N (x, y , z, EA,ℳrv ,N (x1, x2 , x3 ) + EA,ℳrv,N (x2, x3 , x ) + ⋯

y,ℳrv ,N

n nn1

nn2

y,ℳrv ,N

+Ey ℳ

nn1

mn2

nn2

mn1

nn3

+EA,ℳrv,N (xnn2 , xnn1, xn ) + n8)

, rv,N (x

, x , x )

≤ kn Ey ℳ

(xo, x1 , x2 ) + knn1 Ey ℳ

(xo , x1 , x2 ) + ⋯

≥L∗ T ( ℳM,N (x, y, w, EA,ℳrv,N (x1, x2 , x3 ) + EA,ℳrv,N (x2, x3 , x )

, rv,N

+kmn2Ey ℳ

rv,N

o 1 2

+ ⋯ + EA,ℳrv,N (xnn , xnn3, xnn2 ) + 2 ),

ℳM,N (w, z, z , EA,ℳrv,N (xnn3, xnn2 , xnn1 )

mn2

= Ey,ℳrv ,N (xo, x1 , x2 ) k

j=n

⟶ 0.

rv ,N (x , x , x )

+EA,ℳrv ,N (xnn2 , xnn1 , xn ) + 2 ))

≥L∗ T (N(A), N(A)) ≥L∗ N(1)

for every 8 > 0, which implies that

Hence sequence {xn } is a Cauchy sequence.

3 THE MAIN RESULT

Theorem 1: Let {An } be a sequence of mappings Ai of a complete intuitionistic fuzzy metric space (X, ℳM,N , T ) into itself such that, for any three mappings Ai , Aj , Al

m m m

EJ1,ℳrv,N (x1, x2 , xn ) ≤ EA,ℳrv,N (x1, x2 , x3 ) + EA,ℳrv,N (x2, x3 , x )

+ ⋯ + EA,ℳrv ,N (xnn2 , xnn1 , xn ) + n8

ℳM,N (Ai (x), Aj (y), Al (z), ai ,j,l t) ≥L∗ ℳM,N (x, y, z, t)

for some m; here 0 < ai ,j,l < k < 1 for i, ,, = 1, 2, ⋯ , x, y, z ∈

X and t > 0. Then the sequence {An } has a unique common fixed

point in X.

Since 8 > 0 was arbitrary, we have

Proof: Let x

be an arbitrary point in X and define a sequence

EJ1,ℳrv ,N (x1, x2 , xn ) ≤ EA,ℳrv,N (x1, x2 , x3 ) + EA,ℳrv,N (x2, x3 , x )

{x } in X by x

= A (x ), x

= A (x ), ⋯ . Then

we have

1 1 o 2

2 1 3 3 2

+ ⋯ + EA,ℳrv,N (xnn2 , xnn1, xn ).

ℳ (x , x

, x , t) = ℳ

(Am (x ), Am (x ), Am (x ), t)

M,N 1 2 3

M,N 1 o

2 1 3 2

For (ii), we have

≥ ℳ x , x , x , ,

ℳM,N (xn , x, x, 1) >L∗ N(A) ⟺ EA,ℳrv,N (xn , x, x) < 1

L∗ M,N o 1

a1,2,3

m m

for every 1 > 0.

ℳM,N (x2, x3 , x , t) = ℳM,N (A2 (x1), A3 (x2), A

(x3 ), t)

∗ ℳ (x , x , x , )

Lemma 25: Let (X, ℳM,N , T ) be an intuitionistic fuzzy metric

space. If

≥L M,N

1 2 3 a2 3 4

ℳM,N (xn , xnn1 , xnn2 , t) ≥L∗ ℳM,N xo, x1 , x2 , kn

for some k < 1 and n E ℕ then {xn } is a Cauchy sequence.

≥L∗ ℳM,N xo , x1 , x2 , a

and so on. By induction, we have

1,2,3

a2,3,4

Proof: For every A E L\{0L∗ , 1L∗ } and xn E X, we have

EA,ℳrv,N (xn , xnn1 , xnn2 , t)

ℳM,N (xn , xnn1 , xnn2, t) ≥L∗ ℳM,N xo , x1 , x2 ,

for n = 1,2,3 ⋯, which implies

http://www.ijser.org

∏n ai in1 in2

International Journal of Scientific & Engineering Research Volume 4, Issue 1, January-2013 5

ISSN 2229-5518

EA,ℳrv,N (xn , xnn1 , xnn2 , t)

as n → ∞. Therefore, for every t > 0, we have ℳ(x, x, y, t) =

m m

= inf{t > 0 ∶ ℳM,N (xn , xnn1 , xnn2, t) >L ∗ N(A)}

1L∗ , i. e. , x = y. Also Ai (x) = Ai (Ai (x)) = Ai (Ai (x)), i. e., Ai (x) is also a periodic point of Ai . Therefore, x = Ai (x), i. e. , x is a unique common fixed periodic point of the

mappings An for n = 1, 2, ⋯ . This completes the proof.

≤ inf{t > 0 ∶ ℳM,N xo, x1 , x2 , n

i=1

ai,in1,in2

>L ∗ N(A)}

= inf{ ai ,in1,in2 t > 0 ∶ ℳM,N (xo , x1 , x2 , t) >L ∗ N(A)}

n i=1

= ai,in1,in2 inf{t > 0 ∶ ℳM,N (xo, x1 , x2 , t) >L ∗ N(A)}

i=n1

4 CONCLUSIONS

In this paper we have proved a common fixed point theorem for a sequence of mappings for the modified intuitionistic fuzzy metric spaces defined using the notion of continuous t – representable norms. The result can be extended for more
general conditions.

= ai,in1,in2 EA,ℳrv ,N (xo , x1 , x2 , ) ≤ k

i=1

for every λ L \ {0L∗ , 1L∗ }.

EA,ℳrv ,N (xo , x1 , x2 , )

5 REFERENCES

Now, we show that {xn } is a Cauchy sequence. For every

[1] A. George and P. Veeramani, “On Some Results in Fuzzy Metric

∗ ∗

µ L \ {0L∗ , 1L∗ }, there exists γ L \ {0L∗ , 1L∗ } such that

EJ1,ℳrv,N (xn , xnn1 , xm )

≤ Ey,ℳrv ,N (xn , xnn1 , xnn2) + Ey,ℳrv ,N (xnn1 , xnn2 , xnn3 ) + ⋯

+Ey,ℳrv,N (xmn2 , xmn1 , xm )

Spaces,” Fuzzy Sets and Systems, vol. 64, pp. 395-399, 1994.

[2] C. Alaca, D. Turkoglu and C. Yildiz, “Fixed Points in Intuitionistic

Fuzzy Metric Spaces,” Chaos, Solitons and Fractals, vol. 29, pp. 1073 –

1078, 2006.

[3] G. Deschrijver and E.E.Kerre, “On the Relationship between Some

Extensions of Fuzzy Set Theory,” Fuzzy sets and Systems, vol. 23, pp.

227–235, 2003.

≤ kn Ey ℳ

(xo, x1 , x2 ) + knn1 Ey ℳ

(xo , x1 , x2 ) + ⋯

[4] G. Deschrijver, C. Corenelis and E.E.Kerre, “On the Representation of

, rv,N

+kmn2 Ey ℳ

rv,N

o 1 2

Intuitionistic Fuzzy t −norms and t −conorms,” IEEE Transactions on

Fuzzy Systems, vol. 12, pp. 45–61, 2004.

mn2

= Ey,ℳrv ,N (xo, x1 , x2 ) k

j=n

, rv ,N (x , x , x )

⟶ 0.

[5] G. Jungck, “Compatible Mappings and Common Fixed Points” International Journal of Mathematics and Mathematical Sciences, vol. 09, pp 771 – 779, 1986.

[6] H. Adibi, Y. J. Cho, D. O` Regan and R. Saadati, “Common Fixed

As m, n → ∞. Since X is left complete, there is x X such that limn→∞ xn = x. Now we prove that x is a periodic point of Ai for any i = 1, 2, ⋯, we have

ℳM N (x, x, Am (x), t)

Point Theorems in ℒ − Fuzzy Metric Spaces,” Appl. Math. Comp., vol.

182, pp. 820–828, 2006.

[7] I. Kramosil and J. Michalek, “Fuzzy Metric and Statistical Metric

Spaces,” Kybernetica, vol. 11, pp. 326–334, 1975.

m m

≥L∗ T (ℳM,N (x, x, xn , t − kt), ℳM,N (xn , Ai (x), Ai (x), kt))

= T(ℳM,N (x, x, xn , t(1 − k)),

[8] J. Goguen, “ℒ −Fuzzy Sets,” J. Math. Anal. Appl., vol. 18, pp. 145 –

174, 1967.

m m m

ℳM,N (An (xnn1 ), Ai

≥L∗ T (ℳM,N (x, x, xn , t(1 − k)),

(x), Ai

(x), kt))

[9] J.H. Park, “Intuitionistic Fuzzy Metric Spaces,” Chaos, Solitons and

Fractals, vol. 22, pp. 1039 – 1046, 2004.

m m m

ℳM,N (An (xnn1 ), Ai

(x), Ai

(x), an,i,i t))

[10] K.T. Atanassov, “Intuitionistic Fuzzy Sets,” Fuzzy sets and Systems,

≥L∗ T (ℳM,N (x, x, xn , t(1 − k)), ℳM,N (xnn1 , x, x, t))

→ T (1L∗ , 1L∗ ) = 1L∗

vol. 20, pp. 87–96, 1986.

m m [11] M. Grabiec, “Fixed Points in Fuzzy Metric Spaces,” Fuzzy Sets and

as n → ∞. Thus ℳM,N (x, x, Ai (x), t) = 1L∗ and we get Ai (x) =

To show uniqueness, assume that y ≠ x is another periodic

point of Ai . Then we have

Systems, vol. 27, pp. 385 – 389, 1988.

[12] M.S. El Naschie, “A Review of E − Infinity Theory and the Mass

Spectrum of High Energy Particle Physics,” Chaos, Solitons and

Fractals, vol. 19, pp. 209 – 236, 2004.

m m m

1L∗ ≥L∗ ℳM,N (x, x, y, t) = ℳM,N (Ai (x), Ai (x), Aj (y), t)

[13] M.S. El Naschie, “On Two New Fuzzy Kahler Manifolds, Klein

Modular Space and t Hooft Holographic Principles,” Chaos, Solitons and Fractals, vol. 29, pp. 876 – 881, 2006.

≥L∗ ℳM,N (x, x, y, a

)

i,j,l

[14] M.S. El Naschie, “The Idealized Quantum Two - Slit Gedanken

Experiment Revisited - Criticism and Reinterpretation,” Chaos,

≥ ℳ (x, x, y, t ) = ℳ (Am (x), Am (x), Am (y), t/k)

L∗ M,N k M,N i i j

Solitons and Fractals, vol. 27, pp. 843 – 849, 2006.

[15] R. Saadati and J.H. Park, “On the Intuitionistic Fuzzy Topological

Spaces,” Chaos, Solitons and Fractals, vol. 27, pp. 331 –344, 2006.

t t [16] R. Saadati, A. Razani, H. Adibi, “A Common Fixed Point Theorem in

≥L∗ ℳM,N (x, x, y, k 2 ) ≥L∗ · · · ≥L∗ ℳM,N (x, x, y, k n) → 1L∗

ℒ − Fuzzy Metric Spaces,” Chaos, Solitons and Fractals, vol. 33, pp. 358

–363, 2007.

International Journal of Scientific & Engineering Research Volume 4, Issue 1, January-2013 6

ISSN 2229-5518

[17] R Saadati, S. Sedgi and N. Shobe, "Modified Intuitionistic Fuzzy Metric Spaces and Some Fixed Point Theorems," Chaos, Solitons and Fractals, vol. 38, pp. 36- 47, 2008.

[18] RP. Pant, "Common Fixed Points of Non Commuting Mappings," J.

M ath. Anal. Appl., vol. 188, pp. 436- 440, 1994.

[19] S. Sharma, "Common Fixed Point Theorems in Fuzzy Metric Spaces,"

Fuzzy sets and Systems, vol. 127, pp. 345-352, 2002.

[20] V. Gregori and A. Sapena, "On Fixed Point Theorems in Fuzzy

Metric Spaces," Fuzzy sets and Systems, vol. 125, pp. 245-253, 2002.

[21] V. Gregori, S. Romaguera and P. Veereamani, "A Note on

Intuitionistic Fuzzy Metric Spaces," Chaos, Solitons and Fractals, vol.

28, pp. 902- 905, 2006.

[22] Zadeh L.A., "Fuzzy Sets," Inform and Control , vol. 8, pp. 338 - 353,

1965.

IJSER lb)2013

http://www.ijserorq