Fixed Point Theorem in Intuitionistic Fuzzy Metric Spaces Using Compatible Mappings of Type (R)

Fixed Point Theorem in Intuitionistic Fuzzy Metric Spaces Using Compatible Mappings of Type (R)

Oinam Budhichandra Singh

Thambal Marik College, Oinam Pin-795134, Manipur, India

Abstract – In this paper, we prove common fixed point theorem in intuitionistic fuzzy metric space using compatible mappings of type(R). We consider four mappings of which one is continuous. The result generalize many results in the literature.

INTRODUCTION

The concept of intuitionistic fuzzy sets was introduced by Atanassove [2] and studied as a generalization of fuzzy sets. Park [6] defined the intuitionistic fuzzy metric space with the help of continuous t-norms and continuous t- conorms. Later many authors [4]- [5] derived fixed point theorems in intuitionistic fuzzy metric space. Rohen, Singh and Shambu [8] introduced the concept of compatible mappings of type (C) by combining the definitions of compatible and compatible mappings of type (P) and later on it renamed as compatible mappings of type (R).

Key words: fuzzy set, intuitionistic fuzzy set, intuitionistic fuzzy metric space, compatible mappings of type (R), common fixed point.

Now we begin with some definitions

Definition 1.1: A binary operation * : [ 0, 1] X [ 0, 1 ] [ 0, 1 ] is continuous t-norm if * satisfies the following conditions for all a, b, c, d E [ 0, 1 ]

1. * is commutative and associative
2. * is continuous
3. a*1= a
4. a *b c*d whenever a c and b d

Definition 1.2: A binary operation : [ 0, 1 ]X [ 0, 1 ] [ 0, 1 ] is continuous t-conorm if satisfies the following conditions for all a, b, c, d E [ 0, 1 ]

1. is commutative and associative
2. is continuous
3. a 0 =a
4. a b c d whenever a c and b d.

Definition 1.3: A 5-tuple (X, M, N, * , ) is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, * is a continuous t-norm , is a continuous t-conorm and M, N are fuzzy sets

on X2 x ( 0, ) satisfying the following conditions

1. M (x, y, t ) + N ( x, y, t ) 1 for all x, y E X and t > 0
2. M ( x, y, 0 ) =0 for all x, y E X
3. M ( x, y, t ) = 1 for all x, y E X and t > 0 if and only if x = y
4. M ( x, y, t ) = M ( y, x, t ) for all x, y E X and t > 0
5. M ( x, y, t ) * M ( y, z, s ) M ( x, z, t + s ) for all x, y, z E X and s, t > 0
6. For all x, z E X, M ( x, y, *) : [ 0, ] [ 0, 1 ] is left continuous
7. lim M ( x, y, t ) = 1 for all x, y E X and t > 0

   too

8. N ( x, y, 0 ) = 1 for all x, y E X.
9. N ( x, y, t ) = 0 for all x, y E X and t > 0 if and only if x = y.
10. N (x, y, t ) =N (y, x, t ) for all x, y E X and t > 0.
11. N ( x, y, t ) N (y, z, s ) N ( x, z, t+s ) for all x, y, z E X and s, t > 0.
12. For all x, y E X, N (x, y, * ) : [ 0, ] [ 0, 1 ] is right continuous.
13. lim N( x, y, t ) = O for all x, y E X,

    too

    Then ( M N ) is called an intuitionistic fuzzy metric space on X. The functions M( x, y, t ) and N ( x, y, t ) denote the degree of nearness and the degree of non-nearness between x and y w.r.t. t respectively.

    Remark 1.4 Every fuzzy metric space (X, M, * ) is an intuitionistic fuzzy metric space of the form ( X, M, 1-M * ) such that t-norm * and t-conorm are associated as x y =1 ((1-x ) * (1-y )) for all x, y E X.

    Remark1. 5: In intuitionistic fuzzy metric space ( X, M, N, *, ) M ( X, y, *) is non-decreasing and N ( x, y, * ) is non-increasing for all x, y E X.

    Definition 1.6 Let (X , M, N, *, ) be an intuitionistic fuzzy metric space, Then

    1. A sequence {xn} in X is said to be Cauchy sequence if for all t > 0 and p > 0.

       lim M (xn+p,xn,t)=1 and lim N (xn+p,xn,t)=0

       noo noo

    2. A sequence {xn} in X is said to be convergent to appoint xEX if, for all t> 0.

lim M (xn,x,t)=1 and lim N (xn,x,t)=0

noo noo

Definition 1.7 An intuitionistic fuzzy metric space (X , M, N, *, ) is said to be complete if and only if every Cauchy sequence in X is convergent.

Definition 1.8 A pair of self mappings ( A, B) on an intuitionistic fuzzy metric space (X , M, N, *,

• ) is said to be compatible if lim M ( ABxn , BAxn , t) = 1 and if lim N( ABxn , BAxn , t) =0 for

noo noo

all t > 0 , whenever , {xn} is a sequence in X such that lim A xn = lim Bxn = z for some z E X .

noo noo

Definition 1.9 A pair of self mappings ( A, B) on an intuitionistic fuzzy metric space (X , M, N,

*, ) is said to be compatible of type (R) if lim M(ABxn ,BAxn , t ) =1, lim N(ABxn ,BAxn , t )

noo noo

=0 and lim M(AAxn ,BBxn , t ) =1, lim N(AAxn ,BBxn , t ) =0 for all t > 0 , whenever {xn} is a

noo noo

sequence in X such that lim Axn = lim Bxn =z for some z E X .

noo noo

Lemma1.10 Let (X , M, N, *, ) be an intuitionistic fuzzy metric space and for all x , y E X , t > 0 and if for a number k > 1 such that M ( x , y , kt ) M ( x , y, t) and N ( x, y , kt ) N ( x , y , t ) then x = y.

Lemma 1.11 Let (X , M, N, *, ) be an intuitionistic fuzzy metric space and for all x , y E X , t

> 0 and if for a number k > 1 such that M ( yn+2 , yn+1, t) M ( yn+1 , y , kt) ,N ( yn+2 , yn+1, t) N ( yn+1 , y , kt) Then { yn } is a Cauchy sequence in X.

Lemma1.12 Let A and B be compatible self mappings of type (R) on a complet intuitionistic fuzzy metric space (X , M, N, *, ) with a*b =min {a, b} and a b =max { a, b } for all a, b E [ 0, 1 ] If Az =Bz for some z E X then ABz = BAz =AAz =BBz.

2.MAIN RESULTS

Theorem2.1: Let (X , M, N, *, ) be a complete intuitionistic fuzzy metric space with a*b =min

{ a, b } and a b = max { a, b } for all a , b E [ 0, 1 ]. Let A, B, S and T be four self mappings on X satisfying the conditions

1. A(X) T(X), B(X) S(X)
2. One of A, B, S, and T is continuous
3. The pair { A, S } and { B, T } are compatible type (R) on X.
4. There exists k E ( 0, 1 ) such that for every x, y E X and t > 0

M( Ax, By, kt ) M( Sx, Ty, t )* M (Ax, Sy, t ) * M( By, Ty, t ) * M (Ax, Ty, t)

N( Ax, By, kt ) N( Sx, Ty, t ) N (Ax, Sy, t ) N( By, Ty, t ) N (Ax, Ty, t) Then A, B, S and T have a unique common fixed point in X.

Proof:- Let x0 E X from (i) There exists x1 x2 E X such that Ax0 = Tx1 = Bx1 = Sx2. Inductively, we construct sequences { xn } and { yn } in X such that

Ax2n-1 = Tx2n-1 = y2n-1 and Bx2n-1 = Sx2n =y2n for n =1, 2, 3, .Take x =x2n and y = x2n+1 in (iv ) we get M (Ax2n , Bx2n+1 , kt) M ( Sx2n , Tx2n+1, t) * M ( Ax2n, Sx2n, t) * M (Bx2n+1, Tx2n+1, t ) * M (Ax2n, Tx2n+1, t ) M (y2n+1 , y2n+2 , kt ) M ( y2n , y2n+1 , t ) * M ( y2n , t ) *M (y2n+2 , y2n+1 , t )* M( y2n+1 , y2n+2 , t )

M (y2+1 , y2n+2 ,kt ) M ( y2n , y2n+1 , t )* M (y2n , y2n+1 , t ) * M( y2n+1 , y2n+2 , t) M (y2n+1 , y2n+2 , kt ) M(y2n , y2n+1 , t )

And

N (Ax2n , Bx2n+1 , kt) N ( Sx2n , Tx2n+1, t) N ( Ax2n, Sx2n, t) N (Bx2n+1, Tx2n+1, t ) N(Ax2n, Tx2n+1, t ) N (y2n+1 , y2n+2 , kt ) N ( y2n , y2n+1 , t ) N ( y2n , t ) N(y2n+2 , y2n+1 , t ) N( y2n+1 , y2n+2 , t )

N (y2n+1 , y2n+2 ,kt ) N( y2n , y2n+1 , t ) N (y2n , y2n+1 , t ) N( y2n+1 , y2n+2 , t) N (y2n+1 , y2n+2 , kt ) N(y2n , y2n+1 , t )

Similarly ,

M (y2n+2 , y2n+3 , kt) M ( y2n+1 , y2n+2 , t) And

N (y2n+2 , y2n+3 , kt) N ( y2n+1 , y2n+2 , t) Thus , we have

M ( yn+1 , yn+2 , kt) M ( yn ,yn+1 , t) And

N ( yn+1 , yn+2 , kt) N ( yn ,yn+1 , t) For n = 1 , 2 , 3 ,

Therefore , we have

M ( yn ,yn+1 , t) M( yn ,yn+1 ,t ) M ( yn-1 , yn , t

) . M (y1 , y2 , t

) 1

and

q q2 qn

N( yn ,yn+1 , t) N( yn ,yn+1 , t ) N ( yn-1 , yn , t ) . N (y1 , y2 , t

) 0

q q2 qn

When n.

For each > 0 and t > 0 , we can choose n0 E such that M( yn ,yn+1 , t) > 1- and N( yn ,yn+1 , t) for each n n0 .

For m, n E , we suppose m n. Then, we have M ( yn ,ym , t) M ( yn , yn+1 , t ) *

m-n

M ( yn+1 , yn+2 , t

m-n

) *.*M (ym 1 ,ym , t

m-n

) > ( (1- ) * (1 – ). ( m- n ) times * ( 1- ) ) (

1. )

   And

   N ( yn ,ym , t) N ( yn , yn+1 , t )

   m-n

   N ( yn+1 , yn+2 , t

   m-n

   ) .N (ym 1 ,ym , t

   m-n

   ) ( ( ) ( ). ( m- n ) times ( ) ) ( )

   M ( yn ym , t ) > (1 – ) , N( yn ym , t )

   Hence { yn } is a Cauchy sequence in X. As X is complete, { yn } converges to this point z E X. i.e Ax2n = Tx2n+1 z

   Bx2 n = Sx2n+1 z

   Now suppose that A is continuous. Since A and S are compatible type (R) and by Lemma1.12 we get AAx2n and SAx2n Az as n oo

   Now putting x = Ax2n and y = x2n+1 in (i) we get

   M(AAx2n , Bx2n+1 ,kt ) M (SAx2n ,Tx2n+1 , t) * M( AAx2n ,Sx2n+1 , t) * M ( Bx2n+1 ,t ) * M (AAx2n , Tx2n+1 , t ) As n

   M (Az, z ,t ) M (Az, z , t) And

   N(AAx2n , Bx2n+1 ,kt ) N (SAx2n ,Tx2n+1 , t) N ( AAx2n ,Sx2n+1 , t) N ( Bx2n+1 ,t ) N (AAx2n , Tx2n+1 , t ) As n

   N(Az, z ,t ) N(Az, z , t) By Lemma1.10 Az= z AGAIN,

   Putting x= Sx2n , y=x2n+1 in (i) we get

   M(ASx2n , Bx2n+1 ,kt ) M (SSx2n ,Tx2n+1 , t) * M( ASx2n ,Sx2n+1 , t) * M ( Bx2n+1 Tx2n+1,t ) * M (ASx2n , Tx2n+1 , t )

   As n

   M (Sx, z ,t ) M (Sz, z , t) And

   N(ASx2n , Bx2n+1 ,kt ) N (SSx2n ,Tx2n+1 , t) N( ASx2n ,Sx2n+1 , t) N ( Bx2n+1 ,Tx2n+1 t ) N(ASx2n , Tx2n+1

   , t )

   As n

   N (Sx, z ,t ) N (Sz, z , t) By Lemma 1.10 Sz= z Similarly Bz =Tz = z

   Hence z is a common fixed points of A, B , S and T.

   The uniqueness of a common fixed point z follows easily from (i)

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