International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 683

ISSN 2229-5518

Notes On Fixed Point Theorems In Fuzzy

Metric Spaces

Bijendra Singh, Mahendra Singh Bhadauriya

Abstract— Bijendra singh et al [11] proved a common fixed point theorem in fuzzy metric spaces for semi compatible and weak compatible mapping. In this paper we obtain a common fixed point theorem for six self maps for compatible of type (ß) with new implicit contractive condition in fuzzy metric space .

Index TermsCommon fixed points, Compatible maps of type( ß), Compatible Map,Continous maps, Fuzzy metric space, Implicit relation,Weak compatible maps,

—————————— ——————————

1 Introduction

HE concept of fuzzy sets was introduced initially by

Zadeh [13]. Subsequently, several resercheral defined fuzzy metric space in various methods in various ab-

Whenever a c

a, b, c, d [ 0,1]

and b d for all

stract spaces. In this paper ,We deal with the fuzzy metric space defined by Kramosil and Michalek [6] and modified

by George and Veeramani [3]. They obtained that every

Definition 2.2 [12] : The 3-tuple (X, M, *) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-

2

metric space induces a fuzzy meric spaces. Also Grebic [4]

has proved fixed point theorem for fuzzy metric space. In fuzzy metric space, the notion of compatible maps was

norm and M is a fuzzy set on X
lowing conditions :

× ( 0, ) Satisfying the fol-

initiated by Mishra, Sharma and Singh [8]. Recently Jungek and Rhoades [5] introduced and worked on the concept of

weak compatible maps.

(FM 1)

For all x, y, z X

M ( x, y,0) = 0

and s, t > 0

In this paper we prove a fixed point theorem for A, B, S, T,

(FM 2) for all f

> 0 f if and only if x = y

P and Q self maps using implicit relation. Concept of weak compatibility and compatibility of type (ß )of self maps are

(FM 3)

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

characterized to get common fixed points.

(FM 4) M ( x, y, t ) * M ( y, z, s) M ( x + z, t + s)

(FM 5)

M ( x, y,.) : ( 0, ) [ 0,1]

is left

2 Priliminaries and definitions:

continuous

Definition 2.1 [13] : A binary operation *, [0, 1] × [0, 1]

(FM 6)

Lim M ( x, y, t ) = 1

n →∞

[0, 1] is called at norm if ([ 0,1],* )
monoid with unit 1 such that

a * b c * d

is an abelian topological
Note that every metric spaces induces a Fuzzy metric space.

Definition2.3 [11] : Let ( X , d ) be a Metric Space.

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Define a * b = min{

a, b} and

and x, y X

then x = y

M ( x, y, t ) =

and t > 0 .

t

t + d( x, j )

for all x, y X

Proof : Let there exists, k ( 0,1) such that

M ( x, y, kt ) M ( x, y, t )

Then ( X , M ,* ) is a fuzzy metric space. It is called

for all

x, y X

t > 0

the fuzzy metric space induced by d.

Definition 2.4 [3] : A sequence { x n }

in a fuzzy metric
then

M ( x, y, kt ) M ( x, y, t / k ) and so

M ( x, y, kt ) M ( x, y, t / k n ) for

space { X , M ,* ) is defined as Cauchy sequence if for each

positive integer n, taking limit n → ∞

∈> 0 and t > 0 , there exists x N

such that

M ( x, y, k ) 1 and hence x = y

Lemma 2.3 [12] : The only t-norm and satisfying

M ( x n , x m , t ) > 1− ∈ for all n, m x 0

r * r r

for all r [ 0,1] is the minimum t-norm,

The sequence { x n }

x in X iff
is said to be converge to a point
that is

a * b = min{

a, b}

for all a, b [ 0,1]

M ( x n , x, t ) > 1− ∈ for all n, m x 0

A fuzzy metric space ( X , M ,* ) is said to be complete if every Cauchy sequence in it converges to a point in it.

Definition 2.5 [1] : A pair of self mappings (A, S) of

fuzzy metric ( X , M ,* ) is said to be compatible if

Main Theorem

Bijendra singh et al.[11] have proved the following result:

Let ( X , M ,* ) be a complete fuzzy metric space and

lim M ( ASx

n →∞

n , SAx n , t ) 1t > 0 whenever { x n } is

let A, B, S, T, P and Q be mapping from X, into itself
a sequence in X such that
such that the following conditions are satisfied :

lim Sx n n →∞

= lim Ax n n →∞

= x for some x X

(a)

P( X ) ST ( X ), Q( X ) AB )( X )

If the self mapping A and B of a fuzzy metric space

( X , M ,* ) are compatible then they are weakly compatible but its converse is not true.

Lemma 2.1 [9] : Let ( X , M ,* ) be a fuzzy metric space,

(b) either AB or P is continuous
( c) (P, AB) is compatible and (Q, ST) is weakly compatible.
(d) AB = BA, ST = TS, PB = BP, QT = TQ
(e) There exist k ( 0,1) such that for every

then for all x, y X , M ( x, y,.)

is non-decreasing.

x, y X , t > 0

Lemma 2.2 [10] : Let ( X , M ,* ) be a fuzzy metric

space. If there exists k ( 0,1) such that

M ( x, y, kt ) M ( x, y, t )t > 0

M ( Px,Qy, qt) ≥ M ( ABx, STy, t) * M ( Px, ABx, t)

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* M ( Qy, STy , t ) * M ( Px , STy , t )

φ( M ( Px, Qy, kt ), M ( ABx, STy, t ), M ( Px, ABx, t ),

then A, B, S, T, P and Q have a unique common fixed

M (Qy, STy, kt ), M ( Px, STy, t )

) ≥ 0

point in X.
Now we are introducing an implicit-relation and we will prove the above result by using implicit relation in place of
Then A, B, S, T, P and Q have a unique common fixed
point in X.

Proof – Let x 0 X then there exist such that

contractive condition.

A Class of Implicit Relation:

Px 0

= STx 1 = y 0 and

Let φ be the set of all real continuous functions

Qx 1

= ABx 2

= y1

φ( t , t , t , t , t

) : ( R+) 5

R which is

Inductively, we can construct sequences { x n }

and

1 2 3 4 5

{ y n }

in X, such that

non-decreasing in the Ist argument satisfying the following conditions: (φ1) for u, v 0

Px 2 n

= STx 2 n +1 = y2 n

and

φ( u, v, v, u,1) 0 implies u v

Qx 2 n +1

= ABx

2 n +2

= y2 n +1

(φ2) φ( v, u,1,1, u ) 0 or

φ( u,1, u,1, u ) 0 or

for n = 0, 1, 2,……….

Step 1 : We put in contractive condition x = x 2 n , y = x 2 n +1

then we get

φ( u,1,1, u,1) 0 implies u 1

φ( M ( Px

2 n , Qx

2 n +1

, kt ), M ( ABx

2 n , STx

2 n +1

, t ),

Ex.

φ( t 1 , t 2 , t 3 , t 4 , t 5 ) = 13t 1 11t 2 3t 3 5t 4 + t 5 1

M ( Px2 n

M ( Px

, ABx2 n

, STx

, t), M (Qx

, t ) ≥ 0

2 n +1

, STx

2 n +1

, kt ),

Theorem 3.1 : Let ( X , M ,* ) be a complete fuzzy metric space and let A, B, S, T, P and Q be mapping from X into itself such

that the following conditions are satisfied :

2 n 2 n +1

φ( M ( y2 n , y2 n +1 , kt ), M ( y2 n −1 , y2 n , t ), M ( y2 n , y2 n −1 , t )

(a)

P( X ) ST ( X ), Q( X ) AB( X )

M ( y2 n +1 , y2 n , kt ), M ( y2 n , y2 n , t )

) ≥ 0

(b) Either A B or P is continuous.
(c) (P, AB) is compatible of type (ß) and (Q, ST) Is weakly compatible
(d) AB = BA, ST = TS, PB = BP, QT = TR
(e) there exist k ( 0,1) such that for every

x, y X and t > 0

From property of implicit relation

M ( y2 n , y2 n +1 , kt ) M ( y2 n 1 , y2 n , t ) Or M ( y2 n +1 , y2 n , kt ) M ( y2 n , y2 n 1 , t ) Similarly we have

M ( y2 n +2 , y2 n +1 , kt ) M ( y2 n +1 , y2 n , t )

Therefore, for all n, we have

M ( y n +1 , y n , kt ) M ( y n , y n 1 , t )

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To prove that { y n }

is a Cauchy sequence,

φ( M ( PABx2 n , ABABx2 n , t)

M ( PABx2 n , ABABx2 n , t)

M ( PABx2 n , ABABx2 n , t ), M (Qx2 n , STx2 n+1 , kt ),

M ( y

n+1

, yn

, t) ≥ M ( yn

, yn−1

, t / k ) ≥ M ( y

n−1

, yn−2

, t / k 2 )

M (( PABx2 n , STx2 n+1 , t )

) ≥ 0

≥ ………

M ( y, y o

as n → ∞

, t / k n ) 1

Letting n → ∞ , we get

φ( M ( ABz, z, kt ), M ( ABz, z, t ), M ( ABz, ABz, t ),

M ( z, z, t ), M ( ABz, z, t ) ≥ 0

Thus, the result hold for m = 1
By induction hypothesis suppose that result hold for m = p,

M ( yn , yn p +1 , t ) ≥ M ( yn , yn p , t / 2) * M ( yn +1 , yn p +1 , t / 2)

φ( M ( ABz , z, kt ), M ( ABz , z, t ),1,1, M ( ABz , z, t )) 0

Since φ is non-decreasing in first arguments so,

φ( M ( ABz , z, t ), M ( ABz , z, t ),1,1, M ( ABz , z, t ) 0

m = p + 1 .

→ 1*1 = 1

Thus the result holds for
By the property of implicit relation

M ( ABz , z, t ) 1

Hence { y n }

is a Cauchy sequence in X. Which is

ABz = z

complete therefore { y n }

convergence to z i.e. y n

z X .

Step 3 : Putting x = z, y = x 2 n +1 , in contractive con-

Also its subsequences converges to the same point i.e. z X .

{ Qx 2 n +1 } z and { STx 2 n +1 } z

dition, we get

φ( M ( Pz , Qx 2 n +1 , kt ), M ( ABz , STx 2 n +1 , t ), M ( Pz , ABz , t ),

{ Px 2 n } z and { ABx 2 n } z

Case 1 : Suppose AB is continuous.

M ( Qx

2 n +1

, STx

2 n +1

, kt ), M ( Pz , STx

2 n +1

, t )) 0

As AB is continuous,

2

Letting Limit n → ∞ , we get

φ( M ( Pz, z, kt ), M ( z, z, t ), M ( Pz, z, t ),

( AB )

x 2 n ABz

and

M ( z, z, t ), M ( Pz, z, t ) ≥ 0

( AB )Px 2 n ABz

As (P, AB) is compatible pair, we have

P( AB )x 2 n ABz

Step 2 : Putting x = ABx 2 n , y = x 2 n +1 in contrac- tive condition, we have

Since φ is non-decreasing in first argument, so

φ( M ( Pz , z, t ),1, M ( Pz , z, t ),1, M ( Pz , z, t )) 0

By the property of implicit relation

M ( Pz , z, t ) 1

Pz = z

Therefore ABz

= Pz = z

Step 4 : Put x = Bz

and y = x 2 n +1 in contractive

condition we get

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φ( M ( z, Qv , t ) 0

φ( M ( PBz, Qx2 n +1 , kt ), M ( ABBz, STx2 n +1 , t ),

Qv = z

M ( PBz, ABBz, t ), M (Qx2 n +1 , STx2 n +1 , kt ),

STv

= z = Qv

M ( PBz, SAx2 n +1 , t )

) ≥ 0

As ( Q, ST ) is weakly compatible, we have

[As BP = PB, AB = BA so we have

STQv = QSTv

Thus STz = Qz .

P (Bz) = B (Pz) = Bz and
AB (Bz) = BA (Bz) = B (ABz) = Bz] Letting limit n → ∞ ,

φ( M ( Bz, z, kt ), M ( Bz, z, t ), M ( Bz, Bz, t ), M ( z, z, kt ),

Step 6 : Put x = x 2 n , y = z in contractive condition, we get

φ( M ( Px2 n , Qz, kt ), M ( ABx2 n , STz, t), M ( Px2 n , ABx2 n , t ),

M ( Bz, z, t )) ≥ 0

Since φ is non decreasing in first argument, so we can

M (Qz, STz, kt ), M ( Px2 n , STz, t )

) ≥ 0

write φ( M ( Bz , z, t ), M ( Bz , z, t ),1,1, M ( Bz , z, t )) 1

By property of implicit relation
Letting n → ∞ and using equation

{ Qx 2 n +1 } z and { STx 2 n +1 } z

φ( M ( z, Qz, kt ), M ( z, z, t ), M ( z, z, t ),

M ( Bz , z, t ) 1

Bz = z

Step5

M (Qz, z, kt ), M ( z, z, t ) ≥ 0

Now

Bz = z, ABz

= z Az = z

φ( M ( z, Qz, kt ),1,1, M ( z, Qz, kt ),1) 0

M ( z, Qz, t ) 1

therefore Az

= Bz

= Pz = z

(Using property of implicit relation)

Step 5 : As P( X ) ST ( X ) , therefore exist v X

Qz = z

. Such that z = Pz

= STv

. Putting x = x 2 n , y = v ,

Step 7 : Put x = x 2 n , y = Tz

in contractive
we get from contractive condition, we get,

φ( M ( Px2 n , Qv, kt ), M ( ABx2 n , STv, t ), M ( Px2 n , ABx2 n , t )

φ( M ( Px

Condition we get

, QTz, kt ), M ( ABx

, STTz, t), M ( Px

, ABx

, t ),

M (Qv, STv, kt ), M ( Px2 n , STv, t )

) ≥ 0

2 n

M (QTz, STTz, kt ), M ( Px

2 n

, STTz, t ) ≥ 0

2 n 2 n

Letting n → ∞ and using { Px 2 n } z

and { ABx 2 n } z we get

φ( M ( z, Qv, kt ), M ( z, z, t), M ( z, z, t ),

M (Qv, z, kt ), M ( z, z, t )) ≥ 0

φ( M ( z, Qv , t ),1,1, M ( z, Qv , kt ) 0

2 n

As QT = TQ and ST = TS, we have
QTz = TQz = Tz and ST (Tz) = T (STz) = Tz
Letting n → ∞ ,w e get

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φ( M ( z, Tz, kt ), M ( z, Tz, t ), M ( z, z, t ),

M (Tz, Tz, kt ), M ( z, Tz, t )) ≥ 0

Letting n → ∞ , we

φ( M ( Pz, z, kt ), M ( Pz, z, t ), M ( Pz, Pz, t ),

M ( z, z, kt ), M ( pz, z, t )) ≥ 0

φ( M ( z, Tz , kt ), M ( z, Tz , t ),1,1, M ( z, Tz , t )) 0

Since φ is non-decreasing in first argument so

φ( M ( z, Tz , t ), M ( z, Tz , t ),1,1, M ( z, Tz , t )) 0

φ( M ( Pz , z, kt ),1, M ( Pz , z, t ),1,1, M ( Pz , z, t ) 0

Since φ is non decreasing in the first argument so we can write

M ( z, Tz , t ) 1

(Using property of implicit relation)

Tz=z

φ( M ( Pz , z, t ), M ( Pz , z, t ),1,1, M ( Pz , z, t ) 0

M ( Pz , z, t ) 1 (By property of implicit relation)

Now

STz

= Tz

= z Sz

= z hence

Pz = z

Sz = Tz

= Qz = z

Further using Step 5, 6, 7 we get
Combining result, we get

Qz = STz = Sz = Tz = z

Az = Bz

= Pz

= Qz = Tz

= Sz = z

Step 9 : As Q( X ) AB( X ) , there exists w X

Case II : Suppose P is continuous

2

such that z = Qz = ABw

put x = w, y = x 2 n +1 in con-

As P is continuous, P

x 2 n Pz

tractive condition we get

and P( AB )x 2 n Pz

φ( M ( Pw, Qx

2 n +1

, kt ), M ( ABw, STx

2 n +1

, t), M ( Pw, ABw, t),

As (P, AB) is compatible of type (ß) M(PPx n ,(AB)(AB) x n ,t) → 1

2

M (Qx2 n +1 , STx2 n +1 , kt ), M ( Pw, STx2 n +1 , t ) ≥ 0

Letting n → ∞ we get

φ( M ( Pw, z, kt ), M ( z, z, t ), M ( Pw, z, t ),

M(Pz ,(AB)
x n , t ) → 1 when t > 0

2

M ( z, z, kt ), M ( Pw, z, t )) ≥ 0

we have (AB)

x2 n Pz

Step 8 : Put x = Px 2 n , y = x 2 n +1 .

φ( M ( Pw , z, kt ),1, M ( Pw , z, t ),1, M ( Pw , z, t )) 0

φ( M ( PPx 2 n , Qx 2 n +1 , kt ), M ( ABPx 2 n , STx 2 n +1 , t ),

M ( PPx2 n , ABPx2 n , t), M (Qx2 n +1 , STx2 n +1 , kt ),

M ( PPx2 n , STx2 n +1 , t )) ≥ 0

Since φ is non decreasing in the first argument

φ( M ( Pw , z, t ),1, M ( Pw , z, t ),1, M ( Pw , z, t ) 0

M ( Pw , z, t ) 1

(Using property of implicit relation)

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Pw = z = ABw

(a)

P( X ) ⊂ S ( X ),Q( X ) ⊂ A( X )

As (P, AB) is compatible, we have

Pz = ABz

Also from Step 4, we get Bz = z
Thus Az = Bz = Pz = z and we due that z is the common fixed point of the six maps in the case also.

Uniqueness – Let u be another common fixed point of

A, B, S, T, P and Q
Then Au = Bu = Pu = Qu = Su = Tu = u
(b) Either A or P is continuous.
(c) (P, A) is compatible of type ß and (Q, S) is weakly compatible
(d) there exist k ( 0,1) such that for every

x, y X and t > 0

φ( M ( Px, Qy, kt ), M ( Ax, Sy, t ), M ( Px, Ax, t ),

Put x = z, y = u in contractive condition, we get

M (Qy, Sy, kt ), M (Qy, Sy, kt )

) ≥ 0

φ( M ( Pz , Qu , kt ), M ( ABz , STu , t ), M ( Pz , ABz , t ),

unique common fixed point in X.
Then A, S, P and Q have a

M ( Qu , STu , kt ), M ( Pz , STu , t ) 0

Taking n → ∞

φ( M ( z, u, kt ), M ( z, u, t ), M ( z, z, t ),

M (u, u, kt ), M ( z, u, t ) ≥ 0

φ( M ( z, u, kt ), M ( z, u, t ),1,1, M ( z, u, t ) 0

Since φ is non decreasing in first argument so

φ( M ( z, u, t ), M ( z, u, t ),1,1, M ( z, u, t ) 0

M ( z, u, t ) 1

(Using property of implicit relation)

z = u

Therefore z is the unique common fixed point of day maps A, B, S, T, P and Q.

Remark 3.1: If we take B = T = 1, the identify maps on x in the theorem 3.1 ,the condition (b) is satisfied trivially.

Corollary 3.1 : Let ( X , M ,* ) be a complete fuzzy metric space and let A, S, P and Q be mapping from X into itself such that

the following conditions are satisfied :

Remark 3.2 Corollary 3.1 is generalization of the result of cho [ ] in the sense that condion of compatibility of the pairs of self maps ,has been restricted to compatibility of type (ß) and wek compatibility and only one map of the pair is needed to contin- uous.

4 Conclusion

We establish A Common Fixed Point Theorem in Fuzzy Metric Spaces satisfying implicit relations for weakly compatible maps. There are some possible application in engineering, economics in dealing with problems arising in approximation theory, information system. In future scope we can obtain new implicit relation to relax conditions.

Acknowledgment

Our thanks to the Mr. Rakesh Rathore CEO Vikrant insti- tute of technology and management,Gwalior, who have contributed towards making of the paper.

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