An application of Fixed Point Theorems in FuzzyMetric Spaces by ISERP ISERP

Manish Kumar Mishra et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 2, 123 - 129

An application of Fixed Point Theorems in Fuzzy Metric Spaces ( AFPTFMS) Manish Kumar Mishra and Deo Brat Ojha

mkm2781@rediffmail.com, deobratojha@rediffmail.com

Department of Mathematics R.K.G.Institute of Technology Ghaziabad,U.P.,INDIA

Abstract— The aim of this paper is to present common fixed point theorem in fuzzy metric spaces, for four self maps, satisfying implicit relations with Integral Type Inequality. Also, the application of fixed points is studied for the Product spaces.

Recently in 2006, Jungck and Rhoades [13] introduced the concept of

Keywords- Fuzzy metric space,  -chainable fuzzy metric space, compatible mappings, weakly compatible mappings, implicit relation and common fixed point.

Grabiec[14] followed Kramosil and Michalek[9] and he obtained the

the concept of

weakly compatible maps which were found to be more generalized than compatible maps.

INTRODUCTION

version of

Banach contraction principle.

Recently in 2000, B.Singh and M.S.Chauhan[15] brought forward proved some

fuzzy

compatibility in fuzzy metric space. Popa[16]

fixed point

theorems

for

weakly compatible

noncontinous mappings using implicit relations. His work was

Fuzzy set has been defined by Zadeh [1]. Kramosil and Michalek

extended by Imdad[17] who used implicit relations for coincidence

[2],introduced the concept of fuzzy metric space, many authors

commuting property. Singh and Jain[18] extended the result of

extended their views as some George and veera mani [3], Grabiec

Popa[16] in fuzzy metric spaces and Rana[21] proved fixed point

[4], Subramanyan [5],Vasuki[6]. Pant and Jha [7] obtained some

theorems in fuzzy metric space using implicit relation. This paper offers the fixed point theorems on fuzzy

was developed extensively by many authors and used in various

metric spaces, which generalize, extend and fuzzify several known

fields. In 1986 Jungck [8] introduced the notion of compatible maps

fixed point theorems for compatible maps on metric space, by

for a pair of self maps. Several papers have come up involving

making use of implicit relations with Integral Type Inequality . The

compatible maps in proving the existence of common fixed points

condition of - chainable fuzzy metric are characterized to get

both in the classical and fuzzy metric space.

common fixed points. One of its corollaries is applied to obtain

The theory of fixed point equations is one of the preeminent basic

fixed point theorem on product of FM spaces.

IJ A

analogous results proved by Balasubramaniam et al. subsequently, it

tools to handle various physical formulations. Fixed point theorems in

II.

fuzzy mathematics has got a direction of vigorous hope

and vital trust with the study of Kramosil and Michalek[9], who introduced the concept of fuzzy metric space. Later on, this concept of fuzzy metric space was modified by George and Veeramani[10 ]. Sessa[11]

initiated

the

tradition of

improving commutative

condition in fixed point theorems by introducing the notion of weak commuting

property

Further,

Jungck[12]

gave

generalized condition defined as compatibility in metric spaces.

ISSN: 2230-7818

PRELIMINARIES

Definition 2.1 A binary operation  :[0,1] [0,1]  [0,1] is a continuous t – norm if it satisfies the following conditions:

 is communicative and associative; II.  is continuous; I.

 

III. a  a = a for all a 0,1 ;

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Manish Kumar Mishra et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 2, 123 - 129

a  b  c  d whenever a  c a, b, c, d [0,1]

IV.

and

b  d and

Examples of continuous t-norm are

such that

a  b  ab and

  II. a  b  min a, b .  X , M , 

Definition 2.2 A 3-tuple metric space, if X is an arbitrary set,

is called a fuzzy

 is a continuous t2 X  (0,) . Satisfying the norm and M is a fuzzy set on following conditions for each x,y,z  X,s,t  0 , M  x, y, t   0

III.

;

is continuous.

a  b  ab or t M  x, y , t   t  d  x, y 

be a metric space and let

a, b  min{a, b} and let

x, y  X and t  0. Then  X , M ,  is a fuzzy metric M, induced by d is called the standard fuzzy metric. Definition 2.5 : Let a)

 X , M , 

be a fuzzy metric space. Then

x  a sequence n in

X is said to converges to x in X if for each   0 and each t  0 , there exists M  xn , x, t   1   n0  N n  n0 .

such that

for all

It was proved by George and Veeramani[4] that a sequence

 xn  in fuzzy metric space  X , M ,  converges to a point x  X , if and only if M  xn , x, t   1 , for all t  0 .

ISSN: 2230-7818

n 

for some

If the self mapping

A and

 X , M ,  are compatible , then

but its converse is not true.

such

that

x in X .

B of a fuzzy metric space they are weakly compatible

Example 2.1: Let X  [0,8] and a * b  min{a, b} . Let M be the standard fuzzy metric induced by d, where

d ( x, y)  x  y

IJ A

Let

 X , M ,  are called compatible if n  xn  is a sequence in whenever

B of a fuzzy metric space  X , M ,  are called weakly compatible if ABx  BAx, when Ax  Bx for some x  X .

0  r 1 , defined B  x, r , t   { y  X ; M  x, y, t   1  r}.

 X,d

f and g of a fuzzy metric space lim M  fgx, gfx, t   1

Two self mappings

Two self maps A and

 X , M ,  be a metric space. For t  0 there exists B  x, r , t  open ball with center x  X and radius Let

Definition 2.7

n 

;

M  x, y,. :  0,    [0,1]

A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

Definition 2.8 :

M  x, y, t   M  y, z, s   M  x, z, t  s 

IV.

M  x, y, t   M  y, x, t 

for all

lim fxn  lim gxn  x

M  x, y, t   1if and only if x  y

II.

 xn  in

X is said to be Cauchy if for n N each   0 and each t  0 , there exists 0 M  xn , xm , t   1   n, m  n0

b) a sequence

for

x, y  X .

Define two self maps

 X , M ,  as follows:

A and

8  x 0  x  4 Ax   , 4 x8 8

B of a fuzzy metric 0 x4 4 x8

x Bx   8

x  1  1/ n 

Let us consider n , then [A,B] is proved to be not compatible but is weakly compatible. Let

 X , M ,  is called

finite sequence

a metric space and let

x  x0 , x1 , x2 , x3 ......xn  y

  0. A

is called to be

 - chain from x to y if M  xi , xi 1 , t   1   t  0, and i  1, 2,3.......n . A fuzzy metric space

if for all

 X , M ,  is called as  - chainable

x, y  X , there exists a  - chain from x to y.

Lemma 2.9

for all

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 X , M ,  be a fuzzy metric space. If there exists k   0,1 M  x, y, qt   M  x, y, t  such that for all

Example 2.2

Let

x, y  X and t  0 , then x  y .

Then for all

Let 



u, v  0, F  u, v, u, v,1  0

F  u, v, u, v,1  0

F  u,1, u,1, u   0

implies that

2. 3. 4.

d ( x, y )

 (t )dt     (t )dt ,  y x , 0 0 X , Where   ; R  R is a lebesgue integrable mapping which is nonnegative

  0,   (t )dt  0 0

and

such

that,

for

each

. Then f has a unique common fixed

lim f x  z.

IJ A

z  X such that for each x  X ,

n 

Rhoades[20], extended this result by replacing the above condition by the following d ( fx , fy )

 0

1 max{d ( x , y ), d ( x , fx ), d ( y , fy ), [ d ( x , fy )  d ( y , fx )] 2 0

 (t )dt   

 (t )dt

Ojha et al.[19] Let ( X , d ) be a metric space and let f : X  X , F : X  CB( X ) be a single and a f and F are multi-valued map respectively, suppose that occasionally weakly inequality



The pairs compatible.

T X 

commutative (OWC) and satisfy the



J ( Fx , Fy )

 (t ) dt 



ad ( fx , fy ) d P1 ( fx , Fx ), ad ( fx , fy )   P1  P 1 max d ( fy , Fy ), ad ( fx , Fx ) d ( fy , Fy ),  cd P1 ( fx , Fy ) d ( fy , Fx )   

for all x , y in X ,where p  2 is an integer a  0 and 0  c  1 then f and F have unique common fixed point in X.

ISSN: 2230-7818

SX 

are

weakly

is complete.

k   0,1

such that

F M  Ax , By , qt , M  Sx ,Ty ,t , M  Ax , Sx ,t , M  By ,Ty ,t , M  Ax ,Ty ,t 

 (t )dt  0

x , y  X and t  0 , then A, B, S and T have a unique common fixed point in X .

for every

Proof:Let

any

arbitrary

point.

As,

AX  TX , BX  SX x , x  X , such that Ax0  Tx1 and so, there exists 1 2  yn Bx1  Sx2

 xn 

Inductively, we construct the sequences

and

X , such that y2 n  Ax2 n  Tx2 n1 and y2n1  Sx2n2  Bx2 n1 for n  0,1, 2,.... now , using in

condition (iv) with

x  x2 n , y  x2 n1 , we get

F M  Ax2 n , Bx2 n1 , kt , M  Sx2 n ,Tx2 n1 ,t , M  Ax2 n , Sx2 n ,t , M  Bx2 n1 ,Tx2 n1 ,t , M  Ax2 n ,Tx2 n1 ,t 



 (t )dt  0

That is

F M  y2 n , y2 n1 , kt , M  y2 n1 , y2 n ,t , M  y2 n , y2 n1 ,t , M  y2 n1 , y2 n ,t , M  y2 n , y2 n ,t 



 (t )dt  0

That is

F M  y2 n , y2 n1 , kt , M  y2 n1 , y2 n ,t , M  y2 n , y2 n1 ,t , M  y2 n1 , y2 n ,t 



 (t )dt  0

Using condition (a), we have M  y2 n , y2 n1 , kt 



 (t ) dt

 B, S 

and

there exists

-chainable fuzzy metric

d ( fx , fy )



AX  TX and BX  SX  A, T 

u  1.

implies that

be a complete

satisfying the following conditions:

(X,d ) be a complete metric space,  [0,1], f : X  X a mapping such that for each



 X , M , 

uv

F  u, u,1,1, u   0

MAIN RESULTS

space and let A, B, S and T be self-mappings of X ,

Let

summable,

,then

Theorem 3.1 : Let

A Class of implicit relations: be the set of all real 5 F: R R and continuous functions: , nondecreasing in first argument satisfying the following conditions: For

III.

 X , M ,  be a fuzzy metric space. x, y  X , M  x, y,. is non-deceasing. Let

f  t1 , t2 , t3 , t4 , t5   20t1  9t2  6t3  7t4  t5  1

F  .

Lemma 2.9



Let’s consider

 (t )dt 



M  y2 n , y2 n1 , kt 

That is, 0 Similarly, we have



M  y2 n , y2 n1 ,t M  y2 n , y2 n ,t 

M  y2 n , y2 n2 , kt 

 (t )dt 



 (t )dt 



M  y2 n , y2 n1 ,t 

M  y2 n1 , y2 n ,t 

 (t )dt

 (t ) dt

 (t )dt

There fore, for all n even or odd, we have



M  yn , yn1 , kt 

 (t )dt 



M  yn , yn1 ,t 

 (t ) dt

Thus, for any n and t, we have

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Manish Kumar Mishra et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 2, 123 - 129

 (t )dt 



M  yn , yn1 ,t 

Hence

 (t ) dt

z  Bu{ B  S , z  Bu  S , z  Bu  Su  say } Therefore, z  Bu  Su  Tu .

y  To prove that n is a Cauchy sequence, we have M  yn , yn1 , t 



 (t )dt 

 yn1 , yn2 ,t / k 2 

M  yn , yn1 , t / k 

Now

 (t )dt 

 (t ) dt     



 y1 , y0 ,t / k n 



 yn , yn p1 , t 



 (t ) dt 

 (t ) dt  1

 yn , yn p , t / 2 *M  yn1 , yn p1 ,t / 2 

 (t ) dt  1*1  1

Thus, the result holds for m  p  1 . Hence,

 yn 

is a Cauchy

 yn 

sequence in X, which is complete. Therefore converges y  z n , for some z  X . So, it follows that to z, such that  Ax2n  ,Sx2n  ,Bx2n1 and Tx2n1 also converges to z. To  xn  is a Cauchy sequence. prove

M  ym , ym1 , t 

 (t )dt  1 

we have

M  xn , xn1 ,t 



 (t )dt 



for all t  0 and i  1, 2,.......m . Thus

M  y1 , y2 ,t / l M  y2 , y3 ,t / l .......M  ym1 , ym ,t / m 

m  n,



 (t )dt 



F M  z , Bz , kt , M  z ,Tz ,t , M  z , z ,t , M  Bz ,Tz ,t , M  z ,Tz ,t 

 (t )dt  0

Since, F is nondecreasing in the first argument, as well as

z  Tz , since z  T  X  , so we have



F M  z , Bu ,t , M  z , z ,t ,1, M  Bz , z ,t , M  z , z ,t 



F M  z , Bu ,t ,1,1, M  Bz , z ,t ,1

That is ,

 (t )dt  0

 (t ) dt  0



F M  z , Bz ,t 

 (t )dt  1

by (b)

Hence , z  Bz Therefore, z  Bz  Tz . Step III: As,

BX  SX let there exists v  X , such

z  Bz  Sv x  v, y  z in (iv), Put

that



F M  Av , Bz , kt , M  Sv ,Tz ,t , M  Av , Sv ,t , M  Bz ,Tz ,t , M  Av ,Tz ,t 

 (t )dt  0

That is,

F M  Av , z , kt ,1, M  Av , z ,t ,1, M  Av , z ,t 

 (t )dt  0

Since, F is nondecreasing in the first argument, we have

M  xn , xn1 ,t / m  n M  xn1 , xn2 ,t / m  n .......*M  xm1 , xm ,t / m  n 

IJ A

M  xn , xm ,t 



 (t )dt

 1  * 1  * 1  .........* 1   1 

For

Step II : Put x  x2n and y  z we get taking Lim n   ,we get

-chainable, so -chain Since X x x from n to n 1 that is, there exists a finite sequence xn  y1 , y2 ,......... yn  xn 1 such that



 B, S  is weakly compatible, so BSu  SBu , thereby,

Bz  Sz.

as n   . Thus, the result holds for m=1. By introduction hypothesis suppose that result holds for m  p. Now, M

M  yn , yn1 , kt 



 (t )dt

 1  * 1  * 1  .........* 1   1 

 xn  is a Cauchy sequence in X, which is complete.  xn  converges to z  X . Hence its subsequences Therefore  Ax2n  ,Sx2n  ,Bx2n1 and Tx2n1 also converges to z.

Hence,

T X 



F M  Av , z ,t ,1, M  Av , z ,t ,1, M  Av , z ,t 

That is,



F M  Av , z ,t 

 (t )dt  0

 (t )dt  1

…{by (b)}.

z  Av So, . Now A  T  z  Av T or z  Av  Tv .

Therefore, Now as, such

since,

z  Av  Tv .

 A, T  is

weakly compatible, so

ATv  TAv ,

is complete. z T  X  If we take , so there exists u  X , such that z  Tu . x  x and y  u in (iv), so 2n Step I : Put

that Az  Tz . So, combining all the results, we have



Step IV: Put x  Sz and in (iv), we get As F is non decreasing in the first argument, we have

Case I:

F M  Ax2 n , Bu , kt , M  Sx2 n ,Tu ,t , M  Ax2 n , Sx2 n ,t , M  Bu ,Tu ,t , M  Ax2 n ,Tu ,t 

taking Lim



 (t )dt  0

n   ,we get

F M  z , Bu , kt , M  z ,Tu ,t , M  z , z ,t , M  Bz ,Tz ,t , M  z ,Tz ,t 



So,



F M  z , Bu ,t 

 (t ) dt  0

yz

F M  ASz , Bz , kt , M  SSz ,Tz ,t , M  ASz , SSz ,t ,M  Bz ,Tz ,t ,M  ASz ,Tz ,t 

 (t )dt  0

Since, F is nondecreasing in the first argument, so F M  z , Bu ,t ,1,1, M  Bu ,u ,t ,1

Az  Tz  Bz  Sz  z .

 F  F  0

M  Az , z , kt , M  Sz , z ,t , M  Az , Sz ,t , M  z , z ,t , M  Az , z ,t 

M  Az , z , kt , M  z , z ,t , M  Az , z ,t , M  z , z ,t , M  Az , z ,t 

 (t ) dt  1

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by (b)

 (t )dt  0

Since, F is nondecreasing in the first argument, we have

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Manish Kumar Mishra et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 2, 123 - 129 F M  Az , z ,t ,1, M  Az , z ,t ,1, M  Az , z ,t 



F M  Az , z ,t 



 (t ) dt  0

by  b 

 (t ) dt  1

Therefore,

Az  z . Similarly, we can show that Bz  z, Tz  z, and Sz  z Hence Az  Tz  Bz  Sz  z Case II:

SX 

If we take

is complete.

zS X 

, so there exists w  X w X , such

that, z  Tw . The proof is likewise as in Case I. So, similarly, we can

Az  z , Bz  z, Tz  z, and Sz  z . Hence Az  Tz  Bz  Sz  z . Thus, z is the common fixed point of A, B, S and T . show that

Uniqueness: Let w and z be two common fixed points of maps get

A, B, S and T . Put x  z and y  w in (iv), we

F M  Az , Bw , kt , M  Sz ,Tw ,t , M  Az , Sz ,t , M  Bw ,Tw ,t ,M  Az ,Tw ,t 

 (t )dt  0

x, y  X and t  0 .Then A and T have a unique common fixed point in X .  X , M ,   Corollary 2.2 Let

be a complete

F M  z , w , kt , M  z , w ,t , M  z , z ,t , M  w , w ,t , M  z , w ,t 

 (t )dt  0

Since, F is a nondecreasing, in the first argument, therefore, we have



F M  z , w,t , M  z , w,t ,1,1, M  z , w,t 

 (t )dt  0

exists



, such that

F M  Ax , By , kt , M  Sx ,Ty ,t , M  Ax , Sx ,t , M  Sx , By ,2t , M  By ,Ty ,t , M Ty , Ax ,t 

x, y  X and t  0 .Then A, B, S and T have a unique common fixed point in X . Proof: From definition, we have



F M  Sx ,Ty ,t , M  Ax , Sx ,t , M  By ,Ty ,t , M  By , Sx ,2 t , M  Ax ,Ty ,t 

F M  Sx ,Ty ,t , M  Ax , Sx ,t , M  By ,Ty ,t , M  Sx ,Ty ,t , M Ty , By ,t , M  Ax ,Ty ,t 



F M  Sx ,Ty ,t , M  Ax , Sx ,t , M  By ,Ty ,t , M  Ax ,Ty ,t 

Corollary 2.3:

says that the pair

for every ,

IJ A

 Axn  Txn1 .

Remark 2.2: If S = T and A = B, then conditions (i)- (iii)

 A, T  and  B, S  are weakly says that the pairs A X   B  X   T  X  compatible and so, and hence the Ax2n  Tx2 n 1

sequences exists as follows: and Bx2n 1  Sx2n  2 . Theorem 2.1 with S = T = Identity map is: Corollary 2.1: Let

 X , M , 

be a complete



-chainable

A, B, S and T be self-mappings of fuzzy metric space and let X , satisfying (i), (ii) and (iii) of Theorem 2.1 and there k   0,1 exists

, such that

M  Ax , By , kt 

 (t )dt 



M  Sx ,Ty ,t 

 (t )dt

x, y  X and t  0 .Then A, B, S and T have a unique common fixed point in X . Proof: As , we have



M  Sx ,Ty ,t 



 (t ) dt 



M  Sx ,Ty ,t *1

M  Sx ,Ty ,t *M  Ax , Ax ,5 t 



 (t ) dtM

 (t ) dt

M  Sx ,Ty ,t *M  Ax , Sx ,t *M  Sx , By ,2 t *M  By ,Ty ,t , M Ty , Ax ,t 

and hence from Corollary 2.2, unique common fixed point in X. Corollary 2.4:



 (t )dt 



A, B, S and T , we have a

A and B be self-mappings of X , k   0,1

which satisfies the condition M  Ax , By , kt 

 (t ) dt

 X , M ,  be a complete  -chainable

Let

fuzzy metric space and let

ISSN: 2230-7818

A, B, S and T be self-mappings of

X , satisfying (i), (ii) and (iii) of Theorem 2.1 and there k   0,1



AX  TX . In such a situation, the sequence occurs as

 (t ) dt

 X , M ,  be a complete  -chainable

Let

fuzzy metric space and let

weakly compatible and

 (t ) dt

A, B, S and T , we have a

and hence from Theorem 2.1, unique common fixed point in X.

Remark 2.1: If S = T and A = B, then conditions (i)- (iii) is

 (t ) dt



exists

 A, T 

 (t )dt  0

for every ,

z  w . So, z is the unique common fixed point of A, B, S and T .

Thus,

-chainable

fuzzy metric space and let A, B, S and T be self-mappings of X , satisfying (i), (ii) and (iii) of Theorem 2.1 and there k   0,1



 (t )dt  0

for every ,



F M  Ax , By , kt , M  x , y ,t , M  x , Ax ,t , M  y , By ,t , M  y , Ax ,t , M  x , By ,t 



M  x , y ,t 

, such that

 (t ) dt

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Manish Kumar Mishra et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 2, 123 - 129

x, y  X and t  0 .Then A, B, S and T have a unique common fixed point in X . for every ,



M  A x , y , B  u , y , kt , M  A x , y , x ,t , M  B  u ,v ,u ,t , M  x ,u ,t , M  y , v ,t , M  A x , y ,u ,t 

Proof: In corollary 2.3, if we take A=B, so the equation reduces to Grabiec’s fuzzy Banach Centration theorem (see[5])

for all ,

Corollary 2.5:

exactly

 X , M ,  be a complete  -chainable

Let

fuzzy metric space and let A be self-mappings of X , which k   0,1

 (t )dt 



M  x , y ,t 

x, y  X and t  0 .Then A, B, S and T have a unique common fixed point in X .  X , M ,   be a complete

Let

-chainable fuzzy

metric space. For any x, y  X and t  0 , assume that M  x , y ,t   sup M  x0 , x1 ,t *M  x1 , x2 ,t *...........*M  xn1 , xn ,t   M  x , y ,t 

 (t )dt 

Whenever



M   x , y ,t 

M  x , y ,t 

A  w, w  w  B  w, w

M

 (t )dt and

 (t )dt  1  

,such

that

M  x , y ,t 

 (t )dt 



M   x , y ,t 

M  A x , y , B  u , y , kt , M  A x , y , x ,t , M  B  u , y ,u ,t , M  x ,u ,t , M  A x , y ,u ,t 

for all ,

 (t )dt

 X , M  ,  is complete. Consider x  be a To show that  X , M  ,  . Then, for m  n , such Cauchy sequence in

z  y

X such that

A z  y  , y   z  y   B  z  y  , y  For any

 (t )dt  0

x, y, u, v in X . Therefore by Corollary 2.1, we have,

for each y in X, there exists one and only one

is a fuzzy metric satisfying



Then, it is to be prove that



point

Proof: From above relation (i) we have,

 (t ) dt

for every , Proof: :

one

M  Ax , By , kt 



x, y, u, v in X and t  0 .Then there exists

, such that

satisfies the condition

 (t )dt  0



…………………(iii)

y, y '  X by (i), and using relation

M  z  y , z  y ', kt 

 (t )dt 



M  A ( z  y , y ), B ( z  y ' , y ),t 

 (t )dt

that

M  ym1 , ym ,t / m 



 (t )dt 

hence, we have

IJ A M  xn , xm ,t 

 X , M , 

Since



M  xn , z ,t 



 (t )dt 

M   xn , z ,t 



M   xn , xm , t 

 (t ) dt

is complete, there exists

 (t )dt  1  

M   xn , z ,t 

and

 (t )dt 





F M  z  y , z  y ', kt ,1,1,1, M  y , y ',t , M  z  y , z  y ',t 

1  * 1  * 1 .........* 1   1 



we get



M  xn , z ,t 

and

hence

 (t )dt

 (t )dt  1  

 X , M  ,  is complete.

z  X , such that

. Hence

 xn  converges

Since, F is nondecreasing in the first argument, therefore



we have





F M  z  y , z  y ', kt ,1,1,1, M  y , y ',t , M  z  y , z  y ',t 



M  z  y , z  y ' , t 

So,

to z and





 (t )dt 

M z  y , z  y ' ,t / k



A and B be two self-mappings on X  X product , with values in X . If there exists a constant k   0,1



M  y , y ',t *M z  y , z  y ' ,t / k



 (t )dt

Therefore, Corollary 2.4 yields that the map z (.) of X into

w  z  w . w  z  w  a  w, w  B  w, w

itself has exactly one fixed point w in X i.e. Hence, by (ii),

. It

is easily observed that A and B can have only one such common fixed point w in X. REFERENCES [1]

L. A. Zadeh , fuzzy sets, inform control B(1965),338-353.

[2]

I. Kramosil and J. Michalek , Fuzzy metric and statistical metric space, Kybernetika,11(1975), 336-344.

, such that (i)

ISSN: 2230-7818

  1, as n  

 X , M ,  be a complete  -chainable

fuzzy metric space and let

 (t )dt  0

which implies that

IV. AN APPLICATION Now, we shall apply the corollaries 2.1 and 2.4 to establish the following results: Theorem 3.1: : Let

 (t )dt  0

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Manish Kumar Mishra et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 1, Issue No. 2, 123 - 129 [3]

A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy sets and Systems, 64 (1994),395-399.

[4]

M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy sets and Systems, 27(1988), 385-389.

[5]

P.V. Subrahmanyam, Common fixed point theorem in fuzzy metric spaces, Information Sciences,83 (1995) ,105-112.

[6]

R. Vasuki, Common Fixed points for R-weakly computing maps in fuzzy metric spaces, Indian J. Pure Appl. Math., 30,4(1999),419-423.

[7]

R.P. Pant, K. Jha, ‖A remark on common ﬁxed points of four mappings in a fuzzy metric space‖, J. Fuzzy Math. 12(2) (2004), 433437.

[8]

G. Jungck, Compatible mappings and common fixed points, Int. J.Math. & Math. Sci. 9(1986), 771-779.

[9]

O. Kramosil & J Michalek, Fuzzy metric & Statistical metric spaces, Kybernetica, 11 (1975), 326 - 334.

[10] A George & P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets & Systems, 90 (1997), 365-368.

[12] G.Jungck, Compatible mappings Int.J.Math.Math.Sci.,9(1986),771-779.

and common fixed points,

[13] G.Jungck and B.E.Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure and Appl. Math.29 (1998), no.3, 227-238.

[14] M. Grabiec, Fixed point in Fuzzy metric Spaces, Fuzzy sets & systems, 27 (1988), 245-252.

[11] S.Sessa, On weak commutativity condition of mapping in fixed point consideration, Publ.Inst.Math. (Beograd) N,S.,32(46), (1982), 149-153.

[15] Bijendra Singh & M. S. Chauhan, Common fixed points of compatible maps in fuzzy metric space, Fuzzy sets & Systems, 115 (2000), 471– 475. [16] V. Popa, Some fixed point theorems for weakly compatible mappings, Radovi Mathematics 10 (2001), 245 - 252. [17] M. Imdad, S. Kumar & M.S. Khan, Remarks on some fixed point theorems satisfying implicit relations, Radovi Mathematics 11 (2002), 135-143.

IJ A

[18] B. Singh & S . Jain, Semicompatibility & Fixed point theorems in fuzzy metric spaces using implicit relation, Int. J. Math & Math Sc., 16 (2005), 2617-2629.

[19] Deo Brat Ojha, Manish Kumar Mishra and Udayana Katoch,A Common Fixed Point Theorem Satisfying Integral Type for Occasionally Weakly Compatible Maps, Research Journal of Applied Sciences, Engineering and Technology 2(3): 239-244, 2010. [20] Rhoades, B.E.,Two fixed point theorem for mapping satisfying a general contractiv condition of integral type. Int. J. Math. Sci., 3: 2003 40074013. [21] Rana, Dimri and tomar, ― Fixed point theorems in fuzzy metric spaces using implicit relation‖ , International journal of computer applications vol. 8- no.1, 2010, pp. 16-21.

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