# A Related Fixed Point Theorem of Integral Type on Two Fuzzy 2-Metric Spaces

DOI : 10.17577/IJERTV2IS101122

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#### A Related Fixed Point Theorem of Integral Type on Two Fuzzy 2-Metric Spaces

Pheiroijam Suranjoy Singh Sagolband Takyel Kolom Leikai

Imphal West, Manipur – 795001, India.

Abstract

In this paper, a related fixed point theorem is obtained. It extends a result proved by R.K. Namdeo, N.K. Tiwari, B. Fisher and K. Tas [9]. The notion of fuzzy 2-metric spaces satisfying integral type inequalities is used.

Keywords : Fuzzy 2-metric space, fixed point, related fixed point, integral type inequality.

2000 AMS Subject Classification : 47H10, 54H25.

Intoduction

The concept of fuzzy sets was introduced by L. Zadeh [14] in 1965. Fuzzy metric space was introduced by Kramosil and Michalek [7] in 1975. Then, it was modified by George and Veeramani [4] in 1994. Fuzzy has been studied and developed by many mathematicians for many years. Introduction of fuzzy 2-metric space is one of such developments. Gahler [10, 11] investigated 2-metric spaces in a series of his papers. Fuzzy 2-metric space is studied in [6, 8, 12, 13] and many others. Related fixed point is studied in [1, 2, 3, 5, 9] and many more.

Some definitions are stated as follows:

Definition 1.1 : A binary operation : [0, 1] Ã— [0, 1] [0, 1] is called a t – norm in ( [0, 1], ) if follo- wing conditions are satisfied:

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

1. a 1 = a,

2. a b = b a,

3. a b c d whenever a c and b d,

4. a (b c) = (a b) c.

Definition 1.2. The 3-tuple (X, , ) is called a fuzzy 2- metric space if X is an arbitrary set, is a continuous t – norm and is a fuzzy set in X 3 [0, ) satisfying the following conditions:

For all x, y, z, u X and t1, t2, t3 > 0, i. (x, y, z, 0) = 0,

1. (x, y, z, t) = 1, t > 0 and when at least two of the three points are equal,

2. (x, y, z, t) = (y, x, z, t) = (z, x, y, t) ( symmetry about three variables), iv. (x, y, z, t1+t2+t3) (x, y, u, t1) (x, u, z, t2) (u, y, z, t3)

1. (x, y, z,) : [0, ) [0, 1] is left continuous,

2. limt (x, y, z, t) = 1.

Definition 1.3 : Let (X, , ) be a fuzzy 2-metric space. A sequence {xn} X is said to:

1. converge to x in X if and only if

limt (xn, x, a, t) = 1 a X and t > 0.

2. be a Cauchy sequence if and only if limt (xn+p, xn, a, t) = 1 a X, p > 0 and t > 0.

Definition 1.4 : A fuzzy 2-metric space (X, , )

sequence in X is convergent in X.

is aid to be complete if and only if every Cauchy

The following was proved in [9].

Theorem 1.1 : Let (X , d) and (Y, ) be complete metric spaces. Let T be a mapping of X into Y and S

be a mapping of Y into X satisfying the inequalities

d ( Sy, Sy /) d( STx, STx / ) c max{ d (Sy, Sy/ ) (Tx, Tx / ) , d (x / , Sy) ( y / , Tx) ,

d (x, x / ) d (Sy, Sy/ ) , d (Sy, STx) d (Sy/ , STx/ ) }

(Tx, Tx / ) (TSy, TSy/ )

c max{ d (Sy, Sy/ ) (Tx, Tx / ) , d (x / , Sy) ( y / , Tx) ,

( y, y / ) (Tx, Tx / ) , (Tx,TSy) (Tx / , TSy/ ) }

for all x , x / in X and y , y / in Y ,where 0 c < 1. If either S or T is continuous, then ST has a unique fixed point z in X and TS has a unique fixed point w in Y . Further, Tz = w and Sw = z.

Now, theorem 1.1 is extended to two pairs of mappings in integral and fuzzy 2-metric space settings as follows.

Main result

Theorem 2.1 : Let (X, , a, t) and (Y, , a, t) be two complete fuzzy 2-metric spaces. Let A, B be mappings of X into Y and S, T be mappings of Y into X satisfying the inequalities

k(Sy, Ty/ , a, t ) (SAx, TBx/ , a, t)

1

(s) ds 1

min{(Sy, Ty/ , a, t)( Ax, Bx/ , a, t), (x/ , Sy, a, t)( y/ , Ax, a, t) ,

(x, x/ , a, t) (Sy, Ty/ , a, t), (Sy, SAx, a, t) (Ty/ , TBx/, a, t)}

(s) ds

(1)

k( Ax ,Bx/ , a,t)(BSy, ATy/ , a,t)

1

(s) ds

min{(Sy, Ty / , a,t)( Ax, Bx /, a,t), (x /, Sy, a,t)( y /, Ax, a,t),

( y, y /, a,t)( Ax, Bx /, a,t),( Ax, BSy, a,t)(Bx /, ATy /, a,t)}

1

(s) ds

(2)

for all x, x/ in X and y, y/ in Y, where k(0, 1). If A and S or B and T are continuous, then SA and TB have a unique common fixed point z in X and BS and AT have unique common fixed point w in Y. Further, Az = Bz = w and Sw = Tw = z.

Proof : Let x be any arbitrary point in X . We define sequences {xn} and {yn}in X and Y respectively as:

S y2n 1 =

x2n 1 ,

Bx2n 1 y2n ,

Ty2n x2n ,

Ax2n y2n 1 , for n = 1, 2, 3 ,

Applying inequality (1), we get

k(Sy , Ty

, a, t) (SAx , TBx , a, t)

1 2n 1 2n

2n 2n 1

(s) ds

= 1

k 2 (x

2n 1

, x , a, t) 2n

(s) ds

min{(Sy

, Ty

, a,t)( Ax

, Bx

, a,t), (x

, Sy

, a,t)( y

, Ax

, a,t),

2n 1 2n

2n 2n 1

2n 1

2n 1

2n 2n

(x , x

, a,t) (Sy

, Ty

, a,t),(Sy

, SAx

, a,t)(Ty

, TBx

, a,t)}

1

2n 2n1

2n 1 2n

2n 1 2n

2n 2n 1

(s) ds

min{(x

2n 1

, x , a, t)( y

2n

, y

2n 1

, a, t), (x

2n 2n 1

, x

2n 1

, a, t)( y

2n

, y

2n 1

, a, t),

(x

2n

1

, x

2n 1

, a, t) (x

2n 1

, x , a, t), (x

2n 2n 1

, x

2n 1

, a, t) (x

2n

, x , a, t)}

2n

(s) ds

from which it follows that

min{( y

, y , a, t), (x

, x , a, t)}

k(x , x

2n 1 2n

1

, a, t)

(s) ds

1

2n 1 2n

2n 1 2n

(s) ds

(3)

Applying inequality (2), we get

k( Ax , Bx

, a, t)(BSy

, ATy

, a, t)

2n 2n 1

1

2n 1 2n

(s) ds

= 1

k 2 ( y , y

2n 1 2n

, a, t)

(s) ds

min{(Sy

2n 1

, Ty

2n

, a, t)( Ax

2n

, Bx

2n 1

, a, t), (x

2n 1

, Sy

2n 1

, a, t)( y

2n

, Ax

2n

, a, t),

( y , y

2n 1 2n

1

, a, t)( Ax

2n

, Bx

2n 1

, a, t) , ( Ax

2n

, BSy

2n 1

, a, t)(Bx

2n 1

, ATy

2n

, a, t)}

(s) ds

max{(x

, x , a, t) ( y

, y , a, t), (x , x

, a, t)( y , y

, a, t),

2n 1 2n

2n 1 2n

2n 1

2n 1

2n 2n 1

( y

, y , a, t)( y

, y , a, t) ,( y

, y , a, t)( y , y

, a, t)}

1

2n 1 2n

2n 1 2n

2n 1 2n

2n 2n 1

(s) ds

from which it follows that

k( y , y , a, t)

min{( y , y , a, t), (x

, x , a, t)}

2n 1 2n

1

(s) ds

1

2n 1 2n

2n 1 2n

(s) ds

(4)

(3) and (4) can be written as

k(x , x , a, t)

min{( y , y , a, t), (x

, x , a, t)}

n 1 n

1

(s) ds

1

n 1 n

n 1 n

(s) ds

min{( y , y , a, t), (x

, x , a, t)}

k( y , y , a, t)

n 1 n

n 1 n

n 1 n

1

1

(s) ds

which can be again written as

k(x , x , a, t)

min{( y , y , a, t), (x

, x , a, t)}

n 1 n

1

(s) ds 1

n 1 n

n 1 n

(s) ds

(5)

k( y , y , a, t)

min{( y , y , a, t), (x

, x , a, t)}

n 1 n

1

(s) ds 1

n1 n

n 1 n

(s) ds

(6)

From (5) and (6) , by induction, we get

1 min{( y , y , a, t), (x , x , a, t)}

(x , x , a, t)

kn 1 2 1 2

n 1 n

1

( y , y

, a, t)

(s) ds 1

1 min{( y , y , a, t), (x , x , a, t)}

(s) ds

Let

t1

1

t . Now,

p

n 1 n

(s) ds kn 1 2

1

1 2 (s) ds

(xn , xn p , a, t) (xn , xn p , a, t t … p times)

1 (s)ds

1 (s)ds

1 (s)ds

1 (s)ds

1 (s)ds

= 1

1 1 (s)ds

1 (s)ds

1 (s)ds

1

1

1

1

(xn , xn 1, a, t )

(xn1, xn 2, a, t )

(xn p1, xn p , a, t )

1

1

1 min{( y , y , a, t), (x , x , a, t)}

1 min{( y , y , a, t), (x , x , a, t)}

kn 1 2

1

1 2 (s) ds kn p 1 1 2

1

1 2 (s) ds

which implies that

n n p

n n p

(x , x , a, t)

lim (s)ds 1

1

(xn , xn p , a, t) 1

{ xn } is a Cauchy sequence with a limit z in X.

Similarly, { yn } is a Cauchy sequence with a limit w in Y.

Now, on using the continuity of A and S respectively, we get

w = lim y 2n1 = lim Ax 2 n

= Az and z = lim x 2 n

= lim Sy 2 n

= Sw

so that we get

Az = w (7)

Sw = z (8)

From (7) and (8), we get

SAz = z (9)

Again applying inequality (1), we get

k(SAx

2n

1

, TBx

2n 1

, a, t)

(s) ds

min{( Ax , Bx , a, t), ( y

, Ax

, a, t), (x , x , a, t)}

1

2n 2n 1

2n 2n

2n 1 2n

(s) ds

(10)

On letting n , we have

k(Sw, TBz, a, t)

1

( Az , w, a, t)

(s) ds 1

(s) ds

By (7), we have

k(Sw, TBz, a, t)

1

(s) ds 0

k(Sw, TBz, a, t) 1

which implies that

Sw = TBz

and from (8) , we get

z = TBz (11)

From (9) and (11), we get

SAz = z = TBz (12)

Now, (10) gives

k(x

2n 1

1

, Ty

2n

, a, t)

(s) ds

min{( Ax

, Bx

, a, t), ( y

, Ax

, a, t), (x

, x , a, t)}

1

2n 2n 1

2n 2n

2n 1 2n

(s) ds

On letting n , we get

k(z , Tw, a, t)

1

(t) dt 0

k(z , Tw, a, t) 1

which implies that

z = Tw (13)

Again, applying inequality (2), we get

k(BSy

2n 1

1

, ATy

2n

, a, t)

(s) ds

min{(Sy

, Ty

, a, t), (x

, Sy

, a, t), ( y

, y , a, t), ( Ax

, Bx

, a, t)}

1

2n 1 2n

2n 1

2n 1

2n 1 2n

2n 2n 1

(s) ds

(14)

On letting n , we get

k(BSw, ATw, a, t)

1

(s) ds 0

k(BSw, ATw, a, t) 1

which implies that

BSw = ATw (15)

Now, (14) gives

k( y

2n

1

, ATy

2n

, a, t)

(s) ds

min{(Sy

2n 1

1

, Ty

2n

, a, t) , (x

2n 1

, Sy

2n 1

, a, t) , ( y , y

2n 1 2n

, a, t) , ( Ax

2n

, Bx

2n 1

, a, t)}

(s) ds

On letting n , we get

k(w, ATw, a, t)

1

(s) ds 0

k(w, ATw, a, t) 1

which implies that

w = ATw (16)

From (15) and (16), we get

BSw = w = ATw (17)

From (8) and (17) , we get

Bz = w (18)

From (7) and (18), we get

Az = Bz = w (19)

From (8) and (13) , we get

Sw = Tw = z (20)

Similarly, on using the continuity of B and T, the above results hold.

To prove the uniqueness, let SA and TB have a second distinct common fixed point z / in X and BS

and AT have a second distinct common fixed point w / in Y.

Applying inequality (1), we have

k 2 (z , z/ , ,a, t)

1

(s) ds

min{(z , z/ , a, t)( Az , Bz/ , a, t), (z/ , z/ , a, t)(Bz/ , Az, a, t),

(z, z/ , a, t) (z , z/ , a, t), (z/ , z/ , a, t) (z, z, a, t)}

1

(s) ds

k(z , z/ , a, t)

1

(s) ds

min{( Az , Bz/ , a, t), (Bz/ , Az, a, t)}

1 (s) ds

k(z , z/ , a, t)

1

(s) ds

( Az , Bz/ , a, t)

1 (s) ds

(21)

Applying inequality (2), we get

k2 ( Az ,Bz/ , a, t)

1

(s) ds

min{(z, z/ , a, t)( Az, Bz/ , a, t), (z / , z / , a, t)(Bz / , Az, a, t),

( Az, Bz / , a, t)( Az, Bz/ , a, t), ( Az, Bz / , a, t)(Bz / , Az, a, t)}

1

(s) ds

k( Az ,Bz/ , a, t)

1

/

(z , z , a, t)

(z , z , a, t)

(s) ds

1

(s) ds

(22)

From (21) and (22), we get

k2(z, z/ , a, t)

1

(s) ds

(z, z/ , a, t)

1

(s) ds

(z, z/ , a,t)

1

(z, z/ , a,t)

1

(s) ds

(s) ds

1 (z, z/ , a, t)

k 2

1

1 ( z, z/ , a, t)

k 2

1

(s) ds

(s) ds 1

1 (z, z/ , a, t)

k n

(s) ds

(z, z/ , a,t)

1

(s) ds

lim 1

1 (z, z/ , a, t)

k n

(s) ds 1

(z, z/ , a,t) 1

which implies that

z = z / .

This proves the uniqueness of z. Similarly, the uniqueness of w can be proved.

The following corollary is a fuzzy 2-metric space version of theorem 1.1 in integral setting.

Corollary 2.2 : Let (X, , a, t) and (Y, , a, t) be two complete fuzzy 2-metric spaces. Let S be mappings of X into Y and T be mappings of Y into X satisfying the inequalities

k(Ty, Ty/ , a, t ) (TSx, TSx/ , a, t)

1

(s) ds 1

min{(Ty, Ty/ , a, t)(Sx, Sx/ , a, t), (x/ , Ty, a, t)( y/ , Sx, a, t) ,

(x, x/ , a, t) (Ty, Ty/ , a, t), (Ty, TSx, a, t) (Ty / , TSx/ , a, t)}

(s) ds

k(Sx,Sx/ , a, t)(STy, STy/ , a, t)

1

(s) ds 1

min{(Ty,Ty/ , a, t)(Sx, Sx/ , a, t), (x/ , Ty, a, t)( y/ , Sx, a, t),

( y, y/ , a, t)(Sx, Sx/ , a, t), (Sx, STy, a, t)(Sx/ , STy/ , a, t)}

(s) ds

for all x, x/ in X and y, y/ in Y ,where k (0, 1). If either S or T is continuous, then TS has a unique fixed point z in X and ST has a unique fixed point w in Y . Further, Sz = w and Tw = z.

Proof : By putting A = B = S and S = T = T in theorem 2.1, the result easily follows.

References

1. Aliouche A. and Fisher B., A related fixed point theorem for two pairs of mappings on two complete metric spaces, Hacettepe Journal of Mathematics and Statistics, Vol. 34 (2005), 39 – 45.

2. Cho Y.J., Kang S.M. and Kim S.S., Fixed points in two metric spaces, Novi Sad Journal Math., 29 (1) (1999), 47- 53.

3. Fisher B. and Rao K.P.R, A related fixed point theorem on three metric space, Haettepe Journal of Mathematics and Statistics, Vol. 36 (2) (2007), 143 – 146.

4. George A. and Veeramani P., On some result in fuzzy metric space, Fuzzy Set and Systems, Vol. 64, (1994) 395-399.

5. Jain S. and Fisher B., A related fixed point theorem for three metric spaces, Hacettepe Journal of Mathematics and Statistics, Vol. 31 (2002), 19 – 24.

6. Jinkyu Han, A common fixed point theorem on fuzzy 2- metric spaces, Journal of the Chungcheong Mathematical Society, Volume 23, No. 4, December 2010.

7. Kramosil O.and Michalek J., Fuzzy metric and statistical metric spaces, Kybernetica Vol. 11 (1975) 326 – 334.

8. N.Thillaigovindan, S.Anita Shanthi and S.Vijayabalaji, Semigroup actions on intuitionistic fuzzy 2- metric spaces, Int. J. Open Problems Compt. Math., Vol. 3, No. 1, March 2010.

9. R.K. Namdeo, N.K. Tiwari, B.Fisher and K. Tas, Related fixed point theorems on two complete and compact metric spaces, Internat. J. Math. and Math. Sci. ,Vol. 21, No. 3 (1998) , 559 564.

10. S. Gahler, (1965), Uber die uniromisieberkeit 2-metric Raume, Math.Nachr. 28, 235 – 244.

11. S. Gahler, (1963), 2-metric Raume and ihre topologische strucktur, Math. Nachr., 26,115 – 148.

12. Surjeet Singh Chauhan and Kiran Utreja, Fixed point theorem in fuzzy 2- metric space using absorbing maps, Research Journal of Pure Algebra, 2 (2) (2012), 77 – 81.

13. Surjeet Singh Chauhan (Gonder) and Kiran Utreja, A common fixed point theorem in fuzzy 2- metric space, Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 2, 85 91, Hikari Ltd, www.m-hikari.com

14. Zadeh, L.A., Fuzzy sets, Information and control, 8 (1965), 338 – 353.