International Journal of Scientific & Engineering Research, Volume 1, Issue 3, December-2010 1

ISSN 2229-5518

On Common Fixed Point For Compatible mappings in Menger Spaces

M. L. Joshi and Jay G. Mehta

Abstract— In this paper the concept of compatible map in menger space has been applied to prove common fixed point theorem. A fixed point theorem for self maps has been established using the concept of compatibility of pair of self maps.

Index Terms— Common fixed point, menger space, compatible maps, weakly compatible maps.

—————————— • ——————————

N 1942 Menger [1] has introduced the theory of prob- abilistic metric spaces in which a distribution function was used instead of non-negative real number as value of the metric. In 1966, Sehgal [2] initiated the study of con- traction mapping theorems in probabilistic metric spaces. Since then several generalizations of fixed point Sehgal and Bharucha-Reid [3], Sherwood [4], and Istratescu and Roventa [5] have obtained several theorems in probabilis- tic metric space. The study of fixed point theorems in probabilistic metric spaces is useful in the study of exis- tence of solutions of operator equations in probabilistic metric space and probabilistic functional analysis. In 2008, Altun and Turkoglu [3] proved two common fixed point theorems on complete PM-space with an implicit relation. The development of fixed point theory in probabilistic metric spaces was due to Schweizer and Sklar [7] played major role in development of fixed point theory in proba- bilistic metric spaces. Singh et al. [8] introduced the con- cept of weakly commuting mappings in probabilistic me- tric spaces. The concept of weakly-compatible mappings

is most general as every commuting pair is R-weakly

Definition 2.1. [11] A mapping F : R�R+ is called a dis- tribution if it is non-decreasing left continuous with inf

{F(t) : t E R} = 0 and sup {F(t) : t E R} = 1. .

We shall denote by L the set of all distribution functions while H will always denote the specific distri- bution function defined by

f0,t � 0

H(t) = �

l1,t > 0

called a continuous t-norm if it satisfies the following con- ditions:

(t-1) t is commutative and associative; (t-2) t(a,1) = a for all a E [0,1];

(t-3) t(a,b) � t(c,d) for a � c , b � d.

The following are the basic t-norms: TM(x,y) = Min{x.y}

TP(x,y) = x·y

TL(x,y) = Max{x+y-1, 0}.

Each t-norm T can be extended [14] (by associativity) in a

unique way taking for ( , ,... ) [0,1]n

commuting, each pair of R-weakly commuting mappings

x1 x2

1

xn E , ( n E N )

is compatible and each pair of compatible mappings is

the values T

(x1 , x2 ) = T(x1 , x2 ) and

weakly compatible but the converse is not true. Kumar

and Chugh [9] established some common fixed point

T n (x , x ...x ) = T(T n-1

(x1 , x2 ...xn ), xn+1 ) for

theorems in metric spaces by using the ideas of pointwise R-weak commutativity and reciprocal continuity of map- pings. A fixed point theorem concerning probabilistic con- tractions satisfying an implicit relation was proved by Mihet [10] in 2005.

The main object of this paper is to obtain fixed point theorems in the setting of Menger space using concept of compatibility.

we recall some definitions and known results.

n :: 2 and xi E[0, 1] , for all i E {1, 2, ...n + 1} .

set X and a function F: X x X �L, where L is the collection

of all distribution functions and the value of F at (u ,v) E X x X is represented by Fu,v. The function Fu,v is assumed to satisfy the following conditions:

(PM – 1) Fu,v(x) = 1, for all x > 0 if and only if u = v; (PM – 2) Fu,v(0) = 0;

(PM – 3) Fu,v = Fv,u ;

(PM – 4) If Fu,v(x) = 1 and Fv,w(x) = 1 then

IJSER © 2010 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 1, Issue 3, December-2010 2

ISSN 2229-5518

Fu,w(x+y) = 1 for all u,v,w in X and x,y > 0 . Definition 2.4. [11] A Menger space is a triplet (X, F, t) where (X,F) is a PM-space and t is a t-norm such that

f2 - x; x E[0,1)

S ( x) = �

l2; x E[1, 2]

the inequality

Let xn = 1 - 1

n

then Axn = 1 - 1

n

and Sxn = 1 + 1

n

(PM – 5) Fu,w(x+y) :: t{ Fu,v(x) , Fv,w(x) } for all

u,v,w in X and x,y > 0 .

Definition 2.5. [11] A sequence {xn} in a Menger space (X,

Thus Axn � 1 and Sxn � 1 and hence x = 1. Also ASxn = 2 and SAxn = 1 + 1 .

n

F, t) is said to converges to a point x in X if and only if for

Now

lim F ,

(t) = lim F 1 (t) =

t < 1

1

n

ASx SAx

n + t +

each s > 0 and t > 0, there is an integer M( s )E N such

that

for all t > 0.

n n n

F , ( s ) > 1 – t for all n :: M( s )

Hence A and S are not compatible.

xn x

Again

lim FASx ,Sx (t) = lim F2 ,2 (t) = 1

Definition 2.6. [11] The sequence {xn} is said to be Cauchy

n n

n

sequence if for s > 0 and t > 0, there is an integer

M( s )E N such that

Hence A and S are semi compatible and

lim FSAx , Ax (t) = lim F1+ 1 ,2 (t) = t +1 < 1 for t > 0.

F , ( s ) > 1 – t for all n,m :: M( s )

n n n n

xn xm

Definition 2.7. [11] A Menger PM-space (X, F, t) is said to be complete if every Cauchy sequence in X converges to a point in X.

A complete metric space can be treated as a complete

Menger space in the following way:

Lemma 2.1 [11] If (X,d) is a metric space then the metric d induces mappings F: X x X � L , defined by Fp.q = H(x-d(p,q)), p, q E X, where H(k) = 0 for k � 0 and

H(k) = 1 for k > 0.

Further if, t : [0,1] x [0,1] � [0,1] is defined by

Therefore it is clear that S, A are not semi compatible.

Now we will show that the semi compatible pair (A, S) is also weakly compatible .

Now coincidence points of A and S are in [1, 2]. Therefore for any x in [1, 2], we have

Ax = Sx = 2 and AS(x) = 2 = SA(x) and A(2) = 2 = S(2) Thus (A, S) is weakly compatible.

Theorem 1. Theorems, Let (X , F , t) be a complete Menger

t(a,b) = min{a,b}. Then (X, F, t) is a Menger space. It is

space with continuous t -norm t and let

h : X X ,

complete if (X,d) is complete.

k : X X ,

f : X h(X)

and

g : X k(X) be con-

The space (X, F, t) so obtained is called the induced Men- ger space.

tinuous mapping such that ( f , k) and ( g, h) are compatible

space (X, F, t) are said to be compatible if

pairs. Further, suppose that for all

s > 0 the following inequality holds

x, y E X

and for all

F , (x) � u for all x > 0,

Ffx ,gy (s ) :: Fkx ,hy ( (s ))

ASxn SAxn

whenever {xn} is a sequence in X such that

ASxn , SAxn � u for some u in X, as n�oo .

Where

: R+ R+ is an increasing function such that

n

lim

n

(t) = for all

t > 0 . If the sequence {yn }nEN

weakly compatible if they commute at a coincidence point.

Menger space (X, F, t) where t is continuous and

formed by

y2 n-1 = gx2 n-1 = kx2n ,

y2 n = fx2 n = hx2n +1 , n E N

t(x,x) :: x for all x E [0,1] and Sxn , Txn � u for some u

in X. Then TSxn � u provided S is continuous.

is probabilistically bounded for some

x1 E X , then there

(X, F, t) are compatible, then they are weak compatible.

exists a unique common fixed point for the mappings

f , g , h and k .

The converse is not true as seen in following example.

dition.

} be the sequence satisfying the given con-

t

Let Fx , y =

t + d ( x, y)

for all x and y in X and t > 0.

We shall show that {yn }nEN is a Cauchy sequence.

For that, we shall show that

Define:

f x; x E[0,1)

A( x) = �

and

lim

m ,p

Fy ,y

(s ) = H(s ) , for every s E R .

l2; x E[1, 2]

If m = 2i and p = 2 j - 1 ( let j > i ) then we have

IJSER © 2010 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 1, Issue 3, December-2010 3

ISSN 2229-5518

F (s ) = F

2 2 -1

(s )

:: F

( (s ))

Further,

y i y j

fx2 i ,gx2 j-1

kx2 i ,hx2 j-1

F * * (s ) =

F * *

(s )

:: F

* * ( (s ))

= F ( (s )) :: F

( 2 (s ))

ffy , fy

ffy ,gy

kfy ,hy

fx2 j-2 ,gx2 i-1

kx2 j-2 ,hx2 i-1

= F * * ( (s )

:: …. :: F

( n (s ))

H(s )

=

2 -2 ,

( 2 (s )) :: 2 i

2 -1 0 2 -1-2

ffy , fy

ffy , fy

fx i

gx j

Ffx ,gx ( (s ))

for n for s > 0 , which means that fy* is a com-

D 2 i s .

:: sup

inf

Fy ,y (t) =

( ( ))

{ y }

mon fixed point for the mappings f , g , h and k .

t < 2 i (s ) ,kEN n k

n n=1

Since {yn }nEN is probabilistically bounded, by consider-

ing i and j , we get

that there exists another common fixed point z E Z , therefore we get,

lim D

( 2 i (s )) = (s ) .

F * (s ) = F *

(s ) :: F *

( (s )) = F *

( (s ))

i { yn }n=1

fy ,z

ffy ,gz

kfy ,hz

ffy ,gz

By repeating this process, we can prove a similar result for m = 2i - 1 and p = 2 j .

:: ….. ::

Fffy* ,z (

n (s ))

H(s )

If m and p are both even or both odd, we pro-

for n for

s > 0 , which means that fy* is a

ceed as follows.

s s

unique common fixed point for the mappings f , g , h

and k .

F (s ) :: t(F

2 2

( ), F

+ +

( ))

y i y j

y2 i ,y2 i 1 2

y2 i 1 ,y2 j 2

t( H(s ), H(s )) = H(s ) .

s s

2 -1 2 -1 2 1 2 2 2 1

Hence the theorem. 0

y i y j

y i- ,y i 2

y i ,y j- 2 4 C

t( H(s ), H(s )) = H(s ) .

If i and j , for all

Thus we have proved that is {yn }nEN

s > 0 .

a Cauchy sequence

In this paper, we have described common fixed point theorems for four mappings in Menger space by compa- tibility. This idea can be implemented in the other metric

spaces.

in X which means that there exists y* E X

such that

lim y

= y* .

n We are thankful to Prof. L. N. Joshi, Retd. Prof in mathe-

To prove that fy* = gy* = hy* = ky*

follows.

, we proceed as

matics, D.K.V. Science College, Jamnagar and Prof. J. N. Chauhan, Head, Department of Mathematics, M. and N.

fy* =

f lim kx =

lim fkx =

lim kfx =

Virani Science College, Rajkot for their cooperation in the

n 2n n

k lim fx2 = ky .

n

2n n

2 n preparation of this paper. We are also thankful to the nu-

merous referees for their helpful and valuable comments.

gy* =

g lim hx

= lim ghx

= lim hgx

= REFERENCES

n 2 n+1 n

*

2 n+1 n

2n+1

[1] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U.S.A. 28

h lim gx2 +1 = hy .

n

(1942), 535–537.

[2] V.M. Sehgal, Some fixed point theorems in function analysis

Since

F * * (s )

:: F

* * ( (s )) =

F * * ( (s ))

and probability, Ph.D dissertation, Wayne State Univ. Michigan

fy ,gy

ky ,hy

fy ,gy

(1966).

:: …… ::

Ffy* ,gy* (

n (s ))

H(s ) . For all

s > 0 .

[3] V.M. Sehgal and A.T. Bharucha-Reid, Fixed points of contrac- tion mappings on probabilistic metric spaces, Math. Systems Theory 6 (1972), 97–102.

Thus fy* = gy* = hy* = ky* .

The point fy* is a fixed point for the mapping f , g , h and k . We shall show this for the mapping f . The proof for the mappings g , h and k is similar.

From compatibility of ( f , k) we obtain that

[4] H. Sherwood, Complete probabilistic metric spaces, Z. wahr- scheinlichkeits theorie and verw. Grebiete 20 (1971), 117–128.

[5] V.I. Istratescu and I. Sacuiu, Fixed point theorem for contraction mappings on probabilistic metric spaces, Rev. Roumaine Math. Pures. Appl. 18 (1973), 1375–1380.

[6] I. Altun and D. Turkoglu, Some fixed point theorems on fuzzy

metric spaces with implicit relations, Commun. Korean Math. Soc., 23(1)(2008), 111-124.

kfy* =

kf lim y = lim kfy = lim fky =

fk lim y

[7] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North

n n

n

= fky* = ffy* .

n n n

Holland(Amsterdam, 1983).

[8] S.L. Singh, B.D. Pant and R. Talwar, Fixed points of weakly commuting mappings on Menger spaces, Jnanabha 23 (1993),

IJSER © 2010 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 1, Issue 3, December-2010 4

ISSN 2229-5518

115–122.

[9] S. Kumar and R. Chugh, Common fixed point theorems using minimal commutativity and reciprocal continuity conditions in metric spaces, Sci. Math. Japan 56 (2002), 269–275.

[10] D. Mihet, A generalization of a contraction principle in proba-

bilistic metric spaces, Part II, Int. J. Math. Math. Sci. 2005 (2005),

729–736.

[11] S.N. Mishra, Common fixed points of compatible mappings in

PM-spaces, Math. Japon. 36(2), (1991), 283-289.

[12] A. Jain and B. Singh, Common fixed point theorem in Menger space through compatible maps of type (A), Chh. J. Sci. Tech. 2 (2005), 1-12.

[13] B. Singh and S. Jain, Semi-compatibility and fixed point theo- rem in Menger space, Journal of the Chungcheong Mathemati- cal Society 17 (1), (2004), 1-17.

[14] E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer

Academic Publishers, Trend in Logic 8, Dordrecht (2000).

[15] R. A. Rashwan & A. Hedar: On common fixed point theorems of compatible mappings in Menger spaces. Demonstratio Math.

31 (1998), no. 3, 537–546.

[16] G. Jungck & B. E. Rhoades: Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math. 29 (1998), no. 3,

227–238.

IJSER © 2010 http://www.ijser.org