common fixed point theorem in probabilistic …...fixed point theory in pm spaces can be considered...

Int. Journal of Math. Analysis, Vol. 8, 2014, no. 32, 1561 - 1569

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http://dx.doi.org/10.12988/ijma.2014.46167

Common Fixed Point Theorem in

Probabilistic Metric Spaces

1 R. K. Gujetiya,

2 Mala Hakwadiya * and

3 Dheeraj Kumari Mali

1 Department of Mathematics

Govt. P. G. College, Neemuch, India

2 , 3 Pacific Academy of Higher Education

and Research University Udaipur, Rajasthan, India

*Corresponding author

Copyright © 2014 R. K. Gujetiya, Mala Hakwadiya and Dheeraj Kumari Mali. This is an open

access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is

properly cited.

Abstract

In this paper we prove common fixed point theorem for six mapping with

compatibility of probabilistic metric space.

Mathematics Subject Classification: 47H10, 54H25

Keywords: Common fixed point, Probabilistic metric Space, Semi-compatible,

Weakly commuting mapping, weakly compatible

1562 R. K. Gujetiya, Mala Hakwadiya and Dheeraj Kumari Mali

1. Introduction

The notion of probabilistic metric space is introduced by Menger in 1942 [3] and

the first result about the existence of a fixed point of a mapping which is defined

on a Menger space is obtained by Sehgel and Barucha-Reid. Fixed point theory in

PM spaces can be considered as a part of Probabilistic Analysis, which is a very

dynamic area of mathematical research. Recently, a number of fixed point

theorems for single valued and multivalued mappings in menger PM space have

been considered by many authors [1], [7], [5], [4]. In 1998, Jungck [2] introduced

the concept weakly compatible maps and proved many theorems in metric

space. In this paper we will generalize the result of R. Singh[6].

2. Preliminaries

Definition 2.1 [6] Let R denote the set of reals and the non-negative reals.

A mapping F : R → is called a distribution function if it is non

decreasing left continuous with F (t) = 0 and F (t ) = 1.

Definition 2.2: [6] A probabilistic metric space is an ordered pair (X, F) where X

is a nonempty set, L be set of all distribution function and F: X × X → L . We

shall denote the distribution function by F (p, q) or ; p, q X and (x)

will represents the value of F (p, q) at x R . The function F(p, q) is assumed to

satisfy the following conditions:

1. (x) = 1 for all x > 0 if and only if p = q

2. (x) = 0 for every p, q ϵ X

3. (x) = (x) for every p, q ϵ X

4. (x) = 1 and (y) = 1 then (x + y) = 1 for every p, q, r ϵ X.

In metric space (X, d) , the metric d induces a mapping F: X × X → L such that

(x) = = H (x - d (p, q)) for every p,q X and x R, where H is the

distribution function defined as ( ) {

.

Definition 2.3: [6] A mapping ∗: [0, 1] [0, 1] → [0, 1] is called t-norm if

1. ( ∗ ) , 2.( ∗ ) . 3. (a * b)= (b * a) 4. ( ∗ ) ≥ ( ∗ ) for c ≥ a, d ≥ b, 5. (( ∗ ) ∗ ) = ( ∗ ( ∗ ))

Common fixed point theorem 1563

Example: (i) ( ∗ ) = ab (ii) ( ∗ ) = min (a, b)

(iii) ( ∗ ) = max (a + b – 1 ; 0)

Definition 2.4: [6] A Menger space is a triplet (X, F, *) where (X, F) a PM-

space and Δ is is a t-norm with the following condition (x + y) ≥ (x) *

(x) The above inequality is called Menger’s triangle inequality.

Example: Let ( ∗ ) ( ) ( ) and

(x) = { ( )

where H(x) = {

Then (X, F, * ) is a Menger space.

Definition 2.5: [6] Let (X, F, *) be a Menger space. If u x, > 0, (0, 1),

then an ( , ) neighbourhood of u, denoted by ( ) is defined as ( )

ϵ ( ) . If (F, X, *) be a Menger space with the continuous t-

norm t, then the family ( ) ( ) of neighbourhood

induces a Hausdorff topology on X and if ( ∗ )= 1, it is metrizable.

Definition 2.6: [6] A sequence{ } in (X, F, *) is said to be convergent to a point

if for every ϵ > 0 and > 0, there exists an integer N = N( ) such

that ( ) for all or equivalently ( ) for all .

Definition 2.7: [6] A sequence{ } in (X, F, *) is said to be Cauchy sequence if

for every ϵ > 0 and > 0, there exists an integer N = N( ) such that

( ) for all .

Definition 2.8: [6] A Menger space (X, F, *) with the continuous t-norm Δ is said to be complete if every Cauchy sequence in X converges to a point in X.

Definition 2.9: [6] A coincidence point (or simply coincidence) of two mappings is a point in their domain having the same image point under both mappings.

Formally, given two mappings we say that a point x in X is a

coincidence point of f and g if ( ) ( ) .

Definition 2.10: [6] Let (X, F, *) be a Menger space.Two mappings

are said to be weakly compatible if they commute at the coincidence point, i.e.,


the pair is weakly compatible pair if and only if implies that

Definition 2.11: [6] Let (X, F, *) be a Menger space.Two mappings

are said to be Semi compatible if ( ) for all t > 0 whenever { }is a

sequence in X such that for some p in X as n→∞. It follows that (g,

f) is semi compatible and fy = gy imply fgy = gfy by taking { } = y and x = fy

= gy.

Lemma 2.12: [9] Let { } be a sequence in Menger space (X, F, *) where * is

continuous and ( ∗ ) for all x [0, 1]. If there exists a constant k (0, 1)

such that x > 0 and n ϵ N; ( )

( ), then { } is a

Cauchy sequence.

Lemma 2.13: [10] If (X, d) is a metric space, then the metric d induces a

mapping F: X × X → L, defined by (x) = = H (x - d (p, q)) for every p ,

q X and x R. further more if ∗ is defined by

( ∗ ) = min (a, b), then (X, F, *) is a Menger space. It is complete if (X, d) is

complete. The space (X, F, *) so obtained is called the induced Menger space.

Lemma 2.14: [8] Let (X, F, *) be a Menger space. If there exists a constant k

(0, 1) such that ( ) ( ) for all x, y ϵ X and t > 0 then x = y.

Theorem 2.15: [6] Let (X, F,*) be a complete Menger space where * is

continuous and ( ∗ ) for all t ϵ [0,1]. Let A, B, S, and T be mappings from

X into itself such that

(1) A(X) S(X) and B(X) T(X)

(2) S or T is continuous.

(3) The pair (S,A) and (T,B) are semi-compatible

(5) There exists a number k ϵ (0,1) such that

( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ) For all

x,y ϵ X, ϵ (0,2) and t > 0,then A, B, S, and T have a unique common fixed

point.


3. Main Result

Theorem 3.1: Let ( X , F , * ) be a complete Menger space where * is

continuous and (t * t) ≥ t for all t ϵ [0,1]. Let A, B, S, T, U and V be mappings

from X into itself such that

(1) U(X) AB(X) and V(X) ST(X)

(2) AB or ST is continuous.

(3) The pair (AB, U) and (ST, V) are semi-compatible

(4) AB = BA, ST = TS, UB = BU, TV = VT.

(5) There exists a number k ϵ (0,1) such that

( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( ) )

For all x, y ϵ X, ϵ (0,2) and t > 0, then A, B, S, T, U and V have a unique

common fixed point in X.

Proof: Since U(X) AB(X) for any there exists a point such that

.Since V(X) ST(X) for this point we can choose a point

such that . Inductively we can find a sequence as follows

and For n=0, 1, 2,

3,…(5)

for all t > 0 and with q ϵ (0,1) we have

( )

( )

≥ ( ) ∗

( ) ∗ ( ) ∗

( ) ∗

( ) )

≥ ( ) ∗

( ) ∗ ( ) ∗

( ) ∗

( ) )

≥ ( ) ∗

( ) ∗ ( )

Since t-norm is continuous ,letting q 1,we have

( ) ≥

( ) ∗ ( )

Similarly, ( ) ≥

( ) ∗ ( )

Similarly, ( ) ≥

( ) ∗ ( )

Therefore, ( ) ≥

( ) ∗ ( ) for all n ϵ N.

Consequently, ( ) ≥

( ) ∗ ( ) for all n ϵ N.

Repeated application of this inequality will imply that, i ϵ N

( ) ≥

( ) ∗ ( )≥....≥

( ) ∗ ( )


Since ( ) → 1 as i → ∞, it follows that

( ) ≥

( ) for all n ϵ N.

Consequently, ( ) ≥

( ) for all n ϵ N.

Therefore by lemma (2.12) , is a Cauchy sequence in X. Since X is

complete, converges to a point z ϵ X. Since

and are sub sequences of , they also converge to the point z.

i.e., as n→∞ and →z, →z and →z.

Case I: Since AB is continuous. In this case we have ABU →ABz,

ABAB →ABz. Also (U, AB) is semi compatible, we have UAB →ABz .

Step I: Let x = AB , y = with in (5), we get

( ) ≥

( ) ∗ ( ) ∗

( ) ∗

( ) ∗

( ) )

Letting n→∞, we get

( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( )

( ) ≥ ( ) which implies that ABz = z.

Step II: By putting x = z, y = with in (5) we get

( ) ≥

( ) ∗ ( ) ∗ ( ) ∗

( ) ∗

( )

Letting n→∞ we get, ( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( )

( ) ≥ ( ) which implies that Uz = z.

Step III: Now putting x = Bz and y = with in (5), we get

( ) ≥

( ) ∗ ( ) ∗ ( ) ∗

( ) ∗

( )

Since BU = UB, AB = BA, so we have U(Bz) = B(Uz) = Bz and AB(Bz) =

B(ABz) = Bz. Letting n→∞, we get

( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( )

( ) ≥ ( ) which implies that Bz = z. Therefore ABz = z = Bz = Uz

Thus Az = z = Bz = Uz. Hence z is the common fixed point of A , B and U.

Case II: Since ST is continuous . In this case we have

ST →STz , ST(ST )→STz.


Also (V, ST) is semi-compatible, we have VST →STz.

Step IV: Let x = , y = ST with in (5), we get

( ) ≥

( ) ∗ ( ) ∗

( ) ∗

( ) ∗

( )


( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( )

( ) ≥ ( ). So we get STz = z

Step V: By putting x = , y =z with in (5), we get

( ) ≥ ( ) * ( ) * ( )* (t)* (t)


( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( )

( ) ≥ ( ) which implies that Vz = z.

Step VI: Now putting x = , y = Tz with in (5), we get

( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗

( )

Since VT = TV ; ST = TS, so we have VTz = TVz = Tz and ST(Tz) = T(STz) =

Tz. Letting n→∞, we get

( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( )

( ) ≥ ( ) which implies that z = Tz = Vz = STz.Therefore z = Tz = Vz

= Sz Hence z is the common fixed point of T, V and S. So combining the above

result we get Az = Bz = Uz = Sz = Tz = Vz = z, i.e., z is a common fixed point in

X. Hence A, B, S, T, U and V have a common fixed point in X .

Uniqueness : Let w be another common fixed point of A, B, S, T, U and V.

Then putting x = z and y = w and in (5), we get

( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( )

( ) ≥ ( ) ∗ ( ) ∗ ( ) ∗ ( ) ∗ ( )

( ) ≥ ( ) which implies that z = w.

Therefore z is a unique common fixed point of A, B, S, T, U and V in X.


Acknowledegment: The Authors are thankful to the anonymous referees for his

valuable suggestions for the improvement of this paper.

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Received: June 15, 2014

common fixed point theorem in probabilistic …...fixed point theory in pm spaces can be considered...

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