Vol 2, No 1 (2011)

(1)

International Journal of Mathematical Archive-2(1), Jan. - 2011, Page: 163 -168

Available online through

_{www.ijma.info}

International Journal of Mathematical Archive- 2 (1), Jan. – 2011 163

GENERALIZED R-WEAKLY COMMUTING MAPPINGS IN NON- ARCHIMEDEAN MENGER SPACE

Piyush Tripathi and Ajita pathak

Amity University

(Deptt. of Amity School of Engineering and Technology) Viraj Khand 5 Gomti Nagar Lucknow (U.P.) India Email*: [email protected], [email protected]

(Received on: 30-12-10; Accepted on: 08-01-11)

---

ABSTRACT:

W

e have initiated the concept of generalized R- weakly commuting mappings in non- Archimedean probabilistic metric space for the first time. In fact Sessa initiated weakly commuting mappings in a metric space, Singh and Pant defined the same idea in more general setting of probabilistic metric space. Common fixed point theorems have been obtained by using the concept of generalized R- weakly commuting mappings in non- Archimedean Menger probabilistic metric space in the present paper. --- 1. INTRODUCTION:

The existence of fixed point theorems for mappings in probabilistic metric space have been obtained by Lee [5], Istratescu [4], Hadzic [3], Singh and Pant [6], [7] Chang [1], and Cho, Sik, Ha and Chang [2] etc. S Sessa [8] has given the concept of weakly commuting mappings and has obtained some fixed point theorems in metric space.

Using the above said concept of Sessa [S 5] was generalized by Singh and Pant [6] by introducing commuting mappings in probabilistic metric space. The above mentioned idea forced us to introduce the definition of generalized R- weakly commuting mappings in non- Archimedean probabilistic metric space. As a consequence of this definition we have obtained some common fixed point theorems in non- Archimedean Menger probabilistic metric space.

NOTE: Through out this paper we consider (X,F,t) a complete non- Archimedean Menger probabilistic metric space of type

C

_g introduced in [2].

DEFINITION [6]: Two self-mappings f and g on a probabilistic metric

space X will be called weakly commuting if

F

_{fgp gfp}_,

( )

x

≥

F

_{fp gp}_,

( )

x

∀

p

∈

X

and

x

>

0

.

DEFINITION: Two self mappings f and g on a non- Archimedean probabilistic metric space X will be called generalized R- weakly commuting if there exist a real number

R > 0 such that

g F

(

_{fgp gfq}_,

(

Rx

))

≤

g F

(

_{fp gq}_,

( )) ,

x

∀

p q

∈

X

and

x

>

0

.

The following lemma proved by Cho, Sik, Ha and Chang [2].

LEMMA [2]: Let

{

p

_n

}

be a sequence in X such that

1

,

lim

( ) 1

0

n n

n→∞

F

p p+

x

= ∀

x

>

. If the sequence

{

p

n

}

is not a Cauchy sequence in X, then there exist

ε

0

>

0 ,

t

0

>

0

, two

sequence { } and { }

m

_i

n

_i of positive integers such that

,, 0 0 1,, 0 0

( )

>

1 and

as

( ) (

( ))

(1

) and (

( )

(1

)

mi ni mi ni

i i i

p p p p

i m

n

i

ii g F

t

g

ε

g F

t

g

ε

−

+

→ ∞

>

−

≤

−

REMARK: If sequence{

p

_n

}

is not a Cauchy sequence in X and

lim

(

)

0

1

, +

=

∞

→

g

F

pnPn

x

n , then

1 1

0 , 0 , 0 , 0

(1

)

(

( ))

(

( ))

(

( ))

mi ni mi mi mi ni

p p p p p p

g

ε

g F

t

g F

t

g F

t

− −

−

<

≤

+

.

Taking

i

→ ∞

,

lim

(

_,

( ))

₀

(1

₀

)

mi ni

i→∞

g F

p p

t

=

g

−

ε

(1)

---

(2)

Jan.-2011, Page: 163-168 Again,

1 1 1 1

, 0 , 0 , 0 , 0

(

( ))

(

( ))

(

( ))

(

( ))

mi ni mi mi mi ni ni ni

p p p p p p p p

g F

t

g F

t

g F

t

g F

t

− − − −

≤

+

and

1, 1 0 1, 0 , 0 , 1 0

(

( ))

(

( ))

(

( ))

(

( ))

mi ni mi mi mi ni ni ni

p p p p p p p p

g F

t

g F

t

g F

t

g F

t

− − − −

≤

+

. Taking

i

→ ∞

,

1, 1 0 0

lim

(

( ))

(1

)

mi ni

i

g F

p p

t

g

ε

− −

→∞

=

−

(2) Also,

1

1, 1 0 1, 0 , 0 , 0

(

( ))

(

( ))

(

( ))

(

( ))

ni mi ni ni ni mi mi mi

p p p p p p p p

g F

t

g F

t

g F

t

g F

t

+

− + −

≤

+

and

1 1 1 1

, 0 , 0 , 0 , 0

(

( ))

(

( ))

(

( ))

(

( ))

mi ni ni ni ni mi mi mi

p p p p p p p p

g F

t

g F

t

g F

t

g F

t

− − + +

≤

+

.

Taking

i

→ ∞

and from (1), (2) we have

1, 1 0 0

lim

(

( ))

(1

)

ni mi

i

g F

p p

t

g

ε

− +

→∞

=

−

(3)

At last

1 1

, 0 , 0 , 0

(

( ))

(

( ))

(

( ))

mi ni mi ni ni ni

p p p p p p

g F

t

g F

t

g F

t

− −

≤

+

and

1 1

, 0 , 0 , 0

(

( ))

(

( ))

(

( ))

mi ni mi ni ni ni

p p p p p p

g F

t

g F

t

g F

t

−

≤

+

− .

As

i

→ ∞

and from (1), (2)

We have

1

, 0 0

lim

(

( ))

(1

)

mi ni

i→∞

g F

p p₋

t

=

g

−

ε

(4)

1.2 LEMMA [2]: If

ϕ

:[0, )

∞ →

[0, )

∞

is a function such that

ϕ

is upper semi continuous from the right and

ϕ

( )

t

<

t

for all

t > 0, then

( ) For all

0, lim

n

( )

0, where

n

( ) is the -th iteration of ( ).

n

a

t

≥

_→∞

ϕ

t

=

ϕ

t

n

ϕ

t

( ) If { }

b

t

_n is a non-decreasing sequence of real numbers and

t

_n₊₁

≤

ϕ

( ),

t

_n

n

=

1, 2,...

then

lim

_n_→∞

t

_n

=

0

. In particular, if

t

≤

ϕ

( ) for all

t

≥

0, then

t

=

0

.

2 MAIN RESULTS:

2.1 THEOREM: Suppose (X,F,t) be a complete non- Archimedean Menger space and

f

,

h

:

X

→

X

be two R- weakly commuting mappings satisfying,

(1)

∀

x

>

0

,

g

(

F

_fp_,_fq

(

x

))

≤

ϕ

(

g

(

F

_hp_,_hq

(

x

)

)), where

ϕ

:[0, )

∞ →

[0, )

∞

is a function such that

ϕ

is upper semi continuous from the right and

ϕ

( )

t

<

t

for all t > 0.

(2)

f

(

X

)

⊂

h

(

X

)

and f is continuous.Then f and h have unique common fixed point.

PROOF: Let

p

₀

∈

X

, choose

p

₁

∈

X

such that

f

(

p

_o

)

=

h

(

p

₁

)

, because

f

(

X

)

⊂

h

(

X

)

, so we can construct a sequence

{

p

_n

}

such that

f

(

p

_n

)

=

h

(

p

_n₊₁

)

, n = 1,2……….

Now,

(

))

(

1

, +

x

≤

ϕ

F

g

n n fp

fp

g

(

F

hpn,hpn+1

(

x

)

))

g

(

F

fpn,fpn+1

(

x

))

≤

ϕ

(

g

(

F

fpn−1,fpn

(

x

)

)), so by lemma 1.2

1

,

lim

(

( ))

0

n n

n→∞

g F

fp fp+

x

=

.

{

fp

_n

}

is a Cauchy sequence. If {

fp

_n} is not a Cauchy sequence then

∃

∈

₀

>

0 ,

t

₀

>

0

and set of positive integers

{

m

_i

},

{

n

_i

}

and then we can apply the above remark for the sequence {

fp

_n}.We get

, 1 0 0

lim

(

( ))

(1

)

mi ni

i→∞

g F

fp fp+

t

=

g

−

ε

and

lim

(

₀

))

(

1

₀

),

,

=

−

ε

∞

→

g

F

fpmi fpn_i

t

g

(3)

Jan.-2011, Page: 163-168

, 1 , 1 , 1

, 1 1,

0 0 0

0 0 0 0

(

( ))

( (

( )))

(

( ))

(

( ))

(

( )) taking

we get (1

)

(1

)

mi ni mi ni mi ni

mi ni mi ni

fp fp hp hp hp hp

fp fp fp fp

g F

t

g F

t

g F

t

g F

t

g F

t

i

g

ϕ

ε

+ + +

+ −

=

<

→ ∞

−

<

−

Which is not possible so{

fp

_n

}

is a Cauchy sequence.

Since (X,F,t) is complete,

fp

_n

→

z

∈

X

,

hp

_n

→

z

. Due to continuity of f,

ffp

_n

→

fz

and

fhp

_n

→

fz

. Since f and h are R- weakly commuting so,

(

_,

(

))

n n fhp hfp

g F

Rx

≤

g

(

F

_,

(

x

)

n nhp

fp ,

∀

x

>

0

, taking

n

→

∞

we get,

fz

hfp

Rx

F

g

x

F

g

Rx

F

g

_fz_hfp _z_z _fz_hfp _n

n

(

))

≤

(

))

=

0 ,

∀

>

0 (

(

))

=

0 →

(

_, _, _,

z is a common fixed point of f and h, first we prove that z = fz otherwise

g

(

F

_,

(

x

))

≤

ϕ

(

n

n ffp

fp

g

(

F

hpn,hfpn

(

x

)

)),

∀

x

>

0

, taking

n

→

∞

we get

g

(

F

_z_,_fz

(

x

))

≤

ϕ

(

g

(

F

_z_,_fz

(

x

)

))

<

(

g

(

F

_z_,_fz

(

x

)

)), which is not possible so z = fz

Again, since

f

(

X

)

⊂

h

(

X

)

so

∃

z

₁

∈

X

such that

z

=

fz

=

hz

₁, then

(

))

(

1

,

x

≤

ϕ

F

g

_ffp _fz

n

g

(

F

hfpn,hz1

(

x

)

)), taking

n

→

∞

we get

(

))

(

1

,

x

≤

ϕ

F

g

_fz_fz

g

(

F

_fz_,_fz

(

x

)

)) = 0 so

fz

=

fz

₁

=

hz

₁

=

z

Now,

g

(

F

_fz_,_hz

(

Rx

))

=

ϕ

(

)

1 1,

Rx

F

g

_fhz _hfz ))

(

)

0

1

1,

=

≤

g

F

_fz _hz

x

g

(

F

_fz_,_hz

(

Rx

)

=

0 fz

=

hz

=

z

Therefore z is a common fixed point of f and h. For uniqueness suppose

z

₁

,

z

₂ are two common fixed point of f and h. Then

)).

(

))

(

))

(

)))

(

))

(

))

(

2 1 2

1 2

1,

x

g

F

,

x

g

F

,

x

g

F

,

x

g

F

,

x

g

F

,

x

F

g

_z _z

=

_fz _fz

≤

ϕ

_hz _hz _z _z

≤

ϕ

_z _z

<

_z _z

Which is not possible so

z

₁

=

z

₂

f

,

h

:

X

→

X

be two R- weakly commuting mappings satisfying:

(1)

g

(

F

_fp_,_fq

(

x

))

≤

ϕ

(

max{

g

(

F

_fp_,_hp

(

x

))

,

g

(

F

_fq_,_hq

(

x

)),

g

(

F

_hp_,_hq

(

x

)),

g

(

F

_fp_,_fq

(

x

))

}) (2)

f

(

X

)

⊂

h

(

X

)

and f is continuous.

Then f and h have unique common fixed point.

PROOF: Since

f

(

X

)

⊂

h

(

X

)

, so we can construct a sequence

{

p

_n

}

such that

f

(

p

_n

)

=

h

(

p

_n₊₁

)

,

n = 1,2… First we show that {

fp

_n} is a Cauchy sequence,

(

))

(

1

, +

x

≤

ϕ

F

g

n n fp

fp max{

g

(

F

fpn,hpn

(

x

))

,

g

(

F

fpn+1,hpn+1

(

x

)),

g

(

F

hpn,hpn+1

(

x

)),

g

(

F

fpn,fpn+1

(

x

))

})

(

))

(

1

, +

x

≤

ϕ

F

g

n n fp

fp max{

g

(

F

fpn,fpn−1

(

x

))

,

g

(

F

fpn+1,fpn

(

x

)),

g

(

F

fpn−1,fpn

(

x

)),

g

(

F

fpn,fpn+1

(

x

))

})

(

))

(

1

, +

x

≤

ϕ

F

g

n n fp

fp max{

g

(

F

fpn,fpn−1

(

x

))

,

g

(

F

fpn,fpn+1

(

x

))

}).

If

≤

−

(

))

(

1

,

x

F

g

n n fp

fp

g

(

F

fpn,fpn+1

(

x

))

then

g

(

F

fpn,fpn+1

(

x

))

≤

ϕ

(

g

(

F

fpn,fpn+1

(

x

)))

,so by lemma 15.1.2

0 ))

(

lim

1

, +

=

∞

→

g

F

fpn fpn

x

n , again if

g

(

F

fpn,fpn−1

(

x

))

≥

g

(

F

fpn,fpn+1

(

x

))

then

g

(

F

fpn,fpn+1

(

x

))

≤

ϕ

(

g

(

F

fpn,fpn+1

(

x

)))

so again by lemma 5.1.2

lim

(

))

0

1

, +

=

∞

→

g

F

fpn fpn

x

(4)

Jan.-2011, Page: 163-168

{

fp

_n} is a Cauchy sequence. Suppose {

fp

_n}is not a Cauchy sequence then

∃

∈

₀

>

0 ,

t

₀

>

0 }

{

},

{

m

_i

n

_i and then we can apply the above remark for the sequence {

fp

_n}. We get,

lim

(

₀

))

(

1

₀

),

,

=

−

ε

∞

→

g

F

fpmi fpn_i

t

g

i

lim

(

₀

))

(

1

₀

),

1

, +

=

−

ε

∞

→

g

F

t

g

i n i m fp fp i

Now,

ϕ

≤

+

(

))

(

_, ₀

1

t

F

g

i m i n P

p Max {

g

(

F

fpni+1,hPni+1

(

t

0

),

g

(

F

fpmi,hPmi

(

t

0

),

g

(

F

hpni+1,hPmi

(

t

0

),

g

(

F

fpni+1,fPmi

(

t

0

)

}

≤

ϕ

+

(

))

(

_, ₀

1

t

F

g

i m i n P

p Max{

g

(

F

fpni+1,fPni

(

t

0

),

g

(

F

fpmi,fPmi−1

(

t

0

),

g

(

F

fpni,fPmi−1

(

t

0

),

g

(

F

fpni+1,fPmi

(

t

0

)

}

Taking

i

→

∞

we get,

g

(

1 −

ε

₀

)

≤

ϕ

(max {

0 ,

g

(

1 −

ε

₀

),

g

(

1 −

ε

₀

)

})

≤

ϕ

(

g

(

1 −

ε

₀

))

i.e.

g

(

1 −

ε

₀

)

<

g

(

1 −

ε

₀

)

which is not possible hence {

fp

_n}is a Cauchy sequence.

fp

_n

→

z

∈

X

,

hp

_n

→

z

ffp

_n

→

fz

and

fhp

_n

→

fz

. Since f and h are R- weakly commuting so, as theorem 2.1

hfp

_n

→

fz

.

z is a common fixed point of f and h, first we prove that z = fz otherwise,

, , , , ,

(

( ))

(max{ (

( )), (

( ))})

n n n n n n n n n n

fp ffp fp hp ffp hfp hp hfp fp ffp

g F

x

≤

ϕ

g F

x

g F

x

g F

x

g F

x

.

Taking

n

→ ∞

we get,

g

(

F

_z_,_fz

(

x

)

≤

ϕ

(max{

g

(

F

_z_,_z

(

x

)),

g

(

F

_fz_,_fz

(

x

)),

g

(

F

_z_,_fz

(

x

)),

g

(

F

_z_,_fz

(

x

))})

i.e.

g

(

F

_z_,_fz

(

x

))

≤

ϕ

(

g

(

F

_z_,_fz

(

x

)))

<

g

(

F

_z_,_fz

(

x

))

, which is not possible so z = fz.

Again, since

f

(

X

)

⊂

h

(

X

)

so

∃

z

₁

∈

X

such that

z

=

fz

=

hz

₁. Again we show that

z

=

fz

=

hz

₁=

fz

₁, otherwise

))}),

(

)),

(

)),

(

)),

(

(max{

))

(

1 1

1 , , , ,

,

x

g

F

x

g

F

x

g

F

x

g

F

x

F

g

_ffp _fz _ffp _hfp _fz _hz _hfp _hz _ffp _fz

n n

n

≤

ϕ

n

→

∞

1 1 1 1 1

, , , , ,

(

_{fz fz}

( ))

(max{ (

_{fz fz}

( )), (

_{fz hz}

( )), (

_{fz fz}

( )), (

_{fz fz}

( ))}),

g F

x

≤

ϕ

g F

x

g F

x

g F

x

g F

x

since

z

=

fz

=

hz

₁,

))

(

)))

(

))

(

))})

(

,

0 (max{

))

(

1 1

1 , , , ,

,

x

g

F

x

g

F

x

g

F

x

g

F

x

F

g

_z _fz

≤

ϕ

_z_fz _z _fz

≤

ϕ

_z _fz

<

_z _fz .

z

=

fz

=

hz

₁=

fz

₁.

Again,

z

hz

fz

Rx

F

g

x

F

g

Rx

F

g

Rx

F

g

(

_fz_,_hz

(

))

=

(

_fhz_,_hfz

(

))

≤

(

_fz_,_hz

(

))

=

0 (

_fz_,_hz

(

)

=

0 =

=

1

1 1

1

Therefore z is a common fixed point of f and h. For uniqueness suppose

z

₁

,

z

₂are two common fixed point of f and h, then

))})

(

)),

(

)),

(

)),

(

(max{

))

(

))

(

2 1 2

1 2

2 1

1 2

1,

x

g

F

,

x

g

F

,

x

g

F

,

x

g

F

,

x

g

F

,

x

F

g

_z _z

=

_fz _fz

≤

ϕ

_fz _hz _fz _hz _hz _hz _fz _fz

))

(

)))

(

))

(

))})

(

,

0 (max{

))

(

2 1 2

1 2

1,

x

g

F

,

x

g

F

,

x

g

F

,

x

g

F

,

x

F

g

z z

≤

ϕ

z z z z

≤

ϕ

z z

<

z z

z

₁

=

z

₂

f

,

φ

,

h

:

X

→

X

are three mappings satisfying

)

1 (

The pairs

(

f

,

φ

)

and

(

f

,

h

)

are generalized R –weakly commuting. (2)

f

(

X

)

⊂

φ

(

X

),

f

(

X

)

⊂

h

(

X

)

and f is continuous.

(3)

g

(

F

_fp_,_fq

(

x

))

≤

ϕ

(

max{

g

(

F

_fp_,_hq

(

x

))

,

g

(

F

_fp_,φ_q

(

x

)),

g

(

F

_hp_,_fp

(

x

)),

g

(

F

φ_p_,_fp

(

x

))

}),

∀

x

>

0

(4 ) If

∃

p

,

q

∈

X

such that

φ

p

=

hq

=

t

then

φ

q

=

hp

=

t

(5)

Jan.-2011, Page: 163-168

PROOF: Since

f

(

X

)

⊂

φ

(

X

),

f

(

X

)

⊂

h

(

X

)

so we can construct a sequence{

p

_n

}

by using (4) as

...

,...

2 ,

1 ,

1

=

−

p

hp

n

fp

_n

φ

_n _n . First we show that {

fp

For x > 0,

1 1 1

, , , , ,

(

( ))

(max{ (

( )), (

( ))})

n n n n n n n n n n

fp fp fp hp fp p hp fp p fp

g F

x

ϕ

g F

x

g F

_φ

x

g F

x

g F

_φ

x

+

≤

+ +

1 1 1 1

, , , ,

(

( ))

(max{0, (

( ))})

(

( ))

( (

( )))

n n n n n n n n

fp fp fp fp fp fp fp fp

g F

x

ϕ

g F

x

g F

x

ϕ

g F

x

+

≤

− +

≤

−

so by lemma 5.1.2

lim

(

))

0

1

, +

=

∞

→

g

F

fpn fpn

x

n for all x > 0 .

{

fp

_n} is a Cauchy sequence. Suppose {

fp

_n} is not a Cauchy sequence then

∃

∈

₀

>

0 ,

t

₀

>

0 }

{

},

{

m

_i

n

_i and then we can apply the above remark for the sequence {

fp

_n}.

We get,

lim

(

₀

))

(

1

₀

),

,

=

−

ε

∞

→

g

F

fpmi fpn_i

t

g

i

lim

i→∞

g F

(

fpmi,fpni+1

( ))

t

0

=

g

(1

−

ε

0

)

.

Again,

))})

(

)),

(

)),

(

)),

(

(max{

))

(

₀ ₀ ₀ ₀ ₀

, ,

1 , 1

, 1

,

t

g

F

t

g

F

t

g

F

t

g

F

t

F

g

i m i m i

m i m i

n i m i

n i

m fp fp hp fp p hp fp p fp

fp +

≤

ϕ

+ φ + φ

(

₀

))

(max{

(

₀

)),

0 })

, 1

,

t

g

F

t

F

g

i n i m i

n i

m fp fp fp

fp +

≤

ϕ

, taking

i

→

∞

we get,

)

1 (

))

1 (

(

)

1 (

−

ε

₀

≤

ϕ

g

−

ε

₀

<

g

−

ε

₀

g

,

which is not possible hence {

fp

_n

→

z

∈

X

,

hp

_n

→

z

,

φ

p

_n

→

z

ffp

_n

→

fz

,

fhp

_n

→

fz

and

fz

p

f

φ

_n

→

. Since The pairs

(

f

,

φ

)

and

(

f

,

h

)

are generalized R –weakly commuting so as above theorem 5.2.1

fz

hfp

_n

→

and

φ

fp

_n

→

fz

.

z is a common fixed point of f,

φ

and h, first we prove that z = fz otherwise,

0 ))}),

(

)),

(

)),

(

)),

(

(max{

))

(

F

_,

x

≤

g

F

_,

x

g

F

_,

x

g

F

_,

x

g

F

_,

x

∀

x

>

g

n n n

n n

n ffp fp hfp fp fp hp fp p fp

fp

ϕ

φ φ

Taking

n

→

∞

we get,

g

(

F

_z_,_fz

(

x

))

≤

ϕ

(max{

g

(

F

_z_,_fz

(

x

)),

g

(

F

_z_,_z

(

x

))}),

∀

x

>

0

i.e.

g

(

F

_z_,_fz

(

x

))

≤

ϕ

(

g

(

F

_z_,_fz

(

x

)))

<

g

(

F

_z_,_fz

(

x

)),

∀

x

>

0

, which is not possible so z = fz.

Since

f

(

X

)

⊂

φ

(

X

),

f

(

X

)

⊂

h

(

X

)

so

∃

z

₁

,

z

₂

∈

X

such that

z

=

fz

=

hz

₁and

z

=

φ

z

₂

=

fz

i.e. by the given condition (4)

z

=

φ

z

₂

=

hz

₁

=

φ

z

₁

=

hz

₂

=

fz

. Again we show that

z

=

fz

=

hz

₁

=

φ

z

₂

=

fz

₁

=

fz

₂, for this

0 ))}),

(

)),

(

)),

(

)),

(

(max{

))

(

_, _, _, _, _,

1 1

1

x

≤

g

F

x

g

F

x

g

F

x

g

F

x

∀

x

>

F

g

n n n

n n

n

n fz ffp hz ffp z hfp ffp fp ffp

ffp

ϕ

φ φ

Taking

n

→

∞

, we have

0 ))

(

))

0 (

))

(

))})

(

)),

(

(max{

))

(

1 1

1 , , , ,

,

x

≤

g

F

x

g

F

x

g

F

x

≤

g

F

x

=

F

g

_fz_fz

ϕ

_fz_z _fz_fz _fz_fz

ϕ

_fz_fz

i.e.

z

=

fz

=

fz

₁ similarly we can prove

z

=

fz

=

fz

₂i.e.

z

=

fz

=

hz

₁

=

φ

z

₂

=

fz

₁

=

fz

₂

Since The pairs

(

f

,

φ

)

and

(

f

,

h

)

are generalized R –weakly commuting so as above theorem

z

=

fz

=

hz

and

z

fz

z

=

φ

, i.e. z is a common fixed point of f,

φ

and h. For uniqueness suppose

z

₁

,

z

₂are two common fixed point of f,

φ

and

(6)

Jan.-2011, Page: 163-168

))})

(

)),

(

)),

(

)),

(

(max{

))

(

))

(

1 1 1

1 2

1,

x

g

F

,

x

g

F

,

x

g

F

,

x

g

F

,

x

g

F

,

x

F

g

_z _z

=

_fz _fz

≤

ϕ

_fz _hz _fz _φ_z _hz _fz _φ_z _fz

2 1 ,

,

(

))

(max{

(

)),

0 })

(

))

0 (

2 1 2

1 2

1

x

g

F

x

g

F

x

z

F

g

_z _z

≤

ϕ

_z _z _z _z

=

2.4 THEOREM: Suppose (X,F,t) be a complete Menger probabilistic metric space. Suppose

f

,

φ

_i

:

X

→

X

are n +1 mappings (i = 1, 2 ….n), satisfying,

, , ,

( )

(max{

( ),

( )} ,

and

0, (

1, 2... )

i i

fp fq fp q p fp

a

F

x

≤

ϕ

F

_φ

x F

_φ

x

∀

p q

∈

X

x

>

i

=

n

( ) ( )

b f X

⊂

φ

_i

( ),

X i

=

1, 2,... and is continuous.

n

f

1 2 1 2 1 2

( ) The pairs ( , ),

1, 2,... are generalized R - weakly commuting mappings.

(d) If

,

,...

such that

...

then ( ( ))

( (

)),...

( (

))

i

n n n

c

f

i

n

u u

u

X

u

t

u

t

φ

φ σ

=

∃

∈

=

where

σ

:{ ,

x x

₁ ₂

... }

x

_n

→

{ ,

x x

₁ ₂

... }

x

_n are any mapping. Then there exist a unique fixed point of mappings

f

,

φ

_i, (i = 1, 2……….n).

Proof is similar to as theorem 2.3.

REFERENCERS:

1. Chang S. S. Basic theory and application of probabilistic metric spaces (1),Appl. Math. And Mech. (Eng. Edition) 9 (1988), No.2, 123-133.

2. Cho Y. J., Ha K. S. and Chang S. S. Common fixed point theorems for compatible mappings of type (A) in non-Archimedean Menger PM-spaces, Math. Japon. 46 (1) (1997), 169-179.

3. Hadži O. A generalization of the contraction principle in probabilistic metric spaces, Review Research Prirod. Mat. Fak. Novi Sad 10 (1980), 13-21.

4. Istr tescu V. I. On generalized complete probabilistic metric spaces, Sem. Probab. Metric Space, Univ. Of Timisoara, No. 25 (1974).

5. Lee B. S. Fixed point theorems in probabilistic metric spaces,Math. Japon. 45(1) (1997), 89- 96. 6. Singh S. L. and Pant B. D. Coincidence theorems,Math. Japon. 31 (1986), 783-789.

7. Singh S. L. and Pant B. D. Common fixed points of weakly commuting mappings on non –Archimedean Menger spaces, The Vikram Math. J. 6 (1985), 27-31.