Vol 3, No 4 (2012)

(1)

Available online through

ISSN 2229 – 5046

International Journal of Mathematical Archive- 3 (4), April – 2012 1606

RANDOM FIXED POINT OF MULTIVALUED OPERATERS IN POLISH SPACE

**Smriti Mehta*, Dr. AD Singh**

*

, Dr. PL Sanodia

*

and Dr. Vanita Ben Dhagat

Truba Institute of engineering and Information technology Bhopal, India *Govt. M.V.M. College Bhopal, India

Email:

(Received on: 09-02-12; Accepted on: 27-02-12)

________________________________________________________________________________________________

ABSTRACT

In this paper we introduced some more results of random fixed point for one, two and four mappings in Polish Spaces

by using rational contraction.

Keywords: Polish Space, Random multivalued operator, Random fixed point.

AMS Subject Classification: 47H10, 54H25.

________________________________________________________________________________________________

1. INTRODUCTION:

Random fixed point theorem for contraction mappings in polish spaces and random fixed point theorems are of fundamental importance in probabilistic functional analysis. Their study was initiated by the prague school of probabilistic with work of spacek (15) and Hans (5,6).For example survey are refer to Bharucha-Reid (4), Itoh(8) proved several random fixed point theorems and gave their applications to random differential equations in Banach Spaces. Badshah and Gagrani proved existence of common Random fixed points of two Random multivalued operators on Polish spaces

The result of Hardy and Rogers [7] further extended by Wong[16], showing that two self mappings of S and T on a complete metric space satisfying a contractive type condition have a common fixed point. Recently, Beg and Azam [1] further extended it to the case of a pair of multivalued mappings satisfying a more general contractive type condition. In this section we gave a further generalized result of Beg and Shahzad [3] by using fractional inequality

2. PRELIMINARIES

Let (X, d) be a polish space that is a saparable complete metric space and ( Ω, ∑) be a measurable space with ∑ sigma algebra of subsets of Ω. Let 2X be the family of all non-empty subsets of X and CB(X) the family of all nonempty closed subsets of X. A mapping

T: Ω → 2X is called measurable if, for each open subset C of X,

1 1

( )

,

( ) {

:

( )

}.

T

−

C

∈∑

wh

re T

e

−

C

=

ω

∈Ω

T

ω

∩ ≠

C

ϕ

A mapping ξ:Ω→ X is called a measurable selector of a measurable mapping T: Ω → 2X if ξ is measurable and

ξ(ω) ∈ T(ω) for each ω∈ Ω. A mapping f : Ω ×X → X is said to be a random operator if, for each fixed x ∈ X, f(., x) : Ω → X is measurable. A measurable mapping ξ ∶ Ω → X is a random fixed point of a random multivalued operator T: Ω ×X → CB(X) (f : Ω ×X → X) if ξ(ω) ∈ T �ω,ξ(ω)� [ξ(ω) = f �ω,ξ(ω)�]for each ω∈Ω. Let T : Ω ×X

→CB(X) be a random operator and {ξn} a sequence of measurable mappings

ξ

_n

:

Ω

→

X

.

The sequence {ξn} is said

to be asymptotically T-regular if

d

( ( ), ( ,

ξ ω

_n

T

ω ξ ω

_n

( )))

→

0.

3. MAIN RESULT:

Theorem 3.1: Let X be Polish Space. Let T, S: Ω × X → CB(X) be two continuous random multivalued operators. If there exists measurable mappings a, b, c, d, e: Ω → (0, 1), such that

________________________________________________________________________________________________

(2)

IN POLISH SPACE/ IJMA- 3(4), April-2012, Page: 1606-1611

[

]

[

]

[

]

[

]

( ( , ), ( , ))

( ) ( , )

( )

( , ( , ))

( )

( , ( , ))

( )

( , ( , ))

( , ( , )) ( , ( , ))

( )

( , ( , ))

( )

( , )

H S

x T

y

a

d x y

b

d x S

x

d y T

y

c

d x T

y

d y S

x

d

d y S

x

d x S

x

d x S

x d y T

y

e

d y T

y

d x T

y

f

d x y

ω

≤

+





+

_

_





,

, , , ,

( )

( ) 1,

for each x y X

∈

ω

∈Ω

and a b c d e R with a

∈

+

ω

+

b

ω

+

c

ω

+

d

ω

+

e

ω

<

Then there exists a common fixed point of S and T (Here H represents the hausdroff metric on CB(X) induced by the metric d)

Proof:Let ξ0: Ω→X be an arbitrary measurable mapping and choose a measurable mapping

1

:

X such that

1

( )

S

( ,

0

( ))

for each

.

Then for each

ξ

Ω→

ξ ω

∈

ω ξ ω

ω

∈Ω

ω

∈Ω

[

]

[

]

[

]

0 1 0 1

0 0 1 1

0 1 1 0

1 0 0 0

1 1

( ( ,

( )), ( , ( )))

( ) ( ( ), ( ))

( )

( ( ), ( ,

( )))

( ( ), ( , ( )))

( )

( ( ), ( , ( )))

( ( ), ( ,

( )))

( )

( ( ), ( ,

( )))

( ( ), ( ,

( )))

( )

( ( ), ( , ( )))

H S

T

a

d

b

d

S

d

T

c

d

T

d

S

d

S

d

S

e

d

T

ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

≤

+

[

+

₀ ₁

]

0 0 1 1

0 1

( ( ), ( , ( )))

( ( ), ( ,

( ))) ( ( ), ( , ( )))

( )

( ( ), ( ))

d

T

d

S

d

T

f

d

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω ξ ω





+

_

_





It further implies, than there exists a measurable mapping

2

:

X such that

2

( )

T

( , ( ))

1

for each

and

ξ

Ω→

ξ ω

∈

ω ξ ω

ω

∈Ω

1 2 0 1

( ( ),

( ))

( ( ,

( )), ( , ( ))

d

ξ ω ξ ω

=

H S

ω ξ ω

T

ω ξ ω

[

]

[

]

[

]

1 2 0 1

0 0 1 1

0 1 1 0

1 0 0 0

1 1 0

( ( ),

( ))

( ) ( ( ), ( ))

( )

( ( ), ( ,

( )))

( ( ), ( , ( )))

( )

( ( ), ( , ( )))

( ( ), ( ,

( )))

( )

( ( ), ( ,

( )))

( ( ), ( ,

( )))

( )

( ( ), ( , ( )))

( ( ), (

d

w

a

d

b

d

S

d

T

c

d

T

d

S

d

S

d

S

e

d

T

d

T

ξ

ω

ξ ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

≤

+

[

+

₁

]

0 0 1 1

0 1

, ( )))

( ( ), ( ,

( ))) ( ( ), ( , ( )))

( )

( ( ), ( ))

d

S

d

T

f

d

ω ξ ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω ξ ω





+

_

_





[

]

[

]

[

]

0 1 0 1 1 2

0 2 1 1 1 1 0 1

0 1 1 2

1 2 0 2

0 1

( ) ( ( ), ( ))

( ) [ ( ( ), ( ))

( ( ),

( ))]

( )

( ( ),

( ))

( ( ), ( ))

( )

( ( ), ( ))

( ( ), ( )) ( ( ),

( ))

( )

( ( ),

( ))

( ( ),

( )

( ( ), (

a

d

b

d

c

d

e

d

f

d

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ

(3)

[

]

[

]

0 1 0 1 1 2

0 1 1 2 0 1

1 2 0 1 1 2

1 2

( ) ( ( ), ( ))

( ) [ ( ( ), ( ))

( ( ),

( ))]

( )

( ( ), ( ))

( ( ),

( ))

( ) ( ( ), ( ))

( )

( ( ),

( ))

( ( ), ( )

( ( ),

( ))

( ) ( ( ),

( ))

a

d

b

d

c

d

e

d

f

d

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

=

+

0 1 1 2

[ ( )

( )

( )] ( ( ), ( ))

[ ( )

( ) 2 ( )

( )] ( ( ),

( ))

a

b

c

d

e

d

b

c

e

f

d

ω

ξ ω ξ ω

ω

ξ ω ξ ω

=

+

1 2 0 1

( )

( ( ),

( ))

( ( ), ( ))

1 ( )

( ) 2 ( )

( )

a

b

c

d

e

d

b

c

e

f

ω

ξ ω ξ ω

ω

+

≤

−

1 2 0 1

( ( ),

( ))

( ( ), ( ))

d

ξ ω ξ ω

≤

k d

ξ ω ξ ω

( )

1 ( )

( )

( ) 1

1 ( )

( ) 2 ( )

( )

a

b

c

d

e

where k

because a

b

c

d

e

b

c

e

f

ω

_ω

ω

+

=

<

+

<

−

By Beg and Shahjad [3, Lemma 3] in the same manner there exists a measurable mapping

ξ

₃

:

Ω→

X

such that

3

( )

S

( ,

2

( ))

ξ ω

∈

ω ξ ω

for each

ω

∈Ω

and

d

( ( ),

ξ ω ξ ω

₂ ₃

( ))

=

H T

( ( , ( )), ( ,

ω ξ ω

₁

S

ω ξ ω

₂

( ))

[

]

[

]

[

]

2 3 1 2 1 1 2 2

1 2 2 1

2 1 1 1

2 2 1

( ( ), ( )) ( ) ( ( ), ( )) ( ) ( ( ), ( , ( ))) ( ( ), ( , ( )))

( ) ( ( ), ( , ( ))) ( ( ), ( , ( )))

( ) ( ( ), ( , ( ))) ( ( ), (

d a d b d T d S

c d S d T

d d T d T

e d S d S

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω

ω ξ ω

ξ ω

≤ + +

+ +

+

[

+ 2

]

1 1 2 2

1 2 , ( ))) ( ( ), ( , ( ))) ( ( ), ( , ( ))) ( ) ( ( ), ( )) w

d T d S

f

d

ξ ω

ω ξ ω

ξ ω

ω ξ ω

ω

ξ ω ξ ω

 

+ _ _

 

1 2 1 2 2 3

1 3 2 2 2 2 1 2

1 2 2 3

2 3 1 3

1

( ) ( ( ),

( ))

( )[ ( ( ),

( ))

(

( ),

( ))]

( )[ ( ( ),

( ))

(

( ),

( ))]

( )[ (

( ),

( ))

( ( ),

( ))]

( ( ),

( )) (

( ),

( ))

( )[ (

( ),

( ))

( ( ),

( ))]

( )

(

a

d

b

d

c

d

e

d

f

d

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ ω ξ ω

ω

ξ

≤

+

2

( ),

ω ξ ω

( ))













2 3 1 2

(

( ),

( ))

( ( ),

( ))

d

ξ ω ξ ω

≤

k d

ξ ω ξ ω

2

0 1

(

( ),

( ))

k d

ξ ω ξ ω

≤

Similarly, proceeding in the same way, by induction we produce a sequence of measurable mapping

ξ

_n

:

Ω →

X

such that for

γ

>

0

and any

ω

∈Ω

,

ξ

2γ+1

( )

ω

∈

S

( ,

ω ξ ω ξ

2γ

( )) ,

2γ+2

( )

ω

∈

T

( ,

ω ξ

2γ+1

( ))

ω

and

d

(

ξ ω ξ

n

( ),

n+₂

( ))

ω

≤

k d

(

ξ

n−₁

( ),

ω ξ ω

n

( ))...

≤

k d

n

(

ξ ω ξ ω

₀

( ),

₁

( ))

1 1 2 1

1 1 0 1 2 1 0 1

(

( ),

( ))

(

( ),

( ))

(

( ),

( )) ...

(

( ),

( ))

[

...

] ( ( ), ( ))

[1

...

] ( ( ), ( ))

n m n n n n m m

n n m

n m n

d

k

d

k

d

ξ ω ξ ω

ξ ω ξ

ω

ξ

ω ξ

ω

ξ

ω ξ ω

ξ ω ξ ω

(4)

0 1

(

( ),

( ))

( ( ), ( ))

0 ,

.

1

n

n m

k

d

as

m n

k

ξ ω ξ ω

≤

ξ ω ξ ω

→

→∞

−

It follows that {𝜉𝜉_𝑛𝑛(𝜔𝜔)} is a Cauchy sequence and there exists a measurable mapping ξ:Ω→X such that 𝜉𝜉_𝑛𝑛(𝜔𝜔)→𝜉𝜉(𝜔𝜔) for each 𝜔𝜔 𝜖𝜖 Ω.

It further implies that 𝜉𝜉2𝛾𝛾+1(𝜔𝜔) → 𝜉𝜉(𝜔𝜔)and ξ 𝜉𝜉2𝛾𝛾+2(𝜔𝜔) → 𝜉𝜉(𝜔𝜔).

Thus we have for any 𝜔𝜔 𝜖𝜖 Ω,

2 2 2 2

( ( ), ( , ( )))

( ( ),

( ))

(

( ), ( , ( )))

d

ξ ω

S

ω ξ ω

≤

d

ξ ω ξ

γ+

ω

+

d

ξ

γ+

ω

S

ω ξ ω

2 2 2 1

( ( ),

( ))

( ( ,

( )), ( , ( ))

d

ξ ω ξ

γ+

ω

H T

ω ξ

γ+

ω

S

ω ξ ω

≤

+

2 2 2 1

2 1 2 1

( ( ), ( , ( )))

( ( ),

( ))

( ) (

( ), ( ))

( )

(

( ), ( ,

( )))

( ( ), ( , ( )))

( )

(

( ), ( , ( )))

( ( ), ( ,

( )))

( )

( ( ), ( ,

( )))

(

( ), (

d

S

d

a

d

b

d

S

d

T

c

d

T

d

S

d w d

S

d

S

γ γ

ξ ω

ω ξ ω

ξ ω ξ

ω

ξ

ω ξ ω

ω

ξ

ω

ω ξ

ω

ξ ω

ω ξ ω

ω

ξ

ω

ω ξ ω

ξ ω

ω ξ

ω

ξ ω

ω ξ

ω

ξ

ω

+ +

≤

+





+

_

+

_





+

_

+

_

+

₂ ₁

2 1

2 1 2 1

2 1

,

( )))

( )

( ( ), ( , ( )))

(

( ), ( , ( )))

(

( ), ( ,

( ))) ( ( ), ( , ( )))

( )

(

( ), ( ))

e w d

T

d

T

d

S

d

T

f

d

γ

γ γ

γ

ξ

ω

ξ ω

ω ξ ω

ξ

ω

ω ξ ω

ξ

ω

ω ξ

ω

ξ ω

ω ξ ω

ω

ξ

ω ξ ω

+

+ +

+













+

_

+

_





+

_

_









,

Letting

γ

→ ∞

we have

( ( ), ( , ( ))) 0

d

ξ ω

S

ω ξ ω

≤

( )

( , ( ))

.

Hence

ξ ω

∈

S

ω ξ ω

for

ω

∈Ω

,

Similarly for any

ω

∈Ω

2 1 2 1

( ( ), ( , ( )))

( ( ),

( ))

( ( ,

( )), ( , ( ))

d

ξ ω

T

ω ξ ω

≤

d

ξ ω ξ

γ+

ω

+

H S

ω ξ

γ+

ω

T

ω ξ ω

( ( ), ( , ( ))) 0

d

ξ ω

T

ω ξ ω

≤

( )

( , ( ))

.

Therefore

ξ ω

∈

T

ω ξ ω

for

ω

∈Ω

Theorem 3.2: Let X be Polish Space. Let Si, Tj: Ω × X → CB(X) be sequence of random multivalued operators. If there exists measurable mappings a, b, c, d, e: Ω → (0, 1), such that

[

]

( ( , ),

( , ))

( ) ( , )

( )

( ,

( , ))

( ,

( , ))

( )

( ,

( , ))

( ,

( , ))

( )

( ,

( , ))

( ,

( , ))

( ,

( , )) ( ,

( , ))

( )

( ,

( , ))

( ,

( , ))

( )

( , )

i j i j

j i i i

i j

j j

H S

x T

y

a

d x y

b

d x S

x

d y T

y

c

d x T

y

d y S

x

d

d y S

x

d x S

x

d x S

x d y T

y

e

d y T

y

d x T

y

f

d x y

ω





≤

+

_

+

_





+

_

+

_

+













+

_

+

_

+

















,

, , , ,

( )

( ) 1, ,

1, 2... ...

(5)

Proof: Similar to the proof of the theorem 4.9 putting S =Si and T = Tj.

Theorem 3.3: Let X be Polish Space. Let T: Ω × X → CB(X) be a continuous random multivalued operator. If there exists measurable mappings a, b, c, d, e: Ω → (0, 1), such that

[

]

[

]

[

]

[

]

( ( , ), ( , ))

( ) ( , )

( )

( , ( , ))

( )

( , ( , ))

( )

( , ( , ))

( , ( , )) ( , ( , ))

( )

( , ( , ))

( )

( , )

H T

x T

y

a

d x y

b

d x T

x

d y T

y

c

d x T

y

d y T

x

d

d y T

x

d x T

x

d x T

x d y T

y

e

d y T

y

d x T

y

f

d x y

ω

≤

+





+

_

_





,

, , , ,

( )

( ) 1,

for each x y X

∈

ω

∈Ω

and a b c d e R with a

∈

+

ω

+

b

ω

+

c

ω

+

d

ω

+

e

ω

<

Then there exists a common fixed point of T (Here H represents the hausdroff metric on CB(X) induced by the metric d)

Proof: Similar to the proof of the theorem 4.9 putting S = T.

Theorem 3.4: Let X be Polish Space. Let Tj: Ω × X → CB(X) be a sequence of continuous random multivalued operator. If there exists measurable mappings a, b, c, d, e: Ω → (0, 1), such that

[

]

( ( , ),

( , ))

( ) ( , )

( )

( , ( , ))

( ,

( , ))

( )

( ,

( , ))

( , ( , ))

( )

( , ( , ))

( ,

( , ))

( , ( , )) ( ,

( , ))

( )

( ,

( , ))

( ,

( , ))

( )

( , )

i j i j

j i i i

i j

j j

H T

x T

y

a

d x y

b

d x T

x

d y T

y

c

d x T

y

d y T

x

d

d y T

x

d x S

x

d x T

x d y T

y

e

d y T

y

d x T

y

f

d x y

ω





≤

+

_

+

_





+

_

+

_

+













+

_

+

_

+

















,

, , , ,

( )

( ) 1, ,

1, 2... ...

for each x y X

∈

ω

∈Ω

and a b c d e R with a

∈

+

ω

+

b

ω

+

c

ω

+

d

ω

+

e

ω

<

i j

=

n

∞

Then there exists a common fixed point of T (Here H represents the hausdroff metric on CB(X) induced by the metric d)

Proof: Similar to the proof of the theorem 4.10 putting Si = Tj.

4. BIBLIOGRAPHY

[1] I. Beg and A. Azam, J. Austral. Math. Soc. Ser. A, 53 (1992) 313-26.

[2] I. Beg and N. Shahzad, Nonlinear Anal., 20 (1993) 835-47.

[3] I. Beg and N. Shahzad, J. Appl. Math. and Stoch. Anal. 6 (1993) 95-106.

[4] A. T. Bharucha-Reid,“Random Integral Equations”, Academic press, New York, 1972.

[5] O. Hans, Reduzierende, Czech. Math. J. 7 (1957) 154-58.

[6] O. Hans, Random operator equations, Proc. 4th Berkeley Symp. Math. Statist. Probability (1960), Vol. II, (1961) 180-202.

[7] G. E. Hardy and T. D. Rogers, Canad. Math. Bull., 16 (1973) 201-6.

[8] S. Itoh, Pacific J. Math. 68 (1977) 85-90.

(6)

[10] K. Kuratowski and C. Ryll-Nardzewski, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 397-403.

[11] T. C. Lin, Proc. Amer. Math. Soc.103 (1988) 1129-35.

[12] N. S. Papageorgiou, Proc. Amer. Math. Soc. 97(1986) 507-514.

[13] B. E. Rhoades, S. Sessa, M. S. Khan and M. Swaleh, J. Austral. Math. Soc. (Ser. A) 43 (1987) 328-46.

[14] V. M. Sehgal and S. P. Singh, Proc. Amer. Math. Soc., 95(1985) 91-94.

[15] A. Spacek, Zufallige Gleichungen, Czechoslovak Math. J. 5 (1955) 462-66.

[16] C. S. Wong, Pacific J. Math., 48 (1973) 299-312.