Vol 6, No 11 (2015)

Available online throug

ISSN 2229 – 5046

INITIAL VALUE PROBLEM OF THIRD ORDER RANDOM DIFFERENTIAL INCLUSION

D. S. PALIMKAR*

Department of Mathematics,

Vasantrao Naik College, Nanded-431603(M.S.), India.

G. K. PATIL

Department of Mathematics,

ACS College, Shankar Nagar Tq.: Biloli Dist.: Nanded (M.S.), India.

(Received On: 15-10-15; Revised & Accepted On: 16-11-15)

ABSTRACT

I

n this paper, we prove the existence of solution for the initial value problem of third order random differential inclusion through random fixed point theory. We claim that our result is new to the theory of random differential inclusions.

Keywords: Random differential inclusion, random solution, caratheodory condition. AMS Mathematics Subject Classifications: 60H25, 47H10.

1. STATEMENT OF THE PROBLEM

Let

(

Ω_,,

µ

)

be a complete

σ

-finite measure space and let

R

be the real and and let _{J =}

[

_0,T

]

be a closed and bounded interval in

R

. Consider the initial value problem of third order ordinary random differential inclusion (in short RDI),

( )

(

)

(

)

(

)

(

)

0

( ) (

)

1

( ) (

)

2

( )

''' ,

, ( , ),

, ( , ),

, ( , ),

a.e.

0,

, ' 0,

, '' 0,

x

t

F t x t

G t x t

H t x t

t

J

x

q

x

q

x

q

ω

ω ω

ω ω

ω ω

ω

ω

ω

ω

ω

ω

∈

+

+

_{∈ }



=

=

=

_

(1.1) for all

ω

∈Ω

, where

0

,

1

,

2

:

q q q

Ω →

R

is measurable, _{, :}

( )

p

F G J× × Ω →R  R .

By a random solution for the RDI (1.1) we mean a measurable function _{x:Ω →AC J R}

(

_,

)

satisfying for each

ω

∈Ω

_, x'''

₍

t,

ω

₎

=v₁

₍

t,

ω

₎

+

v

2

(

t

,

ω

)

+

v

3

(

t

,

ω

)

∀ ∈

t

J

and

x

(

0,

ω

)

=

q

0

( ) (

ω

, ' 0,

x

ω

)

( )

(

)

( )

1

, '' 0,

2

q

ω

x

ω

q

ω

=

=

for some measurable functions 1

(

)

1, 2, 3: ,

v v v Ω →L J R with

(

)

(

)

1

,

, ( ,

),

v

t

ω

∈

F t x t

ω ω

a.e.

t

∈

J

and

( ) (

)

2

,

, ( , ),

v t

ω

∈

G t x t

ω ω

,

v t

3

( )

,

ω

∈

H t x t

(

, ( , ),

ω ω

)

a.e.

t

∈

J

, where

_{AC J R}

(

_,

)

is the space of absolutely continuous real-valued functions on J. The RDI (1.1) has not been discussed earlier in the literature. In this paper, we prove the existence of solution for the initial value problem of third order random differential inclusion (1.1) through random fixed point theory.

2. AUXIALARY RESULTS

we need the following lemma to prove the result.

Lemma 2.1(Dhage [4]): Let

(

Ω_,,

µ

)

be a complete

σ

−

_finite

measure space and let X be separable Banach space. If

_,

_,

_:

( )

cl

F G H

Ω →



X

are three multi-valued random operators, then the sum F+G+H defined by

(

F

+

G

+

H

)( )

ω

=

F

( )

ω

+

G

( )

ω

+

H

( )

ω

is again a multi-valued random operator on

Ω

.

© 2015, IJMA. All Rights Reserved 88 Corollary 2.2: Let

(

Ω

_,



_,

µ

)

be a complete

σ

-finite measure space and let

[

_{a b,}

]

be a random interval in a separable Banach space X. Let

_,

_,

_:

[

_,

]

( )

cl

A B C

Ω ×

a b

→



X

be three right monotone increasing multi-valued random operators satisfying for each

ω

∈Ω_,

(a)

_A

( )

ω

is a multi-valued contraction, (b)

_B

( )

ω

,

C

( )

ω

are completely continuous, and

(c)

_A

( )

ω

_x

+

_B

( )

ω

_x

+

_C

( )

ω

_x

∈

[

_{a b}

_,

]

for all

x

∈

[ ]

a b

,

.

Furthermore, if the cone K in X is normal, then the random operator inclusion

_x

∈

_A

( )

ω

_x

+

_B

( )

ω

_x

+

_C

( )

ω

_x

has a random solution in

[ ]

a b

,

.

3. EXISTENCE RESULTS

We seek the solutions of RDI (1.1) in the function space _{C J R}

(

_,

)

of continuous real-valued functions on J. Define a norm

|| ||

⋅

in

_{C J R}

(

_,

)

by _{sup ( )}

t J

x x t

∈

= and the order relation

≤

in _{C J R}

(

_,

)

by

_x

≤

_y

⇔

_y

− ∈

_x

_K

_,

where the cone K in _{C J R}

₍

_,

₎

is defined by

(

)

{

, | ( ) 0 for all

}

.

K = x∈C J R x t ≥ t∈J

For any measurable function_{x:Ω →C J R}

(

_,

)

, let

(

)

(

)

(

)

{

}

1 1

( )

( )

,

( ,

)

,

, ( ,

),

a.e.

.

F

S

_ω

x

=

v

∈



Ω

L J R

v t

ω

∈

F t x t

ω ω

t

∈

J

_(3.1)

This is our set of selection functions. The integral of the random multi-valued function F is defined as

(

)

{

(

)

1

}

( )

0

, ( ,

),

0

,

:

( )

t t

F

F s x s

ω ω

ds

=

v s

ω

ds v

∈

S

_ω

x

∫

∫

.

We need the following definitions in the sequel.

Definition 3.1: A multi-valued function

_:

( )

cp

F J

× × Ω →

R



R

is called Caratheodory if for each

ω

∈Ω

, (i) _{t F t x}

(

_{, ,}

ω

)

is measurable for each

x

∈



, and

(ii)

_x



_{F t x}

(

_{, ,}

ω

)

is an upper semi-continuous almost everywhere for

t

∈

J

.

Again, a Caratheodory multi-valued function F is called

L

1-Caratheodory if for each real number r > 0 there exists a measurable function 1

(

)

: ,

r

h Ω →L J R such that for each

ω

∈Ω

(

, ,

)

sup

{

:

(

, ,

)

}

_r

(

,

)

a.e.

F t x w

_

=

u u

∈

F t x

ω

≤

h

t

ω

t

∈

J

for all

x

∈

R

with

x

≤

r

Furthermore, a Caratheodory multi-valued function F is called

L

1_X-Caratheodory if (iii) there exists a measurable function

_h

_:

Ω →

_L

1

(

_{J R}

_,

)

such that

(

, ,

)

(

,

)

a.e.

F t x

ω

≤

h t

ω

t

∈

J

 for all

x

∈

R

,and the function h is called a growth function of

F on

J

× × Ω

R

.

Definition 3.2: A multi-valued function _:

( )

cp

F J× × Ω →R P R is called s-Caratheodory if for each

ω

∈Ω

,

(i)

_{( ,}

_t

ω

₎



_{F t x}

(

_{, ,}

ω

)

is measurable for each

x

∈

R

, and

(ii)

x



F t x

(

, ,

ω

)

is an Hausdorff continuous almost everywhere for

t

∈

J

.

Furthermore, a s-Caratheodory multi-valued function F is called s-

L

1- Caratheodory if (iii)for each real number r > 0 there exists a measurable function _: 1

(

_,

)

r

h Ω →L J R such that for each

w

∈Ω

(iv)

_{F t x}

(

_{, ,}

ω

)

=

_sup

{

_{u u}

_:

∈

_{F t x}

(

_{, ,}

ω

)

}

≤

_{h t}

_r

( )

_,

ω

_a.e.

_t

∈

_J

Then we have the following lemmas which are well-known in the literature.

Lemma 3.1 (Lasota and Opial [8]): Let E be a Banach space. If dim (E) <

∞

and _:

( )

c

F J×E× Ω →_ E is 1

L

- Caratheodory, then 1

( )

( )

0

F

S

_ω

x

≠ /

for each

x

∈

E

.

Lemma 3.2 (Lasota and Opial [8]): Let E be a Banach space, F a Caratheodory multi-valued operator with

1

( )

0

f

S

_ω

≠ /

, and 1

(

)

(

)

:

L

J E

,

→

C J E

,



be a continuous linear mapping. Then the composite operator

(

)

(

)

1

( )

:

,

,

( ,

)

F bd cl

S

_ω

C J E

→

C J E







is a closed graph operator on

_{C J E}

(

_,

)

×

_{C J E}

(

_,

)

.

Lemma 3.3 (Caratheodory theorem [5]): Let E be a Banach space. If

_:

( )

p

F J

×

E

→



E

is s-Caratheodory, then the multi-valued mapping

( , )

t x



F t x t

(

, ( )

)

is jointly measurable for each measurable function

x J

:

→

E

. We consider the following set of hypotheses in the sequal.

( )

A

1 The multi-valued mapping

(

t

,

ω

)



F t x

(

, ,

ω

)

is jointly measurable for all

x

∈

R

.

( )

A

2

F t x w

(

, ,

)

is closed and bounded for each

( )

t

,

ω

∈ × Ω

J

and

x

∈

R

.

( )

A3 F is integrably bounded on

J

× Ω ×

R

.

( )

A

4 There is a function

(

(

)

)

1

,

L

J R

,

∈

Ω





such that for each

ω

∈Ω

,

d

H

(

F t x

(

, ,

ω

) (

,

F t y

, ,

ω

)

)

≤



(

t

,

ω

)

x

−

y

a.e.

t

∈

J

for all

x y

,

∈

R

.

( )

A

5 The multi-valued mapping

( )

1 ( )

F

x



S

_ω

x

is right monotone increasing in _x∈_{C J R}

(

_,

)

almost everywhere for

t

∈

J

. and

( )

B

1 The multi-valued mapping

(

t

,

ω

)



G t x t

(

, ( ,

ω ω

),

)

,

(

_t

_,

ω

)



_{H t x t}

(

_{, ( ,}

ω ω

_),

)

are jointly measurable for all measurable

x

:

Ω →

C J R

( ,

)

.

( )

B

2

G t x

(

, ,

ω

)

_,

H t x

(

, ,

ω

)

are closed and bounded for each

(

t

,

ω

)

∈ × Ω

J

and

x

∈

R

.

( )

B

3 G ,

H

are 1

L

-Caratheodory.

( )

B4 The multi-valued mapping

( )

1 ( ) F

x



S

_ω

x

is right monotone increasing in

_x

∈

_{C J R}

(

_,

)

almost everywhere for

t

∈

J

.

( )

B

5 RDI (1.1) has a strict lower random solution a and a strict upper random solution b with

a

≤

b

on

J

× Ω

.

MAIN RESULT

Theorem 3.1: Assume that the hypotheses

( )

( )

1 5

A − A and

( )

( )

1 5

B − B hold. If

( )

1

1,

L

ω

<



then the RDI (1.1) has a random solution in [a, b] defined on J× Ω

Proof: Let

_X

=

_{C J R}

(

_,

)

. Define a random order interval

[ , ]

a b

in X which is well defined in view of hypothesis

( )

B

5 . Now the RDI (1.1) is equivalent to the random integral inclusion,

(

)

( )

( )

( )

(

(

)

)

2 2

0 1 2 ₀

(

)

'''

,

,

,

,

2

2

t

t

s

x

t

ω

∈

q

ω

+

q

ω ω

+

q

ω

ω

+

∫

−

F s x s

ω ω

ds

(

)

2

0

(

)

, ( ,

),

,

.

2

t

t

s

G s x s

ω ω

ds

t

J

−

+

_∫

∈

(3.2) for all

ω

∈Ω

. Define three multi-valued operators _{A B C, , :}Ω ×

[

_{a b,}

]

→

( )

_X



 by

( )

{

(

) ( )

( )

(

)

1

}

0 ₀ 1 1 ( )

,

|

,

t

,

,

_F

( )

A

ω

x

= ∈

u



Ω

X

u t

ω

=

q

ω

+

∫

v s

ω

ds v

∈

S

_ω

x

(

1

)

1

S

F( )ω

( )

x

© 2015, IJMA. All Rights Reserved 90

And

( )

{

(

) ( )

( )

( )

(

)

1

}

0 1 ₀ 2 2 ( )

,

|

,

t

,

,

_F

( )

B

ω

x

= ∈

u



Ω

X

u t

ω

=

q

ω

+

q

ω ω

+

∫

v

s

ω

ds v

∈

S

_ω

x

(

1

)

2

S

F( )ω

( )

x

=



_

And

( )

(

) ( )

( )

( )

( )

2

(

)

1

0 1 2 ₀ 3 3 ( )

,

|

,

,

,

( )

2

t

F

C

ω

x

=





u

∈

Ω

X

u t

ω

=

q

ω

+

q

ω ω

+

q

ω

ω

+

v s

ω

ds v

∈

S

_ω

x









∫



(

1

)

3

S

F( )ω

( )

x

=





Where,

(

1

(

)

)

(

(

)

)

1, 2, 3: Ω,L J R, → Ω,C J R,

     are continuous operators defined by

(

)

( )

(

)

1 1

,

0 ₀ 1

,

,

t

v t

ω

=

q

ω

+

∫

v s

ω

ds



( )

( )

( )

(

)

2 2

,

0 1 ₀ 2

,

,

t

v t

ω

=

q

ω

+

q

ω ω

+

∫

v

s

ω

ds



(3.5) And

( )

( )

( )

( )

( )

2

3 3

,

0 1 2 ₀ 3

,

.

2

t

v t

ω

=

q

ω

+

q

ω ω

+

q

ω

ω

+

∫

v s

ω

ds



(3.6)

Clearly, the operators _A

( )

ω

and

B

( )

ω

are well defined in view of hypotheses

( )

3

A

and

( )

3

B . We will show that

( )

A

ω

,_B

( )

ω

and

_C

( )

ω

satisfy all the conditions of Corollary 2.2.

Step-I: First, we show that A is closed valued multi-valued random operator on _{Ω ×}

[

_{a b,}

]

. Observe that the operator

( )

A

ω

is equivalent to the composition 1 1SF w( )

 of two operators on 1

(

)

, ,

L J R where

(

1

)

1: Ω,L(J R, ) → X

  is the continuous operator. To show A

( )

ω has closed values, it then suffices to prove that the composition operator 1

1



S

F(ω)



has closed values on [a, b]. Let

_x

∈

_{[ , ]}

_{a b}

be arbitrary and let

{ }

n

v be a

sequence in 1 ( ) F

S _ω (x) converging to v in measure. Then, by the definition of

S

F1( )ω , vn

(

t,

ω

)

∈F t x t

(

,

(

,

ω ω

)

,

)

a.e. for

t

∈

J

.

Since

_{F t x t}

(

_{, ( , ),}

ω ω

)

is closed,

_{v t}

(

_,

ω

)

∈

_{F t x t}

(

_{, ( ,}

ω ω

_),

)

a.e. for

_t

∈

_J

. Hence, 1 ( )

F

v

∈

S

_ω (x). As a result, 1

( )( ) G

S _ω x is a closed set inL J R1

(

,

)

for each

ω

∈Ω

. From the continuity of



₁

,

it follows that

(

1

)

1



S

F(ω)

( )

x



is a closed set in X. Therefore,

A

( )

ω

is a closed-valued multi-valued operator on [a, b] for each

ω

∈Ω

. Next, we show that

A

( )

ω

is a multi-valued random operator on [a, b]. First, we show that the multi-valued mapping

(

)

1

( )

,

x

S

_{F ω}

( )

x

ω



is measurable. Let

(

1

₍

₎

)

, ,

f∈ Ω L J R be arbitrary. Then we have

(

)

{

1

}

1

( ) ( )

,

_F

( )

in f

( )

( )

_L

:

_F

( )

d f S

_ω

x

=

f

ω

−

h

ω

h

∈

S

_ω

x

{

(

)

(

)

}

( ) 0

inf

T

f t

,

ω

h t

,

ω

dt h

:

S

_Fω

( )

x

=

_∫

−

∈

{

(

)

(

)

}

0 inf , : , ( , ),

T

f t

ω

z z F t x t

ω ω

dt

=

_∫

− ∈

(

) (

)

0

( , ) ,

, ( , ),

.

T

d f t

ω

F t x t

ω ω

dt

=

_∫

But by hypothesis

( )

1

A , the mapping

( )

t

,

ω



F t x

(

, ,

ω

)

is measurable, and by (A₄), the mapping

( , , )

x



F t x

ω

is Hausdorff continuous. Hence by Caratheodory theorem, the map

_{( ,}

_t

ω

₎



_{F t x t}

(

_{, ( ,}

ω ω

_),

)

is measurable for all measurable function_x:Ω →_{C J R( , )}. It is known that the multi-valued mapping

(

, ( , ,

)

zd z F t x

ω

is continuous, Hence the mapping multi-valued mapping

(

t x

, , ,

ω

z

)

_{d z F t x}

(

_,

(

_{, ,}

ω

)

)

measurable. Hence we deduce that the mapping

(

_{t x, ,}

ω

_,f

)



_{d f t}

(

_{( , ),}

ω

_{F t x t}

_{( , ( , ),}

ω ω

)

is measurable from

(

)

1

,

J

×

X

× Ω ×

L

J R

into R+. Now the integral is the limit of the finite sum of measurable functions, and so,

(

1

)

( ) , _F ( )

d f S _ω x is measurable. As a result, the multi-valued mapping (.,.) 1 ( )( ) F

S _⋅

Define a function

φ

on

J

× × Ω

X

by

(

)

(

1

)

(

)

1 ( ) ₀

, , _F ( )( ) t , ( , , ) .

t x S _ω x t F s x s ds

φ

ω

=  =

_∫

ω ω

We shall show that

φ

(

_{t x}

_{, ,}

ω

)

is continuous in t in the Hausdorff metric on

R

. Let

{ }

t

_n be a sequence in converging to

t

∈

J

.

Then we have

(

)

(

)

(

, ,

,

, ,

)

H n

d

φ

t

x

ω φ

t x

ω

(

)

(

)

(

)

[ ]

( ) (

)

(

)

(

)

[ ]

( ) (

)

(

)

(

)

[ ]

( )

( )

(

)

[ ]

( )

( ) (

)

0 0

[0, ] 0,

[0, ] 0,

[0, ] 0,

[0, ] 0,

,

,

,

,

, ( ,

),

, ( ,

),

,

( )

, ( ,

),

, ( ,

),

,

( )

, ( ,

),

, ( ,

),

,

0

n

n

n

n

n

t t

H

H _J t _J t

H _J t _J t

t t

J

t r

t J

d

F s x s

ds

F s x s

ds

d

s F s x s

ds

s F s x s

ds

d

s F s x s

ds

s F s x s

ds

s

s

F s x s

ds

s

s h

s

ds

ω ω

ω ω

ω ω

ω ω

ω ω

ω ω

ω ω

ω





=



_



_





=

=

=

−

=

−

→

∫

∫

∫

∫

∫

∫

∫

∫



















as

n

→ ∞

.

Thus the multi-valued mapping

_t



φ

(

_{t x}

_{, ,}

ω

)

is continuous, and hence, and by Lemma 3.3, the mapping

(

)

(

)

0

, ,

, ( ,

),

t

t x

ω



∫

F s x s

ω ω

ds

is measurable. Consequently,

_A

( )

ω

is a random multi-valued operator on [a, b]. Similarly, it can be shown that

( )

B ω _,C

( )

ω is a closed-valued multi-valued operator on [a, b] and the mapping

(

t x

, ,

ω

)



∫

₀t

G s x s

(

, ( , ),

ω ω

)

ds

is measurable. Again, since the sum of two measurable multi-valued functions is measurable, the mapping

(

t x

, ,

ω

)



( )

( )

( )

2

0 1 2

2

q

ω

+q

ω ω

+q

ω

ω

+

(

)

0 , ( , ), t

G s x sω ω ds

∫

is measurable.

Step-II: Next we show that _A

( )

ω

is a multi-valued contraction on X. Let

x y

,

∈

X

be any two element and let

( )( )

1 .

u ∈A ω x Then

u

₁

∈

X

and

(

)

(

)

1 1

0

,

,

t

u t

ω

=

∫

v

s

ω

ds

for some 1

1 F( )

( )

v

∈

S

_ω

x

. Since

(

) (

)

(

, ( ,

),

,

, ( ,

),

)

(

,

)

( ,

)

( ,

) ,

H

d

F t x t

ω ω

F t y t

ω ω

≤



t

ω

x t

ω

−

y t

ω

We obtain that there exists a

_w

∈

_{F t y t}

(

_,

(

_,

ω ω

)

_,

)

such that

(

)

(

) (

)

(

)

1

,

,

,

,

.

v t

ω

−

w

≤



t

ω

x t

ω

−

y t

ω

Thus, the multi-valued operator U defined by

(

)

1

( )( )

( )( )

( )

,

_F

,

U t

ω

=

S

_ω

y

t

∩

K

ω

t

where

( )( )

{

(

)

(

)

(

)

}

1

,

,

( ,

)

,

K

ω

t

=

w v

t

ω

−

w

≤



t

ω

x t

ω

−

y t

ω

has nonempty values and is measurable. Let

v

₂ be a measurable selection function for U. Then there exists

(

)

(

)

2

,

,

,

v

∈

F t y t

ω ω

with

(

)

(

)

(

) (

)

(

)

1

,

2

,

,

,

,

,

© 2015, IJMA. All Rights Reserved 92

Define

(

)

(

)

2

,

₀ 2

,

.

t

u

t

ω

=

∫

v

s

ω

ds

It follows that

( )( )

2

u

∈

A

ω

y

and

(

)

(

)

(

)

(

)

1 2 1 2

0 0

,

,

,

,

t t

u

t

ω

−

u

t

ω

≤

∫

v

s

ω

ds

−

∫

v

s

ω

ds

(

)

(

)

(

) (

)

(

)

( )

1

( )

( )

1 2

0

0

,

,

,

,

,

.

t

t

L

v

s

v

s

ds

t

x s

y s

ds

x

y

ω

ω

ω

ω

ω

ω

ω

ω

≤

−

≤

−

≤

−

∫

∫





Taking the supremum over t, we obtain

( )

( )

( )

1

( )

( )

1 2 _L

u

ω

−

u

ω

≤



ω

x

ω

−

y

ω

.

From this and the analogous inequality obtained by interchanging the role of

x

and

y

we obtain

(

( )( ), ( )( )

)

( ) 1 ( ) ( )

H L

d A

ω

x A

ω

y ≤ 

ω

x

ω

−y

ω

,

for all

x y

,

∈

X

.

This shows that

_A

( )

ω

is multi-valued random contraction on X, since

( )

1

1

L

ω

<



for each

ω

∈Ω

.

Step-III: Next, we show that _B

( )

ω is completely continuous for each

ω

∈Ω

. First, we show that

B

( )(

ω

[ , ]

a b

)

is compact for each

ω

∈Ω

. Let

{

y

_n

( )}

ω

be a sequence in _{B( ) [ , ]ω}

(

_{a b}

)

for some

ω

∈Ω

. We will show that

( )

{

y

n

ω

}

has a cluster point. This is achieved by showing that

{

y

n

( )

ω

}

is uniformly bounded and equi-continuous sequence in X.

Case-I: First, we show that

{

( )

}

n

y ω is uniformly bounded sequence. By the definition of

{

y

n

( )

ω

}

, we have a

( )

1 ( )

( )

n G

v

ω

∈

S

_ω

x

for some

_x

∈

[

_{a b}

_,

]

such that

(

)

0

( )

1

( )

2

( )

2

(

)

0

,

,

,

.

2

t

n n

y

t

ω

=

q

ω

+

q

ω ω

+

q

ω

ω

+

∫

v

s

ω

ds

t

∈

J

Therefore,

(

)

( )

( )

( )

(

)

( )

( )

( )

(

(

)

)

( )

( )

( )

( )

1

2

0 1 2

0 2

0 1 2

0 2

0 1 2

,

,

2

,

,

,

2

2

t

n n

t

n

r _L

y

t

q

q

q

v

s

ds

q

q

q

F s x

s

ds

q

q

q

h

ω

ω

ω

ω ω

ω

ω

ω

ω

ω ω

ω

ω ω

ω

ω

ω ω

ω

ω

≤

+

+

+

≤

+

+

+

≤

+

+

+

∫

∫



for all

t

∈

J

,

where

r

=

a

( )

ω

+

b

( )

ω

. Taking the supremum over t in the above inequality yields,

( )

( )

( )

( )

( )

1

2

0 1 2

2

n r _L

y

ω

≤

q

ω

+

q

ω ω

+

q

ω

ω

+

h

ω

which shows that

{

( )

}

n

y ω is a uniformly bounded sequence in Q

( )(

ω

[ , ]a b

)

.

Next we show that

{

( )

}

n

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

0 0

, , , ,

,

,

, , ,

t

n n n n

t n t

r

y t y v s ds v s ds

v s ds

h s ds

p t p

τ

τ

τ

ω

τ ω

ω

ω

ω

ω

ω

τ ω

− ≤ −

≤

≤

≤ −

∫

∫

∫

∫

where

(

)

(

)

0

,

,

.

t r

p t

ω

=

∫

h

s

ω

ds

From the above inequality, it follows that

y

n

(

t

,

ω

)

−

y

n

(

τ ω

,

)

→

0

as

t

→

τ

. This shows that

{

( )

}

n

y

ω

is an equi-continuous sequence in

_B

( )(

ω

_{[ , ] .}

_{a b}

)

Now

{

( )

}

n

y

ω

is uniformly bounded and equi-continuous for each

ω

∈Ω

, so it has a cluster point in view of Arzela-Ascoli theorem. As a result,

_B

( )

ω

is a compact multi-valued random operator on [a, b] and similarly for

_C

( )

ω

.

Next we show that _B

( )

ω ,

_C

( )

ω

are upper semi-continuous multi-valued random operator on [a, b]. Let

{

( )

}

n

x

ω

be a sequence in X such that

( )

( )

_.

n

x

ω

→

x

_∗

ω

Let

{

( )

}

n

y

ω

be a sequence such that y_n

( )

ω

∈C

( )

ω

x_n and

( )

( )

.

n

y

ω

→y∗

ω

we will show that

y

∗

( )

ω

∈

C

( )

ω

x

∗

.

Since

y

_n

( )

ω

∈

C

( )

ω

x

_n there exists a

( )

1

( )

( )

n G n

v

ω

∈

S

_ω

x

such that

(

)

0

( )

1

( )

2

( )

2

(

)

0

,

,

,

2

t

n n

y

t

ω

=

q

ω

+

q

ω ω

+

q

ω

ω

+

∫

v

s

ω

ds

t

∈

J

.

We must prove that there is a

( )

1

( )

( )

G

v

_∗

ω

∈

S

_ω

x

_∗ such that

(

)

0

( )

1

( )

2

( )

2

(

)

0

,

,

,

.

2

t

y

_∗

t

ω

=

q

ω

+

q

ω ω

+

q

ω

ω

+

∫

v

_∗

s

ω

ds

t

∈

J

Consider the continuous linear operator

(

1

(

)

)

: Ω,L J R,

 

→



(

Ω

,

C J R

( ,

)

)

defined by

(

)

(

)

0

,

,

,

.

t

v t

ω

=

∫

v s

ω

ds

t

∈

J



Now

( )

( )

( )

2

( )

( )

( )

2

0 1 2 0 1 2

( ) (

( ))

( ) (

)

0

2

2

n

y

ω

q

ω

q

ω ω

q

ω

ω

q

ω

y

_∗

ω

q

ω

q

ω ω

q

ω

ω

a

n

s



 



−

+

+

−

−

+

+

→

→ ∞



 





 



From lemma 3.2, it follows that

( )

1

G

S _ω



 is a closed graph operator. Also, from the definition of



, we have

(

)

( )

( )

( )

2

(

1

)

( )

0 1 2 ( )

,

2

n G n

y

t

ω

−

_



q

ω

+

q

ω ω

+

q

ω

ω



_

∈

S

_ω

x









Since

( )

( )

_,

n

y

ω

→

y

_∗

ω

there is a point

( )

1

( )

( )

F

v

_∗

ω

∈

S

_ω

x

_∗ such that

(

)

0

( )

1

( )

2

( )

2

(

)

0

, , , .

2

t

y_∗ t

ω

=_q

ω

+q

ω ω

+q

ω

ω

_+ v_∗ s

ω

ds t∈J

 

∫

© 2015, IJMA. All Rights Reserved 94 Step-VI: Next, we show that

A

( )

ω

is a right monotone increasing and multi-valued random operator on [a, b] into itself for each

ω

∈Ω

. Let

_{x y}

_,

∈

[

_{a b}

_,

]

be such that

x

≤

y

.

Since

( )

5

A

holds, we have that 1

( )

1

( )

( ) ( )

F F

S

_ω

x

≤

S

_ω

y

. Hence _A

( )( )

ω

_{x ≤}i _A

( )( )

ω

_y .Similarly, _B

( )( )

ω

_{x ≤}i _B

( )( )

ω

_y , _C

( )( )

_{ω x ≤}i_C

( )( )

_{ω y} From

( )

5

B , it follows that

_a

≤

_A

( )

ω

_a

+

_B

( )

ω

_a

+

_C

( )

ω

_a

and _A

( )

ω

_b+B

( )

ω

_b+C

( )

ω

_b≤b for all

ω

∈Ω

. Now

A

( )

ω

,

( )

B

ω

and

_C

( )

ω

are monotone increasing, so we have for each

ω

∈Ω

,

( )

( )

( )

i

( )

( )

i

( )

( )

( )

a

≤

A

ω

a

+

B

ω

a

+

C

ω

a

≤

A

ω

x

+

B

ω

x

≤

A

ω

b

+

B

ω

b

+

C

ω

b

≤

b

for all

_x

∈

[ ]

_{a b}

_,

. Hence,

_A

( )

ω

_x

+

_B

( )

ω

_x

+

_C

( )

ω

_x

∈

[

_{a b}

_,

]

for all

x

∈

[ ]

a b

,

.

Thus, the multi-valued random operators _A

_{( )}

_ω ,_B

( )

_ω and _C

( )

ω

satisfy all the conditions of Corollary 2.2 and hence the random operator inclusion _x∈A

( )

ω

_x+B

( )

ω

_x+C

( )

ω

_x has a random solution. This implies that the RDI (1.1) has a random solution on

J

× Ω

. This completes the proof.

REFERENCES

[1]. Aubin J. and A. Cellina, Differential Inclusions, Springer-Verlag, 1984.

[2]. B. C. Dhage, Monotone increasing multi-valued random operators and differential inclusions, Nonlinear Funct. Anal. and Appl. 12(3), 2007 .

[3]. B. C. Dhage, Some algebraic fixed point theorems for multi- valued operators with applications, Discuss. Math. Diff. Incl. Control and Optim. 26, 2006, 5-55.

[4]. B.C.Dhage, Monotone method for discontinuous differential inclusions, Math. Sci. Res. J. 8 (3), 2004, 104-113.

[5]. S. Hu and N.S. Papageorgiou, Hand Book of Multi-valued Analysis Vol.-I, Kluwer Academic Publisher, Dordrechet, Boston, London, 1997.

[6]. S. Itoh, Random fixed point theorems with applications to random differential equations in Banach spaces, J. Math. Anal. Appl. 67, 1979, 261-273.

[7]. K. Kuratowskii and C. Ryll-Nardzeuskii, A general theorem on selectors, Bull. Acad. Pol. Sci. Ser. Math. Sci. Astron. Phy. 13, 1965, 397-403.

[8]. A. Lasota and Z. Opial, An application of Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phy. 13, 1965, 781-786.

[9]. D.S.Palimkar, Existence Theory for Second Order Random Differential Inclusion, International Journal of Advances in Engineering, Science and Technology, Vol. 2, No. 3, 2012.

Source of support: Nil, Conflict of interest: None Declared