Some Existence, Uniqueness And Stability Results Of Nonlocal Random Impulsive Integro-Di¤erential Equations

Some Existence, Uniqueness And Stability Results Of Nonlocal Random Impulsive Integro-Di¤erential Equations

Sayooj Aby Jose

, Weera Yukunthorn

, Juan Eduardo Napoles Valdes

, Hugo Leiva

Received 5 December 2019

Abstract

This paper is concerned with random impulsive integro-di¤erential equations with nonlocal conditions.

At …rst, some su¢ cient conditions which can guarantee existence and uniqueness of mild solution are derived using Banach …xed point theorem. Secondly, combining with Banach …xed point theorem with some inequality techniques, we give stability of the solution. Finally some examples are given to establish the e¤ectiveness of our results.

1 Introduction

The impulsive system has been considered to be one of the most important models in mathematical ecology, and many perfect existence as well as stability results of its modi…ed models have been obtained. For example, in order to maintain the long-term sustainable development of …shery industry, the government puts a lot of little …sh into the sea in spring and allows the …shermen to catch the adult …sh in autumn and winter, which can be described by impulsive di¤erential equations. Also we must choose the impulse perturbation coe¢ cients based on the actual situation, which may oscillate in some ranges or change irregularly.

There will be instantaneous and great changes of population density in the form of perturbations if we take into account the disturbance of environmental factors at certain time moments, which cannot be neglected.

So naturally we can introduce impulsive e¤ects into di¤erential equations (see Bainov and Simeonov [4], Lakshmikantham, Bainov and Simeonov [11] and Saker [17]).

Many authors [9, 22] have studied the existence of solutions of impulsive di¤erential equations of the form

x⁰(t) =f(t; x(t); S(t); T(t)); 0< t < T₀; t6=t_i; (1)

xt0=x0; (2)

x(t_i) =I_i(t_i); i= 1;2; : : : ; p: (3) Guo and Liu [9] also established the existence theorems of maximal and minimal solutions for (1)–(3)with strong conditions provided f is uniformly continuous. Guo and Liu [12], Liu [10] also considered the case when f does not contain integral operator S in (1) and obtained the same conclusion by using monotone iterative technique. Again recently, Liu [10] considered the special case where (1)–(3) has no impulses and Liu [10] obtained a unique solution by using monotone iterative technique with coupled upper and lower quasi-solutions whenf =f(t; x(t); S(t); T(t)). Liu [14] also obtained a similar conclusion. In [9,12,10] the assumptions thatfsatis…es some compactness-type conditions is required. But it is di¢ cult and inconvenient to verify in abstract spaces. By using the successive approximations for the evolution equation with an

Mathematics Sub ject Classi…cations: 35R12, 60H99, 35B35.

yDepartment of Mathematics, Alagappa University, Karaikudi-630 004, India

zRamanujan Centre for Higher Mathematics, Alagappa University, Karaikudi-630 004, India

xFaculty of science and technology, Kanchanaburi Rajabhat University, Kanchanaburi 71000, Thailand

{Department of Mathematics, UNNE-FACENA, Corrientes 3400, Argentina

kSchool of Mathematical Sciences and Computational, University Yachay Tech , San Miguel de Urcuqui, Imbabura- Ecuador

481

unbounded operator A, Rogovchenko [16] studied the existence and uniqueness of the classical solutions.

That is equations of the form

x⁰(t) =Ax(t) +f(t; x(t)); t >0; t6=t_i

with impulsive condition in (2) (3), where A is sectorial operator with some conditions given on the fractional operatorsA ; 0 . Liu [13] studied the existence of mild solutions of the impulsive evolution equation

x⁰(t) =Ax(t) +f(t; x(t)); 0< t < T0; t6=ti

where A is the in…nitesimal generator of C0 semigroup with the impulsive condition in (2) (3)by using semigroup theory.

Most of the published papers on impulsive di¤erential systems deals with the problems related to …xed time impulses. However, actual jumps do not always happen at …xed points but usually at random points.

Recently the properties of solutions to some di¤erential equations with random impulses have been studied [21,2,20,3,8,1].

The existence of solution for non local di¤erential equations have been extensively researched in recent years taking into account the theoretical and practical signi…cance. Byszewski initiated the nonlocal initial conditions for evolution equations [5,6]. There are many applications for nonlocal condition in physics and it is more natural than the classical initial conditionx(0) =x₀. Recently, Sayooj Aby Jose and Venkatesh Usha [19] extended the results of [5, 6] to random impulsive di¤erential equations with non local initial conditions and proved the existence of the solutions by a …xed point theorem.

There are several papers which include the study of impulsive integrodi¤erential equations involving random impulses [18,9,7]. Random impulsive integro-di¤erential equation with non local initial conditions is studied in this paper, hoping that the results obtained will contribute to the area. And it is a well known fact that, if we consider integro-di¤erential equations, in some applications, we will be able to obtain better descriptions of the phenomena under study. Thus, the main objective of this work is to present non local random impulsive integro-di¤erential equations.

This paper is summarized as follows: Section 2 includes some preliminaries. Some hypotheses are included in Section 3. The existence and uniqueness of solution of random impulsive integro-di¤erential equation with nonlocal condition is investigated in section 4. And we have used Lipschitz condition for deriving the main results, followed by stability results in section 5. In the last section two examples are discussed.

2 Preliminaries

Consider a real separable Hilbert space X and a non empty set . Assume that _k is a random variable de…ned from to D_k, whereD_k = (0; d_k)for allk2N(collection of natural numbers) and0< d_k <+1. Also fori; j= 1;2; : : : assume that ifi6=j then i and j are independent with each other. Let be a real constant. Denote< = [ ; T]. Next we consider the nonlocal random impulsive integro di¤erential equations

of the form 8

<

:

x⁰(t) =Ax(t) +f(t; x(t)) +RT

0 f1( ; x(t+ ))d ; t6= _k; t ; x( _k) =b_k( _k)x( _k); k= 1;2;3; : : : ;

x_t0+g(x) =x₀;

(4)

where Ais the in…nitesimal generator of a strongly continuous semi group of bounded linear operatorS(t) in X,f; f₁ :< X !X; b_k :D_k ! <for eachk2N, g:X !X is a given function; ₀=t₀ 2[ ; T]and

k = _{k 1}+ _k for each k2N;here t₀2 < is arbitrary real number. Obviously t0= ₀< ₁< ₂< ₃ < _k< : : :;x( _k ) = lim

t" k

x(t)

according to their path with the normkxk= sup _tjx( )jfor eacht satisfyingt2[ ; T].

Let fBt; t 0g be the simple counting process generated by f ng, that implies fBt tg, also denote Ft as the notation for the -algebra generated byfBt; t 0g. The( ; P;fF^tg)is a probability space. And

the Hilbert space of all fFtg-measurable square integrable random variables with values inX is denoted as L2=L2( ;fF^tg; X).

Let Bdenote Banach spaceB([ ; T]; L2), the family of all fF^tg-measurable random variable with the norm

k k²= sup

t2[ ;T]

Ek k²:

De…nition 1 ([15]) Let A be the in…nitesimal generator of a C0 semigroup S(t). Let u0 2 X and f 2 L¹(0; T;X)with nonlocal condition g(u). Then the functionu2C([0; T];X) is given by

u(t) =S(t)(u0 g(u)) + Z t

0

S(t s)f(s)ds; 0 t T is the mild solution of the initial value problem

u⁰(t) =Au(t) +f(t), 0< t < T;

u(0) +g(x) =u0: (5)

De…nition 2 ([15]) A semigroupfS(t);0 t <1gof bounded linear operators onX is uniformly bounded if there exists a constantK 1 such that

kS(t)k K; fort 0:

De…nition 3 For a givenT 2( ;+1), a stochastic process fx(t)2B; t Tg is called a mild solution to equation (4) in ( ; p;fF^tg), if

(i) x(t)2X isFt-adapted;

(ii)

x(t) =

+1

X

k=0

Yk i=1

bi( i)S(t t0)(x0 g(x)) + Xk i=1

Yk j=i

bj( j) Z _i

i 1

S(t s)f(s; x(s))ds

+ Z t

k

S(t s)f(s; x(s))ds+ Xk i=1

Yk j=i

b_j( _j) Z _i

i 1

S(t s) Z T

0

f₁( ; x(s+ ))d ds

+ Z t

k

S(t s) Z T

0

f₁( ; x(s+ ))d ds I_{[ k;k+1)}(t); t2[ ; T] where

Yn j=m

(:) = 1 asm > n;

Yk j=i

b_j( _i) =b_k( _k)b_{k 1}( _{k 1}): : : b_i( _i) andIA(:)is the index function, i.e.,

I_A(t) =

(1; if t2A;

0; if t =2A

3 Assumptions

In this section, we deals with some hypotheses which are used in our results.

(H1) The function f satis…es the Lipschitz condition. That is, for x; y 2 X and t T there exist constants L0;M0 0such that

Ekf(t; x) f(t; y)k² L0Ekx yk² and Ekf(t;0)k² M0:

(H₂) The function f₁ satis…es the following condition. That is, for x; y 2 X and t T there exist constants L1;M1 0such that

Ek Z T

0

f₁( ; x(t+ )) f₁( ; y(t+ ))d k² L1Ekx(t+ ) y(t+ )k²; Ekf1( ;0)k M1:

(H3) The condition maxi;k Qk

j=1kbj( j)k is uniformly bounded if, there is a constant# >0such that maxi;k

Qk

j=1kb_j( _j)k #; for all _j2D_j; j= 1;2;3; : : : :

(H4) g : X ! X satis…es the Lipschitz condition. That is, for x; y 2 X and t T; there exists a constant L 0 such that

Ekg(x) g(y)k² L kx yk²: (H5)

=K²#²maxf1; #²g(T )² L0+L1+ L

(T )² <1:

4 Existence and Uniqueness

We discuss the existence and uniqueness of the mild solution for the system (4).

Theorem 1 Assume that the hypothesis (H₁)–(H₅) hold. Then the system (4) has a unique mild solution inB.

Proof. Let T be an arbitrary numberT +1 . First we de…ne the nonlinear operator F : B ! B as follows

F x(t) =

+1

X

k=0

Yk i=1

bi( i)S(t t0)(x0 g(x)) + Xk i=1

Yk j=i

bj( j) Z _i

i 1

S(t s)f(s; x(s))ds

+ Z t

k

S(t s)f(s; x(s))ds+ Xk

i=1

Yk j=i

bj( j) Z _i

i 1

S(t s) Z T

0

f1( ; x(s+ ))d ds

+ Z t

k

S(t s) Z T

0

f1( ; x(s+ ))d ds I[ _k;_k+1)(t); t2[ ; T]:

We can prove the continuity ofF easily. Next we will show thatBis mapped intoBunder F.

kF x(t)k²

+1

X

k=0

k Yk i=1

bi( i)kkS(t t0)kkx0 g(x)k

+ Xk i=1

k Yk j=i

bj( j)k Z _i

i 1

kS(t s)f(s; x(s))kds

+ Z t

k

kS(t s)f(s; x(s))kds+ Xk i=1

k Yk j=i

b_j( _j)k Z _i

i 1

S(t s) Z T

0

f₁( ; x(s+ ))d kds

+ Z t

k

kS(t s) Z T

0

f1( ; x(s+ ))d kds I_{[ k;k+1)}(t)

2

2

+1

X

k=0

k Yk i=1

bi( i)k²kS(t t0)k²kx0 g(x)k²I[ _k; _k+1)(t)

+

+1

X

k=0

Xk i=1

k Yk j=i

b_j( _j)k Z _i

i 1

kS(t s)f(s; x(s))kds

+ Z t

k

kS(t s)f(s; x(s))kds I_{[ k; k+1)}(t)

2

+

+1

X

k=0

Xk i=1

k Yk j=i

b_j( _j)k Z _i

i 1

kS(t s) Z T

0

f₁( ; x(s+ ))d kds

+ Z t

k

kS(t s) Z T

0

f1( ; x(s+ ))d kds I_{[ k;k+1)}(t)

2

2K²max

k k

Yk i=1

bi( i)k² kx0 g(x)k²

+2K² max

i;k 1;k Yk j=i

b_i( _j)k²

2 Z t t0

kf(s; x(s))kdsI_{[ k;k+1)}(t)

2

+2K² max

i;k 1;k Yk j=i

bi( j)k

2 Z t t0

k Z T

0

f1( ; x(s+ ))d kdsI_{[ k;k+1)}(t)

2

2K²#²kx0 g(x)k²+ 2K²maxf1; #²g Z t

t0

kf(s; x(s))kds

2

+2K²maxf1; #²g Z t

t0

Z T

0 kf1( ; x(s+ ))kd ds

2

2K²#²kx0 g(x)k²+ 2K²maxf1; #²g(t t0) Z t

t0

kf(s; x(s))k²ds +2K²maxf1; #²g(t t0)

Z t t0

k Z T

0

f1( ; x(s+ ))d kds

2

: Thus we get

EkF x(t)k² 2K²#²kx₀ g(x)k²+ 2K²maxf1; #²g(T ) Z t

t0

Ekf(s; x(s))k²ds +2K²maxf1; #²g(T )

Z t t0

Ek Z T

0

f₁( ; x(s+ ))k²d ds 2K²#²kx₀ g(x)k²+ 4K²maxf1; #²g(T )L0

Z t t0

Ekx(s)k²ds +4K²maxf1; #²g(T )²M0+ 4K²maxf1; #²g(T )²M1

+4K²maxf1; #²g(T )L1

Z t t0

Ekx(s+ )k²ds:

Therefore, sup

t2[ ;T]

EkF x(t)k² 2K²#²kx₀ g(x)k²+ 4K²maxf1; #²g(T )L0

Z t t0

sup

t2[ ;T]

Ekx(s)k²ds +4K²maxf1; #²g(T )²M0+ 4K²maxf1; #²g(T )²M1

+4K²maxf1; #²g(T )L1

Z t t0

sup

t2[ ;T]

Ekx(s+ )k²ds for everyt, t T, HenceF mapsBinto B.

Next we will show that F is a contraction mapping:

kF x(t) F y(t)k²

+1

X

k=0

Yk i=1

kb_i( _i)kkS(t t₀)kk(g(x) g(y))kI_{[ k;k+1)}(t)

2

+

+1

X

k=0

Xk i=1

Yk j=i

kb_i( _j)k Z _i

i 1

kS(t s)kkf(s; x(s)) f(s; y(s))kds

+ Z t

k

kS(t s)kkf(s; x(s)) f(s; y(s))kds I_k;k+1(t)

2

+

+1X

k=0

Xk i=1

Yk j=i

kbj( j)k Z _i

i 1

kS(t s)k Z T

0 kf1( ; x(s+ )) f1( ; y(s+ ))d kds +

Z t

k

kS(t s)kk Z T

0

f1( ; x(s+ )) f1( ; y(s+ ))d kds I_{[ k;k+1)}(t)

2

K² max

k

Yk i=1

kbi( i)k² kg(x) g(y)k²

+K² max

i;k 1;

Yk j=i

kb_j( _j)k

2 Z t t0

kf(s; x(s)) f(s; y(s))kdsI_{[ k; k+1)}(t)

2

+K² max

i;k 1;

Yk j=i

kbj( j)k

2 Z t t0

k Z T

0

f1( ; x(s+ )) f1( ; y(s+ ))d kdsI_{[ k;k+1)}(t)

2

K²#²kg(x) g(y)k²+K²maxf1; #²g(t t0) Z t

t0

kf(s; x(s)) f(s; y(s))k²ds +K²maxf1; #²g(t t0)

Z t t0

k Z T

0

f1( ; x(s+ )) f1( ; y(s+ ))d k²ds:

EkF x(t) F y(t)k² K²#²Ekg(x) g(y)k² +K²maxf1; #²g(T t0)

Z t t0

Ekf(s; x(s)) f(s; y(s))k²ds +K²maxf1; #²g(T t0)

Z t t0

Ek Z T

0

f1( ; x(s+ )) f1( ; y(s+ ))d k²ds K²#²Ekg(x) g(y)k²

+K²maxf1; #²g(T )L0

Z t t0

Ekx(s) y(s)k²ds +K²maxf1; #²g(T )L1

Z t t0

Ekx(s+ ) y(s+ )k²ds:

Taking supremum overt, it follows that

kF x F yk² K²#²L kx yk²+K²maxf1; #²g(T )²L0kx yk²

+K²maxf1; #²g(T )²L1kx yk² K²#²maxf1; #²g(T )²L kx yk² kx yk²;

where =K²#²maxf1; #²g(T )²LandL=L0+L1+ L

(T )². FromH5and0< <1, we get thatF is a contraction mapping. Thus using Banach …xed point theorem we getF has a unique …xed point onB. Hence (4) has a unique mild solution.

Remark 1 Let f :< X !X , f₁ :< X !X andg :X !X satisfy the assumptions(H₁)–(H₅).

Then there exists a unique, global, continuous solutionxto (4) for any initial value(t₀; x₀)with t₀ 0and x02B.

Remark 2 The above theorem is an extension of [19, Theorem 3.1]. Theorem1gives existence and unique- ness of random impulsive integro-di¤ erential equations with nonlocal condition. This solution is practically more useful than the solution of random impulsive di¤ erential equations.

Remark 3 Assume that all hypotheses hold. Then the mild solution for the system (7) without existence of nonlocal condition and the solution is

x(t) =

+1

X

k=0

Yk i=1

bi( i)S(t t0)x0+ Xk i=1

Yk j=i

bj( j) Z _i

i 1

S(t s)f(s; x(s))ds

+ Z t

k

S(t s)f(s; x(s))ds+ Xk i=1

Yk j=i

b_j( _j) Z _i

i 1

S(t s) Z T

0

f₁( ; x(s+ ))d ds

+ Z t

k

S(t s) Z T

0

f₁( ; x(s+ ))d ds I_{[ k;k+1)}(t); t2[ ; T]:

5 Stability

Theorem 2 Let x(t) and x(t)b be solutions of the system (4) with initial value x0 g(x) and xb0 g(bx) respectively. If the assumptions (H1)–(H4) of Theorem 1are satis…ed, then the system (4) is stable in the mean square.

Proof. From assumptions,x(t)andx(t)b are two solutions of the system (4) for everyt2[ ; T]:Then x(t) bx(t) =

+1

X

k=0

Yk i=1

b_i( _i)S(t t₀)(x₀ xb₀) +

+1

X

k=0

Yk i=1

b_i( _i)S(t t₀)(g(x) g(x))b

+ Xk i=1

Yk j=i

bj( j) Z _i

i 1

S(t s) f(s; x(s)) f(s;x(s))b ds

+ Z t

k

S(t s) f(s; x(s)) f(s;bx(s)) ds

+ Xk i=1

Yk j=i

b_j( _j) Z _i

i 1

S(t s) Z T

0

f₁( ; x(s+ )) f₁( ;bx(s+ )) d ds

+ Z t

k

S(t s) Z T

0

f1( ; x(s+ )) f1( ;x(sb + )) d ds I_{[ k;k+1)}(t):

By using(H₁)–(H₄), we get kx(t) bx(t)k² 2

+1

X

k=0

Yk i=1

kb_i( _i)k²kS(t t₀)k²Ekx₀ xb₀k²I_{[ k;k+1)}(t)

+2

+1

X

k=0

Yk i=1

kbi( i)k²kS(t t0)k²Ekg(x) g(x)bk²I_{[ k;k+1)}(t)

+2E

+1

X

k=0

Xk i=1

Yk j=i

kbj( j)k Z _i

i 1

kS(t s)kkf(s; x(s)) f(s;x(s))b kds

+ Z t

k

kS(t s)kkf(s; x(s)) f(s;bx(s))kds I_{[ k; k+1)}(t)

2

+2E

+1

X

k=0

Xk i=1

Yk j=i

kbj( j)k Z _i

i 1

kS(t s)kk Z T

0

f1( ; x(s+ )) f1( ;x(sb + ))d kds

+ Z t

k

kS(t s)kk Z T

0

f₁( ; x(s+ )) f₁( ;bx(s+ )) d kds I_{[ k; k+1)}(t)

2

2K²max

k k

Yk i=1

b_i( _i)k² Ek(x₀ xb₀)k²+ 2K²max

k k

Yk i=1

b_i( _i)k² Ekg(x) g(bx)k²

+2K² max

i;k 1;

Yk j=i

kbj( j)k

2

E Z t

t0

kf(s; x(s)) f(s;bx(s))kdsI_{[ k;k+1)}(t)

2

+2K² max

i;k 1;

Yk j=i

kb_j( _j)k

2

E Z t

t0

k Z T

0

f₁( ; x(s+ )) f₁( ;x(sb + )) d kdsI_{[ k;k+1)}(t)

2

:

Taking supremum overt, it follows that sup

t2[ ;T]kx(t) bx(t)k² 2K²#²Ekx₀ xb₀k²+ 2K²#²Ekg(x) g(x)b k² +2K²maxf1; #²g(T )L0

Z t t0

sup

s2[ ;T]

Ekx(t) x(t)b k²d +2K²maxf1; #²g(T )L1

Z t t0

sup

t2[ ;T]

Ekx(t+ ) x(tb + )k²d :

Using Grownwall inequality, we get sup

t2[ ;T]kx(t) bx(t)k² 2K²#²Ekx₀ bx₀k²exp 2K²max(1; #²)(T )² L Ekx₀ bx₀k²:

where

= 2K²#² exp 2K²max(1; #²)(T )² L and

L=L0+L1+ L (T )²:

Now given" >0, choose = ^" such thatEkx₀ bx₀k²< . Then sup

t2[ ;T]

Ekx(t) bx(t)k ":

Remark 4 Random impulsive integro-di¤ erential equation with local initial condition is a special case of the system (7). So the random impulsive integro-di¤ erential equation with local initial condition is stable in mean square.

6 Example 1

Consider partial random impulsive di¤erential equations 8>

><

>>

:

z_t(t; x) =z_xx(t; x) +F₁(t; z(t; x)); t6= _k; t ; z(x; _k) =q(k) _kz(x; _k); asx24b;

z(t;0) =z(t; ) = 0;

z(t₀; x) +Pq

j=1c_jz(p_j; x) =z₀(x); 0< p₁< p₂< < p_q < T; x2@4b:

(6)

Let4b <ⁿ be a bounded domain with smooth boundary@4b,X =L²(4b), _k be random variable de…ned onD_k (0; d_k)fork2N,d_k 2(0;+1). Also assume that _k follow Erlang distribution and ifi6=j then

i and _j are independent with each other fori; j= 1;2; : : :. Here q is a function of k, _k = _{k 1}+ _k for k2N,t₀2 <⁺.

LetAbe an operator onX byAz= @²z

@x² with the domain D(A) = z2Xjzand @z

@x are absolutely continuous, @²z

@x² 2X; z= 0; z= on@4b :

Thus A generates a strongly continuous semigroup S(t) which is analytic, self adjoint and compact. Fur- thermore the operator A can be represented as

Az= X1 n=1

n²< z; z_n> z_n; z2D(A):

Herezn( ) =

q2Sin(n ); n= 1;2; : : :, forms the orthonormal set of eigenvectors of A. Also for everyz2X;

S(t)z=P₁

n=1e^{( n2t)}< z; zn> zn;which holdskS(t)k e^{( 2(t t0))}; t t0. ThereforeS(t)is a contraction semigroup.

Consider the following assumptions:

(i) f :< X !X;is a continuous function de…ned by

f(t; z)(x) =F₁(t; z(x)); t T; 0 x and also functionf satis…es the Lipschitz condition.

(ii) g:X !X is a continuous function de…ned by g(u)(t) =x0(t)

Xq j=1

cjz(pj; x) 0< p1< p2< < pq < b x2[0; ] where x(s)(t) =z(s; t); 0 t :

(iii) E maxi;k Qk

j=ikq(j)( j)k <1:

Under the conditions, we can de…ne the functionb_k by bk =q(k) k:

Assume that assumptions (i) and (ii) are satis…ed, then the problem (6) becomes an abstract random impulsive di¤erential equation.

Proposition 1 Assume that (H₁)–(H₄) hold. Then there exists a unique mild solution of the system (6) respectively, provided

K²#²maxf1; #²g(T )² L0+ L

(T )² <1 is satis…ed.

Proposition 2 Assume that the conditions of Proposition 1hold. Then the mild solution z of the system (6) is stable in the mean square.

Example 2

Consider partial integro-random impulsive di¤erential equations 8>

>>

<

>>

>:

zt(t; x) =zxx(t; x) +F1(t; z(t; x)) +RT

0 F2( ; z(tsin ; x))d ; t6= _k; t ; z(x; _k) =q(k) kz(x; _k); as x24b;

z(t;0) =z(t; ) = 0;

z(t0; x) +Pq j=1cj ³

pz(pj; x) =z0(x); 0< p1< p2< < pq< T; x2@4b:

(7)

Let4b <ⁿ be a bounded domain with smooth boundary@4b,X =L²(4b), _k be random variable de…ned onD_k (0; d_k)fork2N,d_k 2(0;+1). Also assume that _k follow Erlang distribution and ifi6=j then

i and _j are independent with each other fori; j= 1;2; : : :. Here q is a function of k, _k = _{k 1}+ _k for k2N,t₀2 <⁺.

LetAbe an operator onX byAz= @²z

@x² with the domain D(A) = z2Xjzand @z

@x are absolutely continuous, @²z

@x² 2X; z= 0; z= on@4b :

Thus A generates a strongly continuous semigroup S(t) which is analytic, self adjoint and compact. Fur- thermore the operatorAcan be represented as

Az= X1 n=1

n²< z; zn> zn; z2D(A):

Herez_n( ) =

q2Sin(n ); n= 1;2; : : :, forms the orthonormal set of eigenvectors of A. Also for everyz2X;

S(t)z=P₁

n=1e^{( n2t)}< z; zn> zn;which holdskS(t)k e^{( 2(t t0))}; t t0. ThereforeS(t)is a contraction semigroup.

Consider the following assumptions:

(i) f :< X !X; f1:< X !X is a continuous function de…ned by f(t; z)(x) =F₁(t; z(x)); t T; 0 x ;

f1( ; x(t+ ))d = Z T

0

F2( ; z(tsin ; x))d ; and also functionf andf1 satis…es the Lipschitz condition.

(ii) g:X !X is a continuous function de…ned by g(u)(t) =x0(t)

Xq j=1

cj ³

q

x(pj; t); 0 t ;

where x(s)(t) =z(s; t); 0 t : (iii) E maxi;k

Qk

j=ikq(j)( j)k <1:

Under the conditions, we can de…ne the functionf1; bk by bk =q(k) k and f1( ; x(t+ ))d =

Z T 0

F2( ; z(tsin ; x))d :

Assume that assumptions (i) and (ii) are satis…ed. Then the problem (7) becomes an abstract random impulsive integro-di¤erential equation (4).

Proposition 3 Assume that (H₁)–(H₄) hold. Then there exists a unique mild solution of the system (7) respectively, provided

K²#²maxf1; #²g(T )² L0+L1+ L

(T )² <1 is satis…ed.

Proposition 4 Assume that the conditions of Proposition 3hold. Then the mild solution z of the system (7) is stable in the mean square.

7 Conclusion

We have investigated the existence, uniqueness and stability of an integro-di¤erential system with nonlocal conditions. Here we used contraction mapping principle for proving the existence and uniqueness. Finally some examples are given to show the importance of random implusive di¤erential equations as well as inegro- di¤erential equations with nonlocal conditions. In future we can extend this work to fractional di¤erential equation and the results derived in this paper can be used to analyse the variation in the behaviour of the solution with respect to the variation in the complexity of the system of di¤erential equation considered.

Acknowledgment. The authors would like to thank the anonymous referee for his valuable comments.

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