1. Introduction

In this paper, gives a nonlocal and sufficient condition of the existence of mild solutions for the following neutral stochastic functional integro-differential inclusions with nonlocal conditions:

Where is the infinitesimal generator of a compact, analytic resolvent operator in the Hilbert space . Suppose is a given K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator and denotes the space of all bounded linear operators from in to . Let and be a bounded linear operator. The random variable satisfies and are given functions specified later.

The theory of integro-differential equations or inclusions has become an active area of investigation due to their applications in the fields such as mechanics, electrical engineering, medicine biology, ecology and so on can (see ^{[1, 4, 5]} and references therein). Several authors have established the existence results of mild solutions for these equations (^{[3, 14, 18, 23]} and references therein). In addition, the nonlinear integro-differential equations with resolvent operators serve as an abstract formulation of partial integro-differential equations that arise in many physical phenomena. One can see ^[19] and references therein. The deterministic models often fluctuate due to noise, which is random or at least appears to be so. Therefore, we must move from deterministic problems to stochastic problems. As the generalization of classic impulsive integro-differential equations or inclusions, impulsive neutral stochastic functional integro-differential equations or inclusions have attracted the researchers great interest. And some works have done on the existence results of mild solutions for these equations (^{[15, 20]} and references therein). To the best of our knowledge, there is no work reported on the existence of mild solutions for the impulsive neutral stochastic functional Integro-differential inclusions with nonlocal initial conditions and resolvent operators, and the aim of this paper is to close the gap. In this paper, motivated by the previously mentioned papers, we will study this interesting problem. Sufficient conditions for the existence are given by means of the fixed point theorem for multi-valued mapping due to Dhage ^[7] and the fractional power of operators. Especially, the known results appeared in ^[6] is generalized to the stochastic settings. An example is provided to illustrate the theory.

2. Preliminaries

For more details on this section, We refer the reader to Da prato and Zabczyk ^[24]. throughout the paper and denote two real separable Hilbert spaces. In case without confusion, we just use for the inner product and for the norm.

Let be complete filtered probability space satisfying that contains all -null sets of . An -valued random variable is an -measurable function and the collection of random variables is called a stochastic process. Generally, we just write x(t) instead of and in the space of S. Let be a complete orthonormal basis of . Suppose that is a cylindrical valued wiener process with a finite trace nuclear covariance operator denote which satisfies that So, actually, where are mutually independent one-dimensional standard wiener processes. We assume that is the algebra generated by and . Let and define

If then is called a -Hilbert-Schmidt operator. Let denote the space of all -Hilbert-Schmidt operators The completion of with respect to topology induced by the norm where is a Hilbert space with the above norm topology. Let be infinitesimal generator of a compact, analytic resolvent operator Let denote the Hilbert space of all measurable square integrable random variables with values in Let be the Hilbert space of all square integrable and measurable processes with values in . let denote the family of all measurable, –valued random variables . We use the notations for the family of all subsets of and denote

In what follows, we briefly introduce some facts on multi-valued analysis. For details, ^[5]. A multi-valued map is convex (closed) valued, if is convex (closed) for all . is bounded on bounded sets if (x) is bounded in , for any bounded set B of , that is, is called upper semi continuous (u .s .c. for short) on , if for any , the set is a nonempty, closed subset of H, and if for each open set B of containing , there exists an open neighborhood N of x such that . is said to be completely continuous if is relatively compact, for every bounded subset . If the multi –valued map is completely continuous with nonempty compact values, then is u .s .c. if and only if has a closed graph, i.e., imply . has a fixed point if there is such that A multi-valued map is said to be measurable if for each the mean –square distance between and is measurable.

Definition (2-1) ^[11]

The multi-valued map is said to be -Caratheodory if

i) is measurable for each ;

ii) is u.s.c. for almost all ;

iii) for each , there exists

such that

, for all and for a.e

Lemma (2-1) ^[17]

Let I be a compact interval and a Hilbert space. Let F be an Caratheodory multi-valued map with and let Γ be a linear continuous mapping from to Then, the operator is a closed graph operator in where is known as the selectors set from F, is given by .

Lemma (2-2) ^[10]

Let be a family of deterministic functions with values in (Y, X) The stochastic integral of Φ with respect to is defined by

Lemma (2-3) ^[10]

If satisfies then the above sum in (2.2) is well defined as an -valued random variable and we have

Definition (2-2) ^[2]

A semigroup T(t), of bounded linear operators on a Banach space X is a semigroup of bounded linear operators if: , for every .

Example (2-1), ^[12]

Let , where is a Banach space, and set: . Then

The family is strongly continuous semigroup (semigroup). The following are briefly the most important facts on semigroup theory of bounded linear operators that needed later on.

Theorem (2-1) ^{[8, 18]}

A linear (unbounded) operator A is the generator of a semigroup of contractions if and only if:

(i) A is closed and .

(ii) The resolvent set of A contains and for every ,

Remark (1-1), ^[2]

Let T(t) be a semigroup then there are constants and , such that for . If , is called uniformly bounded Moreover if , it is called a semigroup of contractions. The following are briefly the most import.

Definition (2-3), ^[21]

A one-parameter family of bounded linear operators in the Banach space X is called a strongly continuous cosine family if and only if

i. for all .

ii.

iii. is continuous in on for each fixed .

Definition (2-4), ^[9]

If is a strongly continuous cosine family in X,

i. , associated to the given strongly continuous cosine family, is defined by .

ii. The infinitesimal generator of a cosine family is defined by

Where .

Definition (2-5) ^[8]

Let X be a Banach space, a one-parameter family of bounded linear operators from into is a semigroup of bounded linear operators on if:

1. , where I is the identity operator on .

2. , for every .

Lemma(2-4), ^[22]

Let , be a strongly continuous cosine family on , then there exist constants and such that , for all ,

Theorem (2-2) ^[7]

Let and denote respectively the open and closed balls in a Hilbert space centered at the origin and of radius r and let and two multi-valued operators satisfying

(i) is a contraction, and

(ii) is u.s.c. and completely continuous.

Then, either

(1) the operator inclusion has a solution, or

(2) there exists an with such that for some .

3. Main Result of the Existence and Stability

The following lemma and definition are begging to explain the main results.

Lemma (2-5)

Let be a cosine semigroup and the H-valued function

Then (2.1) has a mixed-stochastic mild solution with , ,

Proof:

Take the H-valued function

Then, different both sides for s and use properties in definition(2.4), we get

Integrate both sides, we get

Definition (2-6)

A bounded function is called mixed-stochastic mild solution of the inclusion system (2.1) if for any , , .

Hypotheses

To investigate the existence of the mixed-stochastic mild solution to the system (2.1), and for the operators we make the following assumption:

1. A is the infinitesimal generator of a compact, analytic resolvent operator , , in the Hilbert space and there exist constants and such that on , .

2. There exist constant such that satisfies the following Lipchitz condition, that is, for any such that the multi-valued map is an Caratheodory function satisfies the following condition:-

i. for each the function is u. s. c , and for each the function is measurable and for each fixed the set

is nonempty.

ii. for some positive numbers

and . Where are positive constants.

3. the map : and there exist positive constants such that

4.

4. Existence of the Inclusion Nonlinear Stochastic Differential System

In this section, the existence of the mixed-stochastic mild solution to the inclusion Problem formulation (2.1) has been discussed.

Theorem (2-3)

Assume the Hypotheses (1-5) are hold . Then for initial value , , and . Then the initial value mixed-stochastic inclusion system (2.1) has mixed -stochastic mild solution .

Proof:

Let the operator defined by

a fixed point of are stochastic- mild solutions of the equation (2.1). Let

We prove that the operators and are satisfy all the condition for theorem(2-2).

Let .

Step(1):

Now to prove that is contraction.

Let from assuming that

we have that

Where ,

Step (2):

Now to prove that is convex for each , let then, there exists such that

Let

and

For each we have

From the condition (3-i) and since

is convex then we have that .

Step (3):

Now to prove maps bounded set into bounded set in B. Indeed, it is enough to show that there exists appositive constant such that for we have if then there exists for each such that

We have that

for the condition the function satisfies from and there exists such that Sup .

Then .

Step (4):

maps bounded set into equicontinuous sets of B. Let . Then we have for each . and there exists such that for each we have.

Where is defined by .

The right- hand side of the above the quality tends to zero as with sufficiently small, also S(t) is a continuous simegroup .

Step (5):

Now to prove is relativity compact in for each .

Where , the set is relatively compact in for . Let and for and there exists such that

Now, we define

for each ,

From definition (2-3), and lemma (2-5) for the cosine simegroup continuous we have

The relative compact sets arbitrarily close to the set then its relative compact in B, thus is a compact multi-valued closed graph.

Step (6):

Now to show that has a closed graph.

Let and we aim to show that indeed, means that there exists such that

there exists , thus

We must prove that there exists such that

Suppose the linear continuous operator .

From lemma (2-1) it follows that is closed graph operator and we have

as , thus

Since , it follows from lemma (2-1) that

That is, there exists a such that

hence has a closed graph.

As in lemma (2-1) Let be alinear continues mapping from to Then, the operator : .

Is a closed graph operator in

Step (7):

The operator inclusion has a solution in . Define an open ball B(0, r) in , where satisfies the inequality given in (5), we that and satisfy all conditions of theorem (2-2) . Therefore , if we can show that the second condition of theorem (2-2) is not true , then , we show that the system (2.1) has Least one mild solution, for . For some with then, we have

thus

is a contradiction to condition (5), thus, has a solution in .

Hence the system(2.1) has at least one mild solution.

5. Example

In this section we will take the following example an fractional partial differential equations

Define by with domain

it is well known that is the infinitesimal generator for a strongly continuous cosine family on has a spectrum the eigenvaluses and eigenvector , with the following

a) is an orthonormal basis of and

b) For and

c) for all

d)

(e)

(f) .

Under the appropriate conditions (1-5) of k, b, g, then theorem (2-3) , ensures the existence of mild solution to problem(2.1).

Asymptotically stable for the mild solution of inclution formulation problem (2.1).

Given as follows:

6. Asymptotically Stable for the Mild Solution of Inclusion Formulation Problem (2.1)

The assymptoically stable of the problem (2.1) has been given in details with necessary and sufficient conditions.

We need to investigate the definition (2-6) on the inclusion problem (2.1).

Theorem (2-4)

Assume the hypotheses (1-5) are hold then the solution has asymptotically stable behivours

Proof

Let and be a two solutions of equation (2.1)

Hence,

such that .

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