1. Introduction

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, On can see ^{[1, 4, 14]} and references therein. Several authors have established the existence results of mild solutions for these equations (see ^{[3, 7, 9, 11, 14]} 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 ^[16] 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 (see ^{[12, 17]} 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 ^[6] and the fractional power of operators. Especially, the known results appeared in ^{[2, 8, 10, 12, 15]} and ^{[5, 6, 11, 16]} are 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 ^[13] 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 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 w 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, one can see ^[10]. A multi-valued map is convex (closed) valued, if is convex (closed)for all . is bounded on bounded sets if is bounded in H, for any bounded set B of H, 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., . 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 (1) (following in ^[5])

Let be a filtered probability space.

(i) The filtration is said to be complete if is a complete and if contains all the null sets.

(ii) The filtration, is said to satisfy the usual hypotheses if it is complete and right continuous, that is where .

Lemma (1) (following in ^[1])

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

Lemma (2), (Ito isometry theorem) (following in ^[20])

Let be the class of functions such that , is measurable, adapted and . Then for every

where is a wiener process.

Definition (2) (following in ^[12])

Let be a constant belonging to (0, 1). A one dimensional fractional Brownian motion of Hurst index is a continuous and centered Gaussian process with covariance function

for .

Remark (1) (following in ^[12])

Let be a one dimensional fractional Brownian motion then

1) .

2) is a covariance function of fractional Brownian motion such that

(1)

3) If the covariance function becomes min .

Remark (2) (following in ^[19])

Let be a one dimensional fractional Brownian motion then for every

1. If Then the increments of are non-correlated, and consequently independent. So is a Wiener Process which denoted further by .

2. If then the increments are positively correlated.

3. If then the increments are negative correlated.

Definition (3) (following in ^[19])

A stochastic process is called self similar if and have the same law.

Remark (3) (following in ^[19])

Fractional Brownian motion of Hurst index is self similar.

In the following sections, we explain the Integration of Deterministic Function with Respect to One Dimensional Fractional Brownian motion

Lemma (3) (following in ^[12])

The one dimensional fractional Brownian motion has the integral representation

(2)

here, is a wiener process and the kernel defined as

(3)

(4)

and is a beta function.

Definition (4) (following in ^[19])

Let the general indicator function be given by

The function f is said to be step function, if there exist a finite number of points , and , such that .

Now, We denote by the set of step functions on . If then by defined above we can write it by the form , where

The integral of a step function with respect to one dimensional fractional Brownian motion is defined , where , , , and is a beta function.

Definition (5) (following in ^[19])

Let the general indicator function be given by

The function f is said to be step function, if there exist a finite number of points , and such that .

Now, We denote by the set of step functions on If then by defined above we can write it by the form , where .

The integral of a step function with respect to one dimensional fractional Brownian motion is defined , where .

Let be the Hilbert space defined as the closure of with respect to the scalar product .

The mapping can be extended to an isometry between and . i.e. the mapping is isometry ^[22].

Remark (4) (following in ^[21])

• If and then by using (Ito isometry theorem), we have

(5)

• If from the equation ( 2.30), we have

(6)

(7)

Lemma (4) (following in ^[11])

For any functions , , then

(i)

(ii)

(8)

From this Lemma above, we obtain

(9)

Remark (5) (following in ^[11])

The space contains the set of functions such that , which includes .

Now, let be the Banach space of all measurable functions on such that

(10)

Lemma (5) (following in ^[18])

Let be the Banach space of all measurable functions on and be the Hilbert space defined as the closure of the set of step functions on . If then. .

As for regard to Integration of Deterministic Function with Respect to Infinite Dimensional Fractional Brownian motion, Let and be two real separable Hilbert spaces and Let be the space of all bounded linear operators from to .

Definition (6) (following in ^[22])

A process with values in separable Hilbert space is called Gaussian if, for every and the real random variable has a normal distribution.

Definition (7) (following in ^[22])

The valued process is said to be an infinite–dimensional fractional Brownian motion or (fractional Brownian motion) if is a centered Gaussian process with covariance COV , where the covariance operator.

Lemma (6) (following in ^[22])

is a fractional Brownian motion if and only if there exists a sequence of real and independent fractional Brownian motion such that

where the series converges in and the orthonormal system in Y.

Rmark (6) (following in ^[22])

Suppose that there exists a complete orthonormal system in . Let be the operator defined by , where are non-negative real numbers with finite trace . The infinite dimensional fractional Brownian motion on Y can be defined by using covariance operator as , where are one dimensional fractional Brownian motions mutually independent on .

In order to defined stochastic integral with respect to the fractional Brownian motion. We introduce the space of all Hilbert-Schmidt operators that is with the inner product is a separable Hilbert space.

Lemma (7) (following in ^[18])

Let be a family of deterministic functions with values in The stochastic integral of with respect to is defined by

(1.14)

Lemma (8) (following in ^[18])

If satisfies then the above sum in lemma (1.13) is well defined as an valued random variable and we have

(11)

3. Problem Formulation

In this section, we study the existence and stability of inclusion equations, type of stochastic dynamical system driven by mixed fractional Brownian motion in a real separable Hilbert space of the following from:

(12)

Where is a generator of cosine semigroup on a Hilbert space and are valued Brownian motion and fractional Brownian motion respectively.

To investigate the existence of the mixed-stochastic mild solution to the system (12), and for the operators A we make the following assumption :

1. A is the infinitesimal generator of a compact, analytic resolve operator in the Hilbert space and there exists constant and such that , on and .

2. There exist constant such that is a continuous function, satisfies the following Lipchitz condition, that is, for any such that

3. a: , is a continuous function and there exists a constant such that for all

4.

5. For the initial condition there exists a positive constants and such that , .

6. The function satisfies For every and there exists such that .

7. The multi-valued map is an Caratheodory function satisfies the following conditions:-

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 each positive number there exists appositive function independent on such that .

iii. .

8.

Where

Lemma(9)

Let be a cosine semigroup and -valued function then the system (12) has a mixed–stochastic mild solution

Proof:

different both sides for S and use properties in Lemma (9), we get

Integrate both sides, we get

(13)

Definition (8)

A bounded function is called mixed solution of the inclusion system (12) if for any

In this section, the existence of mixed-stochastic mild solution in to inclusion problem formulation (12) has been develop.

Theorem(1):

Suppose that conditions(1-8) are hold.

Then for initial value , , such that the intial value mixed-stochastic inclusion problem(1) has mixed-stochastic mild solution .

Proof:

Let the operator defined by

It is clear that the fixed points of are mild solutions of the system (12). Let

(14)

(15)

We prove that the operators and are satisfy all the condition of theorem (1), Let

Step(1):- is a contraction

Let from assuming that

From the assumptions (1-4), we have

By using initial condition (5), and taking supremum over for both sides, we get

We can rewrite last inequality in the following from:

Where

Hence, we obtain

Step (2):- is convex for each if , then there exists from condition , we get

(16)

(17)

Let , then

Since is convex (because has convex values), then, we have

Step (3):- maps bounded sets in to bounded sets in . To show that there exists a positive constant such that for each we have . If then there exists for each , such that

By using Lemma (1.8), Lemma (1.14) and by assumption (1), we obtain

From assumptions (6) and (7-ii) ,we get

(18)

Step(4):- maps bounded sets in to equicontinuous set of , Let such that then for and then,there exists such that for each we have

When the above inequality tends to zero, since S(t) in the uniform operator topology thus the set is equicontinuous.

Step(5):- Now to prove is relatively compact in for each wehre the set is relatively compact in for each . Let and for and , there exists such that

(19)

Now, we define

(20)

for each , thus,

Since sine simegroup operators are a continuous, we have

Then, there exist , we get

By using Lemmas (2), (8), and assumptions (1), (6), (7-ii), we obtain

The relative compact sets arbitrarily close to the set then its relative compact in 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

(21)

There exists , thus

(22)

We must prove that there exists such that

(23)

Suppose the liner continuous operator : .

From lemma (1) it follows that is closed graph operator and we have as , thus

Since it follows from Lemma (1) that,

That is, there exists a such that

Since be a linear continuous mapping from to in Lemma (1). Therefore is a closed graph and u.s.c.

Step (7):-The operator inclusion has a solution in . Define an open ball B(0, r) in , where satisfies the inequality given in (20),we need to show that the system (12) has least one mild solution, for for some with then, we have

By using assumptions (1-3)and (7-i) ,we get

From the assumptions(1),(2)and(3),(6) and using Lemmas (2) and (8), we obtain

From assumption (6), we get

(24)

Thus

Example (1)

Consider the following fractional differential equations

for , ,where, where are a constants.

(1) ,

(2) by

(3) is generator of strongly cosine family on .

(4) The eigenvalues of is and the eigenvectors the set of function is oryhonormal basis of .

(5) For also for all . In addition, for We assume is measurable satisfies

(6) The function is of classes on and for each

(7) The function is continuous and there is such that

(8) The function are continuous and there are appositive constants such that

(9) is a continuous function.

The following theorem investigate the stability of the inclution equation (12) by using Gron will Bellman inequality via cosine dynamical system.

We need to investigate the definition (8) on the inclusion problem (12).

Definition (9)

The solution of the system (12) in said to be stable , if for any there exists a number such that for any other solution of the system (12) satisfying then is said to be asymptotically stable if it stable and if there is a constant such that then

Theorem (2)

Assume the hypotheses (1-9) are hold, and has an asymptotically mild solution

Proof:

Let and be a two solutions of equation (12) such that

(25)

and

(26)

Thus,

Then,

(27)

Where .

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