1. Introduction

In this article we study the neutral second- order abstract differential inclusion problem

(3.1)

Where is a generator of cosine semigroup on a Hilbert space and are valued Brownian motion and fractional Brownian motion respectively with afinit trace nuclear covariance operator . satisfy suitable conditions that will be established later on. The random variable satisfies .

This problem has been studied in case , (^{[1, 2, 3, 16]}). Well-posedness has been established using different fixed point theorems and the theory of strongly continuous cosine families in Banach spaces. We refer the reader to ^{[27, 28]} for a good account on the theory of cosine families.

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 (^{[7, 8, 25]} and references therein). Several authors have established the existence results of mild solutions for these equations (^{[4, 5, 21, 24]}). 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 ^[15] 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 ^{[17, 26]} 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 ^[11] and the fractional power of operators. Especially, the known results appeared in ^{[9, 10]} are generalized to the stochastic settings. An example is provided to illustrate the theory. We refer the reader to Da prato and Zabczyk ^[12]. 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 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

(closed) for all . is bounded on bounded sets if 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 , and if for each open set of containing there exists an open neighborhood of 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.

2. Preliminaries

In this section we present some notation. Assumptions and results needed our proofs later.

Definition (3-1) ^[18]

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 (3-1) ^[19]

If : [0, b] → satisfies then the above sum in

is well defined as an X-valued random variable and we have

(3.2)

Definition (3-2) ^[6]

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

Definition (3-3), ^[14]

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 .

Lemma(3-2), ^[28]

Let , be a strongly continuous cosine (resp,sin) family on X, then there exist constants and such that , for all , for all

Theorem (3-1) ^[11]

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.

To investigate the existence of the mixed-stochastic mild solution to the system (3.1), and for the operators we make the following assumption:

1. is the infinitesimal generator of a cosine semi group , , 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 .

3. The multi-valued maps is an Caratheodory function satisfies the following condition :-

i. is uniformly semi continuous for a. e. for every the function is strongly measurable.

ii. nonempty.

iii. There exists a nonnegative continuous function and a continuous non decreasing positive function such that

4.

5. The function satisfies from and there exists such that .

6. For each the set is relatively compact in.

Lemma (3.3):

Let be a cosine semigroup and the valued function then (3.1) has a mixed–stochastic mild solution

Proof:

different both sides for s and use properties in lemma (3.1), we get

(3.3)

Definition (3.4):

A bounded function is called mixed–stochastic mild solution of the fractional inclusion system (3.1) if for any

4. Existence of the Fractional Stochastic Integro-Differential Inclusion Equations via Cosine Dynamical System

In this section, the existence of the mixed–stochastic mild solution to the stochastic fractional order inclusion problem formulation (3.1) has been discussed.

Theorem (3.2):

Assume the Hypotheses (1-6) are hold with

For the mixed stochastic fractional order system (3.1) with initial conditions

Then has a mixed-stochastic mild solution

Proof:

Let the operator defined by

It is clear that the fixed point of are mild solutions of the system (3.1). Let

(3.4)

(3.5)

We prove that the operators and are satisfy all the condition for theorem (3.1).

Let .

Step(1):- Now to prove that is a contraction.

Let from assuming that

By using Cauchy-Schwarz inequality, we obtain

From the conditions (1),(2) and taking supremum over for both sides, we get

By initial condition , , , we get

Let

Step (2):- Now to prove that is convex for each . Let then, there exists such that such that

(3.6)

(3.7)

Let , then

For each We have

(3.8)

From the condition and since is convex then we have that

Remark (3.1) ^[27]:

We need the following inequalities for complete the proof of existence theorem.

thus

Step (3):- Now to prove that maps bounded sets into bounded set in . Indeed, it is enough to show that there exists appositive constant such that For each . Then there exists for each such that

By using remark (3.1),

From conditions (1) and (3-iii) , we get

By using Lemmas (1-8) and (1-14), we get

Let

were , . Now

(3.9)

If we put,

Let us denote by the right hand side of above then and

Thus

therefore

By taking the integral in both sides, we get

Let

So,

From theorem (3.1), we get

Hence is bounded and then is bounded then the maps bounded sets in to bounded sets in

Step (4):- maps bounded set into equicontinuous sets of , Let therefore each and such that, for each, we have

From this inequality and using remark (3.1), we have

From condition and Lemma (1-8) and (1-14)

The right-hand side of the above inequality tends to zero as with sufficiently small, then

From the last inequality, we have

(3.10)

Since is sine semi group continuous in the uniform operator topology, the set is equicontinuous.

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

(3.11)

(3.12)

For each

From definition (1-20) and remark (3.1) for the sine simegroup continuous, we have

Therefore, letting we can see that there are relative compact sets arbitrarily close to the set is relative compact in .

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

(3.13)

We must prove that there exists such that

(3.14)

Suppose the liner continuous operator

From lemma (1.7) it follows that is closed graph operator and we have

as , thus

Since it follows from Lemma (1.7) that

That is, there exists a such that

By using lemma (1.7) therefore , has a close graph and therefore is 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 (4). We need to show that the system (3.1) has Least one mild solution, for for some with , then, we have

From assumption (1-6), we get

From Lemma (1-8) and (1-14), we obtain

Now,

where,

From integral in both sides, we get

If we put,

Let us denote by the right hand side of above then and

Thus

therefore

By taking the integral in both sides we get

So,

From theorem (3-1), we get

Hence is bounded and then is bounded then the maps bounded sets in to bounded sets into bounded sets in

5. Example (3.1)

Consider the problem

In the space . This problem is the abstract setting of (3.1). To we define the operator with domain

The operator has a discrete spectrum with as eigenvalues and , , as their corresponding normalized eigenvectors. So we may write

Since is positive and self-adjoint in , the operator is the infinitesimal generator of a strongly continuous cosine family which has the form

The associated sine family is

For and , defining the operators

allows us to write (3.12) abstractly as

Under appropriate conditions on which make the conditions (3.1) hold for the corresponding functions . Theorem (3.1) ensures the existence of a mixed stochastic mild solution to problem (3.1). Some special cases of this problem may be found in models of some phenomena with hereditary properties ^{[5, 8, 15]}.

6. Stability for the Mild Solution of Fractional Inclusion Formulation Problem (3.1)

The following theorem investigate the stability of the inclusion equation (3.1) by using Gran will Bellman inequality via cosine dynamical system. We need to investigate the definition (3.1) on the inclusion problem (3.1).

Definition(3.5):

The solution of the system (3.1) in said to be stable, if for any there exists a number such that for any other solution of the system (3.1) satisfying then is said to be asymptotically stable if it stable and if there is a constant such that then

Theorem (3.3):

Assume the hypotheses (1-6) are holds for the system (3.1) with and has an stabile mixed stochastic mild solution.

Proof:

Let and be a two solutions of equation (3.1) such that

and,

Thus,

Then,

Where .

References

[1]	Benchohra. M and Ntouyas .S. K. “Existence of mild solutions of second order initial value problems for delay Integro-differential inclusions with nonlocal conditions”, Mathematica Bohemica, 4 (127) (2002), 613-622.
	In article
[2]	Benchohra. M and Ntouyas. S. K,. “Existence results for the semi-infinite interval for first and second order Integro-differential equations in Banach spaces with nonlocal condition”s, Acta Univ. Palacki. Olomuc, Fac. Rer. Nat. Mathematica 41 (2002), 13-19.
	In article
[3]	Byszewski. L and Laksmikantham. V,. “Theorems about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space”, Appl. Anal. 40 (1) (1991), 11-19.
	In article		View Article
[4]	Balasubramaniam. P,. “Existence of solution of functional stochastic differential inclusions”, Tamkang J. Math. 33 (2002) 35-43.
	In article
[5]	Balasubramaniam. P, Vinayagam. D,. “Existence of solutions of nonlinear neutral stochastic differential inclusions in a Hilbert space”, Stochastic Anal. Appl. 23 (2005) 137-151.
	In article		View Article
[6]	Balakrishnan, A. V., “Applications of Mathematics: Applied Functional Analysis”, by Springer -verlag New York, Inc., (1976(.
	In article
[7]	Chang. Y. K,. “Controllability of impulsive functional differential systems with infinite delay in Banach spaces”, Chaos Solitons Fractals 33 (2007) 1601-1609.
	In article		View Article
[8]	Chang. Y.K, Anguraj. A, Mallika Arjunan. M. M, “Existence results for impulsive neutral functional differential equations with infinite delay”, Nonlinear, Anal. Hybrid Syst. 2 (2008) 209-218.
	In article		View Article
[9]	Chang. Y .K, Nieto. J. J,. “Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators”, Numer. Funct. Anal. Optim. 30 (2009) 227-244.
	In article		View Article
[10]	Cao. Y, Fu. X, “Existence for neutral impulsive differential inclusions with nonlocal conditions”, Nonlinear Anal. 68 (2008) 3707-3718.
	In article		View Article
[11]	Dhage. B. C,. “Multi-valued mappings and fixed points II”, Tamkang J. Math. 37 (2006) 27-46.
	In article
[12]	Da Prato. G, J. Zabczyk. J,. “Stochastic Equations in Infinite Dimensions”, Cambridge University Press, Cambridge, 1992.
	In article
[13]	Diagana, T., “An Introduction to Classical and P-ADIC Theory of Linear Operators and Application”, Nova Science Publishers, (2006).
	In article
[14]	Goldstein J. A.,. “Semigroup of linear operators and applications”, Oxford Univ. Press, New York, 1985.
	In article
[15]	Grimmer. R, Pritchard. A. J,. “Analytic resolvent operators for integral equations in a Banach space”. J. Differential Equations 50 (1983) 234-259.
	In article		View Article
[16]	Hernandez. M. E,. “Existence results for a second order abstract Cauchy problem with nonlocal conditions”, Electr. J. Diff. Eqs, 2005 No. 73 (2005), 1-17.
	In article
[17]	Hino. Y, Murakam. S, Naito. T, “Functional-differential equations with infinite delay”, in: Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991.
	In article
[18]	Lin, A., Hu. L., “Existence results for Impulsive Neutral Stochastic Functional Integro-differential Inclusions with Nonlocal Initial Conditions”, J. Computers and Mathematics with Applications, 59(2010). 64-73.
	In article		View Article
[19]	Li K.. “Stochastic Delay Fractional Evolut- ions Driven by Fractional Brownian Motion”, Mathematical Method in the Applied Sciences, 2014.
	In article
[20]	Lasikcka, I. “Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations”, J. Deferential Equation, 47, pp. 246-272, (1983).
	In article		View Article
[21]	Nouyas. S .K, “Existence results for impulsive partial neutral functional differential inclusions”, Electron. J. Differential Equations 30 (2005) 1-11.
	In article
[22]	Nasser. E. T., “Existence of Mild Solustions For a Neutral Fractional Equation With Fractional Nonlocal Conditions”, E. J, of Differential Equations, Vol. 2012(2012), No.153, pp.1-12.
	In article
[23]	Opial., A. Lasota, Z., “Application of the kakutani-Ky-Fan theorem in the theory of ordinary differential equations or noncompact acyclic-valued map”, Bull. Acad. olon. Sci. Ser.Sci.Math.Astronom.phys. 13(1965 )781-786.
	In article
[24]	Pazy. A,. “Semigroups of linear operators and applications to partial differential equations”, in: Applied Methematical Sciences, vol. 44, Springer Verlag, New York, 1983.
	In article
[25]	Park. J .Y, Balachandtran. K, Annapooran. N, “Existence results for impulsive neutral functional integro-differential equations with infinite delay”, Nonlinear Anal. (2009).
	In article		View Article
[26]	Ren. Y, Hu. L, “Existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays”, Acta Appl. Math. (2009).
	In article
[27]	Travis, C. C. and Webb, G. F., “Compactness, regularity and uniform continuity properties of strongly continuous cosine families “, Houston J.Math.3(4) (1977), 555-567.
	In article
[28]	Travis, C. C. and Webb, G.F.. “Cosine families and abstract nonlinear second order differential equtions”, Acta Math. Acad. Sci. Hungaricae, 32(1978), 76-96.
	In article		View Article