Second-order neutral impulsive stochastic evolution equations with infinite daelay

In this paper, we study a class of second-order neutral impulsive stochasticevolution equations with infinite delay (SNISEEIs in short), in which theinitial value belongs to the abstract space Bh. Sufficient conditions for the existence of themild solutions for SNISEEIs are derived by means of the Krasnoselskii-Schaeferfixed point theorem. Two examples are given to illustrate the obtainedresults.


Introduction
In this paper, we consider the second-order neutral impulsive stochastic evolution equations with infinite delay (SNISEEIs in short) of the following form: d x (t)g(t, x t ) = Ax(t) + f (t, x t ) dt + σ (t, x t ) dw(t), t ∈ J := [, T], () x () = ψ ∈ H, (  ) x(t k ) = I k (x t k ), then (B h , · B h ) is a Banach space []. Let K be another separable Hilbert space with inner product (·, ·) K and norm · K . Suppose {w(t) : t ≥ } is a given K -valued Wiener process with a finite trace nuclear covariance operator Q ≥  defined on a complete probability space ( , F, P) equipped with a normal filtration {F t } t≥ , which is generated by the Wiener process w. We are also employing the same notation · for the operator norm L(K; H), where L(K; H) denotes the space of all bounded linear operators from K into H. Assume that g, f : J × B h → H (i = , ) and σ : J × B h → L Q (K, H) are appropriate mappings specified later. Here, L Q (K, H) denotes the space of all Q-Hilbert-Schmidt operators from K into H, which will be defined in the next section. The initial data φ = {φ(t) : -∞ < t ≤ } is an F  -adapted, B h -valued stochastic process independent of the Wiener process w with finite second moment. ψ is an F  -adapted, H-valued random variable independent of the Wiener process w with finite second moment. I k andĨ k : B h → H are appropriate functions. Moreover, let  = t  < t  < · · · < t m < t m+ = T, be given time points and the symbol ξ (t) represents the jump of the function ξ at t, which is defined by ξ (t) = ξ (t + )ξ (t -). Stochastic partial differential equations (SPDEs in short) with delay have attracted great interest due to their applications in describing many sophisticated dynamical systems in physical, biological, medical and social sciences. One can see [-] and the references therein for details. Moreover, to describe the systems involving derivatives with delay, Hale and Lunel [] introduced the deterministic neutral functional differential equations, which are of great interest in theoretical and practical applications. Taking the environmental disturbances into account, Kolmanovskii and Myshkis [] introduced the neutral stochastic functional differential equations (NSFDEs in short) and gave its applications in chemical engineering and aero elasticity. The investigation of qualitative properties such as existence, uniqueness and stability for NSFDEs has received much attention. One can see [, , -] and the references therein. In addition, impulsive effects exist in many evolution processes in which states are changed abruptly at certain moments of time, involved in such fields as medicine and biology, economics, bioengineering, chemical technology etc. (see [, ] and the references therein).
On the other hand, the study of abstract deterministic second-order evolutions equations governed by the generator of a strongly continuous cosine family was initiated by [] and subsequently studied by [, ]. The second-order stochastic differential equations are the right model in continuous time to account for integrated processes that can be made stationary. For instance, it is useful for engineers to model mechanical vibrations or charge on a capacitor or condenser subjected to white noise excitation through a second-order stochastic differential equations. There are some interesting works that have been done on the second-order stochastic differential equations. To the best of our knowledge, there is no work reported in the literature about SNISEEIs and the aim of this paper is to close this gap. We aim to establish the existence of the mild http://www.advancesindifferenceequations.com/content/2014/1/112 solutions for SNISEEIs by means of the Krasnoselskii-Schaefer fixed point theorem. Two types of stochastic nonlinear wave equations with infinite delay and impulsive effects are provided to illustrate the obtained results.
The paper is organized as follows. In Section , we introduce some preliminaries. In Section , we prove the existence of the mild solutions for SNISEEIs by means of the Krasnoselskii-Schaefer fixed point theorem. In Section , we study the continuous dependence of solutions on the initial values. Two examples are provided in the last section to illustrate the theory.

Preliminaries
In this section, we mention some preliminaries needed to establish our results. For details as regards this section, the reader may refer to Da Prato and Zabczyk [], Fattorini [] and the references therein. Let with e i being a CONS of eigenvectors, and then, w.r.t. this spectral representation of Q the driving Q-Wiener process can be represented as i= are mutually independent onedimensional standard Wiener processes. We assume that F t = σ {w(s) :  ≤ s ≤ t}, which is a σ -algebra generated by w and F T = F . Let ψ ∈ L(K, H) and define If ψ Q < ∞, then ψ is called a Q-Hilbert-Schmidt operator. Let L Q (K, H) denote the space of all Q-Hilbert-Schmidt operators ψ : K → H. The completion L Q (K, H) of L(K, H) with respect to the topology induced by the norm · Q with ψ  Q = (ψ, ψ) is a Hilbert space with the above norm topology.
The collection of all strongly measurable, square-integrable, H-valued random variables, denoted by We say that a function exists, for all k = , . . . , m. In the sequel, we always assume that PC is endowed with the norm To simplify the notations, we put t  = , t n+ = T. For x ∈ PC, we denotex k ∈ C([t k , t k+ ]; L  ( , H)), k = , , . . . , n, given bỹ Now, we consider the space Then we have the following useful lemma appearing in [].
Now, let us recall some facts about cosine families of operators C(t) and S(t) appeared in [, ].
, for all t, s ∈ R, is called a strongly continuous cosine family.
The corresponding strongly continuous sine family To prove our results, we need the following Krasnoselskii-Schaefer type fixed point theorem appearing in [].

It is well known that the infinitesimal generator
Theorem  Let  and  be two operators of H such that (i)  is a strict contraction, and (ii)  is completely continuous. Then either

Existence result
In this section, we aim to give the existence of mild solutions for SNISEEIs ()-(). Firstly, let us propose the definition of the mild solution of SNISEEIs ()-().

Definition  An F t -adapted stochastic process x : (-∞, T] → H is called a mild solution of SNISEEIs ()-() if (i) {x t : t ∈ J} is B h -valued and x(·)| J ∈ PC;
(ii) x(t) ∈ H has càdlàg paths on t ∈ J a.s. and for each t ∈ J, x(t) satisfies the following integral equation: In this paper, we need the following assumptions: (H) The function g : J × B h → H is continuous and there exists a positive constant L g such that The main result of this section is the following theorem. Proof In the sequel, the notation B r (x, Z) stands for the closed ball with center at x and radius r >  in Z, where (Z, · Z ) is a Banach space. Let y : (-∞, T] → H be defined by

Theorem  Assume the conditions (H)-(H) hold and assume that S(t) is compact. Then there exists a mild solution of SNISEEIs
On the space Y = {x ∈ PC : x() = φ()} endowed with the uniform convergence topology, we define the operator : Y → Y by where x is such that x  = φ and x = x on J. From Lemma  and the assumption on φ, we infer that x ∈ PC. Our proof will be split into the following three steps.
Step . In what follows, we prove that there exists r >  such that (B r (y| J , Y )) ⊆ B r (y| J , Y ). In fact, if it is not true, then for each r >  there exist x r ∈ B r (y| J , Y ) and t r ∈ J such that r < E (x r (t r ))y(t r )  . Therefore, from Lemma  and the assumptions, we have where y T = sup ≤s≤T E y(s) . Dividing both sides by r  and taking the limit as r → ∞, In what follows, we aim to show that the operator has a fixed point on B r (y| J , Y ), which implies that ()-() has a mild solution. To this end, we decompose as =  +  , where  ,  are defined on B r (y| J , Y ), respectively, by and for t ∈ J. We will show that  is a contraction and  is completely continuous.
Step .  is a contraction. Let x, y ∈ B r (y| J , Y ). Then, for each t ∈ J, we have Therefore, we get where L  = Ml  [T(TL g + Tr(Q)L σ ) + m k= (L I k + L˜I k )]. Thus, we obtain By (), we see that  is a contraction on B r (y| J , Y ). http://www.advancesindifferenceequations.com/content/2014/1/112 Step .  is completely continuous on B r (y| J , Y ).

Claim   maps bounded sets to bounded sets in B r
In the sequel, r * , r * * are the numbers defined by r * : which shows the desired result of the claim.

Claim  The set of functions
Let ε >  small enough and  < t  < t  . We get which proves that  (B r (y| J , Y )) is equicontinuous on J.
Claim   maps (B r (y| J , Y )) into a precompact set in (B r (y| J , Y )). That is, for each fixed t ∈ J, the set V (t) = {  z(t) : z ∈ (B r (y| J , Y ))} is precompact in (B r (y| J , Y )).
Obviously, V () = {  ()}. Let t >  fixed and for  < ε < t, define Since S(t) is a compact operator, the set V ε (t) = { ε  x(t) : x ∈ (B r (y| J , Y ))} is relatively compact in H for every ε,  < ε < t. Moreover, for each x ∈ (B r (y| J , Y )), we have Therefore, we have and there are precompact sets arbitrary close to the set ). Therefore, from the Arzela-Ascoli theorem, the operator  is completely continuous. From Theorem , we infer that there exists a mild solution for the system ()-().

Examples
In this section, two types of stochastic nonlinear wave equations with infinite delay and impulsive effects are provided to illustrate the theory obtained.