Existence for a Second-Order Impulsive Neutral Stochastic Integrodifferential Equations with Nonlocal Conditions and Infinite Delay

The current paper is concerned with the existence of mild solutions for a class of second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions and infinite delays in a Hilbert space. A sufficient condition for the existence results is obtained by using the Krasnoselskii-Schaefer-type fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators. Finally, an application to the stochastic nonlinear wave equation with infinite delay is given.


Introduction
The theory of impulsive neutral differential equations has been emerging as an important area of investigation in recent years, stimulated by their numerous applications to problems from physics, mechanics, electrical engineering, medicine biology, ecology, and so on. Ordinary differential equations of first and second order with impulses have been treated in several works and we refer the reader to the monographs of Lakshmikantham et al. [1], the papers [2][3][4][5], and the references therein related to this matter. Besides, noise or stochastic perturbation is unavoidable and omnipresent in nature as well as in man-made systems. Therefore, it is of great significance to import the stochastic effects into the investigation of impulsive neutral differential equations. As the generalization of classic impulsive neutral differential equations, impulsive neutral stochastic integrodifferential equations with infinite delays have attracted the researchers' great interest. There are few publications on well-posedness of solutions for these equations (e.g., see, [6][7][8] and the references therein).
Recently, in [9], Cui and Yan proved sufficient conditions for the existence of fractional neutral stochastic integrodifferential equations with infinite delay of the form where 0 < < 1 and denotes the Caputo fractional derivative operator of order by means of Sadovskii's fixed point theorem. And very recently, also thanks to the Sadovskii fixed point theorem combined with a noncompact condition on the cosine family of operators, Arthi and Balachandran [10] established the controllability of the following damped 2 Chinese Journal of Mathematics second-order impulsive neutral functional differential systems with infinite delay: where D is a bounded linear operator on a Banach space with (D) ⊂ ( ).
On the other hand, there has not been very much study of second-order impulsive neutral stochastic functional differential equations with infinite delays, while these have begun to gain attention recently. To be more precise, in [11], Balasubramaniam and Muthukumar discussed on approximate controllability of second-order stochastic distributed implicit functional differential systems with infinite delay. Cui and Yan [12] investigated the existence of mild solutions for impulsive neutral second-order stochastic evolution equations with nonlocal conditions. Mahmudov and McKibben [13] established the results concerning the global existence, uniqueness, approximation, and exact controllability of mild solutions for a class of abstract second-order damped McKean-Vlasov stochastic evolution equations in a real separable Hilbert space. However, up to now, the well-posedness of mild solutions for a class of second-order impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delays in a Hilbert spaces has not been considered in the literature. In order to fill this gap, based on ideas and techniques in the above works, in this paper, we will study the well-posedness of mild solutions for a class of second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions and infinite delays of the form Here, (⋅) is a stochastic process taking values in a real separable Hilbert space H; : ( ) ⊂ H → H is the infinitesimal generator of a strongly continuous cosine family on H. The history : 0 → H, ( ) = ( + ) for ≥ 0, belongs to the phase space B, which will be described in Section 2. Assume that the mappings , : × B × H → H, : × × B → L 0 2 , : × × B → H, = 1, 2, 1 , 2 : H → H, = 1, , : B → B are appropriate functions to be specified later. Furthermore, let 0 < 1 < ⋅ ⋅ ⋅ < < be prefixed points, and Δ ( ) = ( + ) − ( − ) represents the jump of the function at time with determining the size of the jump, where ( + ) and ( − ) represent the right and left limits of ( ) at = , respectively. Similarly ( + ) and ( − ) denote, respectively, the right and left limits of ( ) at . Let ( ) ∈ L 2 (Ω, B) and 1 ( ) be H-valued F -measurable random variables independent of the Wiener process with a finite second moment.
The structure of this paper is as follows. In Section 2, we briefly present some basic notations, preliminaries, and assumptions. The main results in Section 3 are devoted to study the well-posedness of mild solutions for (3) with their proofs. An example is given in Section 4 to illustrate the theory. In the last section, concluding remarks are given.

Preliminaries
In this section, we briefly recall some basic definitions and results for stochastic equations in infinite dimensions and cosine families of operators. For more details on this section, we refer the reader to [14][15][16].
Let (H, ‖ ⋅ ‖ H , ⟨⋅, ⋅⟩ H ) and (K, ‖ ⋅ ‖ K , ⟨⋅, ⋅⟩ K ) denote two real separable Hilbert spaces, with their vector norms and their inner products, respectively. We denote by L(K; H) the set of all linear bounded operators from K into H, which is equipped with the usual operator norm ‖ ⋅ ‖. In this paper, we use the symbol ‖ ⋅ ‖ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let (Ω, F, {F } ≥0 , P) be a complete filtered probability space satisfying the usual condition (i.e., it is right continuous and F 0 contains all P-null sets). Let = ( ( )) ≥0 be a -Wiener process defined on the probability space (Ω, F, {F } ≥0 , P) with the covariance operator such that ( ) < ∞. We assume that there exists a complete orthonormal system { } ≥1 in K, a bounded sequence of nonnegative real numbers such that = , = 1, 2, . . ., and a sequence of independent Brownian motions Let L 0 2 = L 2 ( 1/2 K; H) be the space of all Hilbert-Schmidt operators from 1/2 K to H with the inner product ⟨Ψ, ⟩ L 0 2 = [Ψ * ], where * is the adjoint of the operator .
Let C( , L 2 (Ω, H)) be the Banach space of all continuous maps from to L 2 (Ω, H) satisfying the condition sup ∈ E‖ ( )‖ 2 < ∞. An important subspace is given by L 0 2 (Ω, H) = { ∈ L 2 (Ω, H) : is F 0 -measurable}. Next, to be able to access well-posedness of mild solutions for (3) we need to introduce theory of cosine functions of operators and the second-order abstract Cauchy problem.
It is well known that the infinitesimal generator is a closed, densely defined operator on H, and the following properties hold; see Travis and Webb [16].
where ℎ : → H is an integrable function has been discussed in [17]. Similarly, the existence of solutions of the semilinear second-order abstract Cauchy problem has been treated in [16].
Definition 3. The function (⋅) given by is called a mild solution of (6), and that when ∈ H, (⋅) is continuously differentiable and For additional details about cosine function theory, we refer the reader to [16,17]. Now we define the abstract phase space B. Assume that : is a bounded and measurable function on [− , 0] and If B is endowed with the norm We consider the space where is the restriction of to = ( , +1 ], = 1, . Set ‖ ⋅ ‖ as a seminorm in B defined by Lemma 4 (see [2]). Assume that ∈ B , then for ∈ , ∈ B. Moreover, Next, we present the Krasnoselskii-Schaefer-type fixed point theorem appearing in [19], which is our main tool.
Lemma 5 (see [19]). Let Π 1 and Π 2 be two operators of H such that (a) Π 1 is a contraction, and Then, either
In this paper, we will work under the following assumptions.
(H1) The cosine family of operators { ( )} ∈ on H and the corresponding sine family { ( )} ∈ are compact for > 0, and there exist positive constants , such that for all ∈ , (H2) There exists a positive constant 1 such that for all , ∈ , , ∈ B (H4) For each ( , ) ∈ × , the function 2 ( , , ⋅) : B → H is continuous and for each ∈ B, the function 2 (⋅, ⋅, ) : × → H is strongly measurable. There exists an integrable function : → [0, ∞) and a positive constant such that where (H5) The function : ×B×H → H satisfies the following Carathéodory conditions: (H6) The functions 1 , 2 ∈ C(H, H) and there exist positive constants 1 , 2 such that for all ∈ H, ( ) exists and is continuous. Further, there exists a positive constant ℎ such that (H8) The function : × × B → L(K, H) is continuous and there exists a positive constant such that for all , ∈ and ] ∈ B (H9) The function : B → B is continuous and there exists a positive constant such that for all , ∈ B, ∈ 0 (H10) Assume that the following relationship holds: (1 + 2 1 ) < 1,

Main Results
In this section, we will investigate the existence of mild solutions for a class of second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions and infinite delays in Hilbert spaces. We consider the operator Π : B → B defined by For ∈ B, we defined̃bỹ Theñ∈ B .
Let B 0 = { ∈ B : 0 = 0 ∈ B}. For any ∈ B 0 , we have and thus (B 0 , ‖ ⋅ ‖ ) is a Banach space. Set Then ⊆ B 0 is uniformly bounded, and for ∈ , by Lemma 4, we have Now, we decompose Π as Π = Π 1 + Π 2 , where Obviously, the operator Π having a fixed point is equivalent to Π having one. Now, we will show that the operators Π 1 , Π 2 satisfy all the conditions of Lemma 5.
Proof. Let , V ∈ B 0 . Then, by our assumptions and Lemma 4, for each ∈ , we have Since ‖ 0 ‖ 2 B = 0 and ‖V 0 ‖ 2 B = 0. Taking the supremum over , we obtain By assumption (H10), we conclude that Π 1 is a contraction on B 0 . Thus we have completed the proof of Lemma 7.

Lemma 8. Let the assumptions (H1)-(H10) hold. Then Π 2 is completely continuous.
Proof. The proof of the lemma is long. Therefore it is convenient to divide it into the following four steps.
Finally, by the Arzelá-Ascoli theorem, we can conclude that the operator Π 2 is completely continuous. Thus we have completed the proof of Lemma 8.
In order to study the existence results for system (3), we consider the following nonlinear operator equation: whereΠ is already defined. The following lemma proves that a priori bound exists for the solution of the above equation.
Proof. From (41), by our assumptions, Hölder's inequality and Burkholder-Davis-Gundy's inequality, for ∈ , we have Thus, again by Lemma 4, for every ∈ , we obtain Now, we consider the function defined by Then the function ( ) is nondecreasing in , and we get Consequently, Denoting the right-hand side of the above inequality by ( ).
This implies that which shows that ( ) < ∞. Thus, there exists a constant ( ,Ω, Ω 1 , Ω 2 ) such that ( ) ≤ * , ∈ . So, we get Thus we have completed the proof of Lemma 9. Now, we state the main result of our paper.
Remark 11. In recent years, the stochastic differential equations with Poisson jumps have become very important in modeling the phenomena arising in the fields, such as economics, finance, physics, biology, medicine, and other sciences. It is inspiring that a large number of results about the existence, uniqueness, stability, and invariant measures of stochastic differential equations with Poisson jumps have been reported in the literature. For instance, in [20], Luo and Liu studied the stability of infinite dimensional stochastic evolution with memory and Markovian jumps. Albeverio et al. [21] discussed the existence of global solutions and invariant measures for stochastic differential equations driven by Poisson-type noise with non-Lipschitz coefficients. But there has not been any result on the existence for secondorder impulsive neutral stochastic integrodifferential equations with infinite delays and Poisson jumps. This situation motivates our present research. Therefore, in this remark, we will study the well-posedness for second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions, infinite delays, and Poisson jumps in the form where the functions , , 1 , 2 , , , and 1 , 2 are defined as in Theorem 10; : × H × U → H are appropriate mappings which will be specified later;̃( , V) is a compensated Poisson random measure induced by Poisson point process (⋅), which is independent of the Wiener process and takes values in a measurable space (U, B(U)) with a -finite intensity measure ( V) by ( , V) the Poisson counting measure associated with (⋅); that is, ( , U) = ∑ ∈ , ≤ I U ( ( )), for any measurable set U ∈ B(K − {0}), which denotes the Borel -field of (K − {0}). Let Denote by P 2 ( × U; H) the space of all predictable mappings Then, we may define the H-valued stochastic integral ∫ 0 ∫ U ( , V)̃( , V), which is a centred square-integrable martingale. For the construction of this kind of integral, we refer the reader to Protter [22]. Let ( ) ∈ L 2 (Ω, B) and let 1 ( ) be H-valued Fmeasurable random variable independent of the Wiener process and the Poisson point process (⋅), with finite second moment.
Next, we give the definition of mild solution for (11).
To establish the well-posedness of mild solution for second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions, infinite delays, and Poisson jumps, we need the following assumption.
(H11) For any , ∈ H, and ∈ such that where is a positive constant.
Theorem 13. Assume that the assumptions (H1)-(H11) hold. If then the system (11) has at least one mild solution on .
Proof. By adapting and employing the techniques used in Theorem 10, we can easily prove the conclusion of Theorem 13. Indeed, similar to the above discussion, we consider the mapping Π : B 0 → B 0 defined by Π ( ) = 0, for ∈ 0 and Next, we will show that the operators Π 1 , Π 2 satisfy all the conditions of Lemma 5. Lemma 8 has shown that the operator Π 2 is completely continuous. Moreover, by assumptions (H1)-(H3), (H11), Lemma 4, the Burkholder-Davis-Gundy inequality for pure jump stochastic integral in Hilbert space (Lemma 2.2 in [20]), and assumption (61), for each ∈ , we conclude immediately that Π 1 is contractive.
Finally, by using the same arguments as Theorem 10, we infer that there exists a mild solution of the system (11). This completes the proof of Theorem 13.

Application
As we know, wave equations subject to random excitations have been intensively studied in the last forty years for their applications in physics, relativistic quantum mechanics, or oceanography; see, for instance, [23][24][25][26][27] and the references therein. The stochastic wave equation is one of the fundamental stochastic partial differential equations of hyperbolic type. The well-posedness of its solutions is significantly different from those of solutions to other stochastic partial differential equations, such as the stochastic heat equation or the stochastic Laplace equation. Therefore, in this section, an example on the stochastic nonlinear wave equation will be provided to illustrate the obtained theory. Specifically, we consider the existence of the following impulsive neutral stochastic nonlinear wave equations with nonlocal conditions and infinite delays of the form where ( ) is a standard one-dimensional Wiener process in H defined on a stochastic basis (Ω, F, P), 0 < 1 < 2 < ⋅ ⋅ ⋅ < < , ∈ N, 0 = 0 < 1 < ⋅ ⋅ ⋅ < < are prefixed numbers, and ∈ B. We take H = 2 ([0, ]) with the norm ‖ ⋅ ‖. Define Using (66), one can easily verify that the operators ( ) defined by form a cosine function on H, with associated sine function It is clear that (see [17]) for all ∈ H, ∈ R, (⋅) and ( Then, the system (64) can be written in the abstract form as the system (3). Further, we can impose some suitable conditions on the above defined functions as those in the assumptions (H1)-(H10). Therefore, by Theorem 10, we can conclude that the system (64) has a mild solution on .

Conclusion
In this paper, we have discussed the existence for a class of second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions and infinite delays in a Hilbert space. By using the Krasnoselskii-Schaefer-type fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators, the wellposedness of mild solution for the second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions and infinite delays is obtained. Besides, if the system (3) is added to the Poisson jumps, then we also get the existence of mild solution for second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions, infinite delays, and Poisson jumps. Finally, an example illustrating the applicability of the general theory is also provided.