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.
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. , the papers [2–5], 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, [6–8] and the references therein).
Recently, in , Cui and Yan proved sufficient conditions for the existence of fractional neutral stochastic integrodifferential equations with infinite delay of the form where 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  established the controllability of the following damped second-order impulsive neutral functional differential systems with infinite delay: where is a bounded linear operator on a Banach space with .
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 , Balasubramaniam and Muthukumar discussed on approximate controllability of second-order stochastic distributed implicit functional differential systems with infinite delay. Cui and Yan  investigated the existence of mild solutions for impulsive neutral second-order stochastic evolution equations with nonlocal conditions. Mahmudov and McKibben  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 integrodifferential 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 ; is the infinitesimal generator of a strongly continuous cosine family on . The history , for , belongs to the phase space , which will be described in Section 2. Assume that the mappings , , , , , , are appropriate functions to be specified later. Furthermore, let be prefixed points, and 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 and be -valued -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.
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 [14–16].
Let and denote two real separable Hilbert spaces, with their vector norms and their inner products, respectively. We denote by the set of all linear bounded operators from into , 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 be a complete filtered probability space satisfying the usual condition (i.e., it is right continuous and contains all -null sets). Let be a -Wiener process defined on the probability space with the covariance operator such that . We assume that there exists a complete orthonormal system in , a bounded sequence of nonnegative real numbers such that , and a sequence of independent Brownian motions such that .
Let be the space of all Hilbert-Schmidt operators from to with the inner product , where is the adjoint of the operator .
The collection of all strongly measurable, square-integrable -valued random variables, denoted by , is a Banach space equipped with norm . Let be the Banach space of all continuous maps from to satisfying the condition . An important subspace is given by .
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.
Definition 1. (1) The one-parameter family is said to be a strongly continuous cosine family if the following hold:(i), is the identity operators in ;(ii) is continuous in on for any ;(iii) for all .
(2) The corresponding strongly continuous sine family , associated with the given strongly continuous cosine family , is defined by
(3) The infinitesimal generator of is given by for all .
It is well known that the infinitesimal generator is a closed, densely defined operator on , and the following properties hold; see Travis and Webb .
Proposition 2. Suppose that is the infinitesimal generator of a cosine family of operators . Then, the following hold:(i)there exist a pair of constants and such that and hence, ;(ii), for all ;(iii)there exist such that , .
Thanks to Proposition 2 and the uniform boundedness principle, as a direct consequence we see that both and are uniformly bounded by .
The existence of solutions for the second-order linear abstract Cauchy problem where is an integrable function has been discussed in . Similarly, the existence of solutions of the semilinear second-order abstract Cauchy problem has been treated in .
Definition 3. The function given by is called a mild solution of (6), and that when , 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 . Assume that is a continuous function with . For any , we define If is endowed with the norm then it is clear that is a Banach space .
Let . We consider the space where is the restriction of to , . Set as a seminorm in defined by
Lemma 4 (see ). Assume that , then for , . Moreover,
Next, we present the Krasnoselskii-Schaefer-type fixed point theorem appearing in , which is our main tool.
Lemma 5 (see ). Let and be two operators of such that (a) is a contraction, and(b) is completely continuous. Then, either(i)the operator equation has a solution, or(ii)the set is unbounded for .
Now, we give the definition of mild solution for (3).
Definition 6. An -adapted stochastic process is called a mild solution of (3) on if satisfying and satisfying ; the functions and are integrable on such that the following conditions hold:(i) is a -valued stochastic process;(ii)for arbitrary , satisfies the following integral equation: (iii), , .
In this paper, we will work under the following assumptions.(H1)The cosine family of operators on and the corresponding sine family are compact for , and there exist positive constants , such that for all , (H2)There exists a positive constant such that for all (H3)The function is continuous and there exists a positive constant such that for all ,,
(H4)For each , the function is continuous and for each , the function is strongly measurable. There exists an integrable function and a positive constant such that
where is a continuous nondecreasing function. Assume that the finite bound of is .(H5)The function satisfies the following Carathéodory conditions:(i) is measurable for each ;(ii) is continuous for almost all ;(iii) for almost all , , where and is a continuous increasing function.(H6)The functions and there exist positive constants , such that for all ,(H7)For each , exists and is continuous. Further, there exists a positive constant such that
(H8)The function is continuous and there exists a positive constant such that for all and
(H9)The function is continuous and there exists a positive constant such that for all ,
(H10)Assume that the following relationship holds:
3. 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 defined by
For , we defined by Then .
Let , . It is easy to see that satisfies (14) if and only if satisfies , , and
Let . For any , we have and thus is a Banach space. Set Then is uniformly bounded, and for , by Lemma 4, we have Define the map defined by , for and
Now, we decompose as , where Obviously, the operator having a fixed point is equivalent to having one. Now, we will show that the operators , satisfy all the conditions of Lemma 5.
Lemma 7. Let the assumptions (H1)–(H10) hold. Then is contractive.
Proof. Let . Then, by our assumptions and Lemma 4, for each , we have Since and . Taking the supremum over , we obtain By assumption (H10), we conclude that is a contraction on . Thus we have completed the proof of Lemma 7.
Lemma 8. Let the assumptions (H1)–(H10) hold. Then is completely continuous.
Proof. The proof of the lemma is long. Therefore it is convenient to divide it into the following four steps.
Step 1. maps bounded sets to bounded sets in .
Indeed, it is enough to show that there exists a positive constant such that for each , one has . By our assumptions, Hölder's inequality, and Burkholder-Davis-Gundy's inequality, for , we have Then, for each , we get .
Step 2. maps bounded sets into equicontinuous sets of .
Let , for each . Let . Then, by our assumptions, Hölder's inequality, and Burkholder-Davis-Gundy's inequality, we get The right-hand side of the above inequality which is independent of tends to zero as . Thus, the set is equicontinuous. (Note that we consider only the case ; this proves the equicontinuity for the case where , . Easily we prove the equicontinuity for the case where . And also the other cases and are very simple.)
Step 3. is continuous.
Let , with in . Then, there is a number such that for all and a.e. , so and . In view of (29), for , we have .
By Definition 6, the assumptions (H5)–(H8), for each , and since thanks to the dominated convergence theorem, we obtain that Thus, is continuous.
Step 4. maps onto a precompact set in . That is, for every fixed , the set is precompact in .
It is obvious that is precompact. Let be fixed and a real number satisfying . For , we define the operator Since , , , are compact, it follows that the set is precompact in , for every , . Moreover, also by Hölder's inequality and Burkholder-Davis-Gundy's inequality, for each , we get and there are precompact sets arbitrarily close to the set . Thus, the set is precompact in .
Finally, by the Arzelá-Ascoli theorem, we can conclude that the operator 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: where is already defined. The following lemma proves that a priori bound exists for the solution of the above equation.
Lemma 9. If the assumptions (H1)–(H10) hold. Then, there exists a priori bound such that where depends only on and on the functions and , .
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 . Then , , , and Since , are nondecreasing, for , we get Let Then , , and This implies that which shows that . Thus, there exists a constant such that , . So, we get Thus we have completed the proof of Lemma 9.
Now, we state the main result of our paper.
Theorem 10. Assume that the assumptions (H1)–(H10) hold. Then, the system (3) has at least one mild solution on .
Proof. Let us take the set Then, by Lemma 9, for any , we have , and hence This conclude that is bounded on . Consequently, by Krasnoselskii-Schaefer-type fixed point theorem, there exists a fixed point for on such that . Since , , is a fixed point of the operator which is a mild solution of system (3). The proof for Theorem 10 is thus complete.
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 , Luo and Liu studied the stability of infinite dimensional stochastic evolution with memory and Markovian jumps. Albeverio et al.  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 second- order 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