Stability for stochastic neutral integro-differential equations with infinite delay and Poisson jumps

ABSTRACT This paper investigates a kind of stochastic neutral integro-differential equations with infinite delay and Poisson jumps in the concrete-fading memory-phase space . We suppose that the linear part has a resolvent operator and the nonlinear terms are globally Lipschitzian. We introduce sufficient conditions that ensure the existence and uniqueness of mild solutions by using successive approximation. Moreover, we target exponential stability, including moment exponential stability in -th ( ) and almost surely exponential stability of solutions and their maps. An example illustrates the potential of the main result.


Introduction
Research on stochastic differential equations with delay has received attention over the last few decades because of their appropriateness to describe physical systems subject to delays, such as the ones found in biology, medicine, epidemiology, chemistry, physics, and economics (see Helge et al., 2010;Intissar., 2020;Kostikov & Romanenkov, 2020;Mao, 2007;Trung, 2020 for a brief overview). The qualitative and quantitative properties of solutions of stochastic differential equations with delay, such as the existence, uniqueness, controllability, and stability, have been considered by several authors (see Bouzahir et al., 2017;Dieye et al., 2017;Diop et al., 2014;Taniguchi et al., 2002;Zouine et al., 2020). One point, in particular, has received a lot of attention: the study of existence and asymptotic behavior of mild solutions of some stochastic differential equations on Hilbert spaces, such as the semigroup approach (Taniguchi et al., 2002), comparison theorem (Govindan, 2003), Razumikhin-type theorem (Kai & Yufeng, 2006), analytic technique (Taniguchi, 1998), and Banach fixed-point principle (Diop et al., 2014).
The literature shows many dynamical systems modeled by neutral stochastic partial differential equations 2 (Chen et al., 2014;Cui et al., 2011) [9]. For these equations, some contain the derivatives of delayed states, which differ from stochastic partial differential equations with delays that depend on the present and past states only (for more details on this theory and its applications, see Mao et al., 2017;Yue, 2014). Stochastic integro-differential equations have been intensively studied, with special attention paid to qualitative properties, such as stability, regularity, periodicity, control problems, and optimality conditions (see Dieye et al., 2019;Diop et al., 2014). Due to the existence of an integral term in the equations, we use here the theory of the resolvent operator instead of the strongly continuous semigroups operator (see Grimmer, 1982 for further details).
Yet, most of the researchers dealing with exponential stability have limited their research to finite delay (see Dieye et al., 2017;Diop et al., 2014). Regarding the infinite delay, most investigations have been done for the case of continuous dependence of solutions on the initial value, considering exponential and asymptotic estimates (see, for instance, the papers Cui & Yan, 2012;Mao et al., 2017;Ren & Xia, 2009;Yue, 2014 for an account on phase spaces). It is noteworthy that few contributions exist for characterizing the exponential stability of stochastic equations with infinite delays (see Jiang et al., 2016;Wu et al., 2017). Jiang et al. (2016) showed the exponential stability for a class of second-order neutral stochastic partial differential equations with infinite delays and impulses using the integral inequality technique. Wu et al. (2017) showed boundedness in the mean square and convergence for both solutions and their maps in the phase space C μ by using the Itô formula. Motivated by the above discussion, consider the following neutral stochastic integro-differential equations with infinite delay and Poisson jumps given on the complete probability space ðΩ; F ; PÞ: System (1) holds with u 0 ð: ; þ1Þ � C μ � 7 !H are appropriate functions, the history u t : ðÀ 1; 0�7 !H ; t � 0, such that u t ðθÞ ¼ uðt þ θÞ belongs to the phase space C μ . The process wðtÞ represents a Wiener process on a separable Hilbert space K and Ñ is a compensated Poisson random measure.
To the best of the authors' knowledge, this paper is the first to present a study of the existence and exponential stability of neutral stochastic integrodifferential equations with infinite delay and Poisson jumps. The main contribution of this paper is to find conditions to ensure existence, uniqueness, exponential stability in q-th moment for q � 2 and almost surely exponential stability of solutions and their maps of (1). We show the result using stochastic techniques and the resolvent operator theory, as defined in Grimmer (1982). It is worth mentioning that Diop et al. (2014) studied the system (1) with finite delay. They focused only on the existence of mild solutions and their exponential stability in mean square (Diop et al., 2014). For this reason, our approach can be seen as an extension of the result of Diop et al. (2014) for the infinite delay case.
The organization of this paper is as follows. Some notations and preliminary results are presented in Section 2. The existence and uniqueness of mild solutions for neutral stochastic integro-differential equations with infinite delay are shown in Section 3. Conditions assuring moment exponential stability in the q-th (q � 2) and almost surely exponential stability of the solution uðtÞ, and the solution maps u t , t � 0 are shown in Section 4. Finally, an example that illustrates our results is presented in Section 5.

Notations and preliminary results
Let H and K be two real separable Hilbert spaces (c.f. Taniguchi et al., 2002). We let also LðK ; H Þ be the space of bounded linear operators from K into H associated with k : k to represent the norm operator in H , K , and LðK ; H Þ. We assume that System (1) is equipped with a normal filtrationfF t g t�0 .
Denote by N, the Poisson random measure induced by the σ À finite stationary F t À adapted Poisson point process pð:Þ taking values in a measurable space ð; BðÞÞ, and define the compensated Poisson random measure Ñ as Ñ ðdt; dyÞ ¼ Nðdt; dyÞ À πðdyÞdt, where Nðð0; t� � ΔÞ : for Δ 2 and π is the characteristic measure of N.
Supposed that fωðtÞ; t � 0g represents a K -valued Wiener process which is independent of the Poisson point process on the probability space ðΩ; F ; fF t g t�0 ; PÞ with a positive self-adjoint covariance operator Q. In addition, we suppose that there exists a complete orthonormal system e i in K , a bounded sequence of positive real numbers λ i such that Qe i ¼ λ i e i ; i ¼ 1; 2; . . . , and a sequence fβ i ðtÞg i > 1 of independent standard Brownian motions such that ωðtÞ ¼ P þ1 i¼1 ffi ffi ffi ffi λ i p β i ðtÞe i for t � 0 and F t is the σ-algebra generated by fωðsÞ : 0 � s � tg (see (Taniguchi et al., 2002)). We consider the subspace K 0 ¼ Q 1=2 K of K , it is a Hilbert space equipped with the inner product hu; vi K 0 ¼ hQ À 1=2 u; Q À 1=2 vi K . Let L 0 2 ¼ L 2 ðK 0 ; H Þ be the space of all Hilbert-Schmidt operators from K 0 to H . L 0 2 is a separable Hilbert space endowed with the norm k vk L 0 2 ¼ trððvQ 1=2 ÞðvQ 1=2 Þ � Þ for any v 2 L 0 2 . Hereafter, A and ΥðtÞ are closed linear operators on a Banach space denoted by X, and Y is the Banach space DðAÞ endowed with the graph norm jyj Y :¼ jAyj þ jyj for y 2 Y.
The notations Cð½0; þ1Þ; YÞ, C 1 ð½0; þ1Þ; XÞ and LðY; XÞ represent the space of continuous functions from ½0; þ1Þ into Y, the space of continuously differentiable functions from ½0; þ1Þ into X and the set of bounded linear operators from Y into X, respectively.

Preliminaries on partial integro-differential equations
We now consider the problem with νð0Þ ¼ ν 0 2 X.
The following two conditions, borrowed from Grimmer (1982), are sufficient to assure the existence of solutions for equation (2).
(A 1 ) The operator A is an infinitesimal generator of a C 0 -semigroup on X.
(A 2 ) For all t � 0,ΥðtÞ denotes a closed, continuous linear operator from DðAÞ to X and ΥðtÞ belongs to LðY; XÞ. For any y 2 Y, the map t7 !ΥðtÞy is bounded, differentiable, and its derivative dΥðtÞy=dt is bounded and uniformly continuous on ½0; 1Þ.
We now recall conditions that assure existence of solutions for the deterministic, integro-differential equation with νð0Þ ¼ ν 0 2 X and m : ½0; þ1Þ ! X is a continuous function.
Remark 2. We consider the assumption γðt; 0Þ ¼ f 1 ðt; 0Þ ¼ g 1 ðt; 0Þ ¼ � h 1 ðt; 0; �Þ ¼ 0 for all t � 0, to guarantee that there exists a zero equilibrium solution to the stochastic equation (1). If this assumption does not hold, the equilibrium solution for equation (1) can always be transformed into the zero equilibrium solution of another equation.

Existence and uniqueness
In this section, we present sufficient conditions to guarantee the existence and uniqueness of mild solutions of the equation in (1). To do so, we use the method of successive approximations and some stochastic analysis techniques. Still, we have to develop some new techniques to deal with infinite delay. Hereafter, we replace X by the Hilbert space H in ðA 1 Þ and ðA 2 Þ. Now, we present the following main result.
Proof. The proof of this theorem uses the following sequence of successive approximations that is defined for t � 0 by u n ðtÞ ¼ ϕðtÞ for any n 2 N and for 0 � t � T by for any n � 1 and u 0 ðtÞ ¼ <ðtÞϕð0Þ when 0 � t � T. Take M T ¼ sup 0�t�T k <ðtÞk LðH Þ , from the uniform boundedness M T < 1. The remaining arguments are divided into three main steps.
Step 2: Now we show that u n ; n 2 N is a Cauchy sequence. From the construction of successive approximations, we have u n ðtÞ ¼ u nÀ 1 ðtÞ on ðÀ 1; 0�, for n � 1. For t 2 ½0; T�, we can prove that where I 1 ¼ E k <ðtÞγð0; ϕÞk q . Using ðA 4 Þ, we obtain To show the result for I 2 , we combine ðA 4 Þ with the inequality ða þ bÞ q � 2 qÀ 1 ða q þ b q Þ to obtain Regarding I 3 , combining ðA 5 Þ and the Holder inequality produces Similarly to Step 1, from Lemma 6.2, we have Finally, by employing ðA 5 Þ and Lemma 6.3, we can proceed similarly to obtain Substituting (16)-(20) into (15) results On the other hand, note that k u 1 ðtÞ À u 0 ðtÞk q and recalling that 1 À 10 qÀ 1 K 0 > 0, we can deduce E sup 0�s�t k u 1 ðtÞ À u 0 ðtÞk q �C 5 ðq; TÞ 1 À 10 qÀ 1 K 0 ¼:C 6 ðq; TÞ: By similar arguments as above, we get E k u 2 ðtÞ À u 1 ðtÞk q Indeed, by repeating the iteration as in, for all n � 0, we obtain E sup 0�s�t k u nþ1 ðsÞ À u n ðsÞk q � ðC 7 ðq; TÞtÞ n n!C 6 ðq; TÞ: Therefore, for any m > n � 0, we obtain E k u m ðtÞ À u n ðtÞk q �C 6 ðq; TÞ X mÀ 1 k¼n ðC 7 ðq; TÞtÞ k k! ! 0 as n ! þ1: This argument proves that u n ðtÞ; n � 0 is a Cauchy sequence in L q ðΩ; H Þ: Step 3: Now we prove the existence and uniqueness of the solution of equation (1). One has that u n ðtÞ ! uðtÞ as n ! 1 in L q . The Borel-Cantelli lemma gives us u n ðtÞ uniformly converge to uðtÞ as n ! 1, for t 2 ðÀ 1; T�. Using Assumption ðA 4 Þ and ðA 5 Þ, for all t 2 ½0; T�, we can prove the next inequality holds: k uðsÞ n À uðsÞk q ! 0; as n ! 1: E k u n s À u s k q C μ ds ! 0; as n ! 1: Therefore, we take the limits on both sides of (6) with respect to n to obtain uðtÞ ¼ <ðtÞ½ϕð0Þ þ γð0; ϕÞ� À γðt; u t Þ We can check the uniqueness of the solution by employing the Gronwall lemma, together with a similar argument as that used in the proof of Step 2. This argument completes the proof.
Remark 3. We point out that the local solution exists and it is unique on ðÀ 1; T� for each real number T > 0, then, existence of the solution to equation (1) is global, that is, uðtÞ is defined in ðÀ 1; þ1Þ.

Exponential stability
Here, we use the Gronwall lemma and the properties of the concrete-phase space C μ to obtain the exponential stability for the solutions of the stochastic equation (1) and their maps. Other researchers have studied the stability as well (Dieye et al., 2017(Dieye et al., , 2019, but their results are based on the stochastic convolution, an approach completely detached from ours. Definition 4.1. The mild solution of (1) is said to be q-th moment exponentially stable when ðq � 2Þ if, for any initial value ϕ 2 C μ , F 0 À measurable, there exist two positive real numbers α 1 > 0 and α 2 > 0 such that E k uðtÞk q � α 1 E k ϕ k q C μ expðÀ α 2 tÞ, for all t � 0: For the sake of notational simplicity, we define the function Now, we can introduce the main result of this paper in the following theorem.
Theorem 4.1 Suppose that ðA 3 Þ and all conditions of Theorem 3.1 hold. Suppose in addition that the next two inequalities hold: qμ > λ and C 8 ðqÞ Then, the mild solution uðtÞ and the solution maps u t to equation (1) are q-th moments exponentially stable.
Proof. q-th moment exponential stability of uðtÞ: Combining (5), ðA 3 Þ, ðA 4 Þ and ðA 5 Þ, we can write <ðt À sÞ� h 1 ðs; u s ; zÞÑðds; dzÞk q : (24) Note that and that By Holder inequality, it yields that We note that ð t 0 e À λðtÀ sÞ ds < λ À 1 ; therefore, the last inequality becomes Recalling Lemma 6.2 and Assumption ðA 3 Þ, as before, we use Holder inequality to get that Finally, from Lemma 6.3, Assumptions ðA 3 Þ and ðA 5 Þ and by Holder inequality, we can write � D q M q K 1 ðq À 2ÞK 1 2ðq À 1Þλ If q ¼ 2, the last two inequalities hold true with convention 0 0 :¼ 1. Substituting (25)-(29) into (24), we obtain where C 8 ðqÞ satisfies (22). Multiplying both sides of (30) by e λt yields From properties of the norm k :k C μ (see Appendix), we can write for any t � 0 which implies that Recall that K 0 < 1 10 qÀ 1 and λ À qμ < 0, hence E sup 0�s�t e λs k uðsÞk q which implies that E k uðtÞk q �C 9 ðqÞe ðC 10 ðqÞÀ λÞt : Therefore, from the condition in (23), the result of the q-th moment exponential stability of solution uðtÞ is satisfied. q-th moment exponential stability of u t : For any t � 0, we have multiplying both sides of the last inequality by e λt , we obtain e λt k u t k q C μ �k ϕ k q C μ þ sup 0�s�t e λs k uðsÞk q : Using the Gronwall lemma, we obtain E sup 0�s�t e λs k us k q Cμ � The inequality in (33) assures the exponential stability in q-th moment of the solution maps u t .□

Almost surely exponential stability
Definition 4.2. The mild solution of (1) is said to be almost surely exponentially stable if the following inequality is guaranteed almost surely lim sup t!1 1 t log k uðtÞ k < 0; for any F 0 À measurable initial value ϕ 2 C μ .
We present the main result of this section. 1 t log k u t k C μ � ðC 10 ðqÞ À λÞε q almost surely; for any ε 2 ð0; 1Þ, which implies that uðtÞ and u t are almost surely exponentially stable.
Proof. Now we show ðiÞ for all n � 0. It follows from Theorem 4.1 that E sup n�t�nþ1 k uðtÞk q �C 9 ðqÞe ðC 10 ðqÞÀ λÞðnþ1Þ ¼C 9 ðqÞeC 10 ðqÞÀ λ :e ðC 10 ðqÞÀ λÞn : Let I n be the interval ½n; n þ 1�, for any ε 2 ð0; 1Þ. Define C 10 ðqÞ ¼C 8 ðqÞ 1 À 5 qÀ 1 K 0 : Since, from assumption, 1 À ε > 0 and C 10 ðqÞ À λ < 0, we can use the Markov inequality to write P sup The rightmost term of (34) is bounded from above by P 1 n¼0 e ð1À εÞðC 10 ðqÞÀ λÞn < 1. Therefore, the Borel-Cantelli lemma assures that there exists an integer n 0 such that, for all n � n 0 , sup t2I n k uðtÞk q � e ðC 10 ðqÞÀ λÞnε almost surely: Thus, if t 2 I n and n � n 0 , we get 1 t log k uðtÞk q � 1 n ðC 10 ðqÞ À λÞnε ¼ ðC 10 ðqÞ À λÞε almost surely: It follows that lim sup t!1 1 t log k uðtÞ k� ðC 10 ðqÞ À λÞε q almost surely; which shows that ðiÞ is satisfied. The argument to prove ðiiÞ follows analogous reasoning. Namely, one can use the fact that the solution maps u t are q-th moment exponentially stable and can conclude the result by repeating that previous reasoning. The details are omitted.□ Remark 4. The author of (Grimmer, 1982) presents sufficient conditions for the exponential stability of the resolvent operator ðRðtÞÞ t�0 . The paper (Grimmer, 1982) shows λ and M from the contraction of the C 0semigroup ðSðtÞÞ t�0 and the properties of the function b by using the infinitesimal generator of the translation semigroup.

Example
Set μ > 0 and θ 2 CðR þ ; ðÀ 1; 0�Þ. Consider the following neutral stochastic integro-differential equation with infinite delay and Poisson jumps of the form: k RðtÞ k� Me À λt , see Remark for more details. We now suppose that the next three conditions are valid.
For the case q ¼ 2, by the convention 0 0 ¼ 1, we have the following constants Thus, by Theorems 4.1 and 4.2, these solutions and their maps are mean square exponentially stable and almost surely exponentially stable provided that .

Conclusion
In this paper, we have studied neutral stochastic integrodifferential equations with infinite delay and Poisson jumps under global Lipschitz conditions. In this study, we have used successive approximations to show the existence of mild solutions. We also prove the exponential stability of solutions and their maps. It is uncertain whether our approach copes with weaker conditions, such as local Lipschitz and non-Lipschitz conditions. The results in this paper can be seen as an extension of the ones in (Diop et al., 2014) because we consider the infinite-delay case here; in contrast, the authors of (Diop et al., 2014) have considered the finite delay case.

PUBLIC INTEREST STATEMENT
Stochastic processes are much used to represent mathematical models for phenomena and systems that vary randomly. These phenomena and systems may include the growth of bacterial population, price changes in the stock market, extinction and persistence of diseases, movement of a gas molecule and the number of phone calls. Representing such random phenomena motivates the study of stochastic differential equations. Characterizing the stability of stochastic systems has become a central topic in systems sciences. In this paper, we focus on the stability of stochastic integro-differential equations with noise. ðiiiÞ k u t k Cμ � e À μt k u 0 k Cμ þ e À μt sup 0�s�t e μs k uðtÞ k . ðbÞ

Notes on contributors
The corresponding history t ! u t of the function u in ðaÞ is a C μ -valued continuous function in ½0; TÞ. ðcÞ The space C μ is complete. ðdÞ Suppose that fφ n g is a Cauchy sequence in C μ , and if fφ n ðθÞg converges to φðθÞ for θ on any compact subset of the interval ðÀ 1; 0�, then φ 2 C μ and k φ n À φk Cμ ! 0 as n ! 0.