Exponential behavior of neutral impulsive stochastic integro-di erential equations driven by Poisson jumps and Rosenblatt process

Abstract: In this article, we are concernedwith the neutral impulsive stochastic integro-di erential equations driven by Poisson jumps and Rosenblatt process. By using resolvent operator and some analysis techniques, we ensure existence and uniqueness of solutions. Further, we investigate exponential stability of mild solutions. We have also given an example to illustrate our theoretical results.


Introduction
In the past decades, the theory of nonlinear functional di erential or integro-di erential equations with resolvent operators has become an active research eld due to their applications in many physical phenomena. The resolvent operator is comparable to the semigroup operator for abstract di erential equations in Banach spaces. However, the resolvent operator does not satisfy semigroup properties. The study of deterministic neutral functional di erential equations was initiated by Hale and Mayer [17]. For more details on the theory and their applications, we also refer the readers to Hale and Lunel [18], Kolmanovkii and Nosov [20] and so on. Deterministic and stochastic di erential equations have gained great popularity in the last few years due to their use in many areas, such as physics, electronics, control theory, engineering and economics. Several authors have considered the existence, uniqueness and asymptotic behavior of mild solutions, and many important theory and applications ndings have been obtained. For more details we refer to the papers by Ali et al. [4], El-Borai et al. [11], Gorec and Sathanantham [12], Gupta and Dabas [15], Gupta and al. [16], Laksmikantha [22], Ahmed [2], Ahmed et al. [3], Arthi and Balachandran [5],Gupta et al. [14], Levin et al. [24].
On the other side, fractional Brownian motion was intensively explored due to their applications in various domain. We point out that, a fractional Brownian motion (fBm) of Hurst index H ∈ ( , ) (see [21,26]  We also mention that fBm is not a semimartingale and when H = / , the fBm becomes standard Brownian motion . Further, if H > / , the fBm B H have a long-memory and this property makes it an ideal process to modeling in biology, mathematics nance [7,8] etc. Moreover, the fBm belongs to Hermite family processes, it's selfsimilar which being de ned as limits that appear in the so-called Non-Central Theorem. For d ≥ , they have the following representation When d = the process become a fractional Brownian motion, thus Taqqu [32] have named the Rosenblatt process when d = , which is not Gaussian process but they have stationary increments and long rangedependency. Recently, the Rosenblatt process have attracted attention of many authors due to their properties. For example, Meajima and Tudor [29], Veillette and Taqqu [36] have given many important properties of distributions, Bernet and Tudor [6], Viens and Tudor [35] established the construction of estimator for the self-similarity parameter H. Based on the above works, we investigate the following neutral stochastic functional integro-di erential equations with delay and impulses e ects where A : D(A) ⊂ H → H is the in nitesimal generator of a C -semigroup (T(t)) t≥ of bounded linear operators in a Hilbert space H; for t , Υ(t) a closed linear operator on H, with D(A) ⊂ D(Υ). The impulsive moments t k satisfy the condition < t < t < · · · < t k < · · · , lim k→∞ t k = ∞, and u(t − k ) are the right and left limits of u(t) at t k , respectively which is the jump size of the state . For any t ∈ [ , T] and any continuous function u, the element of PC is de ned by  [25], where the authors used an impulsive integral inequality to prove their result. It should be mentioned that there is no work yet reported on the exponential stability of neutral impulsive stochastic integro-di erential equations driven by Poisson jumps and Rosenblatt process. Motivated by this facts, our main objective is to study the exponential stability for a class of neutral impulsive stochastic integro-di erential equations (1). In this paper, we derive existence and exponential results for the system (1) with the help of resolvent operator and xed point techniques. In the rst result, we obtain the su cient conditions proving existence and uniqueness of the mild solution of (1) by utilizing Banach xed point theorem under Lipschitz conditions on nonlinear terms. While in the second result, we have proved the exponential stability of mild solution via an integral inequality. Our article, expands the usefulness of stochastic integro-di erential equations, since the literature shows results for existence and exponential stability for such equations in the case of semigroup only (see [2,3,11,14,16] and the references therein ). The results obtained improve, extend and complete many other important ones in the literature.
The following is the organization of this paper. We recall some preliminary de nitions and outcomes in Section 2. Section 3 is devoted to investigate existence and uniqueness of mild solution. The exponential stability for the mild solution is also discussed. We give an example in the fourth section to illustrate the results. The last section is dedicated to conclude this paper.

. Rosenblatt process
In this subsection, we recall some basic concepts on the Rosenblatt process as well as the Wiener integral with respect to it. Consider (ξn) n∈Z a stationary Gaussian sequence with mean zero and variance such that its correlation function satis es that R(n) := E(ξ ξn) = n H− k L(n), with H ∈ ( , ) and L is a slowly varying function at in nity. Let g be a function of Hermite rank k, that is, if g admits the following expansion in Hermite polynomials j= g(ξ j ) converges as n → ∞, in the sense of nite dimensional distributions, to the process where the above integral is a Wiener-Itô multiple integral of order k with respect to the standard Brownian motion (B(y)) y∈R and c(H, k) is a positive normalization constant depending only on H and k. The process (Z k H (t)) t≥ is called as the Hermite process and it is H self-similar in the sense that for any c > , (Z k H (ct)) d = (c H Z k H (t)) and it has stationary increments. The fractional Brownian motion (which is obtained from (2) when k = ) is the most used Hermite process to study evolution equations due to its large range of applications. When k = in (2), Taqqu [32] named the process as the Rosenblatt process. The stationarity of increments, self-similarity and long range dependence (see Tindel, Tudor and Viens [33]) were made that the Rosenblatt process is very important in practical applications. However, it is noted that Rosenblatt process is not Gaussian. In fact, due to their properties (long range dependence, self-similarity), the fractional Brownian motion process has large utilization in practical models, for instance in telecommunications and hydrology. So, many researchers prefer to use fractional Brownian motion than other processes because it is Gaussian and it facilitate calculations. However in concrete situations when the Gaussianity is not plausible for the model, one can use the Rosenblatt process. In recent years, there exists many works that investigated on diverse theoretical aspects of the Rosenblatt process. For example, Leonenko and Ahn [23] gave the rate of convergence to the Rosenblatt process in the Non-Central Limit Theorem and the wavelet-type expansion has been presented by Abry and Pipiras [1]. Tudor [34] established, the representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a nite interval and developed the stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus (see also Maejima and Tudor [27]). For more details on Rosenblatt process, we refer the reader to Maejima and Tudor [28,29]), Pipiras and Taqqu [31] and the references therein.
Consider a time interval [ , T] with arbitrary xed horizon T and let Z H (t), t ∈ [ , T] be a onedimensional Rosenblatt process with parameter H ∈ ( , ). According to the work of Tudor [34], the Rosenblatt process with parameter H > can be written as where K H (t, s) is given by The covariance structure of the Rosenblatt process allows to construct Wiener integral with respect to it. We refer to Maejima and Tudor [27] for the de nition of Wiener integral with respect to general Hermite processes and to Kruk, Russo, and Tudor [19] for a more general context (see also Tudor [34]). Note that where the operator I is de ned on the set of functions f : [ , T] → R, which takes its values in the set of functions g : [ , T] → R and is given by Let f be an element of the set E of step functions on [ , T] of the form Then, it is natural to de ne its Wiener integral with respect to Z H as Let H be the set of functions f such that It follows that (see Tudor [34]) It has been proved in Maejima and Tudor [27] that the mapping de nes an isometry from E to L (Ω) and it can be extended continuously to an isometry from H to L (Ω) because E is dense in H. We call this extension as the Wiener integral of f ∈ H with respect to Z H . It is noted that the space H contains not only functions but its elements could be also distributions. Therefore it is suitable to know subspaces |H| of H : space |H| is not complete with respect to the norm . H but it is a Banach space with respect to the norm As a consequence, we have for some constant C(H) > . Let C(H) > stands for a positive constant depending only on H and its value may be di erent in di erent appearances.
De ne the linear operator where K is the kernel of Rosenblatt process in representation (3) Moreover, for f ∈ H, we have Let {Zn(t)} n∈N be a sequence of two-sided one dimensional Rosenblatt process mutually independent on (Ω, F, P). We consider a K-valued stochastic process Z Q (t) given by the following series: Moreover, if Q is a non-negative self-adjoint trace class operator, then this series converges in the space K, that is, it holds that Z Q (t) ∈ L (Ω, K). Then, we say that the above Z Q (t) is a K-valued Q-Rosenblatt process with covariance operator Q. For instance, if {σn} n∈N is a bounded sequence of non-negative real numbers such that Qen = σn en, by assuming that Q is a nuclear operator in K, then the stochastic process is well-de ned as a K-valued Q-Rosenblatt process. [34]). Let φ : Proof. Let {en} n∈N be the complete orthonormal basis of K introduced above. Applying (6) and Hölder inequality, we have

. Partial integro-di erential equations in Banach spaces
In this section, we recall some fundamental results needed to establish our main results. For the theory of resolvent operators we refer the reader to [13]. Throughout this paper, H is a Banach space, A and Υ(t) are closed linear operators on H. Y represents the Banach space D(A) equipped with the graph norm de ned by The notations C([ , +∞); Y), B(Y , H) stand for the space of all continuous functions from [ , +∞) into Y, the set of all bounded linear operators from Y into H, respectively. We consider the following Cauchy problem
The following assumptions are imposed on the system under consideration: (H1) A is the in nitesimal generator of a strongly C -semigroup {S(t)} t≥ on H.

(H2) For all t ≥ , Υ(t) is a closed linear operator from D(A) to H, and Υ(t) ∈ B(Y , H).
For any y ∈ Y, the map t → Υ(t)y is bounded, di erentiable and the derivative t → Υ ′ (t)y is bounded and uniformly continuous on R + .

. Existence of mild solution
In this section, we present and prove the existence and uniqueness of mild solutions of Eq. (1) In order to attain the result, we impose the following assumptions: (H4) There exists a constant K > such that, for ψ j ∈ PC, j = , , the mapping f : [ , +∞) × PC → H satis es the following Lipschitz condition for all t ∈ [ , T]: Now, to prove the existence result of mild solution of Eq.(1), it is su cient to show that Ψ has a xed point.
To this end we subdivide the proof into two steps.
Step 1 : First, we show that the map t → (Ψx)(t) is continuous on the interval [ , T]. Let l be su icently small, for u ∈Λ T and < t < T. We get By De nition 2.3-(ii) , we have lim l→ P = .
By using (ii) of (H3) we obtain that lim l→ P = .
By using Hölder inequality we have: Using Hölder inequality, we have E||f (s, us)|| ds.
Thus, we have lim l→ P = .
By using similar arguments to P and combining assumption (H5) we obtain lim l→ P = .
Also, with the same argument to P and using assumption (H5) we obtain that lim l→ P = .
Similarly, with the same argument to P and using assumption (H5) we obtain that lim l→ P = , lim l→ P = .
For P , application of Lemma 2.1 gives By De nition 2.3-(ii) we have and by De nition 2.3-(i) we have the inequality : and by the Lebesgue dominated convergence theorem, we get lim l→ P = .
By Lemma 2.1 we have and regarding to (H7), we get lim l→ P = .
By using De nition 2.3-(i) we have By combining assumption (H6) and De nition 2.3-(i), we have lim l→ P = .
Therefore,we can conclude lim Hence, the above arguments imply that function t → (Ψx)(t) is continuous on the interval [ , T].
Step 2: In this part of the proof, we will verify that Ψ is contraction mapping inΛ T with some T ≤ T to be speci ed later. Let u, v ∈Λ T and t ∈ [ , T]. By virtue of elementary inequality we obtain By using assumptions (H3)-(H6), De ntion 2.3 together with Hölder's inequality, we get Hence, we have sup where By inequality (9), we have Then there exists < T ≤ T such that < θ(T ) < and the operator Ψ is a contraction onΛ T and hence it has a unique xed point on

. Exponential stability
In this subsection, it is established the exponential stability in the mean square moment of the mild solution for Eq.(1), we need to state the following additional assumptions. (H9)The corresponding resolvent operator (Π(t)) t≥ of Eq. (7) veri es the following : There exist γ > and M > such that ||Π(t)|| ≤ Me −γt , for all t ≥ . (H10)There exist nonnegative real numbers R i ≥ and continuous functions where k := √ R .
Proof. From (10), it is possible to nd a suitable numberl > small enough such that Let assume that µ = γ −l and u(t) be a mild solution of Eq.(1). Then, from (8) we have By assumption (H10) we get where λ = α k .
According to (H9) and (H10) we obtain Employing (H9) , (H10) and Hölder's inequality, we get And we observe that k +k M + +∞ i= ω k < . Thus, the mild solution of Eq.(1) is exponentially stable in mean square moment, since k +κ µ + +∞ k= ω k < and by Lemma 2.2 we have the existence of two positive constants C and r such that E||u(t)|| ≤ Ce −rt for any t ≥ −τ, where θ > is the unique solution to the equation k + k µ−θ e rτ + +∞ k= ω k = and C = max{ν, ν(µ−r) ke rτ } > . This completes the proof of the theorem.

Illustration
This part consist to make an application of the theory studied above. Consider the impulsive neutral stochastic partial integrodi erential equations of the form where Z H Q is Rosenblatt process de ne on probability space (Ω, F, P) , β , β : Let H = Y = L [ , π] with the norm · and en(x) = π sin nx, n = , , · · · . Then (en) n∈N is a complete orthogonal basis in Y. In order to de ne the operator Q : Y → Y, we chose a sequence (σn) n≥ ⊂ R + and set Qen = σn en , and assume that ∞ We suppose that: (i) There exist a positive constant l , < πl < such that β (t, y ) − β (t, y ) ≤ l |y − y |, t ≥ , y , y ∈ R.
We have also that I k (ϕ ) − I k (ϕ ) ≤ d k ϕ − ϕ , where d k = β k , k=1,2,. . . and Thus, all assumptions of Theorem 3.1 are ful lled. Therefore, the existence of a mild solution of Eq. (18) follows. In addition, by Theorem 3.2, we easily see that the mild solution of Eq.(18) is exponentially stable in the 2nd moment.

Conclusion
In this article, we showed existence and unicity of mild solution to Eq.(1) by using Banach xed point theorem. Further, we investigated exponential stability of mild solutions. The last part of this paper is devoted to an example to illustrate our results.