Existence and exponential stability for neutral stochastic integro-differential equations with impulses driven by a Rosenblatt process

Manuscript submitted to doi: AIMS’ Journals Volume 00, Number 0, Xxxx XXXX pp. 000–000 EXISTENCE AND EXPONENTIAL STABILITY FOR NEUTRAL STOCHASTIC INTEGRO–DIFFERENTIAL EQUATIONS WITH IMPULSES DRIVEN BY A ROSENBLATT PROCESS Tom´ as Caraballoa, Carlos Ogouyandjoub, Fulbert Kuessi Allognissodeb, Mamadou Abdoul Diopc aDpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Facultad de Matem´ aticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, Spain bInstitut de Math´ ematiques et de Sciences Physiques, URMPM B.P 613, Porto-Novo, B´ enin bUniversit´ e Gaston Berger de Saint-Louis, UFR SAT D´ epartement de Math´ ematiques, B.P234, Saint-Louis, S´ en´ egal Dedicated to Prof. Dr. Juan J. Nieto on the occasion of his 60th birthday ABSTRACT. The existence and uniqueness of mild solution of an impulsive stochastic system driven by a Rosenblatt process is analyzed in this work by using the Banach fixed point theorem and the theory of resolvent operator developed by R. Grimmer in [12]. Furthermore, the exponential stability in mean square for the mild solution to neutral stochastic integro–differential equations with Rosenblatt process is obtained by establishing an integral inequality. Finally, an example is exhibited to illustrate the abstract theory. 1. Introduction. Integro-differential equations are of great importance in the modeling of several physical phenomena. Their resolution can be done through the theory of resolvent operators (see Grimmer [12]). However, the resolvent operator does not satisfy semigroup properties. The study of the quantitative and qualitative properties of solutions to stochastic neutral differential equations like existence, uniqueness and stability, have been widely examined by many researchers by analyzing various mathematical models in different areas such as mechanics, electronics, control theory, engineering and economics, etc (see [3, 5, 7, 8, 25]). There exist many works dealing with several theoretical aspects of the Rosenblatt process. For example, Leonenko and Ahn [17] proved the rate of convergence to the Rosenblatt process in the Non-Central Limit Theorem. Recently, Sakthivel et al.[26] investigated a class of abstract functional second-order non autonomous stochastic evolution equations driven by Rosenblatt process with index H∈(1 2,1). Several processes of evolution systems have impulsive effects at certain moments of time. This phenomenon is observed in some fields as biology, mechanics and physics 2010 Mathematics Subject Classification. 60H15; 35R60. Key words and phrases. Exponential stability; Resolvent operator; Impulsive neutral stochastic integro– differential equations; Rosenblatt process. This work has been partially supported by FEDER and the Spanish Ministerio de Econom´ ıa y Competitividad project MTM2015-63723-P and the Consejer´ ıa de Innovaci´ on, Ciencia y Empresa (Junta de Andaluc´ ıa) under Proyecto de Excelencia P12-FQM-1492. 1 2 T. CARABALLO, C. OGOUYANDJOU, F. K. ALLOGNISSODE AND M. ABDOUL DIOP (see [16, 27]). The theory of impulsive stochastic integro–differential equations has been receiving much attention recently. However, very few researchers have been interested in the stability of the mild solutions to stochastic integro–differential equations impulsive (see [6, 11, 13]). In this paper, we consider the following neutral stochastic integro–differential equation with impulses of the form dx(t)−g(t,xt,Zt 0 a1(t,s,xs)ds)(1) =Ax(t)−g(t,xt,Zt 0 a1(t,s,xs)ds)dt +Zt 0 ϒ(t−s)x(s)−g(s,xs,Zs 0 a1(t,τ,xτ)dτ)dsdt +f(t,xt,Zt 0 a2(t,s,xs)ds)dt +˜ F(t)dZH Q(t),t∈J,t6=ti, ∆x(ti) = Ii(x(t− i)),t=ti,i=1,2,..., (2) x0(t) = φ(t),−r≤t≤0,(3) where J:= [0,b]and Ais the infinitesimal generator of a strongly continuous semigroup (T(t))t≥0of bounded linear operators in a Hilbert space X with domain D(A),ϒ(t)is a family of closed linear operators on Xwith domain D(ϒ(t)) ⊃D(A),ZH Qis a fractional Brownian motion, φ∈PC :=PC ([−r,0],X) = {ψ:[−r,0]→X,ψ(·)is continuous everywhere except in a finite number of points ˜ tat which ψ(˜ t−)and ψ(˜ t+)exist and ψ(˜ t−) = ψ(˜ t)}. For ψ∈PC ,kψkPC =sups∈[−r,0]kψ(s)k<+∞. The mappings g,f:[0,+∞)×PC ×X→X,a1,a2:[0,+∞)×[0,+∞)×PC →X,˜ F:[0,+∞)→ L0 Q(Y,X)are appropriate continuous functions and will be specified later. The impulsive times tisatisfy 0 =t0<t1<t2< ...,< tm→+∞, which implies that in every interval [0,b]there exists only a finite number of tj.Ii:X→X,∆x(ti)represents the jump in the state xat tidetermining the size of the jump, which is defined by ∆x(ti) = x(t+ i)−x(t− i), where x(t+ i)and x(t− i)are respectively the right and left limits of x(t)at ti. For any continuous function xand any t∈[0,b], we denote by xtthe element of PC defined by xt(θ) = x(t+θ),−r≤θ≤0. The analysis of (1)-(3) driven by fractional Brownian motion when ϒ(t) = 0,for all t≥0, was initiated in Arthi et al. [2], where the authors proved the existence and exponential stability of solutions by using an impulsive integral inequality and a strict contraction principle. Now, in our current paper, we will investigate the existence of solutions and stability problems for the previously mentioned neutral stochastic integro–differential system with impulses driven by a Rosenblatt process, since this problem still has not been considered in the literature. The Rosenblatt process is still of interest in practical applications because of its selfsimilarity, stationarity of increments and long-range dependence. Therefore, it is necessary to consider the impulsive effects for the stability of solutions of neutral stochastic integro–differential equations driven by such a Rosenblatt process. In particular, here we highlight that the results of [2] for fractional Brownian motion can be extended to the case ϒ(t)6=0. The main contributions of this paper are summarized as follows: In this work, a general class of impulsive neutral stochastic integrodifferential equations driven by a Rosenblatt process is considered firstly. Then, using methods of functional analysis, a set of sufficient conditions are proposed ensuring exponential stability or solutions. The results are established with the use of the resolvent operator approach. Our STOCHASTIC INTEGRO–DIFFERENTIAL EQUATIONS WITH IMPULSES 3 paper expands the usefulness of stochastic integro-differential equations, since the literature shows results for existence and exponential stability for such equations in the case of semigroup only. The structure of this paper is organized as follows. In Section 2, we recall some basic definitions and preliminary facts which will be used throughout this work. In Section 3, some results on the existence and uniqueness of mild solutions are established. Section 4 is devoted to the proof of exponential stability of a mild solution in mean square, followed by an illustrative example in Section 5. 2. Preliminaries. Let us start with some basic facts about Rosenblatt process. Also we review some fundamentals about the resolvent operator theory which will be crucial for our study. 2.1. Rosenblatt process. We recall in this subsection, some basic concepts about Rosenblatt processes as well as the Wiener integral with respect to them. Consider (χn)n∈Za stationary Gaussian sequence with mean zero and variance 1 such that its correlation function satisfies that R(n):=E(χ0χn) = n2H−2 kL(n), with H∈(1 2,1)and Lis a slowly varying function at infinity. Let hbe a function of Hermite rank k, that is, if hadmits the following expansion in Hermite polynomials h(x) = ∑ j≥0 cjHj(x),cj=1 j!E(h(χ0Hj(χ0))), then k=minj|cj6=0≥1, where Hj(x)is the Hermite polynomial of degree jgiven by Hj(x) = (−1)jex2 2dj dxje−x2 2. Then, the Non-Central Limit Theorem (see, for example, Dobrushin & Major [10]) says 1 nH∑[nt] j=1h(ξj)converges as n→∞, in the sense of finite dimensional distributions, to the process Zk H(t) = c(H,k)ZRkZt 0 k ∏ j=1 (s−yj)(−1 2+1−H k) +!dsdB(y1)···dB(yk),(4) where the above integral is a Wiener-Itˆ o multiple integral of order kwith respect to the standard Brownian motion (B(y))y∈Rand c(H,k)is a positive normalization constant depending only on Hand k. The process (Zk H(t))t≥0is called as the Hermite process and it is Hself-similar in the sense that for any c>0, (Zk H(ct)) d = (cHZk H(t)) and it has stationary increments. The fractional Brownian motion (which is obtained from (4) when k=1) is the most used Hermite process to study evolution equations due to its large range of applications. When k=2 in (4), Taqqu [29] named the process as Rosenblatt process. The stationarity of increments, self-similarity and long range dependence (see Tindel et al. [30]) imply that Rosenblatt processes become very important in practical applications. Let {ZH(t),t∈[0,T],T>0}be a one-dimensional Rosenblatt process with parameter H∈(1 2,1). According to the work by Tudor [31], the Rosenblatt process with parameter H>1 2can be written as ZH(t) = d(H)Zt 0Zt 0"Zt y1∨y2 ∂KH0 ∂u(u,y1)∂KH0 ∂u(u,y2)du#dB(y1)dB(y2),(5) 4 T. CARABALLO, C. OGOUYANDJOU, F. K. ALLOGNISSODE AND M. ABDOUL DIOP where KH(t,s)is given by KH(t,s) = cHs1 2−HZt s (u−s)H−3/2uH−1/2du for t>s, with cH=sH(2H−1) β(2−2H,H−1 2), β(.,.)denotes the Beta function, KH(t,s) = 0 when t≤s,(B(t),t∈[0,T]) is a Brownian motion, H0=H+1 2and d(H) = 1 H+1qH 2(2H−1)is a normalizing constant. The covariance of the Rosenblatt process {ZH(t),t∈[0,T]}satisfy E(ZH(t)ZH(s)) = 1 2s2H+t2H−|s−t|2H. The covariance structure of the Rosenblatt process allows to construct the Wiener integral with respect to it. We refer to Maejima and Tudor [19] for the definition of Wiener integral with respect to general Hermite processes and to Kruk et al. [15] for a more general context (see also Tudor [31]). Note that ZH(t) = ZT 0ZT 0 I(1[0,t])(y1,y2)dB(y1)dB(y2), where the operator Iis defined on the set of functions f:[0,T]→R, which takes its values in the set of functions h:[0,T]2→R2and is given by I(f)(y1,y2) = d(H)ZT y1∨y2 f(u)∂KH0 ∂u(u,y1)∂KH0 ∂u(u,y2)du. Let ϕbe an element of the set Eof step functions on [0,T]of the form ϕ= n−1 ∑ i=0 ai1(ti,ti+1],ti∈[0,T]. Then, it is natural to define its Wiener integral with respect to ZHas ZT 0 ϕ(u)dZH(u):= n−1 ∑ i=0 ai(ZH(ti+1)−ZH(ti)) = ZT 0ZT 0 ϕ(f)(y1,y2)dB(y1)dB(y2). Let Hbe the set of functions ϕsuch that kϕk2 H:=2ZT 0ZT 0 (I(ϕ)(y1,y2))2dy1dy2<∞. It follows that (see Tudor[31]) kϕk2 H=H(2H−1)ZT 0ZT 0 ϕ(u)ϕ(v)|u−v|2H−2dudv. It has been proved in Maejima and Tudor [19] that the mapping ϕ→ZT 0 ϕ(u)dZH(u) defines an isometry from Eto L2(Ω)and it can be extended continuously to an isometry from Hto L2(Ω)because Eis dense in H. We call this extension as the Wiener integral of ϕ∈Hwith respect to ZH. It is noted that the space Hcontains not only functions but its elements could be also distributions. Therefore it is suitable to know subspaces |H|of H:|H|=nϕ:[0,T]→R|RT 0RT 0|ϕ(u)||ϕ(v)|u−v|2H−2dudv <∞o.The space |H|is STOCHASTIC INTEGRO–DIFFERENTIAL EQUATIONS WITH IMPULSES 5 not complete with respect to the norm k.kHbut it is a Banach space with respect to the norm kϕk2 |H|=H(2H−1)ZT 0ZT 0|ϕ(u)||ϕ(v)|u−v|2H−2dudv. As a consequence, we have L2([0,T]) ⊂L1/H([0,T]) ⊂|H|⊂H. For any ϕ∈L2([0,T]), we have kϕk2 |H|≤2HT2H−1ZT 0|ϕ(s)|2ds and kϕk2 |H|≤C(H)kϕk2 L1/H([0,T]),(6) for some constant C(H)>0. Whenever we write C(H)>0 we mean a positive constant depending only on H, and its value may be different in different places. Define the linear operator K∗ Hfrom Eto L2([0,T]) by (K∗ Hϕ)(y1,y2) = ZT y1∨y2 ϕ(t)∂K ∂t(t,y1,y2)dt, where Kis the kernel of Rosenblatt process in representation (5) K(t,y1,y2) = 1[0,t](y1)1[0,t](y2)Zt y1∨y2 ∂KH0 ∂u(u,y1)∂KH0 ∂u(u,y2)du. Note that (K∗ H1[0,t])(y1,y2) = K(t,y1,y2)1[0,t](y1)1[0,t](y2). The operator K∗ His an isometry between Eto L2([0,T]), which can be extended to the Hilbert space H. In fact, for any s,t∈[0,T]we have K∗ H1[0,t],K∗ H1[0,s]L2([0,T]) =K(t,.,.)1[0,t],K(s,.,.)1[0,s]L2([0,T]) =Zt∧s 0Zt∧s 0 K(t,y1,y2)K(s,y1,y2)dy1dy2 =H(2H−1)Zt 0Zs 0|u−v|2H−2dudv =1[0,t],1[0,s]H. Moreover, for ϕ∈H, we have ZH(ϕ) = ZT 0ZT 0 (K∗ Hϕ)(y1,y2)dB(y1)dB(y2). Let {Zn(t)}n∈Nbe a sequence of two-sided one dimensional Rosenblatt process mutually independent on (Ω,F,P). We consider a K-valued stochastic process ZQ(t)given by the following series: ZQ(t) = ∞ ∑ n=1 zn(t)Q1/2en,t≥0. Moreover, if Qis a non-negative self-adjoint trace class operator, then this series converges in the space K, that is, it holds that ZQ(t)∈L2(Ω,K). Then, we say that the above ZQ(t)is aK-valued Q- Rosenblatt process with covariance operator Q. For instance, if {σn}n∈Nis a bounded sequence of non-negative real numbers such that Qen=σnen, by assuming that Qis a nuclear operator in K, then the stochastic process ZQ(t) = ∞ ∑ n=1 zn(t)Q1/2en= ∞ ∑ n=1 √σnzn(t)en,t≥0, 6 T. CARABALLO, C. OGOUYANDJOU, F. K. ALLOGNISSODE AND M. ABDOUL DIOP is well-defined as a K-valued Q- Rosenblatt process. Definition 2.1. (Tudor [31]). Let ϕ:[0,T]→L0 Q(Y,X)such that ∞ ∑ n=1kK∗ H(ϕQ1/2en)kL2([0,T];H)<∞. Then, its stochastic integral with respect to the Rosenblatt process ZQ(t)is defined, for t≥0, as follows : Zt 0 ϕ(s)dZQ(s):= ∞ ∑ n=1Zt 0 ϕ(s)Q1/2endzn(s)(7) = ∞ ∑ n=1Zt 0Zt 0 (K∗ H(ϕQ1/2en))(y1,y2)dB(y1)dB(y2). Lemma 1. For ψ:[0,T]→L0 Q(Y,X)such that ∑∞ n=1kψQ1/2enkL1/H([0,T];U)<∞holds, and for any a,b∈[0,T]with b >a, we have E   Zb a ψ(s)dZQ(s)    2 ≤c(H)(b−a)2H−1 ∞ ∑ n=1Zb akψ(s)Q1/2enk2ds. If, in addition, ∞ ∑ n=1kψ(t)Q1/2enkis uniformly convergent for t ∈[0,T], then, it holds that E   Zb a ψ(s)dZQ(s)    2 ≤C(H)(b−a)2H−1Zb akψ(s)k2 L0 Q(K,H)ds. Proof. Let {en}n∈Nbe the complete orthonormal basis of Kintroduced above. Applying (7) and H¨ older inequality, we have E   Zb a ψ(s)dZQ(s)    2 =E     ∞ ∑ n=1Zb a ψ(s)Q1/2endzn(s)     2 = ∞ ∑ n=1 E   Zb a ψ(s)Q1/2endzn(s)    2 = ∞ ∑ n=1 H(2H−1)Zb aZb akψ(s)Q1/2enkkψ(t)Q1/2enk|t−s|2H−2dsdt ≤C(H) ∞ ∑ n=1Zb akψ(s)Q1/2enk1/Hds2H ≤C(H)(b−a)2H−1 ∞ ∑ n=1Zb akψ(s)Q1/2enk2ds.(8) STOCHASTIC INTEGRO–DIFFERENTIAL EQUATIONS WITH IMPULSES 7 2.2. Partial integro-differential equations in Banach spaces. In the current section, we recall some definitions, notations and properties needed in the sequel. In what follows, X will denote a Banach space, Aand ϒ(t)are closed linear operators on X.Yrepresents the Banach space D(A), the domain of operator A, equipped with the graph norm |y|Y:=|Ay|+|y|for y∈Y. The notation C([0,+∞);Y)stands for the space of all continuous functions from [0,+∞) into Y. We then consider the following Cauchy problem    ξ0(t) = Aξ(t)+Zt 0 ϒ(t−s)ξ(s)ds for t≥0, ξ(0) = ξ0∈X. (9) Definition 2.2. ([12]) A resolvent operator for Eq. (9) is a bounded linear operator valued function Ψ(t)∈L(X)for t ≥0, satisfying the following properties : (i) Ψ(0) = I and kΨ(t)k≤Neβtfor some constants N and β. (ii) For each x ∈X, Ψ(t)x is strongly continuous for t ≥0. (iii) For x ∈Y,Ψ(·)x∈C1([0,+∞);X)∩C([0,+∞);Y)and Ψ0(t)x=AΨ(t)x+Zt 0 ϒ(t−s)Ψ(s)xds =Ψ(t)Ax +Zt 0 Ψ(t−s)ϒ(s)xds for t ≥0. In the sequel, the assumptions below are important: (H1) Ais the infinitesimal generator of a C0-semigroup (T(t))t≥0on X. (H2) For all t≥0, ϒ(t)is a continuous linear operator from (Y,|·|Y)into (X,|·|X). Moreover, there exists an integrable function c:[0,+∞)→R+such that for any y∈Y, y7→ϒ(t)ybelongs to W1,1([0,+∞),X)and  d dt ϒ(t)yX≤c(t)|y|Yfor y∈Yand t≥0. We recall that Wk,p(Ω) = {u∈Lp(Ω):Dαu∈Lp(Ω),∀|α|≤k}, where Dαuis the weak α-th partial derivative of u. Theorem 1. ([12]) Assume that hypotheses (H1) and (H2) hold. Then Eq. (9) admits a resolvent operator (R(t))t≥0. Theorem 2. ([18]) Assume that hypotheses (H1) and (H2) hold. Let T (t)be a compact operator for t >0. Then, the corresponding resolvent operator Ψ(t)of Eq. (9) is continuous for t >0in the operator norm, namely, for all t0>0, it holds that limh→0kΨ(t0+h)−Ψ(t0)k=0. We recall now some results on the existence of solutions for the following integro–differential equation    ξ0(t) = Aξ(t)+Zt 0 ϒ(t−s)ξ(s)ds +π(t)for t≥0, ξ(0) = ξ0∈X, (10) where π:[0,+∞[→Xis a continuous function. 8 T. CARABALLO, C. OGOUYANDJOU, F. K. ALLOGNISSODE AND M. ABDOUL DIOP Definition 2.3. ([12]) A continuous function ξ:[0,+∞)→X is said to be a strict solution of Eq. (10) if (i) ξ∈C1([0,+∞);X)∩C([0,+∞);Y), (ii) ξsatisfies Eq. (10) for t ≥0. Remark 1. From this definition we deduce that ξ(t)∈D(A), and the function ϒ(t−s)ξ(s)is integrable, for all t >0and s ∈[0,+∞). Theorem 3. ([12]) Assume that (H1)-(H2) hold. If v is a strict solution of Eq. (10), then the following variation of constants formula holds ξ(t) = Ψ(t)v0+Zt 0 Ψ(t−s)π(s)ds for t ≥0.(11) Accordingly, we can establish the following definition. Definition 2.4. ([12]) A function ξ:[0,+∞)→X is called a mild solution of (10), for ξ0∈X, if v satisfies the variation of constants formula (11). Next theorem provides sufficient conditions ensuring the regularity of solutions of Eq. (10). Theorem 4. ([12]) Let π∈C1([0,+∞);X)and let ξbe defined by (11). If ξ0∈D(A), then ξis a strict solution of Eq. (10). The following integral inequality is a key tool in proving the exponential stability of mild solution of the neutral system with impulsive moments, which is stated by Lemma 3.1 in [33]. Lemma 2. For any ω>0, assume that there exist some positive constants αj(j=1,2,3) and βi(i=1,2,···,m)and a function Λ:[−r,+∞)→[0,+∞)such that Λ(t)≤α1e−ωt,for t ∈[−r,0], and Λ(t)≤α1e−ωt+α2sup θ∈[−r,0] Λ(t+θ)+ α3Zt 0 e−ω(t−s)sup θ∈[−r,0] Λ(s+θ)ds +∑ ti<t βie−ω(t−ti)Λ(t− i), for each t ≥0. If α2+α3 ω+∑m i=1βi<1, then Λ(t)≤Me−µtfor t ≥−r, where µ>0is the unique solution to the equation: α2eµr+α3 (ω−µ)eµr+∑m i=1βi=1and M0=max{α1,α1(ω−µ) α3eµr}>0. Remark 2. If βi=0(i=1,2,...)in Lemma 2, then we obtain the results proved in Lemma 3.1 in [5]. Furthermore, if α2=0, then Lemma 2 turns out to be Lemma 3.1 in [6]. 3. Existence and Uniqueness. Before stating and proving the existence and uniqueness of solutions, we will describe now the following conditions on the data of system (1)-(3) which will be imposed in our subsequent results. (H3) There exists a positive constant M, such that for all t≥0, kΨ(t)k≤M. (H4) There exists a positive constant Ma1, such that for all t∈J,κ1,κ2∈PC ,    Zt 0 [a1(t,s,κ1)−a1(t,s,κ2)]ds   ≤Ma1kκ1−κ2k. Also ˜ Ma1=bsup0≤s≤t≤bka1(t,s,0)k. STOCHASTIC INTEGRO–DIFFERENTIAL EQUATIONS WITH IMPULSES 9 (H5) There exists a positive constant Mgsuch that gis X−valued and for all t∈J, κj,φj∈PC ,j=1,2, kg(t,κ1,φ1)−g(t,κ2,φ2)k≤Mg[kκ1−κ2k+kφ1−φ2k]. Also ˜ Mg=supt∈Jkg(t,0,0)kand k=Mg(1+Ma1)<1. (H6) There exists a positive constant Ma2, such that for all t∈J,κ1,κ2∈PC ,    Zt 0 [a2(t,s,κ1)−a2(t,s,κ2)]ds   ≤Ma2kκ1−κ2k. Also ˜ Ma2=bsup0≤s≤t≤bka2(t,s,0)k. (H7) There exists a positive constant Mfsuch that fis X−valued and for all t∈J, κj,φj∈PC ,j=1,2, kf(t,κ1,φ1)−f(t,κ2,φ2)k≤Mf[kκ1−κ2k+kφ1−φ2k]. Also ˜ Mf=supt∈Jkf(t,0,0)kand ˜ M=Mf(1+Ma2). (H8) The function gis continuous on its time variable, i.e., for all κ,φ∈PC , lim t→skg(t,κ,φ)−g(s,κ,φ)k2=0. (H9) The function ˜ F:[0,+∞)→L0 Q(Y,X)satisfies Zt 0k˜ F(s)k2 L0 Q ds <∞,∀t∈[0,b]. For a complete orthonormal basis {an}n∈Nin Y, we have (C.1) ∑∞ n=1k˜ FQ1/2ankL2([0,b];X)<∞. (C.2) ∑∞ n=1|˜ F(t)Q1/2an|Xis uniformly convergent for all t∈[0,b]. (H10) The impulsive functions Ii(i=1,2,...)satisfy the following condition: there exist some positive numbers di(i=1,2,...)such that kIi(κ1)−Ii(κ2)k≤dikκ1−κ2kand kIi(0)k=0, for all κ1,κ2∈PC and ∑+∞ i=1di<∞. We present now the definition of mild solution for the stochastic system (1)-(3). Definition 3.1. An X-valued stochastic process {x(t),t∈[−r,b]}is called a mild solution of the abstract Cauchy problem (1)-(3) if (1) x(·)∈PC ([−r,b],L2(Ω,X)); (2) For t ∈[−r,0],x(t) = φ(t); (3) For t ∈J, x(t)satisfies the following integral equation: x(t) = Ψ(t)[φ(0)−g(0,φ,0)]+g(t,xt,Zt 0 a1(t,s,xs)ds) +Zt 0 Ψ(t−s)f(s,xs,Zs 0 a2(s,τ,xτ)dτ)ds +∑ 0<ti<t Ψ(t−ti)Ii(x(t− i))+Zt 0 Ψ(t−s)˜ F(s)dZH Q(s)P−a.s. Theorem 5. If hypotheses (H1)-(H10) hold for every φ∈PC ,b>0, then system (1)-(3) has a unique mild solution on [−r,b]provided that θ=3M2+∞ ∑ i=1 di2<(1−k)2(12) where k =Mg(1+Ma1)<1. 16 T. CARABALLO, C. OGOUYANDJOU, F. K. ALLOGNISSODE AND M. ABDOUL DIOP 5. Example. In this section we make use of our previous results to study the existence, uniqueness and exponential stability of mild solution to a concrete neutral stochastic partial integro–differential equations with finite delays. Consider the neutral stochastic partial integro–differential equations with impulses of the form dz(t,τ)−σ1t,z(t−ζ,τ),Zt 0 σ2(t,s,z(s−ζ,τ))ds =∂2 ∂ξ2z(t,τ)−σ1t,z(t−ζ,τ),Zt 0 σ2(t,s,z(s−ζ,τ))dsdt +Zt 0 γ(t−s)z(s,τ)−σ1s,z(s−ζ,τ),Zs 0 σ2(t,l,z(l−ζ,τ))dldt +σ3t,z(t−ζ,τ),Zt 0 σ4(t,s,z(s−ζ,τ))dsdt +˜ F(t)dZH Q(t), (30) 0≤τ≤π,t6=ti,t∈[0,+∞), subject to the initial conditions z(t,0) = z(t,π) = 0,0≤t<+∞, ∆z(ti,.)(τ) = l i2z(t− i,τ),t=ti,i=1,2,..., z(t,·) = φ(t,·),−r≤t≤0, where φ(·,·)∈PC ([−r,0],L2[0,π]),σj(·),j=1,2,3,4,are functions defined below, and lis a positive constant, and ˜ F:R+→Ris a continuous function such that ˜ Fsatisfies assumption (H13),ZHdenotes a Rosenblatt process. Let Y=L2([0,π]) and en:=q2 πsin(nx),(n=1,2,3,···).Then (en)n∈Nis a complete orthonormal basis in Y. In order to define the operator Q:Y→Y, we choose a sequence {λn}n≥1⊂R+and set Qen=λnen, and assume that tr(Q) = ∑∞ n=1√λn<∞. Define the process ZHby ZH(t) = ∞ ∑ n=1pλnγH n(t)en, where H∈(1 2,1)and {γH n}n∈Nis a sequence of two-sided one-dimensional Rosenblatt process mutually independent. Define A:D(A)⊂H→Hby A=∂2 ∂z2, with domain D(A) = H2([0,π])∩H1 0([0,π]). Then, Ais the infinitesimal generator of a strongly continuous semigroup {T(t)}t≥0on X, which is given by T(t)φ=∑∞ n=1e−n2t<φ,en>en,φ∈D(A). Let Γ:D(A)⊂X→Xbe the operator defined by Γ(t)(z) = γ(t)Az for t≥0 and z∈D(A), and let the functions σ1(t,x,y),σ2(t,z,z0),σ3(t,x,y),σ4(t,z,z0)be defined as follows: σ1:R+×R×R−→R σ2:R+×R+×R−→R σ3:R+×R×R−→R σ4:R+×R+×R−→R. Moreover, let the following assumptions hold a.s.: (A1) The functions σi,i=1,2,are continuous in the sense that lim t→s|σi(t,x,y)−σi(s,x,y)|2=0,for x,y∈R. STOCHASTIC INTEGRO–DIFFERENTIAL EQUATIONS WITH IMPULSES 17 (A2) There exist non-negative real number q1such that |Zt 0 σ2(t,x,y)ds|2≤q1|y|2,for x,y∈R. (A3) There exist non-negative real number r1such that |Zt 0 σ4(t,x,y)ds|2≤r1|y|2,for x,y∈R. (A4) There exist non-negative real numbers q1,q2≥0 and a continuous function ζ1:[0,+∞)→R+such that |σ1(t,x,y)|2≤q2[|x|2+|y|2]+ζ1(t), for all t≥0 and x,y∈R. (A5) There exist non-negative real numbers r1,r2≥0 and a continuous function ζ2:[0,+∞)→R+such that |σ3(t,x,y)|2=r2[|x|2+|y|2]+ζ2(t,) for all t≥0 and x,y∈R. (A6) There exist non-negative real numbers S1,S2≥0 such that ζj(t)≤Sje−µt,∀t≥0,j=1,2. Let a1(t,s,φ)(τ) = σ2(t,s,φ(θ,τ)), g(t,φ,Zt 0 a1(t,s,φ)ds)(τ) = σ1(t,φ(θ,τ),Zt 0 σ2(t,s,φ(θ,τ))ds), =Z0 −r ν1(θ)φ(θ)(ξ)dθ+Zt 0Z0 −r b2(t)b3(l)φ(l,ξ)dlds, a2(t,s,φ)(τ) = σ4(t,s,φ(θ,τ)), f(t,φ,Zt 0 a2(t,s,φ)ds)(τ) = σ3(t,φ(θ,τ),Zt 0 σ4(t,s,φ(θ,τ))ds), =Z0 −r ˜ b1(t,s,τ,φ(s,τ))ds (31) +Zt 0Z0 −r ˜ b2(s)˜ b3(s,l,τ,φ(l,τ))dlds, where, (1) the function ν1(θ)≥0 is continuous in (−r,0]satisfying Z0 −r ν2 1(θ)dθ<∞,γ1 gZ0 −r ν2 1(θ)dθ1 2 <∞. (2) b2,b3:R→Rare continuous, and γ2 gZ0 −r (b3(s))2ds1 2 <∞. (3) The function ˜ b2:R→Ris continuous and ˜ bi:R→R,i=1,3 are continuous and there exist continuous functions rj:R→R,j=1,2,3,4 such that |˜ b1(t,s,x,y)| ≤ r1(t)r2(s)|y|;(t,s,x,y)∈R4, |˜ b3(t,s,x,y)| ≤ r3(t)r4(s)|y|;(t,s,x,y)∈R4, 18 T. CARABALLO, C. OGOUYANDJOU, F. K. ALLOGNISSODE AND M. ABDOUL DIOP with ˜ Lb 1=R0 −r(r2(s))2ds1 2<∞,˜ Lb 2=R0 −r(r4(s))2ds1 2<∞. Notice that the function gis continuous with respect to variable t. If we put x(t):=x(t)(τ) = z(t,τ)for t≥0 and τ∈[0,π] φ(t)(τ) = z0(t,τ)for t∈[−r,0]and τ∈[0,π], then equation (30) takes the following abstract form dx(t)−g(t,xt,Zt 0 a1(t,s,xs)ds) =Ax(t)−g(t,xt,Zt 0 a1(t,s,xs)ds)dt +Zt 0 B(t−s)x(s)−g(s,xs,Zs 0 a1(t,τ,xτ)dτ)dsdt +f(t,xt,Zt 0 a2(t,s,xs)ds)dt +˜ F(t)dZH Q(t),t∈[0,+∞),t6=ti,(32) ∆x(ti) = Ii(x(t− i)),t=ti,i=1,2,..., x0(t) = φ(t),−r≤t≤0, Moreover, if γis bounded and a C1function such that is derivative γ0is bounded and uniformly continuous, then (H1) and (H2) are satisfied and hence, by Theorem 2.2, Eq. (9) has a resolvent operator (Ψ(t))t≥0on X. STOCHASTIC INTEGRO–DIFFERENTIAL EQUATIONS WITH IMPULSES 19 Moreover for φj∈PC ,j=1,2   gt,φ1,Zt 0 a1(t,s,φ1)ds−gt,φ2,Zt 0 a2(t,s,φ2)ds  L2[0,π] ="Zπ 0 Z0 −r v1(θ)[φ1(θ)(ρ)−φ2(θ)(ρ)]dθ +Zt 0Z0 −r b2(s)b3(l)[φ1(l,ρ)−φ2(l,ρ)]dlds!2 dρ#1/2 ≤√2"Zπ 0Z0 −r v1(θ)[φ1(θ)(ρ)−φ2(θ)(ρ)]dθ2 dρ#1/2 +√2"Zπ 0 Zt 0Z0 −r b2(s)b3(l)[φ1(l,ρ)−φ2(l,ρ)]dlds!2 dρ#1/2 ≤√2"Zπ 0Z0 −r v1(θ)dθ2 dρ#1/2 kφ1−φ2k +b√2"Zπ 0Zt 0 b2 2(s)ds Z0 −r b3(l)[φ1(l,ρ)−φ1(l,ρ)]dl!2 dρ#1/2 ≤√2π"Z0 −r v1(θ)dθ2#1/2 kφ1−φ2k +b√2√πZt 0 b2 2(s)ds1/2 ×Z0 −r b2 3(l)dl1/2 kφ1−φ2k ≤k2kφ1−φ2k where k2=γ1 g+bkb2k∞γ2 g. In the same way we obtain   ft,φ1,Zt 0 a2(t,s,φ1)ds−ft,φ2,Zt 0 a2(t,s,φ2)ds  L2[0,π] ≤l2kφ1−φ2k where l2=kb2k∞˜ Lb 1+k˜ b2k∞kr3kL1˜ Lb 2.Therefore, we may easily verify all the assumptions of Theorem 5 and hence, there exists a mild solution for (30). We assume moreover that there exists β>a>1 and b(t)<1 ae−βtfor all t≥0. Thanks to Lemma 5.2 in [9], we have the following estimates kΨ(t)k≤e−λtwhere λ=1−1 a. Consequently, all the hypotheses of Theorem 6 are fulfilled. Therefore, Eq. (30) possesses a unique mild solution which is exponentially stable provided that pq2(1+q1))+ 4(r2(1+r1)) λ2+4(∑∞ i=1l i2)2 (1−pq2(1+q1))2<1. 6. Conclusion. The interest of this work is the study of neutral impulsive stochastic integro–differential equations driven by a Rosenblatt process in a real separable Hilbert space. The existence, uniqueness and exponential stability were obtained by using the fixed point theorem and the resolvent operators theory. Finally, an example is analyzed to illustrate the effectiveness of the main results. It should be emphasized that system (1)-(3) 20 T. CARABALLO, C. OGOUYANDJOU, F. K. ALLOGNISSODE AND M. ABDOUL DIOP considered in this paper is more general than those in the existing literature. There are two direct issues that require further study. 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