Long time behavior of fractional impulsive stochastic differential equations with infinite delay

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume 00, Number 0, Xxxx XXXX pp. 000–000 LONG TIME BEHAVIOUR OF FRACTIONAL IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY Jiaohui Xu School of Mathematics and Statistics, Xi’an JiaoTong University Xi’an, 710049, PR China Tom´ as Caraballo Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla c/ Tarfia s/n, 41012 Sevilla, Spain (Communicated by the associate editor name) Abstract. This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both K-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter H∈(1/2,1). A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay. Keywords: Impulsive differential equations, fractional derivative, fractional Brownian motion, infinite delay, exponential asymptotic behaviour. 1. Introduction. In the past decades, the theory of impulsive differential equations has become an active area due to its wide applications in several fields such as communications, mechanics, electrical engineering, medicine, biology, etc. For the basic theory of impulsive differential equations, we refer the readers to [2,4,5,6,10,21,27,33] and the references therein. It is well-known that the deterministic models arising in mathematical finance, climate and weather derivatives, often fluctuate due to the presence of some kind of noise. Hence, the stochastic models driven by a Brownian motion or fractional Brownian motion have attracted the researchers great interest (see, e.g., [5, 7, 9, 13, 20, 28, 37]). It is worth mentioning that in [9] the authors established a technical lemma (Lemma 2) which is crucial to the stochastic integral with respect to fractional Brownian motion when considering the Hurst parameter H∈(1/2,1). In our paper, we adopt this point of view to deal with the fractional Brownian motion. On the other hand, the fractional differential equations which are presented in the modeling of many real problems (e.g. in physical phenomena) have been the object of extensive study in order to analyze not only non-random fractional phenomena in physics, but also stochastic processes driven This work has been supported by grant MTM2015-63723-P (MINECO/FEDER, EU) and Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa) under grant 2010/FQM314, and Proyecto de Excelencia P12-FQM- 1492. 1 2 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY by a fractional Brownian motion, see [1, 10, 22, 23, 24, 34] and references therein. Therefore, our main aim in this paper is to address the issue of existence, uniqueness and asymptotic behaviour of mild solutions for the following fractional stochastic impulsive differential equations with infinite delay,            Dα tx(t) = Ax(t) + f(t, xt) + g(t, xt)dB(t) dt +h(t)dBH Q(t) dt , t ≥0, t6=tk,1 2< α < 1, ∆x(tk) = x(t+ k)−x(t− k) = Ik(x(t− k)), k = 1,2,· · · , x(t) = φ(t), t ∈(−∞,0], (1) where Dα tis the Caputo fractional derivative of order 1 2< α < 1, A:D(A)⊆H→His the infinitesimal generator of an α-order fractional compact and analytic operator Tα(t)(t≥0) in a separable Hilbert space H. As usual, B(t) and BH Q(t) denote, respectively, a K-valued Q-cylindrical Brownian motion and fractional Brownian motion defined on a filtered complete probability space (Ω,F,{Ft}t≥0,P). The fixed time tk, where the impulses take place, satisfy 0 = t0< t1<· · · < tk→+∞as k→ ∞. Let Kbe another separable Hilbert space, we consider the functions xt: (−∞,0] →L2(Ω; H) defined by xt(θ) = x(t+θ) for θ∈(−∞,0], which are continuous everywhere except for countable points tk(k∈N+), at which there exist x(t+ k) and x(t− k), and x(tk) = x(t− k) (for each k,x(t+ k) = limh→0x(tk+h) and x(t− k) = limh→0x(tk−h) represent the right-hand and left-hand limits of x(t) at t=tkrespectively). The nonlinear maps f:R+× PC → H,g:R+× PC → L(K,H) and h:R+→L0 Q(K,H) satisfy some appropriate assumptions, where L(K,H), L0 Q(K,H) and PC are different phase spaces which will be described precisely later. Moreover, the impulse function Ik belongs to C(H;H) and φdenotes the initial value to be chosen in an appropriate phase space. The theory of local existence of solutions for impulsive non-stochastic or impulsive stochastic differential equations has experienced a good development up to date. In [24], the local existence of mild solutions for neutral impulsive stochastic integro-differential equations has been investigated. Subsequently, using a Banach fixed point theorem, the existence and uniqueness of local solutions for fractional impulsive differential equations with infinite delay are established in [12]. Henceforth, studies of existence of solutions for impulsive stochastic differential equations have been launched as can be seen, for instance, [33,34,35,36]. Notice that, most of the previous research concerns the case of the local existence of solutions for such equations, there has been little regarding the case of the global existence of solutions except for [10,39], in which the global existence of solutions for fractional impulsive differential equations was obtained. Motivated by the work in [39], we derive the global existence of solutions for problem (1.1). We would like to emphasize that it is meaningful and necessary to discuss the qualitative properties for fractional stochastic differential equations with impulsive perturbations and infinite delay. However, to the best of our knowledge, no results have been reported on the exponential asymptotic behavior of solutions for fractional impulsive stochastic differential equations with infinite delay. Hence the aim of this paper is to prove some results in this field. When studying the exponential decay to zero of solutions to problem (1), without loss of generality, we assume that the α-order fractional solution operator Tα(t)(t > 0) and α-resolvent family Sα(t)(t > 0) are tempered. We organize the rest of this paper as follows. Section 2 presents some basic notations, preliminaries and lemmas which will be used throughout this paper. Based on the properties of the α-order fractional solution operator Tα(t)(t > 0) and α-resolvent family Sα(t)(t > 0), as well as iterate technique, the existence and uniqueness of mild solutions to problem (1) are derived in Section 3. The IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 3 last section is devoted to show the exponential asymptotic behavior of mild solutions for fractional impulsive stochastic differential equations with infinite delay under some appropriate conditions. 2. Preliminaries. In this section we introduce the basic definitions of K-valued Q-cylindrical fractional Brownian motion as well as Brownian motion, also we recall some important definitions and lemmas which will be used in the remainder of this paper. Although the content of this section can be found in several published works, we prefer to include it in our paper to make it more readable and as much self-contained as possible. Throughout this paper, let Hand Kbe two separable Hilbert spaces and L(K,H) be the space of all bounded linear operators from Kto H. For convenience, we will use the same notation k·k to denote the norms in H,Kand L(K,H), and use (·,·) to denote the inner product of Hand Kwithout any confusion. Let (Ω,F,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., right continuous and F0contains all P-null sets of F). Let B= (B(t))t≥0and BH Q= (BH Q(t))t≥0be a K-valued Q-cylindrical Brownian motion and a fractional Brownian motion respectively, defined on (Ω,F,{Ft}t≥0,P) with T rQ < ∞, where Q is a symmetric nonnegative trace class operator from Kinto itself. We assume that there exists a complete orthonormal basis {ek}k≥1in K, a bounded sequence of nonnegative real numbers λk such that Qek=λkek,k= 1,2,· · · . Then for arbitrary t∈[0, T ], B(·), BH Q(·) have the expansions B(t) = ∞ X k=1 pλkβk(t)ek, BH Q(t) = ∞ X k=1 pλkβH k(t)ek, t ≥0, where {βk}k≥1and {βH k}k≥1are, respectively, a sequence of two-sided one-dimensional real valued standard Brownian motions and a sequence of fractional Brownian motions mutually independent on (Ω,F,{Ft}t≥0,P). For ϕ, ψ ∈L(K,H), we define (ϕ, ψ) = Tr[ϕQψ∗], where ψ∗is the adjoint operator of ψ. Then, for any bounded operator ψ∈L(K,H), kψk2 Q=Tr[ψQψ∗] = ∞ X k=1 kpλkψekk2. If kψk2 Q<∞, then ψis called a Q-Hilbert-Schmidt operator, we denote by L0 Q(K,H) the space of all ξ∈L(K,H) such that ξQ1 2is a Hilbert-Schmidt operator. By Proposition 2.8 in [11], if ψis an L(K,H)-valued stochastic process on T×Ω such that ψ(t) is measurable relative to Ftfor all t∈[0, T], and satisfies ZT 0 Ekψ(t)k2dt < ∞, then we have the following property, E    T Z 0 ψ(s)dB(s)    2 ≤Tr(Q) T Z 0 Ekψ(s)k2ds. (2) Lemma 1. ([9])Let ϕ: [0, T]→Qsuch that, for any α,β∈[0, T ]with α > β, Zα β kϕ(s)k2 Qds < ∞. 4 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY Then, E   Zα β ϕ(s)dBH Q(s)    2 ≤cH(2H−1)(α−β)2H−1Zα β kϕ(s)k2 Qds. (3) Below we briefly state the definitions and properties of fractional calculus. Definition 2. ([18,31])The fractional integral of order α > 0with the lower limit 0for a function fis defined as Iα tf(t) = 1 Γ(α) t Z 0 (t−s)α−1f(s)ds, t > 0, where Γ(α)is the Gamma function. Definition 3. ([18,32])The Caputo fractional derivative of order α > 0with the lower limit 0for a function fis defined as Dα tf(t) = 1 Γ(n−α) t Z 0 (t−s)n−α−1f(n)(s)ds =In−α tf(n)(t),for n−1< α < n, n ∈N, where the function f(t)has absolutely continuous derivatives up to order n−1. If 0< α < 1, then Dα tf(t) = 1 Γ(1 −α)Zt 0 (t−s)−αf(1)(s)ds. Obviously, the Caputo fractional derivative of a constant is equal to zero. The Laplace transform of the Caputo fractional derivative of order α > 0is given as L{Dα tf(t); λ}=λαF(λ)− n−1 X k=0 λα−k−1f(k)(0), n −1≤α < n, where F(λ) = R∞ 0e−λtf(t)dt denotes the Laplace transform of a function f(t). If fis an abstract function with values in H, then the integrals which appear in Definition 2 and Definition 3 are taken in the Bochner sense. A measurable function u: [0,∞)→His Bochner integrable if kukis Lebesgue integrable. A two–parameter function of the Mittag-Leffler type is defined by the series expansion Eα,β(z) = ∞ X k=0 zk Γ(αk +β)=1 2πi Z C tα−βet tα−zdt, α, β > 0, z ∈C, where the path Cis a loop which starts and ends at −∞ and encircles the disc |t|≤|z|1 2in the positive sense. For short, Eα(z) = Eα,1(z). It is an entire function which provides a simply generalization of the exponential function E1(z) = ez, and plays a vital role in the theory of fractional differential equations. The most interesting properties of the Mittag-Leffler functions are associated with their Laplace integral, ∞ Z 0 e−λttβ−1Eα,β(ωtα)dt =λα−β λα−ω, Reλ > ω 1 α, ω > 0, and for more details we refer to [31,38]. IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 5 Next we recall some basic definitions concerning the sectorial operator A,α-order fractional solution operator Tα(t) and α-resolvent family Sα(t). Definition 4. ([16])A linear closed densely defined operator Ais said to be sectorial if there are constants ω∈R,θ∈[π 2, π],M > 0, such that the following two conditions are satisfied, (1) σ(A)⊂Pω,θ ={λ∈C:λ6=ω, |arg(λ−ω)|< θ}, (2) kR(λ, A)k ≤ M |λ−ω|, λ ∈Pω,θ. Definition 5. Let Abe a closed and linear operator with domain D(A)defined on H. Let ρ(A)be the resolvent set of A. We say that Ais the generator of an α-resolvent family if there exist ω≥0 and a strongly continuous function Sα:R+→L(H), such that {λα:Re(λ)> ω} ⊂ ρ(A)and (λαI−A)−1y= ∞ Z 0 e−λtSα(t)ydt, Re(λ)> ω, y ∈H, where Sα(t)is called the α-resolvent family generated by A. Definition 6. ([30, 36])A solution operator Tα(t)of (1) is called analytic if Tα(t)admits an analytic extension to a sector Pθ0:= {λ∈C\{0}:|arg λ|< θ0}for some θ0∈(0,π 2]. An analytic solution operator is said to be of analyticity type (ω0, θ0)if for each θ < θ0and ω > ω0, there is a positive constant M=M(ω, θ)such that kTα(t)k ≤ MeωRe(t),t∈Pθ:= {t∈C\{0}:|arg t|< θ}. Denote Aα(ω0, θ0) := {Agenerates analytic solution operators Tα(t)of type (ω0, θ0)}. Definition 7. ([3])A family Tα(t) : R+→L(H)is called an α-order fractional solution operator generated by Aif the following conditions are satisfied: (1) Tα(t)is strongly continuous for t≥0and Tα(0) = I; (2) Tα(t)D(A)⊂D(A)and ATα(t)x=Tα(t)Ax for all x∈D(A)and t≥0; (3) for all x∈D(A),Tα(t)xis a solution of the following operator equation x(t) = x+1 Γ(α) t Z 0 (t−s)α−1Ax(s)ds, t ≥0. Definition 8. An α-order fractional solution operator Tα(t)(t≥0) is called compact if for every t > 0,Tα(t)is a compact operator. Arguing as in the proof of Lemma 3.8 in [15], we obtain the continuity of the α-order fractional solution operator Tα(t) and α-resolvent family Sα(t) in the uniform operator topology for t > 0. Lemma 9. Assume A∈Aα(ω0,θ0), and that the α-order fractional solution operator Tα(t)(t > 0) and the α-resolvent family Sα(t)(t > 0) are compact. Then the following properties are fulfilled: (1) lim h→0kTα(t+h)−Tα(t)k= 0 and lim h→0kSα(t+h)−Sα(t)k= 0 for t > 0; (2) lim h→0+kTα(t+h)−Tα(h)Tα(t)k= 0 and lim h→0+kSα(t+h)−Sα(h)Sα(t)k= 0 for t > 0; (3) lim h→0+kTα(t)−Tα(h)Tα(t−h)k= 0 and lim h→0+kSα(t)−Sα(h)Sα(t−h)k= 0 for t > 0. Lemma 10. ([36])If A∈Aα(ω0,θ0), then for every ω > ω0(ω0∈R+), there exists a constant M=M(ω, θ)such that, for all t > 0, kTα(t)k ≤ Meωt and kSα(t)k ≤ Meωt(1 + tα−1).(4) 6 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY Furthermore, let MT:= sup0≤t≤TkTα(t)k,NT:= sup0≤t≤TMeωt(1 + t1−α). Then we obtain that kTα(t)k ≤ MTand kSα(t)k ≤ NTtα−1.(5) 3. Existence, uniqueness and continuous dependence of mild solutions. In this section, we establish the existence of mild solutions to fractional impulsive stochastic differential equation (1). Before doing this, we first present the abstract phase space PC. Let L2(Ω; H) denote the Banach space of all strongly-measurable, square-integrable H-valued random variables equipped with the norm ku(·)k2 L2=Eku(·)k2, where the expectation Eis defined by Eu =RΩu(·)dP. The abstract phase space PC is defined by PC =ξ: (−∞,0] →L2(Ω; H) is F0-adapted and continuous except in at most a countable number of points {θk},at which there exist ξ(θ+ k) and ξ(θ− k) with ξ(θk) = ξ(θ− k),and sup θ∈(−∞,0] eγθEkξ(θ)k2<∞, for some fixed parameter γ > 0. If PC is endowed with the norm kξkPC =sup θ∈(−∞,0] eγθEkξ(θ)k21 2 , ξ ∈ PC, then, (PC,k·kPC) is a Banach space. In order to establish the main result, we impose the following conditions. (H1)f: [0,∞)× PC → H,g: [0,∞)× PC → L(K,H) are continuous and there exist two functions l1(t), l2(t)∈L∞(R+,R+) such that Ekf(t, x)−f(t, y)k2≤l1(t)kx−yk2 PC, Ekg(t, x)−g(t, y)k2≤l2(t)kx−yk2 PC, for every x,y∈ PC, and almost every t > 0. Moreover, g(t, ·) is measurable relative to Ftfor all t∈[0,∞) satisfying R∞ 0Ekg(t, x)k2dt < ∞. (H2) For h: [0,∞)→Q, there exists a constant q > 1 such that, for every t∈[0, T], Zt 0 kh(s)k2q Qds < ∞. (H3) The functions Ik:L2(Ω; H)→L2(Ω; H) are continuous for each k∈N+, and there exist two positive constants b1, b2>0 such that EkIk(x)k2≤b1Ekxk2+b2,for all x∈L2(Ω; H). There exists a positive constant Nsuch that, for all k∈N+, EkIk(x)−Ik(y)k2≤NEkx−yk2,for all x, y ∈L2(Ω; H). (H4)β= inf k∈N+{tk−tk−1}>0, η= sup k∈N+ {tk−tk−1}<∞. Now, we introduce the definition of mild solutions for problem (1). IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 7 Definition 11. Set Ft=F0for all t∈(−∞,0]. An Ft-adapted stochastic process x: (−∞, T ]→H is called a mild solution of the equation (1) if x(t) = φ(t)for t∈(−∞,0] with φ∈ PC, and for t∈[0, T],x(t)satisfies the integral equation x(t) =                                              Tα(t)φ(0) + Zt 0 Sα(t−s)f(s, xs)ds +Zt 0 Sα(t−s)g(s, xs)dB(s) +Zt 0 Sα(t−s)h(s)dBH Q(s), t ∈[0, t1], Tα(t−t1)(I1(x(t− 1)) + x(t− 1)) + Zt t1 Sα(t−s)f(s, xs)ds +Zt t1 Sα(t−s)g(s, xs)dB(s) + Zt t1 Sα(t−s)h(s)dBH Q(s), t ∈(t1, t2], · · · , Tα(t−tm)(Im(x(t− m)) + x(t− m)) + Zt tm Sα(t−s)f(s, xs)ds +Zt tm Sα(t−s)g(s, xs)dB(s) + Zt tm Sα(t−s)h(s)dBH Q(s), t ∈(tm, T], (6) where tm= max{tk, tk< T, k = 0,1,2,· · · }, and Tα(t) = Eα,1(Atα) = 1 2πi ZΓθ eλt λα−1 λα−Adλ, Sα(t) = tα−1Eα,α(Atα) = 1 2πi ZΓθ eλt 1 λα−Adλ, here the integral contour Γθis oriented counter-clockwise. In the sequel Cdenotes a generic positive constant, which may be different from line to line and even in the same line. Next we are in a position to state the main theorem. Theorem 12. Assume (H1)-(H4)holds true, let A∈Aα(ω0, θ0)with θ0∈(0,π 2]and ω0∈R+, and assume that the α-order fractional solution operator Tα(t)(t > 0) and the α-resolvent family Sα(t)(t > 0) are compact. Assume also that there exist two constants pand qwhich satisfy 1 p+1 q= 1, where 1< p < 1 2(1−α). Then, for every initial value φ∈ PC and every T > 0, the problem (1) has at least one mild solution defined on (−∞, T]. Proof. We start the proof by defining an abstract phase space PCTas follows, for a fixed T > 0, PCT=x(·,·) : (−∞, T ]×Ω→Hsuch that x(t, ·) is Ft-adapted, x(t, ·)∈L2(Ω; H) for all t≤T, x|Jk∈C(Jk;L2(Ω; H)) and sup t∈(−∞,T ] eγtEkx(t)k2<∞, where x|Jkis the restriction of xto Jk= (tk, tk+1], k∈N+. Then the abstract space PCTendowed with the norm kxkPCT=sup t∈(−∞,T ] eγtEkx(t)k21 2 , x ∈ PCT, is a Banach space. Notice that, when considering T= 0, we have PCT=PC0=PC. Now, for φ∈ PC, we define PCT φ={x∈ PCT:x(s) = φ(s), s ≤0}. 8 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY It is clear that PCT φis a closed subset of PCT, and consequently, it is a complete metric subspace of PCT. In order to simplify our presentation, let us abbreviate klik∞by li(i= 1,2) according to the fact li(t)∈L∞(R+,R+), and we write PCφinstead of PCT φfor a fixed T > 0, when no confusion is possible. Let us pick ψ∈ PCφand define                      x0(t) = ψ(t), xn(t) = χ(−∞,0](t)ψ(t) + X k χ(tk,tk+1](t)Tα(t−tk)(Ik(xn−1(t− k)) + xn−1(t− k)) +Zt tk Sα(t−s)f(s, xn−1 s)ds +Zt tk Sα(t−s)g(s, xn−1 s)dB(s) +Zt tk Sα(t−s)h(s)dBH Q(s), t ∈(−∞, T], k = 0,1,2· · · , (7) where I0= 0 and tk< T,χis a characteristic function. To ensure the existence of mild solutions, we split the proof into several steps. Step 1. For all n∈N+,xn(·)∈ PCφ. First, we claim that xn(·) is Ft-adapted for all n∈N+. Obviously, x0(t) = ψ(t)∈ PCφimplies that x0(t) is Ft-adapted for all t∈(−∞, T]. With the help of the continuity of fand Ik, as well as the fact that the limit of measurable functions is a measurable function, it is easy to see that Ik(x0(t− k)) and Rt tkf(s, x0 s)ds are Ft-adapted. In addition, the stochastic integral Rt tkg(s, x0 s)dB(s) is Ft-adapted according to Definition 5.3 in [29]. For the last term Rt tkh(s)dBH Q(s), which is Ftadapted naturally. In conclusion, thanks to the Picard iterations technique, we obtain that x1(t) is also Ft-adapted. By induction, xn(t) is Ft-adapted for all t∈(−∞, T] and n∈N+. Next, we want to prove xn(t)∈L2(Ω; H). Since xn(t) = ψ(t) on (−∞,0], then xn(t) = φ(t) for t∈(−∞,0], and as φ∈ PC, then we have xn(t)∈L2(Ω; H) for all t∈(−∞,0]. Furthermore, for every t∈[0, t1], by (2), (3), (5), (H1)-(H4) and H¨older’s inequality, Ekx1(t)k2≤4M2 Tkψk2 PCφ+ 8N2 T(ηl1+T r(Q)l2)kψk2 PCφ t2α−1 2α−1 + 8N2 T(ηsup s∈[0,t1] kf(s, 0)k2+Tr(Q) sup s∈[0,t1] kg(s, 0)k2)t2α−1 2α−1 + 4N2 TcH(2H−1)t2H−1Zt 0 (t−s)2α−2kh(s)k2 Qds := 4M2 Tkψk2 PCφ+C1kψk2 PCφ t2α−1 2α−1+C2 t2α−1 2α−1+C3t2H−1, (8) where we have used the notation C1= 8N2 T(ηl1+T r(Q)l2), C2= 8N2 T(ηsup s∈[0,T ] kf(s, 0)k2+Tr(Q) sup s∈[0,T ] kg(s, 0)k2), and C3≤4N2 TcH(2H−1)η2(α−1)p+1 2(α−1)p+ 11 pZt 0 kh(s)k2q Qds1 q <∞. IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 9 Notice that, if t+θ∈[0, t1] (where θ∈(−∞,0]), in view of γ > 0, then it follows that eγθEkx1(t+θ)k2≤4M2 Tkψk2 PCφ+C1kψk2 PCφ t2α−1 2α−1+C2 t2α−1 2α−1+C3t2H−1,(9) if t+θ < 0, we have eγθEkx1(t+θ)k2≤e−γteγ(t+θ)Ekψ(t+θ)k2≤e−γtkψk2 PCφ.(10) Combining (9) and (10), kx1 tk2 PC = sup θ∈(−∞,0] eγθEkx1(t+θ)k2 ≤kψk2 PCφ+C2 η2α−1 2α−1+C3η2H−1× 1 X i=0 4M2 T+C1 η2α−1 2α−1i . By induction on n, for t∈[0, t1], we derive that kxn tk2 PC ≤kψk2 PCφ+C2 η2α−1 2α−1+C3η2H−1× n X i=0 4M2 T+C1 η2α−1 2α−1i .(11) For t∈(t1, t2∧T], where t2∧T= min{t2, T}, similar to (8). By (2), (3), (5), (H1)-(H4) and H¨older’s inequality, Ekx1(t)k2≤8M2 T((b1+ 1)kψk2 PCφ+b2) + C1 (t−t1)2α−1 2α−1kψk2 PCφ +C2 (t−t1)2α−1 2α−1+C3(t−t1)2H−1 ≤kψk2 PCφ+ 8M2 Tb2+C2 (t−t1)2α−1 2α−1+C3(t−t1)2H−1 × 1 X i=0 8M2 T(b1+ 1) + C1 (t−t1)2α−1 2α−1i . (12) Using the same argument as in (9) and (10), together with (11) and (12), for t∈(t1, t2∧T], we have the following estimate, kx1 tk2 PC ≤kψk2 PCφ+ 8M2 Tb2+C2 η2α−1 2α−1+C3η2H−1× 1 X i=0 8M2 T(b1+ 1) + C1 η2α−1 2α−1i . By induction on n, for t∈(t1, t2∧T], we deduce kxn tk2 PC ≤kψk2 PCφ+ 8M2 Tb2+C2 η2α−1 2α−1+C3η2H−1× n X i=0 8M2 T(b1+ 1) + C1 η2α−1 2α−1i .(13) In a similar way, combining (11) with (13), for each fixed n∈N+, for all t∈[0, T], we find that kxn tk2 PC ≤kψk2 PCφ+8M2 Tb2+C2 η2α−1 2α−1+C3η2H−1× n X i=0 8M2 T(b1+1)+C1 η2α−1 2α−1i <∞.(14) Taking into account that Ekxn(t)k2≤ kxn tk2 PC, then (14) implies that xn(t)∈L2(Ω; H) for all t∈[0, T]. 16 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY Applying Gronwall’s inequality ( [14,17,25,26]), we derive for t∈(t1, t2∧T], kxt−ytk2 PC ≤21−p(6M2 T(N+ 1) + eA2pη)B1eA2p(t−t1).(40) If T < t2, this shows the assertion. We assume that T > t2, and consequently, Ekx(t2)−y(t2)k2≤21−p(6M2 T(N+ 1) + eA2pη)B1eA2p(t2−t1)=B2.(41) Arguing similarly, we find that for t∈(tk, tk+1 ∧T] with k≥2, kxt−ytk2 PC ≤21−p(6M2 T(N+ 1) + eA2pη)BkeA2p(t−tk),(42) we may as well consider the case tk+1 < T, because the estimate (32) holds true when tk+1 > T with k≥2. Thus Ekx(tk+1)−y(tk+1)k2≤21−p(6M2 T(N+ 1) + eA2pη)BkeA2p(tk+1−tk)=Bk+1.(43) For the sake of convenience, let A3= 21−p(6M2 T(N+ 1) + eA2pη), we deduce the following result by mathematical induction for k≥2, Bk≤A3Bk−1eA2p(tk−tk−1)≤Ak−1 3B1eA2p(tk−t1).(44) Hence, it follows from (42) and (44), and for t∈(tk, tk+1 ∧T], kxt−ytk2 PC ≤A3BkeA2p(t−tk)≤Ak 3B1eA2p(t−t1).(45) In view of the fact that condition (H4) implies that kβ < t < (k+ 1)ηfor t∈(tk, tk+1], taking into account (45), (34) and (40), we deduce for all t∈(0, T ], kxt−ytk2 PC ≤21−p(3M2 T+ 1)kφ−ϕk2 PCeA2ptekln A3 ≤21−p(3M2 T+ 1)kφ−ϕk2 PCe(A2p+ln A3 β)t.(46) The proof is finished.  4. Global existence and asymptotic behavior of mild solutions. In this section, we first prove the global existence of mild solutions to stochastic impulsive differential equations before studying the exponential asymptotic behavior of mild solutions. To start off we state some conditions which will be imposed later. (C1) The closed and linear sectorial operator A∈Aα(ω0, θ0) with θ0∈(0,π 2] and ω0∈R, generates an α-order fractional solution operator Tα(t) and an α-resolvent family Sα(t), on the separable Hilbert space Hwith kTα(t)k ≤ Me−µt,kSα(t)k ≤ Me−µt(1 + tα−1),∀t > 0,where M≥1, Re(µ)∈R+. (C2) There exist two positive constants Cfand Cg, such that, for any x, y ∈ PC and for all t > 0, Ekf(t, x)−f(t, y)k2≤Cfkx−yk2 PC, Ekg(t, x)−g(t, y)k2≤Cgkx−yk2 PC, and Z∞ 0 e2µqskf(s, 0)k2qds < ∞,Z∞ 0 e2µqskg(s, 0)k2qds < ∞, where q > 1 and 1 p+1 q= 1. (C3) In addition to assumption (H2), also suppose that Z∞ 0 e2µqskh(s)k2q Qds < ∞. IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 17 (C4) Under condition (H3), we impose an additional assumption on Ik, that is, for all k∈N+, EkIk(x)k2≤b1Ekxk2,for all x∈L2(Ω; H). 4.1. The global existence of mild solutions in PC.Now we state the global existence of mild solution to the problem (1). Theorem 15. Assume the conditions of Theorem 12 and (C1). Then for every initial value φ∈ PC, there exists a unique solution to problem (1) in the sense of Definition 11, defined on [0,∞). Proof. We derive the local existence and uniqueness of mild solution to (1) by means of the estimates (5) in Theorem 12, where MTand NTare constants depending on Twhich is finite. In order to extend the results to [0,∞), the constants of the estimates (5) must be independent of T. For this aim, we modify the estimates (5) slightly by condition (C1), that is, M0= sup t∈[0,∞) kTα(t)k<∞and N0= sup t∈[0,∞) Me−µt(1 + t1−α)<∞. Replacing MTand NTby M0and N0, separately, in Theorem 12 and Theorem 14, the results still hold true. Now, we are ready to prove the global existence and uniqueness of mild solutions to (1). By Theorem 12 and Theorem 14, we deduce that there exists a unique solution x(1)(t) to the initial value problem (1.1) such that x(1)(t) =            φ(t), t ∈(−∞,0], Tα(t)φ(0) + Zt 0 Sα(t−s)f(s, x(1) s)ds +Zt 0 Sα(t−s)g(s, x(1) s)dB(s) +Zt 0 Sα(t−s)h(s)dBH Q(s), t ∈[0, t1]. Arguing as in the proof of Theorem 12, we derive the existence of x(2)(t) satisfying x(2)(t) =              x(1)(t), t ∈(−∞, t1], Tα(t−t1)(I1(x(1)(t− 1)) + x(1)(t− 1)) + Zt t1 Sα(t−s)f(s, x(2) s)ds +Zt t1 Sα(t−s)g(s, x(2) s)dB(s) + Zt t1 Sα(t−s)h(s)dBH Q(s), t ∈(t1, t2]. Continuing the procedure in this way, we obtain a unique global solution to problem (1) in the sense of Definition 11. This completes the proof.  4.2. Exponential decay of mild solutions in PC.Motivated by the work of Caraballo et al. in [9], we turn our attention in this subsection to prove the exponential asymptotic behavior of the mild solutions to problem (1) in PC. Theorem 16. Let conditions (H4)and (C1)−(C4)hold, assume that γ > 2µ, and 2µq −L−ln w1 β>0,(47) 18 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY where the positive constants Land w1will be explicitly written in the proof. Then, every mild solution x(·)of system (1) with the initial value x0=φ∈ PC satisfies kxtk2q PC ≤C(1 + kφk2q PC)e−(2µq−L−ln w1 β)t,∀t≥0.(48) Proof. We split the proof into three steps. Step 1. By Definition 11, (2), (3), (H4) and (C1)-(C3), we obtain for t∈[0, t1], Ekx(t)k2≤4M2e−2µtkφk2 PC + 16ηM2e−2µt Zt 0 (1 + (t−s)2α−2)e2µs(Cfkxsk2 PC +kf(s, 0)k2)ds + 16Tr(Q)M2e−2µt Zt 0 (1 + (t−s)2α−2)e2µs(Cgkxsk2 PC +kg(s, 0)k2)ds + 8cH(2H−1)η2H−1M2e−2µt Zt 0 (1 + (t−s)2α−2)e2µskh(s)k2 Qds = 4M2e−2µtkφk2 PC + 16M2e−2µt(ηCf+Tr(Q)Cg)Zt 0 (1 + (t−s)2α−2)e2µskxsk2 PCds + 16M2e−2µt Zt 0 (1 + (t−s)2α−2)e2µs(ηkf(s, 0)k2+Tr(Q)kg(s, 0)k2)ds + 8cH(2H−1)η2H−1M2e−2µt Zt 0 (1 + (t−s)2α−2)e2µskh(s)k2 Qds. (49) In view of condition (C3), which ensures the existence of a positive constant L1, using H¨older’s inequality, we derive cH(2H−1)η2H−1Zt 0 (1 + (t−s)2α−2)e2µskh(s)k2 Qds ≤cH(2H−1)η2H−1Zt 0 (1 + (t−s)2α−2)pds1 pZt 0 e2µqskh(s)k2q Qds1 q ≤L1. (50) Observe that condition (C2) ensures the existence of a positive constant L2, by H¨older’s inequality, one has Zt 0 (1 + (t−s)2α−2)e2µskf(s, 0)k2ds ≤Zη 0 (1 + (t−s)2α−2)pds1 pZ∞ 0 e2µqskf(s, 0)k2qds1 q ≤L2.(51) Analogously, one can prove that the following estimate holds true, thanks to condition (C2), with a positive constant L3, Zt 0 (1 + (t−s)2α−2)e2µskg(s, 0)k2ds ≤Zη 0 (1 + (t−s)2α−2)pds1 pZ∞ 0 e2µqskg(s, 0)k2qds1 q ≤L3.(52) IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 19 Replacing (50)-(52) into (49), by H¨older’s inequality, Ekx(t)k2≤4M2e−2µtkφk2 PC + 16M2e−2µt(ηL2+Tr(Q)L3)+8M2e−2µtL1 + 16M2e−2µt(ηCf+Tr(Q)Cg)Zt 0 (1 + (t−s)2α−2)pds1 pZt 0 e2µqskxsk2q PCds1 q ≤4M2e−2µtkφk2 PC + 16M2e−2µtL4+ 16M2e−2µtL5Zt 0 e2µqskxsk2q PCds1 q , (53) where we have used the notation L4=L1 2+ηL2+T r(Q)L3, and L5≤2p−1 p(ηCf+T r(Q)Cg)η+η2(α−1)p+1 2(α−1)p+ 11 p <∞. Note that e(γ−2µ)θ<1 (for θ < 0) since γ > 2µ. For t+θ∈[0, t1], multiplying by eγθ and replacing tby t+θin (53), one has eγθEkx(t+θ)k2≤4M2e−2µ(t+θ)eγθkφk2 PC + 16M2e−2µ(t+θ)eγθL4 + 16M2e−2µ(t+θ)eγθL5Zt+θ 0 e2µqskxsk2q PCds1 q ≤4M2e−2µtkφk2 PC + 16M2e−2µtL4+ 16M2e−2µtL5Zt 0 e2µqskxsk2q PCds1 q , (54) and eγθEkx(t+θ)k2=e−γteγ(t+θ)Ekφ(t+θ)k2≤e−γtkφk2 PC ≤e−2µtkφk2 PC,if t+θ < 0.(55) Therefore, for t∈[0, t1], (54) and (55) lead to kxtk2 PC ≤4M2e−2µtkφk2 PC + 16M2e−2µtL4+ 16M2e−2µtL5Zt 0 e2µqskxsk2q PCds1 q ,(56) thus kxtk2q PC ≤3q−14qM2qe−2µqtkφk2q PC + 3q−142qM2qe−2µqtLq 4 + 3q−142qM2qe−2µqtLq 5Zt 0 e2µqskxsk2q PCds. (57) Multiplying both sides of (57) by e2µqt, e2µqtkxtk2q PC ≤3q−14qM2qkφk2q PC + 3q−142qM2qLq 4+ 3q−142qM2qLq 5Zt 0 e2µqskxsk2q PCds. (58) Gronwall’s inequality implies kxtk2q PC ≤(3q−14qM2qkφk2q PC +L6)e−(2µq−L)t,(59) where we have used the notation L6= 3q−142qM2qLq 4, L = 3q−142qM2qLq 5, 20 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY and, consequently, Ekx(t1)k2≤(3q−14qM2qkφk2q PC +L6)1 qe−2µq−L qt1= (B∗ 1)1 q.(60) Step 2. By Definition 11, (2), (3), (H4) and (C1)-(C4), similar to (49), we obtain for t∈(t1, t2] that Ekx(t)k2≤8M2e−2µ(t−t1)(1 + b1)Ekx(t− 1)k2+ 16M2(ηCf+T r(Q)Cg)e−2µ(t−t1) ×Zt t1 (1 + (t−s)2α−2)e2µ(s−t1)kxsk2 PCds + 16M2e−2µ(t−t1) ×Zt t1 (1 + (t−s)2α−2)e2µ(s−t1)(ηkf(s, 0)k2+Tr(Q)kg(s, 0)k2)ds + 8cH(2H−1)η2H−1M2e−2µ(t−t1)Zt t1 (1 + (t−s)2α−2)e2µ(s−t1)kh(s)k2 Qds. (61) In order to prove the exponential decay to zero of solutions to problem (1), we need the results to the last two terms of (61) including e−(2µ−L q)t1. For this purpose, we do the estimates separately as follows. On the one hand, using the same argument as in (50), condition (C3) ensures the existence of a positive constant which is still denoted by L1for simplicity, such that cH(2H−1)η2H−1Zt t1 (1 + (t−s)2α−2)e(2µ−L q)(s−t1)+ L q(s−t1)kh(s)k2 Qds ≤cH(2H−1)η2H−1e−(2µ−L q)t1Zt t1 (1 + (t−s)2α−2)e2µskh(s)k2 Qds ≤cH(2H−1)η2H−1e−(2µ−L q)t1Zt t1 (1 + (t−s)2α−2)pds1 pZt t1 e2µqskh(s)k2q Qds1 q ≤e−(2µ−L q)t1L1. (62) On the other hand, similar to (51) and (52), using the same technique as in (62), by condition (C2), there are two positive constants still denoted by L2and L3respectively, such that Zt t1 (1 + (t−s)2α−2)e2µ(s−t1)kf(s, 0)k2ds ≤Zt t1 (1 + (t−s)2α−2)pds1 p ×Zt t1 e(2µq−L)(s−t1)eL(s−t1)kf(s, 0)k2qds1 q ≤Zt t1 (1 + (t−s)2α−2)pds1 p ×e−2µq−L qt1Zt t1 e2µqskf(s, 0)k2qds1 q ≤L2e−2µq−L qt1, (63) IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 21 and Zt t1 (1 + (t−s)2α−2)e2µ(s−t1)kg(s, 0)k2ds ≤Zt t1 (1 + (t−s)2α−2)pds1 p ×Zt t1 e(2µq−L)(s−t1)eL(s−t1)kg(s, 0)k2qds1 q ≤Zt t1 (1 + (t−s)2α−2)pds1 p ×e−2µq−L qt1Zt t1 e2µqskg(s, 0)k2qds1 q ≤L3e−2µq−L qt1. (64) Substituting (62)-(64) into (61), and using H¨older’s inequality, Ekx(t)k2≤8M2e−2µ(t−t1)(1 + b1)Ekx(t− 1)k2+ 16M2(ηCf+T r(Q)Cg)e−2µ(t−t1) ×Zt t1 (1 + (t−s)2α−2)pds1 pZt t1 e2µq(s−t1)kxsk2q PCds1 q + 16M2e−2µ(t−t1)(ηL2+T r(Q)L3)e−2µq−L qt1+ 8M2e−2µ(t−t1)L1e−2µq−L qt1 ≤8M2e−2µ(t−t1)(1 + b1)Ekx(t− 1)k2+ 16M2e−2µ(t−t1)L4e−2µq−L qt1 + 16M2e−2µ(t−t1)L5Zt t1 e2µq(s−t1)kxsk2q PCds1 q . (65) Arguing as in the proof of Step 1, due to the fact that e(γ−2µ)θ<1, we obtain for t+θ∈(t1, t2] (where θ∈(−∞,0]) eγθEkx(t+θ)k2≤8M2e−2µ(t+θ−t1)eγθ(1 + b1)Ekx(t− 1)k2+ 16M2e−2µ(t+θ−t1)eγθ ×L4e−2µq−L qt1+ 16M2e−2µ(t+θ−t1)eγθL5Zt+θ t1 e2µq(s−t1)kxsk2q PCds1 q ≤8M2e−2µ(t−t1)(1 + b1)Ekx(t− 1)k2+ 16M2e−2µ(t−t1)L4e−2µq−L qt1 + 16M2e−2µ(t−t1)L5Zt t1 e2µq(s−t1)kxsk2q PCds1 q . (66) It follows from (59) and (60) that, for t∈(t1, t2] and t+θ≤t1, eγθEkx(t+θ)k2≤(3q−14qM2qkφk2q PC +L6)1 qe−2µq−L qt = (3q−14qM2qkφk2q PC +L6)1 qe−2µq−L qt1e−2µq−L q(t−t1) ≤(B∗ 1)1 qeL qηe−2µ(t−t1). (67) 22 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY Hence, (66) and (67) imply that for all t∈(t1, t2], kxtk2 PC ≤8M2(1 + b1) + eL qη(B∗ 1)1 qe−2µ(t−t1)+ 16M2e−2µ(t−t1)L4e−2µq−L qt1 + 16M2e−2µ(t−t1)L5Zt t1 e2µq(s−t1)kxsk2q PCds1 q . (68) Thus kxtk2q PC ≤3q−18M2(1 + b1) + eL qηq B∗ 1e−2µq(t−t1)+ 3q−142qM2qe−2µq(t−t1) ×Lq 4e−(2µq−L)t1+ 3q−142qM2qe−2µq(t−t1)Lq 5Zt t1 e2µq(s−t1)kxsk2q PCds. (69) Multiplying both sides of (69) by e2µq(t−t1), e2µq(t−t1)kxtk2q PC ≤3q−18M2(1 + b1) + eL qηq B∗ 1+ 3q−142qM2qLq 4e−(2µq−L)t1 + 3q−142qM2qLq 5Zt t1 e2µq(s−t1)kxsk2q PCds. (70) Solving the above Gronwall’s inequality yields kxtk2q PC ≤3q−1(8M2(1 + b1) + eL qη)qB∗ 1+L6e−(2µq−L)t1e−(2µq−L)(t−t1),(71) and, consequently, Ekx(t2)k2≤3q−1(8M2(1 + b1) + eL qη)qB∗ 1+L6e−(2µq−L)t11 qe−2µq−L q(t2−t1)= (B∗ 2)1 q.(72) Step 3. The same reasoning as above implies, for t∈(tk, tk+1] with k≥2, kxtk2q PC ≤3q−1(8M2(1 + b1) + eL qη)qB∗ k+L6e−(2µq−L)tke−(2µq−L)(t−tk),(73) and Ekx(tk+1)k2≤3q−1(8M2(1+b1)+eL qη)qB∗ k+L6e−(2µq−L)tk1 q e−2µq−L q(tk+1−tk)= (B∗ k+1)1 q.(74) For convenience, let w1= 3q−1(8M2(1+b1)+eL qη)q. It is obvious w1>2, such that, Pk−2 j=0 wj 1≤ wk−1 1 w1−1≤wk−1 1 w1−w1 2 ≤2wk−2 1. In addition, condition (H4) implies that k−1≤tk−t1 βand kβ < tk. IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 23 Then, for k≥2, the mathematical induction method furnishes that B∗ k≤w1B∗ k−1e−(2µq−L)(tk−tk−1)+L6e−(2µq−L)tk ≤wk−1 1B∗ 1e−(2µq−L)(tk−t1)+L6e−(2µq−L)tk k−2 X j=0 wj 1 ≤B∗ 1e(tk−t1)ln w1 βe−(2µq−L)(tk−t1)+ 2L6e−(2µq−L)tkwk−2 1 ≤B∗ 1e−(2µq−L−ln w1 β)(tk−t1)+ 2L6e−(2µq−L)tke(k−2) ln w1 ≤B∗ 1e−(2µq−L−ln w1 β)(tk−t1)+ 2L6e−(2µq−L)tketk βln w1 ≤B∗ 1e−(2µq−L−ln w1 β)(tk−t1)+ 2L6e−(2µq−L−ln w1 β)tk. (75) Therefore, by (60), (73) and (75) we deduce that, for t∈(tk, tk+1] with k≥2, kxtk2q PC ≤w1B∗ ke−(2µq−L−ln w1 β)(t−tk)+L6e−(2µq−L)t ≤w1B∗ 1e−(2µq−L−ln w1 β)(tk−t1)+ 2L6e−(2µq−L−ln w1 β)tk ×e−(2µq−L−ln w1 β)(t−tk)+L6e−(2µq−L−ln w1 β)t ≤w1(3q−14qM2qkφk2q PC + 3L6)e−(2µq−L−ln w1 β)t ≤C(1 + kφk2q PC)e−(2µq−L−ln w1 β)t, which, thanks to (71) and (59), implies that, for all t≥0, kxtk2q PC ≤C(1 + kφk2q PC)e−(2µq−L−ln w1 β)t.(76) This completes the proof.  Conclusions. We have proved some results concerning the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both K-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter H∈(1/2,1). We also have proved the continuous dependence of mild solutions on initial values and have analyzed the exponential decay to zero of solutions to our fractional stochastic impulsive differential equations with infinite delay. However there are also some interesting points to be analyzed such as the existence and structure of random attractors for this model, since the global asymptotic behavior can be described by such objects (see e.g. [8,19] and the references cited therein). We plan to investigate this topic in the future. Acknowledgment. We would like to thank the anonymous referee for the interesting and helpful comments and suggestions which allowed us to improve the results and presentation of our paper. 24 IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY REFERENCES [1] D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal.,69 (2008), 3692-3705. [2] D. Bahuguna, R. Sakthivel and A. Chadha, Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay, Stoch. Anal. Appl.,35 (2017), 63-88. [3] E.G. Bajlekova, Fractional Evolution Equations in Banach Spaces, University Press Facilities, Eindhoven University of Technology, 2001. [4] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, 2006. [5] T. Blouhi, T. Caraballo and A. Ouahab, Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion, Stoch. Anal. Appl.,34 (2016), 792-834. [6] E.M. Bonotto, M.C. Bortolan, T. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations,262 (2017), 3524-3550. [7] A. Boudaoui, T. Caraballo and A. Ouahab, Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay, Math. Methods Appl. Sci.,39 (2016), 1435-1451. [8] T. Caraballo and P.E. Kloeden, Non-autonomous attractors for integral-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S.,2(2009), 17-36. [9] T. Caraballo, M.J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal.,74 (2011), 3671-3684. [10] A. Chauhan and J. Dabas, Local and global existence of mild solution to an impulsive fractional functional integro-differential equations with nonlocal condition, Commun. Nonlinear Sci. Numer. Simul.,19 (2014), 821-829. [11] R.F. Curtain and P.L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations,10 (1971), 412-430. [12] J. Dabas and A. Chauhan, Existence and uniqueness of mild solutions for an impulsive neutral fractional integro-differential equation with infinite delay, Math. Comput. Modelling,57 (2013), 754-763. [13] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [14] S.S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, 2003. [15] Z.B. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput.,232 (2014), 60-67. [16] M. Haase, The Functional Calculus for Sectorial Operators, Birkh¨auser Basel, 2006. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Spring-Verlag, 1981. [18] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006. [19] P.E. Kloeden, T. Lorenz and M.H. Yang, Attractors in Discrete Time Nonautonomous Dynamical Systems, Springer International Publishing, 2015. [20] B. Øksendal, Stochastic Differential Equations, Springer-Verlag, 1985. [21] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989. [22] Y.J. Li and Y.J. Wang, Uniform asymptotic stability of solutions of fractional functional differential equations, Abstr. Appl. Anal.,5(2013), 685-696. [23] Y.J. Li and Y.J. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, preprint. [24] A.H. Lin, Y. Ren and N.M. Xia, On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators, Math. Comput. Modelling,51 (2010), 413-424. [25] S.Y. Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Inequal. Appl.,1(2013), 549. [26] R.H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Printed in the United States of America, 1976. [27] V.D. Mil’man and A.D. Myskis, On the stability of motion in the prensence of impulses, Sibirsk. Mat. Zh.,1 (1960), 233-237. [28] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-verlag, 2008. [29] X.R. Mao, Stochastic Differential Equations and Applications, Horwood Publication, Chichester, 1997. IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE DELAY 25 [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [31] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [32] J. Pr¨uss, Evolutionary Integral Equations and Applications, Brikhauster, Springer Basel, 1993. [33] Y. Ren, X. Cheng and R. Sakthivel, Impulsive neutral stochastic integral-differential equations with infinite delay fBm, Appl. Math. Comput.,247 (2014), 205-212. [34] R. Sakthivel, P. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal.,81 (2013), 70-86. [35] J.H. Shen and X.Z. Liu, Global existence results of impulsive differential equation, J. Math. Anal. Appl.,314 (2006), 546-557. [36] X.B. Shu, Y.Z. Lai and Y.M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal.,74 (2011), 2003-2011. [37] S. Tindel, C.A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields,127 (2003), 186-204. [38] R.N. Wang, D.H. Chen and T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations,252 (2012), 202-235. [39] Y.J. Wang, F.S. Gao and P.E. Kloeden, On impulsive fractional functional differential equations with weakly continuous nonlinearity and asymptotic hehavior of impulsive fractional delay lattice systems, Electron. J. Differ. Equ.,285 (2017), 1-18. E-mail address:[email protected] E-mail address:[email protected]