International Journal of Analysis and Applications ISSN 2291-8639 Volume 6, Number 1 (2014), 28-43 http://www.etamaths.com ALMOST PERIODIC SOLUTIONS FOR IMPULSIVE FRACTIONAL STOCHASTIC EVOLUTION EQUATIONS TOUFIK GUENDOUZI∗ AND LAMIA BOUSMAHA Abstract. In this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equa- tions involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations. Some known results are improved and gen- eralized. 1. Introduction The study of fractional differential equations has been gaining importance in recent years due to the fact that fractional order derivatives provide a tool for the description of memory and hereditary properties of various phenomena. Due to this fact, the fractional order models are capable to describe more realistic situ- ation than the integer order models. Fractional differential equations have been used in many field like fractals, chaos, electrical engineering, medical science, etc. In recent years, we have seen considerable development on the topics of fractional differential equations, for instance, we refer to the monographs of Abbas et al. [2], Kilbas et al. [14], Miller and Ross [18], Podlubny [20], and the papers [3, 4, 31]. In particular, differential equations with impulsive conditions constitute an im- portant field of research due to their numerous applications in ecology, medicine biology, electrical engineering and other areas of science. Many physical phenomena in evolution processes are modeled as impulsive fractional differential equations and existence results for such equations have been studied by several authors [9, 23, 30]. One of the important problems in the qualitative theory of impulsive differential equations is the existence of almost periodic solutions. At the present time, many results on the existence, uniqueness and stability of these solutions have been ob- tained (see [1, 15, 24, 26] and the references therein). However, only few papers deal with the existence of almost periodic solutions for impulsive fractional differ- ential equations. Recently, Debbouche et al. [11] studied the existence of almost periodic and optimal mild solutions of fractional evolution equations with analytic semigroup in a Banach space. El-Borai et al. [12] established the existence and uniqueness of almost periodic solutions of a class of nonlinear fractional differential equations with analytic semigroup in Banach space, and very recently, Stamov et 2010 Mathematics Subject Classification. 26A33, 34C27, 34G20, 34A37, 35B15. Key words and phrases. Square-mean piecewise almost periodic, Impulsive fractional stochastic differential equations, Analytic semigroup. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 28 IMPULSIVE FRACTIONAL STOCHASTIC EVOLUTION EQUATIONS 29 al. [27] studied the existence of almost periodic solutions for fractional differential equations with impulsive effects. In many cases, deterministic models often fluctuate due to environmental noise, which is random or at least appears to be so. Therefore, we must move from de- terministic problems to stochastic ones. Taking the disturbances into account, the theory of differential equations has been generalized to stochastic case. The exis- tence, uniqueness, stability, controllability and other quantitative and qualitative properties of solutions of stochastic evolution equations have recently received a lot of attention (see [13, 17, 28] and the references therein). The existence of al- most periodic solutions for stochastic differential equations has been discussed in [5, 7, 21]. The existence of almost periodic solutions for impulsive stochastic evolu- tion equations has been reported in [8, 16]. However, up to now the problem for the existence of almost periodic solutions for impulsive fractional stochastic evolution equations have not been considered in the literature. In order to fill this gap, this paper studies the existence of square-mean piecewise almost periodic solutions of the following impulsive fractional stochastic differential equations in the form (1) cDαt x(t) + Ax(t) = F(t,x(t)) + Σ(t,x(t)) dw(t) dt + ∞∑ k=−∞ Gk(x(t))δ(t−τk), t ∈ IR, where the state x(·) takes values in the space L2(IP,H), H is a separable real Hilbert space with inner product (·, ·) and norm ‖·‖; the fractional derivative cDα, α ∈ (0, 1), is understood in the Caputo sense; −A : D(A) ⊂ L2(IP,H) → L2(IP,H) is the infinitesimal generator of an analytic semigroup of a bounded linear operator S(t), t ≥ 0, on L2(IP,H) satisfying the exponential stability; {w(t) : t ≥ 0} is a given K-valued Wiener process with a finite trace nuclear covariance operator Q ≥ 0 defined on a filtered complete probability space (Ω,F,{Ft}t≥0,IP), K is another separable Hilbert space with inner product (·, ·)K and norm ‖ · ‖K; Gk : D(Gk) ⊂ L2(IP,H) → L2(IP,H) are continuous impulsive operators, δ(·) is Dirac’s delta-function, F(t,x) : IR ×L2(IP,H) → L2(IP,H) and Σ(t,x) : IR ×L2(IP,H) → L2(IP,L02(K,H)) are jointly continuous functions (here, L02(K,H) denotes the space of all Q-Hilbert-Schmidt operators from K into H, which is going to be defined below). The structure of this paper is as follows. In Sect. 2, we will recall briefly some preliminaries fact which will be used in paper. Section 3, we establish criteria of the existence of an almost periodic solution and its exponential stability. 2. Preliminaries In this section, we introduce some basic definitions, notation and lemmas which are used throughout this paper. Let (H,‖ · ‖H) and (K,‖ · ‖K) be two real separable Hilbert spaces, and we denote by L(K,H) the set of all linear bound- ed operators from K into H, equipped with the usual operator norm ‖ · ‖. We will use the symbol ‖ · ‖ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let (Ω,F,IP) be a com- plete probability space equipped with a normal filtration {Ft}t≥0 satisfying the usual conditions (i.e., right continuous and F0 containing all IP-null sets). Let {ei}∞i=1 be a complete orthonormal basis of K. Suppose that w = (wt)t≥0 is a 30 GUENDOUZI AND BOUSMAHA cylindrical K-valued Wiener process with a finite trace nuclear covariance oper- ator Q ≥ 0, denote Tr(Q) = ∑∞ i=1 λ̃i = λ̃ < ∞, which satisfies Qei = λ̃iei. So, actually, w(t) = ∑∞ i=1 √ λ̃iwi(t)ei, where {wi(t)}∞i=1 are mutually independent one- dimensional standard Wiener processes. We assume that Ft = σ{w(s) : 0 ≤ s ≤ t} is the σ-algebra generated by w. Let L02 = L2(Q 1 2K,H) be the space of all Hilbert- Schmidt operators from Q 1 2K to H with the inner product (ϕ,φ)L02 = Tr[ϕQφ ∗]. For more details, we refer the reader to Da Prato and Zabczyk [10]. The collection of all measurable, square integrable, H-valued random variables, denoted by L2(IP,H) is a Banach space equipped with norm ‖x(·)‖L2 = (IE‖x(·)‖2) 1 2 , where IE(·) denotes the expectation with respect to the measure IP. Let C(IR,L2(IP,H)) be the Banach space of all continuous maps from IR into L2(IP,H) satisfying the condition supt∈IR IE‖x(t)‖2 < ∞. Let L2F0 (IP,H) denote the family of all F0- measurable, H-valued random variables x(0). Let us now recall some basic definitions and results of fractional calculus. For more details see [14, 18, 20]. Definition 2.1. The fractional integral of order α with the lower limit zero for a function f is defined as Iαf(t) = 1 Γ(α) ∫ t 0 f(s) (t−s)1−α ds, t > 0,α > 0, provided the right-hand side is pointwise defined on [0,∞), where Γ(·) is the gamma function, which is defined by Γ(α) = ∫∞ 0 tα−1e−tdt. Definition 2.2. The Riemann-Liouville fractional derivative of order α > 0, n− 1 < α < n, n ∈ IN, is defined as (R−L)Dα0+f(t) = 1 Γ(n−α) ( d dt )n ∫ t 0 (t−s)n−α−1f(s)ds, where the function f(t) has absolutely continuous derivative up to order (n− 1). Definition 2.3. The Caputo derivative of order α > 0 for a function f : [0,∞) → IR can be written as Dαf(t) = Dα ( f(t) − n−1∑ k=0 tk k! f(k)(0) ) , t > 0,n− 1 < α < n. Remark 2.4. (i) If f(t) ∈Cn[0,∞), then CDαf(t) = 1 Γ(n−α) ∫ t 0 f(n)(s) (t−s)α+1−n ds = In−αf(n)(t), t > 0,n− 1 < α < n. (ii) The Caputo derivative of a constant is equal to zero. (iii) If f is an abstract function with values in H, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochners sense. Let B = {{τk} : τk ∈ IR, τk < τk+1, k ∈ Z} be the set of all sequences unbounded and strictly increasing. We consider the impulsive fractional differential equation (1), and denote by x(t) = x(t; t0,x0), t0 ∈ IR, x0 ∈H, the solution of (1) with the initial condition (2) x(t0) = x0. IMPULSIVE FRACTIONAL STOCHASTIC EVOLUTION EQUATIONS 31 Definition 2.5 ([16]). A stochastic process x : IR → L2(IP,H), is said to be s- tochastically bounded if there exists N > 0 such that IE‖x(t)‖2 ≤ N for all t ∈ IR. Definition 2.6 ([16]). A stochastic process x : IR → L2(IP,H), is said to be s- tochastically continuous in s ∈ IR, if limt→s IE‖x(t) −x(s)‖2 = 0. For {τk} ∈ B and k ∈ Z, let PC(IR,L2(IP,H)) be the space consisting of all stochastically bounded functions φ : IR → L2(IP,H) such that φ(·) is stochastically continuous at t for any t /∈ {τk}, τk ∈ IR, k ∈ Z and φ(τ−k ) = φ(τk). In particu- lar, we introduce the space PC(IR × L2(IP,H),L2(IP,H)) formed by all piecewise stochastically continuous stochastic processes φ : IR ×L2(IP,H) → L2(IP,H) such that for any x ∈ L2(IP,H), φ(·,x) is stochastically continuous at t for any t /∈{τk} and φ(τ−k ,x) = φ(τk,x) for all k ∈ Z, and for any t ∈ IR, φ(t, ·) is stochastically continuous at x ∈ L2(IP,H). Remark 2.7 ([16, 29]). The solution x(t) = x(t; t0,x0) of the problem (1)-(2) is a piecewise stochastically continuous, Ft-adapted measurable process with points of discontinuity at the moments τk, k ∈ Z, at which it is continuous from the left. Definition 2.8 ([26]). The set of sequences {τjk}, τ j k = τk+j − τk, k ∈ Z, j ∈ Z, {τk} ∈ B is said to be equipotentially almost periodic, if for arbitrary � > 0 there exists a relatively dense set B� of IR such that for each κ ∈ B� there is an integer q ∈ Z such that |τk+q − τk −κ| < � for all k ∈ Z. Definition 2.9 ([7]). A stochastic process x ∈ PC(IR,L2(IP,H)) is said to be square-mean picewise almost periodic, if: (i) The set of sequences {τjk}, τ j k = τk+j − τk, k ∈ Z, j ∈ Z, {τk} ∈ B is equipotentially almost periodic. (ii) For any � > 0, there exists a real number δ > 0 such that if the points t′ and t′′ belong to one and the same interval of continuity of x(t) and satisfy the inequality |t′ − t′′| < δ, then IE‖x(t′) −x(t′′)‖2H < �. (iii) For any � > 0, there exists a relatively dense set T such that if τ ∈ T , then IE‖x(t + τ) −x(t)‖2H < �, satisfying the condition |t−τk| > �, k ∈ Z. The elements of T are called �-translation number of x. We denote by AP(IR,L2(IP,H)) the collection of all the square-mean piece- wise almost periodic processes, it thus is a Banach space with the norm ‖x‖∞ = supt∈IR ‖x(t)‖L2 = supt∈IR(IE‖x(t)‖2) 1 2 for x ∈AP(IR,L2(IP,H)). Lemma 2.10 ([16]). Let F ∈AP(IR,L2(IP,H)). Then, R(F), the range of F is a relatively compact set of L2(IP,H). Definition 2.11 ([8]). For {τk} ∈ B, k ∈ Z, the function F(t,x) ∈ PC(IR × L2(IP,H),L2(IP,H)) is said to be square-mean piecewise almost periodic in t ∈ IR and uniform on compact subset of L2(IP,H)) if for every � > 0 and every compact subset K ⊆ L2(IP,H)), there exists a relatively dense subset T of IR such that IE‖F(t + τ,x) −F(t,x)‖2 < �, for all x ∈ K, τ ∈ T , t ∈ IR satisfying |t − τk| > �, k ∈ Z. The collection of all such processes is denoted AP(IR×L2(IP,H),L2(IP,H)). Lemma 2.12 ([16]). Suppose that F(t,x) ∈ AP(IR × L2(IP,H),L2(IP,H)) and F(t, ·) is uniformly continuous on each compact subset K ⊆ L2(IP,H) uniformly 32 GUENDOUZI AND BOUSMAHA for t ∈ IR. That is, for all � > 0, there exists δ > 0 such that x,y ∈ K and IE‖x−y‖2 < δ implies that IE‖F(t,x)−F(t,y)‖2 < � for all t ∈ IR. Then F(·,x(·)) ∈ AP(IR,L2(IP,H)) for any x ∈AP(IR,L2(IP,H)). We obtain the following corollary as an immediate consequence of Lemma 2.12. Corollary 2.13. Let F(t,x) ∈ AP(IR ×L2(IP,H),L2(IP,H)) and F is Lipschitz, i.e., there is a number c > 0 such that IE‖F(t,x) −F(t,y)‖2 ≤ cIE‖x−y‖2, for all t ∈ IR and x,y ∈ L2(IP,H), if for any x ∈AP(IR,L2(IP,H)), then F(·,x(·)) ∈ AP(IR,L2(IP,H)). Definition 2.14. A sequence x : Z → L2(IP,H) is called a square-mean almost periodic sequence if �-translation set of x I(x; �) = {τ ∈ Z : IE‖x(n + τ) −x(t)‖2 < �, for all n ∈ Z} is a relatively dense set in Z for all � > 0. The collection of all square-mean almost periodic sequences x : Z → L2(IP,H) will be denoted by AP(Z,L2(IP,H)). Remark 2.15. If x(n) ∈ AP(Z,L2(IP,H)), then {x(n) : n ∈ Z} is stochastically bounded. Lemma 2.16 ([16]). Assume that F ∈ AP(IR,L2(IP,H)), the sequence {xk : k ∈ Z} is almost periodic in L2(IP,H) and {τjk}, j ∈ Z, is equipotentially almost peri- odic. Then for each � > 0 there are relatively dense sets T�,F,xk of IR and T̂�,F,xk of Z such that the following conditions hold: (i) IE‖F(t + τ)−F(t)‖2 < � for all t ∈ IR, |t−τk| > �, τ ∈ T�,F,xk and k ∈ Z. (ii) IE‖xk+q −xk‖2 < � for all q ∈ T̂�,F,xk and k ∈ Z. (iii) For every τ ∈ T�,F,xk , there exists at least one number q ∈ T̂�,F,xk such that |τqk − τ| < �, k ∈ Z. Consider the linear fractional impulsive stochastic differential equation corre- sponding to (1) (3) cDαt x(t) + Ax(t) = f(t) + σ(t) dw(t) dt + ∞∑ k=−∞ gkδ(t− τk), where f ∈PC(IR,L2(IP,H)), σ ∈PC(IR,L2(IP,L02)) and gk : D(gk) ⊂ L2(IP,H) → L2(IP,H). Let us introduce the following conditions. (C1) The set of sequences {τjk}, τ j k = τk+j − τk, k ∈ Z, j ∈ Z, {τk} ∈ B is equipotentially almost periodic and there exists θ > 0 such that infk τ 1 k = θ. (C2) The function f is in AP(IR,L2(IP,H)) and locally Hölder continuous with points of discontinuity at the moments τk, k ∈ Z at which it is continuous from the left. (C3) The function σ is in AP(IR,L2(IP,L02)) and locally Hölder continuous with points of discontinuity at the moments τk, k ∈ Z at which it is continuous from the left. IMPULSIVE FRACTIONAL STOCHASTIC EVOLUTION EQUATIONS 33 (C4) {gk}, k ∈ Z, of impulsive operators is a square-mean almost periodic se- quence. Lemma 2.17 ([15],[16]). Let the condition (C1) holds. Then (i) There exists a constant p > 0 such that, for every t ∈ IR lim T→∞ ι(t,t + T) T = p. (ii) For each p > 0 there exists a positive integer N such that each interval of length p has no more than N elements of the sequence {τk}, that is, ι(s,t) ≤ N(t−s) + N, where ι(s,t) is the number of points τk in the interval (s,t). The following Lemma is an immediate consequence of Lemma 2.16. Lemma 2.18. Let the conditions (C1)-(C4) hold. Then, for each � > 0 there are relatively dense sets T�,f,σ,gk of IR and T̂�,f,σ,gk of Z such that the following relations hold: (i) IE‖f(t + τ) −f(t)‖2 < �, t ∈ IR, τ ∈ T�,f,σ,gk , |t− τk| > �, k ∈ Z. (ii) IE‖σ(t + τ) −σ(t)‖2 < �, t ∈ IR, τ ∈ T�,f,σ,gk, |t− τk| > �, k ∈ Z. (iii) IE‖gk+q −gk‖2 < �, k ∈ Z, q ∈ T̂�,f,σ,gk. (iv) For each τ ∈ T�,f,σ,gk, ∃q ∈ T̂�,f,σ,gk , such that |τk+q − τk − τ| < �, k ∈ Z. Now, we present the definition of mild solutions for the problem (2)-(3) based on the paper [25]. Definition 2.19. A stochastic process x ∈ PC(J,L2(IP,H)), J ⊂ IR is called a mild solution of the problem (2)-(3) if (i) x0 ∈ L2F0 (IP,H); (ii) x(t) ∈ L2(IP,H) has càdlàg paths on t ∈ J a.s., and it satisfies the follow- ing integral equation (4) x(t) =   T (t− t0)x0 + ∫ t t0 (t−s)α−1S(t−s)f(s)ds + ∫ t t0 (t−s)α−1S(t−s)σ(s)dw(s), t ∈ [t0,τ1], T (t− t0)x0 + T (t− τ1)g1 + ∫ t t0 (t−s)α−1S(t−s)f(s)ds + ∫ t t0 (t−s)α−1S(t−s)σ(s)dw(s), t ∈ (τ1,τ2], ... T (t− t0)x0 + ∑ t0<τk 0, we define the fractional power A−β of the operator a by A−β = 1 Γ(β) ∫ ∞ 0 tβ−1S(t)dt, where A−β is bounded, bijective and Aβ = (A−β)−1, β > 0 a closed linear operator on its domain D(Aβ) and such that D(Aβ) = R(A−β) where R(A−β) is the range of A−β. Furthermore, the subspace D(Aβ) is dense in L2(IP,H) and the expression ‖x‖β = ‖Aβx‖, x ∈D(Aβ), defines a norm on L2(IP,Hβ) := D(Aβ). The following properties are well known. Lemma 2.21 ([19]). Suppose that the preceding conditions are satisfied. Then (i) S(t) : L2(IP,H) →D(Aβ) for every t > 0 and β ≥ 0. (ii) For every x ∈D(Aβ), the following equality S(t)Aβx = AβS(t)x holds. (iii) For every t > 0, the operator AβS(t) is bounded and ‖AβS(t)‖≤ Kβt−βe−λt, Kβ > 0,λ > 0. (iv) For 0 < β ≤ 1 and x ∈D(Aβ), we have ‖S(t)x−x‖≤ Cβtβ‖Aβx‖, Cβ > 0. When −A generates a semi-group with negative exponent, we deduce that if x(t) is a bounded solution of (3) on IR, then we take the limit as t0 → −∞ and using (4), we obtain (see [6]) (5) x(t) = ∫ t −∞ (t−s)α−1S(t−s)f(s)ds+ ∫ t −∞ (t−s)α−1S(t−s)σ(s)dw(s)+ ∑ τk 0. In view of Lemma 2.21 and the definition of the norm in Hβ, we obtain (6) IE‖x(t)‖2β = IE‖A βx(t)‖2 ≤ 3IE ∥∥∥∫ t −∞ (t−s)α−1AβS(t−s)f(s)ds ∥∥∥2 +3IE ∥∥∥∫ t −∞ (t−s)α−1AβS(t−s)σ(s)dw(s) ∥∥∥2 + 3IE∥∥∥ ∑ τk 0, τ ∈ T�,f,σ,gk and q ∈ T̂�,f,σ,gk, k ∈ Z, where the sets T�,f,σ,gk and T̂�,f,σ,gk are defined as in Lemma 2.18. We have IMPULSIVE FRACTIONAL STOCHASTIC EVOLUTION EQUATIONS 37 x(t + τ) −x(t) = (∫ t+τ −∞ (t + τ −s)α−1S(t + τ −s)f(s)ds + ∫ t+τ −∞ (t + τ −s)α−1S(t + τ −s)σ(s)dw(s) + ∑ τk � and Mβ = Kβ [ Γ2(1 −β) λ2(1−β) + Tr(Q)N1 Γ2(1 − 2β) λ2(1−β) + 4N2 ( 1 N β 2 + 1 eβ − 1 )2] . The last inequality implies that x(t) is a square-mean piecewise almost periodic process, so system (2)-(3) has a square-mean piecewise almost periodic solution. The proof is complete. � In order to obtain the existence of square-mean piecewise almost periodic solu- tion to system (1)-(2), we introduce the following conditions: (C5) −A : D(A) ⊆ L2(IP,H) → L2(IP,H) is the infinitesimal generator of an exponentially stable analytic semi-group S(t), t ∈ IR, on L2(IP,H). (C6) F(t,x) ∈ AP(IR ×L2(IP,Hβ),L2(IP,H)) with respect to t ∈ IR uniformly in x ∈ K, for each compact set K ⊆ L2(IP,H), and there exist constants c̃ > 0, 0 < κ < 1, 0 < β < 1, such that IE‖F(t1,x1) −F(t2,x2)‖2 ≤ c̃ ( |t1 − t2|κ + IE‖x1 −x2‖2β ) , 38 GUENDOUZI AND BOUSMAHA where (ti,xi) ∈ IR×L2(IP,Hβ), i = 1, 2. (C7) Σ(t,x) ∈AP(IR ×L2(IP,Hβ),L2(IP,L02)) with respect to t ∈ IR uniformly in x ∈ K, for each compact set K ⊆ L2(IP,H), and there exist constants ĉ > 0, 0 < κ < 1, 0 < β < 1, such that IE‖Σ(t1,x1) − Σ(t2,x2)‖2L02 ≤ ĉ ( |t1 − t2|κ + IE‖x1 −x2‖2β ) , (ti,xi) ∈ IR×L2(IP,Hβ), i = 1, 2. (C8) The sequence {Gk(x)} is almost periodic in k ∈ Z uniformly in x ∈ K ⊆ L2(IP,H), and there exist constants c̄ > 0, 0 < β < 1, such that IE‖Gk(x1) −Gk(x2)‖2 ≤ c̄IE‖x1 −x2‖2β, where x1,x2 ∈ L2(IP,Hβ). Theorem 3.2. Assume that the conditions (C1), (C5)-(C8) are satisfied, then the impulsive fractional stochastic system (1)-(2) admits a unique square-mean piece- wise almost periodic mild solution. Proof. Let B the set of all x ∈ AP(IR,L2(IP,H)) with discontinuities of the first type at the points τk, k ∈ Z, {τk} ∈ B, satisfying the inequality IE‖x‖2 ≤ r, r > 0. Obviously, B is a closed set of AP(IR,L2(P,H)). Define the operator Θ in B by (13) Θx(t) = ∫ t −∞ (t−s)α−1AβS(t−s)F(s,A−βx(s))ds + ∫ t −∞ (t−s)α−1AβS(t−s)Σ(s,A−βx(s))dw(s) + ∑ τk 0, there exists a relatively dense set T such that for τ ∈ T the following property IE‖F(t + τ,A−βx(t + τ)) −F(t,A−βx(t))‖2 < �λ2(1−β) K2βΓ 2(1 −β) IMPULSIVE FRACTIONAL STOCHASTIC EVOLUTION EQUATIONS 39 hold, satisfying the condition |t− τk| > �, for each t ∈ IR and k ∈ Z. By virtue of Lemma 2.21, we have IE‖Θ1x(t + τ) − Θ1x(t)‖2 = IE ∥∥∥∫ t −∞ (t−s)α−1AβS(t−s)[F(s + τ,A−βx(s + τ)) −F(s,A−βx(s))]ds ∥∥∥2 ≤ α2K2β ∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α dηdθ ∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α ×IE‖F(t + τ −η,A−βx(t + τ −η)) −F(t−η,A−βx(t−η))‖2dηdθ ≤ α2K2β (∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α dηdθ )2 ×sup t∈R IE‖F(t + τ,A−βx(t + τ)) −F(t,A−βx(t))‖2 = K2β Γ2(1 −β) λ2(1−β) sup t∈R IE‖F(t + τ,A−βx(t + τ)) −F(t,A−βx(t))‖2 < K2β Γ2(1 −β) λ2(1−β) × �λ2(1−β) K2βΓ 2(1 −β) = �. Hence, Θ1x(·) ∈ B. Similarly, by using condition (C7), since Aβ is closed and Σ(t,x) ∈ AP(IR × L2(IP,Hβ),L2(IP,L02)), we have from Corollary 2.13 that A−βx ∈ B and Σ(·,A−βx(·)) ∈ AP(IR,L2(IP,L02)). Therefore, it follows from Definition 2.9 and Lemma 2.16 that for any � > 0, there exists a relatively dense set T such that for τ ∈ T the following property IE‖Σ(t + τ,A−βx(t + τ)) − Σ(t,A−βx(t))‖2L02 < �λ2−2β N1K 2 βTr(Q)Γ(1 − 2β) hold, satisfying the condition |t− τk| > �, for each t ∈ IR and k ∈ Z. By virtue of Lemma 2.21, for w̃(t) := w(t + τ) −w(τ), we have IE‖Θ2x(t + τ) − Θ2x(t)‖2 = IE ∥∥∥∫ t −∞ (t−s)α−1AβS(t−s)[Σ(s + τ,A−βx(s + τ)) − Σ(s,A−βx(s))]dw̃(s) ∥∥∥2 ≤ α2K2βTr(Q) ∫ ∞ 0 ξ2α(θ) ∫ ∞ 0 θ2(1−β)η2(α−αβ−1)e−2λθη α ×IE‖Σ(t + τ −η,A−βx(t + τ −η)) − Σ(t−η,A−βx(t−η))‖2 L02 dηdθ ≤ α2K2βTr(Q) ∫ ∞ 0 ξ2α(θ) ∫ ∞ 0 θ2(1−β)η2(α−αβ−1)e−2λθη α dηdθ ×sup t∈R IE‖Σ(t + τ,A−βx(t + τ)) − Σ(t,A−βx(t))‖2L02 ≤ K2βTr(Q)N1 Γ(1 − 2β) λ2−2β sup t∈R IE‖Σ(t + τ,A−βx(t + τ)) − Σ(t,A−βx(t))‖2L02 < K2βTr(Q)N1 Γ(1 − 2β) λ2−2β × �λ2−2β N1K 2 βTr(Q)Γ(1 − 2β) = �. Thus, Θ2x(·) ∈ B. And in view of the above, it is clear that Θ maps B into itself. Next, we show that Θ is a contracting operator on B. Let x1,x2 ∈ B. Then , 40 GUENDOUZI AND BOUSMAHA we have IE‖Θx1(t) − Θx2(t)‖2 ≤ 3IE ∥∥∥∫ t −∞ (t−s)α−1AβS(t−s)[F(s,A−βx1(s)) −F(s,A−βx2(s))]ds ∥∥∥2 +3IE ∥∥∥∫ t −∞ (t−s)α−1AβS(t−s)[Σ(s,A−βx1(s)) − Σ(s,A−βx2(s))]dw(s) ∥∥∥2 +3IE ∥∥∥ ∑ τk 0 is sufficiently small and R(θ) is defined as in above. By following similar arguments like those used in (7), we have IE‖Θx1(t) − Θx2(t)‖2 ≤ 3c∗K2β [ Γ2(1 −β) λ2(1−β) +Tr(Q)N1 Γ2(1 − 2β) λ2(1−β) + 4N2 ( 1 N β 2 + 1 eβ − 1 )2] sup t∈IR IE‖x1(t) −x2(t)‖2. Therefore, if c∗ is chosen in the form c∗ ≤ ( 3K2β [ Γ2(1 −β) λ2(1−β) + Tr(Q)N1 Γ2(1 − 2β) λ2(1−β) + 4N2 ( 1 N β 2 + 1 eβ − 1 )2])−1 , we have IE‖Θx1(t) − Θx2(t)‖2 ≤ 3c∗K2β [ Γ2(1 −β) λ2(1−β) + Tr(Q)N1 Γ2(1 − 2β) λ2(1−β) + 4N2 ( 1 N β 2 + 1 eβ − 1 )2] ‖x1 −x2‖2∞, implies that, ‖Θx1 − Θx2‖∞ ≤ √ Λ‖x1 −x2‖∞, Λ = 3c∗K 2 β [ Γ2(1 −β) λ2(1−β) + Tr(Q)N1 Γ2(1 − 2β) λ2(1−β) + 4N2 ( 1 N β 2 + 1 eβ − 1 )2] . Thus, Θ is a contracting operator on B. So by the contraction principle, we conclude that there exists a unique fixed point x for Θ in B, such that x = Θx, that is (14) x(t) = ∫ t −∞ (t−s)α−1AβS(t−s)F(s,A−βx(s))ds + ∫ t −∞ (t−s)α−1AβS(t−s)Σ(s,A−βx(s))dw(s) + ∑ τk 0, for each t ∈ IR, IE‖A−βx(t)‖2 ≤ r. Hence, A−βx ∈ B is mild solution of the problem (1)-(2). � Theorem 3.3. Assume that the conditions (C1), (C5)-(C8) are satisfied, then the impulsive fractional stochastic system (1)-(2) has an exponentially stable almost periodic solution. Proof. Let u(t) be the solution of the following integral equation (15) u(t) = ∫ t −∞ (t−s)α−1AβS(t−s)F(s,A−βu(s))ds + ∫ t −∞ (t−s)α−1AβS(t−s)Σ(s,A−βu(s))dw(s) + ∑ τk