Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion

Manuscript submitted to doi:10.3934/dcdsb.XXXXXX AIMS’ Journals Volume XX, Number 0X, Month XXXX pp. X–XX STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES DRIVEN BY A FRACTIONAL BROWNIAN MOTION Ahmed Boudaoui Laboratory of Mathematics, Univ Sidi Bel Abbes PoBox 89, 22000 Sidi-Bel-Abbes, Algeria Tom´ as Caraballo1 Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080 Sevilla, Spain Abdelghani Ouahab Laboratory of Mathematics, Univ Sidi Bel Abbes PoBox 89, 22000 Sidi-Bel-Abbes, Algeria Abstract. This paper is concerned with the existence and continuous dependence of mild solutions to stochastic differential equations with non–instantaneous impulses driven by fractional Brownian motions. Our approach is based on a Banach fixed point theorem and Krasnoselski-Schaefer type fixed point theorem. 1. Introduction. Stochastic differential equations have many applications in science and engineering, and for this reason, these equations have been receiving much attention over the last decades (see, e.g., [4,8,10,34,15,19,30,29,1] and references therein). Also, in recent years, stochastic differential equations driven by fractional Brownian motions (fBm) have attracted much attention and there are only a few papers published in this field (see, e.g. [9,11,12]). Boudaoui et al. [6] discussed the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fBm with Hurst index H > 1 2. On the other hand, impulsive effects exist in several evolution processes in which states are changed abruptly at certain moments of time, related to fields such as economics, bioengineering, chemical technology, medicine and biology etc (see [20,33,13,27,28]). Boudaoui et al. [8] obtained several new sufficient conditions to ensure the local and global existence and attractivity of mild solutions for stochastic neutral functional differential equations with instantaneous impulses, driven by fractional Brownian motions. 2010 Mathematics Subject Classification. Primary: 34A12, 34A37; Secondary: 35A01, 60H15, 60H20. Key words and phrases. Fractional Brownian motion, fixed point, mild solutions, stochastic functional differential equation. This work has been partially 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. 1Corresponding author 1 2 A. BOUDAOUI, T. CARABALLO AND A. OUAHAB Recently, Hern´andez and O’Regan [18] initially study on Cauchy problems for a new type of first order evolution equations with non–instantaneous impulses. For example, impulses start abruptly at the instant tkand their action continue on a finite time interval (tk, sk]. This type of problem motivates to study certain dynamical changes of evolution processes in pharmacotheraphy [26,14,32]. As a motivation, we can mention a simple situation concerning the hemodynamical behavior of a person who has a decompensation of the glucose level (either high or low level). Then, this person can be prescribed some intravenous insulin to compensate the level. Since the introduction of the drug in the bloodstream and its absorption are gradually continuous processes, we can interpret the above situation as an impulsive action which remains active for a period of time, so a non-instantaneous impulse is taking place. Hence, it is important to take into account the effect of impulses in the investigation of stochastic delay differential equations driven by fBm. In this paper, our main objective is to establish sufficient conditions ensuring existence and continuous dependence of mild solutions to the following first order stochastic impulsive differential equation: dy(t)=[Ay(t) + f(t, yt)]dt +g(t, yt)BH Q(t), t ∈Jk= (sk, tk+1], k = 0,··· , m, (1) y(t) = hk(t, yt)t∈(tk, sk], k = 1,··· , m, (2) y(t) = φ(t)∈ DF0,for a.e. t∈J0= (−∞,0],(3) in a real separable Hilbert space Hwith inner product (·,·) and norm k·k, where A:D(A)⊂ H −→ H is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators {S(t), t ≥0}satisfying kS(t)k2≤M, BH Qis a fractional Brownian motion on a real and separable Hilbert space K, with Hurst parameter H∈(1/2,1), and with respect to a complete probability space (Ω,F,Ft, P) furnished with a family of right continuous and increasing σalgebras {Ft, t ∈J= [0, T ]}satisfying Ft⊂ F. The impulse times tksatisfy 0 = t0=s0< t1≤s1< t2<··· < tm≤sm< tm+1 =T. As for ytwe mean the segment solution which is defined in the usual way, that is, if y(·,·) : (−∞, T ]×Ω→ H,then for any t≥0, yt(·,·) : (−∞,0] ×Ω→ H is given by: yt(θ, ω) = y(t+θ, ω),for θ∈(−∞,0], ω ∈Ω, Before describing the properties fulfilled by the operators f, g, σ and hk, we need to introduce some notation and describe some spaces. In this work, we will use an axiomatic definition of the phase space DF0intro- duced by Hale and Kato [17]. Definition 1.1. DF0is a linear space of family of F0-measurable functions from (−∞,0] into Hendowed with a norm k·kDF0, which satisfies the following axioms: (A-1): If y: (−∞, T]−→ H,T > 0 is such that y0∈ DF0, then for every t∈[0, T) the following conditions hold (i): yt∈ DF0 (ii): ky(t)k ≤ LkytkDF0 (iii): kytkDF0≤K(t) sup{ky(s)k: 0 ≤s≤t}+N(t)ky0kDF0, where L>0 is a constant; K, N : [0,∞)−→ [0,∞), Kis continuous, N is locally bounded and K, N are independent of y(·). (A-2): For the function y(·) in (A-1), ytis a DF0-valued function for t∈[0, T). (A-3): The space DF0is complete. STOCHASTIC DELAY EVOLUTION EQUATIONS WITH IMPULSES 3 Denote e K= sup{K(t) : t∈J}and e N= sup{N(t) : t∈J}. Now, for a given T > 0, we define DFT=y: (−∞, T]×Ω→ H, y|Jk∈C((tk, tk+1],H),for all ω∈Ω,for k= 0, . . . m, y0∈ DF0,and there exist y(t− k) and y(t+ k) with y(tk) = y(t− k), k = 1,··· , m, and sup t∈[0,T ] E(ky(t)k2)<∞, endowed with the norm kykDFT=kφkDF0+ sup 0≤s≤T (Eky(s)k2)1 2, where Jk= (tk, tk+1], k = 0,1,··· , m, and J0= (−∞,0]. Then we will consider our initial data φ∈ DF0. Assume that BH Qis a K-valued fractional Brownian motion with increment covariance given by a non-negative trace class operator Q(see next section for more details), and let us denote by L(K,H) the space of all bounded, continuous and linear operators from Kinto H. Then we assume that hk∈C((tk, sk]×DF0,H) for all k= 1,··· , m,f:J×DF0→ Hand σ:J×DF0→L0 Q(K,H) Here, L0 Q(K,H) denotes the space of all Q-Hilbert- Schmidt operators from Kinto H, which will be also defined in the next section. The plan of this paper is as follows. In Section 2 we introduce notations, definitions, and preliminary facts which are useful throughout the paper. In Section 3 we state and prove our main results by using the Banach fixed point theorem and Krasnoselskii-Schaefer type fixed point theorem [3]. The continuous dependence of mild solutions to problem (1)-(3) is investigated in Section 4. Finally, in Section 5, an example is exhibited to illustrate the applicability of our results. 2. Preliminaries. In this section we introduce notations, definitions, and preliminary facts which will be used throughout this paper. In particular, we consider a fractional Brownian motion as well as the Wiener integral with respect to it. We also establish some important and helpful results for our analysis. Needless to say that we could omit this whole section and refer to other works already published for these preliminaries (see, e.g. [6] and the references therein), but we aim at making this paper as much self contained as possible and this is why we decided to include this material in this section. Recall that (Ω,F,Ft, P) is a complete probability space furnished with a family of right continuous increasing σ-algebras {Ft, t ∈J}satisfying Ft⊂ F. Definition 2.1. Given H∈(0,1),a continuous centered Gaussian process βH= {βH(t), t ∈R},with the covariance function RH(t, s) = E[βH(t)βH(s)] = 1 2(|t|2H+|s|2H−|t−s|2H), t, s ∈R is called a two−sided one−dimensional fractional Brownian motion, and His the Hurst parameter. Now we aim at introducing the Wiener integral with respect to the one-dimensional βH. 4 A. BOUDAOUI, T. CARABALLO AND A. OUAHAB Let T > 0 and denote by Λ the linear space of R−valued step functions on [0, T], that is, ψ∈Λ if ψ(t) = n−1 X i=1 xi1[si,si−1)(t), where t∈[0, T], xi∈Rand 0 = s1< s2<··· < sn=T. For ψ∈Λ we define its Wiener integral with respect to βHas ZT 0 ψ(σ)dβH(σ) = n−1 X i=1 xi(βH(si+1)−βH(si)). Let Hbe the Hilbert space defined as the closure of Λ with respect to the scalar product h1[0,t],1[0,s]iH=RH(t, s). Then, the mapping ψ= n−1 X i=1 xi1[si,si+1)7→ ZT 0 ψ(σ)dβH(σ) is an isometry between Λ and the linear space span {βH(t), t ∈[0, T ]},which can be extended to an isometry between Hand the first Wiener chaos of the fractional Brownian motion spanL2(Ω){βH(t), t ∈[0, T]}(see [31]). The image of an element ψ∈ H by this isometry is called the Wiener integral of ψwith respect to βH.Our next goal is to give an explicit expression of this integral. To this end, consider the kernel KH(t, s) = cHs1/2−HZt s (u−s)H−3/2uH−1/2du, where cH=H(2H−1) B(2−2H,H−1 2)1/2, with B(·,·) denoting the Beta function, and t≤s. It is not difficult to see that ∂KH ∂t (t, s) = cHt s1 2−H(t−s)H−3 2. Consider the linear operator K∗ H: Λ −→ L2([0, T ]) given by (K∗ HΦ)(s) = Zt s Φ(t)∂KH ∂t (t, s)dt. Then (K∗ H1[0,t])(s) = KH(t, s)1[0,t](s), and K∗ His an isometry between Λ and L2([0, T ]) that can be extended to Λ (see [2]). Considering W={W(t), t ∈[0, T ]}defined by W(t) = βH((K∗ H)−11[0,t]), it turns out that Wis a Wiener process and βHhas the following Wiener integral representation: βH(t) = Zt 0 KH(t, s)dW(s). In addition, for any Φ ∈Λ, ZT 0 Φ(s)βH(s)dW(s) = ZT 0 (K∗ HΦ)(t)dW(t) STOCHASTIC DELAY EVOLUTION EQUATIONS WITH IMPULSES 5 if and only if K∗ HΦ∈L2([0, T]). Also denoting L2 H([0, T]) = {Φ∈Λ, K∗ HΦ∈L2([0, T])}, since H > 1/2,we have L1/H([0, T]) ⊂L2 H([0, T]),(4) see [23]. Moreover, the following useful result holds: Lemma 2.2. [24]. For Φ∈L1/H([0, T]), H(2H−1) ZT 0ZT 0|Φ(r)kΦ(u)kr−u|2H−2drdu ≤cHkΦk2 L1/H ([0,T ]). Next we are interested in considering a fractional Brownian motion with values in a Hilbert space and giving the definition of the corresponding stochastic integral. Let Q∈L(K,H) be a non-negative self−adjoint operator. Denote by L0 Q(K,H) the space of all ξ∈L(K,H) such that ξQ1 2is a Hilbert-Schmidt operator. The norm is given by |ξ|2 L0 Q(K,H)=tr(ξQξ∗). Then ξis called a Q-Hilbert-Schmidt operator from Kto H. Let {βH n(t)}n∈Nbe a sequence of two-sided one-dimensional standard fractional Brownian motions mutually independent on (Ω,F, P).When one considers the following series ∞ X n=1 βH n(t)en, t ≥0, where {en}n∈Nis a complete orthonormal basis in K,this series does not necessarily converge in the space K.Thus we consider a K−valued stochastic process BH Q(t) given formally by the following series: BH Q(t) = ∞ X n=1 βH n(t)Q1 2en, t ≥0, which is well-defined as a K-valued Q-cylindrical fractional Brownian motion. Let ϕ: [0, T]7→ L0 Q(K,H) such that ∞ X n=1 kK∗ H(ϕQ1 2en)kL1/H ([0,T ];H)<∞(5) Definition 2.3. Let ϕ: [0, T]−→ L0 Q(K,H) satisfy (5). Then, its stochastic integral with respect to the fractional Brownian motion BH Qis defined, for t≥0,as follows Zt 0 ϕ(s)dBH Q(s) := ∞ X n=1 Zt 0 ϕ(s)Q1/2endβH n(s) = ∞ X n=1 Zt 0 (K∗ H(ϕQ1/2en))(s)dW(s). Notice that if ∞ X n=1 kϕQ1/2enkL1/H ([0,T ];H)<∞,(6) then in particular (5) holds, which follows immediately from (4). 6 A. BOUDAOUI, T. CARABALLO AND A. OUAHAB Lemma 2.4. [11]if ϕ: [0, T]−→ L0 Q(K,H)satisfies ZT 0kϕ(s)k2 L0 Q(K,H)ds < ∞, then the series in (6) is well defined as a H-valued random variable and we have E   Zt 0 ϕ(s)dBH Q(s)    2 ≤2Ht2H−1Zt 0kϕ(s)k2 L0 Q(K,H)ds. Definition 2.5. The map f:J×DF0→ H is said to be L2-Carath´eodory if (i): t7→ f(t, v) is measurable for each v∈ DF0; (ii): v7→ f(t, v) is continuous for almost all t∈J; (iii): for each q > 0, there exists αq∈L1(J, R+) such that Ekf(t, v)k2≤αq(t),for all kvk2 DF0≤qand for a.e. t∈J. Definition 2.6. The map g:J×DF0→L0 Q(K,H) is said to be L2-Carath´eodory if (i): t7→ g(t, v) is measurable for each v∈ DF0; (ii): v7→ g(t, v) is continuous for almost all t∈J; (iii): for each q > 0, there exists α1q∈L1(J, R+) such that Ekg(t, v)k2 L0 Q(K,H)≤α1q(t),for all kvk2 DF0≤qand for a.e. t∈J. The following result is known as Gronwal-Bihari Theorem. Lemma 2.7. [5]Let u, g :J→Rbe positive real continuous functions. Assume there exist c > 0and a continuous nondecreasing function h:R→(0,+∞)such that u(t)≤c+Zt a g(s)h(u(s)) ds, ∀t∈J. Then u(t)≤H−1Zt a g(s)ds,∀t∈J provided Z+∞ c dy h(y)>Zb a g(s)ds. Here H−1refers to the inverse of the function H(u) = Ru c dy h(y)for u≥c. One of the key tools in our approach is the following form of Burton-Kirk’s fixed point theorem Theorem 2.8 (Burton-Kirk’s fixed point theorem [3]).Let Ebe a Banach space, and G1, G2:E→Ebe two operators satisfying: 1. G1is a contraction, and 2. G2is completely continuous Then, either the operator equation y=G1(y) + G2(y)possesses a solution, or the set Ξ = y∈E:λG1(y λ) + λG2(y) = y, for some λ∈(0,1)is unbounded. STOCHASTIC DELAY EVOLUTION EQUATIONS WITH IMPULSES 7 3. Existence of mild solution. In this section, we first establish the existence of mill solutions to stochastic differential equations with non-instantaneous impulses driven by fractional Brownian motions (1)-(3). More precisely, we will formulate and prove sufficient conditions for the existence of solutions to (1)-(3) with infinite delay. In order to establish the results, we will need to impose some of the following conditions. •(H1) There exist constants Lf>0, such that Ekf(t, φ1)−f(t, φ2)k2≤Lfkφ1−φ2k2 DF0 for all φ1, φ2∈ DF0,t∈(sk, tk+1] and k= 0, . . . , m. •(H2) There exist constants Lg>0 such that Ekg(t, φ1)−g(t, φ2)k2 L0 Q(K,H)≤Lgkφ1−φ2k2 DF0 for all φ1, φ2∈ DF0,t∈(sk, tk+1] and k= 0, . . . , m. •(H3) There exist constants Lhk>0,for all φ1, φ2∈ DF0,t∈(tk, sk] and k= 1, . . . , m such that Ekhk(t, φ1)−hk(t, φ2)k2≤Lhkkφ1−φ2k2 DF0 and hk∈C((tk, sk]×DF0,H),for all k= 1,··· , m. •(H4) fand gare a L2-Caratheodory map and for every t∈[0, T] the function t→f(t, yt) and t→g(t, yt), yt∈ DF0are mesurable •(H5) For the initial value φ∈ DF0, there exists a continuous nondecreasing functions ψ, ψ1: [0,∞)→[0,∞) and p, p1∈L1(J, R+) such that Ekf(t, y)k2≤p(t)ψ(kyk2 DF0) and Ekg(t, y)k2 L0 Q(K,H)≤p1(t)ψ1(kyk2 DF0) for a.e. t∈Jand y∈ DF0with K1Ztk+1 sk p(s)ds +K2Ztk+1 sk p1(s)ds < Z∞ K0 ds ψ(s) + ψ1(s)k= 0,··· , m, where K0= 4 e K2ME|b φ(0)|2+ 4 e N2kb φk2 DF0, K1= 8 e K2Mt1, K2= 16 e K2MHt2H−1 1, for t∈[0, t1], and K0=4e K2ME|b φ(0)|2+ 4 e N2kb φk2 DF0 1−12 e K2MLh , K1=12 e K2M(tk+1 −sk) 1−12 e K2MLh , for t∈[sk, tk+1], k = 1,··· , m. K2=24 e K2MHt2H−1 1 1−12 e K2MLh ,for t∈[sk, tk+1], k = 1,··· , m. •(H6) Operator A:D(A)⊂ H → H is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators {S(t)},t∈Jwhich is compact for t > 0 in H. Now, we present the definition of mild solutions to our problem. 8 A. BOUDAOUI, T. CARABALLO AND A. OUAHAB Definition 3.1. Given φ∈ DF0, a H−valued stochastic process {y(t), t ∈(−∞, T ]} is called a mild solution of the problem (1)-(3) if y(t) is measurable and Ft-adapted, for each t > 0, y(t) = φ(t) on (−∞,0], for each t > 0, y(t) = hk(t, yt) for all t∈(tk, sk] and each k= 1,··· , m, and y(t) = S(t)φ(0) + Zt 0 S(t−s)f(s, ys)ds +Zt 0 g(s, ys)dBH Qd(s) for all t∈[0, t1] and y(t) = S(t−sk)hk(sk, ysk) + Zt sk S(t−s)f(s, ys)ds +Zt sk g(s, ys)dBH Qd(s) for all t∈[sk, tk+1] and every k= 1,··· , m. For our main consideration of Problem (1)-(3), a Banach fixed point is used to investigate the existence and uniqueness of solutions for impulsive stochastic differential equations. Theorem 3.2. Assume conditions (H1)-(H3) are satisfied and L0= max{µ1, µ2, µ3}<1, where µ1= 4Me K2(t2 1Lf+ 2Ht2H 1lg),µ2= 2 e K2Lhkand µ3= 6Me K2(Lhk+ Lf(tk+1 −sk)2+ 2H(tk+1 −sk)2HLg). Then, for every initial function φ∈ DF0 there exists a unique associated mild solution y∈ DFTof the problem (1)-(3). Remark 1. It is worth mentioning that the assumption L0<1 in Theorem 3.2 implies some kind of smallness of the functions f, g and hkin comparison with the periods of time when the impulses are active or viceversa. Proof. We first transform problem (1)-(3) into a fixed point one. Consider operator Φ : DFT→ DFTdefined by Φ(y)(t) =                        φ(t), t ∈(−∞,0], S(t)φ(0) + Zt 0 S(t−s)f(s, ys)ds +Zt 0 S(t−s)g(s, ys)dBH Q(s),if t∈[0, t1], hk(t, yt),if t∈(tk, sk], S(t−sk)hk(sk, ysk) + Zt sk S(t−s)f(s, ys)ds +Zt sk S(t−s)g(s, ys)dBH Q(s),if t∈[sk, tk+1], k = 1,2,··· , m. For φ∈ DF0, we define b φby b φ(t) = φ(t), t ∈(−∞,0], S(t)φ(0), t ∈[0, T]; then b φ∈ DFT. Let y(t) = z(t) + b φ(t),−∞ < t ≤T. It is evident that zsatisfies z0= 0,for all t∈(−∞,0], z(t) = Zt 0 S(t−s)f(s, zs+b φs)ds +Zt 0 S(t−s)g(s, zs+b φs)dBH Q(s), t ∈[0, t1], STOCHASTIC DELAY EVOLUTION EQUATIONS WITH IMPULSES 9 z(t) = hk(t, zt+b φt), t ∈(tk, sk] and z(t) = S(t−sk)hk(sk, zsk+b φsk) + Zt sk S(t−s)f(s, zs+b φs)ds +Zt sk S(t−s)g(s, zs+b φs)dBH Q(s),if t∈[sk, tk+1], k = 1,2,··· , m. Set D0 FT={z∈ DFT,such that z0= 0 ∈ DF0}and for any z∈ DF0we have kzkFT=kz0kDF0+ sup t∈J (Ekz(t)k2)1 2= sup t∈J (Ekz(t)k2)1 2, then (D0 FT,k·kFT) is a Banach space. Let the operator b Φ : D0 FT→ D0 FTdefined by b Φ(z) =                          0,if t∈(−∞,0], Zt 0 S(t−s)f(s, zs+b φs)ds +Zt 0 S(t−s)g(s, zs+b φs)dBH Q(s), t ∈[0, t1], hk(t, zt+b φt), t ∈(tk, sk], S(t−tk)hk(sk, zsk+b φsk) + Zt sk S(t−s)f(s, zs+b φs)ds +Zt sk S(t−s)g(s, zs+b φs)dBH Q(s),if t∈[sk, tk+1]. From the assumption it is easy to see that b Φ is well defined. Now we only need to show that b Φ is a contraction mapping. Case 1: For u,v∈ D0 FTand for t∈[0, t1], we have Ekb Φ(u)(t)−b Φ(v)(t)k2 ≤2E   Zt 0 S(t−s)(f(s, us+b φs)−f(s, vs+b φs))ds    2 +2E   Zt 0 S(t−s)(g(s, us+b φs)−g(s, vs+b φs))dBH Q(s)    2 ≤2M(t2 1Lf+ 2Ht2H 1Lg)ku−vk2 DF0 ≤2M(t2 1Lf+ 2Ht2H 1Lg)×2e K2sup 0≤s≤T Eku(s)−v(s)k2 +2 e Nku0k2 DF0+ 2 e Nkv0k2 DF0i, ≤4Me K2(t2 1Lf+ 2Ht2H 1Lg) sup 0≤s≤t1 Eku(s)−v(s)k2 Since ku0k2 DF0= 0, kv0k2 DF0= 0. Taking the supremum over t, we obtain kb Φ(u)−b Φ(v)k2 D0 FT≤4Me K2(t2 1Lf+ 2Ht2H 1Lg)kv−uk2 D0 FT , Case 2: For u,v∈ D0 FTand for t∈(tk, sk], k= 1,2,··· , m, we have Ekb Φ(u)(t)−b Φ(v)(t)k2≤E  hk(t, ut+b φt)−hk(t, vt+b φt)   2 ≤Lhkku−vk2 DF0 ≤2e K2Lhkku−vkD0 FT 16 A. BOUDAOUI, T. CARABALLO AND A. OUAHAB and therefore (12) becomes E|z(t)|2≤Lhkw(t). (13) Using (13) in the definition of w(t), we have that w(t)≤4e K2Lhkw(t)+4e K2ME|b φ(0)|2+ 4 e N2kb φk2 DF0.(14) Thus, we obtain w(t)≤4e K2ME|b φ(0)|2+ 4 e N2kb φk2 DF0 1−4e K2Lhk =Mtk.(15) This implies there is a constant Mtk>0 such that w(t)≤Mtk, t ∈(sk, tk]. So kzt+b φtkDF0≤Mtk, t ∈(sk, tk]. From equation (13) we obtain that E|z(t)|2≤LhkMtk(16) Thus kzk2 DF0 T≤Ltk. Cas 3: For each t∈[sk, tk+1], k = 1 ··· , m z(t) = S(t−sk)hk(sk, zsk+b φsk) + Zt 0 S(t−s)f(s, zs+b φs)ds +Zt 0 S(t−s)g(s, zs+b φs)dBH Q(s). This implies, for each t∈J, Ekz(t)k2≤3MLhkkzsk +b φskk2 DF0+ 3M(tk+1 −sk)Zt 0 p(s)ψ(kzs+b φsk2 DF0)ds + 6MH(tk+1 −sk)2H−1Zt 0 p1(s)ψ1(kzs+b φsk2 DF0)ds. (17) But kzt+b φtk2 DF0≤2(kztk2 DF0+kb φtk2 DF0) ≤4e K2sup s∈[0,T ] E|z(s)|2+ 4 e K2ME|b φ(0)|2+ 4 e N2kφk2 DF0. If we set w(t) the right hand side of the above inequality we have that kzt+b φtk2 DF0≤w(t), and therefore (17) becomes Ekz(t)k2≤3MLhkw(t)+3M(tk+1 −sk)Zt sk p(s)ψ(w(s))ds STOCHASTIC DELAY EVOLUTION EQUATIONS WITH IMPULSES 17 + 6MH(tk+1 −sk)2H−1Zt sk p1(s)ψ1(w)ds. (18) Using (18) in the definition of w, we have that w(t)≤4e K23MLhkw(t)+3M(tk+1 −sk)Zt sk p(s)ψ(w(s))ds + 4MH(tk+1 −sk)2H−1Rt skp1(s)ψ1(w)ds+ 4 e K2ME|b φ(0)|2 +4 e N2kb φk2 DF0. (19) Thus, we obtain w(t)≤¯ K0+¯ K1Zt sk p(s)ψ(w(s))ds +¯ K2Zt sk p1(s)ψ1(w(s))ds. (20) where ¯ K0=4e K2ME|b φ(0)|2+ 4 e N2kb φk2 DF0 1−12 e K2MLhk ,¯ K1=12 e K2M(tk+1 −sk) 1−12 e K2MLhk , ¯ K2=24 e K2MHt2H−1 1 1−12 e K2MLhk . Let us denote by v(t) the right-hand side of inequality (20). Then we have v(0) = ¯ K0, w(t)≤v(t), t ∈[sk, tk+1], k = 1,··· , m and v0(t) = ¯ K1p(t)ψ(w(t)) + ¯ K2p1(t)ψ1(w(t)), t ∈[sk, tk+1]. Using the increasing character of ψand ψ1we obtain v0(t)≤¯ K1p(t)ψ(v(t)) + ¯ K2p1(t)ψ1(v(t)),for a.e. t ∈[sk, tk+1]. This implies, for each t∈J, we have Zv(t) v(0) ds ψ(s) + ψ1(s)≤¯ K1Ztk+1 sk p(s)ds +¯ K2Ztk+1 sk p1(s)ds < Z∞ ¯ K0 ds ψ(s) + ψ1(s). By Lemma 2.7, we have v(t)≤Γ−1 1Ztk+1 sk (K1p(t) + K2p1(t))dt, t ∈[sk, tk+1] where Γ1(x) = Zx ¯ K0 d(u) ψ(u) + ψ1(u). Hence, there exists Mtk+1 >0 such that kzt+b φtk2 DF0< Mtk+1 . From equation (18) we obtain that Ekz(t)k2≤3MLhkMtk+1 + 3M(tk+1 −sk)Ztk+1 sk p(s)ψ(Mtk+1 )ds 18 A. BOUDAOUI, T. CARABALLO AND A. OUAHAB + 6MH(tk+1 −sk)2H−1Ztk+1 sk p1(s)ψ1(Mtk+1 )ds. (21) Thus, there exists Ltk+1 >0 such that kzk2 DF0 T≤Ltk+1 . This implies that the set Ξ is bounded. As a consequence of Theorem 2.8 we deduce that b Φ has a fixed point, since y(t) = z(t) + b φ(t), t ∈(−∞, T]. Then yis a fixed point of the operator Ψ which is a mild solution of the problem (1)-(3). 4. Continuous dependence of mild solutions. In this section we will prove that continuous dependence of mild solutions with respect to the initial data. Theorem 4.1. Suppose that assumptions (H1)−(H3) are fulfilled and the following inequalities hold as well: L2= max{8Me K2(t2 1Lf+Ht2H 1Lg),4Lhke K2,12Me K2(Lhk+t2 1Lf+Ht2H 1Lg)}<1. Then for each φ, φ∗∈ DF0and y(t) = z(t) + b φ(t),y∗(t) = z∗(t) + b φ∗(t)the corresponding mild solutions of the system (1)-(3), the following inequalities hold: kz−z∗k2 D0 FT≤8Mα2(M+ 1)(t2 1Lf+Ht2H 1Lg) 1−8Me K2(t2 1Lf+Ht2H 1Lg)kb φ−b φ∗k2 DF0, t ∈[0, t1], kz−z∗k2 D0 FT≤4Lhkα2(M+ 1) 1−4Lhke K2kb φ−b φ∗k2 DF0, t ∈(tk, sk], k = 1,2,··· , m and kz−z∗k2 D0 FT≤12Mα2(M+ 1)(lhk+t2 1Lf+Ht2H 1Lg) 1−12Me K2(Lhk+t2 1Lf+Ht2H 1Lg)kb φ−b φ∗k2 DF0, for t∈(sk, tk+1], k = 1,2,··· , m. Proof. Estimating for t∈[0, t1], we have Ekz(t)−z∗(t)k2≤2E   Zt 0 S(t−s)[f(s, zs+b φs)−f(s, z∗ s+b φs)]ds    2 +2E   Zt 0 S(t−s)[g(s, zs+b φs)−g(s, z∗ s+b φ∗ s)]dBH Q(s)    2 . But kzt+b φtk2 DF0≤2(kztk2 DF0+kb φtk2 DF0) ≤4e K2sup s∈[0,T ] Ekz(s)k2+ 4 e K2MEkb φ(0)k2+ 4 e N2kφk2 DF0 ≤4e K2sup s∈[0,T ] Ekz(s)k2+ 4α2(M+ 1)kφk2 DF0, where α2= max{e K2,e N2}. Ekz(t)−z∗(t)k2≤2Mt2 1Lf(4 e K2kz−z∗k2 D0 FT + 4α2(M+ 1)kb φ−b φ∗k2 DF0) +2MHt2H 1Lg(4 e K2kz−z∗k2 D0 FT + 4α2(M+ 1)kb φ−b φ∗k2 DF0) STOCHASTIC DELAY EVOLUTION EQUATIONS WITH IMPULSES 19 which implies that kz−z∗k2 D0 FT≤8Mα2(M+ 1)(t2 1Lf+Ht2H 1Lg) 1−8Me K2(t2 1Lf+Ht2H 1Lg)kb φ−b φ∗k2 DF0. For t∈(tk, sk], k= 1,2,··· , m, we have Ekz(t)−z∗(t)k2≤E  hk(t, zt+b φt)−hk(t, z∗ t+b φ∗ t)   2 ≤Lhk(4 e K2kz−z∗k2 D0 FT + 4α2(M+ 1)kb φ−b φ∗k2 DF0) which implies that kz−z∗k2 D0 FT≤4Lhkα2(M+ 1) 1−4Lhke K2kb φ−b φ∗k2 DF0. In a similar way, when t∈[sk, tk+1], we have Ekz(t)−z∗(t)k2≤3EkS(t−sk)[h(sk, zsk+b φsk)−h(sk, z∗ sk+b φ∗ sk)]k2 +3E   Zt sk S(t−s)[f(s, zs+b φs)−f(s, z∗ s+b φ∗ s)]ds    2 +3E   Zt sk S(t−s)[g(s, zs+b φs)−g(s, z∗ s+b φ∗ s)]dBH Q(s)    2 ≤3MLhk(4 e K2kz−z∗k2 D0 FT + 4α2(M+ 1)kb φ−b φ∗k2 DF0) +3Mt2 1Lf(4 e K2kz−z∗k2 D0 FT + 4α2(M+ 1)kb φ−b φ∗k2 DF0) +3MHt2H 1Lg(4 e K2kz−z∗k2 D0 FT + 4α2(M+ 1)kb φ−b φ∗k2 DF0), which implies that kz−z∗k2 D0 FT≤12Mα2(M+ 1)(lhk+t2 1Lf+Ht2H 1Lg) 1−12Me K2(Lhk+t2 1Lf+Ht2H 1Lg)kb φ−b φ∗k2 DF0. 5. An example. Consider the following stochastic partial differential equation with delays and impulsive effects                  ∂ ∂t u(t, ξ) = ∂2 ∂ξ2u(t, ξ) + F(t, ut(., ξ)) +σ(t, ut(., ξ))dBH Q dt ,(t, ξ)∈ m [ i=0 [si, ti+1]×[0, π], u(t, ξ) = Hk(t, ut(., ξ)), k = 1,··· , m, ut(., 0) = ut(., π)=0, t ∈[0, u(t, ξ) = φ(t, ξ),−∞ ≤ t≤0,0≤ξ≤π, (22) where BH Qdenotes a fractional Brownian motion, the impulse times tksatisfy 0 = t0=s0< t1≤s1< t2<··· < tm≤sm< tm+1 =T. As for utwe mean the segment solution which is defined in the usual way, that is, if u(·,·,·):(−∞, T]××[0, π]×Ω→ H,then for any t≥0, ut(·., ·,·):(−∞,0] ×[0, π]×Ω→ H is given by: ut(θ, ξ, ω) = u(t+θ, ξ, ω),for θ∈(−∞,0], ω ∈Ω, F, G : [0, T]×DF0→Rare continuous functions. 20 A. BOUDAOUI, T. CARABALLO AND A. OUAHAB Let y(t)(ξ) = u(t, ξ, .)t∈J, ξ ∈[0, π], hk(t, φ)(ξ) = Hk(t, φ(θ, ξ)), θ ∈(−∞,0], ξ ∈[0, π], k = 1,··· , m, f(t, φ)(ξ) = F(t, φ(θ, ξ)), θ ∈(−∞,0], ξ ∈[0, π], φ(θ)(ξ) = φ(θ, ξ), θ ∈(−∞,0], ξ ∈[0, π], Take K=H=L2([0, π]). We define the operator Aby Au =u00 , with domain D(A) = {u∈ H, u00 ∈ H and u(0) = u(π)=0}. Take K=H=L2([0, π]). We define the operator Aby Au =∂2 ∂ξ2u. with domain D(A) = {u∈ H,∂u ∂ξ ,∂2u ∂ξ2∈ H and u(0) = u(π)=0}. Then, it is well known that Az =− ∞ X n=1 n2hz, enien, z ∈ H, and Ais the infinitesimal generator of a strongly continuous semigroup of bounded linear operators {S(t)}t≥0on H, which is given by S(t)u= ∞ X n=1 e−n2thu, enien,u∈ H, and en(u) = (2/π)1/2sin(nu), n = 1,2,··· , is the orthogonal set of eigenvectors of A. It is well known that {S(t)},t∈J, is compact, and there exists a constant M≥1 such that kS(t)k2≤M. In order to define the operator Q:K −→ K, we choose a sequence {σn}n≥1⊂R+, set Qen=σnen, and assume that tr(Q) = ∞ X n=1 √σn<∞. Define the process BH Q(s) by BH Q= ∞ X n=1 √σnγH n(t)en, where H∈(1/2,1), and {γH n}n∈Nis a sequence of two-sided one-dimensional fractional Brownian motions mutually independent. In the case of t∈(−∞, T]. Assume now that (i): For all k= 0,1··· , m, the function f: [sk, tk+1]×DF0−→ H defined by f(t, u)(·) = F(t, u(·)) is continuous and we impose suitable conditions on F to verify assumption (H1). For example we take F(t, φ) = t+φ 1 + kφkDF0 , t ∈[0, T], φ ∈ DF0. (ii): For all k= 0,1··· , m, the function g: [sk, tk+1]× DF0−→ L0 Q(K,H) defined by g(t, u)(.) = G(t, u(.)) is continuous and it is easy to impose suitable conditions on Gto make assumption (H2) hold. We take G(t, φ) = t2+ sin φ, t ∈[0, T], φ ∈ DF0. STOCHASTIC DELAY EVOLUTION EQUATIONS WITH IMPULSES 21 (iii): For all k= 1,2··· , m, the function hk: (tk, sk]× DF0−→ H defined by hk(t, u)(.) = Hk(t, u(.)) is continuous and it is easy to impose suitable conditions on Hkto make assumption (H3) hold. Hk(t, φ)) = Kkφ, ξ ∈Ω, k = 1,··· , m, t ∈[0, T], φ ∈ DF0, where Kk∈R, k = 1, . . . , m. Thus the problem (22) can be written in the abstract form    dy(t) = A[y(t) + f(t, yt)]dt +g(t, yt)BH Q(t), t ∈Jk= (sk, tk+1], k = 0,··· , m , y(t) = hk(t, yt)t∈(tk, sk], k = 1,··· , m, y(t) = φ(t),for a.e. t∈(−∞,0]. Thanks to these assumptions, it is straightforward to check that (H1)-(H3) hold and thus assumptions in Theorem 3.2 are fulfilled, ensuring that system (22) possesses a mild solution on (−∞, T]. REFERENCES [1] H. M. Ahmed, Semilinear neutral fractional stochastic integro-differential equations with nonlocal conditions, J. Theoret. Probab. 28 (2015), no. 2, 667–680. [2] E. Alos, O. Mazet, D.Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29 (2001), 766–801. [3] C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii, Electron. J. Qual. Theory Differ. Equ. 5(2003), 1–15. [4] J. Bao, Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl. 59 (2010), no. 1, 207–214. [5] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7(1956), 81–94. [6] A. Boudaoui, T. Caraballo, A. Ouahab, Existence of mild solutions to stochastic delay evolution equations with a fractional Brownian motion and impulses, Stoch. Anal. Appl. 33 (2015), no. 2, 244–258. [7] A. Boudaoui, T. Caraballo, A. Ouahab, Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay. Math. Meth. Appl. Sci. 39 (2016), no. 6, 1435–1451. doi: 10.1002/mma.3580. [8] A. Boudaoui, T. Caraballo, A. Ouahab, Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay, Appl. Anal. 95 (2016), no. 9, 2039–2062. [9] B. Boufoussi, S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett. 82 (2012), no. 8, 1549–1558. [10] G. Cao, K. He, X. Zhang, Successive approximations of infinite dimensional SDES with jump, Stoch. Dyn. 5(2005), no. 4, 609–619. [11] T. Caraballo, M. J. Garrido-Atienza, 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. [12] T. Caraballo, Mamadou A. Diop, Neutral stochastic delay partial functional integrodifferential equations driven by a fractional Brownian motion, Front. Math. China 8(2013), no. 4, 745–760. [13] M.M. El-Borai, K. EI-Said EI-Nadi, H.A. Fouad, On some fractional stochastic delay differential equations, Comput. Math. Appl., 59 (2010), no. 3, 1165–1170. [14] G. R. Gautam, J. Dabas, Existence result of fractional functional integrodifferential equation with not instantaneous impulse, Int. J. Adv. Appl. Math. Mech 1.3 (2014), 11–21. [15] T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005), no. 2, 139–154. [16] J. R. Graef, J. Henderson, A. Ouahab, Impulsive differential inclusions. A fixed point approach. De Gruyter Series in Nonlinear Analysis and Applications, 20. De Gruyter, Berlin, 2013. 22 A. BOUDAOUI, T. CARABALLO AND A. OUAHAB [17] J. K. Hale, J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), no. 1, 11–41. [18] E. Hern´andez, D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1641–1649. [19] F. Jiang, Y. Shen, A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl. 61 (2011), no. 6, 1590–1594. [20] V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. [21] X. Li, M. Bohner, An Impulsive Delay Differential Inequality and Applications, Comput. Math. Appl. 64 (2012), no. 6, 1875–1881. [22] X. Li, X. Fu, On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 3, 442–447. [23] Y. Mishura Stochastic Calculus for Fractional Brownian Motion and Related Topics. Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008. [24] D. Nualart The Malliavin Calculus and Related Topics. 2nd ed. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. [26] M. Pierri, D. O’Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comp.,219 (2013), no. 12, 6743– 6749. [27] R. Sakthivel, J. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl. 356 (2009), no. 1, 1–6. [28] A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore 1995. [29] G. Shen, Y. Ren, Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space, J. Korean Statist. Soc. 44 (2015), no. 1, 123–133. doi: 10.1016/j.jkss.2014.06.002. [30] T. Taniguchi, Successive approximations to solutions of stochastic differential equations, J. Differential Equations 96 (1992), no. 1, 152–169. [31] S. Tindel, C. Tudor, F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields,127 (2003), no. 2, 186–204. [32] J. R. Wang, Y. Zhou, Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput. 242 (2014), 649–657. [33] Z. Yan, X. Yan, Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay, Collect. Math. 64 (2013), no. 2, 235–250. [34] Q. Zhu, Asymptotic stability in the pth moment for stochastic differential equations with Levy noise, J. Math. Anal. Appl. 416 (2014), no. 1, 126–142. E-mail address:[email protected] E-mail address:[email protected] E-mail address:agh [email protected]