Exponential behavior of solutions to stochastic integrodifferential equations with distributed delays

Exponential behavior of solutions to stochastic integrodifferential equations with distributed delays Mamadou Abdoul DIOP a1,Tom´as CARABALLO band Abdoul Aziz NDIAYEc aUniversit´e Gaston Berger UFR des Sciences Appliqu´ees et Technologie, D´epartement de Math´ematiques BP 234, Saint-louis, S´en´egal e-mail: mamadou-ab[email protected] bDpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain e-mail: [email protected] cUniversit´e Gaston Berger UFR des Sciences Appliqu´ees et Technologie, D´epartement de Math´ematiques BP 234, Saint-louis, S´en´egal e-mail: [email protected] Abstract: In this work, we study the existence, uniqueness and exponential asymptotic behavior of mild solutions to stochastic integrodifferential delay evolution equations. We assume that the non-delay part generates a C0-semigroup. Keywords: C0semigroup, partial functional differential equations, Brownian motion, finite delay, mild solution, exponential decay in mean square. 1 Introduction In the last two decades, stochastic differential equations have attracted many physicists, mathematicians and engineers, and remarkable contributions have been made to both theory and applications of stochastic partial differential equations with delays. The stochastic functional differential systems appears in many areas of sciences. Such equations arise in the study of stochastic systems in the presence of hereditary influences on the state variable. For example, stochastic integrodifferential systems which cover a large area of system dynamics including reactor dynamics [4, 16, 18], heat transfer by conduction and radiation [15, 19], mathematical modeling of system hysteresis [8, 16], models of transmission of infections or diseases [3], etc. The presence of a stochastic component makes the evolution of the state variable a stochastic process and accounts for intrinsic or external random influences. One of the fundamental problems in such dynamical systems is to establish sufficient conditions under which it exhibits a unique solution. There are many ways to establish such results, for example: (i) Monotone Iterative Techniques, which is 1To whom all correspondence should be sent: mamadou-ab[email protected] 1 based in the method of upper and lower solutions [15], used mainly for boundary value problems, (ii) Topological Methods, which use one of the fixed point theorems such as Schauder-Tychonoff’s [6], and (iii) Construction Methods, such as the Picard successive approximations [17, 2, 1, 11, 13, 14, 20] in which one has to show that the convergence of the approximation process to the original solution. Recently, Caraballo et al. [5] have studied the following class of stochastic delay evolution equations with fractional Brownian motion dx(t) = [Ax(t) + f(t, xt)]dt +g(t)dBH Q(t) for t > 0 x(t) = ϕ(t), t ∈[−r, 0],(0 ≤r < ∞).(1.1) with Hurst parameter H∈(1 2,1). They investigated the existence, uniqueness and exponential behavior of mild solutions to (1.1). In this paper we consider the following stochastic integrodifferential equation,    dx(t) = Ax(t)dt +Zt 0 B(t−s)x(s)ds +F(t, xt)dt +G(t, xt)dw(t),for t∈[0, T] x(s) = ϕ, −r≤s≤0, r ≥0, (1.2) under suitable assumptions on the operator A, the coefficients F, G, and the initial value ϕ. Here w(t) denotes a Brownian motion. The schemes of the proofs are similar to those ones in [5] but with necessary modifications due to the different nature of the problem. To be more precise, we suppose that the non-delay part admits a resolvent operator in the sense given by Grimmer[9], as well as the noise considered in [5] is a fractional Brownian motion while the one in this paper is a Hilbert-valued Wiener process. The main objective of this paper is to investigate existence and uniqueness of mild solutions to the stochastic integrodifferential equation (1.2), and to study its long-time behaviour as well, by using the theory of resolvent operators and Picard type iteration. Recall that the resolvent operator plays an important role in solving Eq. (1.2) in the weak and strict sense, it replaces the role of the C0-semigroup theory. For more details we refer to [10, 9]. The paper is organized as follows: in Section 2 we recall some preliminaries which are used throughout this paper. In Section 3 we state the existence and uniqueness of a mild solution. In Section 4 we study the asymptotic behaviour of (1.2).Finally, Section 5 is devoted to show an example to illustrate the efficiency of the obtained results. 2 Preliminaries 2.1 Wiener process Throughout this paper, let Hand Kbe two real separable Hilbert spaces. We denote by h·,·iH,h·,·iKtheir inner products and by k·kH,k·kKtheir vectors norms, respectively. L(H,K) denote the space of all bounded linear operators from Hinto K, equipped with the usual operator norm k·k. If H=Kwe will simply write L(H) instead of 2 L(H,H). In the sequel, we always use the same symbol k·k to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Moreover, let (Ω,F,{Ft}t≥0,P) be a complete probability space with a normal filtration {Ft}t≥0satisfying the usual conditions (i.e. it is increasing and right-continuous, while F0contains all P−null sets). Let {w(t) : t≥0}denote a K-valued Wiener process defined on the probability space (Ω,F,{Ft}t≥0,P), with covariance operator Q, that is, Ehw(t), xiKhw(s), yiK= (t∧s)hQx, yiK, for all x, y ∈K, where Q is a positive, selfadjoint, trace class operator on K. In particular, we denote by w(t) a K-valued Q-Wiener process with respect to {Ft}t≥0. To define stochastic integrals with respect to the QWiener process w(t), we introduce the subspace K0=Q1/2Kof Kendowed with the inner product hu, viK0=Q−1/2u, Q−1/2vKas a Hilbert space. We assume that there exists a complete orthonormal system {ei}in K, a bounded sequence of positive real numbers λisuch that Qei=λiei, i = 1,2, ... , and a sequence {wi(t)}i>1of independent standard Brownian motions such that w(t) = P+∞ i=1 √λiwi(t)eifor t≥0 and Ft=Fw t, where Fw tis the σ-algebra generated by {w(s) : 0 ≤s≤t}. Let L0 2=L2(K0,H) be the space of all Hilbert–Schmidt operators from K0to H. It turns out to be a separable Hilbert space equipped with the norm kvkL0 2=tr((vQ1/2)(vQ1/2)∗) for any v∈ L0 2. Obviously, for any bounded operator v∈ L0 2, this norm reduces to kvk2 L0 2=tr(vQv∗). 2.2 Partial integro–differential equations in Banach spaces In the present section, we recall some definitions, notations and properties needed in the sequel. In what follows, Hwill denote a Banach space, Aand B(t) are closed linear operators on H.Yrepresents the Banach space D(A), the domain of operator A, equipped with the graph norm |y|Y:= |Ay|+|y|for y∈Y. The notation C([0,+∞); Y) stands for the space of all continuous functions from [0,+∞) into Y. We then consider the following Cauchy problem    v0(t) = Av(t) + Zt 0 B(t−s)v(s)ds for t≥0, v(0) = v0∈H. (2.1) Definition 2.1. ([9]) A resolvent operator for Eq. (2.1) is a bounded linear operator valued function R(t)∈ L(H) for t≥0, satisfying the following properties : (i) R(0) = Iand |R(t)| ≤ Neβt for some constants Nand β. (ii) For each x∈H,R(t)xis strongly continuous for t≥0. (iii) For x∈Y, R(·)x∈C1([0,+∞); H)∩C([0,+∞); Y) and 3 R0(t)x=AR(t)x+Zt 0 B(t−s)R(s)xds =R(t)Ax +Zt 0 R(t−s)B(s)xds for t≥0. For additional details on resolvent operators, we refer the reader to [21, 9]. The resolvent operator plays an important role to study the existence of solutions and to establish a variation of constants formula for non–linear systems. For this reason, we need to know when the linear system (2.1) possesses a resolvent operator. Theorem 2.2 below provides a satisfactory answer to this problem. In what follows we suppose the following assumptions: (H1) A is the infinitesimal generator of a C0-semigroup (T(t))t≥0on H. (H2) For all t≥0, B(t) is a continuous linear operator from (Y, |·|Y) into (H,|·|H). Moreover, there exists an integrable function c: [0,+∞)→R+such that for any y∈Y, y7→ B(t)ybelongs to W1,1([0,+∞),H) and  d dtB(t)yH≤c(t)|y|Yfor y∈Yand t≥0. Theorem 2.2. ([7]) Assume that hypotheses (H1) and (H2) hold. Then Eq. (2.1) admits a resolvent operator (R(t))t≥0. Theorem 2.3. ([12]) Assume that hypotheses (H1) and (H2) hold. Let T(t)be a compact operator for t > 0. Then, the corresponding resolvent operator R(t)of Eq. (2.1) is continuous for t > 0in the operator norm, namely, for all t0>0, it holds that limh→0kR(t0+h)−R(t0)k= 0. In the sequel, we recall some results on the existence of solutions for the following integro–differential equation    v0(t) = Av(t) + Zt 0 B(t−s)v(s)ds +q(t) for t≥0, v(0) = v0∈H, (2.2) where q: [0,+∞[→His a continuous function. Definition 2.4. ([9]) A continuous function v: [0,+∞)→His said to be a strict solution of Eq. (2.2) if (i) v∈C1([0,+∞); H)∩C([0,+∞); Y), (ii) vsatisfies Eq. (2.2) for t≥0. Remark 2.5.From this definition we deduce that v(t)∈D(A), and the function B(t−s)v(s) is integrable, for all t > 0 and s∈[0,+∞). 4 Theorem 2.6. ([9]) Assume that (H1)-(H2) hold. If vis a strict solution of Eq. (2.2), then the following variation of constants formula holds v(t) = R(t)v0+Zt 0 R(t−s)q(s)ds for t≥0.(2.3) Accordingly, we can establish the following definition. Definition 2.7. ([9]) A function v: [0,+∞)→His called a mild solution of (2.2), for v0∈H, if vsatisfies the variation of constants formula (2.3). The next theorem provides sufficient conditions ensuring the regularity of solutions of Eq. (2.2). Theorem 2.8. ([9]) Let q∈C1([0,+∞); H)and vbe defined by (2.3). If v0∈D(A), then vis a strict solution of Eq. (2.2). 3 Existence and uniqueness of mild solution Consider (Ω,F,P) the complete probability space which was introduce in Section 2. Denote Ft=F0, for all t≤0. We denote by C(a, b;L2(Ω; H)) = C(a, b;L2(Ω,F,P;H)) the Banach space of all continuous functions from [a, b] into L2(Ω; H) equipped with the sup norm. Let us also consider two fixed real numbers r≥0 and T > 0. If x∈C(−r, T;L2(Ω; H)), for each t∈[0, T] we denote by xt∈C(−r, 0; L2(Ω; H)) the function defined by xt(s) = x(t+s) for s∈[−r, 0]. In this section we consider the existence and uniqueness of mild solutions to the following stochastic integrodifferential equation with delays:        dX(t) = AX(t) + Zt 0 B(t−s)X(s)ds +f(t, Xt)dt +g(t, Xt)dw(t)t∈[0, T], X(t) = ϕ(t) for t∈[−r, 0], (3.1) where w(t) is the Brownian motion which was introduced in the previous section, the initial data ϕ∈C(−r, 0; L2(Ω; H)) and A:Dom(A)⊂H→His the infinitesimal generator of a strongly continuous semigroup S(·) on H, that is, for t≥0, it holds, |S(t)| ≤ Meρt, M ≥1, ρ ∈R. Let f: [0, T]×C(−r, 0; H)→Hand g: [0, T]×C(−r, 0; H)→ L0 2(K,H) be families of nonlinear operators, defined for almost every t(a.e. t), satisfying (H3) The mapping t∈(0, T)→f(t, ξ)∈His Lebesgue measurable, for a.e. t, and for all ξ∈C(−r, 0; H). 5 (H4) There exists a constant Cf, which is independent of T, such that for any x, y ∈C(−r, T ;H) and t∈[0, T ], Zt 0kf(s, xs)−f(s, ys)k2 Hds ≤CfZt −rkx(s)−y(s)k2 Hds. (H5) ZT 0kf(s, 0)k2 Hds < ∞. (H6) The mapping t∈(0, T)→g(t, ξ)∈His Lebesgue measurable, for a.e. t, and for all ξ∈C(−r, 0; H). (H7) There exists a constant Cgsuch that for any x, y ∈C(−r, T ;H) and t∈[0, T ], Zt 0kg(s, xs)−g(s, ys)k2 L0 2ds ≤CgZt −rkx(s)−y(s)k2 Hds. (H8) ZT 0kg(s, 0)k2 L0 2ds < ∞ Definition 3.1. Let (R(t))t≥0be a resolvent operator for Eq. (2.1). A H-valued process X(t) is called a mild solution of (3.1) if X∈C(−r, T;L2(Ω; H)), X(t) = ϕ(t) for t∈[−r, 0], and for t∈[0, T], satisfies X(t) = R(t)ϕ(0) + Zt 0 R(t−s)f(s, Xs)ds +Zt 0 R(t−s)g(s, Xs)dw(s)P−a.s. (3.2) Theorem 3.2. Suppose that assumptions (H1)-(H7) hold. Then, for every ϕ∈C(−r, 0; L2(Ω; H)), there exists a unique mild solution to (3.1). Proof. We can assume that ρ > 0, otherwise we can take ρ0>0 such that kR(t)k ≤ Meρ0t, for all t≥0. We start by proving the uniqueness of mild solutions. Assume that X, Y ∈C(−r, T;L2(Ω; H)) are two mild solutions of (3.1). Then, EkX(t)−Y(t)k2 H≤2tEZt 0kR(t−s)f(s, Xs)−f(s, Ys)k2 Hds +2tEZt 0kR(t−s)g(s, Xs)−g(s, Ys)k2 L0 2ds ≤2tM2e2ρtEZt 0kf(s, Xs)−f(s, Ys)k2 Hds +2tM2e2ρtEZt 0kg(s, Xs)−g(s, Ys)k2 L0 2ds ≤[2tM2e2ρtCf+ 2tM2e2ρtCg]Zt 0 EkX(s)−Y(s)k2 Hds ≤[2tM2e2ρtCf+ 2tM2e2ρtCg]Zt 0 sup 0≤τ≤s EkX(τ)−Y(τ)k2 Hds, 6 and therefore, since X=Yfor t∈[−r, 0], by taking the supremum in the above inequality, sup 0≤θ≤t EkX(θ)−Y(θ)k2 H≤[2TM2e2ρtCf+ 2TM2e2ρtCg]Zt 0 sup 0≤τ≤s EkX(τ)−Y(τ)k2 Hds. The Gronwall Lemma ensures the uniqueness of solutions. Now we prove the existence of solution to problem (3.1). First of all we check that the well-defined stochastic integral possesses the required regularity. To do that, let us consider h > 0 small enough. We have E   Zt+h 0 R(t+h−s)g(s, xs)dw(s)−Zt 0 R(t−s)g(s, xs)dw(s)    2 ≤2E   Zt 0 (R(t+h−s)−R(t−s))g(s, xs)dw(s)    2 + 2E   Zt+h t R(t−s)g(s, xs)dw(s)    2 := I1+I2 Firstly, I1≤2kR(t+h−s)−R(t−s)kZt 0kg(s, xs)k2 L0 2ds →0 when h→0, thanks to the norm continuty of R(t), t > 0. Secondly, I2≤2M2eρh    Zt+h t g(s, xs)    2 L0 2 ds →0 when h→0. Therefore, the stochastic integral belongs to the space C(−r, T;L2(Ω; H)). We denote X0= 0 and define by recurrence a sequence {Xn}n∈Nof processes as        Xn(t) = R(t)ϕ(0) + Zt 0 R(t−s)f(s, Xn−1 s)ds +Zt 0 R(t−s)g(s, Xn−1 s)dw(s), t ∈[0, T] Xn(t) = ϕ(t), t ∈[−r, 0]. (3.3) The sequence (3.3) is well defined, since X0= 0 ∈C(−r, T;L2(Ω; H)) and given Xn−1∈C(−r, T;L2(Ω; H)), it is not difficult to prove that Xn∈C(−r, T;L2(Ω; H)) as well. Indeed, in order to prove the previous assertion, let us consider again h > 0 sufficently 7 small. Then, EkXn(t+h)−Xn(t)k2 H≤5Ek(R(t+h−s)−R(t−s))ϕ(0)k2 +5E   Zt 0R(t+h−s)−R(t−s)f(s, Xn s)ds    2 H +5E   Zt+h t R(t+h−s)f(s, Xn s)ds    2 H +5E   Zt 0R(t+h−s)−R(t−s)g(s, Xn s)dw(s)    2 H +5E   Zt+h t R(t+h−s)g(s, Xn s)dw(s)    2 H (3.4) := I1+I2+I3+I4+I5. We can easily show easily that EI1→0 when h→0. Now, on the one hand, EI2≤5EZt 0kR(t+h−s)−R(t−s)kkf(s, Xn s)dsk2 H→0 when h→0, thanks to the norm continuty of R(t) and the fact that EZt 0kf(s, Xn s)k2 Hds ≤CfEZt −r Xn−1(s) H+EZt 0kf(s, 0)kHds < ∞, due to Hypotheses (H4) and (H5) and the fact that Xn−1∈C(−r, T;L2(Ω; H)). On the other hand, I3≤5hM2e2ρh Zt+h t f(s, Xn−1 s)−f(s, 0)  2 Hds + 5hM2e2ρh Zt+h tkf(s, 0)k2 Hds ≤5hM2e2ρhCfZt+h −r Xn−1(s)  2 Hds + 5hM2e2ρh Zt+h tkf(s, 0)k2 Hds, so that EI3≤5hM2e2ρhCfZt+h −r E Xn−1(s)  2 Hds + 5hM2e2ρh Zt+h tkf(s, 0)k2 Hds →0 when h→0. As for the term I4we have, EI4≤5EZt 0kR(t+h−s)−R(t−s)kkg(s, Xn s)k2 L0 2ds →0,when h→0, thanks to the norm continuity of R(t) and EZt 0kg(s, Xn s)k2 L0 2ds ≤CgEZt −r Xn−1(s) H+EZt 0kg(s, 0)kL0 2ds < ∞, 8 due to Hypotheses (H6) and (H7) and the fact that Xn−1∈C(−r, T;L2(Ω; H)). For the last term we have, EI5≤5hM2e2ρhCgZt+h −r E Xn−1(s)  2 Hds + 5hM2e2ρh Zt+h tkg(s, 0)k2 L0 2ds →0 when h→0. We want to show now that {Xn}n∈Nis a Cauchy sequence in C(−r, T ;L2(Ω; H)). Firstly, for t∈[0, T] and n∈N, since Xn=Ynon [−r, 0], it holds kXn+1(t)−Xn(t)k2 H≤[2M2te2ρtCf]Zt 0kXn(s)−Xn−1(s)k2 Hds +[2M2te2ρtCg]Zt 0kXn(s)−Xn−1(s)k2 Hds, (3.5) and this implies EkXn+1(t)−Xn(t)k2 H≤[2M2te2ρtCf+ 2M2te2ρtCg]Zt 0 sup 0≤τ≤s EkXn(τ)−Xn−1(τ)k2 Hds. Defining Φn(t) = sup 0≤θ≤t EkXn+1(θ)−Yn+1(θ)k2 H, we have Φn(t)≤C1Zt 0 Φn−1(s)ds for n≥2, for C1= 2TM2e2ρT [Cf+Cg]. Consequently, by iteration we obtain Φn(t)≤Cn−1Tn−1 (n−1)! Φ1(t) for all n≥2 and t∈[0, T]. Since Xn+1(t) = Xn(t),∀t∈[−r, 0], the last estimate implies that {Xn}n∈Nis a solution of (3.1). But, this is straightforward taking into account that Xnis defined by (3.3) and that fand gsatisfies (H3)–(H8), so that, in particular, when n→ ∞, E   Zt 0 R(t−s)(f(s, Xn s−1) −f(s, Xs))ds    2 H≤tM2e2ρtCfZt 0 E Xn−1(s)−X(s)  2 Hds →0, and therefore Xis the unique (mild) solution of (3.1). 4 Exponential decay of solutions in mean square As in this section we are interested in the exponential decay to zero in mean square of the mild solution to (3.1), we shall therefore assume that for each T > 0 and for each ϕ∈C(−r, 0; L2(Ω; H)), problem (3.1) possesses a unique mild solution according to Definition 3.1. We first need to state the following conditions : 9