Existence and uniqueness of solutions for delay stochastic evolution equations

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR DELAY STOCHASTIC EVOLUTION EQUATIONS Tom´as CARABALLO, Mar´ıa J. GARRIDO-ATIENZA and Jos´e REAL Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. e-mails: [email protected] ; [email protected]; [email protected] Abstract Some results on the existence and uniqueness of solutions for stochastic evolution equations containing some hereditary characteristics are proved. In fact, our theory is developed from a variational point of view and in a general functional setting which permit us to deal with several kinds of delay terms in a unified formulation. 1. Introduction and statement of the problem When one wants to model some evolution phenomena arising in physics, biology, engineering, etc., some hereditary characteristics such as after-effect, time-lag, time-delay can appear in the variables. Typical examples can be found in the researches of materials with termal memory, biochemical reactions, population models, etc. (see, for instance, Ruess [10], Wu [11] and references cited therein). This enables us to think that the problem could be better modeled by considering a functional differential equation which takes into account the history of the system. However, in most cases, some kind of randomness can appear in the problem, so that the system should be modeled by a stochastic form of the functional equation. Motivated by these facts, our main purpose in this paper is to analyse the existence and uniqueness of solutions for a class of nonlinear stochastic PDEs with time delays in a variational context, which, in particular, extend and complete the results in Caraballo [2] and Caraballo et al. [4]. Firt of all, we would like to mention that, in the deterministic framework, there exists a wide literature on the existence of different kind of solutions (strong, mild, integral, etc.) to functional differential equations even in the more general context of differential inclusions. It is well worth reading the work by Ruess [10] where one can find a description of the different techniques used to handle this question, in addition to a large list of references concerning these methods (e.g. method of lines, Galerkin approximations, Kato approximants, etc.). However, from a variational point of view, only a few works have been published (Artola [1] for linear and semilinear retarded equations, Caraballo [3] for a more general nonlinear monotone situation in the functional framework, among others). As for the stochastic problem in the variational setting, even much less has been done. As far as we know, only the works by Real [8,9] (in the linear case), Caraballo [2] (nonlinear problem with variable delay) and Caraballo et al. [4] have appeared up to date. But, as the assumptions in these works are rather restrictive so that the operators involved in the equations cannot be general enough, we are now interested in developing a theory which, in particular, contains the previous works and which permits us to prove existence and uniqueness of solution for a wider class of systems. 1 To start off, let us state the abstract framework in which our analysis will be carried out. Let Vand Hbe two real separable Hilbert spaces such that V⊂H≡H∗⊂V∗, where the injections are continuous and dense. We denote by k·k ,|·| and k·k∗the norms in V, H and V∗respectively; by ((·,·)) and (·,·) the scalar products in Vand Hrespectively; and by h·,·i the duality product between V∗and V. Assume that {Ω,F, P}is a complete probability space, equipped with a normal filtration {Ft}t≥0,i.e., F0contains all A∈ F such that P(A) = 0 and Ft=T s>t Fs,∀t≥0.Denote Ft=F0for all t≤0. We suppose also given {W(t)}t≥0, a real valued {Ft} −Wiener process. Given real numbers a<b, and a separable Hilbert space Hwe will denote by I2(a, b;H) the space of all processes X∈L2(Ω ×(a, b),F ⊗ B((a, b)), dP ⊗dt;H) (where B((a, b)) denotes the Borel σ−algebra on (a, b)) such that X(t) is Ft−measurable a.e. t∈(a, b). The space I2(a, b;H) is a closed subspace of L2(Ω ×(a, b),F ⊗ B((a, b)), dP ⊗dt;H). We will denote by C(a, b;H) the Banach space of all continuous functions from [a, b] into Hequipped with sup norm. We will write L2(Ω; C(a, b;H)) instead of L2(Ω,F, dP;C(a, b;H)). Let us also consider two fixed real numbers T > 0 and h > 0. If we consider a function x∈C(−h, T;H), for each t∈[0, T] we will denote by xt∈C(−h, 0; H) the function defined by xt(s) = x(t+s)∀s∈[−h, 0]. Moreover, if y∈L2(−h, T;H), we will also denote by yt∈L2(−h, 0; H), for a.e. t∈(0, T ), the function defined by yt(s) = y(t+s) a.e. s∈(−h, 0). Let A(t, ·) : V→V∗be a family of nonlinear operators defined a.e. t∈(0, T ) and satisfying: (A.1) (Measurability) ∀v∈V, the map t∈(0, T )→A(t, v)∈V∗is Lebesgue measurable. (A.2) (Hemicontinuity) the map θ∈IR → hA(t, u +θv), wi ∈ IR is continuous ∀u, v, w ∈V, and a.e. t∈(0, T). (A.3) (Boundedness) there exists c > 0 such that kA(t, v)k∗≤ckvk ∀ v∈V, a.e. t∈ (0, T). (A.4) (Monotonicity and Coercivity): there exist α > 0 and λ∈IR such that −2hA(t, u)−A(t, v), u −vi+λ|u−v|2≥αku−vk2,∀u, v ∈V, a.e. t ∈(0, T). Let F1: (0, T)×C(−h, 0; H)→V∗and F2: (0, T )×C(−h, 0; V)→V∗be two families of nonlinear operators defined a.e. t∈(0, T) such that: (F1.1) ∀ξ∈C(−h, 0; H),the map t∈(0, T)7−→ F1(t, ξ)∈V∗is Lebesgue measurable, (F1.2) F1(t, 0) = 0,a.e. t∈(0, T), (F1.3) there exists CF1>0 such that 2 kF1(t, ξ)−F1(t, η)k2 ∗≤CF1|ξ−η|2 C(−h,0;H),∀ξ, η ∈C(−h, 0; H),a.e. t ∈(0, T), (F2.1) ∀ξ∈C(−h, 0; V),the map t∈(0, T)7−→ F2(t, ξ)∈V∗is Lebesgue measurable, (F2.2) F2(t, 0) = 0,a.e. t∈(0, T), (F2.3) there exists CF2>0 such that kF2(t, ξ)−F2(t, η)k2 ∗≤CF2kξ−ηk2 C(−h,0;V),∀ξ, η ∈C(−h, 0; V),a.e. t ∈(0, T), (F2.4) there exists KF2>0 such that ∀x, y ∈C(−h, T;V),and ∀t∈[0, T], Zt 0 kF2(s, xs)−F2(s, ys)k2 ∗ds ≤KF2Zt −h kx(s)−y(s)k2ds. Let also G0: (0, T)×C(−h, 0; H)→Hand G1: (0, T)×C(−h, 0; V)→Hbe another two families of nonlinear operators defined a.e. t∈(0, T ) such that: (G0.1) ∀ξ∈C(−h, 0; H),the map t∈(0, T)7−→ G0(t, ξ)∈His Lebesgue measurable, (G0.2) G0(t, 0) = 0,a.e. t∈(0, T), (G0.3) there exists CG0>0 such that |G0(t, ξ)−G0(t, η)|2≤CG0|ξ−η|2 C(−h,0;H),∀ξ, η ∈C(−h, 0; H),a.e. t ∈(0, T), (G1.1) ∀ξ∈C(−h, 0; V),the map t∈(0, T)7−→ G1(t, ξ)∈His Lebesgue measurable, (G1.2) G1(t, 0) = 0,a.e. t∈(0, T), (G1.3) there exists CG1>0 such that |G1(t, ξ)−G1(t, η)|2≤CG1kξ−ηk2 C(−h,0;V),∀ξ, η ∈C(−h, 0; V),a.e. t ∈(0, T), (G1.4) there exists KG1>0 such that ∀x, y ∈C(−h, T;V),and ∀t∈[0, T], Zt 0 |G1(s, xs)−G1(s, ys)|2ds ≤KG1Zt −h kx(s)−y(s)k2ds. We consider the problem 3                      u∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)), u(t) = ψ(0) + Zt 0 A(s, u(s)) ds +Zt 0 (F1(s, us) + F2(s, us) + f(s)) ds +Zt 0 (G0(s, us) + G1(s, us) + g(s)) dW(s), t ∈[0, T], u(t) = ψ(t), t ∈[−h, 0], (P) where f∈I2(0, T;V∗), g∈I2(0, T ;H) and ψ∈I2(−h, 0; V)∩L2(Ω; C(−h, 0; H)) are given. Remark 1.1. It is not difficult to deduce from (F1.1)-(F1.3) that if u∈I2(−h, T ;V)∩ L2(Ω; C(−h, T;H)), the process F1(t, ut) belongs to I2(0, T;V∗). Also, by means of (G0.1)- (G0.3), the process G0(t, ut) belongs to I2(0, T ;H). Remark 1.2. Observe that by (F2.1)-(F2.3), for a given x∈C(−h, T ;V), the function Fx 2: (0, T)→V∗defined by Fx 2(t) = F2(t, xt) a.e.t∈(0, T), belongs to L2(0, T ;V∗). Then, thanks to (F2.4), the mapping Ξ : x∈C(−h, T;V)7→ Fx 2∈L2(0, T;V∗) has a unique extension to a mapping e Ξ which is uniformly continuous from L2(−h, T;V) into L2(0, T;V∗). From now on, we will also write F2(t, xt) = e Ξ(x)(t) for each x∈L2(−h, T;V), and for every x, y ∈L2(−h, T;V) it holds Zt 0 kF2(s, xs)−F2(s, ys)k2 ∗ds ≤KF2Zt −h kx(s)−y(s)k2ds ∀t∈[0, T].(1.1) By a similar argument, we can define G1(t, xt)∈L2(0, T ;H) for each x∈L2(−h, T;V), and ∀x, y ∈L2(−h, T;V) it follows Zt 0 |G1(s, xs)−G1(s, ys)|2ds ≤KG1Zt −h kx(s)−y(s)k2ds ∀t∈[0, T].(1.2) Thus, if u∈I2(−h, T ;V) is given, the process F2(t, ut) belongs to I2(0, T;V∗), the process G1(t, ut) belongs to I2(0, T;H), and, consequently, ∀u, v ∈I2(−h, T;V) we obtain Zt 0 kF2(s, us)−F2(s, vs)k2 ∗ds ≤KF2Zt −h ku(s)−v(s)k2ds ∀t∈[0, T], P −a.s., (1.3) and Zt 0 |G1(s, us)−G1(s, vs)|2ds ≤KG1Zt −h ku(s)−v(s)k2ds ∀t∈[0, T], P −a.s.. (1.4) As a consequence of the preceding remarks, the terms appearing in problem (P) make sense. Now, we are interested in establishing some results on the existence and uniqueness of solution to (P) under some additional assumptions. To this respect, it is worth mentioning that in the absence of hereditary characteristics (i.e. when h= 0), our problem has been solved by Pardoux [6] (see also Da Prato and Zabczyk [5] for a different approach); in the linear case 4 containing variable delays, it has also been treated by Real [9]; Caraballo [2] considered the nonlinear monotone situation with variable delay but for bounded operators Fiand Gi, and finally, Caraballo et al. [4] provided an answer to our problem in the particular situations in which F2≡0, F1(t, ·) is a family of operators from Vinto Hand, what is more important, under stronger assumptions on the family of operators which do not allow us to cover a wide class of applications (e.g. in the case of unbounded operators, essentially the ones containing distributed delays satisfy the assumptions in [4]). Thus, on the one hand, the results we shall obtain can be considered as extensions to the nonlinear case of those obtained in Real [9]. On the other hand, the presence of the term F1and the hypotheses that we shall impose on F2 and G1, permit us, as we have already mentioned, to treat examples which cannot be handled with the results in Caraballo et al. [4]. The paper is organized as follows. In Section 2, we prove a first result on the existence and uniqueness of solution for the problem (P) in the particular case F2≡G1≡0. Then, in Section 3, we establish an existence and uniqueness result for the complete problem. Finally, an example is considered in the last Section to illustrate our results. 2. A first existence and uniqueness result In this section, we shall consider the problem                      u∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)), u(t) = ψ(0) + Zt 0 A(s, u(s)) ds +Zt 0 (F1(s, us) + f(s)) ds +Zt 0 (G0(s, us) + g(s)) dW(s), t ∈[0, T], u(t) = ψ(t), t ∈[−h, 0]. (P0) We can now prove the following result: Theorem 2.1 Assume that hypotheses (A.1)-(A.4),(F1.1)-(F1.3) and (G0.1)-(G0.3) hold. Then, for every ψ∈I2(−h, 0; V)∩L2(Ω; C(−h, 0; H)),f∈I2(0, T;V∗)and g∈I2(0, T;H), there exists a unique solution uto the problem (P0). Proof. Uniqueness of solutions. Assume that u, v ∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)) are two solutions of (P0). Then, Itˆo’s formula and condition (A.4) imply that for all t∈[0, T] |u(t)−v(t)|2+αZt 0 ku(s)−v(s)k2ds ≤λZt 0 |u(s)−v(s)|2ds + 2 Zt 0 hF1(s, us)−F1(s, vs), u(s)−v(s)ids + 2 Zt 0 (G0(s, us)−G0(s, vs), u(s)−v(s)) dW(s) +Zt 0 |G0(s, us)−G0(s, vs)|2ds. 5 Therefore, E·sup 0≤s≤t |u(s)−v(s)|2¸+αE Zt 0 ku(s)−v(s)k2ds ≤2|λ|EZt 0 |u(s)−v(s)|2ds + 2EZt 0 |G0(s, us)−G0(s, vs)|2ds + 4EZt 0 kF1(s, us)−F1(s, vs)k∗ku(s)−v(s)kds + 4E·sup 0≤s≤tZs 0 (G0(θ, uθ)−G0(θ, vθ), u(θ)−v(θ)) dW(θ)¸(2.1) for all t∈[0, T]. Now, we can estimate the terms on the right-hand side of (2.1). On the one hand, 4EZt 0 kF1(s, us)−F1(s, vs)k∗ku(s)−v(s)kds ≤EZt 0·8 αkF1(s, us)−F1(s, vs)k2 ∗+α 2ku(s)−v(s)k2¸ds ≤8 αCF1EZt 0 |us−vs|2 C(−h,0;H)ds +α 2EZt 0 ku(s)−v(s)k2ds ≤8 αCF1EZt 0 sup 0≤r≤s |u(r)−v(r)|2ds +α 2EZt 0 ku(s)−v(s)k2ds. (2.2) On the other hand, Burkholder-Davis-Gundy’s inequality yields that 4E·sup 0≤s≤tZs 0 (G0(θ, uθ)−G0(θ, vθ), u(θ)−v(θ)) dW(θ)¸ ≤12E(sup 0≤s≤t |u(s)−v(s)|·Zt 0 |G0(θ, uθ)−G0(θ, vθ)|2dθ¸1 2) ≤1 2Eµsup 0≤s≤t |u(s)−v(s)|2¶+ 72EZt 0 |G0(θ, uθ)−G0(θ, vθ)|2dθ ≤1 2Eµsup 0≤s≤t |u(s)−v(s)|2¶+ 72CG0EZt 0 |uθ−vθ|2 C(−h,0;H)dθ ≤1 2Eµsup 0≤s≤t |u(s)−v(s)|2¶+ 72CG0EZt 0 sup 0≤r≤θ |u(r)−v(r)|2dθ. (2.3) Thus, (2.1)-(2.3) imply that for all t∈[0, T] 1 2E·sup 0≤s≤t |u(s)−v(s)|2¸+α 2EZt 0 ku(s)−v(s)k2ds ≤·2|λ|+8 αCF1+ 74CG0¸EZt 0 sup 0≤r≤θ |u(r)−v(r)|2dθ. Now, uniqueness follows immediately from Gronwall’s lemma. 6 Existence of solutions: We denote u0≡0, and define by recurrence a sequence {un}n≥1 of processes as solutions to the problem                      un∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)), un(t) = ψ(0) + Zt 0 (A(s, un(s)) −λ 2un(s)) ds +λ 2Zt 0 un−1(s)ds +Zt 0 (F1(s, un−1 s) + f(s)) ds +Zt 0 (G0(s, un−1 s) + g(s)) dW(s), t ∈[0, T], un(t) = ψ(t), t ∈[−h, 0]. (P0 n) Observe that u0≡0∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)), and by Remark 1.1, if un−1∈ I2(−h, T;V)∩L2(Ω; C(−h, T ;H)),it follows that F1(t, un−1 t)∈I2(0, T;V∗),and G0(t, un−1 t)∈ I2(0, T;H),and consequently, from the results in Pardoux [6], there exists a unique un∈ I2(−h, T;V)∩L2(Ω; C(−h, T ;H)) which is a solution of (P0 n). Now, we want to prove that {un}n≥1converges in I2(−h, T;V)∩L2(Ω; C(−h, T;H)) to a process uwhich will be the solution of problem (P0). Applying Itˆo’s formula to the process un+1(t)−un(t), n≥1, and using condition (A.4), we obtain ¯¯un+1(t)−un(t)¯¯2+αZt 0° °un+1(s)−un(s)° °2ds ≤λZt 0 (un+1(s)−un(s), un(s)−un−1(s)) ds + 2 Zt 0F1(s, un s)−F1(s, un−1 s), un+1(s)−un(s)®ds + 2 Zt 0 (G0(s, un s)−G0(s, un−1 s), un+1(s)−un(s)) dW(s) +Zt 0¯¯G0(s, un s)−G0(s, un−1 s)¯¯2ds (2.4) for all t∈[0, T]. Consequently, (2.4) yields E·sup 0≤s≤t¯¯un+1(s)−un(s)¯¯2¸+αE Zt 0° °un+1(s)−un(s)° °2ds ≤2|λ|EZt 0¯¯un+1(s)−un(s)¯¯¯¯un(s)−un−1(s)¯¯ds + 4EZt 0¯¯F1(s, un s)−F1(s, un−1 s), un+1(s)−un(s)®¯¯ds + 4E·sup 0≤s≤tZs 0 (G0(θ, un θ)−G0(θ, un−1 θ), un+1(θ)−un(θ)) dW(θ)¸ + 2EZt 0¯¯G0(s, un s)−G0(s, un−1 s)¯¯2ds. (2.5) Now, observe that 7 2|λ|EZt 0¯¯un+1(s)−un(s)¯¯¯¯un(s)−un−1(s)¯¯ds ≤2β|λ|EZt 0° °un+1(s)−un(s)° °¯¯un(s)−un−1(s)¯¯ds ≤α 3EZt 0° °un+1(s)−un(s)° °2ds +3λ2β2 αEZt 0 sup 0≤θ≤s¯¯un(θ)−un−1(θ)¯¯2ds, (2.6) where β > 0 is a constant such that |v| ≤ βkvk,∀v∈V. On the other hand, thanks to condition (F1.3), we can obtain 4EZt 0¯¯F1(s, un s)−F1(s, un−1 s), un+1(s)−un(s)®¯¯ds ≤4EZt 0° °F1(s, un s)−F1(s, un−1 s)° °∗° °un+1(s)−un(s)° °ds ≤EZt 0·12 α° °F1(s, un s)−F1(s, un−1 s)° °2 ∗+α 3° °un+1(s)−un(s)° °2¸ds ≤12 αCF1EZt 0¯¯un s−un−1 s¯¯2 C(−h,0;H)ds +α 3EZt 0° °un+1(s)−un(s)° °2ds ≤12 αCF1EZt 0 sup 0≤r≤s¯¯un(r)−un−1(r)¯¯2ds +α 3EZt 0° °un+1(s)−un(s)° °2ds. (2.7) In a similar manner as for uniqueness, we can obtain from Burkholder-Davis-Gundy’s inequality that 4E·sup 0≤s≤tZs 0 (G0(θ, un θ)−G0(θ, un−1 θ), un+1(θ)−un(θ)) dW(θ)¸ ≤1 2Eµsup 0≤s≤t¯¯un+1(s)−un(s)¯¯2¶+ 72CG0EZt 0 sup 0≤r≤θ¯¯un(r)−un−1(r)¯¯2dθ. (2.8) Then, we can get from (2.5)-(2.8) and (G0.3), that there exists a positive constant ksuch that for all n≥1 and all t∈[0, T] 1 2E·sup 0≤s≤t¯¯un+1(s)−un(s)¯¯2¸+α 3EZt 0° °un+1(s)−un(s))° °2ds ≤k 2EZt 0 sup 0≤r≤θ¯¯un(r)−un−1(r)¯¯2dθ. (2.9) Now, we define ρn(t) = 1 2E·sup 0≤s≤t¯¯un+1(s)−un(s)¯¯2¸+α 3EZt 0° °un+1(s)−un(s))° °2ds, ∀n≥1,∀t∈[0, T]. Then, (2.9) immediately implies that ρn(t)≤kZt 0 ρn−1(s)ds, ∀n≥1,∀t∈[0, T], 8 and, consequently, by iterating the preceding inequality, we obtain ρn(t)≤kn−1Tn−1 (n−1)! ρ1(T),∀n≥1,∀t∈[0, T].(2.10) Since un+1(t) = un(t),∀t∈[−h, 0], (2.10) implies that {un}n≥1is a Cauchy sequence in I2(−h, T;V)∩L2(Ω; C(−h, T ;H)).Thus, there exists u∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)) such that un→uin I2(−h, T;V)∩L2(Ω; C(−h, T ;H)). Thanks to conditions (F1.3) and (G0.3), we have in particular that F1(t, un t)→F1(t, ut) in I2(0, T;V∗), and G0(t, un t)→G0(t, ut) in I2(0, T;H). Moreover, by (A.3), the sequence {A(t, un(t))}n≥1is bounded in I2(0, T;V∗). Thus, there exist a subsequence {A(t, unk(t))}nk≥1⊂ {A(t, un(t))}n≥1and ξ∈I2(0, T;V∗),such that A(t, unk(t)) * ξ in I2(0, T ;V∗), where *denotes weak convergence. Thus, we can take limits in (P0 nk), and obtain that uis solution of                      u∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)), u(t) = ψ(0) + Zt 0 ξ(s)ds +Zt 0 (F1(s, us) + f(s)) ds +Zt 0 (G0(s, us) + g(s)) dW(s), t ∈[0, T], u(t) = ψ(t), t ∈[−h, 0]. (P00) To simplify the notation, observe that ξis uniquely determined by u, and thus, the whole sequence {A(t, un(t))}n≥1converges weakly to ξin I2(0, T;V∗). In order to prove that uis in fact a solution of problem (P0), we only need to prove that ξ(t) = A(t, u(t)) in (0, T). First of all, applying Itˆo’s formula to |un(t)|2and to |u(t)|2on the interval [0, T], we obtain E|un(T)|2=E|ψ(0)|2+ 2EZT 0 hA(s, un(s)), un(s)ids +λE ZT 0 (un(s), un−1(s)) ds −λE ZT 0 |un(s)|2ds + 2EZT 0F1(s, un−1 s) + f(s), un(s)®ds +EZT 0 |G0(s, un−1 s) + g(s)|2ds (2.12) and E|u(T)|2=E|ψ(0)|2+ 2EZT 0 hξ(s), u(s)ids + 2EZT 0 hF1(s, us) + f(s), u(s)ids +EZT 0 |G0(s, us) + g(s)|2ds. (2.13) 9 and consequently, lim k→∞(xmk−ymk) = 2EZT 0 h−A(t, X(t)) −F2(t, Xt), u(t)idt + 2EZT 0 hη(t)−A(t, X(t)) + σ(t)−F2(t, Xt),−X(t)idt +EZT 0 |G1(t, Xt)|2dt −2EZT 0 (ζ(t), G1(t, Xt)) dt −2αE ZT 0 ((u(t), X(t))) dt +αE ZT 0 kX(t)k2dt. (3.12) Applying Itˆo’s formula to |umk(t)|2on the interval [0, T], E|umk(T)|2≤E|ψ(0)|2+EZT 0 |G1(t, umk t) + g(t)|2dt + 2EZT 0 hA(t, umk(t)) + F2(t, umk t) + f(t), umk(t)idt, and, thus ymk≥E|umk(T)|2−E|ψ(0)|2−EZT 0 |g(t)|2dt +αE ZT 0 kumk(t)k2dt −2EZT 0 (G1(t, umk t), g(t)) dt −2EZT 0 hf(t), umk(t)idt. Letting k→ ∞, lim inf k→∞ ymk≥E|u(T)|2−E|ψ(0)|2−EZT 0 |g(t)|2dt +αE ZT 0 ku(t)k2dt −2EZT 0 (ζ(t), g(t)) dt −2EZT 0 hf(t), u(t)idt. (3.13) Applying once again Itˆo’s formula to |u(t)|2on [0, T], E|u(T)|2=E|ψ(0)|2+EZT 0 |ζ(t) + g(t)|2dt + 2EZT 0 hη(t) + σ(t) + f(t), u(t)idt, and so, from (3.13) lim inf k→∞ ymk≥2EZT 0 hη(t) + σ(t), u(t)idt +EZT 0 |ζ(t)|2dt +αE ZT 0 ku(t)k2dt. (3.14) From (3.12) and (3.14) we have 0≥lim inf k→∞ xmk≥2EZT 0 hη(t)−A(t, X(t)) + σ(t)−F2(t, Xt), u(t)−X(t)idt +EZT 0 |ζ(t)−G1(t, Xt)|2dt +αE ZT 0 ku(t)−X(t)k2dt. (3.15) 16 If we take X(t) = u(t) in (3.15), it follows that ζ(t) = G1(t, ut), t ∈[0, T ].Now, we will set X(t) = u(t)−δZ(t),where δ > 0 and Z∈I2(−h, T;V) is such that Z= 0 in (−h, 0). Then, by (3.15), 0≥2EZT 0 hη(t)−A(t, u(t)−δZ(t)) + σ(t)−F2(t, ut−δZt), δZ(t)idt +EZT 0 |G1(t, ut)−G1(t, ut−δZt)|2dt. (3.16) Dividing by δin (3.16), and letting δ→0,we get by (A.2), (F2.2) and (F2.4), 2EZT 0 hη(t)−A(t, u(t)) + σ(t)−F2(t, ut), Z(t)idt ≤0, and since Z∈I2(0, T;V) is arbitrary, clearly η(t) + σ(t) = A(t, u(t)) + F2(t, ut) in [0, T]. Step 2. Now, we consider problem (P) under the conditions in the theorem. We denote u0≡0, and define by recurrence a sequence {un}n≥1of processes by                                un∈I2(−h, T;V)∩L2(Ω; C(−h, T ;H)), un(t) = ψ(0) + Zt 0 (A(s, un(s)) −λ 2un(s)) ds +λ 2Zt 0 un−1(s)ds +Zt 0 (F1(s, un−1 s) + F2(s, un s) + f(s)) ds +Zt 0 (G0(s, un−1 s) + G1(s, un s) + g(s)) dW(s), t ∈[0, T], un(t) = ψ(t), t ∈[−h, 0]. (Pn) Observe that if un−1∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)), then F1(t, un−1 t)∈I2(0, T;V∗), and G0(t, un−1 t)∈I2(0, T;H).Moreover, the family of operators defined by e A(t, v) = A(t, v)− λ 2v∀v∈V, a.e.t∈(0, T), satisfies conditions (A.1) −(A.5) with λ= 0.Consequently, we can use Step 1 to ensure that problem (Pn) has a unique solution. Now, arguing as in the proof of Theorem 2.1, we can prove that {un}n≥1is a Cauchy sequence in I2(−h, T;V)∩L2(Ω; C(−h, T;H)), and thus, it converges to a process u∈ I2(−h, T;V)∩L2(Ω; C(−h, T ;H)), which will be the solution to (P). In order to obtain our objective, we first apply Itˆo’s formula to the process un+1(t)−un(t), t≥0, n≥1, and using (A.5) we have ¯¯un+1(t)−un(t)¯¯2+αZt 0° °un+1(s)−un(s)° °2ds ≤λZt 0 (un+1(s)−un(s), un(s)−un−1(s)) ds + 2 Zt 0F1(s, un s)−F1(s, un−1 s), un+1(s)−un(s)®ds + 2 Zt 0 (G0(s, un s)−G0(s, un−1 s), G1(s, un+1 s)−G1(s, un s)ds 17 +Zt 0¯¯G0(s, un s)−G0(s, un−1 s)¯¯2ds + 2 Zt 0 (G0(s, un s)−G0(s, un−1 s), un+1(s)−un(s)) dW(s) + 2 Zt 0 (G1(s, un+1 s)−G1(s, un s), un+1(s)−un(s)) dW(s),(3.17) which, together with conditions (F1.4), (G0.4) and (G1.4) imply E¯¯un+1(t)−un(t)¯¯2+αE Zt 0° °un+1(s)−un(s)° °2ds ≤3α 4EZt 0° °un+1(s)−un(s)° °2ds +λ2β2 αZt 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds +4 αCF1Zt 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds +µ3KG1 α+ 1¶CG0Zt 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds, (3.18) where β > 0 is the constant such that |v| ≤ βkvk ∀v∈V. Consequently, (3.18) yields sup 0≤s≤t E¯¯un+1(s)−un(s)¯¯2+α 4EZt 0° °un+1(s)−un(s)° °2ds ≤kZt 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds, (3.19) for all t∈[0, T] and all n≥1, where k=2λ2β2 α+8 αCF1+ 2 µ3KG1 α+ 1¶CG0. Now, if we denote ρn(t) = sup 0≤s≤t E¯¯un+1(s)−un(s)¯¯2+α 4EZt 0° °un+1(s)−un(s)° °2ds, ∀n≥1,∀t∈[0, T], we can deduce from (3.19) that ρn(t)≤(kT)n−1 (n−1)! ρ1(T),∀n≥1,∀t∈[0, T], and thus, ∀n≥1, sup 0≤s≤T E¯¯un+1(s)−un(s)¯¯2+α 4EZT 0° °un+1(s)−un(s)° °2ds ≤(kT)n−1 (n−1)! ρ1(T),(3.20) and, in particular, {un}n≥1is a Cauchy sequence in I2(−h, T;V). Now, in order to prove that {un}n≥1is a Cauchy sequence in L2(Ω; C(−h, T ;H)), we consider again (3.17), take sup 0≤s≤T and, finally, expectation, so that we obtain Eµsup 0≤s≤T¯¯un+1(s)−un(s)¯¯2¶ 18 ≤ |λ|EZT 0¯¯(un+1(s)−un(s), un(s)−un−1(s))¯¯ds + 2EZT 0¯¯F1(s, un s)−F1(s, un−1 s), un+1(s)−un(s)®¯¯ds + 2EZT 0¯¯(G0(s, un s)−G0(s, un−1 s), G1(s, un+1 s)−G1(s, un s)¯¯ds + 2Eµsup 0≤s≤TZs 0 (G0(θ, un θ)−G0(θ, un−1 θ), un+1(θ)−un(θ)) dW(θ)¶ + 2Eµsup 0≤s≤TZs 0 (G1(θ, un+1 θ)−G1(θ, un θ), un+1(θ)−un(θ)) dW(θ)¶ +EZT 0¯¯G0(s, un s)−G0(s, un−1 s)¯¯2ds (3.21) for all t∈[0, T] and all n≥1. Using (F1.4),we obtain 2EZT 0¯¯F1(s, un s)−F1(s, un−1 s), un+1(s)−un(s)®¯¯ds ≤CF1ZT 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds +EZT 0° °un+1(s)−un(s)° °2ds. (3.22) From Burkholder-Davis-Gundy’s inequality, (G0.4) and (G1.4), we have 2Eµsup 0≤s≤TZs 0 (G0(θ, un θ)−G0(θ, un−1 θ), un+1(θ)−un(θ)) dW(θ)¶ ≤1 3Eµsup 0≤s≤T¯¯un+1(s)−un(s)¯¯2¶+ 27CG0ZT 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds, (3.23) EZT 0¯¯G0(s, un s)−G0(s, un−1 s)¯¯2ds ≤CG0ZT 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds, (3.24) 2EZT 0¯¯(G0(s, un s)−G0(s, un−1 s), G1(s, un+1 s)−G1(s, un s)¯¯ds ≤CG0ZT 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds +KG1EZT 0° °un+1(s)−un(s)° °2ds, (3.25) and 2Eµsup 0≤s≤TZs 0 (G1(θ, un+1 θ)−G1(θ, un θ), un+1(θ)−un(θ)) dW(θ)¶ ≤1 3Eµsup 0≤s≤T¯¯un+1(s)−un(s)¯¯2¶+ 27KG1EZT 0° °un+1(s)−un(s)° °2ds. (3.26) Also, |λ|EZT 0 (un+1(s)−un(s), un(s)−un−1(s)) ds ≤1 2EZT 0° °un+1(s)−un(s)° °2ds +λ2β2 2ZT 0 sup 0≤θ≤s E¯¯un(θ)−un−1(θ)¯¯2ds. (3.27) 19 From (3.20)-(3.27), we deduce that {un}n≥1is a Cauchy sequence in L2(Ω; C(−h, T;H)). Thus, there exists usuch that un→uin I2(−h, T;V)∩L2(Ω; C(−h, T;H)). Now, by a similar argument to the one in the proof of theorem 2.1, we can deduce that uis the solution of problem (P). Remark 3.1. The hypothesis concerning the compactness of the injection V⊂Hcan be omitted if, for example, ψ≡0. Remark 3.2. Theorems 2.1. and 3.1. can be extended to the case in which W(t) is an IRn-valued (or Hilbert valued) Wiener process. 4. An example To illustrate our theory, mainly Theorem 3.1, we shall consider the following situation, which cannot be handled with the results in Caraballo [2] or Caraballo et al. [4]. Assume O ⊂ IRnis a bounded open set. Let us set H=L2(O), V =H1 0(O) and V∗=H−1(O). Let φ: [0, T]×IRn→IRnbe a continuous map such that there exists cφ>0 such that |φ(t, x)|IRn≤cφ|x|IRnfor all (t, x)∈[0, T]×IRn, and suppose that (φ(t, x)−φ(t, y)) ·(x−y)≤0∀t∈[0, T ],∀x, y ∈IRn,(4.1) where we denote by ·the escalar product in IRn. It is easy to see that the family of operators A(t, ·) defined by hA(t, u), vi=−ZO ∇u(x)· ∇v(x)dx −ZO φ(t, ∇u(x)) · ∇v(x)dx ∀t∈[0, T],∀u, v ∈V, (4.2) satisfies hypotheses (A.1)-(A.4), with λ= 0 and α≤2. Let us consider now a measurable map k1: [0, T]×IR →IRnand a measurable function ω1: [0, T]→IR such that 0 ≤ω1(t)≤hfor all t∈[0, T ].Suppose that k1(t, 0) = 0,∀t∈[0, T ], and that there exists Lk1>0 such that |k1(t, a)−k1(t, b)|IRn≤Lk1|a−b|,∀t∈[0, T],∀a, b ∈IR.(4.3) Denote by F1(t, ·) the family of operators defined by hF1(t, ξ), vi=−ZO k1(t, ξ(−ω1(t))(x)) · ∇v(x)dx, ∀ξ∈C(−h, 0; H),∀v∈V, (4.4) for each t∈[0, T]. Then, the family F1(t, ·) satisfies assumptions (F1.1)-(F1.4), with CF1=L2 k1and CF1= L2 k1. Consider also k2: [0, T]×IRn→IRn, measurable, and ω2∈C1([0, T]) such that 0 ≤ ω2(t)≤hfor all t∈[0, T], and ω∗ 2= maxt∈[0,T ]ω0 2(t)<1. Suppose that k2(t, 0) = 0,∀t∈ [0, T], and that there exists Lk2>0 such that |k2(t, x)−k2(t, y)|IRn≤Lk2|x−y|IRn,∀t∈[0, T ],∀x, y ∈IRn.(4.5) 20 Denote by F2(t, ·) the family of operators defined by hF2(t, ξ), vi=−ZO k2(t, ∇ξ(−ω2(t))(x)) · ∇v(x)dx, ∀ξ∈C(−h, 0; V),∀v∈V, (4.6) for each t∈[0, T].Then, the family F2(t, ·) satisfies hypotheses (F2.1)-(F2.4), with CF2=L2 k2 and KF2=L2 k2 1−ω∗ 2 . Finally, let l0: [0, T ]×IR →IR and l1: [0, T]×IRn→IR be two measurable functions, such that l0(t, 0) = l1(t, 0) = 0 for all t∈[0, T], and there exist Ll0>0 and Ll1>0 such that |l0(t, a)−l0(t, b)| ≤ Ll0|a−b|,∀t∈[0, T],∀a, b ∈IR,(4.6) and |l1(t, x)−l1(t, y)| ≤ Ll1|x−y|IRn,∀t∈[0, T ],∀x, y ∈IRn.(4.7) Let us also fix ρi: [0, T]→IR, i= 0,1, two measurable functions such that 0 ≤ρi(t)≤hfor all t∈[0, T] and i= 0,1, ρ1∈C1([0, T]), and ρ∗ 1= maxt∈[0,T ]ρ0 1(t)<1. Then, if we define G0(t, ξ)(x) = l0(t, ξ(−ρ0(t))(x)),∀t∈[0, T],∀ξ∈C(−h, 0; H),a.e. x ∈ O,(4.8) and G1(t, ξ)(x) = l1(t, ∇ξ(−ρ1(t))(x)),∀t∈[0, T],∀ξ∈C(−h, 0; V),a.e. x ∈ O,(4.9) it is easy to check that G0satisfies (G0.1)-(G0.4), and that G1satisfies hypotheses (G1.1)- (G1.4), with CG0=CG0=L2 l0,CG1=L2 l1, and KG1=L2 l1 1−ρ∗ 1 . As for hypothesis (A.5), it is fulfilled with b λlarge enough provided 2Lk2 p1−ω∗ 2 +L2 l1 1−ρ∗ 1 <2.(4.10) Consequently, under all the hypotheses above, we can ensure that given ψ∈I2(−h, 0; H1 0(O))∩ L2(Ω; C(−h, 0; L2(O))), f∈I2(0, T;H−1(O)), and g∈I2(0, T;L2(O)), there exists a unique solution u∈I2(−h, T;H1 0(O)) ∩L2(Ω; C(−h, T;L2(O))) to the corresponding problem (P). Such a solution, satisfies, in a generalized sense, the problem                      ∂u(t) ∂t = ∆u(t) + ∇ · (φ(t, ∇u(t))) + ∇ · (k2(t, ∇u(t−ω2(t)))) + ∇ · (k1(t, u(t−ω1(t)))) +f(t) + (l1(t, ∇u(t−ρ1(t))) + l0(t, u(t−ρ0(t))) + g(t)) ∂W (t) ∂t in O × (0, T ), u(0) = 0 on ∂O × (0, T), u(t) = ψ(t) in O × [−h, 0], where, for a vector function ~v = (v1, ..., vn) defined on O, we denote by ∇ ·~v the divergence of ~v defined by ∇ · ~v = n X i=1 ∂vi ∂xi . 21 Acknowledgement. This work has been partly supported by Junta de Andalucia Project FQM314. References [1] M. Artola, Equations et inequations variationnelles a retard, Ann. Sc. Math. Qu´ebec vol. I, no. 2 (1977), 131-152. [2] T. Caraballo, Existence and Uniqueness of Solutions for non-linear Stochastic Partial Differential Equations, Collectanea Mathematica 42(1) (1991), 51-74. [3] T. Caraballo, Nonlinear Partial Functional Differential Equations: Existence and Stability, J. Math. Anal. Appl., to appear. [4] T. 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Kartsatos), Marcel-Dekker, Lecture Notes Pure Appl. Math. 178, New York (1996), 259-288. [11] J. Wu, Theory and applications of partial functional differential equations, SpringerVerlag, New York (1996). 22