Attractors for Parametric Delay Differential Equations and Their Continuous Behavior

ATTRACTORS FOR PARAMETRIC DELAY DIFFERENTIAL EQUATIONS AND THEIR CONTINUOUS BEHAVIOUR PEDRO MAR´ IN–RUBIO Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain E-mail: [email protected] March 18, 2009 Dedicated to the memory of Valery S. Melnik The problem of the continuity of global attractors under minimal assumptions for a general class of parameterized delay differential equations is considered. The theory of equi-attraction developed by Li and Kloeden in [Li & Kloeden, 2004a] is adapted to this framework, so it is proved that the continuity of the attractors with respect to the parameter is equivalent to this equi-attraction property. Keywords: Delay differential equations, global attractors, continuity of attractors, and equiattraction. Mathematics Subject Classifications (2000): 34D45, 37C70, 34K25. 1. Introduction In many fields of science, Biology, Economics, Physics, Chemistry, etc, delay differential equations (DDE for short) are of major interest. Besides the well-posedness of these problems, the qualitative behaviour of DDE includes many features as for general differential equations, although most attention has been paid to stability properties. In any case, even when such results do not hold, it is still useful the study of their long-time behaviour (leading to meaningful situations as permanence, extinction, chaotic phenomena, etc), in particular the study of attractors. There exists a wide literature on this topic, see for instance the seminal monographs [Hale, 1988, Hale & Lunel, 1993] and the references therein, or some recent developments in this field as multi-valued (semi)flows and/or non-autonomous terms in the involved models (e.g. [Caraballo et al., 2005, Caraballo et al., 2007b]). On the other hand, it is a complex matter to understand the structure of the attractor for a general problem and its continuous behaviour, for instance when changes in the model happen. This also indicates somehow robustness of the model (cf. [Hale et al., 1988, Hale & Raugel, 1989]. Many papers are devoted to study continuity properties, being the upper semi continuity one half of the answer, and sometimes the most one can obtain (e.g. cf. [Caraballo & Langa, 2003]). Indeed, the continuity w.r.t. to variations in the original model and the structure of attractors remain as open problems (e.g. see [Carvalho & Langa, 2007, Carvalho et al., 2007, Langa et al., 2007, Efendiev et al., 2005, Gatti et al., 2006] for some recent developments in different situations). 1 2P. Mar´ın-Rubio The theory for dynamical systems associated to DDE obtains similar results too, for instance in the study of approximation of attractors (e.g. [Hines, 1995, Caraballo et al., 2007a]), or when considering changes in the delay length (cf. [Kloeden, 2006]). An early paper on upper semi continuity for a retarded nonlinear PDE is [Boutet de Monvel et al., 1997]. See also [Boutet de Monvel et al., 1998, Chueshov et al., 2005] for the same question but focused on inertial manifolds to deterministic and stochastic problems. Since a middle step between autonomous and non-autonomous models is the appearance of parameters (also as approximation technique), it is natural the study of these properties in parametric systems. In [Mar´ın-Rubio, 2008] the upper semicontinuity of the attractors for dynamical systems associated to DDE with a parametric dependence and without uniqueness was studied. Very recently, Li and Kloeden [Li & Kloeden, 2004a] proved some continuity results for attractors of problems corresponding to different values of a parameter, using the new concept of equi-attraction. This approach translates the problem of the continuity of attractors to another property; in any case, it seems an interesting option to throw light on the question. That work was also adapted to some different situations as a non-autonomous framework with the cocycle formulation (cf. [Li & Kloeden, 2004b]) or even in a multi-valued context (cf. [Li & Kloeden, 2005]). The idea was adapted in [Kloeden & Mar´ın-Rubio, 2008] to an abstract dynamical system related to DDE problems, and applied to a simple DDE just to illustrate its validity. In this paper we consider a general class of parameterized DDE which includes fixed and distributed delays. Each equation is stated in a different state space, and the existence of the global attractor is ensured. Our goal is to study the behaviour of these attractors when varying the parameter. More exactly, we aim to establish continuity results without knowledge of the structure of the attractors, but in the same terms as Li and Kloeden’s theory, relating to the concept of equi-attraction. So, it may be considered the main example to justify and complement our contributions [Kloeden & Mar´ın-Rubio, 2008, Mar´ın-Rubio, 2008]. The structure of the paper is as follows. In Sec. 2 the parametric family of DDEs and main hypotheses are stated. In Sec. 3 some useful estimates and main concepts of dynamical systems and attractor theory are given. Existence of attractors for our family of problems is proved here. Finally, in Sec. 4 the Li and Kloeden’s theory of equi-attraction is recalled, adapted and applied to our model. 2. Statement of the parametric family of DDEs Let us introduce some notation which will be used all through the paper. For a given metric space (X, d), BX(a, r) will denote the open ball of Xwith center aand radius r. In addition, denote the Hausdorff semi distance and Hausdorff distance on X, respectively, by H∗ X(A, B) = sup x∈A d(x, B),and HX(A, B) = max {H∗ X(A, B), H∗ X(B, A)} for any nonempty subsets Aand Bof X. In Rm(m∈N), we denote | · | the Euclidean norm; and for any T > 0 we will denote (CT,k·kT) the Banach space C([−T, 0]; Rm) endowed with the norm kϕkT= supt∈[−T,0] |ϕ(t)|.The usual notation for delay function will be a sub-script: xt(s) = x(t+s) where it has sense. We consider a similar model to that in [Mar´ın-Rubio, 2008]. Concretely, let us state the following assumptions. Hypothesis 2.1. Let Λ⊂Rbe a closed interval, and suppose that positive numbers 0< T∗< T∗, and functions τ, ρ ∈C(Λ; [T∗, T∗]) are given. Consider also the functions F0, F1∈ C(Rm;Rm),and b: [−max Λτ, 0] ×Rm→Rm,measurable w.r.t. its first variable and continuous w.r.t. the second variable, m0, m1∈L1((−max Λτ, 0); R+), and α, β > 0,and k1, k2≥0,such that |b(s, x)| ≤ m1(s)|x|+m0(s), ∀x∈Rm, a.e. s ∈(−max Λτ, 0), hx, F0(x)i≤−α|x|2+β, ∀x∈Rm, |F1(x)|2≤k2 1+k2 2|x|2,∀x∈Rm. Continuity of attractors for parametric DDE 3 Hypothesis 2.2 (Lipschitz character). There exist positive constants LF0, LF1,and Lbsuch that |Fj(x)−Fj(x0)| ≤ LFj|x−x0|for all x, x0∈Rm, j= 0,1,and |b(s, x)−b(s, x0)| ≤ Lb|x−x0|for all x, x0∈Rm,a. e. s∈(−max Λτ, 0). For convenience we introduce the following notation Mλ= max{ρ(λ), τ(λ)}, mi= max ΛZ0 −τ(λ) mi(s)dsfor i= 0,1. Under the above assumptions, consider (for each λ∈Λ) the functional f(λ, ·) : CMλ→Rm given by f(λ, ϕ) = F0(ϕ(0)) + F1(ϕ(−ρ(λ))) +Z0 −τ(λ) b(s, ϕ(s))ds, (1) and the family of DDE x0(t) = f(λ, xt) =F0(x(t))+F1(x(t−ρ(λ))) +Z0 −τ(λ) b(s, x(t+s))ds. (2) Remark 2.3. (a) More general functionals depending on the parameter and/or more different delay terms can be considered. However, for clarity in the presentation, we prefer to restrict to this case. (b) The parametric dependence of the problem will be denoted by upper script (λ) when necessary, i.e. if no confusion is possible, we will just use the notation xfor any solution instead of x(λ). 3. Parameterized semi dynamical systems and their attractors Local existence of solutions for (2) is a well-known result (cf. [Hale & Lunel, 1993]) thanks to the continuity assumptions given in Hypothesis 2.1. With a priori estimates from the following result (proved in [Mar´ın-Rubio, 2008; Lem.4], see also [Caraballo et al., 2005; Lem.34]) we obtain global (and not only local) solutions but no uniqueness. Lemma 3.1. Under the Hypothesis 2.1, consider a local solution xto (2) with initial data in CMλdefined on an interval [0, Tx).Then, there exist positive values A, B, and δsuch that xsatisfies for all t < Tx: eδt|x(t)|2≤ |x(0)|2+Zt 0 eδs(A+Bkxsk2 Mλ)ds. (3) Proof. By the Hypothesis 2.1 and the Young inequality with suitable constants εand ¯ε, it is not difficult to obtain d dt|x(t)|2≤ −δ|x(t)|2+A+Bkxtk2 Mλ, with A= 2β+k2 1 ε+m2 0 ¯ε, B =k2 2 ε+ 2m1,(4) and δgiven by ε+¯ε= 2α−δ, whence (3) follows. Although for the results in [Mar´ın-Rubio, 2008] (multi-valued) it was enough to fulfill model (2) with the Hypothesis 2.1, here we will need a more restrictive structure (semi dynamical systems in a single-valued sense), and uniqueness is required. It is well-known that Hypothesis 2.2 provides uniqueness of solution for the DDE. Definition 3.2. A map S:R+×X→Xis called a semi dynamical system (SDS for short) if for each fixed t≥0,S(t, ·) : X→Xis continuous, and it has a semi-group structure in time, i.e. (a) S(0,·) = Id (identity map) (b) For any pair t1, t2≥0 and for all x∈X, S(t1+t2, x) = S(t1,S(t2, x)). Next proposition follows from standard continuation results (cf. [Hale & Lunel, 1993; Ch.2]). Proposition 3.3. Under Hypotheses 2.1 and 2.2, the following map defines a semi dynamical system: S(λ):R+×CMλ→CMλ (t, ψ)7→ S(t, ψ) = xt with xthe global solution of (2) with x0=ψ. The goal of our study is the following object. 4P. Mar´ın-Rubio Definition 3.4. A nonempty compact subset Aof Xis called a global attractor of a semi dynamical system Sif it is invariant, i.e. S(t, A) = Afor all t∈R+,and attracts bounded subsets Bof X, i.e. lim t→+∞H∗(S(t, B),A) = 0, where S(t, B) = Sb∈BS(t, b). The following two notions suffice in order to have a global attractor (e.g. cf. [Hale, 1988; Ch.3,Sec.4] or [Robinson, 2001; Ch.10]). Definition 3.5. A semi dynamical system S: R+×X→Xis called pointwise dissipative if there exists a bounded set B⊂Xsuch that it attracts the dynamics starting at all single points, i.e. lim t→+∞H∗(S(t, x), B) = 0 ∀x∈X. It is called asymptotically compact if for any bounded set B⊂Xand any sequences {tn}with tn→+∞and {ψ(n)} ⊂ B, the set {S(tn, ψ)}is relatively compact in X. Pointwise dissipativity can be obtained using ideas from Wang & Xu [Wang & Xu, 2003] and Ball [Ball, 2004] already used in [Caraballo et al., 2007b], but here for finite delay (see also Theorem 8 in [Mar´ın-Rubio, 2008]). Theorem 3.6. Assume that Hypotheses 2.1 and 2.2 hold, and α > k2+m1.(5) Then there exist values A, B, and δas in Lemma 3.1 satisfying δ > B. Moreover, the semi dynamical system S(λ)is pointwise attracted by the set B(λ) 0={ψ∈CMλ:kψk2 Mλ≤K:= A δ−B},(6) that is, lim t→+∞H∗ CMλ(S(λ)(t, ϕ), B(λ) 0) = 0 for all ϕ∈ CMλ. Although the proof is very similar to that of [Mar´ın-Rubio, 2008; Th.8], it is included here for the sake of clarity, since it will be used below in other results. Proof. The function g: (0,2α)→Rgiven by g(δ) = δ−2m1−k2 2 2α−δ,which represents δ−Bfrom Lemma 3.1, has its maximum value g(2α−k2) = 2(α−m1− k2)>0.This proves the first claim. For the second part, we proceed in two steps. Step 1: For any R≥1, BCMλ(0,√RK),is positively invariant for the semi dynamical system S(λ)associated to Eq. (2). If not, there must be an initial data ψwith kψk2 Mλ< RK and a solution xof (2) with x0=ψ and a first time t1such that kxt1k2 Mλ=RK, i.e. |x(t1)|2=RK. But from (3) we deduce that |x(t1)|2<e−δt1RK+Zt1 0 e−δ(t1−s)(A+BRK)ds = e−δt1RK +A+BRK δ(1 −e−δt1). Observe that A+BRK δ≤R(A+BK) δ=RK, which is a contradiction with |x(t1)|2=RK. Step 2: The closed ball B(λ) 0=¯ BCMλ(0,√K) attracts any solution of (2). Consider a solution x(·) with initial data ψ∈ CMλwith kψk2 Mλ=ζ≥K(otherwise, the claim holds by Step 1). Thanks to Step 1 we have that |x(t)| ≤ ζfor all t≥0.Therefore, there exists lim sup t→+∞|x(t)|2=σ. So, ∀ε > 0∃T1(ε, λ)>0 : |x(t)|2≤σ+ε∀t≥T1(ε, λ), and kxtk2 Mλ≤σ+ε∀t≥T1(ε, λ) + Mλ.(7) Take now T2(ε) such that e−δtζ+A+Bζ δe−δT2(ε)−e−δt≤ε∀t≥T2(ε). (8) So, for any t≥T2(ε) + T1(ε, λ) + Mλ,from (3), splitting the integral in two parts, |x(t)|2≤e−δt|x(0)|2 +Zt−T2(ε) 0 e−δ(t−s)(A+Bkxsk2 Mλ)ds +Zt t−T2(ε) e−δ(t−s)(A+Bkxsk2 Mλ)ds, Continuity of attractors for parametric DDE 5 applying (8) to the first two addends (thanks to Step 1), and (7) to the last addend, we obtain for all t≥T2(ε) + T1(ε, λ) + Mλ: |x(t)|2≤ε+A+B(σ+ε) δ1−e−δT2(ε). Passing through the limit when εgoes to zero, we deduce that σ= lim sup t→+∞|x(t)|2≤A+Bσ δ, in other words, σ≤A δ−B=K, which finishes the proof. Remark 3.7. In Step 2, T1depends on λ. In principle they are not uniformly bounded in λ, which will have importance in Proposition 4.13 below. The following result is an immediate consequence of the Ascoli-Arzel`a theorem, and its proof is similar to [Caraballo et al., 2005; Prop.10] or [Caraballo et al., 2007b; Prop.2]. Proposition 3.8. Let be given T > 0and a continuous functional h:CT→Rnwhich is bounded (i.e. the image of a bounded set is also bounded), and such that the DDE x0(t) = h(xt)generates a semi dynamical system S.If Ssatisfies the following boundedness condition, ∀R > 0∃M(R)>0such that S(t, BCT(0, R)) ⊂BCT(0, M(R)), then Sis asymptotically compact. We can combine the above results to conclude the existence of attractors for the semi dynamical systems {S(λ)}λ∈Λ. Theorem 3.9. Assume that Hypotheses 2.1 and 2.2 and (5) hold. Then, for each λ∈Λ,(2) generates a semi dynamical system S(λ):R+×CMλ→ CMλ,which possesses a global attractor A(λ),which satisfies a uniform bound (for all λ) on the Euclidean projected space Rm: kψk2 Mλ≤K, ∀ψ∈A(λ), where the constant Kis given in Theorem 3.6. Proof. The pointwise dissipativity of each S(λ)already proved in Theorem 3.6 is fulfilled with the asymptotic compactness since the uniform boundedness condition required in Proposition 3.8 holds from Step 1 in the same theorem. Finally, the value Kof the radius of the pointwise attracting ball, that contains the attractor, is independent of λas (6) and (4) show. 4. Continuous dependence of the attractors on the parameter Our aim now is to show a result of continuous dependence for the obtained attractors A(λ) in Theorem 3.9. Recently, Li and Kloeden [Li & Kloeden, 2004a, Li & Kloeden, 2005] have developed a theory to show equivalent conditions to continuity of parametric attractors (for single valued and multi-valued frameworks respectively). However, it is necessary to adapt these concepts to our framework since the delay affects the phasespace for each λ. We will recover some results in this sense from [Kloeden & Mar´ın-Rubio, 2008]. 4.1. On the Li and Kloeden’s equiattraction theory Definition 4.1. Let {S(λ)}λ∈Λbe a family of semi dynamical systems on X. We say that (a) {S(λ)}is equi-dissipative on Xif there exists a bounded subset Uof Xso that for any bounded subset B⊂X, there exists a TB∈R+ independent of λ∈Λ such that S(λ)(t, B)⊂ U, t ≥TB; (b) {S(λ)}is eventually equi-compact (or uniformly compact for large tin [Li & Kloeden, 2004a]) if for any bounded subset Bof X, there exists a TB∈R+independent of λ∈Λ, such that ∪λ∈ΛS(λ)(t, B) is relatively compact for any t≥TB. The following theorem was proved by Li and Kloeden [Li & Kloeden, 2004a; Th.2.9]. Theorem 4.2. Suppose that {S(λ)}λ∈Λis equidissipative and eventually equi-compact and that A(λ)is the global attractor of S(λ)for λ∈Λ. In addition, suppose that the following conditions are satisfied: 6P. Mar´ın-Rubio (A1) For any t∈R+fixed, S(λ)(t, x)is jointly continuous in (x, λ)on X×Λ. (A2) S(λ)(t, x)is equi-continuous in λfor (t, x) in any bounded subset of R+×X. Then A(λ)is continuous in λwith respect to the Hausdorff distance if and only if {A(λ)}is equiattracting, i.e. for any bounded subset Bof Xand any ε > 0,there is a τ=τ(B, ε)>0independent of λsuch that H∗ X(S(λ)(t, B),A(λ))< ε for all t≥τ and for all λ∈Λ. Remark 4.3. The above equivalence also holds if (A2) is replaced by: (A2’) S(λ)(t, x) is equi-continuous in λfor tin any bounded subset of R+and xin any bounded subset of Sλ∈ΛA(λ). Finally, we recall an additional continuity notion, which will be useful for the following result (cf. [Li & Kloeden, 2004a; Th.2.7]) and below: (A3) For any bounded subset Bof Xand T > 0,S(λ)(t, x) is uniformly continuous in x∈B uniformly w.r.t. λ∈Λ and t≤T, i.e. ∀ε > 0∃δ > 0 : x, y ∈B, d(x, y)< δ ⇒d(S(λ)(t, x),S(λ)(t, y)) < ε, ∀t≤T, λ ∈Λ. Theorem 4.4. Suppose that {S(λ)}λ∈Λis equidissipative, eventually equi-compact, and the assumptions (A1) and (A3) given above hold. Then, if {A(λ)}is continuous in λ, it is also uniformly Lyapunov stable, i.e. for any ε > 0,there exists δ > 0(independent of λ) such that for all λ∈Λ,if d(x, A(λ))< δ, then d(S(λ)(t, x),A(λ))< ε for all t∈R+. 4.2. Embedding on a common state space for DDEs and equi-concepts In order to apply Li and Kloeden’s results to our parametric problem (2) and their attractors A(λ)we need to do some adaptations. We reproduce here main required results from [Kloeden & Mar´ın-Rubio, 2008]. Firstly, it is required a common state space for all the problems independently of the parameter. This means to extend the semi dynamical systems S(λ):R+×CMλ→CMλto another SDS. Theorem 4.5. [cf. [Kloeden & Mar´ın-Rubio, 2008; Th.6]] Let S(τ):R+×Cτ→Cτbe a family of semi dynamical system, with τ∈[T∗, T∗].Then, b S(τ) defines a semi dynamical system on CT∗,where b S(τ)(t, φ)(s):=x(t+s, φ), s ∈[−T∗,0],being x(t, φ):=   φ(t)t∈[−T∗,0], S(τ)(t, φ|[−τ,0])(0) t > 0, Moreover, if S(τ)is jointly continuous w.r.t. (t, φ)∈R+×Cτ,so it is b S(τ). It is clear that we can apply the above result to our family of SDS S(λ).We will denote the extended by SDS b S(λ):R+×CMλ→CMλ. Remark 4.6. When there is a DDE generating the SDS, as it is the case here, it is easy to check that b S(λ)is the SDS associated to the DDE with right hand side b F∈C(CT∗;Rm) defined as b F(b φ) = F(b φ|Mλ). The existence of attractors A(λ)for S(λ)also guaranties the existence of attractors b A(λ)for the extended SDS b S(λ)as an application of the following abstract result (cf. [Kloeden & Mar´ın-Rubio, 2008; Th.7]). Theorem 4.7. Suppose that a semi dynamical system S(τ):R+×Cτ→Cτhas a global attractor A(τ).Then, the extended semi dynamical system b S(τ)given in Theorem 4.5 possesses a global attractor, denoted b A(τ),and it has the following characterization: b A(τ):= ψ∈CT∗:∃entire trajectory ¯ Φ(τ) tof S(τ) in A(τ)with ψ(s) = ¯ φ(s)∀s∈[−T∗,0], where ¯ φ(t)is the projection in Rmof the entire solution ¯ Φ(τ) tdefined by ¯ φ(t) := ¯ Φ(τ) t(0) for all t∈ R. On the other hand, to apply Theorem 4.2, we need to know how equi-concepts are translated from the extended SDS to the original ones. The equi-dissipative notion is not difficult to relate between the original and the extended SDS, passing through the projected trajectories in Rm. Continuity of attractors for parametric DDE 7 Lemma 4.8. [cf. [Kloeden & Mar´ın-Rubio, 2008; Lem.8]] A family of SDS {b S(τ), τ ∈[T∗, T∗]}is equidissipative if and only if there exists a bounded subset Uof Rmsuch that for every bounded subset B of Rmthere exists a TB∈R+, which is independent of τ, such that S(τ)(t, B|[−τ,0])(0) ⊂U∀t≥TBand τ∈[T∗, T∗], where B:= {φ∈CT∗:φ(s)∈B∀s∈[−T∗,0]}. Remark 4.9. As a consequence of the Ascoli-Arzel`a theorem, coming from a DDE, the equi-dissipativity and the compactness of the SDS after an elapsed time T∗implies the eventual equi-compactness. So, it only rests to know how to translate the equi-attraction property. One sensible attempt to translate in an abstract situation the equiattraction property to the original SDS is the following: For every ε > 0 and bounded subset Bof CT∗ there exists Tε,B∈R+independent of τ∈[T∗, T∗] such that H∗ CτS(τ)(t, φ|[−τ,0]),A(τ)< ε for all t≥Tε,B, φ ∈ B, τ ∈[T∗, T∗].(9) The next result was proved in [Kloeden & Mar´ın-Rubio, 2008; Th.11]. Theorem 4.10. Consider S(τ):R+×Cτ→Cτ for τ∈[T∗, T∗],a family of semi dynamical systems, with attractors A(τ),equi-dissipative and equi-attracting in the sense of (9), and satisfying the following condition: (A3’) For any bounded subset Bof CT∗and T > 0, S(τ)(t, x)is uniformly continuous in x∈ B|[−τ,0], uniformly w.r.t. τand t≤T, i.e. ∀ε > 0,∃δ > 0 : x, y ∈ B,kx|[−τ,0] −y|[−τ,0]kτ< δ ⇒ kSτ(t, x|[−τ,0])−S(τ)(t, y|[−τ,0])kτ< ε, ∀t≤T, τ ∈[T∗, T∗].(10) Then b A(τ)is equi-attracting. 4.3. Equi-properties for the DDE model The following result gives the required continuity notions to apply the above results. Let us introduce for commodity the following notation: mτ λ,λ0= min{τ(λ), τ(λ0)}, Mτ λ,λ0= max{τ(λ), τ(λ0)}. Lemma 4.11. Assume Hypotheses 2.1 and 2.2 and (5) hold. Then, (a) For any ϕ∈CMλand ψ∈CMλ0the functional fgiven in (1) satisfies: |f(λ, ϕ)−f(λ0, ψ)|(11) ≤LF0|ϕ(0) −ψ(0)|+LF1|ϕ(−ρ(λ)) −ψ(−ρ(λ0))| +Lbmτ λ,λ0kϕ−ψkmτ λ,λ0 +Z−mτ λ,λ0 −Mτ λ,λ0|b(s, ξ(s))|ds, where ξis ϕor ψdepending on which is the maximum and minimum between τ(λ)and τ(λ0). (b) Properties (A1) and (A2’) hold for the semi dynamical systems b S(λ). Proof. Taking into account the definition of fin (1), one deduces |f(λ, ϕ)−f(λ0, ψ)| =F0(ϕ(0)) −F0(ψ(0)) +F1(ϕ(−ρ(λ))) −F1(ψ(−ρ(λ0))) +Z0 −τ(λ) b(s, ϕ(s))ds−Z0 −τ(λ0) b(s, ψ(s))ds ≤LF0|ϕ(0) −ψ(0)| +LF1|ϕ(−ρ(λ)) −ψ(−ρ(λ0))| +Z0 −mτ λ,λ0|b(s, ϕ(s)) −b(s, ψ(s))|ds +Z−mτ λ,λ0 −Mτ λ,λ0|b(s, ξ(s))|ds. (12) So, one obtains (11). Now, we proceed to check (A1) and (A2’). Consider two functions ϕ, ψ ∈CT∗and the respective solutions x(t) = b S(λ)(t, ϕ)(0) and y(t) = b S(λ0)(t, ψ)(0). Denote z(t) = x(t)−y(t) for t≥ −T∗.Using (12) and the uniform estimate obtained in Step 1 8P. Mar´ın-Rubio in Theorem 3.6, we know that there exists a constant C, depending on max(kϕkT∗,kψkT∗) and b such that |z(t)| ≤ |z(0)|+Zt 0nLF0|x(s)−y(s)| +LF1|x(s−ρ(λ)) −y(s−ρ(λ0))| +Lbmτ λ,λ0kxs−yskmτ λ,λ0 +C|τ(λ)−τ(λ0)|ods ≤ |z(0)|+Zt 0nLF0|x(s)−y(s)| +LF1h|x(s−ρ(λ)) −y(s−ρ(λ))| +|y(s−ρ(λ)) −y(s−ρ(λ0))|i +Lbmτ λ,λ0kxs−yskmτ λ,λ0 +C|τ(λ)−τ(λ0)|ods ≤ |z(0)| + (LF0+LF1+LbT∗)Zt 0kxs−yskT∗ds +LF1Zt 0|y(s−ρ(λ)) −y(s−ρ(λ0))|ds +C|τ(λ)−τ(λ0)|t. If we consider a fixed interval of time [0, T ], thanks to the continuity of ρ, τ, and the solution y (therefore, it is uniformly continuous on [0, T]), it is posible to take a value ε=ε(λ, λ0, ρ, τ, T, y)>0, as small as desired, such that |z(t)|≤|z(0)|+ (LF1+C)εt (13) + (LF0+LF1+LbT∗)Zt 0kzskT∗ds. Now, property (A1) is a consequence of the Gronwall lemma manipulating (13) to change |z(t)|by kz(t)kT∗. In order to obtain (A2) we have to proceed more carefully. We cannot obtain an equicontinuous bound for the term |y(s−ρ(λ)) −y(s− ρ(λ0))|if we deal now not only with a fixed solution and a unique initial data but with a bounded set of initial data. However, we observe (cf. Remark 4.3) that it is enough to prove (A2) for the bounded set composed by the union of all attractors, which is, by the uniform bound, the differential equation, and the Ascoli-Arzel`a theorem an equi-continuous set of functions. This circumvents the difficulty, and one obtains (A2’). Remark 4.12. Following the proof of (A1) above, one deduces that property (A3’) given in Theorem 4.10 (see (10)) also holds. Proposition 4.13. Under the assumptions of Lemma 4.11,{b S(λ)}λ∈Λis equi-dissipative. Proof. We can complete the uniform bound obtained in Theorem 3.9 in order to gain the equidissipativity. We have to prove that there exists a bounded set U ⊂ CT∗,such that for any closed ball B= ¯ BCT∗(0, R)⊂CT∗,there exists a time TB>0,independent of λ, such that for all t≥TB,one has b S(λ)(t, B)⊂ U for all λ∈Λ. Just consider ε > 0 arbitrary, and the value K given in Theorem 3.6. For any λ∈Λ,thanks to the existence of the attractor A(λ),and therefore, that of b A(λ),there exists a time T1(ε, λ, B) (w.l.o.g. bigger than T∗= maxΛMλ) such that b S(λ)(t, B)⊂BCT∗(0,(K+ε)1/2) ∀t≥T1(ε, λ, B).(14) We have to prove that T1(ε, ·,B),as function of λ, is bounded in Λ.Suppose the opposite, then there exist sequences {λn}and {ψ(n)} ⊂ B such that lim n→+∞T1(ε, λn,B) = +∞,and (15) ∃θn< T1(ε, λn,B) : kb S(λn)(θn, ψ(n))k2 T∗≥K+ε. By the compactness of Λ,there exist a value λand a subsequence {λn0} ⊂ {λn}such that limn0→+∞λn0=λ. We will see that (15) comes in contradiction with (14) using this particular value λ. Observe that the uniform bound (Step 1 in Theorem 3.6) of solutions for the DDE’s, the finite delay of all the involved DDE, and the fact that all f(λ, ·) are bounded maps, provide compactness: after the time T∗= maxΛMλ,the set {ϕ(n)} with ϕ(n)=b S(λn)(T∗, ψ(n)),is relatively compact. There exists a subsequence {ϕ(n0)} ⊂ {ϕ(n)}with ϕ(n0)→ϕin CT∗. Continuity of attractors for parametric DDE 9 Now we make the most of property (A1) in a fixed finite interval since we have λn0→λand ϕ(n0)→ϕ. We know by (14) that the solution with initial data ϕsatisfies kx(λ) T1k2 T∗≤K+ε0(for certain value 0 < ε0< ε). Consider the finite time interval [0, T ] with T=T1(ε, λ, B)−T∗(since we have let an elapsed time T∗to get ϕ(n0)). By (A1), the solutions with initial data ϕ(n0), x(λn0) T1stay close of x(λ) T1if n0is big enough. In other words, ∃n0(ε−ε0) : kx(λn0) T1k2 T∗≤K+ (ε+ε0)/2 ∀n0≥n0(ε−ε0). Once again, the positive invariance of any ball proved in Step 1 in Theorem 3.6 points out that this contradicts (15). The main result of this section is the following. Theorem 4.14. Assume that Hypotheses 2.1 and 2.2 and (5) hold. Then, (a) Equi-attraction of the attractors {A(λ)}in the sense of (9) is equivalent to the continuity of the attractors {b A(λ)}. (b) If {b A(λ)}is continuous in λ, then this family is also uniformly Lyapunov stable in the sense given in Theorem 4.4. Proof. Firstly, we have to check the required conditions for Theorem 4.2. These were proved in Lemma 4.11 (recall Remark 4.3 to substitute (A2) by (A2’)), Proposition 4.13, and by the uniform bound in Theorem 3.9, which provides equicompactness (cf. Remark 4.9). The second statement is a consequence of Theorem 4.4 and Theorem 4.10. Acknowledgements The author would like to thank the referees for their helpful comments that improved the presentation of the paper. This work was partially supported by MICINN (Spain) Grant MTM2008-00088 and Junta de Andaluc´ıa Grant P07-FQM-02468. References Ball, J. M. [2004] “Global attractors for damped semilinear wave equations,” Discrete Contin. Dyn. Syst. 10, 31–52. Boutet de Monvel, L., Chueshov, I. & Rezounenko, A. V. [1997] “Long-time behaviour of strong solutions of retarded nonlinear P.D.E.s.,” Comm. Partial Differential Equations 22, 1453–1474. Boutet de Monvel, L., Chueshov, I. & Rezounenko, A. V. [1998] “Inertial manifolds for retarded semilinear parabolic equations,” Nonlinear Anal. 34, 907–925. Caraballo, T., Kloeden, P. & Mar´ın-Rubio, P. [2007] “Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay,” Discrete Contin. Dyn. Syst. Ser. A 19, 177–196. Caraballo, T. & Langa, J. A. [2003] “On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,” Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10, 491–513. Caraballo, T., Mar´ın-Rubio, P. & Valero, J. [2005] “Autonomous and non-autonomous attractors for differential equations with delays,” J. Differential Equations 208, 9–41. Caraballo, T., Mar´ın-Rubio, P. & Valero, J. [2007] “Attractors for differential equations with unbounded delays,” J. Differential Equations 239, 311–342. Carvalho, A. N. & Langa, J. A. [2007] “Nonautonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds,” J. Differential Equations 233, 622–653. Carvalho, A. N., Langa, J. A., Robinson, J. & Su´arez, A. [2007] “Characterization of nonautonomous attractors of a perturbed infinitedimensional gradient system,” J. Differential Equations 236, 570–603. Chueshov, I., Scheutzow, M. & Schmalfuß, B. [2005] Continuity properties of inertial manifolds