DYNAMICAL PROPERTIES OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

. A type of nonautonomous n -dimensional state-dependent delay diﬀerential equation (SDDE) is studied. The evolution law is supposed to satisfy standard conditions ensuring that it can be imbedded, via the Bebutov hull construction, in a new map which determines a family of SDDEs when it is evaluated along the orbits of a ﬂow on a compact metric space. Additional conditions on the initial equation, inherited by those of the family, ensure the existence and uniqueness of the maximal solution of each initial value problem. The solutions give rise to a skew-product semiﬂow which may be noncontinuous, but which satisﬁes strong continuity properties. In addition, the solutions of the variational equation associated to the SDDE determine the Fr´echet diﬀerential with respect to the initial state of the orbits of the semiﬂow at the compatibility points. These results are key points to start using topological tools in the analysis of the long-term behavior of the solution of this type of nonautonomous SDDEs.

1. Introduction. Functional differential equations of state-dependent delay type (SDDEs for short) have been object of active analysis during the last years, due in part to the high theoretical interest of this study, but mainly to the increasing number of models of applied sciences which respond to this pattern: see e.g. Hartung [6], Wu [19], Hartung et al. [7], Mallet-Paret and Nussbaum [13], Barbarossa and Walther [1], He and de la Llave [8], and Krisztin and Rezounenko [12], as well as the many references therein.
In this setting, the regularity properties required on the vector field to guarantee existence, uniqueness, and continuous variation of solutions of initial value problems are much more exigent than in the case of fixed delay or even time-dependent delay differential equations. Especially complex is the nonautonomous case: due to the time-dependence, the solutions do not generate a semiflow on the state space, and more sophisticated tools must be designed in order to use the methods of the topological dynamics in the analysis of the dynamical properties of the solutions. A detailed description of some of these methods can be found in Hale [3] and Sell and You [16]. To establish the bases for the use of these tools in the analysis of nonautonomous SDDEs is the global purpose of this paper.

ISMAEL MAROTO, CARMEN NÚÑEZ AND RAFAEL OBAYA
Let C and W 1,∞ respectively represent the spaces of continuous and Lipschitzcontinuous n-dimensional real functions on [−r, 0]. Hartung analyzes in [5,6] the nonautonomous n-dimensional SDDĖ y(t) = f (t, y(t), y(t− τ (t, y t ))) , t ≥ 0 (1.1) (where y t (s) := y(t + s) for s ∈ [−r, 0]), and the associated initial value problems, given by y 0 = x for x ∈ W 1,∞ . He establishes regularity conditions on the vector field f : R × R n × R n → R n and on the delay τ : R × C → [0, r] guaranteeing the local existence and uniqueness of the solutions y(t, x) of the initial value problem, as well as the fact that the map [−r, 0] → R n , s → y(t + s, x) belongs to W 1,∞ for those values of (t, x) for which it is defined.
We have already mentioned that our global purpose is to describe a scenario on which the methods coming from the topological dynamics can be applied in the analysis of the long-term behavior of the solutions of (1.1). Let us describe our approach. Standard conditions on the temporal variation of the map (f, τ ) : R × R n × R n × C → R n × [0, r] , (t, y 1 , y 2 , v) → (f (t, y 1 , y 2 ), τ (t, v)) (which are satisfied in the uniformly almost-periodic case, but also in much more general situations), ensure that its hull Ω (which is defined as the closure in the compact open topology of the set of time-translated functions (f, τ ) t (s, y 1 , y 2 , v) := (f, τ )(t + s, y 1 , y 2 , v) for t varying in R) is a compact metric space. Its elements are functions ω = (ω 1 , ω 2 ) : R × R n × R n × C → R n × [0, r], (t, y 1 , y 2 , v) → (ω 1 (t, y 1 , y 2 ), ω 2 (t, v)) .
In addition, the map R×Ω → Ω , (t, (ω 1 , ω 2 )) → (ω 1 , ω 2 )·t given by time-translation (i.e., ((ω 1 , ω 2 )·t)(s, y 1 , y 2 , v) = (ω 1 (t + s, y 1 , y 2 ), ω 2 (t + s, v))) defines a continuous flow on Ω. These conditions also ensure that the maps F (ω, y 1 , y 2 ) = ω 1 (0, y 1 , y 2 ) and τ (ω, x) = ω 2 (0, x) for ω = (ω 1 , ω 2 ) are continuous operators: see Hino et al. [9]. This procedure (designed by Bebutov around 1940) takes us to consider the family of nonautonomous SDDEṡ y(t) = F (ω·t, y(t), y(t−τ (ω·t, y t ))) , t ≥ 0 , (1.2) for ω ∈ Ω. Note that the initial equation is included in this one: just take ω = (f, τ ) (which in particular has a dense orbit in Ω). In addition, it turns out that any of the equations of the family satisfies the hypotheses assumed on the initial one. The great advantage of having this family of equations is that its solutions will allow us to define a semiflow of skew-product type on a suitable product space with base Ω. As a matter of fact, we will take a family of the type (1.2) as starting point, without assuming that it comes from the single equation (1.1): Ω will simply be a compact metric space supporting a continuous flow, without further recurrence property (as the existence of a dense orbit on it). In this way, our framework is more general. The conditions that we will impose on F and τ are intended to ensure that each one of the equations of the family satisfied those of [5].
Our first purpose, carried out in Section 3, is to establish a global version of the fundamental Hartung's result: we will show the existence and uniqueness of a maximal solution y(t, ω, x) of the equation (1.2) corresponding to ω with y 0 = x ∈ W 1,∞ , which is defined on a right-open interval [−r, β ω,x ) with 0 < β ω,x ≤ ∞. We will also prove that β ω,x = ∞ if y(t, ω, x) is norm-bounded. As before, it turns out that the map u(t, ω, x) defined by u(t, ω, x)(s) := y(t + s, ω, x) for s ∈ [−r, 0] belongs to W 1,∞ whenever it is defined. We will show that Π : R + × Ω × W 1,∞ → Ω × W 1,∞ , (t, ω, x) → (ω·t, u(t, ω, x)), which is locally defined, determines what we call a pseudo-continuous semiflow: a possibly noncontinuous semiflow but with strong continuity properties, as the continuity of Π : for any fixed time t; and of the section Π ω,x : R + → Ω × W 1,∞ for any (ω, x) satisfying the compatibility condition "x ∈ C 1 ([−r, 0], R n ) andẋ(0 − ) = F (ω, x(0), x(−τ (ω, x)))". For further purposes we represent by C 0 the set of points (ω, x) satisfying this compatibility property. These results can be easily extended to the continuous dependence with respect to parameters. A consequence of the previous properties is that the restriction of Π to any compact Π-invariant set K ⊂ Ω × W 1,∞ defines a global continuous semiflow. Section 3 also describes the Lipschitz variation of the solutions of a particular equation with respect to the initial data.
Our second purpose, carried out in Section 4, concerns the existence and regularity properties of the Fréchet differential of the solutions with respect to the state variable x. We begin by analyzing the properties of the family of (linear) variational equationsż Note that the equation is nonautonomous, linear, and just time-dependent. We begin by analyzing the continuity properties of the maps These properties are one of the key points required to prove that, if z(t, ω, x, v) represents the solution of (1.3) agreeing with v ∈ W 1,∞ in [−r, 0], and w(t, ω, x, v)(s) := z(t + s, ω, x, v) for s ∈ [−r, 0], then the map (t, ω, x, v) → (Π(t, ω, x), w(t, ω, x, v)) defines a new pseudo-continuous semiflow on K × W 1,∞ (linear in this case), where K is any compact Π-invariant subset of Ω × W 1,∞ . The importance of this result relies on the fact that w(t, ω, x, v) = u x (t, ω, x)v; that is, that w(t, ω, x, ·) represents the differential (in the Fréchet sense, as a matter of fact) with respect to the state variable of the Π-semiorbit corresponding to a compatibility point. This last equality concerning the map u x (t, ω, x) : W 1,∞ → W 1,∞ is proved for the local solution by Hartung in [5]. For the sake of completeness we include in Section 4 some steps of the proof adapted to our setting, since they are relevant to understand the regularity properties of the pseudo-continuous semiflow generated by u x . These properties mean that u x (t, ω, x) has full dynamical sense, as we will explain in the next paragraph. An in-deep analysis of some additional regularity properties of u x (t, ω, x) completes the section, and the paper.
In order to see that these results give indeed form to a scenario in which the topological dynamical methods can be applied in the analysis of nonautonomous SDDEs, we mention some of their consequences. The restriction of the pseudo-continuous semiflow Π to any positively invariant compact set K ⊂ Ω × W 1,∞ determines a continuous semiflow on K. If, in addition, the points of K satisfy the compatibility condition previously mentioned, the solutions of the family of linearized equations determines the usually so-called linearized semiflow of Π along the semiorbits of K, namely Π L : These results, also included on Sections 3 and 4, will be the starting point for the analysis of long-term dynamics of the orbits of K, for which we can make use of: the properties of the linear pseudo-continuous semiflow Π L ; and the properties of the continuous discrete semiflows given by the iteration of the continuous map Π L t for any t > 0. These and other questions are developed in [14]. In turn, all these results, combined with techniques of monotone systems (also new in the case of SDDEs) can be applied in the description of applied models, as that of a biological neural network: see [15].
We close this introduction by remarking that some authors consider different formulations providing different properties of regularity. Let us mention some of them. Walther studies in [17,18] autonomous SDDEs defined by a continuously differentiable vector field F : } endowed with the structure of C 1 -manifold. Hartung proves in [6] the existence of the linearized map u x (t, x) : W 1,∞ → C for every x ∈ Ω × W 1,∞ and every v ∈ W 1,∞ when (d/dt)(t − τ (t, u(t, x))) > ρ > 0 for every t. If this inequality is globally satisfied, then the map U → C, (t, x) → u(t, x) is differentiable with respect to the initial data in the complete domain U of F . A similar approach is used by Chen et al. in [2], where the state-dependent delay is supposed to satisfy an ordinary differential equation given by a vector field which is bounded above by a constant ρ * < 1. Properties of regularity of the semiflow are used by Hu and Wu in [10] and by Hu et al. in [11] in order to investigate the Hopf-bifurcation of one-parametric families of SDDEs as well as the global continuation of the periodic solutions. And He and de la Llave use in [8] the parameterization method in order to construct quasi-periodic solutions of quasi-periodic SDDEs, which are defined as the ε-perturbation of an hyperbolic family of ordinary differential equations.
Property (f2) above (with Ω × X instead of Ω) means that u satisfies the cocycle property u(t + l, ω, x) = u(t, ω·l, u(l, ω, x)) whenever the right-hand function is defined. A global skew-product semiflow Π is linear if it takes the form is a bounded linear operator on X; in particular, u(t, ω, x) is linear in x for each (t, ω) ∈ R + × Ω. We end this short section by fixing some notation. Given two Banach spaces (X, · X ) and (Y, · Y ), we represent by Lin(X, Y ) the set of bounded linear maps where | · | is the Euclidean norm in R n . The subset C 1 ⊂ C is given by the functions which have continuous derivative on [−r, 0] (one-sided derivatives at the end points). The set L ∞ is the space of Lebesgue-measurable functions ψ : [−r, 0] → R n which are essentially bounded ; i.e., for which there exists k ≥ 0 such that the set { x ∈ [−r, 0] | |ψ(x)| > k } has zero measure. The norm on L ∞ , which is denoted by · L ∞ , is defined as the inferior of the set of real numbers k ≥ 0 with the previous property. The set W 1,∞ is the Banach space of Lipschitz-continuous functions ψ : 3. State-dependent delay differential equations. Let (Ω, σ, R) be a continuous flow on a compact metric space, and let us represent ω·t = σ(t, ω). Given two maps F : Ω × R n × R n → R n and τ : Ω × C → [0, r], we consider the family of nonautonomous SDDEṡ for ω ∈ Ω. The derivative at t = 0 is the right-hand derivative. It has been explained in the Introduction the way in which one of this families may arise from one of its equations, via the hull procedure. We have also mentioned that if this is the case, at least one of the elements ω ∈ Ω has a dense orbit. But recall that we are not assuming this fact here: we work in the most general case. The conditions on F and τ which we will assume are H1 F : Ω × R n × R n → R n is continuous, and its partial derivatives w.r.t. its second and third arguments exist and are continuous on Ω × R n × R n ; and H2 τ : Ω × C → [0, r] is continuous and differentiable w.r.t. its second argument, and the map D 2 τ : Ω × C → Lin(C, R) is continuous.
Remarks 3.1. 1. Note that H2 ensures the next property: τ is locally Lipschitzcontinuous in the sense that, for every compact subset In order to check this assertion, we take a compact subset K ⊂ Ω × C and note that the setK : whenever (ω, x 1 ) and (ω, x 2 ) belong to K, as asserted.
2. Having in mind the previous remark, it is easy to check that each one of the equations of the family satisfies the conditions A1 and A2 (adapted to our setting) assumed by Hartung in [5]. Therefore, all his local results may be applied. Theorem 3.3 summarizes the dynamical properties of the solutions of the family (3.1). A key role is played by the set of pairs "(equation, initial datum)" which satisfy the compatibility condition, Let us further define C 0 ⊂ Ω × W 1,∞ by (3.2) and and provide U, U, C 0 and U 0 with the respective subspace topologies. Then, (iv) the set U is open in R + × Ω × W 1,∞ and Π satisfies conditions (f1) and (f2) of Section 2 (wherever it makes sense, and with Ω replaced by Ω × W 1,∞ ). (iii) Assume that sup t∈[0,β) u(t, ω, x) C =: c 0 < ∞ for a point (ω, x) ∈ Ω×W 1,∞ and, for contradiction, that β := β ω,x < ∞. We will prove that y(t, ω, x) exists on [−r, β] and satisfies (3.1) on [0, β]: as indicated in (i), this contradicts the definition of β. Recall that the derivatives at the edge points are one-sided.
(iv)&(v) We will first prove these properties under the assumption that F is bounded, which we will remove later.
(vii) In the case that t ≥ r, property (vii) follows from (vi), and if t = 0 the assertion is trivial. So that assume that t ∈ (0, r). Let us take a sequence ((ω m , x m )) in U t with limit ( ω, x) ∈ U t . Let us fix ε > 0 and δ ∈ (0, ε]. We call y m (t) := y(t, ω m , x m ) and y(t) := y(t, ω, x) for t ∈ [−r, t ], and , u m (t) := u(t, ω m , x m ) and u(t) := u(t, ω, x) for t ∈ [0, t ]. According to (v), u( t ) − u m ( t ) C is as small as desired if m is large enough. Therefore, there exists m 0 such that, if m ≥ m 0 , then On the other hand, for t ∈ [0, t ]. It follows easily from the continuity of F and τ guaranteed by H1 and H2 and from (3.6) (which is valid for a δ which can be prefixed from the properties of F and τ ) that |˙ y(t) −ẏ m (t)| ≤ ε/2 for all t ∈ [0, t ] if m is large enough, and clearly the same happens with ˙ x−˙ x m L ∞ . This ensures that ˙ u( t)−u m ( t) L ∞ ≤ ε for large enough m, which together with (3.6) proves the result.
(viii) Let us take a sequence ((t m , ω m , x m )) in U 0 with limit ( t, ω, x) ∈ U 0 and define t 0 , S, y m , y, u m and u as at the beginning of the proof of (vi). Note that and that we already know, by (vii) and (v), that lim m→∞ u( t) − u m ( t) W 1,∞ = 0 and lim m→∞ u m ( t) − u m (t m ) C = 0. Hence, our goal is to prove that It is very easy to check that this property follows from the equicontinuity of the family {ẏ m | m ∈ N} on [−r, t 0 ], which we will prove. It is also easy to deduce from the fact that lim m→∞ẋm =˙ x in C that the family {ẏ m | m ∈ N} is equicontinuous on [−r, 0]. On the other hand, given ε > 0, we conclude by repeating step by step the argument used in the proof of (vi) that there exists δ > 0 such that, if s 1 , s 2 ∈ [0, t 0 ] and |s 1 − s 2 | ≤ δ, then |ẏ m (s 1 ) −ẏ m (s 2 )| ≤ ε all m ∈ N. This means that the family {ẏ m | m ∈ N} is equicontinuous on [0, t 0 ], and hence on [−r, t 0 ], which completes the proof of (viii).
Therefore y * solves the equation (3.1) on [r, 2r]. Consequently, (ẏ k ) converges tȯ y * uniformly on [r, 2r]. Altogether, we have checked the sequence (u(t k + 2r, ω, x)) converges to y * 2r in W 1,∞ , which completes the proof of (ix) and of the theorem. Proof. Note that R + × K ⊂ U and Π(t, ω, x) ∈ K for all (t, ω, x) ∈ R + × K, so that the restriction Π : R + × K → K is well defined and globally defined. And it is easy to check that the topologies induced by · C and · W 1,∞ on K are the same, so that the continuity follows from Theorem 3.3(v).
Remark 3.5. We can repeat the arguments of the proofs of points (v), (vi) and (vii) of Theorem 3.3 in order to prove analogous results on the joint continuity with respect to (t, ω, x, λ) for the solutions of the family of equationsẏ(t) = F (ω·t, y(t), y(t − τ (ω·t, y t , λ), λ) when λ belongs to a Banach space and F and τ satisfy the corresponding jointly continuity properties included in H1 and H2. The details are left to the reader, whom is referred to [5] for a more exhaustive analysis of the regularity properties with respect to parameters of the solution of SDDEs.
We complete this section by analyzing the Lipschitz behaviour of the map u defined in the statement of Theorem 3.3 with respect to the initial condition x. This result is a global version, adapted to our setting, of the local property given by Theorem 1(iv) of [5]. Recall that d Ω represents the distance in Ω.

4.
Differentiability with respect to the initial state. Throughout this section, we assume that H1 and H2 hold, and use the notation established in the previous one. Recall that the compatibility set C 0 and the closely related set U 0 are defined by (ω, x)))} , (4.1) and that we provide them with the topologies induced by those of Ω × W 1,∞ and R + × Ω × W 1,∞ , respectively. It is very easy to deduce from the definition (3.3) of the semiflow Π that Π(U 0 ) = C 0 , which is a fundamental property for what follows. Let us consider the family of (linear) variational equationṡ for (ω, x) ∈ C 0 and v ∈ C. Note that each equation of the family (4.3) is evaluated along one of the positive Π-semiorbits lying on C 0 , and that it is not state-dependent, but just time-dependent. This section presents an analysis of the solutions of this family of delay equations, in the line of that made in Section 3 for the family (3.1). Its importance will be clarified by the properties stated in Corollary 4.3 and Theorem 4.4.
All the results of this section depend on the continuity properties of the maps and C 0 × C → R n , (ω, x, v) → L(ω, x)v , which we analyze in the next proposition. (ii) The map U 0 → Lin(W 1,∞ , R n ), (t, ω, x) → L(Π(t, ω, x)) is continuous.
(iii) Let us fix (ω, x) ∈ C 0 . The map C → R n , v → L(ω, x)v is a bounded linear operator. In addition, for each k > 0, Proof. Recall that H1 and H2 ensure the continuity of τ : Ω × C → R and the existence and continuity of D i F : Ω × R n × R n → Lin(R n , R n ) for i = 2, 3 and of D 2 τ : Ω × C → Lin(C, R).
(i) Let us take a sequence ((ω m , x m )) in C 0 with limit ( ω, x) ∈ C 0 . We will check that L( ω, x)v = lim m→∞ L(ω m , x m )v by proving this property for each one of the terms appearing in the expression of L. So, we write L(ω, x) = L 1 (ω, x)+L 2 (ω, x)+ L 3 (ω, x). We take ε > 0 and v ∈ W 1,∞ with v W 1,∞ = 1, and call τ := τ ( ω, x) and τ m := τ (ω m , x m ). For |L 1 ( ω, x) − L 1 (ω m , x m )|, since |v(0)| ≤ 1, we have so that the continuity of τ and D 2 F shows that it is smaller than ε if m is large enough. Therefore, The norm |L 2 ( ω, x)v − L 2 (ω m , x m )v| is bounded by the sum of two: which is in the same situation as the previous term; and 4) which is smaller than ε for large enough m due to the continuity of τ on C 0 ⊂ Ω×C.
(Incidentally: note that the proof would fail at this point if Lin(W 1,∞ , R n ) were replaced by Lin(C, R n ) as codomain of L.) Altogether, The term L 3 (ω, x) has in turn two factors. For the second one, we have and we can use the continuity of D 2 τ to bound it by ε for large enough m. Finally, and both terms can be easily bounded by ε if m is large enough. It follows easily that L 3 ( ω, x) = lim m→∞ L 3 (ω m , x m ) in Lin(W 1,∞ , R n ). Altogether, we have checked that lim m→∞ L( ω, x) − L(ω m , x m ) Lin(W 1,∞ ,R n ) = 0 , so that (i) is proved.
(iii) The first assertion of (iii) is an easy consequence of the continuity of D 2 F , D 3 F and D 2 τ . The second also follows easily from hypotheses H1 and H2.
(iv) We take a sequence ((ω m , x m , v m )) of points of C 0 × C with limit ( ω, x, v) in U 0 × C, and repeat step by step the proof of (i) (no matter the fact that v m and v belongs to C instead of W 1,∞ ): note that the sequence ( v m C ) is bounded, and that no uniformity in v m is required. The only slightly different point is the analogous of (4.4), which is simpler in the current situation.
The next results (Theorem 4.2 and Corollary 4.3) constitute the analogues of Theorems 3.3 and 3.6 for the family of variational equations (4.3). In particular, we show that this family induces a pseudo-continuous semiflow on K × W 1,∞ , where K is any positively Π-invariant compact subset on C 0 . We will also prove that this It follows easily that w(t, ω, x, v) C ≤ v W 1,∞ + t 0 c w(l, ω, x, v) C , so that the Gronwall lemma ensures that w(t, ω, x, v) C ≤ e c t v W 1,∞ . To finish the proof of (i) is now easy: see for instance the end of the proof of Theorem 3.6(ii). Corollary 4.3. Suppose that conditions H1 and H2 hold. Let K ⊂ C 0 be a positively Π-invariant compact set. We define by (4.5) the function w(t, ω, x, v) for t ∈ R + , (ω, x) ∈ K, and v ∈ C . Then (i) the map is a continuous linear skew-product semiflow with base (K, Π, R + ). (ii) The map satisfies properties (f1) and (f2) with Ω replaced by K × W 1,∞ (for all t ≥ 0 and all l ≥ 0 in the case of (f2)). In addition, The map V 0 K → K × W 1,∞ , t → (Π(t, ω, x), w(t, ω, x, v)) is continuous. Proof. Corollary 3.4 shows that (K, Π, R + ) is a global continuous semiflow. Having this in mind, all the assertions are trivial consequences of Theorem 4.2.
As we anticipated, our next result, Theorem 4.4, will show that, as a matter of fact, Π L : R + × K × W 1,∞ → K × W 1,∞ is the linearized semiflow of Π, in the sense that each one of its semiorbits determine the differential with respect to the state variable of the semiorbits of Π. The first assertion in the theorem is proved (in a slightly different setting) in [5], Theorem 4. For the sake of completeness we give here part of the details of the proof, since they help the reader to understand the dynamical meaning of the function u x (t, ω, x).
Note that the uniformity of the limit (4.8) with respect to the elements of the unit ball means that u x (t, ω, x) is the classical Fréchet differential with respect to the initial state of the function u(t, ω, x), which provides it with full dynamical meaning.
The sets C 0 and U 0 appearing in the next statement are given by (4.1) and (4.2).
Consequently, the map (t, ω, x, v) → u x (t, ω, x)v satisfies all the continuity properties described in Theorem 4.2.