Differentiability in Fréchet spaces and delay differential equations

In infinite-dimensional spaces there are non-equivalent notions of continuous differentiability which can be used to derive the familiar results of calculus up to the Implicit Function Theorem and beyond. For autonomous differential equations with variable delay, not necessarily bounded, the search for a state space in which solutions are unique and differentiable with respect to initial data leads to smoothness hypotheses on the vector functional f in an equation of the general form x′(t) = f (xt) ∈ Rn, with xt(s) = x(t + s) for s ≤ 0, which have implications (a) on the nature of the delay (which is hidden in f ) and (b) on the type of continuous differentiability which is present. We find the appropriate strong kind of continuous differentiability and show that there is a continuous semiflow of continuously differentiable solution operators on a Fréchet manifold, with local invariant manifolds at equilibria.


Introduction
Consider an autonomous delay differential equation with f : U → R n defined on a set of maps (−∞, 0] → R n , and the segment, or history, x t of the solution x at t defined by x t (s) = x(t + s) for all s ≤ 0. A solution on some interval [t 0 , t e ), t 0 < t e ≤ ∞, is a map x : (−∞, t e ) → R n with x t ∈ U for all t ∈ [t 0 , t e ) so that the restriction of x to [t 0 , t e ) is differentiable and Eq.(1.1) holds on this interval.Solutions on the whole real line are defined accordingly.A toy example which can be written in the form (1.1) is the equation Hans-Otto.Walther@math.uni-giessen.dewith functions h : R → R and r : R → [0, ∞).Other examples arise from pantograph equations x (t) = a x(λt) + b x(t) (1.3) with constants a ∈ C, b ∈ R and 0 < λ < 1, and from Volterra integro-differential equations x (t) = t 0 k(t, s)h(x(s))ds (1.4) with k : R n×n → R and h : R n → R n continuous [25].Eq. (1.3) is linear, and both equations (1.3) and (1.4) are non-autonomous.We shall come back to them in Section 9 below.
Building a theory of Eq. (1.1) which (a) covers examples with state-dependent delay like Eq. (1.2) and (b) results in solution operators x 0 → x t , t ≥ 0, which are continuously differentiable begins with the search for a suitable state space.For equations with bounded delay the basic steps of a solution theory were made in [20], starting from the observation that the reformulation of examples of the form (1.2) with a state-dependent delay bounded by some R > 0 as equations of the form (1.1) yields (vector-)functionals on the right hand side which are not even locally Lipschitz continuous on domains in the Banach space of continuous maps on the compact initial interval [−R, 0].In order to obtain an equation with a continuously differentiable functional, so that there is hope for continuous differentiability of solutions with respect to initial data, one must restrict to the Banach space of continuously differentiable functions on the initial interval.This, in turn, means a restriction to solutions which are continuously differentiable everywhere, and not only on the interval where they satisfy Eq. (1.1), as in the by-now well established theory of retarded functional differential equations [2,5].A further observation in [20] is that for a continuously differentiable solution, say, x : [−R, t e ) → R n , the differential equation at t = 0 becomes a compatibility condition on the continuously differentiable initial segment.
According to the preceding remarks the functional f in Eq. (1.1) above should be defined on a subset U of the vector space C 1 = C 1 ((−∞, 0], R n ) of continuously differentiable maps (−∞, 0] → R n .Linearization as in [20] suggests that in the new theory autonomous linear equations with constant delay, like for example, x (t) = −α x(t − 1) will appear, which have solutions on R with arbitrarily large exponential growth at −∞.In order not to loose such solutions we stay with the full space C 1 and work with the topology of locally uniform convergence of maps and their derivatives, which makes C 1 a Fréchet space.
In infinite dimension there a different, non-equivalent generalizations of the canonical notion of continuous differentiability for maps in Euclidean spaces, all of which can be used as a basis for calculus, see for example [19].For maps in Banach spaces, existence of the Fréchet derivatives and continuous dependence with respect to the norm topology on the Banach space of continuous linear mappings is convenient.Without norms, for maps in Fréchet spaces, often continuous differentiability in the sense of Michal [15] is chosen, which means for a continuous map f : V ⊃ U → W, V and W topological vector spaces and U ⊂ V open, that all directional derivatives exist and that the map U × V (u, v) → D f (u)v ∈ W is continuous.In [23][24][25][26], this notion of continuous differentiability was called C 1 MB -smoothness, with reference to [15] and also to work of A. Bastiani [1].In the present paper we prefer to speak of C 1 ζ -smoothness, for a reason which will become apparent below.Let us recommend Part I of Hamilton's paper [6] as an introduction into calculus based on C 1 ζ -smoothness.We return to Eq. (1.1), now for f : ζ -smooth and has the additional extension property that (e) each derivative D f (φ) : C 1 → R n , φ ∈ U, has a linear extension D e f (φ) : C → R n , and the map U × C (φ, χ) → D e f (φ)χ ∈ R n is continuous.
Here C is the Fréchet space of continuous maps (−∞, 0] → R n with the topology of locally uniform convergence.Property (e) is closely related to the earlier notion of being almost Fréchet differentiable from [14], for maps on a Banach space of continuous functions, and it is often satisfied if f comes from an example of a differential equation with state-dependent delay.
The analogue of the compatibility condition from [20] defines the set In [23] we saw that under the hypotheses just mentioned X f , if non-empty, is a C 1 ζ -submanifold of codimension n in C 1 .Notice that X f consists of the segments x t , 0 ≤ t < t e , of all continuously differentiable solutions x : (−∞, t e ) → R n on [0, t e ), 0 < t e ≤ ∞, of Eq. (1.1).It is shown in [23] that these solutions constitute a continuous semiflow (t, x 0 ) → x t on X f , with all solution operators x 0 → x t , t ≥ 0, C 1 ζ -smooth.Let us call the set X f the solution manifold associated with the map f .The motivation for the present study of Eq. (1.1) is the fact that functionals f : C 1 ⊃ U → R n which are C 1 ζ -smooth amd have property (e) are in fact continuously differentiable in a stronger sense.This is the content of Proposition 8.3 below, which guarantees that such functionals f satisfy the conditions stated in the following definition.Definition 1.1.A continuous map f : V ⊃ U → W, V and W topological vector spaces and U ⊂ V open, is said to be continuously differentiable in the sense of Fréchet if all directional derivatives exist, if each map D f (u) : V → W, u ∈ U, is linear and continuous, and if the map D f : U u → D f (u) ∈ L c (V, W) is continuous with respect to the topology β of uniform convergence on bounded sets, on the vector space L c (V, W) of continuous linear maps V → W.
We abbreviate continuous differentiability in the sense of Fréchet by speaking of C 1 βsmoothness.In case V and W are Banach spaces C 1 β -smoothness is equivalent to the familiar notion of continuous differentiability with Fréchet derivatives, see e.g.Proposition 4.2 below.In case V and W are Fréchet spaces C 1 β -smoothness is equivalent to C 1 ζ -smoothness combined with the continuity of the derivative with respect to the topology β on L c (V, W), see e.g.Corollary 3.2 below.For examples of maps C 1 → R n which are C 1 ζ -smooth but not C 1 β -smooth, see [26].
It seems that C 1 β -smoothness of maps in Fréchet spaces which are not Banach spaces has not attracted much attention, compared to C 1 ζ -smoothness and further notions of smoothness [19].For possible reasons, see [1,Chapter II,Section 3].In any case, for the study of Eq. (1.1) the notion of C 1 β -smoothness is useful -and yields, of course, slightly stronger results, compared to the theory based on C 1 ζ -smoothness in [23,24].The present paper shows how to obtain solution manifolds, solution operators, and local invariant manifolds at stationary points, all of them C 1 β -smooth, and discusses Eqs.(1.2)-(1.4)as examples.We mention in this context that we do not touch upon higher order derivatives, in light of the fact that solution manifolds are in general not better than C 1 β -smooth [13].The same holds true for infinite-dimensional invariant manifolds in the solution manifold, like the local stable and center-stable manifolds at stationary points, whereas the finite-dimensional local unstable and center manifolds at stationary points may be k times continuously differentiable, k ∈ N, under appropriate hypotheses on the map f in Eq. (1.1) [10,12].
The present paper is divided into three parts.Part I with Sections 2-8 is about C 1 ζ -and C 1 β -maps in general.Many results in Part I, notably in Sections 2-4, are known in more general settings, see e.g.[9] and the references given there.We present this material in a form which is convenient for the purpose of this paper, and include proofs for convenience.Section 2 introduces topologies on spaces of continuous linear mappings, among them the topologies β and ζ of uniform convergence on bounded sets and on compact sets, respectively.Section 3 about maps in Fréchet spaces characterizes C β -smooth and have property (e).Proposition 9.3 guarantees that the set X f = ∅ is indeed a C 1 β -submanifold of codimension n in the space C 1 .The proof is by Proposition 7.1 on transversality.Sections 10 about segment evaluation maps prepares the construction of solutions to Eq. (1.1) which start from initial data in X f (Section 11).In Section 12 these solutions constitute a continuous semiflow on X f whose solution operators are C 1 β -smooth.Sections 10-12 are analogous to parts of [23], and we only describe how to modify these parts of [23] in order to obtain the present result on the semiflow.
Part III on local invariant manifolds at equilibria is based on [24] about Eq. (1.1) with f only C 1 ζ -smooth.In Sections 13-17 below we explain how to modify constructions in [24], in order to obtain local invariant manifolds at stationary points of the semiflow which are C 1 β -smooth (and not only C 1 ζ -smooth), by means of the transversality and embedding results from Section 7.
The present approach also shows that a technical hypothesis on smoothness which was made in [24] is obsolete.Let us briefly expand on this.An important ingredient in [24] is [23,Proposition 1.2] which says for a map f : C 1 ⊃ U → R n that its smoothness has an implication on the nature of the delay: if f is C 1 ζ -smooth then it is of locally bounded delay in the sense that (lbd) for every φ ∈ U there are a neighbourhood N(φ) ⊂ U and d > 0 such that for all χ, ψ in N(φ) with Property (lbd) is used in [24] in combination with transversality in order to obtain a local stable manifold for Eq.(1.1), with its solution segments defined on (−∞, 0], from a local stable manifold for an associated equation with solution segments x t defined on a compact interval.The local stable manifold for (1.5) stems from [7,Section 3.5], where it was found under the hypothesis that the functional on the right hand side of the delay differential equation considered is -in terms of the present paper -C 1 β -smooth and has an extension property analogous to property (e) above.Without knowing Proposition 8.3 of the present paper, the smoothness properties of f d had to be assumed in [24] as property (d).
For other work on delay differential equations with states x t in Fréchet spaces see [17,18,25].
Notation, preliminaries.The closure of a subset M of a topological space is denoted by M and its interior is denoted by

Part I 2 Preliminaries about topological vector spaces
This section provides a proposition about uniform continuity and introduces the two topologies which are relevant in the sequel.Nothing is new, proofs are included in order to make Part I of the paper more self-contained.
A topological vector space T is a vector space over the real or complex field together with a topology on T which makes addition and multiplication by scalars continuous, with respect to product topologies.We follow [16] and assume in addition that singletons in topological vector spaces are closed subsets.For elementary results about topological vector spaces which below are used without proof consult [16].
Recall that a subset B of a topological vector space T over the field K = R or K = C is bounded if for every neighbourhood N of 0 ∈ T there exists a real r N ≥ 0 with B ⊂ rN for all reals r ≥ r N .The points of convergent sequences form bounded sets, compact sets are bounded.A set Continuous linear maps between topological vector spaces map bounded sets into bounded sets.
Products of topological vector spaces are always equipped with the product topology.

Proposition 2.1 ([25, Proposition 1.2]
). Suppose T is a topological space, W is a topological vector space, M is a metric space with metric d, g : Then g is uniformly continuous on {t} × K in the following sense: For every neighbourhood N of 0 in W there exist a neighbourhood T N of t in T and > 0 such that for all t ∈ T N , all t ∈ T N , all k ∈ K, and all m ∈ M with Let V, W be topological vector spaces over R or C. The vector space of continuous linear maps V → W is denoted by L c = L c (V, W).For a given family F of bounded subsets of V which is closed under finite union and contains all singletons {v} ⊂ V, v ∈ V, a topology τ = τ F on L c (V, W) is defined as follows.For a neighbourhood N of 0 in W and a set B ∈ F consider the set Every finite intersection of such sets U N j ,B j , j ∈ {1, . . ., J}, contains a set of the same kind, because we have F is closed under finite union, and finite intersections of neighbourhoods of 0 in W are neighbourhoods of 0. Then the topology τ is the set of all O ⊂ L c which have the property that for each A ∈ O there exist a neighbourhood N of 0 in W and a set B ∈ F with A + U N,B ⊂ O.It is the easy to show that indeed τ is a topology, with the sets U N,B being neighbourhoods of 0 ∈ L c (V, W).
For the introduction of the topology τ in case of Hausdorff locally convex spaces see for example [9, Section 0.1].
We call a map A from a topological space T into L c τ-continuous at a point t ∈ T if it is continuous at t with respect to the topology τ on L c .

Remark 2.2.
(i) Convergence of a sequence in L c with respect to τ is equivalent to uniform convergence on every set B ∈ F .(Proof: By definition convergence A j → A with respect to τ is equivalent to convergence A j − A → 0 with respect to τ.This means that for each neighbourhood N of 0 in W and for each set B ∈ F there exists j N,B ∈ N so that for all integers j ≥ j N,B , A j − A ∈ U N,B .Or, for all integers j ≥ j N,B and all b ∈ B, (A j − A)b ∈ N. Now the assertion becomes obvious.) (ii) If V and W are Banach spaces and if F consists of all bounded subsets of V then τ is the norm topology on L c (V, W) given by |A| = sup |v|≤1 |Av|.
(iii) In order to verify τ-continuity of a map A : T → L c , T a topological space, at some t ∈ T one has to show that, given a set B ∈ F and a neighbourhood N of 0 in W, there exists a neighbourhood N t of t in T such that for all s ∈ N t we have (A(s) − A(t))(B) ⊂ N.
In case T has countable neighbourhood bases the map A is τ-continuous at t ∈ T if and only if for any sequence T t j → t we have A(t j ) → A(t).For A(t j ) → A(t) we need that given a set B ∈ F and a neighbourhood N of 0 in W, there exists J ∈ N with (A(t j ) − A(t))(B) ⊂ N for all integers j ≥ J.
In the sequel we shall use the previous statement frequently.Proposition 2.3.Singletons {A} ⊂ L c are closed with respect to the topology τ, and L c equipped with the topology τ is a topological vector space.
Proof. 1. (On singletons) Let A ∈ L c be given.We show that L c \ {A} is open.Let S ∈ L c \ {A}.For some b ∈ V, Ab = Sb.For some neighbourhood N of 0 in W, Ab / ∈ Sb + N [16, Theorem 1.12].For all S ∈ U N,{b} we have S b ∈ N, hence (S + S )b ∈ Sb + N, and thereby, S + S = A. Hence S + U N,{b} ⊂ L c \ {A}.
2. (Continuity of addition) Assume S, T in L c and let U N,B be given, N a neighbourhood of 0 in W and B ∈ F .We have to find neighbourhoods of S and T so that addition maps their Cartesian product into S + T + U N,B .As W is a topological vector space there are neighbourhoods N T , N S of 0 in W with N T + N S ⊂ N.For every T ∈ T + U N T ,B and for every S ∈ S + U N S ,B and for every b ∈ B we get 3. (Continuity of multiplication with scalars, in case of vector spaces over C) Let c ∈ C, T ∈ L c .Let a neighbourhood N of 0 in W and a set B ∈ F be given, and consider the neighbourhood U N,B of 0 in L c .There is a neighbourhood N of 0 in W with N + N + N ⊂ N, see e. g. [16, proof of Theorem 1.10].We may assume that N is balanced [16,Theorem 1.14].As TB is bounded there exists r N ≥ 0 such that for reals r ≥ r N , TB ⊂ r N. We infer that for some > 0, (0, )TB ⊂ N. As N is balanced we obtain that for all z ∈ C with 0 < |z| < , For z = 0 we also have zTB ⊂ N, since 0 ∈ N. Because of continuity of multiplication C × W → W there are neighbourhoods U c of 0 in C and N c of 0 in W with Consider T ∈ U N c ,B and c ∈ U c .Observe For every b ∈ B we get which yields the desired continuity at (c, T).
4. The arguments in Part 3 also work for vector spaces over R, with a shorter derivation of the inclusion zTB ⊂ N for reals z ∈ (− , ).
In case that F consists of all bounded subsets of V we write β instead of τ, and in case F consists of all compact subsets of V we write ζ instead of τ.Accordingly we speak of β-continuity and of ζ-continuity.Observe It follows that for a map from a topological space into L c , β-continuity implies ζ-continuity.

smoothness in Fréchet spaces
A Fréchet space F is a locally convex topological vector space which is complete and metrizable.The topology is given by a sequence of seminorms | • | j , j ∈ N, which are separating in the sense that |v| j = 0 for all j ∈ N implies v = 0.The sets form a neighbourhood base at the origin.If the sequence of seminorms is increasing then the sets form a neighbourhood base at the origin.
Products of Fréchet spaces, closed subspaces of Fréchet spaces, and Banach spaces are Fréchet spaces.
A curve in a Fréchet space F is a continuous map c from an interval I ⊂ R of positive length into F.For such a curve and for t ∈ I the tangent vector at t ∈ I is defined as provided the limit exists.As in [6, Part I] the curve is said to be continuously differentiable if it has tangent vectors everywhere and if the map For a continuous map f : V ⊃ U → F, V and F Fréchet spaces and U ⊂ V open, and for u ∈ U and v ∈ V the directional derivative is defined by exist, and that every derivative D f (u By the continuity of D f (u), The preceding proposition in combination with the relationship between the ζ-and βtopologies yields the following.Corollary 3.2.Let V and F be Fréchet spaces, and let a map f : β -smoothness and C 1 ζsmoothness are equivalent, and we simply speak of continuously differentiable maps.For a curve c : I → F on an open interval I ⊂ R this notion of continuous differentiability coincides with the original one for curves on more general intervals.
For continuous maps f : U → F, V, W, F Fréchet spaces and U ⊂ V × W open, partial derivatives are defined in the usual way.For example, D 1 f (v, w) : V → F is given by

C 1 β -maps in Fréchet spaces
In this section V, V 1 , V 2 , F, F 1 , F 2 always denote Fréchet spaces.We begin with a few facts about ζ -maps is linear, and the chain rule holds.We have We turn to C 1 β -smoothness.Proposition 4.2.For Banach spaces V and F and U ⊂ V open a map f : V ⊃ U → F is C 1 β -smooth if and only if there exists a continuous map D f : U → L c (V, F) such that for every u ∈ U and (F) for every > 0 there exists δ > 0 with Proof.We only show that for a Now the continuity of D f at u completes the proof.
Continuous linear maps T : V → F are C 1 β -smooth because they are C 1 ζ -smooth with constant derivative, DT(u) = T for all u ∈ V. Using Corollary 3.2 (i) and continuity of addition and multiplication on L c (V, F) (with the topology β) one obtains from the properties of C 1 ζmaps that linear combinations of C 1 β -maps are C 1 β -maps, that also for C 1 β -maps differentiation is linear, and the integral formula (4.1) holds.If This follows easily from the analogous property for C 1 ζ -maps, by means of the formula for the directional derivatives of f 1 × f 2 and considering neighbourhoods of 0 in F 1 × F 2 which are products of neighbourhoods of 0 in F j , j ∈ {1, 2}.[16, proof of Theorem 1.10].By linearity, for every j ∈ N, 3. Consider the last term.By continuity of Dg( f (u)) at 0 ∈ F, there is a neighbourhood Hence, for all integers j ≥ j 2 , 4. D f (u)B is bounded.Using β-continuity of Dg at f (u) and lim j→∞ f (u j ) = f (u) we find an integer j 3 ≥ j 2 such that for all integers j ≥ j 3 we have 5. Now we use the continuity of W × F (w, h) → Dg(w)h ∈ G at ( f (u), 0).We find a neighbourhood N 3 of 0 in F and an integer j 4 ≥ j 3 such that for all integers j ≥ j 4 we have 6.The β-continuity of D f at u ∈ U yields an integer j N ≥ j 4 such that for all integers j ≥ j N we have 7. For integers j ≥ j N we obtain are linear and continuous, and the maps Proof. 1. Suppose (ii) holds.Then f is C 1 ζ -smooth, and all statements in (i) up to the last one follow from Proposition 4.1 on partial derivatives.In order to deduce the last statement in (i) for k = 1 let a sequence ((u 1j , u 2j )) ∞ j=1 in U be given which converges to some (u 1 , u 2 ) ∈ U. Let a neighbourhood N of 0 in F and a bounded set For each j ∈ N we have Now it becomes obvious how to complete the proof, using the last equation, the statement right before it, and continuity of D 1 f (u 1 , u 2 ).

Proposition 4.1 on partial derivatives applies and yields that
1 in U be given which converges to some (u 1 , u 2 ) ∈ U, as well as a bounded subset B ⊂ V 1 × V 2 and a neighbourhood N of 0 in F. We need to show and for every j ∈ N we have The β-continuity of the map D 1 f yields that the last set is contained in N for j sufficiently large.

Contractions with parameters
The proof of Theorem 5.2 below employs twice the following basic uniform contraction principle.
Proposition 5.1 (See for example [2, Appendix VI, Proposition 1.2]).Let a Hausdorff space T, a complete metric space M, and a map f : T × M → M be given.Assume that f is a uniform contraction in the sense that there exists k ∈ [0, 1) so that d( f (t, x), f (t, y)) ≤ k d(x, y) for all t ∈ T, x ∈ M, y ∈ M, and f (•, x) : T → M is continuous for each x ∈ M. Then the map g : T → M given by g(t) = f (t, g(t)) is continuous.
Notice that the derivative Γ = Dg(t) t of the map g satisfies the equation Proof of Theorem 5.2. 1.A is continuous.So Proposition 5.1 applies to the restriction of A to V × M and yields a continuous map g : Divide by |h| = h.

It follows that each map id
is β-continuous, or equivalently, continuous with respect to the usual norm-topology on L c (B, B).As inversion is continuous we see that also the map is continuous.
3. For all (t, x, t) ∈ V × O B × T and for all x, ŷ in B we have Hence Proposition 5.1 applies to the version of Eq. (5.1) with parameters (t, x, t) ∈ V × O B × T and yields a continuous map γ : or equivalently, This shows that each map γ(t, x, •), .
Consider a neighbourhood N of 0 in B and a bounded set T b ⊂ T. We have to show that for j ∈ N sufficiently large, ( γ(t j , x j ) − γ(t, x))T b ⊂ N.For all j ∈ N and all t ∈ T b we have Now it becomes obvious how to complete the proof, using

, and β-continuity of the partial derivative
due to Proposition 4.4.

Consider the continuous map
which means that the directional derivative Dg(t) t exists and equals ξ(t, t).
So let t ∈ V and t ∈ T be given.Choose a convex neighbourhood N B ⊂ O B of g(t).There exists δ > 0 such that for −δ ≤ h ≤ δ, Notice that for all h ∈ [−δ, δ] and for all θ ∈ [0, 1], With the abbreviation The first term in the last expression converges to 0 as 0 = h → 0. The map is uniformly continuous with value 0 on {0} × [0, 1].This implies that for 0 = h → 0 the last integrand converges to 0 uniformly with respect to θ ∈ [0, 1].Therefore the last integral tends to 0 as 0 = h → 0.

The Implicit Function Theorem
From Theorem 5.2 one obtains the following Implicit Function Theorem.
Theorem 6.1.Let a Fréchet space T, Banach spaces B and E, an open set U ⊂ T × B, a C 1 β -map f : U → E, and a zero (t 0 , x 0 ) ∈ U of f be given.Assume that D 2 f (t 0 , x 0 ) : B → E is bijective.
Then there are open neighbourhoods V of t 0 in T and W of x 0 in B with V × W ⊂ U and a C 1 β -map g : V → W with g(t 0 ) = x 0 and The proof follows the usual pattern, paying attention to and in particular, D 2 R(t 0 , x 0 ) = 0.The map is β-continuous.In order to solve the equation 0 = f (t, x), (t, x) ∈ N T,1 × N B , for x as a function of t, observe that this equation is equivalent to The last expression defines a map The map A is C 1 β -smooth since the linear map D 2 f (t 0 , x 0 )) −1 : E → B is continuous, due to the Open Mapping Theorem.
2. (Contraction) For all t ∈ N T,1 and for all x, x in N B , There are an open neighbourhood N T,2 ⊂ N T,1 of t 0 and δ > 0 such that for all t ∈ N T,2 and all x ∈ B with |x − and thereby and apply Theorem 5.2 to the restriction of Then g(V) ⊂ W. From g(t) = A(t, g(t)) for all t ∈ V we obtain 0 = f (t, g(t)) for these t.

Submanifolds by transversality and embedding
C 1 β -submanifolds of a Fréchet space are defined in the same way as continuously differentiable submanifolds of a Banach space.We begin with the simple facts which are instrumental in the proofs of transversality and embedding results below, and in Parts II-III.
For a subset M of a Fréchet space F the tangent cone of M at x ∈ M is the set T x M of all tangent vectors v = c (0) of continuously differentiable curves c : β -submanifold then the tangent cones of M are closed subspaces of F. For a direct sum decomposition F = G ⊕ H and a C 1 β -diffeomorphism K as before in the definition of a C 1 β -submanifold the map (DK(m)) −1 defines a topological isomorphism from G onto T m M, and K −1 defines an injective map β -submanifold of F and H a Fréchet space, is defined by the property that for all local parametrizations P as above the composition f • P is a C 1 β -map.For h as before and m ∈ M the derivative T m h : T m M → H is defined by T m h(t) = (h • c) (0), for any continuously differentiable curve c : I → F with c(0) = m, c(I) ⊂ M, c (0) = t.The map T m h is linear and continuous.
The restriction of a C 1 β -map on an open subset of F to a C 1 β -submanifold M of F, with range in a Fréchet space H, is a C 1 β -map from M into the target space.
Proposition 7.1.Let a C 1 β -map g : F ⊃ U → G and a C 1 β -submanifold M ⊂ G of finite codimension m be given.Assume that g and M are transversal at a point x ∈ g −1 (M) in the sense that In case dim G = m, M = {g(x)}, and Dg(x) surjective the assertion holds with T g(x) M = {0}.
Proof for M = {g(x)}.1.There are an open neighbourhood We may assume DK(γ) = id since otherwise we can replace K with DK(γ) −1 • K. Then DK(γ) = id maps T γ M onto itself.

By transversality and codim
The projection P : G → Q along T γ M onto Q is linear and continuous (see [16,Theorem 5.16]), and PDK(γ)Dg(x) = PDg(x) is surjective.The preimage For the The restriction Dh(x)| R is an isomorphism.

The C 1
β -map For every y ∈ x + V H + V R , y = x + z + r with z ∈ V H and r ∈ V R , we have As both spaces have the same codimension m they are equal.For every v ∈ F we have Using this we obtain The graph representation of j(W 1 ) now yields that it is a C 1 β -submanifold of F.

Part II 8 Spaces of continuous and differentiable maps
We begin with the Fréchet spaces which are used in the sequel.For n ∈ N and k ∈ N 0 and which define the topology of uniform convergence of maps and their derivatives on compact sets.Analogously we consider the space In case T = 0 we abbreviate and there is an η ∈ N with polynomial components due to the Weierstraß Approximation Theorem.
Proof.It is enough to show that each sequence (φ m ) ∞ 1 in B has a subsequence which converges with respect to the topology on C. For every j ∈ N the set of restrictions φ m | [−j,0] , m ∈ N, is pointwise bounded and equicontinuous.The Theorem of Ascoli and Arzelà yields a subsequence which is uniformly convergent on [−j, 0].Beginning with j = 1 one chooses successively subsequences (φ is a subsequence of each of the former subsequences (up to finitely many indices) and converges uniformly on every interval [−j, 0], j ∈ N, to the continuous map χ : (−∞, 0] → R n given by χ(t) = χ j (t) for −j ≤ t ≤ 0 and j ∈ N. Or, for every j ∈ N, |φ λ(m) − χ| j → 0 as m → ∞.Proof.Let B ⊂ C 1 be bounded.Then each set {φ(t) : φ ∈ B} ⊂ R n , t ≤ 0, is bounded.For every j ∈ N, c j = sup{|φ | j : φ ∈ B} < ∞, and c j is a Lipschitz constant for all restrictions φ| [−j,0] , φ ∈ B. It follows that B is equicontinuous at every t ≤ 0. Proposition 8.1 yields that the closure of B as a subset of C is compact.
Proof.Let φ ∈ U ⊂ C 1 and a neighbourhood of 0 in L c (C 1 , R n ) with respect to the topology β be given, say, a set N W,B as in Section 2, with a neighbourhood W of 0 in R n and a bounded set B ⊂ C 1 .The closure K of B with respect to the topology of C is compact, due to Proposition 8.2, and due to condition (e) the map For all ψ ∈ N and for all χ ∈ B ⊂ K this gives Remark 8.4.See [26] for examples of maps C → R n and β -smooth (and thereby must violate condition (e)).We turn to other spaces and maps which occur in the sequel.The vector space C ∞ = ∩ ∞ k=0 C k will be used without a topology on it.
The differentiation map ∂ k,T : We use various abbreviations, for S < T and d > 0 and k ∈ N 0 : It is easy to see that the linear restriction maps and the linear prolongation maps given by (P d,k φ)(s) = φ(s) for −d ≤ s ≤ 0 and are continuous, and for all d > 0 and k ∈ N 0 , In Part II we also need the closed subspaces Solutions of equations , on some interval I ⊂ R are defined as in case of Eq. (1.1):With J = [−d, 0] or J = (−∞, 0], respectively, they are continuously differentiable maps x : J + I → R n so that x t ∈ U for all t ∈ I and the differential equation holds for all t ∈ I. Observe that x t may denote a map on [−d, 0] or on (−∞, 0], depending on the context. For results on strongly continuous semigroups given by solutions of linear autonomous retarded functional differential equations x (t) = Λx t with Λ : C d → R n linear and continuous, see [2,5].

Examples, and the solution manifold
We begin with the toy example (1.2), with continuously differentiable functions h : R → R and r : R → [0, ∞) ⊂ R. For continuously differentiable functions (−∞, t e ), 0 < t e ≤ ∞, which satisfy Eq. (1.2) for 0 ≤ t < t e this delay differential equation has the form (1.1) for U = C 1 with n = 1 and f = f h,r given by In order to see that f h,r is a composition of C 1 β -maps all defined on open sets of Fréchet spaces it is convenient to introduce the odd prolongation maps P odd : C → C ∞ (with n = 1) and P odd,1 : ∞ (with n = 1) which are defined by the relations (P odd φ)(t) = φ(t) for t ≤ 0, (P odd φ)(t) = −φ(−t) + 2φ(0) for t > 0, and P odd,1 φ = P odd φ for φ ∈ C 1 .Both maps are linear and continuous.With the evaluation map ev ∞,1 : for all φ ∈ C 1 .We also need the evaluation map ev ∞ : C ∞ × R → R given by ev ∞ (φ, t) = φ(t).
Proposition 9.1.The map ev ∞ is continuous and the map ev Proof.Arguing as in the proof of [23,Proposition 2.1] one shows that ev ∞ is continuous and that ev ∞,1 is C 1 ζ -smooth, and that the partial derivatives satisfy the equations in the proposition.It remains to prove that the map In order to prove this, choose j ∈ N with |t| < j and |t k | < j for all k ∈ N. By [16, Theorem 1.37], and it becomes obvious how to complete the proof.
The map ev 1 (•, 0) : C 1 φ → φ(0) ∈ R is linear and continuous, and the evaluation ev The next result says that f h,r satisfies the hypotheses for the results on semiflows and local invariant manifolds in the subsequent sections.Corollary 9.2.For r : R → [0, ∞) ⊂ R and h : R → R continuously differentiable the map f h,r is C 1 β -smooth and has property (e).
Proof.The functions r and h are C 1 β -smooth.The map ev 1 (•, 0) is linear and continuous, hence , by the chain rule (Proposition 4.3) and by C 1 β -smoothness of maps into product spaces.Now use that ev ∞,1 is C 1 β -smooth, due to Proposition 9.1, and apply the chain rule to the composition For each φ ∈ C 1 the term on the right hand side of this equation defines a linear continuation Using that the evaluation ev and differentiation C 1 → C are continuous one finds that the map ζ -smooth and have property (e); by Proposition 8.3 they are in fact C 1 β -smooth.Let us mention that the C 1 β -smooth maps f = f h,r with property (e) from Corollary 9.2, which arise from differential equations with state-dependent delay, are in general not of this kind.In order to see this consider the case h(ξ) = ξ and, say, r(ξ) = ξ 2 , for which and assume there exists a in combination with the continuity of the evaluation ev and with the fact that Choose χ ∈ C with χ(0) = 1 which is not differentiable at t = −1.By assumption, the directional derivative exists, which leads to a contradiction to the non-differentiability of χ at −1.
The pantograph equation (1.3), namely, with constants a ∈ C, b ∈ R and 0 < λ < 1, was extensively studied in [8].For real parameters a, b and arguments t > 0 this is a nonautonomous linear equation with unbounded delay with the projections pr j onto the first and second component, respectively.The map F is C 1 βsmooth, and every continuously differentiable function x : (−∞, t e ) → R, 0 < t e ≤ ∞, which satisfies the pantograph equation for 0 ≤ t < t e also solves the nonautonomous equation The role of the odd prolongation map in the definition of F is more essential than for f h,d above.Here it helps to overcome the obstacle that on one hand F should be defined on an open set containing {0} × C 1 while on the other hand for every t < 0 the term −τ(t) > 0 is not in the domain of data φ ∈ C 1 .
The solutions of Eq. (9.1) can be obtained from the autonomous equation (1.1) with n = 2 and f : in the familiar way: If the continuously differentiable map x : (−∞, t e ) → R satisfies Eq. (9.1) for t 0 ≤ t < t e ≤ ∞ then (s, z) : (−∞, t e − t 0 ) → R 2 given by s(t) = t + t 0 and z(t) = x(t + t 0 ) satisfies the system for 0 ≤ t < t e and s(0) = t 0 .The map f is C 1 β -smooth and has the extension property (e).For the Volterra integro-differential equation (1.4), with k : R n×n → R n and h : R n → R n continuously differentiable the scenario is simpler than in both cases above where delays are discrete.In [25] it is shown that every continuous function (−∞, t e ) → R n , 0 < t e ≤ ∞, which for 0 < t < t e is differentiable and satisfies Eq. (1.4), also satisfies an equation of the form (9.1) for 0 < t < t e , with the C 1 β -map F = F k,h in Eq. (9.1) defined on the space R × C. The associated autonomous equation of the form (1.1) is given by the It follows that the restriction of f k,h to C 1 ((−∞, 0], R n+1 ) is C 1 β -smooth and has property (e), which means that the hypotheses for the theory of Eq. (1.1) in the following sections, with a semiflow on the solution manifold in C 1 ((−∞, 0], R n+1 ), are satisfied.However, in the present case we also get a nice semiflow without recourse to this theory.A result in [25] for Eq.(1.1) with a map f : C ⊃ U → R n which is C 1 β -smooth establishes a continuous semiflow on U, with all solution operators C 1 β -smooth.In the present case, with f = f k,h , the semiflow yields a process of solution operators for the nonautonomous equation (9.1), all of them defined on open subsets of C((−∞, 0], R n ) and C 1 β -smooth.The process incorporates all solutions of the Volterra integro-differential equation.
We return to the general equation (1.1) and begin with the solution manifold It follows that all derivatives Dg(u), u ∈ U, are surjective.Apply Proposition 7.1 to g and M = {0}.

Segment evaluation maps
For the construction of solutions to Eq. (1.1) we need a few facts about the segment evaluation maps All of these maps are linear in the first argument.(i) The maps E T and E 1 T are continuous.
(ii) For every φ ∈ Proof. 1.For assertions (i) and (ii), for the fact that the map and for the formulae for the partial derivatives in assertion (iii) see the proof of [23,Proposition 3.1].It remains to show that the map T × R and a neighbourhood V of 0 in C be given.We may assume for some integer l > 0 and have to show that for j sufficiently large, For every ( φ, t) ∈ C 1 T × R and for all j ∈ N, 2. As the projections from C 1 T × R onto C 1 T and onto R are continuous and linear they map the bounded set B into bounded sets, and we obtain that for some real r > 0 and for all k ∈ N, and σ 1,T,k = sup{| φ| 1,T,k ∈ R : for some t ∈ R, ( φ, t) ∈ B} < ∞.
3. Choose k ∈ N so large that for all s ∈ [−l, 0] and for all j ∈ N, Consider ( φ, t) ∈ B. For each j ∈ N we have Altogether, for every j ∈ N and for all ( φ, t) ∈ B, Using t j → t and |φ j − φ| 1,T,k → 0 as j → ∞ and the uniform continuity of φ on [T − k, T] one finds J ∈ N such that for all integers j ≥ J and for all ( φ, t) ∈ B, It follows that (DE 10 T (φ j , t j ) − DE 10 T (φ, t))B ⊂ V for all integers j ≥ J.

Solutions of the delay differential equation
In the sequel we always assume that U ⊂ C 1 is open and that f : U → R n is C 1 β -smooth and has the property (e).
Following [23] we rewrite the initial value problem as a fixed point equation: Suppose x : (−∞, T] → R n , T > 0, is a solution of Eq. (1.1) on [0, T] with x 0 = φ.Extend φ by φ(t) = φ(0) + tφ (0) to a continuously differentiable function φ : (−∞, T] → R n .Then y = x − φ satisfies y(t) = 0 for t ≤ 0, the curve (−∞, T] s → x s ∈ C 1 is continuous (use x s = E 1 T (x, s) and apply Proposition 10.1 (i)), as well as the curves (−∞, T] s → y s ∈ C 1 and (−∞, T] s → φs ∈ C 1 .For 0 ≤ t ≤ T we get 0T,0 satisfies the fixed point equation where η ∈ C 1 T is the prolongation of η given by η(t) = 0 for all t < 0. In order to find a solution of the initial value problem (11.1) one solves the fixed point equation (11.2) by means of a parametrized contraction on a subset of the Banach space C 1 0T,0 with the parameter φ ∈ U in the Fréchet space C 1 .For φ ∈ X f the associated fixed point η = η φ yields a solution x = η + φ of the initial value problem (11.1).
The application of a suitable contraction mapping theorem, namely, Theorem 5.2, requires some preparation.We begin with the substitution operator [23, Proposition 3.2] guarantees that for 0 < T < ∞ the domain dom T is open and that F T is a (Notice that in order to obtain that F T is C 1 ζ -smooth the chain rule can not be applied, due to lack of smoothness of the map E 1 T .) Proof. 1.Let φ ∈ dom T ⊂ C 1 T , > 0, and a bounded set B ⊂ C 1 T be given.Using the norm on C 0T we have to find a neighbourhood N of φ in C 1 T so that for every ψ ∈ N and for all χ ∈ B, , is β-continuous, hence uniformly β-continuous on the compact set {φ} × [0, T] (see Proposition 2.1).It follows that there is a neighbourhood N of φ in C 1 T such that for every ψ ∈ N and for all s ∈ [0, T] the difference is contained in the neighbourhood for all N U (0),B T of 0 in L c (C 1 , R n ), with U (0) = {x ∈ R n : |x| < }.Finally, we obtain for each ψ ∈ N, s ∈ [0, T], χ ∈ B, The prolongation maps P T : C 1 → C 1 T , 0 < T ≤ ∞, given by P T φ(t) = φ(t) for t ≤ 0, P T φ(t) = φ(0) + tφ (0) for 0 < t ≤ T, and given by Z T φ(t) = φ(t) for 0 ≤ t ≤ T, Z T (φ)(t) = 0 for t < 0, and the integration operators It follows that for every T > 0 the set is open.Let pr 1 and pr 2 denote the projections from C 1 × C 1 0T,0 onto the first and second factor, respectively.Define τ : R n → C 0T by τ(ξ)(t) = ξ.Both projections and τ are continuous linear maps.Using Proposition 11.1, the chain rule, and linearity of differentiation we infer that the map For the derivatives we obtain the following result.
and for 0 ≤ t ≤ T, We now restate [23, Proposition 3.4], which prepares the proof that A T with T > 0 sufficiently small defines a uniform contraction on a small ball in C 1 0T,0 .

The semiflow on the solution manifold
The uniqueness results [23, Propositions 4.5 and 5.1] remain valid, with the same proofs.As in [23, Section 5] we find maximal solutions which are solutions on [0, t φ ) and have the property that any other solution on some interval with left endpoint 0, of the same initial value problem, is a restriction of x φ .The relations In [23,Proposition 5.3] and in its proof the words continuously differentiable can everywhere be replaced by the expression C 1 β -smooth.Thus Σ f is continuous, with each domain an open subset of X f and the time-t-map [23, Proposition 5.5] and its proof remain valid.In the proof of [23, Proposition 6.1] the words continuously differentiable can everywhere be replaced by the expression C 1 β -smooth.This yields

Part III 13 Locally bounded delay
Assume as in Part II that f : and that f has property (e).It is convenient from here on to abbreviate X = X f , Ω = Ω f , and Σ = Σ f .Let a stationary point φ ∈ X of Σ be given, Σ(t, φ) = φ for all t ≥ 0. Then φ is constant.(Proof of this: The solution x of Eq. (1.1) on [0, ∞) with x 0 = φ satisfies x(t) = x t (0) = Σ(t, φ)(0) = φ(0) for all t ≥ 0. For all s < 0 we have x Choose an open neighbourhood N of φ in U and d > 0 according to property (lbd) from Section 1.We restate [24, Proposition 2.1] as follows.
According to [24, Proposition 2.2] f d has property (e).Results from [20,21] apply and show that the equation of the Banach space C 1 d .In the terminology of the present paper, the manifold X d and all solution operators Σ d (t, •), t ≥ 0, with non-empty domain are C 1 β -smooth.The proofs of [24, Propositions 2.3-2.5]remain valid without change.We restate the result as follows.
(ii) For all χ ∈ TφX and for all t ≥ 0, From [7, Sections 3.5 and 4.1-4.3]and from [11] we get local stable, center, and unstable manifolds of

Spectral decomposition of the tangent space
Let Y = TφX.In this section we recall from [24, Section 3] the definitions of the linear stable, center, and unstable spaces of the operators T t : Y → Y, t ≥ 0.
The linear stable space in Y is defined by is linear, and continuous (as its domain is finite-dimensional). at φd , and that it has the following properties (I) and (II), for some β > 0 chosen with z < −β < 0 for all z with z < 0 in the spectrum of the generator of the semigroup on C d .

The local unstable manifold
In this section all segments x t are defined on (−∞, 0].Fix some a > 0 and consider the Banach spaces B a ⊂ C and B 1 a ⊂ C 1 introduced in Section 1.It is easy to see that the linear inclusion maps j 0 : B a → C and j 1 : B 1 a → C 1 are continuous, as well as the restriction and prolongation maps is continuous.Now results from [22] show that X a = {φ ∈ U a : φ (0) = f a (φ)} is a C 1 βsubmanifold of B 1 a , that the solutions of Eq. ( 16.1) define a continuous semiflow Σ a : Ω a → X a on X a , and that there is a local unstable manifold W u a ⊂ X a at the stationary point φ ∈ W u a .W u a is a C 1 β -submanifold of B 1 a consisting of data φ ∈ X a which are solutions of Eq. (16.1) on (−∞, 0] with φ s → φ as s → −∞, and there exist β > γ > 0 and c u > 0 so that (I) |φ s − φ| a,1 ≤ c u e βs |φ − φ| a,1 for all φ ∈ W u a and s ≤ 0, and (II) for every solution ψ ∈ B 1 a of Eq. (16.1) on (−∞, 0] with sup s≤0 |ψ s − φ| a,1 e − γs < ∞ there exists s ψ ≤ 0 with ψ s ∈ W u a for all s ≤ s ψ . In [22] the tangent space TφW u a of W u a at φ is obtained as the vector space T of all maps χ : (−∞, 0] → R n with χ0 = χ ∈ C d,u which for some t > 0 and for all integers j < 0 satisfy χjt = Λ −j χ where Λ : C d,u → C d,u is the isomorphism whose inverse is given by T d,e,t .We have (ii) For every ψ ∈ X which is a solution of Eq.

Local center manifolds
In this section we assume {0} = Y c which is equivalent to {0} = C d,c .
In the sequel we recall the steps which in [24, Section 6] led to a local center manifold at φ which is C There is a decomposition  and the norm given by the preceding supremum.There exists η 1 > 0 so that for every φ ∈ C d,c there is a unique continuously differentiable map x [φ] : R → R n

1 ζ
-smoothness in terms of the topology ζ (therefore name and symbol) and compares it to C 1 β -smoothness.Section 4 provides elements of calculus for C 1 β -maps, including the chain rule.In order to keep Section 4 short we make extensive use of [6, Part I] on C 1 ζ -smoothness.Sections 5-6 deal with parametrized contractions and with the Implicit Function Theorem based on C 1 β -smoothness.. Section 7 contains simple transversality-and embedding results which yield C 1 β -submanifolds of finite dimension or finite codimension.The content of Sections 5-7 is familiar in the Banach space case, and most of it is well-known also in the C 1 ζ -setting [3, 4, 23, 24].Part II with Sections 8-12 is about well-posedness of the initial value problem associated with Eq. (1.1).Section 8 introduces the Fréchet and Banach spaces of continuous and differentiable maps from intervals into Euclidean spaces which will be used in the sequel.We mentioned already Proposition 8.3 which establishes that functionals on the space C 1 with property (e) which are C 1 ζ -smooth also are C 1 β -smooth.Section 9 verifies that for examples of the form (1.2)-(1.4) the associated functionals f on the right hand side of Eq. (1.1) are C 1 ζsmooth and have property (e) -so they are C 1

Recall the notions of C 1 ζProposition 3 . 1 .
-and C 1 β -smoothness from Section 1.It is easy to see that continuously differentiable curves c : I → F on open intervals I ⊂ R are C 1 ζ -smooth and vice versa.Next the notion of C 1 ζ -smoothness from Section 1 is expressed in terms of ζ-continuity.Let a continuous map f : V ⊃ U → W, V and W Fréchet spaces and U ⊂ V open, be given.Suppose that all directional derivatives D These involve the Riemann integral for continuous maps [a, b] → F into a Fréchet space and results from calculus based on C 1 ζ -smoothness which can be found in [6, Sections I.1-I.4].Each derivative D f (u) : V → F, u ∈ U, is linear and continuous.Differentiation of C 1

Theorem 5 . 2 .
Let a Fréchet space T, a Banach space B, open sets V ⊂ T and O B ⊂ B, and a C 1 β -map A : V × O B → B be given.Assume that for a closed set M ⊂ O B we have A(V × M) ⊂ M, and A is a uniform contraction in the sense that there exists k ∈ [0, 1) so that

3 .
(Invariance) By continuity there is an open neighbourhood N T,3 ⊂ N T,2 of t 0 such that |A(t, x 0 ) − A(t 0 , x 0 )| < δ 4 for all t ∈ N T,3 .For all t ∈ N T,3 and x ∈ B with |x − x 0 | ≤ δ this yields

Proof. 1 .
The topology induced by F on the finite-dimensional subspace Y = Dj(b)V of F is given by a norm[16, Section 1.19], and Y has a closed complementary space Z ⊂ F, see[16, Lemma 4.21].The projection P : F → F along Z onto Y is linear and continuous ([16, Theorem 5.16]).The map P • j is C 1 β -smooth and defines a C 1 β -map W → Y.Its derivative at b is an isomorphism V → Y (use Py = y on Y and the injectivity of Dj(b)).The Inverse Mapping Theorem (for maps between finite-dimensional normed spaces) yields a C 1 β -map g : Y ∩ U → V, U open in F and P(j(b)) ∈ Y ∩ U, such that g(P(j(b))) = b, and an open neighbourhood

Proposition 8 . 2 .
The inclusion map C 1 → C maps bounded sets into sets with compact closure.
is linear and continuous.We abbreviate ∂ T = ∂ 1,T and ∂ = ∂ 0 = ∂ 1,0 .The following Banach spaces occur in Parts II and III: For n ∈ N and k ∈ N 0 and reals a < b , C k ([a, b], R n ) denotes the Banach space of k-times continuously differentiable maps [a, b] → R n with the norm given by |φ| [a,b],k = k ∑ κ=0 max a≤t≤b |φ (κ) (t)|.
0)}.In Part III we make use of the Banach space B a , for a > 0 given, of all φ ∈ C with sup t≤0 |φ(t)|e at < ∞, |φ| a = sup t≤0 |φ(t)|e at , and of the Banach space B 1 a of all φ ∈ C 1 with

(
I) There are γ > β, an open neighbourhood Ws d of φd in W s d , and c > 0 such that [0, ∞) × Ws d ⊂ Ω d , and Σ d ([0, ∞) × Ws d ) ⊂ W s d , and for all ψ ∈ Ws d and all t ≥ 0, |Σ d (t, ψ) − φd | d,1 ≤ c e −γt |ψ − φd | d,1 .(II)There exists a constant c > 0 such that each ψ ∈ X d with [0, ∞) × {ψ} ⊂ Ω d ande βt |Σ d (t, ψ) − φd | d,1 < c for all t ≥ 0 belongs to W s d .The codimension of W s d in C 1 d is equal to n + dim Y d,c + dim Y d,u = n + dim C d,c + dim C d,u .As the continuous linear map R d,1 : C 1 → C 1 d is surjective we can apply Proposition 7.1 and obtain an open neighbourhood V of φ in N ⊂ U ⊂ C 1 so that

Proposition 16 . 1 .
TφW u a = Y u .because the maps in the vector space Y u = IC d,u share the property defining the space T and dimY u = dim C d,u = dim T.From a manifold chart for W u a at φ we obtain > 0 and a C 1 β -mapw u a : Y u ( ) → B 1 a , Y u ( ) = {φ ∈ Y u : |φ| a,1 < }, with w u a (0) = φ, w u a (Y u ( ))an open subset of W u a , and Dw u a (0)η = η for all η ∈ Y u .Proposition 7.2 applies to the C 1 β -map j 1 • w u a .So we may assume thatW u = W u ( φ) = j 1 w u a (Y u ( )) is a C 1 β -submanifold of the Fréchet space C 1 with TφW u = j 1 Dw u a (0)Y u = Y u .The proof of [24, Proposition 5.2] remains valid in the present setting.We state the result about the properties of the local unstable manifold W u as follows.(i)Every φ ∈ W u is a solution of Eq. (1.1) on (−∞, 0], with φ s → φ as s → −∞, and for all s ≤ 0, |φ(s) − φ(0)| ≤ c u e βs |φ − φ| a,1 and |φ (s)| ≤ c u e βs |φ − φ| a,1 .

1 .
s , into closed subspaces which defines a projection P 1 d,c :C 1 d → C 1 d onto C d,c, and there is a norm• d,1 on C 1 d which is equivalent to | • | d,1 and whose restriction to C d,c \ {0} is C ∞ -smooth.Next there exists ∆ > 0 with N ∆ = {φ ∈ C 1 d : φ d,1 < ∆} contained in V d so that the restricted remainder map N ∆ φ → g d (φ) − Dg d (0)φ ∈ R n has a global continuation r d,∆ : C 1 d → R n with Lipschitz constant λ = sup φ =ψ r d,∆ (φ) − r d,∆ (ψ) d,1 φ − ψ d,1 <The desired local center manifold at φ ∈ X will be given, up to translation, by segments (−∞, 0] → R n of solutions on R of the equationx (t) = Dg d (0)x t + r d,∆ (x t ) (with segments in C 1 d ) (17.1)which do not grow too much at ±∞.

For η > 0 let C 1 d
,η denote the Banach space of all continuous maps u : R → C 1 d with sup t∈R e −η|t| |u(t)| d,1 < ∞ For every k ∈ K there exist open neighbourhoods T(k) of t in T and δ(k) > 0 such that for all t ∈ T(k) and all m ∈ M Let t ∈ T N , t ∈ T N , k ∈ K, and m ∈ M with d(m, k) < , (t , k) ∈ U, ( t, m) ∈ U be given.For some j ∈ {1, . . ., n}, d(k, k j ) < bounded.Proof: Consider a seminorm | • | 1,j , j ∈ N. Choose an integer k ≥ j + T. The seminorm | • | 1,T,k is bounded on B. For every χ ∈ B and for all s ∈ [0, T] we see from |E 1 T (χ, s)| 1,j = max For every ψ ∈ dom T , χ ∈ B, s ∈ [0, T] we have given by I T φ(t) = 1 Σ(t, φ).(iv) If (t, χ) ∈ Ω d and if x : (−∞, t] → R n given by x(s) = x χ (s) on [−d, t] and by x(s) = (P d,1 χ)(s) for s < −d satisfies {x s : 0 ≤ s ≤ t} ⊂ N then (t, P d,1 χ) ∈ Ω and R d,1 Σ(t, P d,1 χ) = Σ d (t, χ).Proposition 13.2 (iii) shows that φd is a stationary point of the semiflow Σ d .For t ≥ 0 consider the operators T t = D 2 Σ(t, φ) on TφX and T d,t = D 2 Σ d (t, φd ) on Tφ d X d .The proof of [24, Corollary remains valid.We state the result as follows.For (t, φ) ∈ Ω as in Proposition 13.2 (iii) and for all χ s with the linear stable space Y d,s of the strongly continuous semigroup (T d,t ) t≥0 on the tangent space Y d = Tφ d X d ⊂ C 1 d .We have Y d,s = Y d ∩ C d,s with the linear stable space C d,s of the strongly continuous semigroup of solution operators T d,e,t : C d → C d , t ≥ 0, which is defined by the equation v (t) = D e f d ( φd )v t .(14.1) Let C d,c and C d,u denote the finite-dimensional linear center and unstable spaces of the semigroup on C d .Each χ ∈ C d,c ⊕ C d,u uniquely defines an analytic solution v = v χ on R of Eq. (14.1).The injective map The center and unstable spaces in Y are defined as Y c = IC d,c and Y u = IC d,u , respectively.They are finite-dimensional and the maps T t , t ≥ 0, act as isomorphisms on each of them.The stable space Y s is closed and positively invariant under each map T t , t ≥ 0, and we have the decomposition Y= Y s ⊕ Y c ⊕ Y u .Finally, recall the Banach space B 1 a from Section 8 and observeY u ⊂ B 1 a since each v χ , χ ∈ C d,u, and its derivative both have limit 0 at −∞.We begin with the local stable manifold W s d ⊂ X d of the semiflow Σ d at the stationary point φd ∈ X d ⊂ C 1 d as it was obtained in [7].Recall Proposition 4.2.It is easy to see that W s d is a C 1 β -smooth submanifold of the Banach space C 1 d which is locally positively invariant under S d , with tangent space Tφ d W s d = Y d,s