Attraction property of local center-unstable manifolds for differential equations with state-dependent delay

In the present paper we consider local center-unstable manifolds at a stationary point for a class of functional differential equations of the form ẋ(t) = f (xt) under assumptions that are designed for application to differential equations with state-dependent delay. Here, we show an attraction property of these manifolds. More precisely, we prove that, after fixing some local center-unstable manifold Wcu of ẋ(t) = f (xt) at some stationary point φ, each solution of ẋ(t) = f (xt) which exists and remains sufficiently close to φ for all t ≥ 0 and which does not belong to Wcu converges exponentially for t→ ∞ to a solution on Wcu.


Introduction
In the last decade the theory of differential equations with state-dependent delay made significant progress.Apart from other results, the framework developed by Walther in [7,8,9] had a remarkable impact.This series of works is concerned with a class of abstract functional differential equations and contains a proof that under certain mild conditions the solutions of the associated Cauchy problems define a continuous semiflow on a smooth submanifold of a function space.In particular, the resulting semiflow has continuously differentiable solution operators and the linearization of the semiflow along a solution is described by linear variational equations.The vital point of that framework with respect to delay differential equations is the fact that it seems to be typically applicable in cases where the functional differential equation represents an autonomous differential equation with state-dependent delay.Consequently, under the assumption of applicability, one obtains a general setting of smooth dynamical systems for the study of differential equations with state-dependent delay.

E. Stumpf
Nowadays, the semiflow mentioned above is analyzed in various articles and many of its dynamical aspects are well-understood.For instance, a general survey of basic properties together with the linearization process at stationary points as well as the principle of linearized stability is presented in [1].In addition, [1] contains a proof of the existence of local stable and local center manifolds at stationary points.The counterpart of the principle of linearized stability, that is, the principle of linearized instability is discussed in [4].For the existence of continuously differentiable local unstable manifolds at stationary points we refer the reader to [2].The construction of C 1 -smooth local center-unstable manifolds is carried out in [5], whereas the authors of [3] show the existence and smoothness of local center-stable manifolds.
In the present article we address another feature from the dynamical systems theory of semiflows as laid out in the framework [7,8,9]; namely, an attraction property of local centerunstable manifolds obtained in [5].We show that each solution which starts and stays close enough to a stationary point converges exponentially for t → ∞ to a solution on a local center-unstable manifold of the semiflow.In particular, this property provides an asymptotic description of the dynamics of such solutions: for all sufficiently large t they behave like solutions on the considered local center-unstable manifold.However, in order to formulate our main result in detail we have to recall some relevant material.This is done below without presenting proofs.For a deeper discussion of the theory and for the absent proofs we refer the reader to [1,7,8,9].
Throughout this paper, let h > 0, n ∈ N and let • R n denote a norm in R n .Further, we write C for the Banach space of all continuous functions from the interval [−h, 0] into R n , provided with the usual norm ϕ C := sup s∈[−h,0] ϕ(s) R n of uniform convergence.Similarly, let C 1 denote the Banach space of all continuously differentiable functions ϕ : [−h, 0] → R n with the norm ϕ C 1 := ϕ C + ϕ C .Given some function x : I → R n defined on some interval I ⊂ R, and some real t ∈ R with [t − h, t] ⊂ I, the segment x t of x at t is defined by x t (s) := x(t + s), −h ≤ s ≤ 0.
From now on, we consider the functional differential equation given by some function f : U → R n defined on some open neighborhood U ⊂ C 1 of the origin 0 ∈ C 1 and satisfying f (0) = 0.A solution of Eq. (1.1) is either a continuously differentiable function x : [t 0 − h, t e ) → R n with t 0 < t e ≤ ∞ such that x t ∈ U for all t 0 ≤ t < t e and Eq.(1.1) holds for all t 0 < t < t e , or a continuously differentiable function x : R → R n satisfying x t ∈ U and Eq.(1.1) for all t ∈ R, or a continuously differentiable function x : (−∞, t r ] → R n , t r ∈ R, such that x t ∈ U for all t ≤ t r and Eq.(1.1) holds as t < t r .As f (0) = 0 by assumption, it is clear that x(t) = 0, t ∈ R, is a solution of Eq. (1.1) in the sense above.In particular, the subset of C 1 is not empty.We impose that the function f additionally satisfies the following conditions: (S1) f is continuously differentiable, and (S2) for each ϕ ∈ U the derivative D f (ϕ) : C 1 → R n extends to a linear map D e f (ϕ) : C → R n such that the map U × C (ϕ, χ) → D e f (ϕ)χ ∈ R n is continuous.
Then the results of the framework [7,8,9] show that the subset X f of U is a C 1 -smooth submanifold of codimension n.Moreover, for each ϕ ∈ X f there is a unique real t + (ϕ) > 0 and a unique solution x ϕ : [−h, t + (ϕ)) → R n of Eq. (1.1) such that x ϕ 0 = ϕ and x ϕ is not continuable in the forward time direction.For all ϕ ∈ X f and all 0 ≤ t < t + (ϕ) the segments x ϕ t belong to X f , which is therefore called the solution manifold of Eq. (1.1).By assigning F(t, ϕ) := x ϕ t for all (t, ϕ) ∈ Ω where we obtain a continuous semiflow F : Ω → X f with continuously differentiable time-t-maps.
Since x(t) = 0, t ∈ R, is a solution of Eq. (1.1), it is clear that ϕ 0 := 0 ∈ U is a stationary point of the semiflow F such that F(t, 0) = 0 for all t ≥ 0. The linearization of F at ϕ 0 = 0 is the strongly continuous semigroup T = {T(t)} t≥0 of bounded linear operators T(t) := D 2 F(t, 0) on the Banach space The action of an operator T(t), t ≥ 0, on χ ∈ T 0 X f is given by T(t)χ = v χ t , where v χ : [−h, ∞) → R n is the uniquely determined solution of the variational equation v(t) = D f (0)v t with initial value v 0 = χ.The infinitesimal generator G of the strongly continuous semigroup T is given by the linear operator of the space C 2 of all twice continuously differentiable functions from [−h, 0] into R n .The semigroup T is closely related to another strongly continuous semigroup.In order to clarify this point, recall that, due to assumption (S2) on f , the operator D f (0) may be extended to a bounded linear operator D e f (0) : C → R n on C. In particular, the operator L e := D f e (0) defines the linear retarded functional differential equation The solutions of the associated initial value problems

E. Stumpf
We have D(G e ) = T 0 X f and T(t)ϕ = T e (t)ϕ for all t ≥ 0 and all ϕ ∈ D(G e ).
The relation between the semigroups T, T e notably has an effect on the spectra σ(G), σ(G e ) of the two generators G, G e : they coincide as shown in [1].The spectrum σ(G e ) ⊂ C of the generator G e of T e is given by the zeros of a familiar characteristic equation.In particular, it is discrete and contains only eigenvalues of finite rank, that is, all generalized eigenspaces are finite-dimensional.Moreover, for each β ∈ R the intersection σ(G e ) ∩ {λ ∈ C | Re λ > β} is finite.Therefore, the spectral parts σ c (G e ) := {λ ∈ σ(G e ) | Re λ = 0} and σ u (G e ) := {λ ∈ σ(G e ) | Re λ > 0} of σ(G e ) are empty or finite.The associated realified generalized eigenspaces C c and C u are called the center and unstable space of G e , respectively, and each of them is a finite dimensional subspace of C. In contrast, the stable space C s of G e , that is, the realified generalized eigenspace associated with the spectral part is an infinite-dimensional subspace of C. All the spaces C u , C c , and C s are closed and invariant under T e (t), t ≥ 0, and provide the decomposition of the Banach space C. The semigroup T e may be extended to a one-parameter group on each of the two finite dimensional spaces C u , C c , and the decomposition of C also leads to a decomposition of the smaller Banach space C 1 : with the closed subspace With respect to the semigroup T and its generator G, it turns out that both C u and C s belong to D(G e ) = T 0 X f and coincide with the unstable and center space of G, respectively.The stable space of G is given by the intersection C s ∩ T 0 X f and we get the decomposition of the Banach space T 0 X f .All the spaces C u , C c , and C s ∩ T 0 X f are closed in T 0 X f and invariant under the action of the semigroup T. In addition, T is extendable to a one-parameter group on both C u and C c .
After the preparatory steps, we are now in the position to recall the main result from [5] about the existence of local center-unstable manifolds for the semiflow F at the stationary point ϕ 0 = 0.In doing so, we write C cu for the so-called center-unstable space C c ⊕ C u of G.
Then there exist open neighborhoods C cu,0 of 0 in C cu and C 1 s,0 of 0 in C 1 s with N cu := C cu,0 + C 1 s,0 contained in U, and a continuously differentiable map w cu : C cu,0 → C 1 s,0 with w cu (0) = 0 and Dw cu (0) = 0 such that has the following properties.
(i) W cu is a continuously differentiable submanifold of the solution manifold X f of Eq. (1.1) and dim W cu = dim C cu .
(ii) If x : (−∞, 0] → R n is a solution Eq. (1.1) and if x t ∈ N cu for all t ≤ 0, then x t ∈ W cu as t ≤ 0.
(iii) W cu is positively invariant with respect to F relative to N cu ; that is, for all ϕ ∈ W cu and all t > 0 with {F(s, The goal of this paper is to prove the following additional attraction property of local center-unstable manifolds. Theorem 1.2.Under the assumptions of Theorem 1.1, there exists an open neighborhood U A of 0 in U, and reals K A > 0 and η A > 0 with the following property: If ϕ ∈ U A and if the solution x ϕ of Eq. (1.1) does exist for all t ≥ 0 and its segments x ϕ t belongs to U A as long as t ≥ 0, then there is some ψ ∈ X f with x In the next sections, we establish this statement.The main idea of the proof is to consider the global center-unstable manifolds of some smooth modifications of Eq. (1.1) and to show an attraction property for these manifolds -compare Theorem 4.1 -first.This is done constructively by adopting the ideas contained in Vanderbauwhede [6], where a similar result for ordinary differential equations is given.An essential ingredient of the method is to deduce certain integral equations and then to solve these equations by the contraction principle on suitable Banach spaces.Having the attraction property of the global center-unstable manifolds, the main result easily follows by a cut-off technique.
This paper is organized in detail as follows.The next section contains some preliminaries.There, we recall the variation-of-constants formula and some integral operators for inhomogeneous linear functional differential equations.Further, we introduce some smooth modifications of Eq. (1.1) and describe the construction of global center-unstable manifolds.
The third section is devoted to the study of some global semiflows of the modified equations.Apart from the modifications introduced in the second section, in this section we consider further auxiliary modifications of (1.1).
Section 4 begins with a statement about an attraction property of global center-unstable manifolds.Thereafter, we develop step by step a strategy for a proof of this statement.It turns out that the claimed attraction property may be characterized in an alternative way, which notably involves global solutions of certain parameter-dependent integral equations.
In Section 5 we prepare the last arguments for a proof of the attraction of global centerunstable manifolds: we construct parameter-dependent contractions on Banach spaces to solve the parameter-dependent integral equations obtained in Section 4. In addition, we show that the resulting fixed points depend continuously on the parameter.At the end of Section 5, we finally give a proof of the statement claimed at the beginning of Section 4.
The last section contains the proof of our main result.

Preliminaries
In this section we recapitulate some standard facts on delay differential equations and discuss some basics results needed for a proof of the main statement.

Sun-reflexifity
For each t ≥ 0, let (2.2) From the decomposition (1.4) of C 1 , we also get continuous projection operators P u , P c , and P s of C 1 onto subspaces C u , C c , and C 1 s , respectively.By the identification of C and C it easily follows that Finally, in analogy to (2.2), for the action of T on subspaces of T 0 X f we also have (2.3)

Variation-of-constants formula
Next, we are going to recall the variation-of-constants formula for solutions of the inhomogeneous linear differential equation ẋ(t) = L e x t + q(t) (2.4) with a function q : I → R n defined on some interval I ⊂ R. Here, a solution is a continuous function x : I + [−h, 0] → R n satisfying (2.4) for all t ∈ I.In order to state the variationof-constants formula and its properties, we need some preparations and notation.To begin with, let L ∞ ([−h, 0], R n ) denote the Banach space of all measurable and essentially bounded functions from [−h, 0] into R n , equipped with the usual norm • L ∞ of essential least upper bound.Then the product space is isometrically isomorphic to the Banach space C * .After fixing a norm-preserving isomorphism k : Then it follows that the weak-star-integral lies in C. We have We return to Eq. (2.4).If q : I → R n is continuous and if x : I + [−h, 0] → R n is a solution of Eq. (2.4) then the curve u : I τ → x τ ∈ C satisfies the abstract integral equation with Q : I τ → l(q(τ)) ∈ Y * for all s, t ∈ I, s ≤ t.Conversely, if Q : I → Y * is continuous and if u : I → C is a solution of Eq. (2.5) then there exists a continuous x : I + [−h, 0] → R n such that x t = u(t) for all t ∈ I and that x is a solution of Eq. (2.4) on I for q : I τ → l −1 (Q(τ)) ∈ R n .In this way we have a one-to-one correspondence between solutions of Eq. (2.4) and Eq.(2.5).

Preparatory results on inhomogeneous linear equations
Let X denote a Banach space and • X its norm.Then for each η ≥ 0 the linear spaces provided with the weighted supremum norms respectively, become Banach spaces as well.Some of these spaces we will consider repeatedly in the sequel.In order to simplify notation, we shall use the following abbreviations throughout the paper: Moreover, we write P cu := P c + P u for the projection of C 1 along C 1 s and P * for all t ≤ 0, and and for all t ∈ R.
(i) Eq. (2.6) defines a bounded linear map K : as −∞ < s ≤ t < 0, and it is the only one in C 0 η with the property P * cu u(0) = 0.
(ii) Eq. (2.7) defines a bounded linear map K1 : Proof.For the first part of the statement we refer the reader to Proposition 3.2 in [5] where the proof is carried out in full detail.Following these lines, one easily concludes parts (ii) and (iii) of the statement.
Remark 2.3.Under the assumption on η from the last proposition, it is not difficult to see that Moreover, given Q ∈ Y η , u := K η Q is the only solution of Eq. (2.9) in C 1 η with the property P * cu u(0) = 0.

E. Stumpf
(ii) Eq. (2.7) induces a bounded linear mapping Moreover, given Q ∈ Y η , all segments of the solution K 2 η Q of Eq. (2.11) belong to C cu .
Proof.For the proof of the first assertion compare Corollary 3.4 and its proof in [5], whereas the proofs of assertions (ii) and (iii) immediately follows from Proposition 2.2 in combination with Proposition 4.2.1 in Hartung et al. [1].
Remark 2.5.An important ingredient of the proof of the last statement is a smoothing property of the integral equation (2.5).For example, if ).For a proof, compare for instance the proof of Proposition 4.2.1 in Hartung et al. [1].
, where the operators K 1 η and K 2 η are defined as in the last proposition.Then K + η forms a bounded linear operator with Proof.In view of Proposition 2.4, it is clear that the sum K + η of the two bounded linear operators K as well.Furthermore, from the estimates for K 1 η and for K 2 η we get For the remaining part of the assertion, consider for all −∞ < s ≤ t < ∞, which proves the corollary.

Smooth modifications of the nonlinearity and the global center-unstable manifolds of the modified equations
Below we recapitulate some essential ingredients of the proof of Theorem 1.1.In particular, we describe the construction of global center-unstable manifolds for smooth modifications of Eq. (1.1).For the details we refer the reader to [5].Compare also the construction of local center manifolds contained in Hartung et al. [1].
Introducing the maps we may rewrite Eq. (1.1) as ẋ(t) = Lx t + r(x t ) where the right-hand side is separated into a linear and a nonlinear term.It is easily seen that r inherits conditions (S1) and (S2) from f .In particular, we have r(0) = 0 and Dr(0) = 0.In view of dim C cu < ∞, there exists a norm • cu on C cu such that its restriction to In this way we obtain another norm and ρ(t) = 0 for all t ≥ 2, and set For each δ > 0 we introduce by a modification of r that is defined on all of C 1 .Here, ϕ cu denotes the component P cu ϕ of ϕ ∈ C 1 , and analogously ϕ s the component For all sufficiently small δ > 0 the restriction of r δ to a small neighborhood of 0 ∈ C 1 is continuously differentiable, bounded, and has a bounded derivative.More precisely, there exists some is a bounded and continuously differentiable function with a bounded derivative.Furthermore, for all ϕ ∈ {ψ ∈ C 1 | ψ s 1 < δ} we have Next, there is some 0 < δ 1 < δ 0 and a monotone increasing λ : [0, The proof for the existence part of Theorem 1.1 begins with the construction of global center-unstable manifolds for the modified equations where 0 < δ ≤ δ 1 .Recall that these equations are closely related with the integral equations For instance, if x : (−∞, 0] → R n is a continuously differentiable solution of Eq. (2.14), then we obtain a solution of Eq. (2.15) by u : There clearly exists some 0 < δ < δ 1 ensuring With this choice of δ, let us temporarily denote by R : ) and all t ≤ 0. Then R maps C 1 η into Y η and thus induces a mapping which particularly satisfies Attraction property of local center-unstable manifolds η by (S η ϕ)(t) := T e (t)ϕ as ϕ ∈ C cu and t ≤ 0. It easily follows that S η forms a bounded linear operator with (2.17) Using the mappings K η from Proposition 2.4, R δη and S η , we introduce another map Under the given assumptions, for each η has an uniquely determined fixed point u(ϕ) since it forms a contraction of a sufficiently large ball about the origin into itself.The associated solution operator ũη : is globally Lipschitz-continuous, and for each ϕ ∈ C cu the function ũη (ϕ) is a solution of Eq. (2.15) on (−∞, 0] with vanishing C cu component at t = 0. Thus, in view of the one-to-one correspondence of solutions of Eq. (2.14) and Eq. ( 2.15), we see that for each ϕ ∈ C cu there exists a continuously differentiable function x : (−∞, 0] → R n with x t = ũ(ϕ)(t) as t ≤ 0 such that x solves Eq. (2.14) for all t ≤ 0. The global center-unstable manifold of Eq. ( 2.14) at the stationary point 0 ∈ C 1 is now the set We have and for each solution v ∈ C 1 η of Eq. ( 2.15) we have v(t) ∈ W η as t ≤ 0.

Global semiflows of modified equations
The first step towards a proof of our main result Theorem 1.2 will be a similar statement for the modified equations (2.14) and the associated global center-unstable manifolds.As this statement will assert an asymptotic behaviour for t → ∞ of certain solutions of Eq. (2.14), we need some preparations containing, among other things, a discussion about the existence of continuously differentiable solutions for t ≥ 0. This is done below.
To begin with, observe that Eq. (2.14) may be written as with the function f δ : C 1 ϕ → Lϕ + r δ (ϕ).By construction, the set is clearly not empty since we have f δ (0) = f (0) = 0. Nevertheless, f δ does not need to have the properties (S1) and (S2).For this reason, we can not use the results in Walther [7,8,9] in order to conclude the existence of solutions of the initial values problems for t ≥ 0. However, this issue was already addressed by Qesmi and Walther in [3] where the authors prove that for all sufficiently small δ > 0 the initial value problems have uniquely determined solutions.More precisely, the following holds: Proposition 3.1.Let 0 < δ < δ 1 with λ(δ) < 1/5 be given.Then for each ϕ ∈ X δ there exists a unique continuously differentiable solution x : [−h, ∞) → R n of the initial value problem (3.2), and x t ∈ X δ for all t ≥ 0.Moreover, the equations Proof.Compare Corollary 6.2 and Proposition 6.3 in [3].Now, recall once more the one-to-one correspondence between solutions of the differential equation defining the initial value problem (3.2) and solutions of the integral equation (2.15).Fixing some appropriate δ > 0 and any ϕ ∈ X δ and setting u(t) := F δ (t, ϕ) for all t ≥ 0, we first see and then after application of P cu that is, for all 0 ≤ s ≤ t < ∞.
Apart from the global semiflows F δ generated by solutions of (3.2), we will need other auxiliary semiflows.In order to define these semiflows, we first have to study solutions of the modification ẋ of Eq. (3.1) for t ≤ 0 where ϕ ∈ X δ .We show that, for each η ∈ R with c c < η < min{−c s , c u } and all sufficiently small δ > 0, every ϕ ∈ X δ uniquely determines a continuously differentiable solution x : (−∞, 0] → R n of Eq. (3.5) with The proof of this statement is based on a fixed-point argument completely similar to the one used for the construction of the global center-unstable manifolds W η .However, for the convenience of the reader, we carry out the details below.
For the remaining part of this section fix some η ∈ R with We begin with a minor modification of Corollary 4.3 in [5].

E. Stumpf
Next, we consider a slight modification of the bounded linear operator S η used for the construction of W η .Corollary 3.3.For each ϕ ∈ C 1 the curve (−∞, 0] t → T e (t)P cu ϕ ∈ C 1 belongs to C 1 η , and the mapping S η : C 1 → C 1 η defined by (S η ϕ)(t) = T e (t)P cu ϕ for all ϕ ∈ C 1 and all t ≤ 0 is a bounded linear operator with S η ≤ K P cu ( P * c Proof.First, by Corollary 4.5 in [5] it follows that for every ϕ ∈ C 1 the continuous curve (−∞, 0] t → T e (t)P cu ϕ ∈ C 1 is an element of the Banach space C 1 η .Moreover, we have S η = S η • P cu .Hence, as a composition of two bounded linear operators, S η is a bounded linear operator as well, and the estimate (2.17) finally leads to For given ϕ ∈ C 1 consider the constant map Clearly, we have C(ϕ) ∈ C 1 η .Using Corollary 3.4 in [5] and the last two corollaries, we obtain a well-defined map with the bounded linear operator K η : η has exactly one fixed point u = u(ϕ) and the associated solution operator Proof.We mimic the proof of Proposition 4.6 in [5].
We conclude that there is a unique Consequently, û has a global Lipschitz constant as claimed.
For every ϕ ∈ C 1 the fixed point ûη (ϕ) ∈ C 1 η of u = G η (u, ϕ) from the last result is a special solution of the abstract integral equation associated with Eq. (3.5) by the variation-of-constants formula.More precisely, the following holds.Corollary 3.5.For all ϕ ∈ C 1 the mapping ûη (ϕ) from the last proposition is a solution of the abstract integral equation In particular, ûη (ϕ)(0) = P cu ϕ and ûη (ϕ)(t) ∈ C cu for all t ≤ 0.

E. Stumpf
Proof.Following the proof of Corollary 4.7 in [5], consider for given ϕ ∈ C 1 the map By Corollary 3.4 in [5], we conclude that for all −∞ < s ≤ t ≤ 0. In addition, we get P cu ( ûη (ϕ)(0)) = P cu z(0) + P cu ((S η ϕ)(0)) = 0 + P cu T e (0)ϕ = P cu ϕ. ( This shows the assertion. From the one-to-one correspondence between solutions of the abstract integral equation (3.8) and solutions of the differential equation (3.5) we conclude that for each ϕ ∈ C 1 there is a continuously differentiable function x : (−∞, 0] → R n satisfying Eq. (3.5) for all t ≤ 0 and having the properties Moreover, all the segments of x are contained in the center-unstable space C cu , and in view of the uniqueness result from Proposition 3.4 there is no other solution y : (−∞, 0] → R n of Eq. (3.5) having the two properties η (compare also the details in the proof of the next proposition).
Using the solution operator ûη , we define a map Below we prove that F cu η defines a continuous dynamical system on C 1 .
Proposition 3.6.The map F cu η : (−∞, 0] × C 1 → C 1 forms a continuous semiflow.More precisely, F cu η is continuous and satisfies holds.In addition, we find δ2 > 0 such that for each ψ ∈ C 1 with ϕ − ψ C 1 < δ2 we have where Lip( ûη ) is a global Lipschitz constant of ûη due to Proposition 3.4.Set δ := min{ δ1 , δ2 } and consider arbitrary (s, This proves the continuity of F cu η at (t, ϕ). 2. (Proof of the algebraic properties of a semiflow.)To begin with, observe that from the definition of F cu η and the last result it immediately follows that Therefore, the only thing remaining to prove is the additive property of F cu η .For this purpose, let t, ŝ ∈ (−∞, 0] and ϕ ∈ C 1 be given.We have for all t ≤ 0. Accordingly to the last two results, v, w ∈ C 1 η and v(t), w(t) ∈ C cu as t ≤ 0.Moreover, in view of we have v(0) = w(0) and both v and w satisfy Thus, z ∈ C 1 η .In addition, for all s ≤ t ≤ 0 Combining this fact together with R δη (v + C(ϕ)) ∈ Y η due to Corollary 3.2 and [5].Hence, it follows that for all t ≤ 0. In the Banach space C 1 η the last equation reads Therefore, Proposition 3.4 implies In particular, it follows that ûη (ϕ which completes the proof.

An attraction property of the global center-unstable manifolds of the modified equations: the statement and the main idea of the proof
After the preparations in the last sections, we are now in the position to state an attraction property of the global center-unstable manifolds W η of the modified equations (2.14). ) and Then there exist a continuous map H η cu : X δ → W η such that for all (ψ, ϕ) ∈ W η × X δ the following holds: if and only if ψ = H η cu (ϕ).
From now on and until the end of the next section, we suppose that the assumptions of this theorem are satisfied.For a proof we adopt the ideas of Vanderbauwhede [6], where the assertion for the case of ordinary differential equations is discussed.The initial point of this strategy is an alternative characterization of property (4.4).Lemma 4.2.Suppose that F : R × X δ → C 1 is continuous and satisfies (a) F(t, ϕ) = F δ (t, ϕ) for all t ≥ 0 and all ϕ ∈ X δ , and Let ϕ, ψ ∈ X δ be given.Then the following statements are equivalent: (ii) There exists some z as −∞ < s ≤ t < ∞ and ψ = ϕ + z(0).
Consider now z : R → C 1 given by z(t) := v(t) − F(t, ϕ).In view of property (b) and the above, it follows that sup t≤0 e ηt z(t Moreover, combining (a) and (i) we see
2. Suppose now that (ii) holds.Then, in consideration of the one-to-one correspondence between solutions of Eq. (2.14) and of Eq. (2.15) and in consideration of the uniqueness result contained in Proposition 3.1, we have F δ (t, ψ) = F(t, ϕ) + z(t) as t ≥ 0. Using property (a) and the fact z ∈ C 1 η,R , we also infer Hence, it remains to prove ψ ∈ W η .For this purpose, observe that z As, in addition, v is a solution of Eq. (4.5) for all −∞ < s ≤ t ≤ 0, Proposition 4.8 in [5] shows v(0) = ψ ∈ W η and the assertion follows.
In view of the assumptions of the last result, it becomes clear that we need some continuous map F : R × X δ → C 1 with properties (a) and (b) in order to be able to use property (ii) for a proof of Theorem 4.1.Below we construct such a map.The key ingredients here are the global semiflows F δ and F cu η discussed in the last section.Indeed, defining for all ϕ ∈ X δ and all −∞ < s ≤ t < ∞.
Proof. 1. Recall that, by Proposition 3.1, F δ is continuous on [0, ∞) × X δ , and that, by Proposition 3.6, F cu η is continuous on (−∞, 0] × X δ .As for all ϕ ∈ X δ we have it is obvious that F is continuous on all of R × X δ .Moreover, in consideration of the definition, it is also clear that F has property (a) from Lemma 4.2.Next, observe that for each ϕ ∈ X δ we have η due to Proposition 3.4 and the introduction in front of it.For this reason, F( • , ϕ) ∈ C 1 η , which shows that F also has property (b) from Lemma 4.2.
2. It remains to prove that Eq. (4.7) holds.In order to show this, consider ϕ ∈ X δ and −∞ < s ≤ t < ∞.If s ≤ t ≤ 0 then Eq. (4.7) follows from Corollary 3.5: Given ϕ ∈ X δ , we would like to find some z ∈ C 1 η,R such that F( • , ϕ) + z is a solution of Eq. (4.5).For this purpose, we deduce a necessary and sufficient condition for z ∈ C 1 η,R to turn F( • , ϕ) + z into a solution of Eq. (4.5).
for all t ∈ R.

E. Stumpf
This result is motivated by Vanderbauwhede [6,Lemma 5.7].For its proof we need two corollaries, which both are easy consequences of the exponential trichotomy (2.2) of the strongly continuous semigroup T e .Proof.Recall that T e defines a group on the center-unstable space

<∞
for all s, t ∈ R with t − s ≤ 0. As 0 < c c < η < c u , taking the limit for s → ∞ indeed shows lim s→∞ T e (t − s)P * cu z(s) = 0 as claimed.
Corollary 4.6.Let ϕ ∈ X f , z ∈ C 1 η,R , and t ∈ R be given.Then Consequently, using the estimates (2.2) we infer for all s ≤ min{0, t}.Since c s < 0 < η < −c s it becomes clear that Having established the auxiliary results, we are now in position to prove Lemma 4.4.
Proof of Lemma 4.4.We adopt the proof of Lemma 5.7 in Vanderbauwhede [6]. 1. Assume that, given ϕ ∈ X δ and z ∈ C 1 η,R , F( • , ϕ) + z is a globally defined solution of Eq. (4.8) for all −∞ < s ≤ t < ∞.Then, in view of Proposition 4.3 and Eq.(4.7), we get for all −∞ < s ≤ t < ∞.Thus, formula (4.10) holds for all s, t ∈ R as claimed.In particular, this proves that for fixed t ∈ R, we may take the limit for s → ∞ in Eq. (4.10).Then, in consideration of Corollary 4.5, we get Similarly, carrying out the limit process s → −∞ in Eq. ( 4.11) in combination with Corollary 4.6 leads to Hence, it follows that for each t ∈ R.This proves one direction of the assertion.2. Suppose, conversely, that for given ϕ ∈ X δ and z ∈ C 1 η,R Eq. (4.9) holds, and let s ≤ t be given.Obviously, Applying T e (t − s) on both sides of the last equation gives Attraction property of local center-unstable manifolds Next, after adding zero in the way represented above to the right-hand side of Eq. ( 4.12), a simple calculation leads to Hence, by combining the last equation with Eq. (4.7), we finally obtain As s ≤ t were arbitrary given, we conclude that F( • , ϕ) + z is a solution of Eq. (4.8).This completes the proof.Now, consider some fixed ϕ ∈ X δ .If we find some z ∈ C 1 η,R satisfying Eq. (4.9) then Lemma 4.4 implies that F( • , ϕ) + z is a global solution of the abstract integral equation (4.8).Hence, in turn, by application of Lemma 4.2 it follows that ψ := F(0, ϕ) + z(0) = ϕ + z(0) belongs to W η and that ϕ and ψ satisfy (4.4).Therefore, in the next step towards a proof of Theorem 4.1 we would like to solve Eq. (4.9) in C 1 η,R for each given ϕ ∈ X δ .Moreover, under the assumption that these solutions are uniquely determined, in this way we also would obtain a possible choice for the map

The remaining part of the proof for the attraction property of the global center-unstable manifolds
Our next goal is to show that for each fixed ϕ ∈ X δ Eq. (4.9) has a uniquely determined solution in C 1 η,R .This will be done by construction of a parameter-dependent contraction on the Banach space C 1 η,R below.

E. Stumpf
To begin with, observe that Eq. (4.9) may be formally written as Thus, after introducing the mapping r δ : R × we get the representation of Eq. (4.9).Note that the involved map r δ is continuous.Moreover, using (2.12) and (2.13), it easily follows that and In the first instance, representation (5.1) of Eq. (4.9) is purely formal.But next we are going to prove that all the improper integrals on the right-hand side of (5.1) indeed exist.We begin with consideration of the first integral.
Then, in view of the continuity of the maps l, r δ and F( • , ϕ), Q defines a continuous map from R into Y * as well.Furthermore, we claim that Q| (−∞,0] ∈ Y η .Indeed, by Proposition 4.3 we have and thus, by (2.13), For this reason, from Remark 2.3 it follows that u : defines a continuous map from R into C, that its restriction to the interval (−∞, 0] belongs to C 0 η , and that it additionally satisfies Eq. (2.10) for all −∞ < s ≤ t < ∞.Hence, using Remark 2.5 from the second section and Proposition 3.3 in [5], we see that η .This proves the assertion.
Proposition 5.2.Let η > 0 be as in Corollary 5.1, and let Z η denote the map, which assigns to ϕ ∈ X δ the mapping Proof. 1.At first, note that by Proposition 4.3 and Corollary 5.1, for each ϕ ∈ X δ , Z η (ϕ) forms a well-defined continuous map from R into C 1 .Consequently, it remains to prove that for given ϕ ∈ X δ we have sup For this purpose, let ϕ ∈ X δ be given.Then and next we estimate the two terms on the right-hand side separately.
2. (Estimate of sup t≤0 e ηt Z η (ϕ)(t) C 1 .)Using the triangle inequality together with the definition of F and Corollary 5.1, one obtains 3. (Estimate of sup t≥0 e ηt Z η (ϕ)(t) C 1 .)We begin with the observation that, in view of the definition of F and Eq.(3.3), we have for all 0 ≤ s ≤ t < ∞.Indeed, from the representation of Z η (ϕ) derived above it follows that In particular, u(t) = T e (t)u(0) for all t ≥ 0. Next, we claim that, for each t ≥ 0, u(t) lies in the domain D(G e ) of the generator of the semigroup T e .In order to see this, recall once more the one-to-one correspondence between solutions of Eq. (2.4) and Eq.(2.5).The map x : [−h, ∞) → R n given by is continuously differentiable, its segments x t coincide with u(t) for all t ≥ 0, the mapping [0, ∞) t → x t ∈ C 1 is continuous, and additionally x satisfies the differential equation x (t) = L e x t as t ≥ 0. In particular, the last point implies u(t) = x t ∈ D(G e ) = T 0 X f as claimed.
In addition, note that we also have As a consequence, e ηt e c s t u(0 This is the desired conclusion. After analyzing the first improper integral on the right-hand side of Eq. ( 5.1), we now address the existence of the last two.
Proof. 1.As the maps l and r δ are both continuous it is clear that G indeed defines a map from .
Given (ϕ, z) ∈ X δ × C 1 η,R , the existence of the last two integrals on the right-hand side of Eq. (5.1) now follows from Proposition 2.4 and the corollary after it.Indeed, for those (ϕ, z) the sum of the two integrals coincides with the value we obtain a representation of the right-hand side of Eq. (5.1) in the Banach space C 1 η,R .Consequently, given ϕ ∈ X δ , a solution z of Eq. (5.1) in C 1 η,R is a fixed point of the map R η ( • , ϕ).Below we prove that each ϕ ∈ X δ leads to a uniquely determined solution < ∞ due to Proposition 5.2 there clearly is some . Combining this with Corollary 5.3 and assumption (4.2), we conclude that ≤ γ} of the Banach space C 1 η,R .Thus, the Banach contraction principle shows the existence of a unique z = z(ϕ) ∈ C 1 η,R with z = R η (z, ϕ).The function c is clearly continuous.For this reason, under condition (4.2) we have for all η ≤ η < min{−c s , c u } with η − η ≥ 0 small enough.Hence, after fixing some real η ≤ η < min{−c s , c u } with η − η ≥ 0 sufficiently small, one can draw exactly the same conclusion as in Proposition 5.4; that is, for each ϕ ∈ X δ the map R η ( as z ∈ C 1 η,R has a uniquely determined fixed point in C 1 η,R . Our next goal is to show that the fixed point z(ϕ) ∈ C 1 η,R from the last statement depends continuously on ϕ ∈ X δ .For this purpose, we need some auxiliary results.We begin with the proof that the map Z η : X δ → C 1 η,R is continuous.
Proof. 1.Given ϕ, ψ ∈ X δ , we trivially have Hence, it suffices to show Thus sup and, by combining estimate (2.13) with the fact η ≥ η > 0 and Proposition 3.4, we infer that is, Now, observe that from Proposition 2.4 it follows that both K 1 η Q 1 and K 1 η Q 2 are well-defined and belong to C 1 η,R .Further, in view of the last estimate Therefore, ) Define the maps u 1 , u 2 : [0, ∞) → C 1 by u 1 (t) := Z η (ϕ)(t) and u 2 (t) := Z η (ψ)(t), respectively.As shown in part 3 of the proof of Proposition 5.2 both u 1 and u 2 are solutions of u(t) = T e (t − s)u(s) for all 0 ≤ s ≤ t < ∞.Moreover, u(t) := u 1 (t) − u 2 (t) satisfies u(t) = T e (t − s)u(s) for all 0 ≤ s ≤ t < ∞ as well.Following the proof of Proposition 5.2 further, we first see and then, in view of the estimate derived in the last step, 4. By part 2 and part 3 it follows that This shows the continuity of Z η .
Remark 5.7.Observe that in view of the proof the map X δ ϕ → Z η (ϕ) ∈ C 1 η,R is not only continuous but Lipschitz continuous with a global Lipschitz constant.
Proof.Under the given assumptions, a straightforward calculation results in and similarly to sup and therefore Q ∈ Y η,R as claimed.Using the arguments above, especially Eq. (5.4), together with the linearity of the integral operators K i one easily finds u , and so u = 0 ∈ C 1 η,R .For this reason, we conclude that z 1 (t) = z 2 (t) for all t ∈ R, and this finishes the proof.
The following corollary is the last auxiliary result for the proof that the uniquely determined fixed point of R η ( • , ϕ) depends continuously on ϕ ∈ X δ .Corollary 5.9.Suppose that η > η and z ∈ Proof. 1.Let ϕ ∈ X δ and ε > 0 be given.Then, in view of η − η < 0 and z C 1 η,R < ∞ by assumption, we clearly find some R > 0 with Next, recall from Proposition 3.1 that we have Therefore, there is some sufficiently small δ(R, ε) > 0 with the property that both In order to see this claim and so the assertion of the corollary, we show that under given assumptions Using the Lipschitz continuity of r δ given by (2.13), we first deduce that

E. Stumpf
Now we may proceed similarly as in part 4 of the proof of Proposition 5.6 to conclude that Combining these finally yields as claimed.Now we are in the position to state and prove the continuous dependence of the fixed point of the map R η ( • , ϕ) on the parameter ϕ ∈ X δ .Proposition 5.10.Let z η : X δ → C 1 η,R denote the solution operator of the parameter dependent contraction from Proposition 5.4; that is, z η (ϕ) = R η (z η (ϕ), ϕ) for all ϕ ∈ X δ .Then z η is continuous.
After having established all the necessary preparations, we are now able to prove Theorem 4.1 about an attraction property of the global center-unstable manifolds.Consequently, H η cu forms a map from X δ into the global center-unstable manifold W η .Moreover, as a sum of the two continuous maps ϕ → ϕ and ϕ → z η (ϕ) it is clearly continuous as well.
We close this section with a consequence of Theorem 4.1.
Proof.To begin with, observe that in consideration of the definition of H η cu , of Lemma 4.4, and of the uniqueness of solutions we have for all t ≥ 0 and all ϕ ∈ X δ .Next, it is easily seen that H η cu (ϕ) = ϕ for all ϕ ∈ W η .Hence, z η (ϕ)(t) = 0 (5.6) as (t, ϕ) ∈ [0, ∞) × W η .Now let ψ ∈ W η and ε > 0 be given.By the continuity of the map z η due to Proposition 5.10, we clearly find some δ > 0 such that, for all ϕ ∈ X δ with ϕ − ψ C 1 < δ, In the following we use the attraction property of the global center-unstable manifolds obtained in the last sections to give a proof for Theorem 1.2 asserting an attraction property of local center-unstable manifolds.
Given the assumptions of Theorem 1.1, and thus of Theorem 1.2 as well, we clearly find constants η > 0 with c c < η < min{−c s , c u } and 0 < δ < δ 1 such that the conditions of Theorem 4.1 are satisfied.Now, set s,0 , w cu := w η | C cu,0 , and W cu := {ϕ + w cu (ϕ) | ϕ ∈ C cu,0 } , where the map w η is defined by Eq. (2.18).With these definitions Theorem 1.1 follows as shown in [5] in detail.In particular, we have ϕ 0 = 0 ∈ W cu ⊂ W η and r δ (ϕ) = r(ϕ) for all ϕ ∈ N cu .The proof of our main result is now straightforward.
Proof of Theorem 1.2. 1.As N cu ⊂ U is open and both norms • C 1 and • 1 are equivalent, there clearly exists some ε > 0 with {ϕ ∈ C 1 | ϕ C 1 < 2ε} ⊂ N cu .Next, using Corollary 5.11 with ψ = ϕ 0 = 0 ∈ W η we find some 0 < δ < ε such that for all ϕ ∈ X δ with ϕ C 1 < δ we have and so as t ≥ 0. 2. Suppose now that x : [−h, ∞) → R n is a solution of Eq. (1.1) with x t C 1 ≤ δ for all t ≥ 0. Set φ := x 0 and note that r δ (x t ) = r(x t ) for each t ≥ 0, since the segments of x stay in N cu for all t ≥ 0. Hence, we have x t ∈ X δ ∩ X f as t ≥ 0 and x is a solution of the smoothed equation (3.1) as well.In particular, F δ (t, φ) = F(t, φ) for all t ≥ 0.

E. Stumpf
Next, observe that the last inequality of the first part shows that for t ≥ 0 Consequently, all the segments y t = F δ (t, H η cu ( φ)) of the unique solution y : [−h, ∞) → R n of Eq. (3.1) with initial value y 0 = H η cu ( φ) ∈ W η are contained in the neighborhood N cu of ϕ 0 = 0 ∈ C 1 .Therefore, for each t ≥ 0 we have r δ (y t ) = r(y t ) and thus y is also a solution of Eq. (1.1) with segments y t ∈ X δ ∩ X f .In particular, y 0 = H η cu ( φ) ∈ W cu and F(t, H η cu ( φ)) = F δ (t, φ) as t ≥ 0. Now the positive invariance of W cu with respect to F relative to N cu , that is, property (iii) of Theorem 1.1, shows y t = F(t, H η cu (ϕ)) ∈ W cu as t ≥ 0. Furthermore, estimate (6.1) implies for all t ≥ 0.

2 )
induce a strongly continuous semigroup T e = {T e (t)} t≥0 on C. The generator of T e is defined by G e : D(G e ) χ → χ ∈ C on the domain D(G e ) := {χ ∈ C 1 | χ (0) = L e χ}.

1 ≤
z η (ϕ) − z η (ψ) C 1 η,R < ε holds.Hence, in view of Eq. (5.6), it follows that sup t≥0 e ηt z η (ϕ)(t) C 1 = sup t≥0 e ηt z η (ϕ)(t) − z η (ψ)(t) C 1 ≤ z η (ϕ) − z η (ψ) C 1 η,R ≤ εAttraction property of local center-unstable manifolds 43 for all ϕ ∈ X δ satisfying ϕ − ψ C 1 < δ.Combining this with Eq. (5.5) finally shows that for ϕ ∈ X δ with ϕ − ψ C 1 < δ,F δ (t, H η cu (ϕ)) − F δ (t, ϕ) C 1 ≤ e −ηt sup t≥0 e ηt F δ (t, H η cu (ϕ)) − F δ (t, ϕ) C 1 = e −ηt supt≥0 e ηt z η (ϕ)(t) C εe −ηt .6 Proof of Theorem 1.2 T * e (t) denote the adjoint operator of the bounded linear operator T e (t) induced by the solutions of the initial value problems (1.2).The family T * e = {T * e (t)} t≥0 forms a semigroup of bounded linear operators on the dual space C * of C. But in general T * e does not constitute a strongly continuous semigroup on C * with respect to the topology induced by the norm ϕ * C * := sup ϕ C ≤1 |ϕ * (ϕ)|.However, let C denote the set of all ϕ ∈ C * with the property that the curve [0, ∞) t → T * e (t)ϕ ∈ C * is continuous.Then C is a closed subspace of C * and for all t ≥ 0 we have T * e (t)(C ) ⊂ C .As a consequence, the family T e = {T e (t)} t≥0 of operators T e (t) : C ϕ → T * e (t)ϕ ∈ C becomes a strongly continuous semigroup on the Banach space C .Similarly, carrying out the process above with the semigroup T e on C instead of T e on C, we first obtain the family T * (t)} t≥0 of adjoint operators of T e on the dual space C * of C and then the Banach space C ⊂ C * , on which the restriction of T * is strongly continuous.The original Banach space C and semigroup T e are sun-reflexive: There is an isometric linear map j : C → C * such that j(C) = C and T * e = {T * e e (t)(jϕ) = j(T e (t)ϕ) for all t ≥ 0 and all ϕ ∈ C. For simplicity, we identify C with C and omit the embedding operator j in the following.For the spectrum σ(G * e ) of the infinitesimal generator G * e of the semigroup T * e we have σ(G * e ) = σ(G e ).By analogy to the decomposition (1.3) of C, C * can be decomposed as solves Eq. (2.10) for all −∞ < s ≤ t < ∞, and satisfies u(t) ∈ C s as t ∈ R.
1 η ) ⊂ Y η and that R δη is bounded by P * Similarly, from the second inequality of (2.16) we get for all u cu δλ(δ).