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Smooth centre manifolds for impulsive delay diﬀerential equations

Kevin E.M. Church and Xinzhi Liu

1 Introduction

Centre manifold theory has a rich history as one of the fundamental tools in the study of nonlinear dynamical

systems. Broadly speaking, the application of the theory to a given dynamical system near its nonhyperbolic

states permits a reduction of dimension that is locally characteristic of the behaviour of the fully nonlinear

system.

The dynamics of inﬁnite-dimensional systems has been a great source of motivation in the development of

techniques in functional analysis. For instance, (strongly continuous) semigroups of operators are often used

as the building blocks of centre manifolds (and, indeed, other invariant manifolds) for inﬁnite-dimensional

dynamical systems. The body of literature on this topic is vast; for a brief exposure one may consult the

works of Chicone [5], Chow and Lu [6], DaPrato and Lunardi [8], Krizstin [21], Veltz and Fogeras [31], as

well as the textbooks [9, 13].

In recent years, there has been a surge of interest in the dynamics of impulsive diﬀerential equations with

time delays, especially in neural networks, mathematical biology and ecology, as such systems frequently

involve memory eﬀects (discrete or distributed delays), and bursting or discontinuous controls (impulses).

One may consult [2, 22, 24, 27, 33] for background on impulsive diﬀerential equations. While large-scale

emergent behaviour such as synchronization in neural networks can be introduced through pinning algo-

rithms, there is little available in terms of low-dimensional analysis techniques to study the emergence of

classical bifucation patterns. Indeed, analysis of speciﬁc nonlinear impulsive systems with delays appears to

be mostly conﬁned to more static notions such as well-posedness, permanence, existence of global attractors

and binary stability-instability analysis of equilibrium points, with a look toward bifurcation toward perma-

nence of a compact region of the phase space — see [11, 26, 32, 34] for some recent applications to biological

systems. Most dynamic bifurcation analysis at present seems restricted to numerical studies. For instance,

in [35], the largest Lyapunov exponent is used to numerically investigate bifurcations to chaotic attractors

in a three-species food chain model with distributed delay and impulsive control.

In this paper, we establish the theoretical existence, smoothness and reduction principle of centre mani-

folds for a fairly broad class of impulsive delay diﬀerential equations, thereby introducing a classical method

of analysis to this growing ﬁeld of study. It should be mentioned that in the literature, one typically refers

to nonautonomous invariant manifolds (of which the centre manifold is included) as invariant ﬁber bundles.

These are appropriate generalization of invariant manifolds to explicitly time-varying systems that can be

visualized as time-varying manifolds [1]. However, to avoid unnecessary verbiage and to draw a distinction

between them and linear invariant ﬁbre bundles, we will continue to refer to them as centre manifolds.

The structure of the paper is as follows. In Section 2, we provide an imprecise statement of our main

result and elaborate on several of its corollaries – namely, the existence of local centre manifolds for fully

nonautonomous delay diﬀerential equations and ﬁnite-dimensional impulsive systems. We also outline our

method of proof. Section 3 provides background material on impulsive delay diﬀerential equations and some

of the function spaces that will be needed, as well as deﬁnitions speciﬁc to our results. Section 4 is devoted

to the development of a variation-of-constants formula for linear nonhomogeneous impulsive delay diﬀerental

equations that is interesting in its own right, but will be needed extensively after. The existence of Lipschitz

continuous centre manifolds (local and global) is proven in Section 5. A reduction principle (attractivity

properties and restricted dynamics equtions) is established in Section 6. A detour is taken to study periodic

linear systems in Section 7, before establishing the smoothness of the centre manifold in Section 8, where

we also prove that a periodic system necessarily generates a periodic centre manifold. Some examples are

provided in Section 9.

1

2 Statement of results and methodology

This section will be devoted to an informal statement of the main results of this paper, together with a broad

overview of the proofs. We will ultimately be interested in semilinear impulsive delay diﬀerential equations

of the form

˙x=A(t)xt+f(t, xt), t 6=τk

∆x=Bkxt−+gk(xt−), t =τk,(1)

where A(t) : RCR → Rnand B(k) : RCR → Rnare for each t∈Rand k∈Z, bounded linear functionals

acting on the Banach space RCR of uniformly bounded functions φ: [−r, 0] →Rnthat are continuous from

the right and have limits on the left, with r > 0 ﬁnite. Also, f:R× RC R → Rnand g:Z× RCR → Rnare

suﬃciently smooth and vanishing with vanishing ﬁrst derivatives at the origin 0 ∈ RCR, and {τk:k∈Z}is

a sequence of impulse times. We do not require global Lipschitzian conditions on the vector ﬁeld for jump

functional g.

2.1 Statement of the result

Rather imprecisely, the main result of our paper is as follows.

Theorem. Under “reasonable assumptions”, there exists a Lipschitz function C:RCRc→ RCR, with

domain consisting of a time-varying subset RCRc⊂ RCR, with the property that every suﬃciently small

solution of (1) with limited two-sided exponential growth is contained within the graph of C: the local centre

manifold. Moreover, in the absence of unstable components in the linear part of (1), the local centre manifold

attracts nearby solutions. Under certain conditions, the function C:RCRc→ RCR is smooth.

The reasonable assumptions of the theorem include, in particular, a splitting of the phase space RCR

into a time-varying internal direct sum RCRs(t)⊕ RCRc(t)⊕ RCRu(t) of three closed subspaces, which

behave like time-varying stable, centre and unstable subspaces associated to the linear system

˙y=A(t)yt, t 6=τk

∆y=Bkyt−, t =τk.(2)

More precisely, this splitting is a decomposition of RCR as an internal direct sum of three mutually orthogonal

closed stable, centre and unstable ﬁbre bundles over RCR with base space R(equivalently, nonautonomous

sets RCRi⊂R× RCR). In addition, the evolution family U(t, s) : RCR → RCR associated to the linear

system (2) must satisfy certain invertibility and exponential boundedness conditions when restricted to each

factor of the decomposition, deﬁned through the projection operators Pi:RCR → RCRionto the stable,

centre, and unstable ﬁbre bundles. We will later say that the linear part is spectrally separated if these

conditions are satisﬁed.

While presented somewhat abstractly, the spectral conditions are satisﬁed in several important special

cases. For instance, they are satisﬁed when (2) is periodic, as proven in Section 7. The centre manifold is

also smooth in this case.

2.2 Corollary: centre manifolds for ﬁnite-dimensional impulsive systems and

systems with memoryless linear part

Our theorem stated imprecisely in Section 2.1 immediately grants existence and smoothness of local centre

manifolds or invariant ﬁber bundles under similar reasonable assumptions for ordinary impulsive diﬀerential

equations in Euclidean space,

˙x=f(t, x), t 6=τk

∆x=gk(x), t =τk.

It should be noted that there are numerous examples of centre manifold theory for diﬀerence equations

being applied to study periodic systems of impulsive ordinary diﬀerential equations – see [7] for a survey of

2

this method. Despite this, it appears yet to be proven in the literature that such systems possess Ck-smooth

invariant centre ﬁber bundles in general. One result [4] is applicable for impulsive diﬀerential equations in

Banach spaces, but only holds for small nonlinearities and grants C1smoothness. We thus prove prove Ck

smoothness in Euclidean space.

Another useful corollary is the existence and smoothness of the centre manifold for impulsive delay

systems when the linear part is memoryless. That is, systems of the form

˙x=A(t)x+f(t, xt), t 6=τk

∆x=Bkx+gk(xt−), t =τk,

where the nonlinearities vanish and have vanishing derivatives at zero. In this case, the veriﬁcation of spectral

separation can be done on the ﬁnite-dimensional linear part, instead of in the whole inﬁnite-dimensional phase

space. This greatly simpliﬁes calculations.

2.3 Methodology

At its core, our approach to prove the existence and smoothness of local centre manifolds is an adaptation of

the Lyapunov-Perron method used to prove the existence of centre manifolds for various classes of functional

diﬀerential equations without impulses. This programme is carried out successfully in [9, 21, 17, 18], for

example.

The Lyapunov-Perron method makes use of a variation-of-constants formula to reinterpret solutions

of the diﬀerential equation in question as mild solutions of a semilinear integral equation. In the fully

nonautonomous context, this method was used by Chicone [5] to prove a nonautonomous centre manifold

theorem by ﬁrst appealing to the evolution semigroup. The evolution semigroup allows one to eﬀectively

translate the problem into an autonomous setting by enlarging the phase space. Semigroup theory then

provides the requisite variation of constants formula.

To contrast to the approach of Chicone, we work directly with the evolution family associated to (2)

and prove a variation of constants formula that is reminiscent of a classical formula derived by Jack Hale

for functional diﬀerential equations [12]. In the aforementioned reference, Hale proves that solutions of the

inhomogeneous delay diﬀerential equation ˙x=Axt+h(t) satisfy the formal variation of constants formula

xt=T(t−s)xs+Zt

s

T(t−µ)χ0h(µ)dµ,

where T(t) : X→Xis the strongly continuous semigroup associated to the autonomous system ˙x(t) = Axt,

the phase space is X=C([−r, 0],Rn), and χ0: [−r, 0] →Rn×nis deﬁned by χ0(0) = Iand χ0(θ) = 0 for

θ < 0. Strictly speaking, the formula is ill-deﬁned because χ0h(µ) is not in the domain of T(t−µ).

The inconsistencies in Hale’s variation of constants formula can be resolved in several ways, including

adjoint semigroup theory and integrated semigroup theory [14]. We instead opt for a more elementary

approach that is similar to the construction used in [3, 25]. Namely, we work with the phase space RCR of

right-continuous regulated functions at the outset and prove that the nonhomogeneous impulsive functional

diﬀerential equation

˙x=A(t)xt+h(t), t 6=τk

∆x=Bkxt−+r(k), t =τk

(3)

satisﬁes a forward global existence and uniqueness of solutions property, and that its associated homogeneous

equation generates an evolution family U(t, s) : RCR → RCR that is suﬃciently regular to deﬁne and prove

the variation of constants formula

xt=U(t, s)xs+Zt

s

U(t, µ)χ0h(µ)dµ +X

s<τi≤t

U(t, τi)χ0r(i),

where the integral is interpreted in the Pettis (weak) sense. The correctness of this formula is proven in

Section 4.

3

It is interesting to note that the evolution family U(t, s) generally fails to be strongly continuous, further

necessitating the interpretation of the integral in the weak sense. Indeed, the integrand is not even Bochner

measurable, which makes investigations into strong integrability quite diﬃcult.

The weak integral behaves well with respect to composition of bounded linear operators, and as such

commutes with the projection operators Pionto the stable, centre, and unstable ﬁbre bundles associated to

the linearization. This fact is later used to construct, for each s∈R, a bounded linear operator

Kη

s: (h, g)7→ Kη

s(f, g)

mapping inhomogeneities (h, r) satisfying bounded exponential growth conditions onto the unique bounded-

growth solutions of the inhomogeneous equation (3) whose components in the centre ﬁbre bundle vanish at

time s.

The rest of the proof of existence follows loosely the approach taken in, for instance [9, 21, 17, 18].

Namely, one cuts oﬀ the nonlinearities fand gaway from the origin separately in the centre and hyperbolic

directions and constructs a nonlinear ﬁxed-point equation for each s∈R,

u=U(·, s)ϕ+Kη

s◦Rδ(u),

where ϕ∈ RCRc(s) is a component in the centre bundle in the appropriate ﬁber and Rδis the Nemitsky

operator associated to the external direct sum of the nonlinearities fand gfollowing the cutoﬀ procedure.

The ﬁxed point us=us(ϕ) of this equation is then used to deﬁne the global centre manifold at time s.

The ﬁbre bundle deﬁned by “gluing” these manifolds together along the real line deﬁnes the all-time global

centre manifold, and a suitable restriction generates the invariant centre manifold. The proof of invariance

and attraction is similar in spirit to the autonomous delay diﬀerential equations case.

To obtain smoothness, the nonlinearities must be smoothed in a way that makes the Nemitsky operator

smooth in the part of the phase space in which the centre manifold resides. As the centre manifold is time-

varying, the inﬁnite-dimensionality of the space in which it resides makes the smooth renorming of the space

a nontrivial problem to solve. To resolve this, we specialize to linearizations whose evolution families satisfy

a particular decomposability condition that is formally analogous to a Floquet decomposition on the centre

ﬁber bundle. Then, we prove the smoothness of the centre manifold by appealing to methods of contractions

on scales of Banach spaces; see [30, 29, 9] for background on these techniques.

2.4 Notation

The following notation is common to the manuscript. For a subset Z⊂R, the symbol χZwill always denote

the identity-valued indicator function:

χZ(θ) = 0, θ /∈Z

I, θ ∈Z,

with Ithe identity on Rn. The domain of χZwill be either stated or implied. The cardinality of a set S

will be denoted #S. If Vis a normed vector space, the set of bounded linear maps on Vwill be denoted

L(V), and if V1, . . . , Vpare all normed spaces, then the set of bounded p-linear maps from V1× · · · × Vpinto

Vwill be denoted Lp(V1× · · · × Vp, V ). For a function f:T × X→Ywith T ⊆ Rand Banach spaces X, Y ,

Dkf(t, x)∈ Lk(X× · ·· × X, X ) denotes the kth Fr´echet derivative of fin its second variable at the point

(t, x)∈ T × X.

3 Background material

In this section we will collect necessary results on the linear inhomogeneous impulsive retarded functional

diﬀerential equation

˙x=L(t)xt+h(t), t 6=τk(4)

∆x=Bkxt−+rk, t =τk,(5)

4

with the impulse condition ∆x(t) = x(t)−x(t−), where xt−(θ) = x(t+θ) for −r≤θ < 0 and xt−(0) =

limθ→0−x(t+θ), and r > 0 is ﬁnite. We will be working exclusively with spaces of right-continuous regulated

functions; denote

RCR(I , X) = f:I→X:∀t∈I , lim

s→t+f(s) = f(t) and lim

s→t−

f(s) exists,

where X⊆Rnand I⊆R. When Xand Iare closed,

RCRb(I , X) := {f∈ RCR(I , X) : ||f|| <∞}

is a Banach space with the norm ||f|| = supx∈I|f(x)|. We will also at times require the space G(I, X )

of regulated functions from Iinto X; this is merely the set of functions f:I→Xthat posess left- and

right limits at each point, with no continuity sidedness restriction. One may consult [16] for background

on regulated functions. We will write RCR := RCR([−r, 0],Rn) when there is no ambiguity, and note that

since RCRb([−r, 0],Rn) = RCR([−r, 0],Rn), we may identify RCR with its associated Banach space. The

following assumptions will be needed throughout.

H.1 The representation

L(t)φ=Z0

−r

[dθη(t, θ)]φ(θ)

holds, where the integral is taken in the Lebesgue-Stieltjes sense, the function η:R×[−r, 0] →Rn×n

is jointly measurable and is of bounded variation and right-continuous on [−r, 0] for each t∈R, and

such that |L(t)φ| ≤ (t)||φ|| for some :R→Rlocally integrable.

H.2 The sequence τkis monotonically increasing with |τk|→∞as |k|→∞, and the representation

Bkφ=Z0

−r

[dθγk(θ)]φ(θ)

holds for k∈Zfor functions γk: [−r, 0] →Rn×nof bounded variation and right-continuous, such that

|Bk| ≤ b(k).

Remark 3.0.1. Hypothesis H.1–H.2 could in principle be weakened. However, insofar as applied impulsive

diﬀerential equations are concerned, hypothesis H.1 is suﬃcient. Indeed, H.1 includes the case of discrete

time-varying delays: the linear delay diﬀerential equation

˙x=

m

X

k=1

Ak(t)x(t−rk(t))

with rkcontinuous, is associated to a linear operator satisfying condition H.1 with η(t, θ) = PAk(t)H−rk(t)(θ),

where Hz(θ)=1if θ≥zand zero otherwise. It also obviously includes a large class of distributed delays.

Moreover, each of L(t)and Bkis well-deﬁned on G([−r, 0],Rn)and, consequently, on the subspace RCR; see

Lemma 3.1.2.

In what follows, we will introduce several properties of right-continuous regulated functions that will be

needed throughout the paper, in addition to impulsive integral inequalities and some deﬁnitions.

3.1 Properties of regulated functions

We will require some elementary properties of regulated functions. The ﬁrst result will be useful in proving

boundedness of evolution families associated to linear, homogeneous impulsive equations. We have not seen

the following elementary result in the literature, so it will be proven here for completeness.

Lemma 3.1.1. Let r > 0be ﬁnite and let φ∈ RCR([a, b],Rn)for some b≥a+r. With φt: [−r, 0] →Rn

deﬁned in the usual way, t7→ ||φt|| is an element of RCR([a+r, b],R).

5

Proof. Let t∈[a+r, b] be ﬁxed. We will only prove right-continuity, since the proof of the existence of left

limits is similar. It suﬃces to prove that for any decreasing sequence sn↓0, we have ||φt+sn|| → ||φt||.Let

> 0 be given. By right-continuity of φ, for all > 0, there exists δ > 0 such that, if 0 <µ<δ, then

|φ(t+µ)−φ(t)|< . Therefore,

||φt+sn|| = sup

µ∈[−r,0]

|φ(t+µ)| ≤ sup

µ∈[−r,sn]

|φ(t+µ)| ≤ max{||φt||,sup

µ∈[0,sn]

|φ(t+µ)|}

≤max{||φt||,|φ(t)|+} ≤ ||φt|| +,

provided sn< δ. On the other hand, since φis bounded, there exists some sequence xn∈[−r, 0] such that

|φt(xn)| → ||φt||. By passing to a subsequence, we may assume xn→ˆx∈[−r, 0]. If ˆx > −r, then we have

||φt+sn|| ≥ sup

µ∈[−r+sn,0]

|φ(t+µ)|=|φ(ˆx)|=||φt||

provided sn<−ˆx, while if ˆx=−r, we notice that the sequence x0

n=t+−r+snconverges to t+ ˆx, so that

for all > 0, there exists N3>0 such that for n≥N,

||φt+sn|| ≥ |φ(t+sn)≥ ||φt|| − .

Therefore, if we let sN1< δ and sN2<−ˆx, then by setting N= max{N1, N2, N3}, it follows by the above

three inequalities that for n≥N,

−≤ ||φt+sn|| − ||φt|| ≤ .

We conclude ||φt+sn|| converges to ||φt||.

We will eventually need spaces of function f:I→Xthat are diﬀerentiable from the right and whose

right-hand derivatives are elements of RCR(I, X). Speciﬁcally, denote the right-hand derivative by

d+f(t) = lim

→0+

f(t+)−f(t)

and introduce the space

RCR1(I , X) = {f∈ RCR(I , X) : d+f∈ RCR(I , Rn)}.

This space is clearly complete with respect to the norm ||f||1=||f||+||d+f|| when restricted to the subspace

consisting of functions that are || · k|1-bounded.

The next result concerns a dense subspace of G(I, X) and, by extension, the subspace RCR(I , X). The

proof is available in [16]

Lemma 3.1.2. Let Ibe compact. For all f∈ G(I , X), there exists a sequence of step functions fn:I→X

such that ||fn−f|| → 0.

We will need a characterization of the continuous dual of the space RCR, denoted RC R∗. A result from

Tvrdy [28] provides such for the dual of the space of regulated left-continuous scalar-valued functions, and

for our purposes the obvious modiﬁcation that is needed is the following.

Lemma 3.1.3. F∈ RCR∗if and only if there exists q∈Rnand p: [−r, 0] →Rnof bounded variation such

that

F(x) = q∗x(0) + Z0

−r

p∗(t)dx(t),(6)

where the integral is a Perron-Stieltjes integral.

We will need a few convergence and boundedness results for Perron-Stieltjes integrals involving right-

continuous regulated functions and functions of bounded variation. Symmetric arguments to those appearing

in [28] obviously yield the following results; see Theorem 2.8 and Corollary 2.10 therein.

6

Lemma 3.1.4. Let f: [a, b]→Rnbe of bounded variation and g∈ RCR([a, b],Rn). The integral

Rb

af∗(t)dg(t)exists in the Perron-Stieltjes sense, and

Zb

a

f∗(t)dg(t)≤(|f(a)|+|f(b)|+ varb

af)||g||,(7)

where varb

afdenotes the total variation of fon the interval [a, b].

Lemma 3.1.5. Let hn∈ RCR([a, b],Rn)and h∈ RCR([a, b], Rn)be such that ||hn−h|| → 0as n→ ∞.

For any f: [a, b]→Rnof bounded variation, the Perron-Stieltjes integrals Rb

af∗(t)dh(t)and Rb

af∗(t)dhn(t)

exist and

lim

n→∞ Zb

a

f∗(t)dhn(t) = Zb

a

f∗(t)dh(t).(8)

3.2 Integral inequalities

We next provide two inequalities. The ﬁrst is an impulsive Gronwall-Bellman inequality for regulated

functions. The result is similar to Lemma 2.3 of [2], and the proof is omitted. The second one concerns an

elementary estimation of sums of continuous functions at impulses, when the sequence of impulses satisﬁes

a separation condition.

Lemma 3.2.1. Suppose x∈ RCR([s, α],R)satisﬁes the inequality

x(t)≤C+Zt

s

(p(µ)x(µ) + h(µ))dµ +X

s<τi≤t

(bix(τ−

i) + gi) (9)

for some nonnegative integrable function p, integrable and bounded h, nonnegative constants bi,giand c,

and all t∈[s, α]. For t≥s, deﬁne

z(t, s) = exp Zt

s

p(µ)dµY

s<τi≤t

(1 + bi).

Then, µ7→ z(t, µ)is integrable and the following inequality is satisﬁed.

x(t)≤Cz(t, s) + Zt

s

z(t, µ)h(µ)dµ +X

s<τi≤t

z(t, τi)gi.(10)

Lemma 3.2.2. Let f:R→Rbe continuous and suppose {τk}satiﬁes τk+1 −τk≥ξ.

1. If fis increasing, then Ps<τi≤tf(τi)≤1

ξRt+ξ

sf(µ)dµ.

2. If fis decreasing, then Ps<τi≤tf(τi)≤1

ξRt

s−ξf(µ)dµ.

Proof. Let {τ0, . . . , τN}={τk:k∈Z} ∩ (s, t]. If fis increasing, then

X

s<τi≤t

f(τi) =

N

X

i=0

f(τi) = 1

δ

N

X

i=0

f(τi)ξ≤1

ξ

N

X

i=0

f(τ0+iξ)ξ≤1

ξZt+ξ

s

f(µ)dµ.

The decreasing case is similar.

7

3.3 Evolution families, nonautonomous sets and processes

Deﬁnition 3.3.1. Let Xbe a Banach space. An evolution family on Xis a collection of bounded linear

operators {U(t, s)}t≥s∈Ron Xthat satisfy U(t, s) = U(t, v)U(v, s)for all t≥v≥sand U(t, t) = IX.

Nonlinear variants of evolution families are more appropriately deﬁned in terms of nonautonomous sets,

which are a speciﬁc type of ﬁber bundle. This is because insofar as evolution families may serve to deﬁne

solutions to linear Cauchy problems, the interval of existence of a nonlinear Cauchy problems will typically

depend on the initial data. The following deﬁnition is borrowed from Kloeden and Rasmussen [20], with

slightly diﬀerent notation.

Deﬁnition 3.3.2. If Xis a Banach space, a subset M ⊆ R×Xis a nonautonomous set over X. For each

t∈R, the set

M(t) = {x: (t, x)∈ M}

is called the t-ﬁber of M.

Deﬁnition 3.3.3. Aprocess on X is a pair (S, M)where Mis a nonautonomous set over R×Xand

S:M → X, whose action we denote by S(t, (s, x)) = S(t, s)x, and satisﬁes the following.

• {t} × X⊂ M(t)and S(t, t) = IXfor all t∈R.

•S(t, s)x=S(t, v)S(v, s)xwhenever (s, x)∈ M(v)and (v , S(v, s)x)∈ M(t).

Note that the above deﬁnition is diﬀerent, for example, than the one for process appearing in [20], where

processes are deﬁned ﬁrst as (partial) mappings, independent of nonautonomous sets. The reason for our

distinction here is that we want to make precise the notion that a process S(t, s) need not be deﬁned on the

entire Banach space Xfor every pair of time arguments, the way evolution families U(t, s) are.

Deﬁnition 3.3.4. If Mis a nonautonomous set over Xand Yis another Banach space, we will say

a function f:M → Yhas a given regularity property (eg. continuous, Lipschitz continuous, smooth) if

f(t, ·) : Mt→Yhas the same regularity property. That is, regularity is deﬁned ﬁbrewise.

Deﬁnition 3.3.5. Let U(t, s) : X→Xbe a family of bounded linear operators deﬁning a forward process on

a Banach space X— that is, U(t, s) = U(t, v)U(v, s)for all t≥v≥sand U(t, t) = IX. We say that Uis

spectrally separated if there exists a triple (Ps, Pc, Pu)of bounded projection-valued functions Pi:R→ L(X)

with Ps+Pc+Pu=Isuch that the following hold.

1. There exists a constant Nsuch that supt∈R(||Ps(t)|| +||Pc(t)|| +||Pu(t)||) = N < ∞.

2. The projectors are mutually orthogonal; Pi(t)Pj(t)=0for i6=j.

3. U(t, s)Pi(s) = Pi(t)U(t, s)for all t≥sand i∈ {s, c, u}.

4. Deﬁne Ui(t, s)as the restriction of U(t, s)to Xi(s) = R(Pi(s)). The operators Uc(t, s) : Xc(s)→

Xc(t)and Uu(t, s) : Xu(s)→Xu(t)are invertible and we denote Uc(s, t) = Uc(t, s)−1and Uu(s, t) =

Uu(t, s)−1for s≤t.

5. The operators Ucand Uudeﬁne all-time processes on the family of Banach spaces Xc(·)and Xu(·).

Speciﬁcally, the following holds for all t, s, v ∈R.

Uc(t, s) = Uc(t, v)Uc(v, s), Uu(t, s) = Uu(t, v )Uu(v, s).

6. There exist real numbers a < 0< b such that for all > 0, there exists K≥1such that

||Uu(t, s)|| ≤ Keb(t−s), t ≤s(11)

||Uc(t, s)|| ≤ Ke|t−s|, t, s ∈R(12)

||Us(t, s)|| ≤ Kea(t−s), t ≥s. (13)

8

The above deﬁnition is a time-varying version of the spectral decomposition hypotheses associated to the

centre manifold theorem for autonomous delay diﬀerential equations appearing in [9].

Deﬁnition 3.3.6. Let U(t, s) : X→Xbe spectrally separated. The nonautonomous sets

Xi={(t, x) : t∈R, x ∈Xi(t)}

for i∈ {s, c, u}are termed respectively the stable, centre, and unstable bundles associated to U(t, s).

3.4 Existence and uniqueness of solutions for the linear equation

In order to eventually prove the existence of the centre manifold in Section 5, we will need to ﬁrst verify

existence, uniqueness and continuability of solutions.

Lemma 3.4.1. Let h∈ RCR(R,Rn)and let hypotheses H.1–H.2 hold. For all φ∈ RCR and s∈R, there

exists a unique function x∈ RCR([s−r, ∞),Rn)satisfying xs=φand the integral equation

x(t) = (φ(0) + Rt

s[L(µ)xµ+h(µ)]dµ +Ps<τi≤t[Bixτ−

i+ri], t > s

φ(t−s), s −r≤t≤s. (14)

The above lemma follows by hypotheses H.1–2, the Banach ﬁxed-point theorem, Lemma 3.2.1 and typical

continuation methods. It could also be proven by identifying the equation with a generalized ordinary

diﬀerential equation, as in [10]. Under stronger assumptions, one may look at the proof of Theorem 5.1.1

for local existence and uniqueness. Note here that hmay be unbounded on the real line; however, since it is

regulated it is bounded on every compact set [16].

Consider now the homogeneous equation

˙x=L(t)xt, t 6=τk(15)

∆x=Bkxt−, t =τk.(16)

Deﬁnition 3.4.1. Let hypotheses H.1–H.2 hold. For a given (s, φ)∈R× RCR, let t7→ x(t;s, φ)denote

the unique solution of (15)–(16) satisfying xs(·;s, φ) = φ. The function U(t, s) : RCR → RCR deﬁned by

U(t, s)φ=xt(·, s, φ)for t≥sis the evolution family associated to the homogeneous equation (15)–(16).

Lemma 3.4.2. The evolution family satisﬁes the following properties.

1) For s≤t,U(t, s) : RCR → RCR is a bounded linear operator. In particular,

||U(t, s)|| ≤ exp Zt

s

(µ)dµY

s<τi≤t

(1 + b(i)).(17)

2) For s≤v≤t,U(t, s) = U(t, v)U(v, s).

3) For all θ∈[−r, 0],s≤t+θand φ∈ RCR,U(t, s)φ(θ) = U(t+θ, s)φ(0).

4) For all τk> s, one has U(τk, s)=(I+χ0Bk)U(τ−

k, s).1

5) Let C(t, s)denote the evolution family on RCR associated to the “continuous” equation ˙x=L(t)xt.

The following factorization holds:

U(t, s) = C(t, s),[s, t]∩ {τk}k∈Z∈ {{s},∅}

C(t, τk)◦(I+χ0Bk)◦U(τ−

k, s), t ≥τk> s (18)

1Note here that the left limit is the uniform one-point limit. Namely, U(τ−

k, s)φ(θ) = U(τk, s)φ(θ) for θ < 0, while

U(τ−

k, s)φ(0) = U(τk, s)φ(0−).

9

Proof. Property 2) and 3) are immediate consequences of the uniqueness assertion of Lemma 3.4.1 and the

deﬁnition of the evolution family. For property 1), we obtain linearity by noticing that φ7→ x(t;s, φ) is linear

in φfor each t≥sand, consequently, φ7→ xt(·;s, φ) is is also linear. To obtain boundedness, we notice that

by virtue of the integral equation (14), U(t, s)φ(θ) satisﬁes

|U(t, s)φ(θ)| ≤ ||φ|| +Zt+θ

s

|L(µ)U(µ, s)φ|dµ +X

s<τi≤t+θ

|BiU(τ−

i, s)φ|

≤ ||φ|| +Zt

s

(µ)||U(µ, s)φ||dµ +X

s<τi≤t

b(i)||U(τ−

i, s)φ||.

Since the upper bounds are independent of θ, denoting X(t) = U(t, s)φ, we obtain

||X(t)|| ≤ ||φ|| +Zt

s

(µ)||X(µ)||dµ +X

s<τi≤t

b(i)||X(τ−

i)||.

By Lemma 3.1.1, t7→ ||X(t)|is an element of RCR([s−r, ∞),R). Invoking Lemma 3.2.1, we obtain the

desired boundedness (17) of the evolution family and property 1) is proven. Finally, since

U(τk, s)φ(0) = φ(0) + Zτk

s

L(µ)U(µ, s)φdµ +X

s<τi≤τk

BiU(τ−

i, s)φ

=U(τ−

k, s)φ(0) + BkU(τ−

k, s)φ

and U(τ−

k, s)φ(θ) = U(τk, s)φ(θ) for θ < 0, we readily obtain property 4). The veriﬁcation of property 5)

follows by existence and uniqueness of solutions and property 4).

4 The variation of constants formula

Existence, uniqueness and continuability of solutions of the linear inhomogeneous equation (4)–(5) has been

granted by Lemma 3.4.1. From this result we directly obtain a decomposition of solutions.

Lemma 4.0.1. Let h∈ RCR(R,Rn)and let H.1–H.2 hold. Denote t7→ x(t;s, φ;h, r)the solution of

the linear inhomogeneous equation (4)–(5) for inhomogeneities h=h(t)and r=rk, satisfying the initial

condition xs(·;s, φ;h, r) = φ. The following decomposition is valid:

x(t;s, φ;h, r) = x(t;s, φ; 0,0) + x(t;s, 0; h, 0) + x(t;s, 0; 0, r) (19)

With this decomposition, we will now proceed with the derivation of the variation of constants formula.

We prove a pointwise formula in Section 4.1 before lifting the formula into the space RCR in Section 4.2

4.1 Pointwise variation of constants formula

The following lemmas prove representations of the inhomogeneous impulsive term xt(·;s, 0; 0, r) and the

inhomogeneous continuous term xt(·;s, 0; h, 0).

Lemma 4.1.1. Under hypotheses H.1–H.2, one has

xt(·;s, 0; 0, r) = X

s<τi≤t

U(t, τi)χ0ri(20)

Proof. Denote x(t) = x(t;s, 0; 0, r). Clearly, for t∈[s, min{τi:τi> s}, one has xt= 0. Assume without

loss of generality that τ0= min{τi:τi> s}. Then xτ0=χ0r0due to (14). From Lemma 3.4.1 and 3.4.2,

10

we have xt=U(t, τ0)χ0r0for all t∈[τ0, τ1), so we conclude that (20) holds for all t∈[s, τ1). Supposing by

induction that xt=Ps<τi≤tU(t, τi)χ0rifor all t∈[s, τk) for some k≥1, we have

xτk=xτ−

k+χ0Bkxτ−

k+χ0rk

=U(τk, τk−1)xτk−1+χ0rk

=U(τk, τk−1)Ps<τi≤τk−1U(τk−1, τi)χ0ri+χ0rk

=Ps<τi≤τkU(t, τi)χ0ri.

Equality (20) then holds for t∈[τk, τk+1) by applying Lemma 3.4.2. The result follows by induction.

Lemma 4.1.2. Let h∈ RCR(R,Rn). Under hypotheses H.1–H.2, one has

xt(θ;s, 0; h, 0) = Zt

s

U(t, µ)[χ0h(µ)](θ)dµ, (21)

where the integral is deﬁned for each θas the integral of the function µ7→ U(t, µ)[χ0h(µ)](θ)in Rn.

Proof. The proof of this lemma is adapted from the proof of Theorem 16.3 of [12]. Let us denote x(t;s)h=

x(t;s, 0; h, 0). First, we note that operator x(t, s) : RCR([s, t],Rn)→Rnis linear (a consequence of Lemma

3.4.1) for each ﬁxed s≤t, and that it admits an extension to a linear operator ˜x(t, s) : Lloc

1([s, t],Rn)→Rn.

We do not prove this claim, since the proof is essentially identical to how one would prove Lemma 3.4.1. For

w∈[s, t] and denoting ˜xt= [˜x(·, s)h]tfor brevity, we see that

|˜xw(θ)| ≤ Zw+θ

s

|L(µ)˜xµ|dµ +Zw+θ

s

|h(µ)|dµ

≤ |h|1+Zt

s

(µ)||xµ||dµ,

which implies the uniform inequality ||xt|| ≤ |h|1+Rt

s(µ)||˜x||µdµ. Applying Lemma 3.2.1 yields ||˜xt|| ≤

e|`|1|h|1, where |·|1denotes the L1[s, t] norm. Thus, |˜x(t, s)h|=|˜xt(0)| ≤ e|`|1|h|1, so ˜xis bounded. By

classical results of functional analysis, there exists an integrable, essentially bounded n×nmatrix function

µ7→ V(t, s, µ) such that

˜x(t, s)h=Zt

s

V(t, s, µ)h(µ)dµ. (22)

First we show that V(t, s, µ) is independent of s. Let α∈[s, t] and let k∈ L1([s, t],Rn) be such that

k= 0 on [s, α]. Then ˜x(t, s)k=x(t, α)kand x(t, µ)k= 0 for µ∈[s, α]. Thus,

Zt

α

[V(t, s, µ)−V(t, α, µ)]k(µ)dµ = 0

for all k∈ L1([α, t],Rn). Thus, V(t, s, µ) = V(t, α, µ) almost everywhere on [α, t]. Since αis arbitrary, we

have that V(t, s, µ) is independent of s.

Deﬁne V(t, s) = V(t, s, ·) for any t≥sand V(t, s) = 0 for s < t. Let us denote ˜x(t) = ˜x(t, s)hand

Vτ−

i(θ, s) = V(τi+θ, s) when θ < 0 and Vτ−

i(0, s) = V(τ−

i, s). From the integral equation (14) and the

11

representation (22), we have

Zt

s

V(t, µ)h(µ)dµ

=Zt

s

L(µ)˜xµdµ +X

s<τi≤t

Bi˜xτ−

i+Zt

s

h(µ)dµ

=Zt

sZ0

−r

[dθη(µ, θ)]˜x(µ+θ)dµ +X

s<τi≤tZ0

−r

[dθγi(θ)]˜xτ−

i(θ) + Zt

s

h(µ)dµ

=Zt

sZ0

−r

[dθη(µ, θ)] Zµ+θ

s

V(µ+θ, ν )h(ν)dνdµ +X

s<τi≤tZ0

−r

[dθγk(θ)] Zτi+θ

s

Vτ−

i(θ, ν )h(ν)dν +Zt

s

h(µ)dµ

=Zt

sZ0

−r

[dθη(µ, θ)] Zµ

s

V(µ+θ, ν )h(ν)dνdµ +X

s<τi≤tZ0

−r

[dθγk(θ)] Zτi

s

Vτ−

i(θ, ν )h(ν)dν +Zt

s

h(µ)dµ

=Zt

sZt

νZ0

−r

[dθη(ν, θ)]V(µ+θ, ν )h(ν)dµdν +X

s<τi≤tZτi

sZ0

−r

[dθγi(θ)]Vτ−

i(θ, ν )h(ν)dν +Zt

s

h(µ)dµ

=Zt

s

Zt

µZ0

−r

[dθη(ν, θ)]V(ν+θ, µ)h(µ)dν +X

s<τi≤t

χ(−∞,τi](µ)Z0

−r

[dθγk(θ)]Vτ−

i(θ, µ)h(µ) + h(µ)

dµ

=Zt

s

Zt

µZ0

−r

[dθη(ν, θ)]V(ν+θ, µ)dν +X

s<τi≤t

χ(−∞,τi](µ)Z0

−r

[dθγk(θ)]Vτ−

i(θ, µ) + I

h(µ)dµ

=Zt

s

I+Zt

µ

L(µ)Vν(·, µ)dν +X

s<τi≤t

BiVτ−

i(·, µ)

h(µ)dµ.

Since the above holds for all h∈ L1([s, t],Rn), we have that the fundamental matrix V(t, s) satisﬁes

V(t, s) =

I+Zt

s

L(µ)Vµ(·, s)dµ +X

s<τi≤t

BiVτ−

i(·, s), t ≥s

0t < s.

(23)

almost everywhere. By uniqueness of solutions (Lemma 3.4.1, it follows that V(t, s)ξ=U(t, s)[χ0ξ](0) for

all ξ∈Rn. Since ˜x(t, s) is an extension of x(t, s) to the larger space L1([s, t],Rn), representation (22) holds

for h∈ RCR([s, t],Rn). Thus, for all t≥s,

xt(θ;s, 0; h, 0) = x(t+θ, s)h

=Zt+θ

s

V(t+θ, µ)h(µ)dµ

=Zt

s

V(t+θ, µ)h(µ)dµ

=Zt

s

U(t+θ, µ)[χ0h(µ)](0)dµ

=Zt

s

U(t, µ)[χ0h(µ)](θ)dµ,

which is what was claimed by equation (21).

With Lemma 4.0.1 through Lemma 4.1.2 at hand, we arrive at the variation of constants formula.

12

Lemma 4.1.3. Let h∈ RCR(R,Rn). Under hypotheses H.1–H.2, one has the variation of constants formula

xt(θ;s, φ;h, r) = U(t, s)φ(θ) + Zt

s

U(t, µ)[χ0h(µ)](θ)dµ +X

s<τi≤t

U(t, τi)[χ0ri](θ).(24)

4.2 Variation of constants formula in the space RCR

The goal of this section will be to reinterpret the variation of constants formula (24) in such a way that the

integral appearing therein may be thought of as a weak integral in the space RCR. Speciﬁcally, we will show

that the integral may be regarded as a Gelfand-Pettis integral. This form has several advantages, the most

important being it will allow us to later commute bounded projection operators with the integral sign. We

recall the following deﬁnition, which appears in [23].

Deﬁnition 4.2.1. Let Xbe a Banach space and (S, Σ, µ)a measure space. We say that f:S→Xis Pettis

integrable if there exists a set function If: Σ →Xsuch that

ϕ∗If(E) = ZE

ϕ∗fdµ

for all ϕ∗∈X∗and E∈Σ.Ifis the indeﬁnite Pettis integral of f, and If(E)the Pettis integral of fon E.

By abuse of notation, we will often write If(E) = REfdµ when there is no ambiguity. For our purposes,

the following proposition will be of primary usefulness. Its proof is elementary and can be found in numerous

textbooks on functional analysis and integration.

Proposition 4.2.1. The pettis integral posesses the following properties.

•If fis Pettis integrable, then its indeﬁnite Pettis integral is unique.

•If T:X→Xis bounded, then TREfdµ=RE(T f )dµ whenever one of the integrals exists.

•If µ(A∩B) = 0, then RA∪Bfdµ =RAf dµ +RBf dµ.

• || REf dµ|| ≤ RE||f||dµ

Lemma 4.2.1. Let h∈ RCR(R,Rn)and let H.1–H.2 hold. The function U(t, ·)[χ0h(·)] : [s, t]→ RCR is

Pettis integrable for all t≥sand

Zt

s

U(t, µ)[χ0h(µ)]dµ(θ) = Zt

s

U(t, µ)[χ0h(µ)](θ)dµ. (25)

Proof. By Lemma 3.1.3 and the uniqueness assertion of Proposition 4.2.1, if we can show for all p: [−r, 0] →

Rnof bounded variation the equality

Z0

−r

p∗(θ)dZt

s

U(t, µ)[χ0h(µ)](θ)dµ=Zt

sZ0

−r

p∗(θ)dhU(t, µ)[χ0h(µ)](θ)idµ

holds, then Lemma 4.2.1 will be proven. Note that the above is equivalent to

Z0

−r

p∗(θ)dZt

s

V(t+θ, µ)h(µ)dµ=Zt

sZ0

−r

p∗(θ)dhV(t+θ, µ)h(µ)idµ. (26)

We prove the lemma ﬁrst when his a step function. In this case, a consequence of equation (23) is that

θ7→ V(t+θ, µ)h(µ) and µ7→ V(t+θ, µ)h(µ) are piecewise continuous, while Lemma 3.4.1 and Lemma 4.1.2

imply θ7→ Rt

sV(t+θ, µ)h(µ)dµ is also piecewise continuous, all with at most ﬁnitely many discontinuities on

any given bounded set. Conseqently, both integrals in (26) can be regarded as a Lebesgue-Stieltjes integrals,

with Fubini’s theorem granting the desired equality.

13

When h∈ RCR(R,Rn) is an arbitrary right-continuous regulated function, we approximate its restriction

to the inverval [s, t] by a convergent sequence of step functions hnby Lemma 3.1.2. Equation (26) is then

satisﬁed with hreplaced with hn. Deﬁne the functions

Jn(θ) = Zt

s

V(t+θ, µ)hn(µ)dµ, Kn(µ) = Z0

−r

p∗(θ)dhV(t+θ, µ)hn(µ)i,

J(θ) = Zt

s

V(t+θ, µ)h(µ)dµ, K(µ) = Z0

−r

p∗(θ)dhV(t+θ, µ)h(µ)i,

so that R0

−rp∗(θ)dJn(θ) = Rt

sKn(µ)dµ. By Lemma 3.4.2 and elementary integral estimates, Jn→Juni-

formly. The conditions of Lemma 3.1.5 are satisﬁed, and we have the limit

Z0

−r

p∗(θ)dJn(θ)→Z0

−r

p∗(θ)dJ(θ).

Conversely, for each µ∈[s, t], Lemma 3.1.4 applied to the diﬀerence Kn(µ)−K(µ) yields, together with

Lemma 3.4.2,

|Kn(µ)−K(µ)| ≤ (|p(0)|+|p(−r)|+ var0

−rp)Zt

s

exp Zt

y

(ν)dνdy||hn−h||.

Thus, Kn→Kuniformly, and the bounded convergence theorem implies Rt

sKn(µ)dµ →Rt

sK(µ)dµ. There-

fore, equation (26) holds, and the lemma is proven.

With Lemma 4.1.3 and Lemma 4.2.1 at hand, we obtain the variation of constants formula for the linear

inhomogeneous equation (4)–(5) in the Banach space RCR.

Theorem 4.2.1. Let H.1–H.2 hold, and let h∈ RCR(R,Rn). The unique solution t7→ xt(·;s, φ;h, r)∈ RCR

of the linear inhomogeneous impulsive system (4)–(5) with initial condition xs(·;s, φ;h, r) = φ, satisﬁes the

variation-of-constants formula

xt(·;s, φ;h, r) = U(t, s)φ+Zt

s

U(t, µ)[χ0h(µ)]dµ +X

s<τi≤t

U(t, τi)[χ0ri],(27)

where the integral is interpreted in the Pettis sense and may be evaluated pointwise using (25).

As a simple corollary, we can express any solution t7→ xtdeﬁned on [s, ∞) as the solution of an integral

equation.

Corollary 4.2.1.1. Let H.1–H.2 hold, and let h∈ RCR(R,Rn). Any solution t7→ xt∈ RCR of the linear

inhomogeneous impulsive system (4)–(5) deﬁned on the interval [s, ∞)satisﬁes for all t≥sthe equation

xt=U(t, s)xs+Zt

s

U(t, µ)[χ0h(µ)]dµ +X

s<τi≤t

U(t, τi)[χ0ri].(28)

5 Existence of Lipschitz continuous centre manifolds

This section will be devoted to the existence of centre manifolds, a reduction principle, and the derivation

of abstract impulsive diﬀerential equations restricted to the centre manifold.

5.1 Preliminary deﬁnitions and mild solutions of an abstract integral equation

At this stage it is appropriate to introduce several exponentially weighted Banach spaces that will be needed

to construct the centre manifolds. First, denote P C (R,Rn) the set of functions f:R→Rnthat are

14

continuous everywhere except for at times t∈ {τk:k∈Z}where they are continuous from the right and

have limits on the left.

PCη={φ:R→ RCR :φ(t) = ft, f ∈P C(R,Rn),||φ||η= sup

t∈R

e−η|t|||φ(t)|| <∞}

Bη(R,RCR) = {f:R→ RCR :||f||η= sup

t∈R

e−η|t|||f(t)|| <∞}

P Cη(R,Rn) = {f∈P C (R,Rn) : ||f||η= sup

t∈R

e−η|t|||f(t)|| <∞}

Bη

τk(Z,Rn) = {f:Z→Rn:||f||η= sup

k∈Z

e−η|τk||fk|<∞}.

Also, if M ⊂ R× RCR is a nonautonomous set over RCR, we deﬁne the space PCη(R,M) of piecewise-

continuous functions taking values in Mby

PCη(R,M) = {f∈ P Cη:f(t)∈ M(t)}.

If Xηis one of the above spaces, then the normed space Xη,s = (Xη,|| · ||η,s) with norm

||F||η,s =supt∈Re−η|t−s|||F(t)||,dom(F) = R

supk∈Ze−η|τk−s|||F(k)||,dom(F) = Z,

is complete.

Our attention shifts now to the semilinear system

˙x=L(t)xt+f(t, xt), t 6=τk(29)

∆x=Bkxt−+gk(xt−), t =τk,(30)

for nonlinearities f:R× RCR → Rnand gk:RC R → Rn. Additional assumptions on the nonlinearities,

evolution family and sequence of impulses may include the following.

H.3 For each φ∈ RCR([α−r, β],Rn), the function t7→ f(t, φt) is an element of RCR([α, β],Rn).

H.4 The evolution family U(t, s) : RCR → RCR associated to the homogeneous equation (15)–(16) is

spectrally separated.

H.5 φ7→ (t, φ) and φ7→ gk(φ) are Cmfor some m≥1 for each t∈Rand k∈Z, and there exists δ > 0

such that for each j= 0, . . . , m, there exists cj:R→R+locally bounded and a positive sequence

{dj(k) : k∈Z}such that

||Djf(t, φ)−Djf(t, ψ)|| ≤ cj(t)||φ−ψ||,

||Djgk(φ)−Djgk(ψ)|| ≤ dj(k)||φ−ψ||.

for φ, ψ ∈Bδ(0) ⊂ RCR. Also, there exists q > 0 such that ||Djf(t, φ)|| ≤ qcj(t) and ||Djgk(φ)|| ≤

qdj(k) for φ∈Bδ(0).

H.6 f(t, 0) = gk(0) = 0 and Df(t, 0) = Dgk(0) for all t∈Rand k∈Z.

H.7 There exists a constant ξ > 0 such that τk+1 −τk≥ξfor all k∈Z.

Deﬁnition 5.1.1. Amild solution of the semilinear equation (29)–(30) is a function x: [s, T ]→ RC R such

that for all s≤t<T, the function x(t) = xtsatisﬁes the integral equation

x(t) = U(t, s)x(s) + Zt

s

U(t, µ)[χ0f(µ, x(µ))]dµ +X

s<τi≤t

U(t, τi)[χ0g(τi, x(τ−

i))],(31)

and x(t)(θ) = x(t+θ)(0) whenver θ∈[−r.0] satisﬁes t+θ∈[s, T ], where Uis the evolution family associated

to the homogeneous equation (15)–(16) and the integral is interpreted in the Pettis sense.

15

Remark 5.1.1. The right-hand side of equation (31) is well-posed under conditions H.1–H.3 in the sense

that it deﬁnes for s≤t<T, a nonlinear operator from RCR([s−r, t],Rn)into RCR. Note also that for a

function x: [s, T ]→ RCR, we denote x(τ−

i)(θ) = x(τ−

i)(θ)for θ < 0and x(τ−

i)(0) = x(τi)(0−).

If x: [s−r, T )→Rnis a classical solution — that is, xis diﬀerentiable from the right, continuous except at

impulse times τk, continuous from the right on [s−r, T ] and its derivative satisﬁes the diﬀerential equation

(29)–(30) — then t7→ xtis a mild solution. This can be seen by deﬁning the inhomogeneities h(t)≡f(t, xt)

and rk≡gk(xτ−

k), solving the equivalent linear equation (4)–(5) with these inhomogeneities and intial

condition (s, xs)∈R× RCR in the mild sense, and applying Corollary 4.2.1.1. For this reason, we will work

with equation (31) exclusively from now on.

Additionally, one should note that the assumption H.5 implies that the nonlinearities are uniformly locally

Lipschitz continuous. Together with the other assumptions, this implies the local existence and uniqueness

of mild solutions through a given (s, φ)∈R× RCR. Namely, we have the following lemma, which may be

seen as a local, nonlinear version of (3.4.1).

Lemma 5.1.1. Under assumptions H.1–H.5, for all (s, φ)∈R×D, there exists a unique mild solution

x(s,φ): [s, s +α)→ RCR of (31) for some α=α(s, φ)>0, satisfying x(s) = φ. Also, if one deﬁnes the

nonautonomous set

M=[

φ∈RCR [

s∈R[

t∈[s,s+α)

{t}×{s}×{φ},

then S:M → RCR with S(t, s)x=x(s,φ)(t)is a process on RCR.

Combining the discussion following Deﬁnition 5.1.1 with Lemma 5.1.1, it follows that S(t, s) satisﬁes the

following abstract integral equation wherever it is deﬁned.

S(t, s)φ=U(t, s)φ+Zt

s

U(t, µ)χ0f(µ, S(µ, s)φ)dµ +X

s<τi≤t

U(t, τi)χ0g(τi, S(τ−

i, s)φ).(32)

Of use later will be a result concerning the smoothness of the process S:M → RCR. This result is interesting

in its own right and will be useful later in proving the periodicity of centre manifolds; see Theorem 8.3.1.

Theorem 5.1.1. Under hypotheses H.1–H.6, the process S:M → RCR is Ck+1. Also, DS(t, s) =

DS(t, s)φsatisﬁes for t≥sthe abstract integral equation

DS(t, s) = U(t, s) + Zt

s

U(t, µ)χ0Df (µ, S (µ, s)φ)DS(µ, s)dµ +X

s<τi≤t

U(t, τi)χ0Dg(τi, S (τ−

i, s)φ)DS(τ−

i, s).

(33)

Proof. We will prove only that Sis C1, the proof of higher-order smoothness being an essentially identical

albeit notationally cumbersome extension thereof. Let s∈Rbe ﬁxed. Let ψ∈ RCR be given. For given

ν > 0, denote by Bν(ψ) the closed ball centreed at ψwith radius νin RCR.

Introduce for given , δ, ν > 0 the normed vector space (X,δ,ν ,||· ||), where X,δ,ν consists of the functions

φ: [s−r, s +]×Bδ(ψ)→Bν(ψ) such that x7→ φ(t, x) is continuous for each t,φ(t, x)(θ) = φ(t+θ, x)(0)

whenever θ∈[−r, 0] and [t+θ, t]⊂[s−r, s +], and |φ|| <∞for the norm given by

||φ||,δ,ν = sup

t∈[s−r,s+]

||x−ψ||≤δ

||φ(t, x)||.

It can be easily veriﬁed that (X,δ,ν ,|| · ||) is a Banach space. With L(RCR) the bounded linear operators

on RCR, introduce also the space (X,δ ,|| · ||) consisting of functions Φ : [s−r, s +]× RCR → L(RCR)

such that x7→ Φ(t, x) is continuous for each t, Φ(t, x)h(θ) = Φ(t+θ, x)h(0) for all h∈ RCR, and ||Φ|| <∞,

where the norm is ||Φ(t, x)|| = sup||h||=1 ||Φ(t, x)h||,δ,ν . Clearly, (X,δ,|| · ||) is complete.

16

Deﬁne a pair of nonlinear operators

Λ1:X,δ,ν →X,δ,ν,

Λ1(φ)(t, x) = χ[s−r,s)(t)x(t−s) + χ[s,s+](t)U(t, s)x(s) + Zt

s

U(t, s)χ0f(µ, φ(µ, x))dµ

+X

s<τi≤t

U(t, τi)χ0g(τi, φ(τ−

i, x))

Λ2:X,δ ×X,δ →X,δ

Λ2(φ, Φ)(t, x)h=χ[s−r,s)(t)IRCRh+χ[s,s+](t)U(t, s)h+Zt

s

U(t, µ)χ0Df (µ, φ(µ, x))Φ(µ, x)hdµ+

+X

s<τi≤t

U(t, µ)χ0Dg(τi, φ(τ−

i, x)))Φ(τ−

i, x)h

, h ∈ RCR.

By choosing and δsmall enough, Λ1can be shown to be a uniform contraction. Indeed, if we denote

κ= sup||x−ψ||≤2δ||x||, the mean-value theorem grants the estimate

||Λ1(φ)−Λ1(γ)|| ≤ κsup

t∈[s,s+]

Zt

s

||U(t, µ)||c1(µ)dµ +X

s<τi≤t

||U(t, τi)||d1(i)

||φ−γ||

≡κL||φ−γ||

We can always obtain a uniform contraction by taking small enough. Also, note that t7→ Λ1(φ)(t, x)∈

RCR,x7→ Λ1(φ, x) is continuous and Λ1(φ)(t, x)(θ) = Λ1(φ)(t+θ, x)(0). To ensure the appropriate

boundedness, if we denote κ= sup||x−ψ||≤δk0(x), the estimate

||Λ1(φ)−ψ|| ≤ ||φ−ψ|| +κsup

t∈[s,s+]

Zt

s

||U(t, µ)||c0(µ)dµ +X

s<τi≤t

||U(t, τi)||d0(i)

≡δ+κM

implies it is suﬃcient to choose , δ, ν > 0 small enough so that δ+κM< ν. This can always be done

because M→0 as →0 due to H.5 and Lemma 3.4.2.

The continuity of φ7→ Λ2(φ, Φ) for ﬁxed Φ ∈X,δ follows by the estimate

||Λ2(φ, Φ) −Λ2(γ, Φ)|| ≤ Zs+

s

||U(s+, µ)||c1(µ)||(φ(µ, x)−γ(µ, x)||dµ

+X

s<τi≤s+

||U(s+, τi)||d1(i)||φ(τ−

i, x)−γ(τ−

i, x)||

||Φ||.

Also, for each φ∈Bδ(ψ) it is readily veriﬁed that ||Λ2(φ, Φ) −Λ2(φ, Γ)|| ≤ κL||Φ−Γ||, which by previous

choices of , δ, ν > 0 indicates that Φ 7→ Λ2(φ, Φ) is a uniform contraction.

We are ready to prove the statement of the theorem. Denote by (xn, x0

n) the iterates of the map Λ :

X,δ,ν ×X,δ,ν →X,δ,ν ×X,δ,ν deﬁned by Λ(x, x0) = (Λ1(x),Λ2(x, x0)) and initialized at (x0, x0

0) with

x0(t, x) = xand x0

0(t, x) = IRCR. The ﬁber contraction theorem [15] implies convergence (xn, x0

n)→(x, x0).

Note also that Dx0=x0

0. If we suppose Dxn=x0

nfor some n≥0, then for t≥s, Lemma 4.2.1 implies that

17

for each θ∈[−r, 0],

Dxn+1(t, φ)(θ) = D

U(t, s)xn(s, φ)(θ) + Zt

s

U(t, µ)χ0f(µ, xn(µ, φ))(θ)dµ +X

s<τi≤t

U(t, τi)χ0g(τi, xn(τ−

i, φ))(θ)

=D

U(t, s)xn(s, φ)(θ) + Zt

s

V(t+θ, µ)f(µ, xn+1(µ, φ))dµ +X

s<τi≤t

V(t+θ, τi)g(τi, xn+1(τ−

i, φ))

=U(t, s)x0

n(s, φ)(θ) + Zt

s

V(t+θ, µ)Df (µ, xn(µ, φ))x0

n(µ, φ)dµ

+X

s<τi≤t

V(t+θ, τi)Dg(τi, xn(τ−

i, φ))x0

n(τ−

i, φ)

= Λ2(xn, x0

n)(t, φ)(θ)

=x0

n+1(t, φ)(θ),

while for t<s, it is easily checked that Dxn+1(t, φ) = x0

n+1(t, φ).This proves that Dxn+1(θ) = x0

n+1(θ)

pointwise in θ. To prove the result uniformly, we note that the diﬀerence quotient can be written for t≥s

as

1

||h||xn+1(t, φ +h)−xn+1 (t, φ)−x0

n+1(t, φ)h

=Zt

s

U(t, µ)χ0

1

||h||f(µ, xn(µ, φ +h)) −f(µ, xn(µ, φ)) −Df(µ, xn(µ, φ))Dxn(µ, φ)hdµ

+X

s<τi≤t

U(t, τi)χ0

1

||h|||g(τi, xn(τ−

i, φ +h)) −g(τi, xn(τ−

i, φ)) −Dg(τi, xn(τ−

i, φ))Dxn(τ−

i, φ)h.

Since xnis diﬀerentiable by the induction hypothesis, the integrand and summand converge uniformly to

zero as h→0. Thus, xn+1 is diﬀerentiable and Dxn+1 =x0

n+1, so by induction Dxn=x0

nfor each n.

Also, by construction, x0

nis continuous for each nand, being the uniform limit of continuous functions,

x0= limn→∞ x0is continuous. By the fundamental theorem of calculus,

x(φ+h)−x(φ)−x0(φ)h

||h|| = lim

n→∞

xn(φ+h)−xn(φ)−Dxn(φ)h

||h||

= lim

n→∞ Z1

0

1

||h|| [x0

n(φ+ (λ−1)h)−x0

n(φ)] hdλ

=Z1

0

1

||h|| [x0(φ+ (λ−1)h)−x0(φ)] hdλ →0

as h→0. By deﬁnition, xis diﬀerentiable and Dx =x0.

If we deﬁne y(t)φ=x(t, φ) for the ﬁxed point x: [s−r, s +]×Bδ(ψ)→Bν(ψ), then ysatisﬁes

y(t)φ=S(t, s)φfor (t, φ)∈[s, s +]×Bδ(ψ). This can be seen by comparing the ﬁxed point equation

y(t)=Λ1(y)(t, φ) with the abstract integral equation (32). We conclude that Sis C1(ﬁbrewise). The

correctness of equation (33) follows by comparing to the ﬁxed point equation associated to Λ2.

5.2 Bounded solutions of the inhomogeneous linear equation

In this section we will identify a pseudoinverse for η-bounded solutions of the inhomogeneous linear equation

x(t) = U(t, s)x(s) + Zt

s

U(t, µ)[χ0F(µ)]dµ +X

s<τi≤t

U(t, τi)[χ0Gi],−∞ < s ≤t < ∞.(34)

As deﬁned in Deﬁnition 3.3.5, we recall now that RCRc(t) = R(Pc(t)), where Pcis the projection onto the

centre bundle of the linear part of (29)–(30).

18

Lemma 5.2.1. Let η∈(0,min{−a, b})and let H.1, H.2 and H.4 hold. Then,

RCRc(ν) = {ϕ∈ RCR :∃x∈ PCη, x(t) = U(t, s)x(s), x(ν) = ϕ}.(35)

Proof. If ϕ∈ RCRc(ν), then Pc(ν)ϕ=ϕand the function x(t) = U(t, ν )Pc(ν)ϕ=Uc(t, ν)ϕis deﬁned for

all t∈R, satisﬁes x(t) = U(t, s)x(s), x(ν) = ϕ,x(t)(θ) = x(t+θ)(0), and by chosing < η, there exists

K > 0 such that

e−η|t|||x(t)|| ≤ Ke|ν|e−(η−)|t|||ϕ|| ≤ K e|ν|||ϕ||.

Finally, as x(t) = [U(t, s)x(s)(0)]tfor all t∈R, we conclude x∈ PCη.

Conversely, suppose ϕ∈ RCR admits some x∈ P Cηsuch that x(t) = U(t, s)x(s) and x(ν) = ϕ. Let

||x||η=K. We will show that Ps(ν)ϕ=Pu(ν)ϕ= 0, so that ϕ=I ϕ = (Pc(ν) + Ps(ν) + Pu(ν))ϕ=Pc(ν)ϕ,

from which we will conclude ϕ∈ RCRc(ν).

By spectral separation, we have for all ρ<ν,

e−η|ρ|||Ps(ν)ϕ|| =e−η|ρ|||Us(ν, ρ)Ps(ρ)x(ρ)||

≤e−η|ρ|Kea(ν−ρ)||Ps(ρ)|| · ||x(ρ)||

≤KK ea(ν−ρ)||Ps(ρ)||,

which implies ||Ps(ν)ϕ|| ≤ KKeaν ||Ps(ρ)|| exp(η|ρ| − aρ).Since η < −aand ρ7→ ||Ps(ρ)|| is bounded, taking

the limit as ρ→ −∞ we obtain ||Ps(ν)ϕ||| ≤ 0. Similarly, for ρ>ν, we have

e−η|ρ|||Pu(ν)ϕ|| =e−η|ρ|||Uu(ν, ρ)Pu(ρ)x(ρ)||

≤e−η|ρ|Keb(ν−ρ)||Pu(ρ)|| · ||x(ρ)||

≤KK eb(ν−ρ)||Pu(ρ)||,

which implies ||Pu(ν)ϕ|| ≤ KKebν ||Pu(ρ)|| exp(η|ρ| − bρ). Since η <b and ρ7→ ||Pu(ρ)|| is bounded, taking

the limit ρ→ ∞ we obtain ||Pu(ν)ϕ|| ≤ 0. Therefore, Ps(ν)ϕ=Pu(ν)ϕ= 0, and we conclude that

Pc(ν)ϕ=ϕand ϕ∈ RCRc(ν).

Lemma 5.2.2. Let conditions H.1, H.2 and H.4 be satisﬁed. Let h∈ RCR(R,Rn). The integrals

Zt

s

U(t, µ)Pc(µ)[χ0h(µ)]dµ, Zt

v

U(t, µ)Pu(µ)[χ0h(µ)]dµ

are well-deﬁned as Pettis integrals for all s, t, v ∈R, where we deﬁne Ra

bfdµ =−Rb

afdµ when a < b.

Proof. The nontrivial cases are where t≤sand t≤v. For the former, deﬁning H(µ) = χ0h(µ) we have the

string of equalities

Uc(t, s)Pc(s)Zs

t

U(s, µ)H(µ)dµ =Uc(t, s)Zs

t

Uc(s, µ)Pc(µ)H(µ)dµ

=Zs

t

Uc(t, µ)Pc(µ)H(µ)dµ

=Zs

t

U(t, µ)Pc(µ)H(µ)dµ

=−Zt

s

U(t, µ)Pc(µ)H(µ)dµ.

The ﬁrst integral on the left exists due to Lemma 4.2.1 and Proposition 4.2.1. The subsequent equalities

follow by Proposition 4.2.1 and the deﬁnition of spectral separation. The case t≤vfor the other integral is

proven similarly.

19

Deﬁne the (formal) family of linear operators Kη

s:PCη(R,Rn)⊕Bη

τk(Z,Rn)→Bη(R,RCR) by the

equation

Kη

s(F, G)(t) = Zt

s

U(t, µ)Pc(µ)[χ0F(µ)]dµ −Z∞

t

U(t, µ)Pu(µ)[χ0F(µ)]dµ +Zt

−∞

U(t, µ)Ps(µ)[χ0F(µ)]dµ

+

t

X

s

U(t, τi)Pc(τi)[χ0Gi]dτi−

∞

X

t

U(t, τi)Pu(τi)[χ0Gi]dτi+

t

X

−∞

U(t, τi)Ps(τi)[χ0Gi]dτi,

(36)

indexed by s∈R, where the external direct sum PCη,s (R,Rn)⊕Bη,s

τk(Z,Rn) is identiﬁed as a Banach space

with norm ||(f, g)||η,s =||f||η,s +||g||η,s, and the summations are deﬁned as follows:

b

X

a

F(τi)dτi=

X

a<τi≤b

F(τi), a ≤b

−

a

X

b

F(τi)dτi, b < a.

Lemma 5.2.3. Let H.1, H.2, H.4 and H.7 hold, and let η∈(0,min{−a, b}).

1. The function Kη

s:PCη,s(R,Rn)⊕Bη,s

τk(Z,Rn)→Bη,s(R,RC R)with η∈(0,min{−a, b})and deﬁned

by formula (36) is linear and bounded. In particular, the norm satisﬁes

||Kη

s|| ≤ C1

η−1 + e(η−)ξ

ξ+1

−a−η1 + 2e(η−a)ξ

ξ+1

b−η1 + 2e(b+η)ξ

ξ (37)

for some constants Cand independent of s.

2. Kη

shas range in PCη,s and v=Kη

s(F, G)is the unique solution of (34) in PCη,s satisfying Pc(s)v(s) =

0.

3. The expression K∗(F, G)(t) = (I−Pc(t))K0

s(F, G)(t)uniquely deﬁnes, independent of s, a bounded

linear map

K∗:PC0(R,Rn)⊕B0

τk(Z,Rn)→ PC0.

Proof. Let < min{min{−a, b} − η, η}. To show that Kη

sis well-deﬁned, we start by mentioning that all

improper integrals and inifnite sums appearing on the right-hand side of (36) can be interpreted as limits of

well-deﬁned ﬁnite integrals and sums, due to Lemma 4.2.1, Lemma 5.2.2 and Proposition 4.2.1. For brevity,

write

Kη

s(F, G) = Ku,f

1−Kc,F

1+Ku,F

1+Ku,G

2−Kc,G

2+Ks,G

2,

where each term corresponds to the one in (36) in order of appearance.

We start by proving the convergence of the improper integrals. Denote

I(v) = Zv

t

U(v, µ)Pu(µ)[χ0F(µ)]dµ,

and let vk% ∞. We have, for m>nand nsuﬃciently large so that vm>0,

||I(vm)−I(vn)|| ≤ Zvm

vn

KN eb(t−µ)|F(µ)|dµ

≤Zvm

vn

KN eb(t−µ)eηµ||F||ηdµ

=KN ||F||ηebt Zvm

vn

eµ(η−b)dµ

=KN ||F||η

b−ηebt e−vn(b−η)−e−vm(b−η)

≤KN ||F||η

b−ηebte−vn(b−η).

20

Therefore, I(vk)∈ RCR is Cauchy, and thus converges; namely, it converges to the improper integral Ku,F (t).

One can similarly prove that Ks,F (t) converges. For the inﬁnite sums, we employ similar estimates; if we

denote S=Pt<τi<∞||Uu(t, τi)[χ0Gi]|| and assume without loss of generality that τ0= 0, a fairly crude

estimate (that we will later improve) yields

S≤X

t<τi<∞

KN eb(t−τi)eη|τi|||G||η

=X

−|t|<τi≤0

KN ||G||ηebte|τi|(b+η)+X

0<τk<∞

KN ||G||ηebte−(b−η)τi

≤KN ebt |t|

ξe|t|(b+η)+1

1−e−(b−η)ξ||G||η.

Thus, Ku,G(t) converges uniformly. One can show by similar means that Ks,F (t) and Ks,G(t) both converge.

Therefore, Kη

s(F, G)(t)∈ RCR exists. We can now unambigiously state that Kη

sis clearly linear.

Our next task is to prove that ||Kη

s(F, G)]||η,s ≤Q||(F, G)||η,s for constant Qsatisfying the estimate

of equation (37). We will prove the bounds only for ||Ku,F ||η,s ,||Ku,G||η ,s,||Kc,F ||η,s and ||Kc,G||η,s; the

others follow by similar calculations. For t < s, we we have

e−η|t−s|||Ku,F (t)|| ≤ e−η|t−s|Z∞

t

KN eb(t−µ)|F(µ)|dµ

≤eη(t−s)KN Zs

t

eb(t−µ)eη|µ−s|||F||η,sdµ +Z∞

s

eb(t−µ)eη|µ−s|||F||η,sdµ

=eη(t−s)KN ||F||η,s Zs

t

eb(t−µ)eη(s−µ)dµ +Z∞

s

eb(t−µ)eη(µ−s)dµ

=eη(t−s)KN ||F||η,s ebt+ηs e−(b+η)t−e