Abstract

The existence and smoothness of centre manifolds and a reduction principle are proven for impulsive delay differential equations. Several intermediate results of theoretical interest are developed, including a variation of constants formula for linear equations in the phase space of right-continuous regulated functions, linear variational equation and smoothness of the nonautonomous process, and a Floquet theorem for periodic systems. Three examples are provided to illustrate the results.
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 tRand kZ, 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:kZ}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 RCRiR× 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(ts)xs+Zt
s
T(tµ)χ0h(µ)dµ,
where T(t) : XXis 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(tµ).
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)+X
s<τit
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 sR, 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 sR,
u=U(·, s)ϕ+Kη
sRδ(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 ZR, 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 × XYwith T Rand Banach spaces X, Y ,
Dkf(t, x)∈ Lk(X× · ·· × X, X ) denotes the kth Fechet 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 rθ < 0 and xt(0) =
limθ0x(t+θ), and r > 0 is finite. We will be working exclusively with spaces of right-continuous regulated
functions; denote
RCR(I , X) = f:IX:tI , lim
st+f(s) = f(t) and lim
st
f(s) exists,
where XRnand IR. When Xand Iare closed,
RCRb(I , X) := {f∈ RCR(I , X) : ||f|| <∞}
is a Banach space with the norm ||f|| = supxI|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:IXthat 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 tR, and
such that |L(t)φ| ≤ (t)||φ|| for some :RRlocally integrable.
H.2 The sequence τkis monotonically increasing with |τk|→∞as |k|→∞, and the representation
Bkφ=Z0
r
[dθγk(θ)]φ(θ)
holds for kZfor 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(trk(t))
with rkcontinuous, is associated to a linear operator satisfying condition H.1 with η(t, θ) = PAk(t)Hrk(t)(θ),
where Hz(θ)=1if θzand 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 ba+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 sn0, 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 xnˆx[r, 0]. If ˆx > r, then we have
||φt+sn|| ≥ sup
µ[r+sn,0]
|φ(t+µ)|=|φx)|=||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 nN,
||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 nN,
≤ ||φt+sn|| − ||φt|| ≤ .
We conclude ||φt+sn|| converges to ||φt||.
We will eventually need spaces of function f:IXthat 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:IX
such that ||fnf|| → 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∈ RCRif and only if there exists qRnand p: [r, 0] Rnof bounded variation such
that
F(x) = qx(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 ||hnh|| → 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))+X
s<τit
(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 ts, define
z(t, s) = exp Zt
s
p(µ)Y
s<τit
(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)+X
s<τit
z(t, τi)gi.(10)
Lemma 3.2.2. Let f:RRbe continuous and suppose {τk}satifies τk+1 τkξ.
1. If fis increasing, then Ps<τitf(τi)1
ξRt+ξ
sf(µ).
2. If fis decreasing, then Ps<τitf(τi)1
ξRt
sξf(µ).
Proof. Let {τ0, . . . , τN}={τk:kZ} ∩ (s, t]. If fis increasing, then
X
s<τit
f(τi) =
N
X
i=0
f(τi) = 1
δ
N
X
i=0
f(τi)ξ1
ξ
N
X
i=0
f(τ0+)ξ1
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)}tsRon Xthat satisfy U(t, s) = U(t, v)U(v, s)for all tvsand 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
tR, 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 tR.
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, ·) : MtYhas the same regularity property. That is, regularity is defined fibrewise.
Definition 3.3.5. Let U(t, s) : XXbe 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 tvsand 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 suptR(||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 tsand 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 st.
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 K1such that
||Uu(t, s)|| ≤ Keb(ts), t s(11)
||Uc(t, s)|| ≤ Ke|ts|, t, s R(12)
||Us(t, s)|| ≤ Kea(ts), 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) : XXbe spectrally separated. The nonautonomous sets
Xi={(t, x) : tR, 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 sR, there
exists a unique function x∈ RCR([sr, ),Rn)satisfying xs=φand the integral equation
x(t) = (φ(0) + Rt
s[L(µ)xµ+h(µ)]+Ps<τit[Bixτ
i+ri], t > s
φ(ts), s rts. (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 tsis the evolution family associated to the homogeneous equation (15)(16).
Lemma 3.4.2. The evolution family satisfies the following properties.
1) For st,U(t, s) : RCR → RCR is a bounded linear operator. In particular,
||U(t, s)|| ≤ exp Zt
s
(µ)Y
s<τit
(1 + b(i)).(17)
2) For svt,U(t, s) = U(t, v)U(v, s).
3) For all θ[r, 0],st+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}kZ∈ {{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 tsand, 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|+X
s<τit+θ
|BiU(τ
i, s)φ|
≤ ||φ|| +Zt
s
(µ)||U(µ, s)φ||+X
s<τit
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)||+X
s<τit
b(i)||X(τ
i)||.
By Lemma 3.1.1, t7→ ||X(t)|is an element of RCR([sr, ),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<τiτk
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<τit
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<τitU(t, τi)χ0rifor all t[s, τk) for some k1, we have
xτk=xτ
k+χ0Bkxτ
k+χ0rk
=U(τk, τk1)xτk1+χ0rk
=U(τk, τk1)Ps<τiτk1U(τk1, τi)χ0ri+χ0rk
=Ps<τiτkU(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 st, 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(µxµ|+Zw+θ
s
|h(µ)|
≤ |h|1+Zt
s
(µ)||xµ||dµ,
which implies the uniform inequality ||xt|| ≤ |h|1+Rt
s(µ)||˜x||µ. 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)= 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 tsand 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)
=Zt
s
L(µxµ+X
s<τit
Bi˜xτ
i+Zt
s
h(µ)
=Zt
sZ0
r
[dθη(µ, θ)]˜x(µ+θ)+X
s<τitZ0
r
[dθγi(θ)]˜xτ
i(θ) + Zt
s
h(µ)
=Zt
sZ0
r
[dθη(µ, θ)] Zµ+θ
s
V(µ+θ, ν )h(ν)dνdµ +X
s<τitZ0
r
[dθγk(θ)] Zτi+θ
s
Vτ
i(θ, ν )h(ν)+Zt
s
h(µ)
=Zt
sZ0
r
[dθη(µ, θ)] Zµ
s
V(µ+θ, ν )h(ν)dνdµ +X
s<τitZ0
r
[dθγk(θ)] Zτi
s
Vτ
i(θ, ν )h(ν)+Zt
s
h(µ)
=Zt
sZt
νZ0
r
[dθη(ν, θ)]V(µ+θ, ν )h(ν)dµdν +X
s<τitZτi
sZ0
r
[dθγi(θ)]Vτ
i(θ, ν )h(ν)+Zt
s
h(µ)
=Zt
s
Zt
µZ0
r
[dθη(ν, θ)]V(ν+θ, µ)h(µ)+X
s<τit
χ(−∞i](µ)Z0
r
[dθγk(θ)]Vτ
i(θ, µ)h(µ) + h(µ)
=Zt
s
Zt
µZ0
r
[dθη(ν, θ)]V(ν+θ, µ)+X
s<τit
χ(−∞i](µ)Z0
r
[dθγk(θ)]Vτ
i(θ, µ) + I
h(µ)
=Zt
s
I+Zt
µ
L(µ)Vν(·, µ)+X
s<τit
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)+X
s<τit
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 ξRn. 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 ts,
xt(θ;s, 0; h, 0) = x(t+θ, s)h
=Zt+θ
s
V(t+θ, µ)h(µ)
=Zt
s
V(t+θ, µ)h(µ)
=Zt
s
U(t+θ, µ)[χ0h(µ)](0)
=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)+X
s<τit
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:SXis Pettis
integrable if there exists a set function If: Σ Xsuch that
ϕIf(E) = ZE
ϕf
for all ϕXand EΣ.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) = REfwhen 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:XXis bounded, then TREf=RE(T f )whenever one of the integrals exists.
If µ(AB) = 0, then RABf=RAf +RBf .
|| REf || ≤ RE||f||
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 tsand
Zt
s
U(t, µ)[χ0h(µ)](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)=Zt
sZ0
r
p(θ)dhU(t, µ)[χ0h(µ)](θ)i
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)=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)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). By Lemma 3.4.2 and elementary integral estimates, JnJuni-
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)dy||hnh||.
Thus, KnKuniformly, and the bounded convergence theorem implies Rt
sKn(µ)Rt
sK(µ). 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)]+X
s<τit
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 tsthe equation
xt=U(t, s)xs+Zt
s
U(t, µ)[χ0h(µ)]+X
s<τit
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:RRnthat are
14
continuous everywhere except for at times t∈ {τk:kZ}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
tR
eη|t|||φ(t)|| <∞}
Bη(R,RCR) = {f:R→ RCR :||f||η= sup
tR
eη|t|||f(t)|| <∞}
P Cη(R,Rn) = {fP C (R,Rn) : ||f||η= sup
tR
eη|t|||f(t)|| <∞}
Bη
τk(Z,Rn) = {f:ZRn:||f||η= sup
kZ
eη|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 =suptReη|ts|||F(t)||,dom(F) = R
supkZeη|τks|||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([αr, 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 m1 for each tRand kZ, and there exists δ > 0
such that for each j= 0, . . . , m, there exists cj:RR+locally bounded and a positive sequence
{dj(k) : kZ}such that
||Djf(t, φ)Djf(t, ψ)|| ≤ cj(t)||φψ||,
||Djgk(φ)Djgk(ψ)|| ≤ dj(k)||φψ||.
for φ, ψ Bδ(0) ⊂ RCR. Also, there exists q > 0 such that ||Djf(t, φ)|| ≤ qcj(t) and ||Djgk(φ)|| ≤
qdj(k) for φBδ(0).
H.6 f(t, 0) = gk(0) = 0 and Df(t, 0) = Dgk(0) for all tRand kZ.
H.7 There exists a constant ξ > 0 such that τk+1 τkξfor all kZ.
Definition 5.1.1. Amild solution of the semilinear equation (29)(30) is a function x: [s, T ]→ RC R such
that for all st<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))]+X
s<τit
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 st<T, a nonlinear operator from RCR([sr, 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: [sr, T )Rnis a classical solution — that is, xis differentiable from the right, continuous except at
impulse times τk, continuous from the right on [sr, 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 rkgk(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 [
sR[
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)+X
s<τit
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 tsthe abstract integral equation
DS(t, s) = U(t, s) + Zt
s
U(t, µ)χ0Df (µ, S (µ, s)φ)DS(µ, s)+X
s<τit
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 sRbe 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: [sr, 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][sr, s +], and |φ|| <for the norm given by
||φ||,δ,ν = sup
t[sr,s+]
||xψ||≤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 : [sr, 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[sr,s)(t)x(ts) + χ[s,s+](t)U(t, s)x(s) + Zt
s
U(t, s)χ0f(µ, φ(µ, x))
+X
s<τit
U(t, τi)χ0g(τi, φ(τ
i, x))
Λ2:X,δ ×X,δ X,δ
Λ2(φ, Φ)(t, x)h=χ[sr,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<τit
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||xψ||≤2δ||x||, the mean-value theorem grants the estimate
||Λ1(φ)Λ1(γ)|| ≤ κsup
t[s,s+]
Zt
s
||U(t, µ)||c1(µ)+X
s<τit
||U(t, τi)||d1(i)
||φγ||
κL||φγ||
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||xψ||≤δk0(x), the estimate
||Λ1(φ)ψ|| ≤ ||φψ|| +κsup
t[s,s+]
Zt
s
||U(t, µ)||c0(µ)+X
s<τit
||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 M0 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)||
+X
s<τis+
||U(s+, τi)||d1(i)||φ(τ
i, x)γ(τ
i, x)||
||Φ||.
Also, for each φBδ(ψ) it is readily verified that ||Λ2(φ, Φ) Λ2(φ, Γ)|| ≤ κL||ΦΓ||, 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 n0, then for ts, 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)+X
s<τit
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))+X
s<τit
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)
+X
s<τit
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 ts
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)h
+X
s<τit
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 h0. 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+ (λ1)h)x0
n(φ)] hdλ
=Z1
0
1
||h|| [x0(φ+ (λ1)h)x0(φ)] hdλ 0
as h0. By definition, xis differentiable and Dx =x0.
If we define y(t)φ=x(t, φ) for the fixed point x: [sr, 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)]+X
s<τit
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 tR, 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
eη|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 tR, 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,
eη|ρ|||Ps(ν)ϕ|| =eη|ρ|||Us(ν, ρ)Ps(ρ)x(ρ)||
eη|ρ|Kea(νρ)||Ps(ρ)|| · ||x(ρ)||
KK ea(νρ)||Ps(ρ)||,
which implies ||Ps(ν)ϕ|| ≤ KKe||Ps(ρ)|| exp(η|ρ| − ).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
eη|ρ|||Pu(ν)ϕ|| =eη|ρ|||Uu(ν, ρ)Pu(ρ)x(ρ)||
eη|ρ|Keb(νρ)||Pu(ρ)|| · ||x(ρ)||
KK eb(νρ)||Pu(ρ)||,
which implies ||Pu(ν)ϕ|| ≤ KKe||Pu(ρ)|| exp(η|ρ| − ). 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)]
are well-defined as Pettis integrals for all s, t, v R, where we define Ra
bf=Rb
afwhen a < b.
Proof. The nontrivial cases are where tsand tv. 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)=Uc(t, s)Zs
t
Uc(s, µ)Pc(µ)H(µ)
=Zs
t
Uc(t, µ)Pc(µ)H(µ)
=Zs
t
U(t, µ)Pc(µ)H(µ)
=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 tvfor 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)]Z
t
U(t, µ)Pu(µ)[χ0F(µ)]+Zt
−∞
U(t, µ)Ps(µ)[χ0F(µ)]
+
t
X
s
U(t, τi)Pc(τi)[χ0Gi]i
X
t
U(t, τi)Pu(τi)[χ0Gi]i+
t
X
−∞
U(t, τi)Ps(τi)[χ0Gi]i,
(36)
indexed by sR, 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)i=
X
a<τib
F(τi), a b
a
X
b
F(τi)i, 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
η1 + e(η)ξ
ξ+1
aη1 + 2e(ηa)ξ
ξ+1
bη1 + 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) = (IPc(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
1Kc,F
1+Ku,F
1+Ku,G
2Kc,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(tµ)|F(µ)|
Zvm
vn
KN eb(tµ)eηµ||F||η
=KN ||F||ηebt Zvm
vn
eµ(ηb)
=KN ||F||η
bηebt evn(bη)evm(bη)
KN ||F||η
bηebtevn(bη).
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
SX
t<τi<
KN eb(tτi)eη|τi|||G||η
=X
−|t|i0
KN ||G||ηebte|τi|(b+η)+X
0k<
KN ||G||ηebte(bη)τi
KN ebt |t|
ξe|t|(b+η)+1
1e(bη)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
eη|ts|||Ku,F (t)|| ≤ eη|ts|Z
t
KN eb(tµ)|F(µ)|
eη(ts)KN Zs
t
eb(tµ)eη|µs|||F||η,s+Z
s
eb(tµ)eη|µs|||F||η,s
=eη(ts)KN ||F||η,s Zs
t
eb(tµ)eη(sµ)+Z
s
eb(tµ)eη(µs)
=eη(ts)KN ||F||η,s ebt+ηs e(b+η)te