# Numerical dynamics of integrodifference equations:
Hierarchies of invariant bundles in $L^{p}(\Omega)$

Christian Pötzsche Email<EMAIL_ADDRESS>Institut für Mathematik,
Universität Klagenfurt, 9020 Klagenfurt, Austria

###### Abstract

We study how the ”full hierarchy” of invariant manifolds for nonautonomous
integrodifference equations on the Banach spaces of $p$-integrable functions
behaves under spatial discretization of Galerkin type. These manifolds include
the classical stable, center-stable, center, center-unstable and unstable
ones, and can be represented as graphs of $C^{m}$-functions. For kernels with
a smoothing property, our main result establishes closeness of these graphs in
the $C^{m-1}$-topology under numerical discretizations preserving the
convergence order of the method. It is formulated in a quantitative fashion
and technically results from a general perturbation theorem on the existence
of invariant bundles (i.e. nonautonomous invariant manifolds).

###### keywords:

Integrodifference equation, numerical dynamics, dichotomy spectrum,
nonautonomous invariant manifolds, Galerkin method

††articletype: ARTICLE TEMPLATE

65P99, 37J15, 37L25, 47H30, 45M10

##  1 Introduction

Over the last years, integrodifference equations (abbreviated IDEs)

$u_{t+1}=\int_{\Omega}k_{t}(\cdot,y)g_{t}(y,u_{t}(y))\,{\mathrm{d}}y$
($I_{0}$)

became popular and widely used models in theoretical ecology for the temporal
evolution and spatial dispersal of populations having non-overlapping
generations [14]. They constitute an interesting class of infinite-dimensional
dynamical systems and can be seen as a discrete-time counterpart to reaction-
diffusion equations, with whom they share various dynamical features. IDEs
also arise in other fields, such as time-$1$-maps of evolutionary differential
equations or as iterative schemes to solve nonlinear boundary value problems
via an equivalent fixed point formulation (cf. [15, pp. 168–169]). Due to
their origin in ecology, a periodic $t$-dependence in ($I_{0}$) is well-
motivated, but recently also applications requiring more general temporal
forcing became relevant [11].

The long term behavior of IDEs is often illustrated using numerical
simulations, which require to discretize their state space $X$. For this
reason it is a well-motivated question to relate the dynamics of the original
and of the numerically approximated problem. This is a key issue in Numerical
Dynamics [23]. Among the various aspects of this field, we tackle invariant
manifolds. These sets provide the skeleton of the state space for a dynamical
system, since stable manifolds $W_{s}$ might serve as boundary between
different domains of attraction, unstable manifolds $W_{u}$ constitute global
attractors and center manifolds $W_{cs},W_{c},W_{cu}$ capture the essential
dynamics near non-hyperbolic solutions. These sets fulfill the _classical
hierarchy_ (a notion from [5])

$\begin{array}[]{ccccc}W_{s}&\subset&W_{cs}&\subset&X\\\ &&\cup&&\cup\\\
&&W_{c}&\subset&W_{cu}\\\ &&&&\cup\\\ &&&&W_{u}.\end{array}$ (1)

In continuous time, early contributions for autonomous ODEs in
${\mathbb{R}}^{d}$ were [6] (stable and unstable manifolds) and [7] (center
manifolds). Generalizations to infinite-dimensional problems appeared in [1]
(evolutionary PDEs), [9] (delay equations) and [12] (full discretizations of
evolutionary PDEs). The effect of time-discretizations to invariant manifolds
for nonautonomous ODEs in Banach spaces was studied in [13].

Throughout the literature the state space of IDEs typically consists of
continuous or integrable functions [14]. It is all the more important to
understand the behavior of their invariant manifolds in the classical
hierarchy (1) when such equations are spatially discretized. Accordingly the
paper at hand further explores the numerical dynamics of nonautonomous
equations. For related work we refer to [19] on the classical situation of
local stable and unstable manifolds for hyperbolic periodic solutions to
Urysohn IDEs in the space of (Hölder) continuous functions over a compact
habitat. The present approach complements and extends [19] in various aspects:
First, we tackle Hammerstein IDEs ($I_{0}$) in $L^{p}$-spaces rather than over
the continuous functions. Second, we address the complete hierarchy of
invariant manifolds containing also strongly stable, the three types of
center, as well as strongly unstable manifolds and establish convergence
including their derivatives preserving the order of the discretization method.
Third, dealing with general nonautonomous equations requires various
alternative tools such as the dichotomy spectrum [22] (instead of the Floquet
spectrum) and a flexible perturbation theorem for invariant bundles replacing
the Lipschitz inverse function theorem used in [19], whose applicability is
restricted to the hyperbolic situation. We finally point out that such
extended hierarchies of invariant manifolds date back to [5] in the framework
of autonomous ODEs.

The crucial assumption in our analysis are ambient smoothing properties of the
kernels $k_{t}$ paving the way to corresponding error estimates. In order to
minimize technicalities, we deal with globally defined semilinear IDEs having
the trivial solution and therefore obtain global results. By means of well-
known translation and cut-off methods more general and real-life problems
under local assumptions can be adapted to the present setting and then yield
results valid in the vicinity of a given reference solution (cf. Sect. 3.3 or
[18, pp. 256ff, Sect. 4.6]).

This paper is structured as follows: In Sect. 2 we provide the basic
assumptions on nonautonomous Hammerstein IDEs such that they are well-defined
on $L^{p}$-spaces and generate a completely continuous process. In order to
circumvent the pathological smoothness properties of Nemytskii operators on
$L^{p}$-spaces this requires them to satisfy suitable mapping properties into
a larger $L^{q}$-space, $q<p$, being balanced by Hille-Tamarkin conditions on
the kernel (cf. Sect. 2.1). In short, dealing with an IDE ($I_{0}$) of class
$C^{m}$ requires exponents $p>m$. The starting point of our actual analysis
are nonautonomous linear IDEs. Based on exponential dichotomies and the
dichotomy spectrum we present the ’linear algebra’ necessary for our analysis,
where spectral intervals and bundles extend eigenvalue moduli respectively
generalized eigenspaces (cf. [22]) to a time-variant setting. The following
Sect. 3 contains our main results (Thms. 3.3 and 3.6) describing how
nonautonomous invariant manifolds (bundles) and their derivatives up to order
$m-1$ behave under discretization using projection methods. Concrete examples
of projection methods and their applicability in Thms. 3.3 and 3.6 are
discussed in Sect. 4. Finally, for the reader’s convenience, we conclude the
paper with two appendices. App. A contains the central perturbation result
from [17] in a formulation suitable for those not familiar with the calculus
on measure chains. App. B quotes a well-definedness criterion for linear
Fredholm integral operators and (due to the lack of a suitable reference)
provides well-definedness and smoothness properties for Nemytskii operators
between Lebesgue spaces.

#### Notation

Let ${\mathbb{K}}$ be one of the fields ${\mathbb{R}}$ or ${\mathbb{C}}$. A
discrete interval ${\mathbb{I}}$ is the intersection of a real interval with
the integers ${\mathbb{Z}}$ and
${\mathbb{I}}^{\prime}:=\left\\{t\in{\mathbb{I}}:\,t+1\in{\mathbb{I}}\right\\}$.
Given $\ell\in{\mathbb{N}}$ and normed spaces $X,Y$, we write $L_{\ell}(X,Y)$
for the space of bounded $\ell$-linear operators $T:X^{\ell}\to Y$ and
moreover $L_{0}(X,Y):=Y$, as well as $L(X,Y):=L_{1}(X,Y)$. It is handy to
abbreviate $L_{\ell}(X):=L_{\ell}(X,X)$ and $L(X):=L(X,X)$. We write
$N(S):=S^{-1}(0)\subseteq X$ for the kernel, $R(S):=SX\subseteq Y$ for the
range of $S\in L(X,Y)$. Moreover, $\operatorname{id}_{X}$ is the identity map
on $X$, $\sigma(T)$ the spectrum of $T\in L(X)$ and $\left|\cdot\right|$
denotes norms on finite-dimensional spaces. Finally, we write
$\operatorname{lip}f$ for the (smallest) Lipschitz constant of a mapping $f$.

Throughout, let $\Omega\subset{\mathbb{R}}^{\kappa}$ be an open, bounded (and
hence Lebesgue measurable) set and
$\operatorname{diam}\Omega:=\sup_{x,y\in\Omega}\left|x-y\right|$ is its
diameter. For $p\in[1,\infty)$ we introduce the space of
${\mathbb{K}}^{d}$-valued $p$-integrable functions

$\displaystyle L_{d}^{p}(\Omega)$
$\displaystyle:=\biggl{\\{}u:\Omega\to{\mathbb{K}}^{d}\biggl{|}u\text{ is
Lebesgue measurable with
}\int_{\Omega}\left|u(y)\right|^{p}\,{\mathrm{d}}y<\infty\biggr{\\}},$
$\displaystyle L_{d}^{\infty}(\Omega)$
$\displaystyle:=\biggl{\\{}u:\Omega\to{\mathbb{K}}^{d}\biggl{|}u\text{ is
Lebesgue measurable with
}\operatorname*{ess\,sup}_{x\in\Omega}\left|u(y)\right|<\infty\biggr{\\}}$

and equip it with
$\left\|u\right\|_{p}:=\left(\sum_{j=1}^{d}\int_{\Omega}|u_{j}(y)|^{p}\,{\mathrm{d}}y\right)^{1/p}$
resp.
$\left\|u\right\|_{\infty}:=\operatorname*{ess\,sup}_{x\in\Omega}\left|u(x)\right|$
as canonical norms. This yields a strictly decreasing scale
$(L_{d}^{p}(\Omega))_{p\geq 1}$ of Banach spaces. In particular,
$L_{d}^{2}(\Omega)$ is a Hilbert space with inner product $\left\langle
u,v\right\rangle=\sum_{j=1}^{d}\int_{\Omega}u_{j}(y)\overline{v_{j}(y)}\,{\mathrm{d}}y$.
Finally, we write $L^{p}(\Omega):=L_{1}^{p}(\Omega)$.

##  2 Nonautonomous difference equations

Let ${\mathbb{I}}$ be an unbounded discrete interval.

###  2.1 Hammerstein integrodifference equations

We investigate nonautonomous integrodifference equations

$\displaystyle u_{t+1}$ $\displaystyle={\mathscr{F}}_{t}(u_{t}),$
$\displaystyle{\mathscr{F}}_{t}(u)$
$\displaystyle:=\int_{\Omega}k_{t}(\cdot,y)g_{t}(y,u(y))\,{\mathrm{d}}y\quad\text{for
all }t\in{\mathbb{I}}^{\prime},$ ($I_{0}$)

whose right-hand sides are Hammerstein integral operators satisfying the
assumptions:

The _kernels_ $k_{t}:\Omega\times\Omega\to{\mathbb{K}}^{d\times n}$,
$t\in{\mathbb{I}}^{\prime}$, in ($I_{0}$) fulfill _Hille-Tamarkin conditions_
with exponents $p,q\in(1,\infty)$:

  1. $(H_{q,p})$

$k_{t}$ is Lebesgue measurable and with $q^{\prime}>1$ determined by
$\tfrac{1}{q}+\tfrac{1}{q^{\prime}}=1$, we assume
$\int_{\Omega}\biggl{(}\int_{\Omega}\left|k_{t}(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}y\biggr{)}^{p/q^{\prime}}\,{\mathrm{d}}x<\infty\quad\text{for
all }t\in{\mathbb{I}}^{\prime}.$

Then Prop. B.1 guarantees that the linear integral operators

$\displaystyle{\mathscr{K}}_{t}$ $\displaystyle\in
L(L_{n}^{q}(\Omega),L_{d}^{p}(\Omega)),$ $\displaystyle{\mathscr{K}}_{t}v$
$\displaystyle:=\int_{\Omega}k_{t}(\cdot,y)v(y)\,{\mathrm{d}}y\quad\text{for
all }t\in{\mathbb{I}}^{\prime}$ (2)

are well-defined and compact.

Concerning the _growth functions_
$g_{t}:\Omega\times{\mathbb{K}}^{d}\to{\mathbb{K}}^{n}$,
$t\in{\mathbb{I}}^{\prime}$, in ($I_{0}$), for given $m\in{\mathbb{N}}_{0}$
and exponents $p,q\in(1,\infty)$ we assume _Carathéodory conditions_ for all
$0\leq\ell\leq m$:

  1. $(C_{p,q}^{m})$

$D_{2}^{\ell}g_{t}(x,\cdot):{\mathbb{K}}^{d}\to
L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n})$ exists and is continuous for a.a.
$x\in\Omega$, while $D_{2}^{\ell}g_{t}(\cdot,z):\Omega\to
L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n})$ is measurable on $\Omega$ and
with $mq<p$ there exist functions $c\in L^{\tfrac{pq}{p-mq}}(\Omega)$ and
reals $c_{0},\ldots,c_{m}\geq 0$ so that for a.a. $x\in\Omega$ one has

$\left|D_{2}^{\ell}g_{t}(x,z)\right|_{L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n})}\leq
c(x)+c_{\ell}\left|z\right|^{\tfrac{p-\ell q}{q}}\quad\text{for all
}t\in{\mathbb{I}}^{\prime},\,z\in{\mathbb{K}}^{d}.$

Given this, Prop. B.2 and B.4 ensure that the Nemytskii operators

$\displaystyle{\mathscr{G}}_{t}:L_{d}^{p}(\Omega)$ $\displaystyle\to
L_{n}^{q}(\Omega),$ $\displaystyle[{\mathscr{G}}_{t}(u)](x)$
$\displaystyle:=g_{t}(x,u(x))\quad\text{for all
}t\in{\mathbb{I}}^{\prime},\,x\in\Omega$ (3)

are well-defined and in case $mq<p$ also $m$-times continuously
differentiable. In a nutshell, the right-hand sides of ($I_{0}$) satisfy

######  Proposition 2.1 (properties of ${\mathscr{F}}_{t}$).

Let $p,q\in(1,\infty)$ and $m\in{\mathbb{N}}$. If Hypotheses $(H_{q,p})$ and
$(C_{p,q}^{m})$ hold with $mq<p$, then the right-hand sides
${\mathscr{F}}_{t}:={\mathscr{K}}_{t}{\mathscr{G}}_{t}:L_{d}^{p}(\Omega)\to
L_{d}^{p}(\Omega)$ of ($I_{0}$) are well-defined, completely continuous and of
class $C^{m}$ with derivatives
$D^{\ell}({\mathscr{K}}_{t}{\mathscr{G}}_{t})(u)={\mathscr{K}}_{t}D^{\ell}{\mathscr{G}}_{t}(u)$
for all $t\in{\mathbb{I}}^{\prime}$, $0\leq\ell\leq m$ and $u\in
L_{d}^{p}(\Omega)$.

###### Proof.

Let $t\in{\mathbb{I}}^{\prime}$. The Fredholm operators (2) are well-defined
and compact due to Prop. B.1, while the Nemytskii operators (3) are well-
defined, bounded and continuous by Prop. B.2. Then [20, pp. 25–26, Thm. 2.1]
implies that ${\mathscr{K}}_{t}{\mathscr{G}}_{t}$ is completely continuous.
Finally, Prop. B.4 and the Chain Rule applied to the
${\mathscr{K}}_{t}{\mathscr{G}}_{t}$ yield the claimed smoothness assertion. ∎

Under the assumptions of Prop. 2.1 an IDE ($I_{0}$) is well-posed. This means
that for arbitrary initial times $\tau\in{\mathbb{I}}$ and initial states
$u_{\tau}\in L_{d}^{p}(\Omega)$, there are unique _forward solutions_ , i.e.
sequences $(\phi_{t})_{\tau\leq t}$ in $L_{d}^{p}(\Omega)$ satisfying the
_solution identity_ $\phi_{t+1}={\mathscr{F}}_{t}(\phi_{t})$ for $\tau\leq t$,
$t\in{\mathbb{I}}^{\prime}$. A _backward solution_ fulfills the solution
identity for $t<\tau$ and _entire solutions_ $(\phi_{t})_{t\in{\mathbb{I}}}$
satisfy $\phi_{t+1}\equiv{\mathscr{F}}_{t}(\phi_{t})$ on
${\mathbb{I}}^{\prime}$. The _general solution_ to ($I_{0}$) is given by
$\varphi^{0}:\left\\{(t,\tau,u_{\tau})\in{\mathbb{I}}^{2}\times
L_{d}^{p}(\Omega):\,\tau\leq t\right\\}\to L_{d}^{p}(\Omega)$ via the
compositions

$\displaystyle\varphi^{0}(t;\tau,u_{\tau})$
$\displaystyle:=\begin{cases}{\mathscr{F}}_{t-1}\circ\ldots\circ{\mathscr{F}}_{\tau}(u_{\tau}),&\tau<t,\\\
u_{\tau},&t=\tau.\end{cases}$

A subset ${\mathscr{W}}\subseteq{\mathbb{I}}\times L_{d}^{p}(\Omega)$ is
called a _nonautonomous set_ and ${\mathscr{W}}(t):=\\{u\in
L_{d}^{p}(\Omega):\,(t,u)\in{\mathscr{W}}\\}$, $t\in{\mathbb{I}}$, its
$t$-_fiber_. We say ${\mathscr{W}}$ is _forward invariant_ resp.  _invariant_
, provided

${\mathscr{F}}_{t}({\mathscr{W}}(t))\subseteq{\mathscr{W}}(t+1)\quad\text{resp.}\quad{\mathscr{F}}_{t}({\mathscr{W}}(t))={\mathscr{W}}(t+1)\quad\text{for
all }t\in{\mathbb{I}}^{\prime}$

holds. In case all fibers ${\mathscr{W}}(t)$ are linear spaces one speaks of a
_linear bundle_ and it is convenient to write
${\mathscr{O}}:={\mathbb{I}}\times\left\\{0\right\\}$.

###  2.2 Linear integrodifference equations

This section outlines the nonautonomous ’linear algebra’ required to
understand the dynamics of ($I_{0}$) based on linearization [22]. Here, if a
kernel $l_{t}:\Omega\times\Omega\to{\mathbb{K}}^{d\times d}$ satisfies the
Hille-Tamarkin conditions $(H_{p,p})$ with $n=d$, then a linear IDE

$\displaystyle u_{t+1}$ $\displaystyle={\mathscr{L}}_{t}u_{t},$
$\displaystyle{\mathscr{L}}_{t}u$
$\displaystyle:=\int_{\Omega}l_{t}(\cdot,y)u(y)\,{\mathrm{d}}y$ ($L$)

is well-defined with compact ${\mathscr{L}}_{t}\in L(L_{d}^{p}(\Omega))$,
$t\in{\mathbb{I}}^{\prime}$. This yields the _evolution operator_

$\displaystyle\Phi:\left\\{(t,\tau)\in{\mathbb{I}}^{2}:\,\tau\leq t\right\\}$
$\displaystyle\to L(L_{d}^{p}(\Omega)),$ $\displaystyle\Phi(t,\tau)$
$\displaystyle:=\begin{cases}{\mathscr{L}}_{t-1}\cdots{\mathscr{L}}_{\tau},&\tau<t,\\\
\operatorname{id}_{L_{d}^{p}(\Omega)},&\tau=t.\end{cases}$

A linear IDE ($L$) has an _exponential dichotomy_ on ${\mathbb{I}}$ [10, p.
229, Def. 7.6.4], if there exists a projection-valued sequence
$(P_{t})_{t\in{\mathbb{I}}}$ in $L(L_{d}^{p}(\Omega))$ and $K\geq 1$,
$\alpha\in(0,1)$ so that $P_{t+1}{\mathscr{L}}_{t}={\mathscr{L}}_{t}P_{t}$ and
${\mathscr{L}}_{t}|_{N(P_{t})}:N(P_{t})\to N(P_{t+1})$ is an isomorphism for
all $t\in{\mathbb{I}}^{\prime}$ with

$\displaystyle\left\|\Phi(t,s)P_{s}\right\|_{L(L_{d}^{p}(\Omega))}$
$\displaystyle\leq K\alpha^{t-s}\,$
$\displaystyle\left\|\Phi(s,t)[\operatorname{id}_{L_{d}^{p}(\Omega)}-P_{t}]\right\|_{L_{d}^{p}(\Omega)}$
$\displaystyle\leq K\alpha^{t-s}\quad\text{for all }s\leq t.$

This allows us to introduce nonautonomous counterparts to eigenvalue moduli in
terms of the components of the _dichotomy spectrum_

$\Sigma_{\mathbb{I}}({\mathscr{L}}):=\left\\{\gamma>0:\,u_{t+1}=\tfrac{1}{\gamma}{\mathscr{L}}_{t}u_{t}\text{
does not have an exponential dichotomy on }{\mathbb{I}}\right\\}$

for ($L$). The spectrum $\Sigma_{\mathbb{I}}({\mathscr{L}})$ depends on the
interval ${\mathbb{I}}$ such that
$\Sigma_{\mathbb{I}}({\mathscr{L}})\subseteq\Sigma_{\mathbb{Z}}({\mathscr{L}})$.
Under the bounded growth assumption

$a_{0}:=\sup_{t\in{\mathbb{I}}^{\prime}}\biggl{(}\int_{\Omega}\biggl{(}\int_{\Omega}\left|l_{t}(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p}{q^{\prime}}}\,{\mathrm{d}}x\biggr{)}^{\tfrac{1}{p}}<\infty$
(4)

it is contained in $(0,a_{0}]$ and there exists a $\gamma_{0}>0$ with
$(\gamma_{0},\infty)\subseteq(0,\infty)\setminus\Sigma_{\mathbb{I}}({\mathscr{L}})$.
As shown in [22, Cor. 4.13], $\Sigma_{\mathbb{I}}({\mathscr{L}})\neq\emptyset$
is a union of at most countably many intervals which can only accumulate at
some $b_{\infty}\geq 0$. One of the cases holds:

  1. $(S_{1})$

$\Sigma_{\mathbb{I}}({\mathscr{L}})$ consists of finitely many closed
intervals:

    1. Figure 1: Case $(S_{1}^{1})$ with $J$ compact spectral intervals (top) and $(S_{1}^{2})$ with $J+1$ spectral intervals (bottom)
    2. $(S_{1}^{1})$

There exists a $J\in{\mathbb{N}}$ and reals $0<a_{J}\leq
b_{J}<\ldots<a_{1}\leq b_{1}<a_{0}$ with
$\Sigma_{\mathbb{I}}({\mathscr{L}})=\bigcup_{j=1}^{J}[a_{j},b_{j}]$. In this
case we choose the $J$ rates $\gamma_{j}\in(b_{j+1},a_{j})$, $1\leq j<J$, and
$\gamma_{J}\in(0,a_{J})$ (see Fig. 1 (top)).

    3. $(S_{1}^{2})$

There exists a $J\in{\mathbb{N}}_{0}$ and reals $0<b_{J+1}<a_{J}\leq
b_{J}<\ldots<a_{1}\leq b_{1}<a_{0}$ with
$\Sigma_{\mathbb{I}}({\mathscr{L}})=(0,b_{J+1}]\cup\bigcup_{j=1}^{J}[a_{j},b_{j}]$.
Here we choose $J$ rates $\gamma_{j}\in(b_{j+1},a_{j})$, $1\leq j\leq J$ (see
Fig. 1 (bottom)).

Figure 2: Case $(S_{2})$ with infinitely many spectral intervals
$[a_{j},b_{j}]$ accumulating at $b_{\infty}=0$ i.e.
$\sigma_{\infty}=\emptyset$ (top) and accumulating at $b_{\infty}>0$ (bottom)

  2. $(S_{2})$

$\Sigma_{\mathbb{I}}({\mathscr{L}})$ consists of countably infinitely many
intervals: There are strictly decreasing sequences
$(a_{j})_{j\in{\mathbb{N}}}$, $(b_{j})_{j\in{\mathbb{N}}}$ in $(0,\infty)$
such that
$\Sigma_{\mathbb{I}}({\mathscr{L}})=\sigma_{\infty}\cup\bigcup_{j=1}^{\infty}[a_{j},b_{j}],$
where $b_{\infty}<a_{j}\leq b_{j}$ for all $j\in{\mathbb{N}}$,
$\lim_{j\to\infty}a_{j}=b_{\infty}$. Here, $\sigma_{\infty}=\emptyset$ for
$b_{\infty}=0$ (see Fig. 2 (top)) and otherwise
$\sigma_{\infty}=(0,b_{\infty}]$ (see Fig. 2 (bottom)). We choose countably
many rates $\gamma_{j}\in(b_{j+1},a_{j})$ for $j\in{\mathbb{N}}$.

Concerning nonautonomous counterparts to generalized eigenspaces, for each
$\gamma>0$ we define the linear bundles

$\displaystyle\bar{\mathscr{V}}_{\gamma}^{+}$
$\displaystyle:=\left\\{(\tau,u)\in{\mathbb{I}}\times
L_{d}^{p}(\Omega):\,\sup_{\tau\leq
t}\left\|\Phi(t,\tau)u\right\|_{p}\gamma^{\tau-t}<\infty\right\\},$
$\displaystyle\bar{\mathscr{V}}_{\gamma}^{-}$
$\displaystyle:=\left\\{(\tau,u)\in{\mathbb{I}}\times
L_{d}^{p}(\Omega):\,\begin{array}[]{r}\text{there exists a solution
$(\phi_{t})_{t\in{\mathbb{I}}}$ of \eqref{lin} so}\\\ \text{that
$\sup_{\tau\leq t}\left\|\phi_{t}\right\|_{p}\gamma^{\tau-t}<\infty$ and
$\phi_{\tau}=u$}\end{array}\right\\};$

in case $\gamma\in(0,\infty)\setminus\Sigma_{\mathbb{I}}({\mathscr{L}})$, one
denotes $\bar{\mathscr{V}}_{\gamma}^{+}$ as a $\gamma$-_stable_ and
$\bar{\mathscr{V}}_{\gamma}^{-}$ as a $\gamma$-_unstable_ bundle of ($L$).
With rates $\gamma_{j}$ chosen as in $(S_{1})$ or $(S_{2})$ above, we set

$\displaystyle{\mathscr{V}}_{0}^{+}$ $\displaystyle:={\mathbb{I}}\times
L_{d}^{p}(\Omega),$ $\displaystyle{\mathscr{V}}_{0}^{-}$
$\displaystyle:={\mathscr{O}},$ $\displaystyle{\mathscr{V}}_{j}^{+}$
$\displaystyle:=\bar{\mathscr{V}}_{\gamma_{j}}^{+},$
$\displaystyle{\mathscr{V}}_{j}^{-}$
$\displaystyle:=\bar{\mathscr{V}}_{\gamma_{j}}^{-}\quad\text{for }j>0.$

Thanks to [22, Cor. 4.13] the $\gamma$-unstable bundles
$\bar{\mathscr{V}}_{\gamma}^{-}$ are finite-dimensional and:

  1. $(S_{1})$

In both subcases the _spectral bundles_

$\displaystyle{\mathscr{V}}_{0}$ $\displaystyle:={\mathscr{V}}_{0}^{-},$
$\displaystyle{\mathscr{V}}_{j}$
$\displaystyle:={\mathscr{V}}_{j-1}^{+}\cap{\mathscr{V}}_{j}^{-}\neq{\mathscr{O}}\quad\text{for
all }1\leq j\leq J$

are finite-dimensional invariant linear bundles of ($L$) allowing the _Whitney
sum_

${\mathbb{I}}\times
L_{d}^{p}(\Omega)=\bigoplus_{j=0}^{J}{\mathscr{V}}_{j}\oplus{\mathscr{V}}_{J}^{+}.$

  2. $(S_{2})$

The _spectral bundles_

$\displaystyle{\mathscr{V}}_{0}$ $\displaystyle:={\mathscr{V}}_{0}^{-},$
$\displaystyle{\mathscr{V}}_{j}$
$\displaystyle:={\mathscr{V}}_{j-1}^{+}\cap{\mathscr{V}}_{j}^{-}\neq{\mathscr{O}}\quad\text{for
all }j\in{\mathbb{N}}$

are finite-dimensional invariant linear bundles of ($L$) with the _Whitney
sum_

${\mathbb{I}}\times
L_{d}^{p}(\Omega)=\bigoplus_{j=0}^{J}{\mathscr{V}}_{j}\oplus{\mathscr{V}}_{J}^{+}\quad\text{for
all }J\in{\mathbb{N}}.$

The bundle ${\mathscr{V}}_{J}^{-}=\bigoplus_{j=0}^{J}{\mathscr{V}}_{j}$
satisfies
$J\leq\dim{\mathscr{V}}_{J}^{-}=\sum_{j=0}^{J}\dim{\mathscr{V}}_{j}$.

##  3 Invariant bundles under projection methods

Among the various numerical techniques to solve integral equations and thus to
computationally simulate IDEs, we focus on projection methods [3], [4, pp.
446ff]. They are based on a sequence $\Pi_{n}:L_{d}^{p}(\Omega)\to X_{n}^{d}$,
$n\in{\mathbb{N}}$, of bounded projections onto finite-dimensional subspaces
$X_{n}^{d}=X_{n}\times\ldots\times X_{n}\subseteq L_{d}^{p}(\Omega)$. Here
each subspace $X_{n}$ is the span of $d_{n}\in{\mathbb{N}}$ linearly
independent functions $\chi_{1},\ldots,\chi_{d_{n}}:\Omega\to{\mathbb{R}}$ and
it is advantageous to set $X_{0}:=L^{p}(\Omega)$. Under interpolation
conditions (for collocation methods) or orthogonality conditions (for Galerkin
methods), these functions determine projections $\pi_{n}:L^{p}(\Omega)\to
X_{n}$. Given this, we set
$\Pi_{n}:=\operatorname{diag}(\pi_{n},\ldots,\pi_{n})\in L(X_{n}^{d})$ and
refer to Sect. 4 for concrete examples of suitable spaces $X_{n}$ and related
projections $\pi_{n}$.

The resulting spatial discretizations of ($I_{0}$) are difference equations

$\displaystyle u_{t+1}$ $\displaystyle={\mathscr{F}}_{t}^{n}(u_{t}),$
$\displaystyle{\mathscr{F}}_{t}^{n}(u)$
$\displaystyle:=\Pi_{n}{\mathscr{F}}_{t}(u)$ ($I_{n}$)

being well-defined due to Prop. 2.1. and
$\varphi^{n}:\left\\{(t,\tau,u)\in{\mathbb{I}}^{2}\times
L_{d}^{p}(\Omega):\,\tau\leq t\right\\}\to L_{d}^{p}(\Omega)$ denotes their
general solutions.

Throughout the section, we suppose

######  Hypothesis.

Let $m\in{\mathbb{N}}$ and $p,q\in(1,\infty)$ with $mq<p$. Suppose that

  * (i)

$(H_{q,p})$ with a $X(\Omega)$-_smoothing kernel_ , that is, given a subset
$X(\Omega)\subseteq L_{d}^{p}(\Omega)$ for all $t\in{\mathbb{I}}^{\prime}$ the
inclusions ${\mathscr{K}}_{t}L_{d}^{q}(\Omega)\subseteq X(\Omega)$ hold and
there exist $C_{t}\geq 0$ such that

$\left\|{\mathscr{K}}_{t}u\right\|_{X(\Omega)}\leq
C_{t}\left\|u\right\|_{q}\quad\text{for all }u\in L_{n}^{q}(\Omega),$ (5)

  * (ii)

$(C_{p,q}^{m})$ with $c_{2}=\ldots=c_{m}=0$, $g_{t}(x,0)=0$ for all
$t\in{\mathbb{I}}^{\prime}$, a.a. $x\in\Omega$ and there exist measurable
functions $\lambda_{t},\bar{\lambda}_{t}:\Omega\to{\mathbb{R}}_{+}$ with

$\displaystyle\left|D_{2}g_{t}(x,z)-D_{2}g_{t}(x,0)\right|$
$\displaystyle\leq\lambda_{t}(x)\quad\text{for a.a.\ }x\in\Omega\text{ and all
}z\in{\mathbb{K}}^{d},$ (6)
$\displaystyle\left|g_{t}(x,z)-g_{t}(x,\bar{z})\right|$
$\displaystyle\leq\bar{\lambda}_{t}(x)\left|z-\bar{z}\right|\quad\text{for
a.a.\ }x\in\Omega\text{ and all }z,\bar{z}\in{\mathbb{K}}^{d}$ (7)

such that
$\sup_{t\in{\mathbb{I}}^{\prime}}C_{t}\biggl{(}\int_{\Omega}\bar{\lambda}_{t}(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p-q}{p}}<\infty$
and

$\displaystyle L:=$
$\displaystyle\sup_{t\in{\mathbb{I}}^{\prime}}\biggl{(}\int_{\Omega}\biggl{(}\int_{\Omega}\left|k_{t}(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p}{q^{\prime}}}\,{\mathrm{d}}x\biggr{)}^{\tfrac{1}{p}}\biggl{(}\int_{\Omega}\lambda_{t}(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p-q}{p}}<\infty,$
(8)
$\displaystyle\sup_{t\in{\mathbb{I}}^{\prime}}\biggl{(}\int_{\Omega}\biggl{(}\int_{\Omega}\left|k_{t}(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p}{q^{\prime}}}\,{\mathrm{d}}x\biggr{)}^{\tfrac{1}{p}}\biggl{(}\int_{\Omega}\bar{\lambda}_{t}(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p-q}{p}}<\infty,$
(9)

  * (iii)

there exists a function $\Gamma:{\mathbb{R}}_{+}\to(0,\infty)$ with
$\lim_{\varrho\searrow 0}\Gamma(\varrho)=0$ such that the _discretization
error_ satisfies

$\left\|u-\Pi_{n}u\right\|_{p}\leq\Gamma(\tfrac{1}{n})\left\|u\right\|_{X(\Omega)}\quad\text{for
all }n\in{\mathbb{N}},\,u\in X(\Omega).$ (10)

The subsequent Sect. 4 is devoted to specific error estimates (10) for various
function spaces $X(\Omega)$. Moreover, in order to illustrate the smoothing
property (i) we consider two nonsmooth kernels depending on dispersal rates
$\delta_{t}>0$, $t\in{\mathbb{I}}^{\prime}$:

######  Example 3.1 (Laplace kernel).

Consider the _Laplace kernel_ (see [14, pp. 18ff])

$\displaystyle k_{t}:\Omega^{2}$ $\displaystyle\to(0,\infty),$ $\displaystyle
k_{t}(x,y):=\tfrac{\delta_{t}}{2}e^{-\delta_{t}\left|x-y\right|}$

acting on a habitat $\Omega=(-\tfrac{l}{2},\tfrac{l}{2})$, $l>0$. As a
continuous, globally bounded function it satisfies $(H_{q,p})$ with arbitrary
exponents $p,q\in(1,\infty)$ and $d=n=1$. In particular,

$\displaystyle\int_{\Omega}\left(\int_{\Omega}\left|k_{t}(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}y\right)^{p/q^{\prime}}\,{\mathrm{d}}x\leq\int_{\Omega}\left(\int_{\Omega}\left(\tfrac{\delta_{t}}{2}\right)^{q^{\prime}}\,{\mathrm{d}}y\right)^{p/q^{\prime}}\,{\mathrm{d}}x=l^{1+p-1/q}\left(\tfrac{\delta_{t}}{2}\right)^{p}$

and therefore
$\left\|{\mathscr{K}}_{t}\right\|_{L(L^{q}(\Omega),L^{p}(\Omega))}\leq
l^{1+p-1/q}\left(\tfrac{\delta_{t}}{2}\right)^{p}$ for all
$t\in{\mathbb{I}}^{\prime}$. In addition,

$({\mathscr{K}}_{t}u)^{\prime}={\mathscr{K}}_{t}^{(1)}u\quad\text{with}\quad({\mathscr{K}}_{t}^{(1)}u)(x):=-\frac{\delta_{t}^{2}}{2}\int_{\Omega}\operatorname{sgn}(x-y)e^{-\delta_{t}\left|x-y\right|}u(y)\,{\mathrm{d}}y$

for all $x\in\Omega$ and $u\in L^{q}(\Omega)$, as well as
$\|{\mathscr{K}}_{t}^{(1)}\|_{L(L^{q}(\Omega),L^{p}(\Omega))}\leq
l^{1+p-1/q}\left(\tfrac{\delta_{t}^{2}}{2}\right)^{p}$. Hence, we can choose
the smaller space $X(\Omega):=W^{1,p}(\Omega)$ and obtain

$\displaystyle\left\|{\mathscr{K}}_{t}u\right\|_{W^{1,p}(\Omega)}$
$\displaystyle=\left(\left\|{\mathscr{K}}_{t}u\right\|_{p}^{p}+\|{\mathscr{K}}_{t}^{(1)}u\|_{p}^{p}\right)^{1/p}\leq
C_{t}\left\|u\right\|_{q}\quad\text{for all }u\in L^{q}(\Omega),$

with the constants
$C_{t}:=l^{1+1/p-1/(pq)}\tfrac{\delta_{t}}{2}(1+(\tfrac{\delta_{t}}{2})^{p})^{1/p}$.
In conclusion, the Laplace kernel is $W^{1,p}(\Omega)$-smoothing and the
estimate (5) holds.

######  Example 3.2 (root kernel).

The _exponential root kernel_ (see [14, p. 72 for $\alpha=\tfrac{1}{2}$])

$\displaystyle k_{t}:\Omega^{2}$ $\displaystyle\to(0,\infty),$ $\displaystyle
k_{t}(x,y):=\tfrac{\delta_{t}^{2}}{2}e^{-\delta_{t}\left|x-y\right|^{\alpha}}$

with exponent $\alpha\in(0,1]$ may be defined on an open, bounded habitat
$\Omega\subset{\mathbb{R}}^{\kappa}$. Being continuous and globally bounded,
it again fulfills $(H_{q,p})$ with arbitrary $p,q\in(1,\infty)$ and $d=n=1$.
Given $u\in L^{q}(\Omega)$ the Hölder inequality implies

$\left|{\mathscr{K}}_{t}u(x)\right|\stackrel{{\scriptstyle\eqref{defK}}}{{=}}\left|\int_{\Omega}k_{t}(x,y)u(y)\,{\mathrm{d}}y\right|\leq\left(\int_{\Omega}\left|k_{t}(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}y\right)^{1/q^{\prime}}\left\|u\right\|_{q}\leq\frac{\delta_{t}^{2}}{2}\lambda_{\kappa}(\Omega)^{1/q^{\prime}}\left\|u\right\|_{q}$

for all $x\in\Omega$, as well as using the Mean Value Estimate (applied to
$x\mapsto e^{-x}$)

$\displaystyle\left|{\mathscr{K}}_{t}u(x)-{\mathscr{K}}_{t}u(\bar{x})\right|$
$\displaystyle\stackrel{{\scriptstyle\eqref{defK}}}{{=}}\left|\int_{\Omega}[k_{t}(x,y)-k_{t}(\bar{x},y)]u(y)\,{\mathrm{d}}y\right|$
$\displaystyle\leq\left(\int_{\Omega}\left|k_{t}(x,y)-k_{t}(\bar{x},y)\right|^{q^{\prime}}\,{\mathrm{d}}y\right)^{1/q^{\prime}}\left\|u\right\|_{q}$
$\displaystyle\leq\frac{\delta_{t}^{2}}{2}\left(\int_{\Omega}\left|e^{-\delta_{t}\left|x-y\right|^{\alpha}}-e^{-\delta_{t}\left|\bar{x}-y\right|^{\alpha}}\right|^{q^{\prime}}\,{\mathrm{d}}y\right)^{1/q^{\prime}}\left\|u\right\|_{q}$
$\displaystyle\leq\frac{\delta_{t}^{3}}{2}\left(\int_{\Omega}\left|\left|x-y\right|^{\alpha}-\left|\bar{x}-y\right|^{\alpha}\right|^{q^{\prime}}\,{\mathrm{d}}y\right)^{1/q^{\prime}}\left\|u\right\|_{q}$
$\displaystyle\leq\frac{\delta_{t}^{3}}{2}\left(\int_{\Omega}\left|\left|x-y\right|-\left|\bar{x}-y\right|\right|^{\alpha
q^{\prime}}\,{\mathrm{d}}y\right)^{1/q^{\prime}}\left\|u\right\|_{q}$
$\displaystyle\leq\frac{\delta_{t}^{3}}{2}\left(\int_{\Omega}\left|x-\bar{x}\right|^{\alpha
q^{\prime}}\,{\mathrm{d}}y\right)^{1/q^{\prime}}\left\|u\right\|_{q}$
$\displaystyle=\frac{\delta_{t}^{3}}{2}\lambda_{\kappa}(\Omega)^{1/q^{\prime}}\left|x-\bar{x}\right|^{\alpha}\left\|u\right\|_{q}\quad\text{for
all }x,\bar{x}\in\Omega.$

Thus, ${\mathscr{K}}_{t}u$ is Hölder with exponent $\alpha$ and we can choose
$X(\Omega)=C^{\alpha}(\overline{\Omega})$ with

$\left\|{\mathscr{K}}_{t}u\right\|_{C^{\alpha}(\overline{\Omega})}\leq\tfrac{\delta_{t}^{2}}{2}\lambda_{\kappa}(\Omega)^{1/q^{\prime}}\max\left\\{1,\delta_{t}\right\\}\left\|u\right\|_{q}\quad\text{for
all }u\in L^{q}(\Omega).$

In summary, the exponential root kernel is
$C^{\alpha}(\overline{\Omega})$-smoothing and the estimate (5) holds with the
constant
$C_{t}:=\tfrac{\delta_{t}^{2}}{2}\lambda_{\kappa}(\Omega)^{1/q^{\prime}}\max\left\\{1,\delta_{t}\right\\}$
for all $t\in{\mathbb{I}}^{\prime}$.

###  3.1 Pseudo-stable and -unstable bundles

Under the above Hypotheses (i–iii), the right-hand side of ($I_{0}$) is at
least continuously differentiable (cf. Prop. 2.1). Its variational equation
along the trivial solution

$\displaystyle v_{t+1}$ $\displaystyle=D{\mathscr{F}}_{t}(0)v_{t},$
$\displaystyle D{\mathscr{F}}_{t}(0)v$
$\displaystyle:=\int_{\Omega}k_{t}(\cdot,y)D_{2}g_{t}(y,0)v(y)\,{\mathrm{d}}y$
($V_{0}$)

is a linear IDE of the form ($L$) with kernel
$l_{t}(x,y):=k_{t}(x,y)D_{2}g_{t}(y,0)\in{\mathbb{K}}^{d\times d}$, whose
dichotomy spectrum will be denoted by $\Sigma_{\mathbb{I}}$.

Figure 3: The pseudo-stable bundle ${\mathscr{W}}_{j,+}^{0}$ and the pseudo-
unstable bundle ${\mathscr{W}}_{j,-}^{0}$ (black) of ($I_{0}$) intersect along
the trivial solution and perturb as bundles ${\mathscr{W}}_{j,+}^{n}$ and
${\mathscr{W}}_{j,-}^{n}$, $n\geq N$ (grey) for the spatial discretizations
($I_{n}$). Both consist of $t$-fibers being graphs over
${\mathscr{V}}_{j}^{+}$ resp. ${\mathscr{V}}_{j}^{-}$.

######  Theorem 3.3 (pseudo-stable and -unstable bundles under
discretization).

Let Hypotheses (i–iii) hold, $\gamma_{j}>0$, $j\in{\mathbb{N}}_{0}$, be as in
$(S_{1})$ or $(S_{2})$, choose $\alpha_{j}<\beta_{j}$ so that

$\displaystyle\alpha_{j}$ $\displaystyle<\gamma_{j}<\beta_{j},$
$\displaystyle(\alpha_{j},\beta_{j})\cap\Sigma_{\mathbb{I}}$
$\displaystyle=\emptyset$

and let ${\mathscr{V}}_{j}^{+},{\mathscr{V}}_{j}^{-}$ denote the associated
linear bundles of the variational equation ($V_{0}$). Then there exists a
$K\geq 1$ depending only on ($V_{0}$), so that if

$\displaystyle L$ $\displaystyle<\frac{\delta_{\max}}{4K},$
$\displaystyle\delta_{\max}$ $\displaystyle:=\frac{\beta_{j}-\alpha_{j}}{2},$
(11)

$\delta\in(4KL,\delta_{\max}]$ is fixed and
$\Gamma_{j}:=[\alpha_{j}+\delta,\beta_{j}-\delta]$, then there exists a
$N\in{\mathbb{N}}$ such that the following statements are true for $n=0$ and
$n\geq N$ (see Fig. 3):

  1. (a)

In case ${\mathbb{I}}$ is unbounded above, then the _pseudo-stable bundle_

${\mathscr{W}}_{j,+}^{n}:=\left\\{(\tau,u)\in{\mathbb{I}}\times
L_{d}^{p}(\Omega):\,\sup_{\tau\leq
t}\gamma^{\tau-t}\left\|\varphi^{n}(t;\tau,u)\right\|_{p}<\infty\right\\}$

is a forward invariant bundle of ($I_{n}$), which is independent of
$\gamma\in\Gamma_{j}$ and allows the representation

${\mathscr{W}}_{j,+}^{n}=\left\\{(\tau,v+w_{j,+}^{n}(\tau,v))\in{\mathbb{I}}\times
L_{d}^{p}(\Omega):\,(\tau,v)\in{\mathscr{V}}_{j}^{+}\right\\}$

with continuous mappings $w_{j,+}^{n}:{\mathscr{V}}_{j}^{+}\to
L_{d}^{p}(\Omega)$ satisfying $w_{j,+}^{n}(\tau,0)=0$ and

     * $(a_{1})$

there exist $C_{0}^{+}(n),N_{0}^{+}>0$ with
$\lim_{n\to\infty}C_{0}^{+}(n)=\tfrac{K^{2}L}{\delta-2KL}=:C_{0}^{+}(0)$ and
for all $(\tau,v)\in{\mathscr{V}}_{j}^{+}$ holds

$\displaystyle\operatorname{lip}w_{j,+}^{n}(\tau,\cdot)$ $\displaystyle\leq
C_{0}^{+}(n),$
$\displaystyle\left\|w_{j,+}^{n}(\tau,v)-w_{j,+}^{0}(\tau,v)\right\|_{p}$
$\displaystyle\leq N_{0}^{+}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p},$

     * $(a_{2})$

if $\alpha_{j}^{m_{+}}<\beta_{j}$ and
$\delta_{\max}:=\min\Bigl{\\{}\tfrac{\beta_{j}-\alpha_{j}}{2},\alpha_{j}\bigl{(}\sqrt[m_{+}]{\frac{\alpha_{j}+\beta_{j}}{\alpha_{j}+\alpha_{j}^{m_{+}}}}-1\bigr{)}\Bigr{\\}}$,
then the derivatives $D_{2}^{\ell}w_{j,+}^{n}:{\mathscr{V}}_{j}^{+}\to
L_{\ell}(L_{d}^{p}(\Omega))$ exist up to order $\ell\leq m_{+}\leq m$ as
continuous maps and for $1\leq\ell<m_{+}$ there are $N_{\ell}^{+}>0$ such that

$\displaystyle\left\|D_{2}^{\ell}w_{j,+}^{n}(\tau,v)-D_{2}^{\ell}w_{j,+}^{0}(\tau,v)\right\|_{L_{\ell}(L_{d}^{p}(\Omega))}$
$\displaystyle\leq
N_{\ell}^{+}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p}\quad\text{for all
}(\tau,v)\in{\mathscr{V}}_{j}^{+}.$

  2. (b)

In case ${\mathbb{I}}$ is unbounded below, then the _pseudo-unstable bundle_

${\mathscr{W}}_{j,-}^{n}:=\left\\{(\tau,u)\in{\mathbb{I}}\times
L_{d}^{p}(\Omega):\begin{array}[]{l}\text{there exists a solution
}(\phi_{t})_{t\in{\mathbb{I}}}\text{ of \eqref{iden} so}\\\ \text{that
}\sup_{t\leq\tau}\gamma^{\tau-t}\left\|\phi_{t}\right\|_{p}<\infty\text{ and
}\phi_{\tau}=u\end{array}\right\\}$

is a finite-dimensional (unless for spectra $(S_{1}^{1})$ and $j=J$) invariant
bundle of ($I_{n}$), which is independent of $\gamma\in\Gamma_{j}$ and allows
the representation

${\mathscr{W}}_{j,-}^{n}=\left\\{(\tau,v+w_{j,-}^{n}(\tau,v))\in{\mathbb{I}}\times
L_{d}^{p}(\Omega):\,(\tau,v)\in{\mathscr{V}}_{j}^{-}\right\\}\subseteq\begin{cases}{\mathbb{I}}\times
X(\Omega),&n=0,\\\ {\mathbb{I}}\times X_{n}^{d},&n\geq N\end{cases}$

with continuous mappings $w_{j,-}^{n}:{\mathscr{V}}_{j}^{-}\to
L_{d}^{p}(\Omega)$ satisfying $w_{j,-}^{n}(\tau,0)=0$ and

     * $(b_{1})$

there exist $C_{0}^{-}(n),N_{0}^{-}>0$ with
$\lim_{n\to\infty}C_{0}^{-}(n)=\tfrac{K^{2}L}{\delta-2KL}=:C_{0}^{-}(0)$ and
for all $(\tau,v)\in{\mathscr{V}}_{j}^{-}$ holds

$\displaystyle\operatorname{lip}w_{j,-}^{n}(\tau,\cdot)$ $\displaystyle\leq
C_{0}^{-}(n),$
$\displaystyle\left\|w_{j,-}^{n}(\tau,v)-w_{j,-}^{0}(\tau,v)\right\|_{p}$
$\displaystyle\leq N_{0}^{-}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p},$

     * $(b_{2})$

if $\alpha_{j}<\beta_{j}^{m_{-}}$ and
$\delta_{\max}:=\min\Bigl{\\{}\tfrac{\beta_{j}-\alpha_{j}}{2},\beta_{j}\bigl{(}1-\sqrt[m_{-}]{\frac{\alpha_{j}+\beta_{j}}{\beta_{j}+\beta_{j}^{m_{-}}}}\bigr{)}\Bigr{\\}}$,
then the derivatives $D_{2}^{\ell}w_{j,-}^{n}:{\mathscr{V}}_{j}^{-}\to
L_{\ell}(L_{d}^{p}(\Omega))$ exist up to order $\ell\leq m_{-}\leq m$ as
continuous maps and for $1\leq\ell<m_{-}$ there are $N_{\ell}^{-}>0$ such that

$\displaystyle\left\|D_{2}^{\ell}w_{j,-}^{n}(\tau,v)-D_{2}^{\ell}w_{j,-}^{0}(\tau,v)\right\|_{L_{\ell}(L_{d}^{p}(\Omega))}$
$\displaystyle\leq
N_{\ell}^{-}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p}\quad\text{for all
}(\tau,v)\in{\mathscr{V}}_{j}^{-}.$

  3. (c)

If ${\mathbb{I}}={\mathbb{Z}}$ and $L<\frac{\delta}{6K}$, then
${\mathscr{W}}_{j,+}^{n}\cap{\mathscr{W}}_{j,-}^{n}={\mathscr{O}}$ and the
zero solution is the only entire solution $\phi$ to ($I_{n}$) satisfying
$\sup_{t\in{\mathbb{Z}}}\gamma^{-t}\left\|\phi_{t}\right\|_{p}<\infty$ for all
$\gamma\in\Gamma_{j}$.

The discretization bound $N\in{\mathbb{N}}$ under which the bundles
${\mathscr{W}}_{j,+}^{0},{\mathscr{W}}_{j,-}^{0}$ of ($I_{0}$) persist for
$n\geq N$ is large, provided the spectral gap $\beta_{j}-\alpha_{j}$ is small,
the habitat $\Omega\subset{\mathbb{R}}^{\kappa}$ is large (in terms of its
diameter), the Lipschitz constants $\bar{\lambda}_{t}$ or the ‘operator norms’
in (8) are large, or the convergence rate via $\Gamma$ is small.

In case $j=0$ the bundles constructed in Thm. 3.3 reduce to
${\mathscr{W}}_{0,+}^{n}={\mathbb{I}}\times L_{d}^{p}(\Omega)$ and
${\mathscr{W}}_{0,-}^{n}={\mathscr{O}}$. Further specific mention deserves a
dichotomy spectrum of the form $(S_{1}^{1})$ and $j=J$, where we have
${\mathscr{W}}_{J,+}^{n}={\mathscr{O}}$ and
${\mathscr{W}}_{J,-}^{n}={\mathbb{I}}\times X_{n}^{d}$ ($n\geq N$) or
$X(\Omega)$ ($n=0$).

Figure 4: Hyperbolic situation $1\not\in\Sigma_{\mathbb{Z}}$ yielding the
stable bundles ${\mathscr{W}}_{j,+}^{n}$ and the unstable bundles
${\mathscr{W}}_{j,-}^{n}$

######  Remark 3.4 (hyperbolic case).

The trivial solution of ($I_{0}$) is called _hyperbolic_ , if
$1\not\in\Sigma_{\mathbb{Z}}$. In this case we assume $1\in(b_{j+1},a_{j})$
for some $j\in{\mathbb{N}}$ (see Fig. 4).

(1) Then ${\mathscr{W}}_{s}:={\mathscr{W}}_{j,+}^{0}$ is called _stable_ and
${\mathscr{W}}_{u}:={\mathscr{W}}_{j,-}^{0}$ _unstable bundle_ for ($I_{0}$).
The stable bundle contains the _strongly stable bundles_
${\mathscr{W}}_{i,+}^{0}$, $i>j$, while the unstable bundle contains the
finitely many _strongly unstable bundles_ ${\mathscr{W}}_{i,-}^{0}$, $0<i<j$,
of ($I_{0}$).

(2) The dichotomy spectrum $\Sigma_{\mathbb{Z}}$ of ($V_{0}$) behaves upper-
semicontinuously under discretization, that is, if $\Sigma_{\mathbb{Z}}^{n}$
denotes the spectrum of the discretized variational equation

$v_{t+1}=\Pi_{n}D{\mathscr{F}}_{t}(0)v_{t},$ ($V_{n}$)

then for each $\varepsilon>0$ there exists a $\bar{N}\in{\mathbb{N}}$ such
that $\Sigma_{\mathbb{Z}}^{n}\subseteq
B_{\varepsilon}(\Sigma_{\mathbb{Z}}^{0})$ for all $n\geq\bar{N}$. This
guarantees that hyperbolicity is preserved under discretization. In
particular, ${\mathscr{W}}_{i,+}^{0}$ persist as (strongly) stable bundles
${\mathscr{W}}_{i,+}^{n}$, $j\leq i$, and ${\mathscr{W}}_{j,-}^{0}$ persists
as (strongly) unstable bundles ${\mathscr{W}}_{j,-}^{n}$, $0<i\leq j$, of the
discretizations ($I_{n}$) for $n\geq\max\left\\{N,\bar{N}\right\\}$.

######  Remark 3.5 (periodic IDEs).

The periodic situation
$k_{t}=k_{t+\theta}:\Omega\times\Omega\to{\mathbb{K}}^{d\times n}$ and
$g_{t}=g_{t+\theta}:\Omega\times{\mathbb{K}}^{d}\to{\mathbb{K}}^{n}$ for all
$t\in{\mathbb{Z}}$ with some $\theta\in{\mathbb{N}}$ is a relevant special
case of general nonautonomous problems ($I_{0}$). Here the dichotomy spectrum
is

$\Sigma_{{\mathbb{Z}}}=\sqrt[\theta]{\left|\sigma(D{\mathscr{F}}_{\theta}(0)\cdots
D{\mathscr{F}}_{1}(0))\right|}\setminus\left\\{0\right\\}$

and the $\theta$-periodicity of ($I_{0}$) extends to the spatial
discretizations ($I_{n}$), as well as to the bundles
${\mathscr{W}}_{j,+}^{n}$, ${\mathscr{W}}_{j,-}^{n}$ for $n=0$ and $n\geq N$.
For the fibers this means
${\mathscr{W}}_{j,+}^{n}(t+\theta)={\mathscr{W}}_{j,+}^{n}(t)$ and
${\mathscr{W}}_{j,-}^{n}(t+\theta)={\mathscr{W}}_{j,-}^{n}(t)$ for all
$t\in{\mathbb{Z}}$.

###### Proof of Thm. 3.3.

Due to Prop. 2.1 the right-hand side of ($I_{0}$) is of class $C^{m}$ with

${\mathscr{L}}_{t}v:=D{\mathscr{F}}_{t}(0)v=\int_{\Omega}k_{t}(\cdot,y)D_{2}g_{t}(y,0)v(y)\,{\mathrm{d}}y\quad\text{for
all }t\in{\mathbb{I}}^{\prime},\,v\in L_{d}^{p}(\Omega)$

as derivative along the trivial solution. It results from the Lipschitz
condition (7) and [18, p. 363, Prop. C.1.1] that
$\left|D_{2}g_{t}(y,0)\right|\leq\bar{\lambda}_{t}(y)$ for all
$t\in{\mathbb{I}}^{\prime}$ and a.a. $y\in\Omega$. We derive from (9) that the
variational equation ($V_{0}$) satisfies (4) (with kernel
$l_{t}(x,y):=k_{t}(x,y)D_{2}g_{t}(y,0)$) and
$\Sigma_{\mathbb{I}}\subseteq(0,a_{0}]$; so the spectrum is of the form
required in Sect. 2.2.

In order to apply the perturbation Thm. A.1 in $X=L_{d}^{p}(\Omega)$ we first
specify the general difference equation ($\Delta_{\theta}$) as

$u_{t+1}={\mathscr{F}}_{t}(u_{t})+\tfrac{\theta}{\Gamma(\tfrac{1}{n})}\left[{\mathscr{F}}_{t}^{n}(u_{t})-{\mathscr{F}}_{t}(u_{t})\right]={\mathscr{L}}_{t}u_{t}+{\mathscr{N}}_{t}(u_{t})+\theta\bar{\mathscr{N}}_{t}(u_{t})$
(12)

depending on a real parameter $\theta$ and with the functions
${\mathscr{N}}_{t},\bar{\mathscr{N}}_{t}:L_{d}^{p}(\Omega)\to
L_{d}^{p}(\Omega)$,

$\displaystyle{\mathscr{N}}_{t}(u)$
$\displaystyle:={\mathscr{F}}_{t}(u)-D{\mathscr{F}}_{t}(0)u=\int_{\Omega}k_{t}(\cdot,y)h_{t}(y,u(y))\,{\mathrm{d}}y,$
$\displaystyle\bar{\mathscr{N}}_{t}(u)$
$\displaystyle:=\frac{1}{\Gamma(\tfrac{1}{n})}[\Pi_{n}-\operatorname{id}_{L_{d}^{p}(\Omega)}]{\mathscr{F}}_{t}(u)=\frac{1}{\Gamma(\tfrac{1}{n})}\left[\Pi_{n}-\operatorname{id}_{L_{d}^{p}(\Omega)}\right]\int_{\Omega}k_{t}(\cdot,y)g_{t}(y,u(y))\,{\mathrm{d}}y,$

where $h_{t}(y,z):=g_{t}(y,z)-D_{2}g_{t}(y,0)z$. For $\theta=0$ the semilinear
difference equation (12) reduces to the original problem ($I_{0}$), while for
$\theta=\Gamma(\tfrac{1}{n})$ one obtains the spatial discretization
($I_{n}$). We next verify the assumptions of Thm. A.1.

_ad $(H1)$_: According to the choice of $\alpha_{j}<\beta_{j}$, we obtain that
the variational equation ($V_{0}$) along the trivial solution satisfies the
dichotomy estimates (24) with real constants $K\geq 1$, $\alpha=\alpha_{j}$,
$\beta=\beta_{j}$ and an invariant projector $P_{t}\in L(L_{d}^{p}(\Omega))$,
$t\in{\mathbb{I}}$. By the choice of $\gamma\in\Gamma_{j}$ one has
$R(P_{t})={\mathscr{V}}_{j}^{+}(t)$ (if ${\mathbb{I}}$ is unbounded above) and
$N(P_{t})={\mathscr{V}}_{j}^{-}(t)$ (if ${\mathbb{I}}$ is unbounded below) for
all $t\in{\mathbb{I}}$.

_ad $(H2)$_: From Hypothesis (ii) it immediately results that both
${\mathscr{N}}_{t}$ and $\bar{\mathscr{N}}_{t}$ vanish identically in $0$.
Using the Mean Value Theorem we have for all $t\in{\mathbb{I}}^{\prime}$ that

$\displaystyle\left|h_{t}(x,z)-h_{t}(x,\bar{z})\right|$
$\displaystyle=\left|g_{t}(x,z)-g_{t}(x,\bar{z})-D_{2}g_{t}(x,0)(z-\bar{z})\right|$
$\displaystyle=\left|\int_{0}^{1}D_{2}g_{t}(x,\bar{z}+\vartheta(z-\bar{z}))-D_{2}g_{t}(x,0)\,{\mathrm{d}}\vartheta(z-\bar{z})\right|$
$\displaystyle\leq\int_{0}^{1}\left|D_{2}g_{t}(x,\bar{z}+\vartheta(z-\bar{z}))-D_{2}g_{t}(x,0)\right|\,{\mathrm{d}}\vartheta\left|z-\bar{z}\right|\stackrel{{\scriptstyle\eqref{no7s}}}{{\leq}}\lambda_{t}(x)\left|z-\bar{z}\right|$

for a.a. $x\in\Omega$ and $z,\bar{z}\in{\mathbb{K}}^{d}$. Hence, if
${\mathscr{H}}_{t}:L_{d}^{p}(\Omega)\to L_{n}^{q}(\Omega)$ denotes the
Nemytskii operator induced by $h_{t}$, then Cor. B.3 implies for
$t\in{\mathbb{I}}^{\prime}$ that
${\mathscr{N}}_{t}={\mathscr{K}}_{t}{\mathscr{H}}_{t}$ and then

$\displaystyle\left\|{\mathscr{N}}_{t}(u)-{\mathscr{N}}_{t}(\bar{u})\right\|_{p}$
$\displaystyle\leq\left\|{\mathscr{K}}_{t}\right\|_{L(L_{n}^{q}(\Omega),L_{d}^{p}(\Omega))}\left\|{\mathscr{H}}_{t}(u)-{\mathscr{H}}_{t}(\bar{u})\right\|_{q}$
$\displaystyle\stackrel{{\scriptstyle\eqref{propmak1}}}{{\leq}}\biggl{(}\int_{\Omega}\biggl{(}\int_{\Omega}\left|k_{t}(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p}{q^{\prime}}}\,{\mathrm{d}}x\biggr{)}^{\tfrac{1}{p}}\left\|{\mathscr{H}}_{t}(u)-{\mathscr{H}}_{t}(\bar{u})\right\|_{q}$
$\displaystyle\stackrel{{\scriptstyle\eqref{cormag2}}}{{\leq}}\biggl{(}\int_{\Omega}\biggl{(}\int_{\Omega}\left|k_{t}(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p}{q^{\prime}}}\,{\mathrm{d}}x\biggr{)}^{\tfrac{1}{p}}\biggl{(}\int_{\Omega}\lambda_{t}(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p-q}{p}}\left\|u-\bar{u}\right\|_{p}$
$\displaystyle\stackrel{{\scriptstyle\eqref{no63}}}{{\leq}}L\left\|u-\bar{u}\right\|_{p}\quad\text{for
all }u,\bar{u}\in L_{d}^{p}(\Omega).$

Concerning the nonlinearities $\bar{\mathscr{N}}_{t}$ we obtain from the
$X(\Omega)$-smoothing property assumed in Hypothesis (i) that
${\mathscr{F}}_{t}={\mathscr{K}}_{t}{\mathscr{G}}_{t}$ maps into $X(\Omega)$
and thus for $t\in{\mathbb{I}}^{\prime}$ that

$\displaystyle\left\|\bar{\mathscr{N}}_{t}(u)-\bar{\mathscr{N}}_{t}(\bar{u})\right\|_{p}$
$\displaystyle=\tfrac{1}{\Gamma(\tfrac{1}{n})}\left\|[\Pi_{n}-\operatorname{id}_{L_{d}^{p}(\Omega)}][{\mathscr{F}}_{t}(u)-{\mathscr{F}}_{t}(\bar{u})]\right\|_{p}\stackrel{{\scriptstyle\eqref{thmpifb4}}}{{\leq}}\left\|{\mathscr{F}}_{t}(u)-{\mathscr{F}}_{t}(\bar{u})\right\|_{X(\Omega)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{nosmooth}}}{{\leq}}C_{t}\left\|{\mathscr{G}}_{t}(u)-{\mathscr{G}}_{t}(\bar{u})\right\|_{q}\stackrel{{\scriptstyle\eqref{cormag2}}}{{\leq}}C_{t}\biggl{(}\int_{\Omega}\bar{\lambda}_{t}(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p-q}{p}}\left\|u-\bar{u}\right\|_{p}$

for all $u,\bar{u}\in L_{d}^{p}(\Omega)$, yielding $(H2)$ with
$\bar{L}:=\sup_{t\in{\mathbb{I}}^{\prime}}C_{t}\biggl{(}\int_{\Omega}\bar{\lambda}_{t}(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}y\biggr{)}^{\tfrac{p-q}{p}}<\infty$.
We now choose $N\in{\mathbb{N}}$ so large that
$\bar{L}\Gamma(\tfrac{1}{n})\leq L$ holds for all $n\geq N$, which ensures the
inclusion
$\Gamma(\tfrac{1}{n})\in\Theta:=\left\\{\theta\in{\mathbb{R}}:\,\bar{L}\left|\theta\right|\leq
L\right\\}$.

_ad $(H3)$_: From the Chain Rule, for $1<\ell\leq m$ we directly conclude the
derivatives

$\displaystyle D^{\ell}{\mathscr{N}}_{t}(u)$
$\displaystyle={\mathscr{K}}_{t}D^{\ell}{\mathscr{G}}_{t}(u),$ $\displaystyle
D^{\ell}\bar{\mathscr{N}}_{t}(u)$
$\displaystyle=\tfrac{1}{\Gamma(\tfrac{1}{n})}[\Pi_{n}-\operatorname{id}_{L_{d}^{p}(\Omega)}]{\mathscr{K}}_{t}D^{\ell}{\mathscr{G}}_{t}(u)\quad\text{for
all }t\in{\mathbb{I}}^{\prime}$

and $u\in L_{d}^{p}(\Omega)$. If $q_{\ell}:=\tfrac{pq}{p-\ell q}$, then it is
a consequence of Prop. B.4(b) and

$\displaystyle\left\|D^{\ell}{\mathscr{N}}_{t}(u)\right\|_{L_{\ell}(L_{d}^{p}(\Omega))}$
$\displaystyle\leq\left\|{\mathscr{K}}_{t}\right\|_{L(L_{n}^{q}(\Omega),L_{d}^{p}(\Omega))}\left\|D^{\ell}{\mathscr{G}}_{t}(u)\right\|_{L_{\ell}(L_{d}^{p}(\Omega),L_{n}^{q}(\Omega))}$
$\displaystyle\leq\left\|{\mathscr{K}}_{t}\right\|_{L(L_{n}^{q}(\Omega),L_{d}^{p}(\Omega))}\left(\left\|c\right\|_{q_{\ell}}+c_{\ell}\left\|u\right\|_{p}^{q/q_{\ell}}\right)$

combined with $c_{2}=\ldots=c_{m}=0$ that the derivatives
$D^{\ell}{\mathscr{N}}_{t}$ are globally bounded (uniformly in
$t\in{\mathbb{I}}^{\prime}$). Similarly, for all $t\in{\mathbb{I}}^{\prime}$
and $h_{1},\ldots,h_{\ell}\in L_{d}^{p}(\Omega)$ one has

$\displaystyle\left\|D^{\ell}\bar{\mathscr{N}}_{t}(u)h_{1}\cdots
h_{\ell}\right\|_{p}$
$\displaystyle=\tfrac{1}{\Gamma(\tfrac{1}{n})}\left\|[\Pi_{n}-\operatorname{id}_{L_{d}^{p}(\Omega)}]D^{\ell}{\mathscr{F}}_{t}(u)h_{1}\cdots
h_{\ell}\right\|_{p}$
$\displaystyle\stackrel{{\scriptstyle\eqref{thmpifb4}}}{{\leq}}\left\|{\mathscr{K}}_{t}D^{\ell}{\mathscr{G}}_{t}(u)h_{1}\cdots
h_{\ell}\right\|_{X(\Omega)}\stackrel{{\scriptstyle\eqref{nosmooth}}}{{\leq}}C_{t}\left\|D^{\ell}{\mathscr{G}}_{t}(u)h_{1}\cdots
h_{\ell}\right\|_{q}$ $\displaystyle\leq
C_{t}\left(\left\|c\right\|_{q_{\ell}}+c_{\ell}\left\|u\right\|_{p}^{q/q_{\ell}}\right)\left\|h_{1}\right\|_{p}\cdots\left\|h_{\ell}\right\|_{p}\quad\text{for
all }1<\ell\leq m$

and $u\in L_{d}^{p}(\Omega)$, which also yields the global boundedness of
$D^{\ell}\bar{\mathscr{N}}_{t}$ (uniformly in $t\in{\mathbb{I}}^{\prime}$).
After these preparations we can address the proof of statements (a–c):

(a) From Thm. A.1(a) we obtain that (12) has a $\gamma$-stable bundle
${\mathscr{W}}^{+}(\theta)\subseteq{\mathbb{I}}\times L_{d}^{p}(\Omega)$ as
graph of a continuous function $w^{+}(\cdot;\theta):{\mathbb{I}}\times
L_{d}^{p}(\Omega)\to L_{d}^{p}(\Omega)$ for all $\theta\in\Theta$. Then

$\displaystyle{\mathscr{W}}_{j,+}^{0}$ $\displaystyle:={\mathscr{W}}^{+}(0),$
$\displaystyle w_{j,+}^{0}$
$\displaystyle:=w^{+}(\cdot;0)|_{{\mathscr{V}}_{j}^{+}},$
$\displaystyle{\mathscr{W}}_{j,+}^{n}$
$\displaystyle:={\mathscr{W}}^{+}(\Gamma(\tfrac{1}{n})),$ $\displaystyle
w_{j,+}^{n}$
$\displaystyle:=w^{+}(\cdot;\Gamma(\tfrac{1}{n}))|_{{\mathscr{V}}_{j}^{+}}\quad\text{for
all }n\geq N,$

yield the $\gamma$-stable bundles of ($I_{0}$) and ($I_{n}$) resp. the
mappings parametrizing them; in particular, $w_{j,+}^{n}(\tau,0)\equiv 0$ on
${\mathbb{I}}$. Thm. A.1$(a_{2})$ guarantees the Lipschitz conditions

$\displaystyle\operatorname{lip}w_{j,+}^{0}(\tau,\cdot)$
$\displaystyle\stackrel{{\scriptstyle\eqref{noA3}}}{{\leq}}\frac{K^{2}L}{\delta-2KL},$
$\displaystyle\operatorname{lip}w_{j,+}^{n}(\tau,\cdot)$
$\displaystyle\stackrel{{\scriptstyle\eqref{noA3}}}{{\leq}}\frac{K^{2}(L+\Gamma(\tfrac{1}{n})\bar{L})}{\delta-K(L+\Gamma(\tfrac{1}{n})\bar{L})}\quad\text{for
all }\tau\in{\mathbb{I}},\,n\geq N$

and defining $C_{0}^{+}(0):=\tfrac{K^{2}L}{\delta-2KL}$,
$C_{0}^{+}(n):=\tfrac{K^{2}(L+\Gamma(\tfrac{1}{n})\bar{L})}{\delta-K(L+\Gamma(\tfrac{1}{n})\bar{L})}$
yields $\lim_{n\to\infty}C_{0}^{+}(n)=C_{0}^{+}(0)$. Furthermore, the
Lipschitz estimate (27) leads to

$\displaystyle\left\|w_{j,+}^{n}(\tau,v)-w_{j,+}^{0}(\tau,v)\right\|_{p}$
$\displaystyle=\left\|w^{+}(\tau,v;\Gamma(\tfrac{1}{n}))-w^{+}(\tau,v;0)\right\|_{p}$
$\displaystyle\leq\frac{2\delta
K^{3}\bar{L}}{(\delta-4K\bar{L})^{2}}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p}\quad\text{for
all }(\tau,v)\in{\mathscr{V}}_{j}^{+},\,n\geq N.$

If we set $N_{0}^{+}:=\tfrac{2\delta K^{3}\bar{L}}{(\delta-4K\bar{L})^{2}}$,
then this establishes the assertion $(a_{1})$. Under the additional
assumptions imposed for $(a_{2})$ we obtain from Thm. A.1$(a_{3})$ that the
partial derivatives $D_{3}D_{2}^{\ell}w_{+}$ exist for $0\leq\ell<m_{+}$. The
Mean Value Estimate implies

$\displaystyle\left\|D_{2}^{\ell}w_{j,+}^{n}(\tau,v)-D_{2}^{\ell}w_{j,+}^{0}(\tau,v)\right\|_{p}$
$\displaystyle=\left\|D_{2}^{\ell}w^{+}(\tau,v;\Gamma(\tfrac{1}{n}))-D_{2}^{\ell}w^{+}(\tau,v;0)\right\|_{p}$
$\displaystyle\leq\sup_{\theta\in\Theta}\left\|D_{3}D_{2}^{\ell}w^{+}(\tau,v;\theta)\right\|_{p}\Gamma(\tfrac{1}{n})\stackrel{{\scriptstyle\eqref{thmifb6}}}{{\leq}}N^{+}\left\|v\right\|_{p}\Gamma(\tfrac{1}{n})$

for all $(\tau,v)\in{\mathscr{V}}_{j}^{+}$ and $n\geq N$ implying the error
estimate in $(a_{2})$.

(b) This results parallel to (a) using Thm. A.1(b). In particular,
${\mathscr{W}}_{j,-}^{0}:={\mathscr{W}}^{-}(0)$ and
${\mathscr{W}}_{j,-}^{n}:={\mathscr{W}}^{-}(\Gamma(\tfrac{1}{n}))$ are graphs
over the linear bundle ${\mathscr{V}}_{j}^{-}$. Since the right-hand sides of
($I_{0}$) are completely continuous due to Prop. 2.1, we obtain from [15, p.
89, Prop. 6.5] that the derivatives in the variational equation ($V_{0}$) are
compact operators in $L(L_{d}^{p}(\Omega))$. Therefore, [22, Cor. 4.13]
implies that ${\mathscr{V}}_{j}^{-}$ is finite-dimensional, which transfers to
${\mathscr{W}}_{j,-}^{n}$ for $n=0$ and $n\geq N$. Moreover, due to the
invariance
${\mathscr{W}}_{j,-}^{n}(t+1)={\mathscr{F}}_{t}^{n}({\mathscr{W}}_{j,-}^{n}(t))\subseteq
X_{n}^{d}$ for all $t\in{\mathbb{I}}^{\prime}$ one has
${\mathscr{W}}_{j,-}^{n}\subseteq{\mathbb{I}}\times X_{n}^{d}$ for $n\geq N$
and similarly the smoothing property required in Hypothesis (i) implies
${\mathscr{W}}_{j,-}^{0}\subseteq{\mathbb{I}}\times X(\Omega)$.

(c) is a direct consequence of Thm. A.1(c). ∎

###  3.2 Pseudo-center bundles

Complementing Thm. 3.3, we next establish that also the intersections

${\mathscr{V}}_{i,j}:={\mathscr{V}}_{i}^{+}\cap{\mathscr{V}}_{j}^{-}\quad\text{for
}i<j$

and thus the spectral bundles ${\mathscr{V}}_{j}={\mathscr{V}}_{j-1,1}$ of
($V_{0}$) persist under nonlinear perturbations, as well as spatial
discretization as invariant bundles.

######  Theorem 3.6 (pseudo-center bundles under discretization).

Let Hypotheses (i–iii) hold with ${\mathbb{I}}={\mathbb{Z}}$,
$0<\gamma_{j}<\gamma_{i}$, $i,j\in{\mathbb{N}}_{0}$, be as in $(S_{1})$ or
$(S_{2})$, choose $\alpha_{k}<\beta_{k}$ so that

$\displaystyle\alpha_{k}$ $\displaystyle<\gamma_{k}<\beta_{k},$
$\displaystyle(\alpha_{k},\beta_{k})\cap\Sigma_{\mathbb{Z}}$
$\displaystyle=\emptyset\quad\text{ for }k\in\left\\{i,j\right\\}$

and let ${\mathscr{V}}_{i}^{+},{\mathscr{V}}_{j}^{-}$ denote the associated
linear bundles of the variational equation ($V_{0}$). Then there are
$K_{i},K_{j}\geq 1$ depending only on ($V_{0}$), so that if
$K_{\max}:=\\!\displaystyle\max_{k\in\left\\{i,j\right\\}}(K_{k}^{2}+2K_{k})$,

$\displaystyle L$ $\displaystyle<\frac{\delta_{\max}}{2K_{\max}},$
$\displaystyle\delta_{\max}$
$\displaystyle:=\min_{k\in\left\\{i,j\right\\}}\frac{\beta_{k}-\alpha_{k}}{2},$
(13)

$\delta\in(4K_{\max}L,\delta_{\max}]$ is fixed and
$\Gamma_{k}:=[\alpha_{k}+\delta,\beta_{k}-\delta]$ for
$k\in\left\\{i,j\right\\}$, then there exists a $N\in{\mathbb{N}}$ such that
for $n=0$ and $n\geq N$ the _pseudo-center bundle_

$\displaystyle{\mathscr{W}}_{i,j}^{n}$
$\displaystyle:={\mathscr{W}}_{i,+}^{n}\cap{\mathscr{W}}_{j,-}^{n}$
$\displaystyle=\left\\{(\tau,u)\in{\mathbb{Z}}\times
L_{d}^{p}(\Omega):\begin{array}[]{l}\sup_{\tau\leq
t}\gamma_{i}^{\tau-t}\left\|\varphi^{n}(t;\tau,u)\right\|_{p}<\infty\text{ and
there}\\\ \text{exists a solution }(\phi_{t})_{t\in{\mathbb{Z}}}\text{ of
\eqref{iden} such}\\\ \text{that
}\sup_{t\leq\tau}\gamma_{j}^{\tau-t}\left\|\phi_{t}\right\|_{p}<\infty\text{
and }\phi_{\tau}=u\end{array}\right\\}$

is a finite-dimensional (unless for spectra $(S_{1}^{1})$ and $j=J$) invariant
bundle of ($I_{n}$), which is independent of $\gamma_{i}\in\Gamma_{i}$,
$\gamma_{j}\in\Gamma_{j}$ and allows the representation

${\mathscr{W}}_{i,j}^{n}=\left\\{(\tau,v+w_{i,j}^{n}(\tau,v))\in{\mathbb{Z}}\times
L_{d}^{p}(\Omega):(\tau,v)\in{\mathscr{V}}_{i,j}\right\\}\subseteq\begin{cases}{\mathbb{Z}}\times
X(\Omega),&n=0,\\\ {\mathbb{Z}}\times X_{n}^{d},&n\geq N\end{cases}$

with continuous mappings $w_{i,j}^{n}:{\mathscr{V}}_{i,j}\to
L_{d}^{p}(\Omega)$ satisfying $w_{i,j}^{n}(\tau,0)=0$ and

  * (a)

there exist $C_{0}(n),N_{0}>0$ with
$\lim_{n\to\infty}C_{0}(n)=\max_{k\in\left\\{i,j\right\\}}\frac{K_{k}^{2}L}{\delta-(K_{k}^{2}+2K_{k})L}=:C_{0}(0)$
and for all $(\tau,v)\in{\mathscr{V}}_{i,j}$ holds

$\displaystyle\operatorname{lip}w_{i,j}^{n}(\tau,\cdot)$ $\displaystyle\leq
C_{0}(n),$
$\displaystyle\left\|w_{i,j}^{n}(\tau,v)-w_{i,j}^{0}(\tau,v)\right\|_{p}$
$\displaystyle\leq N_{0}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p},$ (14)

  * (b)

if $\alpha_{i}^{m}<\beta_{i}$, $\alpha_{j}<\beta_{j}^{m}$ and

$\delta_{\max}:=\min\Bigl{\\{}\tfrac{\beta_{i}-\alpha_{i}}{2},\tfrac{\beta_{j}-\alpha_{j}}{2},\alpha_{i}\bigl{(}\sqrt[m]{\tfrac{\alpha_{i}+\beta_{i}}{\alpha_{i}+\alpha_{i}^{m}}}-1\bigr{)},\beta_{j}\bigl{(}1-\sqrt[m]{\tfrac{\alpha_{j}+\beta_{j}}{\beta_{j}+\beta_{j}^{m}}}\bigr{)}\Bigr{\\}},$

then the derivatives $D_{2}^{\ell}w_{i,j}^{n}:{\mathscr{V}}_{i,j}\to
L_{\ell}(L_{d}^{p}(\Omega))$ exist up to order $\ell\leq m$ as continuous maps
and for $1\leq\ell<m$ there exist $N_{\ell}>0$ such that

$\displaystyle\left\|D_{2}^{\ell}w_{i,j}^{n}(\tau,v)-D_{2}^{\ell}w_{i,j}^{0}(\tau,v)\right\|_{L_{\ell}(L_{d}^{p}(\Omega))}$
$\displaystyle\leq
N_{\ell}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p}\quad\text{for all
}(\tau,v)\in{\mathscr{V}}_{i,j}.$

The magnitude of $N$ is determined by the same factors as in Thm. 3.3. We
furthermore point out spectra $(S_{1}^{1})$ and $j=J$, in which
${\mathscr{W}}_{i,J}^{n}={\mathscr{W}}_{i,+}^{n}$ for all $i<J$.

######  Remark 3.7 (extended hierarchy).

For dichotomy spectra $\Sigma_{\mathbb{Z}}$ of the form $(S_{1})$ choose
$J\in{\mathbb{N}}$ as in Sect. 2.2, while for spectra $(S_{2})$ let
$J\in{\mathbb{N}}$ be arbitrary. Then it results from their dynamical
characterization and (25) that the bundles constructed in Thms. 3.3 and 3.6
satisfy the following inclusions denoted as _extended hierarchy_

$\begin{array}[]{ccccccccccc}{\mathscr{W}}_{J,+}^{n}&\subset&{\mathscr{W}}_{J-1,+}^{n}&\subset&\ldots&\subset&{\mathscr{W}}_{2,+}^{n}&\subset&{\mathscr{W}}_{1,+}^{n}&\subset&{\mathbb{Z}}\times
L_{d}^{p}(\Omega)\\\ &&\cup&&&&\cup&&\cup&&\cup\\\
&&{\mathscr{W}}_{J-1,J}^{n}&\subset&\ldots&\subset&{\mathscr{W}}_{2,J}^{n}&\subset&{\mathscr{W}}_{1,J}^{n}&\subset&{\mathscr{W}}_{J,-}^{n}\\\
&&&&&&&&\cup&&\cup\\\ &&&&&\ddots&&&\vdots&&\vdots\\\ &&&&&&&&\cup&&\cup\\\
&&&&&&&&{\mathscr{W}}_{1,2}^{n}&\subset&{\mathscr{W}}_{2,-}^{n}\\\
&&&&&&&&&&\cup\\\ &&&&&&&&&&{\mathscr{W}}_{1,-}^{n}.\end{array}$

Figure 5: Nonhyperbolic situation where $1$ is contained in the spectral
interval $[a_{j},b_{j}]$

######  Remark 3.8 (persistence and center bundles).

In a _nonhyperbolic_ situation $1\in[a_{j},b_{j}]$ for some $1<j<J$ (see Fig.
5), ${\mathscr{W}}_{s}={\mathscr{W}}_{j,+}^{0}$ is the stable,
${\mathscr{W}}_{cs}:={\mathscr{W}}_{j-1,+}^{0}$ the _center-stable_ ,
${\mathscr{W}}_{cu}:={\mathscr{W}}_{j,-}^{0}$ the _center-unstable_ and
${\mathscr{W}}_{u}={\mathscr{W}}_{j-1,-}^{0}$ the unstable bundle of
($I_{0}$), while the _center bundle_
${\mathscr{W}}_{c}:={\mathscr{W}}_{j-1,j}^{0}$ completes the _classical
hierarchy_ (cf. (1)). Now a center bundle, by definition, is graph over the
spectral bundle ${\mathscr{V}}_{j}$ associated to the spectral interval
containing $1$. Although the center bundle ${\mathscr{W}}_{c}$ of ($I_{0}$)
persists as invariant bundles ${\mathscr{W}}_{j-1,j}^{n}$ for all $n\geq N$,
in general these sets ${\mathscr{W}}_{j-1,j}^{n}$ fail to be center bundles of
the discretizations ($I_{n}$). This is due to the fact that $1$ needs not to
be contained in the dichotomy spectrum $\Sigma_{\mathbb{Z}}^{n}$ of the
discretized variational equations ($V_{n}$). For instance, one can think of a
singleton interval $[a_{j},b_{j}]=\left\\{1\right\\}$ moving away from $1$
under discretization or the fact that a solid spectral interval
$[a_{j},b_{j}]$ splits into subintervals, which is possible by the upper-
semicontinuity of the spectrum $\Sigma_{\mathbb{Z}}$.

###### Proof of Thm. 3.6.

Above all, choose $i<j$ fixed and we stay in the terminology established in
the proof of Thm. 3.3.

(I) Our present assumptions yield that Thm. A.1 applies to IDEs (12) in
$X=L_{d}^{p}(\Omega)$. Therefore, for each $\theta\in\Theta$ there exists a
$\gamma_{i}$-stable bundle ${\mathscr{W}}^{+}(\theta)$ being a graph of a
mapping $w^{+}(\cdot;\theta):{\mathscr{V}}_{i}^{+}\to L_{d}^{p}(\Omega)$ and a
$\gamma_{j}$-unstable bundle ${\mathscr{W}}^{-}(\theta)$ represented as graph
of $w^{-}(\cdot;\theta):{\mathscr{V}}_{j}^{-}\to L_{d}^{p}(\Omega)$. The fact
that their intersection can be represented as graph of a mapping
$w(\cdot;\theta):{\mathbb{I}}\times L_{d}^{p}(\Omega)\to L_{d}^{p}(\Omega)$
over ${\mathscr{V}}:={\mathscr{V}}_{i,j}$ is shown as in the proof of [18, p.
208, Prop. 4.2.17]. Hence, it remains to establish the convergence statements.
For this purpose, we choose $\theta\in\Theta$ and obtain from Thm. A.1 that

$\displaystyle\operatorname{lip}w^{+}(\tau,\cdot;\theta)$
$\displaystyle\stackrel{{\scriptstyle\eqref{noA3}}}{{\leq}}\frac{2K_{i}^{2}L}{\delta-4K_{i}L}<1,$
$\displaystyle\operatorname{lip}w^{-}(\tau,\cdot;\theta)$
$\displaystyle\stackrel{{\scriptstyle\eqref{noA7}}}{{\leq}}\frac{2K_{j}^{2}L}{\delta-4K_{j}L}<1\quad\text{for
all }\tau\in{\mathbb{Z}}$

due to our strengthened assumptions (13) (in comparison to (11)); for this
note that $\left|\theta\right|\bar{L}\leq L$ due to $\theta\in\Theta$. This
implies that

$\displaystyle T_{\tau}:L_{d}^{p}(\Omega)^{2}\times
L_{d}^{p}(\Omega)\times\Theta$ $\displaystyle\to L_{d}^{p}(\Omega)^{2},$
$\displaystyle T_{\tau}(u,w,v,\theta)$
$\displaystyle:=\begin{pmatrix}w^{+}(\tau,v+w;\theta)\\\
w^{-}(\tau,u+v;\theta)\end{pmatrix}$

is a contraction in its first two arguments $(u,w)\in L_{d}^{p}(\Omega)^{2}$,
uniformly in the parameters $(v,\theta)\in L_{d}^{p}(\Omega)\times\Theta$,
with the Lipschitz constant
$\max_{k\in\left\\{i,j\right\\}}\frac{2K_{k}^{2}L}{\delta-4K_{k}L}<1$. Based
on its unique fixed point
$(\Upsilon_{\tau}^{+},\Upsilon_{\tau}^{-})(v,\theta)\in L_{d}^{p}(\Omega)^{2}$
we define the mapping

$w(\tau,v;\theta):=\Upsilon_{\tau}^{+}(v,\theta)+\Upsilon_{\tau}^{-}(v,\theta)\quad\text{for
all }\tau\in{\mathbb{I}},\,v\in L_{d}^{p}(\Omega),\,\theta\in\Theta,$ (15)

depending only on the projection of $v\in L_{d}^{p}(\Omega)$ onto
${\mathscr{V}}(\tau)$. Given $\theta,\bar{\theta}\in\Theta$ and with
$\Upsilon_{\tau}:=(\Upsilon_{\tau}^{+},\Upsilon_{\tau}^{-})$, using the
Lipschitz estimates (27), (31) from Thm. A.1 we obtain as in the proof of [17,
Thm. 4.2] that there exists a constant $\tilde{N}_{0}\geq 0$ such that

$\left\|\Upsilon_{\tau}(v,\theta)-\Upsilon_{\tau}(v,\bar{\theta})\right\|\leq\tilde{N}_{0}\left\|v\right\|_{p}\left|\theta-\bar{\theta}\right|\quad\text{for
all }(\tau,v)\in{\mathscr{V}}.$ (16)

Referring to (15) this guarantees

$\left\|w(\tau,v;\theta)-w(\tau,v;\bar{\theta})\right\|_{p}\leq
2\tilde{N}_{0}\left\|v\right\|_{p}\left|\theta-\bar{\theta}\right|\quad\text{for
all }(\tau,v)\in{\mathscr{V}},\,\theta,\bar{\theta}\in\Theta.$ (17)

(II) Now suppose that both spectral gap conditions $\alpha_{i}^{m}<\beta_{i}$,
$\alpha_{j}<\beta_{j}^{m}$ hold yielding that $w^{+}(\tau,\cdot)$ and
$w^{-}(\tau,\cdot)$ are of class $C^{m}$ with globally bounded partial
derivatives of order $1\leq\ell\leq m$ w.r.t. $v$ due to (28) and (32). In
order to establish that there exist constants $\tilde{N}_{\ell}\geq 0$ such
that

$\left\|D_{2}D_{1}^{\ell}\Upsilon_{\tau}(v,\theta)\right\|_{L_{\ell}(L_{d}^{p}(\Omega),L_{d}^{p}(\Omega)^{2})}\leq\tilde{N}_{\ell}\left\|v\right\|_{p}\quad\text{for
all }(\tau,v)\in{\mathscr{V}},\,\theta\in\Theta,$ (18)

we proceed by mathematical induction. For $\ell=0$ the bound on
$D_{2}\Upsilon_{\tau}$ results from the Lipschitz condition (16) combined with
[18, p. 363, Prop. C.1.1]. Differentiating the fixed-point identity
$\Upsilon_{\tau}(v,\theta)\equiv T_{\tau}(\Upsilon_{\tau}(v,\theta),v,\theta)$
on ${\mathscr{V}}\times\Theta$ w.r.t. $\theta$ first implies

$D_{2}\Upsilon_{\tau}(v,\theta)\equiv
D_{(1,2)}T_{\tau}(\Upsilon_{\tau}(v,\theta),v,\theta)D_{2}\Upsilon_{\tau}(v,\theta)+D_{4}T_{\tau}(\Upsilon_{\tau}(v,\theta),v,\theta)$
(19)

on ${\mathscr{V}}\times\Theta$ and then thanks to the Product and Chain Rule
(with partially unfolded derivative tree) the derivatives of (19) w.r.t. $v$
of order $1\leq\ell<m$ read as

$\displaystyle D_{2}D_{1}^{\ell}\Upsilon_{\tau}(v,\theta)\equiv$
$\displaystyle\underbrace{D_{(1,2)}T_{\tau}(\Upsilon_{\tau}(v,\theta),v,\theta)}_{\left\|\cdot\right\|<1}D_{2}D_{1}^{\ell}\Upsilon(v,\theta)$
$\displaystyle+D_{v}^{\ell}\left(D_{(1,2)}T_{\tau}(\Upsilon_{\tau}(v,\theta),v,\theta)D_{2}\Upsilon(v,\theta)+D_{4}T_{\tau}(\Upsilon_{\tau}(v,\theta),v,\theta)\right)$

on ${\mathscr{V}}\times\Theta$. Here, the norm bound on
$D_{(1,2)}T_{\tau}(\Upsilon_{\tau}(v,\theta),v,\theta)$ results from the
contraction property of $T_{\tau}(\cdot,v,\theta)$ and again [18, p. 363,
Prop. C.1.1]. Then the second term in the above sum is a sum of products. In
each of them a derivative of $\Upsilon_{\tau}$ w.r.t. $\theta$ occurs exactly
once, which yields a factor containing the term $\left\|v\right\|_{p}$ (note
(29) and (33)), while the remaining factors are derivatives of
$\Upsilon_{\tau}$ w.r.t. $v$, being bounded due to our induction hypothesis
(see (28) and (32)). This establishes (18) from which one has

$\left\|D_{3}D_{2}^{\ell}w(\tau,v;\theta)\right\|_{L_{\ell}(L_{d}^{p}(\Omega),L_{d}^{p}(\Omega))}\leq
2\tilde{N}_{\ell}\left\|v\right\|_{p}\quad\text{for all
}(\tau,v)\in{\mathscr{V}},\,\theta\in\Theta.$ (20)

(III) After the prelude in steps (I) and (II) we finally define

$\displaystyle{\mathscr{W}}_{i,j}^{0}$
$\displaystyle:={\mathscr{W}}^{+}(0)\cap{\mathscr{W}}^{-}(0),$ $\displaystyle
w_{i,j}^{0}$ $\displaystyle:=w(\cdot,0)|_{{\mathscr{V}}},$
$\displaystyle{\mathscr{W}}_{i,j}^{n}$
$\displaystyle:={\mathscr{W}}^{+}(\Gamma(\tfrac{1}{n}))\cap{\mathscr{W}}^{-}(\Gamma(\tfrac{1}{n})),$
$\displaystyle w_{i,j}^{n}$
$\displaystyle:=w(\cdot,\Gamma(\tfrac{1}{n}))|_{{\mathscr{V}}}\quad\text{for
all }n\geq N$

and argue as above, i.e. by referring to [18, p. 208, Prop. 4.2.17] that it
remains to establish the convergence estimates given in (a) and (b) for $n\geq
N$:

(a) Then the error estimate in (14) results from step (I) as for
$(\tau,v)\in{\mathscr{V}}_{i,j}$ one has

$\left\|w_{i,j}^{n}(\tau,v)-w_{i,j}^{0}(\tau,v)\right\|_{p}=\left\|w(\tau,v;\Gamma(\tfrac{1}{n}))-w(\tau,v;0)\right\|_{p}\stackrel{{\scriptstyle\eqref{susy}}}{{\leq}}2\tilde{N}_{0}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p}.$

(b) Using the Mean Value Estimate we derive from step (II) for $1\leq\ell<m$
that

$\displaystyle\left\|D_{2}^{\ell}w_{i,j}^{n}(\tau,v)-D_{2}^{\ell}w_{i,j}^{0}(\tau,v)\right\|_{p}$
$\displaystyle=\left\|D_{2}^{\ell}w(\tau,v;\Gamma(\tfrac{1}{n}))-D_{2}^{\ell}w(\tau,v;0)\right\|_{p}$
$\displaystyle\leq\sup_{\theta\in\Theta}\left\|D_{3}D_{2}^{\ell}w(\tau,v;\theta)\right\|\Gamma(\tfrac{1}{n})\stackrel{{\scriptstyle\eqref{no22}}}{{\leq}}2\tilde{N}_{\ell}\Gamma(\tfrac{1}{n})\left\|v\right\|_{p}$

for all $(\tau,v)\in{\mathscr{V}}_{i,j}$. This establishes claim (b). ∎

###  3.3 Applicability

One cannot expect general IDEs

$\displaystyle u_{t+1}$ $\displaystyle=\tilde{\mathscr{F}}_{t}(u_{t}),$
$\displaystyle\tilde{\mathscr{F}}_{t}(u)$
$\displaystyle:=\int_{\Omega}k_{t}(\cdot,y)\tilde{g}_{t}(y,u(y))\,{\mathrm{d}}y,$
($\tilde{I}_{0}$)

arising in real-world applications [11, 14] to satisfy global assumption such
as (6), (7), the global smallness condition (11), or that the trivial solution
lies in the center of interest. Nonetheless, given any solution
$(\phi_{t})_{t\in{\mathbb{I}}}$ of ($\tilde{I}_{0}$) the _equation of
perturbed motion_

$u_{t+1}=\tilde{\mathscr{F}}_{t}(u_{t}+\phi_{t})-\tilde{\mathscr{F}}_{t}(\phi_{t})=\int_{\Omega}k_{t}(\cdot,y)\bar{g}_{t}(y,u_{t}(y))\,{\mathrm{d}}y$

with growth function
$\bar{g}_{t}(x,z):=\tilde{g}_{t}(x,z+\phi_{t}(x))-\tilde{g}_{t}(x,\phi_{t}(x))$
has the trivial solution. The dynamics in its vicinity is the same as that of
($\tilde{I}_{0}$) near the reference solution $\phi$.

The solutions of ($\tilde{I}_{0}$) being relevant in applications have
(essentially) bounded values $\phi_{t}$. Under this premise a suitable
modification of the growth function
$\bar{g}_{t}:\Omega\times{\mathbb{K}}^{d}\to{\mathbb{K}}^{n}$ allows to apply
Thm. 3.3 and 3.6. For this purpose assume that $\bar{g}_{t}$ satisfies
Hypothesis (ii) with the global conditions (6) and (7) weakened to

  * •

$\lim_{z\to
0}\operatorname*{ess\,sup}_{x\in\Omega}\left|D_{2}\tilde{g}_{t}(x,z+\phi_{t}(x))-D_{2}\tilde{g}_{t}(x,\phi_{t}(x))\right|=0$
for all $t\in{\mathbb{I}}^{\prime}$,

  * •

for each $r>0$ and $t\in{\mathbb{I}}^{\prime}$ there exists a
$\Lambda_{t}(r)\geq 0$ such that

$\left|\tilde{g}(x,z)-\tilde{g}(x,\bar{z})\right|\leq\Lambda_{t}(r)\left|z-\bar{z}\right|\quad\text{for
all }z,\bar{z}\in\bar{B}_{r}(\phi_{t}(x))\text{ and $\mu$-a.a.\ }x\in\Omega.$

Given $\rho>0$, let $\chi_{\rho}:{\mathbb{K}}^{d}\to[0,1]$ denote a _cut-off
function_ , i.e. a $C^{\infty}$-function constant $\chi_{\rho}(z)\equiv 1$ on
$\bar{B}_{\rho}(0)$, $\chi_{\rho}(z)\in(0,1)$ for $z\in
B_{2\rho}(0)\setminus\bar{B}_{\rho}(0)$ and vanishing $\chi_{\rho}(z)\equiv 0$
on the remaining set ${\mathbb{K}}^{d}\setminus B_{2\rho}(0)$ (cf. [18, p.
369, Prop. C.2.16]). First, thanks to this construction the growth functions
$\bar{g}_{t}$ and

$g_{t}(x,z):=\bar{g}_{t}(x,\chi_{\rho}(z)z)=\tilde{g}_{t}(x,\chi_{\rho}(z)z+\phi_{t}(x))-\tilde{g}_{t}(x,\phi_{t}(x))$
(21)

coincide for all $z\in\bar{B}_{\rho}(0)$ and a.a. $x\in\Omega$. Second, by
[18, p. 370, Prop. C.2.17] the radius $\rho>0$ can be chosen so small that
under the balancing conditions (6), (7) the resulting nonlinearities
${\mathscr{N}}_{t},\bar{\mathscr{N}}_{t}:L_{d}^{p}(\Omega)\to
L_{d}^{p}(\Omega)$ in the proof of Thm. 3.3 satisfy the assumptions of Thm.
A.1. This allows to obtain the particular fiber bundles
${\mathscr{W}}_{i,+}^{n},{\mathscr{W}}_{j,-}^{n}$ and
${\mathscr{W}}_{i,j}^{n}$ of the IDE ($I_{0}$), whose growth function $g_{t}$
is given in (21).

In order to transfer this information back to the original IDE
($\tilde{I}_{0}$) we note that the dynamics of ($\tilde{I}_{0}$) in the
vicinity $\\{(t,u)\in{\mathbb{I}}\times
L_{d}^{p}(\Omega):\,\left\|u-\phi_{t}\right\|_{\infty}\leq\rho\\}$ of the
reference solution $\phi$ and the behavior of the modified problem ($I_{0}$)
in the nonautonomous set $\\{(t,u)\in{\mathbb{I}}\times
L_{d}^{p}(\Omega):\,\left\|u\right\|_{\infty}\leq\rho\\}$ are related by a
translation via the solution $\phi$.

##  4 Galerkin discretizations of integrodifference equations

This section demonstrates that the convergence assumption from Hypothesis
(iii) holds for various Galerkin approximations. They are a special case of
general projection methods (cf. [3, Sect. 2] or [4, pp. 448ff]) to discretize
IDEs spatially. Here, let $a<b$.

###  4.1 Galerkin methods on $\Omega=(a,b)$

On the interval $\Omega:=(a,b)$, choose $n\in{\mathbb{N}}_{0}$, the nodes

$\displaystyle a$ $\displaystyle=x_{0}^{n}<x_{1}^{n}<\ldots<x_{m_{n}}^{n}=b$
(22)

and set $m_{0}:=1$ (i.e. one obtains the entire interval $(a,b)$ for $n=0$),

$\displaystyle h_{n}$
$\displaystyle:=\max_{i=1}^{m_{n}}(x_{i}^{n}-x_{i-1}^{n}),$ $\displaystyle
I_{i}^{n}$ $\displaystyle:=[x_{i-1}^{n},x_{i}^{n}]\quad\text{for all }1\leq
i\leq m_{n}.$

We assume throughout that there exists a $C>0$ such that

$\displaystyle h_{n}\leq\tfrac{C}{n}\quad\text{for all }n\in{\mathbb{N}}.$
(23)

#### Piecewise constant approximation

Let $\chi_{I_{i}^{n}}:{\mathbb{R}}\to\left\\{0,1\right\\}$, $1\leq i\leq
m_{n}$, be the characteristic functions over the partition (22). With
$X_{n}:=\operatorname{span}\bigl{\\{}\chi_{I_{1}^{n}},\ldots,\chi_{I_{m_{n}}^{n}}\bigr{\\}}$
one defines the projections (see [2, p. 305, 9.21])

$\Pi_{n}u:=\sum_{i=1}^{m_{n}}\chi_{I_{i}^{n}}(\cdot)\left(\frac{1}{x_{i}^{n}-x_{i-1}^{n}}\int_{x_{i-1}^{n}}^{x_{i}^{n}}u(x)\,{\mathrm{d}}x\right)\quad\text{for
all }u\in L_{d}^{p}(a,b)$

onto the piecewise constant functions $X_{n}^{d}$ and has the error estimate

$\left\|[\operatorname{id}_{L_{d}^{p}(a,b)}-\Pi_{n}]u\right\|_{p}\leq
h_{n}\left\|u^{\prime}\right\|_{p}\quad\text{for all }n\in{\mathbb{N}},\,u\in
W_{d}^{1,p}(a,b).$

This yields a discretization error (10) with the space of weakly
differentiable functions $X(a,b)=W_{d}^{1,p}(a,b)$ having first order
derivatives in $L_{d}^{p}(a,b)$ and $\Gamma(\varrho)=C\varrho$.

#### Continuous piecewise linear approximation

Let $\chi_{i}^{n}:{\mathbb{R}}\to[0,1]$, $0\leq i\leq m_{n}$, denote the hat
functions over the partition (22). For
$X_{n}:=\operatorname{span}\bigl{\\{}\chi_{0}^{n},\ldots,\chi_{m_{n}}^{n}\bigr{\\}}$
and continuous functions $u:(a,b)\to{\mathbb{K}}^{d}$ one defines the
projections

$\Pi_{n}u(x):=\sum_{i=1}^{m_{n}}\chi_{I_{i}^{n}}(x)\left(\frac{x_{i}^{n}-x}{x_{i^{n}}^{n}-x_{i-1}^{n}}u(x_{i-1}^{n})+\frac{x-x_{i-1}^{n}}{x_{i^{n}}^{n}-x_{i-1}^{n}}u(x_{i}^{n})\right)\quad\text{for
all }u\in C_{d}(a,b)$

onto $X_{n}^{d}$. First, for each $l\in\left\\{1,2\right\\}$ one has the error
estimate (cf. [2, p. 309, 9.22])

$\left\|[\operatorname{id}_{L_{d}^{p}(a,b)}-\Pi_{n}]u\right\|_{p}\leq
2h_{n}^{l}\left\|u^{(l)}\right\|_{p}\quad\text{for all
}n\in{\mathbb{N}},\,u\in W_{d}^{l,p}(a,b).$

Hence, (10) holds with $X(a,b)=W_{d}^{l,p}(a,b)$ being the space of functions
whose weak derivatives exist up to order $l\leq 2$ in $L_{d}^{p}(a,b)$ and
$\Gamma(\varrho)=2C^{l}\varrho^{l}$.

Second, for $\alpha\in(0,1]$ and $\alpha$-Hölder functions
$u:[a,b]\to{\mathbb{K}}^{d}$ one establishes the error estimate (s. [4, p.
457, (12.2.4)])

$\left\|[\operatorname{id}_{L_{d}^{p}(a,b)}-\Pi_{n}]u\right\|_{p}\leq(b-a)^{1/p}d^{1/p}h_{n}^{\alpha}\left\|u\right\|_{C_{d}^{\alpha}[a,b]}\quad\text{for
all }n\in{\mathbb{N}},\,u\in C_{d}^{\alpha}[a,b].$

The condition (10) is fulfilled with the ${\mathbb{K}}^{d}$-valued functions
$X(a,b)=C_{d}^{\alpha}[a,b]$ satisfying a Hölder condition with exponent
$\alpha$ and $\Gamma(\varrho)=(b-a)^{1/p}d^{1/p}C^{\alpha}\varrho^{\alpha}$.

###  4.2 Galerkin methods on polygonal domains
$\Omega\subset{\mathbb{R}}^{\kappa}$ with $\kappa=2,3$

Let $\kappa=2,3$ and $\Omega\subset{\mathbb{R}}^{\kappa}$ be a polygonal
domain (open, bounded, connected). For a sequence $(h_{n})_{n\in{\mathbb{N}}}$
in $(0,\infty)$ satisfying (23) suppose the representations

$\Omega=\bigcup_{K\in{\mathfrak{T}}_{n}}K\quad\text{for all }n\in{\mathbb{N}}$

with a sequence of regular triangulations ${\mathfrak{T}}_{n}$, i.e. there
exists a constant $T>0$ such that each family ${\mathfrak{T}}_{n}$ consists of
finitely many polyhedra $K\subseteq\Omega$ and

$\displaystyle\max_{K\in{\mathfrak{T}}_{n}}\operatorname{diam}K$
$\displaystyle\leq h_{n},$
$\displaystyle\max_{K\in{\mathfrak{T}}_{n}}\frac{\operatorname{diam}K}{\sup\left\\{\operatorname{diam}S:\,S\text{
is a ball contained in }K\right\\}}\leq T$

for $n\in{\mathbb{N}}$. Let $P_{k}(K)$ be the space of polynomials of
(maximal) degree $k\in{\mathbb{N}}$ in variables
$x_{1},\ldots,x_{\kappa}\in{\mathbb{R}}$ over $K$ and $X_{n}:=\\{v\in
C(\Omega):\,v|_{K}\in P_{k}(K)\text{ for all }K\in{\mathfrak{T}}_{n}\\}$. For
$1<l\leq k+1$ it results from [21, p. 96, (3.5.9)] that there exists a
$\bar{C}>0$ with

$\left\|[\operatorname{id}_{L_{d}^{2}(\Omega)}-\Pi_{n}]u\right\|_{2}\leq\bar{C}h_{n}^{l}\sqrt{\sum_{\left|\alpha\right|=l}\left\|D^{\alpha}u\right\|_{2}^{2}}\quad\text{for
all }n\in{\mathbb{N}},\,u\in H_{d}^{l}(\Omega).$

Consequently the error estimate (10) holds for functions
$X(\Omega)=H_{d}^{l}(\Omega)$ being weakly differentiable up to order $l\leq
k+1$ with derivatives in $L_{d}^{2}(\Omega)$ and
$\Gamma(\varrho)=\bar{C}C^{l}\varrho^{l}$.

###  4.3 Spectral Galerkin methods

For a Hammerstein IDE ($I_{0}$) let us suppose that the kernels and growth
functions are of the form $k_{t}=k$ and $g_{t}(x,z):=\alpha_{t}g(x,z)$ for all
$t\in{\mathbb{I}}^{\prime}$ with a sequence $\alpha_{t}\in{\mathbb{R}}$ and
functions $k:\Omega\times\Omega\to{\mathbb{K}}^{d\times n}$,
$g:\Omega\times{\mathbb{K}}^{d}\to{\mathbb{K}}^{n}$ such that Hypotheses
(i–iii) are satisfied with $p=2$. The linear operator ${\mathscr{L}}\in
L(L_{d}^{2}(\Omega))$,
${\mathscr{L}}u:=\int_{\Omega}k(\cdot,y)D_{2}g(y,0)u(y)\,{\mathrm{d}}y$ is
compact and in case

$k(x,y)D_{2}g(y,0)=[k(y,x)D_{2}g(x,0)]^{\ast}\quad\text{for a.a.\
}x,y\in\Omega$

also self-adjoint. Consequently there exists a complete orthonormal set
consisting of eigenfunctions $\chi_{i}:\Omega\to{\mathbb{K}}^{d}$ for
${\mathscr{L}}$. Given this, we define the orthogonal projections

$\Pi_{n}u:=\sum_{i=1}^{n}\left\langle
u,\chi_{i}\right\rangle\chi_{i}\quad\text{for all }n\in{\mathbb{N}}$

onto $X_{n}^{d}:=\operatorname{span}\left\\{\chi_{1},\ldots,\chi_{n}\right\\}$
and it is not difficult to arrive at

$\left\|[\operatorname{id}_{L_{d}^{2}(\Omega)}-\Pi_{n}]u\right\|_{2}\leq\sqrt{\sum_{j=n+1}^{\infty}\left|\left\langle
u,\chi_{j}\right\rangle\right|^{2}}\quad\text{for all }n\in{\mathbb{N}}.$

Whence, the convergence function $\Gamma$ in the error estimate (10) depends
on the decay of the Fourier coefficients $\left\langle
u,\chi_{j}\right\rangle$. If the kernel comes from a Sturm-Liouville problem,
then related results are treated in [8, pp. 275, Sect. 5.2].

##  5 Perspectives

An alternative to the $L^{p}$-setting we elaborated on in this paper, is to
employ the strictly decreasing scale of Hölder spaces
$C^{\alpha}(\Omega,{\mathbb{K}}^{d})$ over a compact habitat
$\Omega\subset{\mathbb{R}}^{\kappa}$. For this purpose and exponents
$0<\beta<\alpha\leq 1$ one considers

  * •

Fredholm operators ${\mathscr{K}}_{t}\in
L(C^{\beta}(\Omega,{\mathbb{K}}^{n}),C^{\alpha}(\Omega,{\mathbb{K}}^{d}))$ as
defined in (2),

  * •

Nemytskii operators ${\mathscr{G}}_{t}:C^{\alpha}(\Omega,{\mathbb{K}}^{d})\to
C^{\beta}(\Omega,{\mathbb{K}}^{n})$ as given in (3).

The reason for working with different Hölder exponents are the pathological
mapping properties of the Nemytskii operators ${\mathscr{G}}_{t}$ in case
$\alpha=\beta$. Indeed, then any globally Lipschitz ${\mathscr{G}}_{t}$ is
already affine-linear, cf. [16]. A particular feature of such a Hölder setting
is that persistence and convergence of invariant fibers can be shown for
Nyström discretizations of ($I_{0}$) (see [3, 19]).

##  Appendix A Perturbation of invariant bundles

In this appendix, ${\mathbb{I}}$ is an unbounded discrete interval, while
$(X,\left\|\cdot\right\|)$ denotes a Banach space. We consider nonautonomous
semilinear difference equations

$u_{t+1}={\mathscr{L}}_{t}u_{t}+{\mathscr{N}}_{t}(u_{t})+\theta\bar{\mathscr{N}}_{t}(u_{t})$
($\Delta_{\theta}$)

in $X$ depending on a parameter $\theta\in{\mathbb{R}}$ and having the general
solution $\varphi(\cdot;\theta)$. Suppose that the following assumptions hold:

  * (H1)

${\mathscr{L}}_{t}\in L(X)$, $t\in{\mathbb{I}}^{\prime}$, there exists a
projection-valued sequence $(P_{t})_{t\in{\mathbb{I}}}$ in $L(X)$ and reals
$K\geq 1$, $0<\alpha<\beta$ so that
$P_{t+1}{\mathscr{L}}_{t}={\mathscr{L}}_{t}P_{t}$ and
${\mathscr{L}}_{t}|_{N(P_{t})}:N(P_{t})\to N(P_{t+1})$ is an isomorphism for
all $t\in{\mathbb{I}}^{\prime}$, as well as

$\displaystyle\left\|\Phi(t,s)P_{s}\right\|$ $\displaystyle\leq
K\alpha^{t-s},$
$\displaystyle\left\|\Phi(s,t)[\operatorname{id}_{X}-P_{t}]\right\|$
$\displaystyle\leq K\beta^{s-t}\quad\text{for all }s\leq t.$ (24)

  * (H2)

The identities ${\mathscr{N}}_{t}(0)\equiv\bar{\mathscr{N}}_{t}(0)\equiv 0$ on
${\mathbb{I}}^{\prime}$ hold for mappings
${\mathscr{N}}_{t},\bar{\mathscr{N}}_{t}:X\to X$, $t\in{\mathbb{I}}^{\prime}$,
satisfying the Lipschitz estimates

$\displaystyle L$
$\displaystyle:=\sup_{t\in{\mathbb{I}}^{\prime}}\operatorname{lip}{\mathscr{N}}_{t}<\infty,$
$\displaystyle\bar{L}$
$\displaystyle:=\sup_{t\in{\mathbb{I}}^{\prime}}\operatorname{lip}\bar{\mathscr{N}}_{t}<\infty.$

  * (H3)

Let $m\in{\mathbb{N}}$ and ${\mathscr{N}}_{t},\bar{\mathscr{N}}_{t}$ be
$m$-times continuously differentiable such that

$\displaystyle\sup_{(t,u)\in{\mathbb{I}}^{\prime}\times
X}\left\|D^{\ell}{\mathscr{N}}_{t}(u)\right\|_{L_{\ell}(X)}$
$\displaystyle<\infty,$
$\displaystyle\sup_{(t,u)\in{\mathbb{I}}^{\prime}\times
X}\left\|D^{\ell}\bar{\mathscr{N}}_{t}(u)\right\|_{L_{\ell}(X)}$
$\displaystyle<\infty\quad\text{for all }1<\ell\leq m.$

For our central perturbation result, given $\tau\in{\mathbb{I}}$ and
$\gamma>0$ the linear spaces

$\displaystyle\ell_{\tau,\gamma}^{+}$
$\displaystyle:=\left\\{(\phi_{t})_{\tau\leq t}:\,\phi_{t}\in X\text{ and
}\sup_{\tau\leq t}\gamma^{\tau-t}\left\|\phi_{t}\right\|<\infty\right\\},$
$\displaystyle\ell_{\tau,\gamma}^{-}$
$\displaystyle:=\left\\{(\phi_{t})_{t\leq\tau}:\,\phi_{t}\in X\text{ and
}\sup_{t\leq\tau}\gamma^{\tau-t}\left\|\phi_{t}\right\|<\infty\right\\}$

and $\ell_{\gamma}:=\left\\{(\phi_{t})_{t\in{\mathbb{Z}}}:\,\phi_{t}\in
X\text{ and
}\sup_{t\in{\mathbb{Z}}}\gamma^{-t}\left\|\phi_{t}\right\|<\infty\right\\}$
are due and satisfy

$\displaystyle\ell_{\tau,\gamma}^{+}$
$\displaystyle\subseteq\ell_{\tau,\bar{\gamma}}^{+},$
$\displaystyle\ell_{\tau,\bar{\gamma}}^{-}$
$\displaystyle\subseteq\ell_{\tau,\gamma}^{-}\quad\text{for all
}0<\gamma\leq\bar{\gamma}.$ (25)

These preparations allows us to establish existence, smoothness and
$\theta$-dependence of invariant bundles for ($\Delta_{\theta}$) carrying most
of the technical effort for the results above.

######  Theorem A.1 (existence and perturbation of invariant bundles).

Under the assumptions (H1–H2) with

$\displaystyle L$ $\displaystyle<\frac{\delta_{\max}}{4K},$
$\displaystyle\delta_{\max}$ $\displaystyle:=\frac{\beta-\alpha}{2}$

we choose a fixed $\delta\in(4KL,\tfrac{\beta-\alpha}{2}]$ and set
$\Gamma:=[\alpha+\delta,\beta-\delta]$,
$\Theta:=\left\\{\theta\in{\mathbb{R}}:\,\bar{L}\left|\theta\right|\leq
L\right\\}$. If $\gamma\in\Gamma$, then the following statements are true for
all $\theta\in\Theta$:

  1. (a)

In case ${\mathbb{I}}$ is unbounded above, then the _$\gamma$ -stable bundle_

${\mathscr{W}}^{+}(\theta):=\left\\{(\tau,u)\in{\mathbb{I}}\times
X:\,\varphi(\cdot;\tau,u;\theta)\in\ell_{\tau,\gamma}^{+}\right\\}$

is a forward invariant bundle of ($\Delta_{\theta}$) independent of
$\gamma\in\Gamma$ having the representation
${\mathscr{W}}^{+}(\theta)=\left\\{(\tau,v+w^{+}(\tau,v;\theta))\in{\mathbb{I}}\times
X:\,v\in R(P_{\tau})\right\\}$ with a continuous mapping
$w^{+}:{\mathbb{I}}\times X\times\Theta\to X$ satisfying for all
$\tau\in{\mathbb{I}}$, $u\in X$ that

     * $(a_{1})$

$w^{+}(\tau,0;\theta)=0$ and
$w^{+}(\tau,u;\theta)=w^{+}(\tau,P_{\tau}u;\theta)\in N(P_{\tau})$,

     * $(a_{2})$

$w^{+}:{\mathbb{I}}\times X\times\Theta\to X$ fulfills the Lipschitz estimates

$\displaystyle\operatorname{lip}w^{+}(\tau,\cdot;\theta)$
$\displaystyle\leq\frac{K^{2}(L+\left|\theta\right|\bar{L})}{\delta-2K(L+\left|\theta\right|\bar{L})},$
(26) $\displaystyle\operatorname{lip}w^{+}(\tau,u;\cdot)$
$\displaystyle\leq\frac{2\delta
K^{3}\bar{L}}{(\delta-4K\bar{L})^{2}}\left\|u\right\|,$ (27)

     * $(a_{3})$

if also $(H3)$ holds, $\alpha^{m_{+}}<\beta$ and
$\delta_{\max}:=\min\Bigl{\\{}\tfrac{\beta-\alpha}{2},\alpha\bigl{(}\sqrt[m_{+}]{\frac{\alpha+\beta}{\alpha+\alpha^{m_{+}}}}-1\bigr{)}\Bigr{\\}}$,
then the derivatives $D_{(2,3)}^{\ell}w^{+}:{\mathbb{I}}\times
X\times\Theta\to L_{\ell}(X\times{\mathbb{R}},X)$ exist up to order $\ell\leq
m_{+}\leq m$ as continuous functions and there are $M^{+},N^{+}>0$ such that

$\displaystyle\left\|D_{2}^{\ell}w^{+}(\tau,u;\theta)\right\|_{L_{\ell}(X)}$
$\displaystyle\leq M^{+}\quad\text{for all }1\leq\ell\leq m_{+},$ (28)
$\displaystyle\left\|D_{3}D_{2}^{\ell}w^{+}(\tau,u;\theta)\right\|_{L_{\ell}(X)}$
$\displaystyle\leq N^{+}\left\|u\right\|\quad\text{for all }0\leq l<m_{+}.$
(29)

  2. (b)

In case ${\mathbb{I}}$ is unbounded below, then the _$\gamma$ -unstable
bundle_

${\mathscr{W}}^{-}(\theta):=\left\\{(\tau,u)\in{\mathbb{I}}\times
X:\begin{array}[]{l}\text{there exists a solution
}(\phi_{t})_{t\leq\tau}\in\ell_{\tau,\gamma}^{-}\\\ \text{of \eqref{fbdeq}
satisfying }\phi_{\tau}=u\end{array}\right\\}$

is an invariant bundle of ($\Delta_{\theta}$) independent of $\gamma\in\Gamma$
having the representation
${\mathscr{W}}^{-}(\theta)=\left\\{(\tau,v+w^{-}(\tau,v;\theta))\in{\mathbb{I}}\times
X:\,v\in N(P_{\tau})\right\\}$ with a continuous mapping
$w^{-}:{\mathbb{I}}\times X\times\Theta\to X$ satisfying for all
$\tau\in{\mathbb{I}}$, $u\in X$ that

     * $(b_{1})$

$w^{-}(\tau,0;\theta)=0$ and
$w^{-}(\tau,u;\theta)=w^{-}(\tau,[\operatorname{id}_{X}-P_{\tau}]u;\theta)\in
R(P_{\tau})$,

     * $(b_{2})$

$w^{-}:{\mathbb{I}}\times X\times\Theta\to X$ fulfills the Lipschitz estimates

$\displaystyle\operatorname{lip}w^{-}(\tau,\cdot;\theta)$
$\displaystyle\leq\frac{K^{2}(L+\left|\theta\right|\bar{L})}{\delta-2K(L+\left|\theta\right|\bar{L})},$
(30) $\displaystyle\operatorname{lip}w^{-}(\tau,u;\cdot)$
$\displaystyle\leq\frac{2\delta
K^{3}\bar{L}}{(\delta-4K\bar{L})^{2}}\left\|u\right\|,$ (31)

     * $(b_{3})$

if also $(H3)$ holds, $\alpha<\beta^{m_{-}}$ and
$\delta_{\max}:=\min\Bigl{\\{}\tfrac{\beta-\alpha}{2},\beta\bigl{(}1-\sqrt[m_{-}]{\frac{\alpha+\beta}{\beta+\beta^{m_{-}}}}\bigr{)}\Bigr{\\}}$,
then the derivatives $D_{(2,3)}^{\ell}w^{-}:{\mathbb{I}}\times
X\times\Theta\to L_{\ell}(X\times{\mathbb{R}},X)$ exist up to order $\ell\leq
m_{-}\leq m$ as continuous functions and there are $M^{-},N^{-}>0$ such that

$\displaystyle\left\|D_{2}^{\ell}w^{-}(\tau,u,\theta)\right\|_{L_{\ell}(X)}$
$\displaystyle\leq M^{-}\quad\text{for all }1\leq\ell\leq m_{-},$ (32)
$\displaystyle\left\|D_{3}D_{2}^{\ell}w^{-}(\tau,u,\theta)\right\|_{L_{\ell}(X)}$
$\displaystyle\leq N^{-}\left\|u\right\|\quad\text{for all }0\leq l<m_{-}.$
(33)

  3. (c)

If ${\mathbb{I}}={\mathbb{Z}}$ and $L<\frac{\delta}{6K}$, then
${\mathscr{W}}^{+}(\theta)\cap{\mathscr{W}}^{-}(\theta)={\mathbb{Z}}\times\left\\{0\right\\}$
for $\theta\in\Theta$ and the zero solution is the only solution to
($\Delta_{\theta}$) in $\ell_{\gamma}$ for any growth rate $\gamma\in\Gamma$.

###### Proof.

Apply [17, Thm. 3.3] on the measure chains
${\mathbb{T}}={\mathbb{I}}\subseteq{\mathbb{Z}}$. ∎

##  Appendix B Fredholm and Nemytskii operators on $L^{p}$-spaces

Let $(\Omega,{\mathfrak{A}},\mu)$ be a measure space with $\sigma$-algebra
${\mathfrak{A}}$ and a finite measure $\mu$, while $(Z,\left|\cdot\right|)$
denotes a finite-dimensional Banach space over ${\mathbb{K}}$. For
$p\in[1,\infty)$ we define the space

$\displaystyle L^{p}(\Omega,Z)$ $\displaystyle:=\biggl{\\{}u:\Omega\to
Z\biggl{|}u\text{ is $\mu$-measurable with
}\int_{\Omega}\left|u\right|^{p}\,{\mathrm{d}}\mu<\infty\biggr{\\}}$

of $Z$-valued $p$-integrable functions and equip it with
$\left\|u\right\|_{p}:=\left(\int_{\Omega}\left|u\right|^{p}\,{\mathrm{d}}\mu\right)^{1/p}$
as norm.

Our analysis requires preparations on Fredholm integral operators

${\mathscr{K}}v:=\int_{\Omega}k(\cdot,y)v(y)\,{\mathrm{d}}\mu(y).$

First, for the kernel $k:\Omega\times\Omega\to{\mathbb{K}}^{d\times n}$ we
suppose _Hille-Tamarkin conditions_ : There exist $p,q\in(1,\infty)$ such that

  1. (h1)

$k$ is $\mu\otimes\mu$-measurable,

  2. (h2)

if $q^{\prime}>1$ with $\tfrac{1}{q}+\tfrac{1}{q^{\prime}}=1$, then
$\int_{\Omega}\biggl{(}\int_{\Omega}\left|k(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}\mu(y)\biggr{)}^{p/q^{\prime}}\,{\mathrm{d}}\mu(x)<\infty.$

######  Proposition B.1.

If $(h1$–$h2)$ hold with $p,q>1$, then ${\mathscr{K}}\in
L(L^{q}(\Omega,{\mathbb{K}}^{n}),L^{p}(\Omega,{\mathbb{K}}^{d}))$ is well-
defined and compact with

$\left\|{\mathscr{K}}\right\|_{L(L^{q}(\Omega,{\mathbb{K}}^{n}),L^{p}(\Omega,{\mathbb{K}}^{d}))}\leq\biggl{(}\int_{\Omega}\biggl{(}\int_{\Omega}\left|k(x,y)\right|^{q^{\prime}}\,{\mathrm{d}}\mu(y)\biggr{)}^{\tfrac{p}{q^{\prime}}}\,{\mathrm{d}}\mu(x)\biggr{)}^{\tfrac{1}{p}}.$
(34)

###### Proof.

See [2, pp. 149–150, 5.12] and [2, p. 340, 10.15] for compactness. ∎

Second, let us furthermore consider the Nemytskii operator

$[{\mathscr{G}}(u)](x):=g(x,u(x))\quad\text{for $\mu$-a.a.\ }x\in\Omega$

induced by a growth function
$g:\Omega\times{\mathbb{K}}^{d}\to{\mathbb{K}}^{n}$. Given
$m\in{\mathbb{N}}_{0}$ we assume that for all $0\leq\ell\leq m$ one has the
following _Carathéodory conditions_ :

  1. (c1)

$D_{2}^{\ell}g(x,\cdot):{\mathbb{K}}^{d}\to
L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n})$ exists and is continuous for
$\mu$-a.a. $x\in\Omega$,

  2. (c2)

$D_{2}^{\ell}g(\cdot,z):\Omega\to L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n})$
is $\mu$-measurable on $\Omega$ for all $z\in{\mathbb{K}}^{d}$.

######  Proposition B.2.

Let $p,q\geq 1$, $c\in L^{q}(\Omega)$, $c_{0}\geq 0$ and suppose $(c1$–$c2)$
hold with $m=0$. If the growth estimate

$\left|g(x,z)\right|\leq c(x)+c_{0}\left|z\right|^{\tfrac{p}{q}}\quad\text{for
$\mu$-a.a.~{}$x\in\Omega$ and all }z\in{\mathbb{K}}^{d}$

is satisfied, then ${\mathscr{G}}:L^{p}(\Omega,{\mathbb{K}}^{d})\to
L^{q}(\Omega,{\mathbb{K}}^{n})$ is well-defined, bounded and continuous.

###### Proof.

See [20, p. 63, Thm. 5.1]. ∎

######  Corollary B.3.

Let $1\leq q<p$. If there exists a $\mu$-measurable function
$\lambda:\Omega\to{\mathbb{R}}_{+}$ satisfying

$\left|g(x,z)-g(x,\bar{z})\right|\leq\lambda(x)\left|z-\bar{z}\right|\quad\text{for
$\mu$-a.a.\ }x\in\Omega\text{ and all }z,\bar{z}\in{\mathbb{K}}^{d}$

and $\int_{\Omega}\lambda(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}\mu(y)<\infty$,
then ${\mathscr{G}}:L^{q}(\Omega,{\mathbb{K}}^{d})\to
L^{q}(\Omega,{\mathbb{K}}^{n})$ is globally Lipschitz with

$\operatorname{lip}{\mathscr{G}}\leq\biggl{(}\int_{\Omega}\lambda(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}\mu(y)\biggr{)}^{\tfrac{p-q}{p}}.$
(35)

###### Proof.

Given $u,\bar{u}\in L^{p}(\Omega,{\mathbb{K}}^{d})$, we obtain

$\left\|{\mathscr{G}}(u)-{\mathscr{G}}(\bar{u})\right\|_{q}^{q}=\int_{\Omega}\left|g(y,u(y))-g(y,\bar{u}(y))\right|^{q}\,{\mathrm{d}}\mu(y)\leq\int_{\Omega}\lambda(y)^{q}\left|u(y)-\bar{u}(y)\right|^{q}\,{\mathrm{d}}\mu(y)$

and the Hölder inequality
$|\int_{\Omega}v_{1}(y)v_{2}(y)\,{\mathrm{d}}\mu(y)|\leq\left\|v_{1}\right\|_{r^{\prime}}\left\|v_{2}\right\|_{r}$
for $\tfrac{1}{r}+\tfrac{1}{r^{\prime}}=1$ leads to

$\displaystyle\left\|{\mathscr{G}}(u)-{\mathscr{G}}(\bar{u})\right\|_{q}$
$\displaystyle\leq\left\|\lambda^{q}\right\|_{r^{\prime}}^{1/q}\left\|\left|u(\cdot)-\bar{u}(\cdot)\right|^{q}\right\|_{r}^{1/q}$
$\displaystyle=\biggl{(}\int_{\Omega}\lambda(y)^{qr^{\prime}}\,{\mathrm{d}}\mu(y)\biggr{)}^{\tfrac{1}{qr^{\prime}}}\biggl{(}\int_{\Omega}\left|u(y)-\bar{u}(y)\right|^{qr}\,{\mathrm{d}}\mu(y)\biggr{)}^{\tfrac{1}{qr}}.$

We now choose $r>0$ such that $p=rq$ (note that $q<p$ guarantees $1<r$) and
obtain $qr^{\prime}=\tfrac{p}{p-q}$, as well as $r^{\prime}=\tfrac{r}{r-1}$,
thus

$\left\|{\mathscr{G}}(u)-{\mathscr{G}}(\bar{u})\right\|_{q}\leq\biggl{(}\int_{\Omega}\lambda(y)^{\tfrac{p}{p-q}}\,{\mathrm{d}}\mu(y)\biggr{)}^{\tfrac{p-q}{p}}\left\|u-\bar{u}\right\|_{p},$

which guarantees the Lipschitz condition (35). ∎

######  Proposition B.4.

Let $m\in{\mathbb{N}}$, $p,q\geq 1$ with $mq<p$, $c\in
L^{\tfrac{pq}{p-mq}}(\Omega)$, $c_{0},\ldots,c_{m}\geq 0$ and suppose
$(c1$–$c2)$ hold. If for $0\leq\ell\leq m$ the growth conditions

$\left|D_{2}^{\ell}g(x,z)\right|_{L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n})}\leq
c(x)+c_{\ell}\left|z\right|^{\tfrac{p-\ell q}{q}}\quad\text{for $\mu$-a.a.\
}x\in\Omega\text{ and all }z\in{\mathbb{K}}^{d}$ (36)

are satisfied, then ${\mathscr{G}}:L^{p}(\Omega,{\mathbb{K}}^{d})\to
L^{q}(\Omega,{\mathbb{K}}^{n})$ is of class $C^{m}$ and for each
$0\leq\ell\leq m$, $u\in L^{p}(\Omega,{\mathbb{K}}^{d})$ the following holds:

  1. (a)

$[D^{\ell}{\mathscr{G}}(u)v_{1}\cdots
v_{\ell}](x)=D_{2}^{\ell}g(x,u(x))v_{1}(x)\cdots v_{\ell}(x)$ for $\mu$-a.a.
$x\in\Omega$ and functions $v_{1},\ldots,v_{\ell}\in
L^{q}(\Omega,{\mathbb{K}}^{d})$,

  2. (b)

$\left\|D^{\ell}{\mathscr{G}}(u)\right\|_{L_{\ell}(L^{p}(\Omega,{\mathbb{K}}^{d}),L^{q}(\Omega,{\mathbb{K}}^{n}))}\leq\left\|c\right\|_{\tfrac{pq}{p-\ell
q}}+c_{\ell}\left\|u\right\|_{p}^{\tfrac{p-\ell q}{q}}$.

Due to the lack of a suitable reference we provide an explicit proof.

###### Proof.

Throughout, let $u\in L^{p}(\Omega,{\mathbb{K}}^{d})$ and formally define for
$\mu$-a.a. $x\in\Omega$ that

$\displaystyle{\mathscr{G}}^{\ell}(u)(x)$
$\displaystyle:=D_{2}^{\ell}g(x,u(x))\in
L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n})\quad\text{for all }0\leq\ell\leq
m,$ $\displaystyle[\bar{\mathscr{G}}^{\ell}(u)h](x)$
$\displaystyle:=D_{2}^{\ell}g(x,u(x))h(x)\in
L_{\ell-1}({\mathbb{K}}^{d},{\mathbb{K}}^{n})\quad\text{for all }0<\ell\leq m$

and $h\in L^{p}(\Omega,{\mathbb{K}}^{d})$. First, being $p$-integrable, $u$ is
$\mu$-measurable on $\Omega$. Since $D_{2}^{\ell}g$ satisfy Carathéodory
conditions, because of [20, p. 62, Lemma 5.1] also $x\mapsto
D_{2}^{\ell}g(x,u(x))$ is $\mu$-measurable on $\Omega$. Second, if we set
$q_{\ell}:=\tfrac{pq}{p-\ell q}\geq 1$, then (36) become

$\left|D_{2}^{\ell}g(x,z)\right|\leq
c(x)+c_{\ell}\left|z\right|^{p/q_{\ell}}\quad\text{for all }0\leq\ell\leq
m,\text{$\mu$-a.a.~{}$x\in\Omega$ and all }z\in{\mathbb{K}}^{d}.$

(I) Claim: ${\mathscr{G}}^{\ell}:L^{p}(\Omega,{\mathbb{K}}^{d})\to
L^{q_{\ell}}(\Omega,L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n}))$ is well-
defined and bounded for all $0\leq\ell\leq m$.
Using the growth condition (36) and the $L^{q_{\ell}}$-triangle inequality we
obtain

$\displaystyle\left(\int_{\Omega}\left|D_{2}^{\ell}g(y,u(y))\right|^{q_{\ell}}\,{\mathrm{d}}\mu(y)\right)^{1/q_{\ell}}$
$\displaystyle\leq\left(\int_{\Omega}\left|c(y)+c_{\ell}\left|u(y)\right|^{p/q_{\ell}}\right|^{q_{\ell}}\,{\mathrm{d}}\mu(y)\right)^{1/q_{\ell}}$
$\displaystyle\leq\left\|c\right\|_{q_{\ell}}+c_{\ell}\left\|u\right\|_{p}^{p/q_{\ell}}$

and consequently
$\left\|{\mathscr{G}}^{\ell}(u)\right\|_{q_{\ell}}\leq\left\|c\right\|_{q_{\ell}}+c_{\ell}\left\|u\right\|_{p}^{p/q_{\ell}}$.

(II) Claim: ${\mathscr{G}}^{\ell}:L^{p}(\Omega,{\mathbb{K}}^{d})\to
L^{q_{\ell}}(\Omega,L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n}))$ is
continuous for all $0\leq\ell\leq m$.
Let $(u_{j})_{j\in{\mathbb{N}}}$ be a sequence in
$L^{p}(\Omega,{\mathbb{K}}^{d})$ converging to $u$ w.r.t.
$\left\|\cdot\right\|_{p}$. Because of [2, p. 57, 3.22(1)] this implies
pointwise convergence of a subsequence $\mu$-a.e., that is, w.l.o.g. we can
suppose $\lim_{j\to\infty}u_{j}(x)=u_{j}(x)$ for $\mu$-a.a. $x\in\Omega$,
which shows

${\mathscr{G}}^{\ell}(u_{j})(x)=D_{2}^{\ell}g(x,u_{j}(x))\xrightarrow[j\to\infty]{}D_{2}^{\ell}g(x,u(x))={\mathscr{G}}^{\ell}(u)(x)\quad\text{for
$\mu$-a.a.\ }x\in\Omega.$

The necessity part of Vitali’s convergence theorem [2, p. 57, 3.23] applies to
$(u_{j})_{j\in{\mathbb{N}}}$ and the growth assumptions on $D_{2}^{\ell}g$
guarantee that [2, p. 57, 3.23] can be applied to
${\mathscr{G}}^{\ell}(u_{j})$ with the exponent $q_{\ell}$ rather than $p$.
Thus, the sufficiency part of [2, p. 57, 3.23] shows that
$\lim_{j\to\infty}\left\|{\mathscr{G}}^{\ell}(u_{j})-{\mathscr{G}}^{\ell}(u)\right\|_{q_{\ell}}=0$,
i.e. ${\mathscr{G}}^{\ell}$ is continuous.

(III) Claim: $\bar{\mathscr{G}}^{l+1}:L^{p}(\Omega,{\mathbb{K}}^{d})\to
L\bigl{(}L^{p}(\Omega,{\mathbb{K}}^{d}),L^{q_{\ell}}(\Omega,L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n}))\bigr{)}$
is well-defined and continuous for all $0\leq\ell<m$.
For $h\in L^{p}(\Omega,{\mathbb{K}}^{d})$ we obtain from Hölder’s inequality
that

$\displaystyle\left(\int_{\Omega}\left|D_{2}^{\ell+1}g(y,u(y))h(y)\right|^{q_{\ell}}\,{\mathrm{d}}\mu(y)\right)^{1/q_{\ell}}\leq\left(\int_{\Omega}\left|D_{2}^{\ell+1}g(y,u(y))\right|^{q_{\ell}}\left|h(y)\right|^{q_{\ell}}\,{\mathrm{d}}\mu(y)\right)^{1/q_{\ell}}$
$\displaystyle\leq$
$\displaystyle\left(\left(\int_{\Omega}\left|D_{2}^{\ell+1}g(y,u(y))\right|^{q_{\ell+1}}\,{\mathrm{d}}\mu(y)\right)^{q_{\ell}/q_{\ell+1}}\left\|h\right\|_{p}^{q_{\ell}}\right)^{1/q_{\ell}}=\left\|{\mathscr{G}}^{\ell+1}(u)\right\|_{q_{\ell+1}}\left\|h\right\|_{p}$

and consequently $\bar{\mathscr{G}}^{\ell+1}(u)h\in
L^{q_{\ell}}(\Omega,L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n}))$ results. The
assumptions on $g$ yield the inclusion $\bar{\mathscr{G}}^{(\ell+1)}(u)\in
L\bigl{(}L^{p}(\Omega,{\mathbb{K}}^{d}),L^{q_{\ell}}(\Omega,L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n}))\bigr{)}$,
while the continuity of $\bar{\mathscr{G}}^{\ell}$ is shown as in step (II).

(IV) Claim: ${\mathscr{G}}^{\ell}:L^{p}(\Omega,{\mathbb{K}}^{d})\to
L^{q_{\ell}}(\Omega,L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n}))$ is
continuously differentiable for all $0\leq\ell<m$.
Because $D_{2}^{\ell}g$ is assumed to be continuously differentiable in the
second argument, the Mean Value Theorem and Jensen’s Inequality imply

$\displaystyle\left\|{\mathscr{G}}^{\ell}(u+h)-{\mathscr{G}}^{\ell}(u)-\bar{\mathscr{G}}^{\ell+1}(u)h\right\|_{q_{\ell}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}\left|D_{2}^{\ell}g(y,u(y)+h(y))-D_{2}^{\ell}g(y,u(y))-D_{2}^{\ell+1}g(x,u(y))h(y)\right|^{q_{\ell}}\,{\mathrm{d}}\mu(y)\right)^{1/{q_{\ell}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}\left(\int_{0}^{1}\left|D_{2}^{\ell+1}g(y,u(y)+\vartheta
h(y))-D_{2}^{\ell+1}g(x,u(y))]h(y)\right|\,{\mathrm{d}}\vartheta\right)^{q_{\ell}}\,{\mathrm{d}}\mu(y)\right)^{1/{q_{\ell}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}\int_{0}^{1}\left|D_{2}^{\ell+1}g(y,u(y)+\vartheta
h(y))-D_{2}^{\ell+1}g(x,u(y))]h(y)\right|^{q_{\ell}}\,{\mathrm{d}}\vartheta\,{\mathrm{d}}\mu(y)\right)^{1/{q_{\ell}}}$

and we obtain from Fubini’s theorem along with Hölder’s inequality

$\displaystyle\left\|{\mathscr{G}}^{\ell}(u+h)-{\mathscr{G}}^{\ell}(u)-\bar{\mathscr{G}}^{\ell+1}(u)h\right\|_{q_{\ell}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{1}\int_{\Omega}\left|D_{2}^{\ell+1}g(y,u(y)+\vartheta
h(y))-D_{2}^{\ell+1}g(x,u(y))]h(y)\right|^{q_{\ell}}\,{\mathrm{d}}\mu(y)\,{\mathrm{d}}\vartheta\right)^{1/{q_{\ell}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{1}\int_{\Omega}\left|D_{2}^{\ell+1}g(y,u(y)+\vartheta
h(y))-D_{2}^{\ell+1}g(x,u(y))\right|^{q_{\ell}}\left|h(y)\right|^{q_{\ell}}\,{\mathrm{d}}\mu(y)\,{\mathrm{d}}\vartheta\right)^{1/{q_{\ell}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{1}\left(\int_{\Omega}\left|D_{2}^{\ell+1}g(y,u(y)+\vartheta
h(y))-D_{2}^{\ell+1}g(x,u(y))\right|^{q_{\ell+1}}\,{\mathrm{d}}\mu(y)\right)^{q_{\ell}/{q_{\ell+1}}}\left\|h\right\|_{p}^{q_{\ell}}\,{\mathrm{d}}\vartheta\right)^{1/{q_{\ell}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{1}\left\|{\mathscr{G}}^{\ell+1}(u+\vartheta
h)-{\mathscr{G}}^{\ell+1}(u)\right\|_{L^{q_{\ell+1}}(\Omega,L_{\ell}({\mathbb{K}}^{d},{\mathbb{K}}^{n})))}^{q_{\ell}}\,{\mathrm{d}}\vartheta\right)^{1/q_{\ell}}\left\|h\right\|_{p}$

for $h\in L^{p}(\Omega,{\mathbb{K}}^{d})$. Due to the continuity of
${\mathscr{G}}^{\ell+1}:L^{p}(\Omega,{\mathbb{K}}^{d})\to
L^{q_{\ell+1}}(\Omega,L_{\ell+1}({\mathbb{K}}^{d},{\mathbb{K}}^{n}))$ the
integral tends to zero in the limit $\left\|h\right\|_{p}\to 0$, and hence
${\mathscr{G}}^{\ell}$ is Frechét differentiable in $u$ with derivative
$D{\mathscr{G}}^{\ell}(u)=\bar{\mathscr{G}}^{\ell+1}(u)$ by uniqueness of
derivatives. Its continuity was already shown in step (III).

(V) In particular, this establishes that
${\mathscr{G}}:L^{p}(\Omega,{\mathbb{K}}^{d})\to
L^{q}(\Omega,{\mathbb{K}}^{n})$ is continuously differentiable with
$D{\mathscr{G}}(u)=\bar{\mathscr{G}}^{1}(u)$ as derivative. Then mathematical
induction yields that ${\mathscr{G}}$ is $m$-times continuously differentiable
with the derivatives

$[D^{\ell}{\mathscr{G}}(u)v_{1}\cdots
v_{\ell}](x)=D_{2}^{\ell}g(x,u(x))v_{1}(x)\cdots v_{\ell}(x)\quad\text{for
$\mu$-a.a.\ }x\in\Omega$

and all $v_{1},\ldots,v_{\ell}\in L^{p}(\Omega,{\mathbb{K}}^{d})$,
$1\leq\ell\leq m$, which establishes claim (a). Moreover, the estimates stated
in (b) are a consequence of step (I). ∎

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