# Numerical Dynamics of Integrodifference Equations: Forward Dynamics and
Pullback Attractors111The work of Huy Huynh has been supported by the Austrian
Science Fund (FWF) under grant number P 30874-N35.222Peter E. Kloeden thanks
the Universität Klagenfurt for support and hospitality.

Huy Huynh†, Peter E. Kloeden∗ and Christian Pötzsche† ${\dagger}$ Institut für
Mathematik, Universität Klagenfurt, 9020 Klagenfurt, Austria
<EMAIL_ADDRESS>$\ast$ Mathematisches Institut,
Universität Tübingen, 72076 Tübingen, Germany<EMAIL_ADDRESS>
###### Abstract.

In order to determine the dynamics of nonautonomous equations both their
forward and pullback behavior need to be understood. For this reason we
provide sufficient criteria for the existence of such attracting invariant
sets in a general setting of nonautonomous difference equations in metric
spaces. In addition it is shown that both forward and pullback attractors, as
well as forward limit sets persist and that the latter two notions even
converge under perturbation. As concrete application, we study
integrodifference equation under spatial discretization of collocation type.

###### Key words and phrases:

Forward attractor; Pullback attractor; Urysohn operator; Projection method

###### Key words and phrases:

Integrodifference equation and Pullback attractor and Forward attractor and
Urysohn operator

###### 1991 Mathematics Subject Classification:

37C70; 37G35; 45G15; 65R20; 65P40

##  1\. Introduction

###  1.1. Perturbation of attractors

Obtaining a precise insight into the structure of a global attractor can be
seen as the holy grail in the field of dissipative dynamical systems. This
compact, invariant set attracting each bounded subset of the state space
contains all equilibria, periodic solutions, homo- or heteroclinic connections
and overall the essential dynamics. In general the structure of a global
attractor can be rather complex and therefore it is no surprise that such
objects behave sensitively w.r.t. perturbations and parameter variations [11].
However, under mild conditions global attractors are upper-semicontinuous in
parameters. Criteria for lower semicontinuity and hence for full continuity in
the Hausdorff metric are much harder to prove and often involve conditions
being difficult to verify. Nevertheless it was shown in [16] that global
attractors are at least continuous on a residual subset of the parameter
space.

For nonautonomous dynamical systems [3, 5, 25, 28] the range of possible long
term behaviors, as well as the theory of attractors is understandably more
involved. For instance, in order to obtain a full picture of the dynamics it
might be necessary to consider different attractor concepts simultaneously.
Although the notion of a pullback attractor shares several properties of the
global attractor it is easy to construct systems with totally different
forward dynamics but sharing the same pullback attractor [24]. This deficit
stimulated the development of forward attractors [23]. While a perturbation
theory for pullback attractors can be found in [3, 5] and their continuity on
a residual set is due to [17], related results for forward attractors are
sparse, which might be due to their nonuniqueness and recent origin. An
exception is [27] addressing the upper-semicontinuity of forward limit sets
under time discretizations of finite-dimensional nonautonomous ODEs.

Indeed, particularly important classes of perturbations originate in numerical
analysis, when an equation generating a dynamical system is discretized. It
actually is a key topic in Numerical Dynamics [37] to determine how attractors
behave under discretization. This research was initiated in [21] for temporal
discretizations of autonomous ODEs in ${\mathbb{R}}^{d}$, extensions to
pullback attractors of nonautonomous ODEs can be found in [6], and related
results for semiflows generated by PDEs are due to [12, 22] and [36].

Forward attractors and limit sets of discrete time nonautonomous dynamical
systems in infinite dimensions were constructed and studied in [18] providing
a theoretical foundation. As a continuation, this paper addresses the
persistence of forward attractors and establishes the upper-semicontinuity of
forward limit sets under perturbations. Remaining in the setting of [18] our
criteria are formulated for abstract difference equations in metric spaces. We
believe that they apply to a wide range of evolutionary equations and their
spatial discretizations. Nevertheless, although convergence under perturbation
is proven, at this level of generality we are not able to establish a
particular convergence rate without imposing further assumptions.

###  1.2. Integrodifference equations and spatial discretization

As concrete application serve nonautonomous integrodifference equations (short
IDEs)

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

(see Sect. 3). Such recursions in infinite-dimensional spaces originally stem
from population genetics but gained popularity as models for the dispersal of
species in theoretical ecology over the last decades [29, 30]. Recently also
nonautonomous models were studied in [4, 19], which clearly motivates our
general approach.

The several approaches to spatially discretize IDEs are based on related
techniques for nonlinear integral equations [1]. Among them, in Sect. 3 we
focus on collocation methods approximating continuous functions on a compact
habitat. As concrete collocation spaces we exemplify splines of different
order. In this framework we provide conditions that pullback and forward
attractors, as well as forward limit sets persist under collocation, and
establish convergence of pullback attractors and forward limit sets for
increasingly more accurate discretizations. Convergence rates could not be
derived using our proof techniques, but we experimentally demonstrate in Sect.
4 that quadratic convergence under piecewise linear collocation is preserved.
We close with a construction of forward limit sets for asymptotically
autonomous spatial Ricker equations.

###  1.3. Notation

Let ${\mathbb{R}}_{+}:=[0,\infty)$. A _discrete interval_ ${\mathbb{I}}$ is
the intersection of a real interval with the integers ${\mathbb{Z}}$,
${\mathbb{I}}^{\prime}:=\left\\{t\in{\mathbb{I}}:\,t+1\in{\mathbb{I}}\right\\}$
and in particular

$\displaystyle{\mathbb{I}}_{\tau}^{+}$
$\displaystyle:=\left\\{t\in{\mathbb{I}}:\,\tau\leq t\right\\},$
$\displaystyle{\mathbb{I}}_{\tau}^{-}$
$\displaystyle:=\left\\{t\in{\mathbb{I}}:\,t\leq\tau\right\\}\quad\text{for
}\tau\in{\mathbb{Z}},$

as well as ${\mathbb{N}}_{0}:=\left\\{0,1,2,\ldots\right\\}$.

On a given metric space $(X,d)$, $\overline{A}$ denotes the closure of a
subset $A\subseteq X$, $B_{r}(x):=\left\\{y\in X:\,d(x,y)<r\right\\}$ is the
open ball with center $x\in X$ and radius $r>0$, and $\bar{B}_{r}(x)$ its
closure. We write $\mbox{dist}_{A}(x):=\inf_{a\in A}d(x,a)$ for the _distance_
of a point $x\in X$ to a subset $A\subseteq X$ and $B_{r}(A):=\left\\{x\in
X:\,\mbox{dist}_{A}(x)<r\right\\}$ for its $r$-neighborhood. The _Hausdorff
semidistance_ of bounded and closed subsets $A,B\subseteq X$ is defined as

$\operatorname{dist}\left(A,B\right):=\sup_{a\in A}\mbox{dist}_{B}(a),$

and, with a further closed and bounded subset $C\subseteq X$, one has the
properties

$\displaystyle\operatorname{dist}\left(A,B\right)\leq\operatorname{dist}\left(A,C\right)+\operatorname{dist}\left(C,B\right).$
(1.1)

A mapping ${\mathscr{F}}:X\to X$ is called _bounded_ , if it maps bounded
subsets of $X$ into bounded sets and _globally bounded_ , if
${\mathscr{F}}(X)$ is bounded. A _completely continuous_ map ${\mathscr{F}}$
is continuous and maps bounded sets into relatively compact sets.

Beyond this general terminology, we need to introduce terms commonly used in
the area of nonautonomous dynamics [25]: A subset
${\mathcal{A}}\subseteq{\mathbb{I}}\times X$ with nonempty _fibers_
${\mathcal{A}}(t):=\\{x\in X:\,(t,x)\in{\mathcal{A}}\\}$, $t\in{\mathbb{I}}$,
is called _nonautonomous set_. A nonautonomous set ${\mathcal{A}}$ is said to
have some topological property if each fiber ${\mathcal{A}}(t)$,
$t\in{\mathbb{I}}$, has this property. Furthermore, one speaks of a _bounded_
nonautonomous set ${\mathcal{A}}$, if there exists real $R>0$ and a point
$x_{0}\in X$ such that ${\mathcal{A}}(t)\subseteq B_{R}(x_{0})$ holds for all
$t\in{\mathbb{I}}$. If ${\mathbb{I}}$ is unbounded above, then we define

$\limsup_{t\to\infty}{\mathcal{A}}(t):=\bigcap_{\tau\in{\mathbb{I}}}\overline{\bigcup_{\tau\leq
t}{\mathcal{A}}(t)}.$

##  2\. Nonautonomous difference equations and attractors

Let $(X,d)$ be a complete metric space and $U_{t}\neq\emptyset$,
$t\in{\mathbb{I}}^{\prime}$, be closed subsets of $X$. We consider
nonautonomous difference equations

$\displaystyle\boxed{u_{t+1}={\mathscr{F}}_{t}(u_{t})}$ ($\Delta$)

having continuous right-hand sides ${\mathscr{F}}_{t}:U_{t}\to X$,
$t\in{\mathbb{I}}^{\prime}$.

Given an _initial time_ $\tau\in{\mathbb{I}}$, a _forward solution_
$(\phi_{t})_{\tau\leq t}$ to ($\Delta$) is a sequence satisfying $\phi_{t}\in
U_{t}$ and $\phi_{t+1}={\mathscr{F}}_{t}(\phi_{t})$ for all $\tau\leq
t,t\in{\mathbb{I}}^{\prime}$, whilst an _entire solution_
$(\phi_{t})_{t\in{\mathbb{I}}}$ meets the same identity for all
$t\in{\mathbb{I}}^{\prime}$. The _general solution_
$\varphi(\cdot;\tau,u_{\tau})$ of ($\Delta$) is the unique forward solution
starting at $\tau\in{\mathbb{I}}$ in the _initial state_ $u_{\tau}\in
U_{\tau}$, i.e.

$\displaystyle\varphi(t;\tau,u_{\tau}):=\begin{cases}{\mathscr{F}}_{t-1}\circ\ldots\circ{\mathscr{F}}_{\tau}(u_{\tau}),&\tau<t,\\\
u_{\tau},&\tau=t,\end{cases}$ (2.1)

as long as the iterates stay in $U_{t}$.

From now on, assume that ${\mathscr{F}}_{t}(U_{t})\subseteq U_{t+1}$ holds for
all $t\in{\mathbb{I}}^{\prime}$ and abbreviate

${\mathcal{U}}:=\left\\{(t,u)\in{\mathbb{I}}\times X:\,u\in U_{t}\right\\}.$

Then $\varphi$ satisfies the _process_ (or _two-parameter semi-group_)
_property_

$\varphi(t;s,\varphi(s;\tau,u))=\varphi(t;\tau,u)\quad\text{for all }\tau\leq
s\leq t,\,u\in U_{\tau}.$

A nonautonomous set ${\mathcal{A}}\subseteq{\mathcal{U}}$ is said to be
_positively_ or _negatively invariant_ , if

$\displaystyle{\mathscr{F}}_{t}\left({\mathcal{A}}(t)\right)$
$\displaystyle\subseteq{\mathcal{A}}(t+1),$ $\displaystyle{\mathcal{A}}(t+1)$
$\displaystyle\subseteq{\mathscr{F}}_{t}\left({\mathcal{A}}(t)\right)\quad\text{for
all }t\in{\mathbb{I}}^{\prime}$

holds, respectively. Accordingly, an _invariant_ set ${\mathcal{A}}$ is both
positively and negatively invariant, that is
${\mathscr{F}}_{t}\left({\mathcal{A}}(t)\right)={\mathcal{A}}(t+1)$ for all
$t\in{\mathbb{I}}^{\prime}$.

A bounded nonautonomous set ${\mathcal{A}}\subseteq{\mathcal{U}}$ is called
_absorbing_ , if for each $\tau\in{\mathbb{I}}$ and bounded
${\mathcal{B}}\subseteq{\mathcal{U}}$, there is an _absorption time_
$T=T(\tau,{\mathcal{B}})\in{\mathbb{N}}$ such that

$\varphi(t;\tau,{\mathcal{B}}(\tau))\subseteq{\mathcal{A}}(t)\quad\text{for
all }t,\tau\in{\mathbb{I}},\,t-\tau\geq T.$

In addition, for _pullback_ and _forward absorbing sets_ , the present time
$t$ and the initial time $\tau$ work similarly to those in the properties of
pullback and forward attraction (one notion of time is fixed and the other
tends to infinity or minus infinity, cf. [25, pp. 44–45, Def. 3.3]).

###  2.1. Pullback attractors

Let a discrete interval ${\mathbb{I}}$ be unbounded below. A _pullback
attractor_ ${\mathcal{A}}^{\ast}\subseteq{\mathcal{U}}$ of ($\Delta$) is a
compact and invariant nonautonomous set, which pullback attracts bounded sets
${\mathcal{B}}\subseteq{\mathcal{U}}$, i.e.

$\displaystyle\lim_{s\to\infty}\operatorname{dist}\left(\varphi(\tau;\tau-s,{\mathcal{B}}(\tau-s)),{\mathcal{A}}^{\ast}(\tau)\right)=0\quad\text{for
all }\tau\in{\mathbb{I}}.$

With comprehensive information on pullback attractors given in [3, 5, 25, 33],
our existence condition for pullback attractors is based on the following
notion: A difference equation ($\Delta$) is called _pullback asymptotically
compact_ , if for every $\tau\in{\mathbb{I}}$, every sequence
$(s_{k})_{k\in{\mathbb{N}}}$ in ${\mathbb{N}}$ with $s_{k}\to\infty$ as
$k\to\infty$ and every bounded sequence $(a_{k})_{k\in{\mathbb{N}}}$ with
$a_{k}\in{\mathcal{U}}(\tau-s_{k})$, the sequence $(\varphi(\tau;\tau-
s_{k},a_{k}))_{k\in{\mathbb{N}}}$ in ${\mathcal{U}}(\tau)$ has a convergent
subsequence.

######  Theorem 2.1 (Existence of pullback attractors, cf. [33, p. 19, Thm.
1.3.9]).

Suppose a difference equation ($\Delta$) has a pullback absorbing set
${\mathcal{A}}$. If ($\Delta$) is pullback asymptotically compact, then it
possesses a unique pullback attractor
${\mathcal{A}}^{\ast}\subseteq{\mathcal{A}}$.

The importance of pullback attractors is due to their following dynamical
characterization (see [33, p. 17, Cor. 1.3.4])

${\mathcal{A}}^{\ast}=\left\\{(t,u)\in{\mathcal{U}}:\begin{array}[]{l}\text{there
exists a bounded entire}\\\ \text{solution $\phi$ to \eqref{deq} with
}\phi_{\tau}=u\end{array}\right\\}$ (2.2)

motivating that pullback attractors are a suitable nonautonomous extension of
global attractors [3, p. 20, Thm. 1.9] of autonomous difference equations.

###  2.2. Forward dynamics

On a discrete interval ${\mathbb{I}}$ unbounded above, we assume the right-
hand sides ${\mathscr{F}}_{t}:U\to U$, $t\in{\mathbb{I}}$, of ($\Delta$) act
on and map into a common closed subset $U\neq\emptyset$ of $X$. A _forward
attractor_ ${\mathcal{A}}^{+}\subseteq{\mathcal{U}}$ of ($\Delta$) is a
compact and invariant nonautonomous set, which forward attracts bounded sets
${\mathcal{B}}\subseteq{\mathcal{U}}$, i.e.

$\displaystyle\lim_{s\to\infty}\operatorname{dist}\left(\varphi(\tau+s;\tau,{\mathcal{B}}(\tau)),{\mathcal{A}}^{+}(\tau+s)\right)=0\quad\text{for
all }\tau\in{\mathbb{I}}.$

In contrast to pullback attractors ${\mathcal{A}}^{\ast}$, forward attractors
${\mathcal{A}}^{+}$ need not to be unique (cf. [26, Sect. 4]) and further
properties are given in [7, 18, 23, 25, 26]. Not surprisingly, the existence
of forward attractors is based on suitable compactness properties. For
nonautonomous sets ${\mathcal{A}}\subseteq{\mathcal{U}}$, a difference
equation ($\Delta$) is called

  * •

_${\mathcal{A}}$ -asymptotically compact_, if there exists a nonempty, compact
set $K\subseteq U$ such that $K$ forward attracts ${\mathcal{A}}(\tau)$, i.e.,

$\displaystyle\lim_{s\to\infty}\operatorname{dist}\left(\varphi(\tau+s;\tau,{\mathcal{A}}(\tau)),K\right)=0\quad\text{for
all }\tau\in{\mathbb{I}},$

  * •

_strongly ${\mathcal{A}}$-asymptotically compact_, if there exists a nonempty,
compact set $K\subseteq U$ such that every sequence
$\left((s_{k},\tau_{k})\right)_{k\in{\mathbb{N}}}$ in
${\mathbb{N}}\times{\mathbb{I}}$ with $s_{k}\to\infty$, $\tau_{k}\to\infty$ as
$k\to\infty$ yields

$\lim_{k\to\infty}\operatorname{dist}\left(\varphi(\tau_{k}+s_{k};\tau_{k},{\mathcal{A}}(\tau_{k})),K\right)=0.$

For positively invariant sets ${\mathcal{A}}$, strong
${\mathcal{A}}$-asymptotic compactness implies ${\mathcal{A}}$-asymptotic
compactness (cf. [18, Rem. 4.1]).

In preparation of our next results, let us introduce a set
$\Omega_{{\mathcal{A}}}\subseteq{\mathcal{U}}$ as

$\Omega_{{\mathcal{A}}}(\tau):=\bigcap_{0\leq s}\overline{\bigcup_{s\leq
t}\varphi(\tau+t;\tau,{\mathcal{A}}(\tau))}\quad\text{for all
}\tau\in{\mathbb{I}}$

and the _forward limit set_ for the dynamics of ($\Delta$) starting from
$\Omega_{{\mathcal{A}}}$ is

$\omega_{{\mathcal{A}}}^{+}:=\overline{\bigcup_{\tau\in{\mathbb{I}}}\Omega_{{\mathcal{A}}}(\tau)}\subseteq
U.$

######  Theorem 2.2 (Forward limit sets, cf. [18, Thm. 4.6]).

Suppose a difference equation ($\Delta$) has a bounded and positively
invariant set ${\mathcal{A}}$. If ($\Delta$) is ${\mathcal{A}}$-asymptotically
compact with a compact $K\subseteq U$, then its forward limit set
$\omega_{\mathcal{A}}^{+}\subseteq K$ is nonempty, compact and forward
attracts ${\mathcal{A}}$.

In comparison to Thm. 2.1 a result guaranteeing the existence of forward
attractors is more involved. It requires the assumption
${\mathbb{I}}={\mathbb{Z}}$ and to introduce a further set
${\mathcal{A}}^{\star}\subseteq{\mathcal{U}}$ by its fibers

${\mathcal{A}}^{\star}(\tau):=\bigcap_{0\leq
s}\varphi(\tau;\tau-s,{\mathcal{A}}(\tau-s))\quad\text{for all
}\tau\in{\mathbb{Z}}.$

######  Theorem 2.3 (Forward attractors).

Let ${\mathbb{I}}={\mathbb{Z}}$. Suppose a difference equation ($\Delta$) has
a closed, forward absorbing and positively invariant set ${\mathcal{A}}$. If
($\Delta$) is pullback asymptotically compact, strongly
${\mathcal{A}}$-asymptotically compact with compact $K\subseteq U$ and

  * (i)

for every sequence $\left((s_{k},\tau_{k})\right)_{k\in{\mathbb{N}}}$ in
${\mathbb{N}}\times{\mathbb{Z}}$ with $\tau_{k}\to\infty$ as $k\to\infty$ one
has

$\displaystyle\lim_{k\to\infty}\operatorname{dist}\left(\varphi(\tau_{k}+s_{k};\tau_{k},K),K\right)=0,$

  * (ii)

for all $\varepsilon>0$ and $S\in{\mathbb{N}}$, there exists a $\delta>0$ such
that for all $\tau\in{\mathbb{Z}}$ one has the implication

$\displaystyle\left.\begin{array}[]{c}u,v\in{\mathcal{A}}(\tau)\cup K,\\\
d(u,v)<\delta\end{array}\right\\}\Rightarrow\max_{0<s\leq
S}d(\varphi(\tau+s;\tau,u),\varphi(\tau+s;\tau,v))<\varepsilon,$

  * (iii)

$\omega_{{\mathcal{A}}}^{+}=\limsup_{t\to\infty}{\mathcal{A}}^{\star}(t)$,
i.e. the forward limit sets for the dynamics starting from
$\Omega_{\mathcal{A}}$ and from within ${\mathcal{A}}^{\star}$ coincide

hold, then ${\mathcal{A}}^{\star}\subseteq{\mathcal{A}}$ is a forward
attractor of ($\Delta$).

######  Remark 2.4.

(1) A situation, where an assumption such as (i) trivially holds, is when the
compact set $K\subseteq U$ is positively invariant w.r.t. ($\Delta$).

(2) The assumption (ii) can be interpreted as uniform continuity of every
mapping $\varphi(\tau+s;\tau,\cdot)$ on ${\mathcal{A}}(\tau)\cup K$ and finite
discrete intervals $[\tau,\tau+s]\cap{\mathbb{Z}}$, $\tau\in{\mathbb{I}}$,
$0<s\leq S$. Hence, if we assume that there exists a bounded $B\subseteq U$
satisfying

$\bigcup_{s\in{\mathbb{N}}_{0}}\left({\mathcal{A}}(\tau+s)\cup\varphi(\tau+s;\tau,K)\right)\subseteq
B\quad\text{for all }\tau\in{\mathbb{Z}},$

and a Lipschitz condition with uniform constant $\ell\geq 0$ for each
${\mathscr{F}}_{t}:U\to U$, $t\in{\mathbb{I}}$, on $B$, then
$d(\varphi(\tau+s;\tau,u),\varphi(\tau+s;\tau,v))\leq\ell^{s}d(u,v)$ for all
$u,v\in B$. From this we obtain that (ii) holds true.

###### Proof.

Invariance, being nonempty and compactness of
${\mathcal{A}}^{\star}\subseteq{\mathcal{A}}$ are established in [18, Prop.
3.1], while the forward attraction is due to [18, Cor. 4.19]. ∎

###  2.3. Perturbation of attractors

Rather than sticking to a single problem ($\Delta$), we now proceed to
families of nonautonomous difference equations

$\displaystyle\boxed{u_{t+1}={\mathscr{F}}_{t}^{n}(u_{t})}$ ($\Delta_{n}$)

having continuous right-hand sides ${\mathscr{F}}_{t}^{n}:U_{t}\to X$,
$t\in{\mathbb{I}}^{\prime}$, which are parameterized by
$n\in{\mathbb{N}}_{0}$. The general solution of ($\Delta_{n}$) is denoted by
$\varphi^{n}$.

For $n\in{\mathbb{N}}$ the difference equations ($\Delta_{n}$) may describe
perturbations (concretely, numerical discretizations with accuracy increasing
in $n$) of an initial problem $(\Delta_{0})$. Hence, a crucial concept
allowing us to compare the behavior of solutions to ($\Delta_{n}$),
$n\in{\mathbb{N}}$, to the solutions of $(\Delta_{0})$ is the _local error_

$e_{t}^{n}(u):=d({\mathscr{F}}_{t}^{n}(u),{\mathscr{F}}_{t}(u)).$

We say the difference equations ($\Delta_{n}$), $n\in{\mathbb{N}}$, are
_convergent_ , if there exists a _convergence function_
$\Gamma:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ satisfying $\lim_{\rho\searrow
0}\Gamma(\rho)=0$ such that for every bounded $B\subseteq U_{t}$, there exists
a $C(B)\geq 0$ with

$\displaystyle\sup_{u\in B}e_{t}^{n}(u)\leq
C(B)\Gamma(\tfrac{1}{n})\quad\text{for all
}t\in{\mathbb{I}}^{\prime},\,n\in{\mathbb{N}}.$ (2.3)

One speaks of a _convergence rate_ $\mu>0$, if $\Gamma(\rho)=\rho^{\mu}$.

The following result determines how the local error evolves over time. It is
crucial in order to apply our perturbation results in Thm. 2.6 and 2.7:

######  Proposition 2.5 (Global discretization error).

Let $\tau,T\in{\mathbb{I}}$ with $\tau\leq T$ be fixed, $u\in U_{\tau}$ and
${\mathcal{B}}\subseteq{\mathcal{U}}$ be a bounded nonautonomous set such that

$\displaystyle\varphi^{0}(t;\tau,u)\in{\mathcal{B}}(t)\quad\text{for all
}\tau\leq t\leq T.$ (2.4)

If the difference equations ($\Delta_{n}$), $n\in{\mathbb{N}}$, are convergent
and for every $t\in{\mathbb{I}}^{\prime}$ and bounded $B\subseteq X$ there
exists a real $\ell_{t}(B)\geq 0$ such that

$\displaystyle
d({\mathscr{F}}_{t}^{n}(u),{\mathscr{F}}_{t}^{n}(v))\leq\ell_{t}(B)d(u,v)\quad\text{for
all }n\in{\mathbb{N}},\,u,v\in U_{t}\cap B,$ (2.5)

then for every $\rho>0$ there exists an $N_{0}\in{\mathbb{N}}$ such that for
all $n\geq N_{0}$ one has the inclusion $\varphi^{n}(t;\tau,u)\in
B_{\rho}({\mathcal{B}}(t))$ and the global discretization error

$\displaystyle
d(\varphi^{n}(t;\tau,u),\varphi^{0}(t;\tau,u))\leq\Gamma\left(\tfrac{1}{n}\right)\sum_{l_{1}=\tau}^{t-1}C({\mathcal{B}}(l_{1}))\prod_{l_{2}=l_{1}+1}^{t-1}\ell_{l_{2}}\bigl{(}B_{\rho}({\mathcal{B}}(l_{2}))\bigr{)}$
(2.6)

for all $\tau\leq t\leq T$.

###### Proof.

Let $\rho>0$ and choose $(\tau,u)\in{\mathcal{B}}$, $\tau\leq T$. By induction
over $t\geq\tau$ we establish the existence of a
$N_{0}=N_{0}(T)\in{\mathbb{N}}$ such that both (2.6) and

$\varphi^{n}(t;\tau,u)\in B_{\rho}({\mathcal{B}}(t))\quad\text{for all
}\tau\leq t\leq T,\,n\geq N_{0}$ (2.7)

hold. This is trivially true for $t=\tau$. Assume that (2.6)–(2.7) are
satisfied for some fixed $t<T$ and set
$x_{t}:=d(\varphi^{n}(t;\tau,u),\varphi^{0}(t;\tau,u))$. Using the triangle
inequality,

$\displaystyle x_{t+1}$
$\displaystyle\stackrel{{\scriptstyle\eqref{process}}}{{\leq}}$ $\displaystyle
d\bigl{(}{\mathscr{F}}_{t}^{n}(\underbrace{\varphi^{n}(t;\tau,u)}_{\stackrel{{\scriptstyle\eqref{noind1}}}{{\in}}B_{\rho}({\mathcal{B}}(t))}),{\mathscr{F}}_{t}^{n}(\underbrace{\varphi^{0}(t;\tau,u)}_{\stackrel{{\scriptstyle\eqref{properr1}}}{{\in}}{\mathcal{B}}(t)})\bigr{)}+e_{t}^{n}(\varphi^{0}(t;\tau,u))$
$\displaystyle\stackrel{{\scriptstyle\eqref{properror1}}}{{\leq}}$
$\displaystyle\ell_{t}\bigl{(}B_{\rho}({\mathcal{B}}(t))\bigr{)}x_{t}+e_{t}^{n}(\underbrace{\varphi^{0}(t;\tau,u)}_{\stackrel{{\scriptstyle\eqref{properr1}}}{{\in}}{\mathcal{B}}(t)})$
$\displaystyle\stackrel{{\scriptstyle\eqref{errest}}}{{\leq}}$
$\displaystyle\ell_{t}\bigl{(}B_{\rho}({\mathcal{B}}(t))\bigr{)}x_{t}+C({\mathcal{B}}(t))\Gamma(\tfrac{1}{n})$
$\displaystyle\stackrel{{\scriptstyle\eqref{properr3}}}{{\leq}}$
$\displaystyle\ell_{t}\bigl{(}B_{\rho}({\mathcal{B}}(t))\bigr{)}\Gamma(\tfrac{1}{n})\sum_{l_{1}=\tau}^{t-1}C({\mathcal{B}}(l_{1}))\prod_{l_{2}=l_{1}+1}^{t-1}\ell_{l_{2}}\bigl{(}B_{\rho}({\mathcal{B}}(l_{2}))\bigr{)}+C({\mathcal{B}}(t))\Gamma(\tfrac{1}{n})$
$\displaystyle=$
$\displaystyle\Gamma(\tfrac{1}{n})\sum_{l_{1}=\tau}^{t}C({\mathcal{B}}(l_{1}))\prod_{l_{2}=l_{1}+1}^{t}\ell_{l_{2}}\bigl{(}B_{\rho}({\mathcal{B}}(l_{2}))\bigr{)}$

results. For a sufficiently large $N_{0}$ it holds
$d(\varphi^{n}(t+1;\tau,u),\varphi^{0}(t+1;\tau,u))<\rho$ and the desired
inclusion $\varphi^{n}(t+1;\tau,u)\in B_{\rho}({\mathcal{B}}(t+1))$ results. ∎

Rather than providing suitable conditions on ($\Delta_{n}$) guaranteeing that
the existence of a pullback attractor ${\mathcal{A}}_{0}^{\ast}$ for
$(\Delta_{0})$ implies that also the perturbations ($\Delta_{n}$),
$n\in{\mathbb{N}}$, have pullback attractors ${\mathcal{A}}_{n}^{\ast}$
(_persistence_), on the present abstract level we simply assume that such
attractors ${\mathcal{A}}_{n}^{\ast}$, $n\in{\mathbb{N}}_{0}$, do exist. Then
the next result is a discrete time version of [3, p. 152, Thm. 7.13].

######  Theorem 2.6 (Upper-semicontinuity of pullback attractors).

Let ${\mathbb{I}}$ be unbounded below. If each difference equation
($\Delta_{n}$), $n\in{\mathbb{N}}_{0}$, has a pullback attractor
${\mathcal{A}}_{n}^{\ast}$ and

  * (i)

$\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(\tau)$ is relatively
compact for every $\tau\in{\mathbb{I}}$,

  * (ii)

$\bigcup_{n\in{\mathbb{N}}_{0}}\bigcup_{s\leq\tau}{\mathcal{A}}_{n}^{\ast}(s)$
is bounded for every $\tau\in{\mathbb{I}}$,

  * (iii)

(_convergence condition_) for all $\varepsilon>0$, $S\in{\mathbb{N}}$ and
every compact $K\subseteq X$ there exists an $n_{0}\in{\mathbb{N}}$ such that
for all $\tau\in{\mathbb{I}}$ one has the implication

$n>n_{0}\Rightarrow\max_{0<s\leq S}\sup_{u\in U_{\tau-s}\cap
K}d(\varphi^{n}(\tau;\tau-s,u),\varphi^{0}(\tau;\tau-s,u))<\varepsilon$

hold, then $\lim_{n\to\infty}\max_{t\in
I}\operatorname{dist}\left({\mathcal{A}}_{n}^{\ast}(t),{\mathcal{A}}_{0}^{\ast}(t)\right)=0$
for every bounded $I\subseteq{\mathbb{I}}$.

###### Proof.

Let $n\in{\mathbb{N}}_{0}$. Recall from (2.2) that each fiber
${\mathcal{A}}_{n}^{\ast}(t)\subseteq{\mathcal{U}}(t)$, $t\in{\mathbb{I}}$,
consists of initial values for a bounded entire solutions $\phi^{n}$ to
($\Delta_{n}$). The proof is then divided into two claims.

(I) Claim: _There is a bounded entire solution $\phi^{0}$ to $(\Delta_{0})$
and a subsequence $\left(\phi^{n_{k}}\right)_{k\in{\mathbb{N}}}$ of
$\left(\phi^{n}\right)_{n\in{\mathbb{N}}}$ such that
$\phi^{0}_{t}\in{\mathcal{A}}_{0}^{\ast}(t)$ for all $t\in{\mathbb{I}}$ and
$\left(\phi^{n_{k}}\right)_{k\in{\mathbb{N}}}$ converges to $\phi^{0}$
uniformly on bounded subintervals of ${\mathbb{I}}$._

In fact, $\left(\phi^{n}_{\tau}\right)_{n\in{\mathbb{N}}}$ can be seen as
sequence in
$\overline{\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(\tau)}$ with
$\tau\in{\mathbb{I}}$. Combining this with the fact from (i) that
$\overline{\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(\tau)}$ is
compact yields the existence of an infinite subset
$N_{0}\subseteq{\mathbb{N}}$ so that the subsequence
$\left(\phi^{n}_{\tau}\right)_{n\in N_{0}}$ of
$\left(\phi^{n}_{\tau}\right)_{n\in{\mathbb{N}}}$ converges to a limit
$u_{\tau}\in\overline{\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(\tau)}$.
If we set $\phi^{0}_{t}:=\varphi^{0}(t;\tau,u_{\tau})$ for each
$t\in{\mathbb{I}}_{\tau}^{+}$, then $\left(\phi^{n}\right)_{n\in N_{0}}$
converges to $\phi^{0}$ uniformly on bounded subintervals of
${\mathbb{I}}_{\tau}^{+}$ due to (iii).

Now by mathematical induction, we suppose that there exists a bounded forward
solution $(\phi_{t}^{0})_{s\leq t}$, $s\leq\tau$, to $(\Delta_{0})$ as well as
infinite subsets $N_{s}\subset N_{s-1}$ such that $\left(\phi^{n}\right)_{n\in
N_{s}}$ converges to $\phi^{0}$ uniformly in bounded subintervals of
${\mathbb{I}}_{s}^{+}$. Similarly to the above, since the sequence
$\left(\phi^{n}_{s-1}\right)_{n\in N_{s}}$ is contained in the compact set
$\overline{\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(s-1)}$,
there is an infinite subset $N_{s+1}\subset N_{s}$ so that the subsequence
$\left(\phi^{n}_{s-1}\right)_{n\in N_{s+1}}$ of
$\left(\phi^{n}_{s-1}\right)_{n\in N_{s}}$ is convergent to
$u_{s-1}\in\overline{\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(s-1)}$.
Therefore, $(\phi_{t}^{0})_{s-1\leq t}$ is a bounded solution to
$(\Delta_{0})$ and again by (iii), the sequence $\left(\phi^{n}\right)_{n\in
N_{s+1}}$ converges to $\phi^{0}$ uniformly in bounded subintervals of
${\mathbb{I}}_{s-1}^{+}$.

Repeating this procedure eventually results in a bounded entire solution
$\phi^{0}$ to the equation $(\Delta_{0})$ and thus,
$\phi^{0}_{t}\in{\mathcal{A}}_{0}^{\ast}(t)$ for each $t\in{\mathbb{I}}$ due
to (ii). Moreover, each sequence
$\left(\phi^{n}_{t}\right)_{n\in{\mathbb{N}}}$ in
${\mathcal{A}}_{n}^{\ast}(t)$ has a uniformly convergent subsequence
$\left(\phi^{n_{s}}_{t}\right)_{n_{s}\in N_{s}}$ converging to the limit
$\phi^{0}_{t}\in{\mathcal{A}}_{0}^{\ast}(t)$ for each $t\in{\mathbb{I}}$ as
$n_{s}$ is the $s$-th element of $N_{s}\subseteq{\mathbb{N}}$,
$s\in{\mathbb{N}}$. This completes the proof of the first claim.

(II) Claim:
_$\lim_{n\to\infty}\operatorname{dist}\left({\mathcal{A}}_{n}^{\ast}(t),{\mathcal{A}}_{0}^{\ast}(t)\right)=0$
for all $t\in{\mathbb{I}}$._
By means of contradiction, assume the limit in the claim fails for some
instant $t\in{\mathbb{I}}$, i.e. there are $\varepsilon_{0}>0$ and
$n_{0}\in{\mathbb{N}}$ such that
$\operatorname{dist}\left({\mathcal{A}}_{n_{0}}^{\ast}(t),{\mathcal{A}}_{0}^{\ast}(t)\right)\geq\varepsilon_{0}$
holds. Because ${\mathcal{A}}_{n_{0}}^{\ast}(t)$ is compact, there exists a
bounded entire solution $\phi^{n_{0}}$ with
$\phi_{t}^{n_{0}}\in{\mathcal{A}}_{n_{0}}^{\ast}(t)$ so that
$\mbox{dist}_{{\mathcal{A}}_{0}^{\ast}(t)}(\phi^{n_{0}}_{t})\geq\varepsilon_{0}$,
a contradiction to claim (I) as for every bounded entire solution $\phi^{n}$,
the values $\phi_{t}^{n}\in{\mathcal{A}}_{n}^{\ast}(t)$ have a convergent
subsequence with limit in ${\mathcal{A}}_{0}^{\ast}(t)$.

Because bounded subsets $I\subseteq{\mathbb{I}}$ are finite, the assertion
follows from (II). ∎

In contrast, a related result addressing the forward dynamics inherently
yields the existence of limit sets:

######  Theorem 2.7 (Upper-semicontinuity of forward limit sets).

Let ${\mathbb{I}}$ be unbounded above. Suppose a bounded set
${\mathcal{A}}\subseteq{\mathcal{U}}$ is positively invariant for every
difference equation ($\Delta_{n}$), $n\in{\mathbb{N}}_{0}$. If each
($\Delta_{n}$), $n\in{\mathbb{N}}_{0}$, is strongly
${\mathcal{A}}$-asymptotically compact with a compact $K_{n}\subseteq U$, then
its forward limit set $\omega_{{\mathcal{A}},n}^{+}\subseteq K_{n}$ is
nonempty, compact and forward attracts ${\mathcal{A}}$. If
$K:=\bigcup_{n\in{\mathbb{N}}_{0}}K_{n}$ and moreover

  * (i)

$\omega_{{\mathcal{A}},0}^{+}$ attracts $K$ uniformly in
$\tau\in{\mathbb{I}}$, i.e.,

$\lim_{s\to\infty}\sup_{\tau\in{\mathbb{I}}}\operatorname{dist}\left(\varphi^{0}(\tau+s;\tau,K),\omega_{{\mathcal{A}},0}^{+}\right)=0,$

  * (ii)

for every sequence $\left((s_{k},\tau_{k})\right)_{k\in{\mathbb{N}}}$ in
${\mathbb{N}}\times{\mathbb{I}}$ with $\tau_{k}\to\infty$ as $k\to\infty$ one
has

$\lim_{k\to\infty}\operatorname{dist}\left(\varphi^{n}(\tau_{k}+s_{k};\tau_{k},K_{n}),K_{n}\right)=0\quad\text{for
all }n\in{\mathbb{N}}_{0},$

  * (iii)

for all $\varepsilon>0$, $S\in{\mathbb{N}}$ and $n\in{\mathbb{N}}_{0}$, there
exists a $\delta>0$ such that for all $\tau\in{\mathbb{I}}$ one has the
implication

$\left.\begin{array}[]{c}u,v\in{\mathcal{A}}(\tau)\cup K_{n},\\\
d(u,v)<\delta\end{array}\right\\}\Rightarrow\max_{0<s\leq
S}d(\varphi^{n}(\tau+s;\tau,u),\varphi^{n}(\tau+s;\tau,v))<\varepsilon,$

  * (iv)

(_convergence condition_) for all $\varepsilon>0$, $S\in{\mathbb{N}}$, there
exists an $n_{0}\in{\mathbb{N}}$ such that for all $\tau\in{\mathbb{I}}$ one
has the implication

$\left.\begin{array}[]{c}u\in{\mathcal{A}}(\tau)\cup K,\\\
n>n_{0}\end{array}\right\\}\Rightarrow\max_{0<s\leq
S}d(\varphi^{n}(\tau+s;\tau,u),\varphi^{0}(\tau+s;\tau,u))<\varepsilon$

hold, then
$\lim_{n\to\infty}\operatorname{dist}\left(\omega_{{\mathcal{A}},n}^{+},\omega_{{\mathcal{A}},0}^{+}\right)=0$.

###### Proof.

To begin with, recall from Thm. 2.2 that each ($\Delta_{n}$),
$n\in{\mathbb{N}}_{0}$, has a nonempty, compact, forward limit set
$\omega_{{\mathcal{A}},n}^{+}\subseteq K_{n}$. Moreover, thanks to [18, Thm.
4.10], $\omega_{{\mathcal{A}},n}^{+}$ is _asymptotically negatively invariant_
, i.e., for all $u\in\omega_{{\mathcal{A}},n}^{+}$, $\varepsilon>0$ and $T>0$,
there is an integer $S_{\varepsilon}$ satisfying
$\tau+S_{\varepsilon}-T\in{\mathbb{I}}$ and a point
$u_{\varepsilon}\in\omega_{{\mathcal{A}},n}^{+}$ such that

$d(\varphi^{n}(\tau+S_{\varepsilon};\tau+S_{\varepsilon}-T,u_{\varepsilon}),u)<\varepsilon\quad\text{for
all }\tau\in{\mathbb{I}},\,n\in{\mathbb{N}}_{0}.$

In order to establish
$\lim_{n\to\infty}\operatorname{dist}\left((\omega_{{\mathcal{A}},n}^{+},\omega_{{\mathcal{A}},0}^{+})\right)=0$
we proceed by contradiction. Assume that the above limit does not hold. Then
there is a sequence $(n_{j})_{j\in{\mathbb{N}}}$ in ${\mathbb{N}}$ satisfying
$n_{j}\to\infty$ as $j\to\infty$ and an $\varepsilon_{0}>0$ such that

$\displaystyle
4\varepsilon_{0}\leq\operatorname{dist}\left(\omega_{{\mathcal{A}},n_{j}}^{+},\omega_{{\mathcal{A}},0}^{+}\right)\quad\text{for
all }j\in{\mathbb{N}}.$

By the compactness of $\omega_{{\mathcal{A}},n_{j}}^{+}$, the supremum in the
definition of
$\operatorname{dist}\left(\omega_{{\mathcal{A}},n_{j}}^{+},\omega_{{\mathcal{A}},0}^{+}\right)$
is attained and so is the existence of a point
$w_{j}\in\omega_{{\mathcal{A}},n_{j}}^{+}$ that satisfies

$\displaystyle
4\varepsilon_{0}\leq\operatorname{dist}\left(\omega_{{\mathcal{A}},n_{j}}^{+},\omega_{{\mathcal{A}},0}^{+}\right)=\mbox{dist}_{\omega_{{\mathcal{A}},0}^{+}}(w_{j}).$
(2.8)

On the other hand, from (i), there is a time $T_{0}\in{\mathbb{N}}$ such that

$\displaystyle\operatorname{dist}\left(\varphi^{0}(\tau+s;\tau,K),\omega_{{\mathcal{A}},0}^{+}\right)<\varepsilon_{0}\quad\text{for
all }s\geq T_{0},\,\tau\in{\mathbb{I}}.$ (2.9)

Since $T_{0}>0$ and $w_{j}\in\omega_{{\mathcal{A}},n_{j}}^{+}$, the asymptotic
negative invariance of $\omega_{{\mathcal{A}},n_{j}}^{+}$ yields that there is
an $s_{0}=s_{0}({\varepsilon_{0}})\in{\mathbb{I}}$ with $t_{0}:=\tau+s_{0}$, a
$\tau_{0}:=t_{0}-T_{0}\in{\mathbb{I}}$ and a
$u_{j}=u_{j}(n_{j},T_{0})\in\omega_{{\mathcal{A}},n_{j}}^{+}\subseteq
K_{n_{j}}$ so that

$d\left(w_{j},\varphi^{n_{j}}(t_{0};\tau_{0},u_{j})\right)<\varepsilon_{0}.$

Now notice that as $u_{j}\in{\mathcal{A}}(\tau_{0})\cup K$, with integers
$N=N(\varepsilon_{0},T_{0})>0$ independent of $\tau_{0}$ and
$n_{j}\in{\mathbb{Z}}^{+}_{N}$, our assumption (iv) yields

$\displaystyle
d(\varphi^{n_{j}}(t_{0};\tau_{0},u_{j}),\varphi^{0}(t_{0};\tau_{0},u_{j}))<\varepsilon_{0},$

as well as (2.9) results in

$\displaystyle\mbox{dist}_{\omega_{{\mathcal{A}},0}^{+}}(\varphi^{0}(t_{0};\tau_{0},u_{j}))<\operatorname{dist}\left(\varphi^{0}(t_{0};\tau_{0},K),\omega_{{\mathcal{A}},0}^{+}\right)<\varepsilon_{0}.$

Combining the last three inequalities, we obtain

$\mbox{dist}_{\omega_{{\mathcal{A}},0}^{+}}(w_{j})\leq
d\left(w_{j},\varphi^{n_{j}}(t_{0};\tau_{0},u_{j})\right)+d\left(\varphi^{n_{j}}(t_{0};\tau_{0},u_{j}),\varphi^{0}(t_{0};\tau_{0},u_{j})\right)\\\
+\mbox{dist}_{\omega_{{\mathcal{A}},0}^{+}}(\varphi^{0}(t_{0};\tau_{0},u_{j}))<\varepsilon_{0}+\varepsilon_{0}+\varepsilon_{0}=3\varepsilon_{0},$

a contradiction to (2.8). The proof is done. ∎

##  3\. Integrodifference equations under discretizations

In this Section we apply our abstract perturbation theory for general
difference equations $(\Delta_{0})$ to nonautonomous integrodifference
equations. They are recursions of the form

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

where $\Omega\subset{\mathbb{R}}^{\kappa}$ is assumed to be nonempty and
compact. As state space for ($I_{0}$) we consider an ambient subset of the
${\mathbb{R}}^{d}$-valued continuous functions over $\Omega$, abbreviated by
$C_{d}$ and equipped with the maximum norm
$\left\|u\right\|:=\max_{x\in\Omega}\left|u(x)\right|$.

###  3.1. Basics and collocation

Throughout we suppose the following standing assumption in order to obtain
well-definedness for ($I_{0}$):

######  Hypothesis.

Suppose for all $t\in{\mathbb{I}}^{\prime}$ that a subset
$Z_{t}\subseteq{\mathbb{R}}^{d}$ is nonempty and closed, while a kernel
function $f_{t}:\Omega\times\Omega\times Z_{t}\to{\mathbb{R}}^{d}$ is
continuous and satisfies

  * $(H_{1})$

there exist $\alpha_{t},\beta_{t}:\Omega^{2}\to{\mathbb{R}}_{+}$ measurable in
the second argument with

$\displaystyle a_{t}$
$\displaystyle:=\sup_{x\in\Omega}\int_{\Omega}\alpha_{t}(x,y)\,{\mathrm{d}}y<\infty,$
$\displaystyle b_{t}$
$\displaystyle:=\sup_{x\in\Omega}\int_{\Omega}\beta_{t}(x,y)\,{\mathrm{d}}y<\infty$

and

$\left|f_{t}(x,y,z)\right|\leq\beta_{t}(x,y)+\alpha_{t}(x,y)|z|\quad\text{for
all }x,y\in\Omega,\,z\in Z_{t},$ (3.1)

  * $(H_{2})$

$\int_{\Omega}f_{t}(x,y,u(y))\,{\mathrm{d}}y\in Z_{t+1}$ for all $x\in\Omega$
and $u\in U_{t}$ with

$U_{t}:=\left\\{u\in C_{d}:\,u(x)\in Z_{t}\text{ for all
}x\in\Omega\right\\}.$

The continuity of the kernel functions $f_{t}$ guarantees that the Urysohn
operators

$\displaystyle{\mathscr{F}}_{t}:U_{t}$ $\displaystyle\to C_{d},$
$\displaystyle{\mathscr{F}}_{t}(u)$
$\displaystyle:=\int_{\Omega}f_{t}(\cdot,y,u(y))\,{\mathrm{d}}y$ (3.2)

are completely continuous on the closed sets $U_{t}$ (cf. [31, 35]).
Furthermore, Hypothesis $(H_{2})$ ensures the inclusion
${\mathscr{F}}_{t}(U_{t})\subseteq U_{t+1}$ for all
$t\in{\mathbb{I}}^{\prime}$. In particular, ($I_{0}$) is a special case of
$(\Delta_{0})$ in the metric space $X=C_{d}$, $d(u,v)=\left\|u-v\right\|$.

In order to simulate IDEs ($I_{0}$) on a computer, finite-dimensional
approximations of their right-hand sides (3.2) and of their state space
$C_{d}$ are due. For this purpose, choose linearly independent functions
$\phi_{1},\ldots,\phi_{d_{n}}\in C_{1}$ yielding an _ansatz space_
$X_{n}:=\operatorname{span}\left\\{\phi_{1},\ldots,\phi_{d_{n}}\right\\}$ of
finite dimension $d_{n}$. With a projector $\pi_{n}\in L(C_{1})$ onto the
space $X_{n}$ it is clear that

$\displaystyle\Pi_{n}$ $\displaystyle\in L(C_{d}),$
$\displaystyle\Pi_{n}(u_{j})_{j=1}^{d}$
$\displaystyle:=(\pi_{n}u_{j})_{j=1}^{d}$

is a projector onto the Cartesian product $X_{n}^{d}$. Supplementing this, for
convenience we define $\Pi_{0}:=\operatorname{id}_{C_{d}}$. In the following,
we impose the following stability assumption (cf. [10, p. 50, Def. 4.8]):

######  Hypothesis.

  * $(H_{3})$

The projections $\Pi_{n}$ are uniformly bounded, i.e.

$p:=\sup_{n\in{\mathbb{N}}}\left\|\Pi_{n}\right\|\in[1,\infty),$ (3.3)

  * $(H_{4})$

$\Pi_{n}U_{t}\subseteq U_{t}$ for all $n\in{\mathbb{N}}$ and
$t\in{\mathbb{I}}^{\prime}$.

A _collocation method_ is a spatial discretization of ($I_{0}$) of the form

$\displaystyle u_{t+1}$ $\displaystyle={\mathscr{F}}_{t}^{n}(u_{t}),$
$\displaystyle{\mathscr{F}}^{n}_{t}$
$\displaystyle:=\Pi_{n}{\mathscr{F}}_{t}:U_{t}\to X_{n}^{d}.$ ($I_{n}$)

Thanks to $(H_{4})$ these difference equations are well-defined and fit into
the general framework of ($\Delta_{n}$), $n\in{\mathbb{N}}$. Based on
Hypothesis $(H_{1})$ we next identify absorbing sets for IDEs ($I_{0}$) and
their spatial discretizations.

######  Proposition 3.1 (Absorbing sets for ($I_{n}$)).

Let $\rho>1$ and suppose $(H_{1}$–$H_{4})$ hold.

  1. (a)

If ${\mathbb{I}}$ is unbounded below, and for all $\tau\in{\mathbb{I}}$ one
has

$\displaystyle\lim_{s\to\infty}\prod_{l=\tau-s}^{\tau-1}pa_{l}$
$\displaystyle=0,$ $\displaystyle R_{\tau}$
$\displaystyle:=p\sum_{l_{1}=-\infty}^{\tau-1}b_{l_{1}}\prod_{l_{2}=l_{1}+1}^{\tau-1}pa_{l_{2}}\in(0,\infty)$
(3.4)

and $\sup_{\tau\in{\mathbb{I}}}R_{\tau}<\infty$, then
${\mathcal{A}}:=\left\\{(\tau,u)\in{\mathcal{U}}:\,\left\|u\right\|\leq\rho
R_{\tau}\right\\}$ is pullback absorbing for each collocation discretization
($I_{n}$), $n\in{\mathbb{N}}_{0}$.

  2. (b)

If ${\mathbb{I}}$ is unbounded above, and for all $\tau\in{\mathbb{I}}$ one
has

$\displaystyle\lim_{s\to\infty}\prod_{l=\tau}^{\tau+s-1}pa_{l}$
$\displaystyle=0,$ $\displaystyle R_{\tau}$
$\displaystyle:=p\lim_{t\to\infty}\sum_{l_{1}=\tau}^{t-1}b_{l_{1}}\prod_{l_{2}=l_{1}+1}^{t-1}pa_{l_{2}}\in(0,\infty)$
(3.5)

and $\sup_{\tau\in{\mathbb{I}}}R_{\tau}<\infty$, then
${\mathcal{A}}:=\left\\{(\tau,u)\in{\mathcal{U}}:\,\left\|u\right\|\leq\rho
R_{\tau}\right\\}$ is forward absorbing for each collocation discretization
($I_{n}$), $n\in{\mathbb{N}}_{0}$.

######  Remark 3.2.

(1) When dealing with the initial equation ($I_{0}$), i.e. in case $n=0$, one
can choose $p=1$.

(2) If the reals $a_{t}=:a$, $b_{t}=:b$ obtained from the linear growth
assumption (3.1) are constant in time, then the condition
$a\in[0,\tfrac{1}{p})$ implies that both the pullback and forward absorbing
sets have constant fibers

${\mathcal{A}}=\left\\{(t,u)\in{\mathcal{U}}:\,\left\|u\right\|\leq\tfrac{\rho
pb}{1-pa}\right\\}.$

In particular, for constant functions $\alpha_{t}(x,y)\equiv:\alpha$,
$\beta_{t}(x,y)\equiv:\beta$ in (3.1), then $a=\alpha\mu_{\kappa}(\Omega)$,
$b=\beta\mu_{\kappa}(\Omega)$, where $\mu_{\kappa}(\Omega)>0$ is the Lebesgue
measure of $\Omega\subset{\mathbb{R}}^{\kappa}$.

###### Proof.

Let $t\in{\mathbb{I}}^{\prime}$, $u\in U_{t}$ and $n\in{\mathbb{N}}_{0}$. For
every $x\in\Omega$ we obtain

$\displaystyle\left|{\mathscr{F}}_{t}^{n}(u)(x)\right|$
$\displaystyle\stackrel{{\scriptstyle\eqref{condstab}}}{{\leq}}$
$\displaystyle
p\left|{\mathscr{F}}_{t}(u)(x)\right|\stackrel{{\scriptstyle\eqref{ide0}}}{{\leq}}p\int_{\Omega}\left|f_{t}(x,y,u(y))\right|\,{\mathrm{d}}y$
$\displaystyle\stackrel{{\scriptstyle\eqref{assumpK0}}}{{\leq}}$
$\displaystyle
p\int_{\Omega}\beta_{t}(x,y)\,{\mathrm{d}}y+p\int_{\Omega}\alpha_{t}(x,y)\left|u(y)\right|\,{\mathrm{d}}y\leq
pb_{t}+pa_{t}\left\|u\right\|$

and passing to the supremum over $x\in\Omega$ yields
$\left\|{\mathscr{F}}_{t}(u)\right\|\leq pb_{t}+pa_{t}\left\|u\right\|$. Then
both assertions (a) and (b) for ($I_{n}$) result from [18, Prop. 3.2 resp.
4.16], where the condition $\sup_{\tau\in{\mathbb{I}}}R_{\tau}<\infty$
guarantees boundedness of ${\mathcal{A}}$. ∎

######  Hypothesis.

  * $(H_{5})$

For every $t\in{\mathbb{I}}^{\prime}$ and $r>0$, there exists a function
$\lambda_{t}:\Omega\times\Omega\to{\mathbb{R}}_{+}$ measurable in the second
argument with

$\left|f_{t}(x,y,z)-f_{t}(x,y,\bar{z})\right|\leq\lambda_{t}(x,y)\left|z-\bar{z}\right|\quad\text{for
all }z,\bar{z}\in Z_{t}\cap\bar{B}_{r}(0)$ (3.6)

and $x,y\in\Omega$, where
$\ell_{t}(r):=\sup_{x\in\Omega}\int_{\Omega}\lambda_{t}(x,y)\,{\mathrm{d}}y<\infty$.

######  Lemma 3.3.

Suppose $(H_{1}$–$H_{3})$ and $(H_{5})$ hold. For each
$t\in{\mathbb{I}}^{\prime}$ and $r>0$ one has the Lipschitz estimate

$\left\|{\mathscr{F}}_{t}^{n}(u)-{\mathscr{F}}_{t}^{n}(\bar{u})\right\|\leq
p\ell_{t}(r)\left\|u-\bar{u}\right\|\quad\text{for all
}n\in{\mathbb{N}}_{0},\,u,\bar{u}\in U_{t}\cap\bar{B}_{r}(0).$ (3.7)

###### Proof.

Using (3.3) the claim results from [35, Cor. B.6]. ∎

The local discretization error in collocation discretizations ($I_{n}$)
depends on the smoothness of the image functions
${\mathscr{F}}_{t}(u):\Omega\to{\mathbb{R}}^{d}$. Thereto, given a subspace
$X(\Omega)\subset C_{d}$ one denotes a right-hand side (3.2) as
$X(\Omega)$-_smoothing_ , if ${\mathscr{F}}_{t}(u)\in X(\Omega)$ for all
$t\in{\mathbb{I}}^{\prime}$ and $u\in U_{t}$. For instance, convolution
kernels yield that ${\mathscr{F}}_{t}(u)$ have a higher order smoothness than
the $f_{t}$ itself (cf. [9, pp. 50ff, Sect. 3.4]).

Note that for collocation methods based on polynomial interpolation the
stability condition (3.3) fails even with Chebyshev nodes as collocation
points (see [10, p. 52]). Nevertheless, the stability condition (3.3) holds
when working with spline functions as basis of $X_{n}$. For this purpose, we
restrict to $\Omega=[a,b]$, choose a grid

$a=:x_{0}<x_{1}<\ldots<x_{n}:=b$ (3.8)

and abbreviate $\overline{h}_{n}:=\max_{j=0}^{n-1}(x_{j+1}-x_{j})$,
$\underline{h}_{n}:=\min_{j=0}^{n-1}(x_{j+1}-x_{j})$. Beyond that assume that
the grid (3.8) is extended to a strictly increasing sequence
$(x_{j})_{j\in{\mathbb{Z}}}$.

Let $\beta_{j}^{l}:{\mathbb{R}}\to{\mathbb{R}}_{+}$, $j\in{\mathbb{Z}}$,
denote the _$B$ -splines_ of degree $l\in{\mathbb{N}}$ introduced e.g. in [14,
pp. 242ff] and consider the subspaces

$X_{n}:=\operatorname{span}\left\\{\beta_{-l}^{l},\ldots,\beta_{n-1}^{l}\right\\}\subset
C[a,b].$

######  Example 3.4 (linear splines).

The $B$-splines $\beta_{j}^{1}:{\mathbb{R}}\to[0,1]$, $-1\leq j\leq n$ (the
hat functions), form a basis of the ansatz space consisting of piecewise
linear functions $X_{n}\subset C[a,b]$ having dimension $d_{n}=n+1$. The
spline projections

$\pi_{n}u:=\sum_{j=0}^{n}u(x_{j})\beta_{j-1}^{1}$

fulfill to the interpolation conditions $(\pi_{n}u)(x_{j})=u(x_{j})$, $0\leq
j\leq n$, satisfy $\left\|\pi_{n}\right\|=1$ (see [2, p. 450]), hence $p=1$ in
(3.3), and the error estimate [14, p. 275]

$\displaystyle\left\|u-\pi_{n}u\right\|\leq\tfrac{\overline{h}_{n}^{2}}{8}\left\|u^{\prime\prime}\right\|\quad\text{for
all }n\in{\mathbb{N}},\,u\in C^{2}[a,b].$

Consequently, if ($I_{0}$) is $C^{2}[a,b]^{d}$-smoothing, then an application
to ($I_{n}$) yields

$\displaystyle
e_{t}^{n}(u)\leq\tfrac{\overline{h}_{n}^{2}}{8}\left\|{\mathscr{F}}_{t}(u)^{\prime\prime}\right\|\quad\text{for
all }t\in{\mathbb{I}}^{\prime},\,u\in U_{t}$ (3.9)

for the local discretization error. The same quadratic convergence also holds,
when multidimensional piecewise linear interpolation is used on domains
$\Omega\subset{\mathbb{R}}^{\kappa}$ (see [35, Sect. 3.1.3]).

######  Example 3.5 (quadratic splines).

Given a grid (3.8) we introduce the $n+2$ collocation points

$\displaystyle\xi_{0}$ $\displaystyle:=a,$ $\displaystyle\xi_{j}$
$\displaystyle:=\tfrac{x_{j}+x_{j-1}}{2}\quad\text{for all }0\leq j\leq n,$
$\displaystyle\xi_{n+1}=:$ $\displaystyle b.$

Then the quadratic splines $\beta_{j}^{2}:{\mathbb{R}}\to[0,1]$, $-2\leq j\leq
n$, yield an ansatz space $X_{n}\subset C^{1}[a,b]$ of dimension $d_{n}=n+2$,
where the interpolation conditions $(\pi_{n}u)(\xi_{j})=u(\xi_{j})$, $0\leq
j\leq n+1$, result in a projection $\pi_{n}:C[a,b]\to X_{n}$ satisfying
$\left\|\pi_{n}\right\|\leq 2$ (cf. [20, Cor. 3.2]), thus $p=2$ in (3.3) and

$\left\|u-\pi_{n}u\right\|\leq\tfrac{\overline{h}_{n}^{3}}{24}\bigl{\|}u^{(3)}\bigr{\|}\quad\text{for
all }n\in{\mathbb{N}},\,u\in C^{3}[a,b]$

as interpolation error (see [8, Thm. 6]). Hence, for
$C^{3}[a,b]^{d}$-smoothing equations ($I_{0}$) the local discretization error
becomes

$\displaystyle
e_{t}^{n}(u)\leq\tfrac{\overline{h}_{n}^{3}}{24}\bigl{\|}{\mathscr{F}}_{t}(u)^{(3)}\bigr{\|}\quad\text{for
all }t\in{\mathbb{I}}^{\prime},\,u\in U_{t}.$

######  Example 3.6 (cubic splines).

The cubic splines $\beta_{j}^{3}:{\mathbb{R}}\to[0,1]$, $-3\leq j\leq n$,
establish an ansatz space $X_{n}\subset C^{2}[a,b]$ of dimension $d_{n}=n+3$.
Supplementing the interpolation conditions $(\pi_{n}u)(x_{j})=u(x_{j})$,
$0\leq j\leq n$, by Hermite boundary conditions

$\displaystyle(\pi_{n}u)^{\prime\prime}(a)$
$\displaystyle=u^{\prime\prime}(a),$
$\displaystyle(\pi_{n}u)^{\prime\prime}(b)$
$\displaystyle=u^{\prime\prime}(b)$

leads to a projection $\pi_{n}:C[a,b]\to X_{n}$ with
$\left\|\pi_{n}\right\|\leq
1+\tfrac{3}{2}\tfrac{\overline{h}_{n}}{\underline{h}_{n}}$ (cf. [32, Thm.
3.1]). Whence,
$p:=1+\tfrac{3}{2}\sup_{n\in{\mathbb{N}}}\tfrac{\overline{h}_{n}}{\underline{h}_{n}}<\infty$
implies stability. Furthermore, the interpolation error becomes

$\displaystyle\left\|u-\pi_{n}u\right\|\leq\tfrac{5\overline{h}_{n}^{4}}{384}\bigl{\|}u^{(4)}\bigr{\|}\quad\text{for
all }n\in{\mathbb{N}},\,u\in C^{4}[a,b]$

(see [13, Thm. 4]). In case ($I_{0}$) is $C^{4}[a,b]$-smoothing one obtains

$\displaystyle
e_{t}^{n}(u)\leq\tfrac{5\overline{h}_{n}^{4}}{384}\bigl{\|}{\mathscr{F}}_{t}(u)^{(4)}\bigr{\|}\quad\text{for
all }t\in{\mathbb{I}}^{\prime},\,u\in U_{t}$

as local discretization error.

In conclusion, collocation methods ($I_{n}$), $n\in{\mathbb{N}}$, based on
splines having the order $l\in\left\\{1,2,3\right\\}$ are convergent, provided

  * •

$\lim_{n\to\infty}\overline{h}_{n}=0$ and

  * •

the derivatives ${\mathscr{F}}_{t}(\cdot)^{(l+1)}:U_{t}\to C[a,b]^{d}$ are
bounded functions, that is, they map bounded subsets of $U_{t}$ to bounded
sets uniformly in $t\in{\mathbb{I}}^{\prime}$.

###  3.2. Pullback attractors

Let ${\mathbb{I}}$ be unbounded below. An immediate consequence of Prop. 3.1
is

######  Corollary 3.7.

The pullback absorbing set ${\mathcal{A}}\subseteq{\mathcal{U}}$ from Prop.
3.1(a) is positively invariant for every ($I_{n}$), $n\in{\mathbb{N}}_{0}$.

###### Proof.

Let $n\in{\mathbb{N}}_{0}$, $\tau\in{\mathbb{I}}^{\prime}$ and $u\in
u\cap\bar{B}_{\rho R_{\tau}}(0)$. Thanks to $(H_{2})$ and $(H_{4})$ it remains
to establish the inclusion ${\mathscr{F}}_{\tau}^{n}(u)\subseteq\bar{B}_{\rho
R_{\tau+1}}(0)$. Thereto, we obtain as in the proof of Prop. 3.1 that

$\left\|{\mathscr{F}}_{\tau}^{n}(u)\right\|\leq
pb_{\tau}+pa_{\tau}\left\|u\right\|\leq pb_{\tau}+\rho pa_{\tau}R_{\tau}.$

In combination with

$\displaystyle R_{\tau+1}$
$\displaystyle\stackrel{{\scriptstyle\eqref{propabc1}}}{{=}}p\sum_{l_{1}=-\infty}^{\tau}b_{l_{1}}\prod_{l_{2}=l_{1}+1}^{\tau}pa_{l_{2}}=pa_{\tau}p\sum_{l_{1}=-\infty}^{\tau-1}b_{l_{1}}\prod_{l_{2}=l_{1}+1}^{\tau-1}pa_{l_{2}}+pb_{\tau}\prod_{l_{2}=\tau+1}^{\tau}pa_{l_{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{propabc1}}}{{=}}pa_{\tau}R_{\tau}+pb_{\tau}$

this implies $\left\|{\mathscr{F}}_{\tau}^{n}(u)\right\|\leq\rho
R_{\tau+1}+p(1-\rho)b_{\tau}\leq\rho R_{\tau+1}$. This is the assertion. ∎

######  Theorem 3.8 (Pullback attractors for ($I_{n}$)).

Let $(H_{1}$–$H_{4})$ and the limit relations (3.4) hold for all
$\tau\in{\mathbb{I}}$. If $\sup_{\tau\in{\mathbb{I}}}R_{\tau}<\infty$, then
every ($I_{n}$), $n\in{\mathbb{N}}_{0}$, has a pullback attractor
${\mathcal{A}}_{n}^{\ast}\subseteq{\mathbb{I}}\times C_{d}$. If
${\mathscr{F}}_{t}:U_{t}\to U_{t+1}$ maps bounded subsets of $U_{t}$ into
bounded sets uniformly in $t\in{\mathbb{I}}^{\prime}$ and moreover

  * (i)

$(H_{5})$ is satisfied with
$\ell(r):=\sup_{t\in{\mathbb{I}}^{\prime}}\ell_{t}(r)<\infty$ for all $r>0$,

  * (ii)

the collocation discretizations ($I_{n}$), $n\in{\mathbb{N}}$, are convergent

hold, then $\lim_{n\to\infty}\max_{t\in
I}\operatorname{dist}\left({\mathcal{A}}_{n}^{\ast}(t),{\mathcal{A}}_{0}^{\ast}(t)\right)=0$
on each bounded $I\subseteq{\mathbb{I}}$.

###### Proof.

Let $\rho>1$. We apply Thm. 2.1 and 2.6 to the difference equations ($I_{n}$),
$n\in{\mathbb{N}}_{0}$, in the space $X=C_{d}$ equipped with the metric
$d(u,v):=\left\|u-v\right\|$.

By Prop. 3.1(a) the nonautonomous set
${\mathcal{A}}:=\left\\{(\tau,u)\in{\mathcal{U}}:\,\left\|u\right\|\leq\rho
R_{\tau}\right\\}$ is pullback absorbing for each ($I_{n}$),
$n\in{\mathbb{N}}_{0}$. Furthermore, since the right-hand sides
${\mathscr{F}}_{t}:U_{t}\to C_{d}$ of ($I_{0}$) are completely continuous,
also their discretizations ${\mathscr{F}}_{t}^{n}=\Pi_{n}{\mathscr{F}}_{t}$
share this property for all $t\in{\mathbb{I}}^{\prime}$ and
$n\in{\mathbb{N}}$. Therefore, ($I_{n}$) is pullback asymptotically compact
(see [33, p. 13, Cor. 1.2.22]) and Thm. 2.1 guarantees that every IDE
($I_{n}$), $n\in{\mathbb{N}}_{0}$, has a pullback attractor
${\mathcal{A}}_{n}^{\ast}$ with the claimed properties.

We verify the assumptions of Thm. 2.6:

_ad (i)_ : By the Arzelà-Ascoli theorem [15, p. 44, Thm. 3.3] we have to show
that the union $\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(t)$ is
bounded and equicontinuous for all $t\in{\mathbb{I}}$. Since boundedness is a
consequence of the inclusion
$\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(t)\subseteq{\mathcal{A}}(t)$,
it remains to establish the equicontinuity. Let $n\in{\mathbb{N}}_{0}$. Since
pullback attractors ${\mathcal{A}}_{n}^{\ast}$ are invariant, one has

${\mathcal{A}}_{n}^{\ast}(t+1)={\mathscr{F}}_{t}^{n}({\mathcal{A}}_{n}^{\ast}(t))\subseteq{\mathscr{F}}_{t}^{n}({\mathcal{A}}(t))\quad\text{for
all }t\in{\mathbb{I}}^{\prime}$

and consequently,
$\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(t+1)\subseteq\bigcup_{n\in{\mathbb{N}}_{0}}{\mathscr{F}}_{t}^{n}({\mathcal{A}}(t))$
for all $t\in{\mathbb{I}}^{\prime}$. Because the subsets of equicontinuous
sets are equicontinuous again, it suffices to establish equicontinuity of the
union $\bigcup_{n\in{\mathbb{N}}_{0}}{\mathscr{F}}_{t}^{n}({\mathcal{A}}(t))$
for any $t\in{\mathbb{I}}^{\prime}$.

Thereto, let $\varepsilon>0$. Because ($I_{n}$), $n\in{\mathbb{N}}$, is
assumed to be convergent, there exists an $N\in{\mathbb{N}}$ such that

$e^{n}_{t}(u)=\|{\mathscr{F}}^{n}_{t}(u)-{\mathscr{F}}^{0}_{t}(u)\|<\tfrac{\varepsilon}{4}\quad\text{for
all }u\in{\mathcal{A}}(t),\,n\geq N.$

On the other hand, the image ${\mathscr{F}}^{0}_{t}({\mathcal{A}}(t))$ due to
[15, p. 43, Prop. 3.1] is even uniformly equicontinuous, i.e., there exists a
$\delta>0$ such that

$|x-y|<\delta\quad\Rightarrow\quad|{\mathscr{F}}^{0}_{t}(u)(x)-{\mathscr{F}}^{0}_{t}(u)(y)|<\tfrac{\varepsilon}{4}\quad\text{for
all }x,y\in\Omega.$

Combining both inequalities above, in case $|x-y|<\delta$ it results from the
triangle inequality that

$\displaystyle|{\mathscr{F}}^{n}_{t}(u)(x)-{\mathscr{F}}^{n}_{t}(u)(y)|\leq$
$\displaystyle\,|{\mathscr{F}}^{n}_{t}(u)(x)-{\mathscr{F}}^{0}_{t}(u)(x)|+|{\mathscr{F}}^{0}_{t}(u)(x)-{\mathscr{F}}^{0}_{t}(u)(y)|$
$\displaystyle\leq$
$\displaystyle\,\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{4}<\varepsilon\quad\text{for
all }u\in{\mathcal{A}}(t),\,n\geq N.$

This yields that ${\mathscr{F}}_{t}^{0}({\mathcal{A}}(t))\cup\bigcup_{n\geq
N}{\mathscr{F}}_{t}^{n}({\mathcal{A}}(t))$ is equicontinuous and additionally
including the finitely many relatively compact sets
${\mathscr{F}}_{t}^{n}({\mathcal{A}}(t))$, $1\leq n<N$, preserves
equicontinuity. As a result, every
$\bigcup_{n\in{\mathbb{N}}_{0}}{\mathcal{A}}_{n}^{\ast}(t)$,
$t\in{\mathbb{I}}$, is relatively compact.

_ad (ii)_ : Due to $\bigcup_{n\in{\mathbb{N}}_{0}}\bigcup_{s\leq
t}{\mathcal{A}}_{n}^{\ast}(s)\subseteq{\mathcal{A}}(t)$ for all
$t\in{\mathbb{I}}$ we obtain the claim.

_ad (iii)_ : In order to apply Prop. 2.5 we note our assumption $(H_{5})$ and
Lemma 3.3 imply that for each $r>0$ there exists a $\ell(r)\geq 0$ such that

$\left\|{\mathscr{F}}_{t}^{n}(u)-{\mathscr{F}}_{t}^{n}(v)\right\|\leq
p\ell(r)\left\|u-v\right\|\quad\text{for all
}n\in{\mathbb{N}},\,t\in{\mathbb{I}}^{\prime}$

and $u,v\in U_{t}\cap\bar{B}_{r}(0)$. Hence, the collocation discretizations
($I_{n}$) fulfill (2.5). Now let $\varepsilon>0$,
$\tau\in{\mathbb{I}}^{\prime}$, $S\in{\mathbb{N}}$ and $K\subseteq C_{d}$ be
compact. Given $u\in U_{\tau-s}\cap K$ choose a nonautonomous set
${\mathcal{B}}\subseteq{\mathcal{U}}$ such that

$\varphi^{0}(\tau;\tau-s,u)\in{\mathcal{B}}(\tau)\quad\text{for all }0<s\leq
S.$

Because, by assumption, ${\mathscr{F}}_{t}:U_{t}\to U_{t+1}$ maps bounded
subsets of $U_{t}$ into bounded sets uniformly in $t\in{\mathbb{I}}^{\prime}$,
the set ${\mathcal{B}}$ can be chosen to be bounded. Hence, there exists a
$r>0$ such that ${\mathcal{B}}(t)\subseteq\bar{B}_{r}(0)$ for all
$t\in{\mathbb{I}}$. Now choose $\rho>r$ and due to Prop. 2.5 there exists an
$N_{0}\in{\mathbb{N}}$ such that

$\displaystyle\left\|\varphi^{n}(\tau;\tau-s,u)-\varphi^{0}(\tau;\tau-s,u)\right\|\stackrel{{\scriptstyle\eqref{properr3}}}{{\leq}}$
$\displaystyle\Gamma\left(\tfrac{1}{n}\right)\sum_{l_{1}=\tau-s}^{\tau-1}C({\mathcal{B}}(l_{1}))\prod_{l_{2}=l_{1}+1}^{\tau-1}p\ell(\rho)$
$\displaystyle\leq$
$\displaystyle\Gamma\left(\tfrac{1}{n}\right)C(\bar{B}_{r}(0))\sum_{l_{1}=\tau-s}^{\tau-1}\prod_{l_{2}=l_{1}+1}^{\tau-1}p\ell(\rho)$

for all $0<s\leq S$ and $n>N_{0}$. It is easy to see that the right-hand side
in the above estimate does not depend on $\tau\in{\mathbb{I}}^{\prime}$. Since
our convergence assumption (ii) ensures
$\lim_{n\to\infty}\Gamma(\tfrac{1}{n})=0$, there exists an $n_{0}\geq N_{0}$
such that $n>n_{0}$ implies

$\left\|\varphi^{n}(\tau;\tau-s,u)-\varphi^{0}(\tau;\tau-s,u)\right\|<\varepsilon\quad\text{for
all }u\in U_{\tau-s}\cap K,\,0<s\leq S.$

This finally verifies Thm. 2.6(iii). ∎

###  3.3. Forward dynamics

Let all nonempty, closed $Z:\equiv Z_{t}$ be time-invariant yielding the
constant domains $U:\equiv U_{t}$ on a discrete interval ${\mathbb{I}}$
unbounded above.

The forward dynamics counterpart to Cor. 3.7 has a slightly different
structure:

######  Corollary 3.9.

The forward absorbing set ${\mathcal{A}}\subseteq{\mathcal{U}}$ from Prop.
3.1(b) has constant fibers, i.e. there exists a $R>0$ with $R=R_{\tau}$ for
all $\tau\in{\mathbb{I}}$. Moreover, in case

$pb_{\tau}\leq\rho R(1-pa_{\tau})\quad\text{for all }\tau\in{\mathbb{I}}$
(3.10)

the nonautonomous set ${\mathcal{A}}$ is positively invariant for every
($I_{n}$), $n\in{\mathbb{N}}_{0}$.

###### Proof.

Above all, for $\tau,t\in{\mathbb{I}}$ we define
$r_{\tau}(t):=p\sum_{l_{1}=\tau}^{t-1}b_{l_{1}}\prod_{l_{2}=l_{1}+1}^{t-1}pa_{l_{2}}$
and passing to the limit $t\to\infty$ in

$r_{\tau}(t)=p\sum_{l_{1}=\tau+1}^{t-1}b_{l_{1}}\prod_{l_{2}=l_{1}+1}^{t-1}pa_{l_{2}}+pb_{\tau}\prod_{l_{2}=\tau+1}^{t-1}pa_{l_{2}}=r_{\tau+1}(t)+pb_{\tau}\prod_{l_{2}=\tau+1}^{t-1}pa_{l_{2}}$

implies that $R_{\tau}=R_{\tau+1}$ due to (3.5). With $R:=R_{\tau}$ and $u\in
u\cap\bar{B}_{\rho R}(0)$ we obtain as in the proof of Cor. 3.7 that
$\left\|{\mathscr{F}}_{\tau}^{n}(u)\right\|\leq pb_{\tau}+\rho pa_{\tau}R$ for
all $n\in{\mathbb{N}}_{0}$ and therefore (3.10) yields the assertion. ∎

######  Theorem 3.10 (Forward limit sets for ($I_{n}$)).

Suppose $(H_{1}$–$H_{4})$, the limit relations (3.5) and (3.10) hold for all
$\tau\in{\mathbb{I}}$ yielding the nonautonomous set
${\mathcal{A}}\subseteq{\mathcal{U}}$ from Prop. 3.1(b). If each ($I_{n}$) is
strongly ${\mathcal{A}}$-asymptotically compact with a compact $K_{n}\subseteq
U$, $n\in{\mathbb{N}}_{0}$, then its forward limit set
$\omega_{{\mathcal{A}},n}^{+}\subseteq K_{n}$ is nonempty, compact and forward
attracts ${\mathcal{A}}$. If moreover $K:=\bigcup_{n\in{\mathbb{N}}_{0}}K_{n}$
is bounded, ${\mathscr{F}}_{t}:U\to U$ maps bounded subsets of $U$ into
bounded sets uniformly in $t\in{\mathbb{I}}$ and

  * (i)

$\omega_{{\mathcal{A}},0}^{+}$ attracts $K$ uniformly in
$\tau\in{\mathbb{I}}$, i.e.,

$\lim_{s\to\infty}\sup_{\tau\in{\mathbb{I}}}\operatorname{dist}\left(\varphi^{0}(\tau+s;\tau,K),\omega_{{\mathcal{A}},0}^{+}\right)=0,$

  * (ii)

for every sequence $\left((s_{k},\tau_{k})\right)_{k\in{\mathbb{N}}}$ in
${\mathbb{N}}\times{\mathbb{I}}$ with $\tau_{k}\to\infty$ as $k\to\infty$ one
has

$\lim_{k\to\infty}\operatorname{dist}\left(\varphi^{n}(\tau_{k}+s_{k};\tau_{k},K_{n}),K_{n}\right)=0\quad\text{for
all }n\in{\mathbb{N}}_{0},$

  * (iii)

$(H_{5})$ is satisfied with
$\ell(r):=\sup_{t\in{\mathbb{I}}}\ell_{t}(r)<\infty$ for all $r>0$,

  * (iv)

the collocation discretizations ($I_{n}$), $n\in{\mathbb{N}}$, are convergent

hold, then
$\lim_{n\to\infty}\operatorname{dist}\left(\omega_{{\mathcal{A}},n}^{+},\omega_{{\mathcal{A}},0}^{+}\right)=0$.

###### Proof.

We aim to apply Thm. 2.7 to ($I_{n}$), $n\in{\mathbb{N}}_{0}$. Above all, it
results from the above Cor. 3.9 that ${\mathcal{A}}$ is forward absorbing and
positively invariant for each ($I_{n}$). Then Thm. 2.7 yields the existence of
forward limit sets $\omega_{{\mathcal{A}},n}^{+}$ with the claimed properties
for all $n\in{\mathbb{N}}_{0}$.

It remains to show that the assumptions (iii) and (iv) of Thm. 2.7 hold.

_ad (iii)_ : Choose $r>0$ so large that ${\mathcal{A}}(\tau)\cup
K\subseteq\bar{B}_{r}(0)$ holds for all $\tau\in{\mathbb{I}}$, which is
possible due to the boundedness of ${\mathcal{A}}$ and $K$. By assumption, it
results inductively from Lemma 3.3 that

$\left\|\varphi^{n}(\tau+s;\tau,u)-\varphi^{n}(\tau+s;\tau,v)\right\|\leq(p\ell(r))^{s}\left\|u-v\right\|\quad\text{for
all }n\in{\mathbb{N}}_{0},\,s\in{\mathbb{N}}$

and $\tau\in{\mathbb{I}}$, $u,v\in{\mathcal{A}}(\tau)\cup K_{n}$. Let
$\varepsilon>0$ and $S\in{\mathbb{N}}$ be arbitrarily given. If we choose
$\delta:=\min_{0<s\leq S}\tfrac{\varepsilon}{\ell(r)^{s}}>0$, then
$\left\|u-v\right\|<\delta$ readily leads to the estimate
$\left\|\varphi^{n}(\tau+s;\tau,u)-\varphi^{n}(\tau+s;\tau,v)\right\|<\varepsilon$
for $0<s\leq S$, which implies (iii).

_ad (iv)_ : Let $\varepsilon>0$, $S\in{\mathbb{N}}$ and $\tau\in{\mathbb{I}}$.
As in the proof of Thm. 3.8 one shows that the Lipschitz estimate (2.4) holds
in our present setting with the (local) Lipschitz constant $p\ell(r)$. For
$u\in{\mathcal{A}}(\tau)\cup K$ choose a bounded nonautonomous set
${\mathcal{B}}\subseteq{\mathcal{U}}$ so large that
$\varphi^{0}(t;\tau,u)\in{\mathcal{B}}(t)$ for all $\tau\leq t\leq\tau+S$;
since ${\mathscr{F}}_{t}$ is assumed to be bounded uniformly in
$t\in{\mathbb{I}}$ the nonautonomous set ${\mathcal{B}}$ can be chosen to be
independent of $\tau\in{\mathbb{I}}$. Furthermore, choose $r>0$ so large that
${\mathcal{B}}(t)\subseteq\bar{B}_{r}(0)$ for all $\tau\leq t\leq\tau+S$. For
$\rho>r$, then Prop. 2.5 implies that there is an $N_{0}\in{\mathbb{N}}$ with

$\displaystyle\left\|\varphi^{n}(\tau+s;\tau,u)-\varphi^{0}(\tau+s;\tau,u)\right\|\stackrel{{\scriptstyle\eqref{properr3}}}{{\leq}}$
$\displaystyle\Gamma\left(\tfrac{1}{n}\right)\sum_{l_{1}=\tau}^{\tau+s-1}C({\mathcal{B}}(l_{1}))\prod_{l_{2}=l_{1}+1}^{\tau+s-1}\ell(\rho)$
$\displaystyle\leq$
$\displaystyle\Gamma\left(\tfrac{1}{n}\right)C(\bar{B}_{r}(0))\sum_{l_{1}=\tau}^{\tau+s-1}\prod_{l_{2}=l_{1}+1}^{\tau+s-1}\ell(\rho)$

for all $0<s\leq S$ and $n>N_{0}$. Note that the right-hand side of this
inequality does not depend on $\tau\in{\mathbb{I}}$. Consequently, our
convergence condition $\lim_{n\to\infty}\Gamma(\tfrac{1}{n})=0$ implies that
there exists an $n_{0}\geq N_{0}$ such that $n>n_{0}$ implies

$\left\|\varphi^{n}(\tau+s;\tau,u)-\varphi^{0}(\tau+s;\tau,u)\right\|<\varepsilon\quad\text{for
all }u\in{\mathcal{A}}(\tau)\cap K,\,0<s\leq S$

and therefore Thm. 2.7(iii) is verified. ∎

######  Theorem 3.11 (Persistence of forward attractors for ($I_{n}$)).

Let ${\mathbb{I}}={\mathbb{Z}}$. Suppose $(H_{1}$–$H_{4})$, the limit
relations (3.5) and (3.10) hold for all $\tau\in{\mathbb{Z}}$ yielding the
nonautonomous set ${\mathcal{A}}\subseteq{\mathcal{U}}$ from Prop. 3.1(b). If
each ($I_{n}$) is strongly ${\mathcal{A}}$-asymptotically compact with a
compact $K_{n}\subseteq U$, $n\in{\mathbb{N}}_{0}$, and

  * (i)

for every sequence $\left((s_{k},\tau_{k})\right)_{k\in{\mathbb{N}}}$ in
${\mathbb{N}}\times{\mathbb{Z}}$ with $\tau_{k}\to\infty$ as $k\to\infty$ one
has

$\lim_{k\to\infty}\operatorname{dist}\left(\varphi^{n}(\tau_{k}+s_{k};\tau_{k},K_{n}),K_{n}\right)=0\quad\text{for
all }n\in{\mathbb{N}}_{0},$

  * (ii)

$(H_{5})$ is satisfied with
$\ell(r):=\sup_{t\in{\mathbb{I}}}\ell_{t}(r)<\infty$ for all $r>0$,

  * (iii)

the forward limit sets $\omega_{{\mathcal{A}},n}^{+}$ of ($I_{n}$) satisfy

$\omega_{{\mathcal{A}},n}^{+}=\limsup_{t\to\infty}{\mathcal{A}}_{n}^{\ast}(t)\quad\text{for
all }n\in{\mathbb{N}}_{0}$

with ${\mathcal{A}}_{n}^{\star}(t):=\bigcap_{0\leq
s}\varphi^{n}(t;t-s,{\mathcal{A}}(t-s))$ for all $t\in{\mathbb{Z}}$

hold, then every ($I_{n}$), $n\in{\mathbb{N}}_{0}$, has
${\mathcal{A}}_{n}^{\star}\subseteq{\mathcal{A}}$ as a forward attractor.

###### Proof.

Let $n\in{\mathbb{N}}_{0}$. We verify that ($I_{n}$) satisfies the assumptions
of Thm. 2.3. First of all, Cor. 3.9 ensures that the closed set
${\mathcal{A}}$ is forward absorbing and positively invariant w.r.t.
($I_{n}$). Because each ${\mathscr{F}}_{t}:U\to C_{d}$ is completely
continuous, also the discretizations
${\mathscr{F}}_{t}^{n}=\Pi_{n}{\mathscr{F}}_{t}$ have this property for all
$t\in{\mathbb{Z}}$. Therefore, ($I_{n}$) is pullback asymptotically compact
(see [33, p. 13, Cor. 1.2.22]). Our assumption (i) yields that ($I_{n}$)
fulfills the assumption in Thm. 2.3(i). As in the above proof of Thm. 3.10 one
shows that also Thm. 2.3(ii) holds for ($I_{n}$). Finally, (iii) ensures that
finally the assumption in Thm. 2.3(iii) is satisfied for ($I_{n}$). ∎

##  4\. Illustrations

Let ${\mathbb{I}}$ be a discrete interval and
$\Omega\subset{\mathbb{R}}^{\kappa}$ be a compact habitat. We are discussing
IDEs defined on the (closed) cone

$U=C_{+}(\Omega):=\left\\{u\in C(\Omega):\,0\leq u(x)\text{ for all
}x\in\Omega\right\\}$

of nonnegative continuous functions.

###  4.1. Pullback attractors for generalized Beverton-Holt equations

Consider the scalar nonautonomous IDE

$\displaystyle u_{t+1}$ $\displaystyle={\mathscr{F}}_{t}(u_{t}),$
$\displaystyle{\mathscr{F}}_{t}(u)$
$\displaystyle:=\int_{\Omega}k_{t}(\cdot,y)\frac{\gamma_{t}(y)u(y)}{1+u(y)^{\alpha}}\,{\mathrm{d}}y$
(4.1)

depending on a parameter $\alpha>0$ with continuous kernels
$k_{t}:\Omega^{2}\to{\mathbb{R}}_{+}$ and integrable growth rates
$\gamma_{t}:\Omega\to{\mathbb{R}}_{+}$, $t\in{\mathbb{I}}^{\prime}$.

Figure 1. Shape of generalized Beverton-Holt functions
$g_{\alpha}(z):=\tfrac{z}{1+z^{\alpha}}$ for $\alpha=\tfrac{1}{2}$ (dashed),
$\alpha=1$ (solid) and $\alpha=2$ (dash dotted)

The nonlinearity in this Hammerstein IDE is given by the family of generalized
Beverton-Holt functions $g_{\alpha}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$,
$g_{\alpha}(z):=\tfrac{z}{1+z^{\alpha}}$, whose behavior depends on $\alpha>0$
(see Fig. 1). All these growth functions have in common to be globally
Lipschitz with constant $1$. Consequently, in order to fulfill the assumptions
of Lemma 3.3 we choose $\lambda_{t}(x,y):=k_{t}(x,y)\gamma_{t}(y)$
guaranteeing even a global Lipschitz estimate in (3.7) with

$\ell_{t}:=\sup_{x\in\Omega}\int_{\Omega}k_{t}(x,y)\gamma_{t}(y)\,{\mathrm{d}}y\quad\text{for
all }t\in{\mathbb{I}}^{\prime}.$

In order to establish that the generalized Beverton-Holt equation (4.1) is
dissipative using Prop. 3.1, we suppose that
$(\ell_{t})_{t\in{\mathbb{I}}^{\prime}}$ is bounded and proceed as follows:

  * •

For $\alpha\in(0,1)$ the function $g_{\alpha}$ is unbounded and its tangent in
a point $(\zeta,g_{\alpha}(\zeta))$ for $\zeta>0$ reads as

$z\mapsto\tfrac{1+(1-\alpha)\zeta^{\alpha}}{(1+\zeta^{\alpha})^{2}}(z-\zeta)+\tfrac{\zeta}{1+\zeta^{\alpha}}=\tfrac{\alpha\zeta^{1+\alpha}}{(1+\zeta^{\alpha})^{2}}+\tfrac{1+(1-\alpha)\zeta^{\alpha}}{(1+\zeta^{\alpha})^{2}}z.$

Concerning the estimate (3.1) this allows us to choose

$\displaystyle\alpha_{t}(x,y)$
$\displaystyle:=\tfrac{1+(1-\alpha)\zeta^{\alpha}}{(1+\zeta^{\alpha})^{2}}k_{t}(x,y)\gamma_{t}(y),$
$\displaystyle\beta_{t}(x,y)$
$\displaystyle:=\tfrac{\alpha\zeta^{1+\alpha}}{(1+\zeta^{\alpha})^{2}}k_{t}(x,y)\gamma_{t}(y)$

and consequently

$\displaystyle a_{t}$
$\displaystyle=\tfrac{1+(1-\alpha)\zeta^{\alpha}}{(1+\zeta^{\alpha})^{2}}\ell_{t},$
$\displaystyle b_{t}$
$\displaystyle=\tfrac{\alpha\zeta^{1+\alpha}}{(1+\zeta^{\alpha})^{2}}\ell_{t}.$

If $\zeta>0$ is chosen so large that

$\tfrac{1+(1-\alpha)\zeta^{\alpha}}{(1+\zeta^{\alpha})^{2}}\limsup_{t\to-\infty}\ell_{t}<1\quad\text{on
${\mathbb{I}}$ unbounded below},$

then (4.1) is pullback dissipative, while

$\tfrac{1+(1-\alpha)\zeta^{\alpha}}{(1+\zeta^{\alpha})^{2}}\limsup_{t\to\infty}\ell_{t}<1\quad\text{on
${\mathbb{I}}$ unbounded above}$

yields forward dissipativity of the Beverton-Holt equation (4.1).

  * •

For $\alpha\geq 1$, the function $g_{\alpha}$ is globally bounded with

$\sup_{z\geq
0}g_{\alpha}(z)=\tfrac{1}{\alpha}(\alpha-1)^{1-\tfrac{1}{\alpha}}$

and therefore in (3.1) we can choose

$\displaystyle\alpha_{t}(x,y)$ $\displaystyle:\equiv 0,$
$\displaystyle\beta_{t}(x,y)$
$\displaystyle:=\tfrac{1}{\alpha}(\alpha-1)^{1-\tfrac{1}{\alpha}}k_{t}(x,y)\gamma_{t}(y),$

which results in

$\displaystyle a_{t}$ $\displaystyle\equiv 0,$ $\displaystyle b_{t}$
$\displaystyle=\tfrac{1}{\alpha}(\alpha-1)^{1-\tfrac{1}{\alpha}}\ell_{t}.$

Consequently, the IDE (4.1) is both forward and pullback dissipative.

Figure 2. Sequences of sets containing functions in the fibers
${\mathcal{A}}_{n}^{\ast}(t)$, $0\leq t\leq 10$, for
$\alpha\in\left\\{\tfrac{1}{2},1\right\\}$, respectively from left to right.

Now let ${\mathbb{I}}$ be unbounded below. In any case, we conclude from Thm.
2.1 that the nonautonomous Beverton-Holt equation (4.1) has a pullback
attractor ${\mathcal{A}}_{0}^{\ast}$. In particular, for parameters
$\alpha\in(0,1]$ the growth function $g_{\alpha}$ is strictly increasing and
therefore the right-hand sides of (4.1) are order-preserving. This means that
for all $u,v\in U$ the implication

$u(x)\leq
v(x)\quad\Rightarrow\quad{\mathscr{F}}_{t}^{n}(u)(x)\leq{\mathscr{F}}_{t}(v)(x)\quad\text{for
all }t\in{\mathbb{I}}^{\prime},\,x\in\Omega$

holds. Given a sequence $(\xi_{t})_{t\in{\mathbb{I}}}$ in $C_{+}(\Omega)$ we
introduce the order intervals

$[0,\xi]:=\left\\{(t,v)\in{\mathbb{I}}\times C(\Omega):\,0\leq
v(x)\leq\xi_{t}(x)\text{ for all
}x\in\Omega\right\\}\subseteq{\mathbb{I}}\times C_{+}(\Omega)$

and obtain from [34, Prop. 8] that the pullback attractor
${\mathcal{A}}_{0}^{\ast}$ of (4.1) satisfies the inclusion
${\mathcal{A}}_{0}^{\ast}\subseteq[0,\xi^{0}]$ with a bounded entire solution
$(\xi_{t}^{0})_{t\in{\mathbb{I}}}$ to (4.1). This entire solution is contained
in ${\mathcal{A}}_{0}^{\ast}$ due to the characterization (2.2). In addition,
$\xi^{0}$ is the pullback limit for all sequences starting from ’above’
$\xi^{0}$ (cf. [34, Prop. 8]).

Based on Thm. 3.8 also stable collocation discretizations ($I_{n}$),
$n\in{\mathbb{N}}$, of the generalized Beverton-Holt equation (4.1) possess
pullback attractors ${\mathcal{A}}_{n}^{\ast}$. In order to illustrate
convergence of the pullback attractors ${\mathcal{A}}^{\ast}_{n}$ to
${\mathcal{A}}_{0}^{\ast}$ as $n\to\infty$, we restrict to habitats
$\Omega=[a,b]$ and piecewise linear splines from Ex. 3.4 having the nodes
$x_{j}:=a+j\tfrac{b-a}{n}$, $0\leq j\leq n$, for all $n\in{\mathbb{N}}$. This
results in the spatial discretizations

$\displaystyle u_{t+1}$ $\displaystyle={\mathscr{F}}_{t}^{n}(u_{t}),$
$\displaystyle{\mathscr{F}}_{t}^{n}(u)$
$\displaystyle=\sum_{j=0}^{n}\int_{\Omega}k_{t}(x_{j},y)\frac{\gamma_{t}(y)u(y)}{1+u(y)^{\alpha}}\,{\mathrm{d}}y\beta_{j-1}^{1}.$
(4.2)

In particular, $p=1$ and $\overline{h}_{n}=\tfrac{b-a}{n}$ for all
$n\in{\mathbb{N}}$ and for appropriate kernels $k_{t}$ one obtains the
convergence rate $2$ from (3.9).

Spatial discretization using piecewise linear functions guarantees that also
the right-hand sides ${\mathscr{F}}_{t}^{n}:U\to X_{n}$,
$t\in{\mathbb{I}}^{\prime}$, are order preserving for $\alpha\in(0,1]$. Thus,
[34, Prop. 8] applies to (4.2) as well, and ensures that its pullback
attractors fulfill ${\mathcal{A}}^{\ast}_{n}\subseteq[0,\xi^{n}]$ with bounded
entire solutions $(\xi_{t}^{n})_{t\in{\mathbb{I}}}$ to (4.2) in
${\mathcal{A}}_{n}^{\ast}$.

$n$ | $\alpha=\tfrac{1}{2}$ | $\alpha=1$
---|---|---
16 | 2.112614126300029 | 2.100856100109834
32 | 2.055209004601208 | 2.051123510423984
64 | 2.026777868073563 | 2.025681916479720
128 | 2.013096435137189 | 2.012865860858589
256 | 2.006536458546063 | 2.006433295172438
512 | 2.003256451919377 | 2.003220576642864
1024 | 2.001624177537549 | 2.001610772707442
Table 1. Values of the approximate convergence rates $c(n)$ for
$n\in\left\\{2^{4},\ldots,2^{10}\right\\}$ nodes and parameters
$\alpha\in\left\\{\tfrac{1}{2},1,2\right\\}$.

On this basis, in order to quantify convergence rates of the Hausdorff
distances
$\operatorname{dist}\left({\mathcal{A}}_{n}^{\ast}(t),{\mathcal{A}}_{0}^{\ast}(t)\right)$
between the fibers of the pullback attractors to (4.2) and
${\mathcal{A}}_{0}^{\ast}$ guaranteed by Thm. 3.8, we make use the entire
solutions $\xi^{n}$ and approximate the rates of their convergence to
$\xi^{0}$ as $n\to\infty$. For this purpose, choose the habitat
$\Omega=[-3,3]$, the Laplace kernel
$k_{t}(x,y):=\tfrac{\delta_{t}}{2}e^{-\delta_{t}|x-y|}$ yielding
$C^{2}[-3,3]$-smoothness in the right-hand side of (4.1) for the almost
periodic dispersal rates $\delta_{t}:=2+\sin(\tfrac{t}{3})$, the growth rates
$\gamma_{t}(x):=3-\sin(\tfrac{tx}{5})$ and $n=4096$. In order to evaluate the
remaining integrals in (4.2) we apply the trapezoidal rule and the functions
$\xi_{t}^{n}$ are approximated as pullback limits $\varphi^{n}(t;t-s,\xi)$,
$t\in{\mathbb{I}}$, with $s:=15$ and upper solutions $\xi$. Then the sequence
$c(n):=\log_{2}\tfrac{\|\xi^{n}-\xi^{2n}\|}{\|\xi^{2n}-\xi^{4n}\|}$
approximates the desired convergence rates for $\xi^{n}$ to $\xi^{0}$ given
large values of $n$. This results in the values listed in Tab. 1 which where
illustrated in Fig. 3, respectively. Clearly, the rate $2$ (cf. Ex. 3.4) is
preserved.

Figure 3. Development of the averaged error $\|\xi^{n}-\xi^{2n}\|$ for
$n\in\left\\{2^{4},\ldots,2^{10}\right\\}$ nodes and parameters
$\alpha\in\left\\{\tfrac{1}{2},1,2\right\\}$.

###  4.2. Forward limit sets for asymptotically autonomous Ricker equations

Consider the scalar nonautonomous IDE

$\displaystyle u_{t+1}$ $\displaystyle={\mathscr{F}}_{t}(u_{t}),$
$\displaystyle{\mathscr{F}}_{t}(u)$
$\displaystyle:=\gamma_{t}\int_{\Omega}k(\cdot,y)u_{t}(y)e^{-u_{t}(y)}\,{\mathrm{d}}y+b$
(4.3)

with a continuous kernel $k:\Omega\times\Omega\to{\mathbb{R}}_{+}$, an
inhomogeneity $b\in C_{+}(\Omega)$ and a bounded sequence of growth rates
$(\gamma_{t})_{t\in{\mathbb{I}}}$ in ${\mathbb{R}}_{+}$. For the Ricker growth
function $g:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$, $g(z):=ze^{-z}$ it is
elementary to show

$\displaystyle\left|g(z)\right|$ $\displaystyle\leq\tfrac{1}{e},$
$\displaystyle\left|g(z)-g(\bar{z})\right|$
$\displaystyle\leq\tfrac{1}{e^{2}}\left|z-\bar{z}\right|\quad\text{for all
}z,\bar{z}\in{\mathbb{R}}_{+}.$

Given this, on intervals ${\mathbb{I}}$ unbounded below it is not difficult to
establish that (4.3) has a pullback attractor. However, we are interested in
the forward dynamics of (4.3) under discretization. For this purpose, let us
suppose that ${\mathbb{I}}$ is bounded below, but unbounded above. Moreover,
it is crucial to assume that (4.3) is asymptotically autonomous in the sense
that the coefficient sequence $(\gamma_{t})_{t\in{\mathbb{I}}}$ converges
exponentially. In order to be more precise, let us assume that there exist
reals $\gamma>0$ and $\tilde{K}_{0},\tilde{K}_{1}\geq 1$ such that the
following holds with
$k_{0}:=\sup_{x\in\Omega}\int_{\Omega}k(x,y)\,{\mathrm{d}}y$:

  * •

$\tilde{K}_{0}:=\sup_{s\leq
t}\prod_{l=s}^{t-1}\tfrac{\gamma_{l}}{\gamma}<\infty$ and $\gamma k_{0}<1$,

  * •

$\left|\gamma_{t}-\gamma\right|\leq\tilde{K}_{1}(k_{0}\gamma)^{t}$, which
implies $\lim_{t\to\infty}\gamma_{t}=\gamma$,

  * •

$k_{0}\sup_{t\in{\mathbb{I}}}\gamma_{t}<\tfrac{e^{2}}{1+e^{2}}\tfrac{1-k_{0}\gamma}{\tilde{K}_{0}}$.

As a result, it is shown in [18, Exam. 5.6] that the autonomous limit equation

$\displaystyle u_{t+1}$ $\displaystyle={\mathscr{F}}(u_{t}),$
$\displaystyle{\mathscr{F}}(u)$
$\displaystyle:=\gamma\int_{\Omega}k(\cdot,y)u_{t}(y)e^{-u_{t}(y)}\,{\mathrm{d}}y+b$
(4.4)

has a singleton global attractor $A^{\ast}=\left\\{u^{\ast}\right\\}$ and for
any bounded set $A\subseteq C_{+}(\Omega)$ absorbing for (4.4) the forward
limit set of (4.3) arising from ${\mathcal{A}}:={\mathbb{I}}\times A$ is just
$\omega_{{\mathcal{A}},0}^{+}=\left\\{u^{\ast}\right\\}$.

In addition, it can be shown using [18, Thm. 4.14] that for every bounded
subset $B\subseteq C_{+}(\Omega)$ one has the limit relation

$\lim_{s\to\infty}\sup_{b\in
B}\left\|\varphi^{0}(\tau+s;\tau,b)-{\mathscr{F}}^{s}(b)\right\|=0\quad\text{uniformly
in }\tau\in{\mathbb{I}};$

for this ${\mathbb{I}}$ needs to be bounded below. Consequently, passing to
the least upper bound over $b\in B$ in the inequality

$\inf_{a\in
B}\left\|\varphi^{0}(\tau+s;\tau,b)-{\mathscr{F}}^{s}(a)\right\|\leq\left\|\varphi^{0}(\tau+s;\tau,b)-{\mathscr{F}}^{s}(b)\right\|\quad\text{for
all }b\in B$

leads to the limit relation

$0\leq\operatorname{dist}\left(\varphi^{0}(\tau+s;\tau,B),{\mathscr{F}}^{s}(B)\right)\leq\sup_{b\in
B}\left\|\varphi^{0}(\tau+s;\tau,b)-{\mathscr{F}}^{s}(b)\right\|\xrightarrow[s\to\infty]{}0$

uniformly in the initial time $\tau\in{\mathbb{I}}$. Since the global
attractor $A^{\ast}$ of (4.4) attracts bounded subsets $B\subseteq X$, one
therefore arrives at

$\displaystyle\operatorname{dist}\left(\varphi^{0}(\tau+s;\tau,B),\omega_{{\mathcal{A}},0}^{+}\right)$
$\displaystyle\stackrel{{\scriptstyle\eqref{hs2}}}{{\leq}}$
$\displaystyle\operatorname{dist}\left(\varphi^{0}(\tau+s;\tau,B),{\mathscr{F}}^{s}(B)\right)+\operatorname{dist}\left({\mathscr{F}}^{s}(B),\omega_{{\mathcal{A}},0}^{+}\right)$
$\displaystyle=$
$\displaystyle\operatorname{dist}\left(\varphi^{0}(\tau+s;\tau,B),{\mathscr{F}}^{s}(B)\right)+\operatorname{dist}\left({\mathscr{F}}^{s}(B),A^{\ast}\right)\xrightarrow[s\to\infty]{}0$
(4.5)

uniformly in $\tau\in{\mathbb{I}}$.

Along with the asymptotically autonomous Ricker equation (4.3) we now turn to
their convergent collocation discretizations ($I_{n}$), $n\in{\mathbb{N}}$,
satisfying $(H_{3}$–$H_{4})$. With the abbreviations

$\displaystyle R$
$\displaystyle:=\tfrac{pk_{0}}{e}\sup_{t\in{\mathbb{I}}}\gamma_{t},$
$\displaystyle b_{n}$ $\displaystyle:=\pi_{n}b\in X_{n}$

one can show that the nonautonomous set

$\displaystyle{\mathcal{A}}$
$\displaystyle:=\Bigl{\\{}(t,x)\in{\mathcal{U}}:\,u\in\overline{\bigcup_{n\in{\mathbb{N}}_{0}}{\mathscr{F}}_{t^{\ast}}^{n}(U\cap
B_{R}(b_{n}))}\Bigr{\\}},$ $\displaystyle t^{\ast}$
$\displaystyle:=\begin{cases}t-1,&t>\min{\mathbb{I}},\\\
t,&t=\min{\mathbb{I}}\end{cases}$

is bounded, compact, positively invariant and forward absorbing w.r.t. every
($I_{n}$), $n\in{\mathbb{N}}_{0}$, with absorption time $2$. In particular,
each fiber ${\mathcal{A}}(t)$, $t\in{\mathbb{I}}$, is compact, since every
${\mathscr{F}}_{t^{\ast}}^{n}(U\cap B_{R}(b_{n}))$ is relatively compact and
($I_{n}$), $n\in{\mathbb{N}}$, is convergent.

The above exponential convergence assumption for
$(\gamma_{t})_{t\in{\mathbb{I}}}$ implies that the union
$\bigcup_{t\in{\mathbb{I}}}{\mathscr{F}}_{t}^{n}(U\cap B_{R}(b_{n}))$ is
relatively compact. Thanks to the Lipschitz estimate

$\left\|{\mathscr{F}}_{t}^{n}(u)-{\mathscr{F}}_{t}^{n}(v)\right\|\leq\tfrac{pk_{0}}{e^{2}}\sup_{t\in{\mathbb{I}}}\gamma_{t}\left\|u-v\right\|\quad\text{for
all }n\in{\mathbb{N}}_{0},\,t\in{\mathbb{I}},\,u,v\in U,$

we can apply [18, Thm. 5.5] in order to obtain that the forward limit sets
$\omega_{{\mathcal{A}},n}^{+}$ of ($I_{n}$), $n\in{\mathbb{N}}$, are
asymptotically negatively invariant (cf. the proof of Thm. 2.7).

Because $K:=\bigcup_{n\in{\mathbb{N}}_{0}}K_{n}$ with
$K_{n}:=\overline{\bigcup_{t\in{\mathbb{I}}}{\mathscr{F}}_{t}^{n}(U\cap
B_{R}(b_{n}))}$ is bounded, we obtain from the uniform limit relation (4.5)
established above that the assumption in Thm. 3.10(i) is satisfied. Combining
this with the asymptotic negative invariance of each
$\omega_{{\mathcal{A}},n}^{+}$, $n\in{\mathbb{N}}_{0}$, we conclude as in Thm.
3.10 that the forward limit sets of the collocation discretizations ($I_{n}$),
$n\in{\mathbb{N}}$, fulfill

$\lim_{n\to\infty}\operatorname{dist}\left(\omega_{{\mathcal{A}},n}^{+},\omega_{{\mathcal{A}},0}^{+}\right)=\lim_{n\to\infty}\sup_{a\in\omega_{{\mathcal{A}},n}^{+}}\left\|a-u^{\ast}\right\|=0.$

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