∎

11institutetext: Huy Huynh and Christian Pötzsche22institutetext: Institut für
Mathematik, Universität Klagenfurt, 9020 Klagenfurt, Austria
22email<EMAIL_ADDRESS>33institutetext: Peter E.
Kloeden44institutetext: Mathematisches Institut, Universität Tübingen, 72076
Tübingen, Germany
44email<EMAIL_ADDRESS>
# Forward and pullback dynamics of nonautonomous integrodifference
equations††thanks: The work of Huy Huynh has been supported by the Austrian
Science Fund (FWF) under grant number P 30874-N35.

Basic constructions

Huy Huynh  Peter E. Kloeden  Christian Pötzsche

(Received: / Accepted: date)

###### Abstract

In theoretical ecology, models describing the spatial dispersal and the
temporal evolution of species having non-overlapping generations are often
based on integrodifference equations. For various such applications the
environment has an aperiodic influence on the models leading to nonautonomous
integrodifference equations. In order to capture their long-term behaviour
comprehensively, both pullback and forward attractors, as well as forward
limit sets are constructed for general infinite-dimensional nonautonomous
dynamical systems in discrete time. While the theory of pullback attractors,
but not their application to integrodifference equations, is meanwhile well-
established, the present novel approach is needed in order to understand their
future behaviour.

###### Keywords:

Forward limit set Forward attractor Pullback attractor Asymptotically
autonomous equation Integrodifference equation Urysohn operator

###### MSC:

37C7037C6045G1592D40

††dedicatory: Dedicated to the memory of Russell Johnson

##  1 Introduction

Integrodifference equations not only occur as temporal discretisations of
integrodifferential equations or as time-$1$-maps of evolutionary differential
equations, but are of interest in themselves. First and foremost, they are a
popular tool in theoretical ecology to describe the dispersal of species
having non-overlapping generations (see, for instance, lutscher:19  or
kot:schaeffer:86 ; volkov:lui:07 ; jacobsen:jin:lewis:15 ). While the theory
of Urysohn or Hammerstein integral equations is now rather classical martin:76
, both numerically and analytically, our goal is here to study their iterates
from a dynamical systems perspective. This means one is interested in the long
term behaviour of recursions based on a fixed nonlinear integral operator. In
applications, the iterates for instance represent the spatial distribution of
interacting species over a habitat. One of the central questions in this
context is the existence and structure of an attractor. These invariant and
compact sets attract bounded subsets of an ambient state space $X$ and fully
capture the asymptotics of an autonomous dynamical system hale:88 ;
sell:you:02 . The dynamics inside the attractor can be very complicated and
even chaotic day:junge:mischaikow:04 .

Extending this situation, the main part of this paper is devoted to general
nonautonomous difference equations in complete metric spaces. Their right-hand
side can depend on time allowing to model the dispersal of species in
temporally fluctuating environments bouhours:lewis:16 ; jacobsen:jin:lewis:15
being not necessarily periodic. Thus, the behaviour depends on both the
initial and the actual time. This is why many dynamically relevant objects are
contained in the extended state space ${\mathbb{Z}}\times X$ (one speaks of
nonautonomous sets) johnson:munoz:09 , rather than being merely subsets of the
state space $X$ as in the autonomous case. Furthermore, a complete description
of the dynamics in a time-variant setting necessitates a strict distinction
between forward and pullback convergence johnson:munoz:09 ;
kloeden:rasmussen:10 . For this reason only a combination of several attractor
notions yields the full picture:

  * •

The _pullback attractor_ carvalho:langa:robinson:10 ; kloeden:00 ;
kloeden:rasmussen:10 ; poetzsche:10b  is a compact, invariant nonautonomous
set which attracts all bounded sets from the past. As fixed target problem, it
is based on previous information, at a fixed time from increasingly earlier
initial times. Since it consists of bounded entire solutions to a
nonautonomous system (see (poetzsche:10b, , p. 17, Cor. 1.3.4)), a pullback
attractor can be seen as an extension of the global attractor to nonautonomous
problems and apparently captures the essential dynamics to a certain point.
Meanwhile the corresponding theory is widely developed in discrete and
continuous time. However, pullback attractors reflect the past rather than the
future of systems (see kloeden:poetzsche:rasmussen:11 ) and easy examples
demonstrate that differential or difference equations with identical pullback
attractors might have rather different asymptotics as $t\to\infty$ and
possibly feature limit sets, which are not captured by the pullback dynamics.

  * •

This led to the development of _forward attractors_ , which are also compact
and invariant nonautonomous sets kloeden:rasmussen:10 . This dual concept
depends on information from the future and given a fixed initial time, the
actual time increases beyond all bounds — they are a moving target problem.
Forward attractors are not unique, independent of pullback attractors, but
often do not exist. Nevertheless, we will describe forward attractors using a
pullback construction, even though this has the disadvantage that information
on the system over the entire time axis ${\mathbb{Z}}$ is required.

  * •

Therefore, it was suggested in kloeden:lorenz:16  to work with _forward limit
sets_ , a concept related to the uniform attractor due to vishik:92 . They
correctly describe the asymptotic behaviour of all forward solutions to a
nonautonomous difference equation. These limit sets have forward attraction
properties, but different from pullback and forward attractors, they are not
(even positively) invariant and constitute a single compact set, rather than a
nonautonomous set. Nonetheless, asymptotic forms of positive (and negative)
invariance do hold.

The situation for forward attractors and limit sets is not as well-established
as their pullback counterparts and deserves to be developed for the above
reasons. Their initial construction in kloeden:lorenz:16 ; kloeden:yang:16
requires a locally compact state space, but recent continuous-time results in
cui:kloeden:yang:19 , which extend these to infinite-dimensional dynamical
systems, will be transferred here. We indeed address nonautonomous difference
equations in (not necessarily locally compact) metric and Banach spaces,
introduce the mentioned attractor types and study their properties.

This brings us to our second purpose. The above abstract setting allows
concrete applications to a particularly interesting class of infinite-
dimensional dynamical systems in discrete time, namely integrodifference
equations (IDEs for short). We provide sufficient criteria for the existence
of pullback attractors tailor-made for a quite general class of IDEs. Their
right-hand sides go beyond pure integral operators and might also include
superposition operators, which are used to describe populations having a
sedentary fraction. Such results follow from a corresponding theory of set
contractions contained in (poetzsche:10b, , pp. 15ff), (martin:76, , pp.
79ff). For completely continuous right-hand sides (i.e., Urysohn operators) we
construct forward limit sets and provide an application to asymptotically
autonomous IDEs. We restrict to rather simple IDEs in the space of continuous
functions over a compact domain as state space. More complicated equations and
the behaviour of attractors under spatial discretisation will be tackled in
future papers.

The contents of this paper are as follows: In Sect. 2 we establish the
necessary terminology and provide a useful dissipativity condition for
nonautonomous difference equations. The key notions related to pullback
convergence, i.e., limit sets and attractors are reviewed and established in
Sect. 3. The subsequent Sect. 4 addresses the corresponding notions in forward
time. In detail, it establishes forward limit sets and their (weakened)
invariance properties. For a class of asymptotically autonomous equations it
is shown that their forward limit sets coincide with the global attractor of
the limit equation. Moreover, a construction of forward attractors is
suggested. Finally, in Sect. 5 we provide some applications to various IDEs.
In particular, we illustrate the above theoretical results by studying
pullback attractors and forward limit sets.

#### Notation

Let ${\mathbb{R}}_{+}:=[0,\infty)$. A _discrete interval_ ${\mathbb{I}}$ is
defined as 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 ${\mathbb{N}}_{0}:=\left\\{0,1,2,\ldots\right\\}$.

On a metric space $(X,d)$, $I_{X}$ is the identity map,
$B_{r}(x):=\left\\{y\in X:\,d(x,y)<r\right\\}$ the open ball with center $x\in
X$ and radius $r>0$, and $\bar{B}_{r}(0)$ denotes its closure. We write
$\operatorname{dist}\bigl{(}x,A\bigr{)}:=\inf_{a\in A}d(x,a)$ for the distance
of $x$ from a set $A\subseteq X$ and $B_{r}(A):=\left\\{x\in
X:\,\operatorname{dist}\bigl{(}x,A\bigr{)}<r\right\\}$ for its
$r$-neighbourhood. The _Hausdorff semidistance_ of bounded and closed subsets
$A,B\subseteq X$ is defined as

$\operatorname{dist}\bigl{(}A,B\bigr{)}:=\sup_{a\in A}\inf_{b\in B}d(a,b).$

The _Kuratowski measure of noncompactness_ on $X$ (cf. (martin:76, , pp. 16ff,
I.5)) is denoted by $\chi:{\mathfrak{B}}(X)\to{\mathbb{R}}$, where
${\mathfrak{B}}(X)$ stands for the family of bounded subsets of $X$.

A mapping ${\mathscr{F}}:X\to X$ is said to be _bounded_ , if it maps bounded
subsets of $X$ into bounded sets and _globally bounded_ , if
${\mathscr{F}}(X)$ is bounded. We say a bounded ${\mathscr{F}}$ satisfies a
_Darbo condition_ , if there exists a real constant $k\geq 0$ such that

$\chi\bigl{(}{\mathscr{F}}(B)\bigr{)}\leq k\chi(B)\quad\text{for all
}B\in{\mathfrak{B}}(X).$

The smallest such $k$ is the _Darbo constant_
$\operatorname{dar}\bigl{(}{\mathscr{F}}\bigr{)}\in[0,\infty)$ of
${\mathscr{F}}$. A _completely continuous_ mapping ${\mathscr{F}}$ is bounded,
continuous and satisfies $\operatorname{dar}\bigl{(}{\mathscr{F}}\bigr{)}=0$.

A subset ${\mathcal{A}}\subseteq{\mathbb{I}}\times X$ with $t$-_fibres_
${\mathcal{A}}(t):=\\{x\in X:\,(t,x)\in{\mathcal{A}}\\}$, $t\in{\mathbb{I}}$,
is called _nonautonomous set_. If all fibres ${\mathcal{A}}(t)\subseteq X$,
$t\in{\mathbb{I}}$, are compact, then ${\mathcal{A}}$ is denoted as _compact_
nonautonomous set and we proceed accordingly with other topological notions.
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 $t\in{\mathbb{I}}$.

Finally, on a Banach space $X$, $L(X)$ denotes the space of bounded linear
operators and $\rho({\mathscr{L}})$ is the _spectral radius_ of a
${\mathscr{L}}\in L(X)$.

##  2 Nonautonomous difference equations

Unless otherwise noted, let $(X,d)$ be a complete metric space. We consider
nonautonomous difference equations in the abstract form

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

with continuous right-hand sides ${\mathscr{F}}_{t}:U_{t}\to X$ and defined on
closed sets $U_{t}\subseteq X$, $t\in{\mathbb{I}}^{\prime}$. For an _initial
time_ $\tau\in{\mathbb{I}}$, a _forward solution_ to ($\Delta$) is a sequence
$(\phi_{t})_{\tau\leq t}$ with $\phi_{t}\in U_{t}$ satisfying

$\phi_{t+1}\equiv{\mathscr{F}}_{t}(\phi_{t})$ (1)

for all $\tau\leq t$, $t\in{\mathbb{I}}^{\prime}$, while an _entire solution_
$(\phi_{t})_{t\in{\mathbb{I}}}$ satisfies (1) on ${\mathbb{I}}^{\prime}$. The
unique forward solution starting at $\tau\in{\mathbb{I}}$ in $u_{\tau}\in
U_{\tau}$ is denoted by $\varphi(\cdot;\tau,u_{\tau})$; it is denoted as
_general solution_ to ($\Delta$) and reads as

$\varphi(t;\tau,u_{\tau}):=\begin{cases}{\mathscr{F}}_{t-1}\circ\ldots\circ{\mathscr{F}}_{\tau}(u_{\tau}),&\tau<t,\\\
u_{\tau},&\tau=t,\end{cases}$ (2)

as long as the compositions stay in $U_{t}$. Under the inclusion
${\mathscr{F}}_{t}(U_{t})\subseteq U_{t+1}$, $t\in{\mathbb{I}}^{\prime}$, the
general solution $\varphi(t;\tau,\cdot):U_{\tau}\to U_{t}$ exists for all
$\tau\leq t$ and the _process property_

$\varphi(t;s,\varphi(s;\tau,u))=\varphi(t;\tau,u)\quad\text{for all }\tau\leq
s\leq t,\,u\in U_{\tau}$ (3)

holds; we introduce the nonautonomous set
${\mathcal{U}}:=\left\\{(t,u)\in{\mathbb{I}}\times X:\,u\in U_{t}\right\\}$.

One denotes ($\Delta$) as $\theta$-_periodic_ with some
$\theta\in{\mathbb{N}}$, if ${\mathscr{F}}_{t}={\mathscr{F}}_{t+\theta}$,
$U_{t}=U_{t+\theta}$ and tacitly ${\mathbb{I}}={\mathbb{Z}}$ hold for all
$t\in{\mathbb{Z}}$. In this case the general solution satisfies

$\varphi(t+\theta;\tau+\theta,u_{\tau})=\varphi(t;\tau,u_{\tau})\quad\text{for
all }\tau\leq t,\,(\tau,u_{\tau})\in{\mathcal{U}}$ (4)

yielding a rather tame time-dependence. An _autonomous_ equation ($\Delta$) is
$1$-periodic.

A nonautonomous set ${\mathcal{A}}\subseteq{\mathcal{U}}$ is called
_positively_ or _negatively invariant_ (w.r.t. the difference equation
($\Delta$)), if the respective inclusion

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

holds; an _invariant_ set ${\mathcal{A}}$ is both positively and negatively
invariant, that is,
${\mathscr{F}}_{t}\bigl{(}{\mathcal{A}}(t)\bigr{)}={\mathcal{A}}(t+1)$ for all
$t\in{\mathbb{I}}^{\prime}$. One denotes ${\mathcal{A}}$ as
$\theta$-_periodic_ , if ${\mathcal{A}}(t)={\mathcal{A}}(t+\theta)$ holds for
all $t\in{\mathbb{I}}$ with $t+\theta\in{\mathbb{I}}$.

The next two subsections provide some preparations on nonautonomous difference
equations in Banach spaces $(X,\left\|\cdot\right\|)$:

###  2.1 Semilinear difference equations

Let ${\mathscr{L}}_{t}\in L(X)$, $t\in{\mathbb{I}}^{\prime}$, be a sequence of
bounded linear operators. For a _linear difference equation_

$\boxed{u_{t+1}={\mathscr{L}}_{t}u_{t}}$

we define the _transition operator_
$\Phi:\left\\{(t,\tau)\in{\mathbb{I}}^{2}:\,\tau\leq t\right\\}\to L(X)$ by

$\Phi(t,\tau):=\begin{cases}{\mathscr{L}}_{t-1}\cdots{\mathscr{L}}_{\tau},&\tau<t,\\\
I_{X},&t=\tau.\end{cases}$

Then ($\Delta$) is understood as _semilinear_ , if its right-hand side can be
represented as

${\mathscr{F}}_{t}={\mathscr{L}}_{t}+{\mathscr{N}}_{t}$ (5)

with continuous mappings ${\mathscr{N}}_{t}:U_{t}\to X$,
$t\in{\mathbb{I}}^{\prime}$. The variation of constants formula
(poetzsche:10b, , p. 100, Thm. 3.1.16) yields the general solution of
($\Delta$) in the form

$\varphi(t;\tau,u_{\tau})=\Phi(t,\tau)u_{\tau}+\sum_{s=\tau}^{t-1}\Phi(t,s+1){\mathscr{N}}_{s}\bigl{(}\varphi(s;\tau,u_{\tau})\bigr{)}\quad\text{for
all }\tau\leq t,\,u_{\tau}\in U_{\tau}.$ (6)

The following result will be helpful in the construction of absorbing sets:

###### Lemma 1

Let ${\mathscr{F}}_{t}:U_{t}\to U_{t+1}$ be of semilinear form (5) and suppose
there exist reals $\alpha_{t}\geq 0$, $K\geq 1$ with

$\left\|\Phi(t,s)\right\|\leq K\prod_{r=s}^{t-1}\alpha_{r}\quad\text{for all
}s\leq t.$ (7)

If there exist reals $a_{t}\geq 0$, $b_{t}\geq 0$ such that the nonlinearity
fulfills

$\left\|{\mathscr{N}}_{t}(u)\right\|\leq
b_{t}+a_{t}\left\|u\right\|\quad\text{for all
}t\in{\mathbb{I}}^{\prime},\,u\in U_{t},$ (8)

then the general solution of ($\Delta$) satisfies the estimate

$\left\|\varphi(t;\tau,u_{\tau})\right\|\leq
K\left\|u_{\tau}\right\|\prod_{r=\tau}^{t-1}(\alpha_{r}+Ka_{r})+K\sum_{s=\tau}^{t-1}b_{s}\prod_{r=s+1}^{t-1}(\alpha_{r}+Ka_{r})$
(9)

for all $\tau\leq t$ and $u_{\tau}\in U_{\tau}$.

######  Remark 1 (linear growth)

In case ${\mathscr{L}}_{t}\equiv 0$ on ${\mathbb{I}}^{\prime}$ one can choose
$K=1$, $\alpha_{t}=0$ in (7) and the estimate (9) simplifies to

$\left\|\varphi(t;\tau,u_{\tau})\right\|\leq\left\|u_{\tau}\right\|\prod_{r=\tau}^{t-1}a_{r}+\sum_{s=\tau}^{t-1}b_{s}\prod_{r=s+1}^{t-1}a_{r}\quad\text{for
all }\tau\leq t,\,u_{\tau}\in U_{\tau}.$

###### Proof

Let $\tau\in{\mathbb{I}}$. It is convenient to abbreviate
$e_{\alpha}(t,s):=\prod_{r=s}^{t-1}\alpha_{r}$ and we first assume that
$\alpha_{t}\neq 0$, $t\in{\mathbb{I}}^{\prime}$. Given $u_{\tau}\in U_{\tau}$,
from (6) and (7) we obtain

$\displaystyle\left\|\varphi(t;\tau,u_{\tau})\right\|$ $\displaystyle\leq$
$\displaystyle
Ke_{\alpha}(t,\tau)\left\|u_{\tau}\right\|+K\sum_{s=\tau}^{t-1}e_{\alpha}(t,s+1)\left\|{\mathscr{N}}_{s}(\varphi(s;\tau,u_{\tau}))\right\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{lemabs2}}}{{\leq}}$ $\displaystyle
Ke_{\alpha}(t,\tau)\left\|u_{\tau}\right\|+K\sum_{s=\tau}^{t-1}e_{\alpha}(t,s+1)\bigl{(}b_{s}+a_{s}\left\|\varphi(s;\tau,u_{\tau})\right\|\bigr{)}$

and therefore the sequence
$u(t):=\left\|\varphi(t;\tau,u_{\tau})\right\|e_{\alpha}(\tau,t)$ satisfies

$u(t)\leq
K\left\|u_{\tau}\right\|+K\sum_{s=\tau}^{t-1}b_{s}e_{\alpha}(\tau,s+1)+K\sum_{s=\tau}^{t-1}\frac{a_{s}}{\alpha_{s}}u(s)\quad\text{for
all }\tau\leq t.$

Thus, the Grönwall inequality from (poetzsche:10b, , p. 348, Prop. A.2.1(a))
implies

$u(t)\leq
Ke_{1+\frac{Ka}{\alpha}}(t,\tau)\left\|u_{\tau}\right\|+K\sum_{s=\tau}^{t-1}b_{s}e_{\alpha}(\tau,s+1)e_{1+\frac{Ka}{\alpha}}(t,s+1)$

and consequently

$\left\|\varphi(t;\tau,u_{\tau})\right\|\leq
Ke_{\alpha+Ka}(t,\tau)\left\|u_{\tau}\right\|+K\sum_{s=\tau}^{t-1}b_{s}e_{\alpha+Ka}(t,s+1)\quad\text{for
all }\tau\leq t,$

which is the claimed inequality (9). ∎

###  2.2 Additive difference equations

We now address right-hand sides

${\mathscr{F}}_{t}={\mathscr{G}}_{t}+{\mathscr{K}}_{t},$ (10)

where ${\mathscr{G}}_{t}:U_{t}\to X$ is bounded and continuous, while
${\mathscr{K}}_{t}:U_{t}\to X$, $t\in{\mathbb{I}}^{\prime}$, is assumed to be
completely continuous.

###### Lemma 2

If ${\mathscr{F}}_{t}:U_{t}\to U_{t+1}$ is of additive form (10), then the
general solution of ($\Delta$) satisfies

$\operatorname{dar}\bigl{(}\varphi(t;\tau,\cdot)\bigr{)}\leq\prod_{s=\tau}^{t-1}\operatorname{dar}\bigl{(}{\mathscr{G}}_{s}\bigr{)}\quad\text{for
all }\tau\leq t.$

###### Proof

Since ${\mathscr{F}}_{t}={\mathscr{G}}_{t}+{\mathscr{K}}_{t}:U_{t}\to X$ for
every $t\in{\mathbb{I}}^{\prime}$ is continuous and bounded, their composition
(2) is also continuous and bounded. The estimate for the Darbo constant of
$\varphi(t;\tau,\cdot)$ will be established by mathematical induction. For
$t=\tau$ the assertion is clear, since $\varphi(\tau;\tau,\cdot)=I_{X}$ and
the Lipschitz constant of the identity mapping is $1$; it provides an upper
bound for the Darbo constant (see (martin:76, , p. 81, Prop. 5.3)). For times
$t\geq\tau$, from
$\operatorname{dar}\bigl{(}{\mathscr{K}}_{t}\circ\varphi(t;\tau,\cdot)\bigr{)}=0$,
which holds because ${\mathscr{K}}_{t}$ is completely continuous (cf.
(martin:76, , p. 82, Prop. 5.4)), it follows that

$\displaystyle\operatorname{dar}\bigl{(}\varphi(t+1;\tau,\cdot)\bigr{)}$
$\displaystyle=\operatorname{dar}\bigl{(}{\mathscr{G}}_{t}\circ\varphi(t;\tau,\cdot)+{\mathscr{K}}_{t}\circ\varphi(t;\tau,\cdot)\bigr{)}\leq\operatorname{dar}\bigl{(}{\mathscr{G}}_{t}\circ\varphi(t;\tau,\cdot)\bigr{)}$
$\displaystyle\leq\operatorname{dar}\bigl{(}{\mathscr{G}}_{t}\bigr{)}\operatorname{dar}\bigl{(}\varphi(t;\tau,\cdot)\bigr{)}=\prod_{s=\tau}^{t}\operatorname{dar}\bigl{(}{\mathscr{G}}_{s}\bigr{)}\quad\text{for
all }\tau\leq t$

from (martin:76, , pp. 79–80, Prop. 5.1). This establishes the claim. ∎

##  3 Pullback convergence

In this section, suppose that ${\mathbb{I}}$ is unbounded below and that
${\mathscr{F}}_{t}:U_{t}\to U_{t+1}$, $t\in{\mathbb{I}}^{\prime}$, i.e.,
($\Delta$) generates a process on ${\mathcal{U}}$.

A difference equation ($\Delta$) is said to be _pullback asymptotically
compact_ , if for every $\tau\in{\mathbb{I}}$, every sequence
$(s_{n})_{n\in{\mathbb{N}}}$ in ${\mathbb{N}}_{0}$ with
$\lim_{n\to\infty}s_{n}=\infty$ and every bounded sequence
$(a_{n})_{n\in{\mathbb{N}}}$ with $a_{n}\in{\mathcal{U}}(\tau-s_{n})$, the
sequence $\bigl{(}\varphi(\tau;\tau-s_{n},a_{n})\bigr{)}_{n\in{\mathbb{N}}}$
possesses a convergent subsequence.

###  3.1 Pullback limit sets

The _pullback limit set_ $\omega_{{\mathcal{A}}}\subseteq{\mathcal{U}}$ of a
bounded subset ${\mathcal{A}}\subseteq{\mathcal{U}}$ is given by the fibres

$\omega_{{\mathcal{A}}}(\tau):=\bigcap_{0\leq s}\overline{\bigcup_{s\leq
t}\varphi(\tau;\tau-t,{\mathcal{A}}(\tau-t))}\quad\text{for all
}\tau\in{\mathbb{I}}.$ (11)

For pullback asymptotically compact nonautonomous difference equations
($\Delta$) it is shown in (poetzsche:10b, , p. 14, Thm. 1.2.25) that
$\omega_{{\mathcal{A}}}$ is nonempty, compact, invariant and _pullback
attracts_ ${\mathcal{A}}$, i.e., the limit relation

$\lim_{s\to\infty}\operatorname{dist}\bigl{(}\varphi(\tau;\tau-s,{\mathcal{A}}(\tau-s)),\omega_{{\mathcal{A}}}(\tau)\bigr{)}=0\quad\text{for
all }\tau\in{\mathbb{I}}$ (12)

holds. For positively invariant sets ${\mathcal{A}}$ the defining relation
(11) simplifies to

$\omega_{{\mathcal{A}}}(\tau)=\bigcap_{0\leq
s}\overline{\varphi(\tau;\tau-s,{\mathcal{A}}(\tau-s))}.$ (13)

Therefore, as a fundamental tool for the construction of pullback limit sets
and attractors, as well as for forward attractors in Sect. 4.3, we state

###### Proposition 1

Suppose that ($\Delta$) has a nonempty, positively invariant, closed and
bounded subset ${\mathcal{A}}\subseteq{\mathcal{U}}$. If ($\Delta$) is
pullback asymptotically compact, then the fibres

${\mathcal{A}}^{\star}(\tau):=\bigcap_{0\leq
s}\varphi(\tau;\tau-s,{\mathcal{A}}(\tau-s))\quad\text{for all
}\tau\in{\mathbb{I}}$ (14)

define a maximal invariant, nonempty and compact nonautonomous set
${\mathcal{A}}^{\star}\subseteq{\mathcal{A}}$, which pullback attracts
${\mathcal{A}}$.

If the nonautonomous set ${\mathcal{A}}$ is even compact, then Prop. 1 applies
without the asymptotic compactness assumption.

###### Proof

Since ($\Delta$) generates a continuous process $\varphi$ in discrete time,
the assertion results via an adaption of (kloeden:marin:10, , Prop. 5), where
pullback asymptotic compactness yields that the intersection of the nested
sets in (14) is nonempty. ∎

###  3.2 Pullback attractors

A _pullback attractor_ ${\mathcal{A}}^{\ast}\subseteq{\mathcal{U}}$ of
($\Delta$) is a nonempty, compact, invariant nonautonomous set which pullback
attracts all bounded nonautonomous sets ${\mathcal{B}}\subseteq{\mathcal{U}}$.
Bounded pullback attractors are unique and allow the dynamical
characterisation

${\mathcal{A}}^{\ast}=\left\\{(\tau,u)\in{\mathcal{U}}\,\left|\begin{array}[]{l}\text{there
exists a bounded entire solution}\\\ (\phi_{t})_{t\in{\mathbb{I}}}\text{ of
\eqref{deq} satisfying }\phi_{\tau}=u\end{array}\right.\right\\}$

(cf. (poetzsche:10b, , p. 17, Cor. 1.3.4)). Despite being pullback attracting
nonautonomous sets within ${\mathcal{A}}$, the set ${\mathcal{A}}^{\star}$
constructed in Prop. 1 needs not to be a pullback attractor, since nothing was
assumed outside of ${\mathcal{A}}$. Remedy provides the notion of a _pullback
dissipative_ difference equation ($\Delta$). This means there exists a bounded
set ${\mathcal{A}}\subseteq{\mathcal{U}}$ such that for every
$\tau\in{\mathbb{I}}$ and every bounded nonautonomous set
${\mathcal{B}}\subseteq{\mathcal{U}}$ there is an _absorption time_
$S=S(\tau,{\mathcal{B}})\in{\mathbb{N}}$ such that

$\varphi(\tau;\tau-s,{\mathcal{B}}(\tau-s))\subseteq{\mathcal{A}}(\tau)\quad\text{for
all }s\geq S.$

For a _uniformly pullback dissipative_ equation ($\Delta$) the absorption time
$S$ is independent of $\tau$. One denotes ${\mathcal{A}}$ as a _pullback
absorbing set_.

If ${\mathcal{A}}$ is pullback absorbing, then the set ${\mathcal{A}}^{\star}$
obtained from Prop. 1 becomes a pullback attractor, i.e.,
${\mathcal{A}}^{\star}={\mathcal{A}}^{\ast}$, and one has the characterisation

${\mathcal{A}}^{\ast}=\omega_{{\mathcal{A}}}.$ (15)

A possibility to construct pullback absorbing sets provides

######  Proposition 2 (Pullback absorbing set)

On a Banach space $X$, let $\rho>0$ and ${\mathscr{F}}_{t}:U_{t}\to U_{t+1}$
be of semilinear form (5) satisfying (7), (8). If the limit relations

$\displaystyle\lim_{s\to\infty}\prod_{r=\tau-s}^{\tau-1}(\alpha_{r}+Ka_{r})$
$\displaystyle=0,$ $\displaystyle R_{\tau}$
$\displaystyle:=K\sum_{s=-\infty}^{\tau-1}b_{s}\prod_{r=s+1}^{\tau-1}(\alpha_{r}+Ka_{r})<\infty$

hold for all $\tau\in{\mathbb{I}}$, then the difference equation ($\Delta$) is
pullback dissipative with absorbing set
${\mathcal{A}}:=\left\\{(\tau,u)\in{\mathcal{U}}:\,\left\|u\right\|\leq\rho+R_{\tau}\right\\}$.
In case
$\lim_{s\to\infty}\sup_{\tau\in{\mathbb{I}}}\prod_{r=\tau-s}^{\tau-1}(\alpha_{r}+Ka_{r})=0$
holds, the difference equation ($\Delta$) is uniformly pullback dissipative.

###### Proof

The assertion follows from Lemma 1 by passing over to the pullback limit
$\tau\to-\infty$ in the estimate (9). ∎

A construction of pullback attractors ${\mathcal{A}}^{\ast}$ based on set
contractions, rather than asymptotic compactness, is suitable for later
applications to integrodifference equations (see Sect. 5):

###### Theorem 3.1

If a difference equation ($\Delta$) of additive form (10) is uniformly
pullback dissipative and

$\prod_{s=-\infty}^{T-1}\operatorname{dar}\bigl{(}{\mathscr{G}}_{s}\bigr{)}=0\quad\text{for
some }T\in{\mathbb{I}}$

holds, then there exists a unique bounded pullback attractor of ($\Delta$).

######  Remark 2 (periodic equations)

For $\theta$-periodic difference equations ($\Delta$) and sets
${\mathcal{A}}$, it results from (4) that also the pullback limit sets
$\omega_{\mathcal{A}}$ from (11), the set ${\mathcal{A}}^{\star}$ from Prop. 1
and the pullback attractor ${\mathcal{A}}^{\ast}$ are $\theta$-periodic (cf.
(poetzsche:10b, , pp. 21ff, Sect. 1.4)). Furthermore, Thm. 3.1 applies when
$\prod_{s=0}^{\theta-1}\operatorname{dar}\bigl{(}{\mathscr{G}}_{s}\bigr{)}<1$.

###### Proof

The terminology of poetzsche:10b  and results therein will be used. Let
$\hat{\mathcal{B}}$ denote the family of all bounded sets in
${\mathbb{I}}\times X$. Then Lemma 2 ensures that the general solution
$\varphi(t;\tau,\cdot)$ is $\hat{\mathcal{B}}$-contracting in the sense of
(poetzsche:10b, , p. 15, Def. 1.2.26(i)).

Since ($\Delta$) has a bounded absorbing set ${\mathcal{A}}$, for every
bounded nonautonomous set ${\mathcal{B}}$, there exists an
$S\in{\mathbb{N}}_{0}$ such that
$\varphi(\tau;\tau-s,{\mathcal{B}}(\tau-s))\subseteq{\mathcal{A}}(t)$ holds
for all $s\geq S$. This implies that the $S$-truncated orbit
$\gamma_{\mathcal{B}}^{S}$, fibrewise given by

$\gamma_{\mathcal{B}}^{S}(\tau):=\bigcup_{s\geq
S}\varphi(\tau;\tau-s,{\mathcal{B}}(\tau-s))\subseteq{\mathcal{A}}(\tau)\quad\text{for
all }\tau\in{\mathbb{I}},$

is bounded. Hence, (poetzsche:10b, , p. 16, Prop. 1.2.30) implies that
($\Delta$) is $\hat{\mathcal{B}}$-asymptotically compact, so ($\Delta$) has a
pullback attractor ${\mathcal{A}}^{\ast}$ by (poetzsche:10b, , p. 19, Thm.
1.3.9).

Finally, the pullback attractor ${\mathcal{A}}^{\ast}$ is contained in the
closure of the absorbing set ${\mathcal{A}}$, so is bounded and thus uniquely
determined. ∎

##  4 Forward convergence

In the previous section, we constructed pullback attractors of pullback
asymptotically compact nonautonomous difference equations ($\Delta$) as
pullback limit sets of such absorbing sets. Our next aim is to provide related
notions in forward time. Due to the conceptional difference between pullback
and forward convergence some modifications are necessary, yet.

Above all, this requires a discrete interval ${\mathbb{I}}$ to be unbounded
above. Now the right-hand sides ${\mathscr{F}}_{t}:U_{t}\to U_{t+1}$,
$t\in{\mathbb{I}}$, are defined on a common closed subset $U_{t}=U\subseteq
X$, i.e., the extended state space ${\mathcal{U}}={\mathbb{I}}\times U$ has
constant fibres. Therefore, the general solution
$\varphi:\left\\{(t,\tau,u)\in{\mathbb{I}}\times{\mathcal{U}}:\,\tau\leq
t\right\\}\to U$ is well-defined.

Given a nonautonomous set ${\mathcal{A}}\subseteq{\mathcal{U}}$, a difference
equation ($\Delta$) is said to be

  * •

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

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

  * •

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

$\displaystyle\lim_{n\to\infty}\operatorname{dist}\bigl{(}\varphi(\tau_{n}+s_{n};\tau_{n},{\mathcal{A}}(\tau_{n})),K\bigr{)}=0.$

###### Remark 3

If ${\mathcal{A}}$ is positively invariant, then strong
${\mathcal{A}}$-asymptotic compactness (needed in Thm. 4.3 below) is a
tightening of ${\mathcal{A}}$-asymptotic compactness (required in Thm. 4.2).
Indeed, suppose that the sequence
$(\operatorname{dist}\bigl{(}\varphi(t_{n};\tau,{\mathcal{A}}(\tau)),K\bigr{)})_{n\in{\mathbb{N}}}$
does not converge to $0$. Hence, the strong ${\mathcal{A}}$-asymptotic
compactness of ($\Delta$) and positive invariance of ${\mathcal{A}}$ yields
the contradiction

$\displaystyle\operatorname{dist}\bigl{(}\varphi(t_{n};\tau,{\mathcal{A}}(\tau)),K\bigr{)}$
$\displaystyle=\operatorname{dist}\bigl{(}{\mathscr{F}}_{t_{n}}(\varphi(t_{n}-1;\tau,{\mathcal{A}}(\tau)),K\bigr{)}$
$\displaystyle\leq\operatorname{dist}\bigl{(}{\mathscr{F}}_{t_{n}}({\mathcal{A}}(t_{n}-1)),K\bigr{)}\xrightarrow[n\to\infty]{}0.$

###  4.1 Forward limit sets

Let us investigate the forward dynamics of ($\Delta$) inside a nonautonomous
set ${\mathcal{A}}$. We first capture the forward limit points from a single
fibre ${\mathcal{A}}(\tau)$:

###### Lemma 3

Suppose that ${\mathcal{A}}\neq\emptyset$ is a bounded nonautonomous set. If
($\Delta$) is ${\mathcal{A}}$-asymptotically compact with a compact subset
$K\subseteq U$, then the fibres

$\displaystyle\Omega_{\mathcal{A}}(\tau):=\bigcap_{0\leq
s}\overline{\bigcup_{s\leq
t}\varphi(\tau+t;\tau,{\mathcal{A}}(\tau))}\subseteq K\quad\text{for all
}\tau\in{\mathbb{I}}$ (16)

are nonempty, compact, and forward attract ${\mathcal{A}}(\tau)$, i.e.,

$\lim_{s\to\infty}\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,{\mathcal{A}}(\tau)),\Omega_{\mathcal{A}}(\tau)\bigr{)}=0.$
(17)

An analogous result for pullback limit sets is given in (poetzsche:10b, , p.
9, Lemma 1.2.12).

######  Remark 4 (characterisation of $\Omega_{{\mathcal{A}}}(\tau)$)

The fibres $\Omega_{\mathcal{A}}(\tau)$, $\tau\in{\mathbb{I}}$, consist of
points $v$ such that there is a sequence
$\bigl{(}(s_{n},a_{n})\bigr{)}_{n\in{\mathbb{N}}}$ with
$\lim_{n\to\infty}s_{n}=\infty$, $a_{n}\in{\mathcal{A}}(\tau)$ and

$\lim_{n\to\infty}\varphi(\tau+s_{n};\tau,a_{n})=v.$ (18)

This readily yields the monotonicity
${\mathcal{A}}_{1}\subseteq{\mathcal{A}}_{2}$ $\Rightarrow$
$\Omega_{{\mathcal{A}}_{1}}(\tau)\subseteq\Omega_{{\mathcal{A}}_{2}}(\tau)$
for all $\tau\in{\mathbb{I}}$.

###### Proof

Let $\tau\in{\mathbb{I}}$. Given a sequence
$y_{n}:=\varphi(\tau+s_{n};\tau,a_{n})\in\varphi(\tau+s_{n};\tau,{\mathcal{A}}(\tau))$
with $s_{n}\xrightarrow[n\to\infty]{}\infty$ and
$a_{n}\in{\mathcal{A}}(\tau)$, by the ${\mathcal{A}}$-asymptotic compactness
of ($\Delta$), we obtain

$\displaystyle
0\leq\operatorname{dist}\bigl{(}y_{n},K\bigr{)}\leq\operatorname{dist}\bigl{(}\varphi(\tau+s_{n},\tau,{\mathcal{A}}(\tau)),K\bigr{)}\xrightarrow[n\to\infty]{}0.$

Since $K\subseteq U$ is compact,
$\operatorname{dist}\bigl{(}y_{n},K\bigr{)}=\min_{k\in K}d(y_{n},k)$. This
implies that there exist a sequence $(k_{n})_{n\in{\mathbb{N}}}$ in $K$
satisfying

$\displaystyle
d(y_{n},k_{n})=\operatorname{dist}\bigl{(}y_{n},K\bigr{)}\xrightarrow[n\to\infty]{}0,$

and a subsequence $(k_{n_{j}})_{j\in{\mathbb{N}}}$ converging to $\bar{k}\in
K$. Thus,

$\displaystyle 0\leq d\bigl{(}y_{n_{j}},\bar{k}\bigr{)}\leq
d\bigl{(}y_{n_{j}},k_{n_{j}}\bigr{)}+d\bigl{(}k_{n_{j}},\bar{k}\bigr{)}=\operatorname{dist}\bigl{(}y_{n_{j}},K\bigr{)}+d\bigl{(}k_{n_{j}},\bar{k}\bigr{)}\xrightarrow[j\to\infty]{}0,$

which implies that the subsequence $(y_{n_{j}})_{j\in{\mathbb{N}}}$ converges
to $\bar{k}$. Hence, by the characterisation (18),
$\bar{k}\in\Omega_{\mathcal{A}}(\tau)$, i.e., $\Omega_{\mathcal{A}}(\tau)$ is
nonempty.

Now choose a sequence $(v_{n})_{n\in{\mathbb{N}}}$ in
$\Omega_{\mathcal{A}}(\tau)$. By Rem. 4, for each fixed $n\in{\mathbb{N}}$,
there is a sequence $\bigl{(}(s_{m}^{n},a_{m}^{n})\bigr{)}_{m\in{\mathbb{N}}}$
satisfying $\lim_{m\to\infty}s_{m}^{n}=\infty$ and
$a_{m}^{n}\in{\mathcal{A}}(\tau)$ for all $m\in{\mathbb{N}}$ such that

$\displaystyle\lim_{m\to\infty}\varphi(\tau+s_{m}^{n};\tau,a_{m}^{n})=v_{n},$

i.e., for every $\varepsilon>0$, there is a
$M=M(n,\varepsilon)\in{\mathbb{N}}$ such that

$\displaystyle
d\bigl{(}\varphi(\tau+s_{m}^{n};\tau,a_{m}^{n}),v_{n}\bigr{)}<\varepsilon\quad\text{for
all }m\geq M.$

Since $\lim_{m\to\infty}s_{m}^{n}=\infty$, there is a
$M^{\prime}=M^{\prime}(n)\in{\mathbb{N}}$ satisfying $s_{m}^{n}>n$ for all
$m\geq M^{\prime}$. Pick $\varepsilon=\frac{1}{n}$,
$(\bar{s}_{n})_{n\in{\mathbb{N}}}$ as a subsequence
$(s_{m_{n}}^{n})_{n\in{\mathbb{N}}}$ of $(s_{m}^{n})_{n\in{\mathbb{N}}}$ and
$(\bar{a}_{n})_{n\in{\mathbb{N}}}$ as a subsequence
$(a_{m_{n}}^{n})_{n\in{\mathbb{N}}}$ of $(a_{m}^{n})_{n\in{\mathbb{N}}}$ in
such a way that

$\displaystyle m_{1}$ $\displaystyle=\max\left\\{1,M,M^{\prime}\right\\}+1,$
$\displaystyle m_{n+1}$
$\displaystyle=\max\left\\{1,M,M^{\prime},m_{n}\right\\}+1\quad\text{for all
}n\in{\mathbb{N}}.$

Clearly, $m_{n+1}>\max\left\\{1,M,M^{\prime},m_{n}\right\\}$ for
$n\in{\mathbb{N}}$. Hence, we constructed a sequence
$\bigl{(}(\bar{s}_{n},\bar{a}_{n})\bigr{)}_{n\in{\mathbb{N}}}$ with
$\lim_{n\to\infty}\bar{s}_{n}=\infty$ and $\bar{a}_{n}\in{\mathcal{A}}(\tau)$
such that

$\displaystyle
d\bigl{(}\varphi(\tau+\bar{s}_{n};\tau,\bar{a}_{n}),v_{n}\bigr{)}<\tfrac{1}{n}\quad\text{for
all }n\in{\mathbb{N}}.$

Therefore,

$\displaystyle 0\leq\operatorname{dist}\bigl{(}v_{n},K\bigr{)}$
$\displaystyle\leq
d\bigl{(}\varphi(\tau+\bar{s}_{n};\tau,\bar{a}_{n}),v_{n}\bigr{)}+\operatorname{dist}\bigl{(}\varphi(\tau+\bar{s}_{n};\tau,{\mathcal{A}}(\tau)),K\bigr{)}$
$\displaystyle\leq\tfrac{1}{n}+\operatorname{dist}\bigl{(}\varphi(\tau+\bar{s}_{n};\tau,{\mathcal{A}}(\tau)),K\bigr{)}\xrightarrow[n\to\infty]{}0.$

Similarly as above, because $K$ is compact, there is a subsequence
$(v_{n_{j}})_{j\in{\mathbb{N}}}$ converging to $\bar{v}\in K$. Moreover, since
$\Omega_{\mathcal{A}}(\tau)$ is closed by definition,
$\bar{v}\in\Omega_{\mathcal{A}}(\tau)$, which implies that
$\Omega_{\mathcal{A}}(\tau)$ is compact. Also note that
$\lim_{n\to\infty}\operatorname{dist}\bigl{(}v_{n},K\bigr{)}=0$, so $v_{n}\in
K$, i.e., $\Omega_{\mathcal{A}}(\tau)\subseteq K$.

Suppose that $\Omega_{\mathcal{A}}(\tilde{\tau})$ does not forward attract
${\mathcal{A}}(\tilde{\tau})$ for some $\tilde{\tau}\in{\mathbb{I}}$, i.e.,
there exist a real $\tilde{\varepsilon}>0$ and a sequence
$(\tilde{s}_{n})_{n\in{\mathbb{N}}}$ in ${\mathbb{N}}$ with
$\lim_{n\to\infty}\tilde{s}_{n}=\infty$ and

$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},{\mathcal{A}}(\tilde{\tau})),\Omega_{\mathcal{A}}(\tilde{\tau})\bigr{)}\geq\tilde{\varepsilon}\quad\text{for
all }n\in{\mathbb{N}}.$ (19)

Although the supremum in the Hausdorff semidistance in the left-hand side of
(19) may not be attained due to no condition ensuring that the image
$\varphi(\tilde{\tau}+s_{n};\tilde{\tau},{\mathcal{A}}(\tilde{\tau}))$ is
compact, there still exists a point
$\tilde{y}_{n}:=\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},\tilde{a}_{n})\in\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},{\mathcal{A}}(\tilde{\tau}))$
for each $n\in{\mathbb{N}}$ with $\tilde{a}_{n}\in{\mathcal{A}}(\tilde{\tau})$
such that

$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},{\mathcal{A}}(\tilde{\tau})),\Omega_{\mathcal{A}}(\tilde{\tau})\bigr{)}-\tfrac{\tilde{\varepsilon}}{2}$
$\displaystyle\leq\operatorname{dist}\bigl{(}\tilde{y}_{n},\Omega_{\mathcal{A}}(\tilde{\tau})\bigr{)}$
$\displaystyle\leq\operatorname{dist}\bigl{(}\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},{\mathcal{A}}(\tilde{\tau})\bigr{)},\Omega_{\mathcal{A}}(\tilde{\tau})).$

The above inequalities in fact give
$\operatorname{dist}\bigl{(}\tilde{y}_{n},\Omega_{\mathcal{A}}(\tilde{\tau})\bigr{)}\geq\tfrac{\tilde{\varepsilon}}{2}$
for all $n\in{\mathbb{N}}$. On the other hand, since
$\tilde{y}_{n}\in\varphi(\tilde{\tau}+s_{n};\tilde{\tau},{\mathcal{A}}(\tilde{\tau}))$,
we obtain

$\displaystyle\operatorname{dist}\bigl{(}\tilde{y}_{n},K\bigr{)}\leq\operatorname{dist}\bigl{(}\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},{\mathcal{A}}(\tilde{\tau})),K\bigr{)}\xrightarrow[n\to\infty]{}0$

and thus $\tilde{y}_{n}\in K$. Moreover, since $K$ is compact, there is a
convergent subsequence $(\tilde{y}_{n_{j}})_{j\in{\mathbb{N}}}$ with limit
$\tilde{y}\in K$. This shows $\tilde{y}\in\Omega_{\mathcal{A}}(\tilde{\tau})$
by definition, and thus

$\displaystyle\operatorname{dist}\bigl{(}\tilde{y},\Omega_{\mathcal{A}}(\tilde{\tau})\bigr{)}<\varepsilon\quad\text{for
all }\varepsilon>0,$

a contradiction to (19). Hence, every $\Omega_{\mathcal{A}}(\tau)$ must
forward attract ${\mathcal{A}}(\tau)$. ∎

###### Corollary 1

If in addition ${\mathcal{A}}$ is positively invariant w.r.t. ($\Delta$), then
the inclusions
$\Omega_{\mathcal{A}}(\tau)\subseteq\Omega_{\mathcal{A}}(\tau+1)$ hold for all
$\tau\in{\mathbb{I}}$.

Owing to the positive invariance of ${\mathcal{A}}$, every
$\Omega_{\mathcal{A}}(\tau)$ can also be written as

$\displaystyle\Omega_{\mathcal{A}}(\tau)=\bigcap_{0\leq
s}\overline{\varphi(\tau+s;\tau,{\mathcal{A}}(\tau))}\quad\text{for all
}\tau\in{\mathbb{I}}.$ (20)

Comparing the respective relations (16) and (11), (17) and (12), (20) and (13)
shows that the fibres $\Omega_{{\mathcal{A}}}(\tau)$ are counterparts to the
pullback limit set $\omega_{{\mathcal{A}}}$. However, their invariance
property is missing. This is easily demonstrated by

###### Example 1

Let ${\mathbb{I}}={\mathbb{N}}_{0}$ and $\alpha\in(0,\tfrac{1}{2}]$. The
difference equation $u_{t+1}=\alpha u_{t}+\alpha^{t}$ in ${\mathbb{R}}$
possesses the positively invariant and bounded set
${\mathcal{A}}={\mathbb{N}}_{0}\times[-\tfrac{1}{\alpha},\tfrac{1}{\alpha}]$.
This yields the apparently not even positively invariant sets
$\Omega_{\mathcal{A}}(\tau)=\left\\{0\right\\}$ for all
$\tau\in{\mathbb{N}}_{0}$.

###### Proof

Let $\tau\in{\mathbb{I}}$. Given a point $v\in\Omega_{\mathcal{A}}(\tau)$,
thanks to
${\mathscr{F}}_{\tau}\bigl{(}{\mathcal{A}}(\tau)\bigr{)}\subseteq{\mathcal{A}}(\tau+1)$
and the process property (3), we obtain

$\displaystyle
v\in\varphi(\tau+s;\tau,{\mathcal{A}}(\tau))\subseteq\varphi(\tau+s;\tau+1,{\mathcal{A}}(\tau+1))\subseteq
U\quad\text{for all }s>0.$

This implies
$\Omega_{\mathcal{A}}(\tau)\subseteq\Omega_{\mathcal{A}}(\tau+1)\subseteq U$.
∎

While the fibres $\Omega_{{\mathcal{A}}}(\tau)$ from Lemma 3 yield the long
term behaviour starting from a single fibre of ${\mathcal{A}}$, the following
result addresses all forward limit sets of ($\Delta$) originating from within
an entire nonautonomous set ${\mathcal{A}}$.

######  Theorem 4.1 (Forward $\omega$-limit sets)

Suppose that ${\mathcal{A}}\neq\emptyset$ is positively invariant and bounded.
If ($\Delta$) is ${\mathcal{A}}$-asymptotically compact with a compact subset
$K\subseteq U$, then

$\displaystyle\omega_{{\mathcal{A}}}^{-}$
$\displaystyle:=\bigcap_{\tau\in{\mathbb{I}}}\Omega_{\mathcal{A}}(\tau),$
$\displaystyle\omega_{\mathcal{A}}^{+}$
$\displaystyle:=\overline{\bigcup_{\tau\in{\mathbb{I}}}\Omega_{\mathcal{A}}(\tau)}\subseteq
K$

are nonempty and compact. In particular, $\omega_{{\mathcal{A}}}^{+}$ forward
attracts ${\mathcal{A}}$, i.e.,

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

and called _forward $\omega$-limit set_ of ${\mathcal{A}}$.

Due to Cor. 1, $\omega_{{\mathcal{A}}}^{+}$ is a union over nondecreasing sets
and actually a limit.

######  Remark 5 (characterisation of $\omega_{{\mathcal{A}}}^{+}$)

The forward $\omega$-limit set $\omega_{\mathcal{A}}^{+}$ consists of all
points $v$ such that there is a sequence
$\bigl{(}(s_{n},\tau_{n},a_{n})\bigr{)}_{n\in{\mathbb{N}}}$ with
$\lim_{n\to\infty}\tau_{n}=\infty$, $(\tau_{n},a_{n})\in{\mathcal{A}}$ and
$s_{n}\in{\mathbb{N}}_{0}$ satisfying

$\displaystyle\lim_{n\to\infty}\varphi(\tau_{n}+s_{n};\tau_{n},a_{n})=v.$

######  Remark 6 (periodic equations)

For $\theta$-periodic difference equations ($\Delta$) and sets ${\mathcal{A}}$
the fibres $\Omega_{\mathcal{A}}(\tau)$ are $\theta$-periodic due to (4) and
(16). If ${\mathcal{A}}$ is moreover positively invariant, then
$\Omega_{\mathcal{A}}$ are even constant and thus
$\omega_{\mathcal{A}}^{-}=\omega_{\mathcal{A}}^{+}=\Omega_{\mathcal{A}}(\tau)$
for all $\tau\in{\mathbb{Z}}$.

######  Proof (of Thm. 4.1)

Since $\Omega_{\mathcal{A}}(\tau)$ is nonempty, there exists a point
$v\in\Omega_{\mathcal{A}}(\tau)$ for all $\tau\in{\mathbb{I}}$. This implies
that $v$ is also contained in $\omega_{\mathcal{A}}^{+}$, i.e., the forward
$\omega$-limit set $\omega_{\mathcal{A}}^{+}$ is nonempty. From Lemma 3, we
know that $\Omega_{\mathcal{A}}(\tau)\subseteq K$ for each
$\tau\in{\mathbb{I}}$. This yields $\omega_{\mathcal{A}}^{+}\subseteq K$.
Moreover, since $K$ is compact and $\omega_{\mathcal{A}}^{+}$ is closed,
$\omega_{\mathcal{A}}^{+}$ is also compact. The claimed limit relation is a
consequence of (17) and
$\Omega_{\mathcal{A}}(\tau)\subseteq\omega_{{\mathcal{A}}}^{+}$.

The properties of $\omega_{{\mathcal{A}}}^{-}$ are an immediate consequence of
the fact that $\omega_{{\mathcal{A}}}^{-}$ is an intersection of nested
compact sets (cf. (sell:you:02, , p. 23, Lemma 22.2(5))). ∎

Note that Exam. 1 demonstrates that both the set $\omega_{\mathcal{A}}^{-}$,
as well as the forward limit sets $\omega_{\mathcal{A}}^{+}$ constructed in
Thm. 4.1 are not invariant or even positively invariant. Yet, under additional
assumptions weaker forms of invariance hold:

######  Theorem 4.2 (Asymptotic positive invariance)

Suppose that ($\Delta$) is ${\mathcal{A}}$-asymptotically compact with a
compact subset $K\subseteq U$ for a bounded, positively invariant
${\mathcal{A}}\neq 0$. If for every sequence
$\bigl{(}(s_{n},\tau_{n})\bigr{)}_{n\in{\mathbb{N}}}$ in
${\mathbb{N}}_{0}\times{\mathbb{I}}$ with $\lim_{n\to\infty}\tau_{n}=\infty$,
one has

$\displaystyle\lim_{n\to\infty}\operatorname{dist}\bigl{(}\varphi(\tau_{n}+s_{n};\tau_{n},K),K\bigr{)}=0,$

then the forward $\omega$-limit set $\omega_{\mathcal{A}}^{+}$ is
_asymptotically positively invariant_ , that is, for every strictly decreasing
sequence $\varepsilon_{n}\searrow 0$, there exists a strictly increasing
sequence $T_{n}\nearrow\infty$ in ${\mathbb{I}}$ as $n\to\infty$ such that

$\displaystyle\varphi(\tau+s;\tau,\omega_{\mathcal{A}}^{+})\subseteq
B_{\varepsilon_{n}}\bigl{(}\omega_{\mathcal{A}}^{+}\bigr{)}\quad\text{for all
}T_{n}\leq\tau,\,s\in{\mathbb{N}}_{0}.$ (21)

Recall the definition of the neighborhoods
$B_{\varepsilon_{n}}\bigl{(}\omega_{\mathcal{A}}^{+}\bigr{)}$ and thus (21)
reads as

$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,\omega_{\mathcal{A}}^{+}),\omega_{\mathcal{A}}^{+}\bigr{)}<\varepsilon_{n}\quad\text{for
all }T_{n}\leq\tau,\,s\in{\mathbb{N}}_{0}.$

###### Proof

Suppose by contradiction that there exists a fixed $\varepsilon_{1}>0$ so that
there is a sequence $\bigl{(}(s_{n}^{1},\tau_{n})\bigr{)}_{n\in{\mathbb{N}}}$
with $0\leq s_{n}^{1}=s_{n}^{1}(\varepsilon_{1})\leq
T_{0}(\tau_{n},\varepsilon_{1})$ and $\tau_{n}\to\infty$ as $n\to\infty$
satisfying

$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tau_{n}+s_{n}^{1};\tau_{n},\omega_{\mathcal{A}}^{+}),\omega_{\mathcal{A}}^{+}\bigr{)}\geq\varepsilon_{1}\quad\text{for
all }n\in{\mathbb{N}}.$ (22)

Since $\varphi$ is continuous and $\omega_{\mathcal{A}}^{+}$ is compact due to
Thm. 4.1, $\varphi(\tau_{n}+s_{n}^{1};\tau_{n},\omega_{\mathcal{A}}^{+})$ is
also a compact set. This implies that there exists a

$y_{n}^{1}=y_{n}^{1}(\varepsilon_{1}):=\varphi(\tau_{n}+s_{n}^{1};\tau_{n},w_{n}^{1})\in\varphi(\tau_{n}+s_{n}^{1};\tau_{n},\omega_{\mathcal{A}}^{+})\subseteq\varphi(\tau_{n}+s_{n}^{1};\tau_{n},K)$

with $w_{n}^{1}=w_{n}^{1}(\varepsilon_{1})\in\omega_{\mathcal{A}}^{+}\subseteq
K$ such that

$\displaystyle\operatorname{dist}\bigl{(}y_{n}^{1},\omega_{\mathcal{A}}^{+}\bigr{)}$
$\displaystyle=\operatorname{dist}\bigl{(}\varphi(\tau_{n}+s_{n}^{1};\tau_{n},w_{n}^{1}),\omega_{\mathcal{A}}^{+}\bigr{)}$
$\displaystyle=\operatorname{dist}\bigl{(}\varphi(\tau_{n}+s_{n}^{1};\tau_{n},\omega_{\mathcal{A}}^{+}),\omega_{\mathcal{A}}^{+}\bigr{)}\geq\varepsilon_{1}\quad\text{for
all }n\in{\mathbb{N}}.$

On the other hand, with $y_{n}^{1}\in\varphi(\tau_{n}+s_{n}^{1};\tau_{n},K)$,
by the assumption, we obtain

$\displaystyle
0\leq\operatorname{dist}\bigl{(}y_{n}^{1},K\bigr{)}\leq\operatorname{dist}\bigl{(}\varphi(\tau_{n}+s_{n}^{1};\tau_{n},K),K\bigr{)}\xrightarrow[n\to\infty]{}0,$

implying that $y_{n}^{1}\in K$. Additionally, since the set $K$ is compact,
there is a subsequence $(y_{n_{j}}(\varepsilon_{1}))_{j\in{\mathbb{N}}}$
converging to $\bar{y}_{1}=\bar{y}_{1}(\varepsilon_{1})\in K$. Therefore, by
definition, the inclusion
$\bar{y}_{1}\in\Omega_{\mathcal{A}}(\tau)\subseteq\omega_{\mathcal{A}}^{+}$
leads to $\bar{y}_{1}\in\omega_{\mathcal{A}}^{+}$, i.e.,
$\operatorname{dist}\bigl{(}\bar{y}_{1},\omega_{\mathcal{A}}^{+}\bigr{)}<\varepsilon$
for all $\varepsilon>0$, a contradiction to (22). Thus, for this
$\varepsilon_{1}>0$, there exists an integer $s_{1}=s_{1}(\varepsilon_{1})$
large enough such that

$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tau+s_{1};\tau,\omega_{\mathcal{A}}^{+}),\omega_{\mathcal{A}}^{+}\bigr{)}<\varepsilon_{1}.$

Repeating inductively with $\varepsilon_{n+1}<\varepsilon_{n}$ and
$s_{n+1}(\varepsilon_{n+1})>s_{n}(\varepsilon_{n})$ for all
$n\in{\mathbb{N}}$, we then obtain that $\omega_{\mathcal{A}}^{+}$ is
asymptotically positively invariant. ∎

######  Theorem 4.3 (Asymptotic negative invariance)

Suppose that ($\Delta$) is strongly ${\mathcal{A}}$-asymptotically compact
with a compact subset $K\subseteq U$ for a bounded, positively invariant
${\mathcal{A}}\neq 0$. If for every $\varepsilon>0$ and $T\in{\mathbb{N}}$,
there exists a real $\delta=\delta(\varepsilon,T)>0$ such that for all
$\tau\in{\mathbb{I}}$, one has the implication

$\displaystyle\left.\begin{array}[]{c}u_{0},v_{0}\in{\mathcal{A}}(\tau)\cup
K,\\\ d(u_{0},v_{0})<\delta\end{array}\right\\}\Rightarrow\sup_{0\leq s\leq
T}d\bigl{(}\varphi(\tau+s;\tau,u_{0}),\varphi(\tau+s;\tau,v_{0})\bigr{)}<\varepsilon,$
(25)

then the forward $\omega$-limit set $\omega_{\mathcal{A}}^{+}$ is
_asymptotically negatively invariant_ , that is, for all
$u\in\omega_{\mathcal{A}}^{+}$, $\varepsilon>0$ and $T\in{\mathbb{N}}$, there
are integers $s^{\ast}=s^{\ast}(\varepsilon)$ satisfying
$\tau+s^{\ast}-T\in{\mathbb{I}}$ and
$u_{\varepsilon}^{\ast}\in\omega_{\mathcal{A}}^{+}$ such that

$d\bigl{(}\varphi(\tau+s^{\ast};\tau+s^{\ast}-T,u_{\varepsilon}^{\ast}),u\bigr{)}<\varepsilon.$

###### Proof

Consider reals $\varepsilon>0$ and $T\in{\mathbb{N}}$ and take a point
$u\in\omega_{\mathcal{A}}^{+}$. Thanks to Rem. 5, there is a sequence
$\bigl{(}(s_{n},\tau_{n},a_{n})\bigr{)}_{n\in{\mathbb{N}}}$ with
$T<s_{n}=s_{n}(\varepsilon)\to\infty$, $\tau_{n}\to\infty$ as $n\to\infty$,
$\tau_{n}\in{\mathbb{I}}$, $\tau_{n}+s_{n}-T\in{\mathbb{I}}$ and
$a_{n}=a_{n}(\varepsilon)\in{\mathcal{A}}(\tau_{n})$, and an integer
$N=N_{1}(\varepsilon)$ with

$\displaystyle
d\bigl{(}\varphi(\tau_{n}+s_{n};\tau_{n},a_{n}),u\bigr{)}<\tfrac{\varepsilon}{2}\quad\text{for
all }n\geq N_{1}(\varepsilon).$

Given a sequence
$y_{n}:=\varphi(t^{\ast}_{n}-T;\tau_{n},a_{n})\in\varphi(t^{\ast}_{n}-T;\tau_{n},{\mathcal{A}}(\tau_{n}))\subseteq{\mathcal{A}}(t^{\ast}_{n}-T)$
with $t^{\ast}_{n}=t^{\ast}_{n}(\varepsilon):=\tau_{n}+s_{n}$,
$s_{n}-T\to\infty$, $\tau_{n}\to\infty$ as $n\to\infty$,
$\tau_{n}\in{\mathbb{I}}$ and $a_{n}\in{\mathcal{A}}(\tau_{n})$, by the strong
${\mathcal{A}}$-asymptotic compactness of ($\Delta$), we obtain

$\displaystyle
0\leq\operatorname{dist}\bigl{(}y_{n},K\bigr{)}\leq\operatorname{dist}\bigl{(}\varphi(t^{\ast}_{n}-T;\tau_{n},{\mathcal{A}}(\tau_{n})),K\bigr{)}\xrightarrow[n\to\infty]{}0.$

Since $K$ is compact, $\operatorname{dist}\bigl{(}y_{n},K\bigr{)}=\min_{k\in
K}d(y_{n},k)$. This implies that there exist a sequence
$(k_{n})_{n\in{\mathbb{N}}}$ in $K$ such that

$\displaystyle
d(y_{n},k_{n})=\operatorname{dist}\bigl{(}y_{n},K\bigr{)}\xrightarrow[n\to\infty]{}0,$

and a subsequence $(k_{n_{j}})_{j\in{\mathbb{N}}}$ converging to
$\bar{k}=\bar{k}(\varepsilon)\in K$. Thus,

$\displaystyle 0\leq d\bigl{(}y_{n_{j}},\bar{k}\bigr{)}\leq
d\bigl{(}y_{n_{j}},k_{n_{j}}\bigr{)}+d\bigl{(}k_{n_{j}},\bar{k}\bigr{)}=\operatorname{dist}\bigl{(}y_{n},K\bigr{)}+d\bigl{(}k_{n_{j}},\bar{k}\bigr{)}\xrightarrow[n\to\infty]{}0,$

implying
$y_{n_{j}}:=\varphi(t^{\ast}_{n_{j}}-T;\tau_{n_{j}},a_{n_{j}})\xrightarrow[j\to\infty]{}\bar{k}$
with $t^{\ast}_{n_{j}}:=\tau_{n_{j}}+s_{n_{j}}$. Hence, by Rem. 5, one has
$\bar{k}\in\omega_{\mathcal{A}}^{+}$. Moreover, with
$y_{n_{j}}\in{\mathcal{A}}(t^{\ast}_{n_{j}}-T)$ and $\bar{k}\in K$, by the
assumption, we obtain for an integer $N_{2}(\varepsilon,T)$ large enough,

$\displaystyle
d\bigl{(}\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,y_{n_{j}}),\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,\bar{k})\bigr{)}<\tfrac{\varepsilon}{2}\quad\text{for
all }n_{j}\geq N_{2}(\varepsilon,T).$

Now the triangle inequality and the process property (3) yield

$\displaystyle
d\bigl{(}\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,\bar{k}),u\bigr{)}$
$\displaystyle\leq$ $\displaystyle
d\bigl{(}\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,\bar{k}),\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,y_{n_{j}})\bigr{)}+d\bigl{(}\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,y_{n_{j}}),u\bigr{)}$
$\displaystyle=$ $\displaystyle
d\bigl{(}\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,\bar{k}),\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,y_{n_{j}})\bigr{)}$
$\displaystyle\qquad+d\bigl{(}\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,\varphi(t^{\ast}_{n_{j}}-T;\tau_{n_{j}},a_{n_{j}})),u\bigr{)}$
$\displaystyle=$ $\displaystyle
d\bigl{(}\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,\bar{k}),\varphi(t^{\ast}_{n_{j}};t^{\ast}_{n_{j}}-T,y_{n_{j}})\bigr{)}+d\bigl{(}\varphi(t^{\ast}_{n_{j}};\tau_{n_{j}},a_{n_{j}}),u\bigr{)}$
$\displaystyle<$
$\displaystyle\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{2}=\varepsilon\quad\text{for
all }n\geq N_{1}(\varepsilon),n_{j}\geq N_{2}(\varepsilon,T).$

Setting $u_{\varepsilon}^{\ast}:=\bar{k}\in\omega_{\mathcal{A}}^{+}$, we then
obtain that $\omega_{\mathcal{A}}^{+}$ is asymptotically negatively invariant.
∎

###  4.2 Asymptotically autonomous difference equations

In general it is difficult to obtain the forward limit set
$\omega_{\mathcal{A}}^{+}$ given as limit of the fibres
$\Omega_{\mathcal{A}}(\tau)$ explicitly. This situation simplifies, if
($\Delta$) behaves asymptotically as an autonomous difference equation

$\boxed{u_{t+1}={\mathscr{F}}(u_{t})}$ ($\Delta^{\infty}$)

with right-hand side ${\mathscr{F}}:U\to U$ in a sense to be specified below.
Here, it is common to denote the iterates of ${\mathscr{F}}$ by
${\mathscr{F}}^{s}:U\to U$, $s\in{\mathbb{N}}_{0}$. A maximal, invariant and
nonempty compact set $A^{\ast}\subseteq U$ attracting all bounded subsets of
$U$ is called _global attractor_ of ($\Delta^{\infty}$) (cf. (hale:88, , p.
17)).

For a class of nonautonomous equations ($\Delta$) introduced next, the sets
$\Omega_{\mathcal{A}}(\tau)$, $\tau\in{\mathbb{I}}$, turn out to be constant
and determined by the global attractor $A^{\ast}$ of ($\Delta^{\infty}$).

######  Theorem 4.4 (Asymptotically autonomous difference equations)

Suppose that ($\Delta^{\infty}$) has a bounded absorbing set $A\subseteq U$
and a global attractor $A^{\ast}\subseteq A$. If
${\mathcal{A}}:={\mathbb{I}}\times A$ is a forward absorbing set of ($\Delta$)
and the condition

$\lim_{s\to\infty}\sup_{a\in
A}d(\varphi(\tau+s;\tau,a),{\mathscr{F}}^{s}(a))=0\quad\text{for all
}\tau\in{\mathbb{I}}$ (26)

holds, then $\Omega_{{\mathcal{A}}}(\tau)=A^{\ast}$ for all
$\tau\in{\mathbb{I}}$ and in particular
$\omega_{{\mathcal{A}}}^{-}=\omega_{{\mathcal{A}}}^{+}=A^{\ast}$.

###### Remark 7

Asymptotically autonomous difference equations were also studied in
cui:kloeden:19  in order to show that the fibres ${\mathcal{A}}^{\ast}(\tau)$
of a pullback attractor ${\mathcal{A}}^{\ast}$ to ($\Delta$) converge to the
global attractor $A^{\ast}$ of the limit equation ($\Delta^{\infty}$) as
$\tau\to\infty$. In these results, however, asymptotic autonomy is based on
e.g. the limit relation

$\lim_{\tau\to\infty}d(\varphi(\tau+s;\tau,a_{\tau}),{\mathscr{F}}^{s}(a_{0}))=0\quad\text{for
all }s\in{\mathbb{N}}_{0}$

(see (cui:kloeden:19, , Thm. 1)) with sequences $(a_{n})_{n\in{\mathbb{N}}}$
converging to some $a_{0}$. This condition is clearly different from (26).

###### Proof

Given any $\tau\in{\mathbb{I}}$, we have to show two inclusions:

$(\subseteq)$ Let $v\in\Omega_{\mathcal{A}}(\tau)$. Due to Rem. 4 there exist
sequences $a_{n}\in A$ and $s_{n}\to\infty$ as $n\to\infty$ with

$\lim_{n\to\infty}\varphi(\tau+s_{n};\tau,a_{n})=v$

and it follows from (26) that

$\displaystyle\operatorname{dist}\bigl{(}v,A^{\ast}\bigr{)}$
$\displaystyle\leq$ $\displaystyle
d(v,\varphi(\tau+s_{n};\tau,a_{n}))+d(\varphi(\tau+s_{n};\tau,a_{n}),{\mathscr{F}}^{s_{n}}(a_{n}))+\operatorname{dist}\bigl{(}{\mathscr{F}}^{s_{n}}(a_{n}),A^{\ast}\bigr{)}$
$\displaystyle\leq$ $\displaystyle
d(v,\varphi(\tau+s_{n};\tau,a_{n}))+d(\varphi(\tau+s_{n};\tau,a_{n}),{\mathscr{F}}^{s_{n}}(a_{n}))+\operatorname{dist}\bigl{(}{\mathscr{F}}^{s_{n}}(A),A^{\ast}\bigr{)}\xrightarrow[n\to\infty]{}0,$

since the global attractor $A^{\ast}$ of ($\Delta^{\infty}$) attracts the
absorbing set $A$. This implies that $v\in A^{\ast}$, and since $v$ was
arbitrary, the inclusion $\Omega_{\mathcal{A}}(\tau)\subseteq A^{\ast}$ holds
for $\tau\in{\mathbb{I}}$.

$(\supseteq)$ Conversely, since $A^{\ast}$ is compact, there exists an
$a^{\ast}\in A^{\ast}$ with

$\displaystyle\operatorname{dist}\bigl{(}A^{\ast},\Omega_{{\mathcal{A}}}(\tau)\bigr{)}$
$\displaystyle=\operatorname{dist}\bigl{(}a^{\ast},\Omega_{{\mathcal{A}}}(\tau)\bigr{)}$
$\displaystyle\leq\operatorname{dist}\bigl{(}a^{\ast},\varphi(\tau+s;\tau,A^{\ast})\bigr{)}+\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,A^{\ast}),\Omega_{\mathcal{A}}(\tau)\bigr{)}$

for all $s\in{\mathbb{I}}$ and we separately estimate the two terms on the
right-hand side of this inequality. First, due to the invariance of $A^{\ast}$
there exists $a_{s}^{\ast}\in A^{\ast}$ with
$a^{\ast}={\mathscr{F}}^{s}(a_{s}^{\ast})$ and therefore

$\displaystyle\operatorname{dist}\bigl{(}a^{\ast},\varphi(\tau+s;\tau,A^{\ast})\bigr{)}$
$\displaystyle\leq
d(a^{\ast},\varphi(\tau+s;\tau,a_{s}^{\ast}))=d({\mathscr{F}}^{s}(a_{s}^{\ast}),\varphi(\tau+s;\tau,a_{s}^{\ast}))$
$\displaystyle\leq\sup_{a\in
A^{\ast}}d(\varphi(\tau+s;\tau,a),{\mathscr{F}}^{s}(a))\xrightarrow[s\to\infty]{\eqref{thmaa1}}0.$

Second, from $A^{\ast}\subseteq A={\mathcal{A}}(\tau)$ one has

$\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,A^{\ast}),\Omega_{\mathcal{A}}(\tau)\bigr{)}\leq\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,{\mathcal{A}}(\tau)),\Omega_{\mathcal{A}}(\tau)\bigr{)}\xrightarrow[s\to\infty]{\eqref{lem42b}}0,$

which guarantees the remaining inclusion
$A^{\ast}\subseteq\Omega_{\mathcal{A}}(\tau)$.

Hence, all $\Omega_{\mathcal{A}}(\tau)$ are constant, thus
$\omega_{\mathcal{A}}^{-}=A^{\ast}$ and
$\omega_{\mathcal{A}}^{+}=\overline{A^{\ast}}=A^{\ast}$. ∎

The following simple example illustrates the condition (26):

######  Example 2 (Beverton-Holt equation)

If $\alpha>1$, then it is well known that all solutions to the autonomous
Beverton-Holt equation $v_{t+1}=\tfrac{\alpha v_{t}}{1+v_{t}}$ starting with a
positive initial value converge to $\alpha-1$ (see (lutscher:19, , pp. 13ff)).
We establish that an asymptotically autonomous, but nonautonomous Beverton-
Holt equation

$\displaystyle v_{t+1}$ $\displaystyle=\frac{\tilde{a}_{t}v_{t}}{1+v_{t}},$
$\displaystyle\tilde{a}_{t}$ $\displaystyle:=\frac{f_{t+1}}{f_{t}}\alpha$ (27)

shares this behaviour, whenever the sequence $(f_{t})_{t\in{\mathbb{I}}}$ in
$(0,\infty)$ satisfies

$\lim_{t\to\infty}\sum_{s=0}^{t-1}\frac{f_{\tau+s}}{f_{\tau+t}}\alpha^{s-t}=\frac{1}{\alpha-1}\quad\text{for
all }\tau\in{\mathbb{I}}$ (28)

and grows at most polynomially. For instance, the relation (28) holds for the
sequences $f_{t}=(t+c)^{n}$ with $c\in{\mathbb{R}}$,
$n\in\left\\{1,2,3,4\right\\}$. Indeed, the explicit representation

$\varphi(\tau+t;\tau,a)=\frac{(\alpha-1)a}{\tfrac{\alpha-1}{\alpha^{t}}\tfrac{f_{\tau}}{f_{\tau+t}}+(\alpha-1)a\sum_{s=0}^{t-1}\tfrac{f_{\tau+s}}{f_{\tau+t}}\alpha^{s-t}}$

of the general solution to (27) yields that
$\lim_{t\to\infty}\varphi(t+\tau;\tau,a)=\alpha-1$ holds uniformly in
$a\in[\bar{a},\infty)$ for any $\bar{a}>0$. Consequently, one has

$\displaystyle\sup_{a\geq\bar{a}}\left|\varphi(\tau+t;\tau,a)-F^{t}(a)\right|\leq$
$\displaystyle\sup_{a\geq\bar{a}}\left|\varphi(\tau+t;\tau,a)-(\alpha-1)\right|+\sup_{a\geq\bar{a}}\left|\alpha-1-F^{t}(a)\right|$
$\displaystyle\xrightarrow[t\to\infty]{}0$

and therefore (26) is valid with arbitrary subsets
$A\subseteq[\bar{a},\infty)$.

We continue with two sufficient criteria for the condition (26) to hold.
Thereto we assume in the remaining subsection that $(X,\left\|\cdot\right\|)$
is a Banach space.

######  Theorem 4.5 (Asymptotically autonomous linear difference equations)

Suppose that ${\mathscr{L}}_{t},{\mathscr{L}}\in L(X)$, $b_{t},b\in X$,
$t\in{\mathbb{I}}$, satisfy

$\displaystyle\lim_{t\to\infty}{\mathscr{L}}_{t}$
$\displaystyle={\mathscr{L}},$ $\displaystyle\lim_{t\to\infty}b_{t}$
$\displaystyle=b.$ (29)

If $\rho({\mathscr{L}})<1$, then ($\Delta^{\infty}$) with right-hand side
${\mathscr{F}}(u)={\mathscr{L}}u+b$ and ($\Delta$) with right-hand side
${\mathscr{F}}_{t}(u)={\mathscr{L}}_{t}u+b_{t}$ fulfill the limit relation
(26) on every bounded subset $A\subseteq X$.

###### Proof

We note that (29) implies that ($\Delta$) is uniformly exponentially stable
and that the sequences $({\mathscr{L}}_{t})_{t\in{\mathbb{I}}}$,
$(b_{t})_{t\in{\mathbb{I}}}$ are bounded. Thus, (aulbach:minh:96, , Cor. 5
with $H=\ell^{\infty}({\mathbb{I}},X)$) implies that $\varphi(\cdot;\tau,0)$
is bounded and the representation
$\varphi(t;\tau,u_{\tau})=\Phi(t,\tau)u_{\tau}+\varphi(t;\tau,0)$ for all
$t\geq\tau$, $u_{\tau}\in X$ shows that $\varphi(\cdot;\tau,u_{\tau})$ is
bounded uniformly in $u_{\tau}$ from bounded subsets of $X$. Now it is easy to
see that the difference $\varphi(t;\tau,a)-{\mathscr{F}}^{t-\tau}(a)$ solves
the initial value problem

$\displaystyle w_{t+1}$ $\displaystyle={\mathscr{L}}w_{t}+\tilde{b}_{t},$
$\displaystyle u_{\tau}$ $\displaystyle=0,$

whose inhomogeneity
$\tilde{b}_{t}:=({\mathscr{L}}_{t}-{\mathscr{L}})\varphi(t;\tau,a)+b_{t}-b$
satisfies $\lim_{t\to\infty}\tilde{b}_{t}=0$ uniformly in $a$ from bounded
subsets of $X$. Now using (aulbach:minh:96, , Cor. 5 with
$H=c_{0}({\mathbb{I}},X)$) guarantees that the sequence
$\varphi(t;\tau,a)-{\mathscr{F}}^{t-\tau}(a)$ converges to $0$ as $t\to\infty$
uniformly in $a$ from bounded subsets of $X$, that is, in particular (26)
holds. ∎

######  Theorem 4.6 (Asymptotically autonomous semilinear difference
equations)

Let ${\mathscr{F}}_{t}:U\to U$ be of semilinear form (5) such that

$\left\|\Phi(t,s)\right\|\leq K\alpha^{t-s}\quad\text{for all }s\leq t$ (30)

and

$\left\|{\mathscr{N}}_{t}(u)-{\mathscr{N}}_{t}(\bar{u})\right\|\leq
L\left\|u-\bar{u}\right\|\quad\text{for all }t\in{\mathbb{I}},\,u,\bar{u}\in
U$ (31)

with $K\geq 1$, $\alpha\in(0,1)$ and $L\in(0,\tfrac{1-\alpha}{K})$. If
${\mathscr{L}}\in L(X)$ and ${\mathscr{N}}:U\to X$ satisfy

  * (i)

there exists a $K_{1}\geq 0$ such that
$\left\|{\mathscr{L}}_{t}-{\mathscr{L}}\right\|\leq K_{1}\alpha^{t}$ for all
$t\in{\mathbb{I}}$,

  * (ii)

for every $r>0$ there exists a $K_{2}(r)\geq 1$ such that

$\sup_{u\in U\cap
B_{r}(0)}\left\|{\mathscr{N}}_{t}(u)-{\mathscr{N}}(u)\right\|\leq
K_{2}(r)\alpha^{t}\quad\text{for all }t\in{\mathbb{I}},$

then ($\Delta^{\infty}$) with right-hand side
${\mathscr{F}}(u)={\mathscr{L}}u+{\mathscr{N}}(u)$ and ($\Delta$) fulfill the
limit relation (26) even exponentially on every bounded subset $A\subseteq X$.

The assumption (30) holds in case $\rho({\mathscr{L}})<1$ with
$\alpha\in(\rho({\mathscr{L}}),1)$ (see (przyluski:88, , Thm. 5)).

###### Proof

We proceed in two steps:

(I) Claim: _All solutions to the autonomous equation ( $\Delta^{\infty}$) are
bounded, i.e.,_

$\left\|{\mathscr{F}}^{t}(a)\right\|\leq K\left\|a\right\|+\frac{K}{1-\alpha-
KL}\left\|{\mathscr{N}}(0)\right\|\quad\text{for all }t\geq 0,\,a\in X.$ (32)

This is a consequence of (poetzsche:10b, , p. 155, Thm. 3.5.8(a)).

(II) Let $a\in U$ and we abbreviate $u_{t}:=\varphi(t;\tau,a)$,
$v_{t}:={\mathscr{F}}^{t-\tau}(a)$. Due to step (I) the sequence
$(v_{t})_{t\geq\tau}$ is bounded and we choose $r>0$ so large that
$\left\|v_{t}\right\|<r$ for all $t\in{\mathbb{I}}$. It is easy to see that
the difference $w_{t}:=u_{t}-v_{t}$ satisfies the equation

$w_{t+1}={\mathscr{L}}_{t}w_{t}+({\mathscr{L}}_{t}-{\mathscr{L}})v_{t}+{\mathscr{N}}_{t}(v_{t})-{\mathscr{N}}(v_{t})+{\mathscr{N}}_{t}(u_{t})-{\mathscr{N}}_{t}(v_{t})$
(33)

and fulfills the initial condition $u_{\tau}-v_{\tau}=a-a=0$. Using the
variation of constants formula (poetzsche:10b, , p. 100, Thm. 3.1.16) results

$w_{t}=\sum_{s=\tau}^{t-1}\Phi(t,s+1)\left[({\mathscr{L}}_{s}-{\mathscr{L}})v_{s}+{\mathscr{N}}_{s}(v_{s})-{\mathscr{N}}(v_{s})+{\mathscr{N}}_{s}(u_{s})-{\mathscr{N}}_{s}(v_{s})\right]\quad\text{for
all }t\geq\tau,$

consequently

$\displaystyle\left\|w_{t}\right\|\alpha^{-t}$
$\displaystyle\stackrel{{\scriptstyle\eqref{thmaasl1}}}{{\leq}}$
$\displaystyle\frac{K}{\alpha}\sum_{s=\tau}^{t-1}\alpha^{-s}\left\|({\mathscr{L}}_{s}-{\mathscr{L}})v_{s}+{\mathscr{N}}_{s}(v_{s})-{\mathscr{N}}(v_{s})\right\|+\frac{K}{\alpha}\sum_{s=\tau}^{t-1}\alpha^{-s}\left\|{\mathscr{N}}_{s}(u_{s})-{\mathscr{N}}_{s}(v_{s})\right\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{thmaasl2}}}{{\leq}}$
$\displaystyle\frac{K}{\alpha}\sum_{s=\tau}^{t-1}\alpha^{-s}\left\|({\mathscr{L}}_{s}-{\mathscr{L}})v_{s}+{\mathscr{N}}_{s}(v_{s})-{\mathscr{N}}(v_{s})\right\|+\frac{KL}{\alpha}\sum_{s=\tau}^{t-1}\alpha^{-s}\left\|w_{s}\right\|$

and the Grönwall inequality (poetzsche:10b, , p. 348, Prop. A.2.1(a)) yields

$\left\|w_{t}\right\|\leq\frac{K}{\alpha}\sum_{s=\tau}^{t-1}(\alpha+KL)^{t-s-1}\left\|({\mathscr{L}}_{s}-{\mathscr{L}})v_{s}+{\mathscr{N}}_{s}(v_{s})-{\mathscr{N}}(v_{s})\right\|\quad\text{for
all }\tau\leq t$

If we replace $t$ by $\tau+t$, then it results

$\displaystyle\left\|\varphi(\tau+t;\tau,a)-{\mathscr{F}}^{t}(a)\right\|$
$\displaystyle\leq$
$\displaystyle\frac{K}{\alpha}\sum_{s=\tau}^{t+\tau-1}(\alpha+KL)^{t+\tau-s-1}\left\|({\mathscr{L}}_{s}-{\mathscr{L}})v_{s}+{\mathscr{N}}_{s}(v_{s})-{\mathscr{N}}(v_{s})\right\|$
$\displaystyle\stackrel{{\scriptstyle(i)}}{{\leq}}$
$\displaystyle\frac{K}{\alpha}\sum_{s=\tau}^{t+\tau-1}(\alpha+KL)^{t+\tau-s-1}\bigl{(}K_{1}r\alpha^{s}+\left\|{\mathscr{N}}_{s}(v_{s})-{\mathscr{N}}(v_{s})\right\|\bigr{)}$
$\displaystyle\stackrel{{\scriptstyle(ii)}}{{\leq}}$
$\displaystyle\frac{K\bigl{(}K_{1}r+K_{2}(r)\bigr{)}}{\alpha}(\alpha+KL)^{t+\tau-1}\sum_{s=\tau}^{t+\tau-1}\bigl{(}\tfrac{\alpha}{\alpha+KL}\bigr{)}^{s}\xrightarrow[t\to\infty]{}0$

and therefore even exponential convergence holds in (26). ∎

###  4.3 Forward attractors

In the previous Sect. 3.2 we constructed pullback attractors of nonautonomous
difference equations ($\Delta$) by means of Prop. 1 applied to a pullback
absorbing, positively invariant nonautonomous set. Now it is our goal is to
obtain a corresponding concept in forward time.

Mimicking the approach for pullback attractors, we define a _forward
attractor_ ${\mathcal{A}}^{+}\subset{\mathcal{U}}$ of ($\Delta$) as a
nonempty, compact and invariant nonautonomous set forward attracting every
bounded subset ${\mathcal{B}}\subseteq{\mathcal{U}}$, i.e.,

$\displaystyle\lim_{s\to\infty}\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,{\mathcal{B}}(\tau)),{\mathcal{A}}^{+}(\tau+s)\bigr{)}=0\quad\text{for
all }\tau\in{\mathbb{I}}.$ (34)

As demonstrated in e.g. (kloeden:yang:16, , Sect. 4), forward attractors need
not to be unique. They are Lyapunov asymptotically stable, that is, Lyapunov
stable and attractive in the sense of (34) (see (kloeden:lorenz:16, , Prop.
3.1)). While it is often claimed in the literature that there is no
counterpart to the characterisation (15) of pullback attractors
${\mathcal{A}}^{\ast}$ for forward attractors ${\mathcal{A}}^{+}$ of
nonautonomous equations, a suitable construction will be given now.

A nonautonomous difference equation ($\Delta$) is denoted as _forward
dissipative_ , if there exists a bounded set
${\mathcal{A}}\subseteq{\mathcal{U}}$ such that for every
$\tau\in{\mathbb{I}}$ and bounded ${\mathcal{B}}\subseteq{\mathcal{U}}$ there
is an _absorption time_ $S=S(\tau,{\mathcal{B}})\in{\mathbb{N}}$ such that

$\varphi(\tau+s;\tau,{\mathcal{B}}(\tau))\subseteq{\mathcal{A}}(\tau+s)\quad\text{for
all }s\geq S;$ (35)

one says that ${\mathcal{A}}$ is as a _forward absorbing_ set.

######  Proposition 3 (Forward absorbing set)

On a Banach space $X$, let ${\mathscr{F}}_{t}:U\to U$ be of the semilinear
form (5) satisfying (7), (8) and let $\rho>0$. If the limit relations

$\displaystyle\lim_{s\to\infty}\prod_{r=\tau}^{\tau+s-1}(\alpha_{r}+Ka_{r})$
$\displaystyle=0,$ $\displaystyle R_{\tau}$
$\displaystyle:=K\lim_{t\to\infty}\sum_{s=\tau}^{t-1}b_{s}\prod_{r=s+1}^{t-1}(\alpha_{r}+Ka_{r})<\infty$

hold for all $\tau\in{\mathbb{I}}$ and
$\sup_{\tau\in{\mathbb{I}}}R_{\tau}<\infty$, then the difference equation
($\Delta$) is forward dissipative with absorbing set
${\mathcal{A}}:=\left\\{(t,u)\in{\mathcal{U}}:\,\left\|u\right\|\leq\rho+\sup_{\tau\in{\mathbb{I}}}R_{\tau}\right\\}$.

For constant positive sequences $\alpha_{t}\equiv\alpha$, $a_{t}\equiv a$,
$b_{t}\equiv b$ in (8) satisfying $\alpha+Ka<1$, both the pullback absorbing
set from Prop. 2 and the forward absorbing set from Prop. 3 have constant
fibres and simplify to ${\mathcal{A}}={\mathbb{I}}\times
B_{\rho+\tfrac{Kb}{1-\alpha-aK}}(0)$.

###### Proof

The claim follows readily from relation (9) in Lemma 1. ∎

Using (kloeden:lorenz:16, , Prop. 3.2 with compact replaced by open and
bounded) one shows

###### Proposition 4

Every bounded forward attractor has a nonempty, positively invariant, closed
and bounded forward absorbing set.

First, this Prop. 4 allows us to choose a closed and bounded, positively
invariant set ${\mathcal{A}}\subseteq{\mathcal{U}}$. We then deduce a
nonempty, invariant and compact nonautonomous set
${\mathcal{A}}^{\star}\subseteq{\mathcal{A}}$ from Prop. 1.

Second, the construction of forward attractors requires
${\mathbb{I}}={\mathbb{Z}}$. Different from the pullback situation (with
${\mathcal{A}}$ being pullback absorbing), having an forward absorbing set
${\mathcal{A}}$ does not ensure the forward convergence within
${\mathcal{A}}$, i.e.,

$\lim_{s\to\infty}\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,{\mathcal{A}}(\tau)),{\mathcal{A}}^{\star}(\tau)\bigr{)}=0\quad\text{for
all }\tau\in{\mathbb{Z}}$

and in particular not forward convergence of a general bounded nonautonomous
set ${\mathcal{B}}\subseteq{\mathcal{U}}$ to ${\mathcal{A}}^{\ast}$. This is
because ($\Delta$) may have forward limit points starting in ${\mathcal{A}}$
which are not forward limit points from within ${\mathcal{A}}^{\star}$.
Corresponding examples illustrating this are given in
kloeden:poetzsche:rasmussen:11 .

Now on the one hand, the set of forward $\omega$-limit points for the dynamics
starting in ${\mathcal{A}}^{\star}$ is given by

$\omega_{{\mathcal{A}}}^{\star}:=\bigcap_{\tau\in{\mathbb{Z}}}\overline{\bigcup_{0\leq
s}\varphi(\tau+s;\tau,{\mathcal{A}}^{\star}(\tau))}=\bigcap_{\tau\in{\mathbb{Z}}}\overline{\bigcup_{0\leq
s}{\mathcal{A}}^{\star}(\tau+s)}\subseteq U$

and is nonempty and compact as intersection of nested compact sets. It
consists of all points $u\in U$ such that there is a sequence
$\bigl{(}(s_{n},a_{n})\bigr{)}_{n\in{\mathbb{N}}}$ with
$\lim_{n\to\infty}s_{n}=\infty$ and $a_{n}\in{\mathcal{A}}(\tau+s_{n})$ with
$\tau\in{\mathbb{Z}}$ satisfying

$\displaystyle\lim_{n\to\infty}\varphi(\tau+s_{n};\tau,a_{n})=u.$

On the other hand, the set of forward limit points
$\omega_{{\mathcal{A}}}^{+}$ from within ${\mathcal{A}}$ was constructed in
Thm. 4.1. With ${\mathcal{A}}$ being positively invariant, the chain of
inclusions
$\omega_{{\mathcal{A}}}^{-}\subseteq\omega_{{\mathcal{A}}}^{+}\subseteq K$
holds, while $\omega_{{\mathcal{A}}}^{\star}$ is not necessarily contained in
$\omega_{\mathcal{A}}^{-}$ (see Exam. 3 for an illustration), as well as

$\displaystyle\lim_{t\to\infty}\operatorname{dist}\bigl{(}{\mathcal{A}}^{\star}(t),\omega_{{\mathcal{A}}}^{-}\bigr{)}=0.$

###### Theorem 4.7

Suppose that ($\Delta$) has a positively invariant, closed and bounded set
${\mathcal{A}}\neq\emptyset$. If the assumptions in Prop. 1 and Thms. 4.2–4.3
hold, then the following statements are equivalent:

  1. (a)

${\mathcal{A}}^{\star}$ is _forward attracting_ ${\mathcal{A}}$, that is,

$\displaystyle\lim_{s\to\infty}\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,{\mathcal{A}}(\tau)),{\mathcal{A}}^{\star}(\tau+s)\bigr{)}=0\quad\text{for
all }\tau\in{\mathbb{Z}},$ (36)

  2. (b)

$\omega_{{\mathcal{A}}}^{+}=\omega_{{\mathcal{A}}}^{\star}$.

###### Proof

($\Rightarrow$) Suppose that ${\mathcal{A}}^{\star}$ is forward attracting
from within ${\mathcal{A}}$ and that
$\omega_{{\mathcal{A}}}^{+}\neq\omega_{{\mathcal{A}}}^{\star}$. Since
$\omega_{{\mathcal{A}}}^{\star}\subseteq\omega_{{\mathcal{A}}}^{+}$, there
exists a point
$\tilde{v}\in\omega_{{\mathcal{A}}}^{+}\setminus\omega_{{\mathcal{A}}}^{\star}$,
i.e., there are $\tilde{\tau}\in{\mathbb{Z}}$ and
$\tilde{\varepsilon}=\tilde{\varepsilon}(\tilde{\tau})>0$ such that
$\tilde{v}\in\Omega_{\mathcal{A}}(\tilde{\tau})$ and

$\displaystyle\operatorname{dist}\bigl{(}\tilde{v},\omega_{{\mathcal{A}}}^{\star}\bigr{)}>2\tilde{\varepsilon}.$
(37)

Since $\tilde{v}\in\Omega_{\mathcal{A}}(\tilde{\tau})$, there exists a
sequence $\bigl{(}(\tilde{s}_{n},\tilde{b}_{n})\bigr{)}_{n\in{\mathbb{N}}}$
with $\lim_{n\to\infty}\tilde{s}_{n}=\infty$ and points
$\tilde{b}_{n}\in{\mathcal{A}}(\tilde{\tau})$ satisfying
$\operatorname{dist}\bigl{(}\tilde{v},\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},\tilde{b}_{n})\bigr{)}<\tilde{\varepsilon}$.
Moreover, by the forward attraction of ${\mathcal{A}}^{\star}$, there exists
an $s^{\prime}>0$ such that

$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tilde{\tau}+s^{\prime};\tilde{\tau},{\mathcal{A}}(\tilde{\tau})),{\mathcal{A}}^{\star}(\tilde{\tau}+s^{\prime})\bigr{)}<\tilde{\varepsilon}.$

Combining all of them, we obtain

$\displaystyle\operatorname{dist}\bigl{(}\tilde{v},{\mathcal{A}}^{\star}(\tilde{\tau}+\tilde{s}_{n})\bigr{)}$
$\displaystyle\leq$
$\displaystyle\operatorname{dist}\bigl{(}\tilde{v},\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},\tilde{b}_{n})\bigr{)}+\operatorname{dist}\bigl{(}\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},\tilde{b}_{n}),{\mathcal{A}}^{\star}(\tau+\tilde{s}_{n})\bigr{)}$
$\displaystyle\leq$
$\displaystyle\operatorname{dist}\bigl{(}\tilde{v},\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},\tilde{b}_{n})\bigr{)}+\operatorname{dist}\bigl{(}\varphi(\tilde{\tau}+\tilde{s}_{n};\tilde{\tau},{\mathcal{A}}(\tilde{\tau})),{\mathcal{A}}^{\star}(\tau+\tilde{s}_{n})\bigr{)}$
$\displaystyle<$
$\displaystyle\tilde{\varepsilon}+\tilde{\varepsilon}=2\tilde{\varepsilon}.$

Since $\bigcap_{n\in{\mathbb{N}}}\overline{\bigcup_{m\geq
n}{\mathcal{A}}^{\star}(\tilde{\tau}+\tilde{s}_{n})}\subseteq\omega_{{\mathcal{A}}}^{\star}$
by definition, it then follows

$\displaystyle\operatorname{dist}\bigl{(}\tilde{v},\omega_{{\mathcal{A}}}^{\star}\bigr{)}\leq\operatorname{dist}\left(\tilde{v},\bigcap_{n\in{\mathbb{N}}}\overline{\bigcup_{m\geq
n}{\mathcal{A}}^{\star}(\tilde{\tau}+\tilde{s}_{n})}\right)\leq\operatorname{dist}\bigl{(}\tilde{v},{\mathcal{A}}^{\star}(\tilde{\tau}+\tilde{s}_{n})\bigr{)}<2\tilde{\varepsilon},$

a contradiction to (37). Hence,
$\omega_{{\mathcal{A}}}^{+}=\omega_{{\mathcal{A}}}^{\star}$ holds.

($\Leftarrow$) Suppose that
$\omega_{{\mathcal{A}}}^{+}=\omega_{{\mathcal{A}}}^{\star}$, i.e.,
$\operatorname{dist}\bigl{(}\omega_{{\mathcal{A}}}^{+},\omega_{{\mathcal{A}}}^{\star}\bigr{)}<\varepsilon$
for all $\varepsilon>0$, and that ${\mathcal{A}}^{\star}$ is not forward
attracting from within ${\mathcal{A}}$, i.e., there exist a real
$\tilde{\varepsilon}>0$ and a sequence $(\tilde{s}_{n})_{n\in{\mathbb{N}}}$ in
${\mathbb{N}}_{0}$ with $\lim_{n\to\infty}\tilde{s}_{n}=\infty$ and

$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tau+\tilde{s}_{n};\tau,{\mathcal{A}}^{\star}(\tau)),{\mathcal{A}}(\tau+\tilde{s}_{n})\bigr{)}\geq
2\tilde{\varepsilon}\quad\text{for all }n\in{\mathbb{N}}.$

Although there is no condition ensuring the set
$\varphi(\tau+s_{n};\tau,{\mathcal{A}}(\tau))$ is compact, which means the
supremum in the Hausdorff semidistance may not be attained, there still exists
a point
$\tilde{y}_{n}:=\varphi(\tau+s_{n};\tau,\tilde{b}_{n})\in\varphi(\tau+s_{n};\tau,{\mathcal{A}}(\tau))$
for all $n\in{\mathbb{N}}$ and $\tilde{b}_{n}\in{\mathcal{A}}(\tau)$ such that

$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tau+\tilde{s}_{n};\tau,{\mathcal{A}}(\tau)),{\mathcal{A}}^{\star}(\tau+\tilde{s}_{n})\bigr{)}-\tilde{\varepsilon}$
$\displaystyle\leq\operatorname{dist}\bigl{(}\tilde{y}_{n},{\mathcal{A}}^{\star}(\tau+\tilde{s}_{n})\bigr{)}$
$\displaystyle\leq\operatorname{dist}\bigl{(}\varphi(\tau+\tilde{s}_{n};\tau,{\mathcal{A}}(\tau)\bigr{)},{\mathcal{A}}^{\star}(\tau+\tilde{s}_{n})).$

The above inequalities in fact give
$\operatorname{dist}\bigl{(}\tilde{y}_{n},{\mathcal{A}}^{\star}(\tau+\tilde{s}_{n})\bigr{)}\geq\tilde{\varepsilon}$
for all $n\in{\mathbb{N}}$. Moreover, take a point
$\tilde{a}_{n}\in{\mathcal{A}}^{\star}(\tau+\tilde{s}_{n})$, then

$\displaystyle
d(\tilde{y}_{n},\tilde{a}_{n})\geq\operatorname{dist}\bigl{(}\tilde{y}_{n},{\mathcal{A}}(\tau+\tilde{s}_{n})\bigr{)}\geq\tilde{\varepsilon}\quad\text{for
all }n\in{\mathbb{N}}.$

On the other hand, by assumptions and definitions, ($\Delta$) is
${\mathcal{A}}$-asymptotically compact and both $\tilde{y}_{n}$ and
$\tilde{a}_{n}$ are in $\varphi(\tau+s_{n};\tau,{\mathcal{A}}(\tau))$ for all
$\tau\in{\mathbb{Z}}$, so

$\displaystyle\operatorname{dist}\bigl{(}\tilde{y}_{n},K\bigr{)}$
$\displaystyle\leq\operatorname{dist}\bigl{(}\varphi(\tau+\tilde{s}_{n};\tau,{\mathcal{A}}(\tau)),K\bigr{)}\xrightarrow[n\to\infty]{}0,$
$\displaystyle\operatorname{dist}\bigl{(}\tilde{a}_{n},K\bigr{)}$
$\displaystyle\leq\operatorname{dist}\bigl{(}\varphi(\tau+\tilde{s}_{n};\tau,{\mathcal{A}}(\tau)),K\bigr{)}\xrightarrow[n\to\infty]{}0,$

implying that both $\tilde{y}_{n}$ and $\tilde{a}_{n}$ are in $K$ as well.
Additionally, since $K$ is compact, there are convergent subsequences
$(\tilde{y}_{n_{j}})_{j\in{\mathbb{N}}}$ with limit $\tilde{y}\in K$ and
$(\tilde{a}_{n_{j}})_{j\in{\mathbb{N}}}$ with limit $\tilde{a}\in K$. This
implies
$\tilde{y}\in\Omega_{\mathcal{A}}(\tau)\subseteq\omega_{{\mathcal{A}}}^{+}$
and $\tilde{a}\in\omega_{{\mathcal{A}}}^{\star}$ by definitions. Combining
this with $d(\tilde{y}_{n},\tilde{a}_{n})\geq\tilde{\varepsilon}$ for all
$n\in{\mathbb{N}}$, we arrive at the contradiction

$\displaystyle\operatorname{dist}\bigl{(}\omega_{{\mathcal{A}}}^{+},\omega_{{\mathcal{A}}}^{\star}\bigr{)}\geq\operatorname{dist}\bigl{(}\tilde{y}_{n},\omega_{{\mathcal{A}}}^{\star}\bigr{)}\geq
d(\tilde{y},\tilde{a})\geq\tilde{\varepsilon}\quad\text{for all
}n\in{\mathbb{N}},$

to the assumption. Thus, ${\mathcal{A}}^{\star}$ is forward attracting from
within ${\mathcal{A}}$. ∎

###### Corollary 2

Suppose in addition that ${\mathcal{A}}\subseteq{\mathcal{U}}$ is forward
absorbing. If $\omega_{{\mathcal{A}}}^{+}=\omega_{{\mathcal{A}}}^{\star}$
holds, then ${\mathcal{A}}^{\star}$ is a forward attractor of ($\Delta$).

###### Proof

Due to Prop. 1 the set ${\mathcal{A}}^{\star}$ is already nonempty, compact,
invariant and thus it suffices to show that ${\mathcal{A}}^{\star}$ is forward
attracting. Thereto, suppose that ${\mathcal{B}}\subseteq{\mathcal{U}}$ is
bounded and choose $\tau\in{\mathbb{Z}}$ arbitrarily. With the forward
absorption time $S\in{\mathbb{N}}$ we obtain from Thm. 4.7 that

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau,{\mathcal{B}}(\tau)),{\mathcal{A}}^{\star}(\tau+s)\bigr{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{semigroup}}}{{=}}$
$\displaystyle\operatorname{dist}\bigl{(}\varphi\bigl{(}\tau+s;\tau+S,\varphi(\tau+S,\tau,{\mathcal{B}}(\tau))\bigr{)},{\mathcal{A}}^{\star}(\tau+s)\bigr{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{fordiss}}}{{\leq}}$
$\displaystyle\operatorname{dist}\bigl{(}\varphi(\tau+s;\tau+S,{\mathcal{A}}(\tau+S)),{\mathcal{A}}^{\star}(\tau+s)\bigr{)}\xrightarrow[s\to\infty]{\eqref{thm49a}}0$

and this yields the assertion. ∎

We close this section with a simple, yet illustrative example:

######  Example 3 (Beverton-Holt equation)

Given reals $0<\alpha_{-},\alpha_{+}$ we consider the asymptotically
autonomous Beverton-Holt equation

$\displaystyle v_{t+1}$ $\displaystyle=\frac{\tilde{a}_{t}v_{t}}{1+v_{t}},$
$\displaystyle\tilde{a}_{t}$ $\displaystyle:=\begin{cases}\alpha_{-},&t<0,\\\
\alpha_{+},&0\leq t\end{cases}$ (38)

in $U={\mathbb{R}}_{+}$ having the general solution

$\varphi(t;\tau,v_{\tau})=\frac{v_{\tau}\prod_{r=\tau}^{t-1}\tilde{a}_{r}}{1+v_{\tau}\sum_{s=\tau}^{t-1}\prod_{r=\tau}^{s-1}\tilde{a}_{r}}\quad\text{for
all }\tau\leq t,\,0\leq v_{\tau}.$

It possesses the absorbing set
${\mathcal{A}}={\mathbb{Z}}\times[0,\max\left\\{\alpha_{-},\alpha_{+}\right\\}+1]$
and the forward $\omega$-limit set
$\omega_{\mathcal{A}}^{+}=[0,\max\left\\{0,\alpha_{+}-1\right\\}]$. Depending
on the constellation of the parameters $\alpha_{-},\alpha_{+}$ one obtains the
following capturing the forward dynamics:
$\alpha_{-},\alpha_{+}$ ${\mathcal{A}}^{\ast}={\mathcal{A}}^{\star}$
$\omega_{{\mathcal{A}}}^{\star}$ $\omega_{{\mathcal{A}}}^{-}$
$\omega_{{\mathcal{A}}}^{+}$ $\alpha_{-},\alpha_{+}\leq 1$
$\left\\{0\right\\}$ $\left\\{0\right\\}$ $\left\\{0\right\\}$ $\alpha_{-}\leq
1<\alpha_{+}$ $\left\\{0\right\\}$ $\left\\{0\right\\}$ $[0,\alpha_{+}-1]$
$\alpha_{+}\leq 1<\alpha_{-}$ $\left\\{0\right\\}$ $\left\\{0\right\\}$
$\left\\{0\right\\}$ $1<\alpha_{-}<\alpha_{+}$ $[0,\alpha_{+}-1]$
$[0,\alpha_{-}-1]$ $[0,\alpha_{+}-1]$ $1<\alpha_{+}\leq\alpha_{-}$
$[0,\alpha_{+}-1]$ $[0,\alpha_{+}-1]$ $[0,\alpha_{+}-1]$
For $\alpha_{+}\leq 1$ all fibres
$\Omega_{\mathcal{A}}(\tau)=\left\\{0\right\\}$ are constant. For
$\alpha_{+}>1$ two cases arise:

  * •

$1\leq\alpha_{-}<\alpha_{+}$: Solutions starting in $\alpha_{+}+1$ at time
$\tau<0$ first decay until time $t=0$ and then increase again, which yields

$\Omega_{{\mathcal{A}}}(\tau)=\begin{cases}[0,\varphi(0;\tau,\alpha_{+}+1)],&\tau<0,\\\
[0,\alpha_{+}-1],&0\leq\tau.\end{cases}$

  * •

$1\leq\alpha_{+}\leq\alpha_{-}$: Solutions starting in $\alpha_{+}+1$ decay to
$\alpha_{+}-1$ and thus the fibres are constant
$\Omega_{\mathcal{A}}(\tau)\equiv[0,\alpha_{+}+1]$ on ${\mathbb{Z}}$.

Except for $\alpha_{-}\leq 1<\alpha_{+}$, where the pullback and forward
dynamics of (38) differ, Cor. 2 applies and yields that the pullback attractor
${\mathcal{A}}^{\ast}={\mathcal{A}}^{\star}$ is the forward attractor
${\mathcal{A}}^{+}$.

As a conclusion, in case ${\mathcal{A}}$ is a positively invariant, forward
absorbing nonautonomous set this section provided two concepts to capture the
forward dynamics of ($\Delta$), namely the limit set
$\omega_{{\mathcal{A}}}^{+}$ from Thm. 4.1 and the forward attractor
${\mathcal{A}}^{+}={\mathcal{A}}^{\star}$ constructed in Cor. 2. On the one
hand, the limit set $\omega_{{\mathcal{A}}}^{+}\subseteq U$ is asymptotically
positively invariant, forward attracts and is contained in all other sets with
these properties. It depends only on information in forward time. On the other
hand, the forward attractor ${\mathcal{A}}^{+}\subseteq{\mathcal{U}}$ shares
these properties, but is actually invariant. Its construction is based on
information on the entire axis ${\mathbb{Z}}$ and more restrictively, relies
on the condition $\omega_{{\mathcal{A}}}^{+}=\omega_{{\mathcal{A}}}^{\star}$
from Thm. 4.7(b). The latter might be hard to verify in concrete examples,
unless rather strict assumptions like asymptotic autonomy hold cui:kloeden:19
.

##  5 Integrodifference equations

The above abstract results will now be applied to nonautonomous IDEs. For this
purpose let $(\Omega,{\mathfrak{A}},\mu)$ be a measure space satisfying
$\mu(\Omega)<\infty$. Suppose additionally that $\Omega$ is equipped with a
metric such that it becomes a compact metric space.

We consider the Banach space $X=C(\Omega,{\mathbb{R}}^{d})$ of continuous
${\mathbb{R}}^{d}$-valued functions over $\Omega$ equipped with the norm
$\left\|u\right\|_{0}:=\max_{x\in\Omega}\left|u(x)\right|.$ If
$Z\subseteq{\mathbb{R}}^{d}$ is a nonempty, closed set, then
$U:=\bigl{\\{}u:\Omega\to Z\mid u\in C(\Omega,{\mathbb{R}}^{d})\bigr{\\}}$ is
a complete metric space. Furthermore, we have
${\mathcal{U}}={\mathbb{I}}\times U$, where ${\mathbb{I}}$ is an unbounded
discrete interval.

Given functions $g_{t}:\Omega\times Z\to{\mathbb{R}}^{d}$ and
$k_{t}:\Omega^{2}\times Z\to{\mathbb{R}}^{d}$, the _Nemytskii operator_
${\mathscr{G}}_{t}:U\to C(\Omega,{\mathbb{R}}^{d})$ is defined by

${\mathscr{G}}_{t}(u)(x):=g_{t}(x,u(x))\quad\text{for all
}(t,x)\in{\mathbb{I}}^{\prime}\times\Omega$

and the _Urysohn integral operators_ ${\mathscr{K}}_{t}:U\to
C(\Omega,{\mathbb{R}}^{d})$ by

${\mathscr{K}}_{t}(u)(x):=\int_{\Omega}k_{t}(x,y,u(y))\,{\mathrm{d}}\mu(y)\quad\text{for
all }(t,x)\in{\mathbb{I}}^{\prime}\times\Omega.$

With these operators, a nonautonomous difference equation ($\Delta$) of the
additive form (10) is called an _integrodifference equation_ and explicitly
reads as

$\boxed{u_{t+1}(x)=g_{t}(x,u_{t}(x))+\int_{\Omega}k_{t}(x,y,u_{t}(y))\,{\mathrm{d}}\mu(y)}\quad\text{for
all }(t,x)\in{\mathbb{I}}^{\prime}\times\Omega.$ ($I_{g}$)

Such problems are well-motivated from applications:

  * •

For an integrodifferential equation
$D_{1}u(t,x)=\int_{\Omega}f(t,x,y,u(t,y))\,{\mathrm{d}}y$ with, e.g., a
continuous kernel function
$f:{\mathbb{R}}\times\Omega^{2}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$, the
forward Euler discretisation with step-size $h>0$ gives the IDE

$u_{t+1}(x)=u_{t}(x)+h\int_{\Omega}f(ht,x,y,u_{t}(y))\,{\mathrm{d}}y\quad\text{for
all }(t,x)\in{\mathbb{I}}^{\prime}\times\Omega$

matching ($I_{g}$) with a compact $\Omega\subset{\mathbb{R}}^{\kappa}$ and the
Lebesgue measure $\mu$.

  * •

Population genetics or ecological models of the form

$u_{t+1}(x)=(1-\vartheta)g(x,u_{t}(x))+\vartheta\int_{\Omega}f(x,y,u_{t}(y))\,{\mathrm{d}}y\quad\text{for
all }(t,x)\in{\mathbb{I}}^{\prime}\times\Omega$

are investigated in volkov:lui:07 , where $\vartheta\in[0,1]$ is a parameter
and e.g. continuous functions
$g:\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$,
$f:\Omega^{2}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$. These problems are of
the from ($I_{g}$) with a compact $\Omega\subset{\mathbb{R}}^{\kappa}$ and the
Lebesgue measure $\mu$.

  * •

Let the compact set $\Omega\subset{\mathbb{R}}^{\kappa}$ be countable,
$\eta\in\Omega$ and $w_{\eta}\geq 0$ be reals. Then
$\mu(\Omega^{\prime}):=\sum_{\eta\in\Omega^{\prime}}w_{\eta}$ defines a
measure on the family of all countable subsets
$\Omega^{\prime}\subset{\mathbb{R}}^{\kappa}$. The assumption
$\sum_{\eta\in\Omega}w_{\eta}<\infty$ ensures that $\mu(\Omega)<\infty$.
W.r.t. the resulting $\mu$-integral
$\int_{\Omega}u\,{\mathrm{d}}\mu=\sum_{\eta\in\Omega}w_{\eta}u(\eta)$ the IDE
($I_{g}$) becomes

$u_{t+1}(x)=g_{t}(x,u_{t}(x))+\sum_{\eta\in\Omega}w_{\eta}k_{t}(x,\eta,u_{t}(\eta))\quad\text{for
all }(t,x)\in{\mathbb{I}}^{\prime}\times\Omega.$

Such difference equations occur as _Nyström methods_ with _nodes_ $\eta$ and
_weights_ $w_{\eta}$ as used in numerical discretizations and simulations
atkinson:92  of IDEs ($I_{g}$).

Hypothesis: For every $t\in{\mathbb{I}}^{\prime}$ we suppose:

  * $(H_{1})$

The function $g_{t}:\Omega\times Z\to{\mathbb{R}}^{d}$ is such that
$g_{t}(\cdot,z):\Omega\to{\mathbb{R}}^{d}$ is continuous and there exist reals
$\gamma_{t},\ell_{t}\geq 0$ with

$\displaystyle\left|g_{t}(x,z)\right|$ $\displaystyle\leq\gamma_{t},$
$\displaystyle\left|g_{t}(x,z)-g_{t}(x,\bar{z})\right|$
$\displaystyle\leq\ell_{t}\left|z-\bar{z}\right|\quad\text{for all
}x\in\Omega,\,z,\bar{z}\in Z.$

  * $(H_{2})$

The kernel function $k_{t}:\Omega\times\Omega\times Z\to{\mathbb{R}}^{d}$ is
such that $k_{t}(x,\cdot,z):\Omega\to{\mathbb{R}}^{d}$ is measurable for all
$x\in\Omega$, $z\in Z$, and the following holds for almost all $y\in\Omega$:
$k_{t}(x,y,\cdot):Z\to{\mathbb{R}}^{d}$ is continuous for all $x\in\Omega$ and
the limit

$\lim_{x\to x_{0}}\int_{\Omega}\sup_{z\in
Z\cap\bar{B}_{r}(0)}\left|k_{t}(x,y,z)-k_{t}(x_{0},y,z)\right|\,{\mathrm{d}}\mu(y)=0\quad\text{for
all }r>0$

holds uniformly in $x_{0}\in\Omega$.

  * $(H_{3})$

There exists a function $\kappa_{t}:\Omega^{2}\to{\mathbb{R}}_{+}$, measurable
in the second argument with
$\sup_{x\in\Omega}\int_{\Omega}\kappa_{t}(x,y)\,{\mathrm{d}}\mu(y)<\infty$ and
for almost all $y\in\Omega$ one has

$\left|k_{t}(x,y,z)\right|\leq\kappa_{t}(x,y)\quad\text{for all
}x\in\Omega,\,z\in Z.$

Then the Nemytskii operator ${\mathscr{G}}_{t}:U\to
C(\Omega,{\mathbb{R}}^{d})$ satisfies

$\displaystyle\left\|{\mathscr{G}}_{t}(u)\right\|_{0}$
$\displaystyle\leq\gamma_{t},$
$\displaystyle\left\|{\mathscr{G}}_{t}(u)-{\mathscr{G}}_{t}(\bar{u})\right\|_{0}$
$\displaystyle\leq\ell_{t}\left\|u-\bar{u}\right\|_{0}\quad\text{for all
}u,\bar{u}\in U,$ (39)

while the Urysohn operators ${\mathscr{K}}_{t}:U\to
C(\Omega,{\mathbb{R}}^{d})$ are globally bounded by

$\displaystyle\left\|{\mathscr{K}}_{t}(u)\right\|_{0}\leq\rho_{t}:=\sup_{x\in\Omega}\int_{\Omega}\kappa_{t}(x,y)\,{\mathrm{d}}\mu(y)\quad\text{for
all }u\in U$ (40)

and completely continuous due to (martin:76, , p. 166, Prop. 3.2).

In the following we tacitly suppose ${\mathscr{F}}_{t}(U)\subseteq U$ for all
$t\in{\mathbb{I}}^{\prime}$.

######  Proposition 5 (dissipativity for ($I_{g}$))

If $(H_{1}$–$H_{3})$ with

$\displaystyle\sup_{t\in{\mathbb{I}}^{\prime}}\gamma_{t}$
$\displaystyle<\infty,$
$\displaystyle\sup_{t\in{\mathbb{I}}^{\prime}}\rho_{t}$ $\displaystyle<\infty$
(41)

hold, then the bounded and closed set

$\displaystyle{\mathcal{A}}$
$\displaystyle:=\left\\{(t,u)\in{\mathcal{U}}:\,\left\|u\right\|_{0}\leq\gamma_{t^{\ast}}+\rho_{t^{\ast}}\right\\},$
$\displaystyle t^{\ast}$
$\displaystyle:=\begin{cases}t-1,&t>\min{\mathbb{I}},\\\
t,&t=\min{\mathbb{I}}\end{cases}$

is positively invariant, forward absorbing (if ${\mathbb{I}}$ is unbounded
above), pullback absorbing (if ${\mathbb{I}}$ is unbounded below) w.r.t.
($I_{g}$) with absorption time $1$.

###### Proof

Clearly the set ${\mathcal{A}}$ is closed and due to (41) also bounded. Let
$t,\tau\in{\mathbb{I}}$ with $\tau<t$. Thus, $t^{\ast}=t-1$ and our
assumptions readily imply that

$\displaystyle\left\|\varphi(t;\tau,u)\right\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{process}}}{{=}}$
$\displaystyle\left\|{\mathscr{F}}_{t-1}(\varphi(t-1;\tau,u))\right\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{rhsadd}}}{{\leq}}$
$\displaystyle\left\|{\mathscr{G}}_{t-1}(\varphi(t-1;\tau,u))\right\|+\left\|{\mathscr{K}}_{t-1}(\varphi(t-1;\tau,u))\right\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{noA}}}{{\leq}}$
$\displaystyle\gamma_{t-1}+\left\|{\mathscr{K}}_{t-1}(\varphi(t-1;\tau,u))\right\|\stackrel{{\scriptstyle\eqref{noB}}}{{\leq}}\gamma_{t^{\ast}}+\rho_{t^{\ast}}\quad\text{for
all }(\tau,u)\in{\mathcal{U}}$

and consequently
$\varphi(t,\tau,{\mathcal{U}}(\tau))\subseteq{\mathcal{A}}(t)$ holds for all
$\tau<t$. Thanks to ${\mathcal{A}},{\mathcal{B}}\subseteq{\mathcal{U}}$ for
any bounded ${\mathcal{B}}$ this inclusion guarantees that ${\mathcal{A}}$ is
positively invariant, but also forward and uniformly absorbing with absorption
time $S=1$. ∎

######  Theorem 5.1 (pullback attractor for ($I_{g}$))

Let ${\mathbb{I}}$ be unbounded below. If $(H_{1}$–$H_{3})$ are satisfied with
(41) and there exists a $T\in{\mathbb{I}}$ such that
$\prod_{s=-\infty}^{T-1}\ell_{s}=0$ hold, then the IDE ($I_{g}$) has a unique
and bounded pullback attractor ${\mathcal{A}}^{\ast}\subseteq{\mathcal{A}}$.

###### Proof

We aim to apply Thm. 3.1 to ($I_{g}$). Thereto, Prop. 5 guarantees that
($I_{g}$) is uniformly pullback absorbing. Moreover, since the Lipschitz
constant $\ell_{t}$ of the Nemytskii operator ${\mathscr{G}}_{t}$ is an upper
bound for its Darbo constant and because ${\mathscr{K}}_{t}$ is completely
continuous, the assertion follows. ∎

Without further assumptions not much can be said about the detailed structure
of the pullback attractor ${\mathcal{A}}^{\ast}$. Nevertheless, in case the
functions $g_{t},k_{t}$ satisfy monotonicity assumptions in the second resp.
third argument, it is possible to construct “extremal” solutions in the
attractor poetzsche:15 . We illustrate this in

Figure 1: Pullback convergence to the fibre
$\phi_{\tau}^{+}:\Omega\to{\mathbb{R}}_{+}$ ($\tau=10$, initial function
$u_{0}(x)\equiv 4$, left) and sequence of sets containing the pullback
attractor ${\mathcal{A}}^{\ast}$ ($0\leq t\leq 10$, right) for
$\vartheta=\tfrac{1}{4}$.

######  Example 4 (spatial Beverton-Holt equation)

Let $\vartheta\in[0,1]$ and $a_{t}:\Omega\to(0,\infty)$,
$t\in{\mathbb{I}}^{\prime}$, be continuous functions describing the space- and
time-dependent growth rates and a compact habitat $\Omega$. The spatial
Beverton-Holt equation

$u_{t+1}(x)=(1-\vartheta)\frac{a_{t}(x)u_{t}(x)}{1+u_{t}(x)}+\vartheta\int_{\Omega}k(x,y)\frac{a_{t}(y)u_{t}(y)}{1+u_{t}(y)}\,{\mathrm{d}}\mu(y)$
(42)

for all $(t,x)\in{\mathbb{I}}^{\prime}\times\Omega$ fits into the framework of
($I_{g}$) with $Z={\mathbb{R}}_{+}$, $U=C(\Omega,{\mathbb{R}}_{+})$,

$\displaystyle g_{t}:\Omega\times{\mathbb{R}}_{+}$
$\displaystyle\to{\mathbb{R}}_{+},$ $\displaystyle g_{t}(x,z)$
$\displaystyle:=\frac{a_{t}(x)z}{1+z},$ $\displaystyle
k_{t}:\Omega\times\Omega\times{\mathbb{R}}_{+}$
$\displaystyle\to{\mathbb{R}}_{+},$ $\displaystyle k_{t}(x,y,z)$
$\displaystyle:=k(x,y)\frac{a_{t}(y)z}{1+z}$

and a continuous kernel function $k:\Omega\times\Omega\to(0,\infty)$. Then
$(H_{1}$–$H_{3})$ hold with

$\displaystyle\gamma_{t}$ $\displaystyle=(1-\vartheta)\alpha_{t},$
$\displaystyle\ell_{t}$ $\displaystyle=(1-\vartheta)\alpha_{t},$
$\displaystyle\kappa_{t}(x,y)$
$\displaystyle=\vartheta\alpha_{t}k(x,y)\quad\text{for all }x,y\in\Omega$

and $\alpha_{t}:=\max_{x\in\Omega}a_{t}(x)$. If
$\lim_{t\to-\infty}(1-\vartheta)^{T-t}\prod_{s=t}^{T-1}\alpha_{s}=0$ holds for
some $T\in{\mathbb{I}}$, then Thm. 5.1 yields the existence of a pullback
attractor ${\mathcal{A}}^{\ast}\subseteq{\mathcal{U}}$ for (42). Since the
functions
$g_{t}(x,\cdot),k_{t}(x,y,\cdot):{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ are
strictly increasing more can be said on the structure of
${\mathcal{A}}^{\ast}$. As in (poetzsche:15, , Prop. 8), there exists an
“extremal” entire solution $\phi^{+}$ being pullback attracting from above
such that

${\mathcal{A}}^{\ast}\subseteq\left\\{(t,u)\in{\mathcal{U}}:\,0\leq
u(x)\leq\phi_{t}^{+}(x)\text{ for all }x\in\Omega\right\\}.$

We illustrate both the pullback convergence to the solution $\phi^{+}$, as
well as the sets containing solutions in the pullback attractor
${\mathcal{A}}^{\ast}$ in Fig. 1, where $\Omega=[-\pi,\pi]$ is equipped with
the $1$-dimensional Lebesgue measure, $a_{t}(x):=3-\sin\tfrac{tx}{10}$
(artificial) and the Laplace kernel
$k(x,y):=\tfrac{a}{2}e^{-a\left|x-y\right|}$ for the dispersal rate $a=10$.

The remaining section addresses forward attraction. For simplicity we restrict
to the class of Urysohn IDEs

$\boxed{u_{t+1}(x)=\int_{\Omega}k_{t}(x,y,u_{t}(y))\,{\mathrm{d}}\mu(y)}\quad\text{for
all }(t,x)\in{\mathbb{I}}^{\prime}\times\Omega.$ ($I_{0}$)

Hypothesis: For every $t\in{\mathbb{I}}^{\prime}$ we suppose:

  * $(H_{4})$

For all $r>0$ there exists a function
$\lambda_{t}:\Omega^{2}\to{\mathbb{R}}_{+}$, measurable in the second argument
with
$\sup_{x\in\Omega}\int_{\Omega}\lambda_{t}(x,y)\,{\mathrm{d}}\mu(y)<\infty$
and for almost all $y\in\Omega$ one has

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

######  Proposition 6 (dissipativity for ($I_{0}$))

If $(H_{2}$–$H_{3})$ with $R:=\sup_{t\in{\mathbb{I}}}\rho_{t}<\infty$ hold,
then the bounded and compact nonautonomous set

$\displaystyle{\mathcal{A}}$
$\displaystyle:=\bigl{\\{}(t,u)\in{\mathcal{U}}:\,u\in\overline{{\mathscr{K}}_{t^{\ast}}(U\cap
B_{R}(0))}\bigr{\\}},$ $\displaystyle t^{\ast}$
$\displaystyle:=\begin{cases}t-1,&t>\min{\mathbb{I}},\\\
t,&t=\min{\mathbb{I}}\end{cases}$

is positively invariant, forward absorbing w.r.t. ($I_{0}$) with absorption
time $2$.

###### Proof

Let $t\in{\mathbb{I}}^{\prime}$ with $t-1\in{\mathbb{I}}^{\prime}$ and thus
$t^{\ast}=t-1$. Since the Urysohn operators ${\mathscr{K}}_{t-1}$ are
completely continuous, the fibres
${\mathcal{A}}(t)=\overline{{\mathscr{K}}_{t-1}(U\cap B_{R}(0))}$ are compact.
Thanks to

$\left\|{\mathscr{K}}_{t}(u)\right\|\stackrel{{\scriptstyle\eqref{noB}}}{{\leq}}\rho_{t}\leq
R\quad\text{for all }u\in U$

it follows that ${\mathcal{A}}$ is bounded. Moreover,
${\mathscr{K}}_{t}({\mathcal{A}}(t))\subseteq{\mathscr{K}}_{t}(U\cap
B_{R}(0))={\mathcal{A}}(t+1)$ holds for all $t\in{\mathbb{I}}^{\prime}$ and
${\mathcal{A}}$ is positively invariant. Furthermore, from the inclusion

$\varphi(t;\tau,u)\stackrel{{\scriptstyle\eqref{process}}}{{=}}{\mathscr{K}}_{t-1}(\underbrace{\varphi(t-1;\tau,u)}_{\in
U\cap B_{R}(0)})\in{\mathcal{A}}(t)\quad\text{for all }t-\tau\geq 2,\,u\in U$

we deduce that ${\mathcal{A}}$ is absorbing. ∎

######  Theorem 5.2 (forward limit set for ($I_{0}$))

Suppose that $(H_{2}$–$H_{3})$ hold with additionally
$R:=\sup_{t\in{\mathbb{I}}}\rho_{t}<\infty$. If
$\bigcup_{t\in{\mathbb{I}}}{\mathscr{K}}_{t}(U\cap B_{R}(0))$ is relatively
compact and ${\mathcal{A}}$ is the forward absorbing set from Prop. 6, then
the following are true:

  1. (a)

$\omega_{{\mathcal{A}}}^{+}$ is asymptotically positively invariant,

  2. (b)

$\omega_{{\mathcal{A}}}^{+}$ is asymptotically negatively invariant, provided
$(H_{4})$ is satisfied with

$\sup_{\tau\leq
t<\tau+T}\prod_{s=\tau}^{t-1}\int_{\Omega}\lambda_{s}(x,y)\,{\mathrm{d}}\mu(y)<\infty\quad\text{for
all }\tau\in{\mathbb{I}},\,T\in{\mathbb{N}}.$ (43)

The relative compactness of the union
$\bigcup_{t\in{\mathbb{I}}}{\mathscr{K}}_{t}(U\cap B_{R}(0))$ holds for
instance, if the kernel functions $k_{t}$ stem from a finite set or the images
${\mathscr{K}}_{t}(U\cap B_{R}(0))$ form a nonincreasing/nondecreasing
sequence of sets.

###### Proof

By assumption the set
$K:=\overline{\bigcup_{t\in{\mathbb{I}}}{\mathscr{K}}_{t}(U\cap B_{R}(0))}$ is
compact and this implies that ($I_{0}$) is strongly
${\mathcal{A}}$-asymptotically compact.

(a) By construction of $K\subseteq U$ the assertion results from Thm. 4.2.

(b) Let $u,\bar{u}\in{\mathcal{A}}(s)\cup K$ and choose $r>0$ so large that
${\mathcal{A}}(s)\cup K\subseteq B_{r}(0)$. We conclude

$\displaystyle\left|{\mathscr{K}}_{s}(u)(x)-{\mathscr{K}}_{s}(\bar{u})(x)\right|$
$\displaystyle\leq\int_{\Omega}\left|k_{s}(x,y,u(y))-k_{s}(x,y,\bar{u}(y))\,{\mathrm{d}}\mu(y)\right|$
$\displaystyle\leq\int_{\Omega}\lambda_{s}(x,y)\,{\mathrm{d}}\mu(y)\left\|u-\bar{u}\right\|_{0}\quad\text{for
all }(s,x)\in{\mathbb{I}}\times\Omega$

from assumption $(H_{4})$. After passing to the least upper bound over
$x\in\Omega$ it follows that
$\sup_{x\in\Omega}\int_{\Omega}\lambda_{s}(x,y)\,{\mathrm{d}}\mu(y)$ is a
Lipschitz constant for ${\mathscr{K}}_{s}$ on ${\mathcal{A}}(s)\cup K$. Hence,
the assumption (43) implies (25) and therefore Thm. 4.3 yields the claim. ∎

The above results do apply to the following

######  Example 5 (spatial Ricker equation)

Suppose that the compact $\Omega\subseteq{\mathbb{R}}^{\kappa}$ is equipped
with the $\kappa$-dimensional Lebesgue measure $\mu$ and that $\mu(\Omega)>0$
holds. Let $(\alpha_{t})_{t\in{\mathbb{I}}^{\prime}}$ denote a bounded
sequence of positive reals, $k:\Omega\times\Omega\to{\mathbb{R}}_{+}$ be
continuous and $(b_{t})_{t\in{\mathbb{I}}^{\prime}}$ be a bounded sequence in
$C(\Omega,{\mathbb{R}}_{+})$, $t\in{\mathbb{I}}^{\prime}$. The spatial Ricker
equation

$u_{t+1}(x)=\alpha_{t}\int_{\Omega}k(x,y)u_{t}(y)e^{-u_{t}(y)}\,{\mathrm{d}}y+b_{t}(x)\quad\text{for
all }(t,x)\in{\mathbb{I}}^{\prime}\times\Omega$ (44)

fits in the framework of ($I_{0}$) with $Z={\mathbb{R}}_{+}$ and the kernel
function

$k_{t}(x,y,z):=\alpha_{t}k(x,y)ze^{-z}+\tfrac{b_{t}(x)}{\mu(\Omega)}\quad\text{for
all }x,y\in\Omega,\,z\in{\mathbb{R}}_{+};$

hence, (44) is defined on the cone $U:=C(\Omega,{\mathbb{R}}_{+})$. If
${\mathbb{I}}$ is unbounded below, then (44) possesses a pullback attractor
${\mathcal{A}}^{\ast}\subseteq{\mathcal{U}}$; see Fig. 2 for an illustration.

For our subsequent analysis it is convenient to set
$\gamma:=\sup_{x\in\Omega}\int_{\Omega}k(x,y)\,{\mathrm{d}}y$. We begin with
some preparatory estimates. Above all, (44) satisfies the assumption $(H_{4})$
with $\lambda_{t}(x,y)=\alpha_{t}k(x,y)$, which guarantees the global
Lipschitz condition

$\left\|{\mathscr{K}}_{t}(u)-{\mathscr{K}}_{t}(\bar{u})\right\|\leq\alpha_{t}\gamma\left\|u-\bar{u}\right\|\quad\text{for
all }u,\bar{u}\in U.$ (45)

If we represent the right-hand side of (44) in semilinear form (5) with

$\displaystyle{\mathscr{L}}_{t}u$
$\displaystyle:=\alpha_{t}\int_{\Omega}k(\cdot,y)u(y)\,{\mathrm{d}}y,$
$\displaystyle{\mathscr{N}}_{t}(u)$
$\displaystyle:=\alpha_{t}\int_{\Omega}k(\cdot,y)(e^{-u(y)}-1)u(y)\,{\mathrm{d}}y+b_{t},$

then $\left\|{\mathscr{L}}_{t}\right\|=\alpha_{t}\gamma$ holds, as well as the
global Lipschitz condition

$\left\|{\mathscr{N}}_{t}(u)-{\mathscr{N}}_{t}(\bar{u})\right\|_{0}\leq\alpha_{t}(1+\tfrac{1}{e^{2}})\gamma\left\|u-\bar{u}\right\|_{0}\quad\text{for
all }u,\bar{u}\in U.$ (46)

In order to obtain information on the forward attractor, we suppose that
${\mathbb{I}}={\mathbb{Z}}$ and that (44) is asymptotically autonomous in
forward time, i.e., there exist $\alpha_{+}>0$ and $b\in
C(\Omega,{\mathbb{R}}_{+})$ such that

$\displaystyle\lim_{t\to\infty}\alpha_{t}$ $\displaystyle=\alpha_{+}\in[0,1),$
$\displaystyle\lim_{t\to\infty}b_{t}$ $\displaystyle=b.$

If $\alpha_{+}\gamma<1$, then it follows from the contraction mapping
principle and (45) that the autonomous limit equation

$u_{t+1}(x)=\alpha_{+}\int_{\Omega}k(x,y)u_{t}(y)e^{-u_{t}(y)}\,{\mathrm{d}}y+b(x)\quad\text{for
all }(t,x)\in{\mathbb{I}}^{\prime}\times\Omega$ (47)

has a unique, globally attractive fixed-point $u^{\ast}\in U$.

We choose ${\mathbb{I}}={\mathbb{N}}_{0}$ and an absorbing set $A\subseteq U$
of the limit equation (47) such that ${\mathcal{A}}={\mathbb{I}}\times A$ is
forward absorbing w.r.t. (44). If we assume $\sup_{s\leq
t}\prod_{r=s}^{t-1}\frac{\alpha_{r}}{\alpha_{+}}\leq K$, then the growth
estimate (30) holds with $\alpha=\alpha_{+}\gamma$ due to

$\left\|\Phi(t,s)\right\|\leq\prod_{r=s}^{t-1}\left\|{\mathscr{L}}_{r}\right\|=(\alpha_{+}\gamma)^{t-s}\prod_{r=s}^{t-1}\tfrac{\alpha_{r}}{\alpha_{+}}\leq
K(\alpha_{+}\gamma)^{t-s}\quad\text{for all }s\leq t.$

It follows from (46) that every nonlinearity ${\mathscr{N}}_{t}:U\to
C(\Omega)$ has the Lipschitz constant
$L:=(1+\tfrac{1}{e^{2}})\gamma\sup_{t\geq 0}\alpha_{t}$. Consequently, if
furthermore $(\alpha_{t})_{t\in{\mathbb{I}}}$ converges exponentially to
$\alpha_{+}$ with rate $\alpha$, then Thm. 4.6 applies under the assumption

$(1+\tfrac{1}{e^{2}})\gamma\sup_{t\geq 0}\alpha_{t}<\tfrac{1-\alpha}{K}$ (48)

and thus (26) holds. Hence, we derive from Thm. 4.4 the relations

$\Omega_{{\mathcal{A}}}(\tau)=\omega_{\mathcal{A}}^{-}=\omega_{\mathcal{A}}^{+}=\left\\{u^{\ast}\right\\}\quad\text{for
all }\tau\in{\mathbb{N}}_{0}.$

If we concretely define

$\displaystyle\alpha_{t}$
$\displaystyle:=\begin{cases}\alpha_{+}(1+\alpha^{t}),&t\geq 0,\\\
\alpha_{-},&t<0,\end{cases}$ $\displaystyle b_{t}$
$\displaystyle:=\begin{cases}b,&t\geq 0,\\\ 0,&t<0,\end{cases}$

then this implies the elementary estimate

$\displaystyle\prod_{r=s}^{t-1}\frac{\alpha_{r}}{\alpha_{+}}=\exp\bigl{(}\sum_{r=s}^{t-1}\ln(1+\alpha^{r})\bigr{)}\leq\exp\bigl{(}\sum_{r=s}^{t-1}\alpha^{r}\bigr{)}\leq\exp\bigl{(}\tfrac{1}{1-\alpha}\bigr{)}\quad\text{for
all }s\leq t.$

Hence, we set $K:=\exp\bigl{(}\tfrac{1}{1-\alpha}\bigr{)}$ and (48) simplifies
to
$2(1+\tfrac{1}{e^{2}})\alpha\exp\bigl{(}\tfrac{1}{1-\alpha}\bigr{)}<1-\alpha$.
This assumption can be fulfilled for $\alpha=\alpha_{+}\gamma$ sufficiently
close to $0$, which requires the kernel data $\gamma$ or the asymptotic growth
rate $\alpha_{+}$ to be small.

Even more concretely, on the habitat $\Omega=[-L,L]$ with some real $L>0$ we
again consider the Laplace kernel $k(x,y):=\tfrac{a}{2}e^{-a\left|x-y\right|}$
with dispersal rate $a>0$, which yields $\gamma=1-e^{-aL}$. In this framework,
an illustration of the forward limit set
$\omega_{\mathcal{A}}^{+}=\left\\{u^{\ast}\right\\}$ and subfibres of the
pullback attractor ${\mathcal{A}}^{\ast}$ is given in Fig. 2. Here, both the
pullback attractor ${\mathcal{A}}^{\ast}$ and the forward limit set
$\omega_{\mathcal{A}}^{+}$ capture the long term behaviour of (44).

Figure 2: Functions contained in the fibres ${\mathcal{A}}^{\ast}(t)$ of the
pullback attractor over the times $-10\leq t\leq 10$ (blue) and the forward
limit set
${\mathbb{N}}_{0}\times\omega_{{\mathcal{A}}}^{+}={\mathbb{N}}_{0}\times\left\\{u^{\ast}\right\\}$
(red) for the spatial Ricker equation (44) with Laplace kernel ($a=2$,
$L=10$), $\alpha_{+}=0.12$, $\alpha_{-}=14$ and the constant inhomogeneity
$b(x):=5\cos\tfrac{x}{8}$
More detailed, depicted are the $4$-periodic orbits (blue) of the spatial
Ricker equation, which is autonomous for $t<0$. In addition, the fibres
${\mathcal{A}}^{\ast}(t)$ also contain $0$, a nontrivial fixed point and a
$2$-periodic orbit.

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