# A general approach to nonautonomous shadowing for nonlinear dynamics

Lucas Backes  Departamento de Matemática, Universidade Federal do Rio Grande
do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, RS, Brazil.
<EMAIL_ADDRESS>and  Davor Dragičević  Department of Mathematics,
University of Rijeka, Croatia<EMAIL_ADDRESS>
###### Abstract.

Given a nonautonomous and nonlinear differential equation

(0.1) $x^{\prime}=A(t)x+f(t,x)\quad t\geq 0,$

on an arbitrary Banach space $X$, we formulate very general conditions for the
associated linear equation $x^{\prime}=A(t)x$ and for the nonlinear term
$f:[0,+\infty)\times X\to X$ under which the above system satisfies an
appropriate version of the shadowing property. More precisely, we require that
$x^{\prime}=A(t)x$ admits a very general type of dichotomy, which includes the
classical hyperbolic behaviour as a very particular case. In addition, we
require that $f$ is Lipschitz in the second variable with a sufficiently small
Lipschitz constant. Our general framework enables us to treat various settings
in which no shadowing result has been previously obtained. Moreover, we are
able to recover and refine several known results. We also show how our main
results can be applied to the study of the shadowing property for higher order
differential equations. Finally, we conclude the paper by presenting a
discrete time versions of our results.

###### Key words and phrases:

shadowing, nonautonomous systems, nonlinear dynamics, dichotomies

###### 2010 Mathematics Subject Classification:

Primary: 37C50, 34D09; Secondary: 34D10

##  1\. Introduction

It is a very well known fact that, in general, it might be very complicated or
even impossible to explicitly find a solution of a given differential
equation. Nevertheless, such complicated equations may be models to some
important classes of dynamics arising naturally in science and thus it is
necessary to study them. A possible approach to the study this kind of systems
is the following: instead of looking for exact solutions of these differential
equations, we could look for their approximate solutions.

Over the last decades, computational power has reached a high state of
development allowing one to construct very good approximate solutions of a
given differential equation over large intervals of time. However, the
information given by these approximate solutions will be useful only if they
correspond to actual solutions of the original equation, i.e. only if there
exists a true orbit of the system with a slightly different initial condition
that stays near the approximate trajectory. The systems exhibiting this
property are said to have the _shadowing property_.

The aim of the present paper is to present general conditions under which a
nonautonomous system has the shadowing property. More precisely, given a
nonautonomous and nonlinear differential equation

(1.1) $x^{\prime}=A(t)x+f(t,x)\quad t\geq 0,$

on an arbitrary Banach space $X$, we present very general conditions for the
associated linear equation $x^{\prime}=A(t)x$ and for the nonlinear term
$f:[0,+\infty)\times X\to X$ under which (1.1) has a version of the shadowing
property.

Following our recent paper [2] (which in turn is inspired by recent works on
the so-called quasi-shadowing property in the context of smooth dynamics [14,
16, 17, 18, 24]), we assume that $X$ splits into three directions: the stable,
the unstable and the central direction. All three directions are invariant
under the action of the dynamics associated with the linear equation
$x^{\prime}=A(t)x$ and finally, when restricted to the sum of the stable and
the unstable direction, the dynamics admits the so-called
$(\mu,\nu)$-dichotomy. Then, if $f$ is Lipschitz in the second variable with a
sufficiently small Lipschitz constant, we prove that close to each approximate
solution $t\mapsto y(t)$ of (0.1), one can find a map $x\mapsto x(t)$ which is
a solution of (0.1) up to the movement along the central direction. In the
particular case when the central direction vanishes, our result says that in a
vicinity of each approximate solution of (0.1), we can find its exact
solution. In other words, (0.1) has the shadowing property.

The importance of our results stems from our general framework. As we already
emphasized, we assume that the linear equation $x^{\prime}=A(t)x$ (when
restricted to the sum of the stable and the unstable direction) admits the so-
called $(\mu,\nu)$-dichotomy. This notion was introduced and studied by Bento
and Silva [7, 8] and includes the classical notion of an exponential dichotomy
[15] as a very particular case. In addition, it also extends the notion of a
polynomial dichotomy [6, 9] as well as the notion of a generalized dichotomy
as introduced by Muldowney [20]. It also includes several classes of
asymptotic behaviour that haven’t been studied previously. We refer to [7, 8]
for a detailed discussion.

Hence, by adapting this framework, we are able to simultaneously recover and
refine several known results as well as to discuss the shadowing property in
various settings in which no such results were previously available. When
$x^{\prime}=A(t)x$ admits an exponential or a generalized exponential
dichotomy, we basically recover our previous results obtained in [2, 3].
However, to the best of our knowledge all previously known results (either for
continuous or discrete time) related to the shadowing of (0.1) (or its
discrete time version) belong to this category. Indeed, the case when the
nonlinear part vanishes and when the linear part is either constant or
periodic was treated in [5, 10, 12, 13], while in our previous works [1, 2, 3]
we dealt with the general case. On the other hand, there are no previous
shadowing-type results in the setting when $x^{\prime}=A(t)x$ admits a
$(\mu,\nu)$-dichotomy (or even a polynomial dichotomy for example).

Consequently, our paper aims to open a new direction of research. Indeed,
while in the context of the smooth dynamics shadowing is related to some form
of hyperbolicity [21, 23], we show that in the setting of nonautonomous
dynamics it is possible to obtain shadowing-type results for the classes of
dynamics which don’t exhibit hyperbolic behaviour.

Our techniques build on our previous works [2, 3], which in turn are inspired
by analytic proofs of the classical shadowing lemma [19, 22]. An important
feature of the present paper is that in comparison with our previous works, we
deal with the case of continuous time directly, without passing to the
associated discrete time.

The paper is organized as follows. In Section 2 we introduce the setting we
are going to work on and the conditions on $x^{\prime}=A(t)x$ and $f$ that we
are going to work with. In Section 3 we present the main result of the paper
along with its proof, while in Section 4 we present examples of settings to
which our results are applicable as well as their relation with the results
already available in the literature. Section 5 is dedicated to the
applications of our result to higher order differential equations while in
Section 6 a discrete time version of our main result is presented.

##  2\. Preliminaries

Let $X=(X,|\cdot|)$ be an arbitrary Banach space. By $\mathcal{B}(X)$ we will
denote the space of all bounded linear operators on $X$. The operator norm on
$\mathcal{B}(X)$ will be denoted by $\|\cdot\|$.

For a continuous map $A\colon[0,+\infty)\to\mathcal{B}(X)$, we consider a
linear nonautonomous equation given by

(2.1) $x^{\prime}=A(t)x,\quad t\geq 0.$

By $T(t,s)$, $t,s\geq 0$ we will denote the evolution family associated to
(2.1). We recall that for fixed $s$ and $v\in X$, $t\mapsto T(t,s)v$ is a
solution of (2.1) such that $x(s)=v$. Then, we have the following properties:

  * •

$T(t,t)={\mathbb{I}}$ for $t\geq 0$, where ${\mathbb{I}}$ denotes the identity
operator on $X$;

  * •

$T(t,s)T(s,r)=T(t,r)$ for $t,s,r\geq 0$.

For a continuous map $f\colon[0,+\infty)\times X\to X$, we consider a
semilinear equation given by

(2.2) $x^{\prime}=A(t)x+f(t,x),\quad t\geq 0.$

Let us now introduce some standing assumptions. Let
$\mu\colon[0,+\infty)\to\mathbb{R}$ be a strictly increasing and
differentiable map such that

$\mu(0)=1\quad\text{and}\quad\lim_{t\to+\infty}\mu(t)=+\infty.$

In addition, let $\nu\colon[0,+\infty)\to[1,+\infty)$ be an arbitrary map. We
assume the following conditions:

  1. (1)

there exist three families of projections $P^{i}(t)$, $t\geq 0$,
$i\in\\{1,2,3\\}$ such that

$P^{1}(t)+P^{2}(t)+P^{3}(t)={\mathbb{I}},\quad t\geq 0;$

  2. (2)

for $i,j\in\\{1,2,3\\}$, $i\neq j$ and $t\geq 0$,

$P^{i}(t)P^{j}(t)=0;$

  3. (3)

for arbitrary $t,s\geq 0$ and $i\in\\{1,2,3\\}$, we have that

$T(t,s)P^{i}(s)=P^{i}(t)T(t,s);$

  4. (4)

there exist $D,\lambda>0$ and $d\geq 0$ such that

(2.3) $\|T(t,s)P^{1}(s)\|\leq
D\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\nu(s)^{d}\quad\text{for
$t\geq s$,}$

and

(2.4) $\|T(t,s)P^{2}(s)\|\leq
D\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\nu(s)^{d}\quad\text{for
$t\leq s$;}$

  5. (5)

there exist $c\geq 0$ such that

(2.5)
$|f(t,x)-f(t,y)|\leq\frac{c\mu^{\prime}(t)}{\mu(t)\nu(t)^{d}}|x-y|,\quad\text{for
$t\geq 0$ and $x,y\in X$.}$

For $t\geq 0$, set

$E^{s}(t)=\operatorname{Im}P^{1}(t),\ E^{u}(t)=\operatorname{Im}P^{2}(t)\
\text{and}\ E^{c}(t)=\operatorname{Im}P^{3}(t).$

Furthermore, we set

$|x|_{t}:=\sup_{s\geq
t}\bigg{(}|T(s,t)x|\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{\lambda}\bigg{)}\quad\text{for
$t\geq 0$ and $x\in E^{s}(t)$,}$ $|x|_{t}:=\sup_{s\leq
t}\bigg{(}|T(s,t)x|\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{\lambda}\bigg{)}\quad\text{for
$t\geq 0$ and $x\in E^{u}(t)$,}$

and

$|x|_{t}:=\frac{\mu(t)}{\mu^{\prime}(t)}|x|,\quad\text{for $t\geq 0$ and $x\in
E^{c}(t)$.}$

Finally, for an arbitrary $x\in X$ let

$|x|_{t}:=|P^{1}(t)x|_{t}+|P^{2}(t)x|_{t}+|P^{3}(t)x|_{t}.$

It is easy to see that $|\cdot|_{t}$ is a norm on $X$ for each $t\geq 0$. We
will need the following simple auxiliary result.

######  2.1 Lemma.

The following properties hold:

  * •

for $t\geq 0$ and $x\in E^{s}(t)\cup E^{u}(t)$,

(2.6) $|x|\leq|x|_{t};$

  * •

for $t\geq s$ and $x\in X$,

(2.7) $|T(t,s)P^{1}(s)x|_{t}\leq
D\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\nu(s)^{d}|x|;$

  * •

for $t\leq s$ and $x\in X$,

(2.8) $|T(t,s)P^{2}(s)x|_{t}\leq
D\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\nu(s)^{d}|x|.$

###### Proof.

Observe that (2.6) follows trivially. Let us now prove (2.7). Using (2.3), we
have that

$\begin{split}|T(t,s)P^{1}(s)x|_{t}&=\sup_{r\geq
t}\bigg{(}|T(r,s)P^{1}(s)x|\bigg{(}\frac{\mu(r)}{\mu(t)}\bigg{)}^{\lambda}\bigg{)}\\\
&=\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\sup_{r\geq
t}\bigg{(}|T(r,s)P^{1}(s)x|\bigg{(}\frac{\mu(r)}{\mu(s)}\bigg{)}^{\lambda}\bigg{)}\\\
&\leq\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\sup_{r\geq
s}\bigg{(}|T(r,s)P^{1}(s)x|\bigg{(}\frac{\mu(r)}{\mu(s)}\bigg{)}^{\lambda}\bigg{)}\\\
&\leq
D\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\nu(s)^{d}|x|.\end{split}$

Similarly, one can establish (2.8). ∎

##  3\. Main result

We are now in a position to establish our main result.

######  3.1 Theorem.

Assume that

(3.1) $q:=c(2D+1)+2cD\lambda^{-1}<1.$

Then, there exists $C>0$ with the property that for any $\delta>0$ and an
arbitrary differentiable map $y\colon[0,+\infty)\to X$ satisfying

(3.2)
$|y^{\prime}(t)-A(t)y(t)-f(t,y(t))|\leq\frac{\delta\mu^{\prime}(t)}{\mu(t)\nu(t)^{d}}\quad\text{for
$t\geq 0$,}$

there exists a differentiable map $x\colon[0,+\infty)\to X$ such that:

  * •

$P^{3}(t)(x(t)-y(t))=0$ for $t\geq 0$;

  * •

$\sup_{t\geq 0}|x(t)-y(t)|\leq C\delta$;

  * •

$x^{\prime}(t)-A(t)x(t)-f(t,x(t))\in\operatorname{Im}P^{3}(t)$ for $t\geq 0$;

  * •

for $t\geq 0$,

(3.3) $|x^{\prime}(t)-A(t)x(t)-f(t,x(t))|\leq
C\delta(2D+1)\nu(t)^{d}\max\bigg{\\{}1,\frac{\mu^{\prime}(t)}{\mu(t)}\bigg{\\}}.$

######  3.2 Remark.

We observe that this result may be seen as a version of the shadowing
property. Indeed, Theorem 3.1 says that given a “pseudo-trajectory” $t\mapsto
y(t)$ of (2.2), there exists a map $t\mapsto x(t)$ which is a solution of
(2.2) up to moving it in the central direction that “shadows” $t\mapsto y(t)$
and agrees with it in the central direction $E^{c}(t)$. In particular,
whenever $P^{3}(t)=0$, the map $t\mapsto x(t)$ is an actual solution of (2.2)
and we get the usual shadowing property (see Corollary 3.3). Moreover, if
$P^{3}(t)\neq 0$ then one cannot expect that, in general, the map $t\mapsto
x(t)$ is a solution of (2.2) (see [11]) and in this case condition (3.3) gives
us an estimate for the deviation of $x(t)$ from being such a solution.

###### Proof of Theorem 3.1.

Let

$\mathcal{Y}:=\bigg{\\{}z\colon[0,+\infty)\to X:\text{$z$ is continuous and}\
\|z\|_{\infty}:=\sup_{t\geq 0}|z(t)|_{t}<+\infty\bigg{\\}}.$

Then, $(\mathcal{Y},\|\cdot\|_{\infty})$ is a Banach space. For
$z\in\mathcal{Y}$, we (formally) set

$\begin{split}(\mathcal{T}z)(t)&=-P^{3}(t)(A(t)y(t)+f(t,y(t)+({\mathbb{I}}-P^{3}(t))z(t))-y^{\prime}(t))\\\
&\phantom{=}+\int_{0}^{t}T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds\\\
&\phantom{=}-\int_{t}^{\infty}T(t,s)P^{2}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds,\end{split}$

$t\geq 0$. Observe that it follows from (2.7) and (3.2) that

$\begin{split}&\bigg{|}\int_{0}^{t}T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s))-y^{\prime}(s))\,ds\bigg{|}_{t}\\\
&\leq\int_{0}^{t}|T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s))-y^{\prime}(s))|_{t}\,ds\\\
&\leq
D\int_{0}^{t}\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\nu(s)^{d}|A(s)y(s)+f(s,y(s))-y^{\prime}(s)|\,ds\\\
&\leq
D\delta\int_{0}^{t}\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}\,ds\\\
&=\frac{D\delta}{\lambda}\bigg{(}1-\frac{1}{\mu(t)^{\lambda}}\bigg{)},\end{split}$

and thus

(3.4) $\sup_{t\geq
0}\bigg{|}\int_{0}^{t}T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s))-y^{\prime}(s))\,ds\bigg{|}_{t}\leq\frac{D\delta}{\lambda}.$

Similarly, using (2.8) and (3.2) we have that

$\begin{split}&\bigg{|}\int_{t}^{\infty}T(t,s)P^{2}(s)(A(s)y(s)+f(s,y(s))-y^{\prime}(s))\,ds\bigg{|}_{t}\\\
&\leq
D\int_{t}^{\infty}\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\nu(s)^{d}|A(s)y(s)+f(s,y(s))-y^{\prime}(s)|\,ds\\\
&\leq
D\delta\int_{t}^{\infty}\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}\,ds\\\
&=\frac{D\delta}{\lambda},\end{split}$

and consequently

(3.5) $\sup_{t\geq
0}\bigg{|}\int_{t}^{\infty}T(t,s)P^{2}(s)(A(s)y(s)+f(s,y(s))-y^{\prime}(s))\,ds\bigg{|}_{t}\leq\frac{D\delta}{\lambda}.$

In addition, we have

(3.6) $\begin{split}&|P^{3}(t)(A(t)y(t)+f(t,y(t))-y^{\prime}(t))|_{t}\\\
&=\frac{\mu(t)}{\mu^{\prime}(t)}|P^{3}(t)(A(t)y(t)+f(t,y(t))-y^{\prime}(t))|\\\
&\leq\frac{\mu(t)}{\mu^{\prime}(t)}(2D+1)\nu(t)^{d}|A(t)y(t)+f(t,y(t))-y^{\prime}(t)|,\end{split}$

and therefore

(3.7) $\sup_{t\geq
0}|P^{3}(t)(A(t)y(t)+f(t,y(t))-y^{\prime}(t))|_{t}\leq\delta(2D+1).$

Observe that in (3.6), we have used (see (2.3) and (2.4)) that

$|P^{3}(t)x|\leq|x|+|P^{1}(t)x|+|P^{2}(t)x|\leq|x|+2D\nu(t)^{d}|x|\leq(2D+1)\nu(t)^{d}|x|,$

for $t\geq 0$ and $x\in X$. From (3.4), (3.5) and (3.7), we conclude that

(3.8) $\|\mathcal{T}0\|_{\infty}\leq\bar{D}\delta,$

where

$\bar{D}:=2D+2D\lambda^{-1}+1>0.$

Take now arbitrary $z_{1},z_{2}\in\mathcal{Y}$. By (2.5), (2.6) and (2.7), we
have that

$\begin{split}&\bigg{|}\int_{0}^{t}T(t,s)P^{1}(s)(f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{1}(s))-f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{2}(s)))\,ds\bigg{|}_{t}\\\
&\leq
D\int_{0}^{t}\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\nu(s)^{d}|f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{1}(s))-f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{2}(s))|\,ds\\\
&\leq
Dc\int_{0}^{t}\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}|({\mathbb{I}}-P^{3}(s))z_{1}(s)-({\mathbb{I}}-P^{3}(s))z_{2}(s)|\,ds\\\
&\leq
Dc\int_{0}^{t}\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}(|P^{1}(s)(z_{1}(s)-z_{2}(s))|+|P^{2}(s)(z_{1}(s)-z_{2}(s))|)\,ds\\\
&\leq
Dc\int_{0}^{t}\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}(|P^{1}(s)(z_{1}(s)-z_{2}(s))|_{s}+|P^{2}(s)(z_{1}(s)-z_{2}(s))|_{s})\,ds\\\
&\leq
Dc\int_{0}^{t}\bigg{(}\frac{\mu(t)}{\mu(s)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}|z_{1}(s)-z_{2}(s)|_{s}\,ds\\\
&=\frac{Dc}{\lambda}\bigg{(}1-\frac{1}{\mu(t)^{\lambda}}\bigg{)}\|z_{1}-z_{2}\|_{\infty}\\\
&\leq\frac{Dc}{\lambda}\|z_{1}-z_{2}\|_{\infty},\end{split}$

and thus

(3.9) $\begin{split}&\sup_{t\geq
0}\bigg{|}\int_{0}^{t}T(t,s)P^{1}(s)(f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{1}(s))-f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{2}(s)))\,ds\bigg{|}_{t}\\\
&\leq\frac{Dc}{\lambda}\|z_{1}-z_{2}\|_{\infty}.\end{split}$

Similarly,

$\begin{split}&\bigg{|}\int_{t}^{\infty}T(t,s)P^{2}(s)(f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{1}(s))-f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{2}(s)))\,ds\bigg{|}_{t}\\\
&\leq
D\int_{t}^{\infty}\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\nu(s)^{d}|f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{1}(s))-f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{2}(s))|\,ds\\\
&\leq
Dc\int_{t}^{\infty}\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}|({\mathbb{I}}-P^{3}(s))z_{1}(s)-({\mathbb{I}}-P^{3}(s))z_{2}(s)|\,ds\\\
&\leq
Dc\int_{t}^{\infty}\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}(|P^{1}(s)(z_{1}(s)-z_{2}(s))|+|P^{2}(s)(z_{1}(s)-z_{2}(s))|)\,ds\\\
&\leq
Dc\int_{t}^{\infty}\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}(|P^{1}(s)(z_{1}(s)-z_{2}(s))|_{s}+|P^{2}(s)(z_{1}(s)-z_{2}(s))|_{s})\,ds\\\
&\leq
Dc\int_{t}^{\infty}\bigg{(}\frac{\mu(s)}{\mu(t)}\bigg{)}^{-\lambda}\frac{\mu^{\prime}(s)}{\mu(s)}|z_{1}(s)-z_{2}(s)|_{s}\,ds\\\
&\leq\frac{Dc}{\lambda}\|z_{1}-z_{2}\|_{\infty},\end{split}$

and therefore

(3.10) $\begin{split}&\sup_{t\geq
0}\bigg{|}\int_{t}^{\infty}T(t,s)P^{2}(s)(f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{1}(s))-f(s,y(s)+({\mathbb{I}}-P^{3}(s))z_{2}(s)))\,ds\bigg{|}_{t}\\\
&\leq\frac{Dc}{\lambda}\|z_{1}-z_{2}\|_{\infty}.\end{split}$

Finally,

$\begin{split}&|P^{3}(t)(f(t,y(t)+({\mathbb{I}}-P^{3}(t))z_{1}(t))-f(t,y(t)+({\mathbb{I}}-P^{3}(t))z_{2}(t)))|_{t}\\\
&=\frac{\mu(t)}{\mu^{\prime}(t)}|P^{3}(t)(f(t,y(t)+({\mathbb{I}}-P^{3}(t))z_{1}(t))-f(t,y(t)+({\mathbb{I}}-P^{3}(t))z_{2}(t)))|\\\
&\leq\frac{\mu(t)}{\mu^{\prime}(t)}(2D+1)\nu(t)^{d}|f(t,y(t)+({\mathbb{I}}-P^{3}(t))z_{1}(t))-f(t,y(t)+({\mathbb{I}}-P^{3}(t))z_{2}(t))|\\\
&\leq
c(2D+1)|({\mathbb{I}}-P^{3}(t))z_{1}(t)-({\mathbb{I}}-P^{3}(t))z_{2}(t)|\\\
&\leq c(2D+1)|z_{1}(t)-z_{2}(t)|_{t},\end{split}$

and hence

(3.11) $\begin{split}&\sup_{t\geq
0}|P^{3}(t)(f(t,y(t)+({\mathbb{I}}-P^{3}(t))z_{1}(t))-f(t,y(t)+({\mathbb{I}}-P^{3}(t))z_{2}(t)))|_{t}\\\
&\leq c(2D+1)\|z_{1}-z_{2}\|_{\infty}.\end{split}$

From (3.9), (3.10) and (3.11), we conclude that

(3.12) $\|\mathcal{T}z_{1}-\mathcal{T}z_{2}\|_{\infty}\leq
q\|z_{1}-z_{2}\|_{\infty}.$

Set

$C:=\frac{\bar{D}}{1-q},$

and let

$\mathcal{B}:=\\{z\in\mathcal{Y}:\|z\|_{\infty}\leq C\delta\\}.$

We claim that $\mathcal{T}(\mathcal{B})\subset\mathcal{B}$. Indeed, take an
arbitrary $z\in\mathcal{B}$. It follows from (3.8) and (3.12) that

$\begin{split}\|\mathcal{T}z\|_{\infty}&\leq\|\mathcal{T}0\|_{\infty}+\|\mathcal{T}z-\mathcal{T}0\|_{\infty}\\\
&\leq\bar{D}\delta+q\|z\|_{\infty}\\\ &\leq\bar{D}\delta+qC\delta\\\
&=C\delta,\end{split}$

and thus $\mathcal{T}z\in\mathcal{B}$. From (3.1) and (3.12), we conclude that
$\mathcal{T}\rvert_{\mathcal{B}}\colon\mathcal{B}\to\mathcal{B}$ is a
contraction, and therefore it has a unique fixed point $z\in\mathcal{B}$.
Hence, for each $t\geq 0$ we have that

(3.13)
$\begin{split}z(t)&=-P^{3}(t)(A(t)y(t)+f(t,y(t)+({\mathbb{I}}-P^{3}(t))z(t))-y^{\prime}(t))\\\
&\phantom{=}+\int_{0}^{t}T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds\\\
&\phantom{=}-\int_{t}^{\infty}T(t,s)P^{2}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds.\end{split}$

Set

$\begin{split}\bar{z}(t)&=\int_{0}^{t}T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds\\\
&\phantom{=}-\int_{t}^{\infty}T(t,s)P^{2}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds,\end{split}$

for $t\geq 0$. Observe that $\bar{z}(t)=({\mathbb{I}}-P^{3}(t))z(t)$, $t\geq
0$. Moreover, for $t\geq r\geq 0$ we have that

$\begin{split}&\bar{z}(t)-T(t,r)\bar{z}(r)\\\
&=\int_{0}^{t}T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds\\\
&\phantom{=}-\int_{0}^{r}T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds\\\
&\phantom{=}-\int_{t}^{\infty}T(t,s)P^{2}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds\\\
&\phantom{=}+\int_{r}^{\infty}T(t,s)P^{2}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds,\end{split}$

and consequently

$\begin{split}&\bar{z}(t)-T(t,r)\bar{z}(r)\\\
&=\int_{r}^{t}T(t,s)P^{1}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds\\\
&\phantom{=}+\int_{r}^{t}T(t,s)P^{2}(s)(A(s)y(s)+f(s,y(s)+({\mathbb{I}}-P^{3}(s))z(s))-y^{\prime}(s))\,ds.\end{split}$

By differentiating, we obtain that

$\bar{z}^{\prime}(t)=A(t)\bar{z}(t)+(P^{1}(t)+P^{2}(t))(A(t)y(t)+f(t,y(t)+\bar{z}(t))-y^{\prime}(t)),$

for $t\geq 0$. This easily implies that

(3.14)
$\begin{split}(y+\bar{z})^{\prime}(t)&=A(t)(y(t)+\bar{z}(t))+f(t,y(t)+\bar{z}(t))\\\
&\phantom{=}-P^{3}(t)(A(t)y(t)+f(t,y(t)+\bar{z}(t))-y^{\prime}(t)),\end{split}$

for $t\geq 0$. We define $x\colon[0,+\infty)\to X$ by

$x(t)=y(t)+\bar{z}(t),\quad t\geq 0.$

Then,

$P^{3}(t)(x(t)-y(t))=P^{3}(t)\bar{z}(t)=P^{3}(t)({\mathbb{I}}-P^{3}(t))z(t)=0,$

for $t\geq 0$. Moreover, using (2.6) we have that

$|\bar{z}(t)|=|(P^{1}(t)+P^{2}(t))z(t)|\leq|P^{1}(t)z(t)|_{t}+|P^{2}(t)z(t)|_{t}\leq|z(t)|_{t}\leq\|z\|_{\infty},$

for each $t\geq 0$. Thus,

$\sup_{t\geq 0}|x(t)-y(t)|=\sup_{t\geq 0}|\bar{z}(t)|\leq C\delta.$

In addition, (3.14) implies that
$x^{\prime}(t)-A(t)x(t)-f(t,x(t))\in\operatorname{Im}P^{3}(t)$ for $t\geq 0$.
It remains to establish (3.3). We have that

$\begin{split}&|x^{\prime}(t)-A(t)x(t)-f(t,x(t))|\\\
&=|-P^{3}(t)(A(t)y(t)+f(t,y(t)+\bar{z}(t))-y^{\prime}(t))|\\\
&=|-P^{3}(t)(A(t)y(t)+f(t,y(t)+({\mathbb{I}}-P^{3}(t))z(t))-y^{\prime}(t))|\\\
&=|z(t)-\bar{z}(t)|\\\ &=|z(t)-({\mathbb{I}}-P^{3}(t))z(t))|\\\
&=|P^{3}(t)z(t)|\\\ &\leq(2D+1)\nu(t)^{d}|z(t)|\\\
&\leq(2D+1)\nu(t)^{d}\max\bigg{\\{}1,\frac{\mu^{\prime}(t)}{\mu(t)}\bigg{\\}}|z(t)|_{t}\\\
&\leq
C\delta(2D+1)\nu(t)^{d}\max\bigg{\\{}1,\frac{\mu^{\prime}(t)}{\mu(t)}\bigg{\\}},\end{split}$

where we used that

$\begin{split}|z(t)|&\leq|P^{1}(t)z(t)|+|P^{2}(t)z(t)|+|P^{3}(t)z(t)|\\\
&\leq|P^{1}(t)z(t)|_{t}+|P^{2}(t)z(t)|_{t}+\frac{\mu^{\prime}(t)}{\mu(t)}|P^{3}(t)z(t)|_{t}\\\
&\leq\max\bigg{\\{}1,\frac{\mu^{\prime}(t)}{\mu(t)}\bigg{\\}}(|P^{1}(t)z(t)|_{t}+|P^{2}(t)z(t)|_{t}+|P^{3}(t)z(t)|_{t})\\\
&=\max\bigg{\\{}1,\frac{\mu^{\prime}(t)}{\mu(t)}\bigg{\\}}|z(t)|_{t},\end{split}$

for $t\geq 0$. Thus, (3.3) holds and the proof of the theorem is completed. ∎

Obviously, Theorem 3.1 is most interesting in the particular case when
$P^{3}(t)=0$ for $t\geq 0$. Indeed, in this case we can in the vicinity of
each approximate solution of (2.2) construct an exact solution of (2.2). More
precisely, we have the following result.

######  3.3 Corollary.

Assume that $P^{3}(t)=0$ for each $t\geq 0$ and that (3.1) holds. Then, there
exists $C>0$ with the property that for any $\delta>0$ and an arbitrary
differentiable map $y\colon[0,+\infty)\to X$ satisfying (3.2), there exists a
solution $x\colon[0,+\infty)\to X$ of (2.2) such that

(3.15) $\sup_{t\geq 0}|x(t)-y(t)|\leq C\delta.$

###### Proof.

The desired conclusion follows directly from Theorem 3.1. ∎

We observe that Corollary 3.3 is in particular applicable to a linear equation
(2.1).

######  3.4 Corollary.

There exists $C>0$ with the property that for any $\delta>0$ and an arbitrary
differentiable map $y\colon[0,+\infty)\to X$ satisfying

$|y^{\prime}(t)-A(t)y(t)|\leq\frac{\delta\mu^{\prime}(t)}{\mu(t)\nu(t)^{d}}\quad\text{for
$t\geq 0$,}$

there exists a solution $x\colon[0,+\infty)\to X$ of (2.1) such that (3.15)
holds.

###### Proof.

It sufficies to apply Corollary 3.3 in the case when $f(t,x)=0$ for $t\geq 0$
and $x\in X$. Observe that in this case (2.5) holds with $c=0$ and thus (3.1)
is obviously satisfied. ∎

##  4\. Examples

In this section we discuss a variety of settings to which our results are
applicable. In addition, we discuss the relationship between Theorem 3.1 and
the results already available in the literature.

###  4.1. The case of exponential dichotomies

This situation corresponds to the case when $P^{3}(t)=0$, $\mu(t)=e^{t}$ and
$\nu(t)=1$, $t\geq 0$. Observe that in this setting, (2.5) reduces to

$|f(t,x)-f(t,y)|\leq c|x-y|,\quad\text{for $t\geq 0$ and $x,y\in X$.}$

Moreover, (3.2) is equivalent to

$|y^{\prime}(t)-A(t)y(t)-f(t,y(t))|\leq\delta\quad\text{for $t\geq 0$,}$

Hence, Corollary 3.3 is a analogous to [3, Theorem 5] except that in the
present paper, we consider dynamics on $[0,+\infty)$ instead of $\mathbb{R}$.
We stress that the proof of [3, Theorem 5] is done by passing from the
continuous time dynamics to the associated discrete time dynamics, while in
the present paper we developed a direct approach.

###  4.2. The case of partial exponential dichotomies

In this setting we continue to assume that $\mu(t)=e^{t}$ and $\nu(t)=1$,
$t\geq 0$. However, we don’t require that $P^{3}(t)=0$. In this context,
Theorem 3.1 represents a continuous time version of [2, Theorem 2.]. We stress
that in [2, Section 5.], a different continuous time version of [2, Theorem
2.] (see [2, Theorem 4.]) was proposed and again obtained by suitable
discretization of time. However, Theorem 3.1 presents a more natural
continuous time version of [2, Theorem 2.] (when compared to [2, Theorem 4.]),
since $x$ given by Theorem 3.1 is a solution of (2.2) up to the movement in
the central direction (i.e. along images of $P^{3}(t)$). We emphasize that
this is precisely what corresponds to [2, Theorem 2.] in the case of discrete
time.

We discuss a particular example to which Theorem 3.1 is applicable in this
setting.

######  4.1 Example.

Take $X=\mathbb{R}^{3}$ and set

$A(t)=\begin{pmatrix}-(1+2t)&0&0\\\ 0&1+2t&0\\\ 0&0&0\end{pmatrix},\quad t\geq
0.$

Then, it is easy to show that (2.3) and (2.4) holds with $D=\lambda=1$ and

$P^{1}(x_{1},x_{2},x_{3})=(x_{1},0,0)\quad\text{and}\quad
P^{2}(x_{1},x_{2},x_{3})=(0,x_{2},0),$

$(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$. Take $c\geq 0$ and set

$f(t,x)=c\sin|x|,\quad(t,x)\in[0,+\infty)\times\mathbb{R}^{3}.$

Obviously, (2.5) holds. Hence, Theorem 3.1 is applicable whenever $c$ is
sufficiently small.

###  4.3. The case of tempered exponential dichotomies

Here, we take $P^{3}(t)=0$ and $\mu(t)=e^{t}$, $t\geq 0$. In addition, we
assume that $\nu$ is tempered, i.e. that

(4.1) $\lim_{t\to+\infty}\frac{1}{t}\ln\nu(t)=0.$

Let us discuss a particular example.

######  4.2 Example.

Take $X=\mathbb{R}$ and set

$A(t)=-1+\frac{\cos t}{\sqrt{1+t}}-\sqrt{1+t}\sin t,\quad t\geq 0.$

One can easily show that

$T(t,s)=e^{-(t-s)+\sqrt{1+t}\cos t-\sqrt{1+s}\cos s},\quad t,s\geq 0.$

Obviously,

$T(t,s)\leq e^{-(t-s)+\sqrt{1+t}+\sqrt{1+s}},\quad t\geq s\geq 0.$

Since

$\sqrt{1+t}-\sqrt{1+s}\leq\frac{1}{2}(t-s)\quad t\geq s\geq 0,$

we have that

$T(t,s)\leq e^{-\frac{1}{2}(t-s)+2\sqrt{1+s}},\quad t\geq s\geq 0.$

Therefore, (2.3) and (2.4) hold with $\lambda=\frac{1}{2}$, $D=1$,
$P^{1}(t)={\mathbb{I}}$, $P^{2}(t)=0$, $d=2$ and $\nu(t)=e^{\sqrt{1+t}}$
(which obviously satisfies (4.1)). Take $c\geq 0$ and set

$f(t,x)=\frac{c}{\nu(t)^{d}}\sin x,\quad(t,x)\in[0,+\infty)\times\mathbb{R}.$

We see that (2.5) holds. Again, Theorem 3.1 is applicable whenever $c$ is
sufficiently small.

In this context, Corollary 3.3 is similar to the results established in [4],
although in that paper we dealt with random dynamics.

###  4.4. The case of nonuniform polynomial dichotomies

Here we take $\mu(t)=\nu(t)=t+1$ and $P^{3}(t)=0$ for $t\geq 0$ (see [7]).
Hence, (2.5) reads as

(4.2) $|f(t,x)-f(t,y)|\leq\frac{c}{(1+t)^{d+1}}|x-y|,\quad\text{for $t\geq 0$
and $x,y\in X$,}$

while (3.2) means that

$|y^{\prime}(t)-A(t)y(t)-f(t,y(t))|\leq\frac{\delta}{(1+t)^{d+1}}\quad\text{for
$t\geq 0$.}$

To the best of our knowledge, in the present context, Corollary 3.3 represents
a first result discussing the shadowing property for (2.2) when (2.1) admits a
polynomial dichotomy.

For the sake of completeness we discuss a concrete example in this setting to
which Corollary 3.3 is applicable.

######  4.3 Example.

Take $X=\mathbb{R}^{2}$, $d>0$ and $a<0\leq b$. Furthermore, for $t\geq 0$ set

$A(t)=\begin{pmatrix}a(t)&0\\\ 0&b(t)\end{pmatrix},$

where

$a(t)=\frac{a}{t+1}+\frac{d}{2(t+1)}(\cos t-1)-\frac{d}{2}\ln(1+t)\sin t$

and

$b(t)=\frac{b}{t+1}-\frac{d}{2(t+1)}(\cos t-1)+\frac{d}{2}\ln(1+t)\sin t.$

It is proved in [7, Example 2.1.] that (2.3) and (2.4) hold with $D=1$,
$\lambda=\min\\{-a,b\\}>0$ and

$P^{1}(x,y)=(x,0)\quad\text{and}\quad
P^{2}(x,y)=(0,y),\quad(x,y)\in\mathbb{R}^{2}.$

Take $c\geq 0$ and set

$f(t,x)=\frac{c}{(1+t)^{d+1}}\sin|x|,\quad(t,x)\in[0,+\infty)\times\mathbb{R}^{2}.$

Clearly, (4.2) holds and thus Corollary 3.3 is applicable for $c$ sufficiently
small.

###  4.5. The case of $(\mu,\nu)$-dichotomies

The notion of $(\mu,\nu)$-dichotomy is introduced by Bento and Silva [9] and
corresponds to our general setting with $P^{3}(t)=0$. We observe that this
setting includes the ones discussed in Subsections 4.1 and 4.4 as particular
cases. We refer to [7] for particular examples of this behaviour (which don’t
belong to the classes of examples discussed in Subsections 4.1 and 4.4).
Again, Corollary 3.3 represents a first shadowing result for (2.2) in the case
when (2.1) admits a $(\mu,\nu)$-dichotomy.

##  5\. Equations of higher order

The purpose of this section is to indicate that our results apply for
differential equations of higher order.

We take continuous functions $A,B\colon[0,+\infty)\to\mathcal{B}(X)$ and a
continuous function $f\colon[0,+\infty)\times X\times X\to X$. We consider a
nonlinear differential equation of second order given by

(5.1) $x^{\prime\prime}=A(t)x^{\prime}+B(t)x+f(t,x^{\prime},x)\quad t\geq 0,$

and the associated linear equation

(5.2) $x^{\prime\prime}=A(t)x^{\prime}+B(t)x,\quad t\geq 0.$

Let $X^{2}=X\times X$. Then, $X^{2}$ is a Banach space with respect to the
norm $|(x_{1},x_{2})|^{\prime}=|x_{1}|+|x_{2}|$, $(x_{1},x_{2})\in X^{2}$. For
$t\geq 0$, we define $C(t)\colon X^{2}\to X^{2}$ by

$C(t)(x_{1},x_{2})=(A(t)x_{1}+B(t)x_{2},x_{1}),\quad(x_{1},x_{2})\in X^{2}.$

We consider a linear differential equation on $X^{2}$ given by

(5.3) $z^{\prime}=C(t)z.$

######  5.1 Corollary.

Assume that (5.3) admits a $(\mu,\nu)$-dichotomy (see Section 4.5).
Furthermore, suppose that there exists $c\geq 0$ such that

(5.4)
$|f(t,x_{1},x_{2})-f(t,x_{1}^{\prime},x_{2}^{\prime})|\leq\frac{c\mu^{\prime}(t)}{\mu(t)\nu(t)^{d}}(|x_{1}-x_{1}^{\prime}|+|x_{2}-x_{2}^{\prime}|),$

for $t\geq 0$ and $(x_{1},x_{2}),(x_{1}^{\prime},x_{2}^{\prime})\in X^{2}$.
Then, for every $c$ sufficiently small, there exists $C>0$ with the property
that for any $\delta>0$ and an arbitrary twice differentiable map
$y\colon[0,+\infty)\to X$ satisfying

(5.5) $\sup_{t\geq
0}|y^{\prime\prime}(t)-A(t)y^{\prime}(t)-B(t)y(t)-f(t,y^{\prime}(t),y(t))|\leq\frac{\delta\mu^{\prime}(t)}{\mu(t)\nu(t)^{d}},$

there exists a solution $x\colon[0,+\infty)\to X$ of (5.1) such that

(5.6) $\sup_{t\geq 0}|x(t)-y(t)|\leq C\delta.$

###### Proof.

We define $w\colon[0,+\infty)\to X^{2}$ by

$w(t)=(y^{\prime}(t),y(t)),\quad t\geq 0.$

Furthermore, we define $g\colon[0,+\infty)\times X^{2}\to X^{2}$ by

$g(t,x_{1},x_{2})=(f(t,x_{1},x_{2}),0),\quad(x_{1},x_{2})\in X^{2}.$

By (5.4), we have that

(5.7)
$|g(t,x_{1},x_{2})-g(t,x_{1}^{\prime},x_{2}^{\prime})|^{\prime}\leq\frac{c\mu^{\prime}(t)}{\mu(t)\nu(t)^{d}}|(x_{1},x_{2})-(x_{1}^{\prime},x_{2}^{\prime})|^{\prime},$

for $t\geq 0$ and $(x_{1},x_{2}),(x_{1}^{\prime},x_{2}^{\prime})\in X^{2}$.
Moreover, it follows from (5.5) that

(5.8) $\sup_{t\geq
0}|w^{\prime}(t)-C(t)w(t)-g(t,w(t))|^{\prime}\leq\frac{\delta\mu^{\prime}(t)}{\mu(t)\nu(t)^{d}}$

Then, for $c$ sufficiently small, we can apply Corollary 3.3 and conclude that
there exists a differentiable map $z=(z_{1},z_{2})\colon[0,+\infty)\to X^{2}$
such that

(5.9) $z^{\prime}(t)=C(t)z(t)+g(t,z(t))\quad t\geq 0,$

and

(5.10) $\sup_{t\geq 0}|z(t)-w(t)|^{\prime}\leq C\delta,$

where $C>0$ comes from Corollary 3.3. From (5.9), it follows easily that
$x:=z_{2}$ is a solution of (5.1). Moreover, (5.10) implies that (5.6) holds.
The proof is completed. ∎

######  5.2 Remark.

In the particular case when $X=\mathbb{C}$, $\mu(t)=e^{t}$, $\nu(t)=1$, $t\geq
0$ and with $A$ and $B$ being periodic, the result similar to Corollary 5.1
was obtained in [12, Theorem 2.12.]. Indeed, the only difference is that in
[12], the authors consider dynamics on $\mathbb{R}$ while we consider dynamics
on $[0,+\infty)$.

######  5.3 Remark.

In Corollary 5.1, we have for the sake of simplicity focused on equations of
order $2$. However, it is clear that an analogous result can be formulated for
equations of any order.

##  6\. The case of discrete time

In this section we present a discrete time version of Theorem 3.1. We start by
introducing some useful notations. Let $(X,|\cdot|)$ and
$(\mathcal{B}(X),\|\cdot\|)$ be as in Section 2. We denote by ${\mathbb{N}}$
the set of all natural numbers, including $0$, while ${\mathbb{N}}^{\ast}$
denotes the set of all natural numbers excluding $0$. Given a sequence
$(A_{n})_{n\in\mathbb{N}}$ of bounded linear operators in $\mathcal{B}(X)$ and
$m,n\in\mathbb{N}$, let us consider

${\mathcal{A}}(m,n)=\begin{cases}A_{m-1}\cdots A_{n}&\text{for $m>n$;}\\\
{\mathbb{I}}&\text{for $m=n$.}\\\ \end{cases}$

Let $\mu=(\mu_{n})_{n\in\mathbb{N}}$ be a strictly increasing sequence such
that

$\mu_{0}\geq 1\quad\text{and}\quad\lim_{n\to+\infty}\mu_{n}=+\infty.$

Moreover, let $\nu=(\nu_{n})_{n\in\mathbb{N}}$ be an arbitrary sequence with
$\nu_{n}\geq 1$ for every $n\in{\mathbb{N}}$.

All along this section we are going to assume that the following conditions
are satisfied:

  1. (1)

there exist three families of projections $P^{i}_{n}$, $n\in\mathbb{N}$,
$i\in\\{1,2,3\\}$ such that

$P^{1}_{n}+P^{2}_{n}+P^{3}_{n}={\mathbb{I}},\quad n\in{\mathbb{N}};$

  2. (2)

for $i,j\in\\{1,2,3\\}$, $i\neq j$ and $n\in{\mathbb{N}}$,

$P^{i}_{n}P^{j}_{n}=0;$

  3. (3)

for every $n\in{\mathbb{N}}$ and $i\in\\{1,2,3\\}$, we have that

$A_{n}P^{i}_{n}=P^{i}_{n+1}A_{n};$

  4. (4)

$A_{n}\rvert_{\operatorname{Im}P^{2}_{n}}\colon\operatorname{Im}P^{2}_{n}\to\operatorname{Im}P^{2}_{n+1}$
is an invertible operator for each $n\in{\mathbb{N}}$;

  5. (5)

there exist $D,\lambda>0$ and $d\geq 0$ such that

(6.1) $\|{\mathcal{A}}(m,n)P_{n}^{1}\|\leq
D\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\lambda}\nu_{n}^{d}\quad\text{ for
}m\geq n$

and

(6.2) $\|{\mathcal{A}}(m,n)P_{n}^{2}\|\leq
D\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\lambda}\nu_{n}^{d}\quad\text{ for
}m\leq n$

where

${\mathcal{A}}(m,n):=\big{(}{\mathcal{A}}(n,m)\rvert_{\operatorname{Im}P^{2}_{m}}\big{)}^{-1}\colon\operatorname{Im}P^{2}_{n}\to\operatorname{Im}P^{2}_{m},$

for $m\leq n$.

For $n\in{\mathbb{N}}$, set

$E^{s}_{n}=\operatorname{Im}P^{1}_{n},\ E^{u}_{n}=\operatorname{Im}P^{2}_{n}\
\text{and}\ E^{c}_{n}=\operatorname{Im}P^{3}_{n}.$

Furthermore, we set

$|x|_{n}:=\sup_{m\geq
n}\bigg{(}|{\mathcal{A}}(m,n)x|\bigg{(}\frac{\mu_{m}}{\mu_{n}}\bigg{)}^{\lambda}\bigg{)}\quad\text{for
$n\in{\mathbb{N}}$ and $x\in E^{s}_{n}$,}$ $|x|_{n}:=\sup_{m\leq
n}\bigg{(}|{\mathcal{A}}(m,n)x|\bigg{(}\frac{\mu_{n}}{\mu_{m}}\bigg{)}^{\lambda}\bigg{)}\quad\text{for
$n\in{\mathbb{N}}$ and $x\in E^{u}_{n}$,}$

and

$|x|_{n}:=\left(\frac{\mu^{\lambda}_{n+1}}{\mu^{\lambda}_{n+1}-\mu_{n}^{\lambda}}\right)|x|,\quad\text{for
$n\in{\mathbb{N}}$ and $x\in E^{c}_{n}$.}$

Finally, for an arbitrary $x\in X$ let

$|x|_{n}:=|P^{1}_{n}x|_{n}+|P^{2}_{n}x|_{n}+|P^{3}_{n}x|_{n}.$

It is easy to see that $|\cdot|_{n}$ is a norm on $X$ for every
$n\in{\mathbb{N}}$. Moreover, by repeating the arguments in the proof of Lemma
2.1, one can establish the following result.

######  6.1 Lemma.

The following properties hold:

  * •

for $n\in{\mathbb{N}}$ and $x\in E^{s}_{n}\cup E^{u}_{n}$,

(6.3) $|x|\leq|x|_{n};$

  * •

for $n\geq m$ and $x\in X$,

(6.4) $|{\mathcal{A}}(n,m)P^{1}_{m}x|_{n}\leq
D\bigg{(}\frac{\mu_{n}}{\mu_{m}}\bigg{)}^{-\lambda}\nu_{m}^{d}|x|;$

  * •

for $n\leq m$ and $x\in X$,

(6.5) $|{\mathcal{A}}(n,m)P^{2}_{m}x|_{n}\leq
D\bigg{(}\frac{\mu_{m}}{\mu_{n}}\bigg{)}^{-\lambda}\nu_{m}^{d}|x|.$

###  6.1. Main result for the case of discrete time

Let $f_{n}:X\to X$ be a sequence of maps and suppose there exist $c\geq 0$
such that

(6.6) $|f_{n}(x)-f_{n}(y)|\leq
c\left(\frac{\mu^{\lambda}_{n+1}-\mu_{n}^{\lambda}}{\mu_{n+1}^{\lambda}\nu_{n}^{d}}\right)|x-y|,\quad\text{for
$n\in{\mathbb{N}}$ and $x,y\in X$.}$

The next result is a version of Theorem 3.1 in the present setting.

######  6.2 Theorem.

Suppose that

(6.7) $q:=c(4D+1)<1.$

Then, there exists $C>0$ with the property that for any $\delta>0$ and an
arbitrary sequence $\mathbf{y}=(y_{n})_{n\in{\mathbb{N}}}\subset X$ satisfying

(6.8)
$|y_{n+1}-A_{n}y_{n}-f_{n}(y_{n})|\leq\delta\left(\frac{\mu^{\lambda}_{n+1}-\mu_{n}^{\lambda}}{\mu_{n+1}^{\lambda}\nu_{n}^{d}}\right)\quad\text{for
every $n\in{\mathbb{N}}$,}$

there exists a sequence $\mathbf{x}=(x_{n})_{n\in{\mathbb{N}}}\subset X$ such
that:

  * •

$P^{3}_{n}(x_{n}-y_{n})=0$ for every $n\in{\mathbb{N}}$;

  * •

$\sup_{n\in{\mathbb{N}}}|x_{n}-y_{n}|\leq C\delta$;

  * •

$x_{n+1}-A_{n}x_{n}-f_{n}(x_{n})\in\operatorname{Im}P^{3}_{n}$ for every
$n\in{\mathbb{N}}$;

  * •

for every $n\in\mathbb{N}$,

(6.9) $|x_{n+1}-A_{n}x_{n}-f_{n}(x_{n})|\leq C\delta(2D+1)\nu_{n}^{d}.$

######  6.3 Remark.

As its continuous time counterpart, Theorem 6.2 may be seen as a version of
the shadowing property for the nonlinear dynamics

$x_{n+1}=A_{n}x_{n}+f_{n}(x_{n}),\quad n\in{\mathbb{N}}.$

######  6.4 Remark.

We emphasize that we don’t require operators $A_{n}$ to be invertible. Indeed,
it is only assumed that $A_{n}\rvert_{E_{n}^{u}}\colon E_{n}^{u}\to
E_{n+1}^{u}$ is invertible for each $n\in\mathbb{N}$.

###### Proof of Theorem 6.2.

We follow closely the proof of Theorem 3.1.

Let

$\mathcal{Y}:=\bigg{\\{}\mathbf{z}=(z_{n})_{n\in\mathbb{N}}\subset X:\
\|\mathbf{z}\|_{\infty}:=\sup_{n\in{\mathbb{N}}}|z_{n}|_{n}<+\infty\bigg{\\}}.$

Then, $(\mathcal{Y},\|\cdot\|_{\infty})$ is a Banach space. For
$\mathbf{z}\in\mathcal{Y}$, we set $(\mathcal{T}\mathbf{z})_{0}=0$ and

$\begin{split}(\mathcal{T}\mathbf{z})_{n}&=-P^{3}_{n-1}(A_{n-1}y_{n-1}+f_{n-1}(y_{n-1}+({\mathbb{I}}-P^{3}_{n-1})z_{n-1})-y_{n})\\\
&\phantom{=}+\sum_{m=0}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\\\
&\phantom{=}-\sum_{m=n}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\end{split}$

for all $n\in{\mathbb{N}}^{\ast}$.

We start observing that it follows from (6.4) and (6.8) that

$\begin{split}&\bigg{|}\sum_{m=0}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m})-y_{m+1})\bigg{|}_{n}\\\
&\leq\sum_{m=0}^{n-1}|{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m})-y_{m+1})|_{n}\\\
&\leq
D\sum_{m=0}^{n-1}\bigg{(}\frac{\mu_{n}}{\mu_{m}}\bigg{)}^{-\lambda}\nu_{m}^{d}|A_{m}y_{m}+f_{m}(y_{m})-y_{m+1}|\\\
&\leq
D\delta\sum_{m=0}^{n-1}\bigg{(}\frac{\mu_{n}}{\mu_{m}}\bigg{)}^{-\lambda}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m+1}^{\lambda}}\right)\\\
&=D\delta\frac{1}{\mu_{n}^{\lambda}}\sum_{m=0}^{n-1}\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m}^{-\lambda}\mu_{m+1}^{\lambda}}\\\
&\leq
D\delta\frac{1}{\mu_{n}^{\lambda}}\sum_{m=0}^{n-1}\left(\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}\right)\\\
&\leq D\delta\bigg{(}1-\frac{1}{\mu_{n}^{\lambda}}\bigg{)},\end{split}$

for $n\in\mathbb{N}^{*}$ and thus

(6.10)
$\sup_{n\in{\mathbb{N}}^{\ast}}\bigg{|}\sum_{m=0}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m})-y_{m+1})\bigg{|}_{n}\leq
D\delta.$

Similarly, using (6.5) and (6.8) we have that

$\begin{split}&\bigg{|}\sum_{m=n}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(A_{m}y_{m}+f_{m}(y_{m})-y_{m+1})\bigg{|}_{n}\\\
&\leq
D\sum_{m=n}^{+\infty}\bigg{(}\frac{\mu_{m}}{\mu_{n}}\bigg{)}^{-\lambda}\nu_{m}^{d}|A_{m}y_{m}+f_{m}(y_{m})-y_{m+1}|\\\
&\leq
D\delta\sum_{m=n}^{+\infty}\bigg{(}\frac{\mu_{m}}{\mu_{n}}\bigg{)}^{-\lambda}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m+1}^{\lambda}}\right)\\\
&=D\delta\mu_{n}^{\lambda}\sum_{m=n}^{+\infty}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m}^{\lambda}\mu_{m+1}^{\lambda}}\right)\\\
&=D\delta\mu_{n}^{\lambda}\sum_{m=n}^{+\infty}\left(\frac{1}{\mu_{m}^{\lambda}}-\frac{1}{\mu_{m+1}^{\lambda}}\right)\\\
&=D\delta,\end{split}$

and consequently

(6.11)
$\sup_{n\in{\mathbb{N}}}\bigg{|}\sum_{m=n}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(A_{m}y_{m}+f_{m}(y_{m})-y_{m+1})\bigg{|}_{n}\leq
D\delta.$

Moreover, we have

(6.12) $\begin{split}&|P^{3}_{n}(A_{n}y_{n}+f_{n}(y_{n})-y_{n+1})|_{n}\\\
&=\left(\frac{\mu^{\lambda}_{n+1}}{\mu_{n+1}^{\lambda}-\mu_{n}^{\lambda}}\right)|P^{3}_{n}(A_{n}y_{n}+f_{n}(y_{n})-y_{n+1})|\\\
&\leq\left(\frac{\mu^{\lambda}_{n+1}}{\mu_{n+1}^{\lambda}-\mu_{n}^{\lambda}}\right)(2D+1)\nu_{n}^{d}|A_{n}y_{n}+f_{n}(y_{n})-y_{n+1}|,\end{split}$

and therefore

(6.13)
$\sup_{n\in{\mathbb{N}}}|P^{3}_{n}(A_{n}y_{n}+f_{n}(y_{n})-y_{n+1})|_{n}\leq\delta(2D+1).$

Observe that in (6.12), we have used (see (6.1) and (6.2)) that

$|P^{3}_{n}x|\leq|x|+|P^{1}_{n}x|+|P^{2}_{n}x|\leq|x|+2D\nu_{n}^{d}|x|\leq(2D+1)\nu_{n}^{d}|x|,$

for every $n\in{\mathbb{N}}$ and $x\in X$. Combining (6.10), (6.11) and
(6.13), we conclude that

(6.14) $\|\mathcal{T}\mathbf{0}\|_{\infty}\leq\bar{D}\delta,$

where

$\bar{D}:=4D+1>0.$

Now, let
$\mathbf{z}=(z_{n})_{n\in\mathbb{N}},\mathbf{w}=(w_{n})_{n\in\mathbb{N}}\in\mathcal{Y}$
be arbitrary. By (6.6), (6.3) and (6.4), we have that

$\begin{split}&\bigg{|}\sum_{m=0}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})w_{m}))\bigg{|}_{n}\\\
&\leq
D\sum_{m=0}^{n-1}\bigg{(}\frac{\mu_{n}}{\mu_{m}}\bigg{)}^{-\lambda}\nu_{m}^{d}|f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})w_{m})|\\\
&\leq
Dc\sum_{m=0}^{n-1}\bigg{(}\frac{\mu_{n}}{\mu_{m}}\bigg{)}^{-\lambda}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m+1}^{\lambda}}\right)|({\mathbb{I}}-P^{3}_{m})z_{m}-({\mathbb{I}}-P^{3}_{m})w_{m}|\\\
&\leq
Dc\sum_{m=0}^{n-1}\bigg{(}\frac{\mu_{n}}{\mu_{m}}\bigg{)}^{-\lambda}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m+1}^{\lambda}}\right)(|P^{1}_{m}(z_{m}-w_{m})|+|P^{2}_{m}(z_{m}-w_{m})|)\\\
&\leq
Dc\sum_{m=0}^{n-1}\bigg{(}\frac{\mu_{n}}{\mu_{m}}\bigg{)}^{-\lambda}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m+1}^{\lambda}}\right)(|P^{1}_{m}(z_{m}-w_{m})|_{m}+|P^{2}_{m}(z_{m}-z_{m})|_{m})\\\
&\leq
Dc\frac{1}{\mu_{n}^{\lambda}}\sum_{m=0}^{n-1}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m}^{-\lambda}\mu_{m+1}^{\lambda}}\right)|z_{m}-w_{m}|_{m}\\\
&\leq
Dc\frac{1}{\mu_{n}^{\lambda}}\sum_{m=0}^{n-1}\left(\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}\right)\|\mathbf{z}-\mathbf{w}\|_{\infty}\\\
&\leq Dc\|\mathbf{z}-\mathbf{w}\|_{\infty},\end{split}$

and thus

(6.15)
$\begin{split}&\sup_{n\in{\mathbb{N}}^{\ast}}\bigg{|}\sum_{m=0}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})w_{m})\bigg{|}_{n}\\\
&\leq Dc\|\mathbf{z}-\mathbf{w}\|_{\infty}.\end{split}$

Similarly,

$\begin{split}&\bigg{|}\sum_{m=n}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})w_{m}))\bigg{|}_{n}\\\
&\leq
D\sum_{m=n}^{+\infty}\bigg{(}\frac{\mu_{m}}{\mu_{n}}\bigg{)}^{-\lambda}\nu_{m}^{d}|f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})w_{m})|\\\
&\leq
Dc\sum_{m=n}^{+\infty}\bigg{(}\frac{\mu_{m}}{\mu_{n}}\bigg{)}^{-\lambda}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m+1}^{\lambda}}\right)|({\mathbb{I}}-P^{3}_{m})z_{m}-({\mathbb{I}}-P^{3}_{m})w_{m}|\\\
&\leq
Dc\sum_{m=n}^{+\infty}\bigg{(}\frac{\mu_{m}}{\mu_{n}}\bigg{)}^{-\lambda}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m+1}^{\lambda}}\right)(|P^{1}_{m}(z_{m}-w_{m})|+|P^{2}_{m}(z_{m}-w_{m})|)\\\
&\leq
Dc\sum_{m=n}^{+\infty}\bigg{(}\frac{\mu_{m}}{\mu_{n}}\bigg{)}^{-\lambda}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m+1}^{\lambda}}\right)(|P^{1}_{m}(z_{m}-w_{m})|_{m}+|P^{2}_{m}(z_{m}-w_{m})|_{m})\\\
&\leq
Dc\mu_{n}^{\lambda}\sum_{m=n}^{+\infty}\left(\frac{\mu^{\lambda}_{m+1}-\mu_{m}^{\lambda}}{\mu_{m}^{\lambda}\mu_{m+1}^{\lambda}}\right)|z_{m}-w_{m}|_{m}\\\
&\leq
Dc\mu_{n}^{\lambda}\sum_{m=n}^{+\infty}\left(\frac{1}{\mu_{m}^{\lambda}}-\frac{1}{\mu_{m+1}^{\lambda}}\right)\|\mathbf{z}-\mathbf{w}\|_{\infty}\\\
&\leq Dc\|\mathbf{z}-\mathbf{w}\|_{\infty},\end{split}$

and therefore

(6.16)
$\begin{split}&\sup_{n\in{\mathbb{N}}}\bigg{|}\sum_{m=n}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})w_{m}))\bigg{|}_{n}\\\
&\leq Dc\|\mathbf{z}-\mathbf{w}\|_{\infty}.\end{split}$

Finally,

$\begin{split}&|P^{3}_{n}(f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})z_{n})-f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})w_{n}))|_{n}\\\
&=\left(\frac{\mu_{n+1}^{\lambda}}{\mu^{\lambda}_{n+1}-\mu_{n}^{\lambda}}\right)|P^{3}_{n}(f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})z_{n})-f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})w_{n}))|\\\
&\leq\left(\frac{\mu_{n+1}^{\lambda}}{\mu^{\lambda}_{n+1}-\mu_{n}^{\lambda}}\right)(2D+1)\nu_{n}^{d}|f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})z_{n})-f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})w_{n})|\\\
&\leq c(2D+1)|({\mathbb{I}}-P^{3}_{n})z_{n}-({\mathbb{I}}-P^{3}_{n})w_{n}|\\\
&\leq c(2D+1)|z_{n}-w_{n}|_{n},\end{split}$

and hence

(6.17)
$\begin{split}&\sup_{n\in{\mathbb{N}}}|P^{3}_{n}(f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})z_{n})-f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})w_{n}))|_{n}\\\
&\leq c(2D+1)\|\mathbf{z}-\mathbf{w}\|_{\infty}.\end{split}$

Thus, combining (6.15), (6.16) and (6.17), we conclude that

(6.18) $\|\mathcal{T}\mathbf{z}-\mathcal{T}\mathbf{w}\|_{\infty}\leq
q\|\mathbf{z}-\mathbf{w}\|_{\infty}.$

Set

$C:=\frac{\bar{D}}{1-q},$

and consider

$\mathcal{B}:=\\{\mathbf{z}\in\mathcal{Y}:\|\mathbf{z}\|_{\infty}\leq
C\delta\\}.$

By arguing as in the proof of Theorem 3.1, it follows easily from (6.14) and
(6.18) that $\mathcal{T}\mathbf{z}\in\mathcal{B}$. From (6.7) and (6.18), we
conclude that $\mathcal{T}\rvert_{\mathcal{B}}\colon\mathcal{B}\to\mathcal{B}$
is a contraction, and therefore it has a unique fixed point
$\mathbf{z}=(z_{n})_{n\in{\mathbb{N}}}\in\mathcal{B}$. Hence, for each
$n\in{\mathbb{N}}^{\ast}$ we have that

(6.19)
$\begin{split}z_{n}&=-P^{3}_{n-1}(A_{n-1}y_{n-1}+f_{n-1}(y_{n-1}+({\mathbb{I}}-P^{3}_{n-1})z_{n-1})-y_{n})\\\
&\phantom{=}+\sum_{m=0}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\\\
&\phantom{=}-\sum_{m=n}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1}).\end{split}$

Now, let us consider $\bar{\mathbf{z}}=(\bar{z}_{n})_{n\in{\mathbb{N}}}$ given
by $\bar{z}_{0}=0$ and

$\begin{split}\bar{z}_{n}&=\sum_{m=0}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\\\
&\phantom{=}-\sum_{m=n}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\end{split}$

for every $n\in{\mathbb{N}}^{\ast}$. Observe that
$\bar{z}_{n}=({\mathbb{I}}-P^{3}_{n})z_{n}$, $n\in{\mathbb{N}}$. Moreover, for
every $n>k>0$ we have that

$\begin{split}\bar{z}_{n}-{\mathcal{A}}(n,k)\bar{z}_{k}&=\sum_{m=0}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\\\
&\phantom{=}-\sum_{m=0}^{k-1}{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\\\
&\phantom{=}-\sum_{m=n}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\\\
&\phantom{=}+\sum_{m=k}^{+\infty}{\mathcal{A}}(n,m)P^{2}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\end{split}$

and consequently

$\begin{split}\bar{z}_{n}-{\mathcal{A}}(n,k)\bar{z}_{k}&=\sum_{m=k}^{n-1}{\mathcal{A}}(n,m)P^{1}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1})\\\
&\phantom{=}+\sum_{m=k}^{n-1}{\mathcal{A}}(n,m)P^{2}_{m}(A_{m}y_{m}+f_{m}(y_{m}+({\mathbb{I}}-P^{3}_{m})z_{m})-y_{m+1}).\end{split}$

Therefore,

$\bar{z}_{n+1}=A_{n}\bar{z}_{n}+\left(P^{1}_{n}+P_{n}^{2}\right)\left(A_{n}y_{n}+f_{n}(y_{n}+\bar{z}_{n})-y_{n+1}\right)$

for every $n\in{\mathbb{N}}$. This easily implies that

(6.20)
$\begin{split}y_{n+1}+\bar{z}_{n+1}&=A_{n}(y_{n}+\bar{z}_{n})+f_{n}(y_{n}+\bar{z}_{n})\\\
&\phantom{=}-P^{3}_{n}(A_{n}y_{n}+f_{n}(y_{n}+\bar{z}_{n})-y_{n+1}),\end{split}$

for all $n\in{\mathbb{N}}$. We define $\mathbf{x}=(x_{n})_{n\in{\mathbb{N}}}$
by

$x_{n}=y_{n}+\bar{z}_{n},\quad n\in{\mathbb{N}}.$

Then,

$P^{3}_{n}(x_{n}-y_{n})=P^{3}_{n}\bar{z}_{n}=P^{3}_{n}({\mathbb{I}}-P^{3}_{n})z_{n}=0,$

for $n\in{\mathbb{N}}$. Moreover, using (6.3) we have that

$|\bar{z}_{n}|=|(P^{1}_{n}+P^{2}_{n})z_{n}|\leq|P^{1}_{n}z_{n}|_{n}+|P^{2}_{n}z_{n}|_{n}\leq|z_{n}|_{n}\leq\|\mathbf{z}\|_{\infty},$

for each $n\in{\mathbb{N}}$. Thus,

$\sup_{n\in{\mathbb{N}}}|x_{n}-y_{n}|=\sup_{n\in{\mathbb{N}}}|\bar{z}_{n}|\leq
C\delta.$

In addition, (6.20) implies that
$x_{n+1}-A_{n}x_{n}-f_{n}(x_{n})\in\operatorname{Im}P^{3}_{n}$ for all
$n\in{\mathbb{N}}$. Finally, we have that

$\begin{split}&|x_{n+1}-A_{n}x_{n}-f_{n}(x_{n})|\\\
&=|-P^{3}_{n}(A_{n}y_{n}+f_{n}(y_{n}+\bar{z}_{n})-y_{n+1})|\\\
&=|-P^{3}_{n}(A_{n}y_{n}+f_{n}(y_{n}+({\mathbb{I}}-P^{3}_{n})z_{n})-y_{n+1})|\\\
&=|z_{n}-\bar{z}_{n}|\\\ &=|z_{n}-({\mathbb{I}}-P^{3}_{n})z_{n}|\\\
&=|P_{n}^{3}z_{n}|\\\ &\leq(2D+1)\nu_{n}^{d}|z_{n}|\\\
&\leq(2D+1)\nu_{n}^{d}|z_{n}|_{n}\\\ &\leq
C\delta(2D+1)\nu_{n}^{d},\end{split}$

since

$\begin{split}|z_{n}|&\leq|P_{n}^{1}z_{n}|+|P_{n}^{2}z_{n}|+|P_{n}^{3}z_{n}|\\\
&\leq|P_{n}^{1}z_{n}|_{n}+|P_{n}^{2}z_{n}|_{n}+\frac{\mu_{n+1}^{\lambda}-\mu_{n}^{\lambda}}{\mu_{n+1}^{\lambda}}|P_{n}^{3}z_{n}|_{n}\\\
&\leq|P_{n}^{1}z_{n}|_{n}+|P_{n}^{2}z_{n}|_{n}+|P_{n}^{3}z_{n}|_{n}\\\
&=|z_{n}|_{n}.\end{split}$

Hence, (6.9) holds and the proof of the theorem is completed. ∎

######  6.5 Remark.

One can now easily formulate and prove the appropriate versions of Corollaries
3.3 and 3.4 in the present setting.

######  6.6 Remark.

As in the continuous time case, Theorem 6.2 may be applied to several settings
that we list below:

  * •

the case of exponential dichotomies which corresponds to $\mu_{n}=e^{n}$,
$\nu_{n}=1$ and $P_{n}^{3}=0$ for $n\in\mathbb{N}$;

  * •

the case of partial exponential dichotomies (see [2]) that corresponds to
$\mu_{n}=e^{n}$ and $\nu_{n}=1$ for $n\in\mathbb{N}$;

  * •

the case of tempered exponential dichotomies that corresponds to
$\mu_{n}=e^{n}$, $P_{n}^{3}=0$ for $n\in\mathbb{N}$ and with tempered $\nu$,
i.e.

$\lim_{n\to+\infty}\frac{1}{n}\ln\nu_{n}=0;$

  * •

the case of nonuniform polynomial dichotomies (see [6]) which corresponds to
$\mu_{n}=\nu_{n}=1+n$ and $P_{n}^{3}=0$ for $n\in\mathbb{N}$;

  * •

the case of $(\mu,\nu)$-dichotomies (see [8]) that corresponds to our general
setting with $P_{n}^{3}=0$, $n\in\mathbb{N}$;

Consequently, Theorem 6.2 gives us an unified approach to the shadowing
property in all these settings and, in particular, we can recover some of the
results from [1, 2, 3, 4] and from references therein. On the other hand, we
observe that it does not generalize all the results in [1, 2, 3, 4] because in
those papers we studied a very general version of the shadowing property
called $B$-Lipschitz shadowing property, where $B$ is some Banach sequence
space. The size of a pseudotrajectory and its deviation from an exact
trajectory is then measured via the norm on $B$. When $B=l^{\infty}$ is the
space of all bounded sequences equipped with a supremum norm, we have the
usual shadowing property as studied in present paper. However, if we take $B$
to be some other sequence space (for example $l^{p}$, $1\leq p<\infty$), we
get different versions of the standard shadowing property that are not
necessarily covered by Theorem 6.2.

We end by remarking that, as in the case of continuous time, Thereom 6.2 is a
first shadowing result that deals with the case when $(A_{n})_{n}$ admits a
polynomial, or more generally, a $(\mu,\nu)$-dichotomy.

Acknowledgements. L.B. was partially supported by a CNPq-Brazil PQ fellowship
under Grant No. 306484/2018-8. D.D. was supported in part by Croatian Science
Foundation under the project IP-2019-04-1239 and by the University of Rijeka
under the projects uniri-prirod-18-9 and uniri-pr-prirod-19-16.

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Characterising the Higgs boson with ATLAS data from Run 2 of the LHC The Higgs
boson was discovered by the ATLAS and CMS Collaborations in 2012 using data
from Run 1 of the Large Hadron Collider (2010$-$2012). In Run 2 (2015$-$2018),
about 140 fb-1 of proton–proton collisions at a centre-of-mass energy of 13
TeV were collected by the ATLAS experiment. This review presents the most
important Run 2 results obtained by the ATLAS Collaboration regarding the
properties of the Higgs boson and its interactions with other particles. The
performed studies significantly enhance the understanding of the Higgs boson,
while hunting for deviations from the predictions of the Standard Model of
particle physics. HIGG-2023-11 CERN-EP-2024-087 Physics Reports

###### Contents

  1. 1 Introduction
    1. 1.1 The Higgs mechanism
    2. 1.2 SM predictions for Higgs boson production and decay

##  1 Introduction

One of the main motivations for building the Large Hadron Collider (LHC) was
to investigate the nature of electroweak symmetry breaking. After the
discovery of the Higgs boson [HIGG-2012-27, CMS-HIG-12-028] by the ATLAS and
CMS Collaborations [PERF-2007-01, CMS-CMS-00-001], the focus has shifted to
the measurement of its properties. Ever since, an important goal has been to
determine whether the Higgs boson can be a portal to so-far undiscovered
phenomena.

This report summarises the expansive Higgs boson analysis programme performed
by the ATLAS Collaboration with the data recorded during Run 2 of the LHC
(2015$-$2018) and discusses the wealth of exciting results available at the
time of its publication. Unless otherwise specified, all results are based on
the full Run 2 data sample, corresponding to 140 fb-1 of proton–proton ($pp$)
collisions recorded with the ATLAS detector [DAPR-2021-01]. For detailed
discussions of technical aspects of the conducted analyses, the reader is
referred to the original publications, which are cited throughout the report.
Searches for exotic Higgs boson decays and additional Higgs bosons are covered
elsewhere [HDBSreport].

###  1.1 The Higgs mechanism

The Standard Model of particle physics (SM) is a relativistic quantum field
theory based on gauge symmetries that describes the elementary particles and
their interactions [Weinberg:1967tq, Glashow:1961tr, Salam:1968rm,
tHooft:1972tcz]. Seemingly forbidden by gauge invariance, massive $W$ and $Z$
bosons are enabled by electroweak symmetry breaking. In its simplest form,
electroweak symmetry breaking is realised through the introduction of a
complex doublet scalar field, which manifests itself in a neutral scalar
boson, the Higgs boson [Higgs:1964ia, Higgs:1964pj, Englert:1964et,
Guralnik:1964eu, Kibble:1967sv]. While the Higgs field potential itself is
invariant under the SU(2) transformation, its shape allows for spontaneous
symmetry breaking, which keeps the gauge symmetries intact but leads to a
vacuum expectation value that is non-zero and non-invariant under SU(2).
Interactions between the Higgs field and the fermions, the Yukawa
interactions, give rise to the masses of the fermions. The Yukawa couplings
and the Higgs boson mass are free parameters in the SM.

The LHC offers a unique opportunity to study the Higgs sector and therefore
the nature of electroweak symmetry breaking. With the measurement of the Higgs
boson mass, the SM Higgs sector is fully determined by experimentally measured
parameters. Measurements of the Higgs boson couplings to other particles and
the Higgs boson self-coupling are of particular importance. The Yukawa sector
describes a completely new boson–fermion interaction that is the only
interaction in the SM that distinguishes between particles of different masses
and therefore flavours. The Higgs boson self-coupling can be probed through
double Higgs boson production and is an important parameter of the Higgs field
potential, which is related to the structure of the vacuum and governs the
properties of the electroweak phase transition in the early universe
[Degrassi:2012ry, Reichert:2017puo].

The SM Higgs boson is a scalar boson: spin zero and its interactions are even
under charge–parity (CP) conjugation. In $pp$ collisions, the main Higgs boson
production mechanisms predicted by the SM are gluon–gluon fusion (ggF),
vector-boson fusion (VBF), and associated production with vector bosons
($VH$), top- or bottom-quark pairs ($t\bar{t}H$, $b\bar{b}H$), or a single top
quark ($tH$). The different production processes can be tagged through
dedicated selection criteria, taking advantage of the particular signatures of
the different final states. They are shown, together with the main predicted
decay processes, in Figure 1. The corresponding diagrams for double Higgs
boson production are shown in Figure 2. The expected width of the SM Higgs
boson for a mass of 125 GeV is 4.1 MeV [deFlorian:2016spz].

Figure 1: Top: Higgs boson production via (a) gluon–gluon fusion, (b) vector-
boson fusion, and (c) associated production with vector bosons, (d) top- or
bottom-quark pairs, or (e) a single top quark. Bottom: Higgs boson decays into
(f) a pair of vector bosons, (g) a pair of photons or a $Z$ boson and a
photon, (h) a pair of quarks, and (i) a pair of charged leptons
[ATLAS:2022vkf].

Figure 2: Examples of leading-order Feynman diagrams for Higgs boson pair
production in the (a and b) $\text{ggF }HH$ and (c, d and e) $\text{VBF }HH$
modes.

###  1.2 SM predictions for Higgs boson production and decay

Complementing the measurement programme at the LHC, there is an intense effort
ongoing to calculate Higgs boson production cross-sections and branching
ratios, as well as their associated uncertainties, to the best possible
accuracy in various phase space regions. Furthermore, Monte Carlo (MC)
simulations are developed to optimise the experimental data analysis and to
compare observed and expected results. These differential predictions are more
challenging and therefore only available at lower precision than the total
production cross-sections.

SM Higgs boson production cross-sections are calculated for the different
production modes shown in Figure 1 (a)-(e). The total cross-sections are
listed in Table 1.2, including the achieved accuracy in the electroweak (EW)
and strong (QCD) coupling constant, i.e., tree-level or leading order (LO),
next-to-leading order (NLO), next-to-next-to leading order (NNLO), etc. They
are calculated assuming a Higgs boson mass of 125.09 GeV. This value is based
on the combined Higgs boson mass measurement by the ATLAS and CMS
Collaborations in the LHC Run 1 [HIGG-2014-14].

Table 1:  SM predictions of Higgs boson production cross-sections for a Higgs
boson mass of 125.09 GeV in $pp$ collisions at a centre-of-mass energy of
$\sqrt{s}=13$ TeV. The values are based on Refs. [Anastasiou:2016cez,
Anastasiou:2015ema, Dulat:2018rbf, Harlander:2009mq, Harlander:2009bw,
Harlander:2009my, Pak:2009dg, Actis:2008ug, Actis:2008ts, Bonetti:2018ukf,
Aglietti:2004nj, Bagnaschi:2011tu, Hamilton:2015nsa] for ggF, Refs.
[Ciccolini:2007jr, Ciccolini:2007ec, Bolzoni:2010xr] for VBF, Refs.
[Brein:2012ne, Harlander:2018yio, Brein:2003wg, Brein:2011vx,
Altenkamp:2012sx, Harlander:2014wda, Denner:2014cla, Ciccolini:2003jy,
Denner:2011id] for $WH$ and $ZH$, Refs. [Beenakker:2002nc, Dawson:2003zu,
Yu:2014cka, Frixione:2015zaa] for $t\bar{t}H$, Refs. [Dawson:2003kb,
Dittmaier:2003ej, Harlander:2011aa] for $b\bar{b}H$, and Refs.
[Demartin:2015uha, Demartin:2016axk] for $tH$. They are compiled in Ref.
[deFlorian:2016spz].

Process | N3LO | NLO |  $48.5$${}^{+2.7}_{-3.6}$ | $87.2$
---|---|---|---|---
VBF | approx. NNLO | NLO |  $3.78$$\mathrm{p}\mathrm{m}0.08$ | $6.8$
$qq\rightarrow$WH | NNLO | NLO |  $1.37$$\mathrm{p}\mathrm{m}0.03$ | $2.44$
$qq\rightarrow$ZH | NNLO | NLO |  $0.76$${}^{+0.01}_{-0.02}$ | $1.35$
$gg\rightarrow$ZH | NLO | LO |  $0.12$${}^{+0.03}_{-0.02}$ | $0.22$
$t\mathrm{bar}{t}H$ | NLO | NLO |  $0.51$${}^{+0.03}_{-0.05}$ | $0.92$
$b\mathrm{bar}{b}H$ | NNLO | LO |  $0.49$${}^{+0.10}_{-0.12}$ | $0.88$
$tH$ | NLO | LO |  $0.092$${}^{+0.006}_{-0.011}$ | $0.16$

The SM predicts the double Higgs boson production cross-section to be more
than a thousand times smaller than single Higgs boson production:
$\sigma_{\text{ggF }HH}=31.0^{+2.1}_{-7.2}$ fb [Grazzini:2018bsd,
Heinrich:2019bkc, deFlorian:2016spz, Dawson:1998py, Borowka:2016ehy,
Baglio:2018lrj, Bonciani:2018omm, deFlorian:2013jea, Shao:2013bz,
deFlorian:2015moa, Baglio:2021], computed at NNLO in QCD including
approximated mass top effects, and $\sigma_{\text{VBF
}HH}=1.73\mathrm{p}\mathrm{m}0.04$ fb, at N3LO in QCD, for a Higgs boson mass
of 125 G eV [Dreyer:2018, Baglio:2013, Ling:2014].

Branching fractions, or decay probabilities for the different Higgs boson
decays, as depicted in Figure 1 (f)-(i), also depend on the Higgs boson mass.
For a mass of 125.09 G eV , the SM predicts that decays into $W$ bosons have a
branching fraction of 22??