\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 135, pp. 1--27.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu \newline
ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/135\hfil
 Boundary-value problems on the half-line]
{Boundary-value problems for nonautonomous nonlinear systems
 on the half-line}

\author[J. R. Morris \hfil EJDE-2011/135\hfilneg]
{Jason R. Morris}

\address{Jason R. Morris \newline
Department of Mathematics\\
The College at Brockport\\
State University of New York\\
350 New Campus Drive, Brockport, NY 14420, USA}
\email{jrmorris@brockport.edu}

\thanks{Submitted October 4, 2011. Published October 17, 2011.}
\subjclass[2000]{34B40, 34B15, 34D09, 46E15, 47H11, 47N20}
\keywords{Ordinary differential equation; half-line;
infinite interval; \hfill\break\indent
boundary and initial value problem;
Fredholm operator; degree theory; exponential dichotomy; \hfill\break\indent
properness; a priori bounds}

\begin{abstract}
 A method is presented for proving the existence of solutions for
 boundary-value problems on the half line.  The problems under study
 are nonlinear, nonautonomous systems of ODEs with the possibility
 of some prescribed value at $t=0$ and with the condition that
 solutions decay to zero as  $t$ grows large.  The method relies
 upon a topological degree for proper  Fredholm maps.
 Specific conditions are given to ensure that the  boundary-value
 problem corresponds to a functional equation that involves  an
 operator with the required smoothness, properness, and Fredholm
 properties (including a calculable Fredholm index).
 When the Fredholm index is zero and the solutions are bounded
 \emph{a priori}, then a solution exists.  The method is applied
 to obtain new existence results for systems of the form
 $\dot{v}+g(t,w)=f_1(t)$ and $\dot{w}+h(t,v)=f_2(t)$.
\end{abstract}

\maketitle 
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

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\section{Introduction}

Let $F=\map{F(t,z)}{[0,\infty)\times\mathbb{R}^N}{\mathbb{R}^N}$
be a given function and $P$
be the projection of $\mathbb{R}^N$ onto $X_1$ along a given splitting
$\mathbb{R}^N=X_1\oplus X_2$.  This paper concerns the existence of solutions
to problems of the type
\begin{equation}\label{eq:the-problem}
\begin{gathered}
    \dot{u}(t)+F(t,u(t))=f(t) \quad \text{for all } t\ge0,\\
    Pu(0)=\xi,\\
    \lim_{t\to\infty} u(t)=0,
\end{gathered}
\end{equation}
where $f\in C_0$ and $\xi\in X_1$ are given.

By choice of $X_1$ and $X_2$, different kinds of initial conditions
are obtained.  For example, the choice $X_1=\mathbb{R}^N$ and $X_2=\{0\}$
results in the usual initial value problem.   If $X_1=\{0\}$ and
$X_2=\mathbb{R}^N$, there is no initial condition.  Using the usual
representation of a second (or higher) order system by a larger
first-order system, this framework also accommodates a second (or higher)
order equation or system with a variety of different initial conditions
including those of Dirichlet, Neumann, or mixed type.

This kind of problem was recently approached by Rabier and Stuart in the two
papers \cite{Rabier-and-Stuart-(I)-MR2129786} and
\cite{Rabier-and-Stuart-(II)-MR2133394}.  As the authors point out
in~\cite{Rabier-and-Stuart-(I)-MR2129786}, boundary-value problems on infinite
intervals have been studied by many.  In particular, Andres, Gabor, and
Gorniewicz~\cite{andres-gabor-gorniewicz-MR1603870} contains many references.
The book by Agarwal and O'Regan~\cite{agarwal-o'regan-MR1845855} contains
another good survey of such problems.  As a rule of thumb, most results in the
literature deal with very specific problems and/or involve weaker conditions as
$t\to \infty$.

To discuss more recent results, some more detail is needed.  In
\cite{Rabier-and-Stuart-(I)-MR2129786} and
\cite{Rabier-and-Stuart-(II)-MR2133394} the authors prove the existence of a
solution in the Sobolev space $W^{1,p}\bigl((0,\infty),\mathbb{R}^N\bigr)$, under
appropriate conditions on $F$.  The authors Rabier and Stuart use a degree
argument.  Because the problem is posed on an unbounded interval, the underlying
operator is not a compact perturbation of the identity and therefore the
Leray-Schauder degree cannot be used.  Instead, a degree for proper $C^1$ maps
of Fredholm index zero is employed.  This degree was developed for $C^2$ maps
by Fitzpartick, Pejsachowicz, and
Rabier~\cite{Fitzpatrick-and-Pejsachowicz-and-Rabier-MR1162430}, and later
extended to $C^1$ maps by Pejsachowicz and
Rabier~\cite{Pejsachowicz-and-Rabier-MR1676979}.  Because of this degree
argument, prominent roles are obviously played by the Fredholm and properness
properties, and also by the issue of finding \emph{a priori} bounds on
solutions.

This work continued in the present author's dissertation~\cite{me}, in which
existence is obtained in the space $C_0^1\bigl([0,\infty),\mathbb{R}^N\bigr)$ of
continuously differentiable functions that tend to zero (along with their
derivatives)as $t\to\infty$.  In practice, this allows for simpler \emph{a
priori} bounds analysis than is possible in the Sobolev space setting.
Moreover, the author removed a key assumption from
\cite{Rabier-and-Stuart-(I)-MR2129786} and
\cite{Rabier-and-Stuart-(II)-MR2133394}, namely that $F(t,u)$ have an autonomous
limit $F^\infty(u)$ as $t\to\infty$.

More recently, Ev\'{e}quoz~\cite{Evequoz1-MR2547710,Evequoz2-MR2649373}
considers problems of the form
%\label{eq:the-Evequoz-problem}
\begin{gather*}
\dot{u}(t)+F(t,u(t),\xi,\lambda)=0 \quad \text{for all } t\ge0,\\
Pu(0)=\phi(\xi,\lambda),\\
\lim_{t\to\infty} u(t)=0,
\end{gather*}
in which global continuation in the real parameter $\lambda$
is explored, in both the
continuous and Sobolev settings.  This further expands the range of
applicability of the technique.  In particular, a more complicated dependence on
$\xi$ is allowed, as well as apparently more flexibility (via $\lambda$) in the
path of solutions departing from the trivial solution.  These results are
applied by the same author in~\cite{Evequoz3-MR2677830} to a third order ODE
that arises in the study of free convection boundary layers in porous media.

In this paper, we will first provide all of the necessary background from the
present author's dissertation~\cite{me}.  These arguments are elaborated in some
cases and simplified in others.  Since this material has not been published
elsewhere, we include all such background for completeness and ease of reference
in this and future work.

Once this is complete, we prove the existence of solutions to a class of
boundary value problems of the form
\begin{equation}\label{prob: example}
\begin{gathered}
    \dot{v}+g(t,w)=f_1,\\
    \dot{w}+h(t,v)=f_2,\\
    v(0)=\xi,\\     v(\infty)=w(\infty)=0\,.
\end{gathered}
\end{equation}
A version of this problem was treated
in~\cite{Rabier-and-Stuart-(II)-MR2133394}, in which the nonlinearities
$g$ and $h$ were assumed to have autonomous limits as $t\to\infty$,
and in which restrictions were placed on the magnitudes of the
derivatives $D_w g$ and $D_v h$.  Our results remove these restrictions.

We now briefly indicate the overall arrangement of this paper;
each section includes its own detailed introduction and orientation.
We approach problem~\eqref{eq:the-problem} by writing it as
${\mathcal{F}}(u)=(f,\xi)$, where
\begin{equation}\label{eqn:map-defn}
{\mathcal{F}}(u):=(\dot{u}+F(\cdot,u),Pu(0)).
\end{equation}
In Section 2, we provide the functional setting for ${\mathcal{F}}$
 and prove that under suitable hypotheses concerning $F$, that
${\mathcal{F}}$ is a $C^1$ map of Banach spaces.

In Section 3, we turn to the question of properness.  To determine whether
${\mathcal{F}}$ is proper on closed, bounded subsets of the domain, we provide a
condition that involves checking for solutions of certain limits
of~\eqref{eq:the-problem} obtained by letting $t\to\infty$ in a suitable
topology.

In Section 4, we discuss the desired Fredholm property, along with some
connections with exponential dichotomies and with the properness property.

Having paved the way for the use of topological degree for proper $C^1$ maps of
Fredholm index zero, in Section 5 we prove several existence theorems.  This
shows how to use the results of Sections 2, 3, and 4 (along with \emph{a priori}
bounds for solutions) to prove the existence of solutions
to~\eqref{eq:the-problem}.

In Section 6, we turn to the specific problem~\eqref{prob: example}, showing
that under suitable conditions of $g$ and $h$ that this problem always has a
solution.

Throughout this paper, we will use the following notation and definitions.
Given an interval $I\subseteq\mathbb{R}$, we will denote by $C_{\rm b}(I)$ the
Banach space of all continuous $\mathbb{R}^N$-valued functions on $I$, with the
usual supremum norm
\[
\Norm{u}{\infty}=\sup_{t\in I}\abs{u(t))}.
\]
(Of course any convenient norm $\abs{\xi}$ can be used in $\mathbb{R}^N$.)
We will denote
by $C_{\rm b}^1(I))$ the Banach space consisting of those functions in $C_{\rm
b}(I)$ with bounded derivative.  For this space, we use the norm \[
\Norm{u}{1, \infty}=\Norm{u}{\infty}+\Norm{\dot{u}}{\infty} \] We denote by
$C_0(I)$ the closed subspace of $C_{\rm b}(I)$ that consists of that functions
that tend to zero as $t\to\infty$.  We denote by $C_0^1(I)$ the closed subspace
of $C_{\rm b}^1(I)$ that consists of those functions such that both $u(t)$ and
$\dot{u}(t)$ tend to zero as $t\to\infty$.

Almost always, we will use $I=[0,\infty)$.  In those cases, we will simply
write $C_{\rm b}$, $C_{\rm b}^1$, etc.  Otherwise, we will explicitly
specify the interval, by writing $C_0^1(\mathbb{R})$, $C_{\rm b}((-\infty,0])$,
etc.

Given a function such as
$\map{F=F(t,z)}{[0, \infty)\times\mathbb{R}^N}{\mathbb{R}^N}$,
we will often have need of the so-called Nemytski\u{\i} operator $N_F$
associated to $F$.  The operator $N_F$ acts on functions
$\map{u=u(t)}{[0,\infty)}{\mathbb{R}^N}$ through composition, as follows.
\[
N_F(u)=v,\quad\text{where }v(t)=F(t,u(t)).
\]


\section{Smoothness of the Nemytskii Operator}


In this section we provide the function setting and we give the conditions
on $F=F(t,z)$ that ensure that the induced map ${\mathcal{F}}$
from~\eqref{eqn:map-defn} is a $C^1$ map of Banach spaces.

Let $\map{F=F(t,z)}{[0, \infty)\times\mathbb{R}^N}{\mathbb{R}^N}$ be a
function that satisfies the following conditions:
\begin{gather}\label{eq:F-hyp-1}
    F\text{ is continuous, with }\lim_{t\to\infty} F(t,0)=0,\\
\label{eq:F-hyp-2}
    D_z F\text{ exists and is continuous on }[0,\infty)\times\mathbb{R}^N,
\end{gather}
and for each compact subset $K$ of $\mathbb{R}^N$,
\begin{equation}\label{eq:F-hyp-3}
    F\text{ and }D_z F\text{ are BUC on } [0, \infty)\times K,
\end{equation}
where ``BUC'' means ``bounded and uniformly continuous''.

\begin{lemma}\label{lem:continuity-of-DF}
Let $\map{G=G(t,z)}{[0, \infty)\times\mathbb{R}^N}{\mathbb{R}^N}$ be 
bounded and uniformly continuous on $[0, \infty)\times K$, for every compact 
subset $K$ of $\mathbb{R}^N$.  Then $G$ has the following properties:
\begin{enumerate}
  \item \label{item:buc-property-a} For each $u\in C_{\rm b}$ and each 
  $\epsilon>0$, there is $\delta>0$ such that
\begin{equation*}
    \abs{G(t,u(t))-G(t, v(t))}<\epsilon
\end{equation*}
for all $v\in C_{\rm b}$ such that $\Norm{u-v}{\infty}<\delta$.
  \item \label{item:buc-property-b} For each $u\in C_0$,
\begin{equation*}
    \lim_{t\to\infty} G(t,u(t))-G(t,0)=0.
\end{equation*}
\end{enumerate}
\end{lemma}

\begin{proof} \ref{item:buc-property-a}
Let $R=1+\Norm{u}{\infty}$.  By assumption, the function $G$ is
uniformly continuous on $[0,\infty)\times\cb{0}{R}$.  Thus,
there is $\delta_1>0$ such that
\begin{equation*}
    \abs{G(t,x)-G(t, y)}<\epsilon
\end{equation*}
as long as $x,y\in\cb{0}{R}$ with $\abs{x-y}<\delta_1$.
It follows at once that the choice $\delta=\min(1, \delta_1)$ is sufficient.

\ref{item:buc-property-b} By assumption, the function $G$ is uniformly
continuous on $[0,\infty)\times\cb{0}{1}$.  Thus, there is $0<\delta<1$
such that
\begin{equation*}
    \abs{G(t,x)-G(t, 0)}<\epsilon
\end{equation*}
as long as $\abs{x}<\delta<1$.  Since $\lim_{t\to\infty} u(t)=0$,
part \ref{item:buc-property-b} is proved.
\end{proof}

\begin{theorem}\label{thm:smoothness}
Suppose that $F$ satisfies \eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
and \eqref{eq:F-hyp-3}.  Then the Nemytski\u{\i} operator $N_{F}$
is a well defined $C^1$ map from $C_0^1$ to $C_0$, and
$DN_{F}(u)v=N_{G}(u)v$ (where $G=D_z F$ and where the
multiplication of $N_{G}(u)$ by $v$ is pointwise in $t$).
\end{theorem}

\begin{proof}
Let $u\in C_0^1$ be given.  To see that $N_{F}(u)\in C_0$,
consider a sequence $t_n\to t$ in $[0,\infty)$.  It follows that
$u(t_n)\to u(t)$ since $u$ is continuous, and hence that
$F(t_n,u(t_n))\to F(t,u(t))$. This shows that $N_{F}(u)$ is
continuous on $[0,\infty)$. To see that $N_{F}(u)(t)\to 0$ as
$t\to\infty$, let $\epsilon>0$ be given. Since $u\in C_0$ and $F$
is uniformly continuous on
$[0,\infty)\times\cb{0}{\Norm{u}{\infty}}$ by  \eqref{eq:F-hyp-3},
one finds for all sufficiently large $t$ that
$\abs{F(t,u(t))-F(t,0)}<\epsilon$. By \eqref{eq:F-hyp-1}, this is
enough to show that $N_{F}(u)\in C_0$.

The next claim to verify is that $N_{F}$ is differentiable, with
$DN_{F}=N_{G}$.  Let $u$ and $v$ be chosen members of $C_0^1$. By
the using the mean value theorem once for each $t\ge 0$,
\begin{equation*}
    N_{F}(u+v)-N_{F}(u)-N_{G}(u)v=N_{G}(u+\tau v)v-N_{G}(u)v,
\end{equation*}
where the function $0\le \tau\le 1$ of $t$ depends on the choice of
$v$ (and of course on the choice of $u$).  It thus follows from
Lemma~\ref{lem:continuity-of-DF}~\ref{item:buc-property-a} that
\begin{equation*}
    N_{F}(u+v)-N_{F}(u)-N_{G}(u)v= \operatorname{o}(\Norm{v}{\infty})
  = \operatorname{o}(\Norm{v}{1,\infty})
\end{equation*}
in $C_0$ as $v$ tends to zero in $C_0^1$.

Finally, to check that $DN_{F}$ is continuous, let $u\in C_0^1$
and $\epsilon>0$ be given.  It follows from
Lemma~\ref{lem:continuity-of-DF}~\ref{item:buc-property-a} that
\begin{equation*}
    \Norm{N_{G}(u)v-N_{G}(w)v}{\infty}<\epsilon
\end{equation*}
for all unit vectors $v\in C_0^1$, provided only that
$\Norm{u-w}{1,\infty}$ (or even $\Norm{u-w}{\infty}$) is
sufficiently small.  This proves that $DN_{F}(u)$ varies
continuously in $\bl{C_0^1}{C_0}$ with respect to $u\in C_0^1$.
\end{proof}

\begin{remark} \label{rem: alt nemy} \rm
An examination of the above proof reveals that $N_{F}$ is also of
class $C^1$ as a map from $C_0$ into itself, and also as a map
from $C_{\rm b}$ into itself.
\end{remark}

\begin{remark} \rm
With only the obvious changes, the result holds when $[0,\infty)$ is
replaced by any closed interval, and $(-\infty,\infty)$ in particular.
\end{remark}

We have the following corollary for the operator
$\map{{\mathcal{F}}}{C_0^1}{C_0\times X_1}$ defined by
\begin{equation}\label{eq: Phi}
{\mathcal{F}}(u):=(\dot{u}+N_{F}(u),Pu(0))\quad\text{for }u\in
C_0^1.
\end{equation}

\begin{corollary}\label{cor: smooth}
In the situation of Theorem~\ref{thm:smoothness}, the map ${\mathcal{F}}$ 
is of class $C^1$.  In addition, if we set $G:=D_z F$, then
\[
D{\mathcal{F}}(u)v=(\dot{v}+N_{G}(u)v, Pv(0)).
\]
\end{corollary}
\begin{proof}
Differentiation is a continuous linear map from $C_0^1$ into $C_0$, and 
the evaluation of $v$ at $t=0$ followed by the linear projection $P$ 
is a continuous linear map from $C_0^1$ into $X_1$.  Therefore, 
Corollary~\ref{cor: smooth} is a direct result of Theorem~\ref{thm:smoothness}.
\end{proof}


\section{Properness}

In this section, we establish a necessary and sufficient condition for 
${\mathcal{F}}$ to be proper on the closed, bounded subsets of $C_0^1$.  
The essential idea is the following.  When $\seq{u_n}$ is a bounded 
sequence in $C_0^1$ such that ${{\mathcal{F}}(u_n)}$ converges in 
$C_0\times X_1$, we are to show
that $\seq{u_n}$ has a convergent subsequence in $C_0^1$.  To find a 
convergent subsequence of $\seq{u_n}$, we show that the sequence forms a 
relatively compact set, by the use of a result from
 Rabier~\cite{Rabier-(Ascoli)-MR2032233} that characterizes the relatively 
 compact subsets of $C_0$.
To use this result, one must show that the sequence $\seq{u_n(t)}$ tends to 
zero uniformly (with respect to $n$) as $t\to\infty$.

This ``equi-decay'' is characterized
in~\cite{Rabier-(Ascoli)-MR2032233} by a condition involving
sequences of the form $\seq{u_n(\cdot+\xi_n)}$, where
$\xi_n\to\infty$.  It is this temporal translation to infinity
that ultimately brings one to the condition (Theorem
\ref{thm:proper}) that no equation of the form
$\dot{u}+N_{E}(u)=0$ have a nonconstant $C^1$ solution, where $E$
is any uniform--on--compacta limits of temporal translations of
$F$.  It is a noteworthy artifact of the translation to infinity
that this condition involves problems on the whole line, even
though the original problem is posed on the half line.

\begin{remark} \label{rem:eec} \rm
In this section we will follow the following convention.
Any vector valued function (in particular, any real valued function)
that depends on a variable $t\ge 0$ is extended to negative values
of $t$ whenever convenient, by using the even extension.
Of course, this convention is only used where needed and never for a
function whose domain already includes negative values of $t$.

For example, let $u\in C_0^1$ and $\xi_n\to\infty$ in $\mathbb{R}$.   Then the
sequence $\seq{u(t+\xi_n)}$ is well defined for all  $t\in\mathbb{R}$ under this
convention, and this sequence can hence have  a well-defined pointwise or
uniform--on--compacta limit on all  of $\mathbb{R}$.  Notice that this kind of
sequence is locally eventually  independent of the choice of extension (since
$t+\xi_n$ is eventually positive), and so any limit function is independent of
the choice of extension.  This is characteristic of our later use of this ``even
extension convention'', which will occur with minimal further comment.  The
purpose of this perhaps distracting convention is to avoid the alternate
distraction of a prominently displayed extension operator.
\end{remark}

\subsection{Topological preliminaries}

Let $Z$ be a nonempty subset of $\mathbb{R}^N$, and define the following closed
subspace of $C_{\rm b}(\mathbb{R})=C_{\rm b}(\mathbb{R},\mathbb{R}^N)$:
\begin{equation}\label{eq:C_-dfn}
    C_{Z}(\mathbb{R})=C_{Z}(\mathbb{R}, \mathbb{R}^N)
    :=\big\{u\in C_{\rm b}(\mathbb{R}):\lim_{t\to\pm\infty}
    \operatorname{dist}(u(t),Z)=0\big\},
\end{equation}
consisting of those functions $u=u(t)$ converging to $Z$ as $\abs{t}$
tends to infinity.  We define the space $C_{Z}\subset C_{\rm b}$ of
functions defined on $[0, \infty)$ by requiring that
$ \lim_{t\to\infty}\operatorname{dist}(u(t),Z)=0$.
It is clear that $C_0=C_{\{0\}}$ and that $C_{Z}\subseteq C_{W}$ if and only
if  $\bar{Z}\subset \bar{W}$ (with equality if and only if $\bar{Z}=\bar{W}$).

We say that $Z$ is \emph{totally disconnected}\label{dfn:totally disconnected}
if the connected components of $Z$ are singletons.  Examples include finite
sets, sequences (with or without their limit point), and Cantor-like sets.
If $Z\subset\mathbb{R}$, then $Z$ is totally disconnected if and only
if $Z$ does not contain an interval.

If $\mathcal{H}$ is a subset of $C_{\rm b}$ or $C_{\rm b}(\mathbb{R})$ and
if $I$ is a subset of the corresponding domain $[0,\infty)$ or $\mathbb{R}$
respectively, then $\mathcal{H}(I):=\set{u(t)}{u\in\mathcal{H}, t\in I}$,
the union of the direct images of $I$ under members of $\mathcal{H}$.

Here is the result that we will use from
Rabier~\cite{Rabier-(Ascoli)-MR2032233}, followed by a simple corollary
adapted for use on the half line.

\begin{lemma}[{Rabier~\cite[Corollary~\textup{7}]{Rabier-(Ascoli)-MR2032233}}]
\label{lem:ascoli-new-form}
A subset $\mathcal{H}$ of $C_0(\mathbb{R})$ is
relatively compact if and only if the following three conditions hold:
\begin{enumerate}
\item The set $\mathcal{H}(\mathbb{R})$ is bounded.
\item The set $\mathcal{H}$ is uniformly equicontinuous.
\item There is a compact and totally disconnected subset $Z$ of $\mathbb{R}^N$ 
with the following property.  If $\tilde{u}\in C_{\rm b}(\mathbb{R})$ and 
there are sequences $\seq{u_n}\subset\mathcal{H}$ and
   $\seq{\xi_n}\subset \mathbb{R}$ such that
   $\abs{\xi_n}\to\infty$ and
   \begin{equation*}
  \lim_{n\to\infty} u_n(t+\xi_n)=\tilde{u}(t) \quad \text{for all } t\in \mathbb{R},
\end{equation*}
   then $\tilde{u}(\mathbb{R})\subset Z$.
\end{enumerate}
\end{lemma}

\begin{corollary}\label{cor:ascoli-new-form-halfline}
A subset $\mathcal{H}$ of $C_0$ is
relatively compact if and only if the following three conditions hold:
\begin{enumerate}
\item The set $\mathcal{H}([0,\infty))$ is bounded.\label{item:first-ascoli}
\item The set $\mathcal{H}$ is uniformly equicontinuous.\label{item:second-ascoli}

\item There is a compact and totally disconnected subset $Z$ of $\mathbb{R}^N$ 
with the following property.  If $\tilde{u}\in C_{\rm b}(\mathbb{R})$ and 
there are sequences $\seq{u_n}\subset\mathcal{H}$ and
   $\seq{\xi_n}\subset \mathbb{R}$ such that
   $\xi_n\to\infty$ and
   \begin{equation*}
  \lim_{n\to\infty} u_n(t+\xi_n)=\tilde{u}(t) \quad \text{for all } t\in \mathbb{R},
\end{equation*}
   then $\tilde{u}(\mathbb{R})\subset Z$.\label{item:third-ascoli}
\end{enumerate}
\end{corollary}

\begin{proof} Use the even extension of the functions in $C_0$.  Now apply
Lemma~\ref{lem:ascoli-new-form}.  Also see the final paragraph of Section~2
in~\cite{Rabier-(Ascoli)-MR2032233}, where such generalizations are mentioned.
\end{proof}

\begin{remark} \rm In~\cite{Rabier-(Ascoli)-MR2032233}
Lemma~\ref{lem:ascoli-new-form} is proved in a more general setting.  For one
thing, the functions in $C_0$ are allowed to take values in a general metric
space, and item~\ref{item:first-ascoli} is that $\mathcal{H}(\mathbb{R})$ should
be relatively compact.  This suggests the question of whether
item~\ref{item:third-ascoli} is necessary when the metric space is
$\mathbb{R}^N$.  The example $\mathcal{H}=\set{u_n}{n\in\mathbb{N}}$ where
$u_n(t)=\min\bigl(1, \max(n-t,0)\bigr)$ for $t\ge0$ shows that
item~\ref{item:third-ascoli} is indeed necessary.
\end{remark}

\begin{remark} \rm\label{rem:seq}
When using Lemma~\ref{lem:ascoli-new-form} or
Corollary~\ref{cor:ascoli-new-form-halfline}, one often finds that the set
$\mathcal{H}$ consists of the terms of a sequence $\seq{v_n}$.  If so, one may
assume that $\seq{u_n}$ is a subsequence of $\seq{v_n}$ when checking
condition~\ref{item:third-ascoli}.
\end{remark}

If $\seq{T_n}$ is a sequence of continuous functions from a metric spaces $M$
into a metric space $N$, we will write\footnote{As suggested by the usual name
``compact-open'' for the resulting topology.}
$T=\operatorname{co-lim}_{n\to\infty} T_n$ if the sequence $\seq{T_n}$ converges
uniformly to $T$ on each compact subset of $M$.  In each use of this notation,
$M$ and $N$  will be clear from context.

\begin{lemma}\label{lem:f-omega-set-is-nonempty}
Suppose that $F$ satisfies~\eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
and \eqref{eq:F-hyp-3}.  Let $\seq{\xi_n}\subset\mathbb{R}$ be a
sequence such that $\xi_n\to\infty$ and put $E_n(t,z):=F(t+\xi_n,z)$.
Then there exist a subsequence $\seq{E_{n_k}}$ and a function
$\map{E}{\mathbb{R}\times\mathbb{R}^N}{\mathbb{R}^N}$ such that
\begin{enumerate}

\item $E=\operatorname{co-lim}_{k\to\infty} E_{n_k}$,
\item $E$ satisfies~\eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2}, and \eqref{eq:F-hyp-3} 
(with $[0, \infty)$ replaced by $\mathbb{R}$ and $F$ replaced by $E$), and
\item $D_z E=\operatorname{co-lim}_{k\to\infty} D_z E_{n_k}$.
\end{enumerate}
\end{lemma}

\begin{proof}
  The classical Ascoli-Arzela Theorem applies for each compact subset $K$ of
$\mathbb{R}\times\mathbb{R}^N$.  (The boundedness of
$\set{E_n(\alpha)}{n\in\mathbb{N}, \alpha\in K}$ and
and the equicontinuity of $\seq{E_n\big|_K}$ follow from~\eqref{eq:F-hyp-3}.)
Apply the
Ascoli-Arzela theorem recursively for an increasing sequence of compact sets
that exhausts $\mathbb{R}\times\mathbb{R}^N$.  This diagonal process
yields a function $E$ and a subsequence $\seq{E_{n_k}}$ such that
$E=\operatorname{co-lim}_{k\to\infty} E_{n_k}$.
Repeat this process with $\seq{D_z E_{n_k}}$ to obtain a further
subsequence (still denoted $\seq{E_{n_k}}$) and a function $H$ such that
$ H=\operatorname{co-lim}_{k\to\infty} D_z E_{n_k}$.
As usual, since the convergence is uniform on compact sets, $D_z E$
exists and $D_z E=H$.   Thus, $E$ satisfies~\eqref{eq:F-hyp-2}.
It is now easy to verify that ~\eqref{eq:F-hyp-1} (with $E(t,0)\equiv0$)
and \eqref{eq:F-hyp-3} are inherited by $E$ from $F$.
\end{proof}

\begin{definition} \rm \label{defn:omega-set}
Given a function $F$ that satisfies~\eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
and \eqref{eq:F-hyp-3}, we define the \emph{omega limit set}
\footnote{This terminology is suggested by its use in the linear
setting in~\cite{Sacker-and-Sell-1978-MR0501182}.} of $F$ by
\[
\omega(F)=\set{E}{E=\operatorname{co-lim} E_{n}\text{ for some sequence }
\xi_n\to\infty}.
\]
\end{definition}

Here, $E_n(t,z):=F(t+\xi_n,z)$, just as in
Lemma~\ref{lem:f-omega-set-is-nonempty}.  It is a
corollary of Lemma~\ref{lem:f-omega-set-is-nonempty} that $\omega(F)$
is nonempty.  Before moving on, several examples may be helpful.
The proofs of the following claims are left as exercises
(but may also be found in~\cite{me}).

\begin{example} \rm
The pointwise limit $F^\infty(x)=\lim_{t\to\infty} F(t,x)$ exists if and 
only if $\omega(F)$ is the singleton $\{F^\infty\}$ (which includes the 
autonomous situation $F(t,x)=F^\infty(x)$).  This is the case considered 
in~\cite{Rabier-and-Stuart-(I)-MR2129786, Rabier-and-Stuart-(II)-MR2133394}.
\end{example}

\begin{example} \rm
Suppose $F$ is periodic or asymptotically periodic, in the sense that
\[
\lim_{t\to\infty}\abs{F(t,x)-G(t,x)}=0
\]
pointwise in $x$, for some $G$ that is periodic in $t$.  Then
\[
\omega(F)=\set{G(\cdot+\tau,\cdot)}{0\le \tau< T},
\]
where $T$ is a period of $G$.
\end{example}

\begin{example} \rm
Suppose that for all $x\in\mathbb{R}^N$ and all $R>0$,
\[\lim_{t\to\infty} \sup_{0\le \tau\le R} \abs{F(t,x)-F(t+\tau,x)}=0.\]
In this case we will say that $F$ is \emph{asymptotically autonomous}.
This is the case if and only if $\omega(F)$ consists only of
autonomous functions $G(t,x)=G(x)$.  If $F$ is $C^1$, a sufficient
(but not necessary) condition for asymptotic autonomy is that
$\lim_{t\to\infty} D_t F(t,x)=0$ pointwise in $x$.
\end{example}

\begin{example} \rm
Suppose $F$ is of the quasilinear form $F(t,x)=A(t)q(x)$ where
$A$ is a $d\times d$ matrix function and $q(x)\in\mathbb{R}^N$.
Let $\omega(A)$ denote the set of all uniform-on-compact-intervals
limits of sequences $\seq{A(\cdot+\tau_k)}$ where $\tau_k\to\infty$.
Then $\omega(F)$ consists of all quasilinear functions $G(t,x)=B(t)q(x)$
where $B\in\omega(A)$.  When $F$ is quasilinear, the preceding examples
correspond respectively to the cases that $A(t)$ has a limit as
$t\to\infty$, or that $A$ is asymptotically periodic, or that $A$ is
asymptotically constant.  In this last case, note that
\[
\omega(A)=\cap_{n\in\mathbb{N}}\overline{A\bigl([n, \infty)\bigr)}.
\]
\end{example}

\subsection{Properness via solutions of omega limit equations}

The following definition is of key importance in Theorem~\ref{thm:proper}.
Recall that the definition of the term ``totally disconnected''
is provided on page~\pageref{dfn:totally disconnected}.

\begin{definition} \rm \label{dfn:admissible-omega-set}
   Assume that $F$ satisfies~\eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
and \eqref{eq:F-hyp-3}.  Let $S$ denote the set of all (bounded)
functions $u\in C_{\rm b}^1(\mathbb{R})$ such that $\dot{u}+E(t,u)=0$
on $(-\infty,\infty)$, for some $E$ in the omega-limit set $\omega(F)$.
 We will say that $F$ has an admissible omega-limit set provided that
$S$ consists only of constant functions, and that these constants
form a compact and totally disconnected subset of $\mathbb{R}^N$.
\end{definition}

When $F$ has an admissible omega-limit set, if ``$t$ goes to infinity''
in the equation $\dot{u}+F(t,u)=f$, no resulting equation has a nonconstant
solution that is bounded on $\mathbb{R}$.  This will be the key to proving
Theorem~\ref{thm:proper}.

It is useful to record one more definition, if only to highlight the
connection between the admissibility of the omega-limit set and the
third item in Corollary~\ref{cor:ascoli-new-form-halfline}.

\begin{definition} \rm \label{defn:Z(F)}
Given a function $F$ that satisfies~\eqref{eq:F-hyp-1},
\eqref{eq:F-hyp-2}, and \eqref{eq:F-hyp-3}, we define the \emph{omega zero set}
of $F$ by
\[
Z(F):=\set{z\in\mathbb{R}^N}{E(\cdot,z)=0\text{ for some }E\in\omega(F)}.
\]
\end{definition}

\begin{remark} \rm
Notice that $u(t)=c$ is a constant solution of $\dot{u}+E(t,u)=0$
(for some $E\in\omega(F)$) if and only if $c\in Z(F)$. This shows
that if $F$ has an admissible omega-limit set, then the set $S$ that
is mentioned in Definition\ref{dfn:admissible-omega-set} coincides with $Z(F)$.
\end{remark}

\begin{remark} \rm
Notice that $Z(F)$ includes all $z\in\mathbb{R}^N$ such that
$\lim_{t\to\infty} F(t,z)=0$.  However, $Z(F)$ may contain other points.
As an illustration, consider $F(t,z)=(\sin{\sqrt{t}})z$.  Then $z=0$ is
the only point such that $\lim_{t\to\infty} F(t,z)=0$, but $Z(F)=\mathbb{R}^N$
because $0\in\omega(F)$.
\end{remark}

For Theorem~\ref{thm:proper}, recall that
$\map{{\mathcal{F}}}{C_0^1}{C_0\times X_1}$ is defined by
\begin{equation*}
    {\mathcal{F}}(u):=(\dot{u}+N_{F}(u),Pu(0)),
\end{equation*}
and that it follows from Theorem~\ref{thm:smoothness} that
${\mathcal{F}}\in C^1\bigl(C_0^1,C_0\times X_1\bigr)$ and that
\begin{equation*}
    D{\mathcal{F}}(u)v=(\dot{v}+D_z F(\cdot,u)v,Pv(0))\quad \text{for }u,v\in C_0^1.
\end{equation*}

\begin{theorem}\label{thm:proper}
Assume that $F$ satisfies~\eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
\eqref{eq:F-hyp-3}, and that $F$ has an admissible omega-limit set.
Then ${\mathcal{F}}$ is proper on each subset of $C_0^1$ that is
closed and bounded.
\end{theorem}

\begin{proof}
Let $\seq{u_n}$ be a bounded sequence in $C_0^1$ such that
$\seq{(f_n, \xi_n)}:=\seq{{\mathcal{F}}(u_n)}$ is convergent in
$C_0\times X_1$.  We are to prove that $\seq{u_n}$ has a
$C_0^1$-convergent subsequence.  To do so, we will first use
Corollary~\ref{cor:ascoli-new-form-halfline} to find a $C_0$-convergent
 subsequence; take $\mathcal{H}=\set{u_n}{n\in\mathbb{N}}$
(see Remark~\ref{rem:seq}).

Item~\ref{item:first-ascoli} of Corollary~\ref{cor:ascoli-new-form-halfline}
follows immediately from the boundedness of $\seq{u_n}$ in $C_0^1$.
Item~\ref{item:second-ascoli} also follows from this boundedness.
To see this, since $\seq{u_n}$ is bounded in $C_0^1$, the sequence
$\seq{\dot{u}_n}$ of derivatives is equibounded on $[0,\infty)$.
Therefore, all of the functions in $\mathcal{H}$ are uniformly
Lipshitz on $[0,\infty)$, and share a common Lipshitz constant.
This implies that $\mathcal{H}$ is uniformly equicontinuous.

For item~\ref{item:third-ascoli}, take $Z=Z(F)$ (see Definition~\ref{defn:Z(F)})
which is compact and totally disconnected by
Definition~\ref{dfn:admissible-omega-set}.  Let
${u}\in C_{\rm b}(\mathbb{R})$ and a subsequence $\seq{u_{n_k}}$ of
$\seq{u_n}$ be given.  Let $\seq{\xi_k}$ be a sequence of real numbers
such that $\lim_{k\to\infty}\xi_k=\infty$.  Put $v_k(t):=u_{n_k}(t+\xi_k)$.
Assuming that $\seq{v_k}$ converges pointwise to ${u}$, we are to show
that ${u}(\mathbb{R})\subset Z(F)$.

We make several observations:
\begin{itemize}
\item[(i)] Since $\seq{f_n}\subset C_0$, the translated sequence
$\seq{f_{n_k}(\cdot+\xi_k)}$ converges to zero as $k\to\infty$,
uniformly on compact intervals.
\item[(ii)] By passing to a subsequence and relabeling, we may assume
that the convergence of $\seq{v_k}$ to ${u}$ is uniform on compact
intervals: ${u}=\operatorname{co-lim}_{k\to\infty} v_k$.  This follows
from the boundedness and equicontinuity of $\seq{v_k}$.
\item[(iii)] By again passing to a subsequence, there is some
$E\in\omega(F)$ such that $E=\operatorname{co-lim}_{k\to\infty} E_k$,
 where $E_k(t,z)=F(t+\xi_k,z)$.  See Lemma~\ref{lem:f-omega-set-is-nonempty}
 and Definition~\ref{defn:omega-set}.
\end{itemize}
It now follows readily from (ii) and (iii) that the sequence
$\seq{N_{E_k}(v_k)}$ converges to $\seq{N_{E}({u})}$ uniformly on
compact intervals; we note that
$N_{E_k}(v_k)(t)=E_k(t,v_k(t))=F(t+\xi_k,u_{n_k}(t+\xi_k))$ and
$N_{E}({u})(t)=E(t,{u}(t))$.  As a result, the sequence
$\seq{\dot{v}_k}$ converges uniformly on compact sets to
$-E(t,{u})$.  Indeed,
\begin{align*}
    \dot{v}_k(t)&=\dot{u}_{n_k}(t+\xi_k)\\
      &=f_{n_k}(t+\xi_k)-F(t+\xi_k,u_{n_k}(t+\xi_k))\\
      &=f_{n_k}(t+\xi_k)-E_k(t,v_k(t)),\\
\end{align*}
which converges to $-E(t,u(t))$ as $k\to\infty$, uniformly on compact sets.
But uniform convergence of $\seq{\dot{v}_k}$ on an interval implies
that the limit function $u$ is differentiable and that $\seq{\dot{v}_k}$
converges to $\dot{{u}}$.  Therefore,
\[
\dot{{u}}(t)+ E(t,{u}(t))=0.
\]
Since $u$ is bounded, it follows (from the admissibility of the
omega-limit set of $F$) that $u$ is a constant function, say $u(t)\equiv z$.
Therefore, $E(t, z)=E(t, u(t))=-\dot{u}(t)=0$, so that
$u(\mathbb{R})=\{z\}\subset Z(F)$ by Definition~\ref{defn:Z(F)}.

Having completed the verification of all three items in
Corollary~\ref{cor:ascoli-new-form-halfline}, we conclude that
$\seq{u_n}$ has a subsequence (again denoted $\seq{u_n}$) that
converges in $C_0$ to a limit $u^\infty$.  To complete the proof,
recall that $N_{F}$ is continuous from $C_0$ to itself
(Remark~\ref{rem: alt nemy}). Therefore, the sequence
$\seq{\dot{u}_n}=\seq{-N_{F}(u_n)}$ converges to
$-N_{F}(u^\infty)$ in $C_0$.  In particular, $u^\infty$ is
differentiable and $\seq{\dot{u}_n}$ converges to $\dot{u}^\infty$
in $C_0$. Since (the subsequence) $\seq{u_n}$ is therefore
convergent in $C_0^1$, the proof is complete.
\end{proof}


\section{The Fredholm property}\label{sec:fredholm}

\subsection{Linear systems}

Let $\map{A=A(t)}{[0, \infty)}{\mathbb{R}^{d\times d}}$ be a bounded and
continuous matrix function.  In this section, we develop the Fredholm
properties of the linear operator $\map{D_A}{C_0^1}{C_0}$ defined by
\[
D_A u(t):=\dot{u}(t)+A(t)u(t),
\]
as well as the augmented linear operator
$\map{\Lambda}{C_0^1}{C_0\times X_1}$ defined by
\[
\Lambda u=(D_A u,Pu(0)).
\]
Before we begin, we have a few remarks concerning autonomous systems
(meaning that $A(t)\equiv A$ is constant).  In this case, the
Fredholm property and index is determined by the spectrum of $A$ together
with the dimensions of the associated invariant subspaces of $\mathbb{R}^N$.
If $A$ has any eigenvalue on the imaginary axis, then the range of $D_A$ is
not closed in $C_0$, whence neither $D_A$ nor $\Lambda$ is Fredholm.
Otherwise both operators are Fredholm.  Moreover, the index of $D_A$ is
the algebraic count of eigenvalues of $A$ having positive real part.
The index of $\Lambda$ is diminished from the index of $D_A$ by exactly
the dimension of $X_1$.

These facts are proved in the Sobolev space setting in
\cite[Section 2]{Rabier-and-Stuart-(II)-MR2133394}.  That paper then reduces
the nonautonomous case to the autonomous case by using a limit
$A^\infty=\lim_{t\to\infty} A(t)$.  We do not assume the existence of such a
limit, and because of this our arguments will be significantly different.
In the case $A^\infty$ exists, our results reduce to those
of~\cite[Theorem~4.1]{Rabier-and-Stuart-(II)-MR2133394}, albeit with respect
to spaces of smooth functions rather than Sobolev spaces.

When there is no limit for $A(t)$ as $t\to\infty$, the situation is a bit
more subtle.  Merely to know the spectrum of $A(t)$ is no longer sufficient.
In fact, the operator $\Lambda$ turns out to be Fredholm exactly when
the matrix $A=A(t)$ admits an \emph{exponential dichotomy} on $[0, \infty)$.
This was proved by Palmer~\cite{Palmer-1984-MR764125,Palmer-1988-MR958058}.
Palmer did not consider spaces of functions that tend to zero as $t\to\infty$,
so we will provide proofs herein.

With respect to the linear system $D_A u:=\dot{u}+Au=0$, let $U=U(t)$ denote 
the fundamental matrix solution of the system $\dot{U}+AU=0$  with $U(0)=I$.  
We recall that $A$ admits an exponential dichotomy on $[0,\infty)$ if there 
exist a projection $\Pi$ and positive constants $K$ and $\alpha$ such that
\begin{equation}\label{ineq:exp forward}
\abs{U(t)\Pi U(s)^{-1}}\le K e^{-\alpha(t-s)}
\end{equation}
for all $t\ge s\ge 0$ and
\begin{equation}\label{ineq:exp back}
\abs{U(t)(I-\Pi)U(s)^{-1}}\le K e^{-\alpha(s-t)}
\end{equation}
for all $s\ge t\ge 0$.

It is well known that the range of $\Pi$ is uniquely determined in the sense
that if $A$ also admits an exponential dichotomy with projection $\Pi'$,
then $\operatorname{rge}\Pi'=\operatorname{rge}\Pi$.  Conversely, if
$\Pi'$ is any projection with
$\operatorname{rge}\Pi'=\operatorname{rge}\Pi$, then $A$ admits an
exponential dichotomy with projection $\Pi'$.  (This converse is untrue if
one replaces $[0,\infty)$ by the whole real line.)   Also, $\operatorname{rge}
\Pi$ coincides with the subspace of $\mathbb{R}^N$ consisting of those initial
data $\xi\in\mathbb{R}^N$ such that the solution $u(t)=U(t)\xi$ of $D_A u=0$,
$u(0)=\xi$ remains bounded as $t\to\infty$.  By \eqref{ineq:exp forward} a
solution with initial data in $\operatorname{rge}\Pi$ will tend to $0$
exponentially as $t\to\infty$.  In contrast, \eqref{ineq:exp back} shows that a
solution with initial data outside of $\operatorname{rge}\Pi$ will tend to
infinity exponentially as $t\to\infty$.  For these properties (and others) see
Coppel~\cite{Coppel-MR0481196} or Massera and
Sch\"affer~\cite{massera-schaffer-MR0212324}.

Our first result in this section is that the Fredholm property of the operator
$D_A$ is equivalent to the property that $A$ admit an exponential dichotomy on
$[0, \infty)$, and that the Fredholm index is the same as the rank of any
projection associated with the exponential dichotomy.  In order to prove this
result, it will help to know that when $D_A$ has closed range, it follows $D_A$
is onto $C_0$ and that a certain kind of \emph{a priori} bound exists on
solutions to $D_A u = f$.  That is the content of the following lemma:

\begin{lemma}\label{lem: closed range}
Assume that the operator $\map{D_A}{C_0^1}{C_0}$ has closed range.
Let $V_1$ be the subspace of $\mathbb{R}^N$ consisting of the initial
values $u(0)$ of bounded solutions $u$ to the homogeneous equation $D_A u=0$,
and let $V_2$ be any direct complement of $V_1$.  Then
\begin{enumerate}
\item The operator $D_A$ is onto $C_0$.
\item There is a positive constant $r$ such that for all $f\in C_0$,
one has the estimate
\begin{equation}\label{ineq: a bound}
\Norm{u}{\infty}\le r\Norm{f}{\infty},
\end{equation}
where $u$ is the unique $u\in C_0^1$ such that $D_A u=f$ and $u(0)\in V_2$.
\end{enumerate}
\end{lemma}

\begin{proof}
Since the subspace $C_{0}$ of $C_0$ consisting of compactly supported
functions is dense in $C_0$, it is enough to prove that the range of
$D_A$ contains $C_{0}$.  Let $f\in C_{0}$, and suppose that $f$ is
supported in $[0, T]$.  Let
\[
\xi=-\int_0^T U(s)^{-1}f(s)\,\mathrm{d}s,
\]
and let
\[
u(t)=U(t)\Big(\xi+\int_0^t U(s)^{-1}f(s)\,\mathrm{d}s\Big).
\]
Then $u$ is supported in $[0, T]$ and $D_A u=f$.  This completes the proof
of the first assertion.

To prove the second assertion, let $f\in C_0$ be given.  Since we now
know that $D_A$ is surjective, let $v\in C_0^1$ be such that $D_A v=f$.
 Let $\Pi$ denote the projection onto $V_1$ along $V_2$.  There is a unique
$w\in C_0^1$ such that $D_A w=0$ and $w(0)=\Pi v(0)$.  It follows that
if $u=v-w$, then $D_A u=f$ and $u(0)\in V_2$.  To see that $u$ is unique
in $C_0^1$, the difference of two such functions will be a \emph{bounded}
solution to the homogeneous equation $D_A u=0$ whose initial value lies
in $V_2$.  By definition of $V_2$, this initial value must be zero.

We will denote by $\map{S}{C_0}{C_{\rm b}}$ the linear map that carries
 $f$ into $u$.  We will use the closed graph theorem to prove that $S$ is
bounded, and the proof will then be complete.  Take a sequence
$\seq{(f_n,u_n)}$ in the graph of $S$ and suppose that this sequence
converges in $C_0\times C_{\rm b}$ to some $(f,u)$.  Fix $t>0$.
Using the uniform convergence of $\seq{(f_n,u_n)}$ to $(f,u)$ and the uniform
continuity of $A$ on $[0,t]$, we have
\begin{align*}
u(t)-u(0)&=\lim_{n\to\infty} u_n(t)-u_n(0)\\
  &=\lim_{n\to\infty} \int_0^t -A(s)u_n(s)-f_n(s)\,\mathrm{d}s\\
  &=\int_0^t -A(s)u(s)-f(s)\,\mathrm{d}s.
\end{align*}
Upon differentiating with respect to $t$, we find that $D_A u=f$.
Since it also happens that $u(0)=\lim_{n\to\infty} u_n(0)\in V_2$,
it follows that $Sf=u$ as desired.
\end{proof}

Of course, the second assertion in Lemma~\ref{lem: closed range} continues
to hold if $D_A$ is already known to be surjective.  Here is the first
main result in this section.

\begin{theorem}\label{thm:lin-Fredholm}
Assume that $\map{A=A(t)}{[0, \infty)}{\mathbb{R}^{d\times d}}$ is bounded
and continuous.  Then the operator $\map{D_A}{C_0^1}{C_0}$ defined by
$D_A u(t)=\dot{u}(t)+A(t)u(t)$ is a Fredholm operator if and only if $A$
admits an exponential dichotomy on $[0,\infty)$.
In this case, $D_A$ is surjective and $\dim\ker D_A=\dim\operatorname{rge}\Pi$
so that
\[
\operatorname{ind} D_A=\dim\operatorname{rge} \Pi,
\]
where $\Pi$ is any projection associated with the exponential dichotomy.
Additionally, for all $f\in C_0$,
\begin{equation}\label{eq: range condition}
\set{u(0)}{u\in C_0^1\text{ and }\dot{u}+Au=f}
=\operatorname{rge}\Pi -\int_0^\infty
(I-\Pi)U(s)^{-1}f(s)\,\mathrm{d}s.
\end{equation}
\end{theorem}

\begin{proof}
First assume that $D_A$ is Fredholm.  In particular, the range of $D_A$
is closed in $C_0$.  According to
Coppel~\cite[(Proposition 3 on page 22)]{Coppel-MR0481196}, to prove
that $A$ admits an exponential dichotomy it is sufficient to show
that the equation $\dot{u}(t)+A(t)=f(t)$ has a bounded solution
(not necessarily in $C_0^1$) whenever $f\in C_{\rm b}$.
 So let $f\in C_{\rm b}$ be given, and let $\seq{f_n}$ be a sequence
of continuous functions on $[0,\infty)$ such that for each $n\in\mathbb{N}$,
\begin{enumerate}
  \item $f_n$ agrees with $f$ on $[0,n]$,
  \item $f_n$ is supported in $[0,n+1]$, and
  \item $\abs{f_n(t)}\le\abs{f(t)}$ on $[0, \infty)$.
\end{enumerate}
Since these functions are compactly supported, they are all in $C_0$.
Let $V_1$ and $V_2$ be as in Lemma~\ref{lem: closed range}, and
let $u_n$ be the unique solution to $D_A u_n=f_n$ such that $u_n(0)\in V_2$.
 According to Lemma~\ref{lem: closed range}, the sequence $\seq{u_n}$ is
bounded because the sequence $\seq{f_n}$ is bounded.  There is hence a
subsequence $\seq{u_{n_k}}$ such that the initial values $\xi_k=u_{n_k}(0)$
converge to some $\xi\in V_2$.  Define
\[
u(t)=U(t)\Big(\xi+\int_0^t U(s)^{-1}f(s)\,\mathrm{d}s\Big)
\]
so that $D_A u=f$.  For the desired application of the mentioned result
from Coppel~\cite{Coppel-MR0481196}, it remains to show that $u$ is
bounded on $[0,\infty)$.  Fix $t>0$.  For all $n>t$, we have $f_n(t)=f(t)$.
Therefore for all $n_k>t$,
\[
u_{n_k}(t)-u(t)=U(t)(\xi_k-\xi).
\]
In particular, $u(t)=\lim_{k\to\infty} u_{n_k}(t)$.  Because the sequence
$\seq{u_n}$ is (uniformly) bounded, this shows that $u$ is bounded on
$[0, \infty)$.

Conversely, assume that $A$ admits an exponential dichotomy on
 $[0, \infty)$.  For each $f\in C_0$ and $\xi\in\mathbb{R}^N$,
introduce the notation
\[
u=u_{f,\xi}(t):=U(t)\Big(\xi+\int_0^t
U(s)^{-1}f(s)\,\mathrm{d}s\Big)
\]
for the solution to the initial value problem $D_A u=f$, $u(0)=\xi$,
regardless of whether this solution is in $C_0^1$.  Recall that the map
$S$ that carries $\xi$ into $u_{0,\xi}$ is an isomorphism of $\mathbb{R}^N$
onto the vector space of all solutions to the homogeneous equation $D_A u=0$.
The kernel of $\map{D_A}{C_0^1}{C_0}$ is thus isomorphic to the set of all
$\xi\in\mathbb{R}^N$ such that $S\xi=u_{0,\xi}\in C_0^1$.  We claim that
in fact
\[
\ker D_A=S(\operatorname{rge}\Pi).
\]
Indeed, the defining properties~\eqref{ineq:exp forward}
and~\eqref{ineq:exp back} of exponential dichotomy imply that $u_{0,\xi}$
has exponential decay as $t\to\infty$ when $\xi\in\operatorname{rge}\Pi$,
and exponential growth otherwise.  Therefore, $u_{0,\xi}\in C_0^1$ if
and only if $\xi\in\operatorname{rge}\Pi$.

We next consider the range of $D_A$ in $C_0$.  For each choice of $f$ and
$\xi$, we decompose $u_{f,\xi}$ along the projection $\Pi$ as follows:
\begin{align*}
    u_{f,\xi}&=U(t)(\Pi+I-\Pi)\Big(\xi+\int_0^tU(s)^{-1}f(s)\,\mathrm{d}s\Big)\\
       &=U(t)\Pi\xi + \int_0^t U(t)\Pi U(s)^{-1}f(s)\,\mathrm{d}s\\
       &\quad +U(t)\Big((I-\Pi)\xi+\int_0^t (I-\Pi)U(s)^{-1}f(s)\,\mathrm{d}s\Big)\\
       &=: g_1(t)+g_2(t)+g_3(t).
\end{align*}
First, $g_1=u_{0,\Pi\xi}\in C_0^1$.  Second, let $\epsilon>0$ and let $T>0$
be such that $\abs{f(t)}<\epsilon$ when $t>T$.  Because
of~\eqref{ineq:exp forward}, when $t>T$
\begin{align*}
    \abs{g_2(t)}&\le\int_0^t Ke^{-\alpha{t-s}}\abs{f(s)}\,\mathrm{d}s\\
    &\le K\Norm{f}{\infty}\int_0^T e^{-\alpha(t-s)}\,\mathrm{d}s + K\epsilon\int_T^t
    e^{-\alpha(t-s)}\,\mathrm{d}s\\
    &=K\alpha^{-1}\bigl(\Norm{f}{\infty}(e^{\alpha(T-t)}-e^{-\alpha t})
    +\epsilon(1-e^{\alpha(T-t)})\bigr)
\end{align*}
For sufficiently large $t$, this expression is no more than
$2K\alpha^{-1}\epsilon$.  Since $\epsilon>0$ was arbitrary, this shows
that $g_2\in C_0$.  As for $g_3$, note first that
\[
\eta:=\int_0^\infty (I-\Pi)U(s)^{-1}f(s)\,\mathrm{d}s
\]
is a well-defined element of $\operatorname{rge}(I-\Pi)$; this is due
to~\eqref{ineq:exp back} and the boundedness of $f$.  Indeed,
\[
\abs{(I-\Pi)U(s)^{-1}}\le K^{-1}e^{-\alpha s}
\]
so that the integrand decays exponentially as $s\to\infty$.
Notice now that
\begin{align*}
 g_3(t)&=U(t)\Big((I-\Pi)\xi+\eta-\int_t^\infty
  (I-\Pi)U(s)^{-1}f(s)\,\mathrm{d}s\Big)\\
 &=U(t)(I-\Pi)(\xi+\eta)-\int_t^\infty U(t)(I-\Pi)U(s)^{-1}
f(s)\,\mathrm{d}s\\
\end{align*}
Because of~\eqref{ineq:exp back}, we have
\begin{align*}
    \big|\int_t^\infty U(t)(I-\Pi)U(s)^{-1}f(s)\,\mathrm{d}s\big|
&\le \int_t^\infty Ke^{-\alpha(s-t)}\abs{f(s)}\,\mathrm{d}s\\
&\le K\alpha^{-1}\sup_{s\ge t}\abs{f(s)},
\end{align*}
which converges to zero as $t\to\infty$.  Therefore, $g_3$  will
be in $C_0$ if and only if
$\lim_{t\to\infty}U(t)(I-\Pi)(\xi+\eta)=0$.  For this, it is
necessary and sufficient that one has
$\xi+\eta\in\operatorname{rge}\Pi$.  Since $\xi=u_{f,\xi}(0)$,
this is the content of assertion~\eqref{eq: range condition}.  In
any case, it is possible to choose $\xi$ (say $\xi=-\eta$) so that
$g_3$ is in $C_0$.  In that case, $u_{f,\xi}=g_1+g_2+g_3$ is in
$C_0$ as well.  Because $A$ is bounded and $\dot{u}=-Au+f$, it
follows that $u\in C_0^1$.  This shows that $D_A$ is surjective,
which completes the proof.
\end{proof}

Notice that in the first part of the proof,  we used the fact that
Fredholm maps have closed range (by definition), but we used no
other property of Fredholm maps.  Therefore, as soon as $D_A$ is
known to have closed range, it follows that $D_A$ is Fredholm (and
that $A$ admits an exponential dichotomy on $[0,\infty)$).  This
observation amounts to the following corollary, which will be used
later.

\begin{corollary}\label{cor: closed range}
Assume that $\map{A=A(t)}{[0, \infty)}{\mathbb{R}^{d\times d}}$
is bounded and continuous and that the operator $\map{D_A}{C_0^1}{C_0}$
defined by $D_A u(t)=\dot{u}(t)+A(t)u(t)$ has closed range.
Then $D_A$ is Fredholm.
\end{corollary}

We next consider the Fredholm properties of the differential
operator with evaluation at zero.

\begin{theorem}\label{thm:lin-Fredholm-with-evaluation}
Assume that $\map{A=A(t)}{[0, \infty)}{\mathbb{R}^{d\times d}}$
is bounded and continuous.  Let $P$ be any linear projection in
$\mathbb{R}^N$.  Then the operator $\map{\Lambda}{C_0^1}{C_0\times
X_1}$ defined by $\Lambda u=(D_A u,Pu(0))$ is a Fredholm operator
if and only if $A$ admits an exponential dichotomy on
$[0,\infty)$.  In this case,
\begin{gather*}
\ker\Lambda = \set{U(\cdot)\xi}{\xi\in\operatorname{rge}\Pi\cap\ker P},\\
    \operatorname{rge}\Lambda
 = \set{(f,\eta)\in C_0\times{X}_1}{\eta+\int_0^\infty
(I-\Pi)U(s)^{-1}f(s)\,\mathrm{d}s\in \operatorname{rge}\Pi
+\ker P},
\end{gather*}
and $\operatorname{ind}\Lambda = \dim\operatorname{rge}\Pi - \dim{X}_1$,
 where $\Pi$ is any projection associated to the exponential dichotomy
admitted by $A$.
\end{theorem}

\begin{proof}
We can append the zero map to $D_A$ without changing the Fredholm
property and index, as long as the target space is only trivially enlarged;
this results in the map
\[
\map{(D_A,0)}{C_0^1}{C_0\times\{0\}}.
\]
If we now enlarge the target space to $C_0\times{X}_1$, the codimension
of the range increases by $\dim{X}_1$.  The map $\Lambda$ is then
a finite rank perturbation of the result; recall that neither the
Fredholm property nor the index are affected by perturbations of
finite rank (nor even compact perturbations).  The end result in
changing $D_A$ into $\Lambda$ is that the Fredholm index decreases
by $\dim{X}_1$, the only exception to this being that neither
operator is Fredholm.

Next, $u\in\ker\Lambda$ if and only if $u\in C_0^1$ with $D_A u=0$
and $Pu(0)=0$.  By Theorem~\ref{thm:lin-Fredholm} this is the case
if and only if $u(t)=U(t)\xi$ (so that $D_A=0$) and
$\xi\in\operatorname{rge}\Pi$ (so that $u\in C_0^1$), and
$u(0)\in\ker P$.

Finally, by Theorem~\ref{thm:lin-Fredholm}, $(f,\eta)$ is in the
range of $\Lambda$ if and only if $\eta=P\xi$ and
$\xi\in\operatorname{rge}\Pi-\int_0^\infty
(I-\Pi)U(s)^{-1}f(s)\,\mathrm{d}s.$  Since $\eta=P\xi$ means that
$\eta$ differs from $\xi$ by a vector in $\ker P$, this proves the
claimed characterization of $\operatorname{rge}\Lambda$.
\end{proof}

\begin{corollary}\label{cor:lin-Fredholm-index-zero}
In the situation of Theorem~\ref{thm:lin-Fredholm-with-evaluation},
the operator $\Lambda$ is Fredholm of index zero from $C_0^1$
into $C_0\times{X}_1$ if and only if $\Pi$ and $P$ have the same rank.
In this case, the  following are equivalent:
\begin{enumerate}
\item The map $\Lambda$ is an isomorphism.
\item $\operatorname{rge}\Pi\cap\ker P=\{0\}$.
\item $\mathbb{R}^N=\operatorname{rge}\Pi\oplus\ker P$.
\end{enumerate}
\end{corollary}

\begin{proof}
In the situation of Theorem~\ref{thm:lin-Fredholm-with-evaluation}
the Fredholm index of $\Lambda$ is
$\dim\operatorname{rge}\Pi-\dim{X}_1$.  Of course, this is zero if
and only if $\Pi$ and $P$ have the same rank.  Furthermore, a map
of Fredholm index zero is an isomorphism if and only if the map
has trivial kernel.  By Theorem~\ref{thm:lin-Fredholm-with-evaluation},
this kernel is $\operatorname{rge}\Pi\cap\ker P$.  Finally, under the
assumption that $P$ and $\Pi$ have the same rank, the conditions
that $\operatorname{rge}\Pi\cap\ker P$ and
$\mathbb{R}^N=\operatorname{rge}\Pi\oplus\ker P$ are equivalent.
\end{proof}
By drawing upon what is known about exponential dichotomies, we
can use Corollary~\ref{cor:lin-Fredholm-index-zero} to quickly
deduce a variety of specific conditions that are sufficient for
$\Lambda$ to be Fredholm of index zero.  We present several as
examples. Except where an alternate citation is given, all of
these can be verified by consulting (for
example)~\cite{Coppel-MR0481196} or
\cite{massera-schaffer-MR0212324}.

\begin{example} \rm
If $A^\infty=\lim_{t\to\infty} A(t)$ exists (which includes the
constant case), then $\Lambda$ is Fredholm if and only if
$A^\infty$ has no eigenvalues on the imaginary axis.  In this
case, $\Lambda$ has index zero if and only if the rank of $P$ is
equal to the algebraic count of eigenvalues of $A^\infty$ that
have positive real part.
\end{example}

\begin{example} \rm
If $A$ is asymptotically autonomous, then $\Lambda$ is Fredholm
if and only if the eigenvalues of $A$ are eventually bounded away
from the imaginary axis.  In this case, $\Lambda$ has index zero
if and only if the rank of $P$ is equal to the algebraic count of
eigenvalues that stay to the right of the imaginary axis.  We must
remark that this kind of condition is not valid when $A$ is not
asymptotically autonomous.
\end{example}

\begin{example} \rm
By the usual Floquet theory (see Hsieh and  Sibuya~\cite[pages
87-89]{hsieh-sibuya-MR1697415}), the study of periodic systems can
be reduced to that of autonomous systems.  As a result, if $A$ has
period $T$, then $\Lambda$ is Fredholm if an only if $U(T)$ has no
eigenvalues of unit modulus.  In this case, $\Lambda$ has index
zero if and only if the rank of $P$ is equal to the algebraic
count of eigenvalues of $U(T)$ with modulus greater than unity.
In fact, this example extends to asymptotically periodic systems.
\end{example}

\begin{example} \rm \label{ex:lyapunov}
The operator $\Lambda$ is Fredholm if and only if there are a
bounded, continuously differentiable Hermitian matrix function
$H=H(t)$ and a constant $\beta>0$ such that
\[
H(t)A(t) + A(t)^\ast H(t) - \dot{H}(t)\ge \beta I
\]
for a.e. $t\ge 0$, in the sense of quadratic forms on
$\mathbb{R}^N$. (See Coppel~\cite{Coppel-MR0481196}.)  In this
case (see~\cite[Corollary 4.4]{MR2225801}), the algebraic count
$d$ of positive eigenvalues of $H(t)$ is eventually independent of
$t$, and $\Lambda$ has index zero if and only if the rank of $P$
is equal to $d$.
\end{example}

\subsection{Properness and the Fredholm property}

There is an interesting and practical connection between
Theorems~\ref{thm:proper} and~\ref{thm:lin-Fredholm}.  The
connection is due to Yood's Criterion, which states that a bounded
linear map of Banach spaces is proper on the closed and bounded
subsets of the domain if and only if the kernel of the map is
finite dimensional and the range of the map is closed.  This leads
to a test for the Fredholm property and index.  To use this test,
one can replace $A$ by any element of $\omega(A)$, which may be
easier to work with.  We have the following theorem.

\begin{theorem}\label{thm:yood}
Assume that $A$ is a bounded, uniformly continuous $N\times N$
matrix function on $[0, \infty)$.  Assume that for all
$B\in\omega(A)$, there are no bounded, nontrivial solutions to
$\dot{u}+B(t)u=0$ on $(-\infty,\infty)$.  Then the dimension of
the kernel of $D_B\colon C_0^1\to C_0$ is independent of the
choice of $B\in\omega(A)$.  Moreover, $\Lambda$ is Fredholm of
index $\dim\ker D_B-\dim{X}_1$.
\end{theorem}

\begin{proof}
By assumption, the set $S$ in
Definition~\ref{dfn:admissible-omega-set}  is $S=\{0\}$, which is
certainly compact and totally disconnected.  Therefore, $A$ has an
admissible omega-limit set, and it follows from
Theorem~\ref{thm:proper} that $\Lambda$ is proper on the subsets
of $C_0^1$ that are closed and bounded.  Thus, Yood's criterion
guarantees that $\Lambda$ has closed range in $C_0\times{X}_1$.
It follows from Theorem~\ref{thm:lin-Fredholm} (via
Corollary~\ref{cor: closed range}) that $D_A$ is a surjective
Fredholm operator of index $k=\dim\ker D_A$ and that $A$ admits an
exponential dichotomy on $[0,\infty)$ with associated projection
of rank $k=\dim\ker D_A$.  By
Theorem~\ref{thm:lin-Fredholm-with-evaluation}, the map $\Lambda$
is Fredholm of dimension $\dim\ker D_A-\dim{X}_1$.

To complete the proof, we appeal to Remark~4 in
Sacker~\cite{Sacker-MR543706}, in which Sacker explains that under
the current hypotheses, $\dim\ker D_B=\dim\ker D_A$ for all
$B\in\omega(A)$.
\end{proof}

\begin{remark} \rm
In Theorem~\ref{thm:yood}, it is assumed that there are no
bounded, \emph{nontrivial} solutions to $\dot{u}+B(t)u=0$, while
Definition~\ref{dfn:admissible-omega-set} prohibits bounded
\emph{nonconstant} solutions.  This is because
Theorem~\ref{thm:yood} concerns a linear system, so that the set
$S$ in Definition~\ref{dfn:admissible-omega-set} is automatically
a vector space.  Thus, for $S$ to be compact, it is necessary to
require that $S=\{0\}$.
\end{remark}


\subsection{Nonlinear systems}

Recall that $\mathbb{R}^N=X_1\oplus X_2$ is a given decomposition
of $\mathbb{R}^N$ with associated projection $P$ onto $X_1$ along
$X_2$.  If $F$ satisfies \eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
and \eqref{eq:F-hyp-3} recall once again that we define an
operator $\map{{\mathcal{F}}}{C_0^1}{C_0\times X_1}$ by
\begin{equation*}
    {\mathcal{F}}(u):=(\dot{u}+N_{F}(u),Pu(0)),
\end{equation*}
and that it follows from Theorem~\ref{thm:smoothness} that
${\mathcal{F}}\in C^1\bigl(C_0^1,C_0\times X_1\bigr)$ with
\begin{equation*}
    D{\mathcal{F}}(u)v=(\dot{v}+D_z F(\cdot,u)v,Pv(0))\quad
\text{for }u,v\in C_0^1.
\end{equation*}

\begin{lemma}\label{lem:compact-perturbation}
Assume that $F$ satisfies \eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
and \eqref{eq:F-hyp-3}.  Then for each $u\in C_0^1$, the bounded
linear operator $D{\mathcal{F}}(u)-D{\mathcal{F}}(0)$ is compact.
\end{lemma}

\begin{proof}
Let $\seq{v_n}$ be a bounded sequence in $C_0^1$.  To show that
$\bigl(D{\mathcal{F}}(u)-D{\mathcal{F}}(0)\bigr)v_n =\bigl(D_z
F(\cdot,u)-D_z F(\cdot,0)\bigr)v_n,0)$ has a convergent
subsequence in $C_0\times X_1$ is to show that
\begin{equation*}
    \seq{w_n}:= \seq{\bigl(D_z F(\cdot,u)-D_z F(\cdot,0)\bigr)v_n}
\end{equation*}
has a convergent subsequence in $C_0$.  Firstly, note that for
each $N\in\mathbb{N}$, the (restriction of) the sequence
$\seq{v_n}$ is bounded and uniformly continuous on $[0,N]$.  Thus,
by the  Ascoli-Arzela Theorem and a diagonal sequence argument,
there is a subsequence of $\seq{v_n}$ (again denoted $\seq{v_n}$)
and a function $v\in C_{\rm b}$ such that $v_n\to v$ uniformly on
compact intervals.

We will now show that $\seq{w_n}$ is a Cauchy sequence in $C_0$,
and is hence convergent there.  Let $\epsilon>0$.  The sequence
$\seq{v_n}$ is uniformly bounded on
$[0,\infty)\times\mathbb{R}^N$.  According to
Lemma~\ref{lem:continuity-of-DF}~\ref{item:buc-property-b},
$\lim_{t\to\infty} \bigl(D_z F(t,u(t))-D_z F(t,0)\bigr)=0$.  These
two facts imply that there is $T>0$ such that
$\abs{w_n(t)}<\epsilon/4$ for all $n\in\mathbb{N}$ and all $t>T$.
Thus,
\begin{equation}\label{ineq:large-t}
    \abs{w_n(t)-w_m(t)}<\epsilon/2,\quad\text{for all }
n,m\in\mathbb{N}\text{ and }t>T.
\end{equation}
On the other hand, $\seq{v_n}$ is uniformly convergent  (hence
uniformly Cauchy) on $[0,T]$.  Since $\bigl(D_z F(t,u(t))-D_z
F(t,0)\bigr)$ is bounded on $[0,T]$, the sequence $\seq{w_n}$ is
also uniformly Cauchy on $[0,T]$.  With~\eqref{ineq:large-t}, this
implies that for some $N\in\mathbb{N}$,
\begin{equation*}
    \abs{w_n(t)-w_m(t)}<\epsilon,\quad\text{for all }n,m>N
\text{ and }t\ge 0.
\end{equation*}
That is to say, the sequence $\seq{w_n}$ is Cauchy in $C_0$,  as
advertised.
\end{proof}

\begin{theorem}\label{thm:nonlin-Fredholm}
Let $F$ satisfy \eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},  and
\eqref{eq:F-hyp-3}.  Then $\map{{\mathcal{F}}}{C_0^1}{C_0\times
X_1}$ is a Fredholm operator if and only if $D_z F(\cdot,0)$
admits an exponential dichotomy.  In this case,
$\operatorname{ind} {\mathcal{F}}=0$ if and only if $\dim X_1$
equals the common rank of the projections associated to this
exponential dichotomy.
\end{theorem}

\begin{proof}
Recall that by definition, the nonlinear operator ${\mathcal{F}}$
is Fredholm if and only if $D{\mathcal{F}}(u)$ is Fredholm for
some $u$.  In this case, $D{\mathcal{F}}(u)$ is Fredholm for all
$u$, and the index is independent of $u$, and the Fredholm index
of the nonlinear operator ${\mathcal{F}}$ is defined to be this
common value of $\operatorname{ind} D{\mathcal{F}}(u)$.

Lemma~\ref{lem:compact-perturbation} shows that the Fredholm
property and index (which do not vary under compact perturbations)
of $D{\mathcal{F}}(u)$ agrees with that of $D{\mathcal{F}}(0)$.
An application of Corollary~\ref{cor:lin-Fredholm-index-zero} with
$A(t)=D_z F(t,0)$ completes the proof.
\end{proof}

\section{Existence theorems}

In the first result the hypotheses are relatively abstract.   This
is where the topological degree argument is provided.  Subsequent
results will have more concrete (though less general) hypotheses.

\begin{lemma}\label{lem:exist}
Assume that ${\mathcal{F}}$ is of class $C^1$ from $C_0^1$ to
$C_0\times{X}_1$.  Assume moreover that ${\mathcal{F}}$ is
Fredholm of index zero and is proper on the subsets of $C_0^1$
that are closed and bounded.  Assume as well that for a given pair
$(f,\xi)\in C_0\times {X}_1$, that \emph{a priori} bounds exist in
the sense that the set
\begin{equation}\label{eq:bounds}
\set{u\in C_0^1}{{\mathcal{F}}(u)=(sf,s\xi)\text{ for some }
0\le s\le 1}
\end{equation}
is norm bounded in $C_0^1$.  Finally, assume that ${\mathcal{F}}$
is odd.

Then there exists $u\in C_0^1$ such that ${\mathcal{F}}(u)=(f,\xi)$.
\end{lemma}

\begin{proof}
Let $R>0$ be a norm bound for the set defined in~\eqref{eq:bounds},
and let $B$ be the open ball of radius $R+1$ and center $0$ in $C_0^1$.  
By assumption, $\map{{\mathcal{F}}}{C_0^1}{C_0\times{X}_1}$ is a $C^1$ map 
of Fredholm index zero that is proper on the closure of $B$.  
All of this ensures that ${\mathcal{F}}$ is \emph{$B$-admissible}, in the 
sense of \cite[Definition 4.1]{Pejsachowicz-and-Rabier-MR1676979}.

Next, the choice of $B$ ensures that  $(sf,s\xi)\in
C_0\setminus{\mathcal{F}}(\partial B)$ for all $0\le s\le 1$.  As
it is introduced in~\cite[Corollary
5.5]{Pejsachowicz-and-Rabier-MR1676979}, the absolute degree
$\abs{d}({\mathcal{F}},B,(sf,s\xi))$ is a well-defined nonnegative
integer for all $0\le s\le 1$.  Introduce the homotopy
$\map{h}{[0,1]\times C_0^1}{C_0\times {X}_1}$ by
\[
h(s,u):={\mathcal{F}}(u)-(sf,s\xi).
\]
To conclude that the absolute degree is invariant along $h$,  we
must verify first that $h$ is \emph{$B$-admissible}, in the sense
of~\cite[Definition 4.2]{Pejsachowicz-and-Rabier-MR1676979}. It is
clear that $h$ is $C^1$.  To see that $h$ is Fredholm of index
one, note that $Dh(s,u)$ is a rank one perturbation of the linear
map $L:=(0,D{\mathcal{F}}(u))$ from $\mathbb{R}\times C_0^1$ into
$C_0\times{X}_1$.  Thus, the Fredholm properties of $Dh(s,u)$
coincide with those of $L$.  Now, $L$ has the same range and
target space as $D{\mathcal{F}}(u)$, but the kernel of $L$ is
$\mathbb{R}\times\ker D{\mathcal{F}}(u)$.  Since
$D{\mathcal{F}}(u)$ is assumed to be Fredholm of index zero, it
follows that $L$ is Fredholm of index one, as desired.

For the $B$-admissibility of $h$, we must also verify that  the
restriction of $h$ to $[0,1]\times\overline{B}$ is proper.  That
is to say, the preimage of a compact set in $C_0\times{X}_1$
should have compact intersection with $[0,1]\times\overline{B}$.
Suppose that $\{(s_n, u_n)\}$ is a sequence in
$[0,1]\times\overline{B}$ such that $\{h(s_n,u_n)\}$ converges in
$C_0\times {X}_1$, say to a point $(g,\eta)$.  Since $[0,1]$ is
compact, we may assume that ${s_n}$ converges to some $s_0$.  By
definition of $h$, we see that the sequence $\{\mathcal{F}(u_n)\}$
converges to $(g+s_0 f,\eta+s_0 \xi) $.  It follows at once from
the assumed properness of ${\mathcal{F}}$ on the subsets of
$C_0^1$ that are closed and bounded that $ \{u_n\}$ possesses a
convergent subsequence.  Having verified that $h$ is
$B$-admissible, it now follows from~\cite[Theorem
5.1]{Pejsachowicz-and-Rabier-MR1676979} that
$\abs{d}(h(s,\cdot),B,(0,0))$ is independent of $0\le s\le 1$.

By Borsuk's Theorem, the assumed oddness of ${\mathcal{F}}$
implies that this degree is nonzero when $s=0$.  By homotopy
invariance it follows that $\abs{d}(h(1,\cdot),B,(0,0))$ is
nonzero.  The normalization property of the degree implies the
existence of some $u\in B$ such that
${\mathcal{F}}(u)-(f,\xi)=(0,0)$.  This completes the proof.
\end{proof}

There are various tools available to ensure that the degree  is
nonzero at the $s=0$ point of the homotopy $h$ that is used in the
above proof.  In Lemma~\ref{lem:exist}, the assumption that
${\mathcal{F}}$ be odd can be dispensed with if the degree is
known to be nonzero. For this, one sufficient pair of conditions
is that there be no nonzero solution $u\in C_0^1$ to the
homogeneous equation ${\mathcal{F}}(u)=(0,0)$, and the zero
solution be regular in the sense that $D{\mathcal{F}}(0)$ is an
isomorphism.  Briefly, the condition is that the trivial solution
to ${\mathcal{F}}(u)=(0,0)$ be both unique and regular.

In that case, it follows from the definition of the degree at
regular values that $\abs{d}({\mathcal{F}},B,(0,0))=1$.  An
additional relevance of this situation is that isomorphisms are
automatically Fredholm of index zero.  Since
Lemma~\ref{lem:compact-perturbation} ensures the compactness of
$D{\mathcal{F}}(u)-D{\mathcal{F}}(0)$ for all $u\in C_0^1$, it
follows from the invariance of the Fredholm property under compact
perturbations that $\mathcal{F}$ is Fredholm of index zero.  All
of this results in the following variation of
Lemma~\ref{lem:exist}:

\begin{corollary}\label{cor:exist2}
Assume that ${\mathcal{F}}$ is of class $C^1$ from $C_0^1$ to
$C_0\times{X}_1$.  Assume also that ${\mathcal{F}}$ is proper on
the subsets of $C_0^1$ that are closed and bounded.  Assume as
well that for a given pair $(f,\xi)\in C_0\times {X}_1$, that
\emph{a priori} bounds exist in the sense that the
set~\eqref{eq:bounds} is norm bounded in $C_0^1$.  Finally, assume
that there is no nonzero solution $u\in C_0^1$ to the homogeneous
equation ${\mathcal{F}}(u)=(0,0)$, and that $D{\mathcal{F}}(0)$ is
an isomorphism.

Then there exists $u\in C_0^1$ such that ${\mathcal{F}}(u)=(f,\xi)$.
\end{corollary}

\begin{proof}
The proof of Lemma~\ref{lem:exist} needs to be modified only
according to the above remarks concerning nonzero degree and the
Fredholm property.  In particular, recall that compact
perturbations of linear maps of Fredholm zero are again Fredholm
of index zero, and that ${\mathcal{F}}$ is Fredholm of index zero
if $D{\mathcal{F}}(u)$ is Fredholm of index zero at each $u\in C_0^1$.
\end{proof}

\begin{remark} \rm\label{rem:alternate-path}
There is no harm done (but perhaps no practical gain made)  in
replacing the linear path from $(0,0)$ to $(f,\xi)$
in~\eqref{eq:bounds} by any $C^1$ path from $(0,0)$ to $(f,\xi)$.
\end{remark}

Before stating the next theorem, it may be helpful to bring in the
following definition:

\begin{definition} \rm \label{defn:a-priori-bounds}
Let $f\in C_0$ and $\xi\in{X}_1$ be given.  Let $S$ be the set of
all $u\in C_0^1$ such that
\begin{equation}\label{eq:the-s-problem}
\begin{gathered}
    \dot{u}(t)+F(t,u(t))=sf(t) \quad \text{for all } t\ge 0,\\
    Pu(0)=s\xi
\end{gathered}
\end{equation}
for some $0\le s\le 1$.  If $S$ is norm-bounded in $C_0^1$, we say
that the pair $(f,\xi)$ \emph{satisfies the a priori bounds
condition} for $F$.
\end{definition}

\begin{theorem}\label{thm:exist1}
Assume that $F$ satisfies \eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
and \eqref{eq:F-hyp-3} and that $F$ has an admissible omega-limit
set.  Assume that $(f,\xi)\in C_0\times{X}_1$ satisfies the
\emph{a priori} bounds condition for $F$, and that the homogeneous
system associated to~\eqref{eq:the-problem} has both uniqueness
and regularity of the trivial solution.  Then there is at least
one $u\in C_0^1$ to solve~\eqref{eq:the-problem}.
\end{theorem}

\begin{proof}
We apply Corollary~\ref{cor:exist2}.  It follows from
Corollary~\ref{cor: smooth} that ${\mathcal{F}}$ (as defined
in~\eqref{eq: Phi}) is a $C^1$ map from $C_0^1$ to
$C_0\times{X}_1$.  It follows from Theorem~\ref{thm:proper} that
${\mathcal{F}}$ is proper on the subsets of $C_0^1$ that are
closed and bounded.  According to
Definition~\ref{defn:a-priori-bounds}, the \emph{a priori} bounds
condition of Corollary~\ref{cor:exist2} is satisfied.

Finally, to say that the homogeneous system associated
to~\eqref{eq:the-problem} has both uniqueness and regularity of
the trivial solution means precisely that the remaining
requirements of Corollary~\ref{cor:exist2} are met as well.  We
conclude that there is indeed at least one $u\in C_0^1$ to
solve\eqref{eq:the-problem}.
\end{proof}

\begin{remark} \rm
As per Remark~\ref{rem:alternate-path}, there is no harm in
replacing the linear path from $(0,0)$ to $(f,\xi)$
in~\eqref{eq:the-s-problem} by any $C^1$ path from $(0,0)$ to
$(f,\xi)$.
\end{remark}

Perhaps less useful in practice, the following version  (which
avoids the isomorphism condition) is worth recording:

\begin{theorem}\label{thm:exist2}
Assume that $F$ satisfies \eqref{eq:F-hyp-1}, \eqref{eq:F-hyp-2},
and \eqref{eq:F-hyp-3} and that $F$ has an admissible omega-limit
set. Assume that $(f,\xi)\in C_0\times{X}_1$ satisfies the \emph{a
priori} bounds condition for $F$.  Finally, assume that
${\mathcal{F}}$ is Fredholm of index zero and that $F(t,\cdot)$ is
odd for each $t\ge 0$. Then there is at least one $u\in C_0^1$ to
solve\eqref{eq:the-problem}.
\end{theorem}

\begin{proof}
This result follows from Lemma~\ref{lem:exist} instead of
Corollary~\ref{cor:exist2}.
\end{proof}

\begin{remark} \rm
In applications of Theorem~\ref{thm:exist2}, one must verify that
${\mathcal{F}}$ is Fredholm of index zero.  By
Lemma~\ref{lem:compact-perturbation} it is enough to check that
$D{\mathcal{F}}(0)$ is Fredholm of index zero, for which in turn
it is sufficient to know that $D{\mathcal{F}}(0)$ is an
isomorphism.  In this context, the assumed oddness replaces the
uniqueness of the trivial solution to the homogeneous problem.

Alternately, one can use the results of Section~\ref{sec:fredholm}
to verify that ${\mathcal{F}}$ is Fredholm of index zero.
According to Theorem~\ref{thm:nonlin-Fredholm}, it is necessary
and sufficient that $D_z F(\cdot,0)$ admit an exponential
dichotomy with associated projection of the same rank as $P$.
There are a number of examples given in Section~\ref{sec:fredholm}
of how to do this.  Of particular interest in the context of this
paper is that it is sufficient to set $A=D_z F(\cdot, 0)$ and to
check that for all $B\in\omega(A)$, that there are no bounded,
nontrivial solutions to $\dot{u}+B(t)u=0$ on $(-\infty,\infty)$.
In this case, according to Theorem~\ref{thm:yood} it remains only
to check that for any $B\in\omega(A)$ (and hence all
$B\in\omega(A)$) that the dimensions of $\ker D_A$ and of $X_1$
are equal.

In principle, this approach allows for applications of
Theorem~\ref{thm:exist2} even when the trivial solution to the
homogeneous system associated to~\eqref{eq:the-problem} is neither
unique nor regular.
\end{remark}


\section{Example}

This section provides a new example that builds upon the results
in \cite[Example 7.2, culminating in Theorem
7.3]{Rabier-and-Stuart-(II)-MR2133394}.  (For a variety of other
examples, please see~\cite{me}.)

Let $I$ be any closed interval containing zero, possibly as an
endpoint.  Let $g=g(t,s)\colon I\times\mathbb{R}\to\mathbb{R}$ and
$h=h(t,s)\colon I\times\mathbb{R}\to \mathbb{R}$ be two
real-valued functions with the following properties.  (Of course,
this means that each of the following is true also upon replacing
$g$ by $h$, possibly with different constants.)
\begin{gather}
  g(t,0)=0,\label{eq:g1}\\
  D_s g\text{ exists and is continuous on }I\times\mathbb{R},
 \label{eq:g2} \\
 g, \text{ and }D_s g\text{ are B.U.C. on }I\times K\text{ for each
compact interval }K,\label{eq:g3}\\
 D_s g\text{ is non-negative},\label{eq:g4} \\
  \inf_{t\in I }D_s g(t, 0)>0,\label{eq:regular}
\end{gather}
and finally, there are positive constants $\alpha$ and
$s^\ast$ such that for all $t\in I$,
\begin{equation}
g(t,s)/s\ge \alpha \text{ whenever }\abs{s}>s^\ast.\label{eq:blowup}
\end{equation}
Of course, the significance of conditions~\eqref{eq:g1}-\eqref{eq:g3}
are that $g$ and $h$ satisfy~\eqref{eq:F-hyp-1}-\eqref{eq:F-hyp-3}
(with $N=1$, with each of $g$ and $h$ in place of $F$, and with $I$
in place of $[0,\infty)$).  Also, if condition~\eqref{eq:blowup}
holds for all $\alpha>0$ (so that $s^\ast=s^\ast(\alpha)$),
then $g$ and $h$ are super-linear.

The following are a few simple properties that will be used later.

\begin{lemma}\label{lem:simple-properties}
Continue to assume that $g\colon I\times\mathbb{R}\to\mathbb{R}$
satisfies conditions~\eqref{eq:g1}-\eqref{eq:blowup}. Then
  \begin{enumerate}
    \item $g(t,s)$ has the same sign as $s$.\label{item:simp-prop-sign}
    \item $sg(t,s)$ and $g(t,s)/s$ are both positive when
     $s\neq 0$.\label{item:simp-prop-sg-pos}
    \item If $s>0$, then $\inf_{t\in I} g(t,s)>0$.\label{item:simp-prop-strict-pos}
    \item If $s<0$, then $\sup_{t\in I} g(t,s)<0$.\label{item:simp-prop-strict-neg}
    \item The function $g^\ast(s):=\inf_{t\in I} g(t,s)$ is monotone increasing.
    \label{item:simp-prop-monotone}
    \item $ \lim_{\abs{s}\to\infty} \abs{g(t,s)}=\infty$ uniformly in $t\in I$.
    \label{item:simp-prop-blow-up}
  \end{enumerate}
\end{lemma}

\begin{proof}
  \begin{enumerate}
    \item Since $g(t,0)=0$, this follows from the non-negativity of $D_s g$,
     along with the strict positivity of $D_s g(t,0)$.
    \item This follows from part (a).
    \item This follows from~\eqref{eq:regular}.
    \item This also follows from~\eqref{eq:regular}.
    \item Let $s_2>s_1$, and choose $\epsilon>0$.  Let $t_1$ and $t_2$ be chosen 
    so that both $\abs{g^\ast(s_1)-g(t_1,s_1)}$ and 
    $\abs{g^\ast(s_2)-g(t_2,s_2)}$ are smaller than $\epsilon$.  Then
        \begin{align*}
            g^\ast(s_2)-g^\ast(s_1)
&\ge g(t_2,s_2)-g(t_1,s_1)-2\epsilon\\
&=\bigl(g(t_2,s_2)-g(t_2,s_1)\bigr)+\bigl(g(t_2,s_1)-g(t_1,s_1)\bigr)
 -2\epsilon\\
&\ge 0+g^\ast(s_1)-g(t_1,s_1)-2\epsilon\\
&\ge -3\epsilon,
        \end{align*}
 where we have used the fact that for fixed $t_2$, the function
$g(t_2,s)$ is monotone increasing in $s$. Since $\epsilon>0$ was arbitrary, 
this shows that $g^\ast(s_2)-g^\ast(s_1)\ge 0$, as desired.
    \item This follows from~\eqref{eq:blowup}.
  \end{enumerate}
\end{proof}

Before stating and proving the main existence result, it helps
to prove two lemmas that concern blowup and \emph{a priori} bounds.

\begin{lemma}\label{lem:blow-up}
Let $g=g(t,s)$ and $h=h(t,s)$ be two real valued functions on
$I\times\mathbb{R}$ that satisfy
conditions~\eqref{eq:g1}-\eqref{eq:regular}.
Let $u=(v,w)$ be any nontrivial $C^1$ solution to the homogeneous
problem
\begin{equation}\label{eq:the-lemma-problem}
\begin{gathered}
    \dot{v}+g(t,w)=0,\\
    \dot{w}+h(t,v)=0.
\end{gathered}
\end{equation}
We take $u$ to be extended from $t=0$ as far as possible as a solution,
 perhaps to all of $I$.
\begin{enumerate}
  \item If $v(0)w(0)\le 0$ and $[0,\infty)\subseteq I$, then $u$ blows up as 
  $t\to\infty$ (possibly in finite time).
  \item If $v(0)w(0)\ge 0$, and $(-\infty, 0]\subseteq I$, then $u$ blows up 
  as $t\to -\infty$ (possibly in finite time).
\end{enumerate}
\end{lemma}

\begin{proof}
At most one of $v(0)$ and $w(0)$ is zero, and it is no loss of
generality to assume that $w(0)\neq 0$.  Otherwise, we can exchange
the names of $g$ and $h$.  This exchange results in a corresponding
exchange in the names of $v$ and $w$.

We first consider (a).  We examine the case that $w(0)>0$ and
$v(0)\le 0$; the remaining case has a similar proof.
With reference to Lemma~\ref{lem:simple-properties}
part~\ref{item:simp-prop-sign}, we have
\[
  \dot{v}(0)=-g(0,w(0))<0,
\]
  and
\[
  \dot{w}(0)=-h(0,v(0))\ge 0.
\]
Thus $v$ is decreasing and $w$ is \emph{not} decreasing at $t=0$.
Let $J$ be the set of all $t\ge 0$ such that $w>0$ on $[0,t)$.
Note that in $J$, we have $\dot{v}(t)=-g(t,w)<0$ so that $v$ is
decreasing on $J$.  In particular, $v$ is non-positive on $J$.
Since $\dot{w}(t)=-h(t,v)$, it follows that $w$ is non-decreasing
on $J$.  Unless the solution blows up in finite time (in which
case there is nothing to prove), this shows that
$[0,\infty)\subseteq J$.  Therefore, $w\ge w(0)$ on $[0,\infty)$,
from which it follows that $\dot{v}(t)=-g(t,w(t))\le -g(t,
w(0))\le -\inf_{t\ge 0} g(t, w(0))<0$.  (See
Lemma~\ref{lem:simple-properties},
part~\ref{item:simp-prop-strict-pos}.)  Since $\dot{v}$ is
negative and bounded away from zero, this proves that $v(t)\to
-\infty$ as $t\to\infty$.

We now consider (b).  It is once again without loss of generality that
$w(0)\neq 0$.  We consider the case that $w(0)>0$ and $v(0)\ge 0$;
once again, the omitted case is very similar.  We let $J$ be the set
of all $t\le 0$ such that $w$ is positive on $(t,0]$.
By an analysis similar to that of part (a),  we find that
$(-\infty,0]\subseteq J$ and  $\dot{v}(t)$ is therefore positive
and bounded away from zero on $(-\infty, 0]$.  This proves that
$v(t)\to\infty$ as $t\to -\infty$.
\end{proof}

The next lemma says that if $u=(v,w)$ is a $C_0^1$ solution,
then neither $v$ nor $w$ can become too large relative to the other.

\begin{lemma}\label{lem:bounds}
Let $g=g(t,s)$ and $h=h(t,s)$ be two real valued functions on
$[0,\infty)\times\mathbb{R}$, and assume that $g$ and $h$ satisfy
conditions~\eqref{eq:g1}-\eqref{eq:blowup}.  Let $f_1,f_2\in C_0$,
and let $K\ge 0$ be given.  There exists $R=R(K,f_1,f_2)\ge 0$ with
the following property.  Let $u=(v,w)$ be any $C_0^1$ solution to
the nonhomogeneous problem
\begin{equation}\label{eq:the-other-lemma-problem}
\begin{gathered}
    \dot{v}+g(t,w)=\sigma f_1,\\
    \dot{w}+h(t,v)=\sigma f_2,\\
\end{gathered}
\end{equation}
 for some $0\le \sigma\le 1$.  For all $t_0\ge 0$, if one of $\abs{v(t_0)}$ 
 or $\abs{w(t_0)}$ is no larger than $K$, then the other is no larger than $R$.
\end{lemma}

\begin{proof}
Fix $0\le \sigma\le 1$ and $t_0\ge 0$.  Let $u=(v,w)$ be any
$C_0^1$ solution to~\eqref{eq:the-other-lemma-problem}.  By the
symmetry of~\eqref{eq:the-other-lemma-problem}, it suffices to
assume that $\abs{v(t_0)}\le K$ and prove that $\abs{w(t_0)}\le
R$.  To be clear, we need to find $R=R(K,f_1,f_2)$ such that
$\abs{w(t_0)}\le R$. It is important to ensure that the choice of
$R$ does not depend on the choice of solution $u$, parameter
$\sigma$, nor time $t_0$.

We first consider the case that $v(t_0)$ and $w(t_0)$ are both
nonnegative.  We will show that if $w(t_0)$ is too large relative
to $K$, $f_1$, and $f_2$, then $u\not\in C_0^1$.  It follows from
Lemma~\ref{lem:simple-properties},
part~\ref{item:simp-prop-blow-up} that there is $s_1$ such that
$\inf_{t\ge 0} g(t,s)\ge 1$ for all $s>s_1$.  Suppose that
$w(t_0)>s_1$.  (Otherwise, just take $R>s_1$.)

Define $h^\ast(s):=\inf_{t\ge 0}h(t,s)$.  Let $J$ be the set of
all $t\ge t_0$ such that both $h^\ast(v(\cdot))>-\norm{f_2}-1$ and
$w>s_1$ on the interval $[t_0,t]$.  Because $t_0\in J$, it follows
that $J$ is an interval $[t_0,a)$ for some $t_0<a\le \infty$.
Notice that as $t$ increases, $t$ remains in $J$ only so long as
$v$ and $w$ are large enough.  We will find an upper bound for $a$
by showing that $v$ is ultimately decreasing.  However, this is
sure to occur only after $f_1$ has become negligible.

To this end, let $t_1\ge t_0$ satisfy $\abs{f_1(t)}\le 1/2$ for
all $t>t_1$.  Then for all $t\in J$,
\[
\dot{v}(t)=\sigma f_1(t)-g(t,w(t))\le \sigma f_1(t)-1
\leq  \begin{cases}
 \norm{f_1}, & \text{if $t\le t_1$;} \\
 -1/2, & \text{if $t\ge t_1$.}
 \end{cases}
\]
  Therefore, for all $t\in J$,
\[
  v(t)\leq \begin{cases}
 K+(t-t_0)\norm{f_1}, & \text{if $t\le t_1$;} \\
 K+(t_1-t_0)\norm{f_1}-(1/2)(t-t_1), & \text{if $t>t_1$.}
\end{cases}
\]
Since $h(t,v(t))\ge h^\ast(v(t))>-\norm{f_2}-1$ in $J$, it follows
that $J$ contains only values of $t>t_1$ (if any) such that
\[
  h\bigl(t, K+t_1\norm{f_1}-(1/2)(t-t_1)\bigr)>-\norm{f_2}-1.
\]
  With reference to items~\ref{item:simp-prop-sign}
and~\ref{item:simp-prop-blow-up} of Lemma~\ref{lem:simple-properties}
(with $h$ in place of $g$), this implicitly bounds the right
endpoint $a$ of $J$, in a way that depends only on $K$, $f_1$, and
$f_2$.

Now we can estimate $w(t)$ when $t\in J$.  Notice that
\[
  \sup_{t\in J} h(t,v)\le M:=\sup_{t\in J} h\bigl(t, K+t_1\norm{f_1}\bigr)<\infty.
\]
  Note that $M$ depends only on $K$, $f_1$, and $f_2$.  When $t\in J$,
\begin{equation}\label{ineq: w}
\begin{aligned}
 w(t)&=w(t_0)+\int_{t_0}^t \dot{w}(\tau)\,\mathrm{d}\tau \\
     &= w(t_0)+\int_{t_0}^t \sigma f_2(\tau)-h(\tau,v(\tau))
 \,\mathrm{d}\tau\\
     &\ge w(t_0)-(a-t_0)(\norm{f_2}+M).
  \end{aligned}
\end{equation}
  Now, what would happen if $w(t_0)$ were too large?
From inequality~\eqref{ineq: w} it follows that if
\[
w(t_0)\ge s_1+(a-t_0)(\norm{f_2}+M)+1,
\]
then $w(t)\ge s_1+1$ for all $t\in J$.  By definition of $J$,
it follows that $h^\ast(v(a))=-\norm{f_2}-1$.  (Recall that
$a<\infty$).  Now let's look at the solution $(v,w)$ as $t$ increases
beyond $a$.  Let $J'$ be the set of all $t\ge a$ such that both $w>s_1$
and $h^\ast(v(\cdot))<-\norm{f_2}$ on $[a,t)$.  Note that theses
inequalities hold at $t=a$, so that $J'$ is nonempty.
Let $a'=\sup J'$.  Notice that for all $t\in J'$,
\[
\dot{v}(t)=\sigma f_1(t)-g(t,w)\le -1/2.
\]
Therefore, it is necessary that $a'<\infty$, lest $v$ be unbounded
and the solution $u=(v,w)$ fails to be in $C_0^1$.  However,
the definition of $J'$ allows for only two possibilities
concerning $a'$.  The first is that $w(a')=s_1$.
This is impossible, because $w(a)>s_1$ and
$\dot{w}=\sigma f_2-h(t,v)>0$ on $J'$.  The only remaining possibility
is that $h^\ast(v(a'))=-\norm{f_2}$.  However, $v$ is decreasing
in $J'$, so item~\ref{item:simp-prop-monotone} implies that
$h^\ast(v(\cdot))$ is non-increasing in $J'$.  Since
$h^\ast(v(a))=-\norm{f_2}-1$, this is a contradiction.
We conclude that for $R=s_1+(a-t_0)(\norm{f_2}+M)+1$, if $w(t_0)>R$
then $u=(v,w)\not\in C_0^1$.  This completes the proof in case $v(t_0)$
and $w(t_0)$ are both non-negative.

Still assuming that $v(t_0)\ge 0$, we next consider those solutions
such that $w(t_0)<0$.  We now let $s_1<0$ be such that
$-\norm{f_1}-g(t,s)>2$ for all $t\ge t_0$ and all $s<s_1$.
Suppose that $w(t_0)<s_1$.  This time we take $J$ to be the set
of all $t\ge t_0$ such that $-\norm{f_1}-g(\cdot,w(\cdot))>1$
on $[t_0,t]$.  For all $t\in J$,
\[
\dot{v}(t)=\sigma f_1-g(t,w(t))>1,
\]
so that $v(t)>v(t_0)+(t-t_0)\ge t-t_0$ and so
\[
h(t,v(t))>h(t,t-t_0).
\]
According to item~\ref{item:simp-prop-blow-up} of
Lemma~\ref{lem:simple-properties} there is $t_1\ge t_0$ such that
$h(t,t-t_0)\ge\norm{f_2}+1$ for all $t\ge t_1$.  Thus, for all
$t\in J$,
\begin{align*}
w(t)&=w(t_0)+\int_{t_0}^t \sigma f_2(\tau)-h(\tau,v(\tau))\,\mathrm{d}\tau\\
&\leq \begin{cases}
 w(t_0)+(t-t_0)\norm{f_2}, & \text{if $t\le t_1$;} \\
 w(t_0)+(t_1-t_0)\norm{f_2}-(t-t_1), & \text{if $t\ge t_1$}
\end{cases}\\
&\le w(t_0)+(t_1-t_0)\norm{f_2}.
\end{align*}
This shows that if $w(t_0)\le s_1-(t_1-t_0)\norm{f_2}$, then
$w(t)\le s_1$ for all $t\in J$.  It follows that
 $-\norm{f_1}-g(t,w(t))>2$ for all $t\in J$, so that $J=[t_0,\infty)$.
Since $\dot{v}>1$ on $J$, the solution $u=(v,w)$ is unbounded and
is not in $C_0^1$.  This proves that no $C_0^1$ solution $u=(v,w)$
 satisfies $w(t_0)<s_1-(t_1-t_0)\norm{f_2}$.  The proof is complete
in the case that $v(t_0)\ge 0$.

The argument when $v(t_0)\le 0$ is similar, in principle. However,
it is probably more efficient to use the following reflection
argument.  Let $p(t,s)=-g(t,-s)$, and $q(t,s)=-h(t,-s)$. Note that
$D_s p(t,s)=D_s g(t,-s)$ and $D_s q(t,s)=D_s h(t,-s)$, so that $p$
and $q$ are seen to satisfy
conditions~\eqref{eq:g1}-\eqref{eq:blowup}.  Also, the following
are equivalent:
\begin{itemize}
  \item The pair $(v,w)=(x,y)$ solves \begin{equation}
\begin{gathered}
    \dot{v}+g(t,w)=\sigma f_1,\\
    \dot{w}+h(t,v)=\sigma f_2,\\
\end{gathered}
\end{equation}
  \item The pair $(v,w)=(-x,-y)$ solves\begin{equation}
\begin{gathered}
    \dot{v}+p(t,w)=-\sigma f_1,\\
    \dot{w}+q(t,v)=-\sigma f_2,\\
\end{gathered}
\end{equation}
\end{itemize}
Therefore, if $v(t_0)\le 0$, we can apply the case that has
already been proved to the reflected problem to obtain the desired
 bound.
\end{proof}

With the help of the preceding technical lemmas, it remains  only
to see how Theorem~\ref{thm:exist1} can be applied.

\begin{theorem}\label{thm:main-example}
Let $g$ and $h$ be real-valued functions on
$[0,\infty)\times\mathbb{R}$ that satisfy
conditions~\eqref{eq:g1}-\eqref{eq:blowup}. Let
$\xi\in\mathbb{R}$, and let $f_1, f_2\in C_0$ be such that
$\norm{f_1}+\norm{f_2}<\alpha$, where $\alpha$ is the bound that
appears in condition~\eqref{eq:blowup}.  Then the system
  \begin{equation}\label{eq:the-example-problem}
\begin{gathered}
    \dot{v}+g(t,w)=f_1,\\
    \dot{w}+h(t,v)=f_2,\\
    v(0)=\xi
\end{gathered}
\end{equation}
has at least one solution $(v,w)\in C_0^1$.
\end{theorem}

\begin{proof}
To apply Theorem~\ref{thm:exist1}, we set
$u=\left[\begin{smallmatrix} v \\ w  \end{smallmatrix}\right]$,
  $F(t,z)=F(t,u)=\left[\begin{smallmatrix} g(t,w) \\ h(t,v) 
  \end{smallmatrix}\right]$, $f=\left[\begin{smallmatrix} f_1 \\ f_2 
  \end{smallmatrix}\right]$, and 
  $P\left[\begin{smallmatrix} s_1 \\ s_2 \\ \end{smallmatrix}\right]=s_1$.  
  Using the variable $z=\left[\begin{smallmatrix} s_1 \\ s_2 
  \end{smallmatrix}\right]$, it follows that
  \[
  D_z F(t,z)=\begin{bmatrix}
               0 & D_s g(t,s_2) \\
               D_s h(t,s_1) & 0 \\
             \end{bmatrix}.
  \]
It follows immediately from the conditions placed upon $g$ and $h$
that $F$ satisfies~\eqref{eq:F-hyp-1},~\eqref{eq:F-hyp-2},
and~\eqref{eq:F-hyp-3}.

Next, we must show that $F$ has an admissible omega-limit set.
Let $E$ be any member of $\omega(F)$.  It follows that
$E=\left[\begin{smallmatrix} \tilde{g} \\ \tilde{h} \\
\end{smallmatrix}\right]$ for some $\tilde{g}\in\omega(g)$ and
some $\tilde{h}\in\omega(h)$, and that these are functions that
satisfy conditions~\eqref{eq:g1}-\eqref{eq:blowup} with
$I=\mathbb{R}$.   (Some of these conditions are verified via
Lemma~\ref{lem:f-omega-set-is-nonempty}; the rest follow easily
using the uniform convergence on compact sets.)  Assume that $u\in
C_{\rm b}^1(\mathbb{R})$ is a (bounded) solution to
$\dot{u}+E(t,u)=0$.  For the admissibility of the omega-limit set
of $F$, we are to show that $u$ is constant, and that the set of
all such constant functions (over all choices of $E\in\omega(F)$)
forms a compact, totally disconnected subset of $\mathbb{R}^2$.
Note that $\dot{u}+E(t,u)=0$ means that
\begin{gather*}
    \dot{v}+\tilde{g}(t,w)=0,\\
    \dot{w}+\tilde{h}(t,v)=0.
\end{gather*}
Since $\tilde{g}$ and $\tilde{h}$ satisfy
conditions~\eqref{eq:g1}-\eqref{eq:blowup}, the two parts of
Lemma~\ref{lem:blow-up} together imply that the bounded $(v,w)$
must be the trivial solution.  This proves that $u$ must be zero,
which is indeed constant.  Moreover, the set of all possible
constant solutions is again just $\{0\}$, which is compact and
totally disconnected.  This verifies the admissibility of the
omega-limit set of $F$.

In our application of Theorem~\ref{thm:exist1}, we next verify
that the homogeneous system
\begin{gather*}
    \dot{v}+g(t,w)=0,\\
    \dot{w}+h(t,v)=0\\
    v(0)=0
\end{gather*}
has both uniqueness and regularity of the trivial solution in
$C_0^1$.  Uniqueness follows from the first part of
Lemma~\ref{lem:blow-up}.  For regularity, we must show that the
linearized system
\begin{gather*}
    \dot{v}+D_s g(t,0)w=0,\\
    \dot{w}+D_s h(t,0)v=0\\
    v(0)=0
\end{gather*}
has only the trivial solution in $C_0^1$.  Notice that the maps
$s\mapsto D_s g(t,0)s$ and $s\mapsto D_s h(t,0)s$ satisfy
conditions~\eqref{eq:g1}-\eqref{eq:blowup}.  In particular, the
fact that $g$ and $h$ satisfy condition~\eqref{eq:regular} is what
ensures that $D_sg $ and $D_s h$ satisfy both~\eqref{eq:regular}
and~\eqref{eq:blowup}.  Therefore, the first part of
Lemma~\ref{lem:blow-up} implies that the linearized system has no
nontrivial solutions in $C_0^1$.

Finally, we must verify that $(f,\xi)$ satisfies the  \emph{a
priori} bounds condition; see
Definition~\ref{defn:a-priori-bounds}.  Thus, let $S$ be the set
of all pairs $(v,w)\in C_0^1$ such that
\begin{gather*}
    \dot{v}+g(t,w)=\sigma f_1,\\
    \dot{w}+h(t,v)=\sigma f_2,\\
    v(0)=\sigma \xi
\end{gather*}
for some $0\le \sigma\le 1$.  We are to show that $S$ is norm
bounded in $C_0^1$.

Let $(v,w)\in S$, and let $r(t)=v(t)^2+w(t)^2$. Since $r\in
C_0^1$, this function achieves its maximum at some $t_0\ge 0$ at
which $u=(v,w)$ also achieves its maximum Euclidean norm, which we
will use to establish the \emph{a priori} bounds.  If $t_0=0$,
then Lemma~\ref{lem:bounds} provides an \emph{a priori} bound
$R(K,f_1,f_2)$ for $w(0)$, where $K=\xi$.  In this case,
\begin{equation}\label{ineq:co-bounds1}
\sup_{t\ge 0} \abs{u(t)}=\abs{u(0)}=\sqrt{v(0)^2+w(0)^2}
\le \sqrt{K^2+R^2}.
\end{equation}
Before using inequality~\eqref{ineq:co-bounds1}, we derive an
analogous inequality in case $t_0>0$. If at least one of $v(t_0)$
and $w(t_0)$ is no greater than $K=1$ in absolute value, we use
Lemma~\ref{lem:bounds} to bound the other by $R=R(1,f_1,f_2)$.  It
follows that
\begin{equation}\label{ineq:co-bounds2}
  \sup_{t\ge 0} \abs{u(t)}=\abs{u(t_0)}=\sqrt{v(t_0)^2+w(t_0)^2}
\le \sqrt{1+R^2}.
\end{equation}
It remains to consider the case that both $\abs{v(t_0)}$ and
$\abs{w(t_0)}$ are greater than $1$, while $t_0>0$.  Since $r$
attains its maximum at the interior point $t_0>0$, it follows that
$\dot{r}(t_0)=0$.  Thus,
\[
v(t_0)\dot{v}(t_0)+w(t_0)\dot{w}(t_0)=0,
\]
from which it follows that
\[
v(t_0)\bigl(\sigma f_1(t_0)-g(t_0,w(t_0))\bigr)+w(t_0)
\bigl(\sigma f_2(t_0)-h(t_0,v(t_0))\bigr)=0.
\]
Division by $v(t_0)w(t_0)\neq 0$ results in
\begin{equation}\label{eq:almost}
\frac{g(t_0,w(t_0))}{w(t_0)}+\frac{h(t_0,v(t_0))}{v(t_0)}
=\frac{\sigma f_1(t_0)}{w(t_0)}+\frac{\sigma f_2(t_0)}{v(t_0)}
\end{equation}
Because $\abs{v(t_0)}$ and $\abs{w(t_0)}$ are greater than $1$,
the expression on the right side of equation~\eqref{eq:almost} is
bounded by $\norm{f_1}+\norm{f_2}$ in absolute value.  On the
other hand, both terms of the left side of~\eqref{eq:almost} are
positive; this follows from item~\ref{item:simp-prop-sign} of
Lemma~\ref{lem:simple-properties}.  Thus, each of the two terms is
independently bounded by $\norm{f_1}+\norm{f_2}$, which is assumed
to be less than the bound $\alpha$ from~\eqref{eq:blowup}; this
implies that both $\abs{v(t_0)}$ and $\abs{w(t_0)}$ are no greater
than the value $s^\ast$ from~\eqref{eq:blowup}.  Together with
inequalities~\eqref{ineq:co-bounds1} and~\eqref{ineq:co-bounds2}
from the other two cases, we have verified that there is a
constant $R=R(\xi,f_1,f_2)$ such that
\[
\sup_{t\ge 0} \abs{u(t)}\le R
\]
for all $u\in S$.  This proves that $S$ is norm bounded in $C_0$.
To achieve a bound in the norm of $C_0^1$, it remains only to use
the equations
\begin{gather*}
    \dot{v}=\sigma f_1-g(t,w),\\
    \dot{w}=\sigma f_2-h(t,v).
\end{gather*}
Having bounded $v$ and $w$, the right sides of these equations
serve to bound $\dot{v}$ and $\dot{w}$.  This completes the
verification that $F$ has an admissible omega-limit set, which in
turn completes the application of Theorem~\ref{thm:exist1}.  We
conclude that~\eqref{eq:the-example-problem} has at least one
solution in $C_0^1$, as advertised.
\end{proof}

As a simple corollary, if $g$ and $h$ are strictly super-linear,
then we need not assume that $f_1$ and $f_2$ are small.
Specifically, we have the following corollary.

\begin{corollary}
Let $g$ and $h$ be real-valued functions on
$[0,\infty)\times\mathbb{R}$ that satisfy
conditions~\eqref{eq:g1}-\eqref{eq:blowup}. However, assume
moreover that $g$ and $h$ are super linear, in the sense
that~\eqref{eq:blowup} holds for all $\alpha>0$ (meaning that
$s^\ast=s^\ast(\alpha)$).

Let $\xi\in\mathbb{R}$, and let $f_1, f_2\in C_0$ be arbitrary.
Then the system
  \begin{equation}
\begin{gathered}
    \dot{v}+g(t,w)=f_1,\\
    \dot{w}+h(t,v)=f_2,\\
    v(0)=\xi
\end{gathered}
\end{equation}
has at least one solution $(v,w)\in C_0^1$.
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:main-example}, with
$\alpha=\norm{f_1}+\norm{f_2}+1$.
\end{proof}

\begin{remark} \rm
Some of the results that appear in this paper are elaborations
of arguments that first appeared in the author's Ph. D. dissertation,
completed under the guidance of Professor Patrick J. Rabier
at The University of Pittsburgh.
\end{remark}

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\end{document}