\documentclass[reqno]{amsart}
\usepackage{amscd}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 88, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2003/88\hfil Harmonic solutions \dots]
{Harmonic solutions to perturbations of
periodic separated variables ODEs on manifolds}
\author[Marco Spadini\hfil EJDE--2003/88\hfilneg]
{Marco Spadini}
\address{Dipartimento di Matematica Applicata,
Universit\`a di Firenze, Via S.\ Marta 3 - 50139 Firenze, Italy}
\email{spadini@dma.unifi.it}
\date{}
\thanks{Submitted April 17, 2003. Published August 25, 2003.}
\subjclass[2000]{34C25, 34C40}
\keywords{Ordinary differential equations on manifolds, multiplicity of
\hfill\break\indent
periodic solutions}
\begin{abstract}
We study the set of harmonic solutions to perturbed periodic
separated variables ordinary differential equations on manifolds.
As an application, a multiplicity result is deduced.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\section{Introduction}
In this paper we shall investigate the structure of the set of harmonic
solutions to perturbed periodic separated variable ordinary differential
equations on manifolds.
More precisely, let $M$ be an $m$-dimensional boundaryless differentiable
manifold embedded in $\mathbb{R}^k$. We consider equations of the form
\begin{equation}\label{nope}
\dot x=a(t)g(x),
\end{equation}
where $g:M\to\mathbb{R}^k$ is a continuous tangent vector field and
$a:\mathbb{R}\to\mathbb{R}$
is a continuous $T$-periodic function, $T>0$ given, with nonzero average
\[
\bar a:=\frac{1}{T}\int_0^T a(t)\,\mathrm{d}{t},
\]
and investigate via topological methods the structure of the set of harmonic
(i.e., $T$-periodic) solutions to perturbed equations of the form:
\[
\dot x=a(t)g(x)+\lambda\phi(x),\quad\lambda\geq 0,
\]
with $\phi:M\to\mathbb{R}^k$ a given tangent vector field.
Speaking loosely, we shall prove, under appropriate conditions, the existence
of a connected ``branch'' of $T$-periodic \emph{solution pairs} $(\lambda,x)$
of this equation, with the property that its closure is not contained in any
compact set and meets $g^{-1}(0)$ for $\lambda=0$.
Actually, the methods discussed in this paper shall let us treat withouth
any additional effort the more general case when the perturbation is allowed
to be time-dependent and periodic with the same period of $a$. In other
words, we shall consider, for $T>0$ given, the set of $T$-periodic solutions
to the following parametrized differential equation
\begin{equation}\label{zero}
\dot x=a(t)g(x)+\lambda f(t,x), \quad \lambda\geq 0,
\end{equation}
where $f:\mathbb{R}\times M\to\mathbb{R}^k$ and $g:M\to\mathbb{R} ^k$ are tangent
vector fields
on $M$, $a:\mathbb{R}\to\mathbb{R}$ and $f$ are $T$-periodic in $t$.
Therefore, our discussion will be applicable to the particular case of
periodic perturbations of autonomous ODEs. This corresponds,
in our notation, to $a(\cdot)$ constant (and nonzero). The more general
situation considered in this paper yields a generalization of the results
of \cite{FS1,FS2} in which $a(t)\equiv 1$.
As an application we provide a multiplicity result for equation \eqref{zero}
on compact boundaryless manifolds. Roughly speaking, we shall prove that when
$g$ has $n-1$ zeros at which the linearized unperturbed equation satisfies an
appropriate ``non-$T$-resonance'' condition, and the sum of the indices of
$g$ at these zeros differs from the Euler-Poincar\'e characteristic of $M$,
then \eqref{zero} has at least $n$ solutions of period $T$ for $\lambda>0$
sufficiently small. This fact will be proved via a combination of local
and global results about the set of $T$-periodic solutions of \eqref{zero}.
The multiplicity results so obtained are of topological nature: they could
not, in general, be deduced via implicit function or variational methods.
\section{Notation and preliminary results}
We begin by recalling some facts about the function spaces used in the
sequel. Let $M\subset\mathbb{R}^k$ be a differentiable manifold and $T>0$ a given
real number. The metric subspace $C_T(M)$ of $C_T(\mathbb{R}^k)$ consisting of
all the $T$-periodic continuous functions $x:\mathbb{R}\to M$ is not complete
unless $M$ is closed in $\mathbb{R}^k$. However, $C_T(M)$ is always locally
complete. This fact is a consequence of the following remark: since $M$ is locally
compact, given $x\in C_T(M)$, there exists a relatively compact open subset of $M$
containing the image $x([0,T])$ of $x$.
Note that, since $a(t)$ is not identically zero, a point $p\in M$
corresponds to a constant solution to \eqref{nope} if and only if $g(p)=0$.
This motivates the ensuing definitions.
Let $f$ and $g$ be as in \eqref{zero}. A pair
$(\lambda,p)\in [0,\infty)\times M$ is a \emph{starting point} (of
$T$-periodic solutions) if the Cauchy problem
\begin{equation}\label{due}
\dot x=a(t)g(x)+\lambda f(t,x)\,,\quad
x(0)=p
\end{equation}
has a $T$-periodic solution. A starting point $(\lambda,p)$ is \emph{trivial}
if $\lambda=0$ and $p\in g^{-1}(0)$.
Although the concept of starting point is essentially finite-dimensional,
there is an infinite-dimensional notion strictly correlated to it: that of
\emph{$T$-pair}. We say that a pair $(\lambda,x)\in [0,\infty)\times C_T(M)$
is a $T$-pair if $x$ satisfies \eqref{due}. If $\lambda=0$ and $x$ is
constant, then $(\lambda,x)$ is said \emph{trivial}.
Denote by $X\subset [0,\infty)\times C_T(M)$ the set of the $T$-pairs of
\eqref{due} and by $S\subset [0,\infty)\times M$ the set of the starting
points. Note that, as a closed subset of a locally complete space, $X$ is
locally complete.
One can show that, no matter whether or not $M$ is closed in $\mathbb{R}^k$, the
subset $X$ of $[0,\infty) \times C_T(M)$ consisting of all the $T$-pairs of
\eqref{zero} is always closed and locally compact. Moreover, by the
Ascoli-Arzel\`a Theorem, when $M$ is closed in $\mathbb{R}^k$, any bounded closed
set of $T$-pairs is compact.
\smallskip
As in \cite{FS2}, we tacitly assume some natural identifications. That is,
we will regard every space as its image in the following diagram of closed
embeddings:
\begin{equation}\label{e2.2}
\begin{CD}
[ 0,\infty) \times M @>>> [ 0,\infty ) \times C_T(M) \\
@AAA @AAA \\
M @>>> C_T(M),
\end{CD}
\end{equation}
where the horizontal arrows are defined by regarding any point $p$ in $M$ as
the constant map $\hat p(t) \equiv p$ in $C_T(M)$, and the two vertical
arrows are the natural identifications $p \mapsto (0,p)$ and $x\mapsto (0,x)$.
According to these embeddings, if $\Omega $ is an open subset of
$[0,\infty)\times C_T(M)$, by $\Omega \cap M$ we mean the open subset of $M$
given by all $p\in M$ such that the pair $(0,p)$ belongs to $\Omega$. If $U$
is an open subset of $[0,\infty)\times M$, then $U\cap M$ represents the open
set $\{p \in M : (0,p) \in U\}$.
Observe that any $p\in g^{-1}(0)$ can be seen --in the sense specified above-- as a
$T$-periodic solution of the unperturbed equation \eqref{nope}.
\begin{remark}\label{remuno} \rm
The map $h:X\to S$ given by $(\lambda,x)\mapsto\big(\lambda,x(0)\big)$ is
continuous and onto. Notice that, if $(\lambda,x)$ is trivial, then so is
$\big(\lambda,x(0)\big)$.
In case $f$ and $g$ are $C^1$, $h$ is also one to one. Furthermore, by the
continuous dependence on initial data, we get the continuity of
$h^{-1}:S\to X$. Clearly trivial solution pairs correspond to trivial
starting points under this homeomorphism.
\end{remark}
We now recall some basic facts about the topological degree of tangent
vector fields on manifolds and about the fixed point index.
Let $w: M \to \mathbb{R}^k$ be a continuous tangent vector field on $M$, and let
$V$ be an open subset of $M$ in which we assume $w$ admissible for the
degree, that is $w^{-1}(0)\cap V$ compact. Then, one can associate to the
pair $(w,V)$ an integer, $\deg(w,V)$, called the \emph{degree (or
characteristic) of the vector field} $w$ in $V$, which, roughly speaking,
counts (algebraically) the number of zeros of $w$ in $V$ (see e.g.\
\cite{H, Mi} and references therein). When $M=\mathbb{R}^k$, $\deg(w,W)$ is just the
classical Brouwer degree, $\deg(w,W,0)$, of $w$ at $0$ in any bounded open
neighborhood $W$ of $w^{-1}(0) \cap V$ whose closure is in $V$. Moreover,
when $M$ is a compact manifold, the celebrated Poincar\'e-Hopf Theorem
states that $\deg (w,M)$ coincides with the Euler-Poincar\'e characteristic
of $M$ and, therefore, is independent of $w$.
We recall that when $p$ is an isolated zero of $w$, the index $\mathop{\rm i}(w,p)$
of $w$ at $p$ is defined as $\deg (w,V)$, where $V$ is any isolating open
neighborhood of $p$. If $w$ is $C^1$ and $p$ is a non-degenerate zero
of $w$ (i.e.\ the Fr\'echet derivative $w'(p):T_pM \to \mathbb{R}^k$ is injective),
then $p$ is an isolated zero of $w$, $w'(p)$ maps $T_pM$ onto itself, and
$\mathop{\rm i}(w,p)=\mathop{\rm sign}\det w'(p)$ (see e.g.\ \cite{Mi}).
Let $V$ be an open subset of $M$, and let $\Psi:V\to M$ be continuous. The
map $\Psi$ is said to be admissible (for the fixed point index) on $V$ if its
set of fixed points is compact. In these conditions it is defined an integer,
called the \emph{fixed point index} of $\Psi$ in $V$ and denoted by
$\mathop{\rm ind}(\Psi,V)$, which satisfies all the
classical properties of the Brouwer degree: solution, excision, additivity,
homotopy invariance, normalization etc. A detailed exposition of this matter
can be found, for example, in \cite{N} and references therein. The following
fact deserves to be mentioned: if $M$ is an open subset of $\mathbb{R}^m$, then
$\mathop{\rm ind}(\Psi,V)$ is just the Brouwer degree of $I-\Psi$ in $V$ at $0$,
where $I-\Psi$ is defined by $(I-\Psi)(x)=x-\Psi(x)$.
Let $\gamma:\mathbb{R}\times M\to\mathbb{R}^k$ be a time-dependent tangent vector
field. We will denote by $P_\tau^\gamma$, $\tau\in\mathbb{R}$, the local (Poincar\'e)
$\tau$-translation operator associated to the equation
\begin{equation}\label{geneq}
\dot x=\gamma(t,x).
\end{equation}
One has $P^\gamma_\tau(p)=P^\gamma(\tau,p)$ where the map $P^\gamma:W\to M$
is defined on an open set $W\subset\mathbb{R}\times M$ containing $\{0\}\times M$,
with the property that, for any $p\in M$, the curve $t\mapsto P^\gamma(t,p)$
is the maximal solution of \eqref{geneq} such that $P^\gamma(0,p)=p$.
Therefore, given $\tau\in\mathbb{R}$, the domain of $P^\gamma_\tau$ is the open set
consisting of those points $p\in M$ for which the maximal solution of
\eqref{geneq}, starting from $p$ at $t=0$ is defined up to $\tau$.
Let $V$ be an open subset of $M$, and let $T>0$ be given. Assume that the
solutions of \eqref{nope} are defined in $[0,T]$ for any initial point
$p\in V$, and that $\mathop{\rm ind}\big(P_T^{ag},V\big)$ is well defined. This
clearly implies that $g^{-1}(0)$ is compact, thus $\deg(g,V)$ is defined as well.
\medskip\noindent\textbf{Notation.}
For the sake of simplicity, we shall often denote by $P_t(\lambda,\cdot)$
(instead of by $P^{ag+\lambda f}_t$) the $t$-translation operator associated
to \eqref{zero}.
\medskip
We shall make use of the following result of \cite{FS1}:
\begin{theorem}\label{T.2.1}
Let $\gamma :M\to\mathbb{R} ^k$ be a tangent vector field on a boundaryless
differentiable manifold $M\subset\mathbb{R} ^k$ and $V$ a relatively compact open
subset of $M$. Let $T>0$ be given and assume that, for any $p\in\overline{V}$,
the solution of the Cauchy problem
\[
\dot x=\gamma(x),\quad x(0)=p,
\]
is defined on $[0,T]$. If the translation operator $P^\gamma_T$ associated
to $\dot x=\gamma(x)$ is fixed point free on $\partial V$, then
\[
\mathop{\rm ind}(P^\gamma_T,V)=\deg(-\gamma,V).
\]
\end{theorem}
\begin{remark}\label{traslop} \rm
Let $g:M\to\mathbb{R}^k$ be a $C^1$ tangent vector field, and let
$a:\mathbb{R}\to\mathbb{R}$ be continuous, $T$-periodic with
$1/T\int_0^Ta(t)\,\mathrm{d}{t}=1$. Take any $p\in M$ and consider the Cauchy
problems
\begin{subequations}\label{traslop_p}
\begin{gather}\label{traslop_p1}
\dot x=g(x),\quad x(0)=p; \\
\label{traslop_p2}
\dot x=a(t)g(x),\quad x(0)=p\,.
\end{gather}
\end{subequations}
Denote by $x:I\to M$ and $\xi:J\to M$, $I\subset\mathbb{R}$ and $J\subset\mathbb{R}$
intervals, the (unique) maximal solution of \eqref{traslop_p1} and of
\eqref{traslop_p2} respectively. Clearly, if $\int_0^\tau a(s)\,\mathrm{d}{s}\in I$
for all $\tau\in[0,t]$, then
\[
\xi(t)=x\Big( \int_0^ta(s)\,\mathrm{d}{s}\Big);
\]
hence, $t\in J$.
Moreover, by a standard maximality argument, one can prove that $t\in J$
implies $\int_0^t a(s)\,\mathrm{d}{s}\in I$.
In particular, if $T\in J$, then $\int_0^Ta(s)\,\mathrm{d}{s}=T\in I$. When
this happens, one has $\xi(T)=x(T)$. In other words, if $P_T^{ag}(p)$ is
defined, then so is $P_T^g(p)$ and, in this case, $P_T^g(p)=P_T^{ag}(p)$.
Note also that when $a(t)>0$ for any $t\in [0,T]$ (or, equivalently,
$a(t)<0$ for any $t\in [0,T]$) the function $t\mapsto\int_0^t a(s)\,\mathrm{d}{s}$
is monotone, hence invertible. In particular, $T\in J$ is equivalent to
$T\in I$.
\end{remark}
Using Remark \ref{traslop} we obtain easily the following consequence of
Theorem \ref{T.2.1}.
\begin{corollary}\label{inpf}
Let $g:M\to\mathbb{R}^k$ be a $C^1$ tangent vector field, and let
$a:\mathbb{R}\to\mathbb{R}$ be
continuous, $T$-periodic with $1/T\int_0^Ta(t)\,\mathrm{d}{t}=1$. Given an open
subset $U$ of $M$, if $\mathop{\rm ind}(P_T^{ag},U)$ is well defined, then so is
$\mathop{\rm ind}(P_T^g,U)$ and
\[
\mathop{\rm ind}(P_T^{ag},U)=\mathop{\rm ind}(P_T^g,U)=\deg(-g,U).
\]
\end{corollary}
\begin{remark}\label{traslo1} \rm
Observe that if $p\in M$ is such that $p=P_T^{ag}(p)$, then any $q$ in the
image of the map $t\mapsto P_t^{ag}(p)$ is in the image of
$t\mapsto P_T^g(p)$. This means that it is an initial point of a $T$-periodic
orbit of $\dot x=g(x)$. Therefore $q$ has the property that
$q=P_T^g(q)=P_T^{ag}(q)$.
\end{remark}
\section{Main result}
Let $f:\mathbb{R} \times M\to \mathbb{R} ^k$, $g: M\to\mathbb{R} ^k$ and
$a:\mathbb{R}\to\mathbb{R}$ be as in \eqref{zero}. In the sequel, given
$X\subset\mathbb{R} \times M$ and $\lambda\in\mathbb{R}$,
we will denote the slice $\{x\in M:(\lambda,x)\in X\}$ by the symbol
$X_\lambda$.
By known properties of differential equations, the set
$V\subset [0,\infty)\times M$, given by
\[
V=\Big\{ (\lambda,p): \text{the solution $x(\cdot)$ of \eqref{zero}
satisfying $x(0)=p$ is defined in $[0,T]$}\Big\},
\]
is open. Thus it is locally compact. Clearly $V$ contains the set $S$ of all
starting points of \eqref{zero}. Observe that $S$ is closed in $V$, even if it
could be not so in $[0,+\infty)\times M$. Therefore $S$ is locally compact.
Let $U$ be an open subset of $V$. Since $S\cap U$ is open in $S$, it is
locally compact as well.
We will also use the following global connectivity result (see \cite{FP4}).
\begin{lemma}\label{L.3.1}
Let $Y$ be a locally compact metric space and let $Y_0$ be a compact subset of
$Y$. Assume that any compact subset of $Y$ containing $Y_0$ has nonempty
boundary. Then $Y\setminus Y_0$ contains a not relatively compact component
whose closure (in $Y$) intersects $Y_0$.
\end{lemma}
We now prove a result that, when $a$ is a nonzero constant, reduces to
Theorem 3.1 in \cite{FS1}.
\begin{theorem}\label{T.3.1}
Let $a:\mathbb{R}\to\mathbb{R}$ be a continuous function, and let
$f:\mathbb{R}\times M\to\mathbb{R} ^k$ and
$g:M\to\mathbb{R}^k$ be two $C^1$ tangent vector fields on the boundaryless manifold
$M\subset\mathbb{R}^k$. Assume also that $f$ and $a$ are $T$-periodic, and the
average $\bar a$ of $a$ is nonzero.
Denote by $S$ the set of the starting points for \eqref{zero} and let $U$ be
an open subset of $[0,\infty)\times M$. Assume that $\deg(g,U\cap M)$ is well
defined and nonzero.
Then the set $(S\cap U)\setminus\big(\{0\}\times g^{-1}(0)\big)$ of the nontrivial
starting points (in $U$) of \eqref{zero} admits a connected subset whose
closure in $S\cap U$ meets $\{0\}\times g^{-1}(0)$ and is not compact.
\end{theorem}
\begin{proof}
Since $\bar a\neq 0$, one has that
\[
\deg\Big(\frac{1}{\bar a}g(\cdot),U\cap M\Big)=
(\mathop{\rm sign}\bar a)^m\deg(g,U\cap M)\neq 0,
\]
where $m$ is the dimension of $M$. Hence, replacing if necessary $g$ with
$\bar a g$ and $a$ with $a/\bar a$, we shall assume
$1/T\int_0^T a(s)\,\mathrm{d}{s}=1$.
Since $\deg(g,U\cap M)\neq 0$, $\big(\{0\}\times g^{-1}(0)\big)\cap U$ is nonempty.
Thus $S\cap U$ is nonempty as well. The assertion follows applying Lemma
\ref{L.3.1} to the pair
\[
(Y,Y_0)=\Big(S\cap U , \big(\{0\}\times g^{-1}(0)\big)\cap U\Big).
\]
In fact, if $\Sigma$ is a component as in the assertion of Lemma \ref{L.3.1}
its closure (in $S\cap U$) meets $\{0\}\times g^{-1}(0)$ and is not compact. Assume
by contradiction that there exists a compact subset $C$ of $S\cap U$,
containing $\big(\{0\}\times g^{-1}(0)\big)\cap U$ and with empty boundary in
$S\cap U$. Thus, $C$ is a relatively open subset of $S\cap U$. As a
consequence, $S\cap U\setminus C$ is closed in $S\cap U$, so the distance,
$\delta=\mathop{\rm dist}(C,S\cap U\setminus C)$, between $C$ and
$S\cap U\setminus C$ is nonzero (recall that $C$ is compact). Consider the set
\[
W=\Big\{(\lambda,p)\in U:
\mathop{\rm dist}\big((\lambda,p),C\big)<\frac{\delta}{2}\Big\},
\]
that, clearly, does not meet $S\cap U\setminus C$.
For simplicity, given $s\in [0,+\infty)$, we put
\[
W_s=\big\{p\in M:(s,p)\in W\big\}.
\]
Because of the compactness of $S\cap W=C$, there exists $\lambda_0>0$ such
that $W_{\lambda_0}=\emptyset$. Moreover, the set
\[
\big\{(\lambda,p)\in W:P_T(\lambda,p)=p\big\}
\]
is compact. Then, from the generalized homotopy property of the fixed point
index (see e.g.\ \cite{N}),
\[
0=\mathop{\rm ind}\big(P_T(\lambda_0,\cdot),W_{\lambda_0}\big)=
\mathop{\rm ind}\big(P_T(\lambda,\cdot),W_\lambda\big),
\]
for all $\lambda\in [0,\lambda_0]$. Observe that our contradictory assumption
implies that $P_T^{ag}$ is fixed point free on the boundary of $W_0$,
therefore $\mathop{\rm ind}\big(P_T^{ag},W_0\big)$ is well defined. Applying the
excision property of the degree and Corollary \ref{inpf}, we get
\begin{align*}
\mathop{\rm ind}\big(P_T^{ag},W_0\big) &= \mathop{\rm ind}\big(P_T^g,W_0\big)\\
&= (-1)^m\deg(g,W_0)=(-1)^m\deg (g,U\cap M)\neq 0,
\end{align*}
contradicting the previous formula.
\end{proof}
We are now in a position to state and prove our main result. It is, basically,
an infinite-dimensional version of Theorem \ref{T.3.1} that, when $a$ is a
nonzero constant, reduces to Theorem 3.3 in \cite{FS2}.
\begin{theorem}\label{tuno}
Let $a:\mathbb{R}\to\mathbb{R}$ be a continuous function and let
$f:\mathbb{R}\times M\to\mathbb{R}^k$ and
$g:M\to\mathbb{R}^k$ be two continuous tangent vector fields on the boundaryless
manifold $M\subset\mathbb{R}^k$. Assume that $f$ and $a$ are $T$-periodic, with
average $\bar a\neq 0$. Let $\Omega$ be an open subset of
$[0,\infty)\times C_T(M)$, and assume that the degree $\deg(g,\Omega\cap M)$
is well-defined and nonzero.
Then there exists a connected set $\Gamma$ of nontrivial $T$-pairs in
$\Omega$ whose closure in $[0,\infty) \times C_T(M)$ meets $ g^{-1}(0)\cap\Omega$ and
is not contained in any compact subset of $\Omega$. In particular, if $M$ is
closed in $\mathbb{R}^k$ and $\Omega =[0,\infty )\times C_T(M)$, then $\Gamma$ is
unbounded.
\end{theorem}
\begin{proof}
As in the proof of Theorem \ref{T.3.1} we shall assume, without loss of
generality, that $1/T\int_0^T a(s)\,\mathrm{d}{s}=1$.
Let $X$ denote the set of $T$-pairs of \eqref{zero}. Since $X$ is
closed, it is enough to show that there exists a connected set $\Gamma$ of
nontrivial $T$-pairs in $\Omega$ whose closure in $X\cap\Omega$ meets
$ g^{-1}(0)$ and is not compact.
\smallskip
Assume first that $f$ and $g$ are smooth. Denote by $S$ the set of all
starting points of \eqref{zero}, and take
\[
\tilde S=\big\{ (\lambda ,p)\in S:\text{the solution of \eqref{due} is
contained in $\Omega$} \big\} .
\]
Obviously $\tilde S$ is an open subset of $S$, thus we can find an open
subset $U$ of $V$ such that $S\cap U=\tilde S$, where $V$ is the set
of all the pairs $(\lambda ,p)$ such that the solution of \eqref{due} is
defined in $[0,T]$. We have that
\[
g^{-1}(0)\cap\Omega = g^{-1}(0)\cap\tilde S= g^{-1}(0)\cap U,
\]
thus $\deg (g,U\cap M)=\deg (g,\Omega\cap M)\neq 0$. Applying Theorem
\ref{T.3.1}, we get the existence of a connected set
$\Sigma\subset\big(S\cap U\big)\setminus g^{-1}(0)$ such that its closure in
$S\cap U$ is not compact and meets $ g^{-1}(0)$. Let $h:X\to S$ be the map which
assigns to any $T$-pair $(\lambda ,x)$ the starting point
$\big(\lambda ,x(0)\big)$. By Remark \ref{remuno}, $h$ is a homeomorphism and
trivial $T$-pairs correspond to trivial starting points under $h$. This
implies that $\Gamma=h^{-1}(\Sigma)$ satisfies the requirements.
\medskip
Let us remove the smoothness assumption on $g$ and $f$. Take
$Y_0= g^{-1}(0)\cap\Omega$ and $Y=X\cap\Omega$. We have only to prove
that the pair $(Y,Y_0)$ satisfies the hypothesis of Lemma \ref{L.3.1}.
Assume the contrary. We can find a relatively open compact subset $C$ of
$Y$ containing $Y_0$. Thus there exists an open subset $W$ of $\Omega$ such
that the closure $\overline{W}$ of $W$ in $[0,\infty)\times C_T(M)$ is contained in
$\Omega$, $W\cap Y=C$ and $\partial W\cap Y=\emptyset$. Since $C$ is
compact and $[0,\infty)\times M$ is locally compact, we can choose $W$ in such
a way that the set
\[
\Big\{ \big(\lambda,x(t)\big) \in[0,\infty)\times M:(\lambda ,x)\in W,\;
t\in [0,T]\Big\}
\]
is contained in a compact subset $K$ of $[0,\infty)\times M$. This implies
that $W$ is bounded with complete closure in $\Omega$ and $W\cap M$ is a
relatively compact subset of $\Omega \cap M$. In particular $g$ is nonzero on
the boundary of $W\cap M$ (relative to $M$). By known approximation results,
there exist sequences $\{g_i\}$ of smooth tangent vector fields uniformly
approximating $g$ on $M$. For $i\in\mathbb{N}$ large enough, we get
\[
\deg (g_i,W\cap M)=\deg (g,W\cap M).
\]
Furthermore, by excision,
\[
\deg (g,W\cap M)=\deg (g,\Omega \cap M)\neq 0.
\]
Therefore, given $i$ large enough, the first part of the proof can be
applied to the equation
\begin{equation}
\dot x=a(t)g_i(x)+\lambda f_i(t,x), \label{duei}
\end{equation}
where $\{f_i\}$ is a sequence of smooth $T$-periodic tangent vector fields
uniformly approximating $f$ on $K$.
Let $X_i$ denote the set of $T$-pairs of \eqref{duei}. There exists a connected
subset $\Gamma_i$ of $\Omega\cap X_i$ whose closure in $\Omega$ meets
$g_i^{-1}(0)\cap W$ and is not contained in any compact subset of $\Omega $.
Let us prove that, for $i$ large enough, $\Gamma_i\cap\partial W\neq\emptyset$.
It is sufficient to show that $X_i\cap \overline{W}$ is compact. In fact, if
$(\lambda,x)\in X_i\cap \overline{W}$ we have, for any $t\in [0,T]$,
\[
\left\| \dot x(t)\right\| \leq \max \big\{ \| a(\tau)g(p)+\mu f(\tau
,p)\| :(\mu ,p)\in K\;,\;\tau \in [0,T]\big\} .
\]
Hence, by Ascoli's theorem, $X_i\cap \overline{W}$ is totally bounded and,
consequently, compact, since $X_i$ is closed and $\overline{W}$ is complete.
Thus, for $i$ large enough, there exists a $T$-pair
$(\lambda_i,x_i)\in\Gamma_i\cap\partial W$ of \eqref{duei}. Again by Ascoli's
theorem, we may assume that $x_i\to x_0$ in $C_T(M)$ and $\lambda
_i\to \lambda _0$ with $(\lambda _0,x_0)\in \partial W$. Therefore
\[
\dot x_0(t)=a(t)g\big(x_0(t)\big) +\lambda _0f\big( t,x_0(t)\big)
,\quad t\in \mathbb{R} .
\]
Hence $(\lambda_0,x_0)$ is a $T$-pair in $\partial W$. This contradicts the
assumption $\partial W\cap Y=\emptyset $.
It remains to prove the last assertion. Let $M$ be closed. There exists a
connected set $\Gamma$ of $T$-pairs of \eqref{zero} whose closure is
not compact and meets $ g^{-1}(0)$. We need to show that $\Gamma$ is unbounded.
Assume the contrary. As we already observed, when $M$ is closed any bounded
closed set of $T$-pairs is compact. Thus the closure of $\Gamma$ in
$[0,\infty)\times C_T(M)$ is compact. This yields a contradiction.
\end{proof}
Note that the connected set of $T$-pairs of Theorem \ref{tuno} can be
completely contained in the slice $\{0\}\times C_T(M)$, as in the
following simple example where $M=\mathbb{R}^2$, $T=2\pi$, $a(t)\equiv 1$ and
$\Omega=[0,\infty)\times C_{2\pi}(\mathbb{R}^2)$:
\[
\dot x=y,\,\quad
\dot y=-x+\lambda\sin t
\]
\begin{corollary}
Let $M\subset\mathbb{R}^k$ be a compact boundaryless manifold with $\chi(M)\neq 0$.
Take $a$, $g$ and $f$ as in Theorem \ref{tuno}. Then there exists an unbounded
connected set $\Gamma$ of $T$-pairs whose closure meets $ g^{-1}(0)$ and is such that
\begin{equation}\label{proj}
\pi_1(\Gamma)=[0,\infty),
\end{equation}
where $\pi_1$ denotes the projetion onto the first factor of
$[0,\infty)\times C_T(M)$.
\end{corollary}
\begin{proof}
Take $\Omega=[0,\infty)\times C_T(M)$, so that $\Omega\cap M=M$. By the
Poincar\'e-Hopf theorem
\[
\deg(g,\Omega\cap M)=\deg(g,M)=\chi(M)\neq 0.
\]
Theorem \ref{tuno} yields the existence of an unbounded connected set $\Gamma$
of $T$-pairs whose closure meets $ g^{-1}(0)$. Since $C_T(M)$ is bounded,
\eqref{proj} holds.
\end{proof}
\section{Applications to multiplicity results}
Note that in the previous section, where only ``global'' properties of the
set of $T$-pairs were studied, we merely require the average of the
function $a$ to be nonzero. In this section, where we look also at ``local''
behaviour, we will need to require explicitly that
\[
\frac{1}{T} \int_0^T a(s)\,\mathrm{d}{s}=1.
\]
As we have seen in the proof of Theorem \ref{T.3.1}, this can be assumed
without any loss of generality.
Below, we shall obtain a multiplicity result. In order to do that we will need
to consider also the behavior of the set of $T$-pairs near $ g^{-1}(0)$.
Loosely speaking, in order to find multiplicity results for the periodic
solutions of \eqref{zero} it is necessary to avoid the somehow degenerate
situation when the ``branch'' of $T$-pairs ``sticks'' to the manifold. We
first tackle this problem from an abstract viewpoint.
We need some notation. Let $Y$ be a metric space and $C$ a subset of
$[0,\infty) \times Y$. Given $\lambda \ge 0$, we denote by $C_{\lambda}$ the
slice $\big\{ y \in Y : (\lambda,y) \in C \big\}$. In what follows, $Y$
will be identified with the subset $\{0\}\times Y$ of $[0,\infty) \times Y$.
\begin{definition}[\cite{FPS}] \rm
Let $C$ be a subset of $[0,\infty) \times Y$. We say that a subset $A$ of
$C_0$ is an \emph{ejecting set} (for $C$) if it is relatively open in $C_0$
and there exists a connected subset of $C$ which meets $A$ and is not included
in $C_0$.
\end{definition}
We shall simply say that $q \in C_0$ is an \emph{ejecting point} if $\{q\}$ is
an ejecting set. In this case, $\{q\}$ being open in $C_0$, it is clearly
isolated in $C_0$.
In \cite{FPS} the following theorem which relates ejecting sets and
multiplicity results was proved.
\begin{theorem} \label{t2.4}
Let $Y$ be a metric space and let $C$ be a locally compact subset of
$[0,\infty) \times Y$. Assume that $C_0$ contains $n$ pairwise disjoint
ejecting sets, $n-1$ of which are compact. Then, there exists $\delta > 0$
such that the cardinality of $C_\lambda$ is greater than or equal to $n$ for
any $\lambda \in [0,\delta)$.
\end{theorem}
Let $p$ be a zero of $g$. We give a condition which ensures that $p$
(regarded as the trivial $T$-pair $(0,\hat p)$, where $\hat p$ denotes the
function $\hat p(t)\equiv p$) is an ejecting point for the set $X$ of the
$T$-pairs of \eqref{zero}.
\begin{definition} \rm
A point $p\in g^{-1}(0)$ is said \emph{$T$-resonant} provided that
\begin{enumerate}
\item $g$ is $C^1$ in a neighborhood of $p$;
\item the only $T$-periodic solution of the linearized equation at $p$ (on
$T_pM$)
\begin{equation}\label{lnrzzt}
\dot\xi=a(t)g'(p)\xi
\end{equation}
is trivial (i.e., $\xi(t)\equiv 0$).
\end{enumerate}
\end{definition}
\begin{remark} \rm
If $g$ is $C^1$ in a neighborhood of $p\in g^{-1}(0)$, the $T$-resonancy condition at
$p$ can be read on the spectrum $\sigma \big(g'(p)\big)$ of the endomorphism
$g'(p):T_pM\to T_pM$.
In fact, as one can easily check,
\[
\xi(t)=e^{\int_0^ta(s)\,\mathrm{d}{s}\, g'(p)}u
\]
is the solution of \eqref{lnrzzt} with initial condition
$\xi(0)=u$, $u\in T_pM$. Therefore, $u\in T_pM$ is a starting point for a
periodic solution of \eqref{lnrzzt} if and only if
\[
u\in\ker \Big(I-e^{Tg'(p)}\Big),
\]
where $I:T_pM\to T_pM$ denotes the identity (we are assuming
$1/T\,\int_0^Ta(t)\,\mathrm{d}{t}=1$). Thus $p$ is $T$-resonant if and only
if, for some $n\in\mathbb{Z}$
\[
\frac{2n\pi i}{T}\in\sigma\big(g'(p)\big).
\]
Observe also that if $p\in g^{-1}(0)$ is non-$T$-resonant then the fixed point index
of the Poincar\'e $T$-translation operators associated to the two following
linearized equations at $p$: $\dot y=a(t)g'(p)y$ and $\dot y=g'(p)y$, coincide
with $\mathop{\rm i} (-g,p)$.
\end{remark}
\begin{lemma}\label{isontres}
Assume that $g$ is $C^1$ in a neighborhood of a non-$T$-resonant $p\in g^{-1}(0)$.
Then, $p$ (regarded as a trivial $T$-pair) is an ejecting point for the set
$X$ of the $T$-pairs of \eqref{zero}.
\end{lemma}
\begin{proof}
Observe first that, since $p$ is non-$T$-resonant, it is an isolated zero
of $g$, and there exists a neighborhood $V$ of $p$ such that
$g^{-1}(0)\cap \overline{V}=\{p\}$ and
\[
\deg(g,V)=\mathop{\rm i}(g,p)=\mathop{\rm sign}\det g'(p)\neq 0.
\]
Therefore, taking
\[
\Omega=[0,\infty)\times C_T(V)\subset [0,\infty)\times C_T(M),
\]
one has $\deg(g,\Omega\cap M)=\deg(g,V)\neq 0$. Thus, Theorem \ref{tuno}
yields the existence of a connected set $\Gamma$ of $T$-pairs for \eqref{zero}
whose closure in $\Omega$ contains $p$ and is not compact.
We now prove that, for $V$ small enough and with compact closure
$\overline{V}$, no $T$-periodic solution to \eqref{nope} touches the boundary
$\partial V$ of $V$.
Assume by contradiction that this is not the case. Take a sequence $\{V_n\}$
of open neighborhoods of $p$ such that $\bigcap_{n\in\mathbb{N}}V_n=\{p\}$ and
$\overline{V_{n+1}}\subset V_n$ for all $n\in\mathbb{N}$. Then, there exists a
sequence $\{x_n\}$ of $T$-periodic solutions to \eqref{nope} with the property
that $x_n([0,T])\cap\partial V_n\neq\emptyset$. By Remark \ref{traslo1} we can
assume $x_n(0)\in\partial V_n$. Clearly, due to Remark \ref{traslop}, it is
also not restrictive to assume $x_n(0)\neq x_m(0)$ for $m\neq n$. Put
\[
p_n=x_n(0),\quad\text{and}\quad u_n=\frac{p_n-p}{|p_n-p|}.
\]
Clearly $p_n\to p$. We can assume $u_n\to u\in T_pM$.
Since $g$ is $C^1$, it is known that $P_T^{ag}(\cdot)$ is differentiable.
Define $\Phi:M\to\mathbb{R}^k$ by $\Phi(q)=q-P_T^{ag}(q)$. Clearly $\Phi$ is
differentiable and $\Phi(p_n)=0$, hence
\[
\Phi'(p)u=\lim_{n\to\infty}\frac{\Phi(p_n)-\Phi(p)}{|p_n-p|}=0.
\]
On the other hand, $\Phi'(p)v=v-[P_T^{ag}]'(p)v$ for any $v\in T_pM$. One can
easily verify that the map $\alpha:t\mapsto [P_t^{ag}]'(p)v$ satisfies the
following Cauchy problem
\[
\dot\alpha(t)=a(t)g'(p)\alpha(t)\,,\quad
\alpha(0)=u.
\]
Since $p$ is non-$T$-resonant, $\Phi'(p)u=\alpha(0)-\alpha(T)\neq 0$. This is
a contradiction.
We now prove that $p$ is an ejecting point for $X$. Clearly, if $\Gamma$ is
contained in $\{0\}\times C_T(M)$, then it must be contained into
$\{0\}\times C_T(\overline{V})$ since no $T$-periodic solution to
\eqref{nope} touches $\partial V$. Let us prove that this is impossible.
Assume the contrary. Then, $\Gamma$, as a bounded set of $T$-pairs is totally
bounded. Moreover, $\{0\}\times C_T(\overline{V})$ being complete, the
closure of $\Gamma$ is compact. This proves that $\Gamma$ cannot be contained
in $\{0\}\times C_T(M)$. The assertion follows.
\end{proof}
We are now in a position to establish a multiplicity result for forced
oscillations.
\begin{theorem}\label{multi0}
Let $M$ be a compact boundaryless manifold, and take continuous tangent vector
fields $f:\mathbb{R}\times M\to\mathbb{R} ^k$ and $g:M\to\mathbb{R}^k$, a
continuous function $a:\mathbb{R}\to\mathbb{R}$, and let $f$ and $a$ be
$T$-periodic with $1/T\int_0^Ta(t)\,\mathrm{d}{t}=1$. Then, if $g$ has $n-1$,
$n> 1$, non-$T$-resonant zeros $p_1$,\ldots,$p_{n-1}$ with
\[
\sum_{k=1}^{n-1}\mathop{\rm i}(p_k,g)\neq\chi(M),
\]
there are at least $n$ solutions of period $T$ of equation \eqref{zero} for
$\lambda$ sufficiently small.
\end{theorem}
\begin{proof}
Since $p_1$,\ldots,$p_{n-1}$ are non-$T$-resonant, there exist neighborhoods
$V_1$,\ldots,$V_{n-1}$ such that
\[
\overline{V_i}\cap g^{-1}(0)=\{p_i\}\quad\text{for $i=1,\ldots,n-1$}
\]
Clearly, by excision, $\deg(g,V_i)=\mathop{\rm i}(g,p_i)$, for $i=1,\ldots,n-1$.
Define
\[
V_0=M\setminus\bigcup_{i=1}^{n-1}\overline{V_i}.
\]
By the Poincar\'e-Hopf Theorem, $\deg(g,M)=\chi(M)$. The additivity
property of the degree yields
\[
\deg(g,V_0)=\chi(M)-\sum_{i=1}^{n-1}\mathop{\rm i} (p_i,g)\neq 0
\]
Define
\[
\Omega=[0,\infty)\times C_T(V_0)\subset[0,\infty)\times C_T(M).
\]
Theorem \ref{tuno} implies that $ g^{-1}(0)\cap V_0$ is an ejecting set of
the set of $T$-pairs for \eqref{zero}. The assertion now follows from
Lemma \ref{isontres} and Theorem \ref{t2.4}.
\end{proof}
In the following example we exibit a tangent vector field $g$ to the unit
sphere $S^2$ centered at the origin of $\mathbb{R}^3$ with the property that,
for any $T>0$, only one of its two zeros can be $T$-resonant. Theorem
\ref{multi0} implies that any small enough $T$-periodic perturbation of
equation
\[
\dot x=a(t)g(x),
\]
where $a:\mathbb{R}\to\mathbb{R}$ is any $T$-periodic continuous function with
average equal to $1$, has at least two $T$-periodic solutions.
\begin{example} \rm
Take $M=S^2\subset\mathbb{R}^3$ and let $g$ be the tangent vector field given by
\[
(x,y,z)\mapsto e^z\big(-xz,-yz,1-z^2\big).
\]
That is, $g$ is the gradient on the manifold $M=S^2$ of the functional
$(x,y,z)\mapsto e^z$.
Note that $g$ has the ``poles'' $\mathrm{N}=(0,0,1)$ and
$\mathrm{S}=(0,0,-1)$ as its only two zeros, and
$\sigma\big(g'(\mathrm{N})\big)=\{-e\}$ and
$\sigma\big(g'(\mathrm{S})\big)=\{e^{-1}\}$.
Then, for any $T>0$ for which $\mathrm{N}$ is $T$-resonant, $\mathrm{S}$ is
non-$T$-resonant. Consequently, for any $T>0$, any $T$-periodic
$a:\mathbb{R}\to[0,\infty)$ with $\bar a=1$, and any $T$-periodic
$f:\mathbb{R}\times M\to\mathbb{R}^3$, there exists $\lambda_0>0$ such that
\eqref{zero} admits two $T$-periodic solutions for $\lambda\in [0,\lambda_0)$.
\end{example}
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\end{document}