\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2001(2001), No. 36, pp. 1--9.\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 2001 Southwest Texas State University.}
\vspace{1cm}}
\begin{document}
\title[\hfilneg EJDE--2001/36\hfil Multiplicity for scalar
differential equations]
{Multiplicity of forced oscillations for scalar differential equations}
\author[M. Furi, M. P. Pera, \& M. Spadini\hfil EJDE--2001/36\hfilneg]
{Massimo Furi, Maria Patrizia Pera, \& Marco Spadini}
\address{ Massimo Furi, Maria Patrizia Pera, \& Marco Spadini\hfill\break
Dipartimento di Matematica Applicata ``G. Sansone'',
Via S. Marta 3, 50139 Firenze, Italy }
\email{furi@dma.unifi.it\quad pera@dma.unifi.it\quad
spadini@dma.unifi.it}
\date{}
\thanks{Submitted December 31, 2000. Published May 21, 2001.}
\subjclass[2000]{34C25, 34C60}
\keywords{Forced oscillations, ordinary differential equations,\hfill\break\indent
multiplicity of periodic solutions.}
\begin{abstract}
We give, via topological methods, multiplicity results for small
periodic perturbations of scalar second order differential
equations. In particular, we show that the equation
\[
\ddot{x} = g(x)+\varepsilon f(t,x,\dot x),
\]
where $g$ is $C^1$ and $f$ is
continuous and periodic in $t$, has $n$ forced oscillations,
provided that $g$ changes sign $n$ times ($n>1$).
\end{abstract}
\maketitle
\newenvironment{caso}[2]{\smallskip\noindent\textbf{Case~#1~}(\emph{#2}).~}{\smallskip}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\numberwithin{equation}{section}
\section{Introduction}
Despite the illusory simplicity of the equations considered, the problems of
existence and multiplicity of periodic solutions for periodically forced
second order scalar autonomous differential equations have been the subject
of extensive studies. Classical references for this topic are e.g.\ the
books \cite{Le,RSC,SC}. Currently, although the research is now often
pursued with different methods, the activity in this field is still vigorous
and it is impossible to give here a complete account even of the most recent
results in this field. We confine ourselves to mentioning some papers, as
for instance \cite{DIZ,DMM,FL,MRZ,Or,Za} along with their references, where
various problems of existence and multiplicity of forced oscillations are
investigated using different methods.
In this paper we consider parametrized second order differential equations on
$\mathbb{R}$ of the form (the simpler case when damping is present
was treated in
\cite{FPS2}):
\begin{equation}
\ddot{x} = g(x)+\lambda f(t,x,\dot x),\quad \lambda \geq 0,
\label{moto}
\end{equation}
where $g:\mathbb{R} \to \mathbb{R}$ and $f:\mathbb{R}^3 \to
\mathbb{R}$ are continuous. We are
interested in the \emph{forced oscillations} of \eqref{moto}; that is, those
periodic solutions with the same period as the forcing term $f$. We shall
prove (Theorem \ref{nosci1}) that if the function $g$ changes sign
$n$ times, $n>1$, then, for small values of $\lambda$, \eqref{moto} admits
\emph{at least} $n$ forced oscillations, provided that the uniqueness of
solutions of the Cauchy problem for the unperturbed equation $\ddot x = g(x)$
holds.
In the case $n = 1$ the result is true with an extra assumption on the
unperturbed equation (Theorem \ref{nosci}). Namely, the
non-$T$\mbox{-}isochronism of $\ddot x = g(x)$ is the key condition ensuring
the existence of forced oscillations for the perturbed equation \eqref{moto}.
Such a condition extends to the nonlinear case the non-$T$\mbox{-}resonance
hypothesis for the linear equation
\[
\ddot x = -ax + \lambda \sin \omega t \,.
\]
Theorems \ref{nosci} and \ref{nosci1} are obtained combining an analysis of
the periodic orbits of the unperturbed equation $\ddot x = g(x)$ with a
point-set topology result (Lemma \ref{wybgen}) which gathers some
previously known connectivity results (see e.g.\ \cite{Al, Ku} and \cite{FP}),
which are in the spirit of the so called Wyburn Lemma.
We note in passing that our results give only a \emph{lower bound} which might
be well below the actual number of forced oscillations of \eqref{moto}.
In fact, it follows from a theorem of \cite{DZ} that when
$\lim_{x \to \pm\infty}g(x)/x = +\infty$, $f$ depends only on $t$ and has a
minimal period $T>0$, then \eqref{moto} admits infinitely many
$T$\mbox{-}periodic solutions for any $\lambda>0$ (see also e.g.\
\cite{Mi,Mo} for the case $n = 1$).
The results described in this paper are not merely obtained by specializing
to $\mathbb{R}$ the techniques that were exhibited in \cite{FPS,FPS2}
for the more
general case of ODEs on manifolds. The spirit of Theorems \ref{nosci} and
\ref{nosci1} is indeed quite different from that of those papers which
depended essentially on a different kind of connectivity result (Proposition
\ref{oldcon}).
\section{Ejecting sets and $T$-pairs}
We will denote by $C_T^1(\mathbb{R})$ the Banach space of all the
$T$\mbox{-}periodic
$C^1$ maps $x:\mathbb{R} \to \mathbb{R}$ with the usual $C^1$ norm.
A pair $(\lambda,x) \in [0,\infty) \times C_T^1(R)$ is called a
\emph{$T$\mbox{-}pair for the second\mbox{-}order equation} \eqref{moto} if
$x$ is a solution of \eqref{moto} corresponding to $\lambda $. In particular
we will say that $(\lambda,x)$ is \textit{trivial} if $\lambda = 0$ and $x$ is
constant. Note that, in general, there may exist nontrivial $T$\mbox{-}pairs
of \eqref{moto} even for $\lambda = 0$.
One can show that the set of $T$\mbox{-}pairs of \eqref{moto} is
always a closed, locally compact subset of $[0,\infty) \times
C_T^1(\mathbb{R})$ (see
e.g.\ \cite{FP} or \cite{FS1}). Moreover, any bounded closed subset of
$T$\mbox{-}pairs is compact.
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{comdi}
\begin{CD}
\left[ 0,\infty \right) \times \mathbb{R} @>>> \left[ 0,\infty \right) \times
C^1_T(\mathbb{R})\\ @AAA @AAA \\
\mathbb{R} @>>> C^1_T(\mathbb{R}),
\end{CD}
\end{equation}
where the horizontal arrows are defined by regarding any point $p$
in $\mathbb{R}$ as the constant map $\hat p(t) \equiv p$ in
$C^1_T(\mathbb{R})$, and the two
vertical arrows are the natural identifications $p \mapsto (0,p)$ and $x
\mapsto (0,x)$.
According to these embeddings, we say that $Z \subset \left[
0,\infty \right) \times C^1_T(\mathbb{R})$ meets $K\subset\mathbb{R}$
if there exists a point
$p \in K$ such that the pair $(0,\hat p)$ belongs to $Z$. In this case we
will say that $p$ belongs to $Z$.
\medskip
We will make use of the following consequence of Corollary 4.4 in \cite{FS2}
regarding the global structure of the set of $T$\mbox{-}pairs of \eqref{moto}.
\begin{theorem}\label{indicenz}
If $g$ changes sign at some (isolated) zero $p \in \mathbb{R}$, then
\eqref{moto} admits a connected set of nontrivial $T$\mbox{-}pairs whose
closure meets $p$ and is either unbounded or intersects $g^{-1}(0) \setminus
\{p\}$.
\end{theorem}
\medskip According to the fact that a $T$-pair might have $\lambda = 0$, it
turns out that the connected set in Theorem \ref{indicenz} might be entirely
contained in the slice $\{ 0 \}\times C_T^1(\mathbb{R})$.
This happens for instance for the following equation (with $T = 2\pi$):
\[
\ddot x = -x+\lambda \sin t,\quad \lambda\geq 0.
\]
In order to find multiplicity results for the forced oscillations of
\eqref{moto}
we need to avoid such a ``degenerate'' situation. We tackle this problem from
an abstract viewpoint.
We need some notation. Let $S$ be a metric space and let $C$ be a subset of
$[0,\infty )\times S$. Given $\lambda \geq 0$, we denote by $C_{\lambda}$ the
slice $\big\{ y \in S \mid (\lambda,y) \in C \big\}$. In what follows, $S$
will be identified with the subset $\{0\}\times S$ of $[0,\infty)\times S$.
\begin{definition}
Let $C$ be a subset of $[0,\infty )\times S$. 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 $p \in C_0$ is an ejecting point if $\{p\}$ is an
ejecting set. In this case $p$ is clearly isolated in $C_0$.
Using compactness arguments, it is not difficult to show the following
lemma which relates ejecting sets and multiplicity results.
\begin{lemma} \label{conn2}
Let $S$ be a metric space and let $C$ be a locally compact subset of
$[0,\infty )\times S$. Assume that $C_0$ contains $n$ pairwise disjoint
compact ejecting sets for $C$. Then, there exists $\delta>0$ such that the
cardinality of $C_\lambda$ is greater than or equal to $n$, for $\lambda
\in [0,\delta)$.
\end{lemma}
In fact, in \cite{FPS} we proved the following stronger result.
\begin{proposition}\label{oldcon}
Let $S$ be a metric space and let $C$ be a locally compact subset of
$[0,\infty )\times S$. 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 $\lambda \in [0,\delta)$.
\end{proposition}
In \cite{FPS} we also provided examples showing that in the above proposition
the compactness assumption on $n-1$ ejecting sets cannot be dropped.
\medskip
In the sequel, we shall need an extension (Lemma \ref{wybgen} below) of
the following well-known connectivity result (see e.g.\ \cite{Al} and
\cite[chapter V]{Ku}).
\begin{lemma}\label{wyburn}
Let $Y$ be a compact Hausdorff space and let $Y_1$ and $Y_2$ be
disjoint closed subsets
of $Y$. Then either there exists a connected subset of
$Y\setminus\left( Y_1 \cup
Y_2 \right)$ whose closure intersects $Y_1$ and $Y_2$ or there exist disjoint
compact and open sets $K_1$ and $K_2$ in $Y$ such that $K_1 \supset Y_1$ and
$K_2\supset Y_2$.
\end{lemma}
The following result reduces to Lemma \ref{wyburn} when $n=2$
and $Y$ is compact. Besides, when $n=1$ one gets Lemma 1.4 of \cite{FP}.
\begin{lemma}\label{wybgen}
Let $Y$ be a locally compact Hausdorff topological space and let
$Y_1,\allowbreak \ldots,\allowbreak Y_n$,
$n\geq 1$, be pairwise disjoint compact subsets of $Y$. Then the
following alternative
holds:
\begin{enumerate}
\item
there exists $n$ pairwise disjoint compact open subsets $A_1,\ldots, A_n$
of $Y$ containing $Y_1,\ldots, Y_n$ respectively;
\item
there exists a connected set of $Y \setminus \bigcup_{i=1}^nY_i$ whose closure
in $Y$ meets $\bigcup_{i=1}^nY_i$ and has one of the following properties:
\begin{enumerate}
\item it is not compact;
\item meets at least two different $Y_i$'s.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
We distinguish two cases.
\begin{caso}{1}{$Y$ is compact}In this case the proof is by induction on $n$.
For $n=1$, the assertion holds clearly true since one can take
$A_1=Y$. If $n>1$
assume the assertion true for $n-1$. Apply Lemma \ref{wyburn} to the
two disjoint
closed subsets $Y_n$ and $\bigcup_{i=1}^{n-1}Y_i$ of $Y$. Then,
either there exists
a connected set whose closure meets $Y_n$ and one of the remaining $Y_i$'s, or
there exist disjoint compact open sets $K\supset Y_n$ and $H\supset
\bigcup_{i=1}^{n-1}Y_i$. In the latter case, the assertion follows applying
the inductive assumption to the $n-1$ sets $Y_1,\ldots, Y_{n-1}$ contained
in the compact
space $H$.
\end{caso}
\begin{caso}{2}{$Y$ is not compact}Let $\widehat{Y} = Y \cup \{\infty\}$ be
the one-point compactification of $Y$. Then, applying Case 1 to the Hausdorff
space $\widehat{Y}$ we have that either there exist $n+1$ pairwise disjoint
compact open neighborhoods of $Y_1,\ldots, Y_n,\{\infty\}$ (so that
alternative (1) of the assertion holds) or there exists a connected subset
$C$ of
\[
\widehat{Y}\setminus\Big(\bigcup_{i=1}^nY_i \cup \{\infty\}\Big)
\]
whose closure meets two different sets among $Y_1,\ldots,
Y_n,\{\infty\}$. In this
case, alternative (2) of the assertion holds. In fact, there are two
possibilities (not
mutually exclusive) corresponding to (a) and (b) of (2):
\begin{itemize}
\item the closure of $C$ meets $\{\infty\}$ and (at least) one of the
$Y_i$'s, and, thus, the closure of $C$ in $Y$ cannot be compact;
\item the closure of $C$ meets $Y_i$ and $Y_j$ for some $i\neq j$.
\end{itemize}
\end{caso}
The proof is now complete.
\end{proof}
\section{Application to second order scalar equations}
In this section we will be concerned with the scalar equation
\begin{equation}\label{eq.1}
\ddot x = g(x)+\lambda f(t,x,\dot x),\quad \lambda\geq 0,
\end{equation}
where $g: \mathbb{R} \to \mathbb{R}$ and $f: \mathbb{R}^3 \to
\mathbb{R}$ are continuous and $f$ is
$T$\mbox{-}periodic in $t$ (the period $T>0$ is given).
Consider the unperturbed equation
\begin{equation}\label{zerolev}
\ddot x = g(x).
\end{equation}
We shall always assume the uniqueness of the solutions of the Cauchy problems
associated with \eqref{zerolev}.
The potential energy is any primitive
$V$ of $-g$, while the total mechanical energy, which is a first integral for
\eqref{zerolev}, is
\[
E(s,v) = \frac{v^2}{2}+V(s).
\]
\begin{lemma}\label{img}
If $x(\cdot)$ is a periodic solution of \eqref{zerolev}, its image
$[\alpha_x,\beta_x]$ is such that
\begin{equation}\label{buca}
V(s)\leq V\big(\alpha_x\big) = V\big(\beta_x\big),
\end{equation}
for any $s \in [\alpha_x,\beta_x]$. Consequently, if $p$ is an isolated
zero of $g$ which is not a local minimum point of $V$, then there exists a
neighborhood $U$ of $p$ with the property that there are no periodic
solutions of \eqref{zerolev} with image in $U$ different from the constant
$\hat p(t)\equiv p$.
\end{lemma}
\begin{proof}
Let $ x $ be a periodic solution of period $T$ and let $t_0,t_1 \in [0,T]$ be
such that
$ x (t_0) = \alpha_ x $ and $ x (t_1) = \beta_ x $. Clearly, one has
$\dot x (t_0) = 0 = \dot x (t_1)$, therefore
\[
V\big(\alpha_ x \big) = E\big( x (t_0),\dot x(t_0) \big) =
E\big( x (t_1),\dot x (t_1)\big) = V\big(\beta_ x \big).
\]
This yields the inequality \eqref{buca}, since for any $t \in [0,T]$
\[
V\big(\alpha_ x \big) = V\big(\beta_ x \big) =
\frac{\big(\dot x (t)\big)^2}{2}+V\big( x (t)\big)\geq V\big( x (t)\big).
\]
Let us prove the last assertion. Let $p$ be an isolated zero of $g$ which is
not a local minimum point for $V$. Since $V'(s) = -g(s)$, there exists a
neighborhood $U$ of $p$ which does not contain minimum points of $V$. Then
the inequality \eqref{buca} yields the assertion.
\end{proof}
\begin{lemma}\label{npi}
Let $\mathcal{G} \subset C_T^1(\mathbb{R})$ be a connected set of solutions of
\eqref{zerolev} containing a zero $p$ of $g$ in which $g$ changes sign.
Then one has $\alpha_x \leq p \leq \beta_x$ for any $x \in \mathcal{G}$,
where $\alpha_x$ and $\beta_x$ are as in Lemma \ref{img}. Moreover, if one of
the two intervals
\[
A_p:=\{\alpha_x\mid x \in \mathcal{G}\}, \quad
B_p:=\{\beta_x\mid x \in \mathcal{G}\}
\]
is nontrivial, then the other is
nontrivial as well, and $V$ is decreasing on $A_p$ and increasing on
$B_p$. In addition, $A_p \cap B_p = \{p\}$.
\end{lemma}
\begin{proof}
If $p$ is not a local minimum point for $V$, Lemma \ref{img} implies that
$\mathcal{G}$ reduces to $\{p\}$, and the assertions hold trivially.
Assume, therefore, that $p$ is a local minimum point for $V$.
Let us prove first that $\alpha_x\leq p$ for any $x \in \mathcal{G}$.
Put $D = \{x \in \mathcal{G}\mid\alpha_x \leq p\}$. Clearly $D$ is nonempty
as it contains $p$, and is closed since $x\mapsto\alpha_x$ is continuous. Let
us show that $D$ is open in $\mathcal{G}$. It is enough to prove that, given
$\bar x \in D$, if $\alpha_{\bar x} = p$ then $\bar x$ lays in the interior of
$D$. Assume by contradiction $\bar x$ not in the relative interior of $D$ in
$\mathcal{G}$. Since $p$ is a local minimum point of $V$, the inequality
\eqref{buca} of Lemma \ref{img} yields $\beta_{\bar x} = p$, so that $\bar
x(t) \equiv p$. Let $\delta>0$ be such that $g(s)\neq 0$ in $(p,p+\delta)$.
The fact that $\bar x$ is not in the relative interior of $D$ implies the
existence of $x \in \mathcal{G}$, sufficiently close to $\bar x$, such that
$p < \alpha_x$. As $\bar x$ is constant, by the continuity of the map $x
\mapsto \beta_x$, we may assume that $\beta_x < p + \delta$. Since $V$ is
strictly increasing in $(p, p+\delta)$, again by \eqref{buca}, one has
$\alpha_x = \beta_x$. Therefore $x(t)$ is constantly equal to some constant
$q \in (p,p+\delta)$. Consequently $g(q)=0$, which is a contradiction. Thus
$D$ is open in $\mathcal{G}$, and $\alpha_x\leq p$ for any $x \in
\mathcal{G}$, as claimed. Analogously $p \le \beta_x$ for all $x \in
\mathcal{G}$.
Suppose now that $A_p$ does not reduce to $\{p\}$.
Let us prove first that $V$ is decreasing in $A_p$. Assume by
contradiction that this is not the case. Then there exist $a$, $b$
$ \in A_p$, $a < b$, such that $V(a) < V(b)$. Since
$V$ is $C^1$, there exist $s_1,s_2 \in (a,b)$, with $s_10$ in $[s_1,s_2]$. Consequently, given $x \in \mathcal{G}$ such
that $\alpha_x=s_1$, by Lemma \ref{img} one has $\beta_x=s_1$ as
well. Thus $x$ is a constant solution of \eqref{zerolev}. This implies
$V'(s_1)=0$, which is a contradiction.
The fact that if $B_p$ is nontrivial then $V$ is increasing in $B_p$
can be proved in an analogous way.
Let us show now that if $A_p$ is nontrivial, then so is
$B_p$. Since
$p$ is an isolated zero of
$g$ in which $V$ attains a local minimum, there exists a left
neighborhood of $p$ where $V$ is strictly decreasing. Consequently
there exists $x \in \mathcal{G}$ such that
$V(\alpha_x) = V(\beta_x)>V(p)$, and $B_p$ is nontrivial. A symmetric
argument shows that when $B_p$ is nontrivial so is $A_p$.
It remains to show that the interval $A_p \cap B_p$, which clearly
contains $p$, reduces to $\{p\}$. In fact, if this were not the case, the
function $V$ would be both decreasing and increasing in $A_p \cap B_p$,
contradicting the assumption that $g$ changes sign at $p$.
\end{proof}
We say that the equation \eqref{zerolev} is \emph{$T$\mbox{-}isochronous} if
all its solutions are $T$\mbox{-}periodic.
\begin{lemma}\label{tisoc}
Let $\mathcal{G}$, $p$, $A_p$ and $B_p$ be as in Lemma \ref{npi}.
Assume the equation \eqref{zerolev} non-$T$\mbox{-}isochronous. Then
$A_p \cup B_p\neq\mathbb{R}$.
\end{lemma}
\begin{proof}
Assume by contradiction $A_p \cup B_p = \mathbb{R}$. By Lemma \ref{npi}, $V$ is
decreasing on $A_p$ and increasing on $B_p$. Therefore $V$ attains its
minimum at $p$. Let $\xi:J\to\mathbb{R}$ be a maximal non-$T$\mbox{-}periodic
solution of \eqref{zerolev} (recall that equation \eqref{zerolev} is assumed
non-$T$\mbox{-}isochronous). Since $V$ is bounded from below, then $|\dot
\xi(\cdot)|$ is bounded. This clearly implies that both the interval $J$ and
the image of $\xi$ coincide with $\mathbb{R}$.
Denote by $K$ the total energy of $\xi$, i.e.\
\[
E\left(\xi(t),\dot\xi(t)\right) =
\frac{1}{2}[\dot\xi(t)]^2+V\big(\xi(t)\big) =
K,\quad\textrm{for all $t \in
\mathbb{R}$}.
\]
Let us show that the total energy of any $x \in \mathcal{G}$ is bounded
above by $K$. To this end, take $x \in \mathcal{G}$ and let $t_0 \in
\mathbb{R}$ be
such that $\dot x(t_0) = 0$. Since $\xi(\mathbb{R}) = \mathbb{R}$,
there exists $t_1 \in \mathbb{R}$
such that $\xi(t_1) = x(t_0)$. Therefore,
\[
K = \frac{1}{2}[\dot\xi(t_1)]^2+V\big(\xi(t_1)\big)\geq V\big(x(t_0)\big).
\]
Since $\dot x(t_0) = 0$, $V\big( x(t_0)\big)$ represents the total energy of
the solution
$x$. Consequently,
\[
E\left( x(t),\dot x(t)\right) = V\big( x(t_0)\big)\leq K,\quad\textrm{for
all $t \in \mathbb{R}$}
\]
as claimed.
Again the fact that $V$ is bounded from below implies the existence of a
constant $H$ such that $|\dot x(t)|\leq H$ for all $t \in
\mathbb{R}$. By integrating
and recalling that, as proved in Lemma \ref{npi}, $p$ belongs to the image of
$x$, one obtains
\[
|x(t)-p|\leq HT\quad\textrm{for all $t \in \mathbb{R}$}.
\]
Thus $A_p \cup B_p\subset [p-HT,p+HT]$, contradicting the assumption.
\end{proof}
\begin{lemma}\label{comparamo}
Assume that the $T$\mbox{-}isochronism
property does not hold for \eqref{zerolev}. Let $\mathcal{G} \subset
C^1_T(\mathbb{R})$ be a connected component of the set of
$T$-periodic solutions
of \eqref{zerolev} containing a zero $p$ of $g$ in which $g$ changes
sign. Then $\mathcal{G}$ is compact and does not meet any zeros of $g$
different from $p$.
\end{lemma}
\begin{proof}
Let us prove first that $\mathcal{G}$ is compact. Define, as before, the
intervals
\[
A_p:=\{\alpha_x\mid x \in \mathcal{G}\}, \quad
B_p:=\{\beta_x\mid x \in \mathcal{G}\}.
\]
Since, by assumption, \eqref{zerolev} is non-$T$\mbox{-}isochronous, by
Lemma \ref{tisoc} we have $A_p \cup B_p\neq\mathbb{R}$.
Since by Lemma \ref{npi} one has $A_p \cap B_p = \{p\}$, at least one of
the two intervals, say $B_p$, is bounded. Given any $x \in \mathcal{G}$, one
has
\[
E\big(x(t),\dot x(t) \big) = E(\beta_x,0) = V(\beta_x) \leq \sup
V(B_p) < +\infty \,.
\]
Therefore one has
\[
\left|\dot x(t)\right|\leq k:=\sqrt{2\big(\sup V(B_p)-\inf V(B_p)\big)},
\]
and, since $p \in x\big([0,T]\big)$,
\[
\left| x(t)-p\right|\leq Tk,\quad\textrm{for any $t \in [0,T]$.}
\]
Thus $\mathcal{G}$ is bounded in $C^1_T(\mathbb{R})$. The compactness of
$\mathcal{G}$ now follows from the fact that it is a closed subset of the set
of $T$-periodic solutions of \eqref{zerolev}.
Finally, if $q \in g^{-1}(0)$
belongs to $\mathcal{G}$, then $\alpha_q = \beta_q = q$. Consequently, by
Lemma
\ref{npi}, one has $q = p$.
\end{proof}
\medskip
The following theorem clears the way to the main result of this section.
\begin{theorem} \label{nosci}
Assume that in equation \eqref{eq.1} the force $g$ satisfies the
following conditions:
\begin{itemize}
\item
$g$ changes sign at $n$ zeros;
\item
the $T$\mbox{-}isochronism property does not hold.
\end{itemize}
Then there exists $\delta>0$ such that \eqref{eq.1} has at least $n$ forced
oscillations for $\lambda \in [0,\delta)$.
\end{theorem}
\begin{proof}
Let $p_1,\ldots, p_n$ be zeros at which $g$ changes sign. We shall
prove that there exist $n$ pairwise disjoint compact ejecting sets
for the set $X$ of $T$-pairs of \eqref{eq.1} containing
$p_1,\ldots, p_n$. Then, the assertion will follow from Lemma \ref{conn2}.
By Lemma \ref{comparamo} we get that alternative (2) in Lemma \ref{wybgen}
does not hold, therefore there exist $n$ pairwise disjoint compact and open
subsets $A_1,\ldots, A_n$ of the slice
\[
X_0 = \{x \in C_T^1(\mathbb{R}) \mid x \textrm{ is a solution of
\eqref{zerolev}}\}
\]
containing $p_1,\ldots, p_n$ respectively.
Since $g$ changes sign at the points $p_1,\ldots, p_n$, thus, by Theorem
\ref{indicenz}, for any $i=1,\ldots, n$, there exists a connected set
$\Gamma^i$ of nontrivial $T$\mbox{-}pairs for \eqref{eq.1} whose closure
$\overline{\Gamma^i}$ meets $p_i$ and is either non-compact or intersects
$g^{-1}(0)\setminus\{ p_i\}$. By Lemma \ref{comparamo} we get
$\overline{\Gamma^i} \not\subset X_0$. This implies that all the $A_i$'s
are ejecting sets for $X$.
\end{proof}
We observe that the non-$T$\mbox{-}isochronism assumption in Theorem
\ref{nosci} is not very restrictive.
In fact (see e.g.\ \cite{LS}), the unique odd continuous function $g$ for
which $T$\mbox{-}isochronism holds is
\[
g(s) = -\left(\frac{2\pi}{T}\right)^2s.
\]
See also \cite{Ur} along with the references therein, where a discussion on
isochronal oscillations around a zero of $g$ can be found.
If the function $g$ changes sign at least two times, then the potential
energy has a maximum point $p$. Thus, given a neighborhood $U$ of $p$, the
solutions of \eqref{zerolev} starting close to $p$ (at $t = 0$) remain in $U$
for all $t \in [0,T]$ (recall we are assuming the uniqueness of the solutions
of the Cauchy problem). Consequently, by Lemma \ref{img}, the equation
\eqref{zerolev} is not $T$-isochronous. We can therefore state our
main multiplicity result:
\begin{theorem}\label{nosci1}
Assume that in equation \eqref{eq.1} the force $g$ changes sign at $n>1$
zeros. Then there exists $\delta>0$ such that \eqref{eq.1} has at least $n$
forced oscillations for $\lambda \in [0,\delta)$.
\end{theorem}
\medskip
Observe that when, in equation \eqref{eq.1}, $g^{-1}(0)$ consists of isolated
points and $g$ changes sign at infinitely many zeros, Theorem \ref{nosci1}
implies that, given $n \in \mathbb{N}$, there exists $\delta_n > 0$ such that
\eqref{eq.1} admits at least $n$ forced oscillations for any $\lambda \in
[0,\delta_n)$. This does not mean that one necessarily has infinitely many
$T$\mbox{-}periodic solutions for small $\lambda>0$, as illustrated by the
following equation where we set $T=2\pi$
\begin{equation}\label{ultimoex}
\ddot x = \sin x+\lambda(x-\dot x).
\end{equation}
Clearly, due to the friction effect of the term $-\lambda\dot x$, the only
possible $T$\mbox{-}periodic solutions are the zeros of $\sin x + \lambda
x$. Thus \eqref{ultimoex} has a finite number of $T$\mbox{-}periodic
solutions for any $\lambda>0$.
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\end{document}