\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 $$\label{nope} \dot x=a(t)g(x),$$ 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 $$\label{zero} \dot x=a(t)g(x)+\lambda f(t,x), \quad \lambda\geq 0,$$ 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 $$\label{due} \dot x=a(t)g(x)+\lambda f(t,x)\,,\quad x(0)=p$$ 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: $$\label{e2.2} \begin{CD} [ 0,\infty) \times M @>>> [ 0,\infty ) \times C_T(M) \\ @AAA @AAA \\ M @>>> C_T(M), \end{CD}$$ 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 $$\label{geneq} \dot x=\gamma(t,x).$$ 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 $$\dot x=a(t)g_i(x)+\lambda f_i(t,x), \label{duei}$$ 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 $$\label{proj} \pi_1(\Gamma)=[0,\infty),$$ 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 $T_pM$) $$\label{lnrzzt} \dot\xi=a(t)g'(p)\xi$$ 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} \begin{thebibliography}{0} \bibitem{FP4}\textsc{M. Furi and M. P. Pera}; \textit{A continuation principle for periodic solutions of forced motion equations on manifolds and applications to bifurcation theory}, Pacific J. Math \textbf{160} (1993), 219-244. \bibitem{FPS}\textsc{M. Furi, M. P. Pera, and M. Spadini}; \textit{Forced oscillations on manifolds and multiplicity results for periodically perturbed autonomous systems}, Journal of computational and applied mathematics \textbf{113} (2000), 241--254. \bibitem{FS1}\textsc{M. Furi and M. Spadini}, \textit{On the fixed point index of the flow and applications to periodic solutions of Differential equations on manifolds}, Boll. 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