\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 62, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/62\hfil
Second order delay differential equations on manifolds]
{Forced oscillations for delay motion equations on manifolds}

\author[P. Benevieri, A. Calamai, M. Furi, M. P. Pera\hfil EJDE-2007/62\hfilneg]
{Pierluigi Benevieri, Alessandro Calamai, \\
Massimo Furi, Maria Patrizia Pera}  % in alphabetical order

\address{Pierluigi Benevieri \newline
Dipartimento di Matematica Applicata ``Giovanni Sansone''\\
Universit\`a degli Studi di Firenze \\
Via S. Marta 3 \\
I-50139 Firenze, Italy}
\email{pierluigi.benevieri@unifi.it}

\address{Alessandro Calamai \newline
Dipartimento di Scienze Matematiche \\
Universit\`a Politecnica delle Marche\\
Via Brecce Bianche \\
I-60131 Ancona, Italy}
\email{calamai@math.unifi.it, calamai@dipmat.univpm.it}

\address{Massimo Furi \newline
Dipartimento di Matematica Applicata ``Giovanni Sansone''\\
Universit\`a degli Studi di Firenze \\
Via S. Marta 3 \\
I-50139 Firenze, Italy}
\email{massimo.furi@unifi.it}

\address{Maria Patrizia Pera \newline
Dipartimento di Matematica Applicata ``Giovanni Sansone''\\
Universit\`a degli Studi di Firenze \\
Via S. Marta 3 \\
I-50139 Firenze, Italy}
\email{mpatrizia.pera@unifi.it}

\thanks{Submitted July 29, 2006. Published April 26, 2007.}
\subjclass[2000]{34K13, 37C25}
\keywords{Delay differential equations; Forced oscillations;
periodic solutions; \hfill\break\indent
compact manifolds; Euler-Poincar\'e
characteristic; fixed point index}

\begin{abstract}
 We prove an existence result for $T$-periodic solutions of a
 $T$-periodic second order delay differential equation on a
 boundaryless compact manifold with nonzero Euler-Poincar\'e
 characteristic. The approach is based on an existence result
 recently obtained by the authors for first order delay differential
 equations on compact manifolds with boundary.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction} \label{Introduction}

Let $M \subseteq \mathbb{R}^k$ be a smooth boundaryless manifold and let
\[
f: \mathbb{R} \times M \times M \to \mathbb{R}^k
\]
be a continuous map which is $T$-periodic in the first variable and tangent to
$M$ in the second one; that is,
\[
f(t+T,q,\tilde q) = f(t,q,\tilde q) \in T_qM, \quad \forall\,
(t,q,\tilde q) \in \mathbb{R} \times M \times M,
\]
where $T_qM \subseteq \mathbb{R}^k$ denotes the tangent space of $M$ at $q$.
Consider the following second order delay differential equation on $M$:
%
\begin{equation} \label{intro-eq}
x_\pi''(t) = f(t,x(t),x(t-\tau))-\varepsilon x'(t),
\end{equation}
%
where, regarding \eqref{intro-eq} as a motion equation,
%
\begin{enumerate}
\item
     $x''_\pi(t)$ stands for the tangential part
     of the acceleration $x''(t) \in \mathbb{R}\sp{k}$ at the point $x(t)$;
\item
     the frictional coefficient $\varepsilon$ is a positive real constant;
\item
$\tau > 0$ is the delay.
\end{enumerate}

In this paper we prove that equation \eqref{intro-eq} admits at least one
forced oscillation, provided that the constraint $M$ is compact with nonzero
Euler--Poincar\'e characteristic and that $T \geq \tau$. This generalizes a
theorem of the last two authors regarding the undelayed case (see
\cite{FuPe90}). Our result will be deduced from an existence theorem for
first order delay equations on compact manifolds with boundary recently
obtained by the authors (see \cite[Theorem 4.6]{BCFP1}). The possibility of
reducing \eqref{intro-eq} to the first order equation treated in \cite{BCFP1}
is due to the fact that any second order differential equation on $M$ is
equivalent to a first order system on the tangent bundle $TM$ of $M$. The
difficulty arising from the noncompactness of $TM$ will be removed by
restricting the search for $T$-periodic solutions to a convenient compact
manifold with boundary contained in $TM$. The choice of such a manifold is
suggested by \emph{a priori} estimates on the set of all the possible $T$-periodic
solutions of equation \eqref{intro-eq}. These estimates are made possible by
the compactness of $M$ and the presence of the positive frictional coefficient
$\varepsilon$.

We ask whether or not the existence of forced oscillations holds true even in
the frictionless case, provided that the constraint $M$ is compact with nonzero
Euler-Poincar\'e characteristic. We believe the answer to this question is
affirmative; but, as far as we know, this problem is still unsolved even in
the undelayed case.

An affirmative answer regarding the special case $M = S^2$ (the spherical
pendulum) can be found in \cite{FuPe91t} (see also \cite{FuPe93o} for the
extension to the case $M = S^{2n}$).

We point out that the assumption $T \geq \tau$ is crucial in this paper, since
our result is deduced from Theorem \ref{teo-ramo} below, whose proof, given in
\cite{BCFP1}, is based on the fixed point index theory for locally compact maps
applied to a Poincar\'e-type $T$-translation operator which is a locally
compact map if and only if $T \ge \tau$. In a forthcoming paper we will tackle
the case $0<T<\tau$, in which this operator is not even locally condensing.


\section{Second order delay differential equations on manifolds}
\label{secondorder}


Let, as before, $M$ be a compact smooth boundaryless manifold in $\mathbb{R}^k$.
Given $q\in M$, let $T_qM$ and $(T_qM)\sp{\perp}$ denote, respectively, the
tangent and the normal space of $M$ at $q$. Since $\mathbb{R}\sp{k} = T_qM \oplus
(T_qM)\sp{\perp}$, any vector $u \in \mathbb{R}\sp{k}$ can be uniquely decomposed into
the sum of the \emph{parallel} (or \emph{tangential}) \emph{component $u_\pi\in
T_qM$ of $u$ at $q$} and the \emph{normal component $u_\nu\in (T_qM)\sp{\perp}$
of $u$ at $q$}. By
\[
TM = \{ (q,v)\in \mathbb{R}\sp{k}\times \mathbb{R}\sp{k}:\; q\in M,\;v \in T_qM \}
\]
we denote the \emph{tangent bundle of $M$}, which is a smooth manifold
containing a natural copy of $M$ via the embedding $q \mapsto (q,0)$. The
natural projection of $TM$ onto $M$ is just the restriction (to $TM$ as
domain and to $M$ as codomain) of the projection of $\mathbb{R}\sp{k}\times \mathbb{R}\sp{k}$
onto the first factor.

Given, as in the Introduction, a continuous map $f: \mathbb{R} \times M \times M \to
\mathbb{R}^k$ which is $T$-periodic in the first variable and tangent to $M$ in the
second one, consider the following delay motion equation on $M$:
%
\begin{equation} \label{equ-g}
x_\pi''(t) = f(t,x(t),x(t-\tau))-\varepsilon x'(t),
\end{equation}
%
where
\begin{itemize}
\item[i)]
$x''_\pi (t)$ stands for the parallel component of the acceleration
$x''(t) \in \mathbb{R}\sp{k}$ at the point $x(t)$;
\item[ii)]
the frictional coefficient $\varepsilon$ and the delay $\tau$ are positive
real constants.
\end{itemize}

By a \emph{solution} of \eqref{equ-g} we mean a continuous function
$x: J \to M$, defined on a (possibly unbounded) real interval, with length
greater than $\tau$, which is of class $C^2$ on the subinterval
$(\inf J + \tau,\sup J)$ of $J$ and verifies
\[
x_\pi''(t) = f(t,x(t),x(t-\tau)) - \varepsilon x'(t)
\]
for all $t \in J$ with $t > \inf J + \tau$. A \emph{forced oscillation of
\eqref{equ-g}} is a solution which is $T$-periodic and globally defined on
$J = \mathbb{R}$.

It is known that, associated with $M \subseteq \mathbb{R}\sp{k}$, there exists a
unique smooth map $r: TM \to \mathbb{R}\sp{k}$, called the \emph{reactive force} (or
\emph{inertial reaction}), with the following properties:
\begin{itemize}
\item [(a)] $r(q,v)\in (T_qM)\sp{\perp}$ for any $(q,v)\in TM$;
\item [(b)] $r$ is quadratic in the second
variable;
\item [(c)] any $C\sp{2}$ curve $\gamma: (a,b) \to M$ verifies the condition
\[
\gamma''_{\nu}(t)=r( \gamma(t), \gamma'(t)), \quad \forall t\in (a,b),
\]
i.e., for each $t\in (a,b)$, the normal component $\gamma''_\nu(t)$ of
$\gamma''(t)$ at $\gamma(t)$ equals $r(\gamma(t), \gamma'(t))$.
\end{itemize}

The map $r$ is strictly related to the second fundamental form on $M$ and
may be interpreted as the reactive force due to the constraint $M$.

By condition (c) above, equation \eqref{equ-g} can be equivalently written as
%
\begin{equation} \label{secorcomplete}
x''(t)=r(x(t), x'(t))+f(t,x(t),x(t-\tau))-\varepsilon x'(t).
\end{equation}
%
Notice that, if the above equation reduces to the so-called \emph{inertial
equation}
\[
x''(t)=r(x(t),x'(t)),
\]
one obtains the geodesics of $M$ as solutions.

Equation \eqref{secorcomplete} can be written as a first order differential
system on $TM$ as follows:
\begin{gather*}
x'(t) = y(t)\\
y'(t) = r(x(t), y(t))+f(t,x(t),x(t-\tau))-\varepsilon y(t).
\end{gather*}
This makes sense since the map
%
\begin{equation} \label{defg}
g: \mathbb{R} \times TM \times M \to \mathbb{R}\sp k\times \mathbb{R}\sp k,
\quad g(t,(q,v),\tilde q) = (v,r(q,v)+f(t,q,\tilde q)-\varepsilon v)
\end{equation}
%
verifies the condition $g(t,(q,v),\tilde q) \in T_{(q,v)}TM$ for all
$(t,(q,v),\tilde q) \in \mathbb{R} \times TM \times M$ (see, for example, \cite{Fu}
for more details).

Theorem~\ref{teo-ramo} below, which is a straightforward consequence of
Theorem~4.6 in~\cite{BCFP1}, will play a crucial role in the proof of our
result (Theorem \ref{teo-esistenza}). Its statement needs some preliminary
definitions.

Let $X \subseteq \mathbb{R}^s$ be a smooth manifold with (possibly empty) boundary
$\partial X$. Following \cite{BCFP1}, we say that a continuous map
$F: \mathbb{R} \times X \times X \to \mathbb{R}^s$ is
\emph{tangent to $X$ in the second variable} or,
for short, that $F$ is a \emph{vector field (on $X$)} if
$F(t,p,\tilde p) \in T_p X$ for all
$(t,p,\tilde p) \in \mathbb{R} \times X \times X$.
A vector field $F$ will be said \emph{inward} (to $X$) if for any
$(t,p,\tilde p) \in \mathbb{R} \times \partial X \times X$ the vector
$F(t,p,\tilde p)$ points inward at $p$. Recall that, given
$p \in \partial X$, the set of the tangent
vectors to $X$ pointing inward at $p$ is a closed half-subspace of $T_pX$,
called \emph{inward half-subspace} of $T_pX$ (see e.g.\ \cite{Mi}) and here
denoted $T^-_pX$.

\begin{theorem} \label{teo-ramo}
Let $X \subseteq \mathbb{R}^s$ be a compact manifold with (possibly empty) boundary,
whose Euler--Poincar\'e characteristic $\chi(X)$ is different from zero. Let
$\tau > 0$ and let $F:\mathbb{R} \times X \times X \to \mathbb{R}^s$ be an inward vector field
on $X$ which is $T$-periodic in the first variable, with $T \geq \tau$. Then,
the delay differential equation
%
\begin{equation}
x' (t) = F(t,x(t),x(t-\tau))
\end{equation}
%
has a $T$-periodic solution.
\end{theorem}

The main result of this paper is the following.

\begin{theorem} \label{teo-esistenza}
Assume that the period $T$ of $f$ is not less than the delay $\tau$ and
that the Euler-Poincar\'e characteristic of $M$ is different from
zero. Then, the equation~\eqref{equ-g} has a forced oscillation.
\end{theorem}

\begin{proof}
As we already pointed out, the equation~\eqref{equ-g} is equivalent
to the following first order system on $TM$:
%
\begin{equation} \label{equ-sys}
\begin{gathered}
x'(t) = y(t) \\
y'(t) = r(x(t),y(t))+f(t,x(t),x(t-\tau))-\varepsilon y(t). \\
\end{gathered}
\end{equation}
%
Define $F: \mathbb{R} \times TM \times TM \to \mathbb{R}^k \times \mathbb{R}^k$ by
\[
F(t,(q,v),(\tilde q, \tilde v)) =
(v,r(q,v)+f(t,q,\tilde q)-\varepsilon v).
\]
Notice that the map $F$ is a vector field on $TM$ which is
$T$-periodic in the first variable.

Given $c>0$, set
\[
X_c = (TM)_c = \big\{ (q,v) \in M \times \mathbb{R}^k :
v \in T_q M,\; \|v\|\leq c \big\}.
\]
It is not difficult to show that $X_c$ is a compact manifold
in $\mathbb{R}^k \times \mathbb{R}^k$ with boundary
\[
\partial X_c =
\big\{ (q,v) \in M \times \mathbb{R}^k : v \in T_q M,\; \|v\| = c \big\}.
\]
Observe that
\[
T_{(q,v)}(X_c) = T_{(q,v)}(TM)
\]
for all $(q,v) \in X_c$.
Moreover, $\chi(X_c)=\chi(M)$ since $X_c$ and $M$ are homotopically
equivalent ($M$ being a deformation retract of $TM$).

We claim that, if $c>0$ is large enough, then $F$ is an inward vector
field on $X_c$.
To see this, let $(q,v) \in \partial X_c$ be fixed,
and observe that the inward half-subspace of $T_{(q,v)}(X_c)$ is
\[
T^-_{(q,v)}(X_c) =
\big\{(\dot q, \dot v) \in T_{(q,v)}(TM):
\langle v, \dot v \rangle \leq 0 \big\},
\]
where $\langle \cdot, \cdot \rangle$ denotes the inner product in
$\mathbb{R}^k$.
We have to show that if $c$ is large enough then
$F(t,(q,v),(\tilde q,\tilde v))$ belongs to $T^-_{(q,v)}(X_c)$ for any $t \in
\mathbb{R}$ and $(\tilde q,\tilde v)\in TM$; that is,
\[
\langle v,r(q,v)+f(t,q,\tilde q)-\varepsilon v \rangle =
\langle v,r(q,v) \rangle + \langle v,f(t,q,\tilde q) \rangle -
\varepsilon \langle v,v \rangle \leq 0
\]
for any $t \in \mathbb{R}$ and $(\tilde q,\tilde v)\in TM$.
Now, $\langle v,r(q,v) \rangle = 0$ since $r(q,v)$ belongs to
$(T_qM)\sp{\perp}$. Moreover, $\langle v,v \rangle = c^2$ since
$(q,v) \in \partial X_c$, and
\[
\langle v,f(t,q,\tilde q) \rangle \leq \|v\| \|f(t,q,\tilde q)\|
 \leq K \|v\|,
\]
where
\[
K = \max \big\{\|f(t,q,\tilde q)\|:
(t,q,\tilde q)\in \mathbb{R} \times M \times M \}.
\]
Thus,
\[
\langle v,r(q,v)+f(t,q,\tilde q)-\varepsilon v \rangle \leq
Kc-\varepsilon c^2.
\]
This shows that, if we choose $c>K/\varepsilon$, then $F$ is an inward vector
field on $X_c$, as claimed. Therefore, given $c>K/\varepsilon$, Theorem~\ref{teo-ramo}
implies that system~\eqref{equ-sys} admits a $T$-periodic solution in $X_c$,
and this completes the proof.
\end{proof}


It is evident from this proof that the result holds true even if we replace
\[
f(t,q,\tilde q)-\varepsilon v
\]
by a $T$-periodic force $g(t,(q,v),(\tilde q,\tilde v)) \in T_qM$
satisfying the following assumption:

\begin{quote}
There exists $c>0$ such that
$\langle g(t,(q,v),(\tilde q,\tilde v)), v \rangle \leq 0$
for any
\[
(t,(q,v),(\tilde q, \tilde v)) \in \mathbb{R} \times TM \times TM
\]
such that $\|v\| = c$.
\end{quote}

We point out that, in the above theorem, the condition $\chi(M) \neq 0$
cannot be dropped.
Consider for example the equation
%
\begin{equation} \label{equ-circ}
\theta''(t) = a - \varepsilon \theta'(t), \quad t \in \mathbb{R},
\end{equation}
%
where $a$ is a nonzero constant and $\varepsilon > 0$.
Equation~\eqref{equ-circ} can be regarded as a second order ordinary
differential equation on the unit circle $S^1 \subseteq \mathbb{C}$, where
$\theta$ represents an angular coordinate.
In this case, a solution $\theta(\cdot)$ of~\eqref{equ-circ} is
periodic of period $T>0$ if and only if for some $k\in \mathbb{Z}$ it
satisfies the boundary conditions
\begin{gather*}
\theta (T) - \theta (0) = 2k\pi, \\
\theta' (T) - \theta' (0) =0.
\end{gather*}
Notice that the applied force $a$ represents a nonvanishing autonomous
vector field on $S^1$. Thus, it is periodic of arbitrary period.
However, simple calculations show that there are no $T$-periodic
solutions of~\eqref{equ-circ} if $T \neq 2 \pi \varepsilon/a$.


\begin{thebibliography}{00}

\bibitem{BCFP1}
P.\ Benevieri, A.\ Calamai, M.\ Furi, and M.P.\ Pera,
\textsl{Global branches of periodic solutions for forced delay
differential equations on compact manifolds}, J. Differential
Equations {\bf 233} (2007), 404--416.

\bibitem{Fu}
M.\ Furi, \textsl{Second order differential equations on manifolds and
forced oscillations}, Topological Methods in Differential Equations
and Inclusions, A.\ Granas and M.\ Frigon Eds., Kluwer Acad.\ Publ.\
series C, vol.\ 472, 1995.

\bibitem{FuPe90}
M.\ Furi and M.P.\ Pera, \textsl{On the existence of forced oscillations
for the spherical pendulum}, Boll.\ Un.\ Mat.\ Ital.\ (7) {\bf 4-B}
(1990), 381--390.

\bibitem{FuPe91t}
M.\ Furi and M.P.\ Pera,
\textsl{The forced spherical pendulum does have forced oscillations}.
Delay differential equations and dynamical systems (Claremont, CA, 1990),
176--182, Lecture Notes in Math., 1475,
Springer, Berlin, 1991.

\bibitem{FuPe93o}
M.\ Furi and M.P.\ Pera, \textsl{On the notion of winding number for
closed curves and applications to forced oscillations on
even-dimensional spheres}, Boll.\ Un.\ Mat.\ Ital.\ (7), {\bf 7-A}
(1993), 397--407.

\bibitem{Mi}
Milnor J.W., \textsl{Topology from the differentiable
viewpoint}, Univ. press of Virginia, Charlottesville, 1965.

\end{thebibliography}

\end{document}