% This is example.tex a demonstration file that shows how to code
% an article for Vismath'97.
%
% It is based on the original demonstration file from Springer-Verlag
% for the LaTeX macro package LNCSE (Lecture Notes in Computational 
% Science and Engineering), version 1.0 for LaTeX2e.
 
\documentclass[runningheads]{lncse}
 
\usepackage{makeidx}  % allows for index generation
\usepackage{amsmath}  % if you want to use AMS macros
\usepackage{amsfonts} %
                      % older version would be:  \usepackage{amstex}
\usepackage{epsfig}   % facilitates inclusion of EPS figures 

\makeindex
\begin{document}

%-------------------------------------------------------------------------------
% title

\title{Entire and Meromorphic Solutions of Ordinary Differential Equations}
 

%-------------------------------------------------------------------------------
% abbreviated title (for running head):

\titlerunning{Solutions of Ordinary Differential Equations}
 
%-------------------------------------------------------------------------------
% define label for the first page of your article IF you want to reference 
% this article in another Vismath '97 article using \pageref{begin-...}
%
%\label{begin-yourname}         % uncomment and insert your name 
%                               % see also at the end of the bibliography

%-------------------------------------------------------------------------------
% authors

\author{Ivar Ekeland\inst{1} \and Roger Temam\inst{2}
        Jeffrey Dean \and David Grove \and Craig Chambers 
	\and Kim~B.~Bruce \and Elsa Bertino
       }

%-------------------------------------------------------------------------------
% abbreviated author list (for running head):

\authorrunning{Ivar Ekeland et al.}   
 
%-------------------------------------------------------------------------------
% addresses

\institute{
	   Princeton University, Princeton NJ 08544, USA
	   \and
	   Universit\'{e} de Paris-Sud,
	   Laboratoire d'Analyse Num\'{e}rique, B\^{a}timent 425,\\
	   F-91405 Orsay Cedex, France
	  }

%-------------------------------------------------------------------------------
% typeset the title of the contribution: 

\maketitle

%-------------------------------------------------------------------------------
% name all authors for the index:

\index{Ekeland@Ivar Ekeland}
\index{Dean@Jeffrey Dean}
\index{Grove@David Grove}
\index{Chambers@Craig Chambers}
\index{Bruce@Kim B. Bruce}
\index{Bertino@Elisa Bertino}

%-------------------------------------------------------------------------------
% abstract

\begin{abstract}
   The abstract should summarize the contents of the paper using at least 70
   and at most 150 words. It will be set in 9-point font size and be inset 1.0
   cm from the right and left margins.  There will be two blank lines before
   and after the Abstract. \dots
\end{abstract}

%-------------------------------------------------------------------------------
% your earth-shattering contribution

\section{Fixed-Period Problems: The Sublinear Case}

With this chapter, the preliminaries are over, and we begin the search for
periodic solutions to Hamiltonian systems. All this will be done in the convex
case; that is, we shall study the boundary-value problem
\begin{eqnarray*}
  \dot{x}&=&JH' (t,x)\\
  x(0) &=& x(T)
\end{eqnarray*}
with $H(t,\cdot)$ a convex function of $x$, going to $+\infty$ when
$\left\|x\right\| \to \infty$.


\subsection{Autonomous Systems}

In this section, we will consider the case when the Hamiltonian $H(x)$ is
autonomous. For the sake of simplicity, we shall also assume that it is
$C^{1}$.

We shall first consider the question of nontriviality, within the general
framework of $\left(A_{\infty},B_{\infty}\right)$-subquadratic Hamiltonians. In
the second subsection, we shall look into the special case when $H$ is
$\left(0,b_{\infty}\right)$-subquadratic, and we shall try to derive additional
information.

\subsubsection{The General Case: Nontriviality.}

We assume that $H$ is $\left(A_{\infty},B_{\infty}\right)$-sub\-qua\-dra\-tic
at infinity, for some constant symmetric matrices $A_{\infty}$ and
$B_{\infty}$, with $B_{\infty}-A_{\infty}$ positive definite. Set:
\begin{eqnarray}
\gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ B_{\infty} - A_{\infty} \\
  \lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \
  J \frac{d}{dt} +A_{\infty}\ .
\end{eqnarray}

Theorem~\ref{ghou:pre} tells us that if $\lambda +\gamma < 0$, the
boundary-value problem:
\begin{equation}
\begin{array}{rcl}
  \dot{x}&=&JH' (x)\\
  x(0)&=&x (T)
\end{array}
\end{equation}
has at least one solution
$\overline{x}$, which is found by minimizing the dual
action functional:
\begin{equation}
  \psi (u) = \int_{o}^{T} \left[\frac{1}{2}
  \left(\Lambda_{o}^{-1} u,u\right) + N^{\ast} (-u)\right] dt
\end{equation}
on the range of $\Lambda$, which is a subspace $R (\Lambda)_{L}^{2}$ with
finite codimension. Here
\begin{equation}
  N(x) := H(x) - \frac{1}{2} \left(A_{\infty} x,x\right)
\end{equation}
is a convex function, and
\begin{equation}
  N(x) \le \frac{1}{2}
  \left(\left(B_{\infty} - A_{\infty}\right) x,x\right)
  + c\ \ \ \forall x\ .
\end{equation}


\begin{proposition}
Assume $H'(0)=0$ and $ H(0)=0$. Set:
\begin{equation}
  \delta := \liminf_{x\to 0} 2 N (x) \left\|x\right\|^{-2}\ .
  \label{eq:one}
\end{equation}

If $\gamma < - \lambda < \delta$,
the solution $\overline{u}$ is non-zero:
\begin{equation}
  \overline{x} (t) \ne 0\ \ \ \forall t\ .
\end{equation}
\end{proposition}
%
\begin{proof}
Condition (\ref{eq:one}) means that, for every $\delta ' > \delta$, there is
some $\varepsilon > 0$ such that
\begin{equation}
  \left\|x\right\| \le \varepsilon \Rightarrow N (x) \le
  \frac{\delta '}{2} \left\|x\right\|^{2}\ .
\end{equation}

It is an exercise in convex analysis, into which we shall not go, to show that
this implies that there is an $\eta > 0$ such that
\begin{equation}
  f\left\|x\right\| \le \eta
  \Rightarrow N^{\ast} (y) \le \frac{1}{2\delta '}
  \left\|y\right\|^{2}\ .
  \label{eq:two}
\end{equation}

Critical points of a vector field can be classified according to there topology, 
s.\ Fig.~\ref{CRITICAL}, and play an important role for the overall structure of 
a vector field.  In the context of visualization, critical points have been analyzed 
by J.~Helman and L.~Hesselink, see \cite{helman:RDV,helman:VVF}.
%
\begin{figure}
\centerline{\epsfig{figure=critical.eps,width=1.0\linewidth}}
\caption{Different kinds of ciritical points in a 2D vector field
(sinks, saddle point, improper node, spiral point, center).\label{CRITICAL}}
\end{figure}

Since $u_{1}$ is a smooth function, we will have $\left\|hu_{1}\right\|_\infty
\le \eta$ for $h$ small enough, and inequality (\ref{eq:two}) will hold,
yielding thereby:
\begin{equation}
  \psi (hu_{1}) \le \frac{h^{2}}{2}
  \frac{1}{\lambda} \left\|u_{1} \right\|_{2}^{2} + \frac{h^{2}}{2}
  \frac{1}{\delta '} \left\|u_{1}\right\|^{2}\ .
\end{equation}

If we choose $\delta '$ close enough to $\delta$, the quantity
$\left(\frac{1}{\lambda} + \frac{1}{\delta '}\right)$ will be negative, and we
end up with
\begin{equation}
  \psi (hu_{1}) < 0\ \ \ \ \ {\rm for}\ \ h\ne 0\ \ {\rm small}\ .
\end{equation}

On the other hand, we check directly that $\psi (0) = 0$. This shows that 0
cannot be a minimizer of $\psi$, not even a local one.  So $\overline{u} \ne 0$
and $\overline{u} \ne \Lambda_{o}^{-1} (0) = 0$.\qed
\end{proof}
%
\begin{corollary}
Assume $H$ is $C^{2}$ and $\left(a_{\infty},b_{\infty}\right)$-subquadratic at
infinity. Let $\xi_{1},\allowbreak\dots,\allowbreak\xi_{N}$  be the equilibria,
that is, the solutions of $H' (\xi ) = 0$.  Denote by $\omega_{k}$ the smallest
eigenvalue of $H'' \left(\xi_{k}\right)$, and set:
\begin{equation}
  \omega : = {\rm Min\,} \left\{\omega_{1},\dots,\omega_{k}\right\}\ .
\end{equation}
If:
\begin{equation}
  \frac{T}{2\pi} b_{\infty} <
  - E \left[- \frac{T}{2\pi}a_{\infty}\right] <
  \frac{T}{2\pi}\omega
  \label{eq:three}
\end{equation}
then minimization of $\psi$ yields a non-constant $T$-periodic solution
$\overline{x}$.
\end{corollary}


We recall once more that by the integer part $E [\alpha ]$ of $\alpha \in
\bbbr$, we mean the $a\in \bbbz$ such that $a< \alpha \le a+1$. For instance,
if we take $a_{\infty} = 0$, Corollary 2 tells us that $\overline{x}$ exists
and is non-constant provided that:

\begin{equation}
  \frac{T}{2\pi} b_{\infty} < 1 < \frac{T}{2\pi}
\end{equation}
or
\begin{equation}
  T\in \left(\frac{2\pi}{\omega},\frac{2\pi}{b_{\infty}}\right)\ .
  \label{eq:four}
\end{equation}


\begin{proof}
The spectrum of $\Lambda$ is $\frac{2\pi}{T} \bbbz +a_{\infty}$. The largest
negative eigenvalue $\lambda$ is given by $\frac{2\pi}{T}k_{o} +a_{\infty}$,
where
\begin{equation}
  \frac{2\pi}{T}k_{o} + a_{\infty} < 0
  \le \frac{2\pi}{T} (k_{o} +1) + a_{\infty}\ .
\end{equation}
Hence:
\begin{equation}
  k_{o} = E \left[- \frac{T}{2\pi} a_{\infty}\right] \ .
\end{equation}

The condition $\gamma < -\lambda < \delta$ now becomes:
\begin{equation}
  b_{\infty} - a_{\infty} <
  - \frac{2\pi}{T} k_{o} -a_{\infty} < \omega -a_{\infty}
\end{equation}
which is precisely condition (\ref{eq:three}).\qed
\end{proof}


\begin{lemma}
Assume that $H$ is $C^{2}$ on $\bbbr^{2n} \setminus \{ 0\}$ and that $H'' (x)$
is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local minimizer
$\widetilde{x}$ of $\psi$ has minimal period $T$.
\end{lemma}
%
\begin{proof}
We know that $\widetilde{x}$, or $\widetilde{x} + \xi$ for some constant $\xi
\in \bbbr^{2n}$, is a $T$-periodic solution of the Hamiltonian system:
\begin{equation}
  \dot{x} = JH' (x)\ .
\end{equation}

There is no loss of generality in taking $\xi = 0$. So $\psi (x) \ge \psi
(\widetilde{x} )$ for all $\widetilde{x}$ in some neighbourhood of $x$ in
$W^{1,2} \left(\bbbr / T\bbbz ; \bbbr^{2n}\right)$.

But this index is precisely the index $i_{T} (\widetilde{x} )$ of the
$T$-periodic solution $\widetilde{x}$ over the interval $(0,T)$, as defined in
Sect.~2.6. So
\begin{equation}
  i_{T} (\widetilde{x} ) = 0\ .
  \label{eq:five}
\end{equation}

Now if $\widetilde{x}$ has a lower period, $T/k$ say, we would have, by
Corollary 31:
\begin{equation}
  i_{T} (\widetilde{x} ) =
  i_{kT/k}(\widetilde{x} ) \ge
  ki_{T/k} (\widetilde{x} ) + k-1 \ge k-1 \ge 1\ .
\end{equation}

This would contradict (\ref{eq:five}), and thus cannot happen.\qed
\end{proof}
%
\paragraph{Notes and Comments.}
The results in this section are a refined version of \cite{clar:eke}; the
minimality result of Proposition 14 was the first of its kind.

To understand the nontriviality conditions, such as the one in formula
(\ref{eq:four}), one may think of a one-parameter family $x_{T}$, $T\in
\left(2\pi\omega^{-1}, 2\pi b_{\infty}^{-1}\right)$ of periodic solutions,
$x_{T} (0) = x_{T} (T)$, with $x_{T}$ going away to infinity when $T\to 2\pi
\omega^{-1}$, which is the period of the linearized system at 0.

\begin{table}
\caption{This is the example table taken out of {\it The
         \TeX{}book,} p.\,246
	}
\centering
\begin{tabular}{r@{\quad}rl}
\hline
\multicolumn{1}{l}{\rule{0pt}{12pt}
                   Year}&\multicolumn{2}{l}{World population}\\[2pt]
\hline\rule{0pt}{12pt}
8000 B.C.  &     5,000,000& \\
  50 A.D.  &   200,000,000& \\
1650 A.D.  &   500,000,000& \\
1945 A.D.  & 2,300,000,000& \\
1980 A.D.  & 4,400,000,000& \\[2pt]
\hline
\end{tabular}
\end{table}
%
\begin{theorem} [Ghoussoub-Preiss]\label{ghou:pre}\quad\hskip-12pt
Assume $H(t,x)$ is
$(0,\varepsilon )$-subquadratic at
infinity for all $\varepsilon > 0$, and $T$-periodic in $t$
\begin{equation}
  H (t,\cdot )\ \ \ \ \ {\rm is\ convex}\ \ \forall t
\end{equation}
\begin{equation}
  H (\cdot ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \forall x
\end{equation}
\begin{equation}
  H (t,x)\ge n\left(\left\|x\right\|\right)\ \ \ \ \
  {\rm with}\ \ n (s)s^{-1}\to \infty\ \ {\rm as}\ \ s\to \infty
\end{equation}
\begin{equation}
  \forall \varepsilon > 0\ ,\ \ \ \exists c\ :\
  H(t,x) \le \frac{\varepsilon}{2}\left\|x\right\|^{2} + c\ .
\end{equation}

Assume also that $H$ is $C^{2}$, and $H'' (t,x)$ is positive definite
everywhere. Then there is a sequence $x_{k}$, $k\in \bbbn$, of $kT$-periodic
solutions of the system
\begin{equation}
  \dot{x} = JH' (t,x)
\end{equation}
such that, for every $k\in \bbbn$, there is some $p_{o}\in\bbbn$ with:
\begin{equation}
  p\ge p_{o}\Rightarrow x_{pk} \ne x_{k}\ .
\end{equation}
\end{theorem}
%
\begin{example} [{{\rm External forcing}}]
Consider the system:
\begin{equation}
  \dot{x} = JH' (x) + f(t)
\end{equation}
where the Hamiltonian $H$ is $\left(0,b_{\infty}\right)$-subquadratic, and the
forcing term is a distribution on the circle:
\begin{equation}
  f = \frac{d}{dt} F + f_{o}\ \ \ \ \
  {\rm with}\ \ F\in L^{2} \left(\bbbr / T\bbbz; \bbbr^{2n}\right)\ ,
\end{equation}
where $f_{o} : = T^{-1}\int_{o}^{T} f (t) dt$. For instance,
\begin{equation}
  f (t) = \sum_{k\in \bbbn} \delta_{k} \xi\ ,
\end{equation}
where $\delta_{k}$ is the Dirac mass at $t= k$ and $\xi \in \bbbr^{2n}$ is a
constant, fits the prescription. This means that the system $\dot{x} = JH' (x)$
is being excited by a series of identical shocks at interval $T$.
\end{example}
%
\begin{definition}
Let $A_{\infty} (t)$ and $B_{\infty} (t)$ be symmetric operators in
$\bbbr^{2n}$, depending continuously on $t\in [0,T]$, such that $A_{\infty} (t)
\le B_{\infty} (t)$ for all $t$.

A Borelian function $H: [0,T]\times \bbbr^{2n} \to \bbbr$ is called
$\left(A_{\infty} ,B_{\infty}\right)$-{\it sub\-qua\-dra\-tic at
in\-fi\-ni\-ty} if there exists a function $N(t,x)$ such that:
\begin{equation}
  H (t,x) = \frac{1}{2} \left(A_{\infty} (t) x,x\right) + N(t,x)
\end{equation}
\begin{equation}
  \forall t\ ,\ \ \ N(t,x)\ \ \ \ \
  {\rm is\ convex\ with\  respect\  to}\ \ x
\end{equation}
\begin{equation}
  N(t,x) \ge n\left(\left\|x\right\|\right)\ \ \ \ \
  {\rm with}\ \ n(s)s^{-1}\to +\infty\ \ {\rm as}\ \ s\to +\infty
\end{equation}
\begin{equation}
  \exists c\in \bbbr\ :\ \ \ H (t,x) \le
  \frac{1}{2} \left(B_{\infty} (t) x,x\right) + c\ \ \ \forall x\ .
\end{equation}

Here we would like to insert a color plate. However, only black-and-white
or grey-tone images are possible within the article. All color plates have to
appear in the appendix. Therefore we have to put it at the end and reference it 
not just by its number, but also by its page: see 
Fig.~\ref{NICE} (p.~\pageref{NICE} in the Appendix). 

If $A_{\infty} (t) = a_{\infty} I$ and $B_{\infty} (t) = b_{\infty} I$, with
$a_{\infty} \le b_{\infty} \in \bbbr$, we shall say that $H$ is
$\left(a_{\infty},b_{\infty}\right)$-subquadratic at infinity. As an example,
the function $\left\|x\right\|^{\alpha}$, with $1\le \alpha < 2$, is
$(0,\varepsilon )$-subquadratic at infinity for every $\varepsilon > 0$.
Similarly, the Hamiltonian
\begin{equation}
H (t,x) = \frac{1}{2} k \left\|k\right\|^{2} +\left\|x\right\|^{\alpha}
\end{equation}
is $(k,k+\varepsilon )$-subquadratic for every $\varepsilon > 0$.
Note that, if $k<0$, it is not convex.
\end{definition}
%
\paragraph{Notes and Comments.}
The first results on subharmonics were obtained by Rabinowitz in \cite{rab},
who showed the existence of infinitely many subharmonics both in the
subquadratic and superquadratic case, with suitable growth conditions on $H'$.
Again the duality approach enabled Clarke and Ekeland in \cite{clar:eke:2} to
treat the same problem in the convex-subquadratic case, with growth conditions
on $H$ only.

Recently, Michalek and Tarantello (see \cite{mich:tar} and \cite{tar}) have
obtained lower bound on the number of subharmonics of period $kT$, based on
symmetry considerations and on pinching estimates, as in Sect.~5.2 of this
article.

\clearpage

%-------------------------------------------------------------------------------
% Bibliography

% There are TWO alternatives: using BibTeX (preferred) or using \bibitem directly

% 1. BibTeX version: 
%
% If you want to choose this version uncomment the next two lines and insert the name of the BibTeX file.
%
% \bibliographystyle{vmams}     % choose bibliography style vmams.bst (file provided)
% \bibliography{example}        % use BibTeX file example.bib (BibTeX -> example.bbl)

% 2. \bibitem version:
%
\begin{thebibliography}{8}

\bibitem {clar:eke}
   {\sc F.~Clarke, I.~Ekeland},
   {\em Nonlinear oscillations and boundary-value problems for Hamiltonian systems},
   Arch. Rat. Mech. Anal. {\bf 78} (1982) 315--333.

\bibitem{clar:eke:2}
   {\sc F.~Clarke, I.~Ekeland},
   {\em Solutions p\'{e}riodiques, du p\'{e}riode donn\'{e}e, des \'{e}qua\-tions hamiltoniennes},
   Note CRAS Paris {\bf 287} (1978) 1013--1015.

\bibitem{fran}
   {\sc G.~Francis, J.~M. Sullivan, R.~B. Kusner, K.~A. Brakke, C.~Hartman, and
   G.~Chappell}, {\em The minimax sphere eversion}, Mathematics and Visualization 
   ({\sc K.~Polthier and H.-C.~Hege}, eds.), Springer Verlag, Heidelberg, 1997, 3--20.

\bibitem {helman:RDV}
   {\sc J.~Helman, L.~Hesselink},
   {\em Representation and display of vector field topology in fluid flow data sets},
   Visualization in scientific computing ({\sc G.~M.~Nielson and B.~D.~Shriver}, eds.),
   IEEE Computer Society, 1990, 61--73.

\bibitem {helman:VVF}
   {\sc J.~L.~Helman, L.~Hesselink},
   {\em Visualizing vector field topology in fluid flows"},
   IEEE Computer Graphics and Applications {\bf 11}:3 (1991), 36--46.

\bibitem {mich:tar}
   {\sc R.~Michalek, G.~Tarantello},
   {\em Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems},
   J. Diff. Eq. {\bf 72} (1988) 28--55.

\bibitem {tar}
   {\sc G.~Tarantello},
   {\em Subharmonic solutions for Hamiltonian systems via a $\bbbz_{p}$ pseudoindex theory},
   Annali di Matematica Pura (to appear).

\bibitem {rab}
   {\sc P.~Rabinowitz},
   {\em On subharmonic solutions of a Hamiltonian system},
   Comm. Pure Appl. Math. {\bf 33} (1980) 609--633.

\end{thebibliography}


%-------------------------------------------------------------------------------
% define label for the last page of your article IF you want to reference 
% this article in another Vismath '97 article using \pageref{end-...}
%
%\label{end-yourname}           % uncomment and insert your name 
%                               % see also at the beginning

%-------------------------------------------------------------------------------
% Color Plates

\newpage
{\large \bf Appendix: Color Plates}

\begin{figure}
\centerline{\epsfig{figure=squares.eps,width=0.9\linewidth}}
\caption{Put all your colour graphics in the appendix, so that the high costs
         for printing them can be kept within appropriate limits.
	 You may append two color plates, i.e. two full pages.  \label{NICE}
	}
\end{figure}

%-------------------------------------------------------------------------------

\end{document}
