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+% This is JNSL.DEM the demonstration file of
+% the LaTeX macro package from Springer-Verlag
+% for the Journal of Nonlinear Science
+% version of 16 July 1990
+\def\12{{1\ov 2}}
+\def\al{\alpha}
+\def\Aun{A_\un}
+\def\aun{a_\un}
+\def\bullet{\cdot}
+\def\Bun{B_\un}
+\def\bun{b_\un}
+\def\de{\delta}
+\def\dx{\dot x}
+\def\ep{\varepsilon}
+\def\fa{\forall}
+\def\for{{\rm for}}
+\def\Lai{\Lambda}
+\def\lb{\left[}
+\def\lg{\left\{}
+\def\degr{\hbox{$^\circ$}}
+\def\arcmin{\hbox{$^\prime$}}
+\def\arcsec{\hbox{$^{\prime\prime}$}}
+\def\liminfuu{{\rm lim inf}\,}
+\def\liminfu{\mathop{\vphantom{\tst\sum}\hbox{\liminfuu}}}
+\def\limsupuu{{\rm lim sup}\,}
+\def\limsupu{\mathop{\vphantom{\tst\sum}\hbox{\limsupuu}}}
+\def\lr{\left(}
+\def\lss{\left\|}
+\def\Min{{\rm Min\,}}
+\def\NN{\bbbn}
+\def\ol{\overline}
+\def\om{\omega}
+\def\ov{\over}
+\def\rb{\right]}
+\def\rg{\right\}}
+\def\RRn{\bbbr^{2n}}
+\def\RR{\bbbr}
+\def\rr{\right)}
+\def\rss{\right\|}
+\def\sm{\setminus}
+\def\tst{\textstyle}
+\def\tx{\wt x}
+\def\un{\infty}
+\def\wt{\widetilde}
+\def\ZZ{\bbbz}
+\documentstyle{jns}
+\begin{document}
+
+\title{Haupttitel 14 pt halbfett/Title boldface -- 14/16}
+\titlerunning{Expression of Cellular Oncogenes}
+\subtitle{Untertitel 10 pt halbfett/Subtitle boldface -- 10/11}
+
+\author{Ivar Ekeland\inst{1} and Roger Temam\inst{2}}
+\authorrunning{R. M\"uller}
+
+\institute{Princeton University, Princeton NJ 08544, USA
+\and
+Universit\'e de Paris-Sud,
+Laboratoire d'Analyse Num\'erique, B\^atiment 425,\\
+F-91405 Orsay Cedex, France}
+
+\date{Received June 5, 1989}
+
+\maketitle
+
+\begin{abstract}
+A new variant of the multi-grid algorithms is presented. It uses
+multiple coarse-grid corrections with particularly associated
+prolongations and restrictions. In this paper the robustness with
+respect to anisotropic problems is considered.
+
+\keywords multi-grid method -- coarse--grid correction --
+singular perturbation -- robustness.
+\end{abstract}
+
+\section{The Anisotropic Equation and Standard Multi-Grid Methods}
+\subsection{Introduction}
+Multi-grid methods are known as very fast solvers of a large class of
+discretised partial differential equations. However, the multi-grid
+method cannot be understood as a fixed algorithm. Usually, the
+components of the multi-grid iteration have to be adapted to the given
+problem and sometimes the problems are modified in order to make them
+acceptable for multi-grid methods. In particular, the smoothing
+iteration is the most delicated part of the multi-grid process.
+
+An iteration is called a {\em robust} one, if it works for a sufficient
+large class of problems. Attempts have been made to construct robust
+multi-grid iterations by means of sophisticated smoothing processes\dots
+\newpage
+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 $H(t,\bullet )$ a convex function of $x$, going to $+\un$ when
+$\lss x\rss \to \un$.
+
+\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 $\lr \Aun , \Bun\rr$-subquadratic Hamiltonians. In
+the second subsection, we shall look into the special case when $H$ is
+$\lr 0,\bun\rr$-subquadratic, and we shall try to derive additional
+information.
+\subsubsection{ The General Case: Nontriviality.}
+We assume that $H$ is $\lr \Aun , \Bun \rr$-sub\-qua\-dra\-tic at infinity,
+for some constant symmetric matrices $\Aun$ and $\Bun$, with $\Bun
+-\Aun$ positive definite. Set:
+\begin{eqnarray}
+\gamma :&=&{\rm smallest\ eigenvalue\ of}\ \ \Bun - \Aun \\
+\lambda : &=& {\rm largest\ negative\ eigenvalue\ of}\ \ J {d\ov dt}
++\Aun\ . \end{eqnarray}
+
+Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value
+problem:
+\begin{equation}\begin{array}{rcl} \dx&=&JH' (x)\\
+x(0)&=&x (T)\end{array}\end{equation}
+has at least one solution $\ol x$, which is found by minimizing the dual
+action functional:
+\begin{equation} \psi (u) = \int_o^T \lb \12 \lr \Lai_o^{-1} u,u\rr + N^\ast (-u)\rb
+dt\end{equation}
+
+\noindent on the range of $\Lai$, which is a subspace $R (\Lai )\sb L^2$ with
+finite codimension. Here
+\begin{equation} N(x) := H(x) - \12 \lr \Aun x,x\rr\end{equation}
+is a convex function, and
+\begin{equation} N(x) \le \12 \lr \lr \Bun - \Aun\rr x,x\rr + c\ \ \ \fa x\
+.\end{equation}
+
+\begin{proposition} Assume $H'(0)=0$ and $ H(0)=0$. Set:
+\begin{equation} \de := \liminfu_{x\to 0} 2 N (x) \lss x\rss^{-2}\ .
+\label{eq:one}
+\end{equation}
+
+If $\gamma < - \lambda < \de$, the solution $\ol u$ is non-zero:
+\begin{equation} \ol x (t) \ne 0\ \ \ \fa t\ .\end{equation}
+\end{proposition}
+\begin{proof} Condition (\ref{eq:one}) means that, for every $\de ' >
+\de$, there is some $\ep > 0$ such that
+\begin{equation} \lss x\rss \le \ep \Rightarrow N (x) \le {\de '\ov 2} \lss x\rss^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\lss x\rss \le \eta \Rightarrow N^\ast (y) \le {1\ov 2\de '} \lss
+y\rss^2\ .\label{eq:two}\end{equation}
+
+\begin{figure}
+\vspace{2.5cm}
+\caption{This is the caption of the figure displaying a white eagle and
+a white horse on a snow field}
+\end{figure}
+
+Since $u_1$ is a smooth function, we will have $\lss hu_1\rss_\un \le
+\eta$ for $h$ small enough, and inequality (\ref{eq:two}) will hold,
+yielding thereby:
+\begin{equation} \psi (hu_1) \le {h^2\ov 2} {1\ov \lambda} \lss u_1 \rss_2^2 + {h^2\ov 2}
+{1\ov \de '} \lss u_1\rss^2\ .\end{equation}
+
+If we choose $\de '$ close enough to $\de$, the quantity $\lr {1\ov \lambda}
++ {1\ov \de '}\rr$ will be negative, and we end up with
+\begin{equation} \psi (hu_1) < 0\ \ \ \ \ \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 $\ol u
+\ne 0$ and $\ol u \ne \Lai_o^{-1} (0) = 0$. \qed
+\end{proof}
+\begin{corollary} Assume $H$ is $C^2$ and $\lr \aun
+,\bun\rr$-subquadratic at infinity. Let
+$\xi_1,\allowbreak\dots,\allowbreak\xi_N$ be the
+equilibria, that is, the solutions of $H' (\xi ) = 0$. Denote by $\om_k$
+the smallest eigenvalue of $H'' \lr \xi_k\rr$, and set:
+\begin{equation} \om : = \Min \lg \om_1 , \dots , \om_k\rg\ .\end{equation}
+If:
+\begin{equation} {T\ov 2\pi} \bun < - E \lb - {T\ov 2\pi}\aun\rb < {T\ov
+2\pi}\om\label{eq:three}\end{equation}
+then minimization of $\psi$ yields a non-constant $T$-periodic solution
+$\ol x$.\end{corollary}
+
+We recall once more that by the integer part $E [\al ]$ of $\al \in
+\RR$, we mean the $a\in \ZZ$ such that $a< \al \le a+1$. For instance,
+if we take $\aun = 0$, Corollary 2 tells us that $\ol x$ exists and is
+non-constant provided that:
+
+\begin{equation} {T\ov 2\pi} \bun < 1 < {T\ov 2\pi}\end{equation}
+or
+\begin{equation} T\in \lr {2\pi\ov \om},{2\pi\ov \bun}\rr\ .
+\label{eq:four}\end{equation}
+
+\begin{proof} The spectrum of $\Lai$ is ${2\pi\ov T} \ZZ +\aun$. The
+largest negative eigenvalue $\lambda$ is given by ${2\pi\ov T}k_o +\aun$,
+where
+\begin{equation} {2\pi\ov T}k_o + \aun < 0\le {2\pi\ov T} (k_o +1) + \aun\
+.\end{equation}
+Hence:
+\begin{equation} k_o = E \lb - {T\ov 2\pi} \aun\rb \ .\end{equation}
+
+The condition $\gamma < -\lambda < \de$ now becomes:
+\begin{equation} \bun - \aun < - {2\pi\ov T} k_o -\aun < \om -\aun\end{equation}
+which is precisely condition (\ref{eq:three}).\qed
+\end{proof}
+
+\begin{lemma} Assume that $H$ is $C^2$ on $\RRn \sm \{ 0\}$ and
+that $H'' (x)$ is non-de\-gen\-er\-ate for any $x\ne 0$. Then any local
+minimizer $\tx$ of $\psi$ has minimal period $T$.\end{lemma}
+\begin{proof} We know that $\tx$, or $\tx + \xi$ for some constant $\xi
+\in \RRn$, is a $T$-periodic solution of the Hamiltonian system:
+\begin{equation} \dx = JH' (x)\ .\end{equation}
+
+There is no loss of generality in taking $\xi = 0$. So $\psi (x) \ge
+\psi (\tx )$ for all $\tx$ in some neighbourhood of $x$ in $W^{1,2} \lr
+\RR / T\ZZ ; \RRn\rr$.
+
+But this index is precisely the index $i_T (\tx )$ of the $T$-periodic
+solution $\tx$ over the interval $(0,T)$, as defined in Sect.~2.6. So
+\begin{equation} i_T (\tx ) = 0\ .\label{eq:five}\end{equation}
+
+Now if $\tx$ has a lower period, $T/k$ say, we would have, by Corollary
+31:
+\begin{equation} i_T (\tx ) = i_{kT/k}(\tx ) \ge ki_{T/k} (\tx ) + 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 $x_T$, $T\in
+\lr 2\pi\om^{-1}, 2\pi \bun^{-1}\rr$ of periodic solutions, $x_T (0) =
+x_T (T)$, with $x_T$ going away to infinity when $T\to 2\pi \om^{-1}$,
+which is the period of the linearized system at 0.
+
+\begin{table}
+\caption[ ]{Observational results from NGC 4827}
+\begin{flushleft}
+\renewcommand{\arraystretch}{1.2}
+\begin{tabular}{llllllllllllll}
+\hline
+ & & \multicolumn{3}{l}{RA (1950)} &\ & \multicolumn{3}{l}{Dec (1950)}
+ & $S$ & Pol & \% & $\log P$\\
+\cline{3-5}\cline{7-9}
+ & &(h) &(m) & (s) & & (\degr) & (\arcmin) & (\arcsec)
+ & (mJy) & (mJy) & & (W Hz$^{-1}$)\\
+\hline
+Core & (5 GHz) & 12 & 54 & 18.0 & & 27 & 26 & 56.2
+ & 8 & & & 21.64 \\
+Total&(327 MHz)& & & & & & &
+ & 210 & & & 23.13 \\
+ &(1.4 GHz)& & & & & & &
+ & 57 & 1.3 & 2 & 22.49 \\
+ & (5 GHz) & & & & & & &
+ & 26 & 0.73 & 3 & 22.15 \\
+\hline
+\end{tabular}
+\renewcommand{\arraystretch}{1}
+\end{flushleft}
+\end{table}
+\begin{theorem} [(Ghoussoub-Preiss)] Assume $H(t,x)$ is
+$(0,\ep )$-subquadratic at
+infinity for all $\ep > 0$, and $T$-periodic in $t$
+\begin{equation} H (t,\bullet )\ \ \ \ \ {\rm is\ convex}\ \ \fa t\end{equation}
+\begin{equation} H (\bullet ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \fa x
+\end{equation}
+\begin{equation} H (t,x)\ge n\lr \lss x\rss\rr\ \ \ \ \ {\rm with}\ \ n (s)s^{-1}\to
+\un\ \ {\rm as}\ \ s\to \un\end{equation}
+\begin{equation} \fa \ep > 0\ ,\ \ \ \exists c\ :\ H(t,x) \le {\ep\ov 2}\lss x\rss^2 +
+c\ .\end{equation}
+
+Assume also that $H$ is $C^2$, and $H'' (t,x)$ is positive definite
+everywhere. Then there is a sequence $x_k$, $k\in \NN$, of $kT$-periodic
+solutions of the system
+\begin{equation} \dx = JH' (t,x)\end{equation}
+such that, for every $k\in \NN$, there is some $p_o\in\NN$ with:
+\begin{equation} p\ge p_o\Rightarrow x_{pk} \ne x_k\ .
+\end{equation}
+\qed
+\end{theorem}
+\begin{example} [{\rm(External forcing)}] Consider the system:
+\begin{equation} \dx = JH' (x) + f(t)\end{equation}
+where the Hamiltonian $H$ is $\lr 0,\bun\rr$-subquadratic, and the
+forcing term is a distribution on the circle:
+\begin{equation} f = {d\ov dt} F + f_o\ \ \ \ \ {\rm with}\ \ F\in L^2 \lr \RR / T\ZZ
+; \RRn\rr\ ,\end{equation}
+where $f_o : = T^{-1}\int_o^T f (t) dt$. For instance,
+\begin{equation} f (t) = \sum_{k\in \NN} \de_k \xi\ ,\end{equation}
+where $\de_k$ is the Dirac mass at $t= k$ and $\xi \in \RRn$ is a
+constant, fits the prescription. This means that the system $\dx = JH'
+(x)$ is being excited by a series of identical shocks at interval $T$.
+\end{example}
+\begin{definition} Let $A_\un (t)$ and $B_\un (t)$ be symmetric
+operators in $\RRn$, depending continuously on $t\in [0,T]$, such that
+$A_\un (t) \le B_\un (t)$ for all $t$.
+
+A Borelian function $H: [0,T]\times \RRn \to \RR$ is called $\lr A_\un
+,B_\un\rr$-{\it subquadratic at infinity} if there exists a function
+$N(t,x)$ such that:
+\begin{equation} H (t,x) = \12 \lr A_\un (t) x,x\rr + N(t,x)\end{equation}
+\begin{equation} \fa t\ ,\ \ \ N(t,x)\ \ \ \ \ {\rm is\ convex\ with\ respect\ to}\
+\ x\end{equation}
+\begin{equation} N(t,x) \ge n\lr \lss x\rss\rr\ \ \ \ \ {\rm with}\ \ n(s)s^{-1}\to
++\un\ \ {\rm as}\ \ s\to +\un\end{equation}
+\begin{equation} \exists c\in \RR\ :\ \ \ H (t,x) \le \12 \lr B_\un (t) x,x\rr + c\ \
+\ \fa x\ .\end{equation}
+
+If $A_\un (t) = a_\un I$ and $B_\un (t) = b_\un I$, with $a_\un \le
+b_\un \in \RR$, we shall say that $H$ is $\lr a_\un
+,b_\un\rr$-subquadratic at infinity. As an example, the function $\lss x
+\rss^\al$, with $1\le \al < 2$, is $(0,\ep )$-subquadratic at infinity
+for every $\ep > 0$. Similarly, the Hamiltonian
+\begin{equation} H (t,x) = \12 k \lss k\rss^2 +\lss x\rss^\al\end{equation}
+is $(k,k+\ep )$-subquadratic for every $\ep > 0$. 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.
+
+\begin{thebibliography}{MT1}
+\bibitem[CE1]{clar:eke} Clarke, F., Ekeland, I.: Nonlinear oscillations
+and
+boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal.
+{\bf 78} (1982) 315--333
+\bibitem[CE2]{clar:eke:2} Clarke, F., Ekeland, I.: Solutions
+p\'eriodiques, du
+p\'eriode donn\'ee, des \'equations hamiltoniennes. Note CRAS Paris {\bf
+287} (1978) 1013--1015
+\bibitem[MT1]{mich:tar} Michalek, R., Tarantello, G.: Subharmonic
+solutions with
+prescribed minimal period for nonautonomous Hamiltonian systems. J.
+Diff. Eq. {\bf 72} (1988) 28--55
+\bibitem[Ta1]{tar} Tarantello, G.: Subharmonic solutions for Hamiltonian
+systems via a $\bbbz_p$ pseudoindex theory. Annali di Matematica Pura
+(to appear)
+\bibitem[Ra1]{rab} Rabinowitz, P.: On subharmonic solutions of a
+Hamiltonian
+system. Comm. Pure Appl. Math. {\bf 33} (1980) 609--633
+\end{thebibliography}
+\end{document}