\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
Tenth MSU Conference on Differential Equations and Computational
Simulations. \newline
\emph{Electronic Journal of Differential Equations},
Conference 23 (2016),  pp. 77--86.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document} \setcounter{page}{77}
\title[\hfilneg EJDE-2016/Conf/23 \hfil Multiplicity of solutions]
{Multiplicity of solutions of relativistic-type systems with
  periodic nonlinearities: a survey}

\author[J. Mawhin \hfil EJDE-2016/conf/23 \hfilneg]
{Jean Mawhin}

\address{Jean Mawhin \newline
Institut de Recherche en Math\'ematique et Physique,
Universit\'e Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium}
\email{jean.mawhin@uclouvain.be}


\thanks{Published March 21, 2016.}
\subjclass[2010]{34C15, 34C25, 58E05}
\keywords{Pendulum-type equations; multiple solutions; critical point theory;
\hfill\break\indent Ljusternik-Schnirelmann category}

\begin{abstract}
 We survey recent results on the multiplicity of $T$-periodic solutions
 of differential systems of the form
 $$
 \Big(\frac{u'}{\sqrt{1 - |u'|^2}}\Big)' + \nabla_u F(t,u) = e(t)
 $$
 when $F(t,u)$ is $\omega_i$-periodic with respect to $u_i$ $(i = 1,\ldots,N)$.
 Several techniques of critical point theory are used.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{remark}[theorem]{Remark}
%\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Motivation: the forced pendulum and corresponding systems}

 The  periodic problem for the forced pendulum equation
 \begin{equation}\label{fpe}
u'' + a \sin u = e(t), \quad u(0) = u(T),\quad  u'(0) = u'(T)
\end{equation}
has been for almost one century a source of inspiration for ordinary 
differential equations and nonlinear functional analysis, and  
a cornerstone for  most nonlinear techniques (see e.g. \cite{Maw0,Maw3}). 
In particular its solutions  are the critical points of the Lagrangian 
action functional
 $$
\mathcal{L}(u) := \int_0^T \Big[\frac{{u'}^2}{2} + a \cos u + eu \Big]\,dt
$$
 in the Sobolev space $H^1_T = \{u \in H^1([0,T]) : u(0) = u(T)\}$.

In 1922,  Hamel \cite{Ham} proved that \emph{for each $e \in C([0,T])$  such that
$$
\overline e := T^{-1}\int_0^T e(t)\,dt = 0,
$$
there exists at least one solution of \eqref{fpe} minimizing  $\mathcal{L}(u)$
over  $T$-periodic $C^1$ functions.} 
This result was rapidly forgotten and, following a renewal of interest 
for the problem, in 1980, due to Castro's application \cite{Cas} of 
some minimax method to \eqref{fpe}, Hamel's theorem was rediscovered 
independently around 1981 by Willem \cite{Wil} and Dancer \cite{Dan} 
in the more natural framework of $H^1_T$. Because of the structure of the equation, 
if $u$ is a solution, the same is true for $u + 2k\pi$ for all $k \in \mathbb{Z}$,
so that two solutions of \eqref{fpe} are called 
\emph{geometrically distinct} if they do not differ by an integer  
multiple of $2\pi$.

As shown by the special case of the unforced pendulum, Hamel's existence 
conclusion is not optimal and, in
 1984,  the following multiplicity result was proved in \cite{MaW1}.

\begin{theorem}\label{mawi1}
 For each $e \in L^1(0,T)$ such that $\overline e = 0,$  problem \eqref{fpe} 
has at least two  geometrically distinct solutions.
 \end{theorem}

The second solution was obtained by a mountain pass type argument between 
a minimizing solution $u_0$ and the other one $u_0 + 2\pi$. 
The unforced case shows that this multiplicity result is optimal 
if no restriction is  made upon $a$ and $T$. An immediate generalization 
of Theorem \ref{mawi1}, based upon the same arguments, holds for $a \sin u$ 
replaced by a Carath\'eodory function $f(t,u)$ such that 
$F(t,u) := \int_0^u f(t,s)\,ds$ is $\omega$ periodic in $u$ for a.e. 
fixed  $t \in [0,T]$, and some $\omega > 0$.

The solutions of the  $N$-dimensional corresponding problem
\begin{equation}\label{fps}
u'' + \nabla_u F(t,u) = e(t), \quad u(0) = u(T), \quad u'(0) = u'(T),
\end{equation}
 where $e \in L^1(0,T;\mathbb R^N)$, $F : [0,T] \times \mathbb{R}^N \to \mathbb{R}$ and
$\nabla_uF : [0,T] \times \mathbb{R}^N \to \mathbb{R}^N$ are Carath\'eodory functions such that
 \begin{equation}\label{periodicity}
 F(t,u + \omega_je_j) = F(t,u)\quad (j = 1,\ldots,N)
 \end{equation}
 for a.e. $t \in [0,T]$, all $u \in \mathbb{R}^N$, and some $\omega_i > 0$
$(i = 1,\ldots,N)$, are the critical points of the Lagrangian action functional
 $$ 
\mathcal{L}_N(u) := \int_0^T \big[\frac{|u'|^2}{2} - F(t,u) + (e|u)\big]\,dt
$$
  in the Sobolev space $H^1_T = \{u \in H^1([0,T],\mathbb{R}^N) : u(0) = u(T)\}$.
Here and in the whole paper, $(\cdot | \cdot)$ denotes the inner product in 
$\mathbb{R}^N$ and $|\cdot|$ the corresponding norm. In 1984, the following result was
proved in \cite{MaW2}.

\begin{theorem}\label{mawi2}
If $F$ satisfies assumption \eqref{periodicity}, then, for each 
$e \in L^1(0,T;\mathbb{R}^N)$ such that $\overline e = 0$, problem \eqref{fps}
has at least two geometrically distinct solutions.
\end{theorem}

\emph{Geometrically distinct} solutions of \eqref{fps} are of course solutions 
whose differences are not of the form $\sum_{i=1}^N k_i\omega_i$ for some 
$(k_1,\ldots,k_N) \in \mathbb{Z}^N$. The proof of Theorem \ref{mawi2} is an
easy extension of the argument of the scalar case. Such a multiplicity 
result is not optimal, as easily seen, and was improved around 1988 
independently by Rabinowitz \cite{Rab},   Chang \cite{Cha},  and  the author 
\cite{Maw1}, who got the following multiplicity conclusion.

\begin{theorem}\label{cmr}
If $F$ satisfies assumption \eqref{periodicity}, then, for each 
$e \in L^1(0,T;\mathbb{R}^N)$ such that $\overline e = 0$, problem \eqref{fps} has at least
 $N+1$ geometrically distinct solutions.
\end{theorem}

Although they present technical differences, the three proofs of this result
 use the fact that  $\mathcal{L}_N(u + \omega_j e^j) 
= \mathcal{L}_N(u)$ $(j = 1,\ldots,n)$ and some  Ljusternik-Schnirelmann 
category arguments.


\section{The relativistic forced pendulum and corresponding systems}

In 2010, it was shown in \cite{BrM1} that the solutions of the `relativistic 
forced pendulum equation', i.e. the solutions of the problem
\begin{equation}\label{rfpe}
\Big(\frac{u'}{\sqrt{1 - u'^2}}\Big)' + a \sin u = e(t), \quad
 u(0) = u(T), \quad u'(0) = u'(T),
\end{equation}
namely the functions $u$ of class $C^1$ on $C([0,T]$ such that 
$\|u'\|_\infty < 1$, $\frac{u'}{\sqrt{1 - u'^2}}$ is absolutely continuous on
 $[0,T]$ and which verify the differential equation in \eqref{rfpe} almost 
everywhere, and the periodic boundary conditions, can be associated to 
the critical points of the action defined by
$$
\mathcal{R}(u) := \int_0^T \big[1 - \sqrt{1 - |u'|^2} + a \cos u + eu \big]\,dt
$$
on the closed convex set
$$
K = \{u \in W^{1,\infty}([0,T]) : u(0) = u(T),\; \|u'\|_\infty \leq 1\},
$$
where $\|\cdot\|_\infty$ denotes the $L^\infty$-norm. In 2011, it was shown 
in \cite{BJM} that those solutions could be seen as well be associated 
to the critical points in the sense of Szulkin \cite{Szu1} of the functional 
given on $C_T = \{u \in C([0,T]) : u(0) = u(T)\}$ by
$$
\mathcal{S}(u) = \Phi(u) + \mathcal{G}(u),
$$
where $\Phi$ is defined on $C([0,T])$ by
$$
\Phi(u) :=  \begin{cases} 
\int_0^T [1 - \sqrt{1 - |u'|^2}]\,dt & \text{if }  u \in W^{1,\infty}([0,T])\\
+ \infty & \text{if }  u \in C([0,T]) \setminus W^{1,\infty}([0,T])
\end{cases}
$$
and $\mathcal{G}$ is defined on $C([0,T])$ by
$$
\mathcal{G}(u) = \int_0^T [a \cos u + eu]\,dt. 
$$
$\Phi$ is convex, proper, lower semi-continuous, and  $\mathcal{G} $ 
of class $C^1$, so that $\mathcal{S}$ has the structure required by Szulkin's 
critical point theory \cite{Szu1}. When $\overline e = 0$, 
$\mathcal R$ and $\mathcal{S}$ are bounded from below, and satisfy a suitable 
version of Palais-Smale condition on their set of definition. Consequently, 
they reach there a minimum, and one can show that such minimum corresponds 
to a solution of \eqref{rfpe} (this is less trivial than in the case 
of \eqref{fpe}). Hence,  the following extension of Hamel's result to 
\eqref{rfpe} follows \cite{BrM1,BJM} : \emph{for each $e \in L^1(0,T)$
 such that $\overline e = 0$, problem \eqref{rfpe} has at least one solution
 minimizing $\mathcal{R}$ on $K$ (or  $\mathcal{S}$ on $C_T$)}. 
Another proof of this existence result, based upon some Hamiltonian 
equivalent formulation (described later in a different context) and a 
saddle point theorem, has been given in 2012 by Man\'asevich and Ward \cite{MaWa}.


Like in the classical case, such a conclusion is not optimal for \eqref{rfpe} and, 
in 2012,  Bereanu and Torres \cite{BeT} have proved the following multiplicity 
result.

\begin{theorem}\label{bet}
For each $e \in C([0,T])$ such that  $\overline e = 0$, problem \eqref{rfpe} 
has at least two geometrically distinct solutions.
\end{theorem}

 Their proof is modeled on the one of \cite{MaW1} for the classical pendulum, 
but technically more involved. One first shows the existence of two positive 
minimizers $u_0, \; u_0 + 2\pi$ of $\mathcal{S}$ on $C_T$, then considers a  
modified problem like in the method of  lower and upper solutions with lower 
solution $\alpha = u_0$, and upper solution $\beta = u_1$. Finally one shows 
that  Szulkin's critical points of  the corresponding modified action are 
solutions of \eqref{rfpe}, and obtains the second solution of the modified 
action by a  mountain pass argument.

In  2012, Fonda and Toader \cite{FoT} and, in 2013,  Mar\`o \cite{Mar} have 
proved Theorem \ref{bet} by applying a Poincar\'e-Birkhoff type fixed point 
theorem to the equivalent Hamiltonian formulation mentioned above, and 
Mar\`o has obtained the supplementary information that 
\emph{one of the solutions is unstable.} An extension of Theorem \ref{bet} 
is easily obtained when  $a \sin u$ is replaced by $ \partial_u F(t,u)$, 
with $F(t,u)$ $\omega$-periodic in $u$.

If $e \in L^1(0,T;\mathbb{R}^N)$, if   the functions $F(t,u)$ and $\nabla_u F(t,u)$
are defined and continuous on $[0,T] \times \mathbb{R}^N$,  and if assumption
\eqref{periodicity} holds, one can consider the periodic problem for a
 relativistic system
\begin{equation}\label{rsfp}
 \Big(\frac{u'}{\sqrt{1 - |u'|^2}}\Big)'  
+ \nabla_u F(t,u) = e(t),\quad u(0) = u(T),\quad u'(0) = u'(T).
 \end{equation}
 Its concept of solution is defined in an analogous way as for \eqref{rfpe}. With now
$$
C_T := \{u \in C([0,T],\mathbb{R}^N) : u(0) = u(T)\},
$$
one defines $\mathcal{S} : C_T \to (-\infty,+\infty]$ by
\begin{equation}\label{defS}
\mathcal{S}(u) =  \Phi(u) + \mathcal{G}(u),
\end{equation}
where
$$
\Phi(u) :=  \begin{cases}  
\int_0^ T [1 - \sqrt{1 - |u'|^2}]\,dt & \text{if }  u \in W^{1,\infty}([0,T],\mathbb{R}^N) \\
+ \infty & \text{if }   u \in C_T \setminus W^{1,\infty}([0,T],\mathbb{R}^N),
\end{cases} 
$$
and
$$
\mathcal{G}(u) = \int_0^T[-F(t,u)+(e|u)]\,dt.
$$
In 2011, using the approach of \cite{BrM1}, the following existence result
 was proved in \cite{BrM2}.

\begin{theorem}\label{brm2}
If $F$ satisfies assumption \eqref{periodicity}, then, for each 
$e \in L^1(0,T;\mathbb{R}^N)$ such that $\overline e = 0$,  problem \eqref{rsfp}
has at least one solution.
\end{theorem}

The extension to systems of the methods used in \cite{BeT,FoT,Mar} 
for a scalar equation seeming difficult, the obtention of
 multiplicity results similar to Theorem \ref{cmr} for system \eqref{rsfp} 
has required different approaches, that we now describe.

\section{A Hamiltonian approach}

The first result was given  in 2012 in \cite{Maw2}.  
It is assumed that $F$ and $\nabla_uF$ exist and are continuous, and, 
for simplicity, we extend them as well as $e$, by T-periodicity, 
to $\mathbb{R} \times \mathbb{R}^N$ and to $\mathbb{R}$ respectively. Setting
$$
v = \frac{u'}{\sqrt{1 - |u'|^2}}
$$
in \eqref{rsfp}, which is equivalent to
$$
 u' = \frac{v}{\sqrt{1 + |v|^2}},
$$
we immediately see that  \eqref{rsfp} is equivalent to the first order problem
\begin{equation}\label{rsfp1}
v' = - \nabla_u F(t,u) + e(t),\quad
u' = \frac{v}{\sqrt{1 + |v|^2}}, \quad
u, v\; T\text{-periodic}.
\end{equation}
Defining $H : \mathbb{R} \times \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ by
$$
 H(t,u,v) :=  \sqrt{1 + |v|^2} - 1\ + F(t,u) - (e(t)|u),
$$
we see that \eqref{rsfp1}  has the Hamiltonian form
\begin{equation}\label{hrsfp}
 v' = - \nabla_u H(t,u,v), \quad  u' = \nabla_v H(t,u,v), \quad  u, v\quad
T\text{-periodic}.
 \end{equation}
The action functional naturally associated to \eqref{hrsfp} is given by
$$ 
\mathcal{H}(v,u) = \int_0^T 
\big[ - (v|u')  + F(t,u) - (e|u) + \sqrt{1 + |v|^2} - 1)\big]\,dt.
$$
If we define  the Sobolev space
$$
H^{1/2}_T = \{(v,u)  \in H^{1/2}([0,T],\mathbb{R}^{2N}) : \text{$v$ and $u$ are
 $T$-periodic}\},
$$
then a standard result (see e.g. \cite{Rab2}) implies that if   
$e \in L^s(0,T;\mathbb{R}^N)$ for some $s > 1$, then
$\mathcal{H} \in C^1(H^{1/2}_T,\mathbb R)$ and its critical points solve 
\eqref{hrsfp}, or, explicitly, \eqref{rsfp1}.
Furthermore, it is easy to check that if  $\overline e = 0$, and $F$ satisfies 
condition \eqref{periodicity}, then
$$ 
\mathcal{H}(v,u_1 + k_1\omega_1,\ldots,u_N + k_N\omega_N) = \mathcal{H}(v,u)
$$
for all $(v,u) \in H^{1/2}_T$ and all $(k_1,\ldots,k_N) \in \mathbb{Z}^N$.
Consequently,  we can consider  $\mathcal{H}$ as  defined on 
${\mathbb T}^N \times E$, where
$$
E = \{(v,u) \in H^{1/2}_T : \overline u = 0\}
$$
 and ${\mathbb T}^N$ is the $n$-torus. It can be shown that 
$E = E^- \oplus E^0 \oplus E^+$ where $E^0 \simeq \mathbb{R}^N$, the linear operator
associated to the quadratic form  $(v,u) \mapsto \int_0^T [-(v|u')]\,dt$ is 
negative definite on $E^-$, positive definite on $E^+$, and
 $$
\mathcal{H}(v,\overline u) \to +\infty \quad \text{for all } 
 \overline u \in \mathbb{R}^N  \text{ when }  |v| \to \infty \text{ in } E^0.
$$
Therefore   $\mathcal{H}$  satisfies the conditions of an abstract saddle 
point theorem for indefinite functionals proved in  1990 by Szulkin \cite{Szu1}, 
and based upon the concept of relative Ljusternik-Schnirelmann  category,
 which implies the following multiplicity result for \eqref{rsfp}.
\begin{theorem}\label{ham}   If $F$ satisfies assumption \eqref{periodicity}, then, 
for each $e \in L^s(0,T;\mathbb{R}^N)$ for some $s > 1$, such that $\overline e = 0$,
problem \eqref{rsfp} has at least   $N+1$  geometrically distinct solutions.
\end{theorem}

As one can see, the proof of Theorem \ref{ham} is technically sophisticated, 
both from the critical point theory side, because $\mathcal{H}$ is an 
indefinite functional, and from the topological side,  because the relative 
category is a more involved and delicate concept than Ljusternik-Schnirelmann 
category.  Hence the result of \cite{Maw2} raises the following  
natural questions:
\begin{enumerate}
\item Can  $e \in L^s$ for some $s > 1$ be replaced by the more natural 
 assumption $e \in L^1$?
\item Can  the result be proved using Lagrangian action and classical category?
\end{enumerate}

\section{A Lagrangian approach}

In  2013, Bereanu and Jebelean \cite{BeJ}  proved Theorem \ref{ham}, when 
$F$, $\nabla_uF$ and $e$ are continuous, through an extension to convex, 
lower semicontinuous perturbations of a $C^1$-functional on a Banach space,
 i.e. to functionals of Szulkin type \cite{Szu1}, of an abstract 
multiplicity result  for some symmetric $C^1$ functionals given
 in \cite[Theorem 4.12]{MaW3}, and motivated by Rabinowitz' approach 
in \cite{Rab}.

Let $X$ be a Banach space with dual $X^*$ and duality mapping 
$\langle \cdot,\cdot \rangle$, $G$ a discrete additive subgroup of $X$ such that
 $\operatorname{span}(G)$ has finite dimension $N$, $\pi : X \to X/G$ the 
canonical projection. So $G \simeq \mathbb{Z}^N$, $X = \mathbb{R}^N \oplus Y$ for some closed
subspace $Y$, $u = \overline u + \widetilde u$, with $\overline u \in \mathbb{R}^N$,
$\widetilde u \in Y$.
$A \subset X$ is \emph{$G$-invariant} if $u+g \in A$ for all 
$u \in A$ and $g \in G$, $f : X \to M$ is \emph{$G$-invariant} if 
$f(u + g) = f(u)$ for all $u \in X$ and $g \in G$. The following assumptions 
are made:
\begin{itemize}
\item[(H1)] $\mathcal{G} \in C^1(X,\mathbb{R})$ is $G$-invariant, $\mathcal{G}'$
takes bounded sets into bounded sets.
\item[ (H2)] $\Psi : X \to (- \infty,+\infty]$ is $G$-invariant, convex, 
lower semicontinuous, with closed non-empty domain 
 $D(\Psi) \supset \{ u \in X : \|\widetilde u\| \leq \rho,\; |\Psi(u)| 
 \leq \rho\}$, $\Psi(0) = 0$, $\Psi(u) = \Psi(\widetilde u)$ for all $u \in X$.
\item[(H3)] Any sequence $(u_n)$ in $X$ with $(\overline {u_n})$ bounded has 
a convergent subsequence.
\end{itemize}
According to \cite{Szu1}, $u \in X$ is a \emph{critical point} of 
$\mathcal{S} = \Psi + \mathcal{G}$ if
$$
\langle \mathcal{G}'(u),v- u \rangle + \Psi(v) - \Psi(u) \geq 0 \quad
\text{for all } v \in X.
$$
Let
$$
K = \{u \in X : u \text{ is a critical point}\}
$$
 be the \emph{critical set} of $\mathcal{S}$, and let 
$K_c = \{u \in K : \mathcal{S}(u)  = c\}$. It is easy to see that 
$\mathcal{S}, \mathcal{S}',K, K_c$ are $G$-invariant. Hence, if $u$ 
is a critical point of $\mathcal{S}$, the same is true for $u + g$ 
for all $g \in G$, and the set $\{u + g : g \in G\}$ is called a 
\emph{critical orbit} of $\mathcal{S}$.

If $\mathcal{N}$ is an open neighborhood of $K_c$ and $\varepsilon > 0$, we set
$$
\mathcal{N}_\varepsilon = \{u \in X \setminus \mathcal{N} :
 |\overline u| \leq 2,\; \mathcal{S}(u) \leq c + \varepsilon\}.
$$
The following equivariant deformation lemma, which combines similar results 
in \cite{MaW3,Szu1},  is essential to prove the multiplicity result.

\begin{lemma}\label{defor} 
Let $c \in \mathbb{R}$ and $\mathcal{N}$ be a $G$-invariant
neighborhood of $K_c$. Then, for each $\varepsilon \in (0,1]$, there exists
 $\varepsilon \in (0,\overline \varepsilon]$, $d_\varepsilon > 0$, 
$\varepsilon' \in (0,\varepsilon]$ and
 $\eta \in C([0,\overline t] \times \mathcal{N}_\varepsilon,X)$, 
with the following properties:
\begin{itemize}
\item[(i)] $\eta(0,\cdot) = id_{\mathcal{N}_\varepsilon}$.

\item[(ii)] $\eta(t,u+g) = \eta(t,u) + g$ for all $(t,u) \in [0,\overline t] 
\times \mathcal{N}_\varepsilon$ and all $g \in G$ with
 $u + g \in \mathcal{N}_\varepsilon$.

\item[(iii)] $\|\eta(t,u) - u\| \leq d_\varepsilon t$ for all 
$(t,u) \in [0,\overline t] \times \mathcal{N}_\varepsilon$.

\item[(iv)] $\mathcal{S}(\eta(t,u)) - \mathcal{S}(u) \leq d_\varepsilon t$ 
for all $(t,u) \in [0,\overline t] \times \mathcal{N}_\varepsilon$.

\item[(v)] $\mathcal{S}(\eta(t,u)) - \mathcal{S}(u) 
\leq - \varepsilon't/2$ for all $(t,u) \in [0,\overline t] \times 
(\mathcal{N}_\varepsilon \cap \mathcal{S}^{-1}([c-\varepsilon,+\infty)))$.

\item[(vi)] if $A \subset \mathcal{N}_\varepsilon $ with 
$c \leq \sup_A \mathcal{S}$, then, for all $t \in [0,\overline t]$,
$$
\sup_A \mathcal{S}(\eta(t,\cdot)) - \sup_A \mathcal{S} \leq - \varepsilon' t/2.
$$
\end{itemize}
\end{lemma}
From this lemma, one can construct a deformation in the quotient space $\pi(X)$.
 Defining, like in \cite{MaW3},
$$
\mathcal{A}_{j} = \{A \subset X : A  \text{ is compact and 
cat}_{\pi(X)}(\pi(A)) \geq j \},
$$
one can check that $\mathcal{A}_j \neq \emptyset$ for each $j = 1,\ldots,N+1$ 
and $\mathcal{A}_j$ is a complete metric space for the Hausdorff distance. 
Furthermore, the function $\sigma : \mathcal{A}_j \to (-\infty,+\infty]$ defined by
$$
\sigma(A) = \sup_{A \in \mathcal{A}_j}\mathcal{S}
$$
is lower semicontinuous and  bounded from below. Ekeland's variational principle 
and a rather standard argument of Ljusternik-Schnirelmann type give the following 
multiplicity result.

\begin{proposition}\label{absmul} 
Under assumptions {\rm (H1)--(H3)}, the functional 
$\mathcal{S} = \Psi + \mathcal{G}$ has at least $N+1$ critical orbits.
\end{proposition}


By applying Proposition \ref{absmul} to the functional 
$\mathcal{S} : C_T \to (-\infty,\infty]$ defined in \eqref{defS} with the group
\begin{equation}\label{defG}
G = \Big\{\sum_{k=1}^N k_i \omega_i e^i : k_i \in \mathbb{Z},\; i = 1,\ldots,N \Big\},
\end{equation}
 one obtains easily the following multiplicity result.

\begin{theorem}\label{beje} 
If $F$  satisfies assumption \eqref{periodicity}, then, for each 
$e \in C([0,T],\mathbb{R}^N)$ such that $\overline e = 0$, problem \eqref{rsfp}
has at least   $N+1$  geometrically distinct solutions.
\end{theorem}

The proof of Proposition \ref{absmul} given in \cite{BeJ} is quite complicated 
and technical, but only uses classical Ljusternik-Schnirelmann category.  
In the following section, we describe a recent approach given in  \cite{JMS},
 which answers positively the two questions of the end of Section 3, 
by obtaining the requested multiplicity result through the use of a modified 
equivalent problem, whose action functional is defined in the classical 
Sobolev space $H^1_T$, and to which Theorem 4.12 of \cite{MaW3} can be 
directly applied.


\section{A modified Lagrangian approach}

Let   $e \in L^1(0,T;\mathbb{R}^N)$,  and assume that $F(t,\cdot)$ and
$\nabla_uF(t,\cdot)$ are continuous  for a.e $t\in[0,T]$, that   $F(\cdot,u)$  
and $\nabla_uF(\cdot,u)$ are measurable  for each $u\in \mathbb{R}^N$, and that there
exists some $\alpha\in L^1(0,T)$ such that
$$
|F(t,u)| + |\nabla_u F(t,u)|\leq \alpha(t)
$$
 for a.e. $t \in [0,T]$ and all $u \in \mathbb{R}^N$. Define, for
$v \in B(1) \subset \mathbb{R}^N$,
$$
\varphi(v) := \frac{v}{\sqrt{1 - |v|^2}},
$$
so that
 $$
\varphi^{-1}(w) = \frac{w}{\sqrt{1 + |w|^2}}\quad \text{for all } w \in \mathbb{R}^N.
$$
Let us introduce a  modification of $\varphi$ inspired by recent papers of 
Coelho \emph{et al} \cite{CCOO,CCR} in  problems of positive solutions with 
Dirichlet conditions, but technically different, by setting
$$
K:=\varphi^{-1}\left(\overline{B}(\sqrt{n}\|\alpha\|_{L^1})\right)\subset B(1),
$$
 fixing $R\in(0,1)$ in such a way that
 $$
\frac{R}{\sqrt{1-R^2}}\geq \sqrt{n}\|\alpha\|_{L^1}, \, K\subset\overline{B}(R),
$$
 and defining the homeomorphism  $\psi : \mathbb{R}^N \to \mathbb{R}^N$ by
$$
\psi(y):= (1-\min\{|y|^2,R^2\})^{-1/2}y,
$$
in such a way that
$$
\psi^{-1}(v) =  \max\left\{(1-R^2)^{1/2},(1 + |v|^2)^{-1/2}\right\}v.
$$

\begin{lemma}\label{mono}
For all $y, z \in \mathbb{R}^N$, one has
$$
(\psi(z)-\psi(y)|z-y)\geq|z-y|^2, \; |\psi(y)| 
\leq \frac{1}{\sqrt{1 - R^2}}|y|.
$$
\end{lemma}

With this $\psi$, let us consider the modified problem
 \begin{equation}\label{mp}
 (\psi (u'))' + \nabla_u F(t,u) = e(t),\quad u(0) = u(T),\quad u'(0) = u'(T)
 \end{equation}
 The choice of $R$ above allows us to prove the following equivalence result.

\begin{lemma}\label{equiv}
$u\in C^1$ is solution of \eqref{rsfp} if and only if  it  is a solution 
of \eqref{mp}.
\end{lemma}

If we define now $\Psi : \mathbb{R}^N \to \mathbb{R}$ by
$$
\Psi(y) := 1 - \frac{1 - \min\{|y|^2,R^2\} + 1 - |y|^2}
{2 \sqrt{1 - \min\{|y|^2,R^2\}}},
$$
it is easy to show that $\Psi$ is of class $C^1$ and that
$$
\psi(y) = \nabla \Psi(y), \quad (1/2)|y|^2\leq\Psi(y)
\leq \frac{1}{\sqrt{1-R^2}}|y|^2
$$
for all $y \in \mathbb{R}^N$. Consequently the functional $\mathcal{M}$ given by
$$  
\mathcal{M}(u):=\int_0^T[\Psi(u') - F(t,u) + (e|u)]\, dt
$$
is well defined and of class $C^1$ on $H^1_T$, and
$$ 
\langle\mathcal{M}'(u),v\rangle=\int_0^T[(\psi(u')|v') 
- (\nabla_u F(t,u) - e|v)]\,dt,
$$
for all $u, v\in H^1_T$,  so that its critical points correspond to the weak, 
and hence to the Carath\'eodory solutions of \eqref{mp}. On the other hand, 
the following version of Palais-Smale condition can be proved.

\begin{lemma}\label{PS}
Each sequence  $(u_n)$ in  $H^1_T$ such that $(\mathcal{M}(u_n))$  is bounded,
 $\mathcal{M}'(u_n)\to 0,$ and $(\overline{u}_n)$  is bounded, contains 
a convergent subsequence.
\end{lemma}

As mentioned above, \cite[Theorem 4.12]{MaW3} is just the version of 
Proposition \ref{absmul} for a $C^1$ functional, and we keep the notations 
of Section 4.  If  $\mathcal{I} \in C^1(X,\mathbb{R})$ is   $G$-invariant,
we introduce the following type of Palais-Smale condition, called the 
\emph{$(PS)_G$-condition} : for each sequence $(u_n)$ in $X$ with 
$(\mathcal{G}(u_n))$  bounded and $\mathcal{G}'(u_n)\to 0$,  $(\pi(u_n))$ 
contains a convergent subsequence. Theorem 4.12 in \cite{MaW3} goes as follows.

 \begin{proposition}\label{maw3}
If the vector space spanned in $X$ by $G$ has finite dimension  $N$, and if 
$\mathcal{I}\in C^1(X,\mathbb{R})$  is $G$-invariant,  satisfies $(PS)_G$-condition,  
and is bounded from below, then  $\mathcal{I}$ has at least $N+1$ critical orbits.
\end{proposition}

The proof of Proposition \ref{maw3} given in \cite{MaW3} is based upon
 Ekeland variational principle and classical Ljusternik-Schnirelmann category. 
As shown in \cite{MaW3}, it provides a proof of Theorem \ref{cmr}  for the 
classical pendulum system. It also implies the corresponding result 
for \eqref{rsfp}.

\begin{theorem}\label{mrfps}
If $F$ satisfies condition \eqref{periodicity},  then, for each 
$e \in L^1(0,T;\mathbb{R}^N)$,  problem \eqref{rsfp} has at least $N+1$ geometrically
 distinct solutions.
\end{theorem}

\begin{proof}[Sketch of the proof]
 By Lemma \ref{equiv}, it suffices to prove that the modified action function 
$\mathcal{M}$ satisfies the conditions of Proposition \ref{maw3}. 
It is easy to see that
$$\mathcal{M}(u)=  \int_0^T[ \Psi(u') - F(t,u) + (e(t)|\widetilde u)]\,dt
 = \mathcal{M}(u + \omega_i e^i)
$$
 for all $i = 1,\ldots,n$ and $u\in H^1_T$. If we define $G$ by \eqref{defG},  
using Lemma \ref{mono} and the Wirtinger and Sobolev inequalities, and denoting 
the $L^p$-norm by $\|\cdot\|_p$, one can show that the inequality
$$
\mathcal{M}(u)\geq \frac{1}{2}\|\widetilde{u}'\|_{2}^2-C_2\|\alpha\|_{1} 
-C_1\|h\|_{1}\|\widetilde{u}'\|_{2}
$$
holds, and that  $\mathcal{M}$ satisfies the $(PS)_G$-condition. 
Then the result follows from Proposition  \ref{maw3}.
\end{proof}

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\end{document}