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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 134, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/134\hfil Existence of periodic solutions]
{Existence of periodic solutions for
 sub-linear first-order Hamiltonian systems}

\author[M. Timoumi \hfil EJDE-2015/134\hfilneg]
{Mohsen Timoumi}

\address{Mohsen Timoumi \newline
 Department of Mathematics,
 Faculty of Sciences, 5000 Monastir, Tunisia}
\email{m\_timoumi@yahoo.com}


\thanks{Submitted March 13, 2014. Published May 15, 2015.}
\subjclass[2010]{34C25}
\keywords{Hamiltonian systems; periodic solutions; saddle point theorem;
\hfill\break\indent  least action principle; sub-linear conditions}

\begin{abstract}
 We prove the existence solutions for the sub-linear first-order Hamiltonian
 system
 $J\dot{u}(t)+Au(t)+\nabla H(t,u(t))=h(t)$
 by using the least action principle and a version of the Saddle Point Theorem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider the first-order Hamiltonian system
\begin{equation}
J\dot{u}(t)+Au(t)+\nabla H(t,u(t))=h(t) \label{e1.1}
\end{equation}
where $A$ is a $(2N\times 2N)$ symmetric matrix,
$H\in C^{1}(\mathbb{R}\times\mathbb{R}^{2N},\mathbb{R})$ is $T$-periodic
 in the first variable $(T>0)$ and $h\in C(\mathbb{R},\mathbb{R}^{2N})$
is $T$-periodic.

When $A=0$ and $h=0$, it has been proved that system $\eqref{e1.1}$ has at least
one $T$-periodic solution by the use of critical point theory and minimax
methods \cite{a1,c1,e1,f1,g1,l1,l2, t1,t3,x1}.
 Many solvability conditions are given, such as the convex condition (see [3,5]),
the super-quadratic condition (see
\cite{a1,f1,l1,l2,l4,r2,t1,x1}),
the sub-linear condition (see \cite{c1,t3}).
 When $A$ is not identically null, the existence of periodic solutions for
\eqref{e1.1} has been studied in \cite{l2,t2}.
In all these last papers, the Hamiltonian is assumed to be super-quadratic.
 As far as the general case ($A$ not identically null) is concerned,
to our best knowledge, there is no research about the existence of
periodic solutions for \eqref{e1.1} when $H$ is sub-linear.
In \cite{c1}, the authors considered the special case $A=0$ and $h=0$ and obtain
the existence of subharmonic solutions for \eqref{e1.1} under the following
assumptions:
\smallskip

\noindent(A1) There exist constants $a,b,c>0$, $\alpha\in [0,1[$,
functions $p\in L^{\frac{2}{1-\alpha}}(0,T;\mathbb{R}^{+})$,
$q\in L^2(0,T;\mathbb{R}^{+})$ and a nondecreasing function
$\gamma\in C(\mathbb{R}^{+},\mathbb{R}^{+})$ with the following properties:
\begin{itemize}
\item[(i)] $\gamma(s+t)\leq c(\gamma(s)+\gamma(t))$ for all
$s,t\in\mathbb{R}^{+}$,

\item[(ii)] $\gamma(t)\leq at^{\alpha}+b$ for all $t\in\mathbb{R}^{+}$,

\item[(iii)] $\gamma(t)\to +\infty\ as\ t\to+\infty$,
such that
\begin{gather*}
|\nabla H(t,x)|\leq p(t)\gamma(|x|)+q(t),\quad \forall x\in\mathbb{R}^{2N},
\text{ a.e. } t\in[0,T];\\
\lim_{|x|\to\infty}\frac{1}{\gamma^2(|x|)}\int^T_0H(t,x)dt=\pm\infty.
\end{gather*}
\end{itemize}
Similarly, in \cite{t3} the author considered the case $A=0$ and $h=0$ and
obtained the existence os subharmonic solutions for \eqref{e1.1} under
the following assumptions:
\smallskip

\noindent(A2)
There exist a positive constant $a$, $g\in L^2(0,T;\mathbb{R})$ and a
 non-increasing function $\omega\in C(\mathbb{R}^{+},\mathbb{R}^{+})$
with the properties:
\begin{gather*}
\liminf_{s\to\infty}\frac{\omega(s)}{\omega(\sqrt{s})} > 0,\\
\omega(s)\to 0,\quad \omega(s)s\to\infty\quad\text{as }s\to\infty,
\end{gather*}
such that
\begin{gather*}
|\nabla H(t,x)|\leq a\omega(|x|)|x|+g(t),\quad
 \forall x\in\mathbb{R}^{2N},\text{ a.e. } t\in [0,T];\\
\frac{1}{[\omega(|x|)|x|]^2}\int^T_0H(t,x)dt \to+\infty
\quad \text{as }|x|\to\infty.
\end{gather*}
In Sections 4,5, we will use the Least Action Principle and a version
of the Saddle Point Theorem to study the existence of periodic solutions
for \eqref{e1.1}, when $A$ and $h$ are not necessary null and $H$
satisfies some more general variants conditions replacing
conditions (A1), (A2).

\section{Preliminaries}

Let $T>0$ and $A$ be a $(2N\times 2N)$ symmetric matrix.
Consider the Hilbert space $H^{1/2}(S^{1},\mathbb{R}^{2N})$
where $S^{1}=\mathbb{R}/(T\mathbb{Z})$ and the continuous  quadratic form $Q$
defined on $E$ by
$$
Q(u)=\frac{1}{2}\int^T_0(J\dot{u}(t)\cdot u(t) +Au(t)\cdot u(t))dt
$$
where $x\cdot y$ is the inner product of
$x,y\in\mathbb{R}^{2N}$. Let us denote by $E^{0}$, $E^{-}$, $E^{+}$
respectively the subspaces of $E$ on which $Q$ is null, negative
definite and positive definite. It is well known that these subspaces
are mutually orthogonal in $L^2(S^{1},\mathbb{R}^{2N})$ and in $E$
with respect to the bilinear form
$$
B(u,v)=\frac{1}{2}\int^T_0(J\dot{u}(t)\cdot v(t)
+Au(t)\cdot v(t))dt,\ u,v\in E
$$
associated with $Q$. If $u\in E^{+}$ and $v\in E^{-}$, then $B(u,v)=0$
and $Q(u+v)=Q(u)+Q(v)$.

For $u=u^{-}+u^{0}+u^{+}\in E$, the expression
$\|u\|=[Q(u^{+})-Q(u^{-})+|u^{0}|^2]^{1/2}$ is an equivalent norm in $E$.
It is well known that the space $E$ is compactly embedded in
$L^{s}(S^{1},\mathbb{R}^{2N})$ for all $s\in [1,\infty[$.
In particular, for all $s\in [1,\infty[$, there exists $\lambda_{s}>0$
such that for all $u\in E$,
\begin{equation}
\|u\|_{L^{s}}\leq\lambda_{s}\|u\|. \label{e2.1}
\end{equation}
Next, we have a version of the Saddle Point Theorem \cite{r1}.

\begin{lemma} \label{lem2.1}
 Let $E=E^{1}\oplus E^2$ be a real Hilbert space with
$E^2=(E^{1})^{\bot}$. Suppose that $f\in C^{1}(E,\mathbb{R})$ satisfies
\begin{itemize}
\item[(a)] $f(u)=\frac{1}{2}\langle Lu,u\rangle +g(u)$ and $Lu=L_1P_1u+L_2P_2u$
with $L_{i}:E^{i}\to E^{i}$ bounded and self-adjoint, $i=1,2$;

\item[(b)] $g'$ is compact;

\item[(c)] There exists $\beta\in\mathbb{R}$ such that
$f(u)\leq\beta$ for all $u\in E^{1}$;

\item[(d)] There exists $\gamma\in\mathbb{R}$ such that $f(u)\geq\gamma$
for all $u\in E^2$.
\end{itemize}
Furthermore, if $f$ satisfies the Palais-Smale condition $(PS)_{c}$
for all $c\geq\gamma$, then $f$ possesses a critical
value $c\in[\gamma,\beta]$.
\end{lemma}

\section{Linear Hamiltonian systems}

 Let $A$ be a $(2N\times 2N)$ symmetric matrix, we consider the linear
Hamiltonian system
\begin{equation}
\dot{x}=JAx. \label{e3.1}
\end{equation}
Let $\lambda_1,\dots ,\lambda_{s}$ be all the distinct eigenvalues of
$B=JA$ and $F_1,\dots ,F_{s}$ be the corresponding root subspaces.
The dimension of the root subspace $F_{\sigma}$ is equal to the multiplicity
$m_{\sigma}$ of the corresponding root $\lambda_{\sigma}$ of the
characteristic equation $det(B-\lambda I_{2N})=0$ $(m_1+\dots +m_{s}=2N)$.
The space $\mathbb{R}^{2N}$ splits into a direct sum of the $B$-invariant
subspaces $F_{\sigma}$:
\begin{equation}
\mathbb{R}^{2N}=F_1\oplus\dots \oplus F_{s}. \label{e3.2}
\end{equation}
Each subspace $F_{\sigma}$ possesses a basis
$(a^{\sigma}_1,\dots ,a^{\sigma}_{m_{\sigma}})$ satisfying
$$
Ba^{\sigma}_1=\lambda_{\sigma}a^{\sigma}_1,\ Ba^{\sigma}_2
=\lambda_{\sigma}a^{\sigma}_2+a^{\sigma}_1,\dots ,
Ba^{\sigma}_{m_{\sigma}}=\lambda_{\sigma}a^{\sigma}_{m_{\sigma}}
+a^{\sigma}_{m_{\sigma}-1}.
$$
The  $(m_{\sigma}\times m_{\sigma})$ matrix
\[
Q_{\sigma}(\lambda_{\sigma})=  \begin{pmatrix}
\lambda_{\sigma} & 1 & 0 & 0 & \dots  & 0 \\
0 & \lambda_{\sigma} & 1 & 0 & \dots  & 0 \\
. & . & . & . & \dots  & . \\
0 & 0 & 0 & \dots  & \lambda_{\sigma} & 1 \\
0 & 0 & 0 &\dots  & 0 & \lambda_{\sigma} \\
\end{pmatrix}
\]
is called an elementary Jordan matrix. We have
$B=SQS^{-1}$
where $Q$ is a direct sum of elementary Jordan matrices
\[
Q=  \begin{pmatrix}
Q_1(\lambda_1) & 0 & 0 & \dots  & 0 \\
0 & Q_2(\lambda_2) & 0 & \dots  & 0 \\
. & . & . & \dots  & . \\
. & 0 & 0 & \dots  & Q_{s}(\lambda_{s}) \\
\end{pmatrix}
= Q_1(\lambda_1)\oplus\dots \oplus Q_{s}(\lambda_{s})
\]
the columns of the matrix $S$,
$$
a^{1}_1,\dots ,a^{1}_{m_1};\ a^2_1,\dots ,a^2_{m_2};
\dots ;\ a^{s}_1,\dots ,a^{s}_{m_{s}}$$
form a basis for $\mathbb{R}^{2N}$ and so $det(S)\neq 0$.

The matrizant of equation \eqref{e3.1} is given by
$$
R(t)=e^{tB}=S[\exp(tQ_1(\lambda_1))\oplus \dots \oplus
\exp(tQ_{s}(\lambda_{s}))]S^{-1}=S e^{tQ}S^{-1}.
$$
then the solution of equation \eqref{e3.1} with initial condition $x(0)$ is
$$
x(t)=e^{tB}x(0).
$$
Therefore to each eigenvalue $\lambda_{\sigma}$ corresponds a group
of $m_{\sigma}$-linearly independent solutions:
\begin{equation}
\begin{gathered}
x^{\sigma}_1(t)=e^{\lambda_{\sigma}t}a^{\sigma}_1\\
x^{\sigma}_2(t)=e^{\lambda_{\sigma}t}(ta^{\sigma}_1+a^{\sigma}_2)\\
\dots \\
x^{\sigma}_{m_{\sigma}}(t)=e^{\lambda_{\sigma}t}
(\frac{1}{(m_{\sigma}-1)!}t^{m_{\sigma}-1}a^{\sigma}_1
+\dots +a^{\sigma}_{m_{\sigma}}).
\end{gathered} \label{e3.3}
\end{equation}
Moreover, combining the solutions of all the groups \eqref{e3.3}
(there are obviously $2N$ in all, since $m_1+\dots +m_{s}=2N$),
we obtain a complete system of linearly independent solutions of  \eqref{e3.1}.
Now, assume that $\lambda_1=0$ is an eigenvalue of $B=JA$ and
let $1\leq m\leq m_1$ be the dimension of the corresponding eigenspace
$E_1$. We can replace the basis $(a^{1}_1,\dots ,a^{1}_{m_1})$
of the root subspace $F_1$ by the basis  $(b^{1}_1,\dots ,b^{1}_{m_1})$
where
$(b^{1}_1,\dots ,b^{1}_{m})$ is a basis of $E_1$, $b^{1}_{j}=a^{1}_{j}$
for $m+1\leq j \leq m_1$ and such that $b^{1}_{m+1}=B b^{1}_{m}$.
To this basis corresponds the group of $2N$ linearly independent solutions:
\begin{equation}
\begin{gathered}
u^{1}_1(t)=b^{1}_1\\
\dots \\
u^{1}_{m}(t)=b^{1}_{m}\\
u^{1}_{m+1}(t)=b^{1}_{m}t+b^{1}_{m+1}\\
\dots \\
u^{1}_{m_1}(t)=\frac{1}{(m_1-m)!}b^{1}_{m}t^{m_1-m}+\dots +b^{1}_{m_1}\\
u^{\sigma}_{k}(t)=x^{\sigma}_{k}(t),\quad
 2\leq\sigma\leq s,\; 1\leq k\leq m_{\sigma}.
\end{gathered}  \label{e3.4}
\end{equation}
A solution $u$ of \eqref{e3.1} may be written in the form
$$
u(t)=\sum^{s}_{\sigma=1}\sum^{m_{\sigma}}_{j=1}
\alpha^{\sigma}_{j}u^{\sigma}_{j}(t).
$$
Let $T>0$ be such that $\lambda_{\sigma}T\notin 2i\pi\mathbb{Z}$ for all
$1\leq\sigma\leq s$. If $u$ is $T$-periodic, then for any
$1\leq \sigma\leq s$, we have
$$
\sum^{m_{\sigma}}_{j=1}\alpha^{\sigma}_{j}u^{\sigma}_{j}(kT)
=\sum^{m_{\sigma}}_{j=1}\alpha^{\sigma}_{j}u^{\sigma}_{j}(0),\quad
 \forall k\in\mathbb{Z}.
$$
It is easy to see that $\alpha^{1}_{j}=0$ for $m+1\leq j\leq m_1$
and $\alpha^{\sigma}_{j}=0$ for $2\leq\sigma\leq s$ and
$1\leq j\leq m_{m_{\sigma}}$. Therefore,
 $u(t)=\sum^{m}_{j=1}\alpha^{1}_{j}b^{1}_{j}$.
Hence the set of $T$-periodic solutions of \eqref{e3.1} is equal to $N(A)$.

\begin{example} \label{examp3.1}\rm Let
\[
A=  \begin{pmatrix}
-12 & 6 & 5 & 1 \\
-2 & 1 & 0 & 1 \\
2 & -1 & 0 & -1\\
2 & -1 & 0 & -1\end{pmatrix}
\]
The characteristic equation corresponding to $B=JA$ is
$\det(JA-XI_4)=X^{3}(X-5)=0$. To the eigenvalue $\lambda_1=0$
 corresponds the eigenspace
$$E_1=\operatorname{span}\{e_1,e_2\}
$$
and the root subspace
$$
F_1=\operatorname{span}\{e_1,e_2,e_3\}
$$
where $e_1=(1,2,0,0)$, $e_2=(1,1,1,1)$, $e_3=(0,0,0,1)$ with
$Be_3=e_2$. To the eigenvalue $\lambda_2=5$ corresponds the root subspace
$$
E_2=F_2=\operatorname{span}\{e_4\},
$$
where $e_4=(0,0,1,0)$. Then we have
$JA=SQS^{-1}$
with
\[
S=  \begin{pmatrix}
1 & 1 & 0 & 0 \\
2 & 1 & 0 & 1 \\
0 & 1 & 0 & 1\\
0 & 1 & 1 & 0\end{pmatrix}, \quad
Q=  \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 5\end{pmatrix}
\]
The matrizant of the corresponding equation \eqref{e3.1} is then
\[
R(t)=SQS^{-1}= S\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & t & 0 \\
0 & 0 & 1 & 0\\
0 & 0 & 0 & e^{5t}\end{pmatrix} S^{-1}.
\]
To the basis $(e_1,e_2,e_3,e_4)$ corresponds the group of
$4$-linearly independent solutions
\begin{equation}
\begin{gathered}
u_1(t)=e_1\\
u_2(t)=e_2\\
u_3(t)=te_2+e_3\\
u_4(t)=e^{5t}e_4.
\end{gathered} \label{e3.5}
\end{equation}
A solution of equation \eqref{e3.1} takes the form
$$
u(t)=\alpha_1u_1(t)+\alpha_2u_2(t)+\alpha_3u_3(t)
+\alpha_4u_4(t)
$$
and it is easy to verify that $u$ is $T$-periodic for $T>0$
if and only if $\alpha_3=\alpha_4=0$, i.e. $u\in N(A)$.
\end{example}

\section{First class of sub-linear Hamiltonian systems}

Consider the first-order Hamiltonian system
\begin{equation}
J\dot{u}(t)+Au(t)+\nabla H(t,u(t))=h(t) \label{eHcal}
\end{equation}
where $A$ is a $(2N\times 2N)$ symmetric matrix,
$H:\mathbb{R}\times\mathbb{R}^{2N}\to\mathbb{R}$ is a continuous function,
$T$-periodic in the first variable $(T>0)$ and differentiable with respect
to the second variable with continuous derivative
$\nabla H(t,x)=\frac{\partial H}{\partial x}(t,x)$,
$h\in C(\mathbb{R},\mathbb{R}^{2N})$ is $T$-periodic and
 $J$ is the standard symplectic matrix
$J=\left(\begin{smallmatrix}
0 & -I\\
I & 0
\end{smallmatrix}\right)$.
Let $\gamma:\mathbb{R}^{+}\to\mathbb{R}^{+}$
be a nondecreasing continuous function satisfying the properties:
\begin{itemize}
\item[(i)] $\gamma(s+t)\leq c(\gamma(s)+\gamma(t))$ for all
$s,t\in\mathbb{R}^{+}$,

\item[(ii)] $\gamma(t)\leq at^{\alpha}+b$ for all $t\in\mathbb{R}^{+}$,

\item[(iii)] $\gamma(t)\to +\infty$  as $t\to+\infty$,

\end{itemize}
where $a,b,c$ are positive constants and $\alpha\in[0,1[$.
Consider the following assumptions
\begin{itemize}
\item[(C1)] $\dim(N(A))=m\geq 1$ and $A$ has no eigenvalue of the form
$ki\frac{2\pi}{T}$ $(k\in \mathbb{N}^{*})$;

\item[(H1)]  There exist two functions
 $p\in L^{\frac{2}{1-\alpha}}(0,T;\mathbb{R}^{+})$ and
$q\in L^2(0,T;\mathbb{R}^{+})$ such that
$$
|\nabla H(t,x)|\leq p(t)\gamma(|x|)+q(t),\quad
 \forall x\in\mathbb{R}^{2N},\text{ a.e. } t\in[0,T].
$$
\end{itemize}
Our main results in this section are the following theorems.


\begin{theorem} \label{thm4.1}
Assume {\rm (C1)} and {\rm  (H1)} hold and
\begin{itemize}
\item[(H2)] $H$ satisfies
\[
\limsup_{|x|\to\infty, x\in N(A)}\frac{|x|}{\gamma^2(|x|)}<+\infty,\quad
\lim_{|x|\to\infty,x\in N(A)}\frac{1}{\gamma^2(|x|)}\int^T_0H(t,x)dt
=+\infty.
\]
\end{itemize}
Then   \eqref{eHcal} possesses at least one $T$-periodic solution.
\end{theorem}

\begin{example} \label{examp4.1} \rm
Let $A$ be the matrix defined in Example \ref{examp3.1} and let
$$
H(t,x)=(\frac{3}{4}T-t)|x|^{8/5},\quad \forall x\in\mathbb{R}^{2N},\;
 \forall t\in [0,T].
$$
Then
$$
|\nabla H(t,x)|=\frac{8}{5}|\frac{3}{4}T-t||x|^{3/5}.
$$
Let $\gamma(t)=t^{3/5}, t\geq 0$. It is clear that properties
(i), (ii), (iii) are satisfied. Moreover, we have
\begin{gather*}
\limsup_{|x|\to\infty, x\in N(A)}\frac{|x|}{\gamma^2(|x|)}
= \limsup_{|x|\to\infty, x\in N(A)}\frac{|x|}{|x|^{\frac{6}{5}}}=0<+\infty,\\
\lim_{|x|\to\infty, x\in N(A)} \frac{1}{\gamma^2(|x|)}\int^T_0H(t,x)dt
=\lim_{|x|\to\infty, x\in N(A)}\frac{\frac{1}{4}T^2|x|^{8/5}}{|x|^{\frac{6}{5}}}
=+\infty
\end{gather*}
Hence, by Theorem \ref{thm4.1}, the corresponding system \eqref{eHcal} possesses at
least one $T$-periodic solution.
\end{example}

\begin{theorem} \label{thm4.2}
Assume {\rm (C1)} and {\rm (H1)} hold and
\begin{itemize}
\item[(H3)] $H$ satisfies
\[
\limsup_{|x|\to\infty, x\in N(A)}\frac{\gamma^2(|x|)}{|x|}<\infty,\quad
\lim_{|x|\to\infty}\frac{1}{|x|}\int^T_0H(t,x)dt=+\infty.
\]
\end{itemize}
Then  \eqref{eHcal} possesses at least one $T$-periodic solution.
\end{theorem}

\begin{theorem} \label{thm4.3}
Assume {\rm (C1)} and {\rm (H1)} hold and
\begin{itemize}
\item[(H4)] $H$ satisfies
\[
\limsup_{|x|\to\infty, x\in N(A)}\frac{\gamma^2(|x|)}{|x|}=0,\quad
\lim_{|x|\to\infty}\frac{1}{|x|}\int^T_0H(t,x)dt>\int^T_0|h(t)|dt.
\]
\end{itemize}
Then  \eqref{eHcal} possesses at least one $T$-periodic solution.
\end{theorem}

\begin{example} \label{examp4.2} \rm
 Let $A$ be the matrix defined in Example \ref{examp3.1} and let
$$
H(t,x)=(\frac{1}{2}T-t)ln^{\frac{3}{2}}(1+|x|^2)+\frac{l(t)|x|^{3}}{1+|x|^2},
\quad \forall x\in\mathbb{R}^{2N},\; \forall t\in [0,T],
$$
where $l\in C([0,T],\mathbb{R}^{+})$ with
$\int^T_0l(t)dt>\int^T_0|h(t)|dt$. Then
\begin{align*}
|\nabla H(t,x)|
&\leq\frac{3}{2}|\frac{1}{2}T-t|\big(\ln(1+|x|^2)\big)^{1/2}
\frac{|x|}{1+|x|^2}+\frac{l(t)(5|x|^{4})+3|x|^2)}{1+2|x|^2+|x|^{4}}\\
&\leq \frac{3}{2}|\frac{1}{2}T-t|\big(\ln(1+|x|^2)\big)^{1/2}
\frac{|x|}{1+|x|^2}+c_1
\end{align*}
where $c_1$ is a positive constant.
Let $\gamma(t)=\big(\ln(1+|t|^2)\big)^{1/2}$, $t\geq 0$.
It is clear that conditions (i), (ii), (iii) are satisfied. Moreover,
\begin{gather*}
\limsup_{|x|\to\infty, x\in N(A)}\frac{\gamma^2(|x|)}{|x|}
= \limsup_{|x|\to\infty, x\in N(A)}\frac{ln(1+|x|^2)}{|x|}=0<+\infty,
\\
\lim_{|x|\to\infty, x\in N(A)}\frac{1}{|x|}\int^T_0H(t,x)dt
=\int^T_0l(t)dt>\int^T_0|h(t)|dt.
\end{gather*}
Hence, by Theorem \ref{thm4.3}, the corresponding system \eqref{eHcal} possesses
at least one $T$-periodic solution.
\end{example}

\begin{theorem} \label{thm4.4}
Assume {\rm (C1)} and {\rm (H1)} hold and
\begin{itemize}
\item[(H5)] $H$ satisfies
\[
\int^T_0h(t)dt\bot N(A),\ \lim_{|x|\to\infty, x\in N(A)}
\frac{1}{\gamma^2(|x|)}\int^T_0H(t,x)dt=+\infty.
\]
\end{itemize}
Then \eqref{eHcal} possesses at least one $T$-periodic solution.
\end{theorem}

Theorem \ref{thm4.4} generalizes the result concerning the existence of periodic
solutions for \eqref{eHcal} in \cite[Theorem 3.1]{c1}.

\begin{example} \label{examp4.3}\rm
Let $A$ be the matrix defined in Example \ref{examp3.1} and let
$$
H(t,x)=(\frac{3}{4}T-t)ln^{\frac{3}{2}}(1+|x|^2)+l(t)\big(\ln(1+|x|^2)\big)^{1/2},
\quad x\in\mathbb{R}^{2N},\; t\in [0,T],
$$
where $l\in C([0,T],\mathbb{R}^{+})$ and $h(t)=c(t)v_1+d(t)v_2$, with
$v_1=(2,-1,0,-1), v_2=(0,0,1,-1) \in(N(A))^{\bot}$,
$c,d\in C(\mathbb{R},\mathbb{R})$. Then $\int^T_0h(t)dt\bot N(A)$ and
$$
|\nabla H(t,x)|\leq\frac{3}{2}|\frac{3}{4}T-t|\big(\ln(1+|x|^2)\big)^{1/2}+ l(t).
$$
Let $\gamma(t)=\big(\ln(1+|x|^2)\big)^{1/2}$, $t\geq 0$.
It is easy to verify that $\gamma$ satisfies conditions (i), (ii), (iii).
Moreover,
$$
\lim_{|x|\to\infty, x\in N(A)}\frac{1}{\gamma^2(|x|)}\int^T_0H(t,x)dt=
\lim_{|x|\to\infty, x\in N(A)}\frac{T^2}{4}\big(\ln(1+|x|^2)\big)^{1/2}=+\infty
$$
Hence, by Theorem \ref{thm4.4}, the corresponding system \eqref{eHcal} possesses at
least one $T$-periodic solution.
\end{example}

\begin{remark} \label{rmk4.1} \rm
Let $u(t)$ be a periodic solution of \eqref{eHcal}, then by replacing
$t$ by $-t$ in \eqref{eHcal}, we obtain
$$
\dot{u}(-t) = JH'(-t,u(-t)).
$$
So it is clear that the function $v(t)=u(-t)$ is a periodic solution of the system
$$
\dot{v}(t) = -JH'(-t,v(t)).
$$
Moreover, $-H(-t,x)$ satisfies (H2)--(H5) whenever $H(t,x)$ satisfies the
following assumptions
\begin{itemize}
\item[(H2')]
\[
\limsup_{|x|\to\infty, x\in N(A)}\frac{|x|}{\gamma^2(|x|)}<+\infty,\quad
 \lim_{|x|\to\infty,x\in N(A}\frac{1}{\gamma^2(|x|)}\int^T_0H(t,x)dt=-\infty;
\]

\item[(H3')]
\[
\limsup_{|x|\to\infty, x\in N(A)}\frac{\gamma^2(|x|)}{|x|}<\infty,\quad
\lim_{|x|\to\infty}\frac{1}{|x|}\int^T_0H(t,x)dt=-\infty;
\]

\item[(H4')]
\[
\limsup_{|x|\to\infty, x\in N(A)}\frac{\gamma^2(|x|)}{|x|}=0,\quad
\lim_{|x|\to\infty}\frac{1}{|x|}\int^T_0H(t,x)dt<-\int^T_0|h(t)|dt;
 \]

\item[(H5')]
\[
\int^T_0h(t)dt\bot N(A),\ \lim_{|x|\to\infty, x\in N(A)}
\frac{1}{\gamma^2(|x|)}\int^T_0H(t,x)dt=-\infty.
\]
\end{itemize}
Consequently, the previous Theorems remains true if we replace
(H2)--(H5) by (H2')--(H5').
\end{remark}

\subsection*{Proofs of Theorems}
Consider the functional
$$
\varphi(u)=\frac{1}{2}\int^T_0(J\dot{u}(t)\cdot u(t)+Au(t)\cdot u(t))dt
+\int^T_0H(t,u(t))]\,dt
-\int^T_0h(t)\cdot u(t))\,dt
$$
Let  $E$ be  the space introduced in Section 2.
By assumption (H1) and the property (ii) of $\gamma$, 
\cite[Proposition B37]{r1}
implies that $\varphi\in C^{1}(E,\mathbb{R})$ and the critical points
of $\varphi$ on $E$ correspond to the $T$-periodic solutions of
\eqref{eHcal}, moreover
$$
\varphi'(u)v=\int^T_0[J\dot{u}(t)+Au(t)+\nabla H(t,u(t))]\cdot v(t)\,dt
-\int^T_0h(t)\cdot v(t)\,dt.
$$

\begin{lemma} \label{lem4.1}
Assume {\rm (H1)} holds. Then for any $(PS)$ sequence
$(u_{n})\subset E$ of the functional $\varphi$, there exists a constant
 $c_0>0$ such that
\begin{equation}
\|\tilde{u}_{n}\|\leq c_0(\gamma(\|u^{0}_{n}\|)+1),\quad
 \forall n\in\mathbb{N}\label{e4.1}
 \end{equation}
where $\tilde{u}_{n}=u^{+}_{n}+u^{-}_{n}=u_{n}-u^{0}_{n}$,
with $u^{0}_{n}\in E^{0}$, $u^{-}_{n}\in E^{-}$, $u^{+}_{n}\in E^{+}$.
\end{lemma}

\begin{proof}
Let $(u_{n})_{n\in\mathbb{N}}$ be a $(PS)$ sequence, i.e. $\varphi(u_{n})$
is bounded and $\varphi'(u_{n})\to 0$ as $n\to\infty$. We have
$$
\varphi'(u_{n})(u^{+}_{n}-u^{-}_{n})=2\|\tilde{u}_{n}\|^2
+\int^T_0\nabla H(t,u_{n})\cdot (u^{+}_{n}-u^{-}_{n})dt
-\int^T_0h(t)\cdot(u^{+}_{n}-u^{-}_{n})dt.
$$
Since $\varphi'(u_{n})\to 0$ as $n\to\infty$, there exists a constant
 $c_2>0$ such that
\[
\big|\varphi'(u_{n})(u^{+}_{n}-u^{-}_{n})\big| \\
\leq c_2\|\tilde{u}_{n}\|,\ \forall n\in\mathbb{N}.
\]
By H\"older's inequality and (H1), we have
\begin{equation}
\begin{aligned}
\big|\int^T_0\nabla H(t,u_{n})\cdot(u^{+}_{n}-u^{-}_{n})dt\big|
&\leq \|\tilde{u}_{n}\|_{L^2}(\int^T_0|\nabla H(t,u_{n})|^2dt)^{1/2}\\
&\leq\|\tilde{u}_{n}\|_{L^2}(\int^T_0[p(t)\gamma(|u_{n}|)+q(t)]dt)^{1/2} \\
&\leq\|\tilde{u}_{n}\|_{L^2}\big[(\int^T_0p^2(t)\gamma^2(|u_{n}|)dt)^{1/2}
 +\|q\|_{L^2}\big].
\end{aligned}\label{e4.2}
\end{equation}
Now, by  nondecreasing condition and the properties (i) and (ii) of $\gamma$,
we have
\begin{align*}
\Big(\int^T_0p^2(t)\gamma^2(|u_{n}|)dt\Big)^{1/2}
&\leq\Big(\int^T_0p^2(t)\gamma^2(|\tilde{u}_{n}|+|u^{0}_{n}|)dt\Big)^{1/2}\\
&\leq c\Big(\int^T_0 [p^2(t)[\gamma(|\tilde{u}_{n}|)
 +\gamma(|u^{0}_{n}|)]^2dt\Big)^{1/2}\\
&\leq c\Big[\Big(\int^T_0p^2(t)\gamma^2(|\tilde{u}_{n}|)dt\Big)^{1/2}
 +\|p\|_{L^2}\gamma(|u^{0}_{n}|\Big)\Big]\\
&\leq c\Big[\Big(\int^T_0p^2(t)(a|\tilde{u}_{n}|^{\alpha}+b)^2dt\Big)^{1/2}
 + \|p\|_{L^2} \gamma(|u^{0}_{n}|)\Big]\\
&\leq c\Big[a\Big(\int^T_0p^2(t)|\tilde{u}_{n}|^{2\alpha}dt\Big)^{1/2}
 + b\|p\|_{L^2}+ \|p\|_{L^2} \gamma(|u^{0}_{n}|)\Big]\\
&\leq c\big[a\|p\|_{L^{\frac{2}{1-\alpha}}}\|\tilde{u}_{n}\|^{\alpha}_{L^2}
 + b\|p\|_{L^2}+\|p\|_{L^2}\gamma(|u^{0}_{n}|)\big].
\end{align*}
On the other hand, by \eqref{e2.1} we have
\begin{align*}
2\|\tilde{u}_{n}\|^2
&\leq|\varphi'(u_{n})(u^{+}_{n}-u^{-}_{n})|
 +|\int^T_0\nabla H(t,u_{n})\cdot(u^{+}_{n}-u^{-}_{n})dt|\\
&\quad +|\int^T_0h(t)\cdot(u^{+}_{n}-u^{-}_{n})|
\leq c_2\|\tilde{u}_{n}\|+\|\tilde{u}_{n}\|_{L^2}
 c\big[a\|p\|_{L^{\frac{2}{1-\alpha}}}\|\tilde{u}_{n}\|^{\alpha}_{L^2}\\
&\quad + b\|p\|_{L^2}
+\|q\|_{L^2}+\|p\|_{L^2}\gamma(|u^{0}_{n}|)\big]+\|h\|_{L^2}\|\tilde{u}_{n}\|_{L^2}\\
&\leq ac\|p\|_{L^{\frac{2}{1-\alpha}}}\lambda^{\alpha+1}_2
 \|\tilde{u}_{n}\|^{\alpha+1}+ [c_1+cb\|p\|_{L^2}\\
&\quad +\|q\|_{L^2}+\|h\|_{L^2}]\lambda_2\|\tilde{u}_{n}\|
+c\lambda_2\|p\|_{L^2}\gamma(|u^{0}_{n}|)\|\tilde{u}_{n}\|.
\end{align*}
Since $0\leq\alpha<1$, we deduce that there exists a constant $c_0>0$
satisfying \eqref{e4.1}.
\end{proof}

We will apply Lemma \ref{lem2.1} to the functional $\varphi$ to obtain critical points.

\begin{lemma} \label{lem4.2}
If {\rm (H1)} holds and $H$ satisfies one of the assumptions
{\rm (H2)--(H5)}, then $\varphi$ satisfies the $(PS)_{c}$ condition for
all $c\in\mathbb{R}$.
\end{lemma}

\begin{proof}
Let $(u_{n})_{n\in\mathbb{N}}$ be a $(PS)_{c}$ sequence, that is $\varphi(u_{n})\to c$
and $\varphi'(u_{n}) \to 0$ as $n\to\infty$. Then there exists a positive
constant $c_3$ such that
$$
|\varphi(u_{n})|\leq c_3,\quad \|\varphi'(u_{n})\|\leq c_3,\quad
\forall n\in\mathbb{N}.
$$
By the Mean Value Theorem and H\"older's inequality, we have
\begin{equation}
\begin{aligned}
&\big|\int^T_0(H(t,u_{n})-H(t,u^{0}_{n}))dt\big|\\
&=\big|\int^T_0\int^{1}_0\nabla H(t,u^{0}_{n}+s\tilde{u}_{n})\cdot
\tilde{u}_{n}\,ds\,dt\big|\\
&\leq\|\tilde{u}_{n}\|_{L^2}\int^{1}_0\Big(\int^T_0| \nabla H(t,u^{0}_{n}
+s\tilde{u}_{n})|^2dt\Big)^{1/2}ds.
\end{aligned}  \label{e4.3}
\end{equation}
As in the proof of Lemma \ref{lem4.1}, we have
\begin{equation}
\begin{aligned}
&\Big(\int^T_0| \nabla H(t,u^{0}_{n}+s\tilde{u}_{n})|^2dt\Big)^{1/2}\\
&\leq ac\|p\|_{L^{\frac{2}{1-\alpha}}}\|\tilde{u}_{n}\|^{\alpha}_{L^2}
 + cb\|p\|_{L^2}+\|q\|_{L^2}+c\|p\|_{L^2}\gamma(|u^{0}_{n}|)\|q\|_{L^2}\big].
\end{aligned} \label{e4.4}
\end{equation}
Therefore, by properties \eqref{e2.1}, \eqref{e4.1}, \eqref{e4.3}, \eqref{e4.4}
and since $0\leq\alpha<1$, there exists a positive constant $c_4$ such that
\begin{equation}
\begin{aligned}
\big|\int^T_0(H(t,u_{n})-H(t,u^{0}_{n}))dt\big|
&\leq c_0(\gamma(|u^{0}_{n}|)+1)[ac\|p\|_{L^{\frac{2}{1-\alpha}}}
 c^{\alpha}_0(\gamma(|u^{0}_{n}|)+1)^{\alpha}\\
&\quad +c\|p\|_{L^2}\gamma(|u^{0}_{n}|)+cb\|p\|_{L^2}+\|q\|_{L^2}]\\
&\leq c_4(\gamma^2(|u^{0}_{n}|)+1).
\end{aligned} \label{e4.5}
\end{equation}
Combining \eqref{e2.1}, \eqref{e4.1}, \eqref{e4.2} and \eqref{e4.5} yields
\begin{equation}
\begin{aligned}
c_3&\geq \varphi(u_{n})\\
&\geq -\|\tilde{u}_{n}\|^2+\int^T_0(H(t,u_{n})-H(t,u^{0}_{n}))dt
 +\int^T_0H(t,u^{0}_{n})dt \\
&\quad -\int^T_0h(t)(\tilde{u}_{n}+u^{0}_{n})dt\\
&\geq -c^2_0(\gamma(|u^{0}_{n})|+1)^2-c_4(\gamma^2(|u^{0}_{n})|)+1)
 -c_0\|h\|_{L^2}(\gamma(|u^{0}_{n})|+1) \\
&\quad -\|h\|_{L^{1}}|u^{0}_{n}|+ \int^T_0H(t,u^{0}_{n})dt\\
&\geq -c_5(\gamma^2(|u^{0}_{n})|)+1)-\|h\|_{L^{1}}|u^{0}_{n}|
 +\int^T_0H(t,u^{0}_{n})dt,
\end{aligned} \label{e4.6}
\end{equation}
where $c_5$ is a positive constant.
\smallskip

\noindent\textbf{Case 1:} $H$ satisfies (H2). By \eqref{e4.6}, we have
$$
c_3\geq \gamma^2(|u^{0}_{n})|)[-c_5-\|h\|_{L^{1}}
\frac{|u^{0}_{n}|}{\gamma^2(|u^{0}_{n})|)}
+\frac{1}{\gamma^2(|u^{0}_{n})|)}\int^T_0H(t,u^{0}_{n})dt]-c_5.
$$
It follows from (H2) that $(u^{0}_{n})$ is bounded.
\smallskip

\noindent\textbf{Case 2:} $H$ satisfies (H3) or (H4). Note that by \eqref{e4.6}
$$
c_3\geq |u^{0}_{n}|[-c_5\frac{\gamma^2(|u^{0}_{n})|)}{|u^{0}_{n})|}
-\|h\|_{L^{1}}+\frac{1}{|u^{0}_{n}|}\int^T_0H(t,u^{0}_{n})dt]-c_5.
$$
Hence (H3) or (H4) implies that $(u^{0}_{n})$ is bounded.
\smallskip

\noindent\textbf{Case 2:} $H$ satisfies (H5).
Since $\int^T_0h(t)dt\bot N(A)$, we get as in \eqref{e4.6}
\begin{equation}
\begin{aligned}
c_3&\geq \varphi(u_{n})\\
&\geq -\|\tilde{u}_{n}\|^2
+\int^T_0(H(t,u_{n})-H(t,u^{0}_{n}))dt +\int^T_0H(t,u^{0}_{n})dt\\
&\quad -\int^T_0h(t)\cdot\tilde{u}_{n}dt\\
&\geq -c_5(\gamma^2(|u^{0}_{n})|)+1)+\int^T_0H(t,u^{0}_{n})dt, \\
&\geq \gamma^2(|u^{0}_{n})|)[-c_5
+\frac{1}{\gamma^2(|u^{0}_{n})|)}\int^T_0H(t,u^{0}_{n})dt]-c_5.
\end{aligned} \label{e4.6b}
\end{equation}
Hence (H5) implies that $(u^{0}_{n})$ is bounded.

In all the above cases, $(u^{0}_{n})$ is bounded. We deduce from Lemma \ref{lem4.1}
that $(u_{n})$ is also bounded in $E$. By a standard argument,
we conclude that $(u_{n})$ possesses a convergent subsequence.
The proof of Lemma \ref{lem4.2} is complete.
\end{proof}

Now, decompose $E=E^{-}\oplus (E^{0}\oplus E^{+})$ and let
$E^{1}=E^{-}$ and $E^2=E^{0}\oplus E^{+}$. Remark that by Section 3,
 we have $E^{0}=N(A)$. We will verify that $\varphi$ satisfies condition c)
of Lemma \ref{lem2.1}.
For $u\in E^{1}$, we have
$$
\varphi(u)= -\|u\|^2+\int^T_0(H(t,u)-H(t,0))dt
+\int^T_0H(t,0)dt -\int^T_0h(t)\cdot u dt.
$$
As in the proof of Lemma \ref{lem4.2},
$$
\big|\int^T_0(H(t,u)-H(t,0))dt\big|
\leq [ac\|p\|_{L^{\frac{2}{1-\alpha}}}\|u\|^{\alpha}_{L^2}
+cb\|p\|_{L^2}+\|q\|_{L^2}]\|u\|_{L^2}.
$$
Hence by \eqref{e2.1}, we deduce
\begin{equation}
\begin{aligned}
\varphi(u)
&\leq -\|u\|^2 +ac\|p\|_{L^{\frac{2}{1-\alpha}}}\lambda^{\alpha+1}_2
 \|u\|^{\alpha+1} +(cb\|p\|_{L^2}\\
&\quad +\|q\|_{L^2}+\|h\|_{L^2})\lambda_2\|u\|+\int^T_0H(t,0)dt.
\end{aligned}\label{e4.7}
\end{equation}
Since $0\leq\alpha< 1$,  \eqref{e4.7} implies that $\varphi(u)\to-\infty$
 as $\|u\|\to\infty$. Hence there exists $\beta\in\mathbb{R}$ such that
$f(u)\leq\beta$ for all $u\in E^{1}$.
Condition (c) of Lemma \ref{lem2.1} is then proved.

Let us verify that $\varphi$ satisfies condition (d) of Lemma \ref{lem2.1}.
In fact, for $u\in E^2=E^{0}\oplus E^{+}$,  as in the proof of
Lemma \ref{lem4.2}, we have
\begin{equation}
\begin{aligned}
&\big|\int^T_0(H(t,u)-H(t,u^{0}))dt\big|\\
&\leq\big[ac\|p\|_{L^{\frac{2}{1-\alpha}}}\|u^{+}\|^{\alpha}_{L^2}
+ cb\|p\|_{L^2}+c\|p\|_{L^2}\gamma(|u^{0}|)+\|q\|_{L^2}\big]\|u^{+}\|^{\alpha}_{L^2}.
\end{aligned} \label{e4.8}
\end{equation}
From \eqref{e2.1} and \eqref{e4.8}, we deduce that
\begin{equation}
\begin{aligned}
\varphi(u)
&\geq\|u^{+}\|^2-ac\|p\|_{L^{\frac{2}{1-\alpha}}}\lambda^{\alpha+1}_2\|u^{+}
 \|^{\alpha+1}-c\|p\|_{L^2}\lambda_2\|u^{+}\|\gamma(|u^{0}|)\\
&\quad -(cb\|p\|_{L^2}+\|q\|_{L^2}+\|h\|_{L^2})\lambda_2\|u^{+}\|
-\int^T_0|h|dt|u^{0}|+\int^T_0H(t,u^{0})dt.
\end{aligned} \label{e4.9}
\end{equation}
For $\epsilon>0$, there exists a constant $C(\epsilon)$ such that
$$
c\|p\|_{L^2}\lambda_2\|u^{+}\|\gamma(|u^{0}|)
\leq\epsilon\|u^{+}\|^2+C(\epsilon)\gamma^2(|u^{0}|).
$$
Taking $\epsilon=1/2$, it follows from \eqref{e4.9} that
\begin{equation}
\begin{aligned}
\varphi(u)
&\geq\frac{1}{2}\|u^{+}\|^2-ac\|p\|_{L^{\frac{2}{1-\alpha}}}
 \lambda^{\alpha+1}_2\|u^{+}\|^{\alpha+1}
-\lambda_2(cb\|p\|_{L^2}+\|q\|_{L^2}\\
&\quad +\|h\|_{L^2})\|u^{+}\| -C(\frac{1}{2})\gamma^2(|u^{0}|)
-\int^T_0|h|dt|u^{0}|+\int^T_0H(t,u^{0})dt.
\end{aligned} \label{e4.10}
\end{equation}
Since $0\leq\alpha<1$, the term
$$
\frac{1}{2}\|u^{+}\|^2-ac\|p\|_{L^{\frac{2}{1-\alpha}}}\lambda^{\alpha+1}_2
\|u^{+}\|^{\alpha+1}
-\lambda_2(cb\|p\|_{L^2}+\|q\|_{L^2}+\|h\|_{L^2})\|u^{+}\|
$$
approaches $+\infty$ as $\|u^{+}\|\to\infty$.
It remains to study the following member of \eqref{e4.10}
$$
\psi(u^{0})=-C(\frac{1}{2})\gamma^2(|u^{0}|)
-\int^T_0|h|dt|u^{0}|+\int^T_0H(t,u^{0})dt.
$$
\smallskip

\noindent\textbf{Case 1:} (H2) holds. We have
$$
\psi(u^{0})\geq\gamma^2(|u^{0}|)\big(-C(\frac{1}{2})
-\int^T_0|h|dt\frac{|u^{0}|}{\gamma^2(|u^{0}|)}
+ \frac{1}{\gamma^2(|u^{0}|)}\int^T_0H(t,u^{0})dt\big).
$$
It follows from (H2) that $\psi(u^{0})\to +\infty$ as $|u^{0}|\to \infty$.
\smallskip

\noindent\textbf{Case 2:} (H3) or (H4) holds. We have
$$
\psi(u^{0})\geq|u^{0}|\big(-C(\frac{1}{2})\frac{\gamma^2(|u^{0}|)}{|u^{0}|}
-\int^T_0|h|dt+\frac{1}{|u^{0}|}\int^T_0H(t,u^{0})dt\big).
$$
It follows from (H3) or (H4) that $\psi(u^{0})\to +\infty$ as
 $|u^{0}|\to \infty$.
\smallskip

\noindent\textbf{Case 3:} (H5) holds. Since $\int^T_0h(t)dt\bot N(A)$, we have
$$
\psi(u^{0})\geq\gamma^2(|u^{0}|)\big(-C(\frac{1}{2})
+ \frac{1}{\gamma^2(|u^{0}|)}\int^T_0H(t,u^{0})dt\big).
$$
It follows from (H5) that $\psi(u^{0})\to +\infty$ as $|u^{0}|\to \infty$.

Therefore, if one of assumptions (H2)--(H5) is satisfied, then
$\varphi(u)\to +\infty$ as $\|u\|\to \infty$.
So there exists a constant $\rho$ such that $\varphi(u)\geq\rho$ for all
$u\in E^2$. Condition d) of Lemma \ref{lem2.1} is satisfied. Moreover, it is well
known that the derivative of the functional
$d(u)=\int^T_0H(t,u)dt-\int^T_0hu dt$ is compact. All the conditions
of Lemma \ref{lem2.1} are satisfied, so $\varphi$ possesses a critical point $u$
which is a $T$-periodic solution of system \eqref{eHcal}

\section{Second class of Hamiltonian systems}

For $A, H$ and $h$ be defined as in Section 4, we have the following result.

\begin{theorem} \label{thm5.1}
Let $\omega\in C(\mathbb{R}^{+},\mathbb{R}^{+})$ be a non-increasing function
 with the following properties:
\begin{itemize}
\item[(a)] $\liminf_{s\to\infty}\frac{\omega(s)}{\omega(\sqrt{s})}>0$,

\item[(b)] $\omega(s)\to 0$ and $\omega(s)s\to+\infty$ as $ s\to\infty$.
\end{itemize}
Assume that $A$ satisfies {\rm (C1)}, and $H$ satisfies
\begin{itemize}
\item[(H6)] There exist a positive constant $a$ and a function
$g\in L^2(0,T;\mathbb{R})$ such that
$$
|\nabla H(t,x)|\leq a\omega(|x|)+g(t),\quad \forall x\in\mathbb{R}^{2N},
\text{ a.e. } t\in[0,T];
$$

\item[(H7)]
$$
\lim_{|x|\to\infty,x\in N(A)}\frac{1}{(\omega(|x|)|x|)^2}\int^T_0H(t,x)dt
=+\infty;
$$

\item[(H8)] There exists $f\in L^{1}(0,T;\mathbb{R})$ such that
$$
H(t,x)\geq f(t),\quad \forall x\in\mathbb{R}^{2N},\text{ a.e. } t\in [0,T].
$$
\end{itemize}
Then system \eqref{eHcal} possesses at least one $T$-periodic solution.
\end{theorem}

The above theorem generalizes \cite[Theorem 1.1]{t3}.


\begin{example} \label{examp5.1}\rm
 Take $\omega(s)=\frac{1}{\ln(2+s^2)},\ s\geq 0,$
$$
H(t,x)=(\frac{1}{2}+\cos(\frac{2\pi}{T}t))\frac{|x|^2}{\ln(2+|x|^2)},\quad
 \forall t\in[0,T],\; \forall x\in\mathbb{R}^{2N}
$$
and let $A$ be the matrix defined in Section 3, $h\in C([0,T],\mathbb{R})$.
Then $A, H, h$ satisfy assumptions of Theorem \ref{thm5.1}.
\end{example}

\subsection*{Proof of Theorem \ref{thm5.1}}
 As in Section 4, we will apply  Lemma \ref{lem2.1} to the functional $\varphi$
defined on the space $E$ introduced in section 2.

\begin{lemma}[\cite{t3}] \label{lem5.1}
Assume {\rm (H6)} and {\rm (H7)} hold, then there exists a non-increasing function
$\theta\in C(]0,+\infty[,\mathbb{R}^{+})$ and a positive constant $c_0$ such that
\begin{itemize}
\item[(i)] $\theta(s)\to 0$ and $\theta(s)s\to\infty$  as $s\to\infty$,

\item[(ii)] $\|\nabla H(t,u)\|_{L^2}\leq c_0(\theta(\|u\|)\|u\|+1)$ for all
$u\in E$

\item[(iii)]
\[
\frac{1}{(\theta(\|u^{0}\|)\|u^{0}\|)^2}\int^T_0H(t,u^{0})dt\to+\infty
\quad\text{as } \|u^{0}\|\to\infty.
\]
\end{itemize}
\end{lemma}

\begin{lemma} \label{lem5.2}
Assume {\rm (H6)} holds. Then for any $(PS)$ sequence of the functional
$\varphi$, there exists a constant $c_1>0$ such that
\begin{equation}
\|\tilde{u}_{n}\|\leq c_1\big(\theta(\|u^{0}_{n}\|)\|u^{0}_{n}\|+1\big).
\label{e5.1}
\end{equation}
\end{lemma}

\begin{proof}
Let $(u_{n})$ be a Palais-Smale sequence, that is $(\varphi(u_{n}))$
is bonded and $\varphi'(u_{n})\to 0, as\ n\to\infty$. We have
$$
\varphi'(u_{n})(u^{+}_{n}-u^{-}_{n})=2\|\tilde{u}_{n}\|^2
+\int^T_0\nabla H(t,u_{n}(t))\cdot (u^{+}_{n}-u^{-}_{n})dt
-\int^T_0h(t)\cdot(u^{+}_{n}-u^{-}_{n})dt.
$$
Since $\theta$ is non-increasing and $\|u\|\geq max(\|\tilde{u}\|,\|u^{0}\|)$,
we have
\begin{equation}
\theta(\|u\|)\leq \min(\theta(\|\tilde{u}\|),\theta(\|u^{0}\|)). \label{e5.2}
\end{equation}
By H\"older's inequality, inequalities \eqref{e2.1}, \eqref{e5.1}, \eqref{e5.2}
and Lemma \ref{lem5.1}, we have
\begin{align*}
&\big|\int^T_0\nabla H(t,u_{n}(t))\cdot(u^{+}_{n}-u^{-}_{n})dt\big| \\
&\leq\|u^{+}_{n}-u^{-}_{n}\|_{L^2}(\int^T_0|\nabla H(t,u_{n})|^2dt)^{1/2}\\
&\leq c_2\|\tilde{u}_{n}\|(\theta(\|u_{n}\|)\|u_{n}\|+1) \\
&\leq c_2\|\tilde{u}_{n}\|\big(\theta(\|\tilde{u}_{n}\|)
 \|\tilde{u}_{n}\|+\theta(\|u^{0}_{n}\|)\|u^{0}_{n}\|+1\big).
\end{align*}
Thus there exists positive constants $c_3, c_4$ such that
\begin{align*}
c_3\|\tilde{u}_{n}\|
&\geq\varphi'(u_{n})(u^{+}_{n}-u^{-}_{n})\\
&\geq 2\|\tilde{u}_{n}\|^2-c_2\|\tilde{u}_{n}\|
\big(\theta(\|\tilde{u}_{n}\|)\|\tilde{u}_{n}\|
+\theta(\|u^{0}_{n}\|)\|u^{0}_{n}\|+1\big)-c_4\|\tilde{u}_{n}\|.
\end{align*}
Hence
$$
c_2\theta(\|u^{0}_{n}\|)\|u^{0}_{n}\|
\geq\|\tilde{u}_{n}\|[2-c_2\|\tilde{u}_{n}\|]-c_3-c_4.
$$
Since $\theta(s)\to 0$ as $s\to\infty$, this implies the existence
 of a constant $c_1$ satisfying \eqref{e5.1}.
\end{proof}

\begin{lemma} \label{lem5.3}
$\varphi$ satisfies the $(PS)_{c}$ condition for all real $c$.
\end{lemma}

\begin{proof}
Let $(u_{n})$ be a $(PS)_{c}$-sequence. Assume that $(u^{0}_{n})$ is unbounded.
Going to a subsequence if necessary, we can assume that
$\|u^{0}_{n}\|\to\infty$ as $n\to\infty$. By the Mean Value Theorem,
H\"older's inequality, inequality \eqref{e2.1} and Lemma \ref{lem5.1} (ii),
there exists a positive constant $c_5$ such that
\begin{equation}
\begin{aligned}
&\big|\int^T_0(H(t,u_{n})-H(t,u^{0}_{n}))dt\big|\\
&=\big|\int^T_0\int^{1}_0\nabla H(t,u^{0}_{n}
 +s\tilde{u}_{n})\cdot\tilde{u}_{n}\,ds\,dt\big|\\
&\leq \|\tilde{u}_{n}\|_{L^2}\int^{1}_0
\Big(\int^T_0|\nabla H(t,u^{0}_{n}+s\tilde{u}_{n})dt  |\Big)^{1/2}ds\\
&\leq c_5\|\tilde{u}_{n}\|\big[\theta(\|u^{0}_{n}\|)\|u^{0}_{n}\|
+ \theta(\|u^{0}_{n}\|)\|\tilde{u}_{n}\|+1\big].
\end{aligned} \label{e5.3}
\end{equation}
Hence by Lemma \ref{lem5.2}, there exists a positive constant $c_{6}$ such that
\begin{equation}
\big|\int^T_0(H(t,u_{n})-H(t,u^{0}_{n}))dt\big|
\leq c_{6}\big([\theta(\|u^{0}_{n}\|)\|\tilde{u}^{0}_{n}\|]^2+1\big).
 \label{e5.4}
\end{equation}
Combining \eqref{e2.1}, \eqref{e5.1} and \eqref{e5.4} yields
$$
\varphi(u_{n})\geq-c_{7}\big([\theta(\|u^{0}_{n}\|)\|\tilde{u}^{0}_{n}\|]^2+1\big)
-\frac{1}{T}\int^T_0|h(t)|dt\|u^{0}_{n}\|+ \int^T_0H(t,u^{0}_{n})dt
$$
where $c_{7}$ is a positive constant.

On the other hand, it is easy
to see that $\lim inf_{s\to\infty}\frac{\theta(s)}{\theta(\sqrt{s})}>0$.
So there exists a positive constant $c_{8}$ such that for $s$ large enough
$\theta(s)\geq c_{8}\theta(\sqrt{s})$. Hence for $n$ large enough
$$
\frac{\|u^{0}_{n}\|}{[\theta(\|u^{0}_{n}\|)\|u^{0}_{n}\|]^2}
\geq\frac{1}{c^2_{8}[\theta(\|u^{0}_{n}\|^{1/2})\|u^{0}_{n}\|^{1/2}]^2} \to 0\quad
\text{as } n\to\infty\,.
$$
Therefore,
\begin{align*}
\varphi(u_{n})
&\geq [\theta(\|u^{0}_{n}\|)\|u^{0}_{n}\|]^2[-c_{7}
 -\frac{1}{T}\int^T_0|h(t)|dt\frac{\|u^{0}_{n}\|}{[\theta(\|u^{0}_{n}\|)
\|u^{0}_{n}\|]^2} \\
&\quad +\frac{1}{[\theta(\|u^{0}_{n}\|)\|u^{0}_{n}\|]^2}
 \int^T_0H(t,u^{0}_{n})dt]-c_{7} \to+\infty
\end{align*}
as $n\to\infty$,
which contradicts the boundedness of $(\varphi(u_{n}))$.
 Hence $(\|u^{0}_{n}\|)$ is bounded, and by Lemma \ref{lem5.2},
 $(u_{n})$ is also bounded. By a standard argument, we conclude that
$(u_{n})$ possesses a convergent subsequence. The proof  is complete.
\end{proof}

Now, for $u=u^{0}+u^{+}\in E^2=E^{0}\oplus E^{+}$, we have as in \eqref{e5.3},
$$
|\int^T_0(H(t,u)-H(t,u^{0}))dt|\leq c_5\|u^{+}\|\big[\theta(\|u^{0}\|)\|u^{0}\|
+ \theta(\|u^{0}\|)\|u^{+}\|+1\big].
$$
Since $c_5\theta(\|u^{0}\|)\|u^{0}\|\|u^{+}\|
\leq\frac{1}{2}\|u^{+}\|^2+2c^2_5[\theta(\|u^{0}\|)\|u^{0}\|]^2$, we obtain
\begin{align*}
\varphi(u)
&\geq(\frac{1}{2}-c_5\theta(\|u^{0}\|))\|u^{+}\|^2-c_5\|u^{+}\|\\
&\quad +\big[\theta(\|u^{0}\|)\|u^{0}\|\big]^2
 \Big(-2c^2_5 -\frac{1}{T}\int^T_0|h(t)|dt
\frac{\|u^{0}\|}{[\theta(\|u^{0}\|)\|u^{0}\|]^2} \\
&\quad +\frac{1}{[\theta(\|u^{0}\|)\|u^{0}\|]^2}\int^T_0H(t,u^{0})dt\Big).
\end{align*}
Since $\theta(s)\to 0$ as $s\to\infty$, there exists $r>0$ such that
$c_5\theta(s)\leq\frac{1}{4}$ for $s\geq r$. Then, if $\|u^{0}\|\geq r$, we have
\begin{align*}
\varphi(u)
&\geq\frac{1}{4}\|u^{+}\|^2-c_5\|u^{+}\|
+[\theta(\|u^{0}\|)\|u^{0}\|]^2(-2c^2_5 \\
&\quad -\frac{1}{T}\int^T_0|h(t)|dt
\frac{\|u^{0}\|}{[\theta(\|u^{0}\|)\|u^{0}\|]^2}
+\frac{1}{[\theta(\|u^{0}\|)\|u^{0}\|]^2}\int^T_0H(t,u^{0})dt).
\end{align*}
then $\varphi(u)\to+\infty$ as $\|u^{0}+u^{+}\|\to\infty$, $\|u^{0}\|\geq r$.

If $\|u^{0}\|\leq r$, we have by (H8) and \eqref{e2.1}
$$
\varphi(u)\geq\|u^{+}\|^2+\int^T_0f(t)dt
 -\frac{r}{T}\int^T_0|h(t)|dt-\lambda_2\|h\|_{L^2}\|u^{+}\|
$$
then $\varphi(u)\to+\infty$ as $\|u^{0}+u^{+}\|\to\infty$, $\|u^{0}\|\leq r$.
Therefore $\varphi(u)\to+\infty$ as $\|u\|\to\infty$, $u\in E^2$.

In $E^{1}$,  as in \cite{t3}, we obtain $\varphi(u)\to-\infty$ as $\|u\|\to\infty$.
Hence, by Lemma \ref{lem2.1}, $\varphi$ possesses at least a critical point
$u$ which is a $T$-periodic solution of \eqref{eHcal}.

\begin{thebibliography}{00}


\bibitem{a1}  T. An;
\emph{Periodic solutions of superlinear autonomous Hamiltonian systems
with prescribed period}, J. Math. Anal. Appl. 323 (2006), pp. 854-863.

\bibitem{c1} A. R. Chouikha, M. Timoumi;
\emph{Subharmonic solutions for nonautonomous sublinear first order
Hamiltonian systems}, Arxiv, 1302-4309 V1 (math.DS) 18, Feb 2013.

\bibitem{e1} I. Ekeland;
\emph{Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz},
J. Diff. Eq. (1979), pp. 523-534.

\bibitem{f1} P-L. Felmer;
\emph{Periodic solutions of superquadratic Hamiltonian systems},
J. Diff. Eq. 102 (1993), pp. 188-307.

\bibitem{g1} M. Gilardi, M. Matzeu;
\emph{Periodic solutions of convex autonomous Hamiltonian systems with a
quadratic growth at the origin and superquadratic at infinity},
Annali di Matematica Pura and Applicata-(IV), vol. CXL VII, pp 21-72.

\bibitem{l1} C. Li, Z-Q. Ou, C-L. Tang;
\emph{Periodic and subharmonic solutions for a class of non-autonomous
 Hamiltonian systems}, Nonlinear Analysis 75 (2012), pp 2262-2272.

\bibitem{l2} S. Li, A. Szulkin;
\emph{Periodic solutions for a class of non-autonomous Hamiltonian systems},
J. Diff. Eq., (1994), pp. 226-238.

\bibitem{l3} S. Li, M. Willem;
\emph{Applications of local linking to critical point theory},
J. Math. Anal. Appl. 189, (1995), pp. 6-32.

\bibitem{l4} S. Luan, A. Mao;
\emph{Periodic solutions for a class of non-autonomous Hamiltonian systems},
Nonlinear Analysis 61, (2005), pp. 1413-1426.

\bibitem{m1} J. Mawhin, M. Willem;
\emph{Critical point theory and Hamiltonian systems},
Applied Mathematical Sciences, 74, Springer, New York, 1989.

\bibitem{r1} P. H. Rabinowitz;
\emph{Minimax methods in critical point theory with applications
to differential equations}, CBMS. Reg. Conf. Ser. Math., vol 65,
 Amer. Math. Soc., Providence, RI (1986).

\bibitem{r2} P. Rabinowitz;
\emph{Periodic solutions of Hamiltonian systems},
Comm. Pure Appl. Math. 31 (1978), pp. 157-184.

\bibitem{t1} M. Timoumi;
\emph{Periodic and subharmonic solutions for a class of non coercive
superqadratic Hamiltonian systems}, Nonl. Dyn. and Syst. Theory, 11(3) (2011),
 pp. 319-336.

\bibitem{t2} M. Timoumi;
\emph{Periodic solutions for non coercive super-quadratic Hamiltonian systems},
Dem. Mathematica, Vol. XI, No 2, (2007), pp. 331-346.

\bibitem{t3} M. Timoumi;
\emph{Subharmonic solutions for first-order Hamiltonian systems},
Elect. J. Diff. Eq., Vol. 2013 (2013), No. 197, pp. 1-14.

\bibitem{x1} X. Xu;
\emph{Periodic solutions for non-autonomous Hamiltonian systems possessing
super-quadratic potentials}, Nonlinear Analysis 51, (2002), pp. 941-955.

\end{thebibliography}
\end{document}