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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 197, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/197\hfil Subharmonic solutions]
{Subharmonic solutions for first-order Hamiltonian systems}

\author[M. Timoumi \hfil EJDE-2013/197\hfilneg]
{Mohsen Timoumi}

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

\thanks{Submitted March 31, 2013. Published September 5, 2013.}
\subjclass[2000]{34C25}
\keywords{Hamiltonian systems; periodic solutions; subharmonic;
\hfill\break\indent minimal periods; generalized saddle point theorem}

\begin{abstract}
 In this article, we study the existence of periodic and subharmonic
 solutions for a class of non-autonomous first-order Hamiltonian systems
 such that the nonlinearity  has a growth at infinity
 faster than $|x|^{\alpha}$, $0\leq\alpha < 1$.
 We also study the minimality of periods for such solutions.
 Our results are illustrated by specific examples.
 The proofs are based on the least action principle and a generalized
 saddle point theorem.
\end{abstract}

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

\section{Introduction and statement of main results}

Consider the non-autonomous first-order Hamiltonian system
\begin{equation}
\dot{x}(t)=JH'(t,x(t)) \label{eH}
\end{equation}
where $J$ is the standard symplectic $(2N\times 2N)$-matrix
$$
J=\begin{pmatrix}
0 & -I\\
I & 0
\end{pmatrix}
$$
$H:\mathbb{R}\times\mathbb{R}^{2N}\to\mathbb{R}$, $(t,x)\to H(t,x)$ 
is a continuous function
$T$-periodic $(T>0)$ in $t$ and differentiable with respect to the second
variable such that the derivative $H'(t,x)=\frac{\partial H}{\partial x}(t,x)$
is continuous on $\mathbb{R}\times\mathbb{R}^{2N}$. 
In this article, we are interested first
in the existence of periodic solutions and then the existence of subharmonics
for the system \eqref{eH}. Assuming that $T>0$ is the minimal period of the
time dependence of $H(t,x)$, by subharmonic solution of \eqref{eH} we mean
a $kT$-periodic solution, where $k$ is any integer; when moreover the
periodic solution is not $T$-periodic we call it a true subharmonic.

Using variational methods, there have been various types of results
concerning the existence of subharmonic solutions for Hamiltonian
systems of first and second order. Most of these works treats the
case of second order. Few results have been obtained for the case of
the first order, include for example
\cite{d1,e1,l1,m2,r2,s2,t3,w1}.
Many solvability conditions are given, such as the convexity condition
(see \cite{e1,m2,t3,w1}),
the super-quadratic condition (see \cite{l1,o1,r2}),
the sub-quadratic condition (see \cite{s2}),
the bounded nonlinearity condition (see \cite{t2}) and the sub-linear
condition (see \cite{d1}).
In particular, under the assumptions that $H$ satisfies
\begin{gather}
|H'(t,x)|\leq f(t),\quad \forall x\in\mathbb{R}^{2N},\text{ a.e. } t\in[0,T],
\label{e1.1} \\
H(t,x)\to +\infty\quad\text{as $|x|\to \infty$, uniformly in $t\in [0,T]$},
 \label{e1.2}
\end{gather}
where $f\in L^{1}(0,T;\mathbb{R}^{+})$. It was shown in \cite{t2} that  \eqref{eH}
has subharmonic solutions. In 2007, Daouas and Timoumi \cite{d1} generalized
these results with a condition of sublinear growth.
Precisely, it was assumed that the nonlinearity satisfies
the  following assumptions: There exist $\alpha\in [0,1[$,
$f\in L^{\frac{2}{1-\alpha}}(0,T;\mathbb{R}^{+})$ and $g\in L^{2}(0,T;\mathbb{R}^{+})$
such that
\begin{equation} \label{e1.3}
|H'(t,x)|\leq f(t)|x|^{\alpha}+g(t),\quad \forall x\in\mathbb{R}^{2N},
\text{ a.e. }t\in[0,T];
\end{equation}
\begin{equation}
\lim_{|x|\to\infty}\frac{1}{|x|^{2\alpha}}\int^{T}_{0}H(t,x)=+\infty; \label{e1.4}
\end{equation}
There exist a subset $C$ of $[0,T]$ with $\operatorname{meas}(C)>0$ and
$h\in L^{1}(0,T;\mathbb{R})$ such that
\begin{equation}
\begin{gathered} \label{e1.5}
\lim_{|x|\to\infty}H(t,x)=+\infty,\quad\text{a.e. } t\in C, \\
H(t,x)\geq h(t),\quad \forall x\in\mathbb{R}^{2N},\text{ a.e. } t\in [0,T].
\end{gathered}
\end{equation}
Under these conditions, subharmonic solutions of \eqref{eH} have been obtained.

In all the results discussed above \cite{d1,t2}, the nonlinearity is required
to grow at infinity at most like $|x|^{\alpha}$. Consider the Hamiltonian
\begin{equation}
H(t,x)=\gamma(t)\frac{|x|^{2}}{\ln(2+|x|^{2})},\quad
 t\in[0,T],\; x\in\mathbb{R}^{2N}\label{e1.6}
\end{equation}
where
$$
\gamma(t)=\begin{cases}
\sin(2\pi t/T), & t\in [0,T/2]\\
0,&t\in[T/2,T],
\end{cases}
$$
It is easy to verify that $H$ does not satisfy \eqref{e1.1}, \eqref{e1.2}
nor \eqref{e1.3}, \eqref{e1.4}, \eqref{e1.5}. In 2012,
Timoumi \cite{t4} established the existence of subharmonic solutions for the
second-order Hamiltonian system
$$
\ddot{u}(t)+V'(t,u(t))=0
$$
under some conditions which cover a case analogous to  \eqref{e1.6}.
The goal of the present article is to obtain the existence of periodic
 and subharmonic solutions for first-order Hamiltonian system \eqref{eH}
under conditions covering the case of \eqref{e1.6} and such that the
nonlinearity  has a growth at infinity faster than $|x|^{\alpha}$,
$0\leq \alpha<1$. Our main results are the following theorems.

\begin{theorem} \label{thm1.1}
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)] $\omega(s)\to 0$, $\omega(s)s\to\infty$ as $s\to\infty$.
\end{itemize}
Assume that $H$ satisfies
\begin{itemize}
\item[(H1)] There exist a positive constant $a$ and a function
$g\in L^{2}(0,T;\mathbb{R})$ such that
$$
|H'(t,x)|\leq a\omega(|x|)|x|+g(t),\quad \forall x\in\mathbb{R}^{2N},
\text{ a.e. } t\in [0,T];
$$
\item[(H2)]
$$
\frac{1}{[\omega(|x|)|x|]^{2}}\int^{T}_{0}H(t,x)dt \to+\infty\quad
\text{as } |x|\to\infty.
$$
\end{itemize}
Then system \eqref{eH} possesses at least one $T$-periodic solution.
\end{theorem}

\begin{example} \label{examp1.1} \rm
Consider the Hamiltonian
$$
H(t,x)=(\frac{1}{2}+\cos(\frac{2\pi}{T}t))\frac{|x|^{2}}{\ln(2+|x|^{2})},\quad
 t\in[0,T],\; x\in\mathbb{R}^{2N},
$$
and let $\omega: \mathbb{R}^{+}\to\mathbb{R}^{+}$ defined by
$$
\omega(s)=\frac{1}{\ln(2+s^{2})},\ s\geq 0.
$$
A simple computation shows that $\omega$ is a continuous non-increasing
function satisfying the conditions $(a), (b)$ and the Hamiltonian $H$
satisfies assumptions (H1), (H2) of Theorem \ref{thm1.1} and does not
satisfy the conditions \eqref{e1.1}, \eqref{e1.2} nor \eqref{e1.3}, \eqref{e1.4}.
\end{example}

\begin{theorem} \label{thm1.2}
 Assume that $H$ satisfies {\rm (H1), (H2)} and
\begin{itemize}
\item[(H3)] There exist a subset $C$ of $[0,T]$ with
$\operatorname{meas}(C)>0$ and a
$T$-periodic function $h\in L^{1}(0,T;\mathbb{R})$ such that
\begin{itemize}
\item[(i)] $\lim_{|x|\to\infty}H(t,x)=+\infty$, a.e. $t\in C$,

\item[(ii)] $H(t,x)\geq h(t)$, for all $x\in\mathbb{R}^{2N}$, a.e. $t\in [0,T]$.
\end{itemize}
\end{itemize}
Then, for all integer $k\geq 1$, Equation \eqref{eH} possesses a $kT$-periodic
solution $x_{k}$ such that
\begin{equation}
\lim_{k\to\infty}\|x_{k}\|_{\infty}=+\infty, \label{e1.7}
\end{equation}
where $\|x\|_{\infty}=\sup_{t\in\mathbb{R}}|x(t)|$.
\end{theorem}

\begin{corollary} \label{coro1.1}
Assume that $H$ satisfies {\rm (H1)} and
\begin{itemize}
\item[(H4)] There exist a subset $C$ of $[0,T]$ with $\operatorname{meas}(C)>0$
and a $T$-periodic function $h\in L^{1}(0,T;\mathbb{R})$ such that
\begin{itemize}
\item[(i)]
$\lim_{|x|\to\infty}\frac{H(t,x)}{[\omega(|x|)|x|]^{2}}
=+\infty$, a.e.  $t\in C$ ,

\item[(ii)] $H(t,x)\geq h(t)$ for all $x\in\mathbb{R}^{2N}$, a.e. $t\in [0,T]$.
\end{itemize}
\end{itemize}
Then the conclusion of Theorem \ref{thm1.2} holds.
\end{corollary}

\begin{example} \label{examp1.2} \rm
Let $\omega: \mathbb{R}^{+}\to\mathbb{R}^{+}$ be defined as in example \ref{examp1.1}.
 It is easy to see that the Hamiltonian $H$ defined in \eqref{e1.6}
satisfies assumptions (H1), (H2) and (H3) of Theorem \ref{thm1.2}
with $C=]0,T/4[$ and does not satisfy the conditions \eqref{e1.1}, \eqref{e1.2}
 nor \eqref{e1.3}, \eqref{e1.4}, \eqref{e1.5}.
\end{example}

Concerning the minimality of periods, we have the following result.

\begin{theorem} \label{thm1.3}
Assume that $H$ satisfies {\rm (H1)} and
\begin{itemize}
\item[(H5)]
$\frac{H'(t,x).x}{[\omega(|x|)|x|]^{2}}\to+\infty$ as
$|x|\to\infty$  uniformly in $t\in [0,T]$.
\end{itemize}
Then, for all integer $k\geq 1$, \eqref{eH} possesses a $kT$-periodic
solution $x_{k}$ such that $\lim_{k\to\infty}\|x_{k}\|_{\infty}=+\infty$.
\end{theorem}

Moreover, if the following assumption holds
\begin{itemize}
\item[(H1')]
If $u(t)$ is a periodic function with minimal period $rT$,
$r$ rational, and $H'(t,u(t))$ is a periodic function with
 minimal period $rT$, then $r$ is necessarily an integer,
\end{itemize}
then, for any sufficiently large prime number $k$, $kT$ is the minimal
 period of $x_{k}$.

Here, $x.y$ denotes the usual inner product of $x,y\in\mathbb{R}^{2N}$.

\begin{example} \label{examp1.3}\rm
The Hamiltonian
$$
H(t,x)=(\frac{3}{2}+\cos(\frac{2\pi}{T}t))\frac{|x|^{2}}{\ln(2+|x|^{2})},
$$
satisfies (H1), (H5) and (H1') with $\omega(s)=\frac{1}{\ln(2+s^{2})}$.
\end{example}

\begin{remark} \label{rmk1.1}\rm
 Let $x(t)$ be a periodic solution of \eqref{eH}. By replacing $t$ by $-t$
in \eqref{eH}, we obtain
$$
J\dot{x}(-t)+H'(-t,x(-t))=0.
$$
So, it is clear that the function $y(t)=x(-t)$ is a periodic solution
of the system
$$
J\dot{y}(t)-H'(-t,y(t))=0.
$$
Moreover, $-H(-t,x)$ satisfies (H2), (H3), (H4), (H5) whenever $H(t,x)$
satisfies respectively the following assumptions:
\begin{itemize}
\item[(H2')]
$$
\frac{1}{[\omega(|x|)|x|]^{2}}\int^{T}_{0}H(t,x)dt \to-\infty\quad
\text{as }|x|\to\infty;
$$
\item[(H3')] There exist a subset $C$ of $[0,T]$ with
$\operatorname{meas}(C)>0$ and a $T$-periodic function
 $h\in L^{1}(0,T;\mathbb{R})$ such that
\begin{itemize}
\item[(i)] $\lim_{|x|\to\infty}H(t,x)=-\infty$, a.e. $t\in C$,
\item[(ii)] $ H(t,x)\leq h(t)$ for all $x\in\mathbb{R}^{2N}$, a.e. $t\in [0,T]$;
\end{itemize}

\item[(H4')] There exist a subset $C$ of $[0,T]$ with $\operatorname{meas}(C)>0$
and a $T$-periodic function $h\in L^{1}(0,T;\mathbb{R})$ such that
\begin{itemize}
\item[(i)] $\lim_{|x|\to\infty}\frac{H(t,x)}{[\omega(|x|)|x|]^{2}}=-\infty$,
 a.e.  $t\in C$,

\item[(ii)] $H(t,x)\leq h(t)$ for all $x\in\mathbb{R}^{2N}$, a.e. $t\in [0,T]$;
\end{itemize}

\item[(H5')] $\frac{H'(t,x).x}{[\omega(|x|)|x|]^{2}}\to-\infty$
 as $|x|\to\infty$, uniformly in $t\in [0,T]$.
\end{itemize}
Consequently, we can replace (H2)--(H5) in the above results
by (H2')--(H5') and obtain the same results.
\end{remark}

\section{Preliminaries}

Let $L^{2} (S^{1},\mathbb{R}^{2N})$ denote the set of $2N$-tuples of $T$-periodic
functions which are square integrable on $S^1 = \mathbb{R}/{T\mathbb{Z}}$.
If $x \in L^2(S^1,\mathbb{R}^{2N})$, then $x$ has a Fourier expansion
$$
x(t) \simeq \sum_{m\in\mathbb{Z}}\exp\big(2\pi mtJ/T\big)\hat x_m
$$
where $\hat x_m \in \mathbb{R}^{2N}$ and $\sum_{m\in\mathbb{Z}}|\hat x_m|^2 < \infty$.
Consider the Sobolev space
$$
E=H^{1/2}_{T}= \{ x\in L^2 (S^1,\mathbb{R}^{2N}):  \|x\|_{H^{1/2}_{T}} < \infty\}
$$
where
$$
\|x\|_{H^{1/2}_{T}} = [\sum_{m\in\mathbb{Z}} (1 + |m|)|\hat x_m|^2]^{1/2}.
$$
It is well known that the space $(E,\|\cdot\|_{H^{1/2}_{T}})$ is a Hilbert space.
For $x\in E$, let
$$
Q(x) = \frac{1}{2}\int^{T}_{0}J \dot{x}(t)\cdot x(t)dt.
$$
Note that $Q$ is a quadratic form and by an easy calculation we obtain
\begin{equation}
Q(x) = -\pi \sum_{m\in \mathbb{Z}}m |\hat{x}_{m}|^2,\quad \forall x\in E. \label{e2.1}
\end{equation}
Therefore, $Q$ is a continuous quadratic form on $E$.

Consider the subspaces of $E$:
\begin{gather*}
E^{0} = \mathbb{R}^{2N},\\
E^{-} = \{ x\in E : x(t) \simeq \sum_{m\geq 1}
\exp\big(2\pi mtJ/T\big){\hat x}_m ,\text{ a.e. } t\in [0,T]\},\\
E^{+} = \{ x\in E : x(t) \simeq \sum_{m\leq -1}
 \exp\big(2\pi mtJ/T\big){\hat x}_m,\text{ a.e. } t\in [0,T] \},
\end{gather*}
here $E^{0}$ denotes the space of constant functions. 
Then $E = E^{0} \oplus E^{+} \oplus E^{-}$. In fact it is not difficult 
to verify that $E^{+}, E^{-}, E^{0}$ are respectively the subspaces
of $E$ on which $Q$ is positive definite, negative definite and null; 
and these subspaces are orthogonal with respect to the bilinear form
$$
B(x,y) = \frac{1}{2}\int^{T}_{0}J\dot{x}(t)\cdot y(t) dt,\quad x,y \in E,
$$
associated with $Q$. If $x\in E^{+}$ and $y\in E^{-}$, then $B(x,y) = 0$ and 
$Q(x+y) = Q(x) + Q(y)$. It is also easy to check that $E^{+}, E^{-}$ 
and $E^{0}$ are mutually orthogonal in $L^{2}(S^{1},\mathbb{R}^{2N})$.
It follows that if $x = x^{+} + x^{-} +x^{0} \in E$, then the expression
\begin{equation}
\|x\| = [Q(x^{+}) - Q(x^{-}) + |x^{0}|^{2}]^{1/2} \label{e2.2}
\end{equation}
is an equivalent norm in $E$. So in the following, we will use the norm
defined in \eqref{e2.2} as the norm for $E$. The subspaces
$E^{+}, E^{-}, E^{0}$ are mutually orthogonal with respect to the
associated inner product.

To prove our main results, the following auxiliary result will be needed.

\begin{proposition}[\cite{r2}] \label{prop2.1}  
For all $s\in [1,\infty[$, the space $E$ is compactly embedded in 
$L^{s}(S^{1},\mathbb{R}^{2N})$. In particular there is a constant $\alpha_{s} > 0$
such that for all $x\in E$,
$$
\|x\|_{L^{s}} \leq \alpha_{s} \|x\|.
$$
\end{proposition}

The following result is a version of the saddle point theorem.

\begin{theorem}[\cite{s2}] \label{thm2.1} \rm
 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)] $f(u)=\frac{1}{2}(Lu,u)+d(u)$ and $Lu=L_{1}P_{1}u +L_{2}P_{2}u$,
  with $L_{i}:E^{i}\to E^{i}$ bounded and self-adjoint, $i=1,2$;

\item[(b)] $d'(u)$ 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{theorem}

\begin{remark} \label{rmk2.1} \rm
In (c) we may replace $E^{2}$ by $\varphi_{1}+E^{2}$, $\varphi_{1}\in E^{1}$. 
Likewise in (b) we may have $\varphi_{2}+E^{1}$, $\varphi_{2}\in E^{2}$, 
in place of $E^{1}$.
\end{remark}

\section{Proofs of theorems} 

By making the change of variables $t\to t/k$,  transforms system \eqref{eH} 
into
\begin{equation}
J\dot{u}(t)+kH'(kt,u(t))=0. \label{eHk}
\end{equation}
Hence, to find $kT$-periodic solutions of \eqref{eH}, it suffices to find
$T$-periodic solutions of $({\mathcal H}_{k})$.
Consider the family of functionals $(\Phi_{k})_{k\in\mathbb{B}}$ defined on the
space $E$ introduced above by
$$
\Phi_{k}(u)=\int^{T}_{0}[\frac{1}{2}J\dot{u}(t)\cdot u(t)+kH(kt,u(t))]dt.
$$
Since $\omega$ is bounded, we have by (H1)
\begin{equation}
|H'(t,x)|\leq a\sup_{s\geq 0}\omega(s)|x|+g(t). \label{e3.1}
\end{equation}
So, by \cite[Proposition B.37]{r2},
$\Phi_{k}\in C^{1}(E,\mathbb{R})$ and critical points of $\Phi_{k}$ on $E$
correspond to the $T$-periodic solutions of $({\mathcal H}_{k})$,
moreover one has
\begin{equation}
\Phi'_{k}(u)v=\int^{T}_{0}[J\dot{u}(t)+kH'(kt,u(t))]\cdot v(t)dt,\quad
 \forall u,v\in E. \label{e3.2}
\end{equation}
The following lemma will be needed for the study of the geometry of the
functionals $\Phi_{k}$.

\begin{lemma} \label{lem3.1} 
Assume that (H1), (H2) hold, then there exist a non-increasing function 
$\theta\in C(]0,\infty[,\mathbb{R}^{+})$ and a positive constant $c_{0}$
satisfying the following conditions
\begin{itemize}
\item[(i)] $\theta(s)\to 0$, $\theta(s)s\to +\infty$ as $s\to\infty$, 
\item[(ii)] $\|H'(t,u)\|_{L^{2}}\leq c_{0}[\theta(\|u\|)\|u\|+1]$ for all 
$u\in E$, 
\item[(iii)] 
$$
\frac{1}{[\theta(|u^{0}|)|u^{0}|]^{2}}\int^{T}_{0}H(t,u^{0})dt \to
+\infty\quad\text{as }|u^{0}|\to +\infty. 
$$
\end{itemize}
\end{lemma}

The proof of the above lemma is similar to that of \cite[Lemma 2.1]{t4},
and it is ommitted here.
Now, we  show that, for every positive integer $k$, one can find a
 critical point $u_{k}$ of $\Phi_{k}$. To this aim, we will apply 
the saddle point theorem to each of the $\Phi_{k}$'s. 
Let us fix $k$ and consider the subspaces $E^{1}=E^{-}$, 
$E^{2}=E^{0}\oplus E^{+}$ of $E$. First, we prove the Palais-Smale.

\begin{lemma} \label{lem3.2} 
Assume that {\rm (H1)} and {\rm (H2)} hold. Then for every integer
 $k\geq 1$, the functional $\Phi_{k}$ satisfies the Palais-Smale condition.
\end{lemma}

\begin{proof}
Let $(u_{n})$ be a sequence of $E$ such that $(\Phi_{k}(u_{n}))$ is bounded 
and $\Phi'_{k}(u_{n})\to 0$ as $n\to\infty$. Let us denote 
$\tilde{u}_{n}=u^{+}_{n}+u^{-}_{n}$. We have
\begin{equation}
\Phi'_{k}(u_{n})(u^{+}_{n}-u^{-}_{n})=2\|\tilde{u}_{n}\|^{2}
+k\int^{T}_{0}H'(kt,u_{n}) \cdot(u^{+}_{n}-u^{-}_{n})dt. \label{e3.3}
\end{equation}
Since $\theta$ is non-increasing and $\|u\|\geq \max(|u^{0}|,\|\tilde{u}\|)$, 
we obtain
\begin{equation}
\theta(\|u\|)\leq \min(\theta(|u^{0}|),\theta(\|\tilde{u}\|)). \label{e3.4}
\end{equation}
So by H\"older's inequality, Proposition \ref{prop2.1}, Lemma \ref{lem3.1} (ii) and
property \eqref{e3.4}, we obtain a positive constant $c_{2}$ such that
\begin{equation}
\begin{aligned}
\big|k\int^{T}_{0}H'(kt,u_{n}).(u^{+}_{n}-u^{-}_{n})dt\big|
&\leq k\|u^{+}_{n}-u^{-}_{n}\|_{L^{2}} [\int^{T}_{0}|H'(kt,u_{n})|^{2}dt]^{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{aligned} \label{e3.5}
\end{equation}
Thus, for $n$ large enough
\begin{align*}
\|\tilde{u}_{n}\|
&\geq\Phi'_{k}(u_{n})(u^{+}_{n}-u^{-}_{n})\\
&\geq 2\|\tilde{u}_{n}\|^{2}-c_{2}\Big[\|\tilde{u}_{n}\|
[\theta(\|\tilde{u}_{n}\|)\|\tilde{u}_{n}\|
+\theta(|u^{0}_{n}|)|u^{0}_{n}|+1\Big]
\end{align*}
and then
\begin{equation}
c_{2}\theta(|u^{0}_{n}|)|u^{0}_{n}|\geq\|\tilde{u}_{n}\|
[2-c_{2}\theta(\|\tilde{u}_{n}\|)]-c_{2}-1. \label{e3.6}
\end{equation}
Assume that $(\|\tilde{u}_{n}\|)$ is unbounded, then by going to a
subsequence, if necessary, we can assume that $\|\tilde{u}_{n}\|\to\infty$
as $n\to\infty$. Since $\theta(s)\to 0$ as $s\to\infty$, we deduce
from \eqref{e3.6} that there exists a positive constant $c_{3}$ such that
\begin{equation}
\|\tilde{u}_{n}\|\leq c_{3}[\theta(|u^{0}_{n}|)|u^{0}_{n}|+1] \label{e3.7}
\end{equation}
for $n$ large enough. Since $\omega$ is bounded, then
$|u^{0}_{n}|\to\infty$ as $n\to\infty$.
Now, by the mean value theorem, H\"older's inequality and Lemma \ref{lem3.1} (ii),
we obtain
\begin{equation}
\begin{aligned}
&|\int^{T}_{0}(H(kt,u_{n})-H(kt,u^{0}_{n}))dt|\\
&=|\int^{T}_{0}\int^{1}_{0}H'(kt,u^{0}_{n}+s\tilde{u}_{n}).\tilde{u}_{n}\,ds\,dt|\\
&\leq \|\tilde{u}_{n}\|_{L^{2}}\int^{1}_{0}
\Big(\int^{T}_{0} |H'(kt,u^{0}_{n}+s\tilde{u}_{n})|^{2}dt\Big)^{1/2}ds\\
&\leq c_{0}\|\tilde{u}_{n}\|_{L^{2}}\int^{1}_{0}[\theta(\|u^{0}_{n}
+s\tilde{u}_{n}\|)\|u^{0}_{n}+s\tilde{u}_{n}\|+1]ds.
\end{aligned} \label{e3.8}
\end{equation}
Since $\theta$ is non-increasing and
$\|u^{0}_{n}+s\tilde{u}_{n}\|\geq |u^{0}_{n}|$ for all $s\in [0,1]$,
we deduce from Proposition \ref{prop2.1}, \eqref{e3.7} and \eqref{e3.8}
that there exists a constant $c_{4}>0$ such that for $n$ large enough
\begin{equation}
\begin{aligned}
k|\int^{T}_{0}(H(kt,u_{n})-H(kt,u^{0}_{n}))dt|
&\leq c_{0}\|\tilde{u}_{n}\|_{L^{2}}
\Big[\theta(|u^{0}_{n}|)|u^{0}_{n}|+ \theta(|u^{0}_{n}|)\|\tilde{u}_{n}\|+1\Big]\\
&\leq c_{4}[\theta(|u^{0}_{n}|)|u^{0}_{n}|]^{2},
\end{aligned} \label{e3.9}
\end{equation}
so there exists a positive constant $c_{5}$ such that for $n$ large enough,
\begin{align*}
\Phi_{k}(u_{n})
&= \|u^{+}_{n}\|^{2}- \|u^{-}_{n}\|^{2}
 +k\int^{T}_{0}(H(kt,u_{n})-H(kt,u^{0}_{n}))dt
 + k\int^{T}_{0}H(kt,u^{0}_{n})dt \\
&\geq -\|\tilde{u}_{n}\|^{2}-c_{4}[\theta(|u^{0}_{n}|)|u^{0}_{n}|]^{2}
 +k\int^{T}_{0}H(kt,u^{0}_{n})dt \\
&\geq[\theta(|u^{0}_{n}|)|u^{0}_{n}|]^{2}\Big[-c_{5}
+\frac{k}{[\theta(|u^{0}_{n}|)|u^{0}_{n}|]^{2}}\int^{T}_{0}H(kt,u^{0}_{n})dt\Big].
\end{align*}
By Lemma \ref{lem3.1}, this implies $\Phi_{k}(u_{n})\to+\infty$ as $n\to\infty$
and contradicts the boundedness of $(\Phi_{k}(u_{n}))$. So $(\|\tilde{u}_{n}\|)$
is bounded.

Assume that $(|u^{0}_{n}|)$ is unbounded. Up to a subsequence, if necessary, 
we can assume that $|u^{0}_{n}|\to\infty$, as $n\to \infty$. 
As above, we obtain for $n$ large enough
\begin{align*}
|k\int^{T}_{0}(H(kt,u_{n})-H(kt,u^{0}_{n}))dt|
&\leq c_{0}\|\tilde{u}_{n}\|_{L^{2}} \big[\theta(|u^{0}_{n}|)|u^{0}_{n}|
 +\theta(|u^{0}_{n}|)\|\tilde{u}_{n}\|+1\big]\\
&\leq c_{6}\theta(|u^{0}_{n}|)|u^{0}_{n}|
\end{align*}
where $c_{6}$ is a positive constant. So there exists a constant $c_{7}>0$ 
such that for $n$ large enough
\begin{equation} \label{e3.10}
\begin{aligned}
\Phi_{k}(u_{n})
&\geq -c_{7}\theta(|u^{0}_{n}|)|u^{0}_{n}|
+k\int^{T}_{0}H(kt,u^{0}_{n})dt \\
&\geq [\theta(|u^{0}_{n}|)|u^{0}_{n}|]^{2}
\Big(-\frac{c_{7}}{\theta(|u^{0}_{n}|)|u^{0}_{n}|}
+\frac{k}{[\theta(|u^{0}_{n}|)|u^{0}_{n}|]^{2}}
\int^{T}_{0}H(kt,u^{0}_{n})dt\Big)
\end{aligned}
\end{equation}
so $\Phi_{k}(u_{n})\to+\infty$ as $n\to\infty$, which also contradicts
the boundedness of $(\Phi_{k}(u_{n}))$. Then $(|u^{0}_{n}|)$ is also
bounded and therefore $(\|u_{n}\|)$ is bounded. By a standard argument,
we conclude that $(u_{n})$ possesses a convergent subsequence.
The proof is complete.
\end{proof}

Now, let $u=u^{0} + u^{+}\in E^{2}=E^{0}\oplus E^{+}$ with $u^{0}\neq 0$, 
then as in \eqref{e3.14} there exists a positive constant $c_{8}$ such that
\begin{equation}
k|\int^{T}_{0}(H(kt,u)-H(kt,u^{0}))dt|
\leq c_{8}\|u^{+}\|\big[\theta(|u^{0}|) |u^{0}|
+ \theta(|u^{0}|)\|u^{+}\|+1\big]. \label{e3.11}
\end{equation}
So
\begin{equation}
\Phi_{k}(u)\geq \|u^{+}\|^{2}-c_{8}\|u^{+}\|\Big[\theta(|u^{0}|)|u^{0}|
+ \theta(|u^{0}|)\|u^{+}\|+1\Big]+ k\int^{T}_{0}H(kt,u^{0})dt. \label{e3.12}
\end{equation}

Let $0<\epsilon<1$, Then we have
\begin{equation}
c_{8}\theta(|u^{0}|)|u^{0}|\|u^{+}\|\leq \frac{1}{\epsilon^{2}}c^{2}_{8}
[\theta(|u^{0}|)|u^{0}|]^{2}+\epsilon^{2}\|u^{+}\|^{2}. \label{e3.13}
\end{equation}
Combining \eqref{e3.12} and \eqref{e3.13}, we obtain
\begin{align*}
\Phi_{k}(u)&\geq\big[1-\epsilon^{2}-c_{8}\theta(|u^{0}|)\big]\|u^{+}\|^{2}-c_{8}\|u^{+}\|
+[\theta(|u^{0}|)|u^{0}|]^{2}\\
&\quad\times \Big[-\frac{c^{2}_{8}}{\epsilon^{2}}
+\frac{k}{[\theta(|u^{0}|)|u^{0}|]^{2}}\int^{t}_{0}H(kt,u^{0})dt\Big]
\end{align*}
so
\begin{equation}
\Phi_{k}(u)\to +\infty\quad\text{as } \|u\|\to\infty,\; u\in E^2. \label{e3.14}
\end{equation}
On the other hand, let $\xi\in\mathbb{R}^{2N}, |\xi|>0$. By the mean value theorem,
 for $u\in E^{1}=E^{-}$, we have
\begin{equation}
\begin{aligned}
&|\int^{T}_{0}(H(kt,u)-H(kt,\xi))dt|\\
&=|\int^{T}_{0}\int^{1}_{0}H'(kt,\xi+s(u-\xi)).(u-\xi)\,ds\,dt|\\
&\leq\|u-\xi\|_{L^{2}}\int^{1}_{0}
\Big[\int^{T}_{0}|H'(kt,\xi+s(u-\xi))|^{2}dt\big]^{1/2}ds\\
&\leq\|u-\xi\|_{L^{2}}\int^{1}_{0}\Big[\int^{T}_{0}
 \Big(a\omega(|\xi+s(u-\xi)|)|\xi+s(u-\xi)|+g(kt)\Big)^{2} dt\Big]^{1/2}ds \\
&\leq\|u-\xi\|_{L^{2}}\Big[a\int^{1}_{0}
 \Big(\int^{T}_{0}[\omega(|\xi+s(u-\xi)|)|\xi+s(u-\xi)|]^{2}dt\Big)^{1/2}ds
 + \|g\|_{L^{2}}\Big].
\end{aligned}\label{e3.15}
\end{equation}
For $s\in[0,1]$,
$$
A(s)=\{t\in [0,1]: |\xi+s(u(t)-\xi)|\geq |\xi|\}.
$$
By a classical calculation as in the proof of Lemma \ref{lem3.1},  we obtain
some positive constants $c_{9}$ and $c(\xi)$ such that
\begin{equation}
k|\int^{T}_{0}(H(kt,u)-H(kt,\xi))dt|
\leq c_{9}\omega(|\xi|)\|u\|^{2}+c(\xi)(\|u\|+1). \label{e3.16}
\end{equation}
This implies
\begin{equation}
\Phi_{k}(u)\leq -\|u\|^{2}+c_{9}\omega(|\xi|)\|u\|^{2}
+c(\xi)(\|u\|+1)+k\int^{T}_{0}H(t,\xi)dt. \label{e3.17}
\end{equation}
Since $\omega(s)\to 0$ as $s\to\infty$, there exists $|\xi| > 0$ such
that $c_{9}\omega(|\xi|)\leq \frac{1}{2}$ and then we have for a constant
$c_{10}>0$,
$$
\Phi_{k}(u)\leq -\frac{1}{2}\|u\|^{2}+c_{10}(\|u\|+1)+k\int^{T}_{0}H(t,\xi)dt;
$$
therefore
\begin{equation}
\Phi_{k}(u)\to -\infty\quad\text{as } \|u\|\to\infty,\; u\in E^{1}.
 \label{e3.18}
\end{equation}
Property \eqref{e3.1} and \cite[Proposition B.37]{r1}
 imply that the derivative of the functional $g_{k}(u)=k\int^{T}_{0}H(kt,u)dt$
is compact. Thus Lemma \ref{lem3.2} and properties \eqref{e3.14}, \eqref{e3.18}
imply that for all integer $k\geq 1$, $\Phi_{k}$ satisfies all the
assumptions of the saddle point theorem and then by Remark \ref{rmk2.1},
$\Phi_{k}$ possesses a critical point $u_{k}\in E$ satisfying
\begin{equation}
C_{k}=\Phi_{k}(u_{k})\geq\inf_{u\in E^{1}}\Phi_{k}(\sqrt{k}\varphi+u)
 \label{e3.19}
\end{equation}
where $\varphi(t)=\frac{1}{\sqrt{\pi}}\exp(\frac{2\pi}{T}tJ)e_{1}$ and
$e_{1}$ design the first element in the standard basis of $\mathbb{R}^{2N}$.
Therefore, for all integer $k\geq 1$, $x_{k}(t)=u_{k}(\frac{t}{k})$ is a
 $kT$-periodic solution for \eqref{eH}. Theorem \ref{thm1.1} and the first part
of Theorem \ref{thm1.2} are proved.


\begin{proof}[Proof of Theorem \ref{thm1.2}] 
It remains to prove \eqref{e1.7}. For this, we will prove that the sequence 
$(u_{k})$ obtained above satisfies
\begin{equation}
\lim_{k\to\infty}\frac{1}{k}\Phi_{k}(u_{k})=+\infty.\label{e3.20}
\end{equation}
This will be done by  using  estimates on the levels $C_{k}$ of $\Phi_{k}$.
For this aim the following two lemmas will be needed.

\begin{lemma}[\cite{t1}]  \label{lem3.3} 
If {\rm (H3)} holds, then for every $\delta > 0$, there exists a measurable 
subset $C_{\delta}$ of $C$ with $\operatorname{meas}(C-C_{\delta})< \delta$ 
such that
$$
H(t,x)\to+\infty\quad\text{as $|x|\to\infty$, uniformly in }t\in C_{\delta}.
$$
\end{lemma}

\begin{lemma} \label{lem3.4} 
Assume that $H$ satisfies {\rm (H3)}, then
\begin{equation}
\lim_{k\to\infty}\inf_{u\in E^{2}}\frac{\Phi_{k}(\sqrt{k}\varphi+u)}{k}
=+\infty. \label{e3.21}
\end{equation}
\end{lemma}

\begin{proof}
Arguing by contradiction, assume that there exist sequences
$k_{n}\to\infty$, $(u_{n})\subset E^{2}$ and a constant $c_{11}$ such that
\begin{equation}
\Phi_{k_{n}}(\sqrt{k_{n}}\varphi+u_{n})\leq k_{n}c_{11},\quad
 \forall n\in\mathbb{N}. \label{e3.22}
\end{equation}
Taking $u_{n}=\sqrt{k_{n}}(u^{+}_{n}+u^{0}_{n})$, we obtain by an easy
calculation
\begin{equation}
\Phi_{k_{n}}(\sqrt{k_{n}}\varphi+u_{n})=k_{n}[\|u^{+}_{n}\|^{2}-1
+\int^{T}_{0}H(k_{n}t,\sqrt{k_{n}}(\varphi+u^{+}_{n}+u^{0}_{n}))dt].
 \label{e3.23}
\end{equation}
On the other hand, by (H3) (ii), we have
\begin{equation}
\int^{T}_{0}H(k_{n}t,\sqrt{k_{n}}(\varphi+u^{+}_{n}+u^{0}_{n}))dt
\geq\int^{T}_{0}h(k_{n}t)dt=\int^{T}_{0}h(t)dt \label{e3.24}
\end{equation}
so there exists a positive constant $c_{12}$ such that
\begin{equation}
\Phi_{k_{n}}(\sqrt{k_{n}}\varphi+u_{n})\geq k_{n}(\|u^{+}_{n}\|^{2}-c_{12}).
\label{e3.25}
\end{equation}
Inequalities \eqref{e3.22} and \eqref{e3.25} imply that $(u^{+}_{n})$
is a bounded sequence in $E$. Up to a subsequence, if necessary, we
can find $u^{+}\in E^{+}$ such that
\begin{equation}
u^{+}_{n}(t)\to u^{+}(t)\quad\text{ as $n\to\infty$, a.e. $t\in[0,T]$}. \label{e3.26}
\end{equation}
We claim that $(u^{0}_{n})$ is also bounded in $E$. Indeed,
if we assume otherwise, then by using a subsequence, if necessary,
\eqref{e3.26} implies that
\begin{equation}
|\sqrt{k_{n}}(\varphi(t)+u^{+}_{n}(t)+u^{0}_{n})|\to\infty
\quad\text{as $n\to\infty$, a.e. $t\in [0,T]$}. \label{e3.27}
\end{equation}
Let $\delta=\frac{1}{2}\operatorname{meas}(C)$ and $C_{\delta}$
be as defined in Lemma \ref{lem3.3}. For all positive integer $n$, let us define
the subset $C^{n}_{\delta}$ of $[0,T]$ by
$$
C^{n}_{\delta}=\frac{1}{k_{n}}\cup^{k_{n}-1}_{p=0}(C_{\delta}+pT).
$$
It is easy to verify that
 $\operatorname{meas}(C^{n}_{\delta})=\operatorname{meas}(C_{\delta})$ and
\begin{equation}
H(k_{n}t,x)\to +\infty\quad\text{as $|x|\to \infty$, uniformly in $t\in [0,T]$}.
\label{e3.28}]
\end{equation}
By (H3) (ii), we have
\begin{align*}
&\int^{T}_{0}H(k_{n}t,\sqrt{k_{n}}(\varphi+u^{+}_{n}+u^{0}_{n}))dt\\
&\geq\int_{C^{n}_{\delta}}H(k_{n}t,\sqrt{k_{n}}(\varphi+u^{+}_{n}+u^{0}_{n}))dt
 +\int_{[0,T]-C^{n}_{\delta}}h(k_{n}t)dt\\
&\geq \int_{C^{n}_{\delta}}H(k_{n}t, \sqrt{k_{n}}(\varphi+u^{+}_{n}+u^{0}_{n}))dt
 -\int^{T}_{o}|h(t)|dt.
\end{align*}
Therefore, by \eqref{e3.27} and \eqref{e3.28}, we obtain
\begin{equation}
\int^{T}_{0}H(k_{n}t,\sqrt{k_{n}}(\varphi+u^{+}_{n}+u^{0}_{n}))dt\to +\infty
\quad\text{as } n\to\infty. \label{e3.29}
\end{equation}
We deduce from \eqref{e3.23} and \eqref{e3.29} that
\begin{equation}
\frac{\Phi_{k_{n}}(\sqrt{k_{n}}\varphi+u_{n})}{k_{n}}\to +\infty
\quad\text{as }n\to\infty \label{e3.30}
\end{equation}
which contradicts \eqref{e3.22} and proves our claim. Hence,
by taking a subsequence, if necessary, we can assume that there exists
$u^{0}\in E^{0}$ such that
\begin{equation}
\varphi(t)+u^{+}_{n}(t)+u^{0}_{n}\to u(t)=\varphi(t)+u^{+}(t)+u^{0}\quad
\text{as  $n\to\infty$, a.e. $t\in [0,T]$}. \label{e3.31}
\end{equation}
By Fourier analysis, we have $u(t)\neq 0$ for almost every $t\in [0,T]$.
Therefore,
\begin{equation}
|\sqrt{k_{n}}(\varphi(t)+u^{+}_{n}(t)+u^{0}_{n})|\to\infty
\quad\text{as $n\to\infty$, a.e. $t\in [0,T]$} \label{e3.32}
\end{equation}
and by using \eqref{e3.23}, \eqref{e3.28} and \eqref{e3.32},
we obtain \eqref{e3.30} as above, which contradicts \eqref{e3.22}.
This concludes the proof of Lemma \ref{lem3.4}.
\end{proof}

By \eqref{e3.19} and Lemma \ref{lem3.4}, we have
\begin{equation}
\frac{C_{k}}{k}\to\infty\ as\ k\to\infty. \label{e3.33}
\end{equation}
We claim that $\|u_{k}\|_{\infty}\to \infty$ as $k\to\infty$.
Indeed, if we suppose otherwise, $(u_{k})$ possesses a bounded subsequence
$(u_{k_p})$. Since
\begin{equation}
\frac{\Phi_{k_p}(u_{k_p})}{k_p}
=-\frac{1}{2}\int^{T}_{0}H'(k_pt,u_{k_p})\cdot u_{k_p}dt
+\int^{T}_{0}H(k_pt,u_{k_p})dt \label{e3.34}
\end{equation}
the sequence $(C_{k_p}/k_p)$ is bounded, which contradicts
 \eqref{e3.33}. Consequently, we have
$\|u_{k}\|_{\infty}\to\infty\ as\ k\to\infty$. For $k\geq 1$, define
$x_{k}(t)=u_{k}(t/k)$. Then $x_{k}$ is a $kT$-periodic solution
of the system \eqref{eH} satisfying:
\begin{equation}
\|x_{k}\|_{\infty}=\|u_{k}\|_{\infty}\to\infty\quad\text{as } k\to\infty \label{e3.35}
\end{equation}
which completes the proof of Theorem \ref{thm1.2}.
\end{proof}

 For the proof of Theorem \ref{thm1.3}, we need the following lemma.

\begin{lemma} \label{lem3.5} 
Let {\rm (H1)} and {\rm (H5)} hold. Then for all $\rho>0$, there exists a constant
 $c_{\rho}\geq 0$ such that for all $x\in\mathbb{R}^{2N}$, $|x|>1$ and for a.e.
$t\in[0,T]$,
\begin{equation}
H(t,x)\geq H(t,0)+\frac{\rho}{2}[\omega(|x|)|x|]^{2}(1-\frac{1}{|x|^{2}})
-c_{\rho}\ln(|x|)-\frac{1}{2}a\sup_{r\geq 0}\omega(r)-g(t).  \label{e3.36}
\end{equation}
\end{lemma}

The proof of the above lemma is similar to that of \cite[Lemma 2.5]{t4}, it is
omitted here.

\begin{proof}[Proof of Theorem \ref{thm1.2}]
Since 
$a_{0}=\liminf_{s\to\infty}{\frac{\omega(s)}{\omega(s^{1/2})}}>0$,
 then for $s$ large enough, we have
\begin{equation}
\frac{1}{\omega(s)}\leq \frac{1}{a_{0}\omega(\sqrt{s})}. \label{e3.37}
\end{equation}
this implies that for $|x|$ large enough
\begin{equation}
\frac{\ln(|x|)}{[\omega(|x|)|x|]^{2}}
\leq \frac{\ln(|x|)}{|x|}\frac{1}{a^{2}_{0}[\omega(|x|^{1/2})
 |x|^{1/2}]^{2}}\to 0\ as\ |x|\to\infty. \label{e3.38}
\end{equation}
Combining \eqref{e3.35}, \eqref{e3.38} we obtain (H4).
By applying Corollary \ref{coro1.1}, we obtain a sequence $(x_{k})$ of $kT$-periodic
solutions of \eqref{eH} such that $\lim_{k\to\infty}\|x_{k}\|_{\infty} =+\infty$.

It remains to prove that, for every sufficiently large prime integer 
$k$, $x_{k}$ has $kT$ as minimal period. 
Let $\Psi_{k}$ be the functional defined on the space $H^{1/2}_{kT}$ by
$$
\Psi_{k}(x)=\frac{1}{2}\int^{kT}_{0}J\dot{x}(t)\cdot x(t)dt
+\int^{kT}_{o}H(t,x(t))dt.
$$
It is easy to see that $u\in E=H^{1/2}_{T}$ is a critical point of 
$\Phi_{k}$ if and only if $x(t)=u(\frac{t}{k})$ belongs to $H^{1/2}_{kT}$
 and is a critical point of $\Psi_{k}$. Let $x_{k}\in H^{1/2}_{kT}$ 
be the critical point of $\Psi_{k}$ associated to the critical point 
$u_{k}$ of $\Phi_{k}$ obtained above, the property \eqref{e3.33} is written as
\begin{equation}
\lim_{k\to\infty}\frac{1}{k}\Psi_{k}(x_{k})=+\infty. \label{e3.39}
\end{equation}
On the other hand, let us denote by $S_{T}$ the set of $T$-periodic
solutions of \eqref{eH}. We claim that $S_{T}$ is bounded in $H^{1/2}_{T}$.
Indeed, assume by contradiction that there exists a sequence
$(x_{n})\subset S_{T}$ such that $\|x_{n}\|\to\infty$ as
 $n\to\infty$. Let $x_{n}=x^{+}_{n}+x^{-}_{n}+x^{0}_{n}$.
Multiplying both sides of the identity
\begin{equation}
J\dot{x}_{n}+H'(t,x_{n}(t))=0 \label{e3.40}
\end{equation}
by $x^{+}_{n}$ and integrating, we obtain
\begin{equation}
2\|x^{+}_{n}\|^{2}+\int^{T}_{0}H'(t,x_{n}(t))\cdot x^{+}_{n}dt=0.\label{e3.41}
\end{equation}
Using Lemma \ref{lem3.1} (ii), we can find a positive constant $c_{13}$ such that
\begin{equation}
\|x^{+}_{n}\|\leq c_{13}(\theta(\|x_{n}\|)\|x_{n}\|+1). \label{e3.42}
\end{equation}
By Lemma \ref{lem3.1} (i) and \eqref{e3.42}, we obtain
\begin{equation}
\frac{\|x^{+}_{n}\|}{\|x_{n}\|}\to 0\ as\ n\to\infty. \label{e3.43}
\end{equation}
Similarly, we have
\begin{equation}
\frac{\|x^{-}_{n}\|}{\|x_{n}\|}\to 0\ as\ n\to\infty. \label{e3.44}
\end{equation}
Taking $y_{n}=\frac{x_{n}}{\|x_{n}\|}$ and using \eqref{e3.43}
and \eqref{e3.44}, we may assume without loss of generality that
$y_{n}\to y_{0}\in E^{0}$, with $|y_{0}|=1$. Since the embedding
$E\to L^{2}$, $u\longmapsto u$ is compact, we can assume, by taking a
subsequence if necessary that
\begin{equation}
y_{n}(t)\to y_{0}\quad\text{as $n\to\infty$, a.e. $t\in [0,T]$}, \label{e3.45}
\end{equation}
and consequently
\begin{equation}
|x_{n}(t)|\to +\infty\quad\text{as $n\to\infty$, a.e. $t\in [0,T]$}. \label{e3.46}
\end{equation}
Fatou's lemma, property (a) of $\omega$ and \eqref{e3.46} imply
\begin{equation}
\int^{T}_{0}[\omega(|x_{n}(t)|)|x_{n}(t)|]^{2}dt\to \infty
\quad\text{as } n\to\infty. \label{e3.47}
\end{equation}
On the other hand, by (H5), for all $\rho>0$, there exists a constant
$c_{\rho}\geq 0$ such that
\begin{equation}
H'(t,x).x\geq \rho[\omega(|x|)|x|]^{2}-c_{\rho} \label{e3.48}
\end{equation}
so we obtain
\begin{equation}
\rho\int^{T}_{0}[\omega(|x_{n}(t)|)|x_{n}(t)|]^{2}dt
\leq \int^{T}_{0}H'(t,x_{n}(t))\cdot x_{n}(t)dt +c_{\rho}T. \label{e3.49}
\end{equation}
Furthermore, by \cite[Proposition 3.2]{m1}, we have
\begin{equation}
\int^{T}_{0}H'(t,x_{n}(t))\cdot x_{n}(t)dt
\leq \frac{T}{2\pi} \int^{T}_{0}|H'(t,x_{n}(t))|^{2}dt \label{e3.50}
\end{equation}
Combining \eqref{e3.49} with \eqref{e3.50}, and using assumption (H1),
we obtain
\begin{equation}
\begin{aligned}
\rho\int^{T}_{0}[\omega(|x_{n}(t)|)|x_{n}(t)|]^{2}dt
&\leq \frac{T}{2\pi} \int^{T}_{0}[a\omega(|x_{n}(t)|)|x_{n}(t)|+ g(t)]^{2}dt
+c_{\rho}T \\
&\leq \frac{T}{\pi}(a^{2}\int^{T}_{0}[\omega(|x_{n}(t)|)|x_{n}(t)|]^{2}dt
+\|g\|^{2}_{L^{2}})+c_{\rho}T.
\end{aligned}\label{e3.51}
\end{equation}
Since $\rho > 0$ is arbitrary chosen, \eqref{e3.51} implies that
$(\int^{T}_{0}[\omega(|x_{n}(t)|)|x_{n}(t)|]^{2}dt)$ is bounded,
which contradicts \eqref{e3.47}. Hence $S_{T}$ is bounded and as a
consequence $\Psi_{1}(S_{T})$ is bounded. Since for any $x\in S_{T}$
 one has $\Psi_{k}(x)=k\Psi_{1}(x)$, then there exists a positive constant
$c_{14}$ such that
\begin{equation}
\frac{|\Psi_{k}(x)|}{k}\leq c_{14} \quad \forall x\in S_{T},\; \forall k\geq 1\,.
\label{e3.52}
\end{equation}
Consequently, by \eqref{e3.39} and \eqref{e3.52}, we deduce that,
for $k$ sufficiently large, $x_{k}\notin S_{T}$. By assumption (H1'),
if $k$ is chosen to be prime, the minimal period of $x_{k}$ has to be $kT$.
This completes the proof of Theorem \ref{thm1.3}.
\end{proof}

\subsection*{Acknowledgments} 
The author would like to thank the anohymous referees for their valuable 
suggestions.

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