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\markboth{\hfil Nontrivial periodic solutions \hfil EJDE--2001/69}
{EJDE--2001/69\hfil Guihua Fei \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations}, 
Vol. {\bf 2001}(2001), No. 69, pp. 1--17. \newline 
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or 
http://ejde.math.unt.edu \newline 
ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Nontrivial periodic solutions of asymptotically
 linear Hamiltonian systems
 %
\thanks{ {\em Mathematics Subject Classifications:} 58E05, 58F05, 34C25.
\hfil\break\indent {\em Key words:} periodic solution,
Hamiltonian systems, Conley index, Galerkin approximation.  \hfil\break\indent
\copyright 2001 Southwest Texas State University.
\hfil\break\indent 
Submitted September 14, 2001. Published November 19, 2001.} }
\date{}
%
\author{Guihua Fei}
\maketitle

\begin{abstract}
 We study the existence of periodic solutions for some
 asymptotically linear Hamiltonian systems. By using the
 Saddle Point Theorem and Conley index theory, we obtain
 new results under asymptotically linear conditions.
\end{abstract}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}

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\section{Introduction}

We consider the Hamiltonian system
\begin{equation}
\dot z = JH'(t,z)   \tag{1.1}
\end{equation}
where $H\in C^2([0,1]\times \mathbb{R}^{2N},\mathbb{R})$ is a 1-periodic
function in $t$, and $H'(t,z)$ denotes the gradient of $H(t, z)$ with
respect to the $z$ variable.
Here $N$ is a positive integer and
$J=\begin{pmatrix} 0 & -I_N \\ I_N & 0 \end{pmatrix} $ is the standard $2N\times 2N$
symplectic matrix. We denote by $(x, y)$ and $|x|$ the usual inner product
and norm in $\mathbb{R}^{2N}$ respectively. The function $H$ satisfies
the following conditions.
\begin{enumerate}
\item[(H1)]  There exist $s\in (1,\infty )$ and $a_1, a_2>0$ such that
$$
|H''(t,z)|\leq a_1|z|^s+a_2, \quad\forall (t,z)\in
\mathbb{R}\times \mathbb{R}^{2N}.
$$

\item[(H2)]  $H'(t,z) = B_{\infty}(t)z+o(|z|)$  as $|z| \to \infty $
uniformly in $t$;

\item[(H3)] $H'(t,z) = B_0(t)z+o(|z|) $ as $|z|\to 0$  uniformly in $t$
where $B_0(t)$ and $B_{\infty}(t)$ are $2N\times 2N$ symmetric matrices,
continuous and 1-periodic in $t$. 
\end{enumerate}
The system (1.1) is called asymptotically
linear because of (H2). Obviously, (H3) implies that 0 is a ``trivial"
solution of $(1.1)$. We are interested in nontrivial 1-periodic solutions
of $(1.1)$.

The existence of periodic solutions of $(1.1)$ has been studied by many authors.
If $B_{\infty}(t)$ is non-degenerate, i.e. $1$ is not a Floquet
multiplier of the linear system $\dot y=JB_{\infty}(t)y$, one can see the
results in \cite{a1,c1,c2,d1,l1,l2,l3}. 
If $B_{\infty}(t)$ is degenerate, some resonance
conditions are needed to control the behavior of
$$
G_{\infty}(t,z)=H(t,z)-\frac 12(B_{\infty}(t)z,z).
$$
When $|G'_{\infty}(t,z)|$ is bounded, under the Landesman-Lazer type condition
or strong resonance condition, (1.1) was studied by \cite{c3,s1}
 for the case that $B_{\infty}(t)$ is constant and by \cite{c5,f1}
 for the case that $B_{\infty}(t)$
is continuous and 1-periodic in $t$. When $|G'_{\infty}(t,z)|$ is not bounded,
\cite{f2,s2,s3} studied the case that $B_{\infty}(t)$ is ``finitely degenerate"
\cite{f2}.

In this paper we shall study the case that $|G'_{\infty}(t,z)|$ is not bounded
and $B_{\infty}(t)$ is continuous and 1-periodic in $t$. We assume the
following conditions for $G_{\infty}(t,z)$.
\begin{enumerate}

\item[(H$4^\pm$)] There exist $c_1, c_2 > 0$ such that
\begin{gather*}
\pm [2G_{\infty}(t,z) - (G'_{\infty}(t,z), z)] \geq c_1|z| - c_2, \quad
\forall (t, z)\in[0,1]\times \mathbb{R}^{2N};\\
G_{\infty}(t,z) \to \pm \infty \quad as \quad |z| \to + \infty .
\end{gather*}

\item[(H$5^\pm$)] There exist $1\leq \alpha <2$, $0<\beta <\alpha /2$, and
$M_1, M_2, L>0$ such that
$$
|G'_{\infty}(t,z)|\leq M_1|z|^{\beta}, \quad \pm G_{\infty}(t,z) \geq M_2|z|^{\alpha},
\quad \forall |z|\geq L .
$$

\item[(H$6^\pm$)] There exist $1\leq \alpha <2$, $0<\beta <\alpha /2$, and
$M_1, M_2, L>0$ such that
$$
|G'_{\infty}(t,z)|\leq M_1|z|^{\beta}, \quad \pm (G'_{\infty}(t,z), z) \geq M_2|z|^{\alpha},
\quad \forall |z|\geq L .
$$
\end{enumerate}
According to \cite{c2,l2,l3}, for a given continuous 1-periodic and symmetric matrix
function $B(t)$, one can assign a pair of integers
$(i,n)\in \mathbb{Z} \times \lbrace 0,\cdots ,2N \rbrace $ to it,
which is called the Maslov-type index of $B(t)$. We denote by
$(i_0,n_0)$ and $(i_{\infty}, n_{\infty})$ the Maslov-type
indices of $B_0(t)$ and $B_{\infty}(t)$ respectively. 
Our main result reads as follows.

\begin{theorem} \label{thm1.1}
Suppose that $H$ satisfies $(H1)-(H3)$. Then $(1.1)$ possesses a
nontrivial 1-periodic solution if one of the following cases occurs:
\begin{enumerate}
\item[(i)] (H$4^{+}$) and $i_{\infty}+n_{\infty}\notin [i_0,i_0+n_0]$.

\item[(ii)] (H4$^-$) and $i_{\infty}\notin [i_0,i_0+n_0]$.

\item[(iii)] (H5$^+$) and $i_{\infty}+n_{\infty}\notin [i_0,i_0+n_0]$.

\item[(iv)] (H$5^-$) and $i_{\infty}\notin [i_0,i_0+n_0]$.

\item[(v)] (H$6^+$) and $i_{\infty}+n_{\infty}\notin [i_0,i_0+n_0]$.

\item[(vi)] (H$6^-$) and $i_{\infty}\notin [i_0,i_0+n_0]$.
\end{enumerate}
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
 (1) If
\begin{equation}
H(t, z) = \frac {7|z|^{2}}{2\ln (e+|z|^2)} ,   \tag{1.2}
\end{equation}
by Theorem 1.1(i) the system (1.1) possesses a nontrivial 1-periodic solution. If
\begin{equation}
H(t, z) = \frac {1}{2}|z|^2 - \frac {|z|^{2}}{\ln (e+|z|^2)} ,   \tag{1.3}
\end{equation}
by Theorem 1.1(ii) the system (1.1) possesses a nontrivial 1-periodic solution.
These examples can not be solved by earlier results, for example those
contained in references. More examples are given in Section 3.

\noindent (2) Our result should be compared with those in \cite{f2,s2,s3}.
First, we do not require that $B_{\infty}(t)$ be constant or 
``finitely degenerate". 
Secondly, the conditions (H$6^{\pm}$) with $\beta = \alpha -1$ are 
required in \cite{f2,s2,s3}.
This means that the results in \cite{f2,s2,s3}
 can not be applied to some cases such as (1.2), (1.3), or
$G_{\infty}(t,z)\sim |z|^{\alpha}\ln (1+|z|^2)$ at infinity. But these cases are
covered by Theorem 1.1 . Notice that $\beta = \alpha -1 < \alpha /2$. Therefore
Theorem 1.1(v)\&(vi) generalizes \cite[Theorem 1.1]{f2}, 
 \cite[Theorem 1.2]{s2}, and  \cite[Theorem 1.2]{s3}.

\noindent (3) The condition $(H5^{\pm})$ is rather close to a condition in 
the paper \cite{s5} by Szulkin and Zou. 
The author thanks the referee for pointing out this.
\end{remark}

The proof of our results is given in Section 2. By using the
the Galerkin approximation method \cite{f1,l1},
Saddle point theorem \cite{c6,m1,r1}, and Morse index estimates 
\cite{g1,l4,s4}, we shall prove Theorem 1.1(i)-(iv).
For Theorem 1.1(v)-(vi), we follow the idea in \cite{f2} and use
Conley index theory \cite{c2} to get our conclusions.


\section{Periodic solutions of Hamiltonian systems}


Let $S^1 =\mathbb{R}/(2\pi \mathbb{Z})$, $E=W^{1/2,2}(S^1 ,\mathbb{R}^{2N} )$.
Recall that $E$ is a Hilbert space with norm
$\| \cdot \|$ and inner product $\langle\cdot,\cdot\rangle$, and $E$ consists
of those $z(t)$ in $L^2(S^1 ,\mathbb{R}^{2N} )$ whose Fourier series
$$
z(t)=a_0+\sum _{n=1}^{\infty}(a_n \cos (2\pi nt)+b_n \sin (2\pi nt))
$$
satisfies
$$
\|z\|^2 = |a_0|^2 +\frac 12 \sum _{n=1}^{\infty} n(|a_n|^2 +
|b_n|^2 )<\infty ,
$$
where $a_j , b_j \in \mathbb{R}^{2N} $. For a given continuous 1-periodic
and symmetric matrix function $B(t)$, we define two selfadjoint
operators $A, B \in \mathcal{L}(E)$ by extending the bilinear forms
\begin{equation}
\langle Ax, y\rangle  = \int^1_0 (-J\dot x , y)\,dt, \quad
\langle Bx, y\rangle = \int^1_0 (B(t)x, y)\,dt    \tag {2.1}
\end{equation}
on $E$. Then $B$ is compact  \cite{l2}. We define
\begin{equation}
f(z)=\frac 12\langle Az,z\rangle -\int^1_0 H(t,z)\,dt  \tag {2.2}
\end{equation}
on $E$. It is well know that $f\in C^2(E, \mathbb{R})$ whenever $H$ satisfies
(H1). Looking for the
solutions of (1.1) is equivalent to looking for the critical points of $f$ 
\cite{c3,d1}.

For $B(t)$, by \cite{c2,l2,l3} 
we can define its Maslov-type index as a pair of integers
$(i(B), n(B))\in \mathbb{Z} \times \{0,\cdots ,2N\}$. Using the Floquet theory, we have
\begin{equation}
n(B) = \dim \ker (A-B).      \tag {2.3}
\end{equation}
Let $B_{\infty}(t)$ be the matrix function in (H2) with the Maslov-type
index $(i_{\infty}, n_{\infty})$, and $B_{\infty}$ be the operator, defined
by (2.1), corresponding to $B_{\infty}(t)$. Then by (2.3) we have
$$
n_{\infty} = \dim \ker (A-B_{\infty}).
$$
Let $\cdots \leq \lambda '_2 \leq \lambda '_1 <0<\lambda _1\leq \lambda _2
\leq \cdots $ be the
eigenvalues of $A-B_{\infty} $, and Let $\{ e'_j \}$ and $\{ e_j \}$ be
the eigenvectors of $A-B_{\infty} $ corresponding to $\{ \lambda '_j \}$ and
$\{ \lambda _j \}$ respectively.
For $m\geq 0$, set
$$
E_0=\ker (A-B_{\infty}),
$$
$$
E_m =E_0\oplus \mathop{\rm span}\{ e_1 ,\cdots ,e_m \}\oplus
\mathop{\rm span}\{ e'_1 ,\cdots ,e'_m \}
$$
and let $P_m $ be the orthogonal
projection from E to $E_m $. Then $\{ P_m \}$ is an
approximation scheme with respect to the operator $A-B_{\infty} $, i.e.
\begin{gather*}
(A-B_{\infty} )P_m =P_m (A-B_{\infty} ),  \\
P_m x \to x \quad as \quad m \to \infty , \quad  \forall x\in E.
\end{gather*}
In the following we denote $T^{\#} = (T_{ImT} )^{-1} $, and we also denote by $M^+ (\cdot )$,
$M^- (\cdot )$ and $M^0 (\cdot )$ the
positive definite, negative definite and null subspaces
of the selfadjoint linear operator defining it, respectively. The following result
was proved in \cite{f1}

\begin{theorem}[\cite{f1}] \label{thm2.1}
  For any continuous 1-periodic and symmetric
matrix function $B(t)$ with the
Maslov-type index $(i_0,n_0)$, there exists an $m^{\ast} >0$ such that
for $m\geq m^{\ast} $ we have
\begin{gather}
\dim M^+_d (P_m(A-B)P_m )=m+i_{\infty} -i_0+n_{\infty} -n_0  \nonumber\\
\dim M^-_d (P_m(A-B)P_m )=m-i_{\infty} +i_0     \tag{2.7}   \\
\dim M^0_d (P_m(A-B)P_m )=n_0 \nonumber
\end{gather}
where $d=\frac 14 \|(A-B)^{\#}\|^{-1}$, $M^+_d(\cdot )$, $M^-_d(\cdot )$ and
$M^0_d(\cdot )$ denote the eigenspaces corresponding to
the eigenvalue $\lambda $ belonging to $[d,+\infty ), (-\infty , -d]$ and
$(-d,d)$ respectively.
\end{theorem}

To prove Theorem 1.1 we need the following definition
and saddle point theorem which were given in \cite{g1}.

\begin{definition}[\cite{g1}] \label{def2.2} \rm
  Let $E$ be a $C^2$-Riemannian manifold,
$D$ be a closed subset of $E$. A family $\mathcal{F}(\alpha )$ is said to be a
homological family of dimension $q$ with boundary $D$ if, for some nontrivial
class $\alpha \in H_q(E,D)$, the family $\mathcal{F}(\alpha )$ is defined by
$$
\mathcal{F}(\alpha )=\lbrace G\subset E: \alpha \quad \text{is in the image of }
i_{\ast}: H_q(G,D)\to H_q(E,D) \rbrace ,
$$
where $i_{\ast}$ is the homomorphism induced by the immersion
$i: G\to E$.
\end{definition}

\begin{theorem}[\cite{g1}] \label{thm2.3}
As in Definition 2.2, for given
$E$, $D$ and $\alpha $, let $\mathcal{F}(\alpha )$ be a homological
family of dimension $q$ with boundary $D$. Suppose that $f\in
C^2(E,R)$ satisfies $(PS)$ condition. Set
$$
c\equiv c(f,\mathcal{F}(\alpha ))= \inf_{G\in \mathcal{F}(\alpha )}\sup_
{w\in G}f(w)
$$
If  $\sup_{w\in D}f(w)<c$ and $f'$ is Fredholm on
$$
\mathcal{K}_c=\lbrace x\in E: f'(x)=0, f(x)=c \rbrace ,
$$
then there exists $x\in \mathcal{K}_c$ such that the Morse indices
$m^-(x)$ and $m^0(x)$ of the functional $f$ at $x$ satisfy
$$
q - m^0(x) \leq m^-(x) \leq q .
$$
\end{theorem}

Let $f$ be defined as (2.2) and $f_m $ be the restriction of $f$ to the space
$E_m $.
We say that $f$ satisfies the $(PS)^{\ast}_c $ condition for $c\in \mathbb{R}$,
if any sequence $\{ x_m \}$ such that
$x_m \in E_m $, $f'_m(x_m)\to 0$ and $f_m(x_m)\to c$ possesses a
subsequence convergent in $E$  \cite{l1}.

\begin{lemma} \label{lm2.4}
 Under the conditions of Theorem 1.1, $f$ satisfies the $(PS)^{\ast}_c $
condition for any $c\in \mathbb{R}.$
\end{lemma}

\paragraph{Proof.} For any given $c\in \mathbb{R}$, let $\{z_{m}\}$ be the
$(PS)_{c}^{*}$ sequence, i.e., for $z_{m}\in E_{m}$,
\begin{equation}
f'(z_{m})\to 0, \quad f_{m}(z_{m})\to c . \tag {2.8}
\end{equation}
We want to show that $\{z_{m}\}$ is bounded in $E$. Then by standard arguments
\cite{l1}, $\{z_{m}\}$
possesses a subsequence convergent in $E$.

Suppose $\{z_{m}\}$ is not bounded and $\| z_{m}\| \to +\infty $ as $m\to +\infty $.
Define
$$
g(z)=\int_{0}^{1}G_{\infty }(t,z)dt, \quad \forall z\in E.
$$
Then $f(z) = \frac{1}{2}\langle (A-B_{\infty })z,z\rangle -g(z)$, for all 
$z\in E.$ By (H2) we know that
$$
\frac{|G_{\infty }'(t,z)|}{|z|}\to 0 \quad \text{ as }
 |z|\to \infty .
$$
This means that, for any $\varepsilon >0$, there exist $M>0$ such that
$$
|G_{\infty }'(t,z)|^{2}\leq \varepsilon |z|^{2}+M.
$$
Therefore,
\begin{align*}
|\langle g'(z_{m}),y\rangle | =& |\int_{0}^{1}(G_{\infty }'(t,z_{m}),y)dt| \\
&\leq \int_{0}^{1}|G_{\infty }'(t,z_{m})||y|dt\leq (\int_{0}^{1}|G_{\infty }'
(t,z_{m})|^{2})^{1/2}\| y\| _{L^{2}} \\
&\leq (\varepsilon \| z_{m}\| _{L^{2}}^{2}+M)^{1/2}\| y\| _{L^{2}}
\leq (\varepsilon \| z_{m}\| ^{2}+M)^{1/2}\| y\| .
\end{align*}
This implies
$$
\lim_{m\to \infty }\frac{\| g'(z_{m})\| }{\| z_{m}\| }
\leq \varepsilon ,\text{ for any }\varepsilon >0 ,
$$
i.e.
\begin{equation}
\frac{\| g'(z_{m})\| }{\| z_{m}\| }\to 0 \ \
\text{ as } \ \ m\to +\infty .   \tag {2.9}
\end{equation}
Write
\begin{align*}
z_{m}&=z_{m}^{+}+z_{m}^{-}+z_{m}^{0} \\
&\in M^{+}(P_{m}(A-B_{\infty })P_{m})\oplus
M^{-}(P_{m}(A-B_{\infty })P_{m})\oplus M_{m}^{0}(A-B_{\infty })P_{m}).
\end{align*}
Then
\begin{align*}
\langle f_{m}'(z_{m}),z_{m}^{+}\rangle 
&=\frac{1}{2}\langle (A-B_{\infty })z_{m}^{+},z_{m}^{+}\rangle
-\langle g'(z_{m}),z_{m}^{+}\rangle \\
&\geq C_{1}\| z_{m}^{+}\| ^{2}-\| g'(z_{m})\| \| z_{m}^{+}\| .
\end{align*}
By (2.8) and (2.9), we have
\begin{equation}
\frac{\| z_{m}^{+}\| }{\| z_{m}\| }\to 0 \quad \text{ as } 
 m\to \infty .        \tag {2.10}
\end{equation}
Similarly, we have
\begin{equation}
\frac{\| z_{m}^{\_}\| }{\| z_{m}\| }\to 0 \quad \text{ as }
 m\to \infty .    \tag {2.11}
\end{equation}

\paragraph{Case(i): } (H$4^{+}$) holds.
\begin{align*}
\langle f_{m}'(z_{m}),z_{m}\rangle -2f_{m}(z_{m})
&=\int_{0}^{1}[2G_{\infty }(t,z_{m})-(G_{\infty }'(t,z_{m}),z_{m})]dt \\
&\geq C_{1}\int_{0}^{1}|z_{m}|dt-C_{2}\\
&\geq C_{1}\int_{0}^{1}|z_{m}^{0}|dt-\int_{0}^{1}C_{1}(|z_{m}^{+}|+|z_{m}^{-}|)dt-C_{2} \\
&\geq C_{3}\| z_{m}^{0}\| - C_4(\| z_{m}^{+}\| +\| z_{m}^{-}\| +1) .
\end{align*}
Here we used the fact that $M^{0}(P_{m}(A-B_{\infty })P_{m})=\ker(A-B_{\infty
})$ is finite dimensional. By (2.8), (2.10) and (2.11), we have
\begin{equation}
\frac{\| z_{m}^{0}\| }{\| z_{m}\| }\to 0 \quad \text{ as }
m\to \infty .    \tag {2.12}
\end{equation}
But this implies the following contradiction,
\begin{equation}
1=\frac{\| z_{m}\| }{\| z_{m}\| }\leq \frac{\| z_{m}^{0}\|
+\| z_{m}^{-}\| +\| z_{m}^{+}\| }{\| z_{m}\| }\to 0 \quad
\text{ as }  m\to +\infty .  \tag {2.13}
\end{equation}
Therefore $\{z_{m}\}$ must be bounded, and $f$ satisfies $(PS)^{\ast}_c $ condition under $(H4^+)$.

\paragraph{Case (ii): } (H4$^-$) holds. Similar to case (i), we have
\begin{align*}
2f_{m}(z_{m})-\langle f_{m}'(z_{m}),z_{m}\rangle
&=\int_{0}^{1}[G_{\infty }'(t,z_{m}),z_{m})-2G_{\infty }(t,z_{m})]dt \\
&\geq C_{1}\int_{0}^{1}|z_{m}|-C_{2}\geq C_{3}\| z_{m}^{0}\|
-C_{4}\| z_{m}^{+}\| +\| z_{m}^{-}\| +1) .
\end{align*}
This implies (2.12) and (2.13). Thus $\{z_{m}\}$ must be bounded, and $f$ 
satisfies $(PS)^{\ast}_c $ condition under (H4$^-$).

Notice that we assume $\|z_m\| \to +\infty $ as $m \to +\infty $. Then
by (2.10) and (2.11), there exist $m_{0}>0$ such that for $m\geq m_{0}$
\begin{equation}
\| z_{m}^{0}\| \geq \| z_{m}^{+}+z_{m}^{-}\| .   \tag {2.14}
\end{equation}
Moreover, if $|G_{\infty }'(t,z_{m})|\leq M_{1}|z|^{\beta }$ for $|z|\geq L$, we will show that
for $m$ large enough
\begin{equation}
\| z_{m}^{+}+z_{m}^{-}\| \leq \varepsilon _{0}\| z_{m}^{0}\| ^{\beta } ,
\tag {2.15}
\end{equation}
where $\varepsilon _{0}>0$ is a constant independent of $m$.
In fact, we have
$$
|G_{\infty }'(t,z)|^{2}\leq M_{1}^{2}|z|^{2\beta }+M_{2} ;
$$
\begin{align*}
|\langle g'(z_{m}),y\rangle |&\leq \int_{0}^{1}|G_{\infty }'(t,z_{m})||y|dt
\leq (\int_{0}^{1}|G_{\infty }'(t,z_{m})|^{2}dt)^{1/2}\| y\| _{L^{2}} \\
&\leq (M_{1}^{2}\| z_{m}\| _{L^{2\beta }}^{2\beta }+M_{2})^{1/2}\| y\| _{L^{2}}
\leq (M_{1}^{2}\| z_{m}\| ^{2\beta }+M_{2})^{1/2}\| y\| .
\end{align*}
This implies that for $m$  large enough
\begin{equation}
\frac{\|g'(z_{m})\|}{\| (z_{m})\| ^{\beta }}\leq M_{3} .  \tag {2.16}
\end{equation}
By (2.8), (2.14) and (2.16), for $m$ large enough, we have
\begin{align*} 
0\leftarrow \|f_{m}'(z_{m})\|&=\|\langle A-B_{\infty
})z_{m}-P_{m}g'(z_{m})\| \\
&\geq \varepsilon _{1}\| z_{m}^{+}+z_{m}^{-}\| -M_{3}\|
z_{m}\| ^{\beta } \\
&\geq \varepsilon _{1}\| z_{m}^{+}+z_{m}^{-}\| -M_{3}2^{\beta
}\| z_{m}^{0}\| ^{\beta } .
\end{align*}
This implies that, for $m$ large enough, (2.15) holds.


\paragraph{Case (iii): } (H5$^+$) holds. By (2.2) and (2.15), 
for $m$ large enough,
\begin{equation}
\begin{aligned}
g(z_{m})&=\frac{1}{2}\langle (A-B_{\infty
})(z_{m}^{+}+z_{m}^{-}),z_{m}^{+}+z_{m}^{-}\rangle -f(z_{m}) \\
&\leq C_{1}\| z_{m}^{+}+z_{m}^{-}\| ^{2}+C_0\leq C_{2}\|
z_{m}^{0}\| ^{2\beta }+C_0 .
\end{aligned}  \tag {2.17}
\end{equation}
On the other hand, by (H5$^+$),
\begin{equation}
g(z_{m})=\int_{0}^{1}G_{\infty }(t,z_{m})dt \geq \int_{0}^{1}M_{2}|z_{m}|^{\alpha }dt-M_{3}
\geq M_{4}\| z_{m}^{0}\| ^{\alpha }-M_{3} . \tag {2.18}
\end{equation}
Notice that $\alpha >2\beta $, we get a contradiction from (2.17) and (2.18).
Therefore $\{z_{m}\}$ is bounded, and $f$ satisfies $(PS)^{\ast}_c $ condition under $(H5^+)$.
Here in (2.18) we used the following claim.

\noindent{\bf Claim: } For $m$ large enough, there exists $\varepsilon _{2} > 0$ such that
\begin{equation}
\int_{0}^{1}|z_{m}|^{\alpha }dt\geq \varepsilon _{2}\| z_{m}^{0}\|
^{\alpha } .   \tag {2.19}
\end{equation}
In fact, for $\alpha >1$, by (2.15) and the fact $\beta <1$, we have
\begin{align*}
\int_{0}^{1}(z_{m},z_{m}^{0})dt 
&\leq \Big(\int_{0}^{1}|z_{m}|^{\alpha }dt\Big)^{1/\alpha }
\Big(\int_{0}^{1}|z_{m}^{0}|^{\frac{\alpha }{\alpha -1}}dt\Big)
^{\frac{\alpha -1}{\alpha }} \\
&\leq C_{\alpha }\Big(\int_{0}^{1}|z_{m}|^{\alpha }dt\Big)^{1/\alpha}\|
z_{m}^{0}\| ;
\end{align*}
\begin{align*}
\int_{0}^{1}(z_{m},z_{m}^{0})dt&=\int_{0}^{1}(z_{m}^{0},z_{m}^{0})dt+
\int_{0}^{1}(z_{m}^{+}+z_{m}^{-},z_{m}^{0})dt \\
&\geq \int_{0}^{1}(z_{m}^{0})^{2}dt-\varepsilon _{3}\|
z_{m}^{+}+z_{m}^{-}\| \| z_{m}^{0}\| \\
&\geq \varepsilon _{4}\| z_{m}^{0}\| ^{2}-\varepsilon _{5}\|
z_{m}^{0}\| ^{1+\beta } \geq \varepsilon _{6}\| z_{m}^{0}\| ^{2},
\end{align*}
for $m$ large enough. This implies (2.19) for $\alpha >1$.

For $\alpha =1$, since $z_{m}^{0}\in \ker (A-B_{\infty })$, we know that
$z_{m}^{0}$ satisfies the linear system
$$
\dot z = JB_{\infty }(t)z.
$$
This implies that $z_{m}^{0}(t)\neq 0, \quad \forall t\in [0,1]$. Therefore
$$
c_{1}\| z_{m}^{0}\| \leq |z_{m}^{0}(t)|\leq c_{2}\|
z_{m}^{0}\| , \quad \forall t\in [0,1],
$$
where $c_{1},c_{2}>0$ are constants independent of $m$ \cite{c5}. Now we have
\begin{align*}
\int_{0}^{1}(z_{m},z_{m}^{0})dt&\leq \int_{0}^{1}|z_{m}||z_{m}^{0}|dt
\leq (\int_{0}^{1}|z_{m}|dt)\| z_{m}^{0}(t)\| _{\infty } \\
&\leq c_{2}\| z_{m}^{0}\| (\int_{0}^{1}|z_{m}|dt) .
\end{align*}
Combining this with the proved fact
$$
\int_{0}^{1}(z_{m},z_{m}^{0})dt\geq \varepsilon _6\| z_{m}^{0}\| ^{2},
$$
we get (2.19) for $\alpha =1$.


\paragraph{Case(iv): } (H$5^-$) holds. Similar to case(iii), we have
\begin{gather*}
-\int_{0}^{1}G_{\infty }(t,z_{m})dt\leq |f_{m}(z_{m})|+|\frac{1}{2}\langle
(A-B_{\infty })z_{m},z_{m}\rangle |
\leq C_{2}\| z_{m}^{0}\| ^{2\beta }+C_0 ; \\
-\int_{0}^{1}G_{\infty }(t,z_{m})dt\geq \int_{0}^{1}(M_{2}|z_{m}|^{\alpha }-M_{3})dt
\geq M_{4}\| z_{m}^{0}\| ^{\alpha }-M_{3}.
\end{gather*}
We get a contradiction because of $\alpha >2\beta $. Thus $\{z_{m}\}$ is
bounded, and $f$ satisfies $(PS)^{\ast}_c $ condition under (H$5^-$).

\paragraph{Case(v): } (H$6^+$) holds. For $m$ large enough, by (2.15) 
and the claim in Case (iii), we have
\begin{equation}
\begin{aligned}
\int_{0}^{1}&(G_{\infty }'(t,z_{m}),z_{m})dt\\
&\leq |-\langle f_{m}'(z_{m}),z_{m}\rangle +\langle (A-B_{\infty
})(z_{m}^{+}+z_{m}^{-}),(z_{m}^{+}+z_{m}^{-})\rangle | \\
&\leq \| z_{m}\| +\varepsilon _{6}\| z_{m}^{+}+z_{m}^{-}\| ^{2}
\leq \| z_{m}^{0}\| +\varepsilon _{0}\| z_{m}^{0}\| ^{\beta
}+\varepsilon _{7}\| z_{m}^{0}\| ^{2\beta } ;
\end{aligned}\tag {2.20}
\end{equation}
\begin{equation}
\int_{0}^{1}(G_{\infty }'(t,z_{m}),z_{m})dt \geq
M_{2}\int_{0}^{1}|z_{m}|^{\alpha }dt-M_{3}
\geq M_{4}\| z_{m}^{0}\| ^{\alpha }-M_{3}.    \tag {2.21}
\end{equation}
We get a contradiction from $\alpha >2\beta $, (2.20) and (2.21).
Thus $\{z_{m}\}$ is bounded, and $f$ satisfies $(PS)^{\ast}_c $ condition
under $(H6^+)$.

\paragraph{Case(vi): } (H$6^-$) holds. Similar to case(v), we have
\begin{gather*}
-\int_{0}^{1}(G_{\infty }'(t,z_{m}),z_{m})dt\leq \|
z_{m}^{0}\| +\varepsilon _{0}\| z_{m}^{0}\| ^{\beta }+\varepsilon
_{7}\| z_{m}^{0}\| ^{2\beta } ; \\
-\int_{0}^{1}(G_{\infty }'(t,z_{m}),z_{m})dt\geq M_{4}\|
z_{m}^{0}\| ^{\alpha }-M_{3}.
\end{gather*}
Then $\alpha >2\beta $ implies that $\{z_{m}\}$ must be bounded,
and $f$ satisfies $(PS)^{\ast}_c $ condition under (H$6^-$).
\hfill$\Box$


\paragraph{Proof of Theorem 1.1} {\bf Case(i) \&  (iii): } 
By a direct computation, (H$4^{+}$) and (H5$^+$) imply that
\begin{equation}
G_{\infty }(t,z)\to +\infty \quad \text{as }  \ |z|\to \infty .
\tag {2.22}
\end{equation}
By (H2), for any $\varepsilon >0$, there exists $M>0$ such that
\begin{equation}
|G_{\infty }(t,z)|\leq \varepsilon |z|^{2}+M .   \tag {2.23}
\end{equation}
For $m>0$, by using the same arguments as in the proof of Lemma 2.4, we know
that $f_{m}$ satisfies (PS) condition. Let
\begin{gather*}
X_{m}=M^{-}(P_{m}(A-B_{\infty })P_{m})\oplus M^{0}(P_{m}(A-B_{\infty })P_{m}), 
\\
Y_{m}=M^{+}(P_{m}(A-B_{\infty })P_{m}).
\end{gather*}
By (2.23), for all $z^{+}\in Y_{m}$, we have
$$
f_{m}(z^{+})=\frac{1}{2}\langle (A-B_{\infty })z^{+},z^{+}\rangle
-\int_{0}^{1}G_{\infty }(t,z^{+})dt\geq C_{1}\| z^{+}\|
^{2}-\varepsilon \| z^{+}\| ^{2}-M .
$$
We can choose $0<\varepsilon \leq C_{1}/2$. Then there is a $\delta
>0$ such that
\begin{equation}
f_{m}(z^{+})\geq -\delta >-\infty , \quad \forall z^{+}\in Y_{m} .  \tag{2.24}
\end{equation}
By (2.22), there exist $M_{0}>0$, such that $G_{\infty }(t,z)\geq -M_{0}$, 
for all $z\in \mathbb{R}^{2N}$.
This implies that, for all $z^{-}\oplus z^{0}\in X_{m}$,
\begin{align*}
f_{m}(z^{-}\oplus z^{0})&=\frac{1}{2}\langle (A-B_{\infty
})z^{-},z^{-}\rangle -\int_{0}^{1}G_{\infty }(t,z^{-}+z^{0})dt \\
&\leq -C_{1}\| z^{-}\| ^{2}+M_{0}\leq -2\delta ,
\end{align*}
if $\| z^{-}\| \geq L=\sqrt{\frac{2\delta +M_{0}}{C_{1}}}.$

Since $M^{0}(P_{m}(A-B_{\infty })P_{m})=M^{0}(A-B_{\infty })$ is a finite dimensional space,
by (2.22) we have that
$$
\int_{0}^{1}G_{\infty }(t,z^{-}+z^{0})dt\to +\infty \quad \text{as } \|
z^{0}\| \to +\infty \quad \text{uniformly for }  \| z^{-}\| \leq L \,.
$$
Thus there exists $L_{1}>0$ such that for $\| z^{0}\| \geq L_{1}$ and
$\| z^{-}\| \leq L$
$$
f_{m}(z^{-}+z^{0})\leq -\int_{0}^{1}G_{\infty }(t,z^{-}+z^{0})dt\leq -2\delta .
$$
Let $Q_{m}=\{z^{-}\oplus z^{0}\in X_m : \quad \| z^{-}+z^{0}\| \leq L+L_{1}\}$. Then we have
\begin{equation}
f_{m}(z)\leq -2\delta , \quad \forall z\in \partial Q_{m}.     \tag{2.25}
\end{equation}
Let $S=Y_{m}$. Then $\partial Q_{m}$ and $S$ homologically link \cite{c6}. Let
$D=\partial Q_{m}$ and $\alpha =[Q_{m}]\in H_{k}(E_{m},D)$ with $k=\dim(X_{m})$.
Then $\alpha $ is nontrivial and $\mathcal{F}(\alpha )$ defined by Definition 2.2 is a
homological family of dimension $k$ with boundary $D$  \cite[p. 84]{c6}.
By Theorem 2.3, (2.24) and (2.25), there exists a
critical point $x_{m}$ of $f_{m}$ such that the Morse
indices $m^{-}(x_{m})$ and $m^{0}(x_{m})$ of $f_{m}$ at $x_{m}$
satisfies
\begin{gather}
\dim X_{m}-m^{0}(x_{m})\leq m^{-}(x_{m})\leq \dim X_{m} ;    \tag {2.26} \\
-\delta \leq f_{m}(x_{m})=c_{m}=c(f_{m},\mathcal{F}(\alpha )) .  \tag {2.27}
\end{gather}
Since $Q_{m}\in \mathcal{F}(\alpha )$, by (2.23) we have
\begin{align*}
-\delta &\leq c_{m}\leq \sup_{z^{-}+z^{0} \in
Q_{m}}f_{m}(z^{-}+z^{0})\\
&\leq \frac{1}{2}\| (A-B_{\infty })\|
(L+L_{1})^{2}+\varepsilon (L+L_{1})^{2}+M=M_{2} ,
\end{align*}
where $\delta $ and $M_{2}$ are constants independent of $m$. Hence passing to a subsequence we have
$$
c_{m}\to c,  \quad  -\delta \leq c\leq M_{2} .
$$
Since $f$ satisfies $(PS)_{c}^{*}$ condition, passing to a subsequence,
there exist $x^{*}\in E$ such that
\begin{equation}
x_{m}\to x^{*}, \quad  f(x^{*})=c, \quad  f'(x^{*})=0 \,.   \tag{2.28}
\end{equation}
By standard arguments, $x^{*}$ is a classical solution of (1.1).

Let $B^{*}(t)=H''(t,x^{*}(t))$ and $B^{*}$ be the operator,
defined by (2.1), corresponding to $B^{*}(t)$. Let $(i^{*},n^{*})$ be the
Maslov-type index of $B^{*}(t).$ It is easy to show that
$$
\| f''(z)-(A-B^{*})\| \to 0 \quad \text{ as } \quad \
\| z-x^{*}\| \to 0 .
$$
Let $d=\frac{1}{4}\| (A-B^{*})^{\#}\| ^{-1}$. Then there exists
$r_{0}>0 $ such that
$$
\| f''(z)-(A-B^{*})\| <\frac{1}{2}d, \quad \forall z\in
V_{r_{0}}=\{z\in E:\| z-x^{*}\| \leq r_{0}\} .
$$
This implies that
\begin{equation}
\dim M^{\pm }(f_{m}''(z))\geq \dim M_{d}^{\pm
}(P_{m}(A-B^{*})P_{m}), \quad \forall z\in V_{r_{0}}\cap E_{m} . \tag{2.29}
\end{equation}
By (2.26), (2.28), (2.29) and Theorem 2.1, there exist $m_{1}>m^{*}$ such that
for $m\geq m_{1}$,
$$
m+n_{\infty }=\dim (X_{m})\geq m^{-}(x_{m})\geq \dim
M_{d}^{-}(P_{m}(A-B^{*})P_{m})=m-i_{\infty }+i^{*} ;
$$
\begin{align*}
m+n_{\infty }&=\dim (X_{m})\leq m^{-}(x_{m})+m^{0}(x_{m}) \\
&\leq \dim [M_{d}^{-}(P_{m}(A-B^{*})P_{m})\oplus M_{d}^{0}(P_{m}(A-B^{*})P_{m})] \\
&=m-i_{\infty }+i^{*}+n^{*} .
\end{align*}
This implies that $i_{\infty }+n_{\infty }\in [i^{*},i^{*}+n^{*}]$, which means that $x^{*}\neq 0$, i.e., $x^{*}$ is
a nontrivial 1-periodic solution of the system (1.1).


\paragraph{Case(ii)\&(iv): } By (H4$^-$) and (H$5^-$) we have
$$
G_{\infty }(t,z)\to -\infty \quad \ \text{ as } \quad \ |z|\to \infty .
$$
Let $X_{m}=M^{-}(P_{m}(A-B_{\infty })P_{m})$ and $Y_{m}=M^{+}(P_{m}(A-B_{\infty })P_{m})\oplus
M^{0}(P_{m}(A-B_{\infty })P_{m})$.
By using similar arguments as in the proof of (2.24) and (2.25)
we have
\begin{gather*}
f_{m}(z^{+}+z^{0})\geq -\delta _1>-\infty , \quad 
 \forall z^{+}+z^{0}\in Y_{m}; \\
f_{m}(z)\leq -2\delta _1, \quad \quad \forall z\in \partial Q_{m},
\end{gather*}
where $Q_{m}=\{z^{-}\in X_{m}: \; \| z^{-}\| \leq L_{2}\}$. Here $\delta
_{1}>0$ and $L_{2}>0$ are constants independent of $m$. By using the same
arguments, one can prove that (2.26)-(2.29) still hold.
By Theorem 2.1 there exist $m_{2}>m^{*}$ such that for $m\geq m_{2}$
$$
m=\dim (X)\geq m^{-}(x_{m})\geq \dim
M_{d}^{-}(P_{m}(A-B^{*})P_{m})=m-i_{\infty }+i^{*};
$$
$$
m=\dim (X)\leq m^{-}(x_{m})+m^{0}(x_{m})\leq m-i_{\infty }+i^{*}+n^{*} .
$$
Therefore, we have $i_{\infty }\in [i^{*},i^{*}+n^{*}]$, which implies $x^{*}\neq 0$, i.e., $x^{*}$ is
a nontrivial 1-periodic solution of the system (1.1).

\paragraph{Case(v): } (H$6^+$) holds. We shall use the same idea as in the proof of
\cite[Theorem 1.1]{f2}. Let
$X=\ker (A-B_{\infty })$, $Y=Im(A-B_{\infty })$, and $P:E\to
X$, $Q:E\to Y$ be the orthogonal projections. By the special
construction of the Galerkin approximation scheme $\{P_{m}\}$, we have
$$
E_{m}=X\oplus P_{m}Y, \quad \ker (P_{m}(A-B_{\infty })P_{m})=X, \ \ Im(P_{m}(A-B_{\infty })P_{m})=P_{m}Y .
$$
For given $m>0$, since $\dim E_{m}<+\infty $, we have
\begin{equation}
\int_{0}^{1}|z|^{\alpha }dt\geq c_{m}\| z\| ^{\alpha },
\quad \forall z\in E_{m},   \tag {2.30}
\end{equation}
where $c_{m}>0$ is a constant which depends on $m$.
Let $\pi $ be the flow of $f_{m}$ in $E_{m}$, generated by
\begin{gather*}
\dot y=-(A-B_{\infty })y+QP_{m}g'(x+y) , \\
\dot x=P(P_{m}g'(x+y)) , \quad \text{for }  \ (x,y)\in X \oplus P_mY=E_{m} .
\end{gather*}
Let
$$
V^{\pm }=\{y^{\pm }\in M^{\pm }(P_{m}(A-B_{\infty })P_{m}): \; 
\| y^{\pm }\| \leq r_{Y}\}, \quad 
W=\{x\in X: \; \|x\| \leq r_{X}\}.
$$
We shall show that there are $r_{Y}>0$ and $r_{X}>0$ such that
$D=(V^{-}\times V^{+})\times W$ is an isolating block of $\pi .$

By using the some arguments as in the proof of (2.16), (H$6^+$) implies that
\begin{equation}
\| g'(z)\| \leq M_{3}\| z\| ^{\beta }, \quad \forall z\in
E, \quad \| z\| \geq L.      \tag{2.31}
\end{equation}
On the other hand, (2.30) and (H$6^+$) also imply that
\begin{equation}
\langle g'(z_{m}),z_{m}\rangle  \geq M_{2}c_{m}\| z_{m}\| 
^{\alpha}-M_{4}, \quad \forall z_{m}\in E_{m}.    \tag {2.32}
\end{equation}
For any $x\in \partial W$, $y\in V^{-}\times V^{+}$, by (2.30)-(2.32) we have
\begin{align*}
\frac{d}{dt}(\frac{1}{2}\| x\| ^{2})|_{t=0}&=\langle x, \dot x\rangle |_{t=0}
=\langle x, g'(x+y)\rangle |_{t=0} \\
&=\langle x+y,g'(x+y)\rangle |_{t=0}-\langle y,g'(x+y)\rangle |_{t=0} \\
&\geq M_{2}c_{m}\| x+y\| ^{\alpha }-M_{4}-M_{3}\| x+y\| ^{\beta }\| y\| \\
&\geq \| x+y\| ^{\beta }[M_{2}c_{m}\| x+y\| ^{\alpha -\beta }-M_{3}\| y\| ]-M_{4} \\
&\geq r_{X}^{\beta }[M_{2}c_{m}r_{X}^{\alpha -\beta }-M_{3}r_{Y}]-M_{4} \\
&\geq r_{X}^{\beta }r_{Y}-M_{4}\geq 1>0,
\end{align*}
provided
\begin{equation}
r_{X}^{\alpha -\beta }=(\frac{M_{3}+1}{M_{2}c_{m}})r_{Y}=c'r_{Y}, \quad
\text{and} \quad r_{X}\geq [c'(M_{4}+1)]^{1/2}+1.    \tag {2.33}
\end{equation}
%
For any $y^{-}\in \partial V^{-}$, $y^{+}\in V^{+}$, $x\in W$, and $y=y^{+}+y^{-}$, by (2.30)-(2.33) we
have
\begin{equation}
\begin{aligned}
\frac{d}{dt}(\frac{1}{2}\| y^{-}\| ^{2})|_{t=0}&=\langle \dot {y}^{-},y^{-}\rangle |_{t=0} \\
&=[-\langle (A-B_{\infty })y^{-},y^{-}\rangle +\langle QP_{m}g'(x+y),y^{-}\rangle ]|_{t=0} \\
&\geq \rho r_{Y}^{2}-M_{3}\| x+y\| ^{\beta }\| y^{-}\|
 \geq \rho r_{Y}^{2}-M_{3}(r_{X}+2r_{Y})^{\beta }r_Y \\
&\geq \rho r_{Y}^{2}-M_{3}[(c'r_{Y})^{\frac{1}{\alpha -\beta }
}+2r_{Y}]^{\beta }r_{Y},  \end{aligned} \tag {2.34}
\end{equation}
where
$$
\rho =\inf _{\|y^-\|=1}|\langle y^{-},(A-B_{\infty })y^{-}\rangle |, 
\quad \text{and} \quad y^{-}\in M^{-}(A-B_{\infty }).
$$
If $\alpha -\beta \geq 1$ and $r_{Y}\geq 1$, by (2.34) we have
$$
\frac{d}{dt}(\frac{1}{2}\| y^{-}\| ^{2})|_{t=0}\geq \rho
r_{Y}^{2}-M_{3}[c^{\prime \frac{1}{\alpha -\beta }}+2]^{\beta }
r_{Y}^{\beta +1}>0 ,
$$
provided
\begin{equation}
r_{Y}\geq (\frac{M_{3}[c^{\prime \frac{1}{\alpha -\beta }}+2]^{\beta }+1}{\rho })
^{\frac{1}{1-\beta }}+1 .    \tag{2.35}
\end{equation}
If $\alpha -\beta <1$ and $r_{Y}\geq 1$, we have
$$
\frac{d}{dt}(\frac{1}{2}\| y^{-}\| ^{2})|_{t=0}\geq \rho r_{Y}^{2}-
M_{3}[c^{\prime \frac{1}{\alpha -\beta }}+2]^{\beta }\cdot r_{Y}^{
\frac{\alpha }{\alpha -\beta }}>0,
$$
provided
\begin{equation}
r_{Y}\geq (\frac{M_{3}[c^{\prime \frac{1}{\alpha -\beta }
}+2]^{\beta }+1}{\rho })^{\frac{\alpha -\beta }{\alpha -2\beta }}+1 .  \tag{2.36}
\end{equation}
Now we can choose $r_{X}>0$ and $r_{Y}>0$ such that (2.33)-(2.36) hold.
Similarly, for any $y^{+}\in \partial V^{+}$, $y^{-}\in V^{-}$, $x\in W$, we have
$$
\frac{d}{dt}(\frac{1}{2}\| y^{+}\| ^{2})|_{t=0}<0.
$$
Therefore $D$ is an isolating block of $\pi $ and
$$
D^{-}=(\partial V^{-}\times V^{+})\times W\cup (V^{-}\times V^{+})\times \partial W.
$$
Follow the same arguments as in the proof of Theorem 1.1 in \cite{f2}, by Conley
index theory, $f$ has a critical point $x^{*}\neq 0$, i.e., $x^{*}$ is
a nontrivial 1-periodic solution of the system (1.1).

\paragraph{Case(vi): } (H$6^-$) holds. Using the same arguments as in the proof of
Case(v), (H$6^-$) implies that
$$
\frac{d}{dt}(\frac{1}{2}\| x\| ^{2})|_{t=0}<0.
$$
Therefore $D$ is an isolating block of $\pi $ and $D^{-}=(\partial V^{-}\times V^{+})\times W$.
By Conley index theory \cite[Theorem 3.3]{f2}, $f$ has a critical point $x^{*}\neq 0$, i.e., $x^{*}$ is
a nontrivial 1-periodic solution of the system (1.1).
We omit the details.


\section{Examples}

In this section, we give some examples which can not be
solved directly by the results in the references.

\paragraph{Example 3.1:} Consider the function given by (1.2), i.e.,
$$
H(t,z)=\frac{7|z|^{2}}{2\ln (e+|z|^{2})}, \quad  \forall t\in [0,1], \;
\forall z\in \mathbb{R}^{2N}.
$$
Then $B_{0}(t)=7I_{2N}$, $B_{\infty }(t)=0.$ By a direct computation,
\begin{gather*}
(i_{0},n_{0})=(3N,0), \quad  (i_{\infty },n_{\infty })=(-N,2N), \\
i_{\infty }+n_{\infty }=N\notin [3N,3N]=[i_{0},i_{0}+n_{0}].
\end{gather*}
Moreover, $G_{\infty }(t,z)=H(t,z)$ satisfies (H$4^{+}$).
By Theorem 1.1(i), the system (1.1) possesses a nontrivial 1-periodic solution.

\paragraph{Example 3.2:} Consider the function  given by (1.3), i.e.,
$$
H(t,z)=\frac {1}{2}|z|^2-\frac{|z|^{2}}{\ln (e+|z|^{2})}, \quad 
 \forall t\in [0,1], \; \forall z\in \mathbb{R}^{2N}.
$$
Then $B_{0}(t)=-I_{2N}$, $B_{\infty }(t)=I_{2N}$. By a direct computation
$$
(i_{0},n_{0})=(-N,0), \ \ (i_{\infty },n_{\infty })=(N,0),
\text{ and } G_{\infty }(t,z)=-\frac{|z|^{2}}{\ln (e+|z|^{2})} .
$$
One can show that (H4$^-$) holds. Theorem 1.1(ii) implies that the system (1.1) has a nontrivial 1-periodic
solution.


\paragraph{Example 3.3:} 
Let $H(t,z)\in C^{2}([0,1]\times \mathbb{R}^{2N},\mathbb{R})$ such that
\begin{gather*}
H(t,z)=\frac{7}{2}|z|^{2} \quad \text{for } |z|\leq 1; \\
H(t,z)=|z|\ln (1+|z|^{2}) \quad \text{for } |z|\geq 100.
\end{gather*}
Then $B_{0}(t)=7I_{2N}$, $B_{\infty }(t)=0$, and $G_{\infty }(t,z)=H(t, z)$ satisfies
(H5$^+$) with $\alpha =1$, $\beta =\frac{1}{4}$ and $L$ being large enough.
By Theorem 1.1(iii), the system (1.1) has a nontrivial 1-periodic solution.

\paragraph{Example 3.4:} 
Let $H(t,z)\in C^{2}([0,1]\times \mathbb{R}^{2N},\mathbb{R})$ such that
\begin{gather*}
H(t,z)=\frac{7}{2}|z|^{2} \quad \text{for }  |z|\leq 1; \\
H(t,z)=|z|^{\frac{4}{3}}\ln (1+|z|^{2}) \quad \text{for }  |z|\geq 100.
\end{gather*}
By a direct computation, $G_{\infty }(t,z)=H(t,z)$ satisfies (H$6^+$).
Thus the system (1.1) has a nontrivial 1-periodic solution by Theorem 1.1(v).

\paragraph{Example 3.5:} 
Let $H(t,z)\in C^{2}([0,1]\times \mathbb{R}^{2N},\mathbb{R})$ such that
\begin{gather*}
H(t,z)=\frac{7}{2}|z|^{2} \quad \text{for } |z|\leq 1; \\
H(t,z)=-|z|^{\frac{4}{3}}\ln (1+|z|^{2}) \quad \text{for }  |z|\geq 100.
\end{gather*}
Then (H$6^-$) holds. By Theorem 1.1(vi), the system (1.1) possesses 
a nontrivial 1-periodic solution.


\paragraph{Acknowledgments:} 
The author wishes to express his sincere thanks to the referee for useful 
suggestions.

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\end{thebibliography}

\noindent\textsc{Guihua Fei}\\
Department of Mathematics and statistics \\
University of Minnesota \\
Duluth, MN 55812, USA. \\
e-mail address:  gfei@d.umn.edu

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