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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 150, pp. 1--16.\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-2015/150\hfil Homoclinic orbits]
{Homoclinic orbits of second-order nonlinear difference equations}

\author[H. Shi, X. Liu, Y. Zhang \hfil EJDE-2015/150\hfilneg]
{Haiping Shi, Xia Liu, Yuanbiao Zhang}

\address{Haiping Shi \newline
Modern Business and Management Department,
Guangdong Construction Vocational Technology Institute,
 Guangzhou 510440, China}
\email{shp7971@163.com}

\address{Xia Liu \newline
Oriental Science and Technology College,
Hunan Agricultural University, Changsha 410128, China\newline
Science College, Hunan Agricultural University, Changsha 410128, China}
\email{xia991002@163.com}

\address{Yuanbiao Zhang \newline
Packaging Engineering Institute, Jinan University,
Zhuhai 519070, China}
\email{abiaoa@163.com}

\thanks{Submitted January 23, 2015. Published June 10, 2015.}
\subjclass[2010]{34C37, 37J45, 39A12}
\keywords{Homoclinic orbits; second order; nonlinear difference equations;
 \hfill\break\indent 
discrete variational methods}

\begin{abstract}
 We establish existence criteria for homoclinic orbits of second-order nonlinear
 difference equations by using the critical point theory in combination with
 periodic approximations.
\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}

Homoclinic orbits play an important role in analyzing the
chaos of dynamical  systems, and have been the subject of
many investigations.
If a system has the transversely intersected homoclinic
orbits, then it must be chaotic. If it has the smoothly connected
 homoclinic orbits, then it cannot stand the perturbation, its
 perturbed system probably produce chaotic phenomenon.
So homoclinic orbits have been extensively
 investigated since the time of Poincar\'e, see  
\cite{GuAWO,GuOA,GuOXA1,GuOXA2,GuOXA3,GuOXA4,Po,Ra2}
 and the references therein.

 Difference equations \cite{Ag,CuF} are closely related to differential equations
in the sense that  a differential equation model is often derived from
a difference equation, and numerical solutions of a differential
equation are obtained by discretizing the differential  equation.
 Therefore, the study of homoclinic orbits 
\cite{ChT1,ChT2,ChT3,ChT4,ChW,DeC,FaZ,Lo1,Lo2,MaG1,MaG2,ZhYC} of difference
equation is meaningful.


Here $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ denote the sets of all natural numbers,
integers and real numbers respectively.
For any $a,b\in \mathbb{Z}$, define
 $\mathbb{Z}(a)=\{a,a+1,\dots\}$, $\mathbb{Z}(a,b)=\{a,a+1,\dots,b\}$ when
$a\leq b$. The symbol $l^2$  denotes the space of  real functions whose second
powers are summable on $\mathbb{Z}$. Also, * denotes the transpose of a
 vector.

 This article considers the existence for homoclinic orbits of second-order
nonlinear difference equation
\begin{equation}\label{e1.1}
Lu(t)=f(t,u(t+T),u(t),u(t-T)),\quad t\in \mathbb{Z}
\end{equation}
containing both advance and retardation. Here the
 operator $L$ is the Jacobi operator
 $$
Lu(t)=a(t)u(t+1)+a(t-1)u(t-1)+b(t)u(t),
$$
 where $a(t)$ and $b(t)$ are real valued for each $t\in \mathbb{Z}$,
$T$  is a given nonnegative integer,
 $f\in C(\mathbb{Z}\times\mathbb{R}^3,\mathbb{R})$, $a(t)$,
 $b(t)$ and $f(t,v_1,v_2,v_3)$ are  $M$-periodic in $t$ for a given positive integer
 $M$. Jacobi operators appear in a variety of applications \cite{Te}.

 We may think of \eqref{e1.1} as being a discrete analogue of the
 second-order nonlinear differential equation
\begin{equation}\label{e1.2}
 Su(s)= f(s,u(s+T),u(s),u(s-T)),\quad s\in \mathbb{R},
\end{equation}
where $S$ is the Sturm-Liouville differential expression,
$f\in C(\mathbb{R}^4,\mathbb{R})$.
 Equations similar in structure to \eqref{e1.2} arise in the
 study of homoclinic orbits \cite{GuOA,GuOXA2,GuOXA3,GuOXA4}
of functional differential equations.

For the case $T=1$, Chen and Fang \cite{ChF}  obtained the existence of periodic
and subharmonic  solutions of the second-order $p$-Laplacian difference equation
 $$
\Delta(\varphi_p(\Delta u(t-1)))+f(t,u(t+1),u(t),u(t-1))=0,\quad t\in \mathbb{Z},
$$
 and Chen and Tang \cite{ChT1} obtained the existence of infinitely many homoclinic
orbits of the fourth-order difference equation
 $$
\Delta^4 u(t-2)+q(t)u(t)=f(t,u(t+1),u(t),u(t-1)),\quad t\in \mathbb{Z}
$$
 containing both advance and retardation.

It is well known that critical point theory is a powerful tool that deals with
the problems  of differential equations 
\cite{CaM,GuAWO,GuOA,GuOXA1,GuOXA2,GuOXA3,GuOXA4,MaW,Ra1}.
Only since 2003, critical point theory has been employed to
establish sufficient conditions on the existence of periodic solutions
for second-order difference  equations \cite{GuY1,GuY2}. 
 Along this direction, Ma and Guo \cite{MaG1} (without periodicity assumption) 
and \cite{MaG2} (with periodicity  assumption) applied variational 
methods to prove the existence of
 homoclinic orbits for the special form of \eqref{e1.1} (with $T=0$).
The Ambrosetti-Rabinowitz condition plays a crucial role  to ensure the
 boundedness of Palais-Smale sequences. This is very crucial in
 applying the critical point theory.

Some special cases of \eqref{e1.1} have been studied by many
researchers via variational methods, see 
\cite{GuY1,GuY2,MaG1,MaG2}.
However, to our best knowledge, the  results on homoclinic orbits of \eqref{e1.1}
are scarce in the  literature.
Since \eqref{e1.1} contains both advance and retardation, there are very
few manuscripts dealing with this subject, the traditional ways of establishing the
functional in \cite{DeC,GuY1,GuY2,Lo1,MaG1,MaG2,MoR} 
are inapplicable to our case.

The main purpose of this article is to give some sufficient conditions
for the existence of a nontrivial homoclinic orbit for \eqref{e1.1}
without the classical Ambrosetti-Rabinowitz condition.
In particular, our results generalize and improve the existing results;
see Remarks \ref{rmk1.3} and \ref{rmk1.4}.
The motivation for the present work stems from the recent papers 
\cite{ChF,ChT4,GuOXA4}. 

 Let
 $$
\underline{\lambda}=\min_{t\in {\mathbb{Z}}(1,M)}(b(t)-|a(t-1)|-|a(t)|),\quad
 \bar{\lambda}=\max_{t\in {\mathbb{Z}}(1,M)}(b(t)+|a(t-1)|+|a(t)|).
$$
In this article we use the following hypotheses:
\begin{itemize}
 \item[(H1)] $b(t)-|a(t-1)|-|a(t)|>0$, for all $t\in \mathbb{Z}$;

 \item[(H2)] there exists a functional
$F(t,v_1,v_2)\in C^1(\mathbb{Z}\times \mathbb{R}^2,\mathbb{R})$
 with $F(t+M,v_1,v_2)=F(t,v_1,v_2)$ and
 it satisfies
 $$
\frac{\partial F(t-T,v_2,v_3)}{\partial v_2}
+\frac{\partial F(t,v_1,v_2)}{\partial v_2}
 =f(t,v_1,v_2,v_3);
$$

\item[(H3)] there exist positive constants $\delta_1$ and
$a_1<\underline{\lambda}/4$ such  that
\[
|F(t,v_1,v_2)|\leq a_1\left(v_1^2+v_2^2\right)
\]
for all $t\in \mathbb{Z}$ and $\sqrt{v_1^2+v_2^2}\leq\delta_1$;

\item[(H4)] there exist constants $\rho_1, c_1>\bar{\lambda}/4$ and $b_1$
such  that
\[
F(t,v_1,v_2)\geq c_1\left(v_1^2+v_2^2\right)+b_1
\]
for all $t\in \mathbb{Z}$ and $\sqrt{v_1^2+v_2^2}\geq\rho_1$;

\item[(H5)]
\[
\frac{\partial F(t,v_1,v_2)}{\partial v_1}v_1+
 \frac{\partial F(t,v_1,v_2)}{\partial v_2}v_2-2F(t,v_1,v_2)>0,
\]
 for all $(t,v_1,v_2)\in \mathbb{Z}\times  \mathbb{R}^2\setminus\{(0,0)\}$;

\item[(H6)]
\[
\frac{\partial F(t,v_1,v_2)}{\partial v_1}v_1+
 \frac{\partial F(t,v_1,v_2)}{\partial  v_2}v_2-2F(t,v_1,v_2)\to+\infty
\]
as $\sqrt{v_1^2+v_2^2}\to+\infty$.
\end{itemize}
Our main results are the following theorem.

\begin{theorem} \label{thm1.1}
Suppose that {\rm (H1)--(H6)} are satisfied.
 Then \eqref{e1.1} has a nontrivial homoclinic orbit.
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
By (H4), it is easy to see
 that there exists a constant $\zeta_1>0$ such that
\begin{itemize}
\item[(H4')]
$F(t,v_1,v_2)\geq c_1\left(v_1^2+v_2^2\right)+b_1-\zeta_1$, for all
$(t,v_1,v_2)\in  \mathbb{Z}\times \mathbb{R}^2$.
\end{itemize}
As a matter of fact, letting
\[
\zeta_1=\max\big\{|F(n,v_1,v_2)-c_1\left(v_1^2+v_2^2\right)-b_1|:
 n\in \mathbb{Z}, \sqrt{v_1^2+v_2^2}\leq \rho_1\big\},
\]
 we can easily get the desired result.
\end{remark}

\begin{remark} \label{rmk1.3} \rm
As a special case of Theorem \ref{thm1.1} with $T=0$ and $a(t)<0$,   
we obtain \cite[Theorem 1.1]{MaG2}.  
\end{remark}

\begin{remark} \label{rmk1.4} \rm
In many studies (see e.g. \cite{GuY1,GuY2,MaG1,MaG2}) of
 second-order difference equations, the following classical Ambrosetti-Rabinowitz
 condition is assumed.
\begin{itemize}
\item[(AR)] There exists a constant $\beta>2$ such that
$0<\beta F(t,u)\leq uf(t,u)$  for all $t\in \mathbb{Z}$ and
$u\in \mathbb{R}\setminus\{0\}$.
\end{itemize}
Note that (H4)--(H6) are much weaker than  (AR). Thus our result improves
 that the existing results.
\end{remark}

For the next theorem, we use the hypotheses:
\begin{itemize}
\item[(H7)] there exist positive constants $\delta_2$ and
$a_2>\frac{\bar{\lambda}}{4}$ such  that
\[
|F(t,v_1,v_2)|\geq a_2\left(v_1^2+v_2^2\right)
\]
 for all $t\in \mathbb{Z}$ and $\sqrt{v_1^2+v_2^2}\leq\delta_2$;

\item[(H8)] there exists a constant $1<\mu<2$ such that
\[
0<\frac{\partial F(t,v_1,v_2)}{\partial v_1}v_1+
 \frac{\partial F(t,v_1,v_2)}{\partial v_2}v_2\leq \mu F(t,v_1,v_2),
\]
 for all $(t,v_1,v_2)\in \mathbb{Z}\times \mathbb{R}^2\setminus\{(0,0)\}$.
\end{itemize}

\begin{theorem} \label{thm1.5}
Suppose that {\rm (H1), (H2), (H7), (H8)}  are satisfied.
 Then \eqref{e1.1} has a nontrivial homoclinic orbit.
\end{theorem}

\begin{remark} \label{rmk1.6} \rm
By (H8), there exist constants $a_3>0$ and
$b_2$ such  that
 \[
F(t,v_1,v_2)\leq a_3\left(v_1^2+v_2^2\right)^{\mu/2}+b_2
\quad \text{for all }t\in \mathbb{Z},
\]
 which implies that there exist constants $\rho_2>0$ and
$c_2<\frac{\underline{\lambda}}{4}$ such  that
\begin{itemize}
\item[(H9)] $F(t,v_1,v_2)\leq c_2\left(v_1^2+v_2^2\right)+b_2$ for all
$t\in \mathbb{Z}$
 and $\sqrt{v_1^2+v_2^2}\geq\rho_2$.
\end{itemize}
\end{remark}

By (H9), it is easy to see
 that there exists a constant $\zeta_2>0$ such that
\begin{itemize}
\item[(H9')]
$F(t,v_1,v_2)\leq c_2\left(v_1^2+v_2^2\right)+b_2+\zeta_2$, for all
$(t,v_1,v_2)\in  \mathbb{Z}\times \mathbb{R}^2$.
\end{itemize}

 The remainder of this paper is organized as follows.
In Section 2, we shall  establish the variational framework associated with
\eqref{e1.1} and transfer the problem of the existence of homoclinic
 orbits of \eqref{e1.1} into that of the existence of critical points of
the corresponding  functional.
Some related fundamental results will also be recalled.
In Section 3, we shall complete the proof of the
 results by using the critical point method.
Finally, in Section 4, we shall give two examples to illustrate the results.


\section{Preliminaries}

To apply the critical point theory, we shall establish the corresponding
variational framework for \eqref{e1.1} and give some lemmas which will
 be of fundamental importance in proving our results. We
 start by giving the basic notation.

 Let $S$ be the set of sequences 
\[
u=\{u(t)\}_{t\in \mathbb{Z}}
=(\dots,u(-t),  \dots,u(-1),u(0),u(1),\dots,u(t),\dots);
\]
that is, 
 $$
S=\{\{u(t)\}: u(t)\in \mathbb{R},\; t\in \mathbb{Z}\}.
$$
For any $u,v\in S$, $a,b\in \mathbb{R}$, $au+bv$ is defined by
 $$
au+bv=\{au(t)+bv(t)\}_{t=-\infty}^{+\infty}.
$$
Then $S$ is a vector space.

 For any given positive integers $M$ and $m$, we define
 $$
E_{m}=\{u\in S|u(t+2mM)=u(t),\; \forall t\in \mathbb{Z}\}.
$$
 Clearly, $E_m$ is isomorphic to $\mathbb{R}^{2mM}$.
 $E_m$ can be equipped with the inner product
\begin{equation}\label{e2.1}
 \left(u,v\right)=\sum^{mM-1}_{t=-mM}u(t)\cdot v(t),\quad \forall u,v\in E_m,
\end{equation}
 by which the norm $\|\cdot\|$ can be induced by
\begin{equation}\label{e2.2}
 \|u\|=\Big(\sum^{mM-1}_{t=-mM}u^2(t)\Big)^{1/2},\quad \forall u\in E_m.
\end{equation}
It is obvious that $E_m$ with the inner product \eqref{e2.1}
is a finite dimensional
 Hilbert space and linearly homeomorphic to $\mathbb{R}^{2mM}$.

 In what follows, we define a norm in $E_m$ by
 $$
\|u\|_{\infty}=\max_{t\in { \mathbb{Z}}(-mM,mM-1)}|u(t)|,\quad
 \forall u\in  E_m.
$$
 For  $u\in E_m$, we define the functional $J_m$ by
\begin{equation}\label{e2.3}
 J_m(u)=\frac{1}{2}\sum_{t=-mM}^{mM-1}Lu(t)\cdot u(t)
-\sum_{t=-mM}^{mM-1}F(t,u(t+T),u(t)).
\end{equation}
 Clearly, $J_m\in C^1(E_m,\mathbb{R})$ and for any 
$u=\{u(t)\}_{t\in { \mathbb{Z}}}\in E_m$, by the periodicity of 
$\{u(t)\}_{t\in { \mathbb{Z}}}$, we can compute the partial derivative as
\begin{equation}\label{e2.4}
 \frac{\partial J_m(u)}{\partial u(t)}=
 Lu(t)-f(t,u(t+T),u(t),u(t-T)),\ \forall t\in \mathbb{Z}(-mM,mM-1).
\end{equation}
Thus, $u$ is a critical point of $J_m$ on $E_m$ if and only if
 $$
Lu(t)=f(t,u(t+T),u(t),u(t-T)),\ \forall t\in \mathbb{Z}(-mM,mM-1).
$$
 Due to the periodicity of $u=\{ u(t)\}_{t\in { \mathbb{Z}}}\in E_m$ and
 $f(t,v_1,v_2,v_3)$ in the first variable $t$, we reduce the
 existence of periodic solutions of \eqref{e1.1} to the existence of
 critical points of $J_m$ on $E_m$. That is, the functional $J_m$
 is just the variational framework of \eqref{e1.1}.

Let $E$ be a real Banach space, $J\in C^1(E,\mathbb{R})$, i.e., $J$ is a
 continuously Fr\'echet-differentiable functional
 defined on $E$. $J$ is said to satisfy the Palais-Smale
 condition (PS condition for short) if any sequence
 $\left\{u(t)\right\}\subset E$ for which $\left\{J\left(u(t)\right)\right\}$ is bounded and
 $J^\prime \left(u(t)\right)\to 0$ $(t\to \infty)$ possesses a
 convergent subsequence in $E$.

 Let $B_\rho$ denote the open ball in $E$ about 0 of radius $\rho$
 and let $\partial B_\rho$ denote its boundary.

\begin{lemma}[Mountain Pass Lemma \cite{Ra1}] \label{lem2.1}
 Let $E$ be a  real Banach space and $J\in  C^1(E,\mathbb{R})$ satisfy 
the PS condition. If $J(0)=0$ and
\begin{itemize}
\item[(J1)] there exist constants $\rho, \alpha>0$ such that
 $J|_{\partial B_\rho}\geq \alpha$, and

\item[(J2)] there exists $e\in E\setminus B_\rho$ such that
 $J(e)\leq 0$.
\end{itemize}
 Then $J$ possesses a critical value $c\geq \alpha$ given by
\begin{equation}\label{e2.5}
 c=\inf_{g\in \Gamma}\max_{s\in[0,1]} J(g(s)),
\end{equation}
where
\begin{equation}\label{e2.6}
 \Gamma =\{g\in C([0,1],E)|g(0)=0,\ g(1)=e\}.
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2.2}
Assume that {\rm (H1)} holds. Then there exist
 constants $\underline{\lambda}$ and $\bar{\lambda}$
 independent of $m$, such that 
\begin{equation}\label{e2.7}
 \underline{\lambda}\|u\|^2\leq\sum_{t=-mM}^{mM-1}Lu(t)\cdot u(t)
\leq\bar{\lambda}\|u\|^2.
\end{equation}
\end{lemma}

\begin{proof}
Let 
$$
\sum_{t=-mM}^{mM-1}Lu(t)\cdot u(t)=(P_mu,u),
$$
 where
$u=(u(-mM),\dots,u(-1),u(0),u(1),\dots,u(mM-1))^\ast$ and
\[
P_m= \left(\begin{smallmatrix}
 b(-mM)& a(-mM)& 0& \dots & 0& a(-mM-1) \\
 a(-mM)& b(-mM+1)& a(-mM+1)& \dots & 0& 0 \\
 \dots &\dots &\dots &\dots &\dots &\dots \\
 0& 0& 0& \dots & b(mM-2)& a(mM-2) \\
 a(mM-1)& 0& 0& \dots & a(mM-2)& b(mM-1)
\end{smallmatrix}
\right)
\]
which is a $2mM\times 2mM$ matrix.
By (H1), $P_m$ is positive definite.  

Let 
$\lambda_{-mM}, \lambda_{-mM+1}, \dots, \lambda_{-1}, \lambda_0, \lambda_1, 
\dots, \lambda_{mM-2},\lambda_{mM-1} $
 be the eigenvalues of $P_m$. Applying matrix theory, we see that
 $\underline{\lambda}\leq \lambda_i \leq \overline{\lambda}$,
 $i\in \mathbb{Z}(-mM,mM-1)$.
 From the definition of the norm $\|\cdot\|$, \eqref{e2.7} is obviously true.
\end{proof}


\section{Proof of main results}
In this section, we shall prove the results stated in Section 1 by using 
the critical point theory.

\subsection{Proof of Theorem \ref{thm1.1}}

\begin{lemma} \label{lem3.1}
Suppose that {\rm (H1), (H2)--(H6)} are  satisfied. Then $J_m$ satisfies 
the {\rm PS} condition.
\end{lemma}

\begin{proof}
Assume that $\{u_j\}_{j\in\mathbb{N}}$ in $E_m$ is a sequence such that
 $\{J_m(u_j)\}_{j\in\mathbb{N}}$ is bounded. Then there exists a constant $K_1>0$
 such that $-K_1\leq J_m(u_j)$.
 By \eqref{e2.7} and (H4'), it is easy to see that
\begin{align*}
-K_1\leq J_m(u_j)
&\leq\frac{\bar{\lambda}}{2}\|u_j\|^2
 -\sum_{t=-mM}^{mM-1}\{c_1[u_j^2(t+T)+u_j^2(t)]+b_1-\zeta_1\}\\
 &=\frac{\bar{\lambda}}{2}\|u_j\|^2-2c_1\|u_j\|^2
 +2mM\left(\zeta_1-b_1\right),\ \forall j\in \mathbb{N}.
\end{align*}
 Therefore,
\begin{equation}\label{e3.1}
 \big(2c_1-\frac{\bar{\lambda}}{2}\big)\|u_j\|^2
\leq 2mM(\zeta_1-b_1)+K_1.
\end{equation}
 Since $c_1>\bar{\lambda}/4$, \eqref{e3.1} implies that
 $\{u_j\}_{j\in\mathbb{N}}$ is bounded in $E_m$. Thus, $\{u_j\}_{j\in\mathbb{N}}$
 possesses a convergence subsequence in $E_m$. The desired result follows.
\end{proof}

\begin{lemma} \label{lem3.2}
Suppose that {\rm (H1)--(H6)} are  satisfied. 
Then for any given positive integer $m$, \eqref{e1.1} possesses 
a $2mM$-periodic solution $u_m\in E_m$.
\end{lemma}

\begin{proof}
In our case, it is clear that $J_m(0)=0$. By Lemma \ref{lem3.1}, $J_m$ satisfies the PS
 condition. By (H3), we have
\begin{align*}
J_m(u)
&\geq \frac{\underline{\lambda}}{2}\|u\|^2-a_1\sum^{mM-1}_{t=-mM}
[u^2(t)+u^2(t+T)]\\
&\geq \frac{\underline{\lambda}}{2}\|u\|^2-2a_1\|u\|^2 \\
&=\big(\frac{\underline{\lambda}}{2}-2a_1\big)\|u\|^2.
\end{align*}
Taking $\alpha_1=(\frac{\underline{\lambda}}{2}-2a_1)\delta_1^2>0$, we obtain
 $$
J_m(u)|_{\partial B_{\delta_1}}\geq\alpha_1>0,
$$
 which implies that $J_m$ satisfies the condition (J1) of the
 Mountain Pass Lemma.

Next, we shall verify the condition (J2) of the Mountain Pass Lemma..
There exists a sufficiently large number $\rho>\max\{\rho_1,\delta_1\}$ such that
\begin{equation}\label{e3.2}
\big(2c_1-\frac{\bar{\lambda}}{2}\big)\rho^2\geq |b_1|.
\end{equation}
 Let $e_m^{(1)}\in E_m$ and
\begin{gather*}
e_m^{(1)}(t)= \begin{cases}
 \rho, & \text{if }t=0,\\
 0, & \text{if } t\in\{j\in\mathbb{Z}:-mM\leq j\leq mM-1\text{ and } j\neq0\},
 \end{cases}
\\
e_m^{(1)}(t+T)= \begin{cases}
 \rho, & \text{if } t=0,\\
 0, & \text{if } t\in\{j\in\mathbb{Z}:-mM\leq j\leq mM-1\text{ and } j\neq0\}.
 \end{cases}
\end{gather*}
 Then
\begin{align*}
&F\big(t,e_m^{(1)}(t+T),e_m^{(1)}(t)\big)\\
&= \begin{cases}
 F(0,\rho,\rho), & \text{if } t=0,\\
 0, & \text{if } t\in\{j\in\mathbb{Z}:-mM\leq j\leq mM-1\text{ and } j\neq0\}.
 \end{cases}
\end{align*}
With \eqref{e3.2} and (H4), we have
\begin{equation}\label{e3.3}
\begin{aligned}
J_m\big(e_m^{(1)}\big)
&=\frac{1}{2}\sum_{t=-mM}^{mM-1}L\big(e_m^{(1)}(t)\big)\cdot\big(e_m^{(1)}(t)\big)\\
&\quad -\sum_{t=-mM}^{mM-1}F\left(t,\big(e_m^{(1)}(t+T)\big),\big(e_m^{(1)}(t)\big)\right)\\
&\leq\frac{\bar{\lambda}}{2}\|e_m^{(1)}\|^2-2c_1\rho^2-b_1\\
& =-\big(2c_1-\frac{\bar{\lambda}}{2}\big)\rho^2-b_1\leq0.
\end{aligned}
\end{equation}
All the assumptions of the Mountain Pass Lemma have been verified.
Consequently, $J_m$ possesses a critical value $c_m$ given by
 \eqref{e2.5} and \eqref{e2.6} with $E=E_m$ and $\Gamma=\Gamma_m$, 
where 
\[
\Gamma_m=\big\{g_m\in C([0,1],  E_m)|g_m(0)=0,\ g_m(1)=e_m^{(1)},
\ e_m^{(1)}\in E_m\backslash B_{\rho}\big\}.
\]
 Let $u_m$ denote the corresponding critical point of $J_m$ on $E_m$. 
Note that $\|u_m\|\neq0$ since $c_m>0$.
\end{proof}

\begin{lemma} \label{lem3.3}
Suppose that {\rm (H1)--(H6)} are satisfied. 
Then there exist positive constants $\delta_1$ and $\eta_1$ independent
 of $m$ such that
\begin{equation}\label{e3.4}
 \delta_1\leq \|u_m\|_\infty\leq \eta_1.
\end{equation}
\end{lemma}

\begin{proof}
The continuity of $F(0,v_1,v_2)$  with respect to the second and third 
variables implies that  there exists a constant $\tau_1>0$ such that 
$|F(0,v_1,v_2)|\leq  \tau_1$ for 
$\sqrt{v_1^2+v_2^2}\leq\delta_1$. It is clear that
\begin{align*}
J_m(u_m)
&\leq\max_{0\leq s\leq1}\Big\{\big|\frac{1}{2}\sum_{t=-mM}^{mM-1}L
\big(se_m^{(1)}(t)\big)\cdot\big(se_m^{(1)}(t)\big)\big|\\
&\quad -\sum_{t=-mM}^{mM-1}F\left(t,se_m^{(1)}(t+T),se_m^{(1)}(t)\right)\Big\}\\
&\leq\frac{\bar{\lambda}}{2}\|e_m^{(1)}\|^2+\tau_1\\
&=\frac{\bar{\lambda}}{2}\rho^2+\tau_1.
\end{align*}
 Let $\xi_1=\frac{\bar{\lambda}}{2}\rho^2+\tau_1$. Then 
$J_m(u_m)\leq\xi_1$, which is a bound independent of  $m$. From \eqref{e2.3} 
and \eqref{e2.4}, we have
\begin{align*}
J_m(u_m)
&=\frac{1}{2}\sum^{mM-1}_{t=-mM}\Big[\frac{\partial F(t-T,u_m(t),u_m(t-T))}
 {\partial v_2}u_m(t)\\
&\quad +\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}u_m(t)\Big]
-\sum^{mM-1}_{t=-mM}F(t,u_m(t+T),u_m(t))
\\
&=\frac{1}{2}\sum^{mM-1}_{t=-mM}\Big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}u_m(t+T)\\
&\quad +\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}u_m(t)\Big]
 -\sum^{mM-1}_{t=-mM}F(t,u_m(t+T),u_m(t))
\\
&\leq \xi_1.
\end{align*}
 By (H5) and (H6), there exists a constant $\eta_1>0$ such  that
\[
\frac{1}{2}\Big(\frac{\partial F(t,v_1,v_2)}{\partial v_1}v_1+
 \frac{\partial F(t,v_1,v_2)}{\partial v_2}v_2\Big)-F(t,v_1,v_2)>\xi_1,
\]
for all $t\in \mathbb{Z}$ and $\sqrt{v_1^2+v_2^2}\geq \eta_1$,
which implies that $|u_m(t)|\leq \eta_1$ for all $t\in \mathbb{Z}$; that is,
 $\|u_m\|_\infty\leq \eta_1$.

 From the definition of $J_m$, we have 
\begin{align*}
0&=\left(J'_m(u_m),u_m\right) \\
&\geq\underline{\lambda}\|u_m\|^2
 -\sum^{mM-1}_{t=-mM}\Big[\frac{\partial F(t-T,u_m(t),u_m(t-T))}
 {\partial v_2}u_m(t)\\
&\quad +\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}u_m(t)\Big].
\end{align*}
This inequality and (H3) yield
\begin{align*}
\underline{\lambda}\|u_m\|^2
&\leq  \sum^{mM-1}_{t=-mM}\Big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}u_m(t+T)\\
&\quad +\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}u_m(t)\Big] \\
&\leq  \Big\{\sum^{mM-1}_{t=-mM}\big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}\big]^2\Big\}^{1/2}\|u_m\|\\
&\quad +  \Big\{\sum^{mM-1}_{t=-mM}
\big[\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}\big]^2\Big\}^{1/2}\|u_m\|.
\end{align*}
 That is,
\begin{align*}
\underline{\lambda}\|u_m\|
&\leq  \Big\{\sum^{mM-1}_{t=-mM}\big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}\big]^2\Big\}^{1/2}\\
&\quad +  \Big\{\sum^{mM-1}_{t=-mM}\big[\frac{\partial 
F(t,u_m(t+T),u_m(t))}{\partial v_2}\big]^2\Big\}^{1/2}.
\end{align*}
 Thus,
\begin{equation}\label{e3.5}
\begin{aligned}
 \underline{\lambda}^2\|u_m\|^2
&\leq  2\sum^{mM-1}_{t=-mM}\big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}\big]^2\\
&\quad +2\sum^{mM-1}_{t=-mM}\big[\frac{\partial 
F(t,u_m(t+T),u_m(t))}{\partial v_2}\big]^2.
\end{aligned}
\end{equation}
From this inequality and (H3), we obtain
\[
\underline{\lambda}^2\|u_m\|^2
\leq2\sum^{mM-1}_{t=-mM}\left[2a_1u_m(t+T)\right]^2
+2\sum^{mM-1}_{t=-mM}\left[2a_1u_m(t)\right]^2=16a_1^2\|u_m\|^2.
\]
Thus, we have $u_m=0$. This  contradicts $\|u_m\|\neq0$, which shows that
 $$
\|u_m\|_\infty\geq\delta_1,
$$
 and the proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
Now we shall give the existence of a nontrivial homoclinic orbit.
Consider the sequence $\{u_m(t)\}_{t\in \mathbb{Z}}$ of $2mM$-periodic
 solutions found in Section 3.1. 
First, by \eqref{e3.4}, for any $m\in \mathbb{N}$, there exists
 a constant $t_m\in\mathbb{Z}$ independent of $m$ such that
\begin{equation}\label{e3.6}
 |u_m(t_m)|\geq \delta_1.
\end{equation}

 Since $a(t)$, $b(t)$ and $f(t,v_1,v_2,v_3)$
 are  $M$-periodic in $t$, $\{u_m(t+jM)\}$ is
 also $2mM$-periodic solution of \eqref{e1.1} (for all $j\in \mathbb{N}$). 
Hence, making such  shifts, we can assume that $t_m\in \mathbb{Z}(0,M-1)$ in 
\eqref{e3.6}. Moreover,
 passing to a subsequence of $m$s, we can even assume that
 $t_m=t_0$ is independent of $m$.

 Next, we extract a subsequence, still denote by $u_m$, such that
 $$
u_m(t)\to u(t),\text{as } m\to\infty,\; \forall t\in  \mathbb{Z}.
$$
 Inequality \eqref{e3.6} implies that $|u(t_0)|\geq \xi$ and, hence,
 $u=\{u(t)\}$ is a nonzero sequence. Moreover,
\begin{align*}
&Lu(t)-f(t,u(t+T),u(t),u(t-T))\\
&=\lim_{m\to \infty}[Lu_m(t)-f(t,u_m(t+T),u_m(t),u_m(t-T))]=0.
\end{align*}
 So $u=\{u(t)\}$ is a solution of \eqref{e1.1}.

 Finally, we show that $u\in l^2$. For $u_m\in E_m$, let
\begin{gather*}
P_m=\big\{t\in \mathbb{Z}: |u_m(t)|<\frac{\sqrt{2}}{2}\delta_1, 
-mM\leq t\leq mM-1\big\},
\\
Q_m=\big\{t\in \mathbb{Z}: |u_m(t)| \geq\frac{\sqrt{2}}{2}\delta_1, 
-mM\leq t\leq mM-1\big\}.
\end{gather*}
Since $F(t,v_1,v_2)\in C^1(\mathbb{Z}\times \mathbb{R}^2,\mathbb{R})$, 
there exist constants $\bar{\xi}>0$, $\underline{\xi}>0$ such that
\begin{gather*}
\max\Big\{\Big[\frac{\partial F(t,v_1,v_2)}{\partial v_1}\Big]^2
+ \Big[\frac{\partial F(t,v_1,v_2)}{\partial v_2}\Big]^2: 
\delta_1\leq\sqrt{v_1^2+v_2^2}\leq\eta_1,\; t\in \mathbb{Z}\Big\}
\leq \bar{\xi},
\\
\begin{aligned}
\min\Big\{&\frac{1}{2}\Big[\frac{\partial F(t,v_1,v_2)}{\partial v_1}v_1+
 \frac{\partial F(t,v_1,v_2)}{\partial v_2}v_2\Big]
-F(t,v_1,v_2):\\
& \delta_1\leq\sqrt{v_1^2+v_2^2}\leq\eta_1,\; t\in \mathbb{Z}\Big\}
\geq \underline{\xi}.
\end{aligned}
\end{gather*}
 For $t\in Q_m$,
\begin{align*}
&\Big[\frac{\partial F(t,u_m(t+T),u_m(t))} {\partial v_1}\Big]^2
+\Big[\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}\Big]^2 \\
&\leq\frac{\bar{\xi}}{\underline{\xi}}
 \Big\{\frac{1}{2}\Big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}u_m(t+T)
 +\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}u_m(t)\Big]\\
 &\quad -F(t,u_m(t+T),u_m(t))\Big\}.
\end{align*}
By \eqref{e3.5}, we have
\begin{align*}
&\underline{\lambda}^2\|u_m\|^2\\
&\leq  2\sum_{t\in P_m}\Big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}\Big]^2
+2\sum_{t\in P_m}\big[\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}\big]^2\\
&\quad +2\sum_{t\in Q_m}\Big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}\Big]^2
+2\sum_{t\in Q_m}\Big[\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}\Big]^2 \\
&\leq 2\sum_{t\in P_m}[2a_1u_m(t+T)]^2
 +2\sum_{t\in P_m}[2a_1u_m(t)]^2\\
&\quad +\frac{\bar{\xi}}{\underline{\xi}}\sum_{t\in Q_m}
\Big\{\frac{1}{2}\Big[\frac{\partial F(t,u_m(t+T),u_m(t))}
 {\partial v_1}u_m(t+T)\\
&\quad +\frac{\partial F(t,u_m(t+T),u_m(t))}{\partial v_2}u_m(t)\Big]
  -F(t,u_m(t+T),u_m(t))\Big\} \\
&\leq 16a_1^2\|u_m\|^2+\frac{\bar{\xi}\xi_1}{\underline{\xi}}.
\end{align*}
Thus,
 $$
\|u_m\|^2\leq \frac{\bar{\xi}\xi_1}{\underline{\xi}
\left(\underline{\lambda}^2-16a_1^2\right)}.
$$
For any fixed $D\in \mathbb{Z}$ and $m$ large enough, we have 
 $$
\sum^D_{t=-D}u_m^2(t)\leq\|u_m\|^2
\leq\frac{\bar{\xi}\xi_1}{\underline{\xi}\left(\underline{\lambda}^2-16a_1^2\right)}.
$$
 Since $\bar{\xi}, \underline{\xi}, \xi_1, \underline{\lambda}$ and $a_1$ 
are constants independent of $m$, passing
 to the limit, we have 
 $$
\sum^D_{t=-D}u^2(t)
\leq \frac{\bar{\xi}\xi_1}{\underline{\xi}\left(\underline{\lambda}^2-16a_1^2\right)}.
$$
By the arbitrariness of $D$, $u\in l^2$. Therefore, $u$ satisfies $u(t)\to 0$ as
 $|t|\to \infty$. The existence of a nontrivial homoclinic orbit is obtained.
\end{proof}

\subsection{Proof Theorem \ref{thm1.5}}
Let 
\begin{equation}\label{e3.7}
 J_m^\ast(u)=-\frac{1}{2}\sum_{t=-mM}^{mM-1}Lu(t)\cdot u(t)+
\sum_{t=-mM}^{mM-1}F(t,u(t+T),u(t)).
\end{equation}
 Then
\begin{equation}\label{e3.8}
 \frac{\partial J_m^\ast(u)}{\partial u(t)}=
 -Lu(t)+f(t,u(t+T),u(t),u(t-T)),
\end{equation}
for all $t\in \mathbb{Z}(-mM,mM-1)$.

\begin{lemma} \label{lem3.4}
Suppose that {\rm (H1), (H2), (H7), (H8)} are
 satisfied. Then $J_m^\ast$ satisfies the PS condition.
\end{lemma}.

\begin{proof}
Assume that $\{u_j\}_{j\in\mathbb{N}}$ in $E_m$ is a sequence such that
 $\{J_m^\ast(u_j)\}_{j\in\mathbb{N}}$ is bounded. Then there exists 
a constant $K_2>0$  such that $-K_2\leq J_m^\ast(u_j)$.
 By \eqref{e2.7} and (H9'), it is easy to see that
\[
-K_2\leq J^\ast_m(u_j)\leq
 -\frac{\underline{\lambda}}{2}\|u_j\|^2+2c_2\|u_j\|^2+2mM\left(\zeta_2+b_2\right),
\quad \forall j\in \mathbb{N}.
\]
 Therefore,
\begin{equation}\label{e3.9}
 -\big(2c_2-\frac{\underline{\lambda}}{2}\big)\|u_j\|^2
\leq 2mM\left(\zeta_2+b_2\right)+K_2.
\end{equation}
 Since $c_2<\underline{\lambda}/4$, \eqref{e3.9} implies that
 $\{u_j\}_{j\in\mathbb{N}}$ is bounded in $E_m$. 
Thus, $\{u_j\}_{j\in\mathbb{N}}$
 possesses a convergence subsequence in $E_m$. The desired result follows.
\end{proof}

\begin{lemma} \label{lem3.5}
Suppose that {\rm (H1), (H2), (H7), (H8)} are  satisfied. 
Then for any given positive integer $m$, \eqref{e1.1} possesses a $2mM$-periodic
 solution $u_m^\ast\in E_m$.
\end{lemma}

\begin{proof}
In our case, it is clear that $J_m^\ast(0)=0$. By Lemma \ref{lem3.4}, 
$J_m^\ast$ satisfies the PS  condition. By (H7), we have
\begin{align*}
J_m^\ast(u)
&\geq -\frac{\bar{\lambda}}{2}\|u\|^2
 +a_2\sum^{mM-1}_{t=-mM}[u^2(t)+u^2(t+T)]\\
&\geq -\frac{\bar{\lambda}}{2}\|u\|^2+2a_2\|u\|^2\\
&=-\big(\frac{\bar{\lambda}}{2}-2a_2\big)\|u\|^2.
\end{align*}
Taking $\alpha_2=-(\frac{\bar{\lambda}}{2}-2a_2)\delta_2^2>0$, we obtain
 $$
J_m^\ast(u)|_{\partial B_{\delta_2}}\geq\alpha_2>0,
$$
which implies that $J_m^\ast$ satisfies the condition (J1) of the
 Mountain Pass Lemma.

Next, we shall verify the condition (J2) of the  Mountain Pass Lemma.
There exists a sufficiently large number $\eta>\max\{\rho_2,\delta_2\}$ such that
\begin{equation}\label{e3.10}
 \big(2c_2-\frac{\bar{\lambda}}{2}\big)\eta^2\geq|b_2|.
\end{equation}
Let $e_m^{(2)}\in E_m$ and
\begin{gather*}
e_m^{(2)}(t)=\begin{cases}
 \eta, & \text{if } t=0,\\
 0, & \text{if } t\in\{j\in\mathbb{Z}:-mM\leq j\leq mM-1\text{ and } j\neq0\},
 \end{cases} \\
e_m^{(2)}(t+T)= \begin{cases}
 \eta, & \text{if } t=0,\\
 0, & \text{if } t\in\{j\in\mathbb{Z}:-mM\leq j\leq mM-1\text{ and }j\neq0\}.
 \end{cases}
\end{gather*}
 Then
\begin{align*}
&F\big(t,e_m^{(2)}(t+T),e_m^{(2)}(t)\big)\\
&= \begin{cases}
 F(0,\eta,\eta), & \text{if } t=0,\\
 0, & \text{if } t\in\{j\in\mathbb{Z}:-mM\leq j\leq mM-1\text{ and } j\neq0\},
 \end{cases}
 \end{align*}
 With \eqref{e3.10} and (H9), we have
\begin{equation}\label{e3.11}
\begin{aligned}
J_m^\ast\big(e_m^{(2)}\big)
&=-\frac{1}{2}\sum_{t=-mM}^{mM-1}L\big(e_m^{(2)}(t)\big)
 \cdot\big(e_m^{(2)}(t)\big)\\
&\quad +\sum_{t=-mM}^{mM-1}F\Big(t,e_m^{(2)}(t+T),e_m^{(2)}(t)\Big)\\
&\leq-\frac{\underline{\lambda}}{2}\|e_m^{(2)}\|^2+2c_2\eta^2+b_2\\
& =-\big(\frac{\underline{\lambda}}{2}-2c_2\big)\eta^2+b_2\leq0.
\end{aligned}
\end{equation}

 All the assumptions of the Mountain Pass Lemma have been verified.
 Consequently, $J_m^\ast$ possesses a critical value $c_m^\ast$ given by
 \eqref{e2.5} and \eqref{e2.6} with $E=E_m$ and $\Gamma=\Gamma_m$, 
where 
\[
\Gamma_m=\big\{g_m\in C([0,1],  E_m)|g_m(0)=0,\; g_m(1)=e_m^{(2)},\;
 e_m^{(2)}\in E_m\backslash B_{\eta}\big\}.
\]
Let $u_m^\ast$ denote the corresponding critical point of $J_m^\ast$ on $E_m$.
 Note that $\|u_m^\ast\|\neq0$ since $c_m^\ast>0$.
\end{proof}

\begin{lemma} \label{lem3.6}
Suppose that {\rm (H1), (H2), (H7), (H8)} are
 satisfied. Then there exist positive constants $\delta_2$ and $\eta_2$ independent
 of $m$ such that
\begin{equation}\label{e3.12}
 \delta_2\leq \|u_m^\ast\|_\infty\leq \eta_2.
\end{equation}
\end{lemma}

\begin{proof}
The continuity of $F(0,v_1,v_2)$
 with respect to the second and third variables implies that
 there exists a constant $\tau_2>0$ such that 
$|F(0,v_1,v_2)|\leq  \tau_2$ for $\sqrt{v_1^2+v_2^2}\leq\delta_2$. 
It is clear that
\begin{equation}\label{e3.13}
\begin{aligned}
|J_m^\ast\left(u_m^\ast\right)|
&\leq\max_{0\leq s\leq1}\Big\{\big|-\frac{1}{2}\sum_{t=-mM}^{mM-1}
L\big(se_m^{(2)}(t)\big)\cdot\big(se_m^{(2)}(t)\big)\big|\\
&\quad +\sum_{t=-mM}^{mM-1}F\left(t,se_m^{(2)}(t+T),se_m^{(2)}(t)\right)\Big\}\\
&\leq\frac{\bar{\lambda}}{2}\|e_m^{(2)}\|^2+\tau_2
\\
&=\frac{\bar{\lambda}}{2}\eta^2+\tau_2.
\end{aligned}
\end{equation}

 Let $\xi_2=\frac{\bar{\lambda}}{2}\eta^2+\tau_2$, we have
that $|J_m^\ast\left(u_m^\ast\right)|\leq\xi_2$, which is a bound independent of
 $m$. Then by \eqref{e3.7} and \eqref{e3.8}, we have
\begin{align*}
\xi_2 & \geq J_m^\ast(u_m)\\
&=-\frac{1}{2}\sum^{mM-1}_{t=-mM}
\Big[\frac{\partial F(t-T,u_m^\ast(t),u_m^\ast(t-T))}
 {\partial v_2}u_m^\ast(t)\\
&\quad +\frac{\partial F(t,u_m^\ast(t+T),u_m^\ast(t))}{\partial v_2}u_m^\ast(t)\Big]
 +\sum^{mM-1}_{t=-mM}F(t,u_m^\ast(t+T),u_m^\ast(t)) \\
&=-\frac{1}{2}\sum^{mM-1}_{t=-mM}
\Big[\frac{\partial F(t,u_m^\ast(t+T),u_m^\ast(t))}
 {\partial v_1}u_m^\ast(t+T)\\
&\quad +\frac{\partial F(t,u_m^\ast(t+T),u_m^\ast(t))}{\partial v_2}u_m^\ast(t)\Big]
+\sum^{mM-1}_{t=-mM}F(t,u_m^\ast(t+T),u_m^\ast(t)) \\
&\geq\big(\frac{2-\mu}{2}\big)\sum^{mM-1}_{t=-mM}F(t,u_m^\ast(t+T),u_m^\ast(t))\,.
\end{align*}
Then
\begin{equation}\label{e3.14}
 \sum^{mM-1}_{t=-mM}F(t,u_m^\ast(t+T),u_m^\ast(t))\leq \frac{2\xi_2}{2-\mu}.
\end{equation}
 Since
\begin{align*}
J_m^\ast(u_m^\ast)
&=-\frac{1}{2}\sum^{mM-1}_{t=-mM}
 \Big[\frac{\partial F(t-T,u_m^\ast(t),u_m^\ast(t-T))}
 {\partial v_2}u_m^\ast(t)\\
&\quad +\frac{\partial F(t,u_m^\ast(t+T),u_m^\ast(t))}{\partial v_2}u_m^\ast(t)\Big]
 +\sum^{mM-1}_{t=-mM}F(t,u_m^\ast(t+T),u_m^\ast(t))\\
&\geq-\xi_2.
\end{align*}
This inequality combined with \eqref{e3.14} gives us
\begin{equation}\label{e3.15}
\begin{aligned}
\frac{1}{2}\underline{\lambda}\|u_m^\ast\|
&\leq\frac{1}{2}\sum^{mM-1}_{t=-mM} 
 \Big[\frac{\partial F(t-T,u_m^\ast(t),u_m^\ast(t-T))}
 {\partial v_2}u_m^\ast(t)\\
&\quad +\frac{\partial F(t,u_m^\ast(t+T),u_m^\ast(t))}{\partial v_2}u_m^\ast(t)\Big]\\
&\leq\sum^{mM-1}_{t=-mM}F(t,u_m^\ast(t+T),u_m^\ast(t))+\xi_2\\
&\leq \frac{(4-\mu)\xi_2}{2-\mu},
\end{aligned}
\end{equation}
\begin{equation}\label{e3.16}
\|u_m^\ast\|\leq \frac{2(4-\mu)\xi_2}{(2-\mu)\underline{\lambda}},
\end{equation}
 whose right-hand side is independent of $m$. Then 
 $\|u_m^\ast\|\leq \eta_2$,
 which implies 
 $$
\|u_m^\ast\|_\infty\leq \eta_2.
$$
From the definition of $J_m^\ast$, we have 
\[
0=\big(J_m^{\ast'}(u_m^\ast),u_m^\ast\big)
 \geq-\bar{\lambda}\|u_m^\ast\|^2
 +\sum^{mM-1}_{t=-mM}f(t,u_m^\ast(t+T),u_m^\ast(t),u_m^\ast(t-T))u_m^\ast(t).
\]
 This inequality combined with (H7) yields
\begin{align*}
\bar{\lambda}\|u_m^\ast\|^2
&\geq  \sum^{mM-1}_{t=-mM}\Big[\frac{\partial F(t-T,u_m^\ast(t),u_m^\ast(t-T))}
 {\partial v_2}u_m^\ast(t) \\
&\quad +\frac{\partial F(t,u_m^\ast(t+T),u_m^\ast(t))}{\partial v_2}u_m^\ast(t)\Big]
\\
&=\sum^{mM-1}_{t=-mM}\Big[\frac{\partial F(t,u_m^\ast(t+T),u_m^\ast(t))}
 {\partial v_1}u_m^\ast(t+T)\\
&\quad +\frac{\partial F(t,u_m^\ast(t+T),u_m^\ast(t))}{\partial v_2}u_m^\ast(t)\Big]\\
&\geq 2a_2\sum^{mM-1}_{t=-mM}\big[\left(u_m^\ast(t+T)\right)^2
+\left(u_m^\ast(t)\right)^2\big] \\
&=4a_2\|u_m^\ast\|^2.
\end{align*}
Thus, we have $u_m^\ast=0$. This
 contradicts $\|u_m^\ast\|\neq0$, which shows that
 $\|u_m^\ast\|_\infty\geq\delta_2$,
 and the proof is complete.
\end{proof}

The proof of Theorem \ref{thm1.5} is done
similarly to the proof of Theorem \ref{thm1.1}. 
We omit it here for simplicity.


\section{Examples}

As an application of Theorems \ref{thm1.1} and \ref{thm1.5}, we give two examples 
that illustrate our main results.

\begin{example} \label{examp4.1} \rm 
Let
\begin{gather*}
f(t,v_1,v_2,v_3)
=\gamma v_2\Big(\frac{v_1^2+v_2^2}{v_1^2+v_2^2+1}
 +\frac{v_2^2+v_3^2}{v_2^2+v_3^2+1}\Big),
\\
F(t,v_1,v_2)=\frac{\gamma}{2}[v_1^2+v_2^2-\ln(v_1^2+v_2^2+1)],
\end{gather*}
where $\gamma>\bar{\lambda}$. If (H1) is satisfied, then it is easy to verify 
that all the assumptions of Theorem \ref{thm1.1} are satisfied.
 Consequently, there exists a nontrivial homoclinic orbit.
\end{example}

\begin{example} \label{examp4.2} \rm
Let
 $$
f(t,v_1,v_2,v_3)= \begin{cases}
 v_2[(v_1^2+v_2^2)^{\frac{\mu}{2}-1}+(v_2^2+v_3^2)^{\frac{\mu}{2}-1}],\\
\quad \text{if } (v_1,v_2)\neq(0,0)\text{ and } (v_2,v_3)\neq(0,0),\\[4pt]
 0, \quad\text{if } (v_1,v_2)=(0,0)\text{ or } (v_2,v_3)=(0,0),
 \end{cases}
 $$
 and
 $$
F(t,v_1,v_2)=\frac{1}{\mu}(v_1^2+v_2^2)^{\mu/2},
$$
 where $1<\mu<2$. If (H1) is satisfied, then it is easy to verify all the 
assumptions of Theorem \ref{thm1.5} are satisfied.
 Consequently, there exists a nontrivial homoclinic orbit.
\end{example}


\subsection*{Acknowledgments} 
This project is supported by the National Natural Science Foundation of China 
(No. 11401121), by the Natural Science Foundation of Guangdong Province 
(No. S2013010014460), and by the Hunan Provincial Natural Science Foundation of
China (No. 2015JJ2075).


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\end{document}