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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 94, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/94\hfil Homoclinic solutions]
{Homoclinic solutions for second-order nonlinear difference
equations with Jacobi operators}

\author[F. Xia \hfil EJDE-2017/94\hfilneg]
{Fei Xia}

\address{Fei Xia \newline
Swan College,
Central South University of Forestry and Technology,
Changsha 410004, China}
\email{xiafeiznlkd@163.com}


\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted August 11, 2016. Published March 30, 2017.}
\subjclass[2010]{34C37, 37J45, 39A12, 47J22}
\keywords{Homoclinic solutions; nonlinear difference equations;
\hfill\break\indent Jacobi operators; critical point theory}

\begin{abstract}
 We obtain sufficient conditions for the existence of a nontrivial
 homoclinic solution to a second-order nonlinear difference equation
 with Jacobi operator. To do this, we use variational methods and
 critical point theory. An example is provided to illustrate our main result. 
\end{abstract}

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

\section{Introduction}
Difference equations, the discrete analogs of
differential equations \cite{DeLZS,DeSX,LiZSD,Ya},
occur  in numerous settings and forms,
both in mathematics itself and in its applications to statistics,
computing, electrical circuit analysis, dynamical systems,
economics, biology and other fields. For the general background of
difference equations, we refer to the monographs \cite{Ag,AhP,CuFR}.

We denote by $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ the sets of
all natural numbers, integers and real numbers respectively. For
$a$, $b$ $\in \mathbb{Z}$, define $\mathbb{Z}(a)=\{a,a+1,\dots\},
\mathbb{Z}(a,b)=\{a,a+1,\dots,b\}$ when $a\leq b$. Moreover, $I$
denotes the identity operator.

In this article, we consider the  second-order nonlinear
difference equation
\begin{equation}\label{e1.1}
Lu(t)-\omega
u(t)=f(t,u(t+\Gamma),\dots,u(t),\dots,u(t-\Gamma)),\quad 
t\in \mathbb{Z}
\end{equation}
containing both advances and retardations. 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 $t\in \mathbb{Z}$,
$\omega\in\mathbb{R}$, $f\in C(\mathbb{R}^{2\Gamma+2},\mathbb{R})$,
$\Gamma$ is a given nonnegative integer, $a(t)$, $b(t)$ and
$f(t,y_\Gamma,\dots,y_0,\dots,y_{-\Gamma})$ are all $M$-periodic
in $t$ for a given positive integer $M$.

Jacobi operators appear in a variety of applications \cite{Te}. They
can be viewed as the discrete analogue of Sturm-Liouville operators
and their investigation has many similarities with Sturm-Liouville
theory. Whereas numerous books about Sturm-Liouville operators have
been written, only few on Jacobi operators exist. In particular,
there are currently fewer researches available which cover some
basic topics (like stability, attractivity, positive solutions,
periodic operators, homoclinic solutions, boundary value problems,
etc.) typically found in textbooks on Sturm-Liouville operators
\cite{Mar}.

We may regard \eqref{e1.1} as being a discrete analog of the
second-order differential equation
\begin{equation}\label{e1.2}
Su(s)-\omega u(s)=f(s,u(s+\Gamma),\dots,u(s),\dots,u(s-\Gamma)),\quad
s\in \mathbb{R},
\end{equation}
where $S$ is the Sturm-Liouville differential expression and
$\omega\in \mathbb{R}$, $\Gamma$ is a given nonnegative integer,
$f\in C(\mathbb{R}^{2\Gamma+2},\mathbb{R})$.

Equation \eqref{e1.2} includes the  equation
\begin{equation}\label{e1.3}
c^2u''(s)=V'(u(s+1)-u(s))-V'(u(s)-u(s-1)),\quad
s\in \mathbb{R}.
\end{equation}
Equations similar in structure to \eqref{e1.3} arise in the study of
the existence of solitary waves of lattice differential equations
and the existence of homoclinic solutions for functional
differential equations, see \cite{GuAWO,GuOWA,SmW} and the
references cited therein.

Assuming that $f(t,0,\dots,0,\dots,0)=0$ for $t\in \mathbb{Z}$,
then $\{u(t)\}_{t\in \mathbb{Z}}=\{0\}$ is a solution of
\eqref{e1.1}, which is called the trivial solution. As usual, we say
that a solution $\{u(t)\}_{t\in \mathbb{Z}}$ of \eqref{e1.1} is
homoclinic (to 0) if \eqref{e1.1} holds. In addition, if
$\{u(t)\}_{t\in \mathbb{Z}}\neq\{0\}$, then $u$ is called a
nontrivial solution.

It is well known that homoclinic solutions (homoclinic orbits) play
a very important role in the study of chaos in dynamical systems. It
has been proved that the system must be chaotic provided it has the
transversely intersected homoclinic solutions. Homoclinic solutions
have been extensively studied since the time of Poincar\'{e}, see
\cite{CheT1,LiZS2,Po,Sh,ShLZ1,ShLZ2,ShZ1,ShZ2,Ya,Zh} and the
references therein. Therefore, it possesses important theoretical
significance and practical value to investigate the existence of
homoclinic solutions of \eqref{e1.1} emanating from zero.

By using the Symmetric Mountain Pass Theorem, Chen and Tang
\cite{CheT2} established some existence criteria to guarantee the
fourth-order difference system containing both advance and
retardation
\begin{equation}\label{e1.4}
\Delta^4 u(t-2)+q(t)u(t)=f(t,u(t+1),u(t),u(t-1)), \quad t\in \mathbb{Z}
\end{equation}
has infinitely many homoclinic solutions.

Deng, Liu, Shi and Zhou \cite{DeLSZ} in 2011 proved the existence of
nontrivial homoclinic solutions for a second-order nonlinear
$p$-Laplacian difference equation
\begin{equation}\label{e1.5}
\Delta(\varphi_p(\Delta
u(t-1)))-\varphi_p(u(t))=\lambda(t)f(t,u(t+1),u(t),u(t-1)), \quad t\in
\mathbb{Z},
\end{equation}
without any assumptions on periodicity using the critical point
theory.

When $\Gamma=1$, \eqref{e1.1} reduces to the  special
equation
\begin{equation}\label{e1.6}
Lu(t)-\omega u(t)=f(t,u(t+1),u(t),u(t-1)),\quad  t\in \mathbb{Z},
\end{equation}
containing both advance and retardation. Liu, Zhang and Shi
\cite{LiZS1} considered the existence of a nontrivial homoclinic
solution for \eqref{e1.6} by using the Mountain Pass Lemma in
combination with periodic approximations.

In 2016,  Shi, Liu and Zhang \cite{ShLZ1} obtained the existence of
a nontrivial homoclinic solution for a second-order $p$-Laplacian
difference equation containing both advance and retardation
\begin{equation}\label{e1.7}
\Delta(\varphi_p(\Delta u(t-1)))
-q(t)\varphi_p(u(t))+f(t,u(t+M),u(t),u(t-M))=0, 
\end{equation}
for $t\in \mathbb{Z}$, by using critical point theory.

 Deng, Chen and Shi in \cite{DeCS}  studied the
existence of homoclinic solutions for second-order discrete
Hamiltonian systems by using the critical point theory. However, to
the best of our knowledge, the results on homoclinic solutions of
second-order nonlinear difference equation \eqref{e1.1} which
contains both many advances and retardations are very scarce in the
literature (see \cite{ShLZ1}), because there are only few known
methods to establish the existence of homoclinic solutions of
discrete systems.

Motivated by the  articles \cite{LiZS1,ShLZ1}, our main purpose
is to establish new criteria for the existence of nontrivial
homoclinic orbits to a class of second-order nonlinear difference
equations which contains both several advances and retardations with
Jacobi operators. Our results do not  suppose that the system
satisfies the well-known global Ambrosetti-Rabinowitz superquadratic
assumption. Some existing results are generalized and improved;
 see  Remarks \ref{rem1.2} and \ref{rem1.3} for details.

Throughout this article, for a function $F$, we let
$F'_i(y_1,\dots,y_i\dots,y_n)$ denote the partial derivative of
$F$ on the $i$ variable. For basic knowledge of variational methods,
the reader is referred to \cite{MawW,Ra}.

Our main results are obtained using the following hypotheses:
\begin{itemize}
\item[(H1)] $a(t)\neq0$, $b(t)-|a(t-1)|-|a(t)|>\omega$, for all $t\in \mathbb{Z}$;

\item[(H2)] there exists a function  $F(t,y_\Gamma,\dots,y_0)$
which is continuously differentiable in the variable from $y_\Gamma$
to $y_0$ for every $t\in \mathbb{Z}$ and satisfies
\begin{gather*}
F(t+M,y_\Gamma,\dots,y_0)=F(t,y_\Gamma,\dots,y_0),\\
\sum_{i=-\Gamma}^0F'_{2+\Gamma+i}(t+i,y_{\Gamma+i},\dots,y_i)
=f(t,y_\Gamma,\dots,y_0,\dots,y_{-\Gamma});
\end{gather*}

\item[(H3)] $\lim_{\varrho\to 0} \frac{f(t,y_\Gamma,\dots,y_0,\dots,y_{-\Gamma})}
{y_0}=0$ for $t\in \mathbb{Z}$,
$\varrho=(\sum_{i=-\Gamma}^\Gamma
y_i^2)^{1/2}$;

\item[(H4)] $\lim_{\delta\to 0} \frac{F(t,y_\Gamma,\dots,y_0)}
{\delta^2}=0$ for $t\in \mathbb{Z}$,
$\delta=(\sum_{i=0}^\Gamma y_i^2)^{1/2}$;

\item[(H5)] $\lim_{\delta\to \infty} \frac{F(t,y_\Gamma,\dots,y_0)}
{\delta^2}=\infty$ for all $t\in \mathbb{Z}$,
$\delta=(\sum_{i=0}^\Gamma y_i^2)^{1/2}$;

\item[(H6)] for any $t\in \mathbb{Z}$, $F(t,0,\dots,0)=0$,
$F(t,y_\Gamma,\dots,y_0)\geq F(t,y_0)\geq0$;

\item[(H7)] for any $r>0$, there exist $p=p(r)>0,\ q=q(r)>0$ and $\nu<2$ such that
\[
\Big(2+\frac{1}{p+q\big(\sum_{i=0}^\Gamma y_i^2\big)^{\frac{\nu}{2}}}\Big)
F(t,y_\Gamma,\dots,y_0)
\leq\sum_{i=-\Gamma}^0F'_{2+\Gamma+i}(t,y_{\Gamma},\dots,y_0)y_{-i},
\]
for all $t\in\mathbb{Z}$, $\big(\sum_{i=0}^\Gamma
y_i^2\big)^{1/2}>r$.
\end{itemize}

\begin{theorem} \label{thm1.1}
Assume that {\rm (H1)--(H7)} are satisfied.
Then \eqref{e1.1} has a nontrivial homoclinic solution.
\end{theorem}

\begin{remark} \label{rem1.2}\rm
Theorem \ref{thm1.1} extends Theorem \ref{thm1.1} in \cite{LiZS1}
which is the special case of our Theorem \ref{thm1.1} by letting
$\Gamma=1$.
\end{remark}


\begin{remark} \label{rem1.3}\rm
In the superquadratic case, almost all the existing results (see
e.g. \cite{DeCS,DeLSZ,GuY,MaG,Sh})  need the following
well-known global Ambrosetti-Rabinowitz superquadratic
condition:
\begin{itemize}
\item[(AR)] there exists a constant $\beta>2$
such that\newline $0<\beta F(t,u)\leq uf(t,u)$ for all $t\in
\mathbb{Z}$ and $u\in \mathbb{R}\setminus\{0\}$. 
\end{itemize}
Note that (H5)--(H7) are much weaker than the Ambrosetti-Rabinowitz condition. 
Therefore, our result improves that the existing ones.
\end{remark}

\begin{theorem} \label{thm1.4}
Assume that {\rm (H1)--(H4)} and the following
assumption are satisfied:
\begin{itemize}
\item[(H8)] $F(t,y_\Gamma,\dots,y_0)\geq0$ and there exists a constant 
$\beta>2$ such that
\[
0<\beta F(t,y_\Gamma,\dots,y_0)\leq\sum_{i=0}^\Gamma F'_{2+i}(t,y_{\Gamma},
\dots,y_0)y_{\Gamma-i},
\]
for all $t\in\mathbb{Z}$,  $(y_{\Gamma},\dots,y_0)\in
\mathbb{R}^{\Gamma+1}\setminus\{(0,\dots,0)\}$.
\end{itemize}
Then \eqref{e1.1} has a nontrivial homoclinic solution.
\end{theorem}

The rest of this article is organized as follows. First, 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. Then, in Section 3, some related
lemmas will be stated. Next, in Section 4, we shall complete the
proof of the results by using variational methods and the critical
point method. Finally, in Section 5, we shall give an example to
illustrate the applicability of the main result.


\section{Variational structure}

To apply the critical point theory, the corresponding
variational framework for equation \eqref{e1.1} is established. We
start by some basic notation for the reader's convenience.

Let $S$ be the vector space of all real sequences of the form
$$
u=\{u(t)\}_{t\in { \mathbb{Z}}}=(\dots,u(-t),
\dots,u(-1),u(0),u(1),\dots,u(t),\dots),
$$
namely
$S=\{\{u(t)\}: u(t)\in \mathbb{R},\, t\in \mathbb{Z}\}$.
Define
$$
E=\big\{u\in S:\sum_{t=-\infty}^{+\infty}[(L-\omega I)u(t)\cdot u(t)]
<+\infty\big\}.
$$
The space is a Hilbert space with the inner product
\begin{equation}\label{e2.1}
\langle u,v\rangle =\sum_{t=-\infty}^{+\infty}[(L-\omega I)u(t)v(t)],\quad
 \forall u,v\in E,
\end{equation}
and the corresponding norm
\begin{equation}\label{e2.2}
\|u\|=\sqrt{\langle u,u\rangle }=\sqrt{\sum_{t=-\infty}^{+\infty}[(L-\omega
I)u(t) u(t))]},\quad \forall u\in E.
\end{equation}
Next, we define
$$
l^2=\big\{u\in S:\sum_{t=-\infty}^{+\infty}u^2(t)<+\infty\big\}, \quad
l^\infty=\big\{u\in S:\sup_{t\in \mathbb{Z}}|u(t)|<+\infty\big\},
$$
and their norms are 
\begin{gather*}
\|u\|_2=\Big(\sum_{t=-\infty}^{+\infty}u^2(t)\Big)^{1/2}, \quad \forall u\in l^2, \\
\|u\|_\infty=\sup_{t\in \mathbb{Z}}|u(t)|,\quad \forall u\in l^\infty,
\end{gather*}
respectively.

For $u\in E$, we define the functional $J$ on $E$ as follows:
\begin{equation}\label{e2.3}
\begin{aligned}
J(u)&:=\sum_{t=-\infty}^{+\infty}\big[\frac{1}{2}(L-\omega I)u(t)\cdot u(t)-
F(t,u(t+\Gamma),\dots,u(t))\big] \\
&\; =\frac{1}{2}\|u\|^2-\sum^{+\infty}_{t=-\infty}F(t,u(t+\Gamma),\dots,u(t)).
\end{aligned}
\end{equation}
The functional $J$ is a well-defined $C^1$ functional on $E$ and
\eqref{e1.1} is easily recognized as the corresponding
Euler-Lagrange equation for $J$. Therefore, we are looking for
nonzero critical points of $J$.

\section{Main lemmas}

To apply variational methods and critical point theory for
the existence of a nontrivial homoclinic solution of
\eqref{e1.1}, we shall state some lemmas which will be used in the
proofs of our main results.

\begin{lemma}[\cite{MawW}] \label{lem3.1}
Let $E$ be a real Banach space with its dual space $E^*$ and suppose
that $J\in C^1(E,\mathbb{R})$ satisfies
$$
\max\{J(0),J(e)\}\leq\eta_0<\eta\leq\inf_{\|u\|=\rho}J(u),
$$
for some $\eta_0<\eta$, $\rho>0$ and $e\in E$ with $\|e\|>\rho$. Let
$c\geq\eta$ be characterized by
$$
c=\inf_{\gamma\in\Upsilon}\max_{0\leq s\leq1}J(\gamma(s)),
$$
where $\Upsilon=\{\gamma\in C([0,1],E):\gamma(0)=0,\ \gamma(1)=e\}$
is the set of continuous paths joining 0 to $e$; then there exists
$\{u_k\}_{k\in \mathbb{N}}\subset E$ such that
$J(u_k)\to c$ and
$(1+\|u_k\|)\|J'(u_k)\|_{E^*}\to 0$ as $k\to \infty$.
\end{lemma}

\begin{lemma}[\cite{LiZS1}] \label{lem3.2}
Assume that {\rm (H1)} holds. Then there exists a constant
$\underline{\lambda}$ such that the following inequalities hold:
\begin{gather}\label{e3.1}
\underline{\lambda}\|u\|_2^2\leq\|u\|^2, \\
\label{e3.2}
\underline{\lambda}\|u\|^2_\infty\leq\|u\|^2,
\end{gather}
where $\underline{\lambda}=\inf_{t\in \mathbb{Z}}(b(t)-\omega-|a(t-1)|-|a(t)|)>0$.
\end{lemma}

\begin{lemma} \label{lem3.3}
Assume that {\rm(H1)--(H7)} are satisfied. Then there exists a
constant $c>0$ and a sequence $\{u_k\}_{k\in \mathbb{N}}$
satisfying
\begin{equation}\label{e3.3}
J(u_k)\to c,\quad  \|J'(u_k)\|(1+\|u_k\|)\to 0, \quad k\to \infty.
\end{equation}
\end{lemma}

\begin{proof}
By (H4), there exists a constant $\rho>0$ such that for any
$\sqrt{y_\Gamma^2+\dots+y_0^2}\leq\rho$,
\begin{equation}\label{e3.4}
F(t,y_\Gamma,\dots,y_0)\leq
\frac{\underline{\lambda}}{4(\Gamma+1)}(y_\Gamma^2+\dots+y_0^2), \quad
\forall t\in \mathbb{Z}.
\end{equation}
If $\|u\|=\sqrt{\underline{\lambda}}\rho:=\eta$, then by
\eqref{e3.2}, $|u(t)|\leq\rho$ for all $t\in \mathbb{Z}$. For any
$u\in E$, $\|u\|=\rho$, it follows from \eqref{e2.3} and \eqref{e3.4}
that
\begin{align*}
J(u)&=\frac{1}{2}\|u\|^2-\sum_{t=-\infty}^{+\infty}F(t,u(t+\Gamma),\dots,u(t))\\
&\geq\frac{1}{2}\|u\|^2-\frac{\underline{\lambda}}{4(\Gamma+1)}\sum^{+\infty}_{t=-\infty}
[u^2(t+\Gamma)+\dots+u^2(t)]\\
&\geq\frac{1}{2}\|u\|^2-\frac{\underline{\lambda}}{4}\|u\|_2^2,\\
&\geq\frac{1}{4}\|u\|^2=\frac{1}{4}\eta^2.
\end{align*}

Let $u_0(0)=1$, $u_0(t)=0$ for $t\neq0$. By (H2), (H3), (H5) and
\eqref{e2.3}, we have
\begin{align*}
J(su_0) & =\frac{s^2}{2}\|u_0\|^2-\sum^{+\infty}_{t=-\infty}
 F(t,su_0(t+\Gamma),\dots,su_0(t))\\
&\leq\frac{s^2}{2}\|u_0\|^2-F(0,su_0(\Gamma),\dots,su_0(0))\\
&\leq s^2\big[\frac{1}{2}\|u_0\|^2-\frac{F(0,su_0(\Gamma),\dots,
su_0(0))}{|su_0(0)|^2}\big] \leq0
\end{align*}
for large enough $s>0$.

 Choose $s_1>1$ such that
$s_1\|u_0\|>\eta$ and $J(s_1u_0)\leq0$. Let $e=s_1u_0$, then 
$e\in E$, $\|e\|>\eta$ and $J(e)\leq0$. By Lemma \ref{lem3.1}, there
exists a constant $c\geq\frac{1}{4}\eta^2$ and a sequence
$\{u_k\}_{k\in \mathbb{N}}\subset E$ such that
\eqref{e3.3} holds.
\end{proof}

\begin{lemma} \label{lem3.4}
Assume that {\rm (H1)--(H7)} are satisfied. Then any
$\{u_k\}_{k\in \mathbb{N}}$ satisfying
\begin{equation}\label{e3.5}
J(u_k)\to c>0,\quad \langle
J'(u_k),u_k\rangle\to0,\quad k\to\infty
\end{equation}
is bounded in $E$.
\end{lemma}

\begin{proof}
It follows from (H4) that there exists a constant $0<\rho<1$ such
that for any $\sqrt{u_k^2(t+\Gamma)+\dots+u_k^2(t)}\leq\rho$,
\begin{equation}\label{e3.6}
|F(t,u_k(t+\Gamma),\dots,u_k(t))|\leq
\frac{\underline{\lambda}}{4(\Gamma+1)}\sum_{i=0}^\Gamma
u_k^2(t+i),\quad \forall t\in \mathbb{Z}.
\end{equation}
For $t\in \mathbb{Z}$, by (H7), we have
\begin{equation}\label{e3.7}
\sum_{i=-\Gamma}^0F'_{2+\Gamma+i}(t,u_k(t+\Gamma),\dots,u_k(t))u_k(t-i)
>2F(t,u_k(t+\Gamma),\dots,u_k(t))\geq0,
\end{equation}
and $t\in \mathbb{Z}$, $\sqrt{u_k^2(t+\Gamma)+\dots+u_k^2(t)}>\rho$,
we have
\begin{equation}\label{e3.8}
\begin{aligned}
&F(t,u_k(t+\Gamma),\dots,u_k(t)) \\
&\leq\Big[p+q(\sum_{i=0}^\Gamma u_k^2(t+i))^{\frac{\nu}{2}}\Big] 
\Big[\sum_{t=-\Gamma}^0F'_{2+\Gamma+i}(t,u_k(t+\Gamma),
\dots,u_k(t))u_k(t-i) \\
&\quad -2F(t,u_k(t+\Gamma),\dots,u_k(t))\Big].
\end{aligned}
\end{equation}
By \eqref{e2.1}, \eqref{e2.3} and \eqref{e3.5}, there exist
constants $C_1$ and $C_2$ such that
\begin{equation}\label{e3.9}
\begin{aligned}
C_1&\geq 2J(u_k)-\langle J'(u_k),u_k\rangle \\
&=\sum_{t=-\infty}^{+\infty}\Big[\sum_{i=-\Gamma}^0F'_{2+\Gamma+i}(t,u_k(t+\Gamma),
 \dots,u_k(t))u_k(t-i) \\
&\quad -2F(t,u_k(t+\Gamma),\dots,u_k(t))\Big]
\end{aligned}
\end{equation}
and
\begin{equation}\label{e3.10}
J(u_k)\leq C_2.
\end{equation}
From \eqref{e2.3}, \eqref{e3.2}, \eqref{e3.6},
\eqref{e3.7}, \eqref{e3.8}, \eqref{e3.9} and \eqref{e3.10} it follows that
\begin{align*}
&\frac{1}{2}\|u_k\|^2 \\
&=J(u_k) +\sum_{t=-\infty}^{+\infty}F(t,u_k(t+\Gamma),\dots,u_k(t))\\
&=J(u_k)+\sum_{t\in\mathbb{Z}\big((\sum_{i=0}^\Gamma
u_k^2(t+i))^{1/2}\leq\rho\big)} F(t,u_k(t+\Gamma),\dots,u_k(t))\\
&\quad +\sum_{t\in\mathbb{Z}\big((\sum_{i=0}^\Gamma
u_k^2(t+i))^{1/2}>\rho\big)} F(t,u_k(t+\Gamma),\dots,u_k(t))\\
&\leq J(u_k)+\frac{\underline{\lambda}}{4(\Gamma+1)}
 \sum_{t\in\mathbb{Z}((\sum_{i=0}^\Gamma
 u_k^2(t+i))^{1/2}\leq\rho)} \sum_{i=0}^\Gamma u_k^2(t+i)\\
&\quad +\sum_{t\in\mathbb{Z}\big((\sum_{i=0}^\Gamma
u_k^2(t+i))^{1/2}>\rho\big)}
\Big[p+q(\sum_{i=0}^\Gamma u_k^2(t+i))^{\frac{\nu}{2}}\Big]\\
&\quad \times
\Big[\sum_{i=-\Gamma}^0F'_{2+\Gamma+i}(t,u_k(t+\Gamma),\dots,u_k(t))u_k(t-i)
-2F(t,u_k(t+\Gamma),\dots,u_k(t))\Big]\\
&\leq C_2+\frac{1}{4}\|u_k\|^2+\sum_{t\in\mathbb{Z}}
\Big[p+q(\sum_{i=0}^\Gamma u_k^2(t+i))^{\frac{\nu}{2}}\Big]\\
&\times
\Big[\sum_{i=-\Gamma}^0F'_{2+\Gamma+i}(t,u_k(t+\Gamma),\dots,u_k(t))u_k(t-i)
-2F(t,u_k(t+\Gamma),\dots,u_k(t))\Big]\\
&\leq C_2+\frac{1}{4}\|u_k\|^2+[p+q(\Gamma+1)\|u_k\|_{\infty}^\nu] \\
&\quad \times
\Big[\sum_{i=-\Gamma}^0F'_{2+\Gamma+i}(t,u_k(t+\Gamma),\dots,u_k(t))u_k(t-i)
-2F(t,u_k(t+\Gamma),\dots,u_k(t))\Big]\\
&\leq C_2+\frac{1}{4}\|u_k\|^2+C_1[p+q(\Gamma+1)\|u_k\|_{\infty}^\nu]\\
&\leq C_2+\frac{1}{4}\|u_k\|^2
+C_1[p+\underline{\lambda}^{-\frac{\nu}{2}}q(\Gamma+1)\|u_k\|^\nu], \quad
k\in \mathbb{N}. %\label{e3.11}
\end{align*} 
Since $\nu<2$, from the above inequality it follows  that
$\{u_k\}_{k\in \mathbb{N}}$ is bounded. The proof is
complete.
\end{proof}

\section{Proof of main results}

In this Section, we shall prove our main results by using the
critical point theory.

\begin{proof}[Proof of Theorem \ref{thm1.1}]
Lemma \ref{lem3.3} implies that the existence of a sequence
$\{u_k\}_{k\in \mathbb{N}}\subset E$ satisfying
\eqref{e3.3}, and so \eqref{e3.5}. By Lemma \ref{lem3.4},
$\{u_k\}_{k\in \mathbb{N}}$ is bounded in $E$. Thus,
combining with \eqref{e3.2}, there exists a constant $C_3>0$ such
that
\begin{equation}\label{e4.1}
\sqrt{\underline{\lambda}}\|u_k\|_\infty\leq\|u_k\|\leq
C_3,\quad  \forall k\in \mathbb{N}.
\end{equation}
Hence, by (H2)--(H4), for $t\in \mathbb{Z}$, with
$\big(\sum_{i=0}^\Gamma u_k^2(t+i)\big)^{1/2}\leq 
\frac{1}{\sqrt{\underline{\lambda}}}C_3$, we have
\begin{equation}\label{e4.2}
\begin{aligned}
&\big|\frac{1}{2}f(t,u_k(t+\Gamma),\dots,u_k(t),\dots,u_k(t-\Gamma))u_k(t)
-F(t,u_k(t+\Gamma),\dots,u_k(t))\big| \\
&\leq\frac{c\underline{\lambda}}{4C_3^2}u_k^2(t)
+\frac{c\underline{\lambda}}{4(\Gamma+1)C_3^2}\sum_{i=0}^\Gamma
u_k^2(t+i).
\end{aligned}
\end{equation}

Define $\varepsilon:=\limsup_{k\to\infty}\|u_k\|_\infty$.
We state that $\varepsilon>0$. For the sake of contradiction, we
assume that $\varepsilon=0$. From (H3), \eqref{e2.3}, \eqref{e3.5}
and \eqref{e4.2}, we have
$$
\begin{aligned}
c&=J(u_k)-\frac{1}{2}\langle J'(u_k),u_k\rangle+o(1)\\
&=\frac{1}{2}\sum_{t=-\infty}^{+\infty}f(t,u_k(t+\Gamma),\dots,u_k(t),
 \dots,u_k(t-\Gamma))u_k(t) \\
&\quad -\sum_{t=-\infty}^{+\infty}F(t,u_k(t+\Gamma),\dots,u_k(t))+o(1)\\
&\leq\frac{c\underline{\lambda}}{4C_3^2}\sum_{t=-\infty}^{+\infty}u_k^2(t)
     +\frac{c\underline{\lambda}}{4(\Gamma+1)C_3^2}\sum_{t=-\infty}^{+\infty}
\sum_{i=0}^\Gamma      u_k^2(t+i)+o(1)\\
&\leq\frac{c\underline{\lambda}}{4C_3^2}\|u_k\|_2^2
    +\frac{c\underline{\lambda}}{4C_3^2}\|u_k\|_2^2+o(1)\\
&\leq\frac{c}{2}+o(1),\quad  k\to\infty.
\end{aligned}
$$
This contradiction shows that $\varepsilon>0$.

First, going to a subsequence if necessary, we can assume that the
existence of $t_k\in\mathbb{Z}$ depending on  $u_k$ such that
\begin{equation}\label{e4.3}
|u_k(t_k)|=\|u_k\|_\infty>\frac{\varepsilon}{2}.
\end{equation}
Hence, making such shifts, we can assume that $t_k\in
\mathbb{Z}(0,M-1)$ in \eqref{e4.3}. Moreover, passing to a
subsequence of $k$s, we can even assume that $t_k=t_0$ is
independent of $k$.

Next, we extract a subsequence, still denote by $u_k$, such that
$$
u_k(t)\to u(t),\quad k\to\infty,\quad  \forall t\in \mathbb{Z}.
$$
Inequality \eqref{e4.3} implies that $|u(t_0)|\geq \xi$
and, hence, $u=\{u(t)\}$ is a nonzero sequence. Moreover,
\begin{align*}
&Lu(t)-\omega
u(t)-f(t,u(t+\Gamma),\dots,u(t),\dots,u(t-\Gamma))\\
&=\lim_{k\to \infty}[Lu_k(t)-\omega u_k(t)
-f(t,u_k(t+\Gamma),\dots,u_k(t),\dots,u_k(t-\Gamma))]\\
&=\lim_{k\to \infty}0=0.
\end{align*}
So $u=\{u(t)\}$ is a solution of \eqref{e1.1}.

Finally, for any fixed $D\in \mathbb{Z}$ and $k$ large enough, we
have 
$$
\sum^D_{t=-D}|u_k(t)|^2\leq \frac{1}{\underline{\lambda}}\|u_k\|^2\leq C_3^2.
$$
Since $C_3^2$ is a constant independent of $k$, passing to the
limit, we have 
$$
\sum^D_{t=-D}|u(t)|^2\leq C_3^2.
$$
Because of  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 solution is obtained.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.4}]
By a proof similar to the one in Theorem \ref{thm1.1} and the
process in \cite{LiZS1}, we can prove Theorem \ref{thm1.4}. For
simplicity, the proof is omitted.
\end{proof}

\section{Example}

As an application of Theorem \ref{thm1.1}, we give an example that
illustrates our main result.
For  $t\in \mathbb{Z}$, assume that
\begin{equation}\label{e5.1}
\begin{aligned}
& u(t+1)+u(t-1)-(2+\omega)u(t) \\
&=3\sum_{j=0}^\Gamma \Big\{2u(t)\ln\Big[1+(\sum_{i=0}^\Gamma u^2(t+i-j))^{1/2}\Big]\\
&\quad +\frac{\big\{\sum_{i=0}^\Gamma u^2(t+i-j)\big)^{1/2}u(t)}
{1+\big(\sum_{i=0}^\Gamma u^2(t+i-j)\big)^{1/2}} \Big\},
\end{aligned}
\end{equation}
where  $\omega<-4$. We have
$a(t)=a(t-1)\equiv 1$, $b(t)\equiv -2$, and
$$
F(t,u(t+\Gamma),\dots,u(t))
=3\sum_{i=0}^\Gamma u^2(t+i)\ln\Big[1+(\sum_{i=0}^\Gamma u^2(t+i))^{1/2}\Big].
$$
Then
\begin{align*}
&\sum_{i=-\Gamma}^0F'_{2+\Gamma+i}(t,u(t+\Gamma),\dots,u(t))u(t-i)\\
&=3\Big[2\sum_{i=0}^\Gamma u^2(t+i)\ln\Big[1+(\sum_{i=0}^\Gamma u^2(t+i))^{1/2}\Big]
+\frac{\big(\sum_{i=0}^\Gamma
u^2(t+i)\big)^{3/2}}{1 +\big(\sum_{i=0}^\Gamma u^2(t+i)\big)^{1/2}}\Big]\\
&\geq\Big[2+\frac{1}{1+(\sum_{i=0}^\Gamma u^2(t+i))^{1/2}}\Big]
F(t,u(t+\Gamma),\dots,u(t))\geq0.
\end{align*}
This shows that (H7) holds with $p=q=\nu=1$. It is easy to verify
that all the assumptions of Theorem \ref{thm1.1} are satisfied.
Consequently, \eqref{e5.1} has a nontrivial homoclinic solution.

\subsection*{Acknowledgements}
The author would like to thank the anonu=ymous referees, Prof. Julio G. Dix, 
and the editors for their careful reading and for making some valuable comments 
and suggestions on the manuscript.

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\end{document}