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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 315, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/315\hfil Existence of homoclinic orbits]
{Existence of homoclinic orbits for a class of nonlinear functional
difference equations}

\author[X. Liu, T. Zhou, H. Shi \hfil EJDE-2016/315\hfilneg]
{Xia Liu, Tao Zhou, Haiping Shi}

\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{Tao Zhou \newline
School of Business Administration,
South China University of Technology,
Guangzhou 510640, China}
\email{zhoutaoscut@hotmail.com}

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

\thanks{Submitted October 3, 2016. Published December 10, 2016.}
\subjclass[2010]{34C37, 37J45, 39A12, 47J30, 58E05}
\keywords{Homoclinic orbit; difference equation; critical point theory}

\begin{abstract}
 By using critical point theory, we prove the existence of a nontrivial
 homoclinic orbit for a class of nonlinear functional difference equations.
 Our conditions on the nonlinear term do not need to satisfy the
 well-known global Ambrosetti-Rabinowitz superquadratic condition.
\end{abstract}

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

\section{Introduction}

The article concerns the nonlinear functional difference equation
\begin{equation}\label{e1.1}
 \Delta(p_n(\Delta u_{n-1})^\delta)-q_nu_n^\delta
+f(n,u_{n+1},u_n,u_{n-1})=0,\quad n\in \mathbb{Z},
\end{equation}
where $\Delta$ is the forward difference operator
 $\Delta u_n=u_{n+1}-u_n$, $\Delta^2 u_n=\Delta(\Delta u_n)$,
 $\delta$ is the ratio of odd positive integers,
$\{p_n\}_{n\in \mathbb{Z}}$ and $\{q_n\}_{n\in \mathbb{Z}}$ are real
 sequences, $f\in C(\mathbb{Z}\times \mathbb{R}^3,\mathbb{R})$,
 $T$ is a positive integer,
 $p_{n+T}=p_n$, $q_{n+T}=q_n$, and
 $f(n+T,v_1,v_2,v_3)=f(n,v_1,v_2,v_3)$.

We denote by $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ the 
natural numbers, integers and real numbers respectively. For
$a$, $b$ $\in \mathbb{Z}$, we define
$\mathbb{Z}(a)=\{a,a+1,\dots\}, \mathbb{Z}(a,b)=\{a,a+1,\dots,b\}$
when $a\leq b$.
In this article we use the following assumptions:
\begin{itemize}
\item[(A1)] $p_n>0$ for $n\in \mathbb{Z}$;
\item[(A2)] $q_n>0$ for $n\in \mathbb{Z}$;
\item[(A3)] there exists a functional
 $F(n,v_1,v_2)\in C^1(\mathbb{Z}\times \mathbb{R}^2,\mathbb{R})$ with
 $F(n+T,v_1,v_2)=F(n,v_1,v_2)$ and satisfies
 $$
\frac{\partial F(n-1,v_2,v_3)}{\partial v_2}
 +\frac{\partial F(n,v_1,v_2)}{\partial v_2}
 =f(n,v_1,v_2,v_3);
$$

\item[(A4)] $\lim_{\varrho\to 0} f(n,v_1,v_2,v_3)/v_2^\delta=0$
for $n\in \mathbb{Z}$,
$\varrho=(v_1^{\delta+1}+v_2^{\delta+1}+v_3^{\delta+1}
)^{\frac{1}{\delta+1}}$;

\item[(A5)] $\lim_{\sigma\to 0} F(n,v_1,v_2)/\sigma^{\delta+1}=0$ for
$n\in \mathbb{Z}$,
$\sigma=(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}$.
\end{itemize}
In general, \eqref{e1.1} can be considered as a discrete
analogue of the second order nonlinear functional differential equation
\begin{equation}\label{e1.2}
 (p(t)\varphi(u'))'+q(t)u(t)+f(t,u(t+1),u(t),u(t-1))=0,\quad
 t\in \mathbb{R}.
\end{equation}
This equation includes the equation
 $$
(p(t)\varphi(u'))'+f(t,u(t))=0,\quad t\in \mathbb{R},
$$
which has arose in the study of fluid dynamics, combustion theory, gas
diffusion through porous media, thermal self-ignition of a chemically active mixture
of gases in a vessel, catalysis theory, chemically reacting systems, and
adiabatic reactor \cite{CasY,EsV}.
Equations similar in structure to \eqref{e1.2} arise in the
study of periodic solutions and homoclinic orbits of functional differential
equations \cite{GuOXA,GuX}.

The theory of nonlinear difference equations has been widely used to
study discrete models appearing in many fields
 such as computer science, economics, neural networks, queuing theory, ecology,
cybernetics, biological systems, optimal control,
 population dynamics, etc. Since the past twenty years, there has been much
progress on the qualitative properties of
 difference equations, which included results on homoclinic orbits, periodic
solutions, boundary value
 problems, stability, attractivity, oscillation and other topics, see for example
 \cite{AgW,AlCE,BR,CaiY,CT,CW,DCS,DLST,DSX,El,GuY,Le1,Le2,LZS1,LZS2,LZS3,LZSD,MaG,SLZ1,SLZ2,SLZ3,Ya1,Ya2,Zh,ZhWY} and
 the references therein.

 In 2004, Zhang, Wang and Yu \cite{ZhWY} obtained necessary and sufficient
conditions for the existence of strictly monotone
 increasing positive solutions of the following equation
\begin{equation}\label{e1.3}
 \Delta(p_n(\Delta u_{n-1})^\delta)+q_nu_n^\delta=0.
\end{equation}

 If $f(n,u_{n+1},u_n,u_{n-1})=f(n,u_n)$, Cai and Yu \cite{CaiY} considered
the nonlinear difference equation of  the type
\begin{equation}\label{e1.4}
 \Delta(p_n(\Delta u_{n-1})^\delta)+q_nu_n^\delta=f(n,u_n),\quad
 n\in \mathbb{Z},
\end{equation}
 using the critical point theory, and they obtained some new results on
the existence of periodic solutions.

 If $q_n\equiv1$, Liu, Zhang and Shi in 2015 \cite{LZS1} and 2016 \cite{SLZ3}
respectively  studied a class of nonlinear difference equation
\begin{equation}\label{e1.5}
 \Delta(p_n(\Delta u_{n-1})^\delta)+f(n,u_{n+1},u_n,u_{n-1})=0,\quad
 n\in \mathbb{Z},
\end{equation}
 has at least three $T$-periodic solutions.

 By using the Symmetric Mountain Pass Theorem, Chen and Wang \cite{CW} established
some existence criteria to guarantee  a class of nonlinear difference equation
\begin{equation}\label{e1.6}
 \Delta(p_n(\Delta u_{n-1})^\delta)-q_nu_n^\delta+f(n,u_n)=0,\quad
 n\in \mathbb{Z},
\end{equation}
has infinitely many homoclinic orbits.  Shi, Liu and Zhang \cite{SLZ1} obtained
the existence of a nontrivial  homoclinic orbit for \eqref{e1.1} based on
the Mountain Pass Lemma in combination with periodic approximations.

 In the superquadratic case, almost all the
results in the literature (see e.g. \cite{CaiY,DLST,LZS2,MaG})
need the  well-known global Ambrosetti-Rabinowitz superquadratic
condition:
\begin{itemize}
\item There exists a constant $\beta>2$ such that
 $0<\beta F(n,u)\leq uf(n,u)$ for all $n\in \mathbb{Z}$ and
$u\in \mathbb{R}\setminus\{0\}$.
\end{itemize}
In this article, we introduce the following  conditions that are weaker
than the superquadratic condition
\begin{itemize}
\item[(A6)] $\lim_{\sigma\to 0} F(n,v_1,v_2)/\sigma^{\delta+1}=\infty$
for $n\in \mathbb{Z}$, $\sigma=(v_1^{\delta+1}+v_2^{\delta+1}
 )^{\frac{1}{\delta+1}}$;

\item[(A7)] for any $n\in \mathbb{Z}$, $F(n,0,0)=0$,
$F(n,v_1,v_2)\geq F(n,v_2)\geq0$;

\item[(A8)] for any $\varrho>0$, there exist $a=a_\varrho>0$,
$b=b_\varrho>0$ and $\nu<\delta+1$ such
 that for all $n\in\mathbb{Z}$, $
(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}>\varrho$,
 $$
\Big[\delta+1+\frac{1}{a+b(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{\nu}{\delta+1}}}\Big]
F(n,v_1,v_2)
\leq \frac{\partial F(n,v_1,v_2)}{\partial v_1}v_1
+\frac{\partial F(n,v_1,v_2)}{\partial v_2}v_2.
$$
\end{itemize}

Our main results read as follows.

\begin{theorem} \label{thm1.1}
 Suppose that {\rm (A1)--(A8)} are satisfied. Then  \eqref{e1.1} possesses
a nontrivial homoclinic orbit.
\end{theorem}

\begin{theorem} \label{thm1.2}
 Suppose that {\rm (A1)--(A5)} and the following assumption are satisfied:
\begin{itemize}
\item[(A9)] $F(n,v_1,v_2)\geq0$ and there exists a constant $\beta>2$ such that
\[
0<\beta F(n,v_1,v_2)
\leq\frac{\partial F(n,v_1,v_2)}{\partial v_1}v_1
 +\frac{\partial F(n,v_1,v_2)}{\partial v_2}v_2,
\]
 for all $(n,v_1,v_2)\in \mathbb{Z}\times\mathbb{R}^2\setminus\{(0,0)\}$.
\end{itemize}
Then \eqref{e1.1} possesses a nontrivial homoclinic orbit.
\end{theorem}

For basic knowledge of variational methods, the reader is referred
to \cite{MaW,Ra}.


\section{Variational structure and some lemmas}

To apply  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
main results. We start by some basic notation.

Let $S$ be the set of sequences $u=(\dots,u_{-n},\dots,u_{-1},u_0,u_1,\dots,u_n,
 \dots)=\{u_n\}_{n=-\infty}^{+\infty}$; that is,
$$
S=\{\{u_n\}: u_n\in \mathbb{R},\, n\in \mathbb{Z}\}.
$$
For  $u,v\in S$, $a,b\in \mathbb{R}$, we define
$$
au+bv=\{au_n+bv_n\}_{n=-\infty}^{+\infty}.
$$
Then $S$ is a vector space.
Define
$$
E=\big\{u\in S: \sum_{n=-\infty}^{+\infty}
[p_n(\Delta u_{n-1})^{\delta+1}
 +q_nu_n^{\delta+1}]<+\infty\big\},
$$
 and for $u\in E$,
\begin{equation}\label{e2.1}
 \|u\|=\Big\{\sum_{n=-\infty}^{+\infty}[p_n(\Delta u_{n-1})^{\delta+1}
 +q_nu_n^{\delta+1}]\Big\}^{\frac{1}{\delta+1}},\quad \forall u\in E.
\end{equation}
 Then $E$ is a uniform convex Banach space with this norm.

 As usual, for $1<s<+\infty$, we set
\[
l^s=\big\{u\in S:\sum_{n=-\infty}^{+\infty}|u_n|^s<+\infty\big\}, \quad
l^\infty=\{u\in S:\sup_{n\in \mathbb{Z}}|u_n|<+\infty\},
\]
with their respective norms
\begin{gather*}
\|u\|_s=\Big(\sum_{n=-\infty}^{+\infty}|u_n|^s\Big)^{1/s},\quad \forall u\in l^s,\\
\|u\|_\infty=\sup_{n\in \mathbb{Z}}|u_n|,\quad \forall u\in l^\infty\,.
\end{gather*}

For  $u\in E$, we define the functional
\begin{equation}\label{e2.2}
 J(u):=\frac{1}{\delta+1}\sum_{n=-\infty}^{+\infty}
[p_n(\Delta u_{n-1})^{\delta+1}  +q_nu_n^{\delta+1}]
-\sum_{n=-\infty}^{+\infty}F(n,u_{n+1},u_n).
\end{equation}
 If (A1)--(A3) hold, then $J\in C^1(E,\mathbb{R})$ and one can easily check that
\begin{equation}\label{e2.3}
\begin{aligned}
 \langle J'(u),v\rangle
&=\sum_{n=-\infty}^{+\infty}
 [p_n(\Delta u_{n-1})^{\delta}\Delta v_{n-1}  +q_nu_n^{\delta}v_n]\\
&\quad -\sum_{n=-\infty}^{+\infty}f(n,u_{n+1},u_n,u_{n-1})v_n,\quad
 \forall u,v\in E.
\end{aligned}
\end{equation}
 Thus, we can compute the partial derivative as
\begin{equation}\label{e2.4}
 \frac{\partial J(u)}{\partial u_n}
 =-\Delta(p_n(\Delta u_{n-1})^\delta)
+q_nu_n^\delta-f(n,u_{n+1},u_n,u_{n-1}),\quad \forall n\in \mathbb{Z}.
\end{equation}
So, the critical points of $J$ in $E$ are the solutions of \eqref{e1.1}
with $u_n\to 0$ as $|n|\to \infty$.

\begin{lemma}[\cite{MaW}] \label{lem2.1}
Let $E$ be a  real Banach space with its dual space $E^*$ and assume 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
$$
c=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq1}J(\gamma(t)),
$$
 where $\Gamma=\{\gamma\in C([0,1],E):\gamma(0)=0,\, \gamma(1)=e\}$
is the set of continuous paths joining  $0$ to $e$;
then there exists $\{u^{(k)}\}_{k\in \mathbb{N}}\subset E$ such
 that\newline
 $J(u^{(k)})\to c$ and
 $(1+\|u^{(k)}\|)\|J'(u^{(k)})\|_{E^*}\to 0$ as $k\to \infty$.
\end{lemma}

\begin{lemma} \label{lem2.2}
For $u\in E$ and $s>1$,
\begin{equation} \label{e2.5}
\underline{q}\|u\|_{\infty}^{\delta+1}\leq\underline{q}
\|u\|_s^{\delta+1}\leq\|u\|^{\delta+1},
\end{equation}
 where  $\underline{q}=\inf_{n\in \mathbb{Z}}q_n$.
\end{lemma}

The proof of above lemma is routine; so we omit it.

\begin{lemma} \label{lem2.4}
Suppose that {\rm (A1)--(A8)}  are satisfied. Then there exists a constant
$c>0$ and a sequence $\{u^{(k)}\}_{k\in \mathbb{N}}$ satisfying
\begin{equation}\label{e2.6}
 J(u^{(k)})\to c,\quad \|J'(u^{(k)})\|(1+\|u^{(k)}\|)\to 0,\quad
 k\to \infty.
\end{equation}
\end{lemma}

\begin{proof}
It follows from (A5) that there exists a constant $\rho>0$ such
that
\begin{equation}\label{e2.7}
 F(n,v_1,v_2)\leq \frac{\underline{q}}{4(\delta+1)}(v_1^{\delta+1}+v_2^{\delta+1}),\quad
 \forall n\in \mathbb{Z},\; (v_1^{\delta+1}+v_2^{\delta+1}
)^{\frac{1}{\delta+1}}\leq\rho.
\end{equation}
 Let $\|u\|=\underline{q}^{\frac{1}{\delta+1}}\rho:=\eta$, combining this and
\eqref{e2.5},  we have $|u_n|\leq\rho$ for all $n\in \mathbb{Z}$.
 Therefore, by \eqref{e2.2} and \eqref{e2.7},  we have
\begin{align*}
J(u)&=\frac{1}{\delta+1}\|u\|^{\delta+1}-\sum_{n=-\infty}^{+\infty}F(n,u_{n+1},u_n)\\
&\geq\frac{1}{\delta+1}\|u\|^{\delta+1}
 -\frac{\underline{q}}{4(\delta+1)}\sum_{n=-\infty}^{+\infty}(u_{n+1}^{\delta+1}+u_n^{\delta+1})\\
&\geq\frac{1}{\delta+1}\|u\|^{\delta+1}-\frac{\underline{q}}{2(\delta+1)}\|u\|_{\delta+1}^{\delta+1},\\
&\geq\frac{1}{\delta+1}\|u\|^{\delta+1}-\frac{1}{2(\delta+1)}\eta^{\delta+1}\\
&=\frac{1}{2(\delta+1)}\eta^{\delta+1},\quad \forall u\in E,\; \|u\|=\rho.\\
\end{align*}
 Choose $u^{(0)}\in E$ such that
$$
u^{(0)}_0=1,\quad u^{(0)}_n=0,\quad \forall n\neq0.
$$
 Then, for $\lambda>0$ large enough, it follows from (A3)--(A6) and \eqref{e2.3} that
\begin{align*}
J(\lambda u^{(0)})
&=\frac{\lambda^{\delta+1}}{\delta+1}\|u^{(0)}\|^{\delta+1}
 -\sum_{n=-\infty}^{+\infty}F(n,\lambda u^{(0)}_{n+1},\lambda u^{(0)}_n)\\
&\leq\frac{\lambda^{\delta+1}}{^{\delta+1}}\|u^{(0)}\|^{\delta+1}-F(0,\lambda
u^{(0)}_{n+1},\lambda u^{(0)}_n)\\
&\leq \lambda^{\delta+1}\Big[\frac{1}{\delta+1}\|u^{(0)}\|^{\delta+1}
 -\frac{F(0,\lambda u^{(0)}_{n+1},\lambda u^{(0)}_n)}
{|\lambda u^{(0)}_0|^{\delta+1}}\Big]\leq0.\\
\end{align*}
 Consequently, we can choose $\lambda_1>1$ such that $\lambda_1\|u^{(0)}\|>\eta$
and $J(\lambda_1u^{(0)})\leq0$.  Define $e=\lambda_1u^{(0)}$, then $e\in E$,
$\|e\|>\eta$ and $J(e)\leq0$.
 From Lemma \ref{lem2.1}, one has that there exists a constant
$c\geq\frac{1}{2(\delta+1)}\eta^{\delta+1}$ and a
 sequence $\{u^{(k)}\}_{k\in \mathbb{N}}\subset E$ such that \eqref{e2.6}
 holds.
\end{proof}

\begin{lemma} \label{lem2.5}
Suppose that {\rm (A1)--(A8)}  are satisfied. Then any sequence
$\{u^{(k)}\}_{k\in \mathbb{N}}$ satisfying
\begin{equation}\label{e2.8}
 J(u^{(k)})\to c>0,\quad \|J'(u^{(k)})\|(1+\|u^{(k)}\|)\to 0,\quad
 k\to \infty
\end{equation}
 is bounded in $E$.
\end{lemma}

\begin{proof}
By (A5), we know that there exists a constant $0<\rho<1$ such that
\begin{equation}\label{e2.9}
 F(n,v_1,v_2)\leq \frac{\underline{q}}{4(\delta+1)}(v_1^{\delta+1}+v_2^{\delta+1}),\quad
\forall n\in \mathbb{Z},\;
 (v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}\leq\rho.
\end{equation}
 It follows from (A3) and (A8) that
\begin{equation}\label{e2.10}
 f(n,v_1,v_2,v_3)v_2>(\delta+1)F(n,v_1,v_2)\geq0,\ \forall n\in \mathbb{Z}
\end{equation}
 and
\begin{equation}\label{e2.11}
F(n,v_1,v_2)
\leq[a+b(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{\nu}{\delta+1}}]
 [f(v_1,v_2,v_3)v_2-(\delta+1)F(n,v_1,v_2)],
\end{equation}
for all $n\in \mathbb{Z}$,
$(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}>\rho$.

 From \eqref{e2.2}, \eqref{e2.3} and \eqref{e2.8}, there exist two constants
$M_1$ and $M_2$ such that
\begin{equation}\label{e2.12}
\begin{aligned}
M_1&\geq (\delta+1)J(u^{(k)})-\langle  J'(u^{(k)}),u^{(k)}\rangle \\
&=\sum_{n=-\infty}^{+\infty}[f(n,u^{(k)}_{n+1},u^{(k)}_n,u^{(k)}_{n-1})u^{(k)}_n
 -(\delta+1)F(n,u^{(k)}_{n+1},u^{(k)}_n)]
\end{aligned}
\end{equation}
 and
\begin{equation}\label{e2.13}
 J(u^{(k)})\leq M_2.
\end{equation}
 By \eqref{e2.3}, \eqref{e2.8}, \eqref{e2.9}, \eqref{e2.10}, \eqref{e2.11},
\eqref{e2.12} and \eqref{e2.13}, we have
\begin{equation}
\begin{aligned}
&\frac{1}{\delta+1}\|u^{(k)}\|^{\delta+1}\\
&=J(u^{(k)})  +\sum_{n=-\infty}^{+\infty} F(n,u^{(k)}_{n+1},u^{(k)}_n)\\
&=J(u^{(k)})
 +\sum_{n\in\mathbb{Z}([(u^{(k)}_{n+1})^{\delta+1}
 +(u^{(k)}_n)^{\delta+1}]^{\frac{1}{\delta+1}}\leq\rho)}
 F(n,u^{(k)}_{n+1},u^{(k)}_n)\\
 &\quad +\sum_{n\in\mathbb{Z}([(u^{(k)}_{n+1})^{\delta+1}
 +(u^{(k)}_n)^{\delta+1}]^{\frac{1}{\delta+1}}>\rho)}
 F(n,u^{(k)}_{n+1},u^{(k)}_n)\\
 &\leq J(u^{(k)})+\frac{\underline{q}}{4(\delta+1)}
 \sum_{n\in\mathbb{Z}([(u^{(k)}_{n+1})^{\delta+1}
 +(u^{(k)}_n)^{\delta+1}]^{\frac{1}{\delta+1}}>\rho)}
 \big[(u^{(k)}_{n+1})^{\delta+1}+(u^{(k)}_n)^{\delta+1}\big]\\
 &\quad +\sum_{n\in\mathbb{Z}([(u^{(k)}_{n+1})^{\delta+1}
 +(u^{(k)}_n)^{\delta+1}]^{\frac{1}{\delta+1}}>\rho)}
 \big\{a+b[(u^{(k)}_{n+1})^{\delta+1}+(u^{(k)}_n)^{\delta+1}
 ]^{\frac{\nu}{\delta+1}}\big\}\\
 &\quad \times [f(n,u^{(k)}_{n+1},u^{(k)}_n,u^{(k)}_{n-1})u^{(k)}_n
 -(\delta+1)F(n,u^{(k)}_{n+1},u^{(k)}_n)]\\
 &\leq M_2+\frac{1}{4}\|u^{(k)}\|^{\delta+1}
 +\sum_{n\in\mathbb{Z}}
 \big\{a+b[(u^{(k)}_{n+1})^{\delta+1}+(u^{(k)}_n)^{\delta+1}
 ]^{\frac{\nu}{\delta+1}}\big\}\\
 &\quad \times \big[f(n,u^{(k)}_{n+1},u^{(k)}_n,u^{(k)}_{n-1})u^{(k)}_n
 -(\delta+1)F(n,u^{(k)}_{n+1},u^{(k)}_n)\big]\\
 &\leq M_2+\frac{1}{4}\|u^{(k)}\|^{\delta+1}+[a+2b\|u^{(k)}\|_{\infty}^\nu]\\
 &\times \big[f(n,u^{(k)}_{n+1},u^{(k)}_n,u^{(k)}_{n-1})u^{(k)}_n
 -(\delta+1)F(n,u^{(k)}_{n+1},u^{(k)}_n)\big]\\
 &\leq M_2+\frac{1}{4}\|u^{(k)}\|^{\delta+1}+M_1[a+2b\|u^{(k)}\|_{\infty}^\nu]\\
&\leq M_2+\frac{1}{4}\|u^{(k)}\|^{\delta+1}
 +M_1[a+2\underline{q}^{-\frac{\nu}{\delta+1}}b\|u^{(k)}\|^\nu],\ k\in \mathbb{N}.
\end{aligned}\label{e2.14}
\end{equation}
Since $\nu<\delta+1$, by \eqref{e2.14}, we have that
$\{u^{(k)}\}_{k\in \mathbb{N}}$ is
 bounded. Hence, the proof is complete.
\end{proof}

\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 From Lemma \ref{lem2.4} there exists a sequence
 $\{u^{(k)}\}_{k\in \mathbb{N}}\subset E$ satisfying \eqref{e2.6},
and so \eqref{e2.8}. Hence, from Lemma
 \ref{lem2.5}, we have that $\{u^{(k)}\}_{k\in \mathbb{N}}$ is bounded in $E$.
 It follows from \eqref{e2.5} that there exists a constant $M_3>0$ such that
\begin{equation}\label{e3.1}
 \underline{q}^{\frac{1}{\delta+1}}\|u^{(k)}\|_\infty\leq\|u^{(k)}\|\leq M_3,
 \forall n\in \mathbb{N}.
\end{equation}
From (A3)--(A5) we have
\begin{equation}\label{e3.2}
\begin{gathered}
\big|\frac{1}{\delta+1}f(n,v_1,v_2,v_3)v_2  -F(n,v_1,v_2)\big| 
\leq\frac{c\underline{q}}{4M_3^{\delta+1}}v_2^{\delta+1}
 +\frac{c\underline{q}}{8M_3^{\delta+1}}(v_1^{\delta+1}+v_2^{\delta+1}),\\
 \text{for all }n\in \mathbb{Z},\;
 (v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}
\leq \frac{1}{\underline{q}^{\frac{1}{\delta+1}}}M_3.
\end{gathered}
\end{equation}

If $\xi:=\limsup_{k\to\infty}\|u^{(k)}\|_\infty=0$.
 Then from (A4), \eqref{e2.2}, \eqref{e2.3} and \eqref{e3.2}, one has
\begin{align*}
 c&=J(u^{(k)})-\frac{1}{\delta+1}\big\langle
 J'(u^{(k)}),u^{(k)}\big\rangle+o(1)\\
 &=\frac{1}{\delta+1}\sum_{n=-\infty}^{+\infty}
 f(n,u^{(k)}_{n+1},u^{(k)}_n,u^{(k)}_{n-1})u^{(k)}_n
 -\sum_{n=-\infty}^{+\infty}F\big(n,u^{(k)}_{n+1},u^{(k)}_n\big)+o(1)\\
 &\leq\frac{c\underline{q}}{4M_3^{\delta+1}}
 \sum_{n=-\infty}^{+\infty}\big(u^{(k)}_n\big)^{\delta+1}
 +\frac{c\underline{q}}{8M_3^{\delta+1}}\sum_{n=-\infty}^{+\infty}
 [(u^{(k)}_{n+1})^{\delta+1}+(u^{(k)}_n)^{\delta+1}]\\
 &\leq\frac{c\underline{q}}{4M_3^{\delta+1}}\|u^{(k)}\|_{\delta+1}^{\delta+1}
 +\frac{c\underline{q}}{4M_3^{\delta+1}}\|u^{(k)}\|_{\delta+1}^{\delta+1}+o(1)\\
 &\leq\frac{c}{2}+o(1),\quad  k\to\infty.
\end{align*}
 This contradiction shows that $\xi>0$.

 First, going to a subsequence if necessary, we can assume that the
 existence of $n^{(k)}\in\mathbb{Z}$ independent of $k$ such that
\begin{equation}\label{e3.3}
 |u_{n^{(k)}}^{(k)}|=\|u^{(k)}\|_\infty>\frac{\xi}{2}.
\end{equation}


Hence, making such  shifts, we can assume that $n^{(k)}\in \mathbb{Z}(0,T-1)$
in \eqref{e3.3}. Moreover,  passing to a subsequence of $k$s, we can even
 assume that  $n^{(k)}=n^{(0)}$ is independent of $k$.

 Next, we extract a subsequence, still denoted by $u^{(k)}$, such that
 $$
u^{(k)}_n\to u_n,\quad k\to\infty,\; \forall n\in  \mathbb{Z}.
$$
 Inequality \eqref{e3.3} implies that $|u_{n^{(0)}}|\geq \xi$ and, hence,
 $u=\{u_n\}$ is a nonzero sequence. Moreover,
\begin{align*}
 &\Delta(p_n(\Delta  u_{n-1})^\delta)-q_nu_n^\delta+f(n,u_{n+1},u_n,u_{n-1})\\
 &=\lim_{k\to \infty}
 \Big[\Delta\Big(p_n\big(\Delta (u^{(k)}_{n-1})\big)^\delta\Big)
 -q_n(u^{(k)}_n)^\delta+f(n,u^{(k)}_{n+1},u^{(k)}_n,u^{(k)}_{n-1})\Big]\\
 &=\lim_{k\to \infty}0=0.
\end{align*}
 So $u=\{u_n\}$ is a solution of \eqref{e1.1}.

 Finally, for any fixed $D\in \mathbb{Z}$ and $k$ large enough, we have 
$$
\sum^D_{n=-D}|u^{(k)}_n|^{\delta+1}
\leq \frac{1}{\underline{q}}\|u^{(k)}\|^{\delta+1}
\leq  M_3^{\delta+1}.
$$
 Since $M_3^{\delta+1}$ is a constant independent of $k$, passing
 to the limit, we have 
$$
\sum^D_{n=-D}|u_n|^{\delta+1}\leq M_3^{\delta+1}.
$$
Since $D$ is arbitrary and $u\in l^{\delta+1}$, the function $u$ satisfies 
$u_n\to 0$ as  $|n|\to \infty$. The proof is complete.
\end{proof}

Theorem \ref{thm1.2} can be proved similarly as in the proof of Theorem \ref{thm1.1} 
and using the process in \cite{LZS2}. For simplicity, we omit the proof.
As an application of the main theorems,  we give two
examples to illustrate our results.


\begin{example} \label{examp.3} \rm
In \eqref{e1.1}, let $p_n>0$, $q_n>0$, and
\begin{align*}
 &f(n,u_{n+1},u_n,u_{n-1}) \\
&=(\delta+1)u_n^\delta
 \ln[1+(u_{n+1}^{\delta+1}+u_n^{\delta+1})^{\frac{1}{\delta+1}}]
 +\frac{(u_{n+1}^{\delta+1}+u_n^{\delta+1})^{\frac{1}{\delta+1}}u_n^\delta}
 {1+(u_{n+1}^{\delta+1}+u_n^{\delta+1})^{\frac{1}{\delta+1}}}\\
 &\quad+(\delta+1)u_n^\delta
 \ln\big[1+(u_n^{\delta+1}+u_{n-1}^{\delta+1})^{\frac{1}{\delta+1}}\big]
 +\frac{(u_n^{\delta+1}+u_{n-1}^{\delta+1})^{\frac{1}{\delta+1}}u_n^\delta}
 {1+(u_n^{\delta+1}+u_{n-1}^{\delta+1})^{\frac{1}{\delta+1}}}.
\end{align*}
 Since
\[
F(n,v_1,v_2) =(v_1^{\delta+1}+v_2^{\delta+1})  \ln[1+(v_1^{\delta+1}
 +v_2^{\delta+1})^{\frac{1}{\delta+1}}],
\]
we have 
\begin{gather*}
\begin{aligned}
&\frac{\partial F(n-1,v_2,v_3)}{\partial v_2}
+\frac{\partial F(n,v_1,v_2)}{\partial v_2} \\
&=(\delta+1)v_2^\delta
 \ln[1+(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}]
 +\frac{(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}v_2^\delta}
 {1+(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}}\\
&\quad +(\delta+1)v_2^\delta
 \ln[1+(v_2^{\delta+1}+v_3^{\delta+1})^{\frac{1}{\delta+1}}]
 +\frac{(v_2^{\delta+1}+v_3^{\delta+1})^{\frac{1}{\delta+1}}v_2^\delta}
 {1+(v_2^{\delta+1}+v_3^{\delta+1})^{\frac{1}{\delta+1}}}\\
\end{aligned} \\
\begin{aligned}
 &\frac{\partial F(n,v_1,v_2)}{\partial v_1}v_1+\frac{\partial F(n,v_1,v_2)}{\partial v_2}v_2\\
 &=(\delta+1)(v_1^{\delta+1}+v_2^{\delta+1})
 \ln\big[1+(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}\big]
 +\frac{(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{\delta+2}{\delta+1}}}
 {1+(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}}\\
 &\geq\Big(\delta+1+\frac{1}{(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{1}{\delta+1}}}\Big)
F(n,v_1,v_2) \\
&\geq 0, \quad \forall n\in\mathbb{Z}.
\end{aligned}
\end{gather*}
This shows that (A8) holds with $a=b=\nu=1$. It is easy to verify all the
 conditions of Theorem \ref{thm1.1} are satisfied.
 By Theorem \ref{thm1.1}, \eqref{e1.1} possesses a
 nontrivial homoclinic orbit.
\end{example}

\begin{example} \label{examp3.4} \rm
In \eqref{e1.1}, let $p_n>0$, $q_n>0$,  $\beta>2$ and
$$
f(n,u_{n+1},u_n,u_{n-1})
=\beta\big[(u_{n+1}^{\delta+1}+u_n^{\delta+1})^{\frac{\beta-\delta-1}{\delta+1}}
 +(u_n^{\delta+1}+u_{n-1}^{\delta+1})^{\frac{\beta-\delta-1}{\delta+1}}\big]
u_n^\delta.
$$
Then we have
$$
F(n,v_1,v_2)=(v_1^{\delta+1}+v_2^{\delta+1})^{\frac{\beta}{\delta+1}}.
$$
 By computations similar to those  in \cite{SLZ1}, it is easy to verify 
all the assumptions of Theorem \ref{thm1.2} are satisfied.
Therefore \eqref{e1.1} possesses a nontrivial homoclinic solution.
\end{example}


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

 \begin{thebibliography}{00}

\bibitem{AgW} R. P. Agarwal, P. J. Y. Wong;
\emph{Advanced Topics in Difference Equations}, Kluwer Academic
 Publishers, Dordrecht, 1997.

\bibitem{AlCE} Z. AlSharawi, J. M. Cushing, S. Elaydi;
\emph{Theory and Applications of Difference Equations and Discrete
 Dynamical Systems}, Springer, New York, 2014.

\bibitem{BR} G. M. Bisci, D. Repov\v{s};
\emph{Existence of solutions for $p$-Laplacian discrete equations},
Appl. Math. Comput., 242 (2014), 454-461.

\bibitem{CaiY} X. C. Cai, J. S. Yu;
\emph{Existence theorems of periodic solutions for
 second-order nonlinear difference equations}, Adv. Difference Equ., 
2008 (2008), 1-11.

\bibitem{CasY} A. Castro, R. Shivaji;
\emph{Nonnegative solutions to a semilinear Dirichlet problem in a
 ball are positive and radially symmetric}, Comm. Partial
 Differential Equations, 14(8-9) (1989), 1091-1100.

\bibitem{CT} P. Chen, X. H. Tang;
\emph{Existence of infinitely many homoclinic orbits for
fourth-order difference systems containing both advance and
retardation}, Appl. Math. Comput., 217(9) (2011), 4408-4415.


\bibitem{CW} P. Chen, Z. M. Wang;
\emph{Infinitely many homoclinic solutions for a class of nonlinear
 difference equations}, Electron. J. Qual. Theory Differ. Equ., (47) (2012), 1-18.

\bibitem{DCS} X. Q. Deng, G. Cheng, H. P. Shi;
\emph{Subharmonic solutions and homoclinic orbits of second order
discrete Hamiltonian systems with potential changing sign}, Comput.
Math. Appl., 58(6) (2009), 1198-1206.

\bibitem{DLST} X. Q. Deng, X. Liu, H. P. Shi, T. Zhou;
\emph{Homoclinic orbits for second order nonlinear $p$-Laplacian
difference equations}, J. Contemp. Math. Anal., 46(3) (2011),
172-181.

\bibitem{DSX} X. Q. Deng, H. P. Shi, X. L. Xie;
\emph{Periodic solutions of second order discrete Hamiltonian
systems with potential indefinite in sign}, Appl. Math. Comput.,
218(1) (2011), 148-156.

\bibitem{El} S. Elaydi;
\emph{An Introduction to Difference Equations}, Springer, New York,
2005.

\bibitem{EsV} J. R. Esteban, J. L. V\'{a}zquez;
\emph{On the equation of turbulent filtration in one-dimensional
porous media}, Nonlinear Anal., 10(11) (1986), 1303-1325.

\bibitem{GuOXA} C. J. Guo, D. O'Regan, Y. T. Xu, R. P. Agarwal;
\emph{Existence of homoclinic orbits for a singular second-order
 neutral differential equation}, J. Math. Anal. Appl., 366 (2011), 550-560.

\bibitem{GuX} C. J. Guo, Y. T. Xu;
\emph{Existence of periodic solutions for a class of second order
differential equation with deviating argument}, J. Appl. Math.
Comput., 28(1-2) (2008), 425-433.

\bibitem{GuY} Z. M. Guo, J. S. Yu;
\emph{The existence of periodic and subharmonic solutions
 of subquadratic second order difference equations}, J. London
 Math. Soc., 68(2) (2003), 419-430.

\bibitem{Le1} J. H. Leng;
\emph{Existence of periodic solutions for a higher order nonlinear
difference equation}, Electron. J. Differential Equations, 2016(134)
(2016), 1-10.

\bibitem{Le2} J. H. Leng;
\emph{Periodic and subharmonic solutions for 2$n$th-order
$\phi_c$-Laplacian difference equations containing both advance and
retardation}, Indag. Math. (N.S.), 27(4) (2016), 902-913.

\bibitem{LZS1} X. Liu, Y. B. Zhang, H. P. Shi;
\emph{Existence of periodic solutions for a class of nonlinear
difference equations}, Qual. Theory Dyn. Syst., 14(1) (2015), 51-69.

\bibitem{LZS2} X. Liu, Y. B. Zhang, H. P. Shi;
\emph{Homoclinic orbits of second order nonlinear functional
difference equations with Jacobi operators}, Indag. Math. (N.S.),
26(1) (2015), 75-87.

\bibitem{LZS3} X. Liu, Y. B. Zhang, H. P. Shi;
\emph{Homoclinic orbits and subharmonics for second order
$p$-Laplacian difference equations}, J. Appl. Math. Comput., 43(1)
(2013), 467-478.

\bibitem{LZSD} X. Liu, Y. B. Zhang, H. P. Shi, X. Q. Deng;
\emph{Periodic and subharmonic solutions for fourth-order nonlinear
 difference equations}, Appl. Math. Comput., 236(3) (2014), 613-620.

\bibitem{MaG} M. J. Ma, Z. M. Guo;
\emph{Homoclinic orbits and subharmonics for
 nonlinear second order difference equations}, Nonlinear Anal.,
 67(6) (2007), 1737-1745.

\bibitem{MaW} J. Mawhin, M. Willem;
\emph{Critical Point Theory and Hamiltonian
 Systems}, Springer: New York, 1989.

\bibitem{Ra} P. H. Rabinowitz;
\emph{Minimax Methods in Critical Point Theory with
 Applications to Differential Equations}, Amer.
 Math. Soc., Providence, RI: New York, 1986.

\bibitem{SLZ1} H. P. Shi, X. Liu, Y. B. Zhang;
\emph{Homoclinic orbits for a class of nonlinear difference
equations}, Azerb. J. Math., 6(1) (2016), 87-102.

\bibitem{SLZ2} H. P. Shi, X. Liu, Y. B. Zhang;
\emph{Homoclinic orbits of second-order nonlinear difference
equations}, Electron. J. Differential Equations, 2015(150) (2015),
1-16.

\bibitem{SLZ3} H. P. Shi, X. Liu, Y. B. Zhang;
\emph{Periodic solutions for a class of nonlinear difference
equations}, Hokkaido Math. J., 45(1) (2016), 109-126.

\bibitem{Ya1} L. W. Yang;
\emph{Existence of homoclinic orbits for fourth-order $p$-Laplacian
difference equations}, Indag. Math. (N.S.), 27(3) (2016), 879-892.

\bibitem{Ya2} L. W. Yang;
\emph{Existence theorems of periodic solutions for second-order
difference equations containing both advance and retardation}, J.
Contemp. Math. Anal., 51(2) (2016), 58-67.

\bibitem{Zh} Q. Q. Zhang;
\emph{Homoclinic orbits for a class of discrete periodic Hamiltonian
systems}, Proc. Amer. Math. Soc., 143(7) (2015), 3155-3163.

\bibitem{ZhWY} R. Y. Zhang, Z. C. Wang, J. S. Yu;
\emph{Necessary and sufficient conditions for the existence of
positive solutions of nonlinear difference equations}, Fields Inst.
Commun.,42 (2004), 385-396.


\end{thebibliography}

\end{document}