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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 183, pp. 1--9.\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-2016/183\hfil Standing waves]
{Standing waves for discrete nonlinear Schr\"odinger equations}

\author[M. Jia \hfil EJDE-2016/183\hfilneg]
{Ming Jia}

\address{Ming Jia \newline
Swan College,
Central South University of Forestry and Technology,
Changsha 410004, China}
\email{jmznlkd@163.com}

\thanks{Submitted June 13, 2016. Published July 11, 2016.}
\subjclass[2010]{39A12, 39A70, 35C08}
\keywords{Standing waves; discrete nonlinear Schr\"odinger equation;
\hfill\break\indent  critical point theory}

\begin{abstract}
 The discrete nonlinear Schr\"odinger equation is a nonlinear
 lattice system that appears in many areas of physics such as
 nonlinear optics, biomolecular chains and Bose-Einstein condensates.
 By using critical point theory, we establish some new sufficient
 conditions on the existence results of standing waves for the
 discrete nonlinear Schr\"odinger equations. We give an appropriate
 example to illustrate the conclusion obtained.
\end{abstract}

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

\section{Introduction}

The discrete nonlinear Schr\"odinger (DNLS) equation is one of the
most important inherently discrete models, having a crucial role in
the modeling of a great variety of phenomena, ranging from
solid-state and condensed-matter physics to biology \cite{Chr}.
Fundamental states supported by the DNLS equations are standing
waves due to their periodic time behavior. This kind of solutions
have been found in the experimental observations \cite{Fl}.

In the past decade, the existence of standing waves of the DNLS
equations has drawn a great deal of interest
\cite{Hu,Mai,Pa,Shi1,Shi2,Shi3,Zha,Zho}. The existence for the
periodic DNLS equations with superlinear nonlinearity \cite{Pa} and
with saturable nonlinearity \cite{Zho} has been studied. And the
existence results of standing waves of the DNLS equations without
periodicity assumptions were established in \cite{Zha}. As for the
existence of the homoclinic orbits of nonlinear Schr\"odinger
equations, we refer to \cite{Che,Ta}.

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

This article considers the DNLS equation
\begin{equation}\label{e1.1}
i\dot{\psi}_n=-\Delta\psi_n+v_n\psi_n-\gamma_nf(\psi_n),\
n\in \mathbb{Z},
\end{equation}
where $\Delta \psi_n=\psi_{n+1}+\psi_{n-1}-2\psi_n$ is discrete
Laplacian operator, $v_n$ and $\gamma_n$ are real valued for each
$n\in \mathbb{Z}$, $f\in C(\mathbb{R},\mathbb{R})$, $f(0)=0$ and the
nonlinearity $f(u)$ is gauge invariant, that is,
\begin{equation}\label{e1.2}
f(e^{i\theta}u)=e^{i\theta}f(u),\ \theta\in \mathbb{R}.
\end{equation}

Since standing waves are spatially localized time-periodic solutions
and decay to zero at infinity, $\psi_n$ has the form
$$
\psi_n=u_ne^{-i\omega t},
$$
and
$$
\lim_{|n|\to\infty}\psi_n=0,
$$
where $\psi_n$ is real valued for each $n\in \mathbb{Z}$ and
$\omega\in\mathbb{R}$ is the temporal frequency. Then \eqref{e1.1}
becomes
\begin{gather}\label{e1.3}
-\Delta u_n+v_nu_n-\omega u_n=\gamma_nf(u_n),\quad n\in \mathbb{Z}, \\
\label{e1.4}
\lim_{|n|\to\infty}u_n=0
\end{gather}
holds, where $|n|$ is the length of index $n$.
Actually, our methods allow us to consider the   more
general equation
\begin{equation}\label{e1.5}
\Delta^k(p_{n-k}\Delta^ku_{n-k})+(-1)^kq_nu_n=(-1)^kf_n(u_n),\quad
k\in \mathbb{Z}(1),\; n\in \mathbb{Z},
\end{equation}
with the same boundary condition \eqref{e1.4}. Here, $\Delta$ is
the forward difference operator \cite{Li4,Sha} 
$\Delta u_n=u_{n+1}-u_n$, $\Delta^k u_n=\Delta(\Delta^{k-1} u_n)$,
$p_n$ and $q_n$ are positive real valued for each $n\in \mathbb{Z}$.
$p_n$, $q_n$ and $f_n(x)$ are all $T$-periodic in $n$ for a given
positive integer $T$. When $k=1, p_n\equiv1$ and
$q_n\equiv\varepsilon_n-\omega$, we obtain \eqref{e1.3}. Naturally,
if we look for standing waves of \eqref{e1.1}, we just need to get
the solutions of \eqref{e1.5} satisfying \eqref{e1.4}.

Peil and Peterson \cite{Pe} in 1994 studied the asymptotic behavior
of solutions of $2k$th-order difference equation
\begin{equation}\label{e1.6}
\sum^{k}_{i=0}\Delta^i(r_i(n-i)\Delta^iu(n-i))=0
\end{equation}
with $r_i(n)\equiv0$ for $1\leq i\leq k-1$.

In 1998, Anderson \cite{An} considered \eqref{e1.6} for 
$n\in \mathbb{Z}(a)$, and obtained a formulation of generalized zeros and
$(k, k)$-disconjugacy for \eqref{e1.6}. Cai and Yu \cite{Ca} in 2007
obtained some criteria for the existence of periodic solutions of
the following difference equation
\begin{equation}\label{e1.7}
\Delta^k(r_{n-k}\Delta^ku_{n-k})+f(n,u_n)=0.
\end{equation}

In 2013, Deng, Liu, Zhang and Shi \cite{DeLZ} studied the existence
of periodic for the following $2n$th-order difference equation
containing both advance and retardation with $p$-Laplacian
\begin{equation}\label{e1.8}
\Delta^n(r_{k-n}\varphi_p(\Delta^nu_{k-1}))
=(-1)^nf(k,u_{k+1},u_k,u_{k-1}),\ k\in \mathbb{Z}.
\end{equation}

Recently, Liu, Zhang and Shi \cite{Li2} established various sets of
sufficient conditions of the nonexistence and existence of solutions
for mixed boundary value problem and gave some new results to the
following 2$n$th-order nonlinear difference equation
\begin{equation}\label{e1.9}
    \Delta^n(\gamma_{i-n+1}\Delta^nu_{i-n})=(-1)^nf(i,u_{i+1},u_i,u_{i-1}),\quad
   n\in \mathbb{Z}(1),\; i\in \mathbb{Z}(1,k),
\end{equation}
by using  critical point theory.

Using critical point theory, Shi and Zhang \cite{Shi1} in 2016
 investigated the   more general equation
\begin{equation}\label{e1.10}
    \Delta^k(r_{n-k}\varphi_p(\Delta^ku_{n-1}))+(-1)^kq_n\varphi_p(u_n)
=(-1)^k\gamma_nf(u_n),\quad
    k\in \mathbb{Z}(1),\; n\in \mathbb{Z},
\end{equation}
and obtained a new result concerning the existence of a standing
wave solution.

As it is well known, critical point theory is a powerful tool to
deal with the homoclinic solutions of differential equations
\cite{Gu} and is used to study homoclinic solutions of discrete
systems in recent years \cite{Al,DeC,DeLS,Li1,Li3}. Our aim in this
paper is to obtain the existence results of standing waves for the
discrete nonlinear Schr\"odinger equations by using critical point
theory. The main idea is to transfer the problem of solutions in $E$
(defined in Section 2) of \eqref{e1.5} into that of critical points
of the corresponding functional. The motivation for the present work
stems from the recent paper \cite{Ya}.

For basic knowledge of variational methods, the reader is referred
to \cite{Maw,Ra}.
Let $F_n(x)=\int_0^x f_n(t)dt$, $t\in \mathbb{R}$. Our main results
are as follows.

\begin{theorem} \label{thm1.1}
 Suppose that the following hypotheses are satisfied:
\begin{itemize}
 \item[(H1)]  $F_n(x)$ is continuously differentiable in $x$ for every 
$n\in \mathbb{Z}$, $F_n(x)\geq0$, $F_n(0)=0$;

 \item[(H2)]  $\lim_{|x|\to 0} \frac{f_n(x)} {|x|}=0$ for $n\in \mathbb{Z}$;

\item[(H3)] $\lim_{|x|\to \infty} \frac{F_n(x)}{x^2}=\infty$ for $n\in \mathbb{Z}$;

\item[(H4)] for any $\varrho>0$, there exist $a=a_\varrho>0$,
$b=b_\varrho>0$ and $\alpha<2$ such
     that for all $n\in\mathbb{Z}$, $|x|>\varrho$,
\begin{gather*}
\Big(2+\frac{1}{a+b|x|^{\alpha/2}}\Big)F_n(x)\leq
f_n(x)x.
\end{gather*}
\end{itemize}
Then \eqref{e1.5} has a nontrivial solution satisfying \eqref{e1.4}.
\end{theorem}

\begin{theorem} \label{thm1.2}
Assume that {\rm (H1), (H2)} and the following hypothesis are
satisfied:
\begin{itemize}
\item[(H5)] there exists $\gamma>2$ such that
\begin{gather*}
0<\gamma F_n(x)\leq xf_n(x),\quad \forall n\in\mathbb{Z},\; 
 x\in \mathbb{R}\setminus\{0\}.
\end{gather*}
\end{itemize}
Then \eqref{e1.5} has a nontrivial solution satisfying \eqref{e1.4}.
\end{theorem}


\section{Variational structure}

To apply the critical point theory, the corresponding variational framework for
 equation \eqref{e1.5} is established. We start by some basic notations 
for the reader's convenience.
 Let $S$ be the vector space of all real sequences of the form
 $$
 u=(\dots,u_{-n},\dots,u_{-1},u_0,u_1,\dots,u_n,
     \dots)=\{u_n\}_{n=-\infty}^{+\infty},
 $$
 namely
 $ S=\{\{u_n\}: u_n\in \mathbb{R},\ n\in \mathbb{Z}\}$.
 Define
 $$
 E=\big\{u\in S: \sum^{+\infty}_{n=-\infty}\left[p_{n-1}(\Delta^ku_{n-1})^2
     +q_nu_n^2\right]<+\infty\big\}.
 $$
 The space is a Hilbert space with the inner product
\begin{equation}\label{e2.1}
\langle u,v\rangle =\sum^{+\infty}_{n=-\infty}(p_{n-1}\Delta^ku_{n-1}
     \Delta^kv_{n-1}+q_nu_nv_n),\quad \forall u,v\in E,
\end{equation}
and the corresponding norm
\begin{equation}\label{e2.2}
 \|u\|=\Big(\sum^{+\infty}_{n=-\infty}\left[p_{n-1}(\Delta^ku_{n-1})^2
     +q_nu_n^2\right]\Big)^{1/2},\quad \forall u\in E.
\end{equation}
On the other hand, we define the real sequence spaces
\begin{equation}\label{e2.3}
l^s=\big\{u\in S:
\|u\|_s=\Big(\sum^{+\infty}_{n=-\infty}|u_n|^s\Big)^{1/s}<+\infty\big\},\
    1\leq s<+\infty,
\end{equation}
with $\|u\|_\infty=\sup_{n\in\mathbb{Z}}|u_n|$
when $s=+\infty$.

Since $u\in E$, it follows that
$\lim_{|n|\to\infty}|u_n|=0$. Hence, there
exists $n^*\in\mathbb{Z}$ such that
$$
\|u\|_\infty=|u_{n^*}|=\max_{n\in\mathbb{Z}}|u_n|.
$$
By \eqref{e2.2}, we have
$$
\|u\|^2\geq\sum_{n\in\mathbb{Z}}q_nu_n^2\geq\underline{q}\sum_{n\in\mathbb{Z}}u_n^2
    \geq\underline{q}\|u\|_\infty^2.
$$
Thus,
\begin{equation}\label{e2.4}
\underline{q}\|u\|_\infty^2\leq\underline{q}\|u\|_2^2\leq\|u\|^2.
\end{equation}
For all $u\in E$,\ define the functional $J$ on $E$ as follows:
\begin{equation}\label{e2.5}
\begin{aligned}
J(u):=&\frac{1}{2}\sum_{n=-\infty}^{+\infty}\big[p_{n-1}(\Delta^ku_{n-1})^2
     +q_nu_n^2\big]-\sum_{n=-\infty}^{+\infty}F_n(u_n) \\
&=\frac{1}{2}\|u\|^2-\sum_{n=-\infty}^{+\infty}F_n(u_n),
\end{aligned}
\end{equation}
then $J\in C^1(E,\mathbb{R})$. By using
$$
\Delta^ku_{n-1}=\sum_{i=0}^k(-1)^i
\binom{k}{i} u_{n+k-i-1},
$$
we can compute the partial derivative as
\begin{equation}\label{e2.6}
\frac{\partial J(u)}{\partial
u_n}=(-1)^k\Delta^k(p_{n-k}\Delta^ku_{n-k})+q_nu_n-f_n(u_n),\quad
k\in \mathbb{Z}(1),\; n\in \mathbb{Z}.
\end{equation}
Thus, the critical points of $J$ in $E$ are solutions of
\eqref{e1.5} satisfying \eqref{e1.4}.


\section{Main lemmas}

To apply variational methods and critical point theory to
study the existence of a nontrivial solution of \eqref{e1.5}
satisfying \eqref{e1.4}, we shall state some lemmas which will be
used in the proofs of our main results.

\begin{lemma}[\cite{Ce}] \label{lem3.1}
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
$\{u^{(m)}\}_{m\in \mathbb{N}}\subset E$ such
that
$J(u^{(m)})\to c$ and
    $(1+\|u^{(m)}\|)\|J'(u^{(m)})\|_{E^*}\to 0$ as $m\to \infty$.
\end{lemma}

\begin{lemma}\label{lem3.2}
Assume that {\rm (H1)--(H4)} are satisfied. Then there exists a constant
$c>0$ and a sequence $\{u^{(m)}\}_{m\in \mathbb{Z}}$
satisfying
\begin{equation}\label{e3.1}
     J(u^{(m)})\to c,\quad \|J'(u^{(m)})\|(1+\|u^{(m)}\|)\to 0,\quad
     m\to \infty.
\end{equation}
\end{lemma}

\begin{proof}
From (H2) there exists $\rho>0$ such that
for any $n\in \mathbb{Z}$ and $|x|\leq\rho$,
\begin{equation}\label{e3.2}
F_n(x)\leq \frac{\underline{q}}{4}x^2.
\end{equation}
Define
$\|u\|=\underline{q}^{1/2}\rho:=\eta$.
For any $n\in \mathbb{Z}$, it follows from \eqref{e2.4} that
$|u_n|\leq\rho$. Consequently, for $u\in E,\ \|u\|=\rho$, it comes
from \eqref{e2.5} and \eqref{e3.2} that
\begin{align*}
J(u)&=\frac{1}{2}\|u\|^2-\sum_{n=-\infty}^{+\infty}F_n(u_n)\\
&\geq\frac{1}{2}\eta^2-\frac{\underline{q}}{4}\sum_{n=-\infty}^{+\infty}u_n^2 \\
&\geq\frac{1}{2}\eta^2-\frac{\underline{q}}{4}\|u\|_2^2,\\
&\geq\frac{1}{2}\eta^2-\frac{1}{4}\eta^2 
=\frac{1}{4}\eta^2.
\end{align*}
Set $u^{(0)}_0=1$, $u^{(0)}_n=0$ for $n\neq0$. Then, by (H1), (H2),
(H4) and \eqref{e2.3}, we have
\begin{align*}
J(\theta u^{(0)})
&=\frac{\theta^2}{2}\|u^{(0)}\|^2
    -\sum_{n=-\infty}^{+\infty}F_n(\theta u^{(0)}_n) \\
&\leq\frac{\theta^2}{2}\|u^{(0)}\|^2-F_0(\theta u^{(0)}_n) \\
&\leq \theta^2\big[\frac{1}{2}\|u^{(0)}\|^2
    -\frac{F_0(\theta u^{(0)}_n)}{|\theta u^{(0)}_0|^2}\big]
\leq 0
\end{align*}
for large enough $\theta>0$.
Thus, we can choose $\bar{\theta}>1$ such that
$\bar{\theta}\|u^{(0)}\|>\eta$ and
$J(\bar{\theta}u^{(0)})\leq0$. Define
$e=\bar{\theta}u^{(0)}$, then $e\in E$, $\|e\|>\eta$ and
$J(e)\leq0$. By Lemma \ref{lem3.1},  there exists
$c\geq\frac{1}{4}\eta^2$ and a sequence
$\{u^{(m)}\}_{m\in \mathbb{Z}}\subset E$ such that
\eqref{e3.1} holds.
\end{proof}

\begin{lemma}\label{lem3.3}
Assume that {\rm (H1)--(H4)} are satisfied. Then any sequence
$\{u^{(m)}\}_{m\in \mathbb{N}}$ satisfying
\begin{equation}\label{e3.3}
J(u^{(m)})\to c>0,\quad
\|J'(u^{(m)})\|(1+\|u^{(m)}\|)\to 0,\quad m\to \infty
\end{equation}
is bounded in $E$.
\end{lemma}

\begin{proof}
From (H3) it follows that there exists $0<\rho<1$ such that for any
$n\in \mathbb{Z}$, $|x|\leq\rho$,
\begin{equation}\label{e3.4}
F_n(x)\leq \frac{\underline{q}}{4}x^2.
\end{equation}
For any $n\in \mathbb{Z}$, by (H4), we have
\begin{equation} \label{e3.5}
f_n(x)x>2F_n(x)\geq0
\end{equation}
and for $|x|>\rho$, we have
\begin{equation}\label{e3.6}
F_n(x)\leq(a+b|x|^{\alpha/2})[f_n(x)x-2F_n(x)].
\end{equation}
By \eqref{e2.5}, \eqref{e2.6} and \eqref{e3.1}, there exist
$\tilde{K}$ and $\hat{K}$ such that
\begin{equation}\label{e3.7}
\begin{aligned}
\tilde{K}
&\geq 2J(u^{(m)})-\langle  J'(u^{(m)}),u^{(m)}\rangle \\
&=\sum_{n=-\infty}^{+\infty}\big[f_n(u^{(m)}_n)u^{(m)}_n-2F_n(u^{(m)}_n)\big]
\end{aligned}
\end{equation}
and
\begin{equation}\label{e3.8}
J(u^{(m)})\leq \hat{K}.
\end{equation}
It comes from \eqref{e2.5}, \eqref{e2.6}, \eqref{e3.3},
\eqref{e3.4}, \eqref{e3.5}, \eqref{e3.6}, \eqref{e3.7} and
\eqref{e3.8} that
\begin{equation}\label{e3.9}
\begin{aligned}
&\frac{1}{2}\|u^{(m)}\|^2\\
&=J(u^{(m)})+\sum_{n=-\infty}^{+\infty}F_n(u^{(m)}_n) \\
&=J(u^{(m)}) +\sum_{n\in\mathbb{Z}(|u^{(m)}_n|\leq\rho)}
     F_n(u^{(m)}_n)+\sum_{n\in\mathbb{Z}(|u^{(m)}_n|>\rho)}
     F_n(u^{(m)}_n) \\
&\leq J(u^{(m)})+\frac{\underline{q}}{4}\sum_{n\in\mathbb{Z}(|u^{(m)}_n|>\rho)}
     |u^{(m)}_n|^2 \\
&\quad +\sum_{n\in\mathbb{Z}(|u^{(m)}_n|>\rho)}
     \{a+b|u^{(m)}_n|^{\alpha/2}\}\big[f_n(u^{(m)}_n)u^{(m)}_n
     -2F_n(u^{(m)}_n)\big]\\
&\leq \hat{K}+\frac{1}{4}\|u^{(m)}\|^2+\sum_{n\in\mathbb{Z}}
     \{a+b|u^{(m)}_n|^{\alpha/2}\}\big[f_n(u^{(m)}_n)u^{(m)}_n
     -2F_n(u^{(m)}_n)\big] \\
&\leq \hat{K}+\frac{1}{4}\|u^{(m)}\|^2+(a+2b\|u^{(m)}\|_{\infty}^\alpha)
     \big[f_n(u^{(m)}_n)u^{(m)}_n
     -2F_n(u^{(m)}_n)\big] \\
&\leq \hat{K}+\frac{1}{4}\|u^{(m)}\|^2
 +\tilde{K}(a+2b\|u^{(m)}\|_{\infty}^\alpha) \\
&\leq \hat{K}+\frac{1}{4}\|u^{(m)}\|^2
     +\tilde{K}(a+2\underline{q}^{-\frac{\alpha}{2}}b\|u^{(m)}\|^\alpha),
\quad  m\in \mathbb{N}.
\end{aligned}
\end{equation}
Combining with $\alpha<2$, \eqref{e3.9} imply that
$\{u^{(m)}\}_{m\in \mathbb{N}}$ is bounded. Hence, the
proof of Lemma \ref{lem3.3} is complete.
\end{proof}

\section{Proof of the main results}
In this Section, we shall prove our main results by using the
critical point method.

\begin{proof}[Proof of Theorem \ref{thm1.1}]
By Lemma \ref{lem3.2},  there exists a sequence
$\{u^{(m)}\}_{m\in \mathbb{N}}\subset E$ satisfying
\eqref{e3.1}, and so \eqref{e3.3}. It follows from Lemma
\ref{lem3.3} that $\{u^{(m)}\}_{k\in \mathbb{N}}$ is
bounded in $E$. Therefore, by \eqref{e2.4}, for all $n\in
\mathbb{N}$, we have there exists $\bar{K}>0$ such that
\begin{equation}\label{e4.1}
\underline{q}^{1/2}\|u^{(m)}\|_\infty\leq\|u^{(m)}\|\leq
\bar{K}.
\end{equation}
For any $n\in \mathbb{Z}$, $|x|\leq \underline{q}^{-1/2}\bar{K}$, by (H2), we have
\begin{equation}\label{e4.2}
\big|\frac{1}{2}f_n(x)x-F_n(x)\big|
\leq \frac{c\underline{q}}{4\bar{K}^2}x^2+\frac{c\underline{q}}{4\bar{K}^2}x^2
=\frac{c\underline{q}}{2\bar{K}^2}x^2.
\end{equation}
By way of contradiction, suppose that
$\xi:=\limsup_{m\to\infty}\|u^{(m)}\|_\infty=0$.
Then, by (H2), \eqref{e2.1}, \eqref{e2.3} and \eqref{e3.2}, we
have
\begin{align*}
c&=J(u^{(m)})-\frac{1}{2}\left\langle
J'(u^{(m)}),u^{(m)}\right\rangle+o(1) \\
&=\frac{1}{2}\sum_{n=-\infty}^{+\infty}f_n(u^{(m)}_n)u^{(m)}_n
     -\sum_{n=-\infty}^{+\infty}F_n(u^{(m)}_n)+o(1) \\
&\leq\frac{c\underline{q}}{2\bar{K}^2}\sum_{n=-\infty}^{+\infty}(u^{(m)}_n)^2\\
&\leq\frac{c\underline{q}}{2\bar{K}^2}\|u^{(k)}\|_2^2+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 $n^{(m)}\in\mathbb{Z}$ independent of $m$ such that
\begin{equation}\label{e4.3}
|u_{n^{(m)}}^{(m)}|=\|u^{(m)}\|_\infty>\frac{\xi}{2}.
\end{equation}
Hence, making such shifts, we can assume that 
$n^{(m)}\in \mathbb{Z}(0,T-1)$ in \eqref{e4.3}. Moreover, passing to a
subsequence of $m$s, we can even assume that $n^{(m)}=n^{(0)}$ is
independent of $m$.

Next, we extract a subsequence, still denote by $u^{(m)}$, such that
$$
u^{(m)}_n\to u_n,\ m\to\infty,\ \forall n\in \mathbb{Z}.
$$
Inequality \eqref{e4.3} implies that $|u_{n^{(0)}}|\geq \xi$ and, hence, 
$u=\{u_n\}$ is a nonzero sequence. Moreover,
\begin{align*}
&\Delta^k(p_{n-k}\Delta^ku_{n-k})+(-1)^kq_nu_n-(-1)^kf_n(u_n) \\
&=\lim_{m\to \infty} \big[\Delta^k(p_{n-k}\Delta^ku_{n-k}^{(m)})
+(-1)^kq_nu_n^{(m)}    -(-1)^kf_n(u_n^{(m)})\big]
&=\lim_{m\to \infty}0=0.
\end{align*}
So $u=\{u_n\}$ is a solution of \eqref{e1.5}  satisfying
\eqref{e1.4}.

Finally, for any fixed $\varsigma\in \mathbb{Z}$ and $m$ large
enough, we have 
$$
\sum^\varsigma_{n=-\varsigma}|u^{(m)}_n|^2
\leq \frac{1}{\underline{q}}\|u^{(m)}\|^2\leq
    \bar{K}^2.
$$
Since $\bar{K}^2$ is a constant independent of $k$, passing to the
limit, we have 
$$
\sum^\varsigma_{n=-\varsigma}|u_n|^2\leq \bar{K}^2.
$$
By the arbitrariness of $\varsigma$, $u\in l^2$. Therefore, $u$
satisfies $u_n\to 0$ as $|n|\to \infty$. The
proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 The techniques of the proof of Theorem \ref{thm1.2} are just the same as 
those carried out in the proof  of \cite{Li1}. We do not repeat them here.
\end{proof}

\section{Example}

As an application of Theorem \ref{thm1.1}, we give an example to
illustrate our main result.
 For $n\in \mathbb{Z}$, $k\in \mathbb{Z}(1)$,
assume that
\begin{equation}\label{e5.1}
\begin{aligned}
&\Delta^k\Big(\big(3+\sin^2\frac{\pi (n-k)}{T}\big)\Delta^ku_{n-k}\Big)
    +(-1)^k(2+\cos^2\frac{\pi n}{T})u_n \\
&=2(9+\sin^2\frac{\pi
n}{T})u_n\ln(1+|u_n|)
+\frac{u_n^3}{(1+|u_n|)|u_n|},
\end{aligned}
\end{equation}
where $T$ is a positive integer. We have
\begin{gather*}
p_{n-k}=\Big(3+\sin^2\frac{\pi (n-k)}{T}\Big),\quad
q_n=\Big(2+\cos^2\frac{\pi n}{T}\Big), \\
F_n(x)=\big(9+\sin^2\frac{\pi n}{T}\big)x^2\ln(1+|x|).
\end{gather*}
Then
$$
f_n(x)x=2(9+\sin^2\frac{\pi n}{T})x^2\ln(1+|x|)
+\frac{x^2|x|}{1+|x|}
\geq \big(2+\frac{1}{1+|x|}\big)F_n(x)\geq0.
$$
This shows that (H4) holds with $a=b=\nu=1$. It is easy to check
all the assumptions of Theorem \ref{thm1.1} are satisfied.
Consequently, \eqref{e5.1} has a nontrivial solution satisfying
\eqref{e1.4}.




\begin{thebibliography}{00}

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

\bibitem{An} D. Anderson;
\emph{A 2$n$th-order linear difference equation}, Comput. Math.
Appl., 2(4) (1998), 521-529.

\bibitem{Ca} X. C. Cai, J. S. Yu;
\emph{Existence of periodic solutions for a 2$n$th-order nonlinear
difference equation}, J. Math. Anal. Appl., 329(2) (2007), 870-878.

\bibitem{Ce} G. Cerami;
\emph{An existence criterion for the critical points on unbounded
manifolds (Italian)}, Istit. Lombardo Accad. Sci. Lett. Rend. A,
112(2) (1978), 332-336.

\bibitem{Che} P. Chen, C. Tian;
\emph{Infinitely many solutions for Schr\"odinger-Maxwell
equations with indefinite sign subquadratic potentials}, Appl. Math.
Comput., 226(1) (2014), 492-502.

\bibitem{Chr} D. N. Christodoulides, F. Lederer, Y. Silberberg;
\emph{Discretizing light behaviour in linear and
     nonlinear waveguide lattices}, Nature, 424 (2003), 817-823.

\bibitem{DeC} 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{DeLS} 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{DeLZ} X. Q. Deng, X. Liu, Y. B. Zhang, H. P. Shi;
\emph{Periodic and subharmonic solutions for a 2$n$th-order
difference equation involving $p$-Laplacian}, Indag. Math. (N.S.),
24(5) (2013), 613-625.

\bibitem{Fl} J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides;
\emph{Observation of two-dimensional discrete solitons in optically
induced nonlinear photonic lattices}, Nature, 422  (2003), 147-150.

\bibitem{Gu} C. J. Guo, D. O'Regan, Y. T. Xu, R. P. Agarwal;
\emph{Homoclinic orbits for a singular second-order
    neutral differential equation}, J. Math. Anal. Appl., 366(2) (2010), 550-560.

\bibitem{Hu} M. H. Huang, Z. Zhou;
\emph{On the existence of ground state solutions of the periodic
discrete coupled nonlinear Schr\"odinger lattice}, J. Appl. Math.,
2013 (2013), 1-8.

\bibitem{Li1} 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{Li2} X. Liu, Y. B. Zhang, H. P. Shi;
\emph{Nonexistence and existence results for a 2$n$th-order discrete
mixed boundary value problem}, Rev. R. Acad. Cienc. Exactas
F\'{\i}s. Nat. Ser. A Math. RACSAM, 109(2) (2015), 303-314.

\bibitem{Li3} 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{Li4} 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{Mai} A. Mai, Z. Zhou;
\emph{Ground state solutions for the periodic discrete nonlinear
Schr\"odinger equations with superlinear nonlinearities}, Abstr.
Appl. Anal., 2013  (2013), 1-11.

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

\bibitem{Pa} A. Pankov;
\emph{Gap solitons in periodic discrete nonlinear Schr\"odinger
equations}, Nonlinearity, 19(1) (2006), 27-41.

\bibitem{Pe} T. Peil, A. Peterson;
\emph{Asymptotic behavior of solutions of a two-term
     difference equation}, Rocky Mountain J. Math., 24(1) (1994), 233-251.

\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{Sha} A. N. Sharkovsky, Y. L. Maistrenko, E. Y. Romanenko;
\emph{Difference Equations and Their Applications}, Kluwer Academic
Publishers: Dordrecht, 1993.

\bibitem{Shi1} H. P. Shi, Y. B. Zhang;
\emph{Standing wave solutions for the discrete nonlinear
Schr\"odinger equations with indefinite sign subquadratic
potentials}, Appl. Math. Lett., 58 (2016), 95-102.

\bibitem{Shi2} H. P. Shi, Y. B. Zhang;
\emph{Existence of breathers for discrete nonlinear Schr\"odinger
equations}, Appl. Math. Lett., 50 (2015), 111-118.

\bibitem{Shi3} H. P. Shi, H. Q. Zhang;
\emph{Existence of gap solitons in periodic discrete nonlinear
Schr\"odinger equations}, J. Math. Anal. Appl., 361(2) (2010),
411-419.

\bibitem{Ta} X. H. Tang;
\emph{Infinitely many solutions for semilinear Schr\"odinger
equations with sign-changing potential and nonlinearity}, J. Math.
Anal. Appl., 401(1) (2013), 407-415.

\bibitem{Ya} 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{Zha} G. P. Zhang;
\emph{Breather solutions of the discrete nonlinear Schr\"odinger
equations with unbounded potentials}, J. Math. Phys., 50 (2009),
013505.

\bibitem{Zho} Z. Zhou, J. S. Yu., Y. M. Chen;
\emph{On the existence of gap solitons in a periodic discrete
nonlinear Schr\"odinger equation with saturable nonlinearity},
Nonlinearity, 23(7) (2010), 1727-1740.

\end{thebibliography}

\end{document}