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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 134, 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-2016/134\hfil Higher-order nonlinear difference equations]
{Existence of periodic solutions for higher-order nonlinear difference equations}

\author[J. Leng \hfil EJDE-2016/134\hfilneg]
{Jianhua Leng}

\address{Jianhua Leng \newline
School of Mathematical and Computer Science,
Yichun University,
Yichun 336000, China}
\email{ljh8375@163.com}

\thanks{Submitted April 23, 2016. Published June 7, 2016.}
\subjclass[2010]{39A23, 47J22}
\keywords{Existence; periodic solutions; higher order; difference equation;
\hfill\break\indent critical point theory}

\begin{abstract}
 In this article, we study a higher-order nonlinear difference equation.
 By using critical point theory, we establish sufficient conditions for
 the existence of periodic solutions.
\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, have attracted the interest of many researchers
in the past twenty years since they provided a natural description
of several discrete models. Such discrete models
occur in numerous settings and forms, both in mathematics and in its applications
to computer science, economics, neural
networks, ecology, cybernetics, biological systems, optimal control,
and population dynamics.
These studies  cover many of the branches of difference equations, such as
stability, attractivity, periodicity, oscillation,
 homoclinic orbits, and boundary value problems
\cite{Ag, CaY, ChF, ChT, DeLZS,GuY1,GuY2,Hu,HuH,LiZS1,LiZS2,LiZS3,
LiZSD1, LiZSD2,Mi,Sh1,Sh2,ShLZD,ShZha,ShZho,ZhYC}.
Only a few papers discuss the  periodic solutions of higher-order difference
equations. Therefore, it is worthwhile to explore this topic.

 Let $\mathbb{N}$, $\mathbb{Z}$ 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 $a<b$.
 Let the symbol * denote the transpose of a vector.
 Moreover, for all $n\in \mathbb{N}$, $|\cdot|$ denotes the Euclidean norm
 in $\mathbb{R}^n$ defined by
 $$
|X|=\Big(\sum_{i=1}^nX_i^2\Big)^{1/2},\quad \forall
X=(X_1,X_2,\dots,X_n)\in\mathbb{R}^n.
$$
This article considers the higher order nonlinear difference equation
\begin{equation} \label{e1.1}
 \sum_{i=0}^nr_i(X_{k-i}+X_{k+i})+f(k,X_{k+\Gamma},\dots,X_k,\dots,X_{k-\Gamma})=0,
\quad n\in \mathbb{N},\ k\in \mathbb{Z},
\end{equation}
where $r_i$ is real valued for  $i\in \mathbb{Z}$, $\Gamma$ is a
nonnegative integer, $m$ is a  positive integer,
 $f=(f_1,f_2,\dots,f_m)^\ast\in C(\mathbb{R}^{2\Gamma+2}
\times \mathbb{R}^m,\mathbb{R})$,
 $f(k,Y_\Gamma,\dots,Y_0,\dots,Y_{-\Gamma})$ is $T$-periodic in $k$
for a given positive integer $T$.

 As usual, a solution ${X_k}$ of \eqref{e1.1} is said to be periodic of period
 $T$ if
$$
X_{k+T}=X_k,\quad \forall k\in \mathbb{Z}.
$$

 If $m=1$, $n=1$, $\Gamma=1$, $r_0=-1$, $r_1=1$, then \eqref{e1.1} can be reduced
to the  second-order difference equation
\begin{equation} \label{e1.2}
 \Delta^2 u_{k-1}=f(k,u_{k+1},u_k,u_{k-1}),\quad  k\in \mathbb{Z}.
\end{equation}
This equation  can be seen as an analogue discrete form of the
second-order functional differential equation
\begin{equation} \label{e1.3}
 \frac{d^2u(t)}{dt^2}=f(t,u(t+1),u(t),u(t-1)),\quad t\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,
periodic solutions and homoclinic orbits
 of functional differential equations, see \cite{GuRA,GuRWA,SmW}.

Migda \cite{Mi} in 2004 studied the existence of nonoscillatory solutions of a
higher order linear difference  equation of the form,
\begin{equation} \label{e1.4}
 \Delta^mu_k+\delta a_ku_{k+1}=0,\quad k\in \mathbb{Z}.
\end{equation}

 In 2007, Cai and Yu \cite{CaY} obtained some criteria for the
 existence of periodic solutions of a 2$n$th-order difference equation
\begin{equation} \label{e1.5}
 \Delta^n(r_{k-n}\Delta^nu_{k-n})+f(k,u_k)=0,\quad  n\in \mathbb{Z}(3),\;
 k\in \mathbb{Z},
\end{equation}
 by using the critical point theory.

Shi and Zhang \cite{ShZha} considered the existence of periodic solutions for the
 2$n$th-order nonlinear difference equation
\begin{equation} \label{e1.6}
 \Delta^n(r_{k-n}\Delta^nu_{k-n})=(-1)^nf(k,u_{k+1},u_k,u_{k-1}),\quad
 n\in \mathbb{Z}(3),\; k\in \mathbb{Z},
\end{equation}
by using the Saddle Point Theorem in combination with variational
 technique. \eqref{e1.6} can be seen a special form of system \eqref{e1.1}
with $m=1$ and  $\Gamma=1$.

When the nonlinear term of \eqref{e1.6} is neither superlinear nor sublinear, Xia,
 Zhang and Shi \cite{LiZS3} obtained some criteria for the existence and
multiplicity of periodic and subharmonic  solutions of \eqref{e1.6}.

If $\Gamma=0$, Hu \cite{Hu} in 2014  and Hu, Huang \cite{HuH} in 2008  applied the critical
point theorem and Lyapunov-Schmidt  reduction respectively to prove the
existence of periodic solution of a higher order difference equation as the
 type
\begin{equation} \label{e1.7}
 \sum_{i=0}^nr_i(X_{k-i}+X_{k+i})+f(k,X_k)=0,\quad n\in \mathbb{N},\; k\in \mathbb{Z}.
\end{equation}

Fixed point theorems in cones have been used 
widely for the existence of periodic solutions of
difference equations, see \cite{Ag}.
Also critical point theory which is a powerful tool have been used for 
differential equations, see \cite{GuRA,GuRWA,GuRXA,MaW,Ra}.
Only since 2003, critical point theory has been employed to
establish sufficient conditions on the existence of periodic solutions of difference
equations. Compared to first-order or second-order difference equations,
 the study of higher-order equations has received considerably less
 attention; see \cite{Ag, CaY, ChF, ChT,DeCS,DeLZS,DeSX,GuY1, GuY2,Hu, HuH,LiZS1,
LiZS2, LiZS3,LiZSD1,LiZSD2,Mi,Sh1,Sh2,ShLZD,ShZha,ShZho,ZhYC}.
 However, to the best of our knowledge, results obtained in the literature on
 the periodic solutions of \eqref{e1.1} are very scarce. Since $f$ in 
\eqref{e1.1} depends on $X_{k+\Gamma}, \dots,
 X_{k}, \dots, X_{k-\Gamma}$, the traditional ways of
 establishing the functional in \cite{DeCS,GuY1, GuY2,Hu, HuH,ZhYC}
are not applicable to our case.
The main purpose of this article is to establish sufficient conditions for
the existence of periodic solutions to \eqref{e1.1}. 
Also some  nonexistence conditions of nontrivial
periodic solutions to \eqref{e1.1} are also presented. 
We remark that such results are scarce  in the literature. 
 On the one hand, we  demonstrate the usefulness
of critical point theory in the study of the existence of periodic solutions of
 difference equations.
On the other hand, we  extend  existing results, as stated in Remarks \ref{rmk1.2} 
and \ref{rmk1.3}.
The motivation for the present work stems from the recent papers
\cite{DeLZS,LiZS3,ShZha}.
For basic knowledge of variational methods, the reader is referred to
\cite{MaW,Ra}.

In this article we use the following hypotheses:
\begin{itemize}
\item[(H1)]  $r_0+\sum_{s=1}^n|r_s|\leq0$, and there exists
 $i\in\{1,2,\dots,T\}$ such that
\[
\sum_{s=0}^nr_s\cos\frac{2is\pi}{T}=0;
\]

\item[(H2)] there exists a function
$F(t,Y_\Gamma,\dots,Y_0)\in C^1(\mathbb{R}^{\Gamma+2}\times
\mathbb{R}^m,\mathbb{R})$ such that
\begin{gather*}
F(t+T,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)] there exists a constant $K_0>0$ for all
$(t,Y_\Gamma,\dots,Y_0)\in \mathbb{R}^{\Gamma+2}$ such that
 $$
\big|\frac{\partial F(t,Y_\Gamma,\dots,Y_0)}{\partial Y_j}\big|
\leq K_0,\ j=1,2,\dots,\Gamma;
$$

\item[(H4)] $F(t,Y_\Gamma,\dots,Y_0)\to+\infty$ uniformly for
$t\in \mathbb{R}$ as $\sqrt{|Y_\Gamma|^2+\dots+|Y_0|^2}\to+\infty$.
\end{itemize}

 \begin{theorem} \label{thm1.1}
 Assume {\rm (H1)--(H4)} and  that $T\geq 2n+1$.
 Then \eqref{e1.1} has at least one $T$-periodic solution.
\end{theorem}

\begin{remark} \label{rmk1.2}\rm
 Assumption (H3) implies that there exists
 a constant $K_1>0$ such that
\begin{itemize}
\item[(H3')] $|F(t,Y_\Gamma,\dots,Y_0)|\leq K_1+K_0(|Y_\Gamma|+\dots+|Y_0|)$
 for all $(t,Y_\Gamma,\dots,Y_0)\in \mathbb{R}^{\Gamma+2}$.
\end{itemize}
\end{remark}

\begin{remark} \label{rmk1.3} \rm
 Theorem \ref{thm1.1} extends \cite[Theorem 1.1]{GuY2} which
 is the special the case when $m=1$, $n=1$, $\Gamma=0$,
 $r_0=-1$ and $r_1=1$.
\end{remark}

\begin{theorem} \label{thm1.4}
 Suppose that {\rm (H2)} and
 the following assumptions are satisfied:
\begin{itemize}
\item[(H1')]  $-r_0+\sum_{s=1}^n|r_s|>0$;

\item[(H5)] $Y_0f(t,Y_\Gamma,\dots,Y_0,\dots,Y_{-\Gamma})>0$,
for $Y_0\neq0$ and all $t\in\mathbb{R}$.
\end{itemize}
 Then \eqref{e1.1} has no nontrivial $T$-periodic solution.
\end{theorem}

The rest of this article 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 periodic
 solutions of \eqref{e1.1} into that of the existence of critical points
of the corresponding  functional.
In Section 3, we shall present some lemmas which will play important roles in
 the proofs of our main results.
In Section 4, we shall complete the proof of the  results by using the critical
point method.

\section{Variational structure}

To apply the critical point theory to study the existence of periodic solutions
of equation \eqref{e1.1},  we shall construct suitable variational structure.
At first, we shall state some basic notation
 and lemmas which will be used in the proofs of our main results.

 Let $S$ be the set of sequences $X=(\dots,X_{-k},\dots,X_{-1},X_0,X_1,\dots,X_k,
 \dots)=\{X_k\}_{k=-\infty}^{+\infty}$,
 where $X_k=(X_{k,1},X_{k,2},\dots,X_{k,m})\in\mathbb{R}^m$.

 For any $X,Y\in S$, $a,b\in \mathbb{R}$, $aX+bY$ is defined by
 $$
aX+bY:=\{aX_k+bY_k\}_{k=-\infty}^{+\infty}.
$$
 Then $S$ is a vector space.
For any positive integer $T$, we define a subspace of $S$ by
 $$
E_{T}=\{X\in S:X_{k+T}=X_k,\;\forall k\in \mathbb{Z}\}.
$$
This subspace is equipped with the inner product
\begin{equation} \label{e2.1}
 \langle X,Y\rangle :=\sum^{T}_{j=1}X_j\cdot Y_j,\ \forall X,Y\in E_{T},\ \
\end{equation}
 and the norm
\begin{equation} \label{e2.2}
 \|X\|:=\Big(\sum^{T}_{j=1}|X_j|^2\Big)^{1/2}\,.
\end{equation}
where $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^m$, and
$X_j\cdot Y_j$ denotes the usual scalar product in $\mathbb{R}^m$.

We define the linear map $M: E_T\to \mathbb{R}^{mT}$ by
\begin{equation} \label{e2.3}
 MX:=\big(X_{1,1},\dots,X_{T,1},X_{1,2},\dots,X_{T,2},
\dots,X_{1,m},\dots,X_{T,m}\big)^\ast,
\end{equation}
 where $X=\{X_k\}$, $X_k=(X_{k,1},X_{k,2},\dots,X_{k,m})^\ast$,
$k\in\mathbb{Z}(1,T)$.
It is easy to see that the map $M$ defined in \eqref{e2.3} is a linear
homeomorphism with $\|X\|=|MX|$, and
 $(E_{T},\langle\cdot,\cdot\rangle)$ is a Hilbert space,
 which can be identified with $\mathbb{R}^{mT}$.

For $X\in E_{T}$, define the functional $J$ on $E_{T}$
 as follows
 $$
J(X):=\frac{1}{2}\sum_{k=1}^{T}\sum_{i=0}^nr_i(X_{k-i}+X_{k+i})X_k
 +\sum_{k=1}^{T}F(k,X_{k+\Gamma},\dots,X_k).
$$
Since $E_{T}$ is linearly homeomorphic to $\mathbb{R}^{mT}$, $J$ can
be viewed as a continuously differentiable functional defined on a
finite dimensional Hilbert space. That is, $J\in C^1(E_T,\mathbb{R})$.
Furthermore, $J'(X)=0$ if and only if
 $$
\frac{\partial J(X)}{\partial X_{k,l}}=0,\quad
 l\in\mathbb{Z}(1,m),\; k\in\mathbb{Z}(1,T).
$$
If we define $X_0:= X_T$, then
\[
\frac{\partial J(X)}{\partial X_{k,l}}
=\sum_{i=0}^nr_i(X_{k-i,l}+X_{k+i,l})+f_l(k,X_{k+\Gamma},\dots,X_k,
 \dots,X_{k-\Gamma}),
\]
for all $ l\in\mathbb{Z}(1,m)$ and $k\in\mathbb{Z}(1,T)$.
 Therefore, $X\in E_T$ is a critical point of $J$, i.e.,
$J'(X)=0$ if and only if
\[
\sum_{i=0}^nr_i(X_{k-i,l}+X_{k+i,l})+f_l(k,X_{k+\Gamma},\dots,
X_k,\dots,X_{k-\Gamma})=0,
\]
for all $ l\in\mathbb{Z}(1,m)$ and $k\in\mathbb{Z}(1,T)$.
That is,
 $$
\sum_{i=0}^nr_i(X_{k-i}+X_{k+i})+f(k,X_{k+\Gamma},\dots,
X_k,\dots,X_{k-\Gamma})=0,\quad  k\in\mathbb{Z}(1,T).
$$

On the other hand, $\{X_k\}_{k\in\mathbb{Z}}\in E_T$ is $T$-periodic in $k$ and
 $f(k,Y_\Gamma,\dots,Y_0,\dots,Y_{-\Gamma})$ is $T$-periodic in $k$.
 So $X\in E_T$ is a critical point of $J$ if and only if
 $$
\sum_{i=0}^nr_i(X_{k-i}+X_{k+i})+f(k,X_{k+\Gamma},\dots,X_k,\dots,
X_{k-\Gamma})=0,\quad \forall k\in\mathbb{Z}.
$$
Thus, we reduce the problem of finding $T$-periodic solutions of \eqref{e1.1}
to that of seeking critical points of the functional $J$ in $E_T$.

For all $X\in E_{T}$ and $T\geq2n+1$, $J$ can be rewritten as
$$
J(X)=-\frac{1}{2}\langle DMX,MX\rangle+\sum_{k=1}^{T}F(k,X_{k+\Gamma},\dots,X_k),
$$
where
$X=\{X_k\}\in E_T$, $X_k=(X_{k,1},X_{k,2},\dots,X_{k,m})^\ast$,
 $k\in\mathbb{Z}(1,T)$, and
\begin{gather*}
D= \begin{pmatrix}
 P& & & 0\\
 & P& & \\
 & & \ddots& \\
 0& & & P\\
 \end{pmatrix}_{mT\times mT},
\\
-P= \begin{pmatrix}
 2r_0& r_1& \dots& r_n& 0&\dots& 0&r_n&\dots& r_1 \\
r_1& 2r_0&r_1   & & & & & & &\vdots
\\
\vdots&\ddots &\ddots&\ddots\\
r_n& & & & & & & & & r_{n-1}\\
0& & & & & & & & & r_n\\
\vdots\\
\vdots\\
0 & & & & & & & & & r_n\\
r_n& & & & & & & & & r_{n-1}\\
\vdots  & & & & &\ddots &\ddots &\ddots&\ddots& r_1 \\
r_1&\dots&r_n &0&\dots&0&r_n&\dots &r_1&2r_0
\end{pmatrix}
\end{gather*}
is a $T\times T$ matrix. Assume that the eigenvalues of $P$ are
$\lambda_1,\lambda_2,\dots,\lambda_{T}$ respectively, and $P$ is
a circulant matrix \cite{JiW} denoted by
\[
 P:=\operatorname{Circ}
\big\{-2r_0,-r_1,-r_2,\dots,-r_n,0,\dots,0,-r_n,-r_{n-1},\dots,-r_2,-r_1\big\}.
\]
By \cite{JiW}, the eigenvalues of $P$ are
\begin{equation} \label{e2.4}
\begin{aligned}
 \lambda_j
&=-2r_0-\sum_{s=1}^nr_s\{\exp i\frac{2j\pi}{T}\}^s
 -\sum_{s=1}^nr_s\{\exp i\frac{2j\pi}{T}\}^{T-s}\\
&=-2\sum_{s=0}^nr_s\cos\big(\frac{2js\pi}{T}\big),
\end{aligned}
\end{equation}
where $j=1,2,\dots,T$.
 By \eqref{e2.4}, we know that
\begin{equation} \label{e2.5}
 -2r_0-2\sum_{s=1}^n|r_s|\leq\lambda_j\leq-2r_0+2\sum_{s=1}^n|r_s|,\quad
 j=1,2,\dots,T.
\end{equation}
It follows from (H1) that the matrix $P$ is semi-positive and $\lambda_j\geq0$
 for all $j\in \mathbb{Z}(1,T)$.
Denote
\begin{gather*}
\lambda_{\max}=\max\{\lambda_j:\lambda_j\neq0, j=1,2,\dots,T\},\\
\lambda_{\rm min}=\min\{\lambda_j:\lambda_j\neq0, j=1,2,\dots,T\}.
\end{gather*}
Let
 $$
H=\ker DM=\{X\in E_{T}|DMX=0\in \mathbb{R}^{mT}\}.
$$
Then
$$
H=\{X\in E_{T}:X=\{B\},\, B\in \mathbb{R}^m\}.
$$
Let $G$ be the direct orthogonal complement of $E_{T}$ to $W$,
 i.e., $E_{T}=G\oplus H$. For convenience, we identify $X\in E_{T}$ with
 $X=(X_1,X_2,\dots,X_{T})^\ast$.

\section{Lemmas}

In this section, we give two lemmas which will play important roles in
the proofs of our main results.

Let $E$ be a real Banach space, $J\in C^1(E,\mathbb{R})$, i.e., $J$ is a
continuously Fr\'{e}chet-differentiable functional
defined on $E$. $J$ is said to satisfy the Palais-Smale
 condition (PS condition for short) if any sequence
 $\{X^{(n)}\}_{n\in\mathbb{N}}\subset E$ for which $\{J(X^{(n)})\}_{n\in\mathbb{N}}$ is bounded and
 $J' (X^{(n)})\to 0\ (n\to \infty)$ 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}[{Saddle Point Theorem \cite{MaW,Ra}}] \label{lem3.1}
Let $E$ be a real Banach space,
 $E=E_1\oplus E_2$, where $E_1\neq \{0\}$ and is finite dimensional. Suppose that
 $J\in C^1(E,\mathbb{R})$ satisfies the PS condition and
\begin{itemize}
\item[(H6)] there exist constants $\sigma,\ \rho>0$ such that
$J|_{\partial B_\rho\cap E_1}\leq  \sigma$;

\item[(H7)]  there exists $e\in B_\rho\cap E_1$ and a constant $\omega\geq \sigma$
 such that $J_{e+E_2}\geq \omega$.
\end{itemize}
 Then $J$ possesses a critical value $c\geq\omega$,
 where
 $$
c=\inf_{h\in \Gamma}\max_{u\in B_\rho\cap E_1} J(h(u)),\,
 \Gamma =\{h\in C(\bar{B}_\rho\cap E_1,E)|h|_{\partial B_\rho\cap
 E_1}=\operatorname{id}\}
$$
 and $\operatorname{id}$ denotes the identity operator.
\end{lemma}

\begin{lemma} \label{lem3.2}
 Assume that {\rm (H1)--(H4)} are satisfied. Then $J$ satisfies the PS condition.
\end{lemma}

\begin{proof}
 Let $\{X^{(n)}\}_{n\in\mathbb{N}}\subset E_T$ be such that
 $\{J(X^{(n)})\}_{n\in\mathbb{N}}$ is bounded and $J'(X^{(n)})\to 0$ as
 $n\to \infty.$\ Then there exists a positive constant $K_2$
 such that $|J(X^{(n)})|\leq K_2$.

 Let $X^{(n)}=V^{(n)}+W^{(n)}\in G+H$. For $n$ large enough, since
\begin{align*}
-\|X\|
&\leq  \langle J'(X^{(n)}),MX\rangle\\
& =-\langle DM(X^{(n)}),MX\rangle
 +\sum_{k=1}^{T}f(k,X_{k+\Gamma}^{(n)},\dots,X_k^{(n)},\dots,X_{k-\Gamma}^{(n)})X_k,
\end{align*}
 combining  (H3) with (H4), we have
\begin{align*}
\langle DM(X^{(n)}),MV^{(n)}\rangle
&\leq  \sum_{k=1}^{T}f(k,X_{k+\Gamma}^{(n)},\dots,X_k^{(n)},\dots,
 X_{k-\Gamma}^{(n)})V_k^{(n)} +\|V^{(n)}\| \\
&\leq (\Gamma+1)K_0\sum_{k=1}^{T}|V_k^{(n)}|+\|V^{(n)}\| \\
&\leq \big[(\Gamma+1)K_0\sqrt{T}+1\big]\|V^{(n)}\|.
\end{align*}
 On the other hand, we know that
 $$
\langle DM(X^{(n)}),MV^{(n)}\rangle
 =\langle DM(V^{(n)}),MV^{(n)}\rangle
 \geq\lambda_{\rm min}\|V^{(n)}\|^2.
$$
Thus, we have
$$
\lambda_{\rm min}\|V^{(n)}\|^2\leq [(\Gamma+1)K_0\sqrt{T}+1]\|V^{(n)}\|.
$$
 The above inequality implies that $\{V^{(n)}\}$ is bounded.

Next, we shall prove that $\{W^{(n)}\}$ is bounded. Since
\begin{align*}
K_2
&\geq J(X^{(n)})
=-\frac{1}{2}\langle DMX^{(n)},MX^{(n)}\rangle
 +\sum_{k=1}^{T}F\big(k,X_{k+\Gamma}^{(n)},\dots,X_k^{(n)}\big) \\
&=-\frac{1}{2}\langle DMV^{(n)},MV^{(n)}\rangle
 +\sum_{k=1}^{T}\Big[F(k,X_{k+\Gamma}^{(n)},\dots,X_k^{(n)})\\
&\quad -F\Big(k,W_{k+\Gamma}^{(n)},\dots,W_k^{(n)}\Big)\Big]
+\sum_{k=1}^{T}F(k,W_{k+\Gamma}^{(n)},\dots,W_k^{(n)}),
\end{align*}
we obtain
\begin{align*}
&\sum_{k=1}^{T}F\Big(k,W_{k+\Gamma}^{(n)},\dots,W_k^{(n)}\Big)\\
&\leq K_2+\frac{1}{2}\langle DMV^{(n)},MV^{(n)}\rangle
 +\sum_{k=1}^{T}\big|F(k,X_{k+\Gamma}^{(n)},\dots,X_k^{(n)})\\
&\quad -F(k,W_{k+\Gamma}^{(n)},\dots,W_k^{(n)})\big|\\
&\leq K_2+\frac{1}{2}\lambda_{\max}\|V^{(n)}\|^2+\sum_{k=1}^{T}
 |\frac{\partial F(k,W_{k+\Gamma}^{(n)}+\theta V_{k+\Gamma}^{(n)},\dots,W_k^{(n)}+
 \theta V_k^{(n)})}{\partial Y_\Gamma} V_{k+\Gamma}^{(n)}\\
&\quad +\dots
 +\frac{\partial F(k,W_{k+\Gamma}^{(n)}
 +\theta V_{k+\Gamma}^{(n)},\dots,W_k^{(n)}
 +\theta V_k^{(n)})}{\partial Y_0} V_k^{(n)}| \\
&\leq K_2+\frac{1}{2}\lambda_{\max}\|V^{(n)}\|^2+(\Gamma+1)K_0\sqrt{T}\|V^{(n)}\|,
\end{align*}
where $\theta\in(0,1)$. It is not difficult to see that
$\{\sum_{k=1}^{T} F(k,W_{k+\Gamma}^{(n)},\dots,W_k^{(n)})\}$ is bounded.

 By (H4), $\{W^{(n)}\}$ is bounded. Otherwise, assume
 that $\|W^{(n)}\|\to+\infty$ as $i\to\infty$.
 Since there exist $B^{(n)}\in \mathbb{R}^m$, $n\in\mathbb{N}$, such that
 $W^{(n)}=(B^{(n)},B^{(n)},\dots,B^{(n)})^\ast\in E_{T}$, then
\[
\|W^{(n)}\|=(\sum_{k=1}^{T}|W_k^{(n)} |^2)^{1/2}
 =(\sum_{k=1}^{T}|B^{(n)}|^2)^{1/2}
 =\sqrt{T}|B^{(n)}|\to+\infty
\]
as $n\to\infty$.
Since $F(k,W_{k+\Gamma}^{(n)},\dots,W_k^{(n)})
 =F(k,B_{k+\Gamma}^{(n)},\dots,B_k^{(n)})$, it follows that
 $F(k,W_{k+\Gamma}^{(n)},\dots,W_k^{(n)})\to+\infty$. This
 contradicts  that
$\{\sum_{k=1}^{T}F(k,W_{k+\Gamma}^{(n)},\dots,W_k^{(n)})\}$
 is bounded. Thus the PS condition is satisfied.
\end{proof}

\section{Proof of main results}

 In this Section, we  prove  Theorems \ref{thm1.1} and \ref{thm1.4},
 by using the critical point method.

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 By Lemma \ref{lem3.2}, we know that $J$ satisfies the PS
 condition. To prove Theorem \ref{thm1.1} by using the Saddle
 Theorem, we shall prove the conditions (H6) and (H7).

 From \eqref{e2.5} and (H3'), for any $V\in G$,
\begin{align*}
J(V)&=-\frac{1}{2}\langle DMV,MV\rangle+\sum_{k=1}^{T}F(k,V_{k+\Gamma},\dots,V_k)\\
&\leq-\frac{1}{2}\lambda_{\rm min}\|V\|^2+TK_1+K_0\sum_{k=1}^{T}
 (|V_{k+\Gamma}|+\dots+|V_k|) \\
&\leq-\frac{1}{2}\lambda_{\rm min}\|V\|^2+TK_1+(\Gamma+1)K_0\sqrt{T}\|V\|\to-\infty
\end{align*}
as $\|V\|\to+\infty$.
 Therefore, it is easy to see that  (H6) is satisfied.

 The rest of the proof is similar to that of \cite[Theorem 1.1]{ShZha},
 but for the sake of completeness,  we give the details.

 In the following, we shall verify the condition (H7). For any $W\in H$,
 $W=(W_1,W_2,\dots,W_{T})^\ast$,
 there exists $B\in\mathbb{R}^m$ such
 that $W_k=B$, for all $k\in\mathbb{Z}(1,T)$.
By (H4), we know that there exists a
 constant $C_0>0$ such that $F(k,B,\dots,B)>0$ for $k\in\mathbb{Z}$ and
 $|B|>\frac{C_0}{\sqrt{\Gamma+1}}$.
Let
$K_3=\min\big\{ F(k,B,\dots,B) :k\in\mathbb{Z},\,|B|\leq  C_0/\sqrt{\Gamma+1}\big\}$,
$K_4=\min\{0,K_3\}$. Then
 $$
F(k,B,\dots,B)\geq K_4,\quad \forall (k,B,\dots,B)\in \mathbb{Z}
\times\mathbb{R}^{\Gamma+1}.
$$
 So we have
 $$
J(W)=\sum_{k=1}^{T}F(k,W_{k+\Gamma},\dots,W_k)
=\sum_{k=1}^{T}F(k,B,\dots,B)\geq  TK_4,\quad \forall W\in H.
$$
Conditions of (H6) and (H7) are satisfied.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.4}]
 It follows from (H1')  that the matrix $P$ is negative semi-positive
and $\lambda_j\leq0$ for all $j\in \mathbb{Z}(1,T)$.
 For the sake of contradiction, assume that \eqref{e1.1} has a nontrivial
 $T$-periodic solution. Then $J$ has a nonzero critical point $X^\star$. Since
 $$
\frac{\partial J}{\partial X_k^\star}=
 \sum_{i=0}^nr_i(X_{k-i}^\star+X_{k+i}^\star)+f(k,X_{k+\Gamma}^\star,
\dots,X_k^\star,\dots,X_{k-\Gamma}^\star),
$$
we obtain
\begin{equation} \label{e4.1}
\begin{aligned}
&\sum_{k=1}^{T}f(k,X_{k+\Gamma}^\star,\dots,X_k^\star,\dots,
X_{k-\Gamma}^\star)X_k^\star\\
&=-\sum_{k=1}^{T}\sum_{i=0}^nr_i(X_{k-i}^\star+X_{k+i}^\star)X_k^\star\\
&=\langle DMX^\star,MX^\star\rangle\leq0.
\end{aligned}
\end{equation}
On the other hand, from (H5) it follows that
\begin{equation} \label{e}
 \sum_{i=1}^{T}f(k,X_{k+\Gamma}^\star,\dots,X_k^\star,\dots,
X_{k-\Gamma}^\star)X_k^\star>0.
\end{equation}
 This contradicts \eqref{e4.1} and hence the proof is complete.
\end{proof}

 \begin{thebibliography}{00}

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

\bibitem{CaY} 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{ChF} P. Chen, H. Fang;
\emph{Existence of periodic and subharmonic solutions for
second-order $p$-Laplacian difference equations}, Adv. Difference
Equ., 2007 (2007), 1-9.

\bibitem{ChT} P. Chen, X. Tang;
\emph{Existence and multiplicity of homoclinic orbits for
2$n$th-order nonlinear difference equations containing both many
advances and retardations}, J. Math. Anal. Appl., 381(2) (2011),
485-505.

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

\bibitem{GuRA} C. J. Guo, D. O'Regan, R. P. Agarwal;
\emph{Existence of multiple periodic solutions for a class of
 first-order neutral differential equations}, Appl. Anal. Discrete Math., 5(1) (2011), 147-158.

\bibitem{GuRWA} C. J. Guo, D. O'Regan, C. J. Wang, R. P. Agarwal;
\emph{Existence of homoclinic orbits of superquadratic first-order
Hamiltonian systems}, Z. Anal. Anwend., 34(1) (2015), 27-41.

\bibitem{GuRXA} C. J. Guo, D. O'Regan, Y. T. Xu, R. P. Agarwal;
\emph{Existence of infinite periodic solutions for a class of
first-order delay differential equations}, Funct. Differ. Equ.,
\textbf{19} (2012), 115-123.

\bibitem{GuY1} Z. M. Guo, J. S. Yu;
\emph{Existence of periodic and subharmonic
 solutions for second-order superlinear difference equations}, Sci. China Math., 46(4) (2003), 506-515.

\bibitem{GuY2} 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{Hu} R. H. Hu;
\emph{Multiplicity of periodic solutions for a higher order
difference equation}, Abstr. Appl. Anal., 2014 (2014), 1-7.

\bibitem{HuH} R. H. Hu, L.H. Huang;
\emph{Existence of periodic solutions of a higher order difference
system}, J. Korean Math. Soc., 45(2) (2008), 405-423.

\bibitem{JiW} M. Y. Jiang, Y. Wang;
\emph{Solvability of the resonant 1-dimensional periodic
$p$-Laplacian equations}, J. Math. Anal. Appl., 370(1) (2010),
107-131.

\bibitem{LiZS1} X. Liu, Y. B. Zhang, H. P. Shi;
\emph{Periodic and subharmonic solutions for fourth-order
$p$-Laplacian difference equations}, Electron. J. Differential
Equations, 2014(25) (2014), 1-12.

\bibitem{LiZS2} 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{LiZS3} X. Liu, Y. B. Zhang, H. P. Shi;
\emph{Periodic and subharmonic solutions for a 2$n$th-order
nonlinear difference equation}, Hacet. J. Math. Stat., 44(2) (2015),
357-368.

\bibitem{LiZSD1} 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{LiZSD2} X. Liu, Y. B. Zhang, H. P. Shi, X. Q. Deng;
\emph{Periodic solutions for fourth-order nonlinear functional
 difference equations}, Math. Methods Appl. Sci., 38(1) (2014), 1-10.

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

\bibitem{Mi} M. Migda;
\emph{Existence of nonoscillatory solutions of some higher order
difference equations}, Appl. Math. E-notes, 4(2) (2004), 33-39.

\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{Sh1} H. P. Shi;
\emph{Boundary value problems of second order nonlinear functional
difference equations}, J. Difference Equ. Appl., 16(9) (2010),
1121-1130.

\bibitem{Sh2} H. P. Shi;
\emph{Periodic and subharmonic solutions for second-order nonlinear
difference equations}, J. Appl. Math. Comput., 48(1-2) (2015),
157-171.

\bibitem{ShLZD} H. P. Shi, X. Liu, Y. B. Zhang, X. Q. Deng;
\emph{Existence of periodic solutions of fourth-order nonlinear
 difference equations}, Rev. R. Acad. Cienc. Exactas F\'{\i}s. Nat. Ser. A Math. RACSAM,
 108(2) (2014), 811-825.

\bibitem{ShZha} H. P. Shi, Y. B. Zhang;
\emph{Existence of periodic solutions for a 2$n$th-order nonlinear
difference equation}, Taiwanese J. Math., 20(1) (2016), 143-160.

\bibitem{ShZho} H. P. Shi, X. J. Zhong;
\emph{Existence of periodic and subharmonic solutions for second
order functional difference equations}, Acta Math. Appl. Sin. Engl.
Ser., 26(2) (2010), 229-240.

\bibitem{SmW} D. Smets, M. Willem;
\emph{Solitary waves with prescribed speed on
 infinite lattices}, J. Funct. Anal., 149(1) (1997), 266-275.

\bibitem{ZhYC} Z. Zhou, J. S. Yu, Y. M. Chen;
\emph{Periodic solutions of a 2$n$th-order nonlinear difference
equation}, Sci. China Math., 53(1) (2010), 41-50.

\end{thebibliography}

\end{document}