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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 25, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2014/25\hfil Periodic and subharmonic solutions]
{Periodic and subharmonic solutions for fourth-order $p$-Laplacian
difference equations}

\author[X. Liu, Y. Zhang, H. Shi \hfil EJDE-2014/25\hfilneg]
{Xia Liu, Yuanbiao Zhang, 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{Yuanbiao Zhang \newline
Packaging Engineering Institute, Jinan University, Zhuhai 519070,
China}
\email{abiaoa@163.com}

\address{Haiping Shi \newline
Modern Business and Management Department,
Guangdong Construction Vocational Technology Institute,
Guangzhou 510450, China}
\email{shp7971@163.com}

\thanks{Submitted November 17, 2013. Published January 14, 2014.}
\subjclass[2000]{39A11}
\keywords{Periodic and subharmonic solution; $p$-Laplacian;
difference equation; \hfill\break\indent discrete variational theory}

\begin{abstract}
 Using critical point theory, we obtain criteria for the
 existence and multiplicity of periodic and subharmonic solutions to
 fourth-order $p$-Laplacian difference equations. The proof is based
 on the Linking Theorem in combination with variational technique.
 Recent results in the literature are generalized and improved.
\end{abstract}

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

\section{Introduction}

Let $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ denote 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 $a<b$.
The symbol * denotes the transpose of a vector.

 In this paper, we consider the  forward and backward difference equation
\begin{equation}\label{e1.1}
\Delta^2\big(\gamma_{n-2}\varphi_p(\Delta^2u_{n-2})\big)
=f(n,u_{n+1},u_n,u_{n-1}),\quad n\in \mathbb{Z},
\end{equation}
where $\Delta$ is the forward difference operator
$\Delta u_n=u_{n+1}-u_n$, $\Delta^2 u_n=\Delta(\Delta u_n)$,
$\gamma_n$ is real valued for each $n\in \mathbb{Z}$,
$\varphi_p(s)$ is the $p$-Laplacian operator
$\varphi_p(s)=|s|^{p-2}s(1<p<\infty)$, 
$f\in C(\mathbb{Z}\times \mathbb{R}^3,\mathbb{R})$,
$\gamma_n$ and $f(n,v_1,v_2,v_3)$ are $T$-periodic in $n$ for a given positive
integer $T$.

We may think of \eqref{e1.1} as a discrete
analogue of the  fourth-order functional differential equation
\begin{equation}\label{e1.2}
\begin{gathered}
 \frac{d^2}{dt^2}\big[ \gamma(t)\varphi_p\big(\frac{d^2u(t)}{dt^2}\big)\big]
=f(t,u(t+1),u(t),u(t-1)),\quad t\in  \mathbb{R}.
\end{gathered}
\end{equation}
This equation includes the  equation
\begin{equation}\label{e1.3}
\begin{gathered}
 u^{(4)}(t)=f(t,u(t)),\quad t\in \mathbb{R},
\end{gathered}
\end{equation}
which is used to model deformations of elastic beams \cite{CH,Ra}.
Equations similar in structure to \eqref{e1.2} arise in the
study of the existence of solitary waves of lattice differential equations, 
see Smets and Willem \cite{TiDG}.


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, ecology, cybernetics,
etc. For the general background of difference equations, one can refer 
to monographs  \cite{Ag,GuY2}. 
Since the last decade, there has been much progress
on the qualitative properties of  difference equations, which included 
results on stability and attractivity \cite{ErXY,KoL,MaHS,ZhZ} and
results on oscillation and other topics, see 
\cite{Ag,AgPO1,AgPO2,AnAH,AvH,AvP,GuY1,GuY2,GuY3,He,JiCOA,LiL,LiuG,TiDG,TiG,YaL,
YuG,YuLG,ZhYC,ZhYG}.

 The motivation of this paper is as follows. It is well known that
critical point theory is a powerful tool that deals with the problems
of differential equations \cite{CH,ChT2,GuOXA,MaW,Ra}.
 Starting in 2003, critical 
point theory has been employed to establish sufficient
conditions on the existence of periodic solutions of difference
equations. Particularly, Guo and Yu \cite{GuOA,GuOXA,GuX}
and Shi et al. \cite{ShLZ}  studied the existence of periodic solutions 
of second-order nonlinear 
difference equations by using the critical  point theory. Compared to 
first-order or second-order difference equations,
the study of higher-order equations, and in particular, fourth-order equations, 
has received considerably less  attention 
(see, for example, \cite{Ag,ChF,ChT1,FaZ,LiL,PeR,PoS,ThA,YaL} 
and the references contained  therein). 
Yan, Liu \cite{YaL} in 1997 and Thandapani, Arockiasamy \cite{ThA}
 in 2001 studied  the  fourth-order difference equation 
\begin{equation}\label{e1.4}
 \Delta^2\big(\gamma_{n}\Delta^2u_{n}\big)+f(n,u_n)=0,\quad n\in \mathbb{Z},
\end{equation}
and obtained criteria for the oscillation and nonoscillation of solutions
for equation \eqref{e1.4}. In 2005, Cai, Yu and Guo \cite{CaYG} have obtained 
some criteria for the  existence of periodic solutions of the fourth-order 
difference equation
\begin{equation}\label{e1.5}
 \Delta^2\left(\gamma_{n-2}\Delta^2u_{n-2}\right)+f(n,u_n)=0,\quad
 n\in \mathbb{Z}.
\end{equation}
 In 1995, Peterson and Ridenhour considered the disconjugacy of
equation \eqref{e1.5} when $\gamma_{n}\equiv 1$ and 
$f(n,u_n)=q_nu_n$ (see \cite{PeR}).
 However, to the best of our knowledge, the
results on periodic solutions of fourth-order $p$-Laplacian difference 
equations are very scarce in the literature. 
We found that \cite{CaYG} is the only paper which deals with the
problem of periodic solutions to fourth-order difference
equation \eqref{e1.5}. Furthermore, since \eqref{e1.1} contains both advance 
and retardation, there are very few manuscripts dealing with this subject. 
The main purpose of this paper is to give some sufficient conditions
for the existence and multiplicity of periodic and subharmonic solutions to
fourth-order $p$-Laplacian difference equations. The main approach used in 
our paper is the variational technique and the Linking Theorem. 
In particular, our results not only generalize the results in the literature 
\cite{CaYG},  but also improve them. In fact, one can see the 
Remarks \ref{rmk1.4} and 
\ref{rmk1.9} for details. The motivation for the present work stems 
from the recent  papers in \cite{ChF,GuOA,GuX}.

Let
 $$
\underline{\gamma}=\min_{n\in \mathbb{Z}(1,T)}\{\gamma_n\},\quad
\overline{\gamma}=\max_{n\in \mathbb{Z}(1,T)}\{\gamma_n\}.
$$
Our main results read as follows.

\begin{theorem} \label{thm1.1}
Assume that the following hypotheses are satisfied:
\begin{itemize}
 \item[(F0)] $\gamma_n>0$ for all $n\in\mathbb{Z}$;
 \item[(F1)] there exists a functional $F(n,v_1,v_2)\in C^1(\mathbb{Z}
 \times \mathbb{R}^2,\mathbb{R})$ with $F(n,v_1,v_2)\geq0$ and it satisfies
\begin{gather*}
F(n+T,v_1,v_2)=F(n,v_1,v_2),\\
\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);
\end{gather*}

\item[(F2)] there exist constants $\delta_1>0$,
 $\alpha\in\big(0,\frac{\underline{\gamma}}{2^{{p}/{2}}p}(c_1/c_2)^p
 \lambda_{\rm min}^p\big)$ such that
 $$
 F(n,v_1,v_2)\leq \alpha\Big(\sqrt{v_1^2+v_2^2}\Big)^p, \quad
\text{for $n\in \mathbb{Z}$ and $v_1^2+v_2^2\leq \delta_1^2$};
$$

\item[(F3)]  there exist constants $\rho_1>0$, $\zeta>0$,
 $\beta\in\big(\frac{\bar{\gamma}}{2^{{p}/{2}}p}(c_2/c_1)^p
 \lambda_{\rm max}^p,+\infty\big)$ such that
$$
 F(n,v_1,v_2)\geq \beta\Big(\sqrt{v_1^2+v_2^2}\Big)^p-\zeta,\quad
\text{for $n\in \mathbb{Z}$ and  $v_1^2+v_2^2\geq \rho_1^2$},
$$
 where $c_1, c_2$ are constants which
 can be referred to \eqref{e2.4}, and $\lambda_{\rm min}, \lambda_{\rm max}$
are constants which
 can be referred to \eqref{e2.7}.
\end{itemize}
Then for any given positive integer $m>0$, Equation \eqref{e1.1} has 
at least three $mT$-periodic solutions.
\end{theorem}

\begin{remark} \label{rmk1.2} \rm 
By (F3) it is easy to see
 that there exists a constant $\zeta'>0$ such that
\begin{itemize}
\item[(F3')] 
$$
F(n,v_1,v_2)\geq \beta\Big(\sqrt{v_1^2+v_2^2}\Big)^p-\zeta',\quad
 \forall(n,v_1,v_2)\in  \mathbb{Z}\times \mathbb{R}^2.
$$
\end{itemize}
 As a matter of fact, let
 $\zeta_1=\max\big\{|F(n,v_1,v_2)-\beta\big(\sqrt{v_1^2+v_2^2}\big)^p+\zeta|:
 n\in \mathbb{Z}, v_1^2+v_2^2\leq \rho_1^2\}$,
$\zeta'=\zeta+\zeta_1$, we
 can easily get the desired result.
\end{remark}

\begin{corollary} \label{coro1.3}
 Assume that  {\rm (F0--(F3)} are satisfied. 
Then for any given positive integer $m>0$, \eqref{e1.1} has at least 
two nontrivial  $mT$-periodic solutions.
\end{corollary}

\begin{remark} \label{rmk1.4} \rm
The statement in in the above corollary is the same as 
 \cite[Theorem 1.1]{CaYG}
\end{remark}

\begin{theorem} \label{thm1.5}
Assume that {\rm (F0), (F1)} and the following conditions are satisfied:
\begin{itemize}
 \item[(F4)] $\lim_{\rho\to 0} \frac{F(n,v_1,v_2)} {\rho^p}=0$,
 $\rho=\sqrt{v_1^2+v_2^2}$ for all $(n,v_1,v_2)\in \mathbb{Z}
 \times \mathbb{R}^2$;

\item[(F5)] there exist constants $\theta>p$ and $a_1>0$, $a_2>0$ such that
 $$
F(n,v_1,v_2)\geq a_1\big(\sqrt{v_1^2+v_2^2}\big)^\theta-a_2,\quad
\forall (n,v_1,v_2)\in \mathbb{Z} \times \mathbb{R}^2.
$$
\end{itemize}
Then for any given positive integer $m>0$, Equation \eqref{e1.1} has at 
least three  $mT$-periodic solutions.
\end{theorem}

\begin{corollary} \label{coro1.6}
 Assume that {\rm (F0), (F1), (F4), (F5)} are satisfied. 
Then for any given positive integer $m>0$, Equation \eqref{e1.1} has at 
least two nontrivial  $mT$-periodic solutions.
\end{corollary}

If $f(n,u_{n+1},u_n,u_{n-1})=q_ng(u_n)$, then \eqref{e1.1} reduces to
the fourth-order nonlinear equation
\begin{equation}\label{e1.6}
 \Delta^2\left(\gamma_{n-2}\varphi_p(\Delta^2u_{n-2})\right)
=q_ng(u_n),\ n\in \mathbb{Z},
\end{equation}
where $g\in C(\mathbb{R},\mathbb{R}), q_{n+T}=q_n>0$, for all 
$n\in \mathbb{Z}$. Then, we have the following results.

\begin{theorem} \label{thm1.7}
Assume that {\rm (F0)} and the following hypotheses are
satisfied:
\begin{itemize}
\item[(G1)] there exists a functional $G(v)\in C^1(\mathbb{R},\mathbb{R})$ 
with $G(v)\geq0$ and it satisfies
 $$
\frac{d G(v)}{d v}=g(v);
$$

\item[(G2)] there exist constants $\delta_2>0$,
 $\alpha\in\big(0,\frac{\underline{\gamma}}{p}(c_1/c_2)^p  \lambda_{\rm min}^p\big)$
such that  $ G(v)\leq \alpha|v|^p$, for $|v|\leq \delta_2$;

\item[(G3)] there exist constants $\rho_2>0$, $\zeta>0$,
 $\beta\in\big(\frac{\bar{\gamma}}{p}(c_2/c_1)^p
 \lambda_{\rm max}^p,+\infty\big)$ such that
$$
 G(v)\geq \beta|v|^p-\zeta,\quad\text{for $|v|\geq \rho_2$},
$$
where $c_1, c_2$ are constants which
 can be referred to \eqref{e2.4}, and $\lambda_{\rm min},\lambda_{\rm max}$
are constants which  can be referred to \eqref{e2.7}.
\end{itemize}
 Then for any given positive integer $m>0$, Equation \eqref{e1.6} has at 
least three $mT$-periodic solutions.
\end{theorem}

\begin{corollary} \label{coro1.8}
 Assume that {\rm (F0), (G1)--(G3)} are satisfied. 
Then for any given positive integer $m>0$, Equation \eqref{e1.6} has
 at least two nontrivial  $mT$-periodic solutions.
\end{corollary}

\begin{remark} \label{rmk1.9} \rm
The statement of above corollary is the same as \cite[Theorem 1.2]{CaYG}.
\end{remark}

The rest of the paper 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 periodic solutions of
\eqref{e1.1} into that of the existence of critical points of the
corresponding functional. Some related fundamental results will also
be recalled. Then, in Section 3, we shall complete the proof of the
results by using the critical point method. Finally, in Section 4,
we shall give an example to illustrate the main result.

\section{Variational structure and some lemmas}

In order to apply the critical point theory, we shall establish the
corresponding
 variational framework for \eqref{e1.1} and give some basic notation
 and useful lemmas.
For the basic knowledge of variational methods, the reader is
referred to \cite{Gu,MaW,PaZ,Ra}.

 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 any $u,v\in S$, $a,b\in \mathbb{R}$, $au+bv$ is defined by
 $$
au+bv=\{au_n+bv_n\}_{n=-\infty}^{+\infty}.
$$
 Then $S$ is a vector space.
For any given positive integers $m$ and $T$, $E_{mT}$ is defined
 as a subspace of $S$ by
 $$
E_{mT}=\{u\in S:u_{n+mT}=u_n,\, \forall n\in \mathbb{Z}\}.
$$
 Clearly, $E_{mT}$ is isomorphic to $\mathbb{R}^{mT}$.
 $E_{mT}$ can be equipped with the inner product
\begin{equation}\label{e2.1}
 \left<u,v\right>=\sum^{mT}_{j=1}u_j v_j,\, \forall u,v\in E_{mT},
\end{equation}
 by which we introduce the norm 
\begin{equation}\label{e2.2}
 \|u\|=\Big(\sum^{mT}_{j=1}u_j^2\Big)^{1/2},\quad \forall u\in E_{mT}.
\end{equation}
 It is obvious that $E_{mT}$ with the inner product \eqref{e2.1} 
is a finite dimensional
 Hilbert space and linearly homeomorphic to $\mathbb{R}^{mT}$.

 On the other hand, we define the norm $\|\cdot\|_r$ on $E_{mT}$ as follows:
\begin{equation}\label{e2.3}
 \|u\|_r=\Big(\sum^{mT}_{j=1}|u_j|^r\Big)^{1/r},
\end{equation}
 for all $u\in E_{mT}$ and $r>1$.

 Since $\|u\|_r$ and $\|u\|_2$ are
 equivalent, there exist constants $c_1, c_2$ such that 
$c_2\geq  c_1>0$, and
\begin{equation}\label{e2.4}
 c_1\|u\|_2\leq\|u\|_r\leq c_2\|u\|_2,\quad \forall u\in E_{mT}.
\end{equation}
 Clearly, $\|u\|=\|u\|_2$. For all $u\in E_{mT}$, define the functional
 $J$ on $E_{mT}$  as follows:
\begin{equation}\label{e2.5}
 J(u)=\sum_{n=1}^{mT}\big[\frac{1}{p}\gamma_{n-1}
|\Delta^2u_{n-1}|^p-F(n,u_{n+1},u_n)\big],
\end{equation}
 where
 $$
\frac{\partial F(n-1,v_2,v_3)}{\partial v_2}
+\frac{\partial F(n,v_1,v_2)}{\partial v_2}
 =f(n,v_1,v_2,v_3).
$$
 Clearly, $J\in C^1(E_{mT},\mathbb{R})$ and for any 
$u=\{u_n\}_{n\in  {\mathbb{Z}}}\in E_{mT}$, by using
 $u_0=u_{mT},\ u_1=u_{mT+1}$, we can compute
 the partial derivative as
 $$
\frac{\partial J}{\partial u_n}=\Delta^2
\left(\gamma_{n-2}\varphi_p(\Delta^2u_{n-2})\right)-f(n,u_{n+1},u_{n},u_{n-1}).
$$
Thus, $u$ is a critical point of $J$ on $E_{mT}$ if and only if
$$
\Delta^2\left(\gamma_{n-2}\varphi_p(\Delta^2u_{n-2})\right)
=f(n,u_{n+1},u_{n},u_{n-1}),\quad \forall n\in \mathbb{Z}(1,mT).
$$
Due to the periodicity of $u=\{u_n\}_{n\in {\mathbb{Z}}}\in E_{mT}$
and
 $f(n,v_1,v_2,v_3)$ in the first variable $n$, we reduce the
 existence of periodic solutions of \eqref{e1.1} to the existence of
 critical points of $J$ on $E_{mT}$. That is, the functional $J$
 is just the variational framework of \eqref{e1.1}.

 Let $P$ be the $mT\times mT$ matrix defined by
 $$
P=  \begin{pmatrix}
 2& -1& 0& \dots & 0& -1 \\
 -1& 2& -1& \dots & 0& 0 \\
 0& -1& 2& \dots & 0& 0 \\
 \dots &\dots &\dots &\dots &\dots &\dots \\
 0& 0& 0& \dots & 2& -1 \\
 -1& 0& 0& \dots & -1& 2\\
 \end{pmatrix}.
$$
By matrix theory, we see that the eigenvalues of $P$ are
\begin{equation}\label{e2.6}
 \lambda_k=2\big(1-\cos (\frac{2k}{mT} \pi) \big),\quad k=0,1,2,\dots,mT-1.
\end{equation}
Thus, $\lambda_0=0$, $\lambda_1>0$, $\lambda_2>0,\dots,\lambda_{mT-1}>0$. 
Therefore,
\begin{equation}\label{e2.7}
\begin{gathered}
 \lambda_{\rm min}=\min\{\lambda_1,\lambda_2,\dots,\lambda_{mT-1}\}
=2\Big(1-\cos (\frac{2}{mT} \pi) \Big),\\
\begin{aligned}
 \lambda_{\rm max}
&=\max\{\lambda_1,\lambda_2,\dots,\lambda_{mT-1}\}\\
&=\begin{cases}
 4,&\text{if $mT$ is even},\\
 2\left(1+\cos (\frac{1}{mT} \pi) \right),
&\text{if $mT$ is odd}.
 \end{cases}
 \end{aligned}
\end{gathered}
\end{equation}

Let
 $$
W=\ker P=\{u\in E_{mT}|Pu=0\in \mathbb{R}^{mT}\}.
$$
 Then
 $$
W=\{u\in E_{mT}|u=\{c\},\ c\in \mathbb{R}\}.
$$
Let $V$ be the direct orthogonal complement of $E_{mT}$ to $W$;
 i.e.; $E_{mT}=V\oplus W$. For convenience, we identify $u\in E_{mT}$ with
 $u=(u_1,u_2,\dots,u_{mT})^\ast.$


 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
 $\{u^{(k)}\}\subset E$ for which $\{J\big(u^{(k)}\big)\}$ is bounded and
 $J' (u^{(k)})\to 0(k\to \infty)$ 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}[Linking Theorem \cite{Ra}] \label{lem2.1}
Let $E$ be a real Banach space,
 $E=E_1\oplus E_2$, where $E_1$ is finite dimensional. Suppose that
 $J\in C^1(E,\mathbb{R})$ satisfies the (PS) condition and\newline
 $(J_1)$ there exist constants $a>0$ and $\rho>0$ such that
 $J|_{\partial B_\rho\cap E_2}\geq a;$\newline
 $(J_2)$ there exists an $e\in \partial B_1\cap E_2$ and a constant 
$R_0\geq \rho$  such that $J|_{\partial Q}\leq 0$,
 where $Q=(\bar{B}_{R_0}\cap E_1)\oplus\{re|0<r<R_0\}$.\newline
 Then $J$ possesses a critical value $c\geq a$, where
 $$
c=\inf_{h\in \Gamma}\sup_{u\in Q} J(h(u)),
$$
and $\Gamma =\{h\in C(\bar{Q},E)\mid h|_{\partial Q}=id\}$,
where $id$ denotes the identity operator.
\end{lemma}

\begin{lemma} \label{lem2.2}
Assume that {\rm (F0), (F1), (F3)} are satisfied. 
Then the functional $J$ is bounded from above in $E_{mT}$.
\end{lemma}

\begin{proof}
By (F3') and \eqref{e2.4}, for any $u\in E_{mT}$,
\begin{align*}
J(u)&=\sum_{n=1}^{mT}\Big[\frac{1}{p}\gamma_{n-1}|\Delta^2u_{n-1}|^p
-F(n,u_{n+1},u_n)\Big]
\\
&\leq \frac{\bar{\gamma}}{p}c_2^p\Big[\sum_{n=1}^{mT}\left(\Delta u_n-\Delta
 u_{n-1}\right)^2\Big]^{p/2}  -\sum_{n=1}^{mT}F(n,u_{n+1},u_n)
\\
&\leq \frac{\bar{\gamma}}{p}c_2^p(x^\ast Px)^{p/2}
 -\sum_{n=1}^{mT}\Big[\beta\big(\sqrt{u_{n+1}^2+u_n^2}\big)^p-\zeta'\Big]
\\
&\leq\frac{\bar{\gamma}}{p}c_2^p\lambda_{\rm max}^{p/2}\|x\|_2^p
 -\beta\Big[\Big(\sum_{n=1}^{mT}
 \big(\sqrt{u_{n+1}^2+u_n^2}\big)^p\Big)^{1/p}\Big]^p+mT\zeta'
\\
&\leq\frac{\bar{\gamma}}{p}c_2^p\lambda_{\rm max}^{p/2}\|x\|_2^p
 -\beta c_1^p\Big[\sum_{n=1}^{mT}\left(u_{n+1}^2+u_n^2\right)\Big]^{p/2}
 +mT\zeta'
\\
&=\frac{\bar{\gamma}}{p}c_2^p\lambda_{\rm max}^{p/2}\|x\|_2^p
 -\beta c_1^p\big(2\|u\|_2^2\big)^{p/2}+mT\zeta'
\\
&\leq \frac{\bar{\gamma}}{p}c_2^p\lambda_{\rm max}^{p/2}\|x\|_2^p
 -2^{p/2}\beta c_1^p\|u\|_2^p+mT\zeta',
\end{align*}
where $x=(\Delta u_1,\Delta u_2,\dots,\Delta  u_{mT})^*$. Since
 $$
\|x\|_2^p=\Big[\sum_{n=1}^{mT}\left(u_{n+1}-u_{n},u_{n+1}-u_{n}\right)\Big]^{p/2}
=\left(u^\ast  Pu\right)^{p/2} \leq\lambda_{\rm max}^{p/2}\|u\|_2^p,
$$
 we have
 $$
J(u)\leq\frac{\bar{\gamma}}{p}c_2^p\lambda_{\rm max}^p\|u\|_2^p
 -2^{p/2}\beta c_1^p\|u\|_2^p+mT\zeta'\leq mT\zeta'.
$$
The proof is complete.
\end{proof}

\begin{remark} \label{rmk2.3} \rm
The case  $mT=1$ is trivial. For the case
 $mT=2$, $P$ has a different form, namely,
 $$
P=\begin{pmatrix}
 2& -2\\
 -2& 2
\end{pmatrix}
$$
 However, in this special case, the argument need not to be changed 
and we omit it.
\end{remark}

\begin{lemma} \label{lem2.4}
Assume that {\rm (F0), (F1), (F3)} are satisfied. Then the functional 
$J$ satisfies the (PS) condition.
\end{lemma}

\begin{proof}
Let $\{J\big(u^{(k)}\big)\}$ be a bounded sequence
from the lower bound;
 i.e., there exists a positive constant $M_1$ such that
 $$
-M_1\leq J\left(u^{(k)}\right),\ \forall k\in \textbf{N}.
$$
 By the proof of Lemma \ref{lem2.2}, it is easy to see that
 $$
-M_1\leq J\big(u^{(k)}\big)\leq 
\big(\frac{\bar{\gamma}}{p}c_2^p\lambda_{\rm max}^p
 -2^{p/2}\beta c_1^p\big)
\|u^{(k)}\|_2^p+mT\zeta',\quad \forall k\in \textbf{N}.
$$
 Therefore,
 $$
\big(2^{p/2}\beta c_1^p -\frac{\bar{\gamma}}{p}c_2^p\lambda_{\rm max}^p\big)
\|u^{(k)}\|_2^p\leq M_1  +mT\zeta'.
$$
 Since $\beta>\frac{\bar{\gamma}}{2^{{p}/{2}}p}(c_2/c_1)^p\lambda_{\rm max}^p$,
it is not difficult to know that  $\left\{u^{(k)}\right\}$
is a bounded sequence in $E_{mT}$. As a consequence, $\left\{u^{(k)}\right\}$
possesses a convergence subsequence in $E_{mT}$. Thus
the (PS) condition is verified.
\end{proof}


\section{Proof of 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}]
Assumptions (F1) and (F2) imply that $F(n,0)=0$ and $f(n,0)=0$
for $n\in\mathbb{Z}$. Then $u=0$ is a trivial $mT$-periodic solution
of \eqref{e1.1}.

 By Lemma \ref{lem2.4}, $J$ is bounded from above on $E_{mT}$. We
define $c_0=\sup_{u\in E_{mT}}J(u)$. The proof of Lemma \ref{lem2.4}
 implies $\lim_{\|u\|_2\to+\infty}J(u)=-\infty$. This
 means that $-J(u)$ is coercive. By the continuity of $J(u)$, there
 exists $\bar{u}\in E_{mT}$ such that $J(\bar{u})=c_0$. Clearly,
 $\bar{u}$ is a critical point of $J$.

We claim that $c_0>0$. Indeed, by (F2), for any
 $u\in V,\ \|u\|_2\leq\delta_1$, we have
\begin{align*}
J(u)&=\sum_{n=1}^{mT}\Big[\frac{1}{p}\gamma_{n-1}|\Delta^2u_{n-1}|^p
-F(n,u_{n+1},u_n)\Big]
\\
&\geq \frac{1}{p}\underline{\gamma}c_1^p
\Big[\sum_{n=1}^{mT}\left(\Delta u_n-\Delta  u_{n-1}\right)^2\Big]^{p/2}
 -\sum_{n=1}^{mT}F(n,u_{n+1},u_n)
\\
&\geq \frac{1}{p}\underline{\gamma}c_1^p(x^\ast Px)^{p/2}
 -\sum_{n=1}^{mT}F(n,u_{n+1},u_n)
\\
&\geq \frac{1}{p}\underline{\gamma}c_1^p\lambda_{\rm min}^{p/2}\|x\|_2^p
 -\alpha\sum_{n=1}^{mT}\big( \sqrt{u_{n+1}^2+u_n^2}\big)^p
\\
&=\frac{1}{p}\underline{\gamma}c_1^p\lambda_{\rm min}^{p/2}\|x\|_2^p
 -\alpha\Big[\Big(\sum_{n=1}^{mT}\big(
 \sqrt{u_{n+1}^2+u_n^2}\big)^p\Big)^{1/p}\Big]^p
\\
&\geq \frac{1}{p}\underline{\gamma}c_1^p\lambda_{\rm min}^{p/2}\|x\|_2^p
 -\alpha c_2^p\Big[\sum_{n=1}^{mT}\left(
 u_{n+1}^2+u_n^2\right)\Big]^{p/2},
\end{align*}
where $x=(\Delta u_1,\Delta u_2,\dots,\Delta u_{mT})^*$. Since
 $$
\|x\|_2^p=\Big[\sum_{n=1}^{mT}\left(u_{n+1}-u_{n},u_{n+1}-u_{n}\right)\Big]^{p/2}
=\left(u^\ast  Pu\right)^{p/2}\geq\lambda_{\rm min}^{p/2}\|u\|_2^p,
$$
we have
$$
J(u)\geq \frac{1}{p}\underline{\gamma}c_1^p\lambda_{\rm min}^p\|u\|_2^p
-\alpha c_2^p\left(2\|u\|_2^2\right)^{p/2}
 =\left(\frac{1}{p}\underline{\gamma}c_1^p\lambda_{\rm min}^p-2^{p/2}
c_2^p\alpha\right)\|u\|_2^p.
$$
 Take $\sigma=\left(\frac{1}{p}\underline{\gamma}c_1^p
\lambda_{\rm min}^p-2^{p/2}c_2^p\alpha\right)\delta_1^p$. Then
 $$
J(u)\geq\sigma,\quad \forall u\in V\cap\partial B_{\delta_1}.
$$
Therefore, $c_0=\sup_{u\in E_{mT}}J(u)\geq \sigma>0$. At the
same time, we have also proved that there exist constants $\sigma>0$ 
and $\delta_1>0$ such that
 $J|_{\partial B_{\delta_1}\cap V}\geq \sigma$. That is to say, $J$
 satisfies the condition $(J_1)$ of the Linking Theorem.

Noting that $\sum_{n=1}^{mT}\gamma_{n-1}\left|\Delta^2u_{n-1}\right|^p=0$, 
for all $u\in W$, we have
 $$
J(u)=\frac{1}{p}\sum_{n=1}^{mT}\gamma_{n-1}|\Delta^2u_{n-1}|^p
-\sum_{n=1}^{mT}F(n,u_{n+1},u_n)
= -\sum_{n=1}^{mT}F(n,u_{n+1},u_n)\leq 0.
$$
Thus, the critical point $\bar{u}$ of $J$ corresponding to the
critical value $c_0$ is a nontrivial $mT$-periodic solution of \eqref{e1.1}.

To obtain another nontrivial $mT$-periodic solution of \eqref{e1.1} different
 from $\bar{u}$, we need to use the conclusion of
 Lemma \ref{lem2.1}. We have known that $J$ satisfies the (PS) condition on
$E_{mT}$. In the following, we shall verify the condition $(J_2)$.

 Take $e\in\partial B_1\cap V$, for any $z\in W$ and $r\in\mathbb{R}$, 
let $u=re+z$. Then
\begin{align*}
J(u)
&=\sum_{n=1}^{mT}\big[\frac{1}{p}\gamma_{n}
|\Delta^2u_{n}|^p-F(n,u_{n+1},u_n)\big]
\\
&\leq\sum_{n=1}^{mT}\big[\frac{\bar{\gamma}}{p}r^p\left|\Delta^2e_{n}\right|^p
-F(n,re_{n+1}+z_{n+1},re_n+z_n)\big]
\\
&\leq \frac{\bar{\gamma}}{p}r^pc_2^p\Big[\sum_{n=1}^{mT}\left(\Delta e_n-\Delta
 e_{n-1}\right)^2\Big]^{p/2}
 -\sum_{n=1}^{mT}F(n,re_{n+1}+z_{n+1},re_n+z_n)
\\
&\leq \frac{\bar{\gamma}}{p}r^pc_2^p(y^\ast Py)^{p/2}
 -\sum_{n=1}^{mT}
 \Big\{\beta\left(\sqrt{(re_{n+1}+z_{n+1})^2+(re_n+z_n)^2}\right)^p
 -\zeta'\Big\}
\\
&\leq \frac{\bar{\gamma}}{p}r^pc_2^p(y^\ast Py)^{p/2}
 -\beta c_1^p\big\{\sum_{n=1}^{mT}\big[(re_{n+1}+z_{n+1})^2+(re_n+z_n)^2
\big]\big\}^{p/2}+mT\zeta'
\\
&\leq\frac{\bar{\gamma}}{p}r^pc_2^p\lambda_{\rm max}^{p/2}\|y\|_2^p
 -\beta c_1^p\Big[2\sum_{n=1}^{mT}\left(re_n+z_n\right)^2\Big]^{p/2}+mT\zeta'
\\
&=\frac{\bar{\gamma}}{p}r^pc_2^p\lambda_{\rm max}^{p/2}\|y\|_2^p
 -\beta c_1^pr^p2^{p/2}-\beta c_1^p2^{p/2}\|z\|_2^p+mT\zeta',
\end{align*}
where $y=(\Delta e_{1},\Delta e_{2},\dots,\Delta e_{mT})^*$. Since
 $$
\|y\|_2^p=\Big[\sum_{n=1}^{mT}\left(e_{n+1}-e_{n},e_{n+1}-e_{n}\right)\Big]^{p/2}
 =\left(e^\ast Pe\right)^{p/2}\leq\lambda_{\rm max}^{p/2},
$$
 we have
 $$
J(u)\leq\Big(\frac{\bar{\gamma}}{p}c_2^p\lambda_{\rm max}^p-\beta c_1^p
 2^{p/2}\Big)r^p-\beta c_1^p2^{p/2}\|z\|_2^p+mT\zeta'
\leq-\beta c_1^p2^{p/2}\|z\|_2^p+mT\zeta'.
$$
 Thus, there exists a positive constant $R_1>\delta_1$ such that
for any $u\in\partial Q$,\ $J(u)\leq 0$, where 
$Q=(\bar{B}_{R_1}\cap W)\oplus\{re:0<r<  R_1\}$. 
By the Linking Theorem, $J$ possesses a critical value $c\geq \sigma>0$, where
 $$
c=\inf_{h\in \Gamma}\sup_{u\in Q} J(h(u)),
$$
and $\Gamma =\{h\in C(\bar{Q},E_{mT})\mid h|_{\partial Q}=id\}$.

 Let $\tilde{u}\in E_{mT}$ be a critical point associated to the
critical value $c$ of $J$, i.e., $J(\tilde{u})=c$. If
$\tilde{u}\neq \bar{u}$, then the conclusion of Theorem \ref{thm1.1}
holds. Otherwise, $\tilde{u}=\bar{u}$. Then
$c_0=J(\bar{u})=J(\tilde{u})=c$; that is,
$\sup_{u\in E_{mT}} J(u)=\inf_{h\in \Gamma}
\sup_{u\in Q} J(h(u)).$ Choosing $h=id$, we have $\sup_{u\in Q}
 J(u)=c_0$. Since the choice of $e\in\partial B_1\cap V$ is
 arbitrary, we can take $-e\in\partial B_1\cap V$. 
Similarly, there exists a positive  number $R_2>\delta_1$, for 
any $u\in\partial Q_1$,\ $J(u)\leq 0$, where $Q_1=(\bar{B}_{R_2}\cap W)
 \oplus\{-re|0<r<R_2\}$.

 Again, by the Linking Theorem, $J$ possesses a critical value 
$c'\geq \sigma>0$, where
 $$
c'=\inf_{h\in \Gamma_1}\sup_{u\in Q_1} J(h(u)),
$$
 and $\Gamma_1=\{h\in C(\bar{Q}_1,E_{mT})\mid h|_{\partial Q_1}=id\}$.

 If $c'\neq c_0$, then the proof is finished. If $c'=c_0$, then 
$\sup_{u\in Q_1} J(u)  =c_0$. Due to the fact $J|_{\partial Q}\leq 0$ 
and $J|_{\partial Q_1}\leq  0$, $J$ attains its maximum at some points 
in the interior of  sets $Q$ and $Q_1$. However, $Q\cap Q_1\subset W$ and
 $J(u)\leq0$ for any $u\in W$. Therefore, there must be a point
 $u'\in E_{mT},\ u'\neq \tilde{u}$ and $J(u')=c'=c_0$. The proof
 is complete.
\end{proof}

Similarly to above argument, we can also prove
Theorems \ref{thm1.5} and \ref{thm1.7}, so their proofs are omitted.
Due to Theorems \ref{thm1.1}, \ref{thm1.5} and \ref{thm1.7}, the
conclusion of Corollaries 1.3, 1.6 and 1.8 are obviously true.

\section{Example}

As an application of Theorem \ref{thm1.1}, we give an example to illustrate
our main result.

\begin{example} \rm Assume that for all $n\in \mathbb{Z}$, 
\begin{equation}\label{e4.1}
\begin{aligned} 
\Delta^2\left(\gamma_{n-2}\varphi_p(\Delta^2u_{n-2})\right)
&=\mu u_n\Big[\Big(3+\sin^2(\pi n/T)\Big)
(u_{n+1}^2+  u_{n}^2)^{\frac{\mu}{2}-1}\\
&\quad +\Big(3+\sin^2\big(\pi (n-1)/T\big)\Big)(u_{n}^2  +u_{n-1}^2)
^{\frac{\mu}{2}-1}\Big],\ \ \ \ \ \ \ \
\end{aligned}
\end{equation}
where $\gamma_n$ is real valued for each $n\in \mathbb{Z}$ and 
$\gamma_{n+T}=\gamma_n>0$, $1<p<+\infty$,\ $\mu>p$,\ $T$ 
is a given positive integer.
We have
\begin{align*}
f(n,v_1,v_2,v_3)
&=\mu v_2\Big[\Big(3+\sin^2(\pi n/T)\Big)(v_1^2+  v_2^2)^{\frac{\mu}{2}-1}\\
&\quad +\Big(3+\sin^2\big(\pi (n-1)/T\big)\Big) 
(v_2^2 +v_3^2)^{\frac{\mu}{2}-1}\Big]
\end{align*}
 and
 $$
F(n,v_1,v_2)=[3+\sin^2(\pi n/T)](v_1^2+  v_2^2)^{\frac{\mu}{2}}.
$$
Then
\begin{align*}
&\frac{\partial F(n-1,v_2,v_3)}{\partial v_2}
 +\frac{\partial F(n,v_1,v_2)}{\partial v_2}\\
&=\mu v_2\Big[\Big(3+\sin^2(\pi n/T)\Big)(v_1^2+  v_2^2)^{\frac{\mu}{2}-1}
+\Big(3+\sin^2\big(\pi (n-1)/T\big)\Big)
(v_2^2  +v_3^2)^{\frac{\mu}{2}-1}\Big].
\end{align*}
 It is easy to verify all the assumptions of Theorem \ref{thm1.1} are satisfied.
 Consequently, for any given positive integer $m>0$, \eqref{e4.1} has at least
 three $mT$-periodic solutions.
\end{example}


\subsection*{Acknowledgments} 
We would like to express our
sincere gratitude to the anonymous referee for a very careful reading of the
paper and for all the insightful comments and valuable suggestions.

This project is supported by Specialized Research Fund for the
Doctoral Program of Higher Eduction of China (No. 20114410110002),
National Natural Science Foundation of China (No. 11101098),
Natural Science Foundation of Guangdong Province (No. S2013010014460),
Science and Research Program of Hunan Provincial Science and
Technology Department (Grant No. 2012FJ4109) and Scientific Research
Fund of Hunan Provincial Education Department (No. 12C0170).


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\end{document}