\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 $a0$ 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 $$\label{e1.6} \Delta^2\left(\gamma_{n-2}\varphi_p(\Delta^2u_{n-2})\right) =q_ng(u_n),\ n\in \mathbb{Z},$$ 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 \label{e2.1} \left=\sum^{mT}_{j=1}u_j v_j,\, \forall u,v\in E_{mT}, by which we introduce the norm $$\label{e2.2} \|u\|=\Big(\sum^{mT}_{j=1}u_j^2\Big)^{1/2},\quad \forall u\in E_{mT}.$$ 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: $$\label{e2.3} \|u\|_r=\Big(\sum^{mT}_{j=1}|u_j|^r\Big)^{1/r},$$ 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 $$\label{e2.4} c_1\|u\|_2\leq\|u\|_r\leq c_2\|u\|_2,\quad \forall u\in E_{mT}.$$ Clearly, $\|u\|=\|u\|_2$. For all $u\in E_{mT}$, define the functional $J$ on $E_{mT}$ as follows: $$\label{e2.5} J(u)=\sum_{n=1}^{mT}\big[\frac{1}{p}\gamma_{n-1} |\Delta^2u_{n-1}|^p-F(n,u_{n+1},u_n)\big],$$ 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 $$\label{e2.6} \lambda_k=2\big(1-\cos (\frac{2k}{mT} \pi) \big),\quad k=0,1,2,\dots,mT-1.$$ Thus, $\lambda_0=0$, $\lambda_1>0$, $\lambda_2>0,\dots,\lambda_{mT-1}>0$. Therefore, \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} 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 $Q=(\bar{B}_{R_0}\cap E_1)\oplus\{re|0\frac{\bar{\gamma}}{2^{{p}/{2}}p}(c_2/c_1)^p\lambda_{\rm max}^p$, 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 $Q=(\bar{B}_{R_1}\cap W)\oplus\{re:00$, 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 $Q_1=(\bar{B}_{R_2}\cap W) \oplus\{-re|00$, 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 $n\in \mathbb{Z}$, \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} where $\gamma_n$ is real valued for each $n\in \mathbb{Z}$ and $\gamma_{n+T}=\gamma_n>0$, $1p$,\ $T$ 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). \begin{thebibliography}{99} \bibitem{Ag} R. P. Agarwal; \emph{Difference Equations and Inequalities: Theory, Methods and Applications}, Marcel Dekker: New York, 1992. \bibitem{AgPO1} R. P. Agarwal, K. Perera, D. O'regan; \emph{Multiple positive solutions of singular and nonsingular discrete problems via variational methods}, Nonlinear Anal., 58(1-2) (2004), pp. 69-73. \bibitem{AgPO2} R. P. Agarwal, K. Perera, D. O'regan; \emph{Multiple positive solutions of singular discrete $p$-Laplacian problems via variational methods}, Adv. 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