\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 296, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{7mm}} \begin{document} \title[\hfilneg EJDE-2017/296\hfil Existence of periodic solutions] {Existence of periodic solutions for subquadratic discrete system involving the p-Laplacian} \author[Q. Jiang, S. Ma Z. Hu \hfil EJDE-2017/??\hfilneg] {Qin Jiang, Shang Ma, Zhihua Hu} \address{Qin Jiang \newline Department of Mathematics, Huanggang Normal University, Hubei 438000, China} \email{jiangqin999@126.com} \address{Sheng Ma \newline Department of Mathematics, Huanggang Normal University, Hubei 438000, China} \email{masheng666@126.com} \address{Zhihua Hu (corresponding author)\newline Department of Mathematics, Huanggang Normal University, Hubei 438000, China} \email{zhihuahu123@126.com} \dedicatory{Communicated by Paul H Rabinowitz} \thanks{Submitted April 6, 2017. Published November 28, 2017.} \subjclass[2010]{39A11, 58E50, 70H05, 37J45} \keywords{Discrete system; periodic solution; p-Laplacian; subquadratic; \hfill\break\indent saddle point theorem} \begin{abstract} An existence theorem on periodic solution is established for a class of nonautonomous discrete system involving the p-Laplacian under a subquadratic growth condition. The conclusion is based on saddle point theorem and variational methods. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of main results} Let $\mathbb{Z}$ be the set of integers. Given $a< b$ in $ \mathbb{Z}$, let $ \mathbb{Z}[a,b]=\{a,a+1,\dots,b\}$ and $T >1$ be a positive integer. In this article, we aim at the existence of periodic solution for the nonlinear discrete system involving the p-Laplacian \begin{equation} \label{e1} \Delta_p u(t-1)+\nabla F(t,u(t))=0,\quad \forall t\in \mathbb{Z} \end{equation} where $\Delta_p$ is the discrete p-Laplacian operator, i.e., $$ \Delta_p u(t-1):= \Delta\phi_p(\Delta u(t-1)) = \phi_p(\Delta u(t))-\phi_p(\Delta u(t-1)), $$ $\phi_p(s)=|s|^{p-2}s(p>1)$, $\Delta$ is the forward difference operator and the function $F:\mathbb{Z}\times \mathbb{R}^N\to R$ is continuously differentiable in $x$ for every $t\in \mathbb{Z}$, $\nabla F(t,x)=\frac{\partial F(t,x)}{\partial x}$. In recent years, many authors were interested in difference equations involving the discrete p-Laplacian operator and have obtained many significant conclusions, see, for instance, the papers \cite{Ag,Ag1,Ag2,At,Bona,Ca2009,D,Fara,Gao2014,Guo1,Guo2,he,jiang,jiang1,r,wang,Xue}. Various methods have been used to deal with the existence of solutions to the discrete boundary value problems, we refer to the fixed point theorems in cones in \cite{jiang}, the lower and upper solution method in \cite{At}, the variational method in \cite{Ag1,Ag2,Bona,Ca2009,D,Fara,Gao2014,Guo1,Guo2,jiang,jiang1,r,wang,Xue}. The variational approach represents an important advance as it allows to prove multiplicity results as well. When $p>1$, via dual least principle, system \eqref{e1} under convex condition was investigated in \cite{he}. Recently, some further improved results have been made in \cite{zhang}. Via Linking theorem, the existence of one nonconstant solutions was established for system \eqref{e1} under superquadratic condition in \cite{luo}. In 2007, in \cite{Xue} the authors constructed a variational setting unlike the one in \cite{Guo1} to study the discrete system \eqref{e1} with $p=2$ under subquadratic condition via saddle point theorem. The result was obtained under the following assumptions: \begin{itemize} \item[(A1)] For a given integer $T>0$, $F(t+T,x)=F(t,x)$ for all $(t,x)\in \mathbb{Z}\times \mathbb{R}^N $; \item[(A2)] There are constants $G_1>0$, $0<\beta<2$ such that $$ (x,\nabla F(t,x))\leq \beta F(t,x) $$ for all $(t,x)\in \mathbb{Z}[1,T]\times \mathbb{R}^N$ and $| x|\geq G_1$; \item[(A3)] $F(t,x)\to +\infty$ as $| x | \to \infty$ for $t\in \mathbb{Z}[1,T]$. \end{itemize} \begin{theorem}[\cite{Xue}] \label{thmA} Suppose that {\rm (A1)--(A3)} are satisfied. Then problem \eqref{e1} possesses at least one periodic solution with period $T$. \end{theorem} Inspired by \cite{luo,wang,Xue}, in the article, we further investigate periodic solutions for system \eqref{e1} under a new subquadratic condition which is more general than (A2). Here $\mathcal{H}$ denotes the space of continuous function space such that for any $\theta \in \mathcal{H}$ there exists constant $M_1 > 0$ for which \begin{itemize} \item[(i)] $\theta(t)>0$ for all $t\in R^+$, \item[(ii)] $ \int_{M_1}^t \frac{1}{s\theta(s)} ds\to +\infty$ as $t\to +\infty$. \end{itemize} Our main result is stated using the following assumptions: \begin{itemize} \item[(A4)] There exist a constant $M_1>0$ and a continuous function $\theta(|x|)\in \mathcal{H}$ with $0<\frac{1}{\theta(|x|)} 0$. Indeed, via (A5), when $l > 0$, (A6) implies \begin{itemize} \item[(A6')] $\sum_{t=1}^T F(t,x)\to +\infty$ as $| x | \to +\infty$. \end{itemize} However, (A5) and (A6') are weaker than (A3). (3) There are functions $F$ fulfilling the conditions of Theorem \ref{thm1.1} but not the assumptions in \cite{Guo1,Guo2,he,jiang1,Xue,zhang}. For example, $$ F(t,x)=g(t)\frac{2+|x|^p}{\ln(2+|x|^2)},\quad \forall (t,x)\in \mathbb{Z}[1,T]\times \mathbb{R}^N. $$ Here $$ g(t)=\begin{cases} \sin(2\pi t/T), & t\in [0,T/2],\\ 0,& t\in [T/2,T]. \end{cases} $$ Put $\theta(|x|) = \ln(2+|x|^2)$. A simple computation shows that $F$ satisfies (A1) and (A4)--(A6) in Theorem \ref{thm1.1}, but it does not meet the corresponding conditions of Theorem \ref{thmA}. \end{remark} \section{Proof of Theorem \ref{thm1.1}} For a given positive integer $T$, we define $$ H_T=\{u:Z\to \mathbb{R}^N : u(t+T)=u(t),\; t\in Z\}. $$ $H_T$ is equipped with the inner product $$ \langle u,v\rangle =\sum_{t=1}^{T}(u(t),v(t)),\quad \forall u,v \in H_T $$ and the norm $$ \| u\| =\Big(\sum ^{T}_{t=1}| u(t)|^p\Big)^{1/p},\quad \forall u\in H_T. $$ One can easily see that $(H_T,\langle \cdot ,\cdot \rangle )$ is a finite dimensional Hilbert space and linear homeomorphic to $R^{NT}$. Define $$ \| u \|_\infty=\max_{t\in \mathbb{Z}[1,T]}| u(t)|. $$ Then there exists a constant $c>0$ such that \begin{equation} \label{e2} \| u \|_\infty\leq c\|u\|. \end{equation} For $u\in H_T$, set $$ \tilde{u}=u-\bar{u} \quad \text{and} \quad \tilde{H}_T=\{u\in H_T: \bar{u}=0 \}. $$ Here $\bar{u}=\sum^{T}_{t=1}u(t)$. Then one knows $$H_T= \tilde{H}_T\oplus \mathbb{R}^N.$$ Furthermore, via \cite{luo}, one gets \begin{equation} \label{e3} \sum^{T}_{t=1} | u(t)|^p \leq \frac{(T-1)^{2p-1}}{T^{p-1}} \sum^{T}_{t=1} |\Delta u|^p,\quad \forall u\in \tilde{H}_T. \end{equation} From reference \cite{luo}, it is known that finding $T$-periodic solution of problem \eqref{e1} is equivalent to seeking the critical point of the following functional $\varphi$ defined on $H_T$, $$ \varphi(u)=\frac{1}{p}\sum_{t=1}^T| \Delta u(t)|^p-\sum_{t=1}^TF(t,u(t)). $$ Subsequently, two important lemmas are stated for the readers convenience. \begin{lemma}[saddle point Theorem \cite{r}] \label{lem2.1} Let $X$ be a Banach space with a direct sum decomposition $X=X_1\oplus X_2$ with dim$X_2<\infty$ and let $\varphi$ be a $C^1$ function on $X$ satisfying the (PS) condition and \begin{itemize} \item[(1)] there exist a constant $r$ and a bounded neighborhood $U$ of 0 in $X_2$, such that $\varphi(u)\leq r$\ for$ \ u\in U\subset X_2$, \item[(2)] there exists a constant $\alpha>r$, such that $\varphi(u)\geq \alpha$ for all $u\in X_1$. \end{itemize} Then $\varphi$ has at least one critical point. \end{lemma} As we know, a deformation lemma can be proved with Cerami's condition (C) in \cite{Ce} by replacing the usual (PS) condition. Then the saddle point theorem is tenable under condition (C). \begin{lemma} \label{lem2.2} Under the conditions of Theorem \ref{thm1.1}, we have \begin{equation} \label{e4} F(t,x)\leq \frac{M_2}{M_1^p} | x|^p G(|x|)+M_2 \end{equation} for all $x\in \mathbb{R}^N$ and $t\in \mathbb{Z}[1,T]$, where $$ M_2=\max\{F(t,x):| x|\leq M_1,\; t\in \mathbb{Z}[1,T]\}, \quad G(|x|)=\exp\Big(- \int_{M_1}^{|x|} \frac{1}{s\theta(s)} ds \Big). $$ \end{lemma} \begin{proof} Put $$ y(s)=F(t,sx),\quad s\geq \frac{M_1}{| x|}. $$ Via (A4), a simple computation yields \begin{equation} \label{e5} \begin{aligned} y'(s)&=\frac{1}{s}(\nabla F(t,sx),sx) \\ &\leq \frac{1}{s}( p- \frac{1}{\theta(s|x|)} ) F(t,sx)\\ &=\frac{1}{s}( p- \frac{1}{\theta(s|x|)} ) y(s) \end{aligned} \end{equation} for all $s\geq M_1/| x|$. Set \begin{equation} \label{e6} h(s):=y'(s)-\frac{1}{s}( p- \frac{1}{\theta(s|x|)} ) y(s). \end{equation} Obviously, $h(s)\leq 0$ for all $s\geq \frac{M_1}{| x|}$. Solving the order linear ordinary differential equation \eqref{e6}, together with the fact $h(s)\leq 0$, one derives $$ y(s) \leq \frac{y(\frac{M_1}{| x|})}{M_1^p}|x|^p s^pG(s|x|), \quad \forall s\geq \frac{M_1}{| x|}. $$ Then, one has \begin{equation} \label{e7} F(t,x)=y(1)\leq \frac{F(t,\frac{M_1x}{| x|})}{M_1^p}|x|^p G(|x|), \quad \forall | x|\geq M_1. \end{equation} Furthermore, one can deduce \begin{equation} \label{e8} F(t,\frac{M_1x}{| x|})\leq M_2 \end{equation} for all $x\in \mathbb{R}^N$ and $t\in \mathbb{Z}[1,T]$. Then via \eqref{e7} and \eqref{e8}, one obtains \begin{eqnarray*} F(t,x)\leq \frac{M_2}{M_1^p}|x|^p G(|x|) +M_2 \end{eqnarray*} for all $x\in \mathbb{R}^N$ and $t\in \mathbb{Z}[1,T]$. \end{proof} \begin{remark} \label{rmk2.3} \rm (1) Employing property (ii) of $\theta$, one knows that $G(|x|) \to 0$ as $|x| \to +\infty$. (2) The function $t^p G(t) $ is increasing in $t$ since the range of $\frac{1}{\theta}$ and $(t^pG(t))'=t^{p-1}G(t)(p-\frac{1}{\theta(t)})> 0$. \end{remark} \begin{proof}[Proof of Theorem \ref{thm1.1}] The proof relies on Lemma \ref{lem2.1} with $X=H_T$, $X_1=\tilde{H}_T$, and $X_2=\mathbb{R}^N$. Firstly, one proves that $\varphi$ satisfies condition (C). Indeed, let $\{u_k\}\subset H_T$ be a sequence such that $\{\varphi(u_k)\}$ is bounded and $$ \|\varphi' (u_k)\|(1+\| u_k\|)\to0\quad \text{as } k\to\infty. $$ Then there exists a constant $M_3>0$ for which $$ |\varphi(u_k)|\leq M_3,\quad \|\varphi' (u_k)\|(1+\| u_k\|)\leq M_3. $$ Via (A4), a straightforward computation yields $$ -M_4+(x,\nabla F(t,x))\leq (p- \frac{1}{\theta(|x|)} ) F(t,x) $$ for all $x\in \mathbb{R}^N$ and $t\in \mathbb{Z}[1,T]$. Here $M_4>0$. Thus, one has \begin{align*} (p+1)M_3 &\geq \|\varphi' (u_k)\|(1+\| u_k\|)-p\varphi(u_k)\\ &\geq \langle \varphi'(u_k),u_k\rangle -p\varphi(u_k)\\ &= \sum_{t=1}^T(pF(t,u_k(t))-(\nabla F(t,u_k(t)),u_k(t)))\\ &\geq \sum_{t=1}^T \frac{F(t,u_k(t))}{\theta(|u_k|)}-M_4T \end{align*} for all $k\in \mathbb{N}$. Then it holds \begin{equation} \label{e9} \sum_{t=1}^T \frac{F(t,u_k(t))}{\theta(|u_k|)}\leq M_5 \end{equation} for all $k\in \mathbb{N}$. Here $M_5=M_4T+(p+1)M_3$. In addition, employing \eqref{e4}, \eqref{e2} and (2) in Remark \ref{rmk2.3}, one has \begin{equation} \label{e10} \begin{aligned} M_3&\geq\varphi(u_k)=\frac{1}{p}\sum_{t=1}^T| \Delta u_k(t)|^p -\sum_{t=1}^TF(t,u_k(t)) \\ &\geq \frac{1}{p}\sum_{t=1}^T| \Delta u_k(t)|^p-\sum_{t=1}^T \Big( \frac{M_2}{M_1^p}|u_k(t)|^p G(|u_k(t)|) +M_2\Big) \\ &\geq \frac{1}{p}\sum_{t=1}^T| \Delta u_k(t)|^p - \frac{M_2}{M_1^p}\sum_{t=1}^T\|u_k\|^p_\infty G(\|u_k\|_\infty)-M_2T \\ &\geq \frac{1}{p}\sum_{t=1}^T| \Delta u_k(t)|^p-M_6\|u_k\|^p G(\|u_k\|)-M_2T \end{aligned} \end{equation} for all $k\in \mathbb{N}$ and some $M_6>0$. Thus by \eqref{e10}, for all $k\in \mathbb{N}$, it holds: \begin{equation} \label{e11} \frac{ M_3}{\|u_k\|^p}\geq\frac{\varphi(u_k)}{\|u_k\|^p} \geq \frac{1}{p}\sum_{t=1}^T \frac{|\Delta u_k(t)|^p}{\|u_k\|^p} -M_6 G(\|u_k\|)-\frac{M_2T}{\|u_k\|^p}. \end{equation} Then one claims that $\{u_k\}$ is bounded. Otherwise, there exists a subsequence of $\{u_k\}$, also denoted by $\{u_k\}$, such that \begin{equation} \label{e12} \| u_k\|\to \infty\quad \text{as } k\to +\infty. \end{equation} Put $v_k=u_k/\|u_k\|$. Obviously, $\|v_k\|=1$ and $\{v_k\}$ is bounded in the finite dimensional space $H_T$. Thus there exist a point $v\in H_T$ and a subsequence of $\{v_k\}$, say $\{v_k\}$, such that $$ v_k\to v \quad \text{in } H_T. $$ Then in light of \eqref{e11}, \eqref{e12} and (2) of Remark \ref{rmk2.3}, one deduces that \begin{equation} \label{e13} \sum_{t=1}^T | \Delta v_k|^p \to \sum_{t=1}^T | \Delta v|^p =0 \quad \text{as } k\to +\infty. \end{equation} This means $| \Delta v(t)|=0$. Consequently, one has $| v(t)|$ is a constant for all $t\in \mathbb{Z}[1,T]$. Then one easily gets $$ T|v|^p=\sum_{t=1}^T |v|^p=\|v\|^p= 1. $$ Thus, it holds $|u_k(t)| \to +\infty$ as $k\to +\infty$ for $t \in \mathbb{Z}[1, T ]$. Then via (A6), one deduces $$ \sum_{t=1}^T \frac{F(t,u_k(t))}{\theta(|u_k(t)|)}\to +\infty\quad \text{as } |u_k(t)| \to +\infty. $$ This is a contradiction to \eqref{e9}. Thus $\{u_k\}$ is bounded. In finite dimensional space $H_T$, $\{u_k\}$ has a convergent subsequence. Thus $\varphi$ satisfies condition (C). Secondly, one proves that $\varphi$ satisfies (1) and (2) in Lemma \ref{lem2.1}. For $u\in \mathbb{R}^N$, since $0 < \frac{1}{\theta(t)}< p$, one obtains $$ \varphi(u)=- \sum_{t=1}^T F(t,u(t)) \leq -\frac{1}{p}\sum_{t=1}^T \frac{F(t,u(t))}{\theta(|u(t)|)} \to -\infty\quad \text{as } |u| \to \infty. $$ Thus one concludes that $\varphi(u)\to-\infty$ as $\| u\|\to\infty$ in $\mathbb{R}^N$. Thus (1) in Lemma \ref{lem2.1} is satisfied. Then, in a similar way to \eqref{e10}, from \eqref{e2}, \eqref{e3} and \eqref{e4}, for any $u\in \tilde{H}_T$, one gets \begin{equation} \label{e14} \begin{aligned} \varphi(u)&=\frac{1}{p}\sum_{t=1}^T| \Delta u(t)|^p -\sum_{t=1}^T F(t,u(t)) \\ &\geq \frac{1}{p}\sum_{t=1}^T| \Delta u(t)|^p -\sum_{t=1}^T\Big( \frac{M_2}{M_1^p}|u(t)|^p G(|u(t)|) +M_2\Big) \\ &\geq \frac{1}{p} \frac{T^{p-1}}{(T-1)^{2p-1}} \sum_{t=1}^T | u(t)|^p- \frac{M_2}{M_1^p}\sum_{t=1}^T\|u\|^p_\infty G(\|u\|_\infty)-M_2T \\ &\geq \frac{T^{p-1}}{p(T-1)^{2p-1}} \|u\|^p-M_6\| u\|^p G\left(\| u\|\right)-M_2T \\ & = \big\{\frac{T^{p-1}}{p(T-1)^{2p-1}}- M_6 G( \| u\|) \big\} \| u\|^p-M_2T. \end{aligned} \end{equation} By (2) in Remark \ref{rmk2.3}, one obtains $$ G( \| u\|) \to 0 \quad \text{as } \| u\|\to +\infty. $$ Then it is easy to get $$ \frac{T^{p-1}}{p(T-1)^{2p-1}}- M_6 G( \| u\|)>0 \quad \text{as } \| u\|\to +\infty. $$ Hence by \eqref{e14}, we get $\varphi(u)\to+\infty$ as $\| u\|\to +\infty$. Thus (2) in Lemma \ref{lem2.1} holds. In light of Lemma \ref{lem2.1}, Theorem \ref{thm1.1} is proved. \end{proof} \textbf{Acknowledgements} The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version. The work is supported by the Science Foundation of Education Department for Hubei Provincial, China (No. D20172905). \begin{thebibliography}{99} \bibitem{Ag} R. P. Agarwal; \emph{Difference equations and inequalities. Theory, methods, and applications.} Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 228. Marcel Dekker, Inc., New York, 2000. \bibitem{Ag1} R. P. Agarwal, K. Perera, D. O'Regan; \emph{Multiple positive solutions of singular and nonsingular discrete problems via variational methods}. Nonlinear Anal., 58 (2004), 69-73. \bibitem{Ag2}R. P. Agarwal, K. Perera, D. O'Regan; \emph{Multiple positive solutions of singular discrete p-Laplacian problems via variational methods.} Adv. Diff. Equ. 2005, 2005: 93-99. \bibitem{At} F. M. Atici, A. Cabada; \emph{Existence and uniquness results for discrete second-order periodic boundary value problems.} Comput. Math. Appl., 45 (2003), 1417-1427. \bibitem{Bona} G. Bonanno, P. Candito; \emph{Infinite many solutions for a class of discrete non-linear boundary value problems.} Appl. Anal., 88(4) (2009), 605-616. \bibitem{Ce} G. Cerami; \emph{An existence criterion for the critical points on unbounded manifolds.} Istit. Lombardo Accad. Sci. Lett. Rend., A 112 (1978), 332-336. \bibitem{Ca2009} A. Cabada, A. Iannizzotto, S. Tersian; \emph{Multiple solutions for discrete boundary value problems.} J. Math. Anal. Appl., 356 (2009), 418-428. \bibitem{D} G. D'Agui, J. Mawhin, A. Sciammett; \emph{Positive solutions for a discrete two point nonlinear boundary value problem with p-Laplacian}. J. Math. Anal. Appl., 447( 2017), 383-397. \bibitem{Fara} F. Faraci, A. Iannizzotto; \emph{Multiplicity theorems for discrete boundary value problems.} Aequationes Math., 74 (2007), 111-118. \bibitem{Gao2014} C. H. Gao; \emph{Solutions to discrete multiparameter periodic boundary value problems involving the p-laplacian via critical point theory.} Acta Mathematica Scientia ,34B (4) (2014), 1225-1236. \bibitem{Guo1} 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), no. 2, 419--430. \bibitem{Guo2} Z. M. Guo, J. S. Yu; \emph{Existence of periodic and subharmonic solutions for second-order superlinear difference equations}, Sci. China Ser. A, 46 (4) (2003), 506-515. \bibitem{he} T. He, W. Chen; \emph{Periodic solutions of second order discrete convex systems involving the p-Laplacian}, Appl. Math. Comput., 206 (1) (2008), 124-132. \bibitem{jiang} D. Jiang, J. Chu, D. O'Regan, R. P. Agarwal; \emph{Positive solutions for continuous and discrete boundary value problems to the one-dimension p-Laplacian}, Math. Inequal. Appl., 7 (2004), 523-534. \bibitem{jiang1} Q. Jiang, S. Ma, Z. H. Hu; \emph{Existence of multiple periodic solutions for second-order discrete Hamiltonian systems with partially periodic potentials.} Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 307, 1-14. \bibitem{luo} Z. M. Luo, X. Y. Zhang; \emph{Existence of nonconstant periodic solutions for a nonlinear discrete system involving the p-laplacian}, Bull. Malays. Math. Sci. Soc. 35 (2012), 373-382. \bibitem{Pankov1} A. Pankov, N. Zakharchenko; \emph{On some discrete variational problems}, Acta. Appl. Math 65 (2005), 295-303. \bibitem{r} P. Rabinowitz; \emph{Minimax methods in critical point theory with applications to differential equations}, in: CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc, Providence. RI, 1986. \bibitem{Ri} B. Ricceri; \emph{On a three critical points theorem.} Arch. Math., 75 (2000), 220-226. \bibitem{wang} Z. Y. Wang, J. Z. Xiao; \emph{On periodic solutions of subquadratic second order non-autonomous Hamiltonian systems}, Appl. Math. Lett. 40 (2015), 72-77. \bibitem{Xue} Y. F. Xue, C. L.Tang; \emph{Existence of a periodic solution for subquadratic second-order discrete Hamiltonian systems}, Nonlinear Anal., 67 (7) (2007), 2072-2080. \bibitem{zhang} X. Y. Zhang; \emph{Notes on periodic solutions for a nonlinear discrete system involving the p-laplacian}, Bull. Malays. Math. Sci. Soc. , 37 (2) (2014), 499-509. \end{thebibliography} \end{document}