\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{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 $\| u\|\to +\infty$. 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