\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{cite} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 99, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/99\hfil Existence of infinitely many periodic solutions] {Existence of infinitely many periodic solutions for second-order nonautonomous Hamiltonian systems} \author[Wen Guan, Da-Bin Wang \hfil EJDE-2015/99\hfilneg] {Wen Guan, Da-Bin Wang} \address{Wen Guan \newline Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China} \email{mathguanw@163.com} \address{Da-Bin Wang (corresponding author) \newline Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China} \email{wangdb96@163.com} \thanks{Submitted November 11, 2014. Published April 14, 2015.} \subjclass[2000]{34C25, 58E50} \keywords{Periodic solutions; Minimax methods; linear; Hamiltonian system; \hfill\break\indent critical point} \begin{abstract} By using minimax methods and critical point theory, we obtain infinitely many periodic solutions for a second-order nonautonomous Hamiltonian systems, when the gradient of potential energy does not exceed linear growth. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and main results} Consider the second-order Hamiltonian system \begin{equation} \begin{gathered} \ddot {u}(t)+\nabla F(t,u(t))=0,\quad\text{a.e. }t\in[0,T],\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0\,. \end{gathered} \label{e1.1} \end{equation} Where $T>0$ and $F:[0,T]\times \mathbb{R}^{N}\to \mathbb{R}$ satisfies the following assumption: \begin{itemize} \item[(A1)] $F(t,x)$ is measurable in $t$ for every $x\in \mathbb{R}^{N}$, continuously differentiable in $x$ for a.e. $t\in[0,T]$, and there exist $a\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, $b\in L^1([0,T],\mathbb{R}^{+})$ such that \[ |F(t,x)|\leq a(|x|)b(t),~ |\nabla F(t,x)|\leq a(|x|)b(t) \] for all $x\in \mathbb{R}^{N}$ and a.e. $t\in[0,T]$. \end{itemize} The existence of periodic solutions for problem \eqref{e1.1} was obtained in \cite{a1,b1,h1,l1,m2,m3,m4,r1,t1,t2,t3,t4,t5,w1,w2,w4,z1,z2} with many solvability conditions by using the least action principle and the minimax methods, such as the coercive type potential condition \cite{b1}, the convex type potential condition \cite{m2}, the periodic type potential conditions \cite{w1}, the even type potential condition \cite{l1}, the subquadratic potential condition in Rabinowitz's sense \cite{r1}, the bounded nonlinearity condition (see \cite{m3}), the subadditive condition (see \cite{t1}), the sublinear nonlinearity condition (see \cite{h1,t3}), and the linear nonlinearity condition (see \cite{m4,t5,z1,z2}). In particular, when the nonlinearity $\nabla F(t,x)$ is bounded; that is, there exists $g(t)\in L^1([0,T],\mathbb{R}^{+})$ such that $|\nabla F(t,x)|\leq g(t)$ for all $x\in\mathbb{R}^{N}$ and a.e. $t\in[0,T]$, and that \[ \int_0^TF(t,x)dt\to\pm\infty\quad\text{as } |x|\to\infty, \] Mawhin and Willem \cite{m3} proved that problem \eqref{e1.1} has at least one periodic solution. Han and Tang \cite{h1,t3} generalized these results to the sublinear case: \begin{equation} |\nabla F(t,x)|\leq f(t)|x|^{\alpha}+g(t)\quad \text{for all $x\in\mathbb{R}^{N}$ and a.e. } t\in[0,T] \label{e1.2} \end{equation} with \begin{equation} |x|^{-2\alpha}\int_0^TF(t,x)dt\to\pm\infty\quad \text{as } |x|\to\infty, \end{equation} where $f(t),g(t)\in L^1([0,T],\mathbb{R}^{+})$ and $\alpha\in[0,1)$. Subsequently, when $\alpha=1$ Zhao and Wu \cite{z1,z2}, and Meng and Tang \cite{m4,t5} proved the existence of periodic solutions for problem \eqref{e1.1}, i.e. $\nabla F(t,x)$ does not exceed linear growth: \begin{equation} |\nabla F(t,x)|\leq f(t)|x|+g(t)\quad\text{for all $x\in\mathbb{R}^{N}$ and~ a.e. } t\in[0,T], \label{e1.3} \end{equation} where $f(t),g(t)\in L^1([0,T],\mathbb{R}^{+})$. On the other hand, there are large number of papers that deals with multiplicity results for this problem. In particular, infinitely many solutions for \eqref{e1.1} are obtained in \cite{a2,w3,z3} when the nonlinearity $F(t,x)$ have symmetry. Since the symmetry assumption on the nonlinearity $F$ has play an important role in \cite{a2,w4,z3}, many authors have paid much attention to weak the symmetry condition and some existence results on periodic solutions have been obtained without any symmetry condition \cite{f1,m1,t6,z4}. Especially, Zhang and Tang \cite{z4} obtained infinitely many periodic solutions for \eqref{e1.1} when \eqref{e1.2} holds and $F$ has a suitable oscillating behaviour at infinity: \begin{gather*} \limsup_{r\to+\infty}\inf_{x\in\mathbb{R}^{N},|x|=r}|x|^{-2\alpha} \int^T_0F(t,x)dt=+\infty, \\ \liminf_{R\to+\infty}\sup_{x\in\mathbb{R}^{N},|x|=R}|x|^{-2\alpha} \int^T_0F(t,x)dt=-\infty, \end{gather*} where $\alpha\in[0,1)$. Motivated by the results mentioned above, especially by ideas in \cite{m4,t5,z1,z2,z4}, in this article, by using the minimax methods in critical point theory, we obtain infinitely many periodic solutions for \eqref{e1.1}. Let $H^1_T$ be a Hilbert space $H^1_T=\big\{u:[0,T]\to\mathbb{R}^{N}: u$ is absolutely continuous, $u(0)=u(T)$ and $\dot{u}\in L^2([0,T],\mathbb{R})\big\}$, with the norm \begin{equation} \|u\|=\Big(\int^T_0|u(t)|^2dt +\int^T_0|\dot{u}(t)|^2dt\Big)^{1/2}, \end{equation} for $u\in H^1_T$. Let \begin{equation} J(u)=\frac{1}{2}\int^T_0|\dot{u}(t)|^2dt-\int^T_0F(t,u(t))dt. \end{equation} It is well known that the function $J$ is continuously differentiable and weakly lower semicontinuous on $ H^1_T$ and the solutions of \eqref{e1.1} correspond to the critical points of $J$ (see \cite{m3}). Our main result is the following theorem. \begin{theorem} \label{thm1.1} Suppose that {\rm (A1)} and \eqref{e1.3} with $\int^T_0f(t)dt<\frac{3}{T}$ hold and \begin{gather} \limsup_{r\to+\infty}\inf_{x\in\mathbb{R}^{N},|x|=r}\int^T_0F(t,x)dt=+\infty, \label{e1.4} \\ \liminf_{R\to+\infty}\sup_{x\in\mathbb{R}^{N},|x|=R}|x|^{-2}\int^T_0F(t,x)dt <-\frac{3T^2}{2\pi^2(12-T\int^T_0f(t)dt)} \int^T_0f^2(t)dt. \label{e1.5} \end{gather} Then \begin{itemize} \item[(i)] There exists a sequence of periodic solutions $\{u_{n}\}$ which are minimax type critical points of functional $J$, and $J(u_{n})\to+\infty$ as $n\to\infty$; \item[(ii)] There exists another sequence of periodic solutions $\{u^{*}_{m}\}$ which are local minimum points of functional $J$, and $J(u^{*}_{m})\to-\infty$ as $m\to\infty$. \end{itemize} \end{theorem} \begin{remark} \label{rmk1.1}\rm \item[(i)] As in \cite{z4}, in this paper we do not assume any symmetry condition on nonlinearity; \item[(ii)] Our main result in this paper extends main result in \cite{z4} corresponding to $\alpha=1$. \end{remark} \section{Proof of main results} For $u\in H^1_T$, let \begin{equation} \overline{u}=\frac{1}{T}\int^T_0u(t)dt, ~\widetilde{u}(t)=u(t)-\overline{u}. \end{equation} The following inequalities are well known (see \cite{m3}): \begin{gather*} \|\tilde{u}\|_{\infty}^2 \leq \frac{T}{12}\|\dot{u}\|_{{L}^2}^2 \quad\text{(Sobolev's ~inequality)},\\ \|\tilde{u}\|_{L^2}^2 \leq\frac{T^2}{4\pi^2}\|\dot{u}\|_{{L}^2}^2 \quad \text{(Wirtinger's ~inequality)}. \end{gather*} For the sake of convenience, we denote \[ M_1=\Big(\int_0^Tf^2(t)dt\Big)^{1/2},\quad M_2=\int_0^Tf(t)dt,\quad M_3=\int_0^Tg(t)dt. \] \begin{lemma} \label{lem2.1} Suppose that $\int^T_0f(t)dt<3/T$ and \eqref{e1.3} hold, then\ \begin{equation} J(u)\to+\infty \quad \text{as } \|u\|\to\infty~ in ~\widetilde{H}^1_T, \end{equation} where $\widetilde{H}^1_T=\{u\in H^1_T\mid \overline{u}=0\}$ be the subspace of $H^1_T$. \end{lemma} \begin{proof} From \eqref{e1.3} and Sobolev's inequality, for all $u$ in $\widetilde{H}^1_T$ we have \begin{align*} J(u)&=\frac{1}{2}\int^T_0|\dot{u}(t)|^2dt-\int^T_0F(t,u(t))dt\\ &\geq \frac{1}{2}\int^T_0|\dot{u}(t)|^2dt -\int^T_0f(t)|u(t)|^2dt-\int^T_0g(t)|u(t)|dt\\ &\geq \frac{1}{2}\int^T_0|\dot{u}(t)|^2dt -\|\widetilde{u}\|^2_{\infty}\int^T_0f(t)dt -\|\widetilde{u}\|_{\infty}\int^T_0g(t)dt\\ &\geq \frac{1}{2}\|\dot{u}\|^2_{L^2} -\frac{T}{12}\|\dot{u}\|^2_{L^2}\int^T_0f(t)dt -\big(\frac{T}{12}\big)^{1/2}\|\dot{u}\|_{L^2}\int^T_0g(t)dt\\ &= \Big(\frac{1}{2}-\frac{T}{12}\int^T_0f(t)dt\Big) \|\dot{u}\|^2_{L^2}-C_1\|\dot{u}\|_{L^2}\,. \end{align*} \indent By Wirtinger's inequality, the norm $\|u\|=\big(\int^T_0|\dot{u}(t)|^2dt\big)^{1/2}$ is an equivalent norm on $\widetilde{H}^1_T$. So, $J(u)\to+\infty$ as $\|u\|\to\infty$ in $\widetilde{H}^1_T$. \end{proof} \begin{lemma} \label{lem2.2} Suppose that \eqref{e1.4} holds. Then there exists positive real sequence $\{a_{n}\}$ such that \[ \lim_{n\to\infty}a_{n}=+\infty,\quad \lim_{n\to\infty}\sup_{u\in\mathbb{R}^{N},|u|=a_{n}}J(u)=-\infty\,. \] \end{lemma} The above lemma follows from \eqref{e1.4}. \begin{lemma} \label{lem2.3} Suppose that $\int^T_0f(t)dt<\frac{3}{T}$, \eqref{e1.3} and \eqref{e1.5} hold. Then there exists positive real sequence $\{b_{m}\}$ such that \[ \lim_{m\to\infty}b_{m}=+\infty, \quad \lim_{m\to\infty}\inf_{u\in H_{b_{m}}}J(u)=+\infty, \] where $H_{b_{m}}=\{u\in\mathbb{R^{N}}:|u|=b_{m}\}\bigoplus\widetilde{H}^1_T$. \end{lemma} \begin{proof} By \eqref{e1.5}, we can choose an $a>3T^2/(12\pi^2-\pi^2TM_2)$ such that \[ \liminf_{r\to+\infty}\sup_{x\in\mathbb{R}^{N},|x|=r}|x|^{-2}\int^T_0F(t,x)dt <-\frac{a}{2}M^2_1. \] For any $u\in H_{b_{m}}$, let $u=\overline{u}+\widetilde{u}$, where $|\overline{u}|=b_{m}$, $\widetilde{u}\in \widetilde{H}^1_T$. So, we have \begin{align*} &\big|\int^T_0F(t,u(t))-F(t,\overline{u})dt\big|\\ &= \big|\int^T_0\int^1_0(\nabla F (t,\overline{u} +s\widetilde{u}(t),\widetilde{u}(t))\,ds\,dt\big|\\ &\leq \int^T_0\int^1_0f(t)|\overline{u} +s\widetilde{u}(t)||\widetilde{u}(t)|\,ds\,dt +\int^T_0\int^1_0g(t)|\widetilde{u}(t)|\,ds\,dt\\ &\leq \int^T_0f(t)\left(|\overline{u}| +\frac{1}{2}|\widetilde{u}(t)|\right)|\widetilde{u}(t)|dt+\int^T_0 g(t)|\widetilde{u}(t)|dt\\ &\leq |\overline{u}|\Big(\int^T_0f^2(t)dt\Big)^{1/2} \Big(\int^T_0|\widetilde{u}(t)|^2dt\Big)^{1/2} +\frac{1}{2}\|\widetilde{u}\|^2_{\infty}\int^T_0f(t)dt +\|\widetilde{u}\|_{\infty}\int^T_0g(t)dt\\ &= M_1|\overline{u}|\|\widetilde{u}\|_{L^2} +\frac{M_2}{2}\|\widetilde{u}\|^2_{\infty}+M_3\|\widetilde{u}\|_{\infty}\\ &\leq \frac{1}{2a}\|\widetilde{u}\|^2_{L^2} +\frac{a}{2}M^2_1|\overline{u}|^2 +\frac{M_2}{2}\|\widetilde{u}^2\|_{\infty}+M_3\|\widetilde{u}\|_{\infty}\\ &\leq \big(\frac{T^2}{8a\pi^2}+\frac{TM_2}{24}\big) \|\dot{u}\|^2_{L^2}+\frac{a}{2}M^2_1|\overline{u}|^2 +\big(\frac{T}{12}\big)^{1/2}M_3\|\dot{u}\|_{L^2} \end{align*} for all $u\in H_{b_{m}}$. Hence we have \begin{align*} J(u)&= \frac{1}{2}\int^T_0|\dot{u}(t)|^2dt -\int^T_0[F(t,u(t))-F(t,\overline{u})]dt-\int^T_0F(t,\overline{u})dt\\ &\geq \big(\frac{1}{2}-\frac{T^2}{8a\pi^2} -\frac{TM_2}{24}\big)\|\dot{u}\|^2_{L^2} -\big(\frac{T}{12}\big)^{1/2}M_3\|\dot{u}\|_{L^2}\\ &-|\overline{u}|^2\Big(|\overline{u}|^{-2}\int^T_0 F(t,\overline{u})dt+\frac{a}{2}M^2_1\Big) \end{align*} for all $u\in H_{b_{m}}$. As $\left(|\overline{u}|^2+\|\dot{u}\|_{L^2}\right)^\frac{1}{2}\to\infty$ if and only if $\|u\|\to\infty$, then the Lemma follows from \eqref{e1.5} and the above inequality. \end{proof} Now prove our main result. \begin{proof}[Proof of Theorem \ref{thm1.1}] Let $B_{a_{n}}$ be a ball in $\mathbb{R}^{N}$ with radius $a_{n}$. Then we define a family of maps \[ \Gamma_{n}=\{\gamma\in C(B_{a_{n}},H^1_T):\gamma\big|_{\partial B_{a_{n}}} =Id\big|_{\partial B_{a_{n}}}\} \] and corresponding minimax values \[ c_{n}=\inf_{\gamma\in\Gamma_{n}}\max_{x\in B_{a_{n}}} J(\gamma(x)). \] It is easy to see that each $\gamma$ intersects the hyperplane $\widetilde{H}^1_T$, i.e., for any $\gamma\in\Gamma_{n}$, $\gamma(B_{}a_{n})\cap\widetilde{H}^1_T\neq\emptyset$. By Lemma \ref{lem2.1}, the functional $J$ is coercive on $\widetilde{H}^1_T$. So, there is a constant $M$ such that \[ \max_{x\in B_{a_{n}}} J(\gamma(x))\geq\inf_{u\in\widetilde{H}^1_T} J(u)\geq M. \] Hence \[ c_{n}\geq\inf_{u\in\widetilde{H}^1_T} J(u)\geq M. \] By Lemma \ref{lem2.2}, for all large value of $n$, \[ c_{n}>\max_{u\in\partial B_{a_{n}}} J(u). \] For such $n$, there exists a sequence $\{\gamma_{k}\}$ in $\Gamma_{n}$ such that \[ \max_{x\in B_{a_{n}}} J(\gamma_{k}(x))\to c_{n}, k\to\infty. \] Applying \cite[Theorem 4.3 and Corollary 4.3]{m3}, we know there exists a sequence $\{v_{k}\}$ in $H^1_T$ such that\ \begin{equation} J(v_{k})\to c_{n}, \operatorname{dist}(v_{k},\gamma_{k}(B_{a_{n}}))\to0, J'(v_{k})\to0, \label{e2.1} \end{equation} as $k\to\infty$. If we can show $\{v_{k}\}$ is bounded, then there is a subsequence, which is still be denote by $\{v_{k}\}$ such that \begin{gather*} v_{k}\rightharpoonup u_{n} \quad \text{weakly in } H^1_T, \\ v_{k}\to u_{n}\quad \text{uniformly in } C([0,T],\mathbb{R}^{N}). \end{gather*} Hence \begin{gather*} \langle J'(v_{k})-J'(u_{n}),v_{k}-u_{n}\rangle\to0, \\ \int^T_0(\nabla F(t,v_{k})-\nabla F(t,u_{n}),v_{k}-u_{n})dt\to0 \end{gather*} as $k\to\infty$. Moreover, it is easy to see that \begin{align*} &\langle J'(v_{k})-J'(u_{n}),v_{k}-u_{n}\rangle\\ &= \|\dot{v_{k}}-\dot{u_{n}}\|^2_{L^2}-\int^T_0(\nabla F(t,v_{k})-\nabla F(t,u_{n}),v_{k}-u_{n})dt, \end{align*} so $\|\dot{v_{k}}-\dot{u_{n}}\|^2_{L^2}\to0$ as $k\to\infty$. Then, it is not difficult to obtain $\|v_{k}-v_{n}\|\to0$ as $k\to\infty$. So, we have \[ J'(u_{n})=\lim_{k\to\infty} J'(v_{k})=0,\quad J(u_{n})=\lim_{k\to\infty} J(v_{k})=c_{n}. \] Thus, $u_{n}$ is critical point and $c_{n}$ is critical value of functional $J$. Now, let us show the sequence $\{v_{k}\}$ is bounded in $H^1_T$. By \eqref{e2.1}, for any large enough $k$, we have \begin{equation} c_{n}\leq\max_{x\in B_{a_{n}}} J(\gamma_{k}(x))\leq c_{n}+1, \end{equation} and we can find $w_{k}\in \gamma_{k}(B_{a_{n}})$ such that $\|v_{k}-w_{k}\|\leq1$. Fix $n$, by Lemma \ref{lem2.3}, we can choose a large enough $m$ such that \[ b_{m}>a_{n}\quad \text{and}\quad \inf_{u\in H_{b_{m}}}> c_{n}+1. \] This implies $\gamma(B_{a_{n}})$ cannot intersect the hyperplane $H_{b_{m}}$ for each $k$. Let $w_{k}=\overline{w}_{k}+\widetilde{w}_{k}$, where $\overline{w}_{k}\in\mathbb{R}^{N}$ and $\widetilde{w}_{k}\in\widetilde{H}^1_T$. Then we have $|\overline{w}_{k}|b_{m}$, then $\gamma(B_{a_{n}})$ intersects the hyperplane $H_{b_{m}}$ for any $\gamma\in\Gamma_{n}$. It follows that \[ \max_{x\in B_{a_{n}}} J(\gamma(x))\geq\inf_{u\in H_{b_{m}}} J(u). \] From this inequality and Lemma \ref{lem2.3} we obtain $\lim_{n\to\infty}c_{n}=+\infty$. Result (i) of Theorem \ref{thm1.1} is obtained. Next we prove (ii). For fixed $m$, define the subset $P_{m}$ of $H^1_T$ by \begin{equation} P_{m}=\{u\in H^1_T:u=\overline{u}+\widetilde{u},|\overline{u}| \leq b_{m},\widetilde{u}\in\widetilde{H}^1_T\}. \end{equation} For $u\in P_{m}$, we have \begin{equation} \begin{aligned} J(u) &= \frac{1}{2}\int^T_0|\dot{u}(t)|^2dt-\int^T_0F(t,u(t))dt\\ &\geq \frac{1}{2}\int^T_0|\dot{u}(t)|^2dt -\int^T_0f(t)|u(t)|^2dt-\int^T_0g(t)|u(t)|dt\\ &\geq \frac{1}{2}\int^T_0|\dot{u}(t)|^2dt -2\int^T_0f(t)[|\overline{u}(t)|^2+|\widetilde{u}(t)|^2]dt -\int^T_0g(t)[|\overline{u}(t)|+|\widetilde{u}(t)|]dt\\ &\geq \frac{1}{2}\|\dot{u}(t)\|^2_{L^2} -\frac{T}{6}\|\dot{u}(t)\|^2_{L^2}\int^T_0f(t)dt-2|\overline u(t)|^2 \int^T_0f(t)dt\\ &\quad -\big(\frac{T}{12}\big)^{1/2}\|\dot{u}(t)\|_{L^2} \int^T_0g(t)dt-|\overline{u}(t)|\int^T_0g(t)dt\\ &= \big(\frac{1}{2}-\frac{T}{6}M_2\big)\|\dot{u}(t)\|^2_{L^2} -\big(\frac{T}{12}\big)^{1/2}M_3\|\dot{u}(t)\|_{L^2}-C_3 \end{aligned} \label{e2.6} \end{equation} Then $J$ is bounded below on $P_{m}$. Let \[ \mu_{m}=\inf_{u\in P_{m}} J(u), \] and $\{u_{k}\}$ be a minimizing sequence in $P_{m}$; that is, \[ J(u_{k})\to\mu_{m}\quad \text{as } k\to\infty. \] By \eqref{e2.6}, $\{u_{k}\}$ is bounded in $H^1_T$. Then there is a subsequence, which is still be denoted by $\{u_{k}\}$, such that \[ u_{k}\rightharpoonup u^{*}_{m} ~\text{weakly in}~ H^1_T. \] Since $P_{m}$ is a convex closed subset of $ H^1_T$, $u^{*}_{m}\in P_{m}$. As $J$ is weakly lower semicontinuous, we have \[ \mu_{m}=\lim_{k\to\infty} J(u_{k})\geq J(u^{*}_{m}). \] Since $u^{*}_{m}\in P_{m}$, $\mu_{m}=J(u^{*}_{m})$. If we can show $u^{*}_{m}$ is in the interior of $P_{m}$, then $u^{*}_{m}$ is a local minimum of functional $J$. In fact, let $u^{*}_{m}=\overline{u}^{*}_{m}+\widetilde{u}^{*}_{m}$. From Lemmas \ref{lem2.2} and \ref{lem2.3}, we see $|\overline{u}^{*}_{m}|\neq b_{m}$ for large $m$, which means that ${u}^{*}_{m}$ is in the interior of $P_{m}$. Since $u^{*}_{m}$ is a minimum of $J$ on $P_{m}$, we have \[ J(u^{*}_{m})=\inf_{u\in P_{m}} J(u)\leq\sup_{|u|=b_{m}} J(u). \] It follows from Lemma \ref{lem2.2} that $J(u^{*}_{m})\to-\infty$ as $m\to\infty$. Therefore, the proof is complete. \end{proof} \begin{thebibliography}{00} \bibitem{a1} Nurbek Aizmahin, Tianqing An; The existence of periodic solutions of non-autonomous second-order Hamiltonian systems, \textit{Nonlinear Analysis}, 74 (2011), 4862-4867. \bibitem{a2} F. Antonacci, P. Magrone; Second order nonautonomous systems with symmetric potential changing sign, \textit{Rendiconti di Matematica e delle sue Applicazioni}, 18 (1988), 367-379. \bibitem{b1} M. S. Berger, M. Schechter; On the solvability of semilinear gradient operator equations, \textit{Adv. Math.}, 25 (1977), 97-132. \bibitem{f1} F. Faraci, R. Livrea; Infinitely many periodic solutions for a second-order nonautonomous system, \textit{Nonlinear Anal.}, 54 (2003), 417-429. \bibitem{h1} Z. Q. Han; 2$\pi$-Periodic solutions to $n$-Duffing systems, Nonlinear Analysis and Its Aplications (Deited by D. J. Guo), Beijing: Beijng Scientific and Technical Publisher, 1994, 182-191. (in Chinese) \bibitem{l1} Y. M. Long; Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, \textit{Nonlinear Anal.}, 24 (1995), 1665-1671. \bibitem{m1} S. W. Ma, Y. X. Zhang; Existence of infinitely many periodic solutions for ordinary $p$-Laplacian systems, \textit{J. Math. Anal. Appl.}, 351 (2009), 469-479. \bibitem{m2} J. Mawhin; Semi-coercive monotone variational problems, \textit{Acad. Roy. Belg. Bull. Cl. Sci.},73 (1987), 118-130. \bibitem{m3}J. Mawhin, M. Willem; Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. \bibitem{m4} Q. Meng, X. H. Tang; Solutions of a second-order Hamiltonian sysytem with periodic boundary conditions, \textit{Comm. Pure Appl. Anal.}, 9 (2010), 1053-1067. \bibitem{r1} P. H. Rabinowitz; On subharmonic solutions of Hamiltonian systems, \textit{Comm. Pure Appl.} Math., 33 (1980), 609-633. \bibitem {t1} C. L. Tang; Periodic solutions of nonautonomous second order systems with ¦Ã- quasisubadditive potential, \textit{J. Math. Anal. Appl.}, 189 (1995), 671-675. \bibitem{t2} C. L. Tang; Periodic solutions of nonautonomous second order systems, \textit{J. Math. Anal.} Appl., 202 (1996), 465-469. \bibitem{t3}C. L. Tang; Periodic solutions of nonautonomous second order systems with sublinear nonlinearity, \textit{Proc. Amer. Math. Soc.}, 126 (1998), 3263-3270. \bibitem{t4}C. L. Tang, X. P. Wu; Periodic solutions for second order systems with not uniformly coercive potentia, \textit{J. Math. Anal. Appl.}, 259, (2001) 386-397. \bibitem{t5} X. H. Tang, Q. Meng; Solutions of a second-order Hamiltonian system with periodic boundary conditions, \textit{Nonlinear Analysis:} Real World Applications, 11 (2010), 3722-3733. \bibitem{t6} Z. L. Tao, C. L. Tang; Periodic and subharmonic solutions of second-order Hamiltonian systems, \textit{J. Math. Anal. Appl.}, 293 (2004), 435-445. \bibitem{w1} M. Willem; Oscillations forces de systmes hamiltoniens, in: Public. Smin. Analyse Non Linaire, Univ. Besancon, 1981. \bibitem{w2} X. Wu; Saddle point characterization and multiplicity of periodic solutions of nonautonomous second order systems, \textit{Nonlinear Anal.}, 58 (2004), 899-907. \bibitem{w3} X. P. Wu, C. L. Tang; Periodic solutions of a class of nonautonomous second order systems, \textit{J. Math. Anal. Appl.}, 236 (1999), 227-235. \bibitem{w4} X. P. Wu, C. L. Tang; Periodic solutions of nonautonomous second-order Hamiltonian systems with even-typed potentials, \textit{Nonlinear Anal.}, 55 (2003), 759-769. \bibitem{z1} F. Zhao, X. Wu; Periodic solutions for a class of non-autonomous second order systems, \textit{J. Math. Anal. Appl.}, 296 (2004), 422-434. \bibitem{z2} F. Zhao, X. Wu; Existence and multiplicity of periodic solution for non-autonomous secondorder systems with linear nonlinearity, \textit{Nonlinear Anal.}, 60 (2005), 325-335. \bibitem{z3} W. M. Zou, S. J. Li; Infinitely many solutions for Hamiltonian systems, \textit{Journal of Differential Equations}, 186 (2002), 141-164. \bibitem{z4} P. Zhang, C. L. Tang; Infinitely many periodic solutions for nonautonomous sublinear second-order Hamiltonian systems, \textit{Abstract and Applied Analysis}, Volume 2010, Article ID 620438, 10 pages. \end{thebibliography} \end{document}