\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 121, 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{9mm}} \begin{document} \title[\hfilneg EJDE-2014/121\hfil Existence and multiplicity of periodic solutions] {Existence and multiplicity of periodic solutions generated by impulses for second-order Hamiltonian system} \author[D. Zhang, Q. Wu, B. Dai \hfil EJDE-2014/121\hfilneg] {Dan Zhang, Qinghua Wu, Binxiang Dai} % in alphabetical order \address{Dan Zhang \newline Institute of Computational Mathematics, Department of Mathematics, Hunan University of Science and Engineering, Yongzhou, 425100 Hunan, China} \email{zhang11dan@126.com} \address{Qinghua Wu \newline Institute of Computational Mathematics, Department of Mathematics, Hunan University of Science and Engineering, Yongzhou, 425100 Hunan, China} \email{jackwqh@163.com} \address{Binxiang Dai (corresponding author) \newline School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China} \email{binxiangdai@126.com} \thanks{Submitted February 18, 2014. Published May 2, 2014.} \subjclass[2000]{34K50, 37K05} \keywords{Impulsive differential equations; critical point theory; \hfill\break\indent periodic solution; variant fountain theorems} \begin{abstract} In this article, we study the existence of non-zero periodic solutions for Hamiltonian systems with impulsive conditions. By using a variational method and a variant fountain theorem, we obtain new criteria to guarantee that the system has at least one non-zero periodic solution or infinitely many non-zero periodic solutions. However, without impulses, there is no non-zero periodic solution for the system under our conditions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we consider the problem \begin{equation} \begin{gathered} \ddot{q}(t)=f(t,q(t)),\quad \text{a.e. }t\in(s_{j-1},s_j),\\ \Delta \dot{q}(s_j)=G_j(q(s_j)),\quad j=1,2,\dots,m, \end{gathered}\label{e1.1} \end{equation} where $j\in \mathbb{Z}$, $q\in \mathbb{R}^n$, $\Delta\dot{q}(s_j)=\dot{q}(s_j^{+})-\dot{q}(s_j^{-})$ with $\dot{q}(s_j^{\pm})=\lim_{t\to s_j^{\pm}}\dot{q}(t)$, $f(t,q)=\operatorname{grad}_{q}F(t,q),F(t,q)\in C^1(\mathbb{R}\times\mathbb{R}^n, \mathbb{R}), g_j(q)=\operatorname{grad}_{q}G_j(q), G_j(q)\in C^1(\mathbb{R}^n, \mathbb{R})$ for each $k\in \mathbb{Z}$ and there exist $m\in \mathbb{N}$ and $T\in \mathbb{R}^+$ such that $0=s_00$ for all $t\in [0,T]$ and $q\in \mathbb{R}^{n}\setminus \{0\}$. \item[(F2)] $f(t,q)=\alpha q +w(t,q)$ for all $t\in [0,T]$ and $q\in \mathbb{R}^{n}$, where $\alpha>\frac{2}{\mu}$ for some $\mu >2$, $w(t,q)=\operatorname{grad}_{q}W(t,q)$ and $W(t,q)\geq \frac{1}{2}w(t,q)q>0$ for all $t\in [0,T]$ and $q\in \mathbb{R}^{n}$. \item[(F3)] $F(t,q)\geq \frac{1}{2}f(t,q)q\geq 0$ for all $t\in [0,T]$ and $q\in \mathbb{R}^{n}$. \item[(G1)] There exists $\mu>2$ such that $g_j(q)q\leq \mu G_j(q)<0$ for all $j=1,2,\dots,m$ and $q\in \mathbb{R}^{n}\setminus \{0\}$. \item[(G2)] $g_j(q)=2q +w_j(q)$, where $w_j(q)=\operatorname{grad}_{q}W_j(q)$ and satisfy that there exists $\mu>2$ such that $ w_j(q)q\leq \mu W_j(q)<0$ for all $j=1,2,\dots,m$ and $q\in \mathbb{R}^{n}$. \end{itemize} By using critical point theory, they obtained the following theorems. \begin{theorem}[{\cite[Theorem 1]{h1}}] \label{thm1.1} If $F$ is $T$-periodic in $t$ and satisfies {\rm (F1)}, $g_j(q)$ satisfies {\rm (G1)} for all $j=1,2,\dots,m$, then \eqref{e1.1} possesses at least one non-zero periodic solution generated by impulses. \end{theorem} \begin{corollary}[{\cite[Corollary 1]{h1}}] \label{coro1.1} If $F$ is $T$-periodic in $t$ and satisfies {\rm (F2)}, $g_j(q)$ satisfies {\rm (G1)} for all $j=1,2,\dots,m$, then \eqref{e1.1} possesses at least one non-zero periodic solution generated by impulses. \end{corollary} \begin{theorem}[{\cite[Theorem 2]{h1}}] If $F$ is $T$-periodic in $t$ and satisfies (F3), $g_j(q)$ satisfies {\rm (G2)} for all $j=1,2,\dots,m$, then \eqref{e1.1} possesses at least one non-zero periodic solution generated by impulses. \end{theorem} They only obtained that system \eqref{e1.1} has at least non-zero periodic solution generated by impulses when $f(t,x)$ is asymptotically linear or sublinear. However, there is no work on studying the existence of infinitely many solutions generated by impulses for system \eqref{e1.1} when $f$ is asymptotically linear or sublinear. As a result, the goal of this article is to fill the gap in this area. By using a variant fountain theorem, the results that the system \eqref{e1.1} has infinitely many periodic solutions generated by impulses will be obtained. Meanwhile, we will also consider some cases which are not included in \cite{h1}. This article is organized as follows. In Section 2, we present some preliminaries. In Section 3, we give the main results and their proofs. Some examples are presented to illustrate our main results in the last section. \section{Preliminaries}\label{sec2} First, we introduce some notation and some definitions. Let $E$ be a Banach space with the norm $\|\cdot\|$ and $E=\overline{\oplus_{j\in N}X_j}$ with $\dim X_j<\infty$ for any $j\in \mathbb{N}$. Set $Y_k=\oplus_{j=0}^{k}X_j$, $Z_k=\overline{\oplus_{j=k}^{\infty}X_j}$, $B_k=\{u\in Y_k:\|u\|\leq \rho_k\}$, $N_k=\{u\in Z_k:\|u\|= r_k\}$, here $ \rho_k>r_k>0$. Consider the following $C^1$-functional $\varphi_{\lambda}:E\to \mathbb{R}$ defined by $$ \varphi_{\lambda}(u)=A(u)-\lambda B(u),\quad \lambda\in[1,2]. $$ Assumed that: \begin{itemize} \item[((C1)] $\varphi_{\lambda}$ maps bounded sets to bounded sets uniformly for $\lambda\in[1,2]$. Furthermore, $\varphi_{\lambda}(-u)=\varphi_{\lambda}(u)$ for all $(\lambda,u)\in[1,2]\times E$. \item[(C2)] $B(u)\geq 0$ for all $u\in E$; $A(u)\to \infty$ or $B(u)\to \infty$ as $\|u\|\to\infty$; or \item[(C2')] $B(u)\leq 0$ for all $u\in E$; $B(u)\to -\infty$ as $\|u\|\to\infty$. \end{itemize} For $k\geq2$, define $\Gamma_k =\{\gamma\in C(B_k,E):\gamma ~\text{is odd}; \gamma| \partial B_k=id\}$, \begin{gather*} c_k(\lambda):= \inf _{u\in \Gamma_k}\max_{u\in B_k}\varphi_{\lambda}(u) \\ b_k(\lambda):= \inf _{u\in Z_k,\|u\|=r_k}\varphi_{\lambda}(u) \\ a_k(\lambda):= \max _{u\in Y_k,\|u\|=\rho_k}\varphi_{\lambda}(u) \end{gather*} \begin{theorem}[{\cite[Theorem 2.1]{z5}}] \label{thm2.1} Assume that {\rm (C1)} and {\rm (C2)} (or {\rm (C2')}) hold. If $b_k(\lambda)> a_k(\lambda)$ for all $\lambda\in[1,2]$, then $c_k(\lambda)\geq b_k(\lambda)$ for all $\lambda\in[1,2]$. Moreover, for a.e. $\lambda\in[1,2]$, there exists a sequence $\{u_{n}^k(\lambda)\}_{n=1}^{\infty}$ such that $\sup_{n}\|u_{n}^k(\lambda)\|<\infty, \varphi'_{\lambda}(u_{n}^k(\lambda))\to 0$ and $\varphi_{\lambda}(u_{n}^k(\lambda))\to c_k(\lambda)$ as $n\to\infty$. In particular, if $\{u_{n}^k\}$ has a convergent subsequence for every $k$, then $\varphi_1$ has infinitely many nontrivial critical points $\{u_k\}\subset E\setminus \{0\}$ satisfying $\varphi_1(u_k)\to 0^{-}$ as $k\to\infty$. \end{theorem} Let $E=\{q:\mathbb{R}\to\mathbb{R}^n\text{ is absolutely continuous, } \dot{q}\in L^{2}((0,T),\mathbb{R}^n) \text{ and } q(t)=q(t+T) \text{ for }t\in \mathbb{R}\}$, equipped with the norm $$ \|q\| =\Big( \int_0^{T}|\dot{q}(t)|^{2}dt\Big)^{1/2}. $$ It is easy to verify that $E$ is a reflexive Banach space. We define the norm in $C([0,T],\mathbb{R}^n)$ as $\|q\|_{\infty}=\max_{t\in [0,T]}|q(t)|$. Since $E$ is continuously embedded into $C([0,T],\mathbb{R}^n)$, then there exists a constant $C>0$ such that \begin{equation} \|q\|_{\infty}\leq C\|q\|,\quad \forall q\in E. \end{equation} For each $q\in E$, consider the functional $\varphi$ defined on $E$ by \begin{equation} \label{e2.2} \begin{aligned} \varphi(q) &=\frac{1}{2}\int_0^{T}|\dot{q}(t)|^{2}dt +\int_0^{T}F(t,q(t))dt+\sum^{m}_{j=1}G_j(q(s_j)) \\ &=\frac{1}{2}\|q\|^{2}+\int_0^{T}F(t,q(t))dt +\sum^{m}_{j=1}G_j(q(s_j)). \end{aligned} \end{equation} Suppose $F$ is $T$-periodic in $t$ and $g_j(s)$ are continuous for $j=1,2,\dots,m$, then $\varphi$ is differentiable at any $q\in E$ and \begin{equation} \label{e2.3} \varphi'(q)p=\int_0^{T}\dot{q}(t)\dot{p}(t)dt +\int_0^{T}f(t,q(t))p(t)dt+\sum^{m}_{j=1}g_j(q(s_j))p(s_j), \end{equation} for any $p\in E$. Obviously, $\varphi'$ is continuous. \begin{lemma}[{\cite[Lemma 1]{h1}}] \label{lem2.1} If $q\in E$ is a critical point of the functional $\varphi$, then $q$ is a $T$-periodic solution of system \eqref{e1.1}. \end{lemma} \begin{definition}[{\cite[P 81]{m1}}] \label{def2.1} \rm Let $E$ be a real reflexive Banach space. For any sequence $ \{q_{n}\}\subset E$, if $ \{\varphi(q_{n})\} $ is bounded and $ \varphi'(q_{n})\to 0 $ as $ n\to \infty$ possesses a convergent subsequence, then we say $ \varphi $ satisfies the Palais-Smale condition (denoted by the PS condition for short). \end{definition} \begin{theorem}[{\cite[Theorem 2.7]{r3}}] \label{thm2.2} Let $E$ be a real Banach space and $\varphi\in C^{1}(E,\mathbb{R})$ satisfying the (PS) condition. If $\varphi$ is bounded from below, then $$ c=\inf_{E}\varphi $$ is a critical value of $\varphi$. \end{theorem} \section{Main results}\label{sec3} In this article, we will use the following conditions: \begin{itemize} \item[(V1)] There exists a constant $\delta_1\in [0,\frac{1}{2TC^2})$ such that $f(t,q)q\geq -\delta_1|q|^2$ for all $t\in [0,T]$ and $q\in \mathbb{R}^{n}$. \item[(V2)] $F(t,q)\geq \frac{1}{\beta}f(t,q)q$ for all $t\in [0,T]$ and $q\in \mathbb{R}^{n}$, where $\beta>1$. \item[(V3)] There exist constants $\delta_2>0$ and $\gamma\in [0,2)$ such that $F(t,q)\geq -\delta_2|q|^\gamma$ for all $t\in [0,T]$ and $q\in \mathbb{R}^{n}$. \item[(S1)] $G_j(q)\geq 0$ for all $j=1,2,\dots,m$ and $q\in \mathbb{R}^{n}$. \item[(S2)] There cannot exist a constant $q$ such that $g_j(q)=0 $ for all $j=1,2,\dots,m$. \item[(S3)] There exist constants $a_j, b_j>0$ and $\gamma_j\in [0,1)$ such that $ |g_j(q)|\leq a_j+b_j|q|^{\gamma_j}$ for all $j=1,2,\dots,m$ and $q\in \mathbb{R}^{n}$. \item[(S4)] There exist constants $a_j, b_j>0$ and $\alpha_j>1$ such that $ |g_j(q)|\leq a_j+b_j|q|^{\alpha_j}$ for all $j=1,2,\dots,m$ and $q\in \mathbb{R}^{n}$. \end{itemize} \begin{theorem} \label{thm3.1} If $F$ is $T$-periodic in $t$ and satisfies {\rm (V1)--(V2)}, $g_j(q)$ satisfies {\rm (S1)--(S2)} for all $j=1,2,\dots,m$, then system \eqref{e1.1} possesses at least one non-zero periodic solution generated by impulses. \end{theorem} \begin{proof} It follows form the conditions (V1)--(V2), (S1)--(S2) and \eqref{e2.3} that \begin{equation} \label{e3.1} \begin{aligned} \varphi(q)&= \frac{1}{2}\int_0^{T}|\dot{q}(t)|^{2}dt +\int_0^{T}F(t,q(t))dt+\sum^{m}_{j=1}G_j(q(s_j)) \\ &\geq \frac{1}{2}\|q\|^2+\frac{1}{\beta}\int_0^{T}f(t,q(t))q(t)dt \\ &\geq \frac{1}{2}\|q\|^2 -\frac{\delta_1}{\beta}\int_0^{T}|q(t)|^2dt \\ &\geq \frac{1}{2}\|q\|^2-\frac{\delta_1T}{\beta}\|q\|_{\infty}^2 \\ &\geq \frac{1}{2}\|q\|^2-\frac{\delta_1TC^2}{\beta}\|q\|^2 \\ &= (\frac{1}{2}-\frac{\delta_1TC^2}{\beta})\|q\|^2. \end{aligned} \end{equation} Since $\delta_1\in [0,1/(2TC^2))$ and $\beta>1$, we have $(\frac{1}{2}-\frac{\delta_1TC^2}{\beta})>0$. The inequality \eqref{e3.1} implies that $\lim _{\|q\|\to\infty}\varphi(q)=+\infty$. So $\varphi$ is a functional bounded from below. Next we prove that $ \varphi$ satisfies the Palais-Smale condition. Let $\{q_{n}\} $ be a sequence in $ E $ such that $ \{\varphi(q_{n})\} $ is bounded and $\varphi'(q_{n})\to 0 $ as $n\to\infty$. Then there exists a constant $ C_0$ such that $ |\varphi (q_{n})|\leq C_0$. We first prove that $ \{q_{n}\} $ is bounded. By \eqref{e3.1}, one has $$ C_0\geq\varphi(q_{n}) \geq(\frac{1}{2}-\frac{\delta_1TC^2}{\beta})\|q_{n}\|^2. $$ Since $(\frac{1}{2}-\frac{\delta_1TC^2}{\beta})>0$, it follows that $ \{ q_{n}\} $ is bounded in $E$. Going if necessary to a subsequence, we can assume that there exists $q\in E$ such that $ q_{n}\rightharpoonup q$ in $ E$, $q_{n}\to q$ on $ C([0,T),\mathbb{R}^{n}) $ as $n\to +\infty$. Hence \begin{gather*} (\varphi'(q_{n})-\varphi'(q))(q_{n}-q) \to 0, \\ \int_0^{T}[f(t,q_{n})-f(t,q)](q_{n}(t)-q(t))dt \to 0, \\ \sum^{m}_{j=1}[g_j(q_{n}(s_j))-g_j(q(s_j)) ](q_{n}(s_j)-q(s_j)) \to 0, \end{gather*} as $n\to +\infty$. Moreover, an easy computation shows that \begin{align*} &(\varphi'(q_{n})-\varphi'(q))(q_{n}-q) \\ &= \|q_{n}-q\|^{2} +\int_0^{T}[f(t,q_{n})-f(t,q)](q_{n}(t)-q(t))dt \\ &\quad +\sum ^{m}_{j=1}[g_j(q_{n}(s_j))-g_j(q(s_j))](q_{n}(s_j)-q(s_j)). \end{align*} So $\|q_{n}-q\|\to 0 $ as $n\to +\infty$, which implies that $ \{q_{n}\} $ converges strongly to $ q $ in $ E$. Therefore, $ \varphi$ satisfies the Palais-Smale condition. According to Theorem \ref{thm2.2}, there is a critical point $q$ of $ \varphi$, i.e. $q$ is a periodic solution of system \eqref{e1.1}. The condition (S2) means $q$ is non-zero. So system \eqref{e1.1} possesses a non-zero periodic solution. Finally, let us verify that system \eqref{e1.2} possesses only a zero periodic solution. Suppose $q(t)$ is a periodic solution of system \eqref{e1.2}, then $q(0) = q(T)$. According to condition (V1), we have \begin{align*} 0&= -\int_0^{T}\ddot{q}q dt+\int_0^{T}f(t,q)qdt \\ &= \int_0^{T}|\dot{q}|^2dt+\int_0^{T}f(t,q)qdt-\dot{q}q|_0^{T} \\ &\geq \|q\|^2-\delta_1\int_0^{T}|q|^2dt \\ &\geq \|q\|^2-\delta_1T\|q\|_{\infty}^2 \\ &\geq \|q\|^2-\delta_1TC^2\|q\|^2 \\ &\geq (1-\delta_1TC^2)\|q\|^2. \end{align*} Since $\delta_1\in [0,1/(2TC^2))$, which implies system \eqref{e1.2} does not possess any non-trivial periodic solution. \end{proof} \begin{remark} \rm Condition (F1) guarantee that the conditions (V1) and (V2) hold, but the reverse is not true. For example, take $$ F(t,q)=-\frac{1}{8TC^2}h(t)q^{2}, $$ where $h:\mathbb{R}\to (0,1]$ is continuous with period $T$. Then \[ q f(t,q)=-\frac{1}{4TC^2}h(t)q^{2}\geq -\frac{1}{4TC^2}|q|^{2}. \] Take $\delta_1=\frac{1}{4TC^2}, \beta=2$. It is easy to check that conditions (V1) and (V2) are satisfied, but (F1) cannot be satisfied. Meanwhile, conditions (S1) and (S2) consider the case that $G_j(s)$ are positive functions. Let $$ G_j(q)=\kappa(j)(q^2+\text{e}^q), $$ where $\kappa:\mathbb{Z}\to \mathbb{R}^{+}$ is positive and $m$-periodic in $\mathbb{Z}$. Obviously, $g_j(s)$ can satisfies conditions (S1) and (S2) for all $j=1,2,\dots,m$. In \cite{h1}, they only consider the case that $G_j(q)< 0$, so we extend and improve Theorem \ref{thm1.1}. \end{remark} Next, we assume that the impulsive functions $g_j(q)$ are sublinear. \begin{theorem} \label{thm3.2} If $F$ is $T$-periodic in $t$ and satisfies {\rm (V1)--(V2)}, $g_j(s)$ satisfies conditions (S2) and (S3) for all $j=1,2,\dots,m$, then system \eqref{e1.1} possesses at least one non-zero periodic solution generated by impulses. \end{theorem} \begin{proof} It follows form the conditions (V1)--(V2), (S2)--(S3) and \eqref{e2.3} that \begin{equation} \label{e3.2} \begin{aligned} \varphi(q) &= \frac{1}{2}\int_0^{T}|\dot{q}(t)|^{2}dt +\int_0^{T}F(t,q(t))dt+\sum^{m}_{j=1}G_j(q(s_j)) \\ &\geq \frac{1}{2}\|q\|^2 -\frac{\delta_1}{\beta}\int_0^{T}|q|^2dt +\sum ^{m}_{j=1}G_j(q(s_j)) \\ &\geq \frac{1}{2}\|q\|^2 -\frac{\delta_1T}{\beta}\|q\|_{\infty}^2 -\sum ^{m}_{j=1}[a_j\|q\|_{\infty}+b_j\|q\|_{\infty}^{\gamma_j+1}] \\ &\geq (\frac{1}{2}-\frac{\delta_1TC^2}{\beta})\|q\|^2 -\sum ^{m}_{j=1}[a_jC\|q\|+b_jC^{\gamma_j+1}\|q\|^{\gamma_j+1}]. \end{aligned} \end{equation} Since $(\frac{1}{2}-\frac{\delta_1TC^2}{\beta})>0$ and $\gamma_j\in[0,1)$, the above inequality implies $\lim_{\|q\|\to\infty}\varphi(q)=+\infty$. So the functional $\varphi$ is bounded from below. Next we prove that $ \varphi$ satisfies the Palais-Smale condition. Let $\{q_{n}\} $ be a sequence in $ E $ such that $ \{\varphi(q_{n})\} $ is bounded and $\varphi'(q_{n})\to 0 $ as $n\to\infty$. Then there exists a constant $ C_0$ such that $ |\varphi (q_{n})|\leq C_0$. We first prove that $ \{q_{n}\} $ is bounded. By \eqref{e3.2}, one has $$ C_0\geq\varphi(q_{n}) \geq(\frac{1}{2}-\frac{\delta_1TC^2}{\beta})\|q_{n}\|^2 -\sum ^{m}_{j=1}[a_jC\|q_{n}\|+b_jC^{\gamma_j+1}\|q_{n}\|^{\gamma_j+1}]. $$ Since $(\frac{1}{2}-\frac{\delta_1TC^2}{\beta})>0$, we know that $ \{ q_{n}\} $ is bounded in $E$. Then, as the proof of Theorem \ref{thm3.1}, we can prove $\varphi$ satisfies the Palais-Smale condition. According to Theorem \ref{thm2.2}, there is a critical point $q$ of $ \varphi$, i.e. $q$ is a periodic solution of system \eqref{e1.1}. The condition (S2) means $q$ is non-zero. Similar to the proof of Theorem \ref{thm3.1}, we know system \eqref{e1.2} does not possess any non-trivial periodic solution. So system \eqref{e1.1} possesses a non-zero periodic solution generated by impulses. \end{proof} \begin{example} \label{examp3.1} \rm Let $$ F(t,q)=h(t)q^{6}, $$ where $h:\mathbb{R}\to \mathbb{R}^{+}$ is continuous with period $T$. Then $f(t,q)=6h(t)q^{5}$. It is easy to check that $f(t,q)$ satisfies the conditions (V1) and (V2). Let $$ G_j(q)=(q^\frac{5}{3}+q+\frac{1}{2}\sin 2q), $$ then $g_j(q)=(\frac{5}{3}q^\frac{2}{3}+1+\cos 2q) =\frac{5}{3}q^\frac{2}{3}+2\cos^{2} q$. Hence, $00$, $qf(t,q)\geq0$, so the conditions (V1) and (V3) are satisfied. We also take $G_j(q)$ the same as in Example \ref{examp3.2}, then according to Theorem \ref{thm3.3}, system \eqref{e1.1} possesses at least one non-zero periodic solution generated by impulses. \end{example} \begin{remark} \label{rmk3.3} \rm Obviously, $F(t,q)$ cannot satisfy the conditions (F1) or (F2). So Example \ref{examp3.2} cannot obtain the existence of non-zero periodic solutions in \cite{h1}. \end{remark} \begin{corollary} \label{coro3.1} If $F$ is $T$-periodic in $t$ and satisfies {\rm (V1)} and {\rm (V3)}, $g_j(q)$ satisfies conditions {\rm (S1)} and {\rm (S2)} for all $j=1,2,\dots,m$, then system \eqref{e1.1} possesses at least one non-zero periodic solution generated by impulses. \end{corollary} \begin{theorem} \label{thm3.4} If $F$ is $T$-periodic in $t$ and satisfies {\rm (F1)}, $g_j(q)$ satisfies conditions {\rm (G1)} and {\rm (S4)} for all $j=1,2,\dots,m$. Moreover if $f(t,q)$, $g_j(q)$ are odd about $q$, then system \eqref{e1.1} possesses infinitely many periodic solutions generated by impulses. \end{theorem} To apply Theorem \ref{thm2.1} and to prove Theorem \ref{thm3.4}, we define the functionals $A, B$ and $\varphi_{\lambda}$ on our working space $E$ by \begin{align*} A(q)&=\frac{1}{2}\|q\|^{2}+\int_0^{T}F(t,q(t))dt,\\ B(q)&=-\sum^{m}_{j=1}G_j(q(s_j)), \end{align*} and \begin{equation} \label{e3.4} \varphi_{\lambda}(q) = A(q)-\lambda B(q) = \frac{1}{2}\|q\|^{2}+\int_0^{T}F(t,q(t))dt +\lambda\sum^{m}_{j=1}G_j(q(s_j)), \end{equation} for all $q\in E$ and $\lambda\in[1,2]$. Clearly, we know that $\varphi_{\lambda}(q)\in C^{1}(E,\mathbb{R})$ for all $\lambda\in[1,2]$. We choose a completely orthonormal basis $\{e_j\}$ of $E$ and define $X_j:=\mathbb{R}e_j$. Then $Z_k, Y_k$ can be defined as that in the beginning of Section 2. Note that $\varphi_1=\varphi$, where $\varphi$ is the functional defined in \eqref{e2.3}. \begin{remark} \label{rmk3.4} \rm If (F1) holds, $f(t,q)$ is $T$-periodic in $t$, then there exist constants $d_1,d_2>0$ such that $$ F(t,q)\leq d_1|q|^2+d_2,\quad \forall q\in\mathbb{R}^n. $$ Assume that (G1) holds, then there exist constants $a,b>0$ such that \begin{gather} G_j(q)\geq -a|q|^\mu,\quad \text{for } 0<|q|\leq 1, \label{e3.5}\\ G_j(q)\leq -b|q|^\mu,\quad \text{for }|q|\geq 1. \label{e3.6} \end{gather} The assumption $g_{j+m}=g_j$ and \eqref{e3.5}-\eqref{e3.6} imply that there exists $M>0$ such that \begin{equation} \label{e3.7} G_j(q)\leq -b|q|^\mu+M,\quad \forall q\in\mathbb{R}^n. \end{equation} \end{remark} \begin{lemma} \label{lem3.1} Under the assumptions of Theorem \ref{thm3.4}, $B(q)\geq 0$ and $A(q)\to \infty$ as $\|q\|\to \infty$ for all $q\in E$. \end{lemma} \begin{proof} By (F1) and (G1), for any $q\in E$, we have \begin{gather*} B(q)=-\sum^{m}_{j=1}G_j(q(s_j))\geq 0, \\ A(q)=\frac{1}{2}\|q\|^{2}+\int_0^{T}F(t,q(t))dt\geq \frac{1}{2}\|q\|^{2}. \end{gather*} Which implies that $A(q)\to \infty$ as $\|q\|\to \infty$. \end{proof} \begin{lemma} \label{lem3.2} Under the assumptions of Theorem \ref{thm3.4}, there exists a sequence $\rho_k>0$ large enough such that $$ a_k(\lambda):=\max_{q\in Y_k,\|q\|=\rho_k}\varphi_{\lambda}(q)\leq0, $$ for all $\lambda\in [1,2]$. \end{lemma} \begin{proof} By (F1) and (G1), for any $q\in Y_k$, we have \begin{equation} \label{e3.8} \begin{aligned} \varphi_{\lambda}(q) &\leq \frac{1}{2}\|q\|^2 +\int_0^{T}(d_1|q|^{2}+d_2)dt+\lambda\sum ^{m}_{j=1}(-b|q|^\mu+C_2) \\ &= \frac{1}{2}\|q\|^2+d_1\int_0^{T}|q|^{2}dt+d_2T-\lambda\sum ^{m}_{j=1}b|q|^\mu+\lambda Mm, \end{aligned} \end{equation} \label{e3.9} Let $q=rw$, $r>0, w\in Y_k$ with $\|w\|=1$, we have \begin{equation} \varphi_{\lambda}(rw)\leq \frac{1}{2}r^2+r^{2}d_1\int_0^{T}|w|^{2}dt+d_2T-\lambda r^\mu\sum ^{m}_{j=1}b|w|^\mu+\lambda Mm. \end{equation} Since $\mu>2$, for $\|q\|=\rho_k=r$ large enough, we have $\varphi_{\lambda}(q)\leq 0$; i.e., $a_k(\lambda):=\max_{q\in Y_k, \|q\|=\rho_k}\varphi_{\lambda}(q)\leq0$ for all $\lambda\in [1,2]$. \end{proof} \begin{lemma} \label{lem3.3} Under the assumptions of Theorem \ref{thm3.4}, there exists $r_k>0, \widetilde{b}_k\to \infty$ such that $$ b_k(\lambda):=\inf_{q\in Z_k, \|q\|=r_k}\varphi_{\lambda}(q)\geq \widetilde{b}_k, $$ for all $\lambda\in [1,2]$. \end{lemma} \begin{proof} Set $\beta_k:=\sup_{q\in Z_k,\|q\|=1}\|q\|_{\infty}$. Then $\beta_k\to0$ as $k\to\infty$. Indeed, it is clear that $0<\beta_{k+1}\leq\beta_k$, so that $\beta_k\to\beta\geq0$, as $k\to\infty$. For every $k\geq0$, there exists $q_k\in Z_k$ such that $\|q_k\|=1$ and $\|q_k\|_{\infty}>\beta_k/2$. By definition of $Z_k$, $q_k\rightharpoonup0$ in $E$. Then it implies that $q_k\to0$ in $C([0,T),\mathbb{R}^{n})$. Thus we have proved that $\beta=0$. By (F1) and (S4), for any $q\in Z_k$, we have \begin{equation} \begin{aligned} \label{e3.10} \varphi_{\lambda}(q) &\geq \frac{1}{2}\|q\|^2+\lambda\sum^{m}_{j=1}G_j(q(s_j)) \\ &\geq \frac{1}{2}\|q\|^2-2\sum^{m}_{j=1}(a_j|q|+b_j|q|^{\alpha_j+1}) \\ &\geq \frac{1}{2}\|q\|^2-2\sum^{m}_{j=1}(a_j\|q\|_{\infty} +b_j\|q\|_{\infty}^{\alpha_j+1}) \\ &\geq \frac{1}{2}\|q\|^2-2\sum^{m}_{j=1}(a_j\beta_k\|q\| +b_j\beta_k^{\alpha_j+1}\|q\|^{\alpha_j+1}). \end{aligned} \end{equation} Let \[ r_k=(8\sum ^{m}_{j=1}(a_j\beta_k +b_j\beta_k^{\alpha_j+1}))^{\frac{1}{1-\alpha_j}}. \] Since $\alpha_j>2$, then $r_k\to \infty$ as $k\to\infty$. Evidently, there exists an positive integer $k_0>\bar{n}+1$ such that $$ r_k>1, \quad \forall k\geq k_0. $$ For any $ k\geq k_0$, let $\|q\|=r_k>1$, we have \begin{align*} 2\sum ^{m}_{j=1}(a_j\beta_k\|q\| +b_j\beta_k^{\alpha_j+1}\|q\|^{\alpha_j+1}) &\leq 2\sum ^{m}_{j=1}(a_j\beta_k +b_j\beta_k^{\alpha_k+1})\|q\|^{\alpha_j+1} \\ &= \frac{1}{4}r_k^{1-\alpha_j}r_k^{\alpha_j+1} = \frac{1}{4}r_k^2. \end{align*} Combining this with \eqref{e3.10}, straightforward computation shows that $$ b_k(\lambda):=\inf_{q\in Z_k, \|q\|=r_k}\varphi_{\lambda}(q)\geq\frac{1}{4}r_k^{2} =\widetilde{b}_k\to\infty, $$ as $k\to\infty$ for all $\lambda\in[1,2]$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3.4}] Evidently, the condition (C1) in Theorem \ref{thm2.1} holds. By Lemmas \ref{lem3.1}, \ref{lem3.2}, \ref{lem3.3} and Theorem \ref{thm2.1}, there exist a sequence $\lambda_{n}\to 1$ as $n\to\infty$, $\{q_{n}(k)\}_{n=1}^{\infty}\subset E$ such that $\varphi_{\lambda_{n}}^{'}(q_{n}(k))\to0, \varphi_{\lambda_{n}}(q_{n}(k))\to c_k\in[\widetilde{b}_k, \widetilde{c}_k]$, where $\widetilde{c}_k=\sup_{q\in B_k}\varphi_1(q) $ For the sake of simplicity, in what follows we set $q_{n}=q_{n}(k)$ for all $n\in \mathbb{N}$. Now we show that $\{q_{n}\}_{n=1}^{\infty}$ is bounded in $E$. Indeed, by (F1), $(g_1)$ and \eqref{e3.4}, we have \begin{align*} \mu\varphi_{\lambda_{n}}(q_{n})-\varphi'_{\lambda_{n}}(q_{n})q_{n} &= (\frac{\mu}{2}-1)\|q_{n}\|^2+\mu\int_0^{T}F(t,q_{n})dt -\int_0^{T}f(t,q_{n})q_{n}dt \\ &\quad + \lambda_{n}[\mu\sum ^{m}_{j=1}G_j(q_{n}(s_j))-\sum ^{m}_{j=1}g_j(q_{n}(s_j))q_{n}(s_j)] \\ &\geq (\frac{\mu}{2}-1)\|q_{n}\|^2 . \end{align*} Since $\mu>2$, the above inequality implied that $\{q_{n}\}$ is bounded in $E$. Finally, we show that $\{q_{n}\}$ possesses a strong convergent subsequence in $E$. In fact, in view of the boundedness of $\{q_{n}\}$, without loss of generality, we may assume $q_{n}\rightharpoonup q_0$ as $n\to\infty$, for some $q_0\in E$, then $q_{n}\to q_0$ on $ C([0,T), \mathbb{R}^n) $. Moreover, an easy computation shows that \begin{align*} &(\varphi'_{\lambda_{n}}(q_{n})-\varphi'_{\lambda_{n}}(q_0))(q_{n}-q_0) \\ &= \|q_{n}-q_0\|^{2} +\int_0^{T}[f(t,q_{n}(t))-f(t,q_0(t))](q_{n}(t)-q_0(t))dt \\ &\quad + \lambda_{n}\sum ^{m}_{j=1}[g_j(q_{n}(s_j))-g_j(q_0(s_j))](q_{n}(s_j)-q_0(s_j)). \end{align*} So $\|q_{n}-q_0\|\to 0 $ as $n\to +\infty$, which implies that $ \{q_{n}\} $ converges strongly to $ q_0 $ in $ E $ and $\varphi'_1(q_0) =0$. Hence, $\varphi=\varphi_1$ has a critical point $q_0$ with $\varphi_1(q_0)\in [\widetilde{b}_k, \widetilde{c}_k]$. Consequently, we obtain infinitely many periodic solutions since $\widetilde{b}_k\to \infty$. Similar to the proof of Theorem \ref{thm3.1}, we know system \eqref{e1.2} does not possess any non-trivial periodic solution. Therefore, all the non-zero periodic solutions we obtain are generated by impulses. \end{proof} \begin{example}\label{examp3.3} \rm Let $F(t,q)=h(t)q^{6/5}$, where $h:\mathbb{R}\to \mathbb{R}^{+}$ is continuous with period $T$. Then $f(t,q)=\frac{6}{5}h(t)q^{1/5}$. It is easy to know that $f(t,q)q=\frac{6}{5}h(t)q^{6/5}>0$ and $\frac{1}{2}f(t,q)q=\frac{3}{5}h(t)q^{6/5}\leq F(t,q)$, so $f(t,q)$ satisfies the condition (F1). Let $$G_j(q)=-\kappa(j)(q^4+q^6),$$ where $\kappa:\mathbb{Z}\to (0,10]$ is positive and $m$-periodic in $\mathbb{Z}$. Then $g_j(q)=-\kappa(j)(4q^3+6q^5)\leq 100(1+|q|^5)$. For $g_j(q)q=-\kappa(j)(4q^4+6q^6)\leq -4\kappa(j)(q^4+q^6)$, take $\mu=4$, it is easy to show that conditions (G1) and (S4) are satisfied. Moreover, $f(t,q), g_j(q)$ are odd about $q$. Therefore, according to Theorem \ref{thm3.4}, system \eqref{e1.1} has infinitely many periodic solutions generated by impulses. \end{example} In \cite{h1}, by Theorem \ref{thm1.1}, Example \ref{examp3.3} we can only have the existence of at least one non-zero periodic solution. In this article, we obtain the existence of infinitely many periodic solutions. \subsection*{Acknowledgements} This work is supported by the construct program of the key discipline in Hunan University of Science and Engineering, the Natural Science Foundation of Hunan Province, China (No. 14JJ3134) and the National Natural Science Foundation of China (No. 11271371). The authors are grateful to the anonymous referees for their useful suggestions. \begin{thebibliography}{00} \bibitem{c1} Vittorio Coti Zelati, I. Ekeland, Eric Sr.; \emph{A variational approach to homoclinic orbits in Hamiltonian system}, Math. Ann. 288 (1990) 133-160. \bibitem{c2} C. De Coster, P. 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