\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 195, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/195\hfil Multiple solutions]
{Multiple solutions for impulsive Hamiltonian systems}

\author[G. Zhou \hfil EJDE-2016/195\hfilneg]
{Guanghui Zhou}

\address{Guanghui Zhou \newline
School of Mathematical Sciences,
Information College,
Huaibei Normal University,
 Huaibei 235000,  China}
\email{163zgh@163.com}

\thanks{Submitted May 31, 2016. Published July 20, 2016.}
\subjclass[2010]{34A36, 49J50}
\keywords{Existence and multiplicity; impulsive Hamiltonian system;
\hfill\break\indent critical points theorem}

\begin{abstract}
 In this article, we study second-order impulsive Hamiltonian systems,
 We obtain some existence and multiplicity results,
 by using a variational method and critical point theorem.
 An example illustrate the feasibility  of our results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In this article, we study  the  second-order Hamiltonian
systems with impulsive effects
\begin{equation}
\begin{gathered}
-\ddot{u}+A(t)u= \lambda b(t) \nabla G(u),\quad \text{a.e. } t\in [0,T],\\
\Delta(\dot{u}^i(t_j))=\dot{u}^i(t_j^+)-\dot{u}^i(t_j^-)
=I_{ij}(u^i(t_j)),\quad i=1,2,\dots,N,j=1,2,\dots,l,\\
u(0)-u(T)=u'(0)-u'(T)=0,
\end{gathered} \label{e1.1}
\end{equation}
where $A : [0, T]\to \mathbb{R}^{N\times N}$ is a continuous map from 
$[0, T]$ to the set of $N$-order symmetric matrices,
$T$ is a real positive number,
$u(t) = (u^1(t), u^2(t), \dots, u^N (t))$, $t_j$, $j = 1, 2, \dots, l$,
are the instants where the impulses occur
and $0 = t_0 < t_1 < t_2 < \dots < t_l < t_{l+1} = T$,
$I_{ij} : \mathbb{R}\to \mathbb{R}$  $(i=1,2,\dots,N; j=1,2,\dots,l)$ are continuous.

Recently, with the development of theory and applications of impulsive
differential systems, there have been  some results considering the existence
and multiplicity of solutions for impulsive problems,
by using variational method (see \cite{b1,b2,b3,c1,c2,c3,l1,n1,s1,s2,t1,z1,z2,z3}).
To obtain the existence and multiplicity of solutions, impulsive
functions $I_{ij}(\cdot)$ of all theorems in \cite{s2,z3}, are required to
satisfy the following conditions
\begin{equation}
I_{ij}(y)y\geq 0 \quad \text{for all } i\in \mathcal{A}=\{1,2,\dots,N\},];
 j\in \mathcal{B}=\{1,2,\dots,l\},\;  y\in \mathbb{R},
\label{e1.2}
\end{equation}
or
\begin{equation}
I_{ij}(y)y\leq 0 \quad \text{for all } i\in \mathcal{A},\;
 j\in \mathcal{B},\;  y\in \mathbb{R}.\label{e1.3}
\end{equation}
However, as Dai pointed out in \cite{b3},  there are many functions which
do not satisfy \eqref{e1.2} or \eqref{e1.3}.
For example, when $N=3$ and $l=2$, impulsive functions of \eqref{e1.1} are
\begin{equation}
I_{ij}(y)=-y+1\quad \text{for  } i=1,2,3; j=1,2,\label{e1.4}
\end{equation}
or  a more complicated case such as
\begin{equation}
I_{ij}(y)=\begin{cases}
\frac{y}{2}+1, & i=1,2,3; j=1,\\
-y & i=1,2; j=2,\\
\frac{y}{2} & i=3, j=2.
\end{cases} \label{e1.5}
\end{equation}
So it is important to consider such case.
Motivated by \cite{b3,b5} and the above facts, we will reconsider
problem \eqref{e1.1} and study the existence
of solutions without assumption \eqref{e1.2} or \eqref{e1.3},
 which show that suitable impulses won't influence the existence of of solutions.

The organization of this article is as follows.
In Section 2, we introduce some definitions and lemmas.
In Section 3, by using critical point theorem  \cite{b6,b7},
we obtain some existence and multiplicity
result of  solutions for \eqref{e1.1}.

\section{Preliminaries}

In this section, we introduce notation, definitions, and preliminary facts.
$A: [0,T] \to  \mathbb{R}^{N\times N}$ is a matrix-valued
function fulfilling the following technical assumptions:
\begin{itemize}
\item[(A1)] $ A(t) = (a_{ij}(t))$ is a symmetric matrix with
$a_{ij}\in L^{\infty}([0, T],\mathbb{R}^+)$ for every $t\in [0, T]$.

\item[(A2)] There exists a positive constant $\mu$ such that
$A(t)\xi\cdot \xi\geq \mu|\xi|^2$ for every
$\xi\in \mathbb{R}^N$ and a.e. $t\in [0, T]$.
\end{itemize}
The set
\begin{align*}
H_T^1=\big\{&u:[0,T]\to \mathbb{R}^N: u\ \text{is absolutely continuous,}\\
 &u(0)=u(T) \text{ and } \dot{u}\in L^2([0,T],\mathbb{R}^N)\big\}
\end{align*}
is a Hilbert space with the usual norm
\[
 \|u\|_{H_T^1}=\Big(\int_0^T(|\dot{u}(t)|^2+|u(t)|^2)dt\Big)^{1/2}.
\]
For every $u,v\in H_T^1$, by $(A1),(A_2)$, we define an inner product
\[
\langle u, v\rangle=\int_0^T[(\dot{u}(t), \dot{v}(t))+(A(t)u(t), v(t))]dt,\quad
 \forall u,v\in H_T^1,
\]
which induces the norm
\[
\|u\|=\Big(\int_0^T(|\dot{u}(t)|^2+A(t)|u(t)|^2)dt\Big)^{1/2}.
\]
As in  \cite{b4,z2}, we have
\begin{equation}
\begin{gathered}
A(t)\xi\cdot\xi=\sum_{j=1}^N\sum_{i=1}^Na_{ij}(t)\xi_i\xi_j
\leq\sum_{j=1}^N\sum_{i=1}^N\|a_{ij}\|_{\infty}|\xi|^2,\\
\sqrt{m}\|u\|_{H_T^1}\leq \|u\| \leq \sqrt{M}\|u\|_{H_T^1},
\end{gathered}\label{e2.1}
\end{equation}
where $m=\min\{1,\mu\}, M=\max\{1, \Sigma_{i,j=1}^N\|a_{ij}\|_{\infty}\},
 \|a_{ij}\|_{\infty}=\max_{t\in[0,T]}|a_{ij}(t)|$.
Since $(H_T^1, \|\cdot\|_{H_T^1})$ is compactly embedded in
 $C([0,T], \mathbb{R}^N)$,
then there is a positive number $\bar{k}$ such that for every $u\in H_T^1$,
\begin{equation}
\|u\|_{\infty}\leq \bar{k}\|u\|,\label{e2.2}
\end{equation}
 Thus
\begin{equation}
\bar{k}\leq k=\sqrt{\frac{2}{m}}\max\{\sqrt{T},\frac{1}{\sqrt{T}}\}.\label{e2.3}
\end{equation}
For any $u,v\in H_T^1$, let
\begin{equation}
\Phi(u)=\frac{1}{2}\|u\|^2+\sum_{j=1}^l\sum_{i=1}^N\int_0^{u^i(t_j)}I_{ij}(s)ds,\quad
\Psi(u)=\int_0^Tb(t)G(u(t))dt.\label{e2.4}
\end{equation}
By standard argument, we see that $\Phi, \Psi$ are G\^ateaux differentiable
at any $u\in H_T^1$ and
\begin{equation}
\begin{gathered}
\langle\Phi'(u), v\rangle=\int_0^T[(\dot{u}(t), \dot{v}(t))+(A(t)u(t), v(t))]dt
+\sum_{j=1}^l\sum_{i=1}^NI_{ij}(u^i(t_j))v^i(t_j),\\
\langle\Psi(u),v\rangle=\int_0^Tb(t)(\nabla G(u(t)),v(t))dt.
\end{gathered}\label{e2.5}
\end{equation}
A critical point of the functional $\Phi-\lambda \Psi$ is a function
$u\in H^1_T$ such that $\Phi'(u)(v)-\lambda\Psi'(u)(v)=0$
for every $v\in H^1_T$, i.e.

\begin{definition} \label{def2.1} \rm
 A function $u\in H_T^1$ is a weak solution of \eqref{e1.1} if
\begin{equation}
\begin{aligned}
&\int_0^T[(\dot{u}(t), \dot{v}(t))+(A(t)u(t), v(t))]dt
+\sum_{j=1}^l\sum_{i=1}^NI_{ij}(u^i(t_j))v^i(t_j)\\
&=\lambda\int_0^Tb(t)(G(u(t)),v(t))dt
\end{aligned} \label{e2.6}
\end{equation}
holds for any $v\in H_T^1$.
\end{definition}

Hence, we can claim that
each critical point of the functional $\Phi-\lambda \Psi$ is a weak
solution to problem \eqref{e1.1}.

\begin{theorem}[{\cite[Theorem 3.3]{b4}}] \label{thm2.1}
Let $X$ be a reflexive real Banach space, let
$\Phi,\Psi: X \to R$ be two G\^ateaux differentiable functionals
such that $\Phi$ is sequentially weakly lower semi-continuous and
coercive and $\Psi$ is sequentially weakly upper semi-contin\-uous. Assume
that
\begin{itemize}

\item[(i)] $\Phi$ is convex;

\item[(ii)] For every $x_1,x_2\in X$ such that $\Psi(x_1)\geq 0$
and $\Psi(x_2)\geq 0$, one has $\inf_{s\in[0,1]}\Psi(sx_1+(1-s)x_2)\geq0$;

\item[(iii)] $\inf_X\Phi=\Phi(0)=\Psi(0)=0$;

\item[(iv)] there are three positive constants $r_1, r_2, r_3$ with
$\inf_X\Phi <r_1 <r_2$ such that, if we put
\begin{gather*}
\varphi^{(1)}(r_i)=\inf_{u\in\Phi^{-1}((-\infty,r_i))}
 \frac{\sup_{v\in\Phi^{-1}((-\infty,r_i))}\Psi(v)-\Psi(u)}{r_i-\Phi(u)},\\
\varphi_2(r_1,r_2)=\inf_{u\in\Phi^{-1}((-\infty,r_1))}
 \sup_{v\in\Phi^{-1}([r_1,r_2))}\frac{\Psi(v)-\Psi(u)}{\Phi(v)-\Phi(u)},\\
\varphi^{(3)}(r_2,r_3)
=\frac{\sup_{u\in\Phi^{-1}((-\infty,r_2+r_3))}\Psi(u)}{r_3},\\
\varphi_3(r_1,r_2,r_3)
=\max\{\varphi^{(1)}(r_1), \varphi^{(1)}(r_2), \varphi^{(3)}(r_2,r_3)\},
\end{gather*}
one has
 $\varphi_3(r_1,r_2,r_3)<\varphi_2(r_1,r_2)$.

\item[(v)] For each
$\lambda\in \Lambda_{r_1,r_2,r_3}:=(\frac{1}{\varphi_2(r_1,r_2)},
\frac{1}{\varphi_3(r_1,r_2,r_3)})$, if we put
$$
\Psi_{\frac{r_3}{\lambda}}(u)
=\begin{cases}
\Psi(u) & \text{if }\Psi(u)\leq \frac{r_3}{\lambda},\\
\frac{r_3}{\lambda}& \text{if } \Psi(u)> \frac{r_3}{\lambda},
\end{cases}
$$
then $\Phi-\lambda \Psi_{\frac{r_3}{\lambda}}$ satisfies the
condition  $(PS)_c$, with $c\in \mathbb{R}$.
\end{itemize}
Then for each $\lambda\in \Lambda_{r_1,r_2,r_3}$, the functional
$\Phi-\lambda \Psi$ admits at least three critical points $u_1, u_2, u_3 \in X$
such that $u_1\in \Phi^{-1}((-\infty,r_1)), u_2\in \Phi^{-1}([r_1,r_2))$ and
$u_3\in \Phi^{-1}((-\infty,r_2+r_3))$.
\end{theorem}

\begin{theorem} [{\cite[Theorem 2.6]{b6}}] \label{thm2.2}
Let $X$ be a reflexive real Banach space, let $\Phi: X \to R$ be a
be a sequentially weakly lower
semi-continuous, coercive and continuously G\^ateaux differentiable functional whose G\^ateaux derivative admits a continuous
inverse on $X^*$, and let $\Psi: X \to R$ be a sequentially weakly upper semi-continuous and continuously
 G\^ateaux differentiable functional
whose G\^ateaux derivative is compact. Assume that there exist $r \in \mathbb{R}$ and $x_0, \bar{x} \in X$,
 with $\Phi(x_0)< r < \Phi(\bar{x})$ and $\Psi(x_0) = 0$, such that
\begin{itemize}
\item[(i)]  $\sup_{x\in\Phi^{-1}((-\infty,r])}\Psi(x)<(r-\Phi(x_0))
\frac{\Psi(\bar{x})}{\Phi(\bar{x})-\Phi(x_0)}$;

\item[(ii)]  for each
\[
\lambda\in \Lambda_{r}:=(\frac{\Phi(\bar{x})
-\Phi(x_0)}{\Psi(\bar{x})},
\frac{r-\Phi(x_0)}{\sup_{x\in\Phi^{-1}((-\infty,r])}\Psi(x)}),
\]
 the functional $\Phi-\lambda \Psi$ is coercive.
\end{itemize}
Then for each $\lambda\in \Lambda_{r}$, the functional $\Phi-\lambda \Psi$
has at least three distinct critical points $u_1, u_2, u_3 \in X$.
\end{theorem}


\begin{theorem}[{\cite[Theorem 2.1]{b7}}] \label{thm2.3}
Let $X$ be a reflexive real Banach space, let $\Phi,\Psi: X \to R$ be
two G\^ateaux differentiable functionals
such that $\Phi$ is sequentially weakly lower semicontinuous and coercive
and $\Psi$ is sequentially weakly upper semicontinuous.
For every $r>\inf_X \Phi$, let us put
$$
\varphi(r):=\inf_{u\in\Phi^{-1}((-\infty,r))}
\frac{\sup_{v\in\Phi^{-1}((-\infty,r))}\Psi(v)-\Psi(u)}{r-\Phi(u)},
$$
and
$$
\gamma:=\liminf_{r\to+\infty}\varphi(r),\quad
\delta:=\liminf_{r\to(\inf_X\Phi)^+}\varphi(r).
$$
Then one has
\begin{itemize}
\item[(a)] For every $r>\inf_X\Phi$ and every $\lambda\in (0,\frac{1}{\varphi(r)})$,
the restriction of the functional $I_{\lambda} =\Phi-\lambda\Psi $ to
$\Phi^{-1}((-\infty, r))$ admits a global
minimum, which is a critical point (local minimum) of $I_{\lambda}$ in $X$.

\item[(b)] If $\gamma<+\infty$ then, for each
$\lambda\in (0,\frac{1}{\gamma})$, the following alternative holds:
Either
\begin{itemize}
\item[(b1)] $I_{\lambda}$ possesses a global minimum, or

\item[(b2)] there is a sequence $\{u_n\}$ of critical points
(local minima) of $I_{\lambda}$ such that $\lim_{n\to \infty}\Phi(u_n)=+\infty$.
\end{itemize}
\item[(c)] If $\delta<+\infty$ then, for each $\lambda\in (0,\frac{1}{\delta})$,
the following alternative holds:
Either
\begin{itemize}
\item[(c1)] there is a global minimum of $\Phi$ which is a local minimum of
$I_{\lambda}$, or

\item[(c2)] there is a sequence of pairwise distinct critical points
(local minima) of $I_{\lambda}$ which weakly converges to a global minimum
of $\Phi$.
\end{itemize}
\end{itemize}
\end{theorem}

As in the proof of \cite[Lemma 5]{b3}, we have the following result.

\begin{lemma} \label{lem2.1}
Suppose that $I_{ij}(y)$ is nondecreasing in $y\in \mathbb{R}$ for all
$ i=1,2,\dots,N$, $j=1,2,\dots,l$. Then
$\phi(u)=\sum_{j=1}^l\sum_{i=1}^N\int_0^{u^i(t_j)}I_{ij}(s)ds$
 is convex in $u\in \mathbb{R}^N$.
\end{lemma}


\section{Existence and multiplicity of solutions}

For convenience, we introduce the  assumption
\begin{itemize}
\item[(A3)] There exist constants $c_{ij} > 0$, $d_{ij} > 0$,
$\gamma_{ij}\in [0,1)$, $i=1,2,\dots,N$, $j=1,2,\dots,l$,
such that
$$
|I_{ij}(y)|\leq c_{ij}+ d_{ij}|y|^{\gamma_{ij}} \quad \text{for all }
 y\in \mathbb{R}.
$$
\end{itemize}

\begin{theorem} \label{thm3.1}
Assume that {\rm (A1)--(A3)} hold.
 $I_{ij}(y)$ is nondecreasing for $y\in \mathbb{R}$, for any
$i=1,2,\dots,N$, $j=1,2,\dots,l$  with $I_{ij}(0)=0$.
Let $G\in C^1(\mathbb{R}^N, R)$ be such that
\begin{itemize}
\item[(1)] $G(\xi)\geq G(0)=0$ for any $\xi\in \mathbb{R}^N$,

\item[(2)] there exist $\rho_2\gg \rho_1>0$ and $\bar{\xi}\in \mathbb{R}^N$ such that

\item[(3)] $\rho_1<|\bar{\xi}|<\sqrt{\frac{4M}{\mu}}\rho_2,$ where $\mu<M$ defined as $(A_2)$ and $\eqref{e2.1}$ respectively;

\item[(4)]
\[
\big(1+\frac{4M}{\mu}\big)\frac{\max_{|\xi|\leq \rho_1}G(\xi)}{\rho_1^2}
+4\frac{\max_{|\xi|\leq \rho_2}G(\xi)}{\rho_2^2}<\frac{G(\bar{\xi})}{|\bar{\xi}|^2}.
\]
\end{itemize}
Then for every $b\in L^1([0,T])\setminus \{0\}$ and for every
$\lambda$ in
\[
\Lambda_{\rho_1,\rho_2,\rho_3}
 :=\Big(\frac{MT}{\|b\|_{L^1}\big(1-\frac{\mu}{M}\big)}
\frac{|\bar{\xi}|^2}{G(\bar{\xi})}, \frac{T}{4\|b\|_{L^1}}
\frac{1}{\max\Big\{\frac{1}{\mu}\frac{\max_{|\xi|\leq \rho_1}G(\xi)}{\rho_1^2},
\frac{1}{M}\frac{\max_{|\xi|\leq \rho_2}G(\xi)}{\rho_2^2}\Big\}}\Big),
\]
Equation \eqref{e1.1} has at least three nontrivial solutions $u_1,u_2,u_3$
 such that $\|u_i\|_{\infty}\leq \rho_2$, $i=1,2$.
\end{theorem}

\begin{proof}
Let $\Phi,\Psi$ be as \eqref{e2.4}. Since $I_{ij}(y)$ is nondecreasing
in $y\in \mathbb{R}$ for any $i=1,2,\dots,N$, $j=1,2,\dots,l$,
 we have
\[
\phi(u)=\sum_{j=1}^l\sum_{i=1}^N\int_0^{u^i(t_j)}I_{ij}(s)ds
\]
which is convex for $u\in \mathbb{R}^N$, from Lemma \ref{lem2.1}.
It is obvious that $\|u\|$ is convex in $u\in \mathbb{R}^N$.
Thus $\Phi(u)$  is convex in $u\in \mathbb{R}^N$.
We also know $\Phi(u)$ is continuously G\^ateaux differentiable and sequentially
weakly lower semi-continuous.
By (A3) and \eqref{e2.2}, we
 have
\begin{align*}
\Phi(u)&= \frac{1}{2}\|u\|^2+\sum_{j=1}^l\sum_{i=1}^N\int_0^{u^i(t_j)}I_{ij}(s)ds\\
&\geq \frac{1}{2}\|u\|^2-\sum_{j=1}^l\sum_{i=1}^N(c_{ij}|u|
 +d_{ij}|u|^{1+\gamma_{ij}})\\
&\geq \frac{1}{2}\|u\|^2-\sum_{j=1}^l\sum_{i=1}^N(c_{ij}\|u\|_{\infty}
 +d_{ij}\|u\|_{\infty}^{1+\gamma_{ij}})\\
&\geq \frac{1}{2}\|u\|^2-\sum_{j=1}^l\sum_{i=1}^N(c_{ij}\bar{k}\|u\|
 +d_{ij}\bar{k}^{1+\gamma_{ij}}\|u\|^{1+\gamma_{ij}})\to +\infty
\end{align*}
as $\|u\|\to +\infty$, i.e. $\Phi$ is obviously coercive.
$\Psi$ is continuously G\^ateaux differentiable with compact
derivative, hence it is sequentially weakly continuous.
In addition, assumption (1) and for every $b\in L^1([0,T])\setminus \{0\}$ imply
that $\Psi(u) \geq 0$ for every $u \in H_T^1$, hence (ii) of 
Theorem \ref{thm2.1} holds.
Again from assumption (1), we get (iii).

Choose $\bar{v}(t)=\bar{\xi}, t\in[0,T]$. Let
$$
r_1=\frac{\mu T\rho_1^2}{4}>0,\quad
r_2=r_3= MT\rho_2^2>0, %(*)
$$
for sufficiently small
\begin{gather*}
\rho_1=\Big(\frac{2\bar{\gamma}\sum_{j=1}^l\sum_{i=1}^Nd_{ij}}{\mu T}
\Big)^{\frac{1}{1-\bar{\gamma}}}>0,\\
 \bar{\rho}_1=\min\Big\{\frac{\mu T\rho_1^2}{8\sum_{j=1}^l\sum_{i=1}^Nc_{ij}},
\Big(\frac{\mu T\rho_1^2}{8\sum_{j=1}^l\sum_{i=1}^Nd_{ij}}
\Big)^{\frac{1}{1+\bar{\gamma}}}\Big\},
\end{gather*}
and sufficiently big
\[
\rho_2\gg \Big(\frac{2\sum_{j=1}^l\sum_{i=1}^N(c_{ij}+d_{ij})}{MT}
\Big)^{^{\frac{1}{1-\bar{\gamma}}}}>0,
\]
where $\bar{\gamma}=\max_{i,j}\{\gamma_{ij}\}$.
Then from Section 2, we have
\begin{gather}
\Phi^{-1}((-\infty,r_1))\subseteq \{u\in C([0,T],
\mathbb{R}^N):\|u\|_{\infty}\leq \rho_1\},\label{e3.1} \\
\Phi^{-1}((-\infty,r_2))\subseteq \Phi^{-1}((-\infty,r_2+r_3))
\subseteq\{u\in C([0,T], \mathbb{R}^N):\|u\|_{\infty}\leq \rho_2\},\label{e3.2}
\end{gather}
then by \eqref{e3.1}, \eqref{e3.2},
we have
\begin{equation}
\begin{aligned}
\varphi^{(1)}(r_i)
&= \inf_{u\in\Phi^{-1}((-\infty,r_i))}
 \frac{\sup_{v\in\Phi^{-1}((-\infty,r_i))}\Psi(v)-\Psi(u)}{r_i-\Phi(u)} \\
&\leq \frac{\sup_{v\in\Phi^{-1}((-\infty,r_i))}\Psi(v)}{r_i}\\
&\leq \begin{cases}
\frac{4\|b\|_{L^1}}{\mu T}\frac{\max_{|\xi|\leq \rho_1}G(\xi)}{\rho_1^2},
& \text{if } i=1,\\
\frac{4\|b\|_{L^1}}{MT}\frac{\max_{|\xi|\leq \rho_2}G(\xi)}{\rho_2^2},
& \text{if } i=2.
\end{cases}
\end{aligned} \label{e3.3}
\end{equation}
and
\begin{equation}
\varphi^{(3)}(r_2,r_3)=\frac{\sup_{u\in\Phi^{-1}((-\infty,r_2+r_3))}\Psi(u)}{r_3}
\leq \frac{4\|b\|_{L^1}}{MT}\frac{\max_{|\xi|
\leq \rho_2}G(\xi)}{\rho_2^2},\label{e3.4}
\end{equation}
By \eqref{e3.3}, \eqref{e3.4}, we have
\begin{equation}
\begin{aligned}
\varphi_3(r_1,r_2,r_3)
&=\max\{\varphi^{(1)}(r_1), \varphi^{(1)}(r_2), \varphi^{(3)}(r_2,r_3)\}\\
&\leq\frac{4\|b\|_{L^1}}{T}\max\Big\{\frac{1}{\mu}
 \frac{\max_{|\xi|\leq \rho_1}G(\xi)}{\rho_1^2},
\frac{1}{M}\frac{\max_{|\xi|\leq \rho_2}G(\xi)}{\rho_2^2}\Big\}.
\end{aligned} \label{e3.5}
\end{equation}
Taking into account (A2), (2) in the assumptions  and (A3), we have
\begin{equation}
\begin{aligned}
\Phi(\bar{v})
&= \frac{1}{2}\|\bar{v}\|^2+\sum_{j=1}^l\sum_{i=1}^N
 \int_0^{\bar{v}^i(t_j)}I_{ij}(s)ds\\
&\geq \frac{\mu T}{2}|\bar{\xi}|^2-\sum_{j=1}^l\sum_{i=1}^N(c_{ij}
 |\bar{\xi}|+d_{ij}|\bar{\xi}|^{1+\gamma_{ij}})\\
&\geq \frac{\mu T}{2}|\bar{\xi}|^2-\sum_{j=1}^l\sum_{i=1}^N(c_{ij}
 |\bar{\xi}|+d_{ij}|\bar{\xi}|^{1+\bar{\gamma}})\\
&\geq \inf_{\bar{\rho}_1\geq x\geq \rho_1}
 \{\frac{\mu T}{2}\rho_1^2-\sum_{j=1}^l\sum_{i=1}^N(c_{ij}x+d_{ij}
 x^{1+\bar{\gamma}})\} \\
&=\frac{\mu T\rho_1^2}{4}=r_1>0,
\end{aligned}\label{e3.6}
\end{equation}
for sufficiently small $\rho_1>0$.

In view of (2), (H1) and \eqref{e2.1}, we  have
\begin{equation}
\begin{aligned}
\Phi(\bar{v})
&= \frac{1}{2}\|\bar{v}\|^2+\sum_{j=1}^l
 \sum_{i=1}^N\int_0^{\bar{v}^i(t_j)}I_{ij}(s)ds\\
&\leq \frac{T}{2}\sum_{j=1}^N\sum_{i=1}^N\|a_{ij}\|_{\infty}
|\bar{\xi}|^2+\sum_{j=1}^l\sum_{i=1}^N(c_{ij}|\bar{\xi}|
 +d_{ij}|\bar{\xi}|^{1+\gamma_{ij}})\\
&\leq \frac{TM}{2}\rho_2^2+\sum_{j=1}^l\sum_{i=1}^N(c_{ij}\rho_2+d_{ij}
 \rho_2^{1+\bar{\gamma}})\leq MT\rho_2^2=r_2,
\end{aligned}\label{e3.7}
\end{equation}
for sufficiently big
\[
\rho_2\gg \Big(\frac{2\sum_{j=1}^l\sum_{i=1}^N
(c_{ij}+d_{ij})}{MT}\Big)^{^{\frac{1}{1-\bar{\gamma}}}}>0,
\]
so by (2), we have
\begin{equation}
r_1<\Phi(\bar{v})<r_2.\label{e3.8}
\end{equation}
Since
\begin{equation}
\sup_{v\in\Phi^{-1}([r_1,r_2))}\frac{\Psi(v)-\Psi(u)}{\Phi(v)-\Phi(u)}
\geq \frac{\Psi(\bar{v})-\Psi(u)}{\Phi(\bar{v})-\Phi(u)},\label{e3.9}
\end{equation}
for $u\in\Phi^{-1}((-\infty,r_1))$, it follows that
\begin{equation}
\varphi_2(r_1,r_2)\geq \inf_{u\in\Phi^{-1}((-\infty,r_1))}
\frac{\Psi(\bar{v})-\Psi(u)}{\Phi(\bar{v})-\Phi(u)}.\label{e3.10}
\end{equation}
Fix $u\in \Phi^{-1}((-\infty,r_1))$, from \eqref{e3.1} and $(g_1)-(g_3)$, we have
$$
\Psi(\bar{v})-\Psi(u)\geq \|b\|_{L^1}\big(G(\bar{\xi})-\max_{|\xi|\leq \rho_1} G(\xi)\big)>0.\label{e3.11}$$
From \eqref{e3.7}, we have
\begin{equation}
0<\Phi(\bar{v})-\Phi(u)\leq\Phi(\bar{v})\leq TM|\bar{\xi}|^2.\label{e3.12}
\end{equation}
Then by \eqref{e3.5}, \eqref{e3.10}-\eqref{e3.12} and (2), we have
\begin{equation}
\begin{aligned}
\varphi_2(r_1,r_2)
&\geq  \frac{\|b\|_{L^1}}{TM}\dfrac{\big(G(\bar{\xi})-\max_{|\xi|\leq
 \rho_1} G(\xi)\big)}{|\bar{\xi}|^2} \\
&>\frac{\|b\|_{L^1}}{MT}\Big(1-\frac{1}{\max\{1,\frac{M}{\mu}\}}\Big)
 \frac{G(\bar{\xi})}{|\bar{\xi}|^2}\\
&= \frac{\|b\|_{L^1}}{MT}\Big(1-\frac{\mu}{M}\Big)
 \frac{G(\bar{\xi})}{|\bar{\xi}|^2} \\
&> \varphi_3(r_1,r_2,r_3),
\end{aligned}\label{e3.13}
\end{equation}
for $M>\mu>0$ ;i.e. that (iv) of Theorem \ref{thm2.1} holds.
Moreover $\Lambda_{\rho_1,\rho_2}\subseteq \Lambda_{r_1,r_2,r_3}$ and
for every $\lambda\in \Lambda_{\rho_1,\rho_2}$.
Assumption $(v)$   of Theorem \ref{thm2.1} is verified as a simple consequence
of the regularity of $\Phi,\Psi$ (see \cite[Remark 3.10]{b7}).
Then from Theorem \ref{thm2.1}, the proof is complete.
\end{proof}

As in the proof of Theorem \ref{thm3.1}, by Theorem \ref{thm2.2}, 
we have the following result.

\begin{theorem} \label{thm3.2}
Assume that the assumptions $(A1),(A_2)$ and
the hypotheses (A3) hold. $I_{ij}(y)$ is nondecreasing in $y\in \mathbb{R}$ 
for any $i=1,2,\dots,N$, $j=1,2,\dots,l$  with $I_{ij}(0)=0$.
Let $G\in C^1(\mathbb{R}^N, R)$ be such that
\begin{itemize}
\item[(1)]  $G(0)=0$;

\item[(2)]  there exist $\rho>0$ and $\bar{\xi}\in \mathbb{R}^N$ such that
\[
\frac{\max_{|\xi|\leq \rho}G(\xi)}{\rho^2}
< \frac{\mu(M-\mu)}{4M^2}\frac{G(\bar{\xi})}{M T|\bar{\xi}|^2};
\]

\item[(3)]  $\limsup_{|\xi|\to \infty}\frac{G(\xi)}{|\xi|^2}
<\frac{\max_{|\xi|\leq \rho}G(\xi)}{\rho^2}$.
\end{itemize}
Then for every $b\in L^1([0,T])\setminus \{0\}$ and for every $\lambda$ in
\[
 \Lambda:=\Big(\frac{MT}{\|b\|_{L^1}(1-\frac{\mu}{M})}
\frac{|\bar{\xi}|^2}{G(\bar{\xi})}, 
\frac{T\mu}{4\|b\|_{L^1}}\frac{\rho^2}{\max_{|\xi|\leq \rho}G(\xi)}
\Big),
\]
\eqref{e1.1} has at least three nontrivial solutions $u_1,u_2, u_3$, where  
$k,0<\mu<M$ are defined as \eqref{e2.3}, (A2), \eqref{e2.1}, respectively.
\end{theorem}


\begin{theorem} \label{thm3.3}
Assume that {\rm (A1)-(A3)} hold. $I_{ij}(y)$ is nondecreasing in
 $y\in \mathbb{R}$ for any $i=1,2,\dots,N,j=1,2,\dots,l$  with $I_{ij}(0)=0$.
Let
$$
\alpha=\liminf_{\rho\to +\infty}\frac{\max_{|\xi|\leq \rho}G(\xi)\xi}{\rho^2},\quad
\beta=\limsup_{\rho\to +\infty}\frac{G(\xi)\xi}{|\xi|^2},
$$
and assume that $\alpha <4\beta$.
 Then for every $b\in L^1([0,T])\setminus \{0\}$ and for every
$\lambda$ in $\Lambda
 :=\big( \frac{\mu T}{4\|b\|_{L^1}\beta}, \frac{\mu T}{4\|b\|_{L^1}\alpha}\big)$,
 \eqref{e1.1} has an unbounded sequence nontrivial solutions.
\end{theorem}

\begin{proof}  For every $b\in L^1([0,T])\setminus \{0\}$,
let $\Phi,\Psi$ be as \eqref{e2.4}, using
Theorem \ref{thm2.3}, from the proof of Theorem \ref{thm3.1}, we have that the 
functionals $\Phi,\Psi$ satisfy the regularity assumptions
required in Theorem \ref{thm2.3}. Let us now verify that
$\gamma=\liminf_{\rho\to +\infty}\varphi(\rho)<+\infty$.

Let $\{\rho_n\}$ be a sequence of positive numbers such that $\rho_n\to +\infty$ 
as $n\to +\infty$ and
\begin{equation}
\lim_{n\to +\infty}\frac{\max_{|\xi|\leq \rho_n}G(\xi)}{\rho^2_n}
=\liminf_{\rho\to +\infty}\frac{\max_{|\xi|\leq \rho}G(\xi)}{\rho^2}.\label{e3.14}
\end{equation}
Let $r_n=\frac{\mu T\rho_n^2}{4}, \forall\ n\in N$, similar to the reasoning of
\eqref{e3.3}, we have
\begin{equation}
\begin{aligned}
\varphi(r_n)
&= \inf_{u\in\Phi^{-1}((-\infty,r_n))}
\frac{\big(\sup_{v\in\Phi^{-1}((-\infty,r_n))}\Psi(v)\big)-\Psi(u)}{r_n-\Phi(u)}
\\
&\leq \frac{\sup_{v\in\Phi^{-1}((-\infty,r_n))}\Psi(v) -G(0)\|b\|{L^1}}{r_n}\\
&\leq \frac{4\|b\|_{L^1}}{\mu T}\frac{\max_{|\xi|\leq \rho_n}G(\xi)-G(0)}{\rho_n^2}\\
&=\frac{4\|b\|_{L^1}}{\mu T}\frac{\max_{|\xi|\leq \rho_n}G(\xi)}{\rho_n^2}.
\end{aligned}\label{e3.15}
\end{equation}
Then
\begin{equation}
0\leq \gamma:=\liminf_{r\to+\infty}\varphi(r)
\leq \frac{4\|b\|_{L^1}}{\mu T}\liminf_{\rho\to+\infty}
\frac{\max_{|\xi|\leq \rho}G(\xi)}{\rho^2}
=\frac{4\|b\|_{L^1}}{\mu T}\alpha<+\infty,\label{e3.16}
\end{equation}
i.e. $\gamma=\liminf_{\rho\to +\infty}\varphi(\rho)
=\frac{4\|b\|_{L^1}}{\mu T}\alpha<+\infty$.
In view of $\alpha < 4\beta$ and \eqref{e3.16}, we get that
$\Lambda \subseteq (0,\frac{1}{\gamma})$.
Now, we verify that the functional $\Phi-\lambda \Psi$ is unbounded
from below for $\lambda\in \Lambda$.
In fact, by the choice of $\lambda$ and the
positivity of $\beta$, one has that there exists a sequence
$\{\xi_n\}\subseteq \mathbb{R}^N$ with $|\xi_n|\to +\infty$ such that
for any $n\in N$,
\begin{equation}
\liminf_{n\to +\infty}\frac{G(\xi_n)}{|\xi_n|^2}
> \frac{1}{\frac{4\|b\|_{L^1}}{\mu T}\lambda},
\label{e3.17}
\end{equation}
i.e. $|\xi_n|\to +\infty$ for any $n\in N$, as $|\xi_n|\to +\infty$,
\begin{equation}
\frac{T}{2}\sum_{j=1}^N\sum_{i=1}^N\|a_{ij}\|_{\infty}
+\sum_{j=1}^l\sum_{i=1}^N(\frac{c_{ij}}{|\xi_n|}
+\frac{d_{ij}}{|\xi_n|^{\bar{\gamma}}})
-\lambda\|b\|_{L^1}\frac{G(\xi_n)}{|\xi_n|^2}<0,
\label{e3.18}
\end{equation}
If we let $v_n(t)=\xi_n$, for all $n\in N$, then for $v_n\in H_T^1$,
using the first equality of \eqref{e2.1},
we have
\begin{equation}
\begin{aligned}
I_{\lambda}
&=\Phi(v_n)-\lambda \Psi(v_n) \\
&= \frac{1}{2}\|v_n\|^2+\sum_{j=1}^l\sum_{i=1}^N\int_0^{v_n^i(t_j)}I_{ij}(s)ds
 -\int_0^Tb(t)G(v_n(t))dt \\
&\leq  \frac{T}{2}\sum_{j=1}^N\sum_{i=1}^N\|a_{ij}\|_{\infty}|\xi_n|^2
+\sum_{j=1}^l\sum_{i=1}^N(c_{ij}|\xi_n|+d_{ij}|\xi_n|^{1+\bar{\gamma}})
-\lambda\|b\|_{L^1}G(\xi_n)\\
&=  |\xi_n|^2\Big[\frac{T}{2}\sum_{j=1}^N\sum_{i=1}^N\|a_{ij}\|_{\infty}
+\sum_{j=1}^l\sum_{i=1}^N(\frac{c_{ij}}{|\xi_n|}
+\frac{d_{ij}}{|\xi_n|^{\bar{\gamma}}})
 -\lambda\|b\|_{L^1}\frac{G(\xi_n)}{|\xi_n|^2}\Big].
\end{aligned}\label{e3.19}
\end{equation}
By \eqref{e3.18} and \eqref{e3.19}, we can conclude that
$I_{\lambda}=\Phi-\lambda \Psi$ is unbounded from below.  Applying (b) of
Theorem \ref{thm2.3}, the proof of Theorem \ref{thm3.3} is complete.
\end{proof}

%\section{Example}

\begin{example} \label{examp4.1} \rm
Let $G\in C^1(R, R)$  be as
\[
G(x)=\begin{cases}
0, &\text{if }\ x\leq0;\\
\frac{\frac{1}{2}e^{e^{x}}-e(x+1)}{16}, &\text{if } 0<x<e^2;\\
\frac{e^{e^{x}}-\frac{1}{2}e^{e^{e^2}}-ex-e}{16}, 
&\text{if } e^2\leq x< e^6;\\
\frac{e^{(e^{e^2}+2)}-e}{4e^{6}}x^3-\frac{e^{(e^{e^2}+2)}-e}{4}e^{12}
+\frac{e^{e^{e^6}}-\frac{1}{2}e^{e^{e^2}}-e^7-e}{16},&\text{if } x\geq e^6.
\end{cases}
\]
Then for suitable $b\in L^1([0,1])\setminus \{0\}$ and for
\[
\lambda\in \Big(\frac{M}{\|b\|_{L^1}\big(1-\frac{\mu}{M}\big)}
\frac{|\bar{\xi}|^2}{G(\bar{\xi})}, 
\frac{1}{4\|b\|_{L^1}}\frac{1}{\max\big\{\frac{1}{\mu}
\frac{\max_{|\xi|\leq \rho_1}G(\xi)}{\rho_1^2},
\frac{1}{M}\frac{\max_{|\xi|\leq \rho_2}G(\xi)}{\rho_2^2}\big\}}\Big),
\]
the problem
\begin{equation}
\begin{gathered}
-\ddot{u}+A(t)u= \lambda b(t) \nabla G(u),\quad \text{a.e. }t\in [0,1],\\
\Delta(\dot{u}^i(t_j))=\dot{u}^i(t_j^+)-\dot{u}^i(t_j^-)=I_{ij}(u^i(t_j)),\quad
 i=1,\; j=1,2,\\
u(0)-u(1)=u'(0)-u'(1)=0,
\end{gathered} \label{e4.1}
\end{equation}
has at least two nontrivial solutions $u_1, u_2$ with $\|u\|_{\infty}\leq 2e^6$,
where $N=1$, $T=1$, $A(t)=1+\frac{\sin (2\pi t)}{20}$,
$\mu=0.95$, $M=1.05$, $\rho_1=1$, $\rho_2=2e^6$, $\bar{\xi}=\frac{99e^6}{100}$,
and
\begin{equation}
I_{ij}(y)=\begin{cases}
\frac{y}{4}+1,  i=1, j=1,\\
-\frac{y}{8}, i=1, j=2.\\
\end{cases}\label{e4.2}
\end{equation}
We easily have
\begin{gather*}
\big(1+\frac{4M}{\mu}\big)\frac{\max_{|\xi|\leq \rho_1}G(\xi)}{\rho_1^2}
=\frac{103}{19}G(1)=\frac{103}{304}(e^e-2),\\
4\frac{\max_{|\xi|\leq \rho_2}G(\xi)}{\rho_2^2}
=4\frac{G(e^6)}{e^{12}}=\frac{e^{e^{e^6}}-\frac{1}{2}e^{e^{e^2}}-e^7-e}{16e^{12}},\\
\frac{G(\bar{\xi})}{|\bar{\xi}|^2}=\frac{e^{e^{\frac{99}{100}e^6}}-\frac{1}{2}e^{e^{e^2}}-ex-e}{16\times\frac{9801e^{12}}{10000}}
=\frac{625}{9801e^{12}}(e^{e^{\frac{99}{100}e^6}}-\frac{1}{2}e^{e^{e^2}}
-\frac{2}{3}e^7-e),
\end{gather*}
then the assumptions of Theorem \ref{thm3.1} are satisfied.
\end{example}

\subsection*{Acknowledgements}
This research was supported by
 the Foundations of Anhui Provincial Department of Education (KJ2016A651).


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\end{document}