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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 55, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/55\hfil Global solutions to boundary-value problems]
{Global structure of solutions to boundary-value problems of impulsive differential
equations}

\author[Y. Niu, B. Yan \hfil EJDE-2016/55\hfilneg]
{Yanmin Niu, Baoqiang Yan}

\address{Yanmin Niu \newline
School of Mathematical Sciences,
Shandong Normal University,
Jinan 250014,  China}
\email{1398958626@qq.com}

\address{Baoqiang Yan (corresponding author) \newline
School of Mathematical Sciences,
Shandong Normal University,
Jinan 250014,  China}
\email{yanbqcn@aliyun.com}

\thanks{Submitted January 5, 2016. Published February 25, 2016.}
\subjclass[2010]{34B09, 34B15, 34B37}
\keywords{Comparison arguments; eigenvalues; global bifurcation theorem;
\hfill\break\indent  multiple solutions; asymptotical behavior of solutions}

\begin{abstract}
 In this article, we study the structure of global solutions to the 
 boundary-value problem
 \begin{gather*}
 -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad
 \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\\
  x(0)=x(1)=0,
 \end{gather*}
 where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$,
 $\Delta x|_{t=1/2}=x(\frac{1}{2}+0)-x(\frac{1}{2})$,
 $\Delta x'|_{t=1/2}=x'(\frac{1}{2}+0)-x'(\frac{1}{2}-0)$,
 and $f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$
 are continuous. By a comparison principle and spectral properties
 of the corresponding linear equations, we prove the existence of
 solutions by using Rabinowitz-type global bifurcation
 theorems, and obtain results on the behavior of positive solutions
 for large $\lambda$ when $f(x)=x^{p+1}$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In this article, we study the structure of global solutions to the 
second-order impulsive differential equation
\begin{equation}
\begin{gathered}
 -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad
 \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\\
  x(0)=x(1)=0,
 \end{gathered}\label{e1.1}
\end{equation}
where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$,
$\Delta x|_{t=1/2}=x(\frac{1}{2}+0)-x(\frac{1}{2})$,
$\Delta x'|_{t=1/2}=x'(\frac{1}{2}+0)-x'(\frac{1}{2}-0)$,
 and $f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$
are continuous.

Impulsive differential equations  arise in the contexts of population
dynamics, infectious diseases models, chemical technology and so on;
 see \cite{c2,f1,l3,o1,s3,t1,y2}.
Because impulsive equations appear in applied mathematics, they
attract a lot of attention.
Many authors studied the existence, uniqueness as well as multiplicity
of solutions, by using the topological degree theory and variational method;
 see \cite{a1,c3,l1,l2,l5,l6,n1,n2,s1,s2,t2,x1,x2,y1,y3,z1}.

An important tool to study the existence of solutions for differential
 equations is Rabinowitz global bifurcation theory; see
\cite{b1,c3,d1,m1,m2,m3,r1,r2,r3}.
But up to now, just a few results have shown on the structure of global solutions
for impulsive differential equations by Rabinowitz global bifurcation theorems.
 Liu and O'Regan \cite{l6} considered the second-order impulsive differential equation
\begin{equation}
\begin{gathered}
x''(t)+ra(t)f(t,x(t))=0, \quad t\in(0, 1),\; t\neq t_i,\\
\Delta x|_{t=t_i}=\alpha_ix(t_i-0), \quad i=1,2,\ldots,k,\\
x(0)=x(1)=0,
\end{gathered} \label{e1.2}
\end{equation}
in which they converted \eqref{e1.2} to the  form
\begin{equation}
\begin{gathered}
 y''(t)+\frac{r}{\Pi_{0<t_i<t}(1+\alpha_i)}a(t)f(t,\Pi_{0<t_i<t}(1+\alpha_i)y(t))=0,
\quad t\in(0, 1),\\
y(0)=y(1)=0.
\end{gathered} \label{e1.3}
\end{equation}
By the known properties of the eigenvalues and eigenfunctions of the linear
equation corresponding to \eqref{e1.3}, using bifurcation techniques,
they obtained the existence of multiple solutions to \eqref{e1.2}.

As we know, Rabinowitz global bifurcation theory can be used effectively
is that the spectral properties of relative linearized operators (especially,
nodal zeros of eigenfunctions) are clear. But impulses lead to the
complexity of eigenvalues and eigenfunctions of the impulsive linear equations,
which have not been analyzed completely. A recent work on this aspect
comes from \cite{w1} in which  Wang and  Yan presented the spectral properties
of the  equation
\begin{equation}
\begin{gathered}
 -x''(t)=\lambda x(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta x|_{t=1/2}=\lambda\beta x(\frac{1}{2}),\quad
 \Delta x'|_{t=1/2}=-\lambda\beta x'(\frac{1}{2}-0),\\
  x(0)=x(1)=0,
 \end{gathered}\label{e1.4}
\end{equation}
and studied the existence of multiple solutions for the relative nonlinear
second-order impulsive differential equations by Rabinowitz global
bifurcation theory.

In this article, we consider the nonlinear second-order impulsive differential
equation \eqref{e1.1}, in which the impulses are different from \eqref{e1.4}.
 This article is organized as follows.
In section 2, a Comparison Principle is established for the second-order
impulsive differential equations.
In section 3, we focus on the linear impulsive equation
\begin{gather*}
 -x''(t)=\lambda a(t)x(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad 
 \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\\
  x(0)=x(1)=0,
 \end{gather*}
where $\lambda$, $\beta_1$, $\beta_{2}$, $a(t)$, $\Delta x$, $\Delta x'$
are introduced as in \eqref{e1.1}, and present the eigenvalues and eigenfunctions
properties.
In section 4, we obtain the solutions of \eqref{e1.1} under various hypotheses
on the asymptotic behaviour of $f$ using the global bifurcation theorem from Section 3.
Finally, with the principal eigenvalue, we discuss the special autonomous case
\begin{equation}
\begin{gathered}
 -x''(t)+f(x)=\lambda mx(t),\quad t\in(0,1),\; t\neq \frac{1}{2},\\
 \Delta x|_{\frac{1}{2}}=\beta_1 x(\frac{1}{2}),\quad 
 \Delta x'|_{\frac{1}{2}}=-\beta_{2} x(\frac{1}{2}),\\
  x(0)=x(1)=0,
 \end{gathered} \label{e1.5}
\end{equation}
where $f(x)=x^{p+1}$ and $p>0$, $m>0$ are real numbers, and
give results on the behavior of positive solutions for the large $\lambda$.
Some ideas come from \cite{f1}.

\section{Preliminaries}

Comparison argument in ODE plays an important role in analyzing some
properties of solutions. Naturally, it necessary to study the relative
Comparison argument to get the properties of solutions of \eqref{e1.1}.

\begin{lemma} \label{lem2.1}
Suppose that $y(t)$ is the solution of
\begin{equation}
\begin{gathered}
 y''+Q(t)y=0,\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta y|_{t=1/2}=\beta_1 y(\frac{1}{2}),\quad
 \Delta y'|_{t=1/2}=-\beta_{2} y(\frac{1}{2}),
 \end{gathered}\label{e2.1}
\end{equation}
and $w(t)$ is the solution of
\begin{equation}
\begin{gathered}
 w''+P(t)w=0,\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta w|_{t=1/2}=\beta_1 w(\frac{1}{2}),\quad
 \Delta w'|_{t=1/2}=-\beta_{2} w(\frac{1}{2}),\\
 \end{gathered}\label{e2.2}
\end{equation}
where $P(t)$ and $Q(t)$ are continuous on the same interval $[0,1]$
with $Q(t)\leq P(t)$. If $\alpha$, $\beta$ are two next null points of $y(t)$,
then there must be at least one null point of $w(t)$ on $[\alpha,\beta]$.
\end{lemma}

\begin{proof}
 Suppose $\alpha$, $\beta$ are two next null points of $y(t)$.
We discuss two cases about them.

(1) $\alpha$, $\beta\in[0,1/2)$ or $\alpha$, $\beta\in(\frac{1}{2},1]$.
In this case, it is easy to obtain the results by Sturm comparison theorem in ODE.

(2) $\alpha<\frac{1}{2}<\beta$. Multiply  \eqref{e2.1} by $w(t)$,
 \eqref{e2.2} by $y(t)$ and subtract each other to obtain
            $$
wy''-w''y+[Q(t)-P(t)]wy=0.
$$
Integrating the equation above from $\alpha$ to $\beta$ yields
$$
\int_{\alpha}^{\beta}(wy''-w''y)\,dt
=\int_{\alpha}^{\beta}[P(t)-Q(t)]wy\,dt;
$$
therefore,
\begin{equation}
\begin{aligned}
&\int_{\alpha}^{\frac{1}{2}-}d(wy'-w'y)
 +\int_{\frac{1}{2}+}^{\beta}d(wy'-w'y)\\
&=\int_{\alpha}^{\frac{1}{2}-}[P(t)-Q(t)]wy\,dt
+\int_{\frac{1}{2}+}^{\beta}[P(t)-Q(t)]wy\,dt.\label{e2.3}
\end{aligned}
\end{equation}
From the impulsive conditions in \eqref{e2.1} and \eqref{e2.2}, we have
\begin{gather*}
w(\frac{1}{2}+)=(1+\beta_1)w(\frac{1}{2}-),\\
w'(\frac{1}{2}+)=w'(\frac{1}{2}-)-\beta_{2} w(\frac{1}{2}-),\\
y(\frac{1}{2}+)=(1+\beta_1)y(\frac{1}{2}-),\\
y'(\frac{1}{2}+)=y'(\frac{1}{2}-)-\beta_{2} y(\frac{1}{2}-),\\
\end{gather*}
which together with $y(\alpha)=y(\beta)=0$ guarantees that the left side of
\eqref{e2.3} satisfies
\begin{equation}
-\beta_1\Big[w(\frac{1}{2}-)y'(\frac{1}{2}-)
 -w'(\frac{1}{2}-)y(\frac{1}{2}-)\Big]
-w(\alpha)y'(\alpha)+w(\beta)y'(\beta). \label{e2.4}
\end{equation}
Denote
$$
F(t)=w(t)y'(t)-w'(t)y(t).
$$
Combining  \eqref{e2.1} with \eqref{e2.2}, one has
 $$
F'(t)=w(t)y''(t)-w''(t)y(t)=[P(t)-Q(t)]w(t)y(t).
$$

Suppose that $w(t)$ has no zero in $(\alpha,\beta)$, and without
loss of generality $y(t)$ is positive in $(\alpha,\beta)$.
Then two cases will be discussed according to $w(t)$.

(a) $w(t)>0$, $t\in(\alpha,\beta)$. In this case, the fact that
$F'(t)=[P(t)-Q(t)]w(t)y(t)>0$ for $t\in[\alpha,\frac{1}{2})$
implies that $F(t)$ increases on $[\alpha,\frac{1}{2})$.
Since
$$
F(\alpha)=w(\alpha)y'(\alpha)-w'(\alpha)y(\alpha)=w(\alpha)y'(\alpha)>0,
$$
one has $F(\frac{1}{2}-)>F(\alpha)>0$. Then \eqref{e2.4} is
 $$
-\beta_1F(\frac{1}{2})-w(\alpha)y'(\alpha)+w(\beta)y'(\beta)<0,
$$
while the right-hand side of \eqref{e2.3} satisfies
 $$
\int_{\alpha}^{1/2-}[P(t)-Q(t)]wy dt
+\int_{1/2+}^{\beta}[P(t)-Q(t)]wydt>0,
$$
which is a contradiction.

(b) $w(t)<0$, $t\in(\alpha,\beta)$. Since $F'(t)=[P(t)-Q(t)]w(t)y(t)<0$
for $t\in[\alpha,\frac{1}{2})$, $F(t)$ decreases on
$[\alpha,\frac{1}{2})$. With $F(\alpha)=w(\alpha)y'(\alpha)<0$, one has
$F(\frac{1}{2}-)<0$. Then \eqref{e2.4} is
 $$-\beta_1F(\frac{1}{2})-w(\alpha)y'(\alpha)+w(\beta)y'(\beta)>0,$$
while the right-hand side of  \eqref{e2.3} satisfies
$$
\int_{\alpha}^{\frac{1}{2}-}[P(t)-Q(t)]wydt
+\int_{\frac{1}{2}+}^{\beta}[P(t)-Q(t)]wydt<0,
$$
which is a contradiction. The proof is complete.
\end{proof}

For convenience, we introduce main lemma and some symbols
in global bifurcation theory.
 Let $\mathcal{F}:\mathcal{E}\to\mathcal{E}_1$ where $\mathcal{E}$
and $\mathcal{E}_1$ are real Banach spaces and $\mathcal{F}$ is continuous.
Suppose that the equation $\mathcal{F}(U)=0$ possesses a simple curve
of solutions $\Psi$ given by $\{U(t)|t\in[a,b]\}$.
If for some $\tau\in(a,b)$, $\mathcal{F}$ possesses zeros not lying on
$\Psi$ in every neighborhood of $U(\tau)$, then $U(\tau)$ is said to
be a bifurcation point for $\mathcal{F}$ with respect to the curve
$\Psi$ (see \cite{r2}).

A special family of such equations has the form
\begin{equation}
u=G(\lambda, u),\label{e2.5}
\end{equation}
where $\lambda\in\mathbb{R}$, $u\in E$ is a real Banach space with the
norm $\|\cdot\|$, and $G: \mathcal{E}\equiv\mathbb{R}\times E\to E$
is compact and continuous. In addition, $G(\lambda,u)=\lambda Lu+H(\lambda,u)$,
where $H(\lambda,u)$ is $o(\|u\|)$ for $u$ near 0 uniformly on bounded
$\lambda$ intervals and $L$ is a compact linear map on $E$.
A solution of \eqref{e2.5} is a pair $(\lambda,u)\in\mathcal{E}$.
The known curve of solutions $\Theta=\{(\lambda,0)|\lambda\in\mathbb{R}\}$
will henceforth be referred to as the trivial solutions.
The closure of the set of nontrivial solutions of \eqref{e2.5} will be
denoted by $\Sigma$. A component of $\Sigma$ is a maximal closed connected subset.

If there exist $\mu\in\mathbb{R}$ and $0\neq v\in E$ such that
$v=\mu Lv$, $\mu $ is said to be a real characteristic value of $L$.
The set of real characteristic values of $L$ will be denoted by $\sigma(L)$.
The multiplicity of $\mu\in\sigma(L)$ is the dimension of
$\cup_{j=1}^\infty N((\mu L-I)^j)$ where $I$ is the identity map on $E$
and $N(P)$ denotes the null space of $P$. Since $L$ is compact, $\mu$ is of
finite multiplicity. It is well known that if $\mu\in\mathbb{R}$,
a necessary condition for $(\mu,0)$ to be a bifurcation point of \eqref{e2.5}
with respect to $\Theta$ is that $\mu\in\sigma(L)$.

\begin{lemma}[\cite{r2}] \label{lem2.2}
 If $\mu\in\sigma(L)$ is simple, then $\Sigma$ contains a component
$\mathbf {C}_{\mu}$ that can be decomposed into two subcontinua
$\mathbf {C}_{\mu}^{+}$, $\mathbf {C}_{\mu}^{-}$ such that for some
neighborhood $\mathbf {B}$ of $(\mu,0)$,
$$
(\lambda,u)\in\mathbf {C}_{\mu}^{+}(\mathbf {C}_{\mu}^{-})\cap\mathbf {B},\quad
\text{and}\quad (\lambda,u)\neq(\mu,0)
$$
implies $(\lambda,u)=(\lambda,\alpha v+w)$ where $\alpha>0 (\alpha<0)$
and $\mid{\lambda-\mu}\mid=o(1)$, $\| w\|=o(\mid\alpha\mid)$, at $\alpha=0$.

Moreover, each of $\mathbf{C}_{\mu}^{+}$, $\mathbf{C}_{\mu}^{-}$ either
\begin{itemize}
\item[(1)] meets infinity in $\Sigma$, or

\item[(2)] meets $(\hat{\mu},0)$ where $\mu\neq\hat{\mu}\in\sigma(L)$, or

\item[(3)] contains a pair of points $(\lambda,u)$, $(\lambda,-\mu)$, $u\neq0$.
\end{itemize}
\end{lemma}

\section{Spectral properties for linear impulsive equations}

To apply Rabinowitz global bifurcation theorems on \eqref{e1.1},
 we need to study the linear equation corresponding to \eqref{e1.1}.
In this section the spectral properties including its eigenvalues,
along with their corresponding algebraic multiplicity, and also eigenfunctions
 structures will be discussed for the linear impulsive problems
\begin{equation}
\begin{gathered}
 -x''(t)=\lambda a(t)x(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad 
 \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\\
  x(0)=x(1)=0,
 \end{gathered}\label{e3.1}
\end{equation}
where $\lambda$, $\beta_1$, $\beta_{2}$, $a(t)$, $\Delta x$, $\Delta x'$
are introduced as in \eqref{e1.1}.

Denote $PC[0,1]=\{x:[0,1]\to\mathbb{R}:x(t)$ is continuous at
$t\neq\frac{1}{2}$, and $x(\frac{1}{2}-0)=\lim_{t\to\frac{1}{2}^{-}}x(t)$,
and $x(\frac{1}{2}+0)=\lim_{t\to\frac{1}{2}^{+}}x(t)$ exist $\}$ with the norm
$$
\| x\|=\sup_{t\in[0,1]}| x(t)|,
$$
and $PC'[0,1]=\{x\in PC[0,1]:x'(t)$ is continuous at $t\neq\frac{1}{2}$,
and $x'(\frac{1}{2}-0)=\lim_{t\to\frac{1}{2}^{-}}x'(t)$, and
$x'(\frac{1}{2}+0)=\lim_{t\to\frac{1}{2}^{+}}x'(t)$ exist $\}$ with the norm
$$
\| x\|_1=\max\{\sup_{t\in[0,1]}| x(t)|,\sup_{t\in[0,1]}| x'(t)|\}.
$$
Let $E=\{x\in PC'[0,1]:x(0)=x(1)=0\}$. It is well known that $E$
is a Banach space with the norm $\|\cdot\|_1$.

\begin{lemma}[\cite{g2}] \label{lem3.1}
The solution $x(t)\in PC[I,\mathbb{R}]\cap C^{2}[I',\mathbb{R}]$  of
\eqref{e3.1} is the same as the solution  $x(t)\in PC'[I,\mathbb{R}]$
of the integral equation
\begin{equation}
x(t)=\begin{cases}
\lambda\int_{0}^{1}G(t,s)a(s)x(s)ds
 -\big(\beta_1-\frac{1}{2}\beta_{2}\big) tx(\frac{1}{2}), &t\in[0,\frac{1}{2}],\\
\lambda\int_{0}^{1}G(t,s)a(s)x(s)ds
 +\big(\beta_1+\frac{1}{2}\beta_{2}\big)
(1-t)x(\frac{1}{2}), &t\in(\frac{1}{2},1],
\end{cases} \label{e3.2}
\end{equation}
where $I=[0,1]$, $I'=I\setminus{\{\frac{1}{2}\}}$, and
$$
G(t,s)=\begin{cases}
s(1-t), & 0\leq s\leq t\leq 1,\\
t(1-s), & 0\leq t\leq s\leq 1.
\end{cases}
$$
\end{lemma}

\begin{lemma} \label{lem3.2}
  All eigenvalues of \eqref{e3.1} are in $\mathbb{R}$.
\end{lemma}

\begin{proof}
Let $\lambda=\alpha+i\gamma$ be an eigenvalue of \eqref{e3.1} and
$u(t)=u_1(t)+iu_{2}(t)$ be an eigenfunction corresponding to $\lambda$.
It is sufficient to prove that $\gamma=0$. Since
$\lambda=\alpha+i\gamma$ and $u(t)=u_1(t)+iu_{2}(t)$ satisfy \eqref{e3.1},
it follows that
\begin{equation}
-u''_1=a(t)(\alpha u_1-\gamma u_{2}),\quad
-u''_{2}=a(t)(\alpha u_{2}+\gamma u_1),\label{e3.3}
\end{equation}
and
\begin{equation}
\begin{gathered}
\Delta u_1|_{t=1/2}=\beta_1 u_1(\frac{1}{2}),\quad
 \Delta u'_1|_{t=1/2}=-\beta_{2} u_1(\frac{1}{2}),\\
\Delta u_{2}|_{t=1/2}=\beta_1 u_{2}(\frac{1}{2}),\quad
 \Delta u'_{2}|_{t=1/2}=-\beta_{2} u_{2}(\frac{1}{2}),\\
u_1(0)=u_1(1)=0,\quad u_{2}(0)=u_{2}(1)=0.
 \end{gathered}\label{e3.4}
\end{equation}
Multiply  \eqref{e3.3} by $u_{2}(t)$ and $u_1(t)$ respectively
and subtract each other to yield that
$$
-u''_1u_{2}+u''_{2}u_1=-a(t)\gamma(u^{2}_1+u^{2}_{2}).
$$
Integrating the equation above from $0$ to $1$ and with condition \eqref{e3.4},
 we have
\begin{equation}
\beta_1 u'_1(\frac{1}{2}-)u_{2}(\frac{1}{2})
-\beta_1 u'_{2}(\frac{1}{2}-)u_1(\frac{1}{2})
=-\int_{0}^{1}a(t)\gamma(u^{2}_1+u^{2}_{2})dt.\label{e3.5}
\end{equation}
Denote $F(t)=u'_1(t)u_{2}(t)-u'_{2}(t)u_1(t)$ for
$t\in[0,\frac{1}{2})$. Since $u_1$, $u_{2}$ satisfy \eqref{e3.3},
it follows that
$$
F'(t)=a(t)\gamma(u^{2}_1(t)+u^{2}_{2}(t)).
$$
If $\gamma>0$, by $a(t)>0$ and $u^{2}_1(t)+u^{2}_{2}(t)>0$,
one has $F'(t)>0$. Then $F(0)=u'_1(0)u_{2}(0)-u'_{2}(0)u_1(0)=0$
implies that $F(\frac{1}{2}-)>0$. The left side of \eqref{e3.5}
is $\beta_1 F(\frac{1}{2}-)>0$, while the right side of it is negative,
which is a contradiction. On the other hand, with the similar discussion,
we deduce the contradiction if $\gamma<0$.
Therefore, $\gamma=0$ and Lemma \ref{lem3.2} holds.
\end{proof}

\begin{theorem} \label{thm3.1} Equation
\eqref{e3.1} possesses an increasing sequence of eigenvalues
$$
0<\lambda_1<\lambda_{2}<\dots<\lambda_{k}<\dots,
 \lim_{k\to+\infty}\lambda_k=+\infty.
$$
And the eigenfunction $u_k$ corresponding to $\lambda_k$ has exactly
$k-1$ nodal zeros in $(0,1)$.
\end{theorem}

The proof of this theorem will be divided into two parts,
in which the methods chosen are different due to the complexity of the impulse.
Part 1 mainly shows the existence of principal eigenvalue and eigenfunction
by Schauder's fixed point theorem, while Part 2 guarantees the existence
of second and subsequent eigenvalues by some tools on uncontinuous Sturm-Liouville
systems.
\smallskip

\noindent\textbf{Part 1. Existence of principal eigenvalue and eigenfunction}
In  \eqref{e3.2}, let
$$
y(t)=\begin{cases}
x(t)+(\beta_1-\frac{1}{2}\beta_{2})tx(\frac{1}{2}), &t\in[0, \frac{1}{2}],\\[4pt]
x(t)-(\beta_1+\frac{1}{2}\beta_{2})(1-t)x(\frac{1}{2}),&t\in(\frac{1}{2}, 1],
\end{cases}
$$
and then $y(t)\in C^{1}[0,1]$. Equation \eqref{e3.2} is transformed into
\begin{equation}
y(t)=\lambda\int_{0}^{1}G(t,s)a(s)H(y(s))ds,\label{e3.6}
\end{equation}
where
\begin{equation}
H(y(t))=\begin{cases}
y(t)-\frac{\beta_1-\frac{1}{2}\beta_{2}}{1+\frac{1}{2}\beta_1
 -\frac{1}{4}\beta_{2}}y(\frac{1}{2})t, & t\in[0,\frac{1}{2}],\\[4pt]
y(t)+\frac{\beta_1+\frac{1}{2}\beta_{2}}{1+\frac{1}{2}\beta_1
 -\frac{1}{4}\beta_{2}}y(\frac{1}{2})(1-t), & t\in(\frac{1}{2},1].
\end{cases} \label{e3.7}
\end{equation}
By Lemma \ref{lem3.1}, the study of \eqref{e3.1} is the same as that of \eqref{e3.6}.
Next we give the main result.

\begin{lemma} \label{lem3.3}
There exist $\lambda^{\ast}>0$ and nonnegative function $x^{\ast}$ satisfying
\eqref{e3.1}.
\end{lemma}

\begin{proof}
Let $D:=\{y\in C[0,1]|\int_{0}^{1}y(s)ds=1,y(t)\geq0,y(t)$
is concave and $y(0)=y(1)=0\}$ and define an operator $A$:
$$
(Ay)(t)=\frac{\int_{0}^{1}G(t,s)a(s)H(y(s))ds}
{\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y(s))\,ds\,dt},\quad \forall y\in D.
$$
We show $(Ay)(t)>0$. In fact, by  \eqref{e3.7}, the sign of $(Ay)(t)$
depends on the sign of $H(y(t))$. Since $y\in D$ and $\beta_1\geq\beta_{2}\geq0$,
it is easy to obtain that for $t\in(\frac{1}{2},1]$
$$
y(t)+\frac{\beta_1+\frac{1}{2}\beta_{2}}{1+\frac{1}{2}
\beta_1-\frac{1}{4}\beta_{2}}y(\frac{1}{2})(1-t)>0.
$$
Define
$$
g(t)=y(t)-\frac{\beta_1
-\frac{1}{2}\beta_{2}}{1+\frac{1}{2}\beta_1
-\frac{1}{4}\beta_{2}}y(\frac{1}{2})t, \quad t\in[0,\frac{1}{2}].
$$
Since $y(t)$ is concave, together with $g(0)=y(0)\geq0$ and
$$
g(\frac{1}{2})=\frac{1}{1+\frac{1}{2}\beta_1-\frac{1}{4}\beta_{2}}y(\frac{1}{2})
\geq 0,
$$
one has $g(t)\geq0$ for $t\in[0,\frac{1}{2}]$. That is to say $(Ay)(t)\geq0$
for $t\in[0,1]$. Thus, we conclude that $D$ is a closed convex set and an
standard argument shows that $A:D\to D$ is continuous.
Furthermore, the following two facts show that $A(D)$ is compact.

(1) $A(D)$ is uniformly bounded. For $y\in D$, since $\int_0^1y(t)dt\leq \|y\|$,
one has $\|y\|\geq 1$. Choose $y(t_0)=\max_{t\in[0,1]}\|y\|$.
For $t\in (0,t_0)$, we have
$y(t)=y(\frac{t}{t_0}t_0+(1-\frac{t}{t_0})0)
\geq \frac{t}{t_0}y(t_0)+(1-\frac{t}{t_0})y(0)=\frac{t}{t_0}\|y\|$.
The same argument shows that
for $t\in (t_0,1)$, we have $y(t)\geq \frac{1-t}{1-t_0}\|y\|$. Hence,
$$
y(t)\geq \frac{t(1-t)}{t_0(1-t_0)}\|y\|\geq t(1-t)\|y\|\geq t(1-t),\quad
 \forall t\in[0,1]
$$
and
$$
\int_0^1y(s)ds\geq\int_0^1\|y\|s(1-s)ds=\|y\|\frac{1}{6},
\text{ i.e.} \|y\|\leq 6.
$$
Next we estimate
$$
\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y(s))\,ds\,dt,\quad
 \forall y\in D.
$$
As $a(t)>0$ is continuous, suppose
$$
0<k\leq a(t)\leq K, t\in[0,1],
$$
and denote
$$
B_1=\frac{\beta_1-\frac{1}{2}\beta_{2}}{1+\frac{1}{2}\beta_1
-\frac{1}{4}\beta_{2}}, B_{2}=\frac{\beta_1
+\frac{1}{2}\beta_{2}}{1+\frac{1}{2}\beta_1
-\frac{1}{4}\beta_{2}}.
$$
With \eqref{e3.7} and $y(t)\geq t(1-t)$ for $t\in[0,1]$, we have
\begin{align*}
&\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y(s))\,ds\,dt\\
&=\int_{0}^{1/2}(1-t)\int_{0}^{t}sa(s)
 \Big(y(s)-B_1y(\frac{1}{2})s\Big)\,ds\,dt\\
&\quad +\int_{0}^{1/2}t\int_{t}^{1/2}(1-s)a(s)
 \left(y(s)-B_1y(\frac{1}{2})s\right)\,ds\,dt\\
&\quad +\int_{0}^{1/2}t\int_{\frac{1}{2}}^{1}(1-s)a(s)
 \left(y(s)+B_{2}y(\frac{1}{2})(1-s)\right)\,ds\,dt\\
&\quad +\int_{\frac{1}{2}}^{1}(1-t)\int_{0}^{1/2}sa(s)
 \left(y(s)-B_1y(\frac{1}{2})s\right)\,ds\,dt\\
&\quad +\int_{\frac{1}{2}}^{1}(1-t)\int_{\frac{1}{2}}^{t}sa(s)
 \left(y(s)+B_{2}y(\frac{1}{2})(1-s)\right)\,ds\,dt\\
&\quad +\int_{\frac{1}{2}}^{1}t\int_{t}^{1}(1-s)a(s)
 \left(y(s)+B_{2}y(\frac{1}{2})(1-s)\right)\,ds\,dt\\
&\geq k\Big[\frac{1}{60}+\frac{5}{384}y(\frac{1}{2})(B_{2}-B_1)\Big],
\end{align*}
where
$$
B_{2}-B_1=\frac{\beta_{2}}{1+\frac{1}{2}\beta_1-\frac{1}{4}\beta_{2}}\geq0
$$
and $y\in D$. Hence,
$$
\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y(s))\,ds\,dt\geq\frac{k}{60}, \quad y\in D.
$$
Since $G(t,s)$ is bounded on $[0,1]\times[0,1]$, by letting $0\leq G(t,s)\leq M$,
we have
\begin{align*}
|(Ay)(t)|
&\leq \frac{60}{k}\big|\int_{0}^{1}G(t,s)a(s)H(y(s))ds\big|\\
&\leq \frac{60}{k}MK\big|\int_{0}^{1}H(y(s))ds\big|,
\end{align*}
where
$$
\big|\int_{0}^{1}H(y(s))ds\big|
=\Big(1+\frac{1}{8}y(\frac{1}{2})(B_{2}-B_1)\Big).
$$
That implies $A(D)$ is uniformly bounded.

(2) $A(D)$ is equicontinuous.
In fact, as discussed in above, it's true for all $y\in D$ that
$$
\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y(s))\,ds\,dt\geq\frac{k}{60}.
$$
Since $G(t,s)$ is continuous on $[0,1]\times[0,1]$, it is uniformly
continuous on $[0,1]\times[0,1]$. So for all $\varepsilon>0$,
there exists $\delta>0$ such that
$$
|G(t,s)-G(t',s)|<C\varepsilon,\quad \forall |t-t'|<\delta,
$$
where
$$
C=\frac{k}{60K\left(1+\frac{1}{8}y(\frac{1}{2})(B_{2}-B_1)\right)}.
$$
And then
\begin{align*}
&|(Ay)(t)-(Ay)(t')|\\
&=\Big|\frac{\int_{0}^{1}G(t,s)a(s)H(y(s))ds}
 {\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y(s))\,ds\,dt}
 -\frac{\int_{0}^{1}G(t',s)a(s)H(y(s))ds}
 {\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y(s))\,ds\,dt}\Big|\\
&=\Big|\frac{\int_{0}^{1}[G(t,s)-G(t',s)]a(s)H(y(s))ds}
 {\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y(s))\,ds\,dt}\Big|\\
&\leq \frac{C\varepsilon\times60K}{k}\int_{0}^{1}H(y(s))ds
=\varepsilon.
\end{align*}

Arezela-Ascoli theorem and (1) (2) imply that $A(D)$ is compact.
According to Schauder fixed point theorem, there exists a
$y^{\ast}\in D$ satisfying $Ay^{\ast}=y^{\ast}$, i.e.
\begin{equation}
\frac{\int_{0}^{1}G(t,s)a(s)H(y^{\ast}(s))ds}
{\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y^{\ast}(s))\,ds\,dt}
=y^{\ast}(t).\label{e3.8}
\end{equation}
Denote $\int_{0}^{1}\int_{0}^{1}G(t,s)a(s)H(y^{\ast}(s))\,ds\,dt
=\frac{1}{\lambda^{\ast}}$ and then \eqref{e3.8} can be written as
$$
y^{\ast}=\lambda^{\ast}\int_{0}^{1}G(t,s)a(s)H(y^{\ast}(s))ds.
$$
Note Lemma \ref{lem3.1} and transformation
$$
y(t)=\begin{cases}
x(t)-\left(\beta_1-\frac{1}{2}\beta_{2}\right)tx(\frac{1}{2}),
 &t\in[0,\frac{1}{2}],\\
x(t)-\left(\beta_1+\frac{1}{2}\beta_{2}\right)(1-t)x(\frac{1}{2}),
&t\in(\frac{1}{2},1],
\end{cases}
$$
we define
$$
x^{\ast}(t)=\begin{cases}
y^{\ast}(t)-\frac{\beta_1-\frac{1}{2}\beta_{2}}{1+\frac{1}{2}\beta_1
 -\frac{1}{4}\beta_{2}}y^{\ast}(\frac{1}{2})t, &t\in[0,\frac{1}{2}],\\[4pt]
y^{\ast}(t)+\frac{\beta_1+\frac{1}{2}\beta_{2}}{1+\frac{1}{2}\beta_1
 -\frac{1}{4}\beta_{2}}y^{\ast}(\frac{1}{2})(1-t), &t\in(\frac{1}{2},1],
\end{cases}
$$
and such $x^{\ast}$ satisfies Lemma \ref{lem3.3}.
\smallskip

\noindent\textbf{Part 2. Existence of second and subsequent eigenvalues}
In this part, we consider uncontinuous Sturm-Liouville equation
\begin{equation}
-x''(t)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2}, \label{e3.9}
\end{equation}
with the boundary value condition
\begin{equation}
x(0)=0, x(1)=0,\label{e3.10}
\end{equation}
and the impulsive conditions
\begin{gather}
x(\frac{1}{2}+)=(1+\beta_1)x(\frac{1}{2}-), \label{e3.11}\\
x'(\frac{1}{2}+)=x'(\frac{1}{2}-)-\beta_{2}x(\frac{1}{2}-). \label{e3.12}
\end{gather}
For each $\lambda\in R$, let $\zeta_{-}(t,\lambda)$ and
$\zeta_{+}(t,\lambda)$ be the solutions of \eqref{e3.9} satisfying
the initial condition
\begin{equation}
\begin{gathered}
\zeta_{-}(0)=0,\zeta'_{-}(0)=-1,\zeta_{+}(\frac{1}{2}+)
=(1+\beta_1)\zeta_{-}(\frac{1}{2}-),\\
\zeta'_{+}(\frac{1}{2}+)=-\beta_{2}\zeta_{-}(\frac{1}{2}-)
+\zeta'_{-}(\frac{1}{2}-),
\end{gathered} \label{e3.13}
\end{equation}
and let $\eta_{+}(t,\lambda)$ and $\eta_{-}(t,\lambda)$ be the solutions
of \eqref{e3.9} satisfying the initial condition
\begin{equation}
\begin{gathered}
\eta_{+}(1)=0,\quad \eta'_{+}(1)=-1,\quad
\eta_{-}(\frac{1}{2}-)=\frac{1}{1+\beta_1}\eta_{+}(\frac{1}{2}+),\\
\eta'_{-}(\frac{1}{2}-)=\frac{1}{1+\beta_1}
\Big[\beta_{2}\eta_{+}(\frac{1}{2}+)+(1+\beta_1)\eta'_{+}(\frac{1}{2}+)\Big].
\end{gathered} \label{e3.14}
\end{equation}
Denote the solution of \eqref{e3.9} with condition \eqref{e3.13} and
 \eqref{e3.14} respectively by
$$
\zeta(t,\lambda)=\begin{cases}
\zeta_{-}(t,\lambda), &t\in(0, \frac{1}{2}),\\
\zeta_{+}(t,\lambda), &t\in(\frac{1}{2}, 1),
\end{cases}
\quad
\eta(t,\lambda)=\begin{cases}
\eta_{-}(t,\lambda), &t\in(0, \frac{1}{2}),\\
\eta_{+}(t,\lambda), &t\in(\frac{1}{2}, 1).
\end{cases}
$$
\end{proof}

By Lemma \ref{lem3.2}, we choose $\lambda$ as a real number.
When $\lambda$ increases, the number of zeros of $\zeta(t,\lambda)$
increases subsequently in view of Lemma \ref{lem2.1}.
Let $\rho_{-}(\lambda)$ be the number of zeros of $\zeta(t,\lambda)$
on $(0,\frac{1}{2})$, $\rho_{+}(\lambda)$ be the number of zeros of
$\zeta(t,\lambda)$ on $(\frac{1}{2},1)$ and
$\rho(\lambda)=\rho_{-}(\lambda)+\rho_{+}(\lambda)$ be the number of
zeros of $\zeta(t,\lambda)$ on $(0,\frac{1}{2})\cup(\frac{1}{2},1)$.
Then we have the following conclusions.

\begin{lemma} \label{lem3.4} \quad
\begin{itemize}
\item[(1)] There exists $A>0$ such that $\rho(\lambda)=0$ for all $\lambda<-A$;
\item[(2)] $\lim_{\lambda\to+\infty}\rho(\lambda)=+\infty$
\end{itemize}
\end{lemma}

\begin{proof}
Denote $M_{-}=\min_{t\in[0,\frac{1}{2}]}a(t)$ and
 $M_{+}=\min_{t\in[\frac{1}{2},1]}a(t)$.

(i) When $\lambda<0$, we first consider the ODE initial value problem
\begin{equation}
\begin{gathered}
 x''(t)+\lambda M_{-}x(t)=0,\quad t\in[0,\frac{1}{2}],\\
  x(0,\lambda)=0,x'(0,\lambda)=-1,
 \end{gathered}\label{e3.15}
\end{equation}
and its solution can be written as
$$
x_{-}(t,\lambda)=\frac{-1}{2\sqrt{-\lambda M_{-}}}e^{\sqrt{-\lambda M_{-}}t}
+\frac{1}{2\sqrt{-\lambda M_{-}}}e^{-\sqrt{-\lambda M_{-}}t}.
$$
The above expression implies that $x_{-}(t,\lambda)<0$ as
$\lambda\to-\infty$, which means $\rho_{-}(\lambda)=0$ by comparison theorem of ODE.

Next, for the another ODE equation,
\begin{equation}
\begin{gathered}
 x''(t)+\lambda M_{+}x(t)=0,\quad t\in[\frac{1}{2},1],\\
 x\big(\frac{1}{2}+,\lambda\big)=(1+\beta_1)x
\big(\frac{1}{2}-,\lambda\big),x'\big(\frac{1}{2}+,\lambda\big)
=x'\big(\frac{1}{2}-,\lambda\big)-\beta_{2} x\big(\frac{1}{2}-,\lambda\big),
 \end{gathered}\label{e3.16}
\end{equation}
the solution  is
\begin{align*}
&x_{+}(t,\lambda)\\
&=\Big(\frac{1+\beta_1}{2}x_{-}\big(\frac{1}{2}-,\lambda\big)
 +\frac{-\beta_{2} x_{-}\big(\frac{1}{2}-,\lambda\big)
+x'_{-}(\frac{1}{2}-,\lambda)}{2\sqrt{-\lambda M_{+}}}\Big)
 e^{\sqrt{-\lambda M_{+}}(t-\frac{1}{2})} \\
&\quad +\Big(\frac{1+\beta_1}{2}x_{-}\big(\frac{1}{2}-,\lambda\big)
-\frac{-\beta_{2} x_{-}\big(\frac{1}{2}-,\lambda\big)
+x'_{-}\big(\frac{1}{2}-,\lambda\big)}{2\sqrt{-\lambda M_{+}}}\Big)
e^{-\sqrt{-\lambda M_{+}}(t-\frac{1}{2})},
\end{align*}
from which we see $x_{+}(t,\lambda)<0$ as $\lambda\to-\infty$
and we conclude that $\rho_{+}(\lambda)=0$.
\smallskip

(ii) When $\lambda>0$, denote $M=\min\{M_{-},M_{+}\}$ and consider  \eqref{e3.15}.
A solution of it has the form
$$
x_{-}(t,\lambda)=\frac{-1}{\sqrt{\lambda M_{-}}}\sin\sqrt{\lambda M_{-}}t,
$$
of which the number of zeros on $(0,\frac{1}{2})$ is
$[\frac{\frac{1}{2}-0}{\pi}\sqrt{\lambda M_{-}}]$ or
$[\frac{\frac{1}{2}-0}{\pi}\sqrt{\lambda M_{-}}]+1$.

Then we turn to  \eqref{e3.16}, while the solution  is
\begin{align*}
x_{+}(t,\lambda)
&=(1+\beta_1)x_{-}\big(\frac{1}{2}-,\lambda\big)
 \cos\sqrt{\lambda M_{+}}\big(t-\frac{1}{2}\big)\\
&\quad -\frac{1}{\sqrt{\lambda M_{+}}}\Big(-\beta_{2}x_{-}
 \big(\frac{1}{2}-,\lambda\big)+x'_{-}\big(\frac{1}{2}-,\lambda\big)\Big)
 \sin\sqrt{\lambda M_{+}}\big(t-\frac{1}{2}\big)\\
&= C\sin\left(\sqrt{\lambda M_{+}}t+\gamma\right).
\end{align*}
We know the number of zeros of $x_{+}(t,\lambda)$ on $(\frac{1}{2},1)$
is $[\frac{1-\frac{1}{2}}{\pi}\sqrt{\lambda M_{+}}]$ or
$[\frac{1-\frac{1}{2}}{\pi}\sqrt{\lambda M_{+}}]+1$.
Notice that
$$
\big[\frac{\frac{1}{2}-0}{\pi}\sqrt{\lambda M_{-}}\big]
+\big[\frac{1-\frac{1}{2}}{\pi}\sqrt{\lambda M_{+}}\big]
>\big[\frac{\frac{1}{2}-0}{\pi}\sqrt{\lambda M}\big]
 +\big[\frac{1-\frac{1}{2}}{\pi}\sqrt{\lambda M}\big].
$$
By a comparison argument, when $\lambda>M$, the number of zeros of
$\zeta(t,\lambda)$ is no less than
$[\frac{\frac{1}{2}-0}{\pi}\sqrt{\lambda M}]
+[\frac{1-\frac{1}{2}}{\pi}\sqrt{\lambda M}]$,
which prove (2). We complete the proof.
\end{proof}

Denote by $\mathfrak{R}_{k}$ ($k=1,2,\dots$) the point set on $\mathbb{R}$:
$$
\mathfrak{R}_1=\{\lambda\in\mathbb{R}:\rho(\lambda)=0\},\quad
\mathfrak{R}_{k}=\{\lambda\in\mathbb{R}:\rho(\lambda)\geq k-1>0\}.
$$
According to Lemma \ref{lem3.4} and the continuity of
$\zeta_{\pm}(t,\lambda)$, $\mathfrak{R}_{k}$ $(k=1,2,\dots)$ are nonempty
and $\mathfrak{R}_{k}(k>1)$ are closed sets with lower bounds.

\begin{lemma} \label{lem3.5} 
 Denote $\mu_{k}=\min{\mathfrak{R}_{k}(k=2,\dots )}$. 
If $\zeta(1,\mu_{k})=0$, then $\zeta(t,\mu_{k})$ has exactly $k-1$ 
zeros in $(0,\frac{1}{2})\cup(\frac{1}{2},1)$.
\end{lemma}

\begin{proof} 
Since $\mu_{k}=\min{\mathfrak{R}_{k}(k=2,\dots )}$, it follows 
that $\rho(\mu_{k})\geq k-1$, which means that $\zeta(t,\mu_{k})$ 
has at least $k-1$ zeros, denoting by $0<t_1<t_{2}<\cdots<t_{k-1}<\cdots$.
Now we prove $t_{k}=1$.

On the contrary, assume $t_{k}<1$. $\zeta'(t_{k},\mu_{k})=0$ is not possible 
in view of uniqueness of initial value problem due to $\zeta(t_{k},\mu_{k})=0$.
Noticing that 
$$
-\zeta''_{\lambda}=a(t)\zeta+\lambda a(t)\zeta_{\lambda}, \quad 
\zeta_{\lambda}(0,\lambda)=\zeta'_{\lambda}(0,\lambda)=0, 
$$
and using the Green formula for $\zeta_{\lambda}(t,\mu_{k})$ and 
$\zeta(t,\mu_{k})$ on $[0,t_{k}]$, we have 
$$
(1+\beta_1)\int_{0}^{1/2}a(t)\zeta^{2}(t,\mu_{k})dt+\int_{\frac{1}{2}}^{t_{k}}
a(t)\zeta^{2}(t,\mu_{k})dt=\zeta_{\lambda}(t_{k},\mu_{k})\zeta'(t_{k},\mu_{k}),
$$
hence
$$
\zeta_{\lambda}(t_{k},\mu_{k})
=\frac{(1+\beta_1)\int_{0}^{1/2}a(t)\zeta^{2}(t,\mu_{k})dt
+\int_{\frac{1}{2}}^{t_{k}}a(t)\zeta^{2}(t,\mu_{k})dt}{\zeta'(t_{k},\mu_{k})}\neq0.
$$
It follows by the implicit function theorem that with $\zeta(t,\lambda)=0$, 
there exists a unique implicit function $t=t(\lambda)$ at the neighbourhood 
of $(t_{k},\mu_{k})$,
which satisfies $t_{k}=t(\mu_{k})$ and
  $$
\frac{dt}{d\lambda}\mid_{(t_{k},\mu_{k})}
=-\frac{\zeta_{\lambda}(t_{k},\mu_{k})}{\zeta'(t_{k},\mu_{k})}
=-\frac{(1+\beta_1)\int_{0}^{1/2}a(t)\zeta^{2}(t,\mu_{k})dt
+\int_{\frac{1}{2}}^{t_{k}}a(t)\zeta^{2}(t,\mu_{k})dt}{\zeta'^{2}(t_{k},\mu_{k})}<0.
$$
Namely, $t=t(\lambda)$ decreases with $\lambda$. Then we select 
$\lambda_{\ast}<\mu_{k}$ satisfying $t_{\ast}=t(\lambda_{\ast})\in(t_{k},b)$ 
such that $\zeta(t_{\ast},\lambda_{\ast})=0$. 
By Lemma \ref{lem2.1}, $\zeta(t,\lambda_{\ast})$ has at least $k-1$ 
zeros in $(a,t_{\ast})$, which implies that $\lambda_{\ast}\in\mathfrak{R}_{k}$. 
This contradicts with $\mu_{k}=\min{\mathfrak{R}_{k}}$. Consequently,
 $t_{k}=1$ and the proof is complete.
\end{proof}

By Lemma \ref{lem3.5}  we deduce the next statements.

\begin{corollary} \label{coro3.6}  
$[\mu_{k},\mu_{k+1})=\mathfrak{R}_{k}\backslash \mathfrak{R}_{k+1},
 \mathfrak{R}_{k}=\cup_{j=k}^{\infty}[\mu_{j},\mu_{j+1})(k=2,\dots )$.
\end{corollary}

\begin{corollary} \label{coro3.7}
 $\rho(\lambda)$ is an increased step function:
 $\rho(\lambda)=k-1(k=1,2,\dots ), \lambda\in[\mu_{k},\mu_{k+1})$.
\end{corollary}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
First it will be proved that \eqref{e3.1} has only one eigenvalue 
in each $[\mu_{n},\mu_{n+1})(n=2,\dots )$, denoting by $\lambda_{n}$, 
and the corresponding eigenfunction $\zeta(t,\lambda_{n})$ has exactly $n-1$ 
zeros in $(0,\frac{1}{2})\cup(\frac{1}{2},1)$. Because 
$\zeta(t,\mu_{n})$ satisfies the boundary value condition $\zeta(1,\mu_{n})=0$, 
it follows from Lemma \ref{lem3.5} that $\lambda_{n}=\mu_{n}$.

Next we will show there is at most one eigenvalue of \eqref{e3.1}, 
denoting by $\lambda_1$, except for $\lambda_{n}$ $(n=2,\dots )$ mentioned above.
And then $\lambda_1\in\mathfrak{R}_1$ and the corresponding eigenfunction
has no zero in $(0,1/2)\cup(1/2,1)$. 
Lemma \ref{lem3.3} guarantees the existence of $\lambda_1$ with
$\lambda_{\ast}=\lambda_1$, so we just need to show the uniqueness 
of $\lambda_1$.
Assume there are two different principal eigenvalues: 
$\lambda_1$, $\lambda'_1$. Using the Green's formula on eigenfunctions
$\zeta(t,\lambda_1)$ and $\zeta(t,\lambda'_1)$, we have
\begin{align*}
&(\lambda_1-\lambda'_1)\Big(\int_{0}^{1/2}\zeta(t,\lambda_1)\zeta(t,
\lambda'_1)dt+\frac{1}{1+\beta_1}
\int_{\frac{1}{2}}^{1}\zeta(t,\lambda_1)\zeta(t,\lambda'_1)dt\Big)\\
&=\zeta(1,\lambda_1)\zeta'(1,\lambda'_1)-\zeta'(1,\lambda_1)
\zeta(1,\lambda'_1)-\zeta(0,\lambda_1)\zeta'(0,\lambda'_1)
+\zeta'(0,\lambda_1)\zeta(0,\lambda'_1).
\end{align*}
The left side of the above equation is not zero, while the right side is zero. 
That is a contradiction.

During the discussion, it is not difficult to find that zeros of
 $\zeta_{-}(t,\lambda)$ and $\zeta_{+}(t,\lambda)$ are nodal zeros 
by Sturm-Liouville theory in ODE. The proof is complete.  
\end{proof}

We remark that result in Theorem \ref{thm3.1} can be obtained for the  equation
\begin{gather*}
 -x''(t)+q(t)x(t)=\lambda a(t)x(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad
 \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}), \\
  x(0)=x(1)=0,
 \end{gather*}
where $q(t)\in C[0,1]$, $q(t)\geq0($ or $q(t)\leq0)$ and $a(t)$,
$\Delta x|_{t=1/2}$ and $\Delta x'|_{t=1/2}$ are defined as before.

\begin{lemma} \label{lem3.8} 
 For each $k\geq 1$, the algebraic multiplicity of eigenvalue $\lambda_k$ is 
$1$.
\end{lemma}

\begin{proof}
 Define operator $K: PC'[0,1]\to PC'[0,1]$ as follows:
$$
(Kx)(t)=\begin{cases}
\int_{0}^{1}G(t,s)a(s)x(s)ds-\frac{1}{\lambda}
\big(\beta_1-\frac{1}{2}\beta_{2}\big) tx(\frac{1}{2}) ,& t\in[0,\frac{1}{2}],\\[4pt]
\int_{0}^{1}G(t,s)a(s)x(s)ds+\frac{1}{\lambda}
\big(\beta_1+\frac{1}{2}\beta_{2}\big)(1-t)x(\frac{1}{2}) ,& t\in(\frac{1}{2},1].
\end{cases}
$$
We just need to prove $\ker(I-\lambda_{k}K)^2\subset \ker(I-\lambda_{k}K)$.

For any $y\in \ker(I-\lambda_{k}K)^2$, from $(I-\lambda_{k}K)^2 y=0$, 
one has that $(I-\lambda_{k}K)y\in ker(I-\lambda_{k}K)$.
Let $\lambda_{k}$ be the $k^{th}$ eigenvalue of \eqref{e3.1} 
and $u_{k}(t)$ be the eigenfunction corresponding to $\lambda_{k}$.
By Theorem \ref{thm3.1}, there exists a $\gamma$ satisfying
$$
(I-\lambda_{k}K)y=\gamma u_{k}(t),\quad t\in[0,1],
$$
which implies that $y$ satisfies
\begin{equation}
\begin{gathered}
y''+\lambda_ka(t)y+\gamma\lambda_{k}a(t)u_{k}(t)=0, \quad
 t\in(0,1),\; t\neq\frac{1}{2},\\
\Delta y|_{t=1/2}=\beta_1 y(\frac{1}{2}),\quad 
\Delta y'|_{t=1/2}=-\beta_{2} y(\frac{1}{2}),\\
y(0)=y(1)=0.
\end{gathered} \label{e3.17}
\end{equation}
Now we prove $\gamma=0$.
Otherwise, assume $\gamma>0$ and notice \eqref{e3.17} when $k=1$.
 With $\lambda_1>0$, $u_1(t)>0$ for $t\in(0,1)$, two cases of $y$
will be discussed.

If $y(t)$ has no zero in $(0,1)$, assume $y(t)>0$ and rewrite \eqref{e3.17} 
in the form
\begin{equation}
\begin{gathered}
y''+\lambda_1a(t)\big[1+\frac{\gamma u_1(t)}{y(t)}\big]y(t)=0, \quad
t\in(0,1), t\neq\frac{1}{2},\\
\Delta y|_{t=1/2}=\beta_1 y(\frac{1}{2}),\quad 
\Delta y'|_{t=1/2}=-\beta_{2} y(\frac{1}{2}),\\
y(0)=y(1)=0.
\end{gathered}\label{e3.18}
\end{equation}
Inequality $1+\frac{\gamma u_1(t)}{y(t)}>1$, and Lemma \ref{lem2.1}, guarantee
 that $y(t)$  has a zero in $(0,1)$, which contradicts with the assumption.

If $y(t)$ has a zero in $(0,1)\backslash{\frac{1}{2}}$, denote it by $t^{\ast}$, 
so that $y(t)>0$ for $t\in(0,t^{\ast})$ and $y(t)<0$ for $t\in(t^{\ast},1)$ 
(such $t^{\ast}$ always exists because it is a nodal zero).
So  \eqref{e3.17} on $(t^{\ast},1)$ can be transformed into \eqref{e3.18}. 
From $1+\frac{\gamma u_1(t)}{y(t)}<1$ and Lemma \ref{lem2.1}, $u_1(t)$
has a zero in $(t^{\ast},1)$, which yields a contradiction.

So $\gamma=0$ and $y\in ker(I-\lambda_{k}K)$. We complete the proof.
\end{proof}

\section{Multiple solutions for nonlinear impulsive differential Equation}

In this section, we consider the problem\
\begin{equation}
\begin{gathered}
 -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad 
 \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\\
 x(0)=x(1)=0,
 \end{gathered}\label{e4.1}
\end{equation}
where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$,
$\Delta x|_{t=1/2}=x(\frac{1}{2}+)-x(\frac{1}{2}-)$,
$\Delta x'|_{t=1/2}=x'(\frac{1}{2}+)-x'(\frac{1}{2}-)$, and
$f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$ are continuous.
In addition, suppose the following assumptions of $f(t,x)$ hold.
\begin{equation}
f(t,0)=0,f(t,s)=o(s) \label{e4.2}
\end{equation}
at the neighbourhood of $0$ and uniformly for all $t\in[0,1]$.
\begin{equation}
s\mapsto\frac{f(t,s)}{s}, \quad  s\mapsto-\frac{f(t,-s)}{s} \label{e4.3}
\end{equation}
strictly increase on $\mathbb{R^{+}}$, for all $t\in[0,1]$.
\begin{equation}
\lim_{s\to\pm\infty}\frac{f(t,s)}{s}=+\infty,  \label{e4.4}
\end{equation}
uniformly for all $t\in[0,1]$.
Denote $\lambda_i$ be the $i^{th}$ eigenvalue of problem
\begin{equation}
\begin{gathered}
 -\phi''(t)=\lambda a(t)\phi(t),\quad t\in(0,1),\quad t\neq\frac{1}{2},\\
 \Delta \phi|_{t=1/2}=\beta_1 \phi(\frac{1}{2}),\quad 
 \Delta \phi'|_{t=1/2}=-\beta_{2} \phi(\frac{1}{2}),\\
  \phi(0)=\phi(1)=0.
 \end{gathered}\label{e4.5}
\end{equation}
For the rest of this article, we suppose that the  initial value problem
\begin{gather*}
 -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad
 \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\\
 x(t_{0})=x'(t_{0})=0,
 \end{gather*}
has the unique trivial solution $x\equiv0$ on $[0,1]$, for any $t_{0}\in[0,1]$.

Before giving main results, it is necessary to give the lower and upper 
solution theorem of \eqref{e4.1}, as a useful tool in the proof of 
the existence of positive solutions.
Now we consider
\begin{equation}
\begin{gathered}
 -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq t_1,\\
 \Delta x|_{t_1}=\beta_1 x(t_1),\quad 
 \Delta x'|_{t_1}=-\beta_{2} x(t_1),\\
  x(0)=x(1)=0,
 \end{gathered}\label{e4.6}
\end{equation}
where $f:D\to R$ is continuous, $D\subseteq[0,1]\times R$ and $t_1\in(0,1)$.
The solution of \eqref{e4.6} is defined as
$x(t)\in PC([0,1],R)\cap PC''((0,1),R)$ satisfying \eqref{e4.6},
where $PC([0,1],R)=\{x:[0,1]\to R:x(t)$ is continuous at
$t\neq t_1$, $x(t_1-0)=\lim_{t\to t_1^{-}}x(t)$ and
$x(t_1+0)=\lim_{t\to t_1^{+}}x(t)$ exist $\}$, and
$PC''((0,1),R)=\{x:(0,1)\to R:x''(t)$ is continuous at
 $t\neq t_1$, $\lim_{t\to t_1^{-}}x''(t)$ and
$ \lim_{t\to t_1^{+}}x''(t)$ exist $\}$.

If $\alpha(t)\in PC([0,1],R)\cap PC''((0,1),R)$ satisfies
\begin{gather*}
 -\alpha''(t)+f(t,\alpha(t))\leq\lambda a\alpha(t),\quad t\in(0,1),\;
 t\neq\frac{1}{2},\\
 \Delta \alpha|_{t_1}=\beta_1 \alpha(t_1),\quad 
 \Delta \alpha'|_{t_1}=-\beta_{2} \alpha(t_1),\\
  \alpha(0)\leq0,\alpha(1)\leq0,
 \end{gather*}
it is a lower solution of \eqref{e4.6}, while the upper solution
 is defined as $\beta(t)$ satisfying
\begin{gather*}
 -\beta''(t)+f(t,\beta(t))\geq\lambda a\beta(t),\quad t\in(0,1),\;
  t\neq\frac{1}{2},\\
 \Delta \beta|_{t_1}=\beta_1 \beta(t_1),\quad 
 \Delta \beta'|_{t_1}=-\beta_{2} \beta(t_1),\\
  \beta(0)\geq0,\beta(1)\geq0.
 \end{gather*}
Denote 
$$
D_{\alpha}^{\beta}=\{(t,x)\in(0,1)\times R, \alpha(t)\leq x\leq\beta(t), t\in(0,1)\},
$$
where $\alpha(t)$, $\beta(t)\in PC([0,1],R)$ with $\alpha(t)\leq\beta(t)$ 
for all $t\in[0,1]$. Then we give the lower and upper solution theorem of
 \eqref{e4.6}.

\begin{lemma} \label{lem4.1} 
 Assume $\alpha(t)$, $\beta(t)$ are the lower and upper solutions respectively 
with $\alpha(t)\leq\beta(t)$ for $t\in(0,1)$, and $D_{\alpha}^{\beta}\subseteq D$. 
Moreover, for each $\lambda\in R$ there exists 
$ h(t,\lambda)\in C([0,1]\times R,R^{+})$ satisfying 
\begin{gather*}
|H(t,x,\lambda)|:= |f(t,x)-\lambda a(t)x(t)|
\leq h(t,\lambda),\quad \forall(t,x)\in D_{\alpha}^{\beta},\\
\int_{0}^{1}s(1-s)h(s,\lambda)<g(\lambda)<+\infty.
\end{gather*}
Then there is at least one solution of \eqref{e4.6} denoted
by $x\overline(t)$ with $\alpha(t)\leq x\overline(t)\leq\beta(t)$.
\end{lemma}

\begin{proof}
 Define auxiliary functions
$$
H^{\ast}(t,x,\lambda)=\begin{cases}
H(t,\alpha(t),\lambda), &x<\alpha(t),\\
H(t,x,\lambda), &\alpha(t)\leq x(t)\leq\beta(t),\\
H(t,\beta(t),\lambda), &x>\beta(t),
 \end{cases}
$$
 where $H(t,x,\lambda)=f(t,x)-\lambda a(t)x(t)$, and
 $$
I^{\ast}(x)=\begin{cases}
I(\alpha(t_1)), &x< \alpha(t_1),\\
I(x), &\alpha(t_1)\leq x\leq\beta(t_1),\\
I(\beta(t_1)), &x>\beta(t_1),
 \end{cases}
 $$
 where $I(x)=x(t_1)$.
It is obvious that
 $$
|H^{\ast}(t,x,\lambda)|<h(t,\lambda),\quad \forall (t,x)\in(0,1)\times R.
$$
Then we consider problem
\begin{equation}
\begin{gathered}
 -x''(t)+H^{\ast}(t,x,\lambda)=0,\quad t\in(0,1),\; t\neq t_1,\\
 \Delta x|_{t_1}=\beta_1 I^{\ast}(x(t_1)),\quad 
 \Delta x'|_{t_1}=-\beta_{2} I^{\ast}(x(t_1)),\\
  x(0)=x(1)=0,
 \end{gathered}\label{e4.7}
\end{equation}
and show that if $x(t)$ is a solution of \eqref{e4.7},
 $\alpha(t)\leq x(t)\leq\beta(t)$ holds,
which implies that $x(t)$ is a solution of  \eqref{e4.6}.

On the contrary, if there is a $t^{\ast}$ such that $x(t^{\ast})<\alpha(t^{\ast})$,
 there are three possible cases for $t^{\ast}$.

(1) Assume $t^{\ast}<t_1$. Let
$r=\inf\{t<t^{\ast}: x(s)<\alpha(s), \forall s\in[t,t^{\ast}], t\in(0,1)\}$ 
and $r'=\sup\{t>t^{\ast}: x(s)<\alpha(s), \forall s\in[t^{\ast},t], t\in(0,1)\}$.
 We discuss two cases about $r'$.

(a) There exists a $t'\in (t^{\ast},t_1]$ satisfying $x(t')=\alpha(t')$.
In this case, it's obvious that $r'\leq t_1$. Then for every $ t\in(r,r')$,
$$
x(r)=\alpha(r), x(r')=\alpha(r'), x(t)<\alpha(t),
 H^{\ast}(t,x,\lambda)=H(t,\alpha(t),\lambda).
$$
Thus, $x(t)$ satisfies
$$ 
-x''(t)+H(t,\alpha(t),\lambda)=0,\quad t\in(r,r').
$$
On the other hand, since $\alpha(t)$ is a lower solution with
$$
-\alpha''(t)+H(t,\alpha(t),\lambda)\leq0,\quad t\in(r,r'),
$$
by denoting $z(t)=\alpha(t)-x(t)$, for every $ t\in(r,r')$, one has
$$ 
z(r)=z(r')=0, z''(t)\geq0.
$$
By maximum principle, $z(t)\leq0$ holds for all $t\in(r,r')$.
 Namely for $ t\in(r,r')$, $\alpha(t)\leq x(t)$, which contradicts 
$\alpha(t^{\ast})> x(t^{\ast})$.

(b) For every $t\in [t^{\ast},t_1]$, $x(t)<\alpha(t)$ holds. In this case,
noticing $\alpha(t_1+0)=\alpha(t_1)+\beta_1\alpha(t_1)> x(t_1)
+\beta_1\alpha(t_1)=x(t_1)+\beta_1 I^{\ast}(x(t_1))=x(t_1+0)$,
we have $r'>t_1$ and for every $ t\in(r,r')$,
 $$
x(r)=\alpha(r), x(r')=\alpha(r'), x(t)<\alpha(t), H^{\ast}
(t,x,\lambda)=H(t,\alpha(t),\lambda).
$$
Since $x(t)$ and $\alpha(t)$ satisfy
\begin{gather*} 
-x''(t)+H(t,\alpha(t),\lambda)=0,\quad t\in(r,r')\setminus{t_1},\\
-\alpha''(t)+H(t,\alpha(t),\lambda)\leq0,\quad t\in(r,r')\setminus{t_1},
\end{gather*}
by letting $z(t)=\alpha(t)-x(t)$, for $ t\in(r,r')$, it holds that
\begin{gather*}
 -z''(t)\geq0,\quad t\in(r,r'),\; t\neq t_1,\\
 \Delta z|_{t_1}=0,\quad 
 \Delta z'|_{t_1}=0,\\
  z(r)=z(r')=0.
 \end{gather*}
Since $-z''(t)\geq0$ for $t\in(r,r')\setminus t_1$ and
$\Delta z'|_{t_1}=0$, $z'(t)$ increases and is continuous on $(r,r')$.
Lagrange mean value theorem, together with $z(r)=0$ and $z(t_1)>0$
guarantees that there exists a $t'\in(r,t_1)$ with $z'(t')>0$.
Similarly, with $z(t_1+0)>0$ and $z(r')=0$, there exists  a $t''\in(t_1,r')$
so that $z'(t'')<0$. This
contradicts to the monotonicity of $z'(t)$.
According to the discussion above, we find case (1) impossible.

(2) Assume $t^{\ast}=t_1$. As before, define
$r=\inf\{t<t^{\ast}: x(s)<\alpha(s), \forall s\in[t,t^{\ast}], t\in(0,1)\}$ 
and $r'=\sup\{t>t^{\ast}: x(s)<\alpha(s), \forall s\in[t^{\ast},t], t\in(0,1)\}$. 
The conclusion on $(r,r')$ is same with the case (b) in (1), which can be excluded.

(3) Assume $t^{\ast}>t_1$. The definitions of $r,r'$ are same as before.
We divide the discussion of $r$ into two cases.

(a') There exists a $t'\in (t_1,t^{\ast}]$ satisfying $x(t')=\alpha(t')$
or $\alpha(t_1+0)=x(t_1+0)$. This case is similar to the $(a)$ in case (1),
which deduces a contradiction.

(b') For every $t\in (t_1,t^{\ast}]$, $x(t)<\alpha(t)$ holds, and so
$x(t_1+0)=x(t_1)+\beta_1I^{\ast}(x(t_1))<\alpha(t_1+0)
=\alpha(t_1)+\beta_1I(\alpha(t_1))$. Then it can be concluded that
$x(t_1)<\alpha(t_1)$. This case is similar to $(b)$ in case (1),
which is impossible.

Combining cases (1), (2) and (3), we conclude that for $ t\in[0,1]$, 
$x(t)\geq\alpha(t)$. With the same method, $x(t)\leq\beta(t)$ is true 
for $t\in[0,1]$.
The fact is that if $x(t)$ is the solution of  \eqref{e4.7}, it must 
be the solution of  \eqref{e4.6}, which has been proved. Next we will 
give the existence of solutions for \eqref{e4.7}.

The solution of \eqref{e4.7} has the form (see \cite{g2})
\begin{align*}
x(t)
&=-\int_{0}^{1}G(t,s)H^{\ast}(s,x(s),\lambda)ds
 +[\beta_1-(t-t_1\beta_{2})]\sum_{0<t_1<t}I^{\ast}(x(t_1))\\
&\quad -t[\beta_1-(1-t_1)\beta_{2}] I^{\ast}(x(t_1)), \quad t\in[0,1],
\end{align*}
where 
$$
G(t,s)=\begin{cases}
s(1-t), & 0\leq s\leq t\leq 1,\\
t(1-s), & 0\leq t\leq s\leq 1.
\end{cases}
$$
For $x\in PC([0,1],R)$ and $\lambda \in R$, define
\begin{align*}
(T_{\lambda}x)(t)
&= -\int_{0}^{1}G(t,s)H^{\ast}(s,x(s),\lambda)ds
+[\beta_1-(t-t_1\beta_{2})]\sum_{0<t_1<t}I^{\ast}(x(t_1))\\
&\quad -t[\beta_1-(1-t_1)\beta_{2}]I^{\ast}(x(t_1)).
\end{align*}
From the dominated convergence theorem, we can check that 
$T_{\lambda}:PC([0,1],R)\to PC([0,1],R)$ is a continuous and bounded 
operator for each $\lambda \in R$. Furthermore, $T_{\lambda}(PC([0,1],R))$ 
is compact.
Indeed, for $t\in[0,1]\backslash t_1$, we have
\begin{align*}
&\big|\frac{d}{dt}(T_{\lambda}x)(t))\big|\\
&\leq \big|\frac{d}{dt}\int_{0}^{t}s(1-t)H^{\ast}ds\big|
 +\big|\frac{d}{dt}\int_{t}^{1}t(1-s)H^{\ast}ds\big|
+M|I^{\ast}(x(t_1))|\\
&\leq \int_{0}^{t}s|H^{\ast}(s,x(s),\lambda)|ds
 +\int_{t}^{1}(1-s)|H^{\ast}(s,x(s),\lambda)|ds+M|I^{\ast}(x(t_1))|\\
&=\gamma_{\lambda}(t)+M|I^{\ast}(x(t_1))|,
\end{align*}
and
\begin{align*}
 \int_{0}^{1}\gamma_{\lambda}(s)ds
 &\leq \lim_{t\to1-}(1-t)\int_{0}^{t}sh(s,\lambda)ds
 +\lim_{t\to1+}\int_{t}^{1}(1-s)h(s,\lambda)ds\\
&\quad +2\int_{0}^{1}s(1-s)h(s,\lambda)ds\\
 &\leq 4\int_{0}^{1}s(1-s)h(s,\lambda)ds\\
 &\leq g(\lambda)
 <+\infty,
\end{align*}
so that for each $\lambda\in R,\ \gamma_{\lambda}(t)\in L^{1}([0,1],R^{+})$.
According to Schauder's fixed point theorem, $T_{\lambda}$ has at least a 
fixed point $\overline{x}\in PC[0,1]$, which implies $\overline{x}(t)$ is 
a solution of  \eqref{e4.7}. The proof is complete. 
\end{proof}

\begin{theorem} \label{thm4.1} 
 Assume that conditions \eqref{e4.2}, \eqref{e4.3} and \eqref{e4.4} hold. 
For $\lambda>\lambda_1$, there exists an unique positive solution of
\eqref{e4.1}. The mapping $\lambda\mapsto x_{\lambda}$ is continuous from 
$(\lambda_1,+\infty)$ to $PC''([0,1],R)$ and the branch
$\{(\lambda,x_{\lambda}),\lambda\in(\lambda_1,+\infty)\}$ bifurcates from
the right of trivial solutions at $(\lambda_1,0)$. Moreover,
$x_{\lambda}$ increases strictly with $\lambda$: if $\lambda<\mu$, then 
$x_{\lambda}<x_{\mu}$. Finally, $x_{\lambda}\to+\infty$, uniformly for 
$t$ in the closed interval $\subseteq (0,1)$ as $\lambda\to+\infty$.
\end{theorem}

\begin{proof} 
For $\rho\in C([0,1],R^{+})$, denote $\nu_i(\rho)$ the $i^{th}$ eigenvalue of problem
\begin{equation}
\begin{gathered}
 -\phi''(t)+\rho\phi=\nu a(t)\phi(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\\
 \Delta \phi|_{t=1/2}=\beta_1 \phi(\frac{1}{2}),\quad 
 \Delta \phi'|_{t=1/2}=-\beta_{2} \phi(\frac{1}{2}),\\
  \phi(0)=\phi(1)=0.
 \end{gathered} \label{e4.8}
\end{equation}
Then $\lambda_i=\nu_i(0)$. By Lemma \ref{lem2.1}, if $0<\rho\leq\widehat{\rho}$
and $\rho\not\equiv\widehat{\rho}$, then
 $$
 \nu_i(\rho)<\nu_i(\widehat{\rho}),\quad \forall i \in \mathbb{N}.
$$

It is true that there is no nontrivial solution of  \eqref{e4.1}, 
if $\lambda\leq \lambda_1$. Indeed, on the contrary, let $x\geq0$
be a solution of \eqref{e4.1}, and then $x$ satisfies \eqref{e4.8} 
with $\rho(t)=\frac{f(t,x(t))}{x(t)}$ if $x(t)>0$, $\rho(t)=0$ 
if $x(t)=0$, and $\nu=\lambda$. By condition \eqref{e4.3}, 
$\rho\geq0$ and $\rho\not\equiv0$ hold, so that
 $\lambda=\nu_1(\rho)>\nu_1(0)=\lambda_1$, which is a contradiction.

We show the existence of solutions of \eqref{e4.1} for $\lambda>\lambda_1$.
Let $\phi_1(t)$ be the prime eigenfunction of \eqref{e4.5} corresponding
to $\lambda_1$, so $\phi_1(t)>0$ for $t\in(0,1)$.
First, it will be verified that, for sufficiently small $\varepsilon>0$, 
$\varepsilon\phi_1$ is a lower solution of \eqref{e4.1}. Indeed,
with $\lambda>\lambda_1$ and condition \eqref{e4.2}, for sufficiently
small $\varepsilon$, we have
\begin{align*}
-\varepsilon\phi_1''(t)+f(t,\varepsilon\phi_1(t))
 -\lambda a(t)\varepsilon\phi_1(t)
&=\lambda_1 a(t)\varepsilon\phi_1(t)-\lambda a(t)\varepsilon\phi_1(t)
 +f(t,\varepsilon\phi_1(t)) \\
&=\Big[(\lambda_1-\lambda)a(t)
 +\frac{f(t,\varepsilon\phi_1(t))}{\varepsilon\phi_1(t)}\Big]
 \varepsilon\phi_1(t)
<0,
\end{align*}
and $\Delta \varepsilon\phi_1|_{t=1/2}=\beta_1\varepsilon\phi_1(\frac{1}{2})$,
$\Delta(\varepsilon\phi_1)'|_{t=1/2}=-\beta_{2} \varepsilon\phi_1(\frac{1}{2})$,
and $\varepsilon\phi_1(0)=\varepsilon\phi_1(1)=0$, which infer that
$\varepsilon\phi_1$ is a lower solution.
Next we will find a upper solution of \eqref{e4.1}. Condition \eqref{e4.4} 
implies that there exists a sufficiently grand constant $M_{\lambda}$ 
for each $\lambda\in R$, so that $f(t,s)\geq\lambda a(t)s $ for 
$s>M_{\lambda}$ and $t\in[0,1]$.
Define
$$
x^{\ast}=\begin{cases}
C_1t+C_{2}, & t\in[0,\frac{1}{2}],\\
C_{3}t+C_{4}, & t\in(\frac{1}{2},1],
\end{cases}
$$
where $C_1$, $C_{2}$, $C_{3}$ and $C_{4}$ can be selected by the
following conditions:
\begin{gather*}
 x^{\ast}(\frac{1}{2}+)=(1+\beta_1)x^{\ast}(\frac{1}{2}-),\\
(x^{\ast})'(\frac{1}{2}+)=(x^{\ast})'(\frac{1}{2}-)
 -\beta_{2} x^{\ast}(\frac{1}{2}-),\\
 x^{\ast}(t)>M_{\lambda},\quad \forall t\in [0,1].
\end{gather*}
Such $x^{\ast}$ defined above satisfies
\begin{gather*}
 -(x^{\ast})''(t)+f(t,x^{\ast})\geq\lambda ax^{\ast}(t),\quad t\in(0,1),\;
  t\neq\frac{1}{2},\\
 \Delta x^{\ast}|_{t=1/2}=\beta_1 x^{\ast}(\frac{1}{2}),\quad 
 \Delta (x^{\ast})'|_{t=1/2}=-\beta_{2} x^{\ast}(\frac{1}{2}),\\
  x^{\ast}(0)\geq0,  x^{\ast}(1)\geq0,
 \end{gather*}
which means $x^{\ast}$ a upper solution of \eqref{e4.1}.
It is easy to check that Lemma \ref{lem4.1} holds, so there exists a positive 
solution of \eqref{e4.1} between $\varepsilon\phi_1$ and $ x^{\ast}$.

Let $x_{\lambda}$ be the maximum positive solution of \eqref{e4.1}
 satisfying $\varepsilon\phi_1\leq x(t)\leq x^{\ast}$.
Then $x_{\lambda}$ is unique. Indeed, assume $x$ be another positive 
solution of \eqref{e4.1} with $x<x_{\lambda}$. Combining this with
 condition \eqref{e4.3}, we have 
$$
\frac{f(\cdot,x)}{x}<\frac{f(\cdot,x_{\lambda})}{x_{\lambda}},
$$ 
while 
$$
\lambda=\nu_1\Big(\frac{f(\cdot,x)}{x}\Big)
=\nu_1\Big(\frac{f(\cdot,x_{\lambda})}{x_{\lambda}}\Big).
$$ 
This is a contradiction.

Assume $\lambda_1<\lambda<\mu$ and $x_{\lambda}$, $x_{\mu}$ be the positive solutions of \eqref{e4.1} corresponding to $\lambda$, $\mu$ respectively. Then
\begin{align*}
-x''_{\lambda}(t)+f(t,x_{\lambda})-\mu ax_{\lambda}(t)
&=\lambda a(t)x_{\lambda}(t)-\mu a(t)x_{\lambda}(t) \\
&=(\lambda-\mu)a(t)x_{\lambda}(t)
<0,
\end{align*}
which indicates that $x_{\lambda}$ is a lower solution of \eqref{e4.1} 
corresponding to $\mu$. By the uniqueness of solutions, one has
 $x_{\lambda}<x_{\mu}$, namely the monotone of $x_{\lambda}$ with $\lambda$. 
The prior estimation of $M_{\lambda}$ and uniqueness of solution of 
\eqref{e4.1}, along with Lemma \ref{lem2.2}, mean that the mapping of 
$\lambda\mapsto x_{\lambda}$ is continuous from $(\lambda_1,+\infty)$
to $PC''([0,1],R)$ and the branch of 
$\{(\lambda,x_{\lambda}), \lambda\in(\lambda_1,+\infty)\}$
bifurcates from the right of trivial solutions at $(\lambda_1,0)$.

Then we prove $x_{\lambda}\to+\infty$, as $\lambda\to+\infty$ uniformly 
for $t$ in the closed interval $\subseteq (0,1)$.
For all $\gamma>0$, define
$$
\Lambda(\gamma)=\lambda_1\|a\|_{C^{0}[0,1]}
+\max_{t\in[0,1]}\frac{f(t,\gamma)}{\gamma}.
$$
Then for all $\gamma>0$, $\Lambda(\gamma)<+\infty$ and $\Lambda(\gamma)$ 
increases with $\gamma$. Here $\phi_1$ is also the prime eigenfunction of
 \eqref{e4.5} and satisfies  $\|\phi_1\|_1=1$.
For  all $\lambda \geq\Lambda(\gamma)$, $\gamma\phi_1$ is a lower solution of
 \eqref{e4.1}. In fact, condition \eqref{e4.3} implies that 
$f(t,\gamma\phi_1)\leq(f(t,\gamma)\setminus\gamma)\gamma\phi_1$,
by which we compute 
$$
-(\gamma\phi_1)''+f(t,\gamma\phi_1)\leq{\lambda_1a(t)
+\frac{f(t,\gamma)}{\gamma}}\gamma\phi_1\leq\Lambda(\gamma)
\gamma\phi_1\leq\lambda \gamma\phi_1.
$$
So $\gamma\phi_1$ is a lower solution of \eqref{e4.1} with
$\gamma\phi_1\leq x_{\lambda}$, and  for all $\gamma>0$ there exists
$\Lambda(\gamma)>0$ telling that 
$$
\lambda \geq\Lambda(\gamma)\Rightarrow\gamma\phi_1\leq x_{\lambda},\quad t\in(0,1).
$$
Since $\phi_1>0$ for $t\in(0,1)$, it can be deduced that
 $x_{\lambda}\to+\infty$, as $\lambda\to+\infty$ uniformly for $t$ in the 
closed interval $\subseteq (0,1)$.
The proof is complete.
\end{proof}

For convenience of the next proof, we give some symbols here. 
Let $E=\{x\in PC'[0,1]:x(0)=x(1)=0\}$. It is well known that $E$ 
is a Banach space with the norm $\|\cdot\|_1$.
Denote $S_{k}$ be the set of functions in $E$ which have exactly $k-1$ 
simple nodal zeros in $(0,1)$ (by a nodal zero we mean the function 
changes sign at the zeros and at a simple nodal zero, the derivative 
of the function is nonzero) and denote
$$
S_{k}^{+}=\{x\in S_{k}; x'(0)>0\}, \quad S_{k}^{-}=-S_{k}^{+}.
$$ 
Finally, let $\Phi_{k}^{\pm}=\mathbb{R}\times S_{k}^{\pm}$ and 
$\Phi_{k}=\mathbb{R}\times S_{k}$.
It is easy to show that for any positive integer $k$, $ S_{k}$, 
$S_{k}^{+}$ and $S_{k}^{-}$ are open in $E$. With the fact that
 \eqref{e4.5} is the linear equation of \eqref{e4.1} at the neighbourhood 
of $x=0$, global bifurcation theorem of Rabinowitz can be applied to 
\eqref{e4.1}.

\begin{theorem} \label{thm4.2} 
Suppose that $f$ satisfies conditions \eqref{e4.2}, 
\eqref{e4.3} and \eqref{e4.4}. Then for $\lambda\leq\lambda_1$, Equation
 \eqref{e4.1} admits trivial solution $x_{0}=0$. 
For $\lambda_{k}<\lambda\leq\lambda_{k+1}$, Equation $\eqref{e4.1}$ possesses 
at least $2k+1$ solutions: $x_{0}=0$, $x_1^{\pm}$, $\ldots x_{k}^{\pm}$,
with $x_{j}^{+}\in S_{j}^{+}$ and $x_{j}^{-}\in S_{j}^{-}$. 
\end{theorem}

\begin{proof} 
The existence of a solution $x\in S_{k}^{\pm}$ for $\lambda>\lambda_{k}$ 
is obtained easily. In fact, $x(t)$ is a solution of \eqref{e4.1} if and only
 if $x(t)$ is a solution of equation
$x=\lambda Lx+H(t,x)$,
where $L$ is defined as 
$$
(Lx)(t)=\begin{cases}
\int_{0}^{1}G(t,s)a(s)x(s)ds
 -\frac{1}{\lambda}\big(\beta_1-\frac{1}{2}\beta_{2}\big)tx(\frac{1}{2}),
 & t\in[0,\frac{1}{2}],\\[4pt]
\int_{0}^{1}G(t,s)a(s)x(s)ds+\frac{1}{\lambda}
 \big(\beta_1+\frac{1}{2}\beta_{2}\big)(1-t)x(\frac{1}{2}),
 & t\in(\frac{1}{2},1],
\end{cases}
$$
and $H(t,x)=\int_{0}^{1}G(t,s)f(s,x(s))ds$ satisfying 
$\lim_{x\to0} H(t,x)/ x=0$.

The global bifurcation theorem of Rabinowitz can be applied for this problem 
and we just need to prove
$\mathbf{C}_{k}^{\pm}\cap({\lambda}\times E)\neq\emptyset$.
By Lemma \ref{lem2.2} and Lemma \ref{lem3.8}, there exists one unbounded continua 
$\mathbf{C}_{k}$ of solutions of \eqref{e4.1}, which bifurcates 
from the right of the trivial solutions at $(\lambda_{k},0)$ and satisfies
$$
(\lambda_{k},0)\in\mathbf{C}_{k}\subset\Phi_{k}\cup\{(\lambda_{k},0)\}.
$$
Moreover, $\mathbf{C}_{k}=\mathbf{C}_{k}^{+}\cup\mathbf{C}_{k}^{-}$ and 
$\mathbf{C}_{k}^{+}\cap\mathbf{C}_{k}^{-}=(\lambda_{k},0)$, 
while $\mathbf{C}_{k}^{\pm}$ are two unbounded continua in $R \times E$ 
satisfying 
$$
\mathbf{C}_{k}^{\pm}\subset\Phi_{k}^{\pm}\cup\{(\lambda_{k},0)\}.
$$
By Theorem \ref{thm4.1}, we know that $\|x\|_1\leq M(\lambda)$ so that
$\mathbf{C}_{k}^{\pm}\subset\{(\lambda,x);\lambda\geq\lambda_{k},\|x\|_1
\leq M(\lambda)\}$ and the projection of $\mathbf{C}_{k}^{\pm}$ on $R$ 
is unbounded belonging to $(\lambda_{k},+\infty)$. 
So $\mathbf{C}_{k}^{\pm}\cap({\lambda}\times E)\neq\emptyset$, and there 
exists at least $2k+1$ solutions of \eqref{e4.1} in $S_{k}^{\pm}$ for 
$\lambda\geq\lambda_{k}$. 
\end{proof}

\section{Behavior of positive solutions of  autonomous impulsive differential
equation for large $\lambda$}

First, we study the existence of the positive solutions to
\begin{equation}
\begin{gathered}
 -x''(t)+x^{p+1}=\lambda mx(t),\quad t\in(0,1),\; t\neq \frac{1}{2},\\
 \Delta x|_{\frac{1}{2}}=\beta_1 x(\frac{1}{2}),\quad 
 \Delta x'|_{\frac{1}{2}}=-\beta_{2} x(\frac{1}{2}),\\
  x(0)=x(1)=0,
 \end{gathered}\label{e5.1}
\end{equation}
where $p>0$, $m>0$ are real numbers. This  is a special case of \eqref{e1.1}
with $f(t,x)=x^{p+1}$ and $a(t)=m$, so we can use the results in section 4 on it.
The behavior of the positive solutions of \eqref{e5.1}, as $\lambda\to\infty$,
will be discussed, on the basis of the comparison argument and the property
of principal eigenvalue of linear equation of \eqref{e5.1}.

By Theorem \ref{thm3.1}, we denote $\sigma_i$ the $i^{th}$ eigenvalue of
\begin{equation}
\begin{gathered}
 -x''(t)=\lambda x(t),\quad t\in(0,1),\quad t\neq \frac{1}{2},\\
 \Delta x|_{\frac{1}{2}}=\beta_1 x(\frac{1}{2}),\quad 
 \Delta x'|_{\frac{1}{2}}=-\beta_{2} x(\frac{1}{2}),\\
  x(0)=x(1)=0.
 \end{gathered}\label{e5.2}
\end{equation}
From Theorem \ref{thm4.1}, \eqref{e5.1} admits a positive solution,
if and only if $\lambda>\sigma_1/m$. Let $u_{\lambda, m}$ be 
such positive solution. In order to study the behavior of 
$u_{\lambda, m}$ as $\lambda\to\infty$, we use the change of variable 
\begin{equation}
u=\lambda^{1/p}v, \label{e5.3}
\end{equation}
which transforms \eqref{e5.1} into
\begin{equation}
\begin{gathered}
 -\frac{1}{\lambda}v''=mv-v^{p+1},\quad t\in(0,1),\quad t\neq \frac{1}{2},\\
 \Delta v|_{\frac{1}{2}}=\beta_1 v(\frac{1}{2}),\quad 
 \Delta v'|_{\frac{1}{2}}=-\beta_{2} v(\frac{1}{2}),\\
  v(0)=v(1)=0.
 \end{gathered} \label{e5.4}
\end{equation}
So the problem of analyzing the behavior of $u_{\lambda, m}$ as
$\lambda\to\infty$ is equivalent to analyzing the behavior of the unique
positive solution of  \eqref{e5.4}, which is denoted by
$\theta_{\lambda, m}$, as $\lambda\to\infty$. The behavior of $\theta_{\lambda, m}$
in the interior of $[0,1]$ is given by the following conclusion.

\begin{theorem} \label{thm5.1} 
 Let $J\subset (0,1)$ be a compact interval. Then 
\begin{equation}
\lim_{\lambda\to+\infty}\theta_{\lambda, m}=m^{1/p} \label{e5.5}
\end{equation}
uniformly in $J$.
\end{theorem}

\begin{proof} Given $\lambda>\frac{\sigma_1}{m}$,
$-\theta''_{\lambda, m}(t_{0})\geq0$ at the point $t_{0}\in I$ where 
$\theta_{\lambda, m}$ achieves its maximum and hence 
\begin{equation}
\theta_{\lambda, m}\leq m^{1/p}, \quad t\in I.  \label{e5.6}
\end{equation}
Then it is sufficient to show that given $\varepsilon>0$, there exists
$\lambda(\varepsilon)$ such that
\begin{equation}
m^{1/p}-\varepsilon\leq \theta_{\lambda, m}(t), \quad t\in J, \label{e5.7}
\end{equation}
for all $\lambda\geq\lambda(\varepsilon)$. To show this, we discuss as follows.
Consider $t_{0}\in J'$ and $J'=[\alpha,\beta]\subset I$. Without loss of
generality, we assume $\frac{1}{2}\in J'$. Then $\sigma_1^{J'}$
(resp. $\varphi^{J'}>0$) will stand for the principal eigenvalue
(resp. eigenfunction) of \eqref{e5.2} with boundary conditions
$\varphi^{J'}(\alpha)=\varphi^{J'}(\beta)=0$.
We consider $\varphi^{J'}(t)$ normalized so that
$\|\varphi^{J'}\|_1=1$. Let $\lambda(t_{0},\varepsilon)>0$ be such that
\begin{equation}
\big[m-\frac{\sigma_1^{J'}}{\lambda(t_{0},\varepsilon)}\big]^{1/p}
\geq m^{1/p}-\frac{\varepsilon}{2}. \label{e5.8}
\end{equation}
Lemma \ref{lem2.1} shows that the mapping $\lambda\to \theta_{\lambda, m}$ is strictly
increasing as far as $\lambda>\frac{\sigma_1^{J'}}{m}$. Thus,
\begin{equation}
\theta_{\lambda, m}\geq\theta_{\lambda(t_{0},\varepsilon), m}
\geq\theta_{\lambda(t_{0},\varepsilon), m}^{J'}, \quad
\lambda\geq\lambda(t_{0},\varepsilon), \label{e5.9}
\end{equation}
where $\theta_{\lambda(t_{0},\varepsilon), m}^{J'}$ is the unique positive
solution to
\begin{equation}
\begin{gathered}
 -\frac{1}{\lambda(t_{0},\varepsilon)}v''
=mv-v^{p+1},\quad t\in J', \; t\neq \frac{1}{2},\\
 \Delta v|_{\frac{1}{2}}=\beta_1 v(\frac{1}{2}),\quad 
 \Delta v'|_{\frac{1}{2}}=-\beta_{2} v(\frac{1}{2}),\\
  v(\alpha)=v(\beta)=0.
 \end{gathered}\label{e5.10}
\end{equation}
It follows easily that
$[m-\frac{\sigma_1^{J'}}{\lambda(t_{0},\varepsilon)}]^{1/p}\varphi^{J'}$
is a lower solution of \eqref{e5.10}. Thus, due the convexity of
$v\to v^{1/p}$ and using the maximum principle, we find that
$$
[m-\frac{\sigma_1^{J'}}{\lambda(t_{0},\varepsilon)}]^{1/p}\varphi^{J'}
\leq\theta_{\lambda(t_{0},\varepsilon), m}^{J'}.
$$
From the inequality and \eqref{e5.9} it follows that
$$
\theta_{\lambda, m}\geq\big[m-\frac{\sigma_1^{J'}}{\lambda(t_{0},\varepsilon)}
\big]^{1/p}\varphi^{J'}, \quad \lambda\geq\lambda(t_{0},\varepsilon).
$$
Hence, from \eqref{e5.8} the following holds
$$
\theta_{\lambda, m}\geq\big[m^{1/p}-\frac{\varepsilon}{2}\big]\varphi^{J'},
$$
for all $\lambda\geq\lambda(t_{0},\varepsilon)$ and $t\in J'$.
Since $\varphi^{J'}(t_{0})=1$ and the local continuity of $\varphi^{J'}$,
there exists $J\subset J'\subset I$ such that
$$
\varphi^{J'}(t)\geq\frac{m^{1/p}-\varepsilon}{m^{1/p}
-\frac{\varepsilon}{2}}, \quad t\in J.
$$
Thus,
$$
\theta_{\lambda, m}\geq m^{1/p}-\varepsilon, \quad t\in J,
$$
for all $\lambda\geq\lambda(t_{0},\varepsilon)$. Then we complete the proof.
\end{proof}

\subsection*{Acknowledgments}
This research is supported by Young Award of Shandong Province
(ZR2013AQ008) and the Fund of Science and Technology Plan of Shandong Province (2014GGH201010).


\begin{thebibliography}{00}

\bibitem{a1} R. P. Agarwal, D. O'Regan;
\emph{Multiple nonnegative solutions for second-order impulsive differential equations},
 Appl. Math. Comput.114 (2000), 51-59.

\bibitem{b1} H. Berestycki;
\emph{Le nombre de solutions de certains problemes semi-lineaires elliptiques}, J
ournal of Functional Analysis 40,1-29(1981).

\bibitem{c1} H. Chen, Z. He;
\emph{Variational approach to some damped Dirichlet problems with impulses}, 
Mathematical Methods in the Applied Sciences, in press, DOI: 10.1002/mma.2777.

\bibitem{c2} M. Choisy, J. F. Guegan, P. Rohani;
\emph{Dynamics of infections diseases and pulse vaccination:teasing apart 
the embedded resonance effects}, Physica D 22 (2006), 26-35.

\bibitem{c3} Y. Cui, J. Sun, Y. Zou;
\emph{Global bifurcation and multiple results for Sturm-Liouville problems}, 
Journal of Computational and Applied Mathematics, 235(2011), 2185-2192.

\bibitem{d1} E. N. Dancer;
\emph{On the Structure of Solutions of Nonlinear Eigenvalue Problems}, 
Indiana University Mathematics Journal, vol.23, (1974), 1069-1076.

\bibitem{f1} J. M. Fraile, J. Lopezgomez, J. C. Delis;
\emph{On the global structure of the set of positive solutions of some semilinear 
elliptic boundary value problems}, Journal of Differential Equations 123 (1995),
 180-212.

\bibitem{g1} S. Gao, L. Chen, J. J. Nieto, A. Torres;
\emph{Analysis of a delayed epidemic model with pulse vaccination and saturation 
incidence}, Vaccine 24 (2006), 6037-6045.

\bibitem{g2} D. Guo, V. Lakshmikantham, Z. Liu;
\emph{Nonlinear integral equations in abstract spaces}, 
Kluwer Academic (Dordrecht and Boston, Mass), 1996.

\bibitem{l1} E. K. Lee, Y. H. Lee;
\emph{Multiple positive solutions of singular gelfand type problem for 
second-order impulsive differential equations}, 
 Appl. Math. Comput. 40 (2004), 307-328.

\bibitem{l2} E. K. Lee, Y. H. Lee;
\emph{Multiple positive solutions of singular two point boundary value problems 
for second-order impulsive differential equations}, 
Appl. Math. Comput. 158 (2004), 745-759.

\bibitem{l3} W. Li, H. Huo;
\emph{Global attractivity of positive periodic solutions for an impulsive delay 
periodic model of respiratory dynamics}, J. Comput. Appl. Math. 174 (2005), 227-238.

\bibitem{l4} X. Lin, D. Jiang;
\emph{Multiple positive solutions of Dirichlet boundary-value problems for 
second-order impulsive differential equations}, J. Math. Anal. Appl. 321 (2006), 
501-514.

\bibitem{l5} Y. Lee, X. Liu;
\emph{Study of singular boundary value problems for second order impulsive 
differential equations}, J. Math. Anal. Appl. 331 (2007), 159-176.

\bibitem{l6} Y. Liu, D. O'Regan;
\emph{Multiplicity results using bifurcation techniques for a class of boundary
 value problems of impulsive differential equations}, 
Commu Nonlinear Sci Nummer Simulat 16 (2011), 1769-1775.

\bibitem{m1} R. Ma;
\emph{Global behavior of the components of nodal solutions of asymptotically 
linear eigenvalue problems}, Appl. Math. Lett. 21 (2008), 754-60.

\bibitem{m2} R. Ma, Y. An;
\emph{Global structure of positive solutions for nonlocal boundary value problems 
involving integral conditions}, Nonlinear Anal. 71 (2009), 4364-76.

\bibitem{m3} R. Ma, D. O'Regan;
\emph{Nodal solutions for second-order m-point boundary value problems 
with nonlinearities across several eigenvalues},
 Nonlinear Anal. TMA 64 (2006), 1562-1577.

\bibitem{n1} J. J. Nieto, D. O'Regan;
\emph{Variational approach to impulsive differential equations}, 
Nonlinear Anal. RWA 10 (2009), 680-690.

\bibitem{n2} J. J. Nieto;
\emph{Variational formulation of a damped Dirichlet impulsive problem}, 
Appl. Math. Lett. 23 (2010), 940-942.

\bibitem{o1} A. d'Onofrio;
\emph{On pulse vaccination strategy in the SIR epidemic model with vertical 
transmission}, Appl. Math. Lett. 18 (2005), 729-32.

\bibitem{r1} B. P. Rynne;
\emph{Infinitely many solutions of superlinear fourth order boundary value problems}, 
Topol. Methods Nonlinear Anal. 19 (2) (2002), 303-312.

\bibitem{r2} P. H. Rabinowitz;
\emph{Some global results for nonlinear eigenvalue problems}, 
J. Funct. Anal. 7 (1971), 487-513.

\bibitem{r3} P. H. Rabinowitz;
\emph{On bifurcation from infinity}, J. Differ. Equations 14 (1973), 462-475.

\bibitem{s1} J. Sun, H. Chen;
\emph{Multiplicity of solutions for a class of impulsive differential equations 
with Dirichlet boundary conditions via  variant fountain theorems}, 
Nonlinear Anal. RWA 11 (5) (2010), 4062-4071.

\bibitem{s2} J. Sun, D. O'Regan;
\emph{Impulsive periodic solutions for singular problems via variational methods}, 
Bulletin of the Australian Mathematical Society 86 (2012), 193-204.

\bibitem{s3} M. Scheffer, et al.;
\emph{Catastrophic shifts in ecosystems}, Nature 413 (2001), 591-596.

\bibitem{t1} S. Tang, L. Chen;
\emph{Density-dependent birth rate, birth pulses and their population dynamics 
consequences}, J. Math. Biol. 44 (2002), 185-199.

\bibitem{t2} Y. Tian, W. Ge;
\emph{Variational methods to Sturm-Liouville boundary value problem for 
impulsive differential equations}, Nonlinear Anal. 72 (1) (2010), 277-287.

\bibitem{w1} Jing Wang, Baoqiang Yan;
\emph{Global properties and multiple solutions for boundary value problems 
of impulsive differential equations}, Electron. J. Diff. Equ., Vol. 2013 (2013),
 No. 171, pp. 1-14.

\bibitem{x1} J. Xiao, J. J. Nieto, Z. Luo;
\emph{Multiplicity of solutions for nonlinear second order impulsive differential 
equations with linear derivative dependence via variational methods}, 
Communications in Nonlinear Science and Numerical Simulation 17 (2012), 426-432.

\bibitem{x2} J. Xiao, J. J. Nieto, Z. Luo;
\emph{Existence of multiple solutions of some second order impulsive differential 
equations}, Topological Methods in Nonlinear Analysis 43 (2014), 287-296.

\bibitem{y1} D. Yang;
\emph{A variational principle for boundary-value problems with non-linear 
boundary conditions}, Electron. J. Diff. Equ., Vol. 2015 (2015), No. 301, pp. 1-6.

\bibitem{y2} J. Yan, A. Zhao, J. J. Nieto;
\emph{Existence and global attractivity of positive periodic solution of periodic 
single-species impulsive Lotka-Volterra systems}, Math. Comput. Model. 40 (2004), 
509-518.

\bibitem{y3} M. Yao, A. Zhao, J. Yun;
\emph{Periodic boundary value problems of second-order impulsive differential 
equations}, Nonlinear Anal. TMA 70 (2009), 262-273.

\bibitem{z1} D. Zhang;
\emph{Multiple Solutions of Nonlinear Impulsive Differential Equations with 
Dirichlet Boundary Conditions via Variational Method}, 
Results in Mathematics 63 (2013), 611-628.

\end{thebibliography}

\end{document}