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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 18, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/18\hfil Solutions for boundary-value problems]
{Solutions to boundary-value problems for second-order
impulsive differential equations \\ at resonance}

\author[Y. Tian\hfil EJDE-2008/18\hfilneg]
{Yu Tian} 

\address{Yu Tian \newline
 School of Science, Beijing University of Posts and
Telecommunications, Beijing 100876, China}
\email{tianyu2992@163.com}

\thanks{Submitted September 6, 2007. Published February 5, 2008.}
\thanks{Supported by grant 10726038 from  Tianyuan Fund of Mathematics
in China}
\subjclass[2000]{34B15, 34B37}
\keywords{Second-order boundary-value problem; Schaefer's theorem;
\hfill\break\indent
impulsive differential equations; two-point boundary condition}

\begin{abstract}
 In this paper, we investigate the existence and uniqueness
 of solutions to  boundary-value problems  for second-order
 impulsive  differential equations at resonance.
 To obtain these results, we apply fixed point methods and
 new differential inequalities.
\end{abstract}

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

\section{Introduction}

We consider the uniqueness for the nonlinear impulsive boundary value
problem (IBVP)
\begin{equation}\label{1}
\begin{gathered}
    x''=f(t, x, x'), \quad t\in [0, T],\,t\neq t_1,\\
    \Delta x(t_1)=I(x(t_1)),\quad \Delta
    x'(t_1)=J(x'(t_1)),\quad t_1\in (0, T),
\end{gathered}
\end{equation}
where $t_1$ is a fixed value and
\begin{equation}\label{2}
    x\in \beta_0,
\end{equation}
where $I\in C(\mathbb{R}^n, \mathbb{R}^n)$,
$J \in C(\mathbb{R}^n, \mathbb{R}^n)$,
$\Delta x(t_1)=x(t_1^+)-x(t_1^-)$,
$\Delta {x}'(t_1)=x'(t_1^+)-x'(t_1^-)$,
where $x^{(i)}(t_1^+)$ (respectively $x^{(i)}(t^-_1)$) denote  the
right limit (respectively left limit) of $x^{(i)}(t)$ at $t=t_1$,
$i=0, 1$.  The function
$f: [0, T]\times \mathbb{R}^n\times \mathbb{R}^n\to
\mathbb{R}^n$ is a $L^2$-Carath\'edory nonlinear function, that is $f$
satisfies
\begin{itemize}
\item[(i)] the map $(y_0, y_1)\to f(t, y_0, y_1)$ is
continuous for a.e. $t\in [0, T]\setminus\{t_1\}$,
\item[(ii)] the map $t\to f(t, y_0, y_1)$ is measurable for
all $(y_0, y_1)\in \mathbb{R}^n\times \mathbb{R}^n$,
\item[(iii)] for each $r>0$, there exists an $\alpha_r\in L^2[0, T]$ such
that  $|f(t, x, y)|\le \alpha_r(t)$  for a.e. $t\in [0, T]$ and
every $(x, y)\in \mathbb{R}^n\times \mathbb{R}^n$.
\end{itemize}
Boundary condition \eqref{2} represents periodic boundary condition
\begin{equation}\label{periodic}
    x(0)=x(T),\quad x'(0)=x'(T)
\end{equation}or the Neumann boundary condition
\begin{equation}\label{neumann}
    x'(0)=x'(T)=0.
\end{equation}

In recent years, impulsive differential equations have been studied
extensively because of its wide application in many fields such as:
chemotherapy; population dynamics; optimal control; ecology;
biotechnology and physics.  We refer the reader to
\cite{c1,l1,l2,l3,n1,n2,o1} and
the references therein for nice examples and applications. For a
general theory on impulsive differential equations, see the
monographs \cite{l1} and \cite{s1}.

There are many authors who have considered the solvability of
boundary value problems with impulses. Nieto \cite{n1} studied
the existence of solutions to the first order periodic problem with
impulse. Chen, Tisdell and Yuan \cite{c1} studied the solvability
of the periodic problem with impulse. Lin and Jiang \cite{l2}
studied the existence of positive solutions for the
 second order Dirichlet boundary value problem with impulse.

For the uniqueness of solutions, there are some papers related
boundary value problems for first order differential equations. In
\cite{o1}, the uniqueness of solutions was obtained by  nonlinear
alternative of Leray-Schauder type in Fr\'echet spaces. In
\cite{z1}, the existence and uniqueness of solutions was obtained
by employing the method of upper and lower solution coupled with the
monotone iterative technique. Nieto and Tisdell \cite[Section 4.2]{n2},
 obtained existence and uniqueness of
solutions to first order IBVPs. Their methods included Schaefer's
fixed-point theorem and differential inequalities.

 As far as we know, there are few authors who study the
uniqueness of solutions for second-order IBVPs. The aim of this work
is to study the uniqueness of solutions for second-order impulsive
differential equations with periodic condition and Neumann
condition.

 This paper is
organized as follows. In section 2, we present some novel
differential inequalities, which are useful to estimate  {\it a
priori} bounds on solutions. In section 3, we devote our attention
to the uniqueness of solution to \eqref{1}, \eqref{2}. The proof of
main results are divided into two parts. First we apply the
Schaefer's fixed point theorem to prove the existence of at least
one solution. Second we prove the uniqueness of solutions by
contradiction.

We note that the main results of this paper are easy to extend to an
arbitrary impulse $I_i, J_i$, $i=1, 2, \dots, p$. However, for
clarity and brevity, we restrict our attention to BVPs with one
impulse.

For the remainder of the section, we introduce notations and
definitions which are used throughout the paper. Let
$J'=[0, T]\setminus\{t_1\}$, $t_0=0$, $t_2=T$. The space
\begin{align*}
PC^1([0, T]; \mathbb{R}^n)
=&\{x:[0, T]\to \mathbb{R}^n: x|_{(t_k,
t_{k+1})}\in C^1(t_k, t_{k+1}),\; k=0, 1, \\
&\quad x(t_1^-)=x(t_1),
x'(t_1^-)=x'(t_1),\; x(t_1^+),\; x'(t_1^+)\mbox{ exist}\}
\end{align*}
is a Banach space with the norm $\|x\|_{PC^1}=\max\{\|x\|_{PC},
\|x'\|_{PC}\}$, where $ \|x\|_{PC}=\sup_{t\in[0, T]}|x(t)|$.
Let
\[
Y=\{x\in PC^1([0, T]; \mathbb{R}^n): x|_{(t_k, t_{k+1})}\in W^{2, 2}(t_k,
t_{k+1}),\; k=0, 1\}.
\]
Clearly,
$Y$ is a Banach space.

 A function $x$ is said to be a solution of \eqref{1}, \eqref{2}, if
$x\in Y$ satisfies \eqref{1}, \eqref{2}.

For $x, y\in \mathbb{R}^n$, we denote by  $\langle x, y\rangle$
the usual inner product and by $|x|$
the norm $(\sum_{i=1}^{n}x_i^2)^{1/2}$. In
addition we denote
$\|x\|_{L^2}=\big(\int_{0}^{T}|x(t)|^2dt\big)^{1/2}$.

\section{Related Lemmas}

\begin{lemma}\label{lem1}
If $x\in Y$, then
$\|x^{(i)}\|_{L^2}\le\frac{2T}{\pi}\|x^{(i+1)}\|_{L^2}$, $i=0, 1$.
\end{lemma}

\begin{proof}
Let
\[
y(t)=\begin{cases}
x(t),&t\in [0, T],\\
x(2T-t),&t\in[T, 2T],\\
-x(-t),& t\in[-T, 0],\\
-x(t+2T),&t\in[-2T, -T].
\end{cases}
\]
Then
\begin{itemize}
\item[(i)] $y, y''$ are odd functions on $[-2T, 2T]$, $y'$ is an even
function on $[-2T, 2T]$;
\item[(ii)] $\int_{-2T}^{2T}y^{(i)}(s)ds=0, i=0, 1, 2$;
\item[(iii)]
$\int_{-2T}^{2T}|y^{(i)}(t)|^2dt=4\int_{0}^{T}|x^{(i)}(t)|^2dt, i=0,
1, 2$.
\end{itemize}
So $y$ has the Fourier expansion
\[
y(t)=\sum_{n=1}^{\infty}b_n\sin\frac{n\pi t}{2T}.
\]
The right hand series converges to $\frac{y(t^+)+y(t^-)}{2}$ at the
points
$t=0$, $t_1$, $2T-t_1$, $-t_1$, $-2T+t_1$, respectively. The Parseval
equality implies
\[
\int_{-2T}^{2T}|y'(t)|^2dt
= \sum_{n=1}^{\infty}\frac{n^2\pi^2}{2T}b_n^2\ge
\frac{\pi^2}{4T^2}\sum_{n=1}^{\infty}2Tb_n^2
=\frac{\pi^2}{4T^2}\int_{-2T}^{2T}|y(t)|^2dt.
\]
So
$\int_{-2T}^{2T}|y(t)|^2dt\le\frac{4T^2}{\pi^2}\int_{-2T}^{2T}|y'(t)|^2dt$.
By (iii), we have $\|x\|_{L^2}\le \frac{2T}{\pi}\|x'\|_{L^2}$.

On the other hand, since (ii) holds, the function $y'$ has the
Fourier expansion
\[
y'(t)=\sum_{n=1}^{\infty}a_n \cos\frac{n\pi t}{2T},
\]
and the right hand series converges to $\frac{y'(t^+)+y'(t^-)}{2}$  at the
points $t=0, t_1, T, 2T-t_1, -t_1, -T, -2T+t_1$. The Parseval
equality implies
\[
\int_{-2T}^{2T}|y''(t)|^2dt
= \sum_{n=1}^{\infty}
\frac{n^2\pi^2}{2T}a_n^2\ge\frac{\pi^2}{4T^2}\sum_{n=1}^{\infty}2Ta_n^2\\
=\frac{\pi^2}{4T^2}\int_{-2T}^{2T}|y'(t)|^2dt.
\]
So
$\int_{-2T}^{2T}|y'(t)|^2dt\le\frac{4T^2}{\pi^2}\int_{-2T}^{2T}|y''(t)|^2dt$.
By (iii), we have $\|x'\|_{L^2}\le \frac{2T}{\pi}\|x''\|_{L^2}$.
 \end{proof}

\begin{lemma}\label{lem2}
If $x\in Y$, then
\begin{enumerate}
\item $\|x\|_{PC}\le T^{1/2}\Gamma \|x'\|_{L^2}$,
\item $\|x^{(i)}\|_{PC}\le
(\frac{2T}{\pi})^{1-i}T^{1/2}\Gamma\|x''\|_{L^2},
i=0, 1$,
\end{enumerate}
where $\Gamma=\frac{2T}{\pi\min\{t_1, T-t_1\}}+1$.
\end{lemma}

\begin{proof}
For $t\in[0, t_1]$. It follows from the mean value theorem that
\[
x(\tau_1)=\frac{1}{t_1}\int_{0}^{t_1}x(s)ds
\]
for some $\tau_1\in[0, t_1]$.
Hence for $t\in [0, t_1]$, using H\"older's inequality, we
have
\begin{align*}
|x(t)|&= \big|x(\tau_1)+\int_{\tau_1}^{t}x'(s)ds\big| \\
&\leq \frac{1}{t_1}\int_{0}^{t_1}|x(s)|ds+\int_{0}^{T}|x'(s)|ds\\
&\leq \frac{1}{t_1}\int_{0}^{t_1}|x(s)|ds+\|x'\|_{L^2}T^{1/2}.
\end{align*}
For $t\in (t_1, T]$, it follows from the mean value theorem that
\[
x(\tau_2)=\frac{1}{T-t_1}\int_{t_1}^{T}x(s)ds
\]
for some $\tau_2\in(t_1, T]$. Hence for $t\in(t_1, T]$,
using H\"older inequality,
\begin{align*}
|x(t)|&= \big|x(\tau_2)+\int_{\tau_2}^{t}x'(s)ds\big|\\
&\leq \frac{1}{T-t_1}\int_{t_1}^{T}|x(s)|ds+\int_{0}^{T}|x'(s)|ds\\
&\leq \frac{1}{T-t_1}\int_{t_1}^{T}|x(s)|ds+\|x'\|_{L^2}T^{1/2}.
\end{align*}
Therefore,
\begin{align*}
\|x\|_{PC}
&\le \frac{1}{\min\{t_1, T-t_1\}}\int_{0}^{T}|x(s)|ds+\|x'\|_{L^2}T^{1/2}\\
&\le \frac{T^{1/2}}{\min\{t_1, T-t_1\}}\|x\|_{L^2}+\|x'\|_{L^2}
T^{1/2}.
\end{align*}
 By Lemma \ref{lem1}, we have
 \[
 \|x\|_{PC}\le T^{1/2}\Gamma \|x'\|_{L^2}.
\]
So 1) holds. Applying Lemma
\ref{lem1} again, we have $\|x\|_{PC}\le
\frac{2T^{\frac{3}{2}}}{\pi}\Gamma\|x''\|_{L^2}$. Similar to the
above process, we have
\[
\|x'\|_{PC}\le \frac{T^{1/2}}{\min\{t_1,
T-t_1\}}\|x'\|_{L^2}+\|x''\|_{L^2}T^{1/2}\le
T^{1/2}\Gamma \|x''\|_{L^2}.
\]
Therefore, 2) holds.
\end{proof}

\section{Main results}

\begin{theorem}\label{th1}
Suppose that there exist $a_i\in C([0, T], R^+), p, q>0$ such that
\begin{equation}\label{th11}
   |f(t, x_0, x_1)-f(t, y_0, y_1)|\le
   \sum_{i=0}^{1}a_i(t)|x_i-y_i|
\end{equation}
for $(t, x_i, y_i)\in[0, T]\times \mathbb{R}^n\times \mathbb{R}^n$,
$i=0, 1$;
   \begin{equation}\label{th12}
    |I(x)-I(y)|\le p|x-y|,\quad |J(x)-J(y)|\le q|x-y|,\quad x, y\in
    \mathbb{R}^n;
   \end{equation}
and
\begin{equation}\label{th13}
    \sum_{i=0}^{1}\|a_i\|_0\big(\frac{2T}{\pi}\big)^{2-i}
+\Big[\big(\frac{2T}{\pi}\big)^2+
    (p+q)\frac{2T^2}{\pi}\Gamma^2\Big]<1,
\end{equation}
where $ \Gamma$ is defined in Lemma \ref{lem2}.
Then \eqref{1}, \eqref{2} has a unique solution in $Y$.
\end{theorem}

\begin{proof}
Consider the following impulsive BVP,
which is equivalent to \eqref{1}--\eqref{2},
\begin{equation}\label{21}
\begin{gathered}
    x''-x= f(t, x, x')-x, \quad t\in[0, T]\setminus\{t_1\}\\
    \Delta x(t_1)=I(x(t_1)),\quad \Delta
    x'(t_1)=J(x'(t_1)),\\
    x\in \beta.
\end{gathered}
\end{equation}
Since the linear boundary value problem
\begin{equation}\label{impuls}
 x''-x=0,    \quad x\in \beta
\end{equation}
has only the zero solution,  there exists a
unique, continuous once-differentiable  Green's function
$G: [0, T]\times [0, T]\to R$ such that \eqref{21} may be
equivalently reformulated as the integral
equation
\begin{align*}
x(t)&=\int_{0}^{T}G(t, s)[f(s, x(s), x'(s))-x(s)]ds\\
&\quad +G(t,t_1)J(x'(t_1))-(\partial G/\partial s)(t, t_1)I(x(t_1)),\quad
t\in[0, T].
\end{align*}
Define the operator $H: PC^1([0, T], \mathbb{R}^n)\to
PC^1([0, T], \mathbb{R}^n)$ by
\begin{equation}\label{15}
\begin{aligned}
(Hx)(t)&=\int_{0}^{T}G(t, s)[f(s, x(s), x'(s))-x(s)]ds\\
&\quad +G(t, t_1)J(x'(t_1))-(\partial G/\partial s)(t, t_1)I(x(t_1)),
\quad t\in[0, T]\,.
\end{aligned}
\end{equation}
Then consider the family of equations
\begin{equation}\label{151}
    x=\lambda Hx,
\end{equation}
$\lambda\in (0, 1)$.
Since $Hx$ actually belongs to $Y$
for each $x\in PC^1([0, T], \mathbb{R}^n)$, $H$ is a compact map. We will
apply Schaefer's fixed point theorem \cite{l4} to prove that
$H$ has at least one fixed point in $PC^1([0, T], \mathbb{R}^n)$.
Since $H: PC^1([0, T], \mathbb{R}^n)\to PC^1([0, T], \mathbb{R}^n)$
is compact, it
remains to verify that all solutions to \eqref{15} are bounded
independently of $\lambda$.

Since $x$ is a solution of $x=\lambda Hx$, then $x\in Y$ satisfies
\begin{equation}\label{212}
\begin{gathered}
    x''-x=\lambda [f(t, x, x')-x],\quad t\in [0, T]\setminus\{t_1\},\\
    \Delta x(t_1)=\lambda I(x(t_1)),\quad \Delta
    x'(t_1)=\lambda J(x'(t_1)),\\
    x\in \beta.
\end{gathered}
\end{equation}
Multiplying by $x''$ and integrating from $0$ to $T$, we have
\begin{equation}\label{8}
   \|x''\|_{L^2}^2\le \lambda \big|\int_{0}^{T}f(t, x(t),
   x'(t))x''(t)dt\big|+(1-\lambda)\big|\int_{0}^{T}x(t)x''(t)dt\big|.
\end{equation}
By \eqref{th11}, we have
\begin{equation}\label{9.0}
\begin{aligned}
& \big|\int_{0}^{T}f(t, x(t),x'(t))x''(t)dt\big|\\
&\le\int_{0}^{T}\left|f(t, x(t), x'(t))-f(t, 0,
0)\right|\times\left|x''(t)\right|dt+\int_{0}^{T}\left|f(t, 0,
0)x''(t)\right|dt\\
&\le\sum_{i=0}^{1}\int_{0}^{T}|a_i(t)x^{(i)}(t)x''(t)|dt+\|f(t,
0, 0)\|_{L^2}\|x''\|_{L^2}\\
&\le\sum_{i=0}^{1}\|a_i\|_0\|x^{(i)}\|_{L^2}\|x''\|_{L^2}+\|f(t,
0, 0)\|_{L^2}\|x''\|_{L^2}.\end{aligned}
\end{equation}
 By the impulsive condition
\begin{equation}\label{9}\begin{aligned}
\big|\int_{0}^{T}x(t)x''(t)dt\big|
&=\big|\sum_{i=0}^{1}x(t)x'(t)|_{t_i^+}^{t_{i+1}^{-}}\big|+
\int_{0}^{T}|x'(t)|^2dt\\
&\le\left|x(T)x'(T)-x(0)x'(0)-\Delta
(x(t_1)x'(t_1))\right|+\int_{0}^{T}|x'(t)|^2dt.\end{aligned}
\end{equation}
The boundary condition \eqref{2}, the impulsive condition, and condition
\eqref{th12} imply
\begin{equation}\label{10}\begin{aligned}
&\big|x(T)x'(T)-x(0)x'(0)-\Delta (x(t_1)x'(t_1))\big|
  +\int_{0}^{T}|x'(t)|^2dt\\
&=\left|x'(t_1^+)\Delta x(t_1)+x(t_1)\Delta
  x'(t_1)\right|+\|x'\|_{L^2}^{2}\\
&=\left|x'(t_1^+)(I(x(t_1))-I(0))+x(t_1)(J(x'(t_1))-J(0))\right|\\
&\quad +\left|x'(t_1^+)I(0)\right| +\left|x(t_1)J(0)\right|
 +\|x'\|_{L^2}^{2}\\
&\le(p+q)\|x\|_{PC}\|x'\|_{PC}+\|x'\|_{L^2}^{2}+\|x\|_{PC}|J(0)|
 +\|x'\|_{PC}|I(0)|.
\end{aligned}\end{equation}
Substituting \eqref{9.0} \eqref{9} \eqref{10} into \eqref{8}, we
have
\begin{align*}
\|x''\|_{L^2}^{2}
&\leq \sum_{i=0}^{1}\|a_i\|_0\|x^{(i)}\|_{L^2}\|x''\|_{L^2}+\|f(t,0,
0)\|_{L^2}\|x''\|_{L^2}\\
&\quad +(p+q)\|x\|_{PC}\|x'\|_{PC}+\|x'\|_{L^2}^{2}
  +\|x\|_{PC}|J(0)|+\|x'\|_{PC}|I(0)|.
\end{align*}
By Lemma \ref{lem1} and Lemma \ref{lem2} we have
\begin{align*}
\|x''\|_{L^2}^{2}
&\leq \sum_{i=0}^{1}\|a_i\|_0
  \big(\frac{2T}{\pi}\big)^{2-i}\|x''\|_{L^2}^{2}
  +\|f(t,0,0)\|_{L^2}\|x''\|_{L^2}\\
&\quad+\big[\big(\frac{2T}{\pi}\big)^2
  +(p+q)\frac{2T^2}{\pi}\Gamma^2\big]\|x''\|_{L^2}^{2}+M
\|x''\|_{L^2}
\end{align*}
for some sufficiently large constant
$M>0$. Condition \eqref{th13} implies that $x''$ is bounded in
$L^2([0, T], \mathbb{R}^n)$.  Lemma \ref{lem2} means that $x$ is bounded in
$PC^1([0, T], \mathbb{R}^n)$. Applying Schaefer's fixed point theorem,
\eqref{1}, \eqref{2} has at least one solution.

Now we will show that \eqref{1}, \eqref{2} has a unique solution. If
$x, y$ are both solutions of \eqref{1}, \eqref{2}. Then $u=x-y$
satisfies
\begin{equation}\label{11}
\begin{gathered}
u''(t)=f(t, x(t), x'(t))-f(t, y(t),
y'(t)),\quad t\in[0, T]\setminus\{t_1\},\\
\Delta u(t_1)=I(x(t_1))-I(y(t_1)),\quad \Delta
u'(t_1)=J(x'(t_1))-J(y'(t_1))\\
u\in \beta_0.
\end{gathered}
\end{equation}
Similar to the above process we have
\[
\|u''\|_{L^2}^{2}\le\sum_{i=0}^{1}\|a_i\|_0
\big(\frac{2T}{\pi}\big)^{2-i}\|u''\|_{L^2}^{2}.
\]
Condition \eqref{th13} implies that $\|u''\|_{L^2}=0$. Lemma
\ref{lem2} means that $u(t)\equiv0$ for $t\in [0, T]$. The proof is
complete.
\end{proof}

\begin{theorem}\label{thm2}
Suppose that the conditions \eqref{th11}
\eqref{th12} in Theorem \ref{th1} hold. Furthermore, we assume that
\begin{equation}\label{098}
 \|a_1\|_0<\frac{1}{T},\quad \sum_{i=0}^{1}\|a_i\|_0
\big(\frac{2T}{\pi}\big)^{2-i}
+\frac{(p+q)T^2\Gamma^2(\|a_0\|_0+1)}{1-T\|a_1\|_0}<1
\end{equation}
Then problem \eqref{1}, \eqref{neumann} has a unique solution in
$Y$.
\end{theorem}

\begin{proof}
According to the proof of Theorem \ref{th1}, we have  that the
function $x\in Y$ satisfies
\begin{equation}\label{2121}
\begin{gathered}
    x''-x=\lambda [f(t, x, x')-x],\quad t\in [0, T]\setminus\{t_1\},\\
    \Delta x(t_1)=\lambda I(x(t_1)),\quad \Delta
    x'(t_1)=\lambda J(x'(t_1)),\\
    x'(0)=x'(T)=0.
\end{gathered}
\end{equation}
Multiplying by $x$ and integrating from $0$ to $T$, we have
\begin{equation}\label{lambda}
    \int_{0}^{T}x''(s)x(s)ds=\int_{0}^{T}\lambda f(s, x(s),
    x'(s))x(s)ds+(1-\lambda)\int_{0}^{T}x^{2}(s)ds.
\end{equation}
By boundary condition \eqref{neumann} and impulsive condition we
have
\begin{equation}\label{3}
    \begin{aligned}\int_{0}^{T}x''(s)x(s)ds&=-\Delta
    (x'(t_1)x(t_1))+x'(T)x(T)-x'(0)x(0)-\int_{0}^{T}[x'(s)]^2ds\\
    &=-\Delta(x'(t_1)x(t_1))-\|x'\|_{L^2}^{2}.
\end{aligned}
\end{equation}
By \eqref{neumann} \eqref{lambda}  we have
\begin{equation}\label{4}
\begin{aligned}
\|x'\|_{L^2}^{2}
&\le |\Delta(x'(t_1)x(t_1))|+\int_{0}^{T}|f(s, x(s), x'(s))x(s)|ds\\
&\le |x'(t_1^+)\Delta x(t_1)+x(t_1)\Delta x'(t_1)|\\
&\quad +\int_{0}^{T}|f(s, x(s), x'(s))-f(s, 0, 0)||x(s)|ds+ \|f(t, 0,
0)\|_{L^2}\|x\|_{L^2}\\
&\le
\left|x'(t_1^+)(I(x(t_1))-I(0))+x(t_1)(J(x'(t_1))-J(0))\right|
  +|x'(t_1^+)I(0)|\\
&\quad +|x(t_1)J(0)|
 +\sum_{i=0}^{1}\int_{0}^{T}|a_i(s)x^{(i)}(s)x(s)|ds+\|f(t,0,
 0)\|_{L^2}\|x\|_{L^2}\\
&\le (p+q)\|x'\|_{PC}\|x\|_{PC}
+\sum_{i=0}^{1}\|a_i\|_{0}\|x^{(i)}\|_{L^2}\|x\|_{L^2}+\|f(t,0,
0)\|_{L^2}\|x\|_{L^2}\\
&\quad +\|x'\|_{PC}|I(0)|+\|x\|_{PC}|J(0)|.
\end{aligned}
\end{equation}

We assume that $\|x'\|_{PC}=|x'(\xi)|$ for $\xi\in [0, T]$. If
$\xi\in [0, t_1]$, integrating from $0$ to $\xi$ on the both sides
of \eqref{2121}, we have
\begin{equation}\label{a}
x'(\xi)=\lambda\int_{0}^{\xi} f(s, x(s),
x'(s))ds+(1-\lambda)\int_{0}^{\xi}x(s)ds.
\end{equation}
If $\xi\in (t_1, T]$, integrating from $\xi$ to $T$ on the both
sides of \eqref{2121} we have
\begin{equation}\label{b}
-x'(\xi)=\lambda\int_{\xi}^{T} f(s, x(s),
x'(s))ds+(1-\lambda)\int_{\xi}^{T}x(s)ds.
\end{equation}
By \eqref{a} \eqref{b} and condition \eqref{th11}, one has
\begin{align*}
\|x'\|_{PC}&\leq \int_{0}^{T}|f(s, x(s),
x'(s))|ds+\int_{0}^{T}|x(s)|ds\\
&\leq \int_{0}^{T}|f(s, x(s), x'(s))-f(s, 0, 0)|ds+\|f(t, 0,
0)\|_{L^2}T^{1/2}+T\|x\|_{PC}\\
&\leq T\sum_{i=0}^{1}\|a_i\|_0\|x^{(i)}\|_{PC}+\|f(t, 0,
0)\|_{L^2}T^{1/2}+T\|x\|_{PC}.
\end{align*}
So
\begin{equation}\label{cc}
\|x'\|_{PC}\le
\frac{T(\|a_0\|+1)}{1-T\|a_1\|_0}\|x\|_{PC}+\frac{\|f(t, 0,
0)\|_{L^2}T^{1/2}}{1-T\|a_1\|_0}.
\end{equation}
Substituting \eqref{cc} into \eqref{4}, and noticing Lemma
\ref{lem1}, Lemma \ref{lem2}, we obtain that
\[
\|x'\|_{L^2}^{2}\le \Big\{\frac{(p+q)T^2\Gamma^2(\|a_0\|_0+1)}{1-T\|a_1\|_0}
+\sum_{i=0}^{1}\|a_i\|_0\big(\frac{2T}{\pi}\big)^{2-i}
\Big\}\|x'\|_{L^2}^{2}+N\|x'\|_{L^2}
\]
holds for sufficiently large
constant  $N>0$. The condition \eqref{098} means that $x'$ is
bounded in $L^2[0, T]$. Lemma \ref{lem2} means that there exists
$L_1>0$ such that $\|x\|_{PC}\le L_1$. \eqref{cc} means that there
exists $L_2>0$ such that $\|x'\|_{PC}\le L_2$. So $x$ is bounded in
$PC^1([0, T], \mathbb{R}^n)$. Applying Schaefer's fixed point theorem,
problem \eqref{1}, \eqref{neumann} has at least one solution.

Now we  show that \eqref{1}--\eqref{neumann} has a unique
solution. If $x, y$ are both solutions,
 then $u=x-y$ satisfies
\begin{equation}\label{11b}
\begin{gathered}
u''(t)=f(t, x(t), x'(t))-f(t, y(t),
y'(t)),\quad t\in[0, T]\setminus\{t_1\},\\
\Delta u(t_1)=I(x(t_1))-I(y(t_1)),\quad \Delta
u'(t_1)=J(x'(t_1))-J(y'(t_1)),\\
u'(0)=u'(T)=0.
\end{gathered}
\end{equation}
Similar to the above process, we have
\[
\|u'\|_{L^2}^{2}\le\Big[(p+q)\frac{T^2\|a_0\|_0\Gamma^2}{1-T\|a_1\|_0}
+\sum_{i=0}^{1}\|a_i\|_0\big(\frac{2T}{\pi}\big)^{2-i}\Big]\|u'\|_{L^2}^{2}.
\]
The condition \eqref{098} implies that $\|u'\|_{L^2}=0$.
Lemma \ref{lem2} gives that $u\equiv 0$ for $t\in [0, T]$.
 So problem \eqref{1}, \eqref{neumann} has a
unique solution in $Y$.
\end{proof}

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\end{document}