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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 24, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/24\hfil Existence of solutions]
{Existence of solutions for second-order impulsive boundary-value problems}

\author[A. Boucherif, A. S. Al-Qahtani, B. Chanane \hfil EJDE-2012/24\hfilneg]
{Abdelkader Boucherif, Ali S. Al-Qahtani, Bilal Chanane} 

\address{Abdelkader Boucherif \newline
King Fahd University of Petroleum and Minerals\\
Department of Mathematics and Statistics\\
P.O. Box 5046, Dhahran 31261, Saudi Arabia}
\email{aboucher@kfupm.edu.sa}

\address{Ali S. Al-Qahtani \newline
King Fahd University of Petroleum and Minerals\\
Department of Mathematics and Statistics\\
P.O. Box 5046, Dhahran 31261, Saudi Arabia}
\email{alitalhan@hotmail.com}

\address{Bilal Chanane \newline
King Fahd University of Petroleum and Minerals\\
Department of Mathematics and Statistics\\
P.O. Box 5046, Dhahran 31261, Saudi Arabia}
\email{chanane@kfupm.edu.sa}

\thanks{Submitted September 12, 2011. Published February 7, 2012.}
\subjclass[2000]{34B37, 34B15, 47N20}
\keywords{Second order boundary value problems; impulse effects; \hfill\break\indent
fixed point theorem}

\begin{abstract}
 In this article we discuss the existence of solutions of
 second-order boundary-value problems subjected to impulsive effects. Our
 approach is based on fixed point theorems.
\end{abstract}

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


\section{Introduction}

Differential equations involving impulse effects arise naturally in the
description of phenomena that are subjected to sudden changes in their
states, such as population dynamics, biological systems, optimal control,
chemotherapeutic treatment in medicine, mechanical systems with impact,
financial systems. For typical examples see \cite{nieto, pandit}. For a
general theory on impulsive differential equations the interested reader can
consult the monographs \cite{bainov, lakshmi, samoilenko}, and the papers
\cite{ahmad, erbe, lakmeche, lee, nieto1, rachunkova, rogov, tomecek} and
the references therein. Our objective is to provide
sufficient conditions on the data in order to ensure the existence of at
least one solution of the  problem
\begin{equation}
\begin{gathered}
(p(t)x'(t))'+q(t)x(t)=F(t,x(t),x'(t)),\quad t\neq t_k,\; t\in [ 0,1], \\
\Delta x(t_k)=U_k(x(t_k),x'(t_k)), \\
\Delta x'(t_k)=V_k(x(t_k),x'(t_k)),\quad k=1,2,\dots ,m, \\
x(0)=x(1)=0,
\end{gathered}  \label{e1}
\end{equation}
where $x\in\mathbb{R}$ is the state variable;
 $F:\mathbb{R}_{+}\times\mathbb{R}^2\to \mathbb{R}$ is a piecewise continuous function;
$U_k$ and $V_k$ represent the jump
discontinuities of $x$ and $x'$, respectively, at $t=$ $t_k\in (0,1)$, 
called impulse moments, with $0<t_1<t_2<\dots <t_m<1$.

\section{Preliminaries}

In this section we introduce some definitions and notations that will be
used in the remainder of the paper.

Let $J$ denote the real interval $[0,1]$. Let $J'=J\backslash
\{t_1$, $t_2,\dots ,t_m\}$. $PC(J)$ denotes the space of all functions $
x:J\to\mathbb{R}$ continuous on $J'$, and for $i=1,2,\dots ,m$, 
$x(t_{i}^{+})= \lim_{\epsilon \to  0+} x(t_{i}+\epsilon )$ and 
$x(t_{i}^{-})=\lim_{\epsilon \to  0} x(t_{i}-\epsilon )$
exist. We shall write $x(t_{i}^{-})=x(t_{i})$. This is a Banach space when
equipped with the sup-norm; i.e., 
$\| x\| _0=\sup_{t\in J} | x(t)| $. Similarly, $PC^1(J)$ is the
space of all functions $x\in PC(J)$, $x$is continuously differentiable on 
$J'$, and for $i=1,2,\dots ,m$, $x'(t_{i}^{+})$ and $x'(t_{i}^{-})$ exist and 
$x'(t_{i})=x'(t_{i}^{-})$. For $x\in PC^1(J)$ we define its norm by
$\| x\|_1=\| x\| _0+\| x'\| _0$.
Then $( PC^1(I),\| \cdot \| _1) $ is a Banach space.

The following linear problem plays an important role in our study.
\begin{equation}
\begin{gathered}
(p(t)x'(t))'+q(t)x(t)=f(t),\quad t\neq t_k,t\in [0,1], \\
\Delta x(t_k)=U_k(x(t_k),x'(t_k)), \\
\Delta x'(t_k)=V_k(x(t_k),x'(t_k)),\quad k=1,2,\dots ,m, \\
x(0)=x(1)=0,
\end{gathered}  \label{e2}
\end{equation}
To study \eqref{e2} we first consider the problem without impulses
\begin{equation}
\begin{gathered}
(p(t)x'(t))'+q(t)x(t)=f(t),\quad t\in [ 0,1] \\
x(0)=x(1)=0.
\end{gathered}  \label{e3}
\end{equation}
We shall assume, throughout the paper, that the following condition holds.
\begin{itemize}
\item[(H0)] 
\begin{itemize} 
\item[(i)] $p\in C^1(J:\mathbb{R}),p(t)\geq p_0>0$, for all $t\in J$.

\item[(ii)] $q\in C(J:\mathbb{R}),q(t)\leq p_0\pi ^2$, for all $t\in J$,
 and $q(t)<p_0\pi ^2$ on a
subset of $J$ of positive measure.
\end{itemize}
\end{itemize}

\begin{lemma}\label{lem1}
If {\rm (H0)} is satisfied, then for any nonzero $x\in C^2(J:\mathbb{R})$ with
$x(0)=x(1)=0$,
\begin{equation*}
\int_0^1\{p(t)(x'(t))^2-q(t)x^2(t)\}dt>0.
\end{equation*}
\end{lemma}

\begin{proof}
The proof of this lemma is presented in \cite{bouch1}. We shall reproduce it
here for the sake of completeness. Since $q(t)\leq p_0\pi ^2$ on a
subset of $J$ of positive measure, we have 
\begin{equation*}
p(t)(x'(t))^2-q(t)x^2(t)>p_0((x'(t))^2-\pi ^2x^2(t)).
\end{equation*}
This inequality yields
\begin{equation*}
\int_0^1\{p(t)(x'(t))^2-q(t)x^2(t)\}dt>p_0\int_0^1\{(x'(t))^2-\pi
^2x^2(t)\}dt.
\end{equation*}
We show that
\begin{equation*}
\mathcal{J}(x)=\int_0^1\{(x'(t))^2-\pi ^2x^2(t)\}dt\geq 0
\end{equation*}
for all functions $x\in C^2(J:\mathbb{R})$ with $x(0)=x(1)=0$. 
The function $u$ that minimizes $\mathcal{J}(x)$
satisfies the Euler-Lagrange equation (see \cite{elsgolts})
\begin{equation*}
u''+\pi ^2u=0,
\end{equation*}
and the boundary conditions $u(0)=u(1)=0$. Then $u(t)=\sin \pi t$ or $
u(t)=0, $ and $\mathcal{J}(u)=0$. Since $\mathcal{J}(x)\geq \mathcal{J}(u)$
it follows that $\mathcal{J}(x)\geq 0$, and so
\begin{equation*}
\int_0^1\{p(t)(x'(t))^2-q(t)x^2(t)\}dt>0.
\end{equation*}
This completes the proof of the lemma.
\end{proof}

\begin{lemma} \label{lem2}
If {\rm (H0)} is satisfied, then the linear problem
\begin{equation}
\begin{gathered}
(p(t)x'(t))'+q(t)x(t)=0 \\
x(0)=x(1)=0.
\end{gathered}  \label{e4}
\end{equation}
has only the trivial solution.
\end{lemma}

\begin{proof}
Assume on the contrary that  \eqref{e4} has a nontrivial solution 
$x_0$. Then   \eqref{e4} implies $[(p(t)x_0'(t))'+q(t)x_0(t)] x_0(t)=0$ which
yields
\begin{align*}
0 &= \int_0^1[ (p(t)x_0'(t))'+q(t)x_0(t)
] x_0(t)\,dt \\
&= \int_0^1[ (p(t)x_0'(t))'] x_0(t)
\,dt+\int_0^1q(t)x_0^2(t)\,dt \\
&= -\int_0^1[ p(t)x_0^{\prime 2}(t)-q(t)x_0^2(t)] \,dt<0.
\end{align*}
This is a contradiction. See Lemma \ref{lem1}. Therefore $x_0\equiv 0$ is
the only solution of \eqref{e4}.
\end{proof}

It is well known that the unique solution of \eqref{e3} is given by
\begin{equation*}
x(t)=\int_0^1G(t,s)f(s)ds,
\end{equation*}
where $G(\cdot ,\cdot ):J\times J\to\mathbb{R}$ is the Green's function 
corresponding to  \eqref{e4}.

\begin{lemma} \label{lem3}
The solution to \eqref{e2} is 
\begin{equation}
\begin{split}
x(t) &= \int_0^1G(t,s)f(s)ds-\sum_{k=1}^{m}\frac{\partial G(t,t_k)}{
\partial s}p(t_k)U_k(x(t_k),x'(t_k))   \\
&\quad+\sum_{k=1}^{m}G(t,t_k)p(t_k)V_k(x(t_k),x'(t_k)).
\end{split}  \label{e5}
\end{equation}
\end{lemma}

\begin{proof}
We shall use of superposition principle and write $x(t)=y(t)+z(t)+w(t)$,
where $y(t)$ solves the problem
\begin{equation}
\begin{gathered}
(p(t)y'(t))'+q(t)y(t)=f(t),\quad t\in J, \\
\Delta y(t_k)=0, \\
\Delta y'(t_k)=0,\quad k=1,2,\dots ,m, \\
y(0)=y(1)=0,
\end{gathered}  \label{e6}
\end{equation}
while $z(t)$ solves the problem
\begin{equation}
\begin{gathered}
(p(t)z'(t))'+q(t)z(t)=0,\quad t\neq t_k,t\in J,\\
\Delta z(t_k)=U_k(x(t_k),x'(t_k)), \\
\Delta z'(t_k)=0,\quad k=1,2,\dots ,m, \\
z(0)=z(1)=0,
\end{gathered}  \label{e7}
\end{equation}
and $w(t)$ solves the problem
\begin{equation}
\begin{gathered}
(p(t)w'(t))'+q(t)w(t)=0,\quad t\neq t_k,t\in J,\\
\Delta w(t_k)=0, \\
\Delta w'(t_k)=V_k(x(t_k),x'(t_k)),\quad k=1,2,\dots ,m, \\
w(0)=w(1)=0.
\end{gathered} \label{e8}
\end{equation}
It is clear that
\begin{equation*}
y(t)=\int_0^1G(t,s)f(s)ds,\quad t\in I.
\end{equation*}
For $k=1,2,\dots ,m$, set
\begin{gather*}
z_k(t)=-\frac{\partial G(t,t_k)}{\partial s}p(t_k)U_k(x(t_k),x'(t_k)),\quad t\in J,
\\
w_k(t)=G(t,t_k)p(t_k)V_k(x(t_k),x'(t_k)),\quad t\in J.
\end{gather*}
Using the properties of Green's function and its derivatives we can prove
that the functions $z_k$ and $w_k,k=1,2,\dots ,m$, are the solutions of
problems \eqref{e7} and \eqref{e8}, respectively. Consequently, 
$x=y+\sum_{k=1}^{m}z_k+\sum_{k=1}^{m}w_k$ is a solution of problem 
\eqref{e2}.
\end{proof}

\section{Nonlinear Problem}

In this section we present our main results on the existence of
solutions for nonlinear boundary-value problems for the second-order 
impulsive control system.
 Consider the problem
\begin{equation}
\begin{gathered}
(p(t)x'(t))'+q(t)x(t)=F(t,x(t),x'(t)),\quad t\neq t_k,t\in J, \\
\Delta x(t_k)=U_k(x(t_k),x'(t_k)), \\
\Delta x'(t_k)=V_k(x(t_k),x'(t_k)),\quad k=1,2,\dots ,m, \\
x(0)=x(1)=0,
\end{gathered} \label{e9}
\end{equation}
where $x\in \mathbb{R}$ is the state variable; 
$F:\mathbb{R}_{+}\times\mathbb{R}^2\to\mathbb{R}$ is a piecewise continuous function; 
$U_k$ and $V_k$ are impulsive functions
representing the jump discontinuities of $x$ and $x'$ at $t\in\{t_1,t_2,\dots ,t_m\}$.

The nonlinear system
\begin{equation}
\begin{gathered}
(p(t)\acute{x}(t)\acute{)}+q(t)x(t)=F(t,x(t),x'(t)) \\
x(0)=x(1)=0,
\end{gathered} \label{e10}
\end{equation}
is equivalent to the nonlinear integral equation
\begin{equation*}
x(t)=\int_0^1G(t,s)F(s,x(s),x'(s))ds,\quad \text{for all }t\in J
\end{equation*}
It follows from Lemma \ref{lem3} that any solution of \eqref{e9} satisfies
\begin{equation}
\begin{split}
x(t) &= \int_0^1G(t,s)F(s,x(s),x'(s))ds-\sum_{k=1}^{m}W(t,t_k)p(t_k)U_k(x(t_k),x'(t_k))\\
&\quad +\sum_{k=1}^{m}G(t,t_k)p(t_k)V_k(x(t_k),x'(t_k)).
\end{split}\label{e11}
\end{equation}
where 
$W(t,t_k)=\frac{\partial G(t,t_k)}{\partial s}$.
Let 
\begin{gather*}
K=\max \{| G(t,s)| :( t,s) \in J\times J\},\quad
L=\max \{| W(t,s)| :( t,s) \in J\times J\}, \\
M=\sup \{| \frac{\partial G(t,s)}{\partial t}| :( t,s) \in J\times J\},\quad
N=\sup \{| \frac{\partial W(t,s)}{\partial t}| :( t,s) \in J\times J\},\\
P=\max \{K,L,M,N\}.
\end{gather*}
For the next theorem we use the following assumptions:
\begin{itemize}
\item[(H1)] $F(\cdot ,\cdot ,\cdot )$ is continuous on $J'$ and
satisfies the Lipschitz condition
\begin{equation*}
| F(t,x_1,y_1)-F(t,x_2,y_2)| \leq \beta (| x_1-y_1| +| x_2-y_2| ).
\end{equation*}

\item[(H2)] $U_k$ and $V_k$ are continuous and satisfy the Lipschitz conditions
\begin{gather*}
| U_k(x_1,y_1)-U_k(x_2,y_2)| \leq c_k(| x_1-y_1| +| x_2-y_2| ), \\
| V_k(x_1,y_1)-V_k(x_2,y_2)| \leq  d_k(| x_1-y_1| +| x_2-y_2| ),
\end{gather*}

\item[(H3)] $2P( \beta+R\sum_{k=1}^{m}c_k+R\sum_{k=1}^{m}d_k) <1$ .

\end{itemize}

\begin{theorem} \label{thm1}
Under assumptions  {\rm (H0)--(H3)}, problem  
\eqref{e9} has a unique solution.
\end{theorem}

\begin{proof}
Define an operator $\omega :PC^1(J)\to  PC^1(J)$ by
\begin{align*}
\omega (x)(t) 
&= \int_0^1G(t,s)F(s,x(s),x'(s))ds-\sum_{k=1}^{m}W(t,t_k)p(t_k)U_k(x(t_k),x'(t_k))\\
&\quad+\sum_{k=1}^{m}G(t,t_k)p(t_k)V_k(x(t_k),x'(t_k)).
\end{align*}
It is clear that any solution of \eqref{e9} is a fixed point of $\omega $ and
conversely any fixed point of $\omega $ is a solution of \eqref{e9}.

 We shall show that $\omega $ is a contraction.
Let $x,y\in PC(J)$, then
\begin{align*}
\| \omega (x)-\omega (y)\| _0
 &\leq \sup_{t\in J}\big\{\int_0^1| G(t,s)| |
F(s,x(s),x'(s))-F(s,y(s),y'(s))| ds \\
&\quad + \sum_{k=1}^{m}| W(t,t_k)| p(t_k)|
U_k(x(t_k),x'(t_k))-U_k(y(t_k),y'(t_k))| \\
&\quad + \sum_{k=1}^{m}| G(t,t_k)| p(t_k)|
V_k(x(t_k),x'(t_k))-V_k(y(t_k),y'(t_k))| \big\}
\\
&\leq \sup_{t\in J}\Big\{\int_0^1| G(t,s)|
(\beta ( \| x-y\| _0+\| x'-y'\| _0) )ds \\
&\quad + R\sum_{k=1}^{m}| W(t,t_k)| c_k( \|
x-y\| _0+\| x'-y'\| _0)
\\
&\quad + R\sum_{k=1}^{m}| G(t,t_k)| d_k( \|x-y\| _0+\| x'-y'\|_0) \Big\}.
\end{align*}
Now, by using (H1) and (H2), we have
\begin{equation}
\| \omega (x)-\omega (y)\| _0\leq \beta K\|
x-y\| _1+RL\sum_{k=1}^{m}c_k\| x-y\|
_1+RK\sum_{k=1}^{m}d_k\| x-y\| _1.  \label{e12}
\end{equation}
We have 
\begin{align*}
\frac{d}{dt}\omega (x)(t) 
&= \int_0^1\frac{\partial G(t,s)}{\partial t}
F(s,x(s),x'(s))ds-\sum_{k=1}^{m}\frac{\partial W(t,t_k)}{\partial
t}U_k(x(t_k),x'(t_k)) \\
&\quad + \sum_{k=1}^{m}\frac{\partial G(t,t_k)}{\partial t}\
V_k(x(t_k),x'(t_k)).
\end{align*}
Let $x,y\in PC(J)$, then
\begin{align*}
\| \frac{d}{dt}\omega (x)-\frac{d}{dt}\omega (y)\| _0
&\leq \sup_{t\in J} \Big\{\int_0^1| \frac{\partial G(t,s)
}{\partial t}\ | | F(s,x(s),x'(s))-F(s,y(s),y'(s))| ds \\
&\quad + \sum_{k=1}^{m}| \frac{\partial W(t,t_k)}{\partial t}
| | U_k(x(t_k),x'(t_k))-U_k(y(t_k),y'(t_k))| \\
&\quad + \sum_{k=1}^{m}| \frac{\partial G(t,t_k)}{\partial t}
| | V_k(x(t_k),x'(t_k))-V_k(y(t_k),y'(t_k))| \Big\}.
\end{align*}
Conditions (H1) and (H2) imply
\begin{equation}
\| \frac{d}{dt}\omega (x)-\frac{d}{dt}\omega (y)\| _0\leq
\beta M\| x-y\| _1+RN\sum_{k=1}^{m}c_k\|
x-y\| _1+RM\sum_{k=1}^{m}d_k\| x-y\| _1.
\label{e13}
\end{equation}
From \eqref{e12} and \eqref{e13} we obtain
\begin{align*}
\| \omega (x)-\omega (y)\| _1 
&= \| \omega (x)-\omega (y)\| _0+\| \frac{d}{dt}\omega (x)-\frac{d}{dt}
\omega (y)\| _0 \\
&\leq \Big( \beta K+RL\sum_{k=1}^{m}c_k+RK\sum_{k=1}^{m}d_k\Big)
\| x-y\| _1 \\
&\quad + ( \beta M+RN\sum_{k=1}^{m}c_k+RM\sum_{k=1}^{m}d_k)
\| x-y\| _1 \\
&\leq 2P\Big( \beta +R\sum_{k=1}^{m}c_k+R\sum_{k=1}^{m}d_k\Big)
\| x-y\| _1
\end{align*}
Condition (H3) implies that $\omega $ is a contraction. By the
Banach fixed point theorem $\omega $ has a unique fixed point $x$, which is
the unique solution of \eqref{e9}.
\end{proof}

For the next Theorem, we use the following assumptions:
\begin{itemize}
\item[(H4)] $F:[0,1]\times\mathbb{R}^2\to\mathbb{R}$ is continuous on $J'$ and
there exists $h:J\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ a Caratheodory function,
 nondecreasing with respect to its second
argument such that
\begin{equation*}
| F(t,x,y)| \leq h(t,| x| +|y| ),\quad\text{a.e. }t\in [ 0,1].
\end{equation*}

\item[(H5)] $U_k$ and $V_k$ are continuous and there exist $a_k>0$ and $
b_k>0 $ such that
\begin{equation*}
| U_k(x(t_k),y(t_k))| \leq a_k, \quad
| V_k(x(t_k),y(t_k))| \leq b_k,\text{ \ }k=1,2,\dots ,m.
\end{equation*}

\item[(H6)] $\lim_{\varrho \to  +\infty } \sup \frac{1}{
\varrho }\big( \int_0^1h(t,\varrho )dt+\sum_{k=1}^{m}R(
a_k+b_k) \big) < 1/(2P)$.
\end{itemize}

\begin{theorem}\label{thm2}
Under assumptions {\rm (H0), (H4)--(H6)}, problem \eqref{e9} has at least one solution.
\end{theorem}

\begin{proof}
The proof is given in two steps.

 \textbf{Step 1.} A priori bound on solutions. Let $x\in PC^1(J)$
be a solution of \eqref{e9}.
\begin{align*}
x(t) &= \int_0^1G(t,s)F(s,x(s),x'(s))ds-\sum_{k=1}^{m}W(t,t_k)p(t_k)U_k(x(t_k),x'(t_k)) \\
&\quad + \sum_{k=1}^{m}G(t,t_k)p(t_k)V_k(x(t_k),x'(t_k)),
\end{align*}
and
\begin{align*}
x'(t) &= \int_0^1\frac{\partial G(t,s)}{\partial t}
F(s,x(s),x'(s))ds-\sum_{k=1}^{m}\frac{\partial W(t,t_k)}{\partial
t}p(t_k)U_k(x(t_k),x'(t_k)) \\
&\quad + \sum_{k=1}^{m}\frac{\partial G(t,t_k)}{\partial t}
p(t_k)V_k(x(t_k),x'(t_k)).
\end{align*}
It is easy to see that
\begin{align*}
| x(t)| &\leq K\int_0^1| F(s,x(s),x'(s))| ds+RL\sum_{k=1}^{m}| U_k(x(t_k),x'(t_k))| \\
&\quad + RK\sum_{k=1}^{m}| V_k(x(t_k),x'(t_k))| ,
\end{align*}
and
\begin{align*}
| x'(t)| &\leq M\int_0^1|
F(s,x(s),x'(s))| ds+RN\sum_{k=1}^{m}|
U_k(x(t_k),x'(t_k))| \\
&\quad + RM\sum_{k=1}^{m}| V_k(x(t_k),x'(t_k))| .
\end{align*}
Conditions (H4), (H5) and (H6) lead to
\begin{equation*}
\| x\| _0+\| x'\| _0\leq
(K+M)\int_0^1h(s,\| x\| _0+\| x'\| _0)ds+\sum_{k=1}^{m}R((L+N)l_k+(K+M)p_k).
\end{equation*}
Since $\| x\| _1=\| x\| _0+\|x'\| _0$ and $h$ is nondecreasing, then
\begin{equation*}
\| x\| _1\leq 2P\int_0^1h(s,\| x\|_1)ds+\sum_{k=1}^{m}R(2Pa_k+2Pb_k),
\end{equation*}
or
\begin{equation*}
\| x\| _1\leq 2P( \int_0^1h(s,\|x\| _1)ds+\sum_{k=1}^{m}R( a_k+b_k) ) .
\end{equation*}
Let $\beta _0=\| x\| _1$.
Then the above inequality gives
\begin{equation}
\frac{1}{2P}\leq \frac{1}{\beta _0}\Big( \int_0^1h(s,\beta
_0)ds+\sum_{k=1}^{m}R( a_k+b_k) \Big) .  \label{e14}
\end{equation}
Condition (H6) implies that there exists $r>0$ such that for all 
$\beta >r$, we have
\begin{equation}
\frac{1}{\beta }\Big( \int_0^1h(s,\beta )ds+\sum_{k=1}^{m}R(
a_k+b_k) \Big) <\frac{1}{2P}.  \label{e15}
\end{equation}
Comparing \eqref{e14} and \eqref{e15} we see that $\beta _0\leq r$ .
Hence we have
$\| x\| _1\leq r$.


\textbf{Step 2.} Existence of solutions.
 Let $\Omega =\{x\in PC^1(J):\| x\| _1<r+1\}$.
Then $\Omega $ is an open convex subset of $PC^1(J)$. Define an operator 
$H $ by
\begin{align*}
H(\lambda ,x)(t) 
&= \lambda \int_0^1G(t,s)F(s,x(s),x'(s))ds
 +\lambda \sum_{k=1}^{m}W(t,t_k)U_k(x(t_k),x'(t_k)) \\
&\quad + \lambda \sum_{k=1}^{m}G(t,t_k)V_k(x(t_k),x'(t_k)),\quad 
 0\leq \lambda \leq 1.
\end{align*}
Then $H(\lambda ,\cdot ):\bar{\Omega}\to  PC^1(J)$ is compact and
has no fixed point on $\partial \Omega $ (see \cite{lakmeche}). It is an
admissible homotopy between the constant map $H(0,\cdot )\equiv 0$\ and $
H(1,\cdot )\equiv \omega $. Since $H(0,\cdot )$\ is essential then $
H(1,\cdot )$\ is essential which implies that $\omega \equiv H(1,\cdot )$
has a fixed point in $\Omega $. This fixed point is a solution of our
problem.
\end{proof}

The following assumptions are used in the next theorem.
\begin{itemize}
\item[(H7)] there exists $g\in L^1(J)$ such that
\begin{equation*}
| F(t,x,y)| \leq g(t)\text{ \ for almost }t\in J,\quad x,y\in\mathbb{R}.
\end{equation*}

\item[(H8)] $U_k:\mathbb{R}^2\to\mathbb{R}$ is continuous and there exists 
$\alpha _k>0$ such that
\begin{equation*}
| U_k(x(t_k),y(t_k))| \leq \alpha _k( \|x\| _0+\| y\| _0) ,\text{ }k=1,2,\dots ,m.
\end{equation*}

\item[(H9)] $V_k:\mathbb{R}^2\to\mathbb{R}$ is continuous and there exists $\beta _k>0$ 
such that
\begin{equation*}
| V_k(x(t_k),y(t_k))| \leq \beta _k( \|x\| _0+\| y\| _0) ,\text{ \ }k=1,2,\dots ,m.
\end{equation*}

\item[(H10)] $2PR\sum_{k=1}^{m}( \alpha _k+\beta _k) <1$.
\end{itemize}

\begin{theorem} \label{thm3}
Under assumptions {\rm  (H0), (H7)--(H10)}, equation \eqref{e5} has at least one solution.
\end{theorem}

\begin{proof}
The proof is given in two steps.

\textbf{Step1.} A priori bound on solutions.
We have
\begin{align*}
x(t) &= \int_0^1G(t,s)F(s,x(s),x'(s))ds-\sum_{k=1}^{m}W(t,t_k)p(t_k)U_k(x(t_k),x'(t_k)) \\
&\quad + \sum_{k=1}^{m}G(t,t_k)p(t_k)V_k(x(t_k),x'(t_k)).
\end{align*}
and
\begin{align*}
x'(t) &= \int_0^1\frac{\partial G(t,s)}{\partial t}
F(s,x(s),x'(s),(Sx)(s))ds+\sum_{k=1}^{m}\frac{\partial W(t,t_k)}{
\partial t}p(t_k)U_k(x(t_k),x'(t_k)) \\
&\quad + \sum_{k=1}^{m}\frac{\partial G(t,t_k)}{\partial t}
p(t_k)V_k(x(t_k),x'(t_k)).
\end{align*}
 It is easy to see that
\begin{align*}
| x(t)| &\leq K\int_0^1| F(s,x(s),x'(s))| ds+RL\sum_{k=1}^{m}| U_k(x(t_k),x'(t_k))| \\
&\quad +RK\sum_{k=1}^{m}|V_k(x(t_k),x'(t_k))| ,
\end{align*}
and
\begin{align*}
| x'(t)| &\leq M\int_0^1|
F(s,x(s),x'(s))| ds+RN\sum_{k=1}^{m}|
U_k(x(t_k),x'(t_k))| \\
&\quad +RM\sum_{k=1}^{m}|V_k(x(t_k),x'(t_k))| .
\end{align*}
From (H7), (H8) and (H9), we obtain
\begin{align*}
\| x\| _0+\| x'\| _0 
&\leq (K+M)\| g\| _{L^1}+\sum_{k=1}^{m}R( L+N)
\alpha _k(\| x\| _0+\| x'\|_0) \\
&\quad +\sum_{k=1}^{m}R( K+M) \beta _k(\| x\|_0+\| x'\| _0).
\end{align*}
Setting $\mu =2PR\sum_{k=1}^{m}( \alpha _k+\beta _k) $, we obtain
\begin{equation*}
\| x\| _1\leq 2P\| g\| _{L^1}+\mu \| x\| _1.
\end{equation*}
Then
$(1-\mu )\| x\| _1\leq 2P\| g\| _{L^1}$.
Using condition (H10) we obtain
\begin{equation*}
\| x\| _1\leq ( \frac{2P}{1-\mu }) \|g\| _{L^1}:=r_1.
\end{equation*}

\textbf{Step 2.} Existence of solutions.
Let $\Omega _1=\{x\in PC^1(J):\| x\| _1<r_1+1\}$.
The rest of the proof is similar to that of Theorem \ref{thm2}, and it
is omitted.
\end{proof}

\subsection*{Acknowledgements} The authors are grateful to the King Fahd University
of Petroleum and Minerals for its support.

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\end{document}