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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 68, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/68\hfil Existence of solutions]
{Existence of solutions for second-order nonlinear impulsive
boundary-value problems}

\author[B. Ahmad\hfil EJDE-2009/68\hfilneg]
{Bashir Ahmad}

\address{Bashir Ahmad \newline
Department of Mathematics\\ 
Faculty of Science, 
King Abdulaziz University\\ 
P.O. Box. 80203, Jeddah 21589, Saudi Arabia}
\email{bashir\_qau@yahoo.com}

\thanks{Submitted October 13, 2008. Published May 19, 2009.}
\subjclass[2000]{34B10, 34B15}
\keywords{Impulsive differential equations; Schaefer's theorem;
\hfill\break\indent
 periodic and anti-periodic boundary conditions;
  existence of solutions}

\begin{abstract}
  We prove the existence of solutions for a second-order nonlinear
  impulsive boundary-value problem by applying Schaefer's fixed
  point theorem.
  Results for periodic and anti-periodic impulsive boundary-value
  problems can be obtained as special cases of the results in
  this article.
\end{abstract}

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

\section{Introduction}

Impulsive boundary-value problems have been extensively studied in
recent years.  The study of impulsive differential equations
provide a natural description of observed evolution processes of
several real world problems in biology, physics, engineering, etc.
For the general theory of impulsive differential equations, we
refer the reader to \cite{l1,r1,s1,z1}. Some recent results for
periodic and anti-periodic nonlinear impulsive boundary-value
problems can be found in \cite{a1,b1,c1,c2,d1,l3,l4,q1,y1,y2,w1}.
Bai and Yang \cite{b1} applied Schaefer's fixed point theorem to
establish the existence of solutions for second-order nonlinear
impulsive differential equations with periodic boundary
conditions. Motivated by the studies in \cite{b1}, we study the
existence of solutions for the  impulsive nonlinear boundary-value
problem
\begin{equation}
\begin{gathered}
u''(t)=f(t,u(t),u'(t)),\quad t \in [0,T], \; t \ne t_{1},\\
u(t_{1}^+)-u(t_{1}^-)=I(u(t_{1})), \quad
u'(t_{1}^+)-u'(t_{1}^-)=J(u(t_{1})),\\
u(0)= \mu u(T),  \quad u'(0)= \mu u'(T),
\end{gathered}
\label{eNP}
\end{equation}
where $f : [0,T] \backslash \{t_1\}\times
\mathbb{R}^{n}\times \mathbb{R}^{n} \to \mathbb{R}^{n}$ is
continuous, $I, J: \mathbb{R}^{n} \to \mathbb{R}^{n}$ are continuous
functions defining the impulse at $t_1 \in (0,T)$ and $\mu$ is a
fixed real number with $|\mu|\ge 1$.
We assume that $f(t_{1}^+, x,y) = \lim_{t \to t_{1}^+}f(t,x,y)$ and
$f(t_{1}^-, x,y) = \lim_{t \to t_{1}^-}f(t,x,y)$ both exist with
$f(t_{1}^-, x,y)=f(t_1, x,y$). For the sake of simplicity (as in
\cite{c2}), we consider only one impulse at $t=t_1 \in (0,T)$. An
arbitrary finite number of
impulses can be addressed similarly.

We remark that the impulsive boundary-value problem
\eqref{eNP} reduces to a periodic boundary-value problem
\cite{b1} for $\mu=1$
and anti-periodic boundary-value problem for $\mu=-1$. Thus,
problem \eqref{eNP} can be regarded as a generalization of periodic and
anti-periodic boundary-value problems.

Let us define the Banach spaces
\begin{gather*}
\begin{aligned}
PC([0,T],\mathbb{R}^{n}) &=\big\{u \in C([0,T] \backslash
\{t_1\}\times \mathbb{R}^{n}),
\text{ $u$ is left continuous at $t=t_1$,}\\
&\quad \text{and the right hand limit $u(t_{1}^+)$ exists}\big\},
\end{aligned}
\\
\begin{aligned}
PC^1([0,T],\mathbb{R}^{n}) &=\big\{u \in PC([0,T],\mathbb{R}^{n}),
\text{ $u'$ is left continuous at $t=t_1$,} \\
&\quad \text{and the right hand limit $u'(t_{1}^+)$ exists}\big\},
\end{aligned}
\end{gather*}
with the norms $\|u\|_{PC}=\sup_{t \in [0,T]}|u(t)|$, and
$\|u\|_{PC^1}=\max\{\|u\|_{PC}, \|u'\|_{PC}\}$, respectively.

A function $u \in PC^1([0,T],\mathbb{R}^{n}) \cap C^2([0,T]
\backslash \{t_1\}\times \mathbb{R}^{n})$ is a solution to
 \eqref{eNP} if it satisfies \eqref{eNP} for all
$t \in [0,T]$.\\ For $\sigma \in PC([0,T],\mathbb{R}^{n})$, $p \ge
0$, $q>0$, consider the  linear impulsive problem
\begin{equation}
\begin{gathered}
u''(t)-pu'(t)-qu(t)+\sigma(t)=0,\quad t \in [0,t],\; t \ne t_{1},\\
u(t_{1}^+)-u(t_{1}^-)=I(u(t_{1})), \quad
u'(t_{1}^+)-u'(t_{1}^-)=J(u(t_{1})),\\
u(0)= \mu u(T), \quad u'(0)= \mu u'(T), \quad
 \mu \in \mathbb{R} \quad (\mu\ne0),
\end{gathered}
\label{eLP}
\end{equation}
 whose associated auxiliary equation has the roots
$$
r_1=\frac{p+\sqrt{p^2+4q}}{2}, \quad
r_2=\frac{p-\sqrt{p^2+4q}}{2}.
$$
In view of $p \ge 0$, $q>0$, it is clear that  $r_1$ and $r_2$ are
respectively positive and negative real numbers. We need the
following lemma for the sequel. The proof of this
lemma is omitted as it can be obtained by direct computations.

\begin{lemma} \label{lem1}
 $u \in PC^1([0,T],\mathbb{R}^{n}) \cap C^2([0,T]
\backslash \{t_1\}\times \mathbb{R}^{n})$ is a solution of \eqref{eLP}
if and only if it satisfies the following impulsive integral equation
\begin{equation}
u(t)=\int_{0}^{T} G(t,s)\sigma(s)ds -G(t,t_1)J(u(t_1))
+W(t,t_1)I(u(t_1)),\label{e1.1}
\end{equation}
 where
$$
G(t,s)=\frac{1}{r_1-r_2}\begin{cases}
\frac{e^{r_1(t-s)}}{\mu e^{r_1T}-1}+\frac{e^{r_2(t-s)}}{1-\mu e^{r_2T}},
&0\le s< t\le T, \\[4pt]
\frac{\mu e^{r_1(T+t-s)}}{\mu e^{r_1T}-1}+\frac{\mu
e^{r_2(T+t-s)}}{1-\mu e^{r_2T}}, &0\le t \le s \le T,
\end{cases}
$$
$$
W(t,s)=\frac{1}{r_1-r_2}\begin{cases}
\frac{r_2e^{r_1(t-s)}}{\mu e^{r_1T}-1}
+\frac{r_1e^{r_2(t-s)}}{1-\mu e^{r_2T}}, &0\le s< t\le T, \\[4pt]
\frac{\mu r_2 e^{r_1(T+t-s)}}{\mu e^{r_1T}-1}+\frac{\mu r_1
e^{r_2(T+t-s)}}{1-\mu e^{r_2T}}, &0\le t \le s \le T,
\end{cases}
$$
 with ($\mu e^{r_1 T} -1) \neq 0 $ and
$(1 - \mu e^{r_2 T})\neq 0$.
\end{lemma}

As $r_1 \ge -r_2>0$ $(p \ge 0, q>0)$,
we find that
\begin{equation}
|G(t,s)|\le |G_1|, \quad
|W(t,s)|\le r_1|G_1|, \quad
|G_t(t,s)|\le r_1|G_1|, \quad
|W_t(t,s)|\le r_1^2|G_1|,
\label{e1.2}
\end{equation}
 where
$$
G_1=\frac{\mu (e^{r_1T}-e^{r_2T})}{(r_1-r_2)(\mu e^{r_1T}-1)
(1-\mu e^{r_2T})}.
$$
 Let
\begin{equation}
H=\max \{|G_1|, r_1|G_1|, r_1^2|G_1|\}. \label{e1.3}
\end{equation}
Define an operator $ \Lambda: PC^1([0,T],\mathbb{R}^{n}) \to
PC([0,T],\mathbb{R}^{n})$ by
\begin{equation}
\begin{aligned}
\Lambda u(t)&=\int_{0}^{T} G(t,s)[-f(s,u(s),u'(s))+pu'(s)+qu(s)]ds\\
&\quad -G(t,t_1)J(u(t_1))+W(t,t_1)I(u(t_1)), \quad t \in [0,T].
\end{aligned}\label{e1.4}
\end{equation}
It follows by Lemma \ref{lem1} that $u$ is a fixed point of the operator
$\Lambda$ if and only if $u$ is a solution of \eqref{eNP}.

In view of the continuity of $f, I, J$, the operators $\Lambda_1,
\Lambda_2$ defined by
\begin{gather*}
\Lambda_1 u(t)=\int_{0}^{T}
G(t,s)\Big[-f(s,u(s),u'(s))+pu'(s)+qu(s)\Big]ds, \quad t \in [0,T], \\
\Lambda_2 u(t)=-G(t,t_1)J(u(t_1))+W(t,t_1)I(u(t_1)), \quad t \in [0,T],
\end{gather*}
are compact. Thus, $\Lambda=\Lambda_1+\Lambda_2$ is a compact
operator.

\section{Existence of solutions}

\begin{theorem} \label{thm1}
 Let $f : [0,T] \backslash \{t_1\}\times
\mathbb{R}^{n}\times \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $I, J:
 \mathbb{R}^{n} \to \mathbb{R}^{n}$ be continuous functions.
If  there exist nonnegative constants
$\alpha, \beta_1, \beta_2, \gamma_1, \gamma_2,  M$ such that
 \begin{itemize}
\item[(A1)] For all $(t,x,y) \in ([0,T]\backslash \{t_1\})\times \mathbb{R}^{n}\times
\mathbb{R}^{n}$,
\[
\|f(t,x,y)-py-qx\| \le 2 \alpha [\langle
x+y, f(t,x,y)\rangle +  \|y\|^2]+M,
\]

\item[(A2)] $\|I(x)\| \le \beta_1 \|x\|+\gamma_1$,
$\|J(x)\| \le \beta_2 \|x\|+\gamma_2$ with $r_1\beta_1 +\beta_2
<1/H$, for all $x \in \mathbb{R}^{n}$.
\end{itemize}
Then  problem \eqref{eNP} has at least one solution.
\end{theorem}

\begin{proof}
 From the preceding section, we know that $u$ is a
fixed point of the operator $\Lambda$ if and only if $u$ is a
solution of \eqref{eNP}. Thus we need to show that the operator
$\Lambda$ (indeed compact) has at least one fixed point.
For that, we apply Schaefer's theorem to show that all the solutions
to the following equation are bounded a priori with the bound
independent of $\lambda$,
\begin{equation}
u=\Lambda \lambda u, \quad \lambda \in (0,1). \label{e2.1}
\end{equation}
Letting $u$ to be a solution of \eqref{e2.1}, we have
\begin{gather*}
u''(t)-pu'(t)-qu(t)=\lambda [f(t,u(t),u'(t))-pu'(t)-qu(t)], \quad
t \in [0,T],\; t \ne t_{1},\\
u(t_{1}^+)-u(t_{1}^-)=\lambda I(u(t_{1})), \quad
u'(t_{1}^+)-u'(t_{1}^-)=\lambda J(u(t_{1})),\\
u(0)= \mu u(T),  \quad u'(0)= \mu u'(T), \quad  \mu \in \mathbb{R}
\quad (|\mu|\ge1).
\end{gather*}
Using  (A1)-(A2) and \eqref{e1.2}-\eqref{e1.3},
we have
\begin{equation}
\begin{aligned}
&\|u(t)\|\\
&= \lambda \|\Lambda u(t)\|\\
&=\|\int_{0}^{T}\lambda G(t,s)\Big[f(s,u(s),u'(s))-pu'(s)-qu(s)\Big]ds\\
&\quad - \lambda G(t,t_1)J(u(t_1))+ \lambda W(t,t_1)I(u(t_1))\|\\
& \le |G_1| \Big[\int_{0}^{T}\lambda \|f(s,u(s),u'(s))-pu'(s)-qu(s)\|ds\\
&\quad +\lambda (\|J(u(t_1))\|+  r_1\|I(u(t_1))\|)\Big]\\
& \le |G_1| \Big[\int_{0}^{T}(2 \alpha(\langle
u(s)+u'(s), \lambda f(s,u(s),u'(s))\rangle + \|u'\|^2)+M)ds\\
&\quad +(r_1\beta_1 +\beta_2)\|u(t_1)\|+r_1\gamma_1+\gamma_2\Big]\\
& = |G_1| \Big[\int_{0}^{T}(2 \alpha(\langle
u(s)+u'(s), \lambda f(s,u(s),u'(s))+(1-\lambda)pu'(s)\\
&\quad +(1-\lambda)q u(s)\rangle + \|u'\|^2)+M)ds-\int_{0}^{T}2 \alpha
\langle u(s)+u'(s), (1-\lambda)pu'(s)\\
&\quad+(1-\lambda)q u(s)\rangle
ds +(r_1\beta_1 +\beta_2)\|u(t_1)\|+r_1\gamma_1+\gamma_2\Big].
\end{aligned} \label{e2.2}
\end{equation}
In view of the fact that $|\mu| \ge 1$, we have
\begin{equation}
\begin{aligned}
&-2 \alpha \int_{0}^{T}\langle u(s)+u'(s),
(1-\lambda)pu'(s)+(1-\lambda)q u(s)\rangle ds\\
&=-2 \alpha (1-\lambda)q\int_{0}^{T}\|u(s)\|^2ds-2 \alpha
(1-\lambda)p\int_{0}^{T}\|u'(s)\|^2ds\\
&\quad + 2 \alpha(1-\lambda)(p+q)\int_{0}^{T}\langle u(s),
u'(s) \rangle ds\\
&\le2 \alpha
(1-\lambda)(p+q)\int_{0}^{T}\langle u(s), u'(s) \rangle ds\\
&=\alpha (1-\lambda)(p+q)\int_{0}^{T}\frac{d}{ds}(\|u(s)\|^2)
ds\\
&=\alpha (1-\lambda)(p+q)(\|u(T)\|^2-\|u(0)\|^2)\\
&\le \alpha (1-\lambda)(p+q)(1-\mu^2)\|u(T)\|^2 \le 0.
\end{aligned} \label{e2.3}
\end{equation}
Using \eqref{e2.3} in \eqref{e2.2}, we obtain
\begin{align*}
&\|u(t)\|\\
&= \lambda \|\Lambda u(t)\|\\
&\le |G_1| \Big[\int_{0}^{T}(2 \alpha(\langle
u(s)+u'(s), \lambda f(s,u(s),u'(s))+(1-\lambda)pu'(s)\\
&\quad +(1-\lambda)q u(s)\rangle + \|u'(s)\|^2)+M)ds+ (r_1\beta_1
+\beta_2)\|u(t_1)\|+r_1\gamma_1+\gamma_2\Big]\\
&= |G_1|\Big[\int_{0}^{T}(2 \alpha(\langle u(s)+u'(s),
u''(s)\rangle+\langle u(s)+u'(s), u'(s)\rangle\\
&\quad -\langle u(s), u'(s)\rangle)+M)ds +(r_1\beta_1
+\beta_2)\|u(t_1)\|+r_1\gamma_1+\gamma_2\Big]\\
& \le |G_1| \Big[\int_{0}^{T}(2 \alpha(\langle u(s)+u'(s),
u''(s)+u'(s)\rangle+M)ds\\
&\quad +(r_1\beta_1 +\beta_2)\|u(t_1)\|
 +r_1\gamma_1+\gamma_2\Big]\\
& = |G_1| \Big[\int_{0}^{T}(\alpha
\frac{d}{ds}(\|u(s)+u'(s)\|^2)+M)ds+(r_1\beta_1 +\beta_2)\|u(t_1)\|
+r_1\gamma_1+\gamma_2\Big]\\
& = |G_1| \Big[\alpha(\|u(T)+u'(T)\|^2-\|u(0)+u'(0)\|^2)+MT\\
&\quad +(r_1\beta_1 +\beta_2)\|u(t_1)\|
+r_1\gamma_1+\gamma_2\Big]\\
& =  |G_1| [\alpha(1-\mu^2)\|u(T)+u'(T)\|^2+MT
+(r_1\beta_1 +\beta_2)\|u(t_1)\|+\gamma_1+\gamma_2\Big]\\
& \le  |G_1| \Big[MT +(r_1\beta_1
+\beta_2)\|u(t_1)\|+r_1\gamma_1+\gamma_2\Big],
\end{align*}
where we have used the fact that $\alpha(1-\mu^2)\|u(T)+u'(T)\|^2
\le 0$ (by the assumption $|\mu| \ge 1$). Taking supremum on
$[0,T]$, we obtain
$$
\sup_{t \in [0,T]}\|u(t)\| \le \frac{|G_1| [MT+r_1\gamma_1+\gamma_2]}
{1-|G_1|(r_1\beta_1 +\beta_2)}.
$$
 Similarly, it can be shown that
$$
\sup_{t \in [0,T]}\|u'(t)\| \le \frac{H [MT+r_1\gamma_1+\gamma_2]}
{1-H(r_1\beta_1 +\beta_2)}.
$$
 Thus, we have
\begin{align*}
\|u\|_{PC^1}
&= \max \{\frac{|G_1| [MT +r_1\gamma_1+\gamma_2]}
{1-|G_1|(r_1\beta_1 +\beta_2)}, \frac{H [MT+r_1\gamma_1+\gamma_2]}
{1-H(r_1\beta_1 +\beta_2)}\}\\
&= \frac{H [MT+r_1\gamma_1+\gamma_2]} {1-H(r_1\beta_1
+\beta_2)},
\end{align*}
which is the desired bound independent
of $\lambda$. Hence, by Schaefer's fixed point theorem \cite{l2}, the
operator $\Lambda$ has at least one fixed point which implies that
the problem \eqref{eNP} has at least one solution.
This completes the proof.
\end{proof}

\subsection*{Example}
Consider the  scalar nonlinear impulsive problem
\begin{equation}
\begin{gathered}
u''(t)=(u(t)+u'(t))^3+u'(t)+2u(t)+2t,\quad t \in [0,1],\; t \ne t_{1},\\
u(t_{1}^+)-u(t_{1}^-)=\frac{1}{6}u(t_{1}), \quad
u'(t_{1}^+)-u'(t_{1}^-)=\frac{1}{8}u(t_{1}),\\
u(0)= \mu u(T), \quad u'(0)= \mu u'(T), \quad
 \mu \in \mathbb{R} \quad (|\mu| \ge 1).
\end{gathered}
\label{e2.4}
\end{equation}
 Here, $T=1$, $f(t,x,y)=(x+y)^3+y+2x+2t$, $p=1$, $q=2$,
$r_1=2$, $r_2=-1$, $\beta_1=1/6$, $\beta_2=1/8$, $\gamma_1=\gamma_2=0$,
$1/H=0.3$. Moreover, for $\alpha =2/3$, $M= 8/3$, we find that
\begin{align*}
& 2 \alpha[(x+y)f(t,x,y)+y^2]+M\\
&=\frac{4}{3}[(x+y)^4+(x+y)^2+x(x+y)+2t(x+y)+y^2]+\frac{8}{3}\\
&=\frac{4}{3}[(x+y)^4+(x+y)^2] +
\frac{4}{3}(x+\frac{1}{2}y)^2+\frac{8}{3}t(x+y)+y^2+\frac{8}{3}\\
& \ge \frac{4}{3}[(x+y)^4+(x+y)^2] +
\frac{4}{3}(x+\frac{1}{2}y)^2-\frac{8}{3}|x+y|+y^2+\frac{8}{3}\\
& = \frac{4}{3}[(x+y)^4+(|x+y|-1)^2 +
(x+\frac{1}{2}y)^2]+y^2+\frac{4}{3}\\
& \ge |x+y|^3+y^2+1, \quad \forall (t,x,y) \in ([0,1] \backslash
\{t_1\}) \times \mathbb{R}\times \mathbb{R}.
\end{align*}
Thus, for all $(t,x,y) \in ([0,1] \backslash \{t_1\}) \times
\mathbb{R}\times \mathbb{R}$,
$$
|f(t,x,y)-2x-y| \le 2 \alpha[(x+y)f(t,x,y)+y^2]+M.
$$
 Hence, the assumptions
(A1)-(A2)  are satisfied. Therefore, by
 Theorem \ref{thm1},  problem \eqref{e2.4} has at least one
solution.


\subsection*{Remarks}
(1) If the function
$f$ does not depend on $u'(t)$, then the assumption (A1) takes
the form
$$
\|f(t,x)-qx\| \le 2 \alpha \langle x, f(t,x)\rangle +M,
 \quad (t,x) \in ([0,T] \backslash \{t_1\})\times
\mathbb{R}^{n}.
$$
 For example,  consider a scalar function
$$
f(t,x)=x^5+x+2t, \quad (t,x) \in ([0,1] \backslash \{t_1\})\times
\mathbb{R}.
$$
 For $\alpha =1/2$, $M= 2$, we obtain
\begin{align*}
2 \alpha \langle x, f(t,x)\rangle +M
&=x^6+x^2+2tx+2 \\
&\ge x^6+x^2-2|x|+2\\
&=x^6+(|x|-1)^2+1 \\
&\ge |x|^5+1, \quad \forall
(t,x) \in ([0,1] \backslash \{t_1\})\times \mathbb{R}.
\end{align*}
Thus, $|f(t,x)-x| \le 2 \alpha x f(t,x)+M$, for all $(t,x) \in
[0,1] \times \mathbb{R}$.
\smallskip

(2) A similar proof follows for a modified form of Theorem \ref{thm1}
obtained by replacing the assumption (A1) by the  condition
$$
\|f(t,x,y)-py-qx\| \le 2 \alpha \langle y, f(t,x,y)\rangle +M,
 \quad (t,x,y) \in ([0,T] \backslash \{t_1\})\times
\mathbb{R}^{n}\times \mathbb{R}^{n}.
$$

(3) The results presented in this paper are new and a variety
of special cases can be recorded by fixing the value of $\mu$. For instance, if we take
$\mu=1$ in the problem \eqref{eNP}, the results for impulsive periodic
boundary-value problems \cite{b1} appear as a special case while
$\mu=-1$ in \eqref{eNP} yields the existence results for anti-periodic
second order boundary-value problems.


\subsection*{Acknowledgments}
The author is grateful to the anonymous reviewer and the editor
for their valuable suggestions and comments that led to the
improvement of the original manuscript.

\begin{thebibliography}{00}

\bibitem{a1} B. Ahmad, J. J. Nieto;
\emph{Existence and approximation of solutions for a class of
nonlinear impulsive functional differential equations with
anti-periodic boundary conditions}. Nonlinear Anal. 69 (2008) 3291--3298.

\bibitem{b1} C. Bai,  D. Yang;
\emph{Existence of solutions for
second-order nonlinear impulsive differential equations with
periodic boundary value conditions}. Bound. Value Probl. 2007, Art.
ID 41589, 13 pp.

\bibitem{c1} Y. Chen, J. J. Nieto, D. O'Regan;
\emph{Anti-periodic solutions for fully
nonlinear first-order differential equations.} Math. Comput. Model.
46 (2007), 1183--1190.

\bibitem{c2} J. Chen, C. C. Tisdell, R. Yuan;
\emph{On the solvability of periodic boundary-value problems
with impulse}. J. Math. Anal. Appl. 331 (2007) 902--912.

\bibitem{d1} W. Ding, Y. Xing, M. Han;
\emph{Antiperiodic boundary-value problems for first order impulsive
functional differential equations}.
Appl. Math. Comput. 186 (2007),  45--53.

\bibitem{l1} V. Lakshmikantham, D. D. Bainov, P. S. Simeonov;
\emph{Theory of Impulsive Differential Equations}.
World Scientific, Singapore, 1989.

\bibitem{l2} N. G. Llyod;
\emph{Degree Theory}. Cambridge Tracts in Mathematics, No.73,
Cambridge University Press, Cambridge, 1978.

\bibitem{l3} Z. Luo,  J. J. Nieto;
\emph{New results of periodic boundary-value problem for impulsive
 integro-differential equations}. Nonlinear Anal. 70 (2009) 2248--2260.

\bibitem{l4} Z. Luo, J. Shen, J. J. Nieto;
\emph{Antiperiodic boundary-value problem
for first order impulsive ordinary differential equations}.
Comput. Math. Appl. 49 (2005), 253--261.

\bibitem{q1} D. Qian, and X. Li;
\emph{Periodic solutions for ordinary differential equations with
sublinear impulsive effects}. J. Math. Anal. Appl. 303 (2005),
288--303.

\bibitem{r1} Y. V. Rogovchenko;
\emph{Impulsive evolution systems: Main results and new
trends}. Dynam. Contin. Discrete Impuls. Systems 3 (1997), 57--88.

\bibitem{s1} A. M. Samoilenko, N. A. Perestyuk;
\emph{Impulsive Differential Equations}.
World Scientific, Singapore, 1995.

\bibitem{y1} X. Yang, J. Shen;
\emph{Periodic boundary-value problems for
second-order impulsive integro-differential equations}. J. Comput.
Appl. Math. 209 (2007) 176--186.

\bibitem{y2} M. Yao, A. Zhao, J. Yan;
\emph{Periodic boundary-value problems of second-order impulsive
differential equations}. Nonlinear Anal. 70 (2009) 262--273.

\bibitem{w1} K. Wang;
\emph{A new existence result for nonlinear first-order
anti-periodic boundary-value problems}. Appl. Math. Letters, 21
(2008) 1149--1154.

\bibitem{z1} S. T. Zavalishchin, A. N. Sesekin;
\emph{Dynamic Impulse Systems. Theory and Applications}.
Kluwer Academic Publishers Group, Dordrecht, 1997.

\end{thebibliography}

\end{document}