\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 67, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/67\hfil Impulsive neutral functional differential
inclusions]
{Impulsive neutral functional differential inclusions  with variable times}

\author[Mouffak Benchohra \&  Abdelghan Ouahabi\hfil EJDE--2003/67\hfilneg]
{Mouffak Benchohra \&  Abdelghani Ouahab}

\address{Laboratoire de Math\'ematiques, Universit\'e de
Sidi Bel Abb\`es\\
BP 89, 22000 Sidi Bel Abb\`es, Alg\'erie}
\email{benchohra@univ-sba.dz}

\date{}
\thanks{Submitted November 27, 2002. Published June 13, 2003.}
\subjclass[2000]{34A37, 34A60, 34K25}
\keywords{Impulsive neutral functional differential inclusions, variable times,
\hfill\break\indent condensing map, fixed point}

\begin{abstract}
  In this paper, we study the existence of solutions for  first and
  second order impulsive neutral functional differential inclusions
  with variable times. Our main tool is a fixed point theorem due to
  Martelli for condensing multivalued maps.
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\numberwithin{equation}{section}
\allowdisplaybreaks

\section{Introduction}

 This  paper concerns the existence of solutions for
 initial-value problems for first and second order neutral functional
 differential inclusions with impulsive effects at variable times. In
 Section 3, we consider the first order initial-value problem (IVP for short)
\begin{gather}\label{e1}
\frac{d}{dt}[y(t)-g(t,y_t)]\in F(t,y_{t}), \quad\mbox{a. e. }
 t\in J=[0,T], \; t\neq \tau_k(y(t)), \; k=1,\ldots,m, \\
\label{e2}
y(t^+)=I_{k}(y(t)), \quad  t=\tau_k(y(t)), \; k=1,\ldots,m, \\
\label{e3}
y(t)=\phi(t), \quad t\in [-r,0],
\end{gather}
where $F: J\times D\to 2^{\mathbb{R}^{n}}$ is a compact convex valued multivalued
map,   $ g: J\times D\to\mathbb{R}^n$ is given function,
$D=\{\psi:[-r,0]\to\mathbb{R}^n; \psi$ is
continuous everywhere except for a finite number of points
$\bar{t}$  at which  $\psi(\bar{t})$ and $\psi(\bar{t}^+)$ exist and
 $\psi(\bar{t}^-)=\psi(\bar{t})\}$,
$\phi\in D$,  $0<r<\infty$, $\tau_k: \mathbb{R}^n\to\mathbb{R}$, $I_k: \mathbb{R}^n\to\mathbb{R}^n$,
$k=1,2,\ldots,m$ are given
functions satisfying some assumptions that will be specified later.

For any function $y$ defined on  $[-r,T]$ and any $t\in J$ we denote by
$y_{t}$ the element of $D$ defined by
$$
y_{t}(\theta)=y(t+\theta), \quad \theta\in [-r,0].
$$
Here $y_{t}(\cdot)$ represents the history of the state from time $t-r$, up to
the present time $t$.
In Section 4, we consider the second order IVP
\begin{gather}\label{e4}
\frac{d}{dt}[y'(t)-g(t,y_t)]\in F(t,y_t), \quad\mbox{a. e. } t\in J=[0,T], \;
 t\neq \tau_k(y(t)),\; k=1,\ldots,m, \\ \label{e5}
y(t^+)=I_{k}(y(t)),\quad  t=\tau_k(y(t)), \; k=1,\ldots,m, \\ \label{e6}
y'(t^{+})=\overline I_{k}(y(t)),\quad  t=\tau_k(y(t)),\; k=1,\ldots,m, \\
\label{e7}
y(t)=\phi(t),\quad  t\in[-r,0],\; y'(0)=\eta,
\end{gather}
where $g, F, I_k$, and $\phi$ are as in the problem
(\ref{e1})--(\ref{e3}), $\overline I_{k}\in C(\mathbb{R}^n,\mathbb{R}^n)$ and
$\eta\in \mathbb{R}^n$.

The theory of impulsive differential equations have become
important  in some mathematical models  of real
processes and phenomena studied in physics, chemical technology,
population dynamics, biotechnology and economics. There has been a
significant development in impulse theory in recent years,
especially in the area of impulsive differential equations and
 inclusions with fixed moments; see
the monographs of Bainov and Simeonov \cite{BaSi},  Lakshmikantham
{\em et al} \cite{LaBaSi}, and Samoilenko and Perestyuk
\cite{SaPe}, the papers of Benchohra {\em et al}
\cite{BeHeNt1}-\cite{BeHeNt4} and the references therein. The
theory of impulsive differential equations with variable time is
relatively less developed due to the difficulties created by  the
state-dependent impulses. Recently,  some interesting extensions
to impulsive  differential equations with variable times have been
done by  Bajo and Liz \cite{BaLi}, Frigon and  O'Regan
\cite{FrOr,FrOr1,FrOr2}, Kaul {\em et al} \cite{KaLaLe}, Kaul and
Liu \cite{KaLi}, \cite{KaLi1}, Lakshmikantham {\em et al}
\cite{LaLeKa}, \cite{LaPaVa}, Liu and Ballinger \cite{LB} and the
references cited therein.

The main theorems of this paper extend the problem (\ref{e1})-(\ref{e3})
considered by Benchohra {\em et al} \cite{BeHeNt1, BeHeNt2,BeHeNt3} when
the impulse times are constant.   Our approach  is based on the
Martelli fixed point theorem \cite{Mar}.

\section{Preliminaries}

In this section, we introduce notation, definitions, and
preliminary facts from multivalued analysis which are used
throughout this paper. Let $(a,b)$ be an open interval.
$AC^i((a,b),\mathbb{R}^n)$ is the space of $i$-times
differentiable functions $y:(a,b) \to \mathbb{R}^n$, whose
$i^{th}$ derivative, $y^{(i)}$, is absolutely continuous.

Let $(X, \|\cdot\|)$ be a Banach space. A multi-valued map
$G:X\to 2^{X}$ has convex (closed) values if $G(x)$ is
convex (closed) for all $x\in X$. $G$ is bounded on  bounded sets
if $G(B)$ is bounded in $X$ for each bounded set $B$ of $X$, i.e.
 $ \sup_{x\in B}\{\sup\{\|y\|: y\in G(x) \}\}<\infty$. $G$ is
called upper semi-continuous (u.s.c.) on $X$ if for each $x_{0}\in
X$ the set $G(x_{0})$ is a nonempty, closed subset of $X$, and if
for each open set $N$ of $X$ containing $G(x_{0})$, there exists
an open neighborhood $M$ of $x_{0}$ such that  $G(M)\subseteq N$.
$G$ is said to be completely continuous if $G(B)$ is relatively
compact for every bounded subset $B\subseteq X$. If the
multi-valued $G$ is completely continuous with nonempty compact
values, then $G$ is u.s.c. if and only if $G$ has a closed graph
(i.e. $x_{n}\to x_{*}$, $y_{n}\to y_{*}$, $y_{n}\in G(x_{n})$
imply $y_{*}\in G(x_{*})$). An upper semicontinuous map $G:X\to
2^X$ is said to be condensing if for any subset $B\subseteq X$
with $\alpha(B)\not=0$, we have $\alpha(G(B))<\alpha(B)$, where
$\alpha$ denotes the Kuratowski measure of noncompacteness. For
properties of the Kuratowski measure, we refer to the book of
Banas and Goebel \cite{BaGo}. We remark that a completely
semicontinuous multivalued map is the easiest example of a
condensing map. $G$ has a fixed point if there is $x\in X$ such
that  $x\in G(x)$. In the following, $CC(\mathbb{R}^n)$ denotes
the set of all nonempty compact, convex subsets of $\mathbb{R}^n$.


\begin{definition} \label{def2.1} \rm
A multi-valued map $F:J\times D\to 2^{\mathbb{R}^{n}}$ is said to
be $L^{1}$-Carath\'eodory  if
\begin{itemize}
\item[(i)] $t\mapsto F(t,u)$ is  measurable for each $y\in D;$
\item[(ii)] $u\mapsto F(t,u)$ is upper semi-continuous for almost all
$t\in J;$
\item[(iii)] For each $q>0$,  there exists $\phi_{q} \in L^{1}(J,\mathbb{R}_{+})$
such that
$$
\|F(t,u)\|=\sup\{|v|: v\in F(t,y)\}\leq\phi_{q}(t)
$$
for all $\|u\|_{D}\leq q$ and for almost all $t\in J$.
\end{itemize}
\end{definition}

For more details on multi-valued maps we refer the reader to the
books of Deimling \cite{Dei}, Gorniewicz \cite{Gor}, Hu and Papageorgiou
\cite{HuPa} and Tolstonogov \cite{Tol}.

For a function $y$ defined on $[-r,T]$ we define the   set
$$
S_{F,y}=\{v\in L^{1}(J,\mathbb{R}^n): \ v(t)\in F(t,y_t) \hbox { for a.e. } t\in J\},
$$
which is known as the set of {\em selection functions}.

The following lemmas are crucial in the proof of our main theorem.

\begin{lemma}[\cite{LaOp}] \label{l1}
Let $I$ be a compact real interval and $X$ be a Banach space. Let $F$ be a
multi-valued map satisfying the Carath\'eodory conditions with the set
of $L^{1}$-selections $S_{F}$ nonempty, and let $\Gamma$ be a linear
 continuous mapping from $L^{1}(I, X)$ to $C(I, X)$.
Then the operator
$$
\Gamma \circ S_{F}:C(I, X)\to CC(C(I, X)), \quad  y\mapsto
(\Gamma \circ S_{F})(y):=\Gamma(S_{F,y})$$
is a closed graph operator in $C(I, X)\times C(I, X)$.
\end{lemma}

\begin{lemma}[\cite{Mar}] \label{l2}
Let $ N:X\to CC(X)$ be an upper semicontinuous and condensing map. If the set
 $$ \mathcal{M}:=\{ y\in X: y\in\lambda N(y) \mbox{ for some } 0<\lambda<1 \}
 $$
is bounded, then $ N$ has a fixed point.
\end{lemma}

To define the solutions of problems (\ref{e1})-(\ref{e3}) and
(\ref{e4})-(\ref{e7}), we shall consider the  space
\begin{align*}
PC&=\big\{y: [0,T]\to\mathbb{R}^n: \hbox{there exist } 0 =t_0 < t_1 < \ldots < t_m <t_{m+1}
= T \\
&\hbox{ such that } t_k = \tau_k(y(t_k)), \; y(t^{-}_{k})
 \hbox{ and }  y(t^{+}_{k}) \hbox{ exist with } y(t_k^-) =y(t_k), \\
& \ k=1,\ldots,m, \hbox{ and } y \in C([t_k,t_{k+1}], \mathbb{R}^n) \; k=0, \ldots,m
\big\}.
\end{align*}
Set $\Omega:=\{y:[-r,T]\to{\mathbb{R}^{n}}: y\in D\cap PC\}$. In
what follows, we will assume that $F$ is an $L^{1}$-Carath\'eodory
function.

\section{First Order Impulsive NFDIs}

Let us start by defining what we mean by a solution of problem
(\ref{e1})--(\ref{e3}).

\begin{definition}\label{d1} \rm
A function  $y\in \Omega\cap\cup_{k=0}^{m}AC((t_{k},t_{k+1}),\mathbb{R}^{n})$
is said to be a solution of (\ref{e1})--(\ref{e3}) if there exists
$v(t)\in F(t,y_t)$  a.e.  $t\in[0,T]$ such that
$\frac{d}{dt}[y(t)-g(t,y_{t})]=v(t)$
a.e. on  $J,\; t\neq \tau_k(y(t))$, $k=1,\ldots,m$, $y(t^+)=I_{k}(y(t))$,
$t=\tau_k(y(t))$, $k=1,\ldots,m$,
$ y(t)=\phi(t)$, $t\in[-r,0]$.
\end{definition}

We are now in  a position to state and prove our existence result for the
problem (\ref{e1})-(\ref{e3}). For the study of this problem we first list
the following hypotheses:
\begin{itemize}
\item[(H1)] The functions $\tau_k\in C^1(\mathbb{R}^n,\mathbb{R})$  for $k=1,\ldots,m$.
Moreover,
$$
0<\tau_1(x)<\ldots<\tau_m(x)
<\tau_{m+1}(x)=T\quad \mbox{for all } x\in \mathbb{R}^n.
$$
\item[(H2)]  There exist constants $c_{k}$, such that
$|I_{k}(x)|\leq c_{k}$,  $k=1,\ldots,m$ for each $x\in \mathbb{R}^n$.
\item[(H3)]  The function $g$ is  completely continuous and
there exist constants  $0\leq d_1<1$, $d_2\geq 0$ such that
$$
|g(t,u)|\leq d_1\|u\|+d_2,\quad    t\in[0,T],\quad  u\in D\,.
$$
\item[(H4)] There exist a continuous nondecreasing function
$\psi:[0,\infty)\to (0,\infty)$, and $p\in L^{1}([0,T],\mathbb{R}_{+})$ such that
$$
\|F(t,u)\|\leq p(t)\psi (\|u\|) \quad \mbox{a.e. }   t\in [0,b],
\hbox{ and each } u\in D
$$
with $\int_1^{\infty}\frac{d\gamma}{\psi(\gamma)}=\infty$

\item[(H5)]  For  all $(t,x)\in [0,T]\times \mathbb{R}^n$ and for all $y_t\in D$ we have
$\langle \tau'_k(x), v(t)\rangle \neq 1$ for $k=1,\ldots,m$,
for all $v\in S_{F,y}$, where $\langle \cdot,\cdot\rangle$ denotes the scalar
product  in $\mathbb{R}^n$.

\item[(H6)] $g$ is a nonnegative function.

\item[(H7)] $\tau_k$  is a non-increasing function and
$I_k(x)\leq x$ for all $x\in R^n, k=1,\ldots,m$.

\item[(H8)]  For all $x\in \mathbb{R}^n$,
$\tau_k(x)<\tau_{k+1}(I_k(x))$ for $k=1,\ldots,m$.
\end{itemize}

\begin{theorem} \label{t1}
Assume that hypotheses (H1)-(H8) hold.
Then the initial-value problem (\ref{e1})--(\ref{e3}) has at least one
solution on $[-r,T]$.
\end{theorem}

The proof of this theorem will be given in several steps.\\
{\bf Step 1:} Consider the problem
\begin{gather}\label{eq1}
\frac{d}{dt}[y(t)-g(t,y_t)]\in F(t, y_t), \quad\mbox{a.e. } t\in [0,T],\\
\label{eq2}
 y(t)=\phi(t),\quad t\in[-r,0].
 \end{gather}
Transform the problem into a  fixed point problem. Consider the operator
$\mathcal{N}: C([-r,T],\mathbb{R}^n)\to 2^{C([-r,T],\mathbb{R}^n)}$
defined as
$\mathcal{N}(y)=\{h\in C([-r,T],\mathbb{R}^n)\}$ where for  $v\in S_{F,y}$,
$$h(t)=\begin{cases} \phi(t), &\mbox{if } t\in[-r,0)\\
\phi(0)-g(0,\phi(0))+g(t,y_t)+\int_{0}^{t}v(s)ds,
&\mbox{if } t\in[0,T].
\end{cases}
$$

 \begin{remark}\label{r1}\rm
 We can easily show  that  the fixed points of
$N$ are solutions to (\ref{eq1})--(\ref{eq2}).
\end{remark}

We shall show that the operator $N$ is  completely continuous.
Using (H3) it suffices to show that the operator $N_1: C([-r,T],\mathbb{R}^n)\to
C([-r,T],\mathbb{R}^n)$
defined as
$N_1(y)=\{h\in C([-r,T],\mathbb{R}^n)\}$, where
$$
h(t)=\begin{cases} \phi(t), &\mbox{if } t\in [-r,0); \\
\phi(0)+\int_{0}^{t}v(s)ds, &\mbox{if } t\in[0,T],
\end{cases}
$$
is completely continuous. The proof will be given in several Claims.

\noindent {\bf Claim 1:} {\em $N_1(y)$ is convex for each $y\in C([-r,T],\mathbb{R}^n)$.}
Indeed, if $v_{1},\ v_{2}$ belong to $N_1(y)$, then there exist $v_{1},
v_{2}\in  S_{F, y}$ such that  for each $t\in J$, we have
$$
h_i(t)=\phi(0)+\int_{0}^{t}v_i(s)ds,\; i=1,2.
$$
Let $0\leq d\leq 1$. Then for each $t\in J$ we have
$$
(dh_{1}+(1-d)h_{2}(t)=\phi(0)+\int_{0}^{t}[dv_{1}(s)+(1-d)v_{2}(s)]ds
$$
Since $ S_{F, y}$ is convex
(because $F$ has convex values) then
$$
dh_{1}+(1-d)h_{2}\in N_1(y).
$$
{\bf Claim 2}: {\em $N_1$ maps bounded sets into bounded sets in
$C([-r,T],\mathbb{R}^n)$.}  Indeed, it is sufficient to show that
for any $q>0$ there exists a positive constant $\ell$ such that
for each $y\in B_{q}=\{y\in C([-r,T],\mathbb{R}^n): \|y\|_{\infty}\leq q \}$
we have $\|N_1(y)\|\leq \ell$. Let $y\in B_{q}$ and
$h\in N_1(y)$ then there exists $v\in S_{F, y}$ such that for each
$t\in J$ we have
$$ h(t)=\phi(0)+\int_{0}^{t}v(s)ds.
$$
Thus,
$$
|h(t)|\leq |\phi(0)|+\int_{0}^{t}|v(s)|ds
\leq \|\phi\|_{\infty}+\|h_{q}\|_{L^{1}}:=\ell.
$$
{\bf Claim 3:} {\em $N_1$ maps bounded sets  into
equicontinuous sets of $C([-r,T],\mathbb{R}^n)$.}
 Let $u_{1}, u_{2}\in J$, $u_{1}<u_{2}$ and $B_{q}$ be a bounded set of
$C(J,\mathbb{R}^N)$ as in Claim 2. Let $y\in B_{q}$ and $h\in N_1(y)$.
Then there exists $v\in S_{F, y}$ such that for each $t\in J$, we have
$$
h(t)=\phi(0)+\int_{0}^{t}v(s)ds
$$
Then
$$
|N_1(y(u_{2}))-N_1(y(u_{1}))|\leq \int_{u_{1}}^{u_{2}}h_q(s)ds.
$$
As $u_{2}\to u_{1}$ the right-hand side of the above
inequality tends to zero.

As a consequence of Claims 2 and 3 and the Arzela-Ascoli theorem
we can conclude that $\mathcal{N}:C(J,\mathbb{R}^N)\to
2^{C(J,\mathbb{R}^N)}$ is a completely continuous  multi-valued
operator, and  therefore, a condensing map.


\noindent{\bf Claim 4:} {\em $ N_1$ has a closed graph.}
 Let $y_{n}\to y_{*}, \ h_{n}\in N_1(y_{n})$,  and  $h_{n} \to h_{*}$.
We shall prove that $h_{*}\in N(y_{*})$.
$h_{n}\in N_1(y_{n})$ means that there exists $v_{n}\in S_{F, y_{n}}$
such that for each $t\in J$,
$$
h_{n}(t)=\phi(0)+\int_{0}^{t}v_{n}(s)ds.
$$
We must prove that there exists $h_{*}\in S_{F, y_{*}}$
such that for each $t\in J$,
$$
h_{*}(t)=\phi(0)+\int_{0}^{t}v_{*}(s)ds.
$$
Clearly,
$$
\bigl\|\bigl(h_{n}-\phi(0)\bigr)
-\bigl(h_{*}-\phi(0)\bigr)
\bigr\|_{\infty} \to 0, \quad \hbox{as } n\to \infty.
$$
Consider the linear continuous operator
$\Gamma: L^{1}(J,\mathbb{R}^n)\to C(J,\mathbb{R}^n)$,
$$
v\mapsto (\Gamma v)(t)= \int_{0}^{t}v(s)ds.
$$
>From Lemma \ref{l1}, it follows that $\Gamma\circ S_{F}$
is a closed graph operator.
Since
$\big(h_{n}(t)-\phi(0)\big) \in \Gamma( S_{F, y_{n}})$,
it follows from Lemma \ref{l1} that for some $v_{*}\in S_{F,v_{*}}$,
$$
h_{*}(t)=\phi(0)+\int_{0}^{t}v_{*}(s)ds\,.
$$


\noindent{\bf Claim 5:}  {\em The following  set is bounded,}
$$
\mathcal{E}(\mathcal{N}):=\{y\in C([-r,T],\mathbb{R}^n): y\in\lambda\mathcal{N}(y),
\hbox{ for some } 0<\lambda<1 \}\,.
$$
Let $y\in \mathcal{E}(\mathcal{N})$. Then there exists $v\in S_{F,y}$ such that
 $y\in\lambda\mathcal{N}(y)$, for some $0<\lambda<1$. Thus,
for each $t\in[0,T]$,
$$
y(t)=\lambda\big(\phi(0)-g(0,\phi)+g(t,y_{t})+
\int_{0}^{t}v(s)\, ds\big).
$$
This implies, by  (H2)--(H4), that for each $t\in J$ we have
$$
|y(t)|\leq \|\phi\|+d_1\|\phi\|+d_1\|y_t\|+2d_{2}+
\int_{0}^{t}p(s)\psi(\|y_{s}\|)ds.
$$
We consider the function
$$
\mu(t)=\sup\{|y(s)|: -r\leq s\leq t \}, \ \ 0\leq t\leq T.
$$
Let $t^{*}\in [-r,t]$ be such that $\mu(t)=|y(t^{*})|$. If $t^{*}\in
J$, by the previous inequality
we have for $t\in J$
$$
\mu(t)\leq \|\phi\|+d_1\|\phi\|+d_1\mu(t)+2d_{2}+
\int_{0}^{t}p(s)\psi(\mu(s))ds.
$$
Thus
$$
\mu(t)\leq\frac{1}{1-d_1}\Big[\|\phi\|+d_1\|\phi\|+2d_{2}
+\int_0^{t}p(s)\psi(\mu(s))ds\Big], \quad t\in J.
$$
If $t^{*}\in[-r,0]$ then $\mu(t)=\|\phi\|$ and the previous inequality
holds.

Let us take the right-hand side of the above inequality to be $v(t)$.
Then
\begin{gather*}
c=v(0)=\frac{1}{1-d_1}(\|\phi\|+d_1\|\phi\|+2d_{2}), \\
\mu(t)\leq v(t), \quad t\in J, \\
v'(t)=\frac{1}{1-d_1}p(t)\psi(\mu(t)), \quad t\in J.
\end{gather*}
Using the nondecreasing character of $\psi$, we obtain
$$
v'(t)\leq \frac{1}{1-d_1}p(t)\psi(v(t)).
$$
This implies that  for each $t\in J$,
$$
\int_{v(0)}^{v(t)}\frac{d\gamma}{\psi(\gamma)}
\leq\frac{1}{1-d_1}\int_{0}^{T}p(s)ds<
\int_{v(0)}^{\infty}\frac{d\gamma}{\psi(\gamma)}.
$$
This inequality implies that there exists a constant $K$ such that
$v(t)\leq K$, $t\in J$, and hence $\mu(t)\leq K$,  $t\in J$.
Since for every $t\in [0,T]$, $\|y_{t}\|\leq\mu(t)$, we have
$$
\|y\|_{\infty}\leq K'=\max\{\|\phi\|,K\},
$$
where $K'$ depends only $T, d_1, d_2$,
and on the functions $p, \phi$ and $\psi$.
This shows that $\mathcal{E}(\mathcal{N})$ is bounded.

Set $X:=C([-r,T],\mathbb{R}^n)$. As a consequence of Lemma \ref{l2} we deduce that
$N$ has a fixed point which is a solution of (\ref{eq1})--(\ref{eq2}).
Denote this solution by $y_1$.
Define the function
$$
r_{k,1}(t)=\tau_{k}(y_1(t))-t\quad \mbox{for } t\geq0.
$$
Hypothesis (H1) implies that
$r_{k,1}(0)\neq 0$ for $k=1,\ldots,m$. If
$r_{k,1}(t)\neq 0$ on $[0,T]$ for $k=1,\ldots,m$;
i.e.,
$$
t\neq \tau_k(y_1(t)) \quad \hbox{on } [0,T]\quad
\hbox{for } k=1,\ldots,m,
$$
then $y_1$ is a solution of the problem (\ref{e1})-(\ref{e3}).

It remains to consider the case when
$r_{1,1}(t)=0$ for some $t\in [0,T]$.
Now since
$r_{1,1}(0)\neq 0$ and $r_{1,1}$ is continuous, there exists
$t_1>0$ such that
$$
r_{1,1}(t_1)=0,  \quad \hbox{and} \quad  r_{1,1}(t)\neq 0 \quad \hbox{for all }
t\in[0,t_1).
$$
Thus by (H1) we have
$r_{k,1}(t)\neq 0$ for all  $t\in[0,t_1)$, and $k=1,\ldots,m$.

\noindent {\bf Step 2:} Consider now the  problem
\begin{gather}\label{eq3}
\frac{d}{dt}[y(t)-g(t,y_t)]\in F(t, y_t), \quad\mbox{a.e. } t\in [t_1,T], \\
\label{eq4}
y(t_1^+)=I_1(y_1(t_1)).
\end{gather}
Transform this problem into a  fixed point problem. Consider the
operator
$N_2: C([t_1,T],\mathbb{R}^n)\to 2^{C([t_1,T],\mathbb{R}^n)}$ defined as
$$
 N_2(y)=\big\{ h\in C([t_1,T],\mathbb{R}^n): h(t)=
I_1(y_1(t_1))-g(t_1,y_{t_1})+g(t,y_t)+\int_{t_{1}}^{t}v(s)ds\big\},
$$
where $v\in S_{F,y}$. As in Step 1 we can show that $N_2$ is
completely continuous, and that the following  set is bounded,
$$
 \mathcal{E}(N_2):=\{y\in C([t_1,T],\mathbb{R}^n):  y\in\lambda N_2(y), \quad\hbox{for some }
 0<\lambda<1 \}\,.
$$
 Set $X:=C([t_1,T],\mathbb{R}^n)$. As a consequence of Martelli's theorem, we
deduce that $N_2$ has a fixed point $y$ which is a solution to
problem (\ref{eq3})--(\ref{eq4}). Denote this solution by $y_2$.
Define
$$ r_{k,2}(t)=\tau_{k}(y_2(t))-t\quad \hbox{for}\ t\geq
t_{1}. $$
If $ r_{k,2}(t)\neq 0$ on $(t_1,T]$ for all $k=1,\ldots,m$,
 then
$$
y(t)=\begin{cases} y_{1}(t), & \mbox{if } t\in [0,t_{1}],\\
y_{2}(t), &\mbox{if }t\in (t_{1},T],
\end{cases}
$$
is a solution of the problem (\ref{e1})--(\ref{e3}).
It remains to consider the case when
$r_{2,2}(t)=0$, for some $t\in (t_{1},T]$.
By (H8), we have
\begin{align*}
r_{2,2}(t_1^+)&= \tau_{2}(y_{2}(t_{1}^{+}))-t_{1} \\
&= \tau_{2}(I_{1}(y_{1}(t_{1})))-t_{1} \\
&> \tau_{1}(y_{1}(t_{1}))-t_{1} \\
&= r_{1,1}(t_{1})=0.
\end{align*}
Since $r_{2,2}$ is continuous, there exists
$t_2>t_1$ such that $r_{2,2}(t_{2})=0$
and $r_{2,2}(t)\neq 0$ for all $ t\in(t_1,t_2)$.
It is clear by (H1) that
$$
r_{k,2}(t)\neq 0 \quad \hbox{for all } t\in(t_1,t_2), \; k=2,\ldots,m.
$$
Suppose now that there is $\bar s\in (t_1,t_2]$ such
that $r_{1,2}(\bar s)=0$.
Consider the function $L_1(t)=\tau_1(y_2(t)-g(t,y_t))-t$.
>From (H6)-(H8) it follows that
\begin{align*}
L_{1}(\bar s)&= \tau_{1}(y_2(\bar s)-g(\bar s,y_{\bar s}))-\bar s\\
&\geq \tau_{1}(y_{2}(\bar s))-\bar s  \\
&= r_{1,2}(\bar s)=0.
\end{align*}
Thus the function $L_{1}$ attains a nonnegative maximum at some point
$s_1\in(t_1,T]$. Since
$$
\frac{d}{dt}[y_2(t)-g(t,y_{2t})]\in F(t,y_{2t}), \quad\mbox{a.e. } t\in (t_{1},T),
$$
then there exists $v(\cdot)\in L^1((t_{1},T))$ with $v(t)\in F(t,y_{2t})$,
 a.e. $t\in (t_{1},T)$ such that
$$
\frac{d}{dt}[y_2(t)-g(t,y_{2t})]=v(t),
$$
then
$$
L'_{1}(s_1)=\tau_1'(y_2(s_1)-g(s_1,y_{2s_1}))
\frac{d}{dt}[y_2(s_1)-g(s_1,y_{s_1})]-1=0.
$$
Therefore,
$$\langle\tau'_1(y_2(s_1)-g(s_1,y_{2s_1})), v(s_1)\rangle=1,
$$
which is a contradiction by (H4).

\noindent{\bf Step 3:} We continue this process and take into account that
$y_{m}:=y\Bigr|_{[t_{m},T]}$
is a solution to the problem
\begin{gather}\label{eq5}
\frac{d}{dt}[y(t)-g(t,y_{t})]\in F(t,y_{t}), \quad\mbox{a.e. } t\in (t_{m},T),\\
\label{eq13}
y(t^+_{m})=I_{m}(y_{m-1}(t_{m})).
\end{gather}
The solution $y$ of the problem  (\ref{e1})-(\ref{e3})
is then defined by
$$
y(t)=\begin{cases}
y_{1}(t), & \mbox{if } t\in [-r,t_{1}],\\
y_{2}(t), & \mbox{if } t\in (t_{1},t_{2}], \\
\dots \\
y_{m}(t), & \mbox{if } t\in (t_{m},T].
\end{cases}
$$
%\end{proof}

\section{Second Order Impulsive NFDIs}

In this section, we study the initial-value problem (\ref{e4})--(\ref{e7}).

\begin{definition}\label{d2} \rm
 A function  $y\in \Omega\cap\cup_{k=0}^{m}AC((t_{k},t_{k+1}),\mathbb{R}^{n})$
is said to be a
solution of (\ref{e4})--(\ref{e7}) if there exists
$v(t)\in F(t,y_t)$  a.e.   $t\in[0,T]$ such that
$\frac{d}{dt}[y'(t)-g(t,y_{t})]=v(t)$
a.e. on  $J,\ t\neq \tau_k(y(t))$, $k=1,\ldots,m$, $y(t^+)=I_{k}(y(t))$,
$t=\tau_k(y(t))$, $k=1,\ldots,m$, $y'(t^{+})=\overline I_{k}(y(t))$,
$t=\tau_k(y(t))$ $k=1,\ldots,m$,
$ y(t)=\phi(t)$, $t\in[-r,0]$ and $y'(0)=\eta$.
\end{definition}

For the next theorem we need the following assumptions:
\begin{itemize}
\item[(A1)] There exist positive constants $\overline d_{k}$ such that
$|\overline I_{k}(x)|\leq\overline d_{k}$, $k=1,\ldots,m$ for each $x\in\mathbb{R}^n$

\item[(A2)] There exists  a continuous nondecreasing function
$\psi:[0,\infty)\to (0,\infty)$
and $p\in L^{1}([0,T],\mathbb{R}_{+})$
such that
$\|F(t,u)\|\leq p(t)\psi (\|u\|)$ a.e. $t\in [0,T]$ and each $u\in D$
with
$$
\int_{1}^{\infty}\frac{d\gamma}{\gamma+\psi(\gamma)}=\infty,
$$
\item[(A3)]  For
all
$(t,\bar s, x)\in [0,T]\times[0,T]\times \mathbb{R}^n$ and all $y_t\in D$ we have,
for all $v\in S_{F,y}$,
$$
\langle \tau'_k(x),\overline I_k(y(\bar s))-g(\bar s,y_{\bar s})+g(t,y_t)
+\int_{\bar s}^{t} v(s)ds\rangle \neq 1\quad
\hbox{for } k=1,\ldots,m\,.
$$

\item[(A4)] For all $x\in \mathbb{R}^n$,
$\tau_k(I_k(x))\leq \tau_k(x)<\tau_{k+1}(I_k(x))$ for $k=1,\ldots,m$.
\end{itemize}

\begin{theorem} \label{thm4.2}
 Assume that  (H1)-(H3)  and (A1)-(A4) are satisfied.
Then the IVP (\ref{e4})--(\ref{e7}) has at least one solution.
\end{theorem}

The proof of this theorem will be given in several steps.

\noindent{\bf Step 1:} Consider the  problem
\begin{gather}\label{eq7}
\frac{d}{dt}[y'(t)-g(t,y_t)]\in F(t, y_t), \quad\mbox{a.e. } t\in [0,T],\\
\label{eq8}
y(t)=\phi(t),\quad  t\in[-r,0], \; y'(0)=\eta\,.
\end{gather}
Transform the problem into a  fixed point problem. Consider the operator
$\overline N_1: C([-r,T],\mathbb{R}^n)\to 2^{C([-r,T],\mathbb{R}^n)}$
defined as
$\overline N_1(y)=\{h\in C([-r,T],\mathbb{R}^{n})\}$ where
$$
h(t)=\begin{cases}
\phi(t),& \mbox{if } t\in [-r,0]; \\
\phi(0)+[\eta-g(0,\phi(0))]t
+\int^{t}_{0}g(s,y_s)ds
+\int_{0}^{t}(t-s)v(s)ds& \mbox{if } t\in [0,T].
\end{cases}
$$
As in Theorem \ref{t1}, we can show that $\overline N_1$
is completely continuous.
Now we prove only that the following set is bounded,
$$
\mathcal{E}(\overline N_1):=\{y\in C([-r,T],\mathbb{R}^n):
y\in\lambda \overline N_1(y), \hbox{ for some } 0<\lambda<1 \}\,.
$$
Let $y\in \mathcal{E}(\overline N_1)$. Then there exists $v\in
S_{F,y}$ such that $y\in\lambda \overline N_1(y)$ for some
$0<\lambda<1$. Thus for each $t\in[0,T]$ we have
$$
y(t)=\lambda\phi(0)+\lambda[\eta-g(0,\phi(0))]t +\lambda\int^{t}_{0}g(s,y_s)ds
+\lambda\int_{0}^{t}(t-s)v(s)ds.
$$
This implies, by (H2), (H3), (A1), and (A2), that for each $t\in [0,T]$ we have
\begin{align*}
|y(t)|
&\leq \|\phi\|+T(|\eta|+\|\phi\|d_1+d_2)+\int^{t}_{0}\!
d_1{\|y_s\|}ds+Td_{2} +\int_{0}^{t}\! (T-s)p(s)\psi(\|y_s\|)ds \\
&\leq \|\phi\|+T(|\eta|+\|\phi\|d_1+2d_2)+\int^{t}_{0}M(s){\|y_s\|}ds
+\int_{0}^{t}M(s)\psi(\|y_s\|)ds,
\end{align*}
where $M(t)=\max\{d_{1}, (T-t)p(t)\}$.
Consider the function
$$
\mu(t)=\sup\{|y(s)|:-r\leq s\leq t\}, \quad 0\leq t\leq T.
$$
Let $t^*\in [-r,t]$ be such that $\mu(t)=|y(t^*)|$.
If $t^*\in J$, by the previous inequality,  for $t\in[0,T]$,  we have
 $$
\mu(t) \leq \|\phi\| +T(|\eta|+\|\phi\|d_1+2d_2)
+\int^{t}_{0}M(s){\mu(s)}ds +\int^t_0M(s)\psi(\mu(s))ds.
 $$
Let us denote the right-hand side of the above inequality to be  $v(t)$.
Then
\begin{gather*}
v(0)=\|\phi\|+T(|\eta|+\|\phi\|d_1+2d_2), \\
v'(t)=M(t)\mu(t)+M(t)\psi(\mu(t)),\ t\in [0,T].
\end{gather*}
 Using the nondecreasing character of $\psi$ we obtain, for a.e. $t\in [0,T]$,
$$
v'(t)\leq M(t)v(t)+M(t)\psi(v(t))
=M(t)[v(t)+\psi(v(t))].
$$
This implies  that, for each $t\in[0,T]$,
$$
\int_{v(0)}^{v(t)}\frac{d\gamma}{\gamma+\psi(\gamma)}\leq\int_{0}^{T}M(s)ds<
\int_{v(0)}^{\infty}\frac{d\gamma}{\gamma+\psi(\gamma)}.
$$
This inequality implies that there exists a constant $b^{*}$ such
that $v(t)\leq b^{*}, \ t\in [0,T]$, and hence $\mu(t)\leq b^{*},
\ t\in[0,T] $. Since for every $t\in [0,T], \|y_{t}\|\leq\mu(t)$,
we have
$$
\|y\|_{\infty}\leq \max\{\|\phi\|,b^{*}\},
$$
where $b^{*}$ depends only $T$ and on the functions $p$ and $\psi$. This
shows that $\mathcal{E}( \overline N_1)$ is bounded.

 Set $X:=C([-r,T],\mathbb{R}^n)$. As a consequence of Martelli's theorem, we
deduce that $\overline N_{1}$ has a fixed point $y$ which is a
solution to (\ref{eq7})--(\ref{eq8}). Denote this solution
by $y_1$. Define the function
$$
r_{k,1}(t)=\tau_{k}(y_1(t))-t\quad \hbox{for}\quad  t\geq0.
$$
Hypothesis (H1) implies that $ r_{k,1}(0)\neq 0$ for $k=1,\ldots,m$.
If
$r_{k,1}(t)\neq 0$ on $[0,T]$ for $k=1,\ldots,m$;  i.e.,
$$
 t\neq \tau_k(y_1(t)) \quad \hbox{on }
[0,T] \quad \mbox{and for } k=1,\ldots,m\,.
$$
Then $y_1$ is a solution of the problem (\ref{e1})-(\ref{e3}).

It remains to consider the case when $r_{1,1}(t)=0$ for some $t\in [0,T]$.
Now since $r_{1,1}(0)\neq 0$
and $r_{1,1}$ is continuous, there exists
$t_1>0$ such that
$$
r_{1,1}(t_1)=0,  \quad \mbox{and} \quad  r_{1,1}(t)\neq 0 \quad  \mbox{for all }
t\in[0,t_1).
$$
Thus,  by (H1) we have
$r_{k,1}(t)\neq 0$ for all $t\in[0,t_1)$ and $k=1,\ldots,m$.

{\bf Step 2:} Consider now the  problem
\begin{gather}\label{eq9}
\frac{d}{dt}[y'(t)-g(t,y_t)]\in F(t, y_t), \quad\mbox{a.e. } t\in [t_1,T],\\
\label{eq10}
y(t_1^+)=I_1(y_1(t_1)), \\
\label{eq11}
y'(t_1^+)=\overline I_1(y_1(t_1)).
\end{gather}
Transform the problem into a  fixed point problem. Consider the operator
$\overline N_2: C([t_1,T],\mathbb{R}^n)\to 2^{C([t_1,T],\mathbb{R}^n)}$
defined as
$\overline N_2(y)=\{ h\in C([t_1,T],\mathbb{R}^n)\}$ where
$$
h(t)=I_1(y_1(t_1))+(t-t_1)\overline
I_1(y_1(t_1))-(t-t_1)g(t_1,y_{t_1}) +\int^t_{t_1}\! g(s,y_s)ds
+\int_{t_{1}}^{t}(t-s)v(s)ds,
$$
with $v\in S_{F,y}$. As in Step 1 we can show that
 $\overline N_2$ is  completely continuous,
and that the following set is bounded,
$$
\mathcal{E}(\overline N_2):=\{y\in C([t_1,T],\mathbb{R}^n):
y\in\lambda\overline N_2(y), \hbox{ for some } 0<\lambda<1 \}.
$$
Set $X:=C([t_1,T],\mathbb{R}^n)$. As a consequence of Martelli's
theorem, we deduce that $N_2$ has a fixed point $y$ which is a
solution to problem (\ref{eq9})--(\ref{eq11}). Denote this
solution by $y_2$. Define
$$
r_{k,2}(t)=\tau_{k}(y_2(t))-t\quad \hbox{for}\ t\geq t_{1}.
$$
If
$r_{k,2}(t)\neq 0$ on $(t_1,T]$ for all $k=1,\ldots,m$,
then
$$
y(t)=\begin{cases}
y_{1}(t), & \mbox{if } t\in [0,t_{1}],\\
y_{2}(t), & \mbox{if } t\in (t_{1},T],
\end{cases}
$$
is a solution of the problem (\ref{e4})--(\ref{e7}).
It remains to consider the case when
$r_{2,2}(t)=0$,  for some $t\in (t_{1},T]$.
By (A4) we have
\begin{align*}
r_{2,2}(t_1^+)&= \tau_{2}(y_{2}(t_{1}^{+}))-t_{1} \\
&= \tau_{2}(I_{1}(y_{1}(t_{1}))-t_{1} \\
&> \tau_{1}(y_{1}(t_{1}))-t_{1} \\
&= r_{1,1}(t_{1})=0.
\end{align*}
Since $r_{2,2}$ is continuous, there exists
$t_2>t_1$ such that $r_{2,2}(t_{2})=0$, and
$r_{2,2}(t)\neq 0$ for all $t\in(t_1,t_2)$.
It is clear by (H1) that
$$
r_{k,2}(t)\neq 0 \quad \hbox{for all }
t\in(t_1,t_2), \quad k=2,\ldots,m.
$$
Suppose now that there is $\bar s\in (t_1,t_2]$ such that
$r_{1,2}(\bar s)=0$.
 From (A4), it follows that
\begin{align*}
r_{1,2}(t_1^+)&= \tau_{1}(y_{2}(t_{1}^{+}))-t_{1} \\
&= \tau_{1}(I_{1}(y_{1}(t_{1})))-t_{1} \\
&\leq \tau_{1}(y_{1}(t_{1}))-t_{1} \\
&= r_{1,1}(t_{1})=0.
\end{align*}
Thus, the function $r_{1,2}$ attains a nonnegative maximum at some point
$s_1\in(t_1,T]$. Since
$$
\frac{d}{dt}[y'_2(t)-g(t, y_{2t})]\in F(t,y_{2t}), \quad\mbox{a.e. }
 t\in (t_{1},T),
$$
there exist $v(\cdot)\in L^1((t_{1},T))$ with $v(t)\in F(t,y_{2t})$,
 a.e. $t\in (t_{1},T)$ such that
$$
y'_2(t)=\overline I_1(y(t_1))-g(t_1,y_{t_1})+g(t,y_{t})
+\int_{t_1}^{t} v(s)ds\,.
$$
Then
$$
r'_{1,2}(s_1)=\tau_1'(y_2(s_1))
\Big(\overline I_1(y(t_1))-g(t_{1},y_{t_{1}})+g(s_1,y_{s_1})
+\int_{t_1}^{s_{1}} v(s)ds\Big)-1=0.
$$
Therefore,
$$
\langle\tau'_1(y_2(s_1)), \overline I_1(y(t_1))-g(t_{1},y_{t_{1}})
+g(s_1,y_{s_1})
+\int_{t_1}^{s_{1}} v(s)ds\rangle=1,
$$
which is a contradiction by (A3).

\noindent{\bf Step 3:} We continue this process by taking into account that
$y_{m}:=y\bigr|_{[t_{m},T]}$
is a solution to the problem
\begin{gather}\label{eq12}
\frac{d}{dt}[y'(t)-g(t,y_{t})]\in F(t,y_{t}), \ a.e.\ t\in (t_{m},T),\\
\label{eq14}
y(t^+_{m})=I_{m}(y_{m-1}(t_{m})), \\
\label{eq15}
y'(t^+_{m})=\overline I_{m}(y_{m-1}(t_{m})).
\end{gather}
The solution $y$ of the problem  (\ref{e4})-(\ref{e7})
is then defined by
$$
y(t)=\begin{cases}
y_{1}(t), &\mbox{if } t\in [-r,t_{1}],\\
y_{2}(t), &\mbox{if } t\in (t_{1},t_{2}], \\
\dots \\
y_{m}(t), & \mbox{if } t\in (t_{m},T]\,.
\end{cases}
$$
%\end{proof}

\begin{thebibliography}{00}

\bibitem{BaGo} J. Banas and K. Goebel,  \textit{Measures of Noncompactnes in Banach
Spaces}, Marcel-Dekker, New York, 1980.

\bibitem{BaLi} I. Bajo and E. Liz,
\textit{Periodic  boundary value problem
for first order differential equations with impulses at variable times},
J. Math. Anal. Appl. {\bf 204} (1996), 65-73.

\bibitem{BaSi} D. D. Bainov and P.S. Simeonov, \textit{Systems with Impulse Effect},
Ellis Horwood Ltd., Chichister, 1989.

\bibitem{BeHeNt1} M. Benchohra, J. Henderson and S. K. Ntouyas, \textit{An existence
result for first order impulsive functional differential equations in Banach
spaces}, Comput. Math. Appl. {\bf 42} (10\&11) (2001), 1303-1310.

\bibitem{BeHeNt2} M. Benchohra, J. Henderson and S. K. Ntouyas, \textit{Impulsive
neutral functional differential equations in Banach spaces},  Appl. Anal.
{\bf 80} (3) (2001), 353-365.

\bibitem{BeHeNt3} M. Benchohra, J. Henderson and S. K. Ntouyas, \textit{Impulsive
neutral functional differential inclusions in Banach spaces},
Appl. Math. Lett. {\bf 15}  (2002), 917-924.

\bibitem{BeHeNt4} M. Benchohra, J. Henderson and S. K. Ntouyas, \textit{Multivalued
impulsive  neutral functional differential inclusions in Banach spaces},
Tamkang J. Math. {\bf 33} (1) (2002), 77-38.

\bibitem{Dei} K. Deimling,  \textit{Multivalued Differential Equations}, Walter De
Gruyter, Berlin-New York, 1992.

\bibitem{Don} Y. Dong, \textit{Periodic boundary value problems for functional
differential equations with impulses},  J. Math. Anal. Appl.  {\bf 210}
(1997), 170-181.

\bibitem{FrOr} M. Frigon and D. O'Regan,
\textit{Impulsive differential equations with variable times},  Nonlinear Anal.
{\bf 26} (1996), 1913-1922.

\bibitem{FrOr1} M. Frigon and D. O'Regan, \textit{First order impulsive initial and
periodic problems with variable moments},  J. Math. Anal. Appl.
{\bf 233} (1999), 730-739.

\bibitem{FrOr2} M. Frigon and D. O'Regan, \textit{Second order Sturm-Liouville
BVP's with impulses at variable moments},   Dynam. Contin. Discrete Impuls.
Systems {\bf8 } (2), (2001),  149-159.

\bibitem{Gor} L. Gorniewicz, \textit{Topological Fixed Point
Theory of Multivalued Mappings, Mathematics and its Applications}, 495,
Kluwer Academic Publishers, Dordrecht, 1999.

\bibitem{HuPa} Sh. Hu and N. Papageorgiou,  \textit{Handbook of Multivalued
Analysis, Volume I: Theory}, Kluwer, Dordrecht, Boston, London, 1997.

\bibitem{KaLaLe} S. K. Kaul, V. Lakshmikantham and S. Leela, \textit{Extremal
solutions, comparison principle and stability criteria for impulsive
differential equations with variable times}, Nonlinear Anal. {\bf 22}
(1994), 1263-1270.

\bibitem{KaLi} S. K. Kaul and X. Z. Liu, \textit{Vector
Lyapunov functions for impulsive
differential systems with variable times},
 Dynam. Contin. Discrete Impuls. Systems {\bf 6} (1999), 25-38.

\bibitem{KaLi1} S. K. Kaul and X. Z. Liu, \textit{Impulsive
integro-differential equations with variable times},
 Nonlinear Studies, {\bf 8} (2001), 21-32.

\bibitem{LaBaSi} V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, \textit{Theory
of Impulsive Differential Equations}, World Scientific, Singapore, 1989.

\bibitem{LaLeKa}  V. Lakshmikantham, S. Leela and S.K. Kaul,
\textit{Comparison principle for impulsive differential equations with variable
times and stability theory},  Nonlinear Anal. {\bf 22}
(1994), 499-503.

\bibitem{LaPaVa}  V. Lakshmikantham, N.S. Papageorgiou, and J. Vasundhara,
\textit{The method of upper and lower solutions and monotone technique for
impulsive differential equations with variable
moments},  Appl. Anal. {\bf 15} (1993), 41-58.

\bibitem{LaOp} A. Lasota and Z. Opial, \textit{An application of the Kakutani-Ky
Fan theorem in the theory
of ordinary differential equations},   Bull. Acad. Pol. Sci. Ser. Sci. Math.
Astronom. Phys. {\bf 13} (1965), 781-786.

\bibitem{LB} X. Liu and G. Ballinger, \textit{Existence and continuability of
solutions for differential equations with delays and state-dependent
impulses},  Nonlinear Anal. {\bf 51} (2002), 633-647.

\bibitem{Mar}  M. Martelli,  \textit{A Rothe's type theorem for non-compact
acyclic-valued map},   Boll. Un. Mat. Ital. {\bf 4} (3), (1975), 70-76.

\bibitem{SaPe} A. M. Samoilenko and N. A. Perestyuk, \textit{Impulsive Differential
Equations, World Scientific}, Singapore, 1995.

\bibitem{Tol} A. A. Tolstonogov, \textit{Differential Inclusions in a Banach Space},
Kluwer Academic Publishers, Dordrecht, 2000.

\end{thebibliography}

\end{document}