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\markboth{M. Benchohra, M. Guedda and M. Kirane}{Impusive semilinear
functional differential equations}
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\begin{document}
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\begin{center}
{\bf IMPULSIVE SEMILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS}
\vskip 1cm
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{\bf M. Benchohra$^{1}$, M. Guedda$^{2}$ and  M. Kirane$^{3}$  }\\
\vskip 0.3cm
$^{1}$ Department of Mathematics, University of Sidi Bel Abbes  \\
BP 89 2000 Sidi Bel Abbes,  Algeria \\
e-mail: benchohra@yahoo.com \\
\vskip 0.2cm
$^{2}$ Universit\'e de Picardie Jules Verne  \\
LAMFA UPRES A 6119, 80 039 Amiens, France \\
e-mail: Mohamed.guedda@u-picardie.fr \\
and\\
\vskip 0.2cm
$^{3}$ Laboratoire de Math\'{e}matiques, P\^{o}le Sciences et Technologies\\
Universit\'e de La Rochelle  \\
Avenue M. Cr\'{e}peau, 17042 La Rochelle Cedex 1, France \\
e-mail: Mokhtar.Kirane@univ-lr.fr \\
\vskip 0.2cm
\end{center}
\begin{abstract}
In this paper the Leray-Schauder nonlinear alternative combined with  the
semigroup theory is used to investigate the existence of
 mild solutions for
first order impulsive semilinear functional differential equations in Banach
spaces.
\end{abstract}


\noindent
{\bf Key words and phrases:} Initial value problem, impulsive functional
differential equations fixed point, nonlinear alternative, mild solution.\\
\noindent
{\bf AMS (MOS) Subject Classifications}: 34A37, 34G20, 35R10 \\
\ \\

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\setcounter{theorem}{0}
\setcounter{remark}{0}
\begin{center}
{\bf 1. \ INTRODUCTION}
\end{center}

This paper is concerned with the existence of mild solutions for the impulsive
semilinear functional differential equation of the form:
\begin{equation}\label{e1}
y'-A(t)y=f(t,y_{t}), \ \ t\in J=[0,b], \ t\not=t_{k}, \ \ k=1,\ldots,m,
\end{equation}
\begin{equation}\label{e2}
y(t_{k}^{+})=I_{k}(y(t_{k}^{-})), \ \ k=1,\ldots,m,
\end{equation}
\begin{equation}\label{e3}
y(t)=\phi(t), \ t\in [-r,0]
\end{equation}
where \ $f:J\times C([-r,0],E)\longrightarrow E \ $ is a
given map, $ \phi\in C([-r,0],E), \  A(t), t\in J$ a linear closed
operator from  a dense subspace $D(A(t))$ of $E$ into $E$ and $E$ a real
Banach space with the norm $|\cdot|, \
0=t_{0}<t_{1}<\ldots
<t_{m}<t_{m+1}=b,
\ I_{k}\in C(E,E) \ (k=1,2,\ldots,m),$ $y(t_{k}^{-})$ and $y(t_{k}^{+})$
represent the left and right limits of $y(t)$ at $t=t_{k}$, respectively.

For any continuous function $y$ defined on  $[-r,b]-\{t_{1},\ldots,t_{m}\}$
and
any $t\in J$, we denote by $y_{t}$ the element of $C([-r,0], E)$ defined by
$$
y_{t}(\theta)=y(t+\theta), \ \ \theta\in [-r,0].$$
Here $y_{t}(.)$ represents the history of the state from time $t-r$, up to
the present time $t$.

Many evolution processes and phenomena experience changes of state abrupt
through short-term perturbations. Since the durations of the perturbations
are negligible in comparison with the duration of each process, it is quite
natural to assume that these perturbations act in terms of impulses. See the
monographs of Lakshmikantham et al. \cite{LaBaSi}, and
Samoilenko and Perestyuk \cite{SaPe} and the papers of Erbe et al.
\cite{ErFrLiWu}, Kirane and  Rogovchenko \cite{KiRo},
Liu et al. \cite{LiSiZh} and Liu and Zhang \cite{LiZh} various properties of
their solutions are studied and a large bibliography is proposed. For the
existence of Caratheodory solutions to the problem
(\ref{e1})-(\ref{e3}) with $A\equiv 0$ we refer the reader to the paper of
Benchohra et al. \cite{BeHeNt1}.

This paper will be divided into three sections. In Section 2 we will recall
briefly some basic definitions and preliminary results which
will be used throughout Section 3. In Section 3 we establish an existence
theorem for (\ref{e1})-(\ref{e3}). Our approach  is based on the nonlinear alternative
of Leray-Schauder type \cite{DuGr} combined with the semigroup
theory \cite{Paz}. In our results we do not assume any type of monotonicity
 condition
on $I_{k}, k=1,\ldots,m $ which is usually the situation in the literature,
see for instance, \cite{ErFrLiWu}, \cite{KiRo} and \cite{LiSiZh}. This paper
extends to the semilinear and the functional cases some results obtained
by Frigon and O'Regan in \cite{FrOr1}.

\setcounter{section}{2}
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\setcounter{remark}{0}
\begin{center}
{\bf 2. \ PRELIMINARIES}
\end{center}
We will briefly recall some basic definitions and preliminary results that
we will use in the sequel. Let $E$ be a real Banach.
$B(E)$ denotes the Banach space of bounded linear
operators from $E$ into $E$.

A measurable function $y:J\longrightarrow E$ is Bochner integrable
if and only if $|y|$ is Lebesgue integrable. For properties of the Bochner
integral, we refer to Yosida \cite{Yos}.

$L^{1}(J,E)$ denotes the Banach space of  functions
$y:J\longrightarrow E$ which are
Bochner integrable normed by
$$
\|y\|_{L^{1}}=\Int_{0}^{b}|y(t)|dt \quad \hbox{for all} \quad y\in
L^{1}(J,E).$$

Let $X$ be a Banach space. A map $G:X\to X$ is said to be completely
continuous if it is continuous and $G(B)$ is relatively compact for
every bounded subset $B\subseteq X$.
\begin{definition} The map $f:J\times C([-r,0],E)\longrightarrow E$ is
said to be an  $L^{1}$-Caratheodory  if
\begin{itemize}
\item[(i)] $t\longmapsto f(t,u)$ is  measurable for each $u\in C([-r,0],E);$
\item[(ii)] $u\longmapsto f(t,u)$ is continuous for almost all
$t\in J;$
\item[(iii)] For each $q>0,$  there exists $h_{q} \in L^{1}(J,\R_{+})$
such that $|f(t,u)|\leq h_{q}(t)$ for all
$\|u\|\leq q$ and for almost all $t\in J$.
\end{itemize}
\end{definition}

In order to define the mild solution of (\ref{e1})-(\ref{e3}) we shall
consider the following space
$$
\matrix{\Omega=\{y&:&[-r,b]\longrightarrow E: y_{k}\in
C(J_{k},E), k=0,\ldots,m \
\hbox{and there exist} \hfill\cr
&&\cr
&& \ y(t^{-}_{k}) \ \hbox{and} \ y(t^{+}_{k}), k=1,\ldots,m \ \hbox{with} \
y(t^{-}_{k})=y(t_{k})\}\hfill\cr}$$ which is a Banach space with the norm
$$
\|y\|_{\Omega}=\Max\{\|y_{k}\|_{\infty}, k=0,\ldots,m \},$$
where $y_{k}$ is the restriction of $y$ to $J_{k}=(t_{k},t_{k+1}], \
k=0,\ldots,m.$

Before we give the definition of a mild solution of problem
(\ref{e1})-(\ref{e3}), we make the following assumption on $A(t)$ in (1):\\

(H1)$\; \; A(.): t\longrightarrow A(t), \ t\in J$ is continuous such that
$$
A(t)y=\Lim_{h\rightarrow 0^{+}}\Frac{T(t+h,t)y-y}{h}, \quad y\in D(A(t)),$$
where $T(t,s)\in B(E)$ for each
$(t,s)\in\gamma:=\{(t,s); 0\leq s\leq t<b\},$ satisfying \par
(i) $T(t,t)=I \, (I$ is the identity operator in $E$),\par
(ii) $T(t,s)T(s,r)=T(t,r)$ for $0\leq r\leq s\leq t<b,$ \par
(iii) the mapping $(t,s)\longmapsto T(t,s)y$ is strongly continuous in
$\gamma$ for each $y\in E$; moreover there exists a positive constant M such that 
for any $(t,s) \in \gamma, \; \vert \vert T(t, s)\vert \vert \leq M.$
\vskip 0.3cm
\begin{definition} A function $y\in \Omega$ is said to be a mild
solution of (\ref{e1})-(\ref{e3})  (see \cite{LiSiZh}) if
$$
y(t)=\left\{\matrix{\phi(t), \ \ \ t\in [-r,0], \hfill\cr
&&\cr
T(t,0)\phi(0)+\Int_{0}^{t}T(t,s)f(s,y_{s})ds, \ \ \ t\in [0,t_{1}],
\hfill\cr
&&\cr
I_{k}(y(t_{k}^{-}))+\Int_{t_{k}}^{t}T(t,s)f(s,y_{s})ds, \ \ \ t\in
J_{k}, \ k=1,\ldots,m.\hfill\cr}\right.$$
\end{definition}
Our main result is based on the following:
\begin{lemma} (Nonlinear Alternative \cite{DuGr}). Let $X$ be a
Banach space with $C\subset X$ convex. Assume $U$ is a relatively open
subset of $C$ with $0\in U$ and
$G:\overline U\longrightarrow C$ is  a compact map. Then either,
\begin{itemize}
\item[{\bf (i)}] \ $G$ has a fixed point in $\overline U$; or
\item[{\bf (ii)}] \ there is a point $u\in\partial U$ and $\lambda\in (0,1)$
with $u=\lambda G(u)$.
\end{itemize}
\end{lemma}
\noindent
\begin{remark} By $\overline U$ and $\partial U$ we denote the closure
of $U$ and the boundary of $U$ respectively.
\end{remark}
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\begin{center}
{\bf 3. \ MAIN RESULT}
\end{center}
We are now in a position to state and prove our existence result  for
the IVP (\ref{e1})-(\ref{e3})
\begin{theorem}
 Let $t_{0}=0, \ t_{m+1}=b.$ Suppose:
\vskip 0.3cm
\begin{itemize}
\item[(H2)] \quad $f:J\times C([-r,0],E)\longrightarrow E$ is an
$L^{1}$-Caratheodory  map;
\vskip 0.3cm
\item[(H3)]  there exist  a continuous nondecreasing function
$\psi:[0,\infty)\longrightarrow (0,\infty)$
and $p\in L^{1}(J,\R_{+})$
such that
$$
|f(t,u)|\leq p(t)\psi (\|u\|) \ \hbox{for a.e.} \  t\in J \ \hbox{and each} \
u\in C([-r,0],E)$$
with
$$
\Int_{t_{k-1}}^{t_{k}}p(s)ds<\Int_{N_{k-1}}^{\infty}\Frac{d \tau}{\psi(\tau)},
\quad k=1,\ldots,m+1;$$
Here \ $N_{0}=M\|\phi\|$ \ and for \ $k=2,\ldots,m+1$ \ we have
$$
N_{k-1}= \Sup_{y\in [-M_{k-2},M_{k-2}]}|I_{k-1}(y)|,
\ \  M_{k-2}=\Gamma_{k-1}^{-1}\Bigl(M\Int_{t_{k-2}}^{t_{k-1}}p(s)ds\Bigr)$$
with
$$
\Gamma_{l}(z)=\Int_{N_{l-1}}^{z}\Frac{d \tau}{\psi(\tau)}, \quad z\geq
N_{l-1}, \ \
l\in\{1,\ldots,m+1\}.$$
\vskip 0.3cm
\item[(H4)] \quad For each bounded set $B\subset C([-r,b],E)$ and for each
$t\in J$ the set
$$
\Bigl\{I_{k}(y(t_{k}^{-}))+\Int_{t_{k}}^{t}T(t,s)f(s,y_{s})ds: y\in B\Bigr\}$$
is relatively compact in $E$, $k=0,\ldots, m$.
\end{itemize}
\vskip 0.3cm
Then  the problem (\ref{e1})-(\ref{e3}) has at least one mild solution $y\in\Omega.$
\end{theorem}
\begin{remark} (i) (H3) is satisfied  if for example
$\psi(u)=u+1,$ \ for $u>0$.  \\
(ii) If $T(t,s), (t,s)\in\gamma$ is compact, then (H4) is satisfied.   \\
(iii) The condition $(H4)$ is also satisfied if the function $f$ maps bounded sets
into relatively compact sets in $C(J,E)$. \\
(iv) If the Banach space $E$ is finite dimentional, then the condition (H4) is
automatically satisfied.

\end{remark}


{\bf Proof of the theorem}. The proof is given  in several steps. \\
{\bf Step 1.} Consider the problem (\ref{e1})-(\ref{e3}) on
$[-r,t_{1}]$
\begin{equation}\label{e4}
y'-A(t)y=f(t,y_{t}),\ \ \hbox {a.e.} \ \ t\in J_{0},
\end{equation}
\begin{equation}\label{e5}
y(t)=\phi(t), \ t\in [-r,0].
\end{equation}
We shall show that the possible mild solutions of (\ref{e4})-(\ref{e5}) are
{\em a priori} bounded, that is
there exists a constant $b_{0}$ such that, if $y\in \Omega$ is a mild
solution of (\ref{e4})-(\ref{e5}), then
$$
\Sup\{|y(t)|:  t\in [-r,0]\cup J_{0}\}\leq b_{0}.$$
Let $y$ be a (possible) solution to (\ref{e4})-(\ref{e5}).  Then  for each
$t\in [0,t_{1}]$
$$
y(t)-T(t,0)\phi(0)=\Int_{0}^{t}T(t,s)f(s,y_{s})ds.$$
>From (H3) we get
$$
|y(t)|\leq M\|\phi\|+M\Int_{0}^{t}p(s)\psi(\|y_{s}\|)ds, \ t\in [0,t_{1}].$$
We consider the function $\mu_{0}$ defined by
$$
\mu_{0}(t)=\Sup\{|y(s)|: -r\leq s\leq t \}, \ \ 0\leq t\leq t_{1}.$$
Let $t^{*}\in [-r,t]$ be such that $\mu_{0}(t)=|y(t^{*})|$. If $t^{*}\in
[0,t_{1}]$, by the previous inequality
we have for $t\in [0,t_{1}]$
$$
\mu_{0}(t)\leq M\|\phi\|+M\Int_{0}^{t}p(s)\psi(\mu_{0}(s))ds. $$
If $t^{*}\in [-r,0]$ then $\mu_{0}(t)=\|\phi\|$ and the previous
inequality holds since $M\geq 1$.

Let us take the right-hand side of the above inequality as $v_{0}(t),$ then we
have
$$
v_{0}(0)=M\|\phi\|=N_{0}, \quad \mu_{0}(t)\leq v_{0}(t), \quad t\in [0,t_{1}]$$
and
$$v_{0}'(t)=Mp(t)\psi(\mu_{0}(t)), \quad t\in [0,t_{1}].$$
Using the nondecreasing character of $\psi$ we get
$$
v_{0}'(t)\leq Mp(t)\psi(v_{0}(t)),\quad t\in [0,t_{1}].$$
This implies  for each $t\in [0,t_{1}]$ that
$$
\Int_{N_{0}}^{v_{0}(t)}\Frac{d \tau}{\psi(\tau)}\leq M\Int_{0}^{t_{1}}p(s)ds.$$
In view of (H3), we obtain
$$
|v_{0}(t^{*})|\leq \Gamma_{1}^{-1}\Bigl(M\Int_{0}^{t_{1}}p(s)ds\Bigr):=M_{0}.$$
Since for every $t\in [0,t_{1}], \|y_{t}\|\leq\mu_{0}(t)$, we have
$$
\Sup_{t\in [-r,t_{1}]}|y(t)|\leq \Max(\|\phi\|,M_{0})=b_{0}.$$
We transform this problem into a fixed point problem.
A mild solution to (\ref{e4})-(\ref{e5}) is a fixed point of the operator
$G:C([-r,t_{1}],E)\longrightarrow C([-r,t_{1}],E)$
defined by
$$
G(y)(t)=\left\{\matrix{\phi(t), \quad
\hfill t\in [-r,0] \cr
\cr
T(t,0)\phi(0)+\Int_{0}^{t}T(t,s)f(s,y_{s})ds, \quad \hfill t\in
J_{0}.\cr}\right.
$$
We shall show that $G$ satisfies the assumptions of Lemma 2.2.
\vskip 0.3cm
{\bf Claim 1}: {\em $G$ sends bounded sets into bounded sets in
$C(J_{0},E)$}.
\vskip 0.3cm
Let $B_{q}:=\{y\in C(J_{0},E): \|y\|_{\infty}=\Sup_{t\in
J_{0}}|y(t)|\leq q \}$ be a bounded set
in $C_{0}(J_{0},E)$ and $y\in B_{q}$. Then for each $t\in J_{0}$
$$
G(y)(t)=T(t,0)\phi(0)+\Int_{0}^{t}T(t,s)f(s,y_{s})ds.$$
Thus for each $t\in J_{0}$ we get
$$
\matrix{|G(y)(t)|&\leq&M\|\phi\|+M\Int_{0}^{t}|f(s,y_{s})|ds \hfill \cr
&& \cr
&\leq& M\|\phi\|+M\psi(q)\|p\|_{L^{1}}. \hfill \cr}$$
\vskip 0.3cm
{\bf Claim 2}: {\em $G$ sends bounded sets in $C(J_{0},E)$ into
equicontinuous sets}.
\vskip 0.3cm
Let $r_{1}, r_{2}\in J_{0}, \ r_{1}<r_{2}$,
$B_{q}$ be a bounded set in
$C_{0}(J_{0},E)$ as in Claim 1 and $y\in B_{q}$. Then
$$
\matrix{|G(y)(r_{2})-G(y)(r_{1})|&\leq & |(T(r_{2},0)\phi(0)-T(r_{1},0)\phi(0)|+
\Bigl|\Int_{0}^{r_{2}}[T(r_{2},s)-T(r_{1},s)]f(s,y_{s})ds\Bigr| \hfill \cr
&& \cr
&& \ + \ \Bigl|\Int_{r_{1}}^{r_{2}}T(r_{1},s)f(s,y_{s})ds\Bigr| \hfill \cr
&& \cr
&\leq & |(T(r_{2},0)\phi(0)-T(r_{1},0)\phi(0)|+
\Int_{0}^{r_{2}}|T(r_{2},s)-T(r_{1},s)|h_{q}(s)ds \hfill \cr
&& \cr
&& \ + \ M\Int_{r_{1}}^{r_{2}}h_{q}(s)ds. \hfill \cr}$$
The right hand side is independent of $y\in B_{q}$ and tends to zero as
$r_{2}-r_{1}\to 0$.

The equicontinuity for the cases $r_{1}<r_{2}\leq 0$ and $r_{1}\leq 0\leq
r_{2}$ are obvious.
\vskip 0.3cm
{\bf Claim 3:} {\em  $G$ is continuous.}
\vskip 0.3cm
Let $\{y_{n}\}$ be a sequence such that $y_{n}\longrightarrow y$ in
$C(J_{0},E).$
Then there is an integer $q$ such that $\|y_{n}\|_{\infty}\leq q$ for all
$n\in\N$ and $\|y\|_{\infty}\leq q$, so $y_{n}\in B_{q}$ and $y\in B_{q}.$

We have then by the dominated convergence theorem
$$
\|G(y_{n})-G(y)\|_{\infty}\leq M
\Sup_{t\in J_{0}}\Bigl[\Int_{0}^{t}|f(s,y_{ns})-f(s,y_{s})|ds\Bigr]
\longrightarrow 0, \ \hbox{as} \ n\to\infty.
$$
Thus $G$ is continuous.

Set
$$
U=\{y\in C([-r,t_{1}],E): \|y\|_{\infty} <b_{0}+1 \}.$$
As a consequence of Claims 1, 2 and 3 and (H4) together with the Arzela-Ascoli
 theorem we can conclude that \ $G:\overline U\longrightarrow C([-r,t_{1}],E)$ is a
completely continuous map.
\vskip 0.3cm

>From the choice of $U$ there is no $y\in\partial U$ such that
$y=\lambda G(y)$ for any $\lambda\in (0,1)$.

As a consequence of Lemma 2.1 we deduce that $G$ has a fixed point
$y_{0}\in\overline U$ which is a mild solution of (\ref{e4})-(\ref{e5}).
\vskip 0.3cm
{\bf Step 2.} Consider now the following problem on $J_{1}:=[t_{1},t_{2}]$
\begin{equation}\label{e6}
y'-A(t)y=f(t,y_{t}),\ \ \hbox {a.e.} \ \ t\in J_{1},
\end{equation}
\begin{equation}\label{e7}
y(t_{1}^{+})=I_{1}(y(t_{1}^{-})).
\end{equation}
Let $y$ be a (possible) mild solution to (\ref{e6})--(\ref{e7}). Then  for each
$t\in [t_{1},t_{2}]$
$$
y(t)-I_{1}(y(t_{1}^{-}))=\Int_{t_{1}}^{t}T(t,s)f(s,y_{s})ds.$$
>From (H3) we get
$$
|y(t)|\leq \Sup_{t\in [-r,t_{1}]}|I_{1}(y_{0}(t^{-}))|
+M\Int_{t_{1}}^{t}p(s)\psi(\|y_{s}\|)ds, \ t\in [0,t_{1}].$$
We consider the function $\mu_{1}$ defined by
$$
\mu_{1}(t)=\Sup\{|y(s)|: t_{1}\leq s\leq t \}, \ \ t_{1}\leq t\leq t_{2}.$$
Let $t^{*}\in [t_{1},t]$ be such that $\mu_{1}(t)=|y(t^{*})|$.
Then we have for $t\in [t_{1},t_{2}]$
$$
\mu_{1}(t)\leq N_{1}+M\Int_{t_{1}}^{t}p(s)\psi(\mu_{1}(s))ds. $$
Let us take the right-hand side of the above inequality as $v_{1}(t),$ then we
have
$$
v_{1}(t_{1})=N_{1}, \quad
\mu_{1}(t)\leq v_{1}(t), \quad t\in [t_{1},t_{2}]$$
and
$$v_{1}'(t)=Mp(t)\psi(\mu_{1}(t)), \quad t\in [t_{1},t_{2}].$$
Using the nondecreasing character of $\psi$ we get
$$
v_{1}'(t)\leq Mp(t)\psi(v_{1}(t)),\quad t\in [t_{1},t_{2}].$$
This implies  for each $t\in [t_{1},t_{2}]$ that
$$
\Int_{N_{1}}^{v_{1}(t)}\Frac{d \tau}{\psi(\tau)}\leq
M\Int_{t_{1}}^{t_{2}}p(s)ds.$$
In view of (H3), we obtain
$$
|v_{1}(t^{*})|\leq
\Gamma_{1}^{-1}\Bigl(M\Int_{t_{1}}^{t_{2}}p(s)ds\Bigr):=M_{1}.$$
Since for every $t\in [t_{1},t_{2}], \|y_{t}\|\leq\mu_{1}(t)$, we have
$$
\Sup_{t\in [t_{1},t_{2}]}|y(t)|\leq M_{1}.$$
A mild solution to (\ref{e6})--(\ref{e7}) is a fixed point of the operator
$G:C(J_{1},E)\longrightarrow C(J_{1},E)$
defined by
$$
G(y)(t):=I_{1}(y(t_{1}^{-}))+\Int_{t_{1}}^{t}T(t,s)f(s,y_{s})ds.$$
Set
$$
U=\{y\in C(J_{1},E): \|y\|_{\infty} <M_{1}+1 \}.$$
As in Step 1 we can show that \ $G:\overline U\longrightarrow C(J_{1},E)$ is
a completely continuous map.

>From the choice of $U$ there is no $y\in\partial U$ such that
$y=\lambda G(y)$ for any $\lambda\in (0,1)$.

As a consequence of Lemma 2.1 we deduce that $G$ has a fixed point
$y_{1}\in\overline U$ which is a mild solution of (\ref{e6})--(\ref{e7}).

{\bf Step 3.} Continue this process and construct solutions
$y_{k}\in C(J_{k},E), k=2,\ldots,m$ to
\begin{equation}\label{e8}
y'-A(t)y=f(t,y_{t}),\ \ \hbox {a.e.} \ \ t\in J_{k},
\end{equation}
\begin{equation}\label{e9}
y(t_{k}^{+})=I_{k}(y(t_{k}^{-})).
\end{equation}
Then
$$
y(t)=\cases{y_{0}(t), \quad &$t\in [-r,t_{1}]$ \hfill \cr
     y_{1}(t), \quad &$t\in (t_{1},t_{2}]$ \hfill \cr
. \cr
. \cr
. \cr
   y_{m-1}(t), \quad &$t\in (t_{m-1},t_{m}]$ \hfill \cr
y_{m}(t), \quad &$t\in (t_{m},b]$ \hfill \cr }
$$
is a mild solution of (\ref{e1})-(\ref{e3}). \ $\Box$
\vskip 0.3cm
%\input{macrfran}
\def\refname{\centerline{\normalsize\bf REFERENCES}}

\begin{thebibliography}{ }
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result for first order impulsive functional differential equations in Banach
spaces, {\em Comput. Math. Appl.} {\bf 42} (2001), 1303-1310.
\bibitem{DuGr} J. Dugundji and A. Granas, Fixed Point Theory, Monografie Mat.
PWN, Warsaw, 1982.
\bibitem{ErFrLiWu} L. Erbe, H.I. Freedman, X.Z. Liu and J.H. Wu,  Comparison
principles for impulsive parabolic equations with applications to models of
singles species growth,
{\em J. Austral. Math. Soc. Ser. B} {\bf 32} (1991), 382-400.
\bibitem{FrOr1} M. Frigon and D. O'Regan, Existence results for first order
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