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

\AtBeginDocument{{\noindent\small 
\emph{Electronic Journal of Differential Equations}, 
Vol. 2008(2008), No. 36, 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  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/36\hfil Existence of solutions]
{Existence of solutions for impulsive neutral functional
differential equations \\ with multiple delays}

\author[M. Lakrib\hfil EJDE-2008/36\hfilneg]
{Mustapha Lakrib}

\address{ Mustapha Lakrib \newline
Laboratoire de Math\' ematiques,
Universit\' e Djillali Liab\`es, 
B.P. 89, 22000 Sidi Bel Abb\`es, Alg\'erie} 
\email{mlakrib@univ-sba.dz}

\thanks{Submitted September 5, 2007. Published March 12, 2008.}
\subjclass[2000]{34A37, 34K40, 34K45} 
\keywords{Neutral functional differential equations; impulses; multiple delay;
\hfill\break\indent fixed point theorem}

\begin{abstract}
 In this paper  an existence result  for  initial value problems
 for first order impulsive neutral functional differential
 equations with multiple delay is proved under weak conditions.
\end{abstract}

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

\section{Introduction}

Impulsive differential equations have become more important in
recent years in some mathematical models of processes and
phenomena studied in physics, optimal control, chemotherapy,
biotechnology, population dynamics and ecology. The reader is
referred to monographs \cite{Ben,LBS,S} and references therein.

In this paper we study the existence of solutions for initial
value problems for first order neutral functional differential
equations, with multiple delays and with impulsive effects, of the
form
\begin{gather}\label{e11}
\begin{gathered}
\frac{d}{dt}[x(t)- f(t, x_t)]=g(t, x_t)+\sum_{i=1}^px(t-\tau_i),\\
\mbox{a.e. } t\in J=[0,1], t\neq t_{k}, k=1,\dots,m,
\end{gathered}\\
\label{e12}
\Delta x|_{t=t_k}=I_{k}(x(t_{k}^{-})), \quad  k=1,\dots,m, \\
\label{e13} x_0=\phi,
\end{gather}
where $f,g:J\times \mathcal{D} \to  \mathbb{R}^n$ are given
functions, $\mathcal{D}$ consists of functions
$\psi:J_0\to\mathbb{R}^n$ such that $\psi$ is continuous
everywhere except for a finite number of points  $s$ at which
$\psi(s^-)$ and $\psi(s^+)$ exist with $\psi(s^-)=\psi(s)$,
$J_0=[-r,0]$, $r=\max\{\tau_i:\ i=1,\dots,p\}$, $\phi\in
\mathcal{D}$, $0=t_{0}<t_{1}<\dots<t_{m}<t_{m+1}=1$, $I_{k}:
\mathbb{R}^n\to \mathbb{R}^n$ and $\Delta
x|_{t=t_k}=x(t_{k}^{+})-x(t_{k}^{-})$, $k=1,2,\dots,m$.

Our method of study is to convert the initial value problem
(\ref{e11})-(\ref{e13}) into equivalent integral equation and
apply the Schaefer's fixed point theorem.

In the literature the existence of solutions for impulsive
differential equations is studied under restrictive conditions on
the impulses $I_k$, $k=1,\dots,m$. In many results, in addition to
continuity, boundedness condition is  often assumed, which is not
fulfilled in some important cases such as for linear impulses.
Here, the only condition on the $I_k$, $k=1,\dots,m$, is
continuity.

Throughout this paper, the terminology and notation are those used
in functional analysis.

By $L^1(J,\mathbb{R}^n)$ we denote the Banach space of measurable
functions $x: J\to \mathbb{R}^n$ which are Lebesgue integrable,
normed by
$$
\|x\|_{L^1}=\int_0^1|x(t)|dt.
$$
For $\psi\in \mathcal{D}$,   the norm of $\psi$ is defined by
$$
\|\psi\|_\mathcal{D}=\sup\{|\psi(\theta)|:\theta\in J_0 \}.
$$

In order to define the solution of problem
(\ref{e11})-(\ref{e13}), we introduce the space
$PC(J,\mathbb{R}^n)$ consisting of functions $x: J\to
\mathbb{R}^n$ such that $x$ is continuous everywhere except for
$t=t_{k}$ at which $x(t^{-}_{k})$ and  $ x(t^{+}_{k})$ exist and
$x(t^{-}_{k})=x(t_{k})$, $k=1,\dots,m$. If we set
$\Omega=\{x:J_1\to\mathbb{R}^n\ /\ x\in \mathcal{D}\cap
PC(J,\mathbb{R}^n)\}$, where $J_1=[-r,1]$, then $\Omega$ is a
Banach space normed by
$$
 \|x\|=\sup\{|x(t)|: t\in J_1\},\quad x\in\Omega.
 $$
Obviously, for any $x\in\Omega$ and any $t\in J$, the history
function $x_t$ defined by $x_{t}(\theta)=x(t+\theta)$, for
$\theta\in J_0$, belongs to $\mathcal{D}$.

Also we denote by $AC((t_{k},t_{k+1}),\mathbb{R}^n)$ the space of
all absolutely continuous functions $x: (t_{k},t_{k+1})\to
\mathbb{R}^n$, $k=0, \dots, m$.

A function $x\in \Omega\cap AC((t_{k},t_{k+1}),\mathbb{R}^n)$,
$k=0,\dots, m$, is said to be a solution of problem
(\ref{e11})-(\ref{e13}) if $x-f(\cdot,x_.)$ is absolutely
continuous on $J\setminus\{t_1,\dots,t_m\}$ and $x$ satisfies the
differential equation  (\ref{e11}) a.e. on
$J\setminus\{t_1,\dots,t_m\}$ and the conditions
(\ref{e12})-(\ref{e13}).

Our main result will be proved using the following fixed point
theorem due to Schaefer \cite{Sch} (see  also \cite[page
29]{Smart}).

\begin{theorem}  \label{t1}
Let $X$ be a normed space and let $\Gamma:X\rightarrow X$ be a
completely continuous map, that is, it is a continuous mapping
which is compact on each bounded subset of $X$. If the set
$\mathcal{E}=\{x\in X: \lambda x=\Gamma x \mbox{ for some
}\lambda>1\}$ is bounded, then $\Gamma$ has a fixed point.
\end{theorem}

\section{Existence result}

In this section we state and prove our existence result for
problem (\ref{e11})-(\ref{e13}), using the following conditions:
\begin{itemize}
\item [(H1)] The function $ f:J\times
\mathcal{D} \to \mathbb{R}^n$ is such that
$$|f(t,x)|\leq  c_1\|x\|_\mathcal{D}+c_2\,\,\,\mbox{for all}\,\,\, t\in
 J\,\,
\hbox{and all}\,\,\, x\in \mathcal{D}$$ where $0\leq c_1<1$ and
$c_2\geq 0$ are some constants.
 \item[(H2)] The function $ g:J\times \mathcal{D} \to
\mathbb{R}^n$ is Carath\'eodory, that is,
\begin{itemize}
\item[(i)] $t\mapsto g(t,x)$ is measurable for each $x\in \mathcal{D}$,
\item[(ii)] $x\mapsto g(t,x)$ is continuous for a.e. $t\in J$.
\end{itemize}
\item [(H3)] There
exist a function $q\in L^1(J,\mathbb{R})$ with $q(t)>0$ for a.e.
$t\in J$ and a continuous and nondecreasing function
$\psi:[0,\infty)\to [0,\infty)$ such that
 $$
  |g(t,x)|\le q(t)\psi(\|x\|_\mathcal{D})\quad
\mbox{for a.e. $ t\in J$ and each $x\in \mathcal{D}$}
$$
 with
\begin{equation}
\label{inequ1}
 \int_{C}^{\infty}\frac{ds}{s+ \psi(s)}=\infty
\end{equation}
where
\[
C=\frac{1}{1-c_1}\Big[\|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}
\tau_i\Big)+2c_2\Big].
\]
\item [(H4)] The functions
 $I_k:\mathbb{R}^n\to \mathbb{R}^n$, $k=1,\dots,m$,
are continuous.
\end{itemize}

\begin{theorem}\label{t2}
 Under assumptions {\rm (H1)--(H4)},
the initial value problem \eqref{e11}--\eqref{e13} has a solution
on $J_1$.
\end{theorem}

\begin{proof}  Transform the problem (\ref{e11})-(\ref{e13}) into a
fixed point problem. Consider the operator $\Gamma:\Omega\to
\Omega$ defined by
\[
\Gamma x(t)=\begin{cases}
\phi(t) &\mbox{for $t\in J_0$},\\[3pt]
\phi(0)-f(0,\phi(0))+f(t,x_{t})+\int_0^t g(s,x_s)ds\\
+\sum_{i=1}^p\int_{-\tau_i}^{0}\phi(s)ds
+\sum_{i=1}^p\int_{0}^{t-\tau_i}x(s)ds \\
+\sum_{0 < t_k < t}I_{k}(x(t_{k}^{-})) &\mbox{for $t\in J$}.
\end{cases}
\]

We shall show that the operator $\Gamma$ satisfies the conditions
of Theorem \ref{t1} with $X=\Omega$. For better readability, we
break the proof into a sequence of steps.

\smallskip
\noindent {\bf Step 1.} We show that $\Gamma$ has bounded values
for bounded sets in $\Omega$. To show this, let $B$ be a bounded
set in $\Omega$. Then there exists a real number $\rho>0$ such
that $\|x\|\le \rho$, for all $x\in B$.

Let $x\in B$ and $t\in J$. After some standard calculations we get
\begin{align*}
|\Gamma x(t)|&\le
\|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}\tau_i\Big)
+2c_2+c_1\|x_t\|_{\mathcal{D}}+
\int_0^1q(s)\psi(\|x_s\|_\mathcal{D})ds\\
&\quad +p \int_0^1|x(s)|ds+\sum_{k=1}^{m}|I_{k}(x(t_{k}^{-}))|\\
&\le
\|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}\tau_i\Big)+2c_2+(c_1+p)\rho\\
&\quad + \psi(\rho)\|q\|_{L^1} +\sum_{k=1}^{m} \sup\{|I_k(u)|:
|u|\leq \rho\}=:\eta.
\end{align*}
If $t\in J_0$, then $|\Gamma x(t)|\le\|\phi\|_{\mathcal{D}}$ and
the previous inequality holds. Hence
$$
\|\Gamma x\|\le\eta, \quad \mbox{for all}\quad x\in B,
$$
that is, $\Gamma$ is bounded on bounded subsets of $\Omega$.

\smallskip
\noindent {\bf Step 2.}
 Next we show that $\Gamma$ maps bounded sets into equicontinuous sets.
Let $B$ be, as in Step 1, a bounded set  and $x\in B$. Let $t$ and
$h\not=0$ be such that $t,t+h\in
J\backslash\{t_{1},\dots,t_{m}\}$. It is not difficult to get
\begin{align*}
&|\Gamma x(t+h)-\Gamma x(t)|\hfill\\
&  \leq
|f(t+h,x_{t+h})-f(t,x_{t})|+\psi(\rho)\int_{t}^{t+h}q(s)ds+
  p\rho h+\sum_{t < t_k<t+ h}|I_{k}(x(t_{k}^{-}))|.
\end{align*}
As $h\to 0$ the right-hand side of the above inequality tends to
zero. This proves the equicontinuity on
$J\backslash\{t_{1},\dots,t_{m}\}$.

It remains to examine the equicontinuity at $t=t_{i}$, $i=1,\dots,
m$. Let $t=t_i$ for some $i\in\{1,\dots, m\}$ and let $h\not=0$ be
such that $\{t_k:k\not=i\}\cap[t_i-|h|,t_i+|h|]=\emptyset$. Then
we have
\begin{align*}
|\Gamma x(t_i+h)-\Gamma x(t_{i})| \leq
|f(t_i+h,x_{t_i+h})-f(t_i,x_{t_i})|+\psi(\rho)\int_{t_i}^{t_i+h}q(s)ds
+p\rho h.
\end{align*}
The right-hand side of the above inequality tends to zero as $h\to
0$. The equicontinuity on $J_0$ follows from the uniform
continuity of $\phi$ on this interval.
\smallskip

\noindent {\bf Step 3.}
 Now we show that $\Gamma$ is continuous.
Let $\{x_n\}\subset \Omega$ be a sequence such that $x_n\to x$. We
will show that $\Gamma x_n\to \Gamma x$. For $t\in J$, we obtain
\begin{equation}\label{equ5}
 \begin{aligned}
 |\Gamma x_n(t)-\Gamma x(t)|
& \leq |f(t,x_{nt})-f(t,x_{t})|
 +\int_{0}^{1}|g(s,x_{ns})-g(s,x_{s})|ds\hfill\\
&\quad+ p\int_{0}^{1}|x_{n}(s))-x(s)|ds
+\sum_{k=1}^{m}|I_{k}(x_n(t_{k}^{-}))-I_{k}(x(t_{k}^{-}))|.
\end{aligned}
\end{equation}
Using (H3) it can easily shown that the function $t\mapsto
g(t,x_{nt})-g(t,x_{t})$ is Lebesgue integrable. By the continuity
of $f$ and $I_k$, $k=1,\ldots,m$, and the dominated convergence
theorem, the right-hand side of inequality (\ref{equ5}) tends to
zero as $n\to\infty$; which completes the proof that $\Gamma$ is
continuous.

As a consequence of Steps 1 to 3, together with the
Arzel\'a-Ascoli theorem, we conclude that $\Gamma$ is completely
continuous.
\smallskip

\noindent{\bf Step 4.} Finally we show that  the set
$\mathcal{E}=\{x\in \Omega: \lambda x=\Gamma x \mbox{ for some
}\lambda>1\}$ is bounded. Let $x\in  \mathcal{E}$ and let
$\lambda>1$ be such that $\lambda x=\Gamma x$. Then
$x|_{[-r,t_1]}$ satisfies, for each $t\in [0,t_1]$,
\begin{align*}
x(t)&=  \lambda^{-1}\big[ \phi(0)-f(0,\phi(0))+f(t,x_{t})+\int_0^t
 g(s,x_s)ds\hfill\\
& \quad +\sum_{i=1}^p\int_{-\tau_i}^{0}\phi(s)ds
+\sum_{i=1}^p\int_{0}^{t-\tau_i}x(s)ds\big].
\end{align*}
It is straightforward to verify that
\begin{equation}
\label{equ15}
\begin{aligned}
 |x(t)| & \leq  \|\phi\|_{\mathcal{D}}\Big(1+c_1
 +\sum_{i=1}^{p}\tau_i\Big)+2c_2  +c_1\|x_t\|_{\mathcal{D}}\\
 &\quad +\int_0^t [q(s)\psi(\|x_s\|_{\mathcal{D}})ds+ p |x(s)|]ds.
 \end{aligned}
\end{equation}

Introduce the function $v_{1}(t)= \max\{|x(s)|: s\in[-r,t]\}$, for
$t\in [0,t_1]$. We have $|x(t)|, \|x_t\|_\mathcal{D}\le v_{1}(t)$
for all $t\in [0,t_1]$ and there is $t^*\in [-r,t]$ such that
$v_{1}(t)=|x(t^*)|$. If $t^*<0$, we have
$v_{1}(t)\leq\|\phi\|_{\mathcal{D}}$ for all $t\in [0,t_1]$. Now,
if $t^*\geq 0$, from (\ref{equ15}) it follows that, for $t\in
[0,t_1]$,
\[
v_{1}(t) \leq
\|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}\tau_i\Big)
+2c_2+c_1v_{1}(t)+\int_0^t[q(s)\psi(v_{1}(s))+p v_{1}(s)]ds
\]
and hence
\[
v_{1}(t)\leq  C_1^{1}+C_1^{2}\int_0^t
Q(s)[\psi(v_{1}(s))+v_{1}(s)]ds
\]
where
$$
C_1^{1}=C_1^{2}\Big[\|\phi\|_{\mathcal{D}}\Big(1+c_1
+\sum_{i=1}^{p}\tau_i\Big)+2c_2\Big], \quad
 C_1^{2}=\frac{1}{1-c_1}
$$
and $Q(t)=\max\{q(t),p\}$, for $t\in[0,t_1]$. Set
$$
w_{1}(t)=C_1^{1}+C_1^{2}\int_0^t
Q(s)[\psi(v_{1}(s))+v_{1}(s)]ds,\quad \hbox{for } t\in [0,t_1].
$$
Then we have $v_{1}(t)\le w_{1}(t)$ for all $t\in [0,t_1]$. A
direct differentiation of $w_{1}$ yields
\begin{gather*}
 w_1'(t)\leq Q(t)[\psi(w_1(t))+w_1(t)],\quad \mbox{a.e. } t\in
[0,t_1]\\
 w_1(0)=C_1^{(1)}.
  \end{gather*}
By integration, this gives
 \begin{equation}
 \label{inequ15}
\int_0^t \frac{w_1'(s)}{\psi(w_1(s))+w_1(s)}\,ds\le  \int_0^t
Q(s)ds\leq \|Q\|_{L^1},\quad t\in [0,t_1].
 \end{equation}
By a change of variables, inequality (\ref{inequ15}) implies
$$
\int_{C_1^{1}}^{w_1(t)}\frac{ds}{\psi(s)+s}\le \|Q\|_{L^1},\quad
t\in [0,t_1].
$$
By (\ref{inequ1}) and the mean value theorem, there is a constant
$M_1=M_1(t_1)>0$ such that $  w_1(t)\le M_1$ for all $t\in
[0,t_1]$, and therefore $v_1(t)\le M_1$, for all $t\in[0,t_1]$. At
last, we choose $M_1$ such that $\|\phi\|_{\mathcal{D}}\leq M_1$
to get
\[
 \max\{|x(t)|:t\in[-r,t_1]\}=v_1(t_1)\le M_1.
\]

Now, consider $x|_{[-r,t_2]}$. It satisfies, for each $t\in
[0,t_2]$,
\begin{align*}
x(t)&=  \lambda^{-1}\big[ \phi(0)-f(0,\phi(0))+f(t,x_{t})+\int_0^t
 g(s,x_s)ds\hfill\\
& \quad +\sum_{i=1}^p\int_{-\tau_i}^{0}\phi(s)ds
+\sum_{i=1}^p\int_{0}^{t-\tau_i}x(s)ds+I_1(x(t_1))\big].
\end{align*}
Therefore,
\begin{equation}
\label{equ17}
\begin{aligned}
 |x(t)|&\leq   \|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}
 \tau_i\Big)+2c_2+c_1\|x_t\|_{\mathcal{D}}\\
 & \quad +\int_0^t
[q(s)\psi(\|x_s\|_{\mathcal{D}})+ p
|x(s)|]ds+\sup\{|I_1(u)|:|u|\leq M_1\}.
\end{aligned}
\end{equation}
Denote $ v_2(t)= \max\{|x(s)|: s\in[-r,t]\}$, for $t\in [0,t_2]$.
Then, for each $t\in [0,t_2]$, we have $|x(t)|,
\|x_t\|_\mathcal{D}\le v_2(t)$. Let $t^*\in [-r,t]$ be such that
$v_2(t)=|x(t^*)|$. In the case $t^*<0$, we have
$v_2(t)\leq\|\phi\|_{\mathcal{D}}$ for all $t\in [0,t_2]$. Now, if
$t^*\geq 0$, then by (\ref{equ17}) we have, for $t\in [0,t_2]$,
\begin{align*}
v_2(t)&\leq
\|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}\tau_i\Big)
+2c_2+c_1v_2(t)+\int_0^t[ q(s)\psi(v_2(s)) +p v_2(s)]ds\\
& \quad +\sup\{|I_1(u)|:|u|\leq M_1\};
\end{align*}
that is,
\[
v_2(t)\leq  C_2^{1}+C_2^{2}\int_0^t Q(s)[\psi(v_2(s))+v_2(s)]ds
\]
where
$$
C_2^{1}=C_2^{2}\Big[\|\phi\|_{\mathcal{D}}\Big(1+c_1+\sum_{i=1}^{p}
\tau_i\Big)+2c_2+\sup\{|I_1(u)|:|u|\leq M_1\}\Big],
$$
$C_2^{2}=1/(1-c_1)$, and $Q(t)=\max\{q(t),p\}$, for $t\in[0,t_2]$.

If we set
$$
w_2(t)=C_2^{1}+C_2^{2}\int_0^t Q(s)[\psi(v_2(s))+v_2(s)]ds,\quad
\hbox{for}\  t\in [0,t_2],
$$
then $v_2(t)\le w_2(t)$ for all $t\in [0,t_2]$ and
\begin{gather*}
 w_2'(t)\leq Q(t)[\psi(w_2(t))+w_2(t)],\quad \mbox{a.e. } t\in
[0,t_2]\\
 w_2(0)=C_2^{1}.\hfill
  \end{gather*}
This yields
 \[
\int_0^t \frac{w_2'(s)}{\psi(w_2(s))+w_2(s)}\,ds\le  \int_0^t
Q(s)ds\leq \|Q\|_{L^1},\quad t\in [0,t_2]
 \]
which  implies
$$
\int_{C_2^{1}}^{w_2(t)}\frac{ds}{\psi(s)+s}\le \|Q\|_{L^1},\quad
t\in [0,t_2].
$$
Again, by (\ref{inequ1}) and the mean value theorem, there is a
constant \mbox{$M_2=M_2(t_1,t_2)>0$} such that $  w_2(t)\le M_2$
for all $t\in [0,t_2]$, and then $v_2(t)\le M_2$, for all
$t\in[0,t_2]$. Finally, if we choose $M_2$ such that
$\|\phi\|_{\mathcal{D}}\leq M_2$, we get
\[
 \max\{|x(t)|:t\in[-r,t_2]\}=v_2(t_2)\le M_2.
\]
Continue this process for $x|_{[-r,t_3]}, \ldots, x|_{J_1}$, we
obtain that there exists a constant $M=M(t_1,\ldots,t_m)>0$ such
that
$$
 \|x\|\le M.
$$
This finish to show that the set $\mathcal{E}$ is bounded in
$\Omega$.

As a result the conclusion of Theorem \ref{t1}  holds and
consequently the initial value problem  (\ref{e11})-(\ref{e13})
has a solution $x$ on $J_1$. This completes the proof.
\end{proof}

We conclude this paper with a discussion on two special cases. In
each one, some of the conditions in Theorem \ref{t2} can be either
removed or weakened.
\smallskip

\noindent\textbf{Case 1:} Consider the initial value problem for
first order impulsive functional differential equations with
multiple delays
\begin{gather}\label{e111}
x'(t)=g(t, x_t)+\sum_{i=1}^px(t-\tau_i),\  \mbox{a.e. } t\in
J=[0,1], t\neq t_{k}, k=1,\dots,m,\\
\label{e112}
\Delta x|_{t=t_k}=I_{k}(x(t_{k}^{-})), \quad  k=1,\dots,m, \\
\label{e113} x_0=\phi,
\end{gather}
derived from problem (\ref{e11})-(\ref{e13}) when $f\equiv 0$. In
this case one obtains the next existence result which is an
immediate corollary of Theorem \ref{t2}.


\begin{theorem}
Under conditions {\rm (H2)--(H4)}, the initial value problem
\eqref{e111}--\eqref{e113} has a solution on $J_1$ if constant $C$
in {\rm (H3)} is replaced by
\[
C=\|\phi\|_{\mathcal{D}}\Big(1+\sum_{i=1}^{p}\tau_i\Big).
\]
\end{theorem}

\noindent\textbf{Case 2:} Without the second term in the right
hand side of (\ref{e111}), problem (\ref{e111})-(\ref{e113}) is an
initial value problem for first order impulsive functional
differential equations
\begin{gather}\label{e1111}
x'(t)=g(t, x_t),\  \mbox{a.e. } t\in J=[0,1], t\neq t_{k},
k=1,\dots,m, \\
\label{e1112}
\Delta x|_{t=t_k}=I_{k}(x(t_{k}^{-})), \quad  k=1,\dots,m, \\
\label{e1113} x_0=\phi.
\end{gather}
The corresponding existence result is as bellow. Its proof is
omitted because it is the same as the proof of Theorem \ref{t2}.


\begin{theorem}
Under conditions {\rm (H2)--(H4)}, the initial value problem
\eqref{e1111}--\eqref{e1113} has a solution on $J_1$ if relation
\eqref{inequ1} in {\rm (H3)} is replaced by
\[
\int_{\|\phi\|_{\mathcal{D}}}^{\infty}\frac{ds}{\psi(s)}=\infty.
\]
\end{theorem}


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\end{thebibliography}
\end{document}