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\markboth{\hfil Existence results  \hfil EJDE--2002/96}
{EJDE--2002/96\hfil H. J. Lee, J. Park, J. Y. Park  \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 96, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Existence results for second-order neutral functional differential
  and integrodifferential inclusions in Banach spaces
 %
\thanks{ {\em Mathematics Subject Classifications:} 34A60, 34K40, 45K05.
\hfil\break\indent
{\em Key words:} Mild solution, neutral functional differential and
 integrodifferential inclusions, \hfil\break\indent  fixed point.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted August 8, 2002. Published November 6, 2002.} }
\date{}
%
\author{Haeng Joo Lee, Jeongyo Park, \& Jong Yeoul Park}
\maketitle

\begin{abstract}
  In this paper, we investigate the existence of mild solutions
  on a compact interval to second order neutral functional
  differential and integrodifferential inclusions in Banach spaces.
  The results are obtained by using the theory of continuous cosine
  families and a fixed point theorem for condensing maps due
  to Martelli.
\end{abstract}

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


\section{Introduction}

In this paper we prove the existence of mild solutions, defined on a compact
interval, for second-order neutral functional differential and
integrodifferential inclusions in Banach spaces.
In Section 3 we consider the second-order neutral functional differential
inclusion
\begin{equation}
\begin{gathered}
  \frac{d}{dt}[y'(t)-g(t,y_t )] \in Ay(t)+F(t,y_t ), \quad
  t \in J=[0,T], \\
   y_0 =\phi, \quad  y'(0)=x_0 ,
\end{gathered}\label{e1}
\end{equation}
where $J_0 =[-r,0]$, $F: J \times C(J_0 ,E) \to 2^E$ is a bounded,
closed, convex valued multivalued map, $g:J \times C(J_0 ,E) \to
E$ is given function, $\phi \in C(J_0 ,E)$,  $x_0 \in E$, and $A$
is the infinitesimal generator of a strongly continuous cosine
family $\{C(t):t \in R\}$ in a real Banach space $E$ with the norm
$|\cdot|$.

For a continuous function $y$ defined on the interval
$J_1 =[-r,T]$ and  $t \in J$, we denote by $y_t$ the element of
$C(J_0 ,E)$ defined by
$$
  y_t (\theta)=y(t+\theta), \quad \theta \in J_0 .
$$
Here $y_t (\cdot)$ represents the history of the state from time $t-r$,
up to the present time $t$.

In Section 4 we investigate the existence of mild solutions for second order
neutral functional integrodifferential  inclusion
\begin{equation}
\begin{gathered}
   \frac{d}{dt}[y'(t)-g(t,y_t )] \in Ay(t)+\int_0^t K(t,s)F(s,y_s )ds,
   \quad t \in J=[0,T], \\
   y_0 =\phi, \quad y'(0)=x_0 ,
\end{gathered} \label{e3}
\end{equation}
where $A, F, g, \phi$ are as in the problem \eqref{e1} and
$K:D \to R$, $D=\{(t,s) \in J \times J : t \geq s\}$.

In many cases it is advantageous to treat the second order abstract
differential equations directly rather than to convert them into first order
systems. A useful tool for the study of abstract second order equations is
the theory of strongly continuous cosine families. Here we use of the basic
ideas from cosine family theory \cite{t1,t2}.

Existence results for differential inclusions on compact intervals, are given
in the papers of Avgerinos and Papageorgiou \cite{a1}, Papageorgiou
\cite{p1,p2}, and
Benchohra \cite{b2,b3} for differential inclusions on noncompact intervals.

This paper is motivated by the recent papers of Benchohra and
Ntouyas \cite{b3,b4,b5} and Ntouyas \cite{n1}. In \cite{b3} second
order functional differential inclusions are studied. In [5,6]
functional differential and integrodifferential inclusions are
studied. In \cite{n1} neutral functional integrodifferential
equations was studied. Here we compose the above results and prove
the existence of mild solutions for problems \eqref{e1} and
\eqref{e3}, relying on a fixed point theorem for condensing maps
due to Martelli \cite{m1}.

\section{Preliminaries}

In this section, we introduce notation, definitions,
and preliminary facts from multivalued analysis
which are used throughout this paper.

Let $C(J,E)$ be the Banach space of continuous
functions from $J$ into $E$ with the norm
$$
  \|y\|_{\infty} :=\sup \{|y(t)|:t \in J  \}.
$$
Let $B(E)$ denote the Banach space of
bounded linear operators from $E$ into $E$.
A measurable
function $y:J \to E$ is Bochner
integrable if and only if $|y|$ is Lebesque
integrable. (For properties of the Bochner
             integral see Yosida \cite{y1}.)

Let $L^1 (J,E)$ denotes the Banach space of
continuous  functions $y:J \to E$ which are
Bochner integrable, with the norm
$$
  \|y\|_{L^1}=\int_0^T | y(t)| dt \quad  \mbox{for all } y \in L^1 (J,E).
$$

Let $(X,\|\cdot\|)$ be a Banach space.
A multivalued map  $G:X \to 2^X$ is convex (closed) valued,
if $G(x)$ is convex (closed) for all $x \in X$.
$G$ is bounded on bounded sets
if $G(D)= \bigcup_{x \in D} G(x)$ is bounded in $X$,
for any bounded set $D$ of $X$, i.e.,
$$
  \sup_{x \in D}\{\mbox{sup}\{\|y\|:y \in G(x)\}\}< \infty.
$$

A map $G$ is called upper semicontinuous 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 $V$ of $X$ containing $G(x_0 )$,
there exists an open neighborhood $A$ of $x_0$
such that $G(A) \subseteq V$.

A map $G$ is said to be completely continuous
if $G(D)$ is relatively compact for every
bounded subset $D \subseteq X$.
If the multivalued map $G$ is completely
continuous  with nonempty compact values,
then $G$ is upper semicontinuous if and only if
$G$ has a closed graph, i.e.,
for $x_n \to x_*$, $y_n \to y_*$,  with $y_n \in Gx_n$
 we have $y_* \in Gx_*$.
The map $G$ has a fixed point if there is $x \in X$ such that $x \in Gx$.

In the following,  $BCC(X)$ denotes the set of
all nonempty bounded closed and convex
subsets  of $X$.
A multivalued map $G:J \to BCC(X)$ is said
to  be measurable
if for each $x \in X$, the distance between $x$ and $G(t)$ is a measurable
function on $J$.
For more details on multivalued maps, see the
books of Deimling \cite{d1} and Hu and Papageorgiou \cite{h1}.

An upper semicontinuous map $G:X \to 2^X $ is said to be condensing if,
for any bounded subset $D \subseteq X$, with $\alpha (D) \not= 0$,
we have
$$
  \alpha (G(D)) <\alpha (D),
$$
where $\alpha$ denotes the Kuratowski measure of noncompactness.
For properties of the Kuratowski measure, we refer to Banas and
Goebel \cite{b1}.

We remark that a completely continuous multivalued map is the easiest example of a condensing map.\\

We say that the family $\{C(t): t \in R\}$ of operators in $B(E)$ is a strongly continuous cosine family if
\begin{itemize}
  \item[(i)]  $C(0)=I$, is the identity operator in $E$
  \item[(ii)] $C(t+s) + C(t-s) = 2C(t)C(s)$  for all $s,t \in R $
  \item[(iii)]  The map $t \to C(t)y$ is strongly continuous
                       for each $y \in X$.
\end{itemize}
The   strongly continuous sine family $\{S(t):t \in R\}$,
associated to the given strongly continuous
cosine family $\{C(t):t\in R\}$, is defined by
$$
 S(t)y=\int_0^t C(s)y\,ds, \quad y \in E, \; t \in R.
$$
The infinitesimal generator $A:E \to E$ of
a  cosine family $\{C(t):t \in R\}$ is defined by
$$
  Ay =\frac{d^2}{dt^2 }C(t)y\Big|_{t=0}.
$$
For more details on strongly continuous
cosine and sine families,  we refer
the reader to the books of Goldstein \cite{g1}
and to the papers of Fattorini \cite{f1,f2}
and of Travis and Webb \cite{t1,t2}.

The considerations of this paper are based on the following fixed point theorem.

\begin{lemma}[\cite{m1}] \label{lm2.1}
Let $X$ be a Banach space and $N:X \to BCC(X) $ be a condensing map.
 If the set
$\Omega :=\{y \in X : \lambda y \in Ny, \mbox{ for some }\lambda >1 \}$
is bounded, then $N$ has a fixed point.
\end{lemma}


\section{Second Order Neutral Differential Inclusions}

In this section we give an existence result for the problem \eqref{e1}.
Let us list the following hypotheses.
\begin{description}
\item[(H1)]  $A$ is the infinitesimal generator of a  strongly continuous
cosine family $C(t)$, $t \in R$, of bounded linear operators from $E$
into itself.

\item[(H2)]  $C(t)$, $t>0$ is compact.

\item[(H3)] $F:J \times C(J_0 ,E) \to BCC(E)$; $(t,u) \to F(t,u)$
is measurable with respect to $t$ for each $u \in C(J_0 ,E)$,
upper semicontinuous with respect to $u$ for each $t \in J$, and
for each  fixed $u \in C(J_0 ,E)$, the set
$$
 S_{F,u} =\{f \in L^1 (J,E) : f(t) \in F(t,u) \mbox{ for a.e. } t \in J \}
$$
is nonempty.

\item[(H4)] The function $g:J \times C(J_0 ,E) \to E$ is completely
continuous and for any bounded set $K$ in $C(J_1 ,E)$,
the set $\{t \to g(t,y_t ):y \in K\}$ is equicontinuous in $C(J,E)$.

\item[(H5)] There exist constants $c_1$ and $c_2$ such that
$$
|g(t,v)| \leq c_1 \|v \|+c_2 ,  \quad  t\in J, \; v \in C(J_0 ,E)
$$

\item[(H6)] $\|F(t,u)\|: =\sup\{|v|:v \in F(t,u) \} \leq p(t)\Psi (\|u\|)$
for almost all $t \in J$ and $u \in C(J_0 ,E)$,
where $p \in L^1 (J, R_+ )$  and $\Psi:R_+ \to (0,\infty)$   is
continuous  and increasing with
$$
  \int_0^T  m(s)ds < \int_c^\infty \frac{ds}{s+\Psi(s)},
$$
where $c=M\|\phi \|+MT[|x_0 |+c_1 \|\phi\|+2c_2 ]$,
$m(t)=\max\{Mc_1 , MTp(t)\}$ and $M=\sup \{| C(t)|:t\in J \}$.
\end{description}


\paragraph{Remark}% 3.1}
(i) If $\dim E<\infty $,
then for each $v \in C(J_0 ,E)$,  $S_{F,u} \not = \phi$
              (see Lasota and Opial \cite{g1}).\\
(ii) $S_{F,u}$ is nonempty if and only if the function $Y:J \to R$ defined by
$$
  Y(t):=\inf \{|v|: v \in F(t,u)\}
$$
belongs to $L^1 (J,R)$ (see Papageorgiou[15]).

In order to define the concept of mild solution for \eqref{e1},
by comparison with abstract Cauchy problem
\begin{gather*}
  y''(t)=Ay(t)+h(t)\\
 y(0)=y_0 , \quad y'(0)=y_1
\end{gather*}
whose properties are well known \cite{t1,t2}, we associate problem
\eqref{e1} to the integral equation
\begin{equation}
 y(t)=C(t)\phi(0)+S(t)[x_0 -g(0,\phi)]+ \int_0^t C(t-s)g(s,y_s )ds
   +\int_0^t S(t-s) f(s)ds,  \label{e5}
\end{equation}
$t \in J$, where
$$
  f \in S_{F,y} =\{f \in L^1 (J,E): f(t) \in F(t,y_t ) \mbox{ for a.e. }
   t \in J\}.
$$


\paragraph{Definition} % 3.1
A function $y:(-r,T) \to E$, $T>0$ is called a mild solution of
the problem \eqref{e1} if $y(t)=\phi(t)$, $t \in [-r,0]$, and
there exists a $v \in L^1 (J, E)$ such that $v(t) \in F(t,y_t )$
a.e. on $J$, and the integral equation \eqref{e5} is satisfied.


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

\begin{lemma}[\cite{l1}] \label{lm3.1}
Let $I$ be a compact real interval, and let $X$ be a Banach space.
Let $F$ be a multivalued map satisfying $(H3)$, and let $\Gamma$ be a
linear continuous mapping from $L^1 (I,X)$ to $C(I,X)$.
Then, the operator
$$
  \Gamma \circ S_F : C(I,X) \to BCC(C(I,X)), \quad
   y \to (\Gamma \circ S_F )(y)=\Gamma (S_{F,y})
$$
 is a closed graph operator in $C(I,X) \times C(I,X)$.
\end{lemma}

Now, we are able to state and prove our main theorem.


\begin{theorem} \label{thm3.1}
Assume that Hypotheses (H1)-(H6) are satisfied.
Then system \eqref{e1} has at least one mild solution on $J_1$.
\end{theorem}

\paragraph{Proof.}
Let $C:=C(J_1 ,E)$ be the Banach space of continuous functions from
$J_1$ into $E$ endowed with the supremum norm
$$
 \| y\|_\infty :=\sup \{ |y(t)|:t \in J_1 \}, \quad \mbox{for } y \in C.
$$
Now we transform the problem into a fixed point problem.
Consider the multivalued map,
$N:C \to 2^{C}$ defined by
$Ny$ the set of functions $h \in C$ such that
$$h(t)= \begin{cases} \phi(t),    &\mbox{if }t \in J_0 \\[2pt]
       C(t)\phi(0)+S(t)[x_0 -g(0,\phi)]\\
       + \displaystyle\int_0^t C(t-s)g(s,y_s )ds
       +\displaystyle\int_0^t S(t-s) f(s)ds, &\mbox{if }t \in J
   \end{cases}
$$
where
$$
f \in S_{F,y} =\{f \in L^1 (J,E): f(t) \in F(t,y_t ) \mbox{ for a.e. }
t \in J\}.
$$
We remark that the fixed points of $N$ are mild solutions to \eqref{e1}.


We shall show that $N$ is completely continuous
with bounded closed convex values
and it is upper semicontinuous.
The proof will be given in several steps.
\\
{\bf Step 1.} $Ny$ is convex for each $y \in C$.
Indeed, if $h_1$, $h_2$ belong to $Ny$,
then there exist $f_1 , f_2 \in S_{F,y} $ such that,
for each $t \in J$ and $i=1,2$, we have
\[
 h_i (t)= C(t)\phi(0) +S(t)[x_0 -g(0, \phi)]+ \int_0^t C(t-s)g(s,y_s )ds \\
+\int_0^t S(t-s) f_i (s)ds.
\]
Let $0 \leq \alpha \leq 1$.
Then, for each $t \in J$, we have
\begin{eqnarray*}
 (\alpha h_1  + (1-\alpha )h_2 )(t) &=& C(t)\phi(0) +S(t)[x_0 -g(0,\phi)]
 + \int_0^t C(t-s)g(s,y_s )ds \\
&& +\int_0^t S(t-s) [\alpha f_1 (s)     +(1-\alpha ) f_2 (s)]\,ds.
\end{eqnarray*}
Since $S_{F,y}$ is convex (because $F$ has convex values),
then
$$
 \alpha h_1 + (1-\alpha )h_2 \in Ny.
$$
{\bf Step 2.} $N$ maps bounded sets into bounded sets in  $C$.
Indeed, it is enough to show that there exists a positive constant
$\ell$ such that, for each $h \in Ny$, $y \in B_q =\{y \in C:
\|y\|_\infty \leq q \}$, one has $\|h\|_\infty \leq \ell$. If $h
\in Ny$, then there exists $f \in S_{F,y }$ such that for each $t
\in J$ we have
\[
 h (t)= C(t)\phi(0) +S(t)[x_0 -g(0,\phi)] +  \int_0^t C(t-s)g(s,y_s )ds \\
 +\int_0^t S(t-s) f(s) ds.
\]
By (H5) and (H6),  we have that, for each $t \in J$,
\begin{eqnarray*}
 | h(t) | &\leq& |C(t)\phi(0)|+|S(t)[x_0 -g(0,\phi)]|
 +\big|\int_0^t C(t-s)g(s,y_s )ds \big|\\
  && + \big| \int_0^t S(t-s) f(s)ds \big| \\
 &\leq& M \|\phi\|+M T[|x_0 |+c_1 \|\phi\|+2c_2  ]
 +Mc_1 \int_0^t \|y_s \|ds \\
&& +MT \sup_{y \in [0,q]} \Psi (y) \big( \int_0^t p(s)ds\big)
\end{eqnarray*}
Then for each $h \in N(B_q )$ we have
\begin{eqnarray*}
\| h \|_\infty &\leq& M\|\phi\|+MT[|x_0 |+c_1 \|\phi\|+2c_2 ]
+Mc_1 \int_0^T \|y_s \|ds \\
&&+MT \sup_{y \in [0,q]} \Psi (y) \big( \int_0^T p(s)ds\big):=\ell.
\end{eqnarray*}

\noindent
{\bf Step 3.}  $N $ maps bounded sets into equicontinuous sets of $C$.
Let $t_1 ,t_2 \in J$, $0<t_1 <t_2$,
and  let $B_q =\{y \in C : \|y\|_\infty \leq q \}$ be a bounded set of
$C(J_1,E)$.
For each $y \in B_q$ and $h \in Ny$,
there exists $f \in S_{F,y}$ such  that for $t \in J$,
\[
h (t)= C(t)\phi(0) +S(t)[x_0 -g(0,\phi)] +  \int_0^t C(t-s)g(s,y_s )ds
+ \int_0^t S(t-s)f(s)\, ds\,.
\]
Thus,
\begin{eqnarray*}
\lefteqn{|h (t_2 )-h(t_1 )|}\\
&\leq& \big|[C(t_2 ) - C(t_1 )]\phi(0) \big| +\big|[S(t_2 )
- S(t_1 )][x_0 -g(0,\phi)]\big| \\
 && + \big| \int_0^{t_2} [C(t_2 -s)-C(t_1 -s)]g(s,y_s )ds\big|
              + \big| \int_{t_1}^{t_2} C(t_1 -s)g(s,y_s )ds\big|\\
 && + \big| \int_0^{t_2} [S(t_2 -s)-S(t_1 -s)]f(s)ds \big| +
 \big| \int_{t_1}^{t_2}S(t_1 -s)f(s)ds \big|\\
 &\leq& |C(t_2 ) - C(t_1 )|\|\phi\| +|S(t_2 ) - S(t_1 )|[|x_0 |+
                               c_1 \|\phi\|+c_2 ] \\
 && + \int_0^{t_2} |C(t_2 -s)-C(t_1 -s)|[c_1 \|y_s \|+c_2]ds\\
 && + \int_{t_1}^{t_2} |C(t_1 -s)|[c_1 \|y_s \|+c_2]ds\\
 && + \int_0^{t_2} |S(t_2 -s)-S(t_1 -s)|\|f(s)\|ds
    + \int_{t_1}^{t_2}|S(t_1 -s)|\|f(s)\|ds.
\end{eqnarray*}
As $t_2 \to t_1 $ the right-hand side of the above inequality tend to zero.
The equicontinuities for the cases $t_1 <t_2 \leq 0$ and $t_1 \leq0 \leq t_2 $
are obvious.
As a consequence of Step 2, Step 3, (H2) and (H4)
together with the Ascoli-Arzela theorem,
we can conclude that $N:C \to 2^C$ is a compact multivalued map,
and therefore, a condensing map.

\noindent{\bf Step 4.}  $N$ has a closed graph.
Let $y_n \to y_*$, $ h_n \in Ny_n$, and $  h_n \to h_*$.
We shall prove that $h_* \in Ny_*$.
$h_n \in Ny_n$ means that there exists $f_n \in S_{F,y_n}$,
such that for $t \in J$,
\[
h_n (t)= C(t)\phi(0)+S(t)[x_0 -g(0,\phi)] +\int_0^t C(t-s)g(s,y_{ns})ds
+\int_0^t S(t-s)f_n (s) ds\,.
\]
We must prove that there exists $f_* \in S_{F,y_* }$ such that for $t \in J$,
\[
h_* (t)= C(t)\phi(0) +S(t)[x_0 -g(0,\phi)] +\int_0^t C(t-s)g(s,y_{*s})ds
 +\int_0^t S(t-s)f_* (s) ds\,.
\]
Clearly, we have that as $n \to \infty$,
\begin{eqnarray*}
  \Big\| \Big(h_n -C(t)\phi(0) -S(t)[x_0 -g(0,\phi)]-\int_0^t
  C(t-s)g(s,y_{ns})ds \Big) &&\\
  -\Big(h_* -C(t)\phi(0) -S(t)[x_0 -g(0,\phi)]-\int_0^t C(t-s)
  g(s,y_{*s})ds\Big)\Big\|_{\infty} &\to& 0\,.
\end{eqnarray*}
Consider the linear and continuous operator $\Gamma :L^1 (J,E) \to C(J,E)$
defined as
\[
f \to \Gamma  (f)(t)=\int_0^t S(t-s)f(s)ds.
\]
From Lemma \ref{lm3.1},
it follows that $\Gamma \circ S_F$ is a closed graph operator.
Moreover, we have that
$$
 h_n (t)-C(t)\phi(0) -S(t)[x_0 -g(0,\phi)] -\int_0^t C(t-s)g(s,y_{ns})ds
 \in \Gamma (S_{F,y_n }).
$$
Since $y_n \to y_*$, it follows from Lemma \ref{lm3.1}
that
$$
 h_* (t)-C(t)\phi(0) -S(t)[x_0 -g(0,\phi)] -\int_0^T C(t-s)g(s,y_{*s})ds
 = \int_0^t S(t-s)  f_* (s) ds
$$
for some $f_* \in S_{F,y_*}$.
Therefore $N$ is a completely continuous multivalued map, upper
semicontinuous with convex closed values.
In order to prove that $N$ has a fixed point, we need one more step.
\\
{\bf Step 5.} The set
$$
\Omega: =\{y \in C:\lambda y \in Ny, \mbox{ for some } \lambda >1\}
$$
is  bounded. Let $y \in \Omega$.
Then $\lambda y \in Ny$ for some $\lambda >1$.
Thus, there exists $f \in S_{F,y}$  such that
\begin{eqnarray*}
 y(t)&=&\lambda^{-1}C(t)\phi(0)+\lambda^{-1}S(t)[x_0 -g(0,\phi)] +\lambda^{-1}\int_0^t C(t-s)g(s,y_s )ds \\
 && +\lambda^{-1}\int_0^t S(t-s)f(s) ds, \quad t \in J.
\end{eqnarray*}
This implies by (H5)-(H6) that for each $t \in J$,
we have
\begin{eqnarray*}
|y(t)| &\leq& M\|\phi\|+MT[|x_0 |+c_1\|\phi\|+2c_2 ] \\
  &&  +Mc_1 \int_0^t \|y_s \|ds +MT\int_0^t p(s)\Psi(\|y_s \|)ds.
\end{eqnarray*}
We consider the function
$$
  \mu (t)=\sup \{|y(s)|:-r \leq s \leq t\}, \ \ \ \ t \in J.
$$
Let $t^* \in [-r,t]$ be such that $\mu(t)=|y(t^*)|$. If $t^* \in J$,
by the previous inequality we have for $t \in J$,
\begin{eqnarray*}
\mu(t)
&\leq& M\|\phi\|+MT[|x_0 |+c_1\|\phi\|+2c_2 ]
+Mc_1 \int_0^{t^*} \|y_s \|ds \\
&&+MT\int_0^{t^* }p(s)\Psi(\|y_s \|)ds\\
  &\leq& M\|\phi\|+MT[|x_0 |+c_1\|\phi\|+2c_2 ]
+Mc_1 \int_0^t \mu(s)ds\\
&&+MT\int_0^t p(s)\Psi(\mu(s))ds.
\end{eqnarray*}
If $t^* \in J_0$, then $\mu(t) \leq \| \phi \|$ and the previous inequality obviously holds.
Let us denote the right-hand side of the
above inequality as  $v(t)$.
Then, we have
\begin{gather*}
 c=v(0)= M\|\phi\|+MT[|x_0 |+c_1\|\phi\|+2c_2 ], \\
 \mu(t) \leq v(t), \quad  t \in J,\\
  v'(t)= Mc_1 \mu(t)+MTp(t)\Psi(\mu(t)), \quad t \in J.
\end{gather*}
Using the nondecreasing character of $\Psi$, we get
\[
  v'(t) \leq Mc_1 v(t)+MTp(t)\Psi(v(t))
 \leq m(t)[v(t)+\Psi (v(t))],  \quad t \in J.
\]
This implies that for each $t \in J$ that
$$
  \int_{v(0)}^{v(t)} \frac{ds}{s+\Psi(s)} \leq  \int_0^T m(s)ds
             <\int_{v(0)}^{\infty} \frac{ds}{s+\Psi(s)}.
$$
This inequality implies that there exists
     a constant   $L$   such that $v(t) \leq L$,  $t \in J$,
and  hence $\mu(t) \leq L$, $t \in J$.
Since for every $t \in J$, $\|y_t \| \leq \mu(t)$, we have
$$
  \|y\|_\infty:=\sup\{|y(t)|:-r \leq t \leq T\} \leq L,
$$
where   $L$ depends only on $T$ and  on the function $p$  and $\Psi$.
This shows that $\Omega$ is  bounded.

Set $X:=C$. As a consequence of Lemma \ref{lm2.1}, we deduce that $N$ has
a fixed point which is a mild solution of the system \eqref{e1}.


\section{Second Order Neutral Integrodifferential Inclusions}

In this section we consider the solvability of the problem \eqref{e3}.
We need the following assumptions

\begin{description}
\item[(H7)] For each $t \in J$, $K(t,s)$ is measurable on $[0,t]$ and
     $$
       K(t)= \mbox{ess sup} \{|K(t,s)|, 0 \leq s \leq t \}
     $$
     is bounded on $J$.
\item[(H8)] The map $t \to K_t $ is continuous from $J$ to
$L^\infty (J, R)$, here $K_t (s)=K(t,s)$.

\item[(H9)] $\|F(t,u)\|: =\sup\{|v|:v \in F(t,u) \} \leq p(t)\Psi (\|u\|)$
        for almost all $t \in J$ and $u \in C(J_0 ,E)$,
        where $p \in L^1 (J, R_+ )$  and $\Psi:R_+ \to (0,\infty)$   is
        continuous  and increasing with
        $$
         \int_0^T  m(s)ds < \int_c^\infty \frac{ds}{s+\Psi(s)},
        $$
        where $c=M\|\phi \|+MT[|x_0 |+c_1 \|\phi\|+2c_2 ]$,
        $m(t)=\max\{Mc_1 , MT^2 \sup_{t \in J}$
        $K(t) p(t)\}$ and $M=\sup \{|C(t)|:t \in J \}$.
\end{description}
We define the mild solution for the problem \eqref{e3} by the integral
equation
\begin{equation}
\begin{aligned}
 y(t)=&C(t)\phi(0)+S(t)[x_0 -g(0,\phi)]+ \int_0^t C(t-s)g(s,y_s )ds \\
 & +\int_0^t S(t-s)\int_0^s K(s,u)f(u)duds, \quad t \in J,
 \end{aligned}\label{e6}
\end{equation}
where
$ f \in S_{F,y} =\{f \in L^1 (J,E): f(t) \in F(t,y_t )
\mbox{ for a.e. }t \in J\}$.


\paragraph{Definition} % 4.1.}
A function $y:(-r,T) \to E$, $T>0$ is called a mild solution of the problem
\eqref{e3} if $y(t)=\phi(t)$, $t \in [-r,0]$, and there exists a
$v \in L^1 (J, E)$ such that $v(t) \in F(t,y_t )$ a.e. on $J$,
 and the integral equation \eqref{e6} is satisfied.

\begin{theorem} \label{thm4.1}
Assume that hypotheses (H1)--(H5), (H7)--(H9) are satisfied.
Then system \eqref{e3} has at least one mild solution on $J_1$.
\end{theorem}

\paragraph{Proof.}
Let $C:=C(J_1 ,E)$ be the Banach space of continuous functions from $J_1$ into $E$
endowed with the supremum norm
$$
 \| y\|_\infty :=\sup \{ |y(t)|:t \in J_1 \}, \mbox{ for } y \in C.
$$
We transform the problem into a fixed point problem.
Consider the multivalued map, $Q:C \to 2^C$ defined by
$Qy$, the set of functions $h \in C$ such that
$$h(t)= \begin{cases} \phi(t),  &\mbox{if } t \in J_0 \\[2pt]
 C(t)\phi(0)+S(t)[x_0 -g(0,\phi)]
 + \displaystyle\int_0^t C(t-s)g(s,y_s )\,ds\\
 + \displaystyle\int_0^t S(t-s) \displaystyle\int_0^s K(s,u)f(u)\,du\,ds,
   &\mbox{if } t \in J ,      \end{cases}
$$
where
$$f \in S_{F,y} =\{f \in L^1 (J,E): f(t) \in F(t,y_t ) \mbox{ for a.e. }
 t \in J\}.
$$
We remark that the fixed points of $Q$ are mild solutions to \eqref{e3}.

As in Theorem 3.1 we can show that $Q$ is completely continuous with
bounded closed convex values and
it is upper semicontinuous, and therefore a condensing map.
We repeat only the Step 5, i.e. we show that
the set
$$
\Omega: =\{y \in C:\lambda y \in Qy, \mbox{ for some } \lambda >1\}
$$
is  bounded. Let $y \in \Omega$.
Then $\lambda y \in Qy$ for some $\lambda >1$.
Thus, there exists $f \in S_{F,y}$  such that
\begin{eqnarray*}
 y(t)&=&\lambda^{-1}C(t)\phi(0)+\lambda^{-1}S(t)[x_0 -g(0,\phi)] +\lambda^{-1}\int_0^t C(t-s)g(s,y_s )ds \\
 && +\lambda^{-1}\int_0^t S(t-s)\int_0^s K(s,u)f(u)\,du \,ds, \quad t \in J.
\end{eqnarray*}
This implies by (H5)-(H6) that for each $t \in J$,
we have
\begin{eqnarray*}
  |y(t)| &\leq& M\|\phi\|+MT[|x_0 |+c_1\|\phi\|+2c_2 ] \\
  && +Mc_1 \int_0^t \|y_s \|ds +MT^2 \sup_{t \in J}K(t)
  \int_0^t p(s)\Psi(\|y_s \|)ds.
\end{eqnarray*}
We consider the function
$$
  \mu (t)=\sup \{|y(s)|:-r \leq s \leq t\}, \quad t \in J.
$$
Let $t^* \in [-r,t]$ be such that $\mu(t)=|y(t^*)|$. If $t^* \in J$,
by the previous inequality we have for $t \in J$,
\begin{eqnarray*}
\mu(t) &\leq& M\|\phi\|+MT[|x_0 |+c_1\|\phi\|+2c_2 ]\\
 && +Mc_1 \int_0^{t^*} \|y_s \|ds +MT^2 \sup_{t \in J}K(t)
  \int_0^{t^* }p(s)\Psi(\|y_s \|)ds\\
  &\leq& M\|\phi\|+MT[|x_0 |+c_1\|\phi\|+2c_2 ]\\
 &&+Mc_1 \int_0^t \mu(s)ds+MT^2 \sup_{t \in J}K(t)\int_0^t p(s)\Psi(\mu(s))ds.
\end{eqnarray*}
If $t^* \in J_0$, then $\mu(t) \leq \| \phi \|$ and the previous
inequality obviously holds.

Let us denote  the right-hand side of the
above inequality as  $v(t)$. Then, we have
\begin{gather*}
  c=v(0)= M\|\phi\|+MT[|x_0 |+c_1\|\phi\|+2c_2 ], \\
  \mu(t) \leq v(t),  \quad  t \in J,\\
  v'(t)= Mc_1 \mu(t)+MT^2 \sup_{t \in J}K(t)p(t)\Psi(\mu(t)), \quad  t \in J.
\end{gather*}
Using the nondecreasing character of $\Psi$, for $t \in J$,
\[
 v'(t) \leq Mc_1 v(t)+MT^2 \sup_{t \in J}K(t)p(t)\Psi(v(t))
 \leq m(t)[v(t)+\Psi (v(t))]\,.
\]
This implies that for each $t \in J$,
$$
  \int_{v(0)}^{v(t)} \frac{ds}{s+\Psi(s)} \leq  \int_0^T m(s)ds
             <\int_{v(0)}^{\infty} \frac{ds}{s+\Psi(s)}.
$$
This inequality implies that there exists
     a constant $L$ such that $v(t) \leq L$,  $t \in J$,
and  hence $\mu(t) \leq L$, $t \in J$.
Since for every $t \in J$, $\|y_t \| \leq \mu(t)$, we have
$$
  \|y\|_\infty:=\sup\{|y(t)|:-r \leq t \leq T\} \leq L,
$$
where   $L$ depends only on $T$ and  on the function $p$  and $\Psi$.
This shows that $\Omega$ is  bounded.

Set $X:=C$. As a consequence of Lemma \ref{lm2.1}, we deduce that
$Q$ has a fixed point and thus system \eqref{e1} is controllable
on $J_1$.\\


\noindent\textbf{Acknowledgment:} This work was supported by Brain
Korea 21, 1999.


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\noindent\textsc{Haeng Joo Lee} (e-mail: leehj@pusan.ac.kr)\\
\textsc{Jeongyo Park} (e-mail: khpjy@hanmail.net)\\
\textsc{Jong Yeoul Park} (e-mail: jyepark@pusan.ac.kr)\\[2pt]
Department of Mathematics, Pusan National University\\
Pusan 609-735, Korea



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