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\markboth{\hfil Neutral functional differential inclusions \hfil EJDE--2000/20}
{EJDE--2000/20\hfil M. Benchohra \& S. K. Ntouyas \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~20, pp.~1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 Existence results for  neutral functional differential and
integrodifferential inclusions  in Banach spaces
\thanks{ {\em Mathematics Subject Classifications:} 34A60, 34G20, 34K40.
\hfil\break\indent
{\em Key words and phrases:} Initial value problems, Convex multivalued map,
\hfil\break\indent
Neutral functional  differential and integrodifferential inclusion.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted October 28, 1999. Published March 12, 2000.} }
\date{}
%
\author{ M. Benchohra \& S. K. Ntouyas }
\maketitle

\begin{abstract}
 In this paper we investigate the existence of solutions on a compact interval
 for the first and second  order initial-value problems for neutral functional
 differential  and integrodifferential inclusions in Banach spaces.
 We shall use of a fixed point theorem for condensing maps introduced by
 Martelli.
\end{abstract}

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

\section{Introduction}

Existence of solutions on compact intervals for neutral functional
differential equations has received much attention in recent years.  We refer
for instance to the books of Erbe, Qingai and Zhang \cite{ErQiZh},
Hale \cite{Hal} and Henderson \cite{Hen}, the paper of
Ntouyas \cite{Nto} and Ntouyas, Sficas and Tsamatos
\cite{NtSfTs}. For other results on
functional differential and integrodifferential equations, we mention for
instance the paper  of Hristova and Bainov \cite{HrBa}, Nieto, Jiang and
Jurang \cite{NiJiJu}, Ntouyas \cite{Nto1}, \cite{Nto2} and Ntouyas and
Tsamatos \cite{NtTs}.

In the above mentioned papers the main tools used for the existence of
solutions are the monotone iterative method combined with upper and lower
solutions or the topological transversality theory of Granas. For more
details on these theories we refer the interesting reader
to the book of
Ladde, Lakshmikantham and Vatsala \cite{LaLaVa} and the monograph of
Dugundji and Granas \cite{DuGr}.

This paper is organized as follows. In section 2, we introduce some
definitions and
preliminary facts from multi-valued analysis which are used later.
In section~3, we give an existence result of solutions on compact
intervals to the initial value problem (IVP for short) of the first order
neutral functional differential inclusion
$$ \displaylines{
\hfill \frac{d}{dt}[y(t)-f(t,y_t)]\in F(t,y_t), \ \ \mbox{a.e.} \ \
t\in J=[0,T], \hfill\llap{(1.1)} \cr
\hfill y_0=\phi,  \hfill\llap{(1.2)}
}$$
where $  F:J\times C(J_0,E)\to 2^{E}$ ($J_0=[-r,0]$) is a
bounded, closed, convex multi-valued map,
$f:J\times C(J_0,E)\to E$, \ $\phi \in C(J_0,E)$,
and $E$ a real Banach space with the norm $|\cdot|$.

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

Section 4 is devoted to the study of the existence of solutions to
 the first order IVP for neutral functional integrodifferential
inclusion of the form
$$ \displaylines{
\hfill \frac{d}{dt}[y(t)-f(t,y_t)]\in \int_0^{t}K(t,s)F(s,y_{s})ds, \ \ t\in
J=[0,T], \hfill\llap{(1.3)} \cr
\hfill y_0=\phi, \hfill\llap{(1.4)}
}$$
where $F$, $f$, $\phi$  are as in the problem (1.1)-(1.2) and
$K:D\to{\mathbb R}$, $D=\{(t,s)\in J\times J: t\geq s \}$.

In Section 5, we give an existence theorem for solutions to the second order
IVP for neutral functional differential inclusions of the form
$$ \displaylines{
\hfill \frac{d}{dt}[y'(t)-f(t,y_t)]\in F(t,y_t), \ \ t\in J=[0,T],
\hfill\llap{(1.5)} \cr
\hfill y_0=\phi, \ y'(0)=\eta. \hfill\llap{(1.6)}
}$$
where $F$, $f$, $\phi$  are as in the problem (1.1)-(1.2) and $\eta\in E.$

The strategy is to reduce the existence of solutions to
problems (1.1)-(1.2), (1.3)-(1.4)  and  (1.5)-(1.6)  to the search for
fixed points of a suitable
multi-valued map on the Banach space $C(J_1, E)$.
To prove the existence of fixed points, we shall rely on a fixed
point theorem for condensing maps introduced by Martelli \cite{Mar}.



\section{Preliminaries}

This section presents notation, definitions, and preliminary facts
from multi-valued 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 Lebesgue integrable. (For properties of  the Bochner
integral see Yosida \cite{Yos}).

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

Let $(X, \|\cdot\|)$ be a Banach space. Then a multi-valued map $G:X\to2^{X}$
is convex (closed) valued if $G(x)$ is  convex (closed)
for all $x\in X$.  $G$ is bounded on bounded sets if $G(B)=\cup_{x\in B}G(x)$
is bounded in $X$ for any 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 $x_*\in X$ the set $G(x_*)$ is a nonempty, closed subset
of $X$, and if for each open set $B$ of $X$ containing $G(x_*)$, there
exists an open neighbourhood $V$ of $x_*$ such that  $G(V)\subseteq B$.

$G$ is said to be completely continuous if $G(B)$ is relatively compact for
every bounded subset $B\subseteq X$.

If the multi-valued map $G$ is completely continuous with nonempty compact
values, then $G$ is u.s.c. if and
only if $G$ has a closed graph (i.e. $x_{n}\to x_*, \
y_{n}\to y_*, \ y_{n}\in Gx_{n}$ imply $y_*\in Gx_*$). \par
$G$ 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 multi-valued map $G:J\to BCC(E)$  is said to be measurable if for
each $x\in E$ the function $Y:J\to{\mathbb R}$ defined by
$$
Y(t)=d(x,G(t))=\inf\{|x-z|: z\in G(t)\}$$
belongs to $L^{1}(J,{\mathbb R}).$ For more details on multi-valued maps
see the books
of Deimling \cite{Dei} and Hu and Papageorgiou \cite{HuPa}.

An upper semi-continuous map $G:X\to X$ is said to be
condensing \cite{BaGo} 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 \cite{BaGo}.

We remark that a completely continuous multi-valued map is the easiest
example of a condensing map.

Our existence results will be proved using the following fixed point result.

\begin{lemma} \label{l1}
\cite{Mar}.  Let $X$ be a Banach space and
$N:X\to BCC(X)$ 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{Existence results for differential inclusions}

In this section we give an existence result for the problem (1.1)-(1.2). For
the study of this problem we first list the following hypotheses:

\paragraph{(H1)} There exists constants $0\leq c_1<1$ and $c_{2}\geq 0$
such that
$$
|f(t,u)|\leq c_1\|u\|+c_{2}, \ \ t\in J, \ u\in C(J_0,E);$$

\paragraph{(H2)} $F:J\times C(J_0,E)\to BCC(E); (t,u)\longmapsto F(t,u)$ is
 measurable with respect to $t$ for each $u\in C(J_0,E)$, u.s.c.
with respect to $u$ for each $t\in J$ and for each fixed
$u\in C(J_0,E)$ the set
$$
S_{F,u}=\Bigl\{g\in L^{1}(J,E): \ g(t)\in F(t,u) \ \hbox {for a.e. } \ t\in
J \Bigr\}$$
is nonempty;

\paragraph{(H3)}  $\|F(t,u)\|:=\sup\{|v|: v\in F(t,u) \}\leq
p(t)\psi(\|u\|)$ for
almost all $t\in J$
and all $u\in C(J_0,E)$, where $p\in L^{1}(J,{\mathbb R}_{+})$ and  \
$\psi:{\mathbb R}_{+}\to (0,\infty)$ is continuous and increasing  with
$$
\int_0^{T}p(s)ds<\int_{c}^{\infty}\frac{d\tau}{\psi(\tau)}; $$
where $c=\frac{1}{1-c_1}[(1+c_1)\|\phi\|+2c_{2}]$;

\paragraph{(H4)} The function $f$ is completely continuous and for any
bounded set
$A\subseteq C(J_1,E)$ the set $\{t\to f(t,y_t): y\in A \}$ is
equicontinuous in $C(J,E)$;

\paragraph{(H5)} For each bounded $B\subset C(J_1,E)$,  $u\in B$ and $t\in J$
the set
$$
\Bigl\{\int_0^{t}g(s)ds: g\in S_{F,u} \Bigr\}$$
is relatively compact.

\begin{remark}
(i) \ If dim$E<\infty$, then for
each $u\in C(J_0,E), \  S_{F,u}\neq \emptyset$ (see Lasota and Opial
\cite{LaOp}).\\
(ii) \ $S_{F,u}$ is nonempty if and only if the function
$Y:J\to {\mathbb R}$ defined by
$$
Y(t):=\inf\{|v|: v\in F(t,u) \}$$
belongs to $L^{1}(J,{\mathbb R})$ (see Papageorgiou \cite{Pap1}).
\end{remark}
\begin{definition} By a solution to the IVP (1.1)-(1.2) it mean a
function $y: J_1\to E$ such that $y_0=\phi, \
y_t\in C(J_0,E)$, the function $y(t)-f(t,y_t)$ is absolutely
continuous and the inclusion (1.1) hold a.e. on $J$.
\end{definition}
The following Lemma is crucial in the proof of our existence results.\par

\begin{lemma} \label{l2}
\cite{LaOp}. Let $I$ be a compact real interval and $X$ be a Banach space.
Let $F$ be a
multivalued map satisfying (H2) and let $\Gamma$ be a linear continuous
mapping from $L^{1}(I,X)$ to $C(I,X)$,
then the operator
$$
\Gamma \circ S_{F}:C(I,X)\to BCC(C(I,X)), \ y\longmapsto
(\Gamma \circ S_{F})(y):=\Gamma(S_{F,y})$$
is a closed graph operator in $C(I,X)\times C(I,X)$.
\end{lemma}

Now, we are in a position to state and prove our main theorem for this section
\begin{theorem}\label{t1}
Assume that hypotheses (H1)-(H5) hold. Then the IVP
(1.1)-(1.2) has at least one solution on $J_1$.
\end{theorem}
\paragraph{Proof.} Let $C(J_1,E)$ be the Banach  space of continuous functions
from
$J_1$ into $E$ endowed with the sup-norm
$$
\|y\|_{\infty}:=\sup\{|y(t)|: t\in [-r,T] \}, \ \mbox{for} \ y\in C(J_1,E).
$$
Transform the  problem into a fixed point problem.
Consider the multivalued map, $N:C(J_{1},E)\longrightarrow 2^{C(J_{1},E)}$
defined by: \par
$$
Ny:=\left\{ h\in C(J_{1},E): h(t)=\left\{\begin{array}{ll} \phi(t), &
\mbox{ if $t\in J_{0}$}\\
\phi(0)-f(0,\phi)+f(t,y_{t})+\int_{0}^{t}g(s)ds, & \mbox {if $t\in J$ }
\end{array}\right.\right\}
$$
where
$$
g\in S_{F,y}=\Bigl\{g\in L^{1}(J,E) : g(t)\in F(t,y_t) \ \ \mbox{for
a.e.} \ \ t\in J \Bigr\}.
$$
\begin{remark}  It is clear that the fixed points of $N$ are  solutions
to (1.1)-(1.2).
\end{remark}
We shall show that $N$ is a completely continuous multivalued map, u.s.c.
with convex closed values. The proof will be given in several steps.

\paragraph{Step 1:} {\em $Ny$ is convex for each $y\in C(J_1,E)$. }
Indeed, if $h_1,\ h_{2}$ belong to $Ny$, then there exist $g_1,
g_{2}\in S_{F,y}$ such that  for each $t\in J$ we have
$$
h_1(t)=\phi(0)-f(0,\phi)+f(t,y_t)+\int_0^{t}g_1(s)ds$$
and
$$
h_{2}(t)=\phi(0)-f(0,\phi)+f(t,y_t)+\int_0^{t}g_{2}(s)ds.$$
Let $0\leq k\leq 1$. Then for each $t\in J$ we have
$$
(kh_1+(1-k)h_{2})(t)=\phi(0)-f(0,\phi)+f(t,y_t)+\int_0^{t}[kg_1(s)+(1-k)
g_{2}(s)]ds.
$$
Since $S_{F,y}$ is convex (because $F$ has convex values)
then
$$
kh_1+(1-k)h_{2}\in Ny
$$
which finish the proof of Step 1. \par
We next shall prove that $N$ is a completely continuous operator.
Using (H4) it suffices to show that the operator $
N_{1}:C(J_{1},E)\longrightarrow 2^{C(J_{1},E)}$ defined by: \par
$$
N_{1}y:=\left\{ h\in C(J_{1},E): h(t)=\left\{\begin{array}{ll}\phi(t), &
\mbox{ if $t\in J_{0}$}\\
\int_{0}^{t}g(s)ds, &\mbox{if $t\in J$ }
\end{array}\right.\right\}
$$
is completely continuous.

\paragraph{Step 2:} {\em $N_1$ maps bounded sets into bounded sets in
$C(J_1,E).$ }
Indeed, it is enough to show that there exists a positive constant $\ell$
such that for each $h\in N_1y, y\in B_{q}=\{y\in C(J_1,E):
\|y\|_{\infty}\leq q \}$ one has
$\|h\|_{\infty}\leq \ell$. \par
If $h\in N_1y$, then there exists $g\in S_{F,y}$ such that for each
$t\in J$ we have
$$
h(t)=\int_0^{t}g(s)ds.$$
By (H3) we have for each $t\in J$
\begin{eqnarray*}
\|h(t)\|&\leq&\int_0^{t}\|g(s)\|ds \\
&\leq& \sup_{y\in [0,q]}\psi(y)\Bigl(\int_0^{t}p(s)ds\Bigr)\\
&\leq& \sup_{y\in [0,q]}\psi(y)\Bigl(\int_0^{t}p(s)ds\Bigr).
\end{eqnarray*}
Then for each $h\in N(B_{q})$ we have
$$
\|h\|_{\infty}\leq \sup_{y\in
[0,q]}\psi(y)\Bigl(\int_0^{T}p(s)ds\Bigr):=\ell.$$

\paragraph{Step 3:} {\em $N_1$ maps bounded sets into equicontinuous sets of
$C(J_1,E)$. }
Let $t_1, t_{2}\in J, t_1<t_{2}$ and
$B_{q}=\{y\in C(J_1,E): \|y\|_{\infty}\leq q\}$ be a bounded set of
$C(J_1,E)$. \par
For each $y\in B_{q}$ and $h\in N_1y$, there exists $g\in S_{F,y}$ such that
$$
h(t)=\int_0^{t}g(s)ds.$$
Thus
\begin{eqnarray*}
\|h(t_{2})-h(t_1)\|&\leq & \Bigl\|\int_{t_1}^{t_{2}}g(s)ds\Bigr\| \\
&\leq & \sup_{y\in [0,q]}\psi(y)\Bigl(\int_{t_1}^{t_{2}}p(s)ds\Bigr).
\end{eqnarray*}
As $t_{2}\to t_1$ the right-hand side of the above inequality
tends to zero. \par
The equicontinuity for the cases $t_1<t_{2}\leq 0$ and $t_1\leq 0\leq
t_{2}$ are
obvious.\par
As a consequence of Step 2, Step 3, (H4) and (H5) together
with the Ascoli-Arzela theorem we can conclude that
\ $N:C(J_1, E)\to 2^{C(J_1, E)}$ is
a compact multivalued map, and therefore, a condensing map.

\paragraph{Step 4:} {\em $N$ 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 $g_{n}\in S_{F,y_{n}}$ such that
$$
h_{n}(t)=\phi(0)-f(0,\phi)+f(t,y_{nt})+\int_0^{t}g_{n}(s)ds, \ \ t\in J.$$
We must prove that there exists $g_*\in S_{F,y_*}$ such that
$$
h_*(t)=\phi(0)-f(0,\phi)+f(t,y_{*t})+\int_0^{t}g_*(s)ds, \ \ t\in J.$$
Since $f$ is continuous we have that
$$
\|(h_{n}-\phi(0)+f(0,\phi)-f(t,y_{nt}))-(h_*-\phi(0)+f(0,\phi)-f(t,y_{*t})) \|
_{\infty} \to 0,$$
as $n\to \infty$.
Consider the linear continuous operator
$$\displaylines{
\Gamma: L^{1}(J,E)\to C(J,E) \cr
g\mapsto \Gamma(g)(t)= \int_0^{t}g(s)ds.
}$$
From Lemma \ref{l2}, it follows that $\Gamma\circ S_{F}$ is a closed graph
operator.
Moreover, we have that
$$
(h_{n}(t)-\phi(0)+f(0,\phi)-f(t,y_{nt})) \in \Gamma(S_{F,y_{n}}).$$
Since $y_{n}\to y^*$, it follows from Lemma \ref{l2} that
$$
(h_*(t)-\phi(0)+f(0,\phi)-f(t,y_t))= \int_0^{t}g_*(s)ds$$
for some $g_*\in S_{F,y^*}$.

\paragraph{Step 5:} {\em The set
$\Omega:=\{y\in C(J_1, E): \lambda y\in Ny \,\, \mbox{for some}\,\,
\lambda>1 \}$
is bounded. }
Let $y\in \Omega.$ Then $\lambda y\in Ny$ for some $\lambda>1$. Thus
there exists
$g\in S_{F,y}$ such that for $t\in J$,
$$
y(t)=\lambda^{-1}\phi(0)-\lambda^{-1}f(0,\phi)+\lambda^{-1}f(t,y_t)+\lambda^{-
1}\int_0^{t}g(s)ds\,.$$
This implies by  (H1), (H3) that for each $t\in J$ we have
$$
\|y(t)\|\leq \|\phi\|+c_1\|\phi\|+2c_{2}+c_1\|y_t\|+
\int_0^{t}p(s)\psi(\|y_{s}\|)ds.
$$
We consider the function $\mu$ defined by
$$
\mu(t)=\sup\{|y(t)|: -r\leq s\leq t \}, \ \ t\in J.$$
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& \|\phi\|+c_1\|\phi\|+2c_{2}+c_1\|y_t\|+
\int_0^{t}p(s)\psi(\|y_{s}\|)ds \\
&\leq&
\|\phi\|+c_1\|\phi\|+2c_{2}+c_1\mu(t)+\int_0^{t}p(s)\psi(\mu(s))ds.
\end{eqnarray*}
Thus
$$
\mu(t)\leq\frac{1}{1-c_1}\Bigl\{(1+c_1)\|\phi\|+2c_{2}+
\int_0^{t}p(s)\psi(\mu(s))ds.\Bigr\}$$
If $t^*\in J_0$ then $\mu(t)=\|\phi\|$ and the previous inequality
holds. \\
Let us take the right-hand side of the above inequality as $v(t)$, then we
have
$$
c=v(0)=\frac{1}{1-c_1}\Bigl\{(1+c_1)\|\phi\|+2c_{2}\Bigr\} \ \
\mbox{and} \ \ \mu(t)\leq v(t), \ \ t\in J.$$
Using the nondecreasing character of $\psi$ we get
$$
v'(t)\leq p(t)\psi(v(t)),\,\, t\in J.$$
This implies  for each $t\in J$ that
$$
\int_{v(0)}^{v(t)}\frac{du}{\psi(u)}\leq \int_0^{T}p(s)ds<
\int_{v(0)}^{\infty}\frac{du}{\psi(u)}.$$
This inequality implies that there exists a constant $b$ such that
$v(t)\leq b, \ t\in J$, and hence $\mu(t)\leq b, \ t\in J$.
Since for every $t\in J, \|y_t\|\leq\mu(t)$, we have
$$
\|y\|_{\infty}:=\sup\{|y(t)|: -r\leq t\leq T \}\leq b,$$
 where $b$ depends only on $T$ and on the functions $p$ and $\psi$.
This shows that $\Omega$ is bounded.

Set $X:=C(J_1,E)$. As a consequence of Lemma \ref{l1} we deduce that
$N$ has a fixed point which is a solution of (1.1)-(1.2).
\hfill$\diamondsuit$


\section{ Existence results for integrodifferential inclusions}

In this section we consider the solvability of IVP (1.3)-(1.4).
Let us state the following hypotheses:

\paragraph{(H6)}  For each $t\in J, \ K(t,s)$ is measurable on  $J$ and
$$
K(t)= \mbox{ess sup} \{|K(t,s)|,  \ \ 0\leq s\leq t \},$$
is bounded on $J$;

\paragraph{(H7)} The map $t\longmapsto K_t$ is continuous from $J$ to
$L^{\infty}(J,{\mathbb R})$; here
$K_t(s)=K(t,s);$

\paragraph{(H8)} $\|F(t,u)\|:=\sup\{|v|: v\in F(t,u) \}\leq
p(t)\psi(\|u\|)$ for
almost all $t\in J$ and all $u \in C(J_0,E)$,
where $p\in L^{1}(J,{\mathbb R}_{+})$ and  \ $\psi:{\mathbb R}_{+}\to
(0,\infty)$ is
continuous and increasing  with
$$
T\sup_{t\in
J}K(t)\int_0^{T}p(s)ds<\int_{c}^{\infty}\frac{d\tau}{\psi(\tau)}; $$
where $c=\frac{1}{1-c_1}\Bigl\{(1+c_1)\|\phi\|+2c_{2}\Bigr\}$;

\paragraph{(H9)} For each bounded $B\subset C(J_1,E)$,  $u\in B$ and $t\in J$
the set
$$
\Bigl\{\int_0^{t}\int_0^{s}K(s,\sigma)g(\sigma)d\sigma ds:
g\in S_{F,u} \Bigr\}$$
is relatively compact.

\begin{definition} By a solution to the IVP (1.3)-(1.4) it mean a
function $y: J_1\to E$ such that $y_0=\phi, \
y_t\in C(J_0,E)$, the function $y(t)-f(t,y_t)$ is absolutely
continuous and the inclusion (1.3) hold a.e. on $J$.
\end{definition}
Now, we are able to state and prove our main theorem.

\begin{theorem}  Assume that hypotheses (H1), (H2), (H4), (H6)-(H9) are
satisfied.
Then the IVP (1.3)-(1.4) has at least one  solution on $J_1$.
\end{theorem}

\paragraph{Proof.} Let $C(J_1,E)$ be the Banach  space of continuous functions
from
$J_1$ into $E$ endowed with the sup-norm
$$
\|y\|_{\infty}:=\sup\{|y(t)|: t\in [-r,T] \}, \ \mbox{for} \ y\in C(J_1,E).
$$
Transform the  problem into a fixed point problem.
Consider the multivalued map, $N:C(J_{1},E)\longrightarrow 2^{C(J_{1},E)}$
defined by: \par
$$
Ny:=\left\{ h\in C(J_{1},E): h(t)=\left\{\begin{array}{ll}\phi(t), & \mbox{
if $t\in J_{0}$}\\[0.2cm]
\phi(0)-f(0,\phi)+f(t,y_{t})\\
\quad +\int_{0}^{t}\int_{0}^{s}K(s,u)g(u)duds, & \mbox{if $t\in J$
}
\end{array}\right.\right\}
$$
where
$$
g\in S_{F,y}=\Bigl\{g\in L^{1}(J,E) : g(t)\in F(t,y_t) \ \ \mbox{for
a.e.} \ \ t\in J \Bigr\}.
$$

\begin{remark}  It is clear that the fixed points of $N$ are solutions
to (1.3)-(1.4).
\end{remark}
As in Theorem \ref{t1} we can show that  $N$ is a completely continuous
multi-valued
map, u.s.c. with convex closed values, and therefore a condensing map.\par
Here we repeat the proof  that the set
$$
\Omega:=\{y\in C(J_1, E): \lambda y\in Ny,  \,\, \mbox{for some}\,\,
\lambda>1 \}$$
is bounded. Let $y\in \Omega.$ Then $\lambda y\in Ny$ for some $\lambda>1$.
Thus
there exists
$g\in S_{F,y}$ such that for $t\in J$,
$$
y(t)=\lambda^{-1}\phi(0)-\lambda^{-1}f(0,\phi)+\lambda^{-1}f(t,y_t)+
\lambda^{-1}\int_0^{t}\int_0^{s}K(s,u)g(u)\,du\,ds.$$
This implies by  (H1), (H6)-(H8) that for each $t\in J$ we have
\begin{eqnarray*}
\|y(t)\|&\leq&
\|\phi\|+c_1\|\phi\|+2c_{2}+c_1\|y_t\|+\Bigl\|\int_0^{t}\int_0^{s}K(s,
u)g(u)\,du\,ds\Bigr\| \\
&\leq&
\|\phi\|+c_1\|\phi\|+2c_{2}+c_1\|y_t\|+\int_0^{t}\int_0^{s}|K(s,u)|p(u
)\psi(\|y_{u}\|)\,du\,ds \\
&\leq& \|\phi\|+c_1\|\phi\|+2c_{2}+c_1\|y_t\|+T\sup_{t\in
J}K(t)\int_0^{t}p(s)\psi(\|y_{s}\|)ds.
\end{eqnarray*}
We consider the function $\mu$ defined by
$$
\mu(t)=\sup\{|y(t)|: -r\leq s\leq t \}, \ \ t\in J.$$
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& \|\phi\|+c_1\|\phi\|+2c_{2}+c_1\|y_t\|+T\sup_{t\in
J}K(t)\int_0^{t}p(s)\psi(\|y_{s}\|)ds \\
&\leq& \|\phi\|+c_1\|\phi\|+2c_{2}+c_1\mu(t)+T\sup_{t\in
J}K(t)\int_0^{t}p(s)\psi(\mu(s))ds.
\end{eqnarray*}
Thus
$$
\mu(t)\leq\frac{1}{1-c_1}\Bigl\{(1+c_1)\|\phi\|+2c_{2}+
T\sup_{t\in J}K(t)\int_0^{t}p(s)\psi(\mu(s))ds\Bigr\}.
$$
If $t^*\in J_0$ then $\mu(t)=\|\phi\|$ and the previous inequality
holds.\par
Let us take the right-hand side of the above inequality as $v(t)$, then we
have
$$
c=v(0)=\frac{1}{1-c_1}\Bigl\{(1+c_1)\|\phi\|+2c_{2}\Bigr\} \ \
\mbox{and} \ \
\mu(t)\leq v(t),\,\, t\in J.$$
Using the nondecreasing character of $\psi$ we get
$$
v'(t)\leq T\sup_{t\in J}K(t)p(t)\psi(v(t)),\,\, t\in J.$$
This implies  for each $t\in J$ that
$$
\int_{v(0)}^{v(t)}\frac{du}{\psi(u)}\leq T\sup_{t\in J}K(t)\int_0^{T}p(s)ds<
\int_{v(0)}^{\infty}\frac{du}{\psi(u)}.
$$
This inequality implies that there exists a constant $b$ such that
$v(t)\leq b, \ t\in J$, and hence $\mu(t)\leq b, \ t\in J$.
Since for every $t\in J, \|y_t\|\leq\mu(t)$, we have
$$
\|y\|_{\infty}:=\sup\{|y(t)|: -r\leq t\leq T \}\leq b,$$
 where $b$ depends only  on $T$ and on the functions $p$ and $\psi$.
This shows that $\Omega$ is bounded.

Set $X:=C(J_1,E)$. As a consequence of Lemma \ref{l1} we deduce that
$N$ has a fixed point which is a solution of (1.3)-(1.4).
\hfill$\diamondsuit$


\section{Second order differential inclusions}

In this section we consider the solvability of IVP (1.5)-(1.6).
For the study of this problem we first list the following hypotheses:



\paragraph{(H10)} $\|F(t,u)\|:=\sup\{|v|: v\in F(t,u) \}\leq
p(t)\psi(\|u\|)$ for
almost all $t\in J$
and all $u\in C(J_0,E)$, where $p\in L^{1}(J,{\mathbb R}_{+})$ and  \
$\psi:{\mathbb R}_{+}\to (0,\infty)$ is continuous and increasing  with
$$
\int_0^{T}M(s)ds<\int_{c}^{\infty}\frac{d\tau}{u+\psi(\tau)}; $$
where $c=\|\phi\|+[|\eta|+c_1\|\phi\|+2c_{2}]T$ and
$M(t)=max\{1,c_1,p(t)\}$;

\paragraph{(H11)} for each bounded  $B\subset  C(J_1,E)$,  $y\in B$ and
$t\in J$ the set
$$
\Bigl\{\int_0^{t}\int_0^{u}g(u)\,du\,ds: g\in S_{F,y} \Bigr\}$$
is relatively compact.

\begin{definition} By a solution to the IVP (1.5)-(1.6) we shall mean a
differentiable function $y: J_1\to E$ such that $y_0=\phi,
y'(0)=\eta,
y_t\in C(J_0,E)$, the function $y'(t)-f(t,y_t)$ is absolutely
continuous and the inclusion (1.5) hold a.e. on $J$.
\end{definition}

Now, we are in a position to state and prove our main theorem in this section.

\begin{theorem}  Assume that hypotheses (H1), (H2), (H4), (H10), (H11) hold.
Then the IVP (1.5)-(1.6) has at least one solution on $J_1$.
\end{theorem}
\paragraph{Proof.}  Let $C(J_1,E)$ be the Banach  space of continuous
functions from
$J_1$ into $E$ endowed with the sup norm
$$
\|y\|_{\infty}:=\sup\{|y(t)|: t\in [-r,T] \}, \ \mbox{for} \ y\in C(J_1,E).
$$
Transform the  problem into a fixed point problem.
Consider the multivalued map, $N:C(J_{1},E)\longrightarrow 2^{C(J_{1},E)}$
defined by: \par
$$
Ny:=\left\{ h\in C(J_{1},E): h(t)=\left\{\begin{array}{ll}\phi(t), & \mbox{
if $t\in J_{0}$}\\[0.2cm]
\phi(0)+[\eta-f(0,\phi)]t\\
+\int_{0}^{t}f(s,y_{s})ds+\int_{0}^{t}\int_{0}^{s}g(u)duds, &
\mbox{if $t\in J$ }
\end{array}\right.\right\}
$$

\begin{remark}  It is clear that the fixed points of $N$ are  solutions
to (1.5)-(1.6).
\end{remark}

As in Theorem \ref{t1} we can show that  $N$ is a completely continuous
multi-valued map, u.s.c.
with convex closed values.

Now we prove only  that the set
$$
\Omega:=\{y\in C(J_1, E): \lambda y\in Ny, \,\, \mbox{for some}\,\,
\lambda>1 \}$$
is bounded.

Let $y\in \Omega.$ Then $\lambda y\in Ny$ for some $\lambda>1$. Thus
there exists
$g\in S_{F,y}$ such that for $t\in J$,
$$
y(t)=\lambda^{-1}\phi(0)+\lambda^{-1}[\eta-f(0,\phi)]t+
\lambda^{-1}\int_0^{t}f(s,y_{s})ds+\lambda^{-1}\int_0^{t}\int_0^{u}g(u)
\,du\,ds\,.$$
This implies by  (H1), (H3) that for each $t\in J$ we have
\begin{eqnarray*}
\|y(t)\| &\leq&\|\phi\|+[|\eta|+c_1\|\phi\|+2c_{2}]T \\
&&+c_1\int_0^{t}\|y_{s}\|ds+
\int_0^{t}\int_0^{s}p(u)\psi(\|y_{u}\|)\,du\,ds\\
&\leq& \|\phi\|+[|\eta|+c_1\|\phi\|+2c_{2}]T \\
&&+\int_0^{t}M(s)\|y_{s}\|ds+
\int_0^{t}M(s)\int_0^{s}\psi(\|y_{u}\|)\,du\,ds\,,
\end{eqnarray*}
where $M(t)=max\{1, c_1, p(t)\}$.
We consider the function $\mu$ defined by
$$
\mu(t)=\sup\{|y(t)|: -r\leq s\leq t \}, \ \ t\in J.$$
Let $t^*\in [-r,t]$ be such that $\mu(t)=|y(t^*)|$. If $t^*\in
J$, by the previous inequality
we have for $t\in J$
$$
\mu(t)\leq \|\phi\|+[|\eta|+c_1\|\phi\|+2c_{2}]T+\int_0^{t}M(s)\mu(s)ds+
\int_0^{t}M(s)\int_0^{s}\psi(\mu(u))\,du\,ds.
$$
If $t^*\in J_0$ then $\mu(t)=\|\phi\|$ and the previous inequality
holds.
Denoting by $u(t)$ the right hand side of the above inequality we have
$$u(0)=\|\phi\|+[|\eta|+c_1\|\phi\|+2c_{2}]T=c, \quad \mu(t)\le u(t),
\quad t\in J$$
and
\begin{eqnarray*}
u'(t)&\le& M(t)\mu(t)+M(t)\int_0^{t}\psi(\mu(s))ds\\
&\le & M(t)\left[u(t)+\int_0^{t}\psi(u(s))ds\right], \quad t\in J.
\end{eqnarray*}
Put
$$v(t)=u(t)+\int_0^{t}\psi(u(s))ds, \quad t\in J.$$
$$u(0)=\|\phi\|+[|\eta|+c_{1}\|\phi\|+2c_{2}]T=c, \quad \mu(t)\le u(t),
\quad t\in J$$
and
\begin{eqnarray*}
v'(t)&=& u'(t)+\psi(u(t))\\
&\le & M(t)[v(t)+\psi(v(t))], \quad t\in J.
\end{eqnarray*}
This implies that  for each $t\in J$,
$$
\int_{v(0)}^{v(t)}\frac{du}{u+\psi(u)}\leq \int_0^{T}M(s)ds<
\int_{v(0)}^{\infty}\frac{du}{u+\psi(u)}\,.$$
This inequality implies that there exists a constant $b$ such that
$v(t)\leq b, \ t\in J$, and hence $\mu(t)\leq b, \ t\in J$.
Since for every $t\in J, \|y_t\|\leq\mu(t)$, we have
$$
\|y\|_{\infty}:=\sup\{|y(t)|: -r\leq t\leq T \}\leq b,$$
 where $b$ depends only on $T$ and on the functions $p$ and $\psi$.
This shows that $\Omega$ is bounded.

Set $X:=C(J_1,E)$. As a consequence of Lemma \ref{l1} we deduce that
$N$ has a fixed point which is a solution of (1.5)-(1.6).
\hfill$\diamondsuit$\smallbreak


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\noindent{\sc M. Benchohra }\\
D\'epartement de Math\'ematiques, Universit\'e de Sidi Bel Abb\`es, \\
 BP 89, 22000 Sidi Bel Abb\`es, Alg\'erie \\
e-mail: benchohra@yahoo.com \smallskip

\noindent{\sc S. K. Ntouyas }\\
Department of Mathematics, University of Ioannina, \\
451 10 Ioannina, Greece\\
e-mail: sntouyas@cc.uoi.gr


\end{document}