\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 50, pp. 1--10.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/50\hfil Existence of viable solutions]
{Existence of viable solutions for nonconvex  differential
inclusions}
\author[M. Bounkhel, T. Haddad\hfil EJDE-2005/50\hfilneg]
{Messaoud Bounkhel, Tahar Haddad}  % in alphabetical order

 \address{Messaoud Bounkhel \hfill\break
King Saud University, College of Science,
Department of Mathematics, Riyadh 11451, Saudi Arabia}
\email{bounkhel@ksu.edu.sa}

\address{Tahar Haddad \hfill\break
University of Jijel, Department of Mathematics,
B.P. 98, Ouled Aissa, Jijel, Algeria}
\email{haddadtr2000@yahoo.fr}

\date{}
\thanks{Submitted December 26, 2004. Published May 11, 2005.}
\subjclass[2000]{34A60, 34G25, 49J52, 49J53}
\keywords{Uniformly regular functions; normal cone; \hfill\break\indent
nonconvex differential inclusions}

\begin{abstract}
 We show the existence result of viable solutions to the
 differential inclusion
 \begin{gather*}
 \dot x(t)\in  G(x(t))+F(t,x(t))  \\
  x(t)\in S \quad \mbox{on } [0,T],
 \end{gather*}
 where  $F: [0,T]\times H\to H$ $(T>0)$ is a continuous
 set-valued mapping, $G:H\to H$  is a Hausdorff upper
 semi-continuous set-valued mapping such that
 $G(x)\subset \partial g(x)$, where $g :H\to \mathbb{R}$ is a regular
 and locally Lipschitz function  and  $S$  is a ball, compact
 subset in  a separable Hilbert space $H$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Porposition}
\newtheorem{rem}[theorem]{Remark}
\newtheorem{coro}[theorem]{Corolloray}

\section{Introduction}

Let $T>0$. It is well known that the solution set of the
differential inclusion
\begin{gather*} % (*)
\dot x(t)\in  G(x(t))  \quad \mbox{ a.e. } \quad [0,T]\\
 x(0)=x_0 \in \mathbb{R}^d,
\end{gather*}
can be empty when the set-valued mapping $G$ is upper
semicontinuous with non\-empty nonconvex values. Bressan, Cellina
and Colombo \cite{bressancellinacolombo},  proved an existence
result fo the above equation  by assuming that  the set-valued
mapping $G$  is  included in the  subdifferential of  a convex
lower semicontinuous  (l.s.c.) function $g:\mathbb{R}^d \to
\mathbb{R}$. This result  has been extended in many ways by many
authors; see for example
\cite{anconacolombo,benabdellah2,benabdellahcastaing,bounkhel1,
morchadi,rossi,Trongducha}. The recent extension of the above
equation was studied by Bounkhel \cite{bounkhel1}, in which the
author proved an existence result of viable solutions
 in the finite dimensional
case for the differential inclusion
\begin{equation}%\label{DI}
\begin{gathered}
\dot x(t)\in G(x(t))+F(t,x(t))\quad\mbox{ a.e. } [0,T]\\
x(t)\in S, \quad \mbox{on  } [0,T].
\end{gathered}  \label{2}
\end{equation}
This extension covers all the other extensions given in the finite
dimensional case.  In the present paper we extend this result to
the infinite dimensional setting.
A function $x(\cdot)$ is called a viable solution if it satisfies 
the differential inclusion and $x(t)\in S$
for all $t\in [0,T]$ and for some closed set $S$.

\section{Uniformly regular functions}

Let $H$ be a real separable Hilbert space. Let us recall
the concept of regularity that will be used in the sequel \cite{bounkhel1}.

\begin{definition}[\cite{bounkhel1}] \label{def1} \rm
Let $f:H\to \mathbb{R}\cup \{+\infty \}$ be a l.s.c. function and
let $O\subset \mathop{\rm dom} f$ be a nonempty open subset.
We will say that $f$ is uniformly regular over $O$ if there exists
a positive number $\beta \geq 0$ such that
for all $x\in O$ and for all $\xi \in \partial ^{P}f(x)$ one has
\begin{equation}
\langle \xi ,x'-x\rangle \leq f(x')-f(x)+\beta \Vert x'-x\Vert^{2}
\quad \mbox{for all } x'\in O.
\end{equation}
\end{definition}

Here $\partial ^{P}f(x)$ denotes the proximal subdifferential of
$f$ at $x$ (for its definition the reader is refereed for instance
to \cite{bounkhelthibault3}). We will say that $f$ is uniformly
regular over closed set $S$ if there exists an open set $O$
containing $S$ such that $f$ is uniformly regular over $O$. The
class of functions that are uniformly regular over sets is so
large. For more details and examples we refer the reader to
\cite{bounkhel1}. The following proposition summarizes some
important properties for uniformly regular locally Lipschitz
functions over sets needed in the sequel. For the proof of these
results we refer the reader to \cite{bounkhel1,bounkhel3}.

\begin{proposition}\label{prop1}
Let $f:H\to \mathbb{R}$ be a locally Lipschitz function and $S$ a
nonempty closed set. If $f$ is uniformly regular over $S$, then
the following hold:
\begin{itemize}
\item[(i)] The proximal subdifferential of $f$ is closed over $S$,
that is, for every $x_{n}\to x\in S$ with $x_{n}\in S$ and
every $\xi _{n}\to \xi $ with $\xi _{n}\in \partial
^{P}f(x_{n})$ one has $\xi \in \partial ^{P}f(x)$

\item[(ii)] The
proximal subdifferential of $f$ coincides with $\partial ^{C}f(x)$
the Clarke subdifferential for any point $x$ (see for instance
\cite{bounkhelthibault3} for the definition of $\partial ^{C}f$)

\item[(iii)] The proximal subdifferential of $f$ is upper
hemicontinuous over $S$, that is, the support function $x\mapsto
\langle v,\partial ^{P}f(x)\rangle $ is u.s.c. over $S$
for every $v\in H$

\item[(iv)] For any absolutely continuous map
$x:[0,T]\to S$ one has
$$ \frac{d}{dt}(f\circ x)(t)=\langle
\partial^C f(x(t));\dot x(t) \rangle.
$$
 \end{itemize}
\end{proposition}

Now we are in position to state and prove our main result in this
paper.

\begin{theorem} \label{thm2.1}
Let  $g:H\to \mathbb{R}$ be a locally Lipschitz function and
$\beta$-uniformly regular over $S\subset H$. Assume that
\begin{itemize}
\item[(i)]   $S$ is nonempty ball compact subset in $H$, that is,
the set $S\cap r \mathbb{B}$ is compact for any $r>0$;
\item[(ii)] $G:H
\to H$ is a Hausdorff $u.s.c$ set valued map with compact
values satisfying $G(x)\subset \partial ^{C}g(x)$ for all $x\in
S$;
\item[(iii)] $F:[0,T]\times H\to  H$ is a continuous set
valued map with compact values;

\item[(iv)] For any $(t,x)\in I\times S$,   the following tangential
condition holds
\begin{equation}
 \liminf_{h\to 0}
\frac{1}{h}e\big(x+h\left[G(x)+F(t,x)\right];S\big)=0\,, \label{3}
\end{equation}
where $ e(A;S):= \sup_{a\in A} d_S(a)$.
\end{itemize}
 Then, for any $x_{0}\in S$ there exists $a\in ]0,T[$ such that
the differential inclusion (\ref{DI}) has a viable solution on
$[0,a]$.
\end{theorem}

\begin{proof}
 Let  $\rho >0$ such that $K_{0}:=S\cap (x_{0}+\rho
B)$ is compact and $g$ is $L$-Lipschitz on $x_{0}+\rho B$. Since
$F$ and $G$ are continuous and the set $I\times K_0$ is compact,
 there exists a positive scalar $M$ such that
\begin{equation}
\| G(x)\| +\| F(t,x)\| \leq M,
\end{equation}
for all $(t,x)\in I\times K_{0}$. Since $(t_{0},x_{0})\in I\times
K_{0}$, then (by $(\ref{3})$)
\[
 \liminf_{h\to 0}\frac{1}{h}e\Big(x_{0}+h\big[G(x_{0})
+F(t_{0},x_{0})\big];S\Big)=0.
\]
Put $\alpha:=\min\{T, \frac{\rho}{M+1},1\}$. Hence for every $m
\ge 1 $ there exists $0<\xi< \frac{\alpha}{2}$ such that
\begin{equation}\label{eq:6}
e\Big(x_{0}+\xi
\left[G(x_{0})+F(t_{0},x_{0})\right];S\Big)<\frac{\xi}{m}.
\end{equation}
Let  $b_{0}\in G(x_{0})+F(t_{0},x_{0})$ and put
$$
\lambda _{0}^{m}:=\max\big\{\xi\in (0,\frac{\alpha}{2}]: \xi \le
T-t_0 \mbox{ and } d_{S}(x_{0}+\xi b_{0})<\frac{\xi}{m} \big\}.
$$
Since $x_0\in S$, we have
\[
d_{S}(x_{0}+\lambda _{0}^{m}b_{0})\leq \lambda _{0}^{m}\Vert
 b_{0}\Vert \le \lambda _{0}^{m} M < M.
\]
So, there exists $\Psi_{0}^{m}\in S \cap {\mathbb{B}} (x_{0}+\lambda
_{0}^{m}b_{0},M+1)$ such that
\[
 \Vert \Psi _{0}^{m}-x_{0}-\lambda_{0}^{m}b_{0}\Vert =d_{S}(x_{0}
+\lambda _{0}^{m}b_{0}),
\]
and so
\[
\| \frac{1}{\lambda _{0}^{m}}[\Psi
_{0}^{m}-x_{0}]-b_{0}\| =\frac{1}{\lambda
_{0}^{m}}d_{S}(x_{0}+\lambda _{0}^{m}b_{0}) < \frac{1}{m}
\]
by (\ref{eq:6}) and the definition of  $\lambda _{0}^{m}$.
Let $w_{0}^{m}:=\frac{\Psi^{m}_{0}-x_{0}}{\lambda _{0}^{m}}$ and
$x_1^m:=x_{0}+\lambda _{0}^{m}w_{0}^{m}\in S$. Thus, we obtain
\begin{equation}
\begin{gathered}
w_{0}^{m}\in G(x_{0})+F(t_{0},x_{0})+\frac{1}{m}B, \\
 \| x_{1}^{m}-x_{0}\| =\lambda _{0}^{m}\| w_{0}^{m}\| <\lambda
_{0}^{m}(M+\frac{1}{m})<\lambda _{0}^{m}(M+1).
\end{gathered} \label{eq:resume1}
\end{equation}
We can choose, a priori, $a<\alpha$ and find $\lambda _{0}^{m} <
a$ such that $0<\lambda _{0}^{m}<a <T$. Then $ \Vert
x_{1}^{m}-x_{0} \Vert <\rho$, that is, $x_{1}^{m}\in (x_{0}+\rho
B)$ and so (\ref{eq:resume1}) ensures $x_{1}^{m}\in S\cap
(x_{0}+\rho B)=K_{0}$.
 We reiterate this process for
constructing  sequences $\{w_{i}^{m}\}_{i}$, $\{t_{i}^{m}\}_{i}$,
$\{\lambda_{i}^{m}\}_{i}$, and $\{x_{i}^{m}\}_{i}$ satisfying for
 some rank  $\nu _{m} \ge 1 $ the following assertions:
\begin{enumerate}
\item[(a)] $0=t_{0}^{m}$,  $t_{\nu _{m}}^{m}\leq a <T$ with
$ t_{i}^{m}=\sum_{k=0}^{i-1}\lambda _{k}^{m}$ for all
$i\in \{1,\dots ,\nu _{m}\}$;

\item[(b)] $x_{i}^{m}=x_{0}+  \sum_{k=0}^{i-1}
\lambda_{k}^{m}w_{k}^{m}$ and
$(t_{i}^{m},x_{i}^{m})\in [ 0,T]\times K_{0}$ for all
$i\in \{0,\dots ,\nu_{m}\}$;

\item[(c)] $w_{i}^{m}\in G(x_{i}^{m})+F(t_{i}^{m},x_{i}^{m})+\frac{1}{m}B$
with $w_{i}^{m}=\frac{\Psi _{i}^{m}-x_{i}^{m}}{\lambda _{i}^{m}}$ and
$\Psi _{i}^{m}\in S \cap \mathbb{B}(x_{i}^{m}+\lambda_{i}^{m}b_{i}^{m},M+1)$
for all $i\in \{0,\dots ,\nu _{m}-1\}$, where
$$
\lambda _{i}^{m}:=\max\{\xi\in (0,\frac{\alpha}{2}]: \xi\le T-t^m_i
\mbox{ and } d_{S}(x^m_{i}+\xi b_{i})<\frac{1}{m} \xi\}(\forall\,
i = 1,\dots ,\nu _{m}-1).
$$
\end{enumerate}
It is easy to see that for $i=1$ the assertions (a), (b), and (c)
are fulfilled. Let now $i\ge 2$. Assume that (a), (b), and (c) are
satisfied  for any $j = 1,\dots ,i$.
If, $a<t_{i+1}^{m}$, then we take $\nu _{m}=i$ and so the process
of iterations is stopped and we get (a), (b), and (c)  satisfied
with
$$t_{\nu _{m}}^{m}\leq a< t_{\nu _{m}+1}^{m} <T.
$$
In the other case, i.e., $t_{i+1}^{m}\leq a$, we define
$x_{i+1}^{m}$ as follows
\[
x_{i+1}^{m}:=x_{i}^{m}+\lambda
_{i}^{m}w_{i}^{m}=x_{0}+\sum_{k=0}^{i}\lambda _{k}^{m}w_{k}^{m}
\]
and so
\[
\| x_{i+1}^{m}-x_{0}\| \leq \sum_{k=0}^{i}\lambda
_{k}^{m}\Vert w_{k}^{m} \Vert  \leq (M+1) \sum_{k=0}^{i}\lambda
_{k}^{m} \leq  t_{i+1}^{m}(M+1) \leq a(M+1) <\rho,
\]
which ensures that $x_{i+1}^{m}\in K_{0}$. Thus the conditions
(a), (b), and (c) are satisfied for $i+1$. Now we have to prove
that this iterative process is finite, i.e., there exists a
positive integer $\nu_m$ such that $$ t^m_{\nu_m}\le a <
t^m_{\nu_m+1}.
$$
Suppose the contrary that is,
$$
 t^m_{i}\le a, \quad \mbox{for all } i\ge 1.
$$
Then the bounded increasing sequence $\{t^m_i\}_i$ converges to
some $\bar t$ such that $\bar t \le a < T$. Hence
$$
\Vert x^m_i-x^m_j\|\le (M+1)|t^m_i-t^m_j|\to 0 \quad \mbox{ as } i,j\to
\infty.
$$
Therefore, the sequence $\{x_{i}\}_{i}$ is a Cauchy sequence and
hence, it converges to some $\bar x\in K_0$.  As
$(\overline{t},\overline{x})\in [ 0,T]\times K_{0}$, by
$(\ref{3})$ and the Hausdorff upper semi-continuity  of $G+F$,
there exist  $\lambda \in (0,T-\overline{t})$, and
an integer $i_{0}\ge 1$ such that for all $i\geq i_{0}$,
\begin{gather}
e\Big(\overline{x}+\lambda
\left[G(\overline{x})+F(\overline{t},\overline{x})\right];S\Big)
\leq  \frac{\lambda }{6m} \label{12bis} \\
{e}\Big(G(x^m_{i})+F(t^m_{i},x^m_{i});G(\overline{x})+F(\overline{t},\overline{x})\Big)
\leq \frac{1 }{12m} \label{13bis}\\
\| x^m_{i}-\overline{x}\|
\leq \frac{\lambda }{6m} \label{14bis}\\
 \bar t- t^m_{i}  \leq  \frac{ \lambda}{2}.\label{15bis}
\end{gather}
 Therefore, for any $b_{i}\in G(x^m_{i})+F(t^m_{i},x^m_{i})$, there exists
(by the definition of the distance function) an element $\bar b$ in
$G(\overline{x})+F(\overline{t},\overline{x})$ such that
$$
\| b_{i} - \bar b\| \leq d(b_i,G(\overline{x
})+F(\overline{t},\overline{x}))+  \frac{1 }{12m}.
$$
Hence this inequality and (\ref{13bis})  yield
$$
\| b_{i} - \bar b\| \leq e\Big(G(x^m_{i})+F(t^m_{i},x^m_{i});G(\overline{x%
})+F(\overline{t},\overline{x})\Big)+  \frac{1 }{12m}\le \frac{1
}{6m}.
$$
This last inequality and the relations (\ref{12bis}) and
(\ref{14bis}) ensure
\begin{gather*}
d_{S}(x^m_{i}+\lambda b_{i}) \leq \| x^m_{i}-\overline{x
}\| +d_{S}(\overline{x}+\lambda \bar b)+\lambda \Vert
b_{i}-\bar b  \Vert \\
\leq \frac{ \lambda }{6m}+e\Big(\overline{x}+\lambda
\left[G(\overline{x})+F(\overline{t},\overline{x})\right];S\Big)+\frac{
\lambda }{6m} \leq \frac{ \lambda }{2m}.
\end{gather*}
On the other hand, by construction and by (\ref{15bis}), we obtain
$$
t^m_{i+1}\le  \overline{t}<t^m_{i}+\lambda \leq T, \mbox{ and
hence } \lambda > t^m_{i+1}-t^m_{i}=\lambda^m_{i}.
$$
Thus, there exists  some $\lambda > \lambda^m_i$ such that
$0<\lambda < T-\bar t\le T-t^m_i$ (for all $i\ge i_0$) and
$ d_S(x^m_i+\lambda b_i)\le
\frac{\lambda}{2m}<\frac{\lambda}{m}$.
 This contradicts the definition of $\lambda^m_{i}$. Therefore, there
is an integer $\nu_m \ge 1$ such that $t_{\nu_m}\leq a<t_{\nu_m +1}$ and for
which the assertions (a), (b), and (c) are fulfilled.

According to what precedes, we have (by (c))
\begin{align*}
 \Vert \Psi _{i}^{m} \Vert
&\le \Vert \Psi _{i}^{m}-(x_{i}^{m}+
\lambda_{i}^{m}b_{i}^{m})\Vert  + \Vert x_{i}^{m}+
\lambda_{i}^{m}b_{i}^{m}\Vert \\
&\le  (M+1)+\Vert x_0 -(x_0 - x_{i}^{m})+ \lambda_{i}^{m}b_{i}^{m}\Vert \\
&\le \Vert x_0 \Vert+ \Vert x_0- x_{i}^{m}\|+\lambda
_{i}^{m}\| b_{i}^{m} \Vert  +(M+1) \\
&\leq  \Vert x_{0} \Vert+\rho+2M+1.
\end{align*}
This implies  $\Psi _{i}^{m}\in K_{1}:=S \cap \mathbb{B}(0,R)$, with
$R:=\Vert x_{0} \Vert+\rho+2M+1$. Note that the ball-compactness
of $S$ ensures the compactness of $K_1$.

 On the other hand, it follows from the assertion (c) that
 \begin{equation}
 w_{i}^{m}-f_{i}^{m}-c_{i}^{m}\in G(x_{i}^{m}), \mbox{ where }  c_{i}^{m}\in \frac{1}{m}B
 \quad \mbox{ and } \quad f_{i}^{m}\in
F(t_{i}^{m},x_{i}^{m}),
\end{equation}
 for all $ i\in \{0,\dots ,\nu _{m}\}$.

\subsection*{Approximate Solutions}
Using the sequences $\{x_{i}^{m}\}_{i}$, $\{t_{i}^{m}\}_{i}$,
$\{f_{i}^{m}\}_{i}$, and $\{c_{i}^{m}\}_{i}$ constructed previously
to construct the step functions
$x_{m}(\cdot)$, $f_{m}(\cdot)$, $c_{m}(\cdot)$, and $\theta_m(\cdot)$
with the following properties:
\begin{enumerate}
\item $x_{m}(t)=x_{i}^{m}+(t-t_{i}^{m})w_{i}^{m}$ on
$[t_{i}^{m},t_{i+1}^{m}]$ for all $i\in \{0,\dots ,\nu _{m}\}$;

\item  $f_{m}(t)=f_{m}(\theta _{m}(t))\in F(\theta
_{m}(t),x_{m}(\theta _{m}(t)))$ on $[0,a]$ with
$$
\theta_{m}(t)=t_{i}^{m} \mbox{ if } t\in [ t_{i}^{m},t_{i+1}^{m}[,
\quad \mbox{for all } i\in \{0,\dots ,\nu _{m}\},\; \theta _{m}(a)=a;
$$

\item  $c_{m}(t) = c_{i}^{m}\in \frac{1}{m}B$   if
$t\in [t_{i}^{m},t_{i+1}^{m}]$, for all $i\in \{0,\dots ,\nu _{m}\}$ and
\begin{equation}
  \lim_{m\to \infty} \sup_{t\in [ 0,a]}
 \Vert c_{m}(t) \Vert =0.
\end{equation}
\end{enumerate}
Then
$$
\| x_{m}(t_{i+1}^{m})-x_{m}(t_{i}^{m})\| =(
t_{i+1}^{m}-t_{i}^{m})\| w_{i}^{m}\| \leq (1+M)(
t_{i+1}^{m}-t_{i}^{m}),
$$
and so, for all $i,j\in \{0,\dots ,\nu _{m}-1\}(i>j)$, we have
\begin{align*}
 \Vert x_{m}(t_{i}^{m})-x_{m}(t_{j}^{m}) \Vert
&\le \sum_{k=j+1}^{i} \|
x_{m}(t_{k}^{m})-x_{m}(t_{k-1}^{m})\| \\
&\le (M+1) \sum_{k=j+1}^{i} (t_{k}^{m}-t_{k-1}^{m})=
(M+1)|t_{i}^{m}-t_{j}^{m}|.
\end{align*}
Also, we have by construction for a.e. $t\in [t_{i}^{m},t_{i+1}^{m}]$
and for all $i\in \{0,\dots ,\nu _{m}\}$
\begin{equation}
 \Vert \dot x_{m}(t) \Vert = \Vert
w_{i}^{m} \Vert  \le M+1.\label{9}
\end{equation}

\subsection*{Convergence of approximate solutions}
We note that the sequence $f_{m}$ can be constructed with the
relative compactness property in the space of bounded functions
(see \cite{Trongducha}). Therefore, without loss of generality we
can suppose that there is a bounded function $f$ such that
\begin{equation}
\lim_{m\to \infty} \sup_{t\in [0,a]} \Vert f_{m}(t)-f(t)
\Vert =0\,.
\end{equation}
Now, we prove that the approximate solutions $x_{m}(.)$ converge
to a  viable solution of \eqref{2}.

It is clear by construction that  $\{x_{m}\}_m$ are Lipschitz
continuous with constant $M+1$ and%
\[
x_{m}(t) = x_{i}^{m}+(t-t_{i}^{m})w_{i}^{m}
=x_{i}^{m}+(\frac{t-t_{i}^{m}}{\lambda _{i}^{m}})(\Psi
_{i}^{m}-x_{i}^{m}).
\]
On the other hand, we have $ 0 \le t - t_{i}^{m} \le
t_{i+1}^{m}-t_{i}^{m}=\lambda_{i}^{m}$ and so $0\le
\frac{t-t_{i}^{m}}{\lambda _{i}^{m}}\leq 1$, and hence we get
\begin{equation}
(\frac{t-t_{i}^{m}}{\lambda _{i}^{m}})(\Psi _{i}^{m}-x_{i}^{m})\in \overline{%
co}[\{0\}\cup (K_{1}-K_{0})].
\end{equation}
Thus,
\begin{equation}
 x_{m}(t) \in K:=K_{0}+\overline{co}[\{0\}\cup
(K_{1}-K_{0})].
\end{equation}
Therefore, since the set $K$ is compact ( because $K_0$ and $K_1$
are compact), then  the assumptions of the Arzela-Ascoli theorem
are satisfied. Hence a subsequence of $x_{m}$ my be extracted
(still denoted $x_{m}$) that converges to an absolutely continuous
mapping $ x:[0,a]\to H$ such that
\begin{equation}
\begin{gathered}
 \lim_{m\to \infty} \max_{t\in   [0,a]}
 \Vert x_{m}(t)-x(t) \Vert =0 \\
\dot x_{m}(.)\rightharpoonup \dot x(.)\mbox{ in the weak topology
of } L^{2}([0,a],H).
\end{gathered}  \label{10}
\end{equation}
Recall now that $f_{m}$ converges pointwise a.e. on $[0,a]$ to
$f$.  Then the continuity of the set-valued mapping $F$ and the
closedness of the set $F(t,x(t))$ entail $f(t)\in F(t,x(t)) $.
Now, it remains to prove that
\begin{equation}
\begin{gathered}
x(t)\in S; \\
-f(t)+x'(t)\in G(x(t))\quad \mbox{a.e.  on } \quad [0,a].
\end{gathered}
\end{equation}
By construction we have $x_{i}^{m}\in K_{0}$ (for all $i \in
\{0,\dots ,\nu _{m}-1\}$). This ensures
\[
d_{K_{0}}(x(t)) \leq  \| x_{m}(t)-x_{i}^{m}\|
+\| x_{m}(t)-x(t)\| \leq \frac{1+M}{m}+\|
x_{m}(t)-x(t)\|
\]
which approaches $0$ as $m$ approaches $\infty$.
The closedness of $K_0$  yields $ d_{K_{0}}(x(t))=0 $ and so
$x(t)\in K_{0}\subset S$.

 By construction, we have for a.e. $t\in [0,a]$
\begin{equation}\label{15}
\dot x_{m}(t)-f_{m}(t)-c_{m}(t)\in G(x_{m}(\theta _{m}(t)))\subset
\partial ^{C}g(x_{m}(\theta _{m}(t)))=\partial ^{P}g(x_{m}(\theta
_{m}(t))),
\end{equation}
 where the above equality follows from the uniform  regularity of
$g$ over $C$ and the part $(ii)$ in Proposition
\ref{prop1}.
We can thus apply Castaing techniques (see
for example \cite{castaing}). The weak convergence (by (\ref{10}))
in $L^2([0,a],H)$ of $\dot x_m(\cdot)$  to $\dot x(\cdot)$ and
Mazur's Lemma  entail
$$
\dot x(t)\in \bigcap_m\overline{co}\{\dot x_k(t):\, k\geq m\},
\quad \mbox{ for a.e.  on } [0,a].
$$
Fix any such $t$ and consider any $\xi\in H$. Then, the last
relation above yields
$$
\big<\xi,\dot x(t)\big>\leq \inf_m\sup_{k\geq m}\big<\xi,\dot
x_m(t)\big>
$$
and hence  Proposition \ref{prop1} part (iii) and (\ref{15}) yield
\begin{align*}
\langle \xi ,\dot x(t) \rangle
&\leq  \lim_m \sup \sigma (\xi ,\partial ^{P}g(x_{m}(\theta _{m}(t)))
  +f_{m}(t)+c_{m}(t)) \\
&\leq  \sigma (\xi ,\partial ^{P}g(x(t))+f(t))\quad
\mbox{for any } \xi \in H,
\end{align*}
So, the convexity and the closedness of the set $\partial
^{P}g(x(t))$ ensure
\begin{equation}
-f(t)+\dot x (t)\in \partial ^{P}g(x(t)).
\end{equation}
Now, since $g$ is uniformly regular over $C$ and $x:[0,a] \to C$
we have
\begin{align*}
\frac{d}{dt}(g\circ x)(t)
&=\langle \partial^{P}g(x(t)),\dot x (t)\rangle\\
&=\langle -f(t)+\dot x(t),\dot  x (t)\rangle \\
&=\| \dot  x (t)\| ^{2}-\langle f(t), \dot x (t)\rangle.
\end{align*}
Consequently,
\begin{equation}
g(x(a))-g(x_{0})=\int_{0}^{a}\| \dot  x (s)\|
^{2}ds-\int_{0}^{a}\left\langle f(s),\dot x (s)\right\rangle ds
\label{11}
\end{equation}%
On the other hand, by (\ref{15})
 and Definition \ref{def1} we have for all $i\in \{0,\dots \nu _{m}-1\}$
\begin{align*}
g(x_{i+1}^{m})-g(x_{i}^{m})
&\geq \langle \dot x_{m}
(t)-f_{i}^{m}-c_{i}^{m},x_{i+1}^{m}-x_{i}^{m}\rangle -\beta
\| x_{i+1}^{m}-x_{i}^{m}\| ^{2} \\
&=\big\langle \dot x_{m}
(t)-f_{m}(t)-c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x_m
(s)ds\big\rangle -\beta \| x_{i+1}^{m}-x_{i}^{m}\| ^{2}\\
&\geq \int_{t_{i}^{m}}^{t_{i+1}^{m}}\| \dot x_{m}
(s)\| ^{2}ds-\int_{t_{i}^{m}}^{t_{i+1}^{m}}\langle
\dot x_{m} (s),f_{m}(s)\rangle ds \\
&\quad-\langle c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot
x_m (s)ds\rangle -\beta (M+1)^{2}(t_{i+1}^{m}-t_{i}^{m})^{2} \\
&\geq \int_{t_{i}^{m}}^{t_{i+1}^{m}}\| \dot x_{m}
(s)\| ^{2}ds-\int_{t_{i}^{m}}^{t_{i+1}^{m}}\langle
\dot x_{m} (s),f_{m}(s)\rangle ds \\
&\quad-\langle c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x_m
(s)ds\rangle -\frac{\beta
(M+1)^{2}}{m}(t_{i+1}^{m}-t_{i}^{m}).
\end{align*}
By adding, we obtain
\begin{equation}
g(x_{m}(t_{\nu _{m}}^{m}))-g(x_{0})\geq \int_{0}^{t_{\nu
_{m}}^{m}}\| \dot x_{m} (s)\|
^{2}ds-\int_{0}^{t_{\nu _{m}}^{m}}\left\langle f_{m}(s),\dot x_{m}
(s)\right\rangle ds-\varepsilon _{1,m}  \label{12}
\end{equation}%
with
$$\varepsilon_{1,m}= \sum_{i=0}^{\nu _{m}-1} %
\langle c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x_m
(s)ds \rangle +\frac{\beta (M+1)^{2}t_{\nu _{m}}^{m}}{m}
$$
and
\begin{equation}
g(x_{m}(a))-g((t_{\nu _{m}}^{m}))\geq \int_{t_{\nu _{m}}^{m}}^{a}
\Vert \dot x_{m} (s) \Vert ^{2}ds-\int_{t_{\nu _{m}}^{m}}^{a}
\left\langle \dot x_{m} (s),f_{m}(s) \right\rangle ds-\varepsilon
_{2,m} \label{13}
\end{equation}
with
 $$
 \varepsilon _{2,m}= \langle c_{\nu _{m}}^{m},\int_{t_{\nu
_{m}}^{m}}^{a}\dot x (s)ds \rangle +\frac{\beta
(M+1)^{2}(a-t_{\nu _{m}}^{m})}{m}.
$$
Therefore, we get
\begin{equation}\label{16}
g(x_{m}(a))-g((x_{0}))\geq \int_{^{0}}^{a} \Vert \dot x_{m}(s)
\Vert ^{2}ds-\int_{0}^{^{a}} \langle f_{m}(s),\dot x_{m}(s)
\rangle ds-\varepsilon_{m}
\end{equation}
where
$$
\varepsilon_m =\varepsilon_{1,m}+\varepsilon_{2,m} =
\sum_{i=0}^{\nu _{m}-1}  \langle
c_{i}^{m},\int_{t_{i}^{m}}^{t_{i+1}^{m}}\dot x (s)ds
\rangle + \langle c_{\nu _{m}}^{m},\int_{t_{\nu
_{m}}^{m}}^{a}\dot x(s)ds \rangle +\frac{\beta
a(M+1)^{2}}{m}.
$$
 Using our construction we get
\begin{align*}
 \vert \varepsilon _{m} \vert
&\leq \sum_{i=0}^{\nu _{m}-1}
\Vert c_{i}^{m} \Vert \int_{t_{i}^{m}}^{t_{i+1}^{m}} \Vert \dot x
(s) \Vert ds+ \Vert c_{\nu _{m}}^{m} \Vert \int_{t_{\nu
_{m}}^{m}}^{a} \Vert \dot x (s) \Vert ds +\frac{\beta
(M+1)^{2}a}{m} \\
&\leq  \sum_{i=0}^{\nu _{m}-1}\frac{1}{m}
(t_{i+1}^{m}-t_{i}^{m})(M+1)+\frac{1}{m}(a-t_{\nu
_{m}}^{m})(M+1)+\frac{ \beta (M+1)^{2}a}{m} \\
&=\frac{(M+1)}{m}\big[ \sum_{i=0}^{\nu _{m}-1} (t_{i+1}^{m}-t_{i}^{m})
+(a-t_{\nu_{m}}^{m}) \big] +\frac{\beta (M+1)^{2}a }{m}\\
&=\frac{(M+1)a}{m}+\frac{\beta (M+1)^{2}a}{m} \to  0
\quad \mbox{ as } m\to \infty.
\end{align*}
We have also
\[
 \lim_{m\to \infty} \int_{0}^{a} \langle
f_{m}(s),\dot x_{m}(s) \rangle ds=\int_{0}^{a} \langle f(s),\dot
x(s) \rangle ds.
\]
Taking the limit superior  in (\ref{16}) when  $m\to
\infty $ we obtain
\begin{equation}
g(x(a))-g(x_{0})\geq  \limsup_m \int_{0}^{a} \Vert \dot x_{m} (s)
\Vert ^{2}ds-\int_{0}^{a} \langle f(s),\dot x (s) \rangle ds.
\end{equation}%
This inequality compared with (\ref{11}) yields
\[
\int_{0}^{a}\| \dot x (s)\| ^{2}ds\geq \limsup_m
\int_{0}^{a}\|\dot x_{m} (s)\| ^{2}ds,
\]
that is,
\begin{equation}
\| \dot x \| _{L^{2}([0,a],H)}^{2}\geq \limsup_m
\| \dot x_{m} \| _{L^{2}([0,a],H)}^{2}.
\end{equation}%
On the other hand the weak $l.s.c$ of the norm ensures
\[
\| \dot  x \| _{L^{2}([0,a],H)}^{2}\leq \liminf_m
\| \dot x_{m} \| _{L^{2}([0,a],H)}^{2}
\]
Consequently, we get
\[
\| \dot x \| _{L^{2}([0,a],H)}= \lim_{m}
\| \dot  x_{m} \| _{L^{2}([0,a],H)}.
\]
Hence there exists a subsequence of $\{\dot x_{m}\}_{_m}$ (still
denoted $\{\dot x_{m}\}_{_m}$)  converges poitwisely a.e on
$[0.a]$ to $\dot x$. \vskip2mm \noindent Since
$$
(x_{m}(t),\dot x_{m} (t)-f_{m}(t)-c_{m}(t))\in gphG, \quad
\mbox{a.e. on }  [0.a],
$$
and as $G$ has a  closed graph, we obtain
$$
(x(t),\dot x (t)-f(t))\in gphG \quad \mbox{a.e.  on } [0.a],
$$
and so
\[
\dot x (t)\in G(x(t))+F(t,x(t))\quad \mbox{a.e.  on }  [0.a]
\]
The proof is complete.
\end{proof}

\begin{rem}\label{r1} {\rm
 An inspection of the proof of Theorem \ref{thm2.1} shows that the
uniformity of the constant $\beta$ was  needed only over the set
$K_0$ and so it was not necessary  over all the set $S$. Indeed,
it suffices to take the uniform regularity of $g$ locally over
$S$, that is, for every point $\bar x\in S$ there exist $\beta\geq
0$ and a  neighborhood $V$ of $x_0$ such that $g$ is uniformly
regular over  $S\cap V$.}
\end{rem}

 We conclude  the paper with two corollaries  of our main result
in Theorem \ref{thm2.1}.

\begin{coro}
Let  $K\subset  H$ be  a nonempty uniformly prox-regular  closed subset of a finite dimensional
 space $H$ and
 $F: [0,T]\times H\to H$ be  a continuous set-valued mapping with compact values.
Then, for any $x_0\in K$ there exists $a\in ]0,T[$ such that the
following differential inclusion
\begin{gather*}
\dot x(t)\in  -\partial^C
d_K(x(t)) + F(t,x(t)) \quad \mbox{\rm a.e. on }[0,a]\\
 x(0)=x_0 \in K,
\end{gather*}
has at  least one  absolutely continuous  solution on $[0,a]$.
\end{coro}

\begin{proof}
In \cite[Theorem 3.4]{bounkhelthibault1} (see also
\cite[theorem 4.1]{bounkhel1}) it is shown that the function $g:=d_K$
is uniformly regular over $K$ and so it is uniformly regular over
some neighborhood $V$ of $x_0\in K$. Thus, by Remark \ref{r1}, we
apply Theorem \ref{thm2.1} with $S=H$ (hence the tangential condition
$(\ref{3})$ is satisfied), $K_0:=V\cap S=V$,  and the set-valued
mapping $G:=\partial^C d_K$ which satisfies the hypothesis of
Theorem \ref{thm2.1}.
\end{proof}

Our second corollary concerns the following differential inclusion
\begin{equation} \label{DI}
\begin{gathered}
\dot x(t)\in  -N^C(S;x(t)) + F(t,x(t)) \quad \mbox{a.e.  }\\
x(t)\in S, \quad \mbox{ for  all $t$ and } x(0)=x_0 \in S.
\end{gathered}
\end{equation}
This type of differential inclusion has been introduced in
\cite{henry} for studying some economic problems.

\begin{coro} Let $H$ be a separable Hilbert space.
Assume that \begin{enumerate}
 \item $F: [0,T]\times H\to H$ is a continuous set-valued mapping with compact
values;
\item  $S$ is a nonempty  uniformly prox-regular closed
subset in $H$;
\item  For any $(t,x)\in I\times S$ the tangential
condition
$$ \liminf_{h\downarrow 0} h^{-1}e\big(x+h(\partial^Cd_S
(x)+F(t,x));S\big)=0,
$$
for any $(t,x)\in I\times S$  holds.
 \end{enumerate}
 Then, for any $x_0\in S$, there exists $a\in ]0,T[$ such that the
differential inclusion \eqref{DI}
 has at lease one absolutely continuous solution on
$[0,a]$.
\end{coro}

\subsection*{Acknowledgement}
The authors would like to thank the anonymous referee for his careful
reading of the paper and for his pertinent suggestions and remarks.

\begin{thebibliography}{9}
\bibitem{anconacolombo} F. Ancona and G. Colombo;
 {\it  Existence of solutions for a class of
non convex differential inclusions}, Rend. Sem. Mat. Univ. Padova, Vol. {\bf 83} (1990), pp. 71-76.

\bibitem{benabdellah2} H. Benabdellah;
{\it Sur une Classe d'equations differentielles multivoques semi
continues superieurement a valeurs non convexes}, S\'em. d'Anal.
convexe, expos\'e No.  6, 1991.

\bibitem{benabdellahcastaing}  H. Benabdellah, C. Castaing and A. Salvadori;
 {\it Compactness and discretization methods for differential inclusions
and evolution problems},
Atti. Semi. Mat. Fis. Modena, Vol. {\bf  XLV}, (1997), pp. 9-51.

\bibitem{bounkhel1}  M. Bounkhel; {\it Existence Results of Nonconvex
Differential Inclusions}, J. Portugaliae Mathematica, Vol.  {\bf
59} (2002), No. 3,   pp.  283-310.

\bibitem{bounkhel3}  M. Bounkhel;
 {\it On arc-wise essentially smooth mappings between Banach spaces},
J. Optimization,  Vol.  \textbf{51} (2002), No. 1, pp. 11-29.

\bibitem{bounkhelthibault3}  M. Bounkhel and L. Thibault;
 {\it On various notions of  regularity of sets},  Nonlinear Anal.:
Theory, Methods and Applications, Vol. {\bf 48} (2002),
No. 2, 223-246.

 \bibitem{bounkhelthibault1}  M. Bounkhel and L. Thibault;
 {\it Further characterizations of regular sets in Hilbert spaces and their applications to nonconvex sweeping process},
 Preprint, Centro de Modelamiento M\`atematico (CMM), Universidad de chile, (2000).

\bibitem{bressancellinacolombo} A. Bressan, A. Cellina, and G. Colombo;
 {\it Upper semicontinuous differential inclusions without convexity},
Proc. Amer. Math. Soc. Vol. {\bf 106}, (1989), pp. 771-775.

\bibitem{castaing} C. Castaing, T.X. DucHa, and M. Valadier;
{\it Evolution equations governed by
the sweeping process}, Set-valued Analysis, Vol. {\bf 1} (1993),
pp.109-139.

\bibitem{henry} C. Henry; {\it An existence theorem for a class of
differential equations with multivalued
right-hand side}, J. Math. Anal. Appl. Vol. {\bf 41}, (1973), pp. 179-186.

\bibitem{morchadi} R. Morchadi and S. Gautier;
 {\it  A viability result for a first order differential
inclusion without convexity}, Preprint, University Pau (1995).

\bibitem{rossi} P. Rossi; {\it Viability for upper semicontinuous
differential inclusions without convexity},
 Diff. and Integr. Equations, Vol. {\bf 5} , (1992), No. 2, pp. 455-459.

\bibitem{Trongducha} X. D. H. Truong;
{\it Existence of Viable solutions of nonconvex differential inclusions},
Atti. Semi. Mat. Fis. Modena, Vol. {\bf XLVII}, (1999), pp.
457-471.
\end{thebibliography}

\end{document}