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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 114, pp. 1--11.\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-2005/114\hfil Carath\'{e}odory perturbation]
{Carath\'{e}odory perturbation of a second-order differential
inclusion with constraints}
\author[S. Amine, R. Morchadi, S. Sajid\hfil EJDE-2005/114\hfilneg]
{Sa\"{\i}da Amine, Radouan Morchadi, Sa\"{\i}d Sajid }

\address{Sa\"{\i}da Amine \hfill\break
U.F.R Mathematics and Applications,
Department of Mathematics, Faculty of Sciences and Techniques
of Mohammedia, BP 146, Mohammedia, Morocco}
\email{aminesaida@hotmail.com}

\address{Radouan Morchadi \hfill\break
U.F.R Mathematics and Applications,
Department of Mathematics, Faculty of Sciences and Techniques
of Mohammedia, BP 146, Mohammedia, Morocco}
\email{morchadi@hotmail.com}

\address{Sa\"{\i}d Sajid \hfill\break
U.F.R Mathematics and Applications,
Department of Mathematics, Faculty of Sciences and Techniques
of Mohammedia, BP 146, Mohammedia, Morocco}
\email{saidsajid@hotmail.com}

\date{}
\thanks{Submitted June 30, 2005. Published October 21, 2005.}
\subjclass[2000]{34A60}
\keywords{Upper semicontinuous multifunction;
 cyclically monotone; \hfill\break\indent
 Carath\'{e}odory function}

\begin{abstract}
We prove the existence of local solutions for the second-order
 viability problem
 $$
 \ddot{x}(t) \in f(t,x(t),\dot{x}(t))+F(x(t),\dot{x}(t)),\quad
 x(t)\in K.
 $$
 Here $K$ is a closed subset of $\mathbb{R}^{n}$, $F$
 is an upper semicontinuous multifunction with compact values
 contained in the subdifferential of a convex proper lower
 semicontinuous function $V$, and $f$ is a Carath\'{e}odory function.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

 \section{Introduction}

 This paper concerns the second-order nonconvex viability
problem
\begin{equation} \label{e1.1}
\begin{gathered}
\ddot{x}(t) \in f(t,x(t),\dot{x}(t))+F(x(t),\dot{x}(t))\\
(x(0),\dot{x}(0))=(x_{0},v_{0})\\
 x(t)\in K.
\end{gathered}
\end{equation}
Where $F$ is a  globally  upper semicontinuous multifunction,
cyclically monotone, i.e. $F(x,y)\subset\partial V(y)$, defined
from $K\times U$ into the subset of all nonempty compact subset of
$\mathbb{R}^{n}$; $f$ is a Carath\'{e}odory function from
$\mathbb{R} \times K \times U$ to $\mathbb{R}^{n}$, where $K$ is a
closed subset of $\mathbb{R}^{n}$, $U$ is an open subset of
$\mathbb{R}^{n}$ and $\partial V$ is the subdifferential of a
lower semicontinuous and convex function from $U$ into
$\mathbb{R}^{n}$.

Existence of solutions of differential inclusions with upper
semicontinuous and cyclically monotone right-hand side was first
established by Bressan, Cellina and Colombo \cite{b2}. It has been
proved the  existence of local solutions of a first order problem
without constraints: $\dot{x}(t)\in F(x(t))$. The approach is
based on some technics related to the subdifferential properties
applied to approximate solutions. To avoid the difficulty of the
weak convergence of the derivatives of such approximate solutions,
authors rely on the basic relation
$$
\frac{d}{dt}(V(x(t)))=\|\dot{x}(t)\| ^{2}.
$$
 Regarding the existence of viable solutions of
second-order upper semicontinuous differential inclusions without
convexity, we refer to Lupulescu \cite{l1,l2}, where two results
are given in this subject: the first deals with the problem
\eqref{e1.1} where $f\equiv 0$; while the second studies problem
\eqref{e1.1}, but without constraint, i.e. $K=\mathbb{R}^{n}$.

 The first work in the second-order viability problem,
was done by Cornet and Haddad \cite{c1}, the authors were subject
concerned by a problem with upper semicontinuous right-hand side,
not cyclically monotone but convex. This research program was
pursued by some works, see \cite{a4,m1}. For the nonconvex case,
existence results may be found in \cite{a1,h1,m2}.

It is known that viability problems require tangential
conditions. So far as we know, for a second-order viability
problem, most of the time, authors use the second-order contingent
set
$$
A_{K}(x,y)=\big\{ z\in \mathbb{R}^{n}:\lim_{h\to 0^{+}} \inf
\frac{d_{K}(x+hy+\frac{1}{2}h^{2}z)}{\frac {h^{2}}{2}}=0\big\}
$$
 introduced by Ben-Tal.
In the present paper we prove the existence solutions to
\eqref{e1.1} assuming the tangential condition:
For all $(t,x,y)\in I\times K\times U$, there exists $w\in F(x,y)$ such
that
$$
 \liminf_{h\to 0^{+}} \frac{1}{h^{2}}d_{K}(x+hy+\frac{h^{2}}{2}w
 +\int_{t}^{t+h}f(\tau ,x,y)d\tau )=0.
$$
This condition is also used by Lupulescu \cite{l1} with $f\equiv 0$.

\section{Notations and statement of the main result}

 Let $\mathbb{R}^{n}$ be the $n$-dimensional Euclidean space
with scalar product $\langle \ ;\ \rangle $ and norm $\| \ \| $.
Let $K$ be a closed subset of $\mathbb{R}^{n}$, $U$ be a nonempty
open subset of $\mathbb{R}^{n}$ and denote $\Omega =K\times U$.
For each $x\in\mathbb{R}^{n}$ we denote by $d_{K}(x)$ the distance
from $x$ to $K$. For $r>0$, $B(x,r)$ stands for the ball centered
at $x$ with radius $r$ and $\overline{B}(x,r)$ its closure, $B$ is
the unit ball of $\mathbb{R}^{n}$.

Let $F$ be a multifunction from $\Omega $ into the set of all
nonempty compact subsets of $\mathbb{R}^{n}$. Let $f$ be a
function from $\mathbb{R}\times \Omega $ into $\mathbb{R}^{n}$ .
Assume that $F$ and $f$ satisfy the following conditions:
\begin{itemize}
\item[(A1)] $F$ is upper semicontinuous, i.e. for all $(x,y)$ and
for every $\varepsilon >0$ there exists $\delta >0$ such that $
F(x',y')\subseteq F(x,y)+\varepsilon B$, whenever $\|(x,y) -
(x',y')\| \leq \delta$;

\item[(A2)] There exists a convex proper and lower semicontinuous
function $V:\mathbb{R}^{n}\to \mathbb{R}^{n}$ such that
 $F(x,y)\subset \partial V(y)$, where $\partial V$ denotes
 the subdifferential of the function $V$;

\item[(A3)] $f: \mathbb{R}\times \Omega \to \mathbb{R}^{n}$ is a
 Carath\'{e}odory function, i.e. for each $(x,y)\in \Omega$,
  $t\to f(t,x,y)$ is measurable and for all $t\in \mathbb{R}$,
  $(x,y)\to f(t,x,y)$ is continuous;

\item[(A4)] There exists $m\in L^{2}(\mathbb{R})$ such that
$\| f(t,x,y)\| \leq m(t)$ for all $(t,x,y)\in \mathbb{R}\times \Omega$;

\item[(A5)] (Tangential condition) For all
$(t,x,v)\in \mathbb{R}\times \Omega$, there exists $w\in F(x,v)$ such
that
$$
\liminf_{h\to 0^{+}}
\frac{1}{h^{2}}d_{K}\Big(x+hv+\frac{h^{2}}{2}w+\int_{t}^{t+h}f(\tau
,x,v)d\tau \Big)=0.
$$
\end{itemize}
Let $(x_{0},y_{0})\in \Omega$. Assuming that  $F$ and $f$
satisfy (A1)--(A5), we shall prove the following result.


\begin{theorem} \label{thm2.1}
There exist  $T>0$  and an absolutely continuous $x:[ 0,T] \to
\mathbb{R}^{n}$  for which $\dot x$  is also absolutely continuous
such that
\begin{gather*}
\ddot{x}(t) \in f(t,x(t),\dot{x}(t))+F(x(t), \dot{x}(t)) \quad
\mbox{a.e. } t\in [0,T]\\
(x(0),\dot{x}(0))=(x_{0},y_{0})\\
x(t)\in K  \quad  \forall t\in [0,T].
\end{gather*}
\end{theorem}

\section{Proof of the main result}

We begin by recalling the following result which
was proved in \cite{b2}, and will be used in the second step
of the proof of the main result.

\begin{lemma} \label{lem3.1}
 Let $V$  be a convex proper lower semicontinuous function such that
$F(x,y)\subset \partial V(y)$,  for any $ (x,y)\in \Omega$.   Then
there exist $r=r_{x,y}>0$   and  $M=M_{x,y}>0$   such that
$$
\| F(x,y)\| =\sup_{z\in F(x,y)}\| z\| \leq M \quad\mbox{on }
B((x,y),r)
$$
and $V$ is Lipschitz continuous on $B(y,r)$  with constant $M$.
\end{lemma}

Let $r$ and $M$ be the real numbers defined in the Lemma above, and
 such that $B(y_{0},r)\subset U$. Choose $T_{1}>0$ such that
\begin{equation} \label{e3.1}
\int_{0}^{T_{1}}(m(s)+M+1)\,ds <\frac{r}{3}.
\end{equation}
Set
\begin{equation} \label{e3.2}
T_{2}=\min \big\{ \frac{r}{3(M+1)}, \frac{2r}{3(
\| y_{0}\| +r)}\big\}.
\end{equation}
In the sequel, we denote by $\Omega _{0}$ the compact
set $(K\times\overline{B}(y_{0},r))\cap
\overline{B}((x_{0},y_{0}),r)$ and choose $T$ such that
\begin{equation} \label{e3.3}
T \in ]0,\min \{ T_{1},T_{2}\}]\,.
\end{equation}
The following result will be used for
proving the viability property of the solutions to \eqref{e1.1}.

\begin{lemma} \label{lem3.2}
 Let $F$  and $f$ satisfy assumptions (A1)--(A5). Then
for each $\varepsilon >0$ there exists $\eta \in ]0,\varepsilon[ $
such that for each $(t,x,v)$ in $[ 0,T] \times \Omega_{0}$, there
exist $w$ in $F(x,v)+\frac{\varepsilon }{T}B$   and $h$ in $[ \eta
,\varepsilon ]$; that is,
$$
\big( x+hv+\frac{h^{2}}{2}w+\int_{t}^{t+h}f(\tau
,x,v)d\tau\big) \in K.
$$
\end{lemma}

\begin{proof}
Let $(t,x,v)\in [ 0,T] \times \Omega_{0}$, let $ \varepsilon >0$.
Since $F$ is upper semicontinuous, then there exists
$ \delta _{(x,v)}>0$ such that
\begin{equation}
F(y,u)\subset F(x,v)+ \frac {\varepsilon}{T} B \quad\forall
(y,u)\in B((x,v),\delta _{(x,v)}).
\label{e3.4}
\end{equation}
On the other hand, for all $(s,y,u)\in [0,T]\times \Omega _{0}$,
by the tangential condition, there exist $h_{(s,y,u)} \in ]
0,\varepsilon ] $ and $c \in F(y,u)$ such that
$$
d_{K}\Big(y+h_{(s,y,u)}u+\frac{h_{(s,y,u)}^{2}}{2}c
+\int_{s}^{s+h_{(s,y,u)}}f(\tau ,y,u)d\tau
\Big)<h_{(s,y,u)}^{2}\frac{\varepsilon }{4T}.
$$
 Consider the subset
\begin{align*}
N(s,y,u)=&\big\{ (l,a,b )\in  \mathbb{R}\times
(\mathbb{R}^{n})^{2} : d_{K}\big(a+h_{(s,y,u)}b
+\frac{h_{(s,y,u)}^{2}}{2}c\\  
&+\int_{l}^{l+h_{(s,y,u)}}f(\tau
,a,b )d\tau \big)<h_{(s,y,u)}^{2}\frac{\varepsilon }{4T}\big\}.
\end{align*}
Since
$\| f(l,a,b )\| \leq m(l)$ for all $(l,a,b)\in \mathbb{R}\times\Omega$,
the dominated convergence theorem shows that the
function
$$
(l,a,b)\to  a+h_{(s,y,u)}b
+\frac{h_{(s,y,u)}^{2}}{2}c +\int_{l}^{l+h_{(s,y,u)}}f(\tau,a,b)d\tau
$$
is continuous. So that, the function
$$
(l,a,b)\to  d_{K}(a+h_{(s,y,u)}b
+\frac{h_{(s,y,u)}^{2}}{2}c +\int_{l}^{l+h_{(s,y,u)}}f(\tau
,a,b )d\tau )
$$
is continuous and consequently the subset $N(s,y,u)$ is open.
Moreover, since $(s,y,u)$ belongs to $N(s,y,u)$, there exists a
ball $B((s,y,u),\eta _{(s,y,u)})$ with radius
$\eta_{(s,y,u)}<\delta_{(x,v)} $ and contained in $N(s,y,u)$.
Therefore, the compactness of $[0,T] \times \Omega _{0}$ implies
that it can be covered by $q$ such balls
$B((s_{i},y_{i},u_{i}),\eta _{(s_{i},y_{i},u_{i})})$. For
simplicity, put
$$
h_{(s_{i},y_{i},u_{i})}:=h_{i},\quad
\eta_{(s_{i},y_{i},u_{i})}:=\eta_{i}, \quad\eta :=
\min_{i=1,\dots ,q} h_{i}>0.
$$
Let $(t,x,v)\in [0,T] \times \Omega _{0}$.
Since $(t,x,v)$ belongs to one of the balls
$B((s_{i},y_{i},u_{i}),\eta_{i})$, there exist $x_{i}\in K$  and
$c_{i}\in F(y_{i},u_{i})$ such that
\begin{align*}
&\big\| c_{i}-\frac{2}{h_{i}^{2}}(x_{i}-x-h_{i}v-\int_{t}^{t+h_{i}}
f(\tau,x,v)\,d\tau)\big\|\\
&\leq \frac{1}{h_{i}^{2}}d_{K}(x+h_{i}v+\frac{h_{i}^{2}}{2}c_{i}
+\int_{t}^{t+h_{i}}f(\tau ,x,v)\,d\tau )
+\frac{\varepsilon }{4T}
\leq \frac{\varepsilon }{2T}.
\end{align*}
 Let us set
$$
w=\frac{2}{h_{i}^{2}}(x_{i}-x-h_{i}v-\int_{t}^{t+h_{i}}f(\tau,x,v)\,d\tau),
$$
then
$$
(x+h_{i}v+\frac{h_{i}^{2}}{2}w+\int_{t}^{t+h_{i
}}f(s\tau,x,v)d\tau)\in K \quad \mbox{and}\quad
\|c_{i}-w\| \leq \frac{\varepsilon }{2T}.
$$
Since
$(t,x,v)\in B((s_{i},y_{i},u_{i}),\eta_{i})$ and
 $\eta_{i}<\delta_{(x,v)}$,
relation \eqref{e3.4} implies
$$
F(y_{i},u_{i})\subset F(x,v)+\frac{\varepsilon }{2T}B;
$$
 so that $w\in F(x,v)+\frac{\varepsilon }{T}B$. Hence
the Lemma is proved.
\end{proof}

 Now, we are able to prove the main result. Our approach
consists of constructing, in a first step, a sequence of
approximate solutions and deduce, in a second step, from available
estimates that a subsequence converges to a solution of \eqref{e1.1}.

\subsection*{Step 1. Construction of approximate solutions}
 Let $(x_{0},y_{0})\in \Omega _{0}$ and
 $\varepsilon >0$. By Lemma \ref{lem3.2}, there exist $\eta >0$, $h_{0}$
 in $[\eta ,\varepsilon ] $ and  $w_{0}$ in
 $F(x_{0},y_{0})+\frac{\varepsilon }{T}B$ such that

$$
\big(x_{0}+h_{0}y_{0}+\frac{h_{0}^{2}}{2}w_{0}+\int_{0}^{h_{0}}f(\tau
,x_{0},y_{0})d\tau \big)\in K.
$$
Put
$$
x_{1}= x_{0}+h_{0}y_{0}+\frac{h_{0}^{2}}{2}w_{0}+\int_{0}^{h_{0}}f(\tau
,x_{0},y_{0})d\tau \quad \mbox{and} \quad y_{1}=y_{0}+h_{0}w_{0}.
$$
Since $w_{0} \in F(x_{0}, y_{0})\subset B(0,M+1)$,
$\| f(t,x_{0},y_{0})\| \leq m(t)$, by \eqref{e3.1}, \eqref{e3.2}, we
obtain
\begin{align*}
\| x_{1}-x_{0}\|
&=\big\|h_{0}y_{0}+\frac{h_{0}^{2}}{2}w_{0}+\int_{0}^{h_{0}}
f(\tau ,x_{0},y_{0})\,d\tau \big\|\\
&\leq T\|y_{0}\| +\frac{T}{2}\| w_{0}\|
+\| \int_{0}^{h_{0}}f(\tau,x_{0},y_{0})\,d\tau \|\\
& \leq T\| y_{0}\| +\int_{0}^{T}(M+1+m(\tau ))d\tau\\
& \leq T\| y_{0}\| +\frac{r}{3}\leq r,
\end{align*}
and
$$
\| y_{1}-y_{0}\| =\| h_{0}w_{0} \| \leq T\| w_{0}\| <\frac{r}{3}<r
$$
and thus $(x_{1},y_{1})\in $ $\Omega _{0}$.
 By induction, for $p\geq 2$ and for every $i=1,\dots ,p-1$,
we construct  $(h_{i}, (x_{i},y_{i}), w_{i})$ in
$[\eta,\varepsilon ]\times\Omega_{0}\times\mathbb{R}^{n}$ such that
$\sum_{i=0}^{p-1}h_{i}\leq T$ and
\begin{gather*}
x_{i}=(x_{i-1}+h_{i-1}y_{i-1}+\frac{h_{i-1}^{2}}{2}w_{i-1}
+\int_{h_{i-2}}^{h_{i-2}+h_{i-1}}f(\tau ,x_{i-1},y_{i-1})\,d\tau) \in K;\\
y_{i}=y_{i-1}+h_{i-1}w_{i-1};\\
w_{i}\in F(x_{i},y_{i})+\frac{\varepsilon }{T}B.
\end{gather*}
Since $h_{i}\in ]\eta,\varepsilon[$ there exists an
integer $s$, such that
$$
\sum_{i=0}^{s-1}h_{i}< T \leq \sum_{i=0}^{s}h_{i}.
$$
 In what follows, choose $\varepsilon$ small such
that
$$
\sum_{i=0}^{s-1}\frac {h_{i}^{2}}{2}\leq
\sum_{i=0}^{s-1}h_{i}< T.
$$
For all $p=1,\dots ,s-1$ define
$(h_{p})_{p}\subset [\eta ,\varepsilon ] $,
$(x_{p},y_{p})_{p}\subset \Omega _{0}$, and $(w_{p})_{p}$ as  follows
\begin{gather*}
x_{p}=(x_{p-1}+h_{p-1}y_{p-1}+\frac{h_{p-1}^{2}}{2}w_{p-1}+
\int_{h_{p-2}}^{h_{p-2}+h_{p-1}}f(\tau
,x_{p-1},y_{p-1})\,d\tau) \in K;\\
y_{p}=y_{p-1}+h_{p-1}w_{p-1};\\
 w_{p}\in F(x_{p},y_{p})+\frac{\varepsilon }{T}B.
\end{gather*}

\noindent\textbf{Claim:}  For  $p=1,\dots ,s-1$, the points
$(x_{p},y_{p})$ are in $\Omega _{0}$.

 Indeed, by definition of $(x_{p},y_{p})$, we have
\begin{gather*}
x_{p}=x_{0}+\sum_{i=0}^{p-1}h_{i}y_{i}
+\sum_{i=0}^{p-1}\frac{h_{i}^{2}}{2}w_{i}+
\int_{0}^{h_{0}}f(\tau ,x_{0},y_{0})\,d\tau
+\sum_{i=1}^{p-1}\int_{\sum_{j=0}^{i-1}h_{j}}^{\sum_{j=0}^{i}h_{j}}f(\tau
,x_{i},y_{i})\,d\tau;\\
y_{p}=y_{p-1}+h_{p-1}w_{p-1};\\
w_{p}\in F(x_{p},y_{p})+\frac{\varepsilon }{T}B.
\end{gather*}
Hence
\begin{equation}
 \| y_{p}-y_{0}\| \leq \|
\sum_{i=0}^{p-1}h_{i}w_{i}\| \leq T(M+1)\leq
\frac{r}{3}\leq r, \label{e3.5}
\end{equation}
and
\begin{align*}
&\| x_{p}-x_{0}\| \\
&=\big\|\sum_{i=0}^{p-1}h_{i}y_{i}+\sum_{i=0}^{p-1}\frac{h_{i}^{2}}{2}w_{i}
+\int_{0}^{h_{0}}f(\tau ,x_{0},y_{0})\,d\tau
+\sum_{i=1}^{p-1}\int_{\sum_{j=0}^{i-1}h_{j}}^{\sum_{j=0}^{i}h_{j}}f(\tau
,x_{i},y_{i})\,d\tau \big\|\\
&\leq (\| y_{0}\|+\frac{r}{3})\sum_{i=0}^{p-1}h_{i}+(M+1)
\sum_{i=0}^{p-1}\frac{h_{i}^{2}}{2}+\int_{0}^{T}m(\tau )\,d\tau.
\end{align*}
Since
$$
\sum_{i=0}^{p-1}h_{i}\leq T \quad\mbox{and}\quad
\sum_{i=0}^{p-1}\frac{h_{i}^{2}}{2}\leq T,
$$
 by \eqref{e3.1}--\eqref{e3.3}, we have
\begin{equation}
\begin{gathered}
\| x_{p}-x_{0}\| \leq (\| y_{0}\|
+\frac{r}{3})T+\int_{0}^{T}(M+1+m(\tau ))\,d\tau ,\\
 \|x_{p}-x_{0}\| \leq T(\| y_{0}\|
+\frac{r}{3})+\frac{r}{3}\leq r;
\end{gathered}\label{e3.6}
\end{equation}
hence $(x_{p},y_{p})_{p}\subset \Omega _{0}$ which proves the claim.

 For any nonzero integer $k$ and for $q=1,\dots ,s$ denote by $h_{q}^{k}$
 a real associated to $\varepsilon =\frac{1}{k}$ and
 $(t,x,y)=(h_{q-1}^{k}, x_{q},y_{q})$ given by Lemma \ref{lem3.2}. Let the
 sequence $(\tau _{k}^{q})_{k}$ defined by
\begin{gather*}
\tau _{k}^{0}=0, \quad \tau _{k}^{s}=T,\\
 \tau _{k}^{q}=h_{0}^{k}+\dots+h_{q-1}^{k};
\end{gather*}
and consider the sequence of functions $(x_{k}(.))_{k}$
defined on each interval $[\tau _{k}^{q-1},\tau_{k}^{q}[ $ by
\begin{gather*}
\begin{aligned}
x_{k}(t)&=x_{q-1}+(t-\tau _{k}^{q-1})y_{q-1}+\frac{(t-\tau
_{k}^{q-1})^{2}}{2}w_{q-1}\\
&\quad+\int_{\tau _{k}^{q-1}}^{t}(t-\tau) f(\tau ,x_{q-1},y_{q-1})d\tau;
\end{aligned}\\
 x_{k}(0)=x_{0}.
\end{gather*}

\subsection*{Step 2. Convergence of approximate solutions}
By the definition of $x_{k}$, for all $t\in [\tau
_{k}^{q-1},\tau _{k}^{q}[$ we have
\begin{gather*}
\dot{x}_{k}(t)=y_{q-1}+(t-\tau _{k}^{q-1})w_{q-1}+\int_{\tau
_{k}^{q-1}}^{t} f(\tau ,x_{q-1 },y_{q-1})d\tau;\\
\ddot{x}_{k}(t)=w_{q-1}+ f(t,x_{q-1 },y_{q-1}).
\end{gather*}
Hence by \eqref{e3.5} and \eqref{e3.6} we have the estimates
\begin{equation}
\| \ddot{x}_{k}(t )\| \leq \| w_{q-1}\| +\| f(t,x_{q-1
},y_{q-1})\| \leq M+1+m(t);\label{e3.7}
\end{equation}
\begin{align*}
\| \dot{x}_{k}(t)\|
&=\| \dot{x}_{k}(\tau
_{k}^{q-1})+\int_{\tau _{k}^{q-1}}^{t}\ddot{x}_{k}(\tau
))d\tau \|;\\
\| \dot{x}_{k}(t)\|&\leq \| y_{q-1}\|
+\big\| \int_{0}^{T}(M+1+m(\tau ))d\tau \big\| \\
&\leq \| y_{q-1}\|+\frac{r}{3}
 \leq \| y_{0}\| +\frac{2r}{3};
\end{align*}
 and
\begin{align*}
 \| x_{k}(t)\|&=\| x_{k}(\tau
_{k}^{q-1})+\int_{\tau _{k}^{q-1}}^{t}\dot{x}_{k}(\tau ))d\tau \| \\
&\leq \| x_{q-1}\| + \int_{0}^{T}(\| y_{0}\| +\frac{2r}{3})d\tau
\\
&\leq \| x_{0}\| +T\| y_{0}\| +(1+\frac{2T}{3})r.
\end{align*}
By \eqref{e3.7} one has
$$
\int_{0}^{T}\| \ddot{x}_{k}(t)\| ^{2}dt\leq
\int_{0}^{T} (M+1+m(t))^{2}dt.
$$
Then the sequence $(\ddot{x}_{k}(.))_{k}$ is bounded in
$L^{2}([0,T] ,\mathbb{R}^{n})$ and $(\dot{x}_{k}(.))_{k}$ is
equiuniformly continuous. Moreover, we see that $(x_{k}(.))_{k}$
is equi-Lipschitzian, hence equiuniformly continuous.
 Therefore, the sequence $(\ddot{x}_{k}(.))_{k}$ is bounded
 in $L^{2}([0,T] ,\mathbb{R}^{n})$,
$(\dot{x}_{k}(.))_{k}$ and $(x_{k}(.)) _{k}$ are bounded in
$C([0,T] ,\mathbb{R}^{n})$ and equiuniformly continuous, hence, by
\cite[Theorem 0.3.4]{a3} there exist a subsequence, still denoted
by $(x_{k}(.))_{k}$ and an absolutely continuous function $x:[0,T]
\to \mathbb{R}^{n}$ such that
\begin{itemize}
\item[(i)] $x_{k}$ converges uniformly to $x$; \item[(ii)]
$\dot{x}_{k}$ converges uniformly to $\dot{x}$; \item[(iii)]
$\ddot{x}_{k}$ converges weakly in $L^{2}([0,T] ,\mathbb{R}^{n})$
to $\ddot{x}$.
\end{itemize}
The family of approximate solutions $x_{k}$ satisfies
the following property.

\begin{proposition} \label{prop3.3}
 For every $t\in [0,T[$  there exits $q\in \{ 1,\dots ,s\}$
such that
$$
\lim_{k\to \infty }d_{grF}\big(x_{k}(t),\dot{x}_{k}(t);
\ddot{x}_{k}(t)-f(t,x_{k}(\tau_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\big)=0
$$
\end{proposition}

\begin{proof} Let $t\in $ $[0,T] $. By
construction of $\tau _{k}^{q}$, there exists $q\in {1,\dots s}$ such
that  $t\in [\tau _{k}^{q-1},\tau _{k}^{q}[ $ and
$(\tau _{k}^{q})_{k}$ converges to $t$.
 Moreover, for $q=1,\dots s$
$$
\ddot{x}_{k}(t)-f(t,x_{k}(\tau_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))
=w_{q-1}\in F(x_{k}(\tau _{k}^{q-1}), \dot{x}_{k}(\tau
_{k}^{q-1}))+\frac{1}{kT}B,
$$
then
\begin{align*}
& \lim_{k\to \infty
}d_{grF}((x_{k}(t),\dot{x}_{k}(t));\ddot{x}_{k}(t)-f(t,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1})))\\
& \leq \lim_{k\to \infty }(\|
x_{k}(t)-x_{k}(\tau _{k}^{q-1})\| +\|
\dot{x}_{k}(t)-\dot{x}_{k}(\tau _{k}^{q-1})\|
+\frac{1}{kT}).
\end{align*}
 Since $\| \ddot{x}_{k}(t)\| \leq M+1+m(t)$ ,
$\| \dot{x}_{k}(t)\| \leq \| y_{0}\|
+\frac{2r}{3}$ and $(\tau _{k}^{q})_{k}$ converges to $t$, it follows that
$$
\lim_{k\to \infty }\| x_{k}(t)-x_{k}(\tau
_{k}^{q-1})\| =\lim_{k\to \infty }\|
\dot{x}_{k}(t)-\dot{x}_{k}(\tau _{k}^{q-1})\| = 0,
$$
hence
$$
 \lim_{k\to \infty }d_{grF}\big((x_{k}(t),\dot{x}_{k}(t));\ddot{x}_{k}(t)
 -f(t,x_{k}(\tau _{k}^{q-1}),
 \dot{x}_{k}(\tau _{k}^{q-1}))\big)=0.
 $$
This completes the proof.
\end{proof}
Since $x_{k}\to x$ uniformly, $\dot{x}_{k}\to \dot{x}$ uniformly,
$\ddot{x}_{k}\to \ddot{x}$ weakly in $L^{2}([0,T]
,\mathbb{R}^{n})$ and $(f(.,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1})))_{k}$ converges to
$f(.,x(.),\dot{x}(.))$ in $L^{2}([0,T];\mathbb{R}^{n})$ and $F$ is
upper semicontinuous, then by \cite[Theorem 1.4.1]{a3}, $x$ is a
solution of the convexified problem
\begin{gather*}
\ddot{x}(t)\in f(t,x(t),\dot{x}(t))+\text{co}(F(x(t),\dot{x}(t)))
 \mbox{ a.e.  on }[0,T]; \\
x(0)=x_{0},\quad \dot{x}(0)=y_{0}.
\end{gather*}
Consequently for all  $t\in [0,T] $  we have
\begin{equation}
\ddot{x}(t)-f(t,x(t),\dot{x}(t))\in
\partial V(\dot{x}(t))\label{e3.8}
\end{equation}

\begin{proposition} \label{prop3.4}
The application $x$ is a solution of \eqref{e1.1}.
\end{proposition}

\begin{proof} By \eqref{e3.8} and \cite[Lemma 3.3]{b3}, we obtain
$$
\frac{d}{dt}(V(\dot{x}(t)))=\langle \ddot{x}(t) , \ddot{x}(t)-f(t,x(t),
\dot{x}(t))\rangle \quad \mbox{a.e in }
[0,T];
$$
therefore,
\begin{equation}
V(\dot{x}(T))-V(y_{0})=\int_{0}^{T}\| \ddot{x}(\tau
)\| ^{2} d\tau -\int_{0}^{T}\langle \ddot{x}(\tau
);f(\tau ,x(\tau ),\dot{x}(\tau ))\rangle d\tau.
\label{e3.9)}
\end{equation}
 On the other hand, for  $q=1,\dots ,s$ and $t\in [\tau
_{k}^{q-1},\tau_{k}^{q} [$,
$$
\left(\ddot{x}_{k}(t)-f(t,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\right)\in
F(x_{k}(\tau _{k}^{q-1}),\dot{x}_{k}(\tau
_{k}^{q-1}))+\frac{1}{kT}B.
$$
Then
$$
\left(\ddot{x}_{k}(t)-f(t,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\right)\in
\partial V(\dot{x}_{k}(\tau _{k}^{q-1})+\frac{1}{kT}B,
$$
hence, there exists $b_{q}$ $\in B$ such that
\begin{equation}
\left(\ddot{x}_{k}(t)-f(t,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau
_{k}^{q-1}))+\frac{1}{kT}b_{q}\right)\in \partial
V(\dot{x}_{k}(\tau _{k}^{q-1}).\label{e3.10}
\end{equation}
 Properties of the subdifferential of a convex function
imply that for every $z$ in $\partial V(\dot{x}_{k}(\tau
_{k}^{q-1})$, we have
\begin{equation}
V(\dot{x}_{k}(\tau _{k}^{q}))-V(\dot{x}_{k}(\tau
_{k}^{q-1}))\geq  \langle \dot{x}_{k}(\tau _{k}^{q}) -
\dot{x}_{k}(\tau _{k}^{q-1}) ; z\rangle. \label{e3.11}
\end{equation}
Then by \eqref{e3.10}
\begin{align*}
&V(\dot{x}_{k}(\tau_{k}^{q}))-V(\dot{x}_{k}(\tau _{k}^{q-1}))\\
&\geq \langle \dot{x}_{k}(\tau _{k}^{q})- \dot{x}_{k}(\tau
_{k}^{q-1});\ddot{x}_{k}(t)-f(t,x_{k}(\tau _{k}^{q-1}),
\dot{x}_{k}(\tau _{k}^{q-1}))+\frac{1}{kT}b_{q}\rangle ;
\end{align*}
thus
\begin{align*}
&V(\dot{x}_{k}(\tau_{k}^{q}))-V(\dot{x}_{k}(\tau _{k}^{q-1}))\\
&\geq\langle \int_{\tau _{k}^{q-1}}^{\tau _{k}^{q}}\ddot{x}_{k}(\tau )
d\tau  ; \ddot{x}_{k}(t)-f(t,x_{k}(\tau_{k}^{q-1}),
\dot{x}_{k}(\tau_{k}^{q-1}))+ \frac{1}{kT}b_{q}\rangle \,.
\end{align*}
Since $\ddot{x}_{k}$ is constant in
$[\tau_{k}^{q-1},\tau_{k}^{q}[$, it follows that
\[
\begin{aligned}
V(\dot{x}_{k}(\tau _{k}^{q}))-V(\dot{x}_{k}(\tau_{k}^{q-1}))
&\geq \int_{\tau _{k}^{q-1}}^{\tau_{k}^{q}}\langle \ddot{x}_{k}(\tau ) ;
 \ddot{x}_{k}(\tau ) \rangle d\tau \\
&\quad-\int_{\tau _{k}^{q-1}}^{\tau
_{k}^{q}}\langle \ddot{x}_{k}(\tau ) ;  f(t,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle d\tau\\
&\quad+ \int_{\tau _{k}^{q-1}}^{\tau _{k}^{q}}\langle \ddot{x}_{k}(\tau ) ;
\frac{1}{kT}b_{q}\rangle d\tau;
\end{aligned}
\]
hence we have
\begin{equation}
\begin{aligned}
&V(\dot{x}_{k}(T))-V(y_{0})\\
&\geq \int_{0}^{T}\| \ddot{x}_{k}(\tau
)\| ^{2}d\tau - \sum_{q=1}^{s}\int_{\tau
_{k}^{q-1}}^{\tau _{k}^{q}}\langle \ddot{x}_{k}(\tau ) ; f(\tau
,x_{k}(\tau _{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle d\tau \\
&\quad +\sum_{q=1}^{s}\frac{1}{kT}\int_{\tau _{k}^{q-1}}^{\tau
_{k}^{q}}\langle \ddot{x}_{k}(\tau );b_{q}\rangle d\tau.
\end{aligned} \label{e3.12}
\end{equation}

\noindent \textbf{Claim:}  The sequence
$\big(\sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau
_{k}^{q}}\langle \ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle d\tau )_{k}$
converges to
$\int_{0}^{T}\langle\ddot{x}(\tau ) ; f(\tau ,x(\tau ),\dot{x}(\tau
))\rangle d\tau$.

\begin{proof} Since
$[0,T]=\bigcup_{q=1}^{s}[\tau _{k}^{q-1}, \tau _{k}^{q}]$,
we have
\begin{align*}
&\big\| \sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau
_{k}^{q}}\langle\ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle d\tau
  - \int_{0}^{T}\langle \ddot{x}(\tau ) ; f(\tau ,x(\tau ),\dot{x}(\tau
))\rangle d\tau \big\|\\
& =
\| \sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau
_{k}^{q}}(\langle \ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle
 - \langle \ddot{x}(\tau ) ;
f(\tau,x(\tau ),\dot{x}(\tau ))\rangle )d\tau \| \\
&\leq
\sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau
_{k}^{q}}\| \langle \ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle
 - \langle \ddot{x}(\tau ) ;
f(\tau,x(\tau ),\dot{x}(\tau ))\rangle \| d\tau\,.
\end{align*}
Since
\begin{align*}
&\sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau _{k}^{q}}\| \langle
\ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle
 - \langle \ddot{x}(\tau ) ;
f(\tau,x(\tau ),\dot{x}(\tau ))\rangle \| d\tau \\
&\leq \sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau _{k}^{q}}\|
\langle \ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle
 - \langle \ddot{x}_{k}(\tau
 ) ; f(\tau ,x_{k}(\tau ),\dot{x}_{k}(\tau ))\rangle \| d\tau \\
&\quad +
 \sum_{q=1}^{s}\int_{\tau
_{k}^{q-1}}^{\tau _{k}^{q}}\| \langle \ddot{x}_{k}(\tau ) ; f(\tau
,x_{k}(\tau ),\dot{x}_{k}(\tau ))\rangle
 - \langle \ddot{x}_{k}(\tau ) ; f(\tau
,x(\tau ),\dot{x}(\tau ))\rangle \| d\tau \\
&\quad +
 \sum_{q=1}^{s}\int_{\tau
_{k}^{q-1}}^{\tau _{k}^{q}}\| \langle \ddot{x}_{k}(\tau ) ; f(\tau
,x(\tau ),\dot{x}(\tau ))\rangle
 - \langle \ddot{x}(\tau ) ; f(\tau ,x(\tau),\dot{x}(\tau ))\rangle
 \| d\tau \\
&= \sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau _{k}^{q}}\| \langle
\ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle
 -\langle \ddot{x}_{k}(\tau) ; f(\tau ,x_{k}(\tau ),\dot{x}_{k}(\tau ))
\rangle \| d\tau \\
&\quad + \int_{0}^{T}\| \langle \ddot{x}_{k}(\tau ) ; f(\tau
,x_{k}(\tau ),\dot{x}_{k}(\tau ))\rangle -
\langle \ddot{x}_{k}(\tau) ; f(\tau ,x(\tau ),\dot{x}(\tau ))\rangle \| d\tau\\
&\quad+ \int_{0}^{T}\| \langle \ddot{x}_{k}(\tau ) ; f(\tau
,x(\tau ),\dot{x}(\tau ))\rangle - \langle \ddot{x}(\tau ) ;
f(\tau ,x(\tau ),\dot{x}(\tau ))\rangle \| d\tau,
\end{align*}
it follows that
\begin{align*}
&\big\| \sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau
_{k}^{q}}\langle\ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle d\tau
  - \int_{0}^{T}\langle \ddot{x}(\tau ) ; f(\tau ,x(\tau ),\dot{x}(\tau
))\rangle d\tau \big\|\\
&= \sum_{q=1}^{s}\int_{\tau _{k}^{q-1}}^{\tau _{k}^{q}}\| \langle
\ddot{x}_{k}(\tau ) ; f(\tau ,x_{k}(\tau
_{k}^{q-1}),\dot{x}_{k}(\tau _{k}^{q-1}))\rangle
 -\langle \ddot{x}_{k}(\tau) ; f(\tau ,x_{k}(\tau ),\dot{x}_{k}(\tau ))
\rangle \| d\tau \\
&\quad + \int_{0}^{T}\| \langle \ddot{x}_{k}(\tau ) ; f(\tau
,x_{k}(\tau ),\dot{x}_{k}(\tau ))\rangle -
\langle \ddot{x}_{k}(\tau) ; f(\tau ,x(\tau ),\dot{x}(\tau ))\rangle \| d\tau\\
&\quad+ \int_{0}^{T}\| \langle \ddot{x}_{k}(\tau ) ; f(\tau
,x(\tau ),\dot{x}(\tau ))\rangle - \langle \ddot{x}(\tau ) ;
f(\tau ,x(\tau ),\dot{x}(\tau ))\rangle \| d\tau\,.
\end{align*}
Since $f$ is a Carath\'{e}odory function, $x_{k}$ and
$\dot{x}_{k}$ are uniformly lipschitz continuous,
$\|\ddot{x}_{k}(s)\| \leq M+1+m(s)$, $m\in
L^{2}([0,T],\mathbb{R}^{n})$, $x_{k}\to x$, $\dot{x}_{k}\to
\dot{x}$ uniformly and $\ddot{x}_{k}$ $\to \ddot{x}$ weakly in
$L^{2}([0,T] ,\mathbb{R}^{n})$ then the last term converges to 0.
Hence the claim is proved.
\end{proof}

Since
$$
\lim_{k\to \infty
}\sum_{q=1}^{s}\frac{1}{k}\int_{\tau _{k}^{q-1}}^{\tau
_{k}^{q}}\langle \ddot{x}_{k}(\tau ) ; b_{q}\rangle d\tau =0,
$$
by passing to the limit as $k\to \infty $
in \eqref{e3.12} and using the continuity of the function $V$ on the
ball $B(y_{0},r)$, we obtain the estimate
$$
V(\dot{x}(T))-V(y_{0})\geq \lim_{k\to \infty }\sup
\int_{0}^{T}\| \ddot{x}_{k}(\tau )\| ^{2}d\tau
-\int_{0}^{T}<\ddot{x}(\tau );f(\tau ,x(\tau
),\dot{x}(\tau )>\,d\tau.
$$
Moreover, by  \eqref{e3.8}, we have
$$
\| \ddot{x}\| _{2}^{2}\geq \lim_{k\to
\infty }\sup \| \ddot{x}_{k}\| _{2}^{2},
$$
and by the weak lower semicontinuity of the norm, it
follows that
$$
\| \ddot{x}\| _{2}^{2}\leq \lim_{k\to \infty }\inf \| \ddot{x}_{k}\| _{2}^{2}.
$$
Hence $\lim_{k\to \infty }\|\ddot{x}_{k}\| _{2}^{2}=\| \ddot{x}\|
_{2}^{2}$, i.e. $((\ddot{x}_{k}))_{k}$ converges to $\ddot{x}$
strongly in $L^{2}([0,T] ,\mathbb{R}^{n})$. So that there exists a
subsequence $\ddot{x}_{k}$ which converges pointwisely almost
every where to $\ddot{x}$. In view of Proposition \ref{prop3.3},
we conclude that
$$
 d_{grF}(x(t),\dot{x}(t),\ddot{x}(t)-f(t,x(t),\dot{x}(t)))=0\quad
 \mbox{a.e. }t\in [0,T].
$$
Since the graph of $F$ is closed, we have
$$
\ddot{x}(t)\in f(t,x(t),\dot{x}(t))+F(x(t),\dot{x}(t))\
\mbox{a.e. } t\in [0,T] .
$$
Finally, let $t\in [0,T] $. Recall that
there exits $(\tau _{k}^{q})_{k}$ such that
$\lim_{k\to \infty }$ $\tau _{k}^{q}=t$ for all
$t\in [0,T] $. Since $\underset{k\to \infty
}{\lim }\| x(t)-x_{k}(\tau _{k}^{q})\| =0$, $x_{k}(\tau
_{k}^{q})\in K$ and $K$ is closed, by passing to the limit for
$k\to \infty $ we obtain $x(t)\in K$. This completes the
proof.
\end{proof}

\subsection*{Acknowledgements}
 The authors would like to thank the anonymous referee for
 reading carefully the original manuscript and for giving us
 some suggestions.

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\end{document}