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\markboth{\hfil A viability result \hfil EJDE--2002/76}
{EJDE--2002/76\hfil Vasile Lupulescu \hfil}

\begin{document}

\title{\vspace{-1in}
\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 76, pp. 1--12. \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} \\
%
 A viability result for second-order differential inclusions 
%
\thanks{\emph{Mathematics Subject Classifications:} 34G20, 47H20. 
\hfil\break\indent
{\em Key words:} second-order contingent set, subdifferential, 
viable solution. 
\hfil\break\indent
\copyright 2002 Southwest Texas State University. 
\hfil\break\indent
Submitted December 9, 2001. Revised March 26, 2002. Published August 20, 2002.} }

\date{}
\author{Vasile Lupulescu}
\maketitle

\begin{abstract}
  We prove a viability result for the second-order differential inclusion 
  \[
  x''\in F(x,x'),\quad (x(0), x'(0))=(x_0,y_0)\in Q:=K\times \Omega, 
  \]
  where $K$ is a closed and $\Omega$ is an open subsets of $\mathbb{R}^m$,
  and is an upper semicontinuous set-valued map with compact values,  
  such that $F(x,y) \subset \partial V(y)$, for some convex proper  
  lower semicontinuous function $V$.
\end{abstract}

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

\section{Introduction}

Bressan, Cellina and Colombo \cite{b2} proved the existence of local
solutions to the Cauchy problem 
\[
x'\in F(x) ,\quad x(0) =\xi \in K,
\]
where $F$ is an upper semicontinuous, cyclically monotone, and compact
valued multifunction. While Rossi \cite{r2} proved a viability result for
this problem. On the other hand, for the second order differential inclusion 
\[
x''\in F(x,x') ,\quad x(0) =x_0,\quad x'(0)=y_0,
\]
existence results were obtained by many authors \cite{a1,a4,c1,c2,m1,s1}).
In \cite{l1}, existence results are proven for the case when $F(.,.)$ is an
upper semicontinuous set-valued map with compact values, such that $F(x,y)
\subset \partial V(y)$ for some convex proper lower semicontinuous function 
$V$.

The aim of this paper is to prove a viability result for the second-order
differential inclusion 
\[
x''\in F(x,x') ,\quad (x(0) ,x'(0)
)=(x_0,y_0)\in Q:=K\times \Omega ,
\]
where $K$ is a closed and $\Omega $ is an open subsets of $\mathbb{R}^m$, and 
$F:Q\subset \mathbb{R}^{2m}\to 2^{\mathbb{R}^m}$ is an upper semicontinuous
set-valued map with compact values, such that $F(x,y) \subset \partial V(y) 
$, for some convex proper lower semicontinuous function $V$.

\section{Preliminaries and statement of main result}

Let $\mathbb{R}^m$ be the m-dimensional Euclidean space with scalar product 
$\langle .,.\rangle $ and norm $\| .\| $. For $x\in \mathbb{R}^m$ and 
$\varepsilon >0$ let 
\[
B_{\varepsilon }(x)=\{y\in \mathbb{R}^m:\| x-y\| <\varepsilon \}
\]
be the open ball, centered at $x$ with radius $\varepsilon $, and let 
$\overline{B}_{\varepsilon }(x)$ be its closure. Denote by $B$ the open unit
ball $B=\{ x\in \mathbb{R}^m:\| x\| <1\}$.

For $x\in \mathbb{R}^m$ and for a closed subsets $A\subset \mathbb{R}^m$ we denote
by $d(x,A)$ the distance from $x$ to $A$ given by 
\[
d(x,A) =\inf \{ \| x-y\| :y\in A\} .
\]

Let $V:\mathbb{R}^m\to \mathbb{R}$ be a proper lower semicontinuous convex
function. The multifunction $\partial V:\mathbb{R}^m\to 2^{\mathbb{R}^m}$
defined by 
\[
\partial V(x)=\{\xi \in \mathbb{R}^m:V(y)-V(x)\geqslant \langle \xi ,y-x\rangle
,\; \forall y\in \mathbb{R}^m\}
\]
is called subdifferential (in the sense of convex analysis) of the function 
$V$.

We say that a multifunction $F:\mathbb{R}^m\to 2^{\mathbb{R}^m}$ is upper
semicontinuous if for every $x\in \mathbb{R}^m$ and every $\varepsilon >0$
there exists $\delta >0$ such that 
\[
F(y) \subset F(x) +B_{\varepsilon }(0) , \quad \forall y\in B_{\delta }(x) .
\]

This definition of the upper semicontinuous multifunction is less
restrictive than the usual (see Definition 1.1.1 in \cite{a3} or Definition
1.1 in \cite{d1}). Actually such a property is called ($\varepsilon $, 
$\delta $)-upper semicontinuity (see Definition 1.2 in \cite{d1}) and it is
only equivalent to the upper semicontinuity for compact-valued
multifunctions (see Proposition 1.1 in \cite{d1}).

For $K\subset \mathbb{R}^m$ and $x\in K$ denote by $T_{K}(x)$ the Bouligand's
contingent cone of $K$ at $x$, defined by 
\[
T_{K}(x)=\big\{v\in \mathbb{R}^m:\liminf_{h\to 0+} \frac{d(x+hv,K)}{h}=0\big\}.
\]

For $K\subset \mathbb{R}^m$ and $(x,y)\in K\times \mathbb{R}^m$ we denote by 
$T_{K}^{(2)}(x,y)$ the second-order contingent set of $K$ at $(x,y)$
introduced by Ben-Tal \cite{b1} and defined by 
\[
T_{K}^{(2)}(x,y)=\big\{v\in \mathbb{R}^m:\liminf_{h\to 0+} \frac{d(x+hy+%
\frac{h^{2}}{2}v,K)}{h^{2}/2}=0\big\}.
\]
We remark that if $T_{K}^{(2)}(x,y)$ is non-empty then, necessarily, $y\in
T_{K}(x)$.

Moreover (see \cite{a4}, \cite{c2}, \cite{m1}), if $F$ is upper
semicontinuous with compact convex values and if $x:[ 0,T] \to \mathbb{R%
}^m$ is a solution of the Cauchy problem 
\[
x''\in F(x,x'),\quad x(0)=x_0,\quad x'(0)=y_0,
\]
such that $x(t)\in K$, $\forall t\in \lbrack 0,T]$, then 
\[
(x(t),x'(t))\in \mathop{\rm graph}(T_{K}),\quad \forall t\in [0,T),
\]
hence, in particular, $(x_0,y_0)\in \mathop{\rm graph}(T_{K})$.

For a multifunction $F:Q:=K\times \Omega \subset \mathbb{R}^{2m} \to 2^{%
\mathbb{R}^m}$ and for any $(x_0,y_0)\in \mathop{\rm graph}(T_{K})$ we consider
the Cauchy problem 
\begin{equation}
x''\in F(x,x') ,\quad (x(0) ,x'(0)
)=(x_0,y_0)\in Q  \label{1}
\end{equation}
under the following assumptions:

\begin{enumerate}
\item[(H1)]  $K$ is a closed and $\Omega $ and open subset of $\mathbb{R}^{m}$,
such that 
\[
Q:=K\times \Omega \subset \mathop{\rm graph}(T_{K})
\]

\item[(H2)]  $F$ is an upper semicontinuous compact valued multifunction
such that 
\[
F(x,y)\cap T_{K}^{(2)}(x,y)\neq \varnothing ,\quad \forall (x,y)\in Q;
\]

\item[(H3)]  There exists a proper convex and lower semicontinuous function 
$V:$ $\mathbb{R}^{m}\to \mathbb{R}$ such that 
\[
F(x,y)\subset \partial V(y),\quad \forall (x,y)\in Q.
\]
\end{enumerate}

\paragraph{Remark.}

A convex function $V:\mathbb{R}^m\to \mathbb{R}$ is continuous in the
whole space $\mathbb{R}^m$ (Corollary 10.1.1 in \cite{r1}) and almost
everywhere differentiable (Theorem 25.5 in \cite{r1}). Therefore, (H3)
strongly restricts the multivaluedness of $F$.

\paragraph{Definition.}

%1.
By viable solution of the problem \eqref{1} we mean any absolutely
continuous function $x:[0,T]\to \mathbb{R}^m$ with absolutely
continuous derivative $x'$ such that $x(0)=x_0$, $x(0)=y_0$, 
\begin{gather*}
x''(t)\in F(x(t),x'(t))\quad a.e.\text{ on }[0,T], \\
(x(t),x'(t))\in Q \quad \forall t\in [0,T].
\end{gather*}

Our main result is the following:\medskip

\begin{theorem} \label{thm1}
If $F:Q\subset \mathbb{R}^{2m}\to 2^{\mathbb{R}^m}$
 and $V:\mathbb{R}^m\to \mathbb{R}$
satisfy assumptions (H1)--(H3), then then for every
$(x_0,y_0)\in Q$  there exist $T>0$ and
$x:[0,T]\to \mathbb{R}^m$, a viable solution of the problem
\eqref{1}.
\end{theorem}

\section{Proof of the main result}

We start this section with the following technical result, which will be
used to prove the main result.

\begin{lemma} \label{lm2}
Assume $Q=K\times \Omega \subset \mathbb{R}^{2m}$
satisfies (H1), $F:Q\to 2^{\mathbb{R}^m}$ satisfies
(H2), $Q_0\subset Q$ is a compact subset and $(x_0,y_0)\in Q_0$.
Then for every $k\in \mathbb{N}^{\ast }$ there exist
$h_{k}^{0}\in (0,\frac{1}{k}]$ and $u_{k}^{0}\in \mathbb{R}^m$
such that
\begin{gather*}
x_0+h_{k}^{0}y_0+\frac{(h_{k}^{0}) ^{2}}{2}u_{k}^{0}\in K,
(x_0,y_0,u_{k}^{0})\in \mathop{\rm graph}(F)+\frac{1}{k}(B\times B\times B).
\end{gather*}
\end{lemma}

\paragraph{Proof.}

Let $(x,y)\in Q$ be fixed. Since by (H2), $F(x,y)\cap T_{K}^{(2)}(x,y)\neq
\emptyset$, there exists $v=v_{(x,y)}\in F(x,y)$ such that 
\[
\liminf_{h\to 0_+} \frac{d(x+hy+\frac{h^{2}}{2}v,K)}{h^{2}/2}=0.
\]
Hence, for every $k\in \mathbb{N}^{\ast }$ there exists $h_{k}=h_{k}(x,y)\in (0,%
\frac{1}{k}]$ such that 
\begin{equation}
d(x+h_{k}y+\frac{h_{k}^{2}}{2}v,K)<\frac{h_{k}^{2}}{4k}.  \label{2}
\end{equation}
By the continuity of the map $(a,b)\to d(a+h_{k}b+\frac{h_{k}^{2}}{2}%
v,K)$ it follows that 
\[
N(x,y)=\big\{(a,b):d(a+h_{k}b+\frac{h_{k}^{2}}{2}v,K)<\frac{h_{k}^{2}}{4k}%
\big\}
\]
is an open set and, by \eqref{2}, it contains $(x,y)$. Then there exists 
$r:=r(x,y) \in (0,\frac{1}{k})$ such that $B_{r}(x,y)\subset N(x,y)$. Since 
$Q_0$ is compact there exists a finite subset $\{ (x_{j},y_{j})\in
Q:1\leqslant j\leqslant m\} $ such that 
\[
Q_0\subset \bigcup_{j=1}^mB_{r_{j}}(x_{j},y_{j}).
\]
We set 
\[
h_0(k) :=\min \{ h_{k}(x_{j},y_{j}) :j\in \{ 1,\dots ,m\} \} .
\]

Since $(x_0,y_0)\in Q_0$, there exists $j_0\in \{1,2,\dots m\}$ such that 
\begin{equation}
(x_0,y_0)\in B_{r_{j_0}}(x_{j_0},y_{j_0})\subset N(x_{j_0},y_{j_0}).
\label{3}
\end{equation}
Denote by $h_{k}^{0}:=h_{k}(x_{j_0},y_{j_0})$ and remark that, by \eqref{2}
and \eqref{3}, one has $h_{k}^{0}\in [ h_0(k) ,\frac{1}{k}] $ and there
exists $z_0\in K$ such that we have that 
\[
\frac{d(x_0+h_{k}^{0}y_0+\frac{(h_{k}^{0}) ^{2}}{2}v_0,z_0)}{(h_{k}^{0})
^{2}/2}\leqslant \frac{d(x_0+h_{k}y_0+\frac{(h_{k}^{0}) ^{2}}{2}v_0,K)}{%
(h_{k}^{0}) ^{2}/2}+\frac{1}{2k}<\frac{1}{k},
\]
hence 
\begin{equation}
\| \frac{z_0-x_0-h_{k}^{0}y_0}{(h_{k}^{0}) ^{2}/2}-v_0\| <\frac{1}{k}.
\label{4}
\end{equation}
Let 
\[
u_{k}^{0}:=\frac{z_0-x_0-h_{k}^{0}y_0}{(h_{k}^{0}) ^{2}/2}.
\]
Then 
\[
x_0+h_{k}^{0}y_0+\frac{(h_{k}^{0}) ^{2}}{2}u_{k}^{0}\in K.
\]
By \eqref{4} and \eqref{3} we get successively: 
\begin{gather*}
\| u_{k}-v_0\| <\frac{1}{k}, \\
d((x_0,y_0) ,(x_{j_0},y_{j_0})) \leqslant r_{j_0}<\frac{1}{k},
\end{gather*}
hence $(x_0,y_0,u_{k}^{0})\in \mathop{\rm graph}(F) +\frac{1}{k}(B\times
B\times B)$. \hfill$\square$

\paragraph{Proof of Theorem \ref{thm1}}

Let $(x_0,y_0)\in Q\subset \mathop{\rm graph}(T_{K})$. Since $\Omega \subset 
\mathbb{R}^m$ is an open subset, there exist $r>0$ such that $\overline{B}%
_{r}(y_0)\subset \Omega$.

We set $Q_0:=\overline{B}_{r}(x_0,y_0)\cap (K\times \overline{B}_{r}(y_0)).$
Since $Q_0$ is a compact set, by the upper semicontinuity of $F$ and
Proposition 1.1.3 in \cite{a3}, we have that 
\[
F(Q_0):=\bigcup_{(x,y)\in Q_0}F(x,y)
\]
is a compact set, hence there exists $M>0$ such that: 
\[
\sup \{ \| v\| :v\in F(x,y),\quad (x,y)\in Q_0\} \leqslant M.
\]
Let 
\begin{equation}
T=\min \big\{\frac{r}{2(M+1) },\sqrt{\frac{r}{M+1}}, \frac{r}{2(\| y_0\| +1)}%
\big\}.  \label{5}
\end{equation}
We shall prove the existence of a viable solution of the problem \eqref{1}
defined on the interval $[0,T]$. Since $(x_0,y_0)\in Q_0$ then, by Lemma \ref
{lm2}, there exist $h_{k}^{0}\in [ h_0(k) ,\frac{1}{k}] $ and $u_{k}^{0}\in 
\mathbb{R}^m$ such that 
\[
x_0+h_{k}^{0}y_0+\frac{1}{2}(h_{k}^{0})^{2}u_{k}^{0}\in K
\]
and $(x_0,y_0,u_{k}^{0})\in \mathop{\rm graph}(F) +\frac{1}{k}(B\times
B\times B)$. Define 
\begin{equation}
\begin{aligned}
x_{k}^{1}:=&x_0+h_{k}^{0}y_0+\frac{1}{2}(h_{k}^{0})^{2}u_{k}^{0}; \\
y_{k}^{1}:=&y_0+h_{k}^{0}u_{k}^{0}. \end{aligned}  \label{6}
\end{equation}
We remark that if $h_{k}^{0}<T$ then 
\begin{gather*}
\| x_{k}^{1}-x_0\| \leqslant h_{k}^{0}\| y_0\| + \frac{1}{2}%
(h_{k}^{0})^{2}\| u_{k}^{0}\| <h_{k}^{0}\| y_0\| +\frac{1}{2}%
(h_{k}^{0})^{2}(M+1) , \\
\| y_{k}^{1}-y_0\| =h_{k}^{0}\| u_{k}^{0}\| <h_{k}^{0}(M+1) ,
\end{gather*}
and by the choice of $T$ we get 
\[
\| x_{k}^{1}-x_0\| <r,\quad \| y_{k}^{1}-y_0\| <r.
\]
Therefore $(x_{k}^{1},y_{k}^{1}) \in Q_0$ and by Lemma \ref{lm2}, there
exist $h_{k}^{1}\in [ h_0(k) ,\frac{1}{k}] $ and $u_{k}^{1}\in \mathbb{R}^m$
such that 
\begin{gather*}
x_{k}^{1}+h_{k}^{1}y_{k}^{1}+\frac{1}{2}(h_{k}^{1})^{2}u_{k}^{1}\in K, \\
(x_{k}^{1},y_{k}^{1},u_{k}^{1})\in \mathop{\rm graph}(F) +\frac{1}{k}%
(B\times B\times B).
\end{gather*}
We claim that, for each $k\in \mathbb{N}^{\ast }$, there exist $m(k)\in\mathbb{N}%
^{\ast }$ and $h_{k}^{p}$, $x_{k}^{p}$, $y_{k}^{p}$, $u_{k}^{p}$, such that
for every $p\in \{ 2,\dots ,m(k)-1\} $, we have that:

\begin{enumerate}
\item[(i)]  $\sum_{j=0}^{m(k)-1}h_{k}^{j}\leqslant
T<\sum_{j=0}^{m(k)}h_{k}^{j}$

\item[(ii)]  
\begin{align*}
x_{k}^{p}=& x_{k}^{0}+\big(\sum_{i=0}^{p-1}h_{k}^{i}\big)y_{0}+\frac{1}{2}%
\sum_{i=0}^{p-1}(h_{k}^{i})^{2}u_{k}^{i}+\sum_{i=0}^{p-2}%
\sum_{j=i+1}^{p-1}h_{k}^{i}h_{k}^{j}u_{k}^{i}, \\
y_{k}^{p}=& y_{k}^{0}+\sum_{i=0}^{p-1}h_{k}^{i}u_{k}^{i};
\end{align*}

\item[(iii)]  $(x_{k}^{p},y_{k}^{p})\in Q_{0}$

\item[(iv)]  $(x_{k}^{p},y_{k}^{p},u_{k}^{p})\in \mathop{\rm graph}(F)+\frac{%
1}{k}(B\times B\times B)$.
\end{enumerate}

If $h_{k}^{0}+h_{k}^{1}\geq T$ then we set $m(k) =1$. Assume that 
$h_{k}^{0}+h_{k}^{1}<T$ and define 
\begin{equation}
\begin{aligned} x_{k}^{2}:=&
x_{k}^{1}+h_{k}^{1}y_{k}^{1}+\frac{1}{2}(h_{k}^{1})^{2}u_{k}^{1}, \\
y_{k}^{2}:=& y_{k}^{1}+h_{k}^{1}u_{k}^{1}. \end{aligned}  \label{7}
\end{equation}
Then by \eqref{6} and \eqref{7} we have that 
\begin{align*}
x_{k}^{2}:=&x_{k}^{0}+(h_{k}^{0}+h_{k}^{1})y_{k}^{0}+\frac{1}{2}%
(h_{k}^{0})^{2}u_{k}^{1}+(h_{k}^{1})^{2}u_{k}^{1}+h_{k}^{0}h_{k}^{1}u_{k}^{0},
\\
y_{k}^{2}:=&y_{k}^{0}+h_{k}^{0}u_{k}^{0}+h_{k}^{1}u_{k}^{1}
\end{align*}
and since $h_{k}^{0}+h_{k}^{1}\leq T$ and 
\begin{align*}
\| x_{k}^{2}-x_0\| \leqslant &(h_{k}^{0}+h_{k}^{1})\| y_{k}^{0}\| +\frac{1}{2%
}(h_{k}^{0})^{2}\| u_{k}^{0}\| + \frac{1}{2}(h_{k}^{1})^{2}\| u_{k}^{1}\|
+h_{k}^{0}h_{k}^{1}\| u_{k}^{0}\| \\
<&(h_{k}^{0}+h_{k}^{1})\| y_{k}^{0}\| +\frac{1}{2}%
(h_{k}^{0}+h_{k}^{1})^{2}(M+1) ,
\end{align*}
it follows 
\[
\| x_{k}^{2}-x_0\| <r,\| y_{k}^{2}-y_0\| <r,
\]
hence $(x_{k}^{2},y_{k}^{2}) \in Q_0$.

Assume that $h_{k}^{q}$ $x_{k}^{q}$, $y_{k}^{q}$ $u_{k}^{q}$, have been
constructed for $q\leqslant p$ satisfying (ii)--(iv) and that we construct 
$h_{k}^{p+1}$, $x_{k}^{p+1}$, $y_{k}^{p+1}$, $u_{k}^{p+1}$ satisfying such
properties. Since $(x_{k}^{p},y_{k}^{p}) \in Q_0$, by lemma 2, there exist 
$h_{k}^{p}\in [ h_0(k) ,\frac{1}{k}] $ and $u_{k}^{p}\in \mathbb{R}^m$ such that 
\begin{gather*}
x_{k}^{p}+h_{k}^{p}y_{k}^{p}+\frac{1}{2}(h_{k}^{p})^{2}u_{k}^{p}\in K, \\
(x_{k}^{p},y_{k}^{p},u_{k}^{p})\in \mathop{\rm graph}(F)+\frac{1}{k}(B\times
B\times B).
\end{gather*}
If $h_{k}^{0}+h_{k}^{1}+\dots +h_{k}^{p}\geqslant T$ then we set $m(k) =p$.
Assume that $h_{k}^{0}+h_{k}^{1}+\dots +h_{k}^{p}<T$ and define 
\begin{equation}
\begin{aligned}
x_{k}^{p+1}:=&x_{k}^{p}+h_{k}^{p}y_{k}^{p}+%
\frac{1}{2}(h_{k}^{p})^{2}u_{k}^{p}, \\
y_{k}^{p+1}:=&y_{k}^{p}+h_{k}^{p}u_{k}^{p}. \end{aligned}  \label{8}
\end{equation}
Then, by the above equations and (ii), we obtain that 
\begin{eqnarray*}
x_{k}^{p+1} &=&x_{k}^{p}+h_{k}^{p}y_{k}^{p}+\frac{1}{2}
(h_{k}^{p})^{2}u_{k}^{p}=x_{k}^{0}+\big(\sum_{i=0}^{p-1}h_{k}^{i}\big) y_0 +%
\frac{1}{2}\sum_{i=0}^{p-1}(h_{k}^{i})^{2}u_{k}^{i} \\
&&+\frac{1}{2}\sum_{i=0}^{p-1}(h_{k}^{i})^{2}u_{k}^{i}
+\sum_{i=0}^{p-2}\sum_{j=i+1}^{p-1}h_{k}^{i}h_{k}^{j}u_{k}^{i}
+h_{k}^{p}\sum_{i=0}^{p-1}h_{k}^{i}u_{k}^{i} +\frac{1}{2}%
(h_{k}^{p})^{2}u_{k}^{p} \\
&=&x_{k}^{0}+\big(\sum_{i=0}^{p}h_{k}^{i}\big) y_0+\frac{1}{2}
\sum_{i=0}^{p}(h_{k}^{i})^{2}u_{k}^{i}+\sum_{i=0}^{p-1}
\sum_{j=i+1}^{p}h_{k}^{i}h_{k}^{j}u_{k}^{i}
\end{eqnarray*}
and 
\[
y_{k}^{p+1}:=y_{k}^{p}+h_{k}^{p}u_{k}^{p}=y_{k}^{0}+%
\sum_{i=0}^{p-1}h_{k}^{i}u_{k}^{i}+h_{k}^{p}u_{k}^{p}=y_{k}^{0}+%
\sum_{i=0}^{p}h_{k}^{i}u_{k}^{i}.
\]
Therefore, 
\begin{eqnarray*}
\| x_{k}^{p+1}-x_0\| &\leqslant &\big(\sum_{i=0}^{p}h_{k}^{i}\big) \| y_0\| +%
\frac{1}{2} \sum_{i=0}^{p}(h_{k}^{i})^{2}\| u_{k}^{i}\|
+\sum_{i=0}^{p-1}\sum_{j=i+1}^{p}h_{k}^{i}h_{k}^{j}\| u_{k}^{i}\| \\
&\leqslant &\big(\sum_{i=0}^{p}h_{k}^{i}\big) \| y_0\| +\frac{M+1}{2}\big(%
\sum_{i=0}^{p}h_{k}^{i}\big) ^{2}
\end{eqnarray*}
and 
\[
\| y_{k}^{p+1}-x_0\| \leqslant \sum_{i=0}^{p}h_{k}^{i}\| u_0^{i}\| \leqslant
(M+1)\big(\sum_{i=0}^{p}h_{k}^{i}\big).
\]
Since $\sum_{i=0}^{p}h_{k}^{i}<T$ one obtains that 
\[
\| x_{k}^{p+1}-x_0\| <r,\quad \| y_{k}^{p+1}-x_0\| <r,
\]
hence $(x_{k}^{p+1},y_{k}^{p+1}) \in Q_0$.

We remark that this iterative process is finite because $h_{k}^{p}\in
[h_0(k) ,\frac{1}{k}] $, implies the existence of an integer $m(k)$ such
that 
\[
h_{k}^{0}+h_{k}^{1}+\dots +h_{k}^{m(k)-1}\leqslant
T<h_{k}^{0}+h_{k}^{1}+\dots +h_{k}^{m(k)-1}+h_{k}^{m(k)}.
\]

By (iv), for every $k\in \mathbb{N}^{\ast }$ and every $p\in \{ 0,1,\dots ,m(k)
\} $ there exists $(a_{k}^{p},b_{k}^{p},v_{k}^{p})\in \mathop{\rm graph}(F)$
such that 
\begin{equation}
\| x_{k}^{p}-a_{k}^{p}\| <\frac{1}{k},\quad \|y_{k}^{p}-b_{k}^{p}\| <\frac{1%
}{k},\quad \| u_{k}^{p}-v_{k}^{p}\|<\frac{1}{k};  \label{9}
\end{equation}
hence, 
\begin{equation}
\begin{gathered} \| x_{k}^{p}\| \leqslant \| x_{k}^{p}-x_0\| +\| x_0\|
\leqslant \frac{1}{k}+\| x_0\| \leqslant 1+\| x_0\| , \\ \| y_{k}^{p}\|
\leqslant \| y_{k}^{p}-y_0\| +\| y_0\| \leqslant \frac{1}{k}+\| y_0\|
\leqslant 1+\| y_0\| , \\ \| u_{k}^{p}\| \leqslant \| u_{k}^{p}-v_{k}^{p}\|
+\| v_{k}^{p}\| \leqslant \frac{1}{k}+M\leqslant 1+M. \end{gathered}
\label{10}
\end{equation}
Let us set 
\[
t_{k}^{p}=h_{k}^{0}+h_{k}^{1}+\dots +h_{k}^{p-1},\quad t_{k}^{0}=0.
\]
We remark that for all $k\in \mathbb{N}^{\ast}$ and all $p\in \{ 1,\dots ,m(k)\}
$, we have 
\begin{equation}
t_{k}^{p}-t_{k}^{p-1}<\frac{1}{k}\quad\text{and}\quad
t_{k}^{m(k)-1}\leqslant T<t_{k}^{m(k)}.  \label{11}
\end{equation}
For each $k\geqslant 1$ and for $p\in \{ 1,\dots ,m(k)\} $ we set 
$I_{k}^{p}=[t_{k}^{p-1},t_{k}^{p}]$ and for $t\in I_{k}^{p}$ we define 
\begin{equation}
x_{k}(t)=x_{k}^{p-1}+(t-t_{k}^{p-1})y_{k}^{p-1}+\frac{1}{2}
(t-t_{k}^{p-1})^{2}u_{k}^{p-1}.  \label{12}
\end{equation}
Then 
\begin{equation}
\begin{gathered} x_{k}'(t)=y_{k}^{p-1}+(t-t_{k}^{p-1})u_{k}^{p-1},\quad
\forall t\in I_{k}^{p}, \\ \\ x_{k}''(t)=u_{k}^{p-1},\quad \forall t\in
I_{k}^{p}, \end{gathered}  \label{13}
\end{equation}
hence, by \eqref{10}, for all $t\in \lbrack 0,T]$, we obtain 
\begin{equation}
\begin{aligned} \| x_{k}''(t)\| \leqslant & \|u_{k}^{p-1}\| <M+1 \\ \|
x_{k}'(t)\| \leqslant &\| y_{k}^{p-1}\| +(t-t_{k}^{p})\| u_{k}^{p}\| <\|
y_0\| +M+2 \\ \| x_{k}(t)\| \leqslant &\| x_{k}^{p-1}\| +(t-t_{k}^{p-1})\|
y_{k}^{p-1}\| +\frac{1}{2}(t-t_{k}^{p-1})^{2} \| u_{k}^{p-1}\| \\ \leqslant&
\| x_0\| +\|y_0\| +M+3. \end{aligned}  \label{14}
\end{equation}
Moreover, for all $t\in \lbrack 0,T]$ we have that 
\[
(x_{k}(t),x_{k}'(t),x_{k}''(t))\in
(x_{k}^{p},y_{k}^{p},u_{k}^{p})+\frac{\| y_0\| +M+2}{k}B\times \frac{M+1}{k}%
B\times \{0\};
\]
hence, by (iv), we have 
\begin{equation}
(x_{k}(t),x_{k}'(t),x_{k}''(t))\in \mathop{\rm graph}%
(F)+\varepsilon (k)(B\times B\times \{0\}),  \label{15}
\end{equation}
where $\varepsilon (k)\to 0$ when $k\to \infty .$ Then, by %
\eqref{12}, \eqref{13} and \eqref{14}, we obtain that $(x_{k}^{\prime%
\prime})_{k}$ is bounded in $L^{2}([0,T],\mathbb{R}^m)$, $(x_{k}')_{k}$
and $(x_{k})_{k}$ are bounded in $C([0,T],\mathbb{R}^m)$ and equi-Lipschitzian,
hence, by Theorem 0.3.4 in \cite{a3} there exist a subsequence (again
denoted by $(x_{k})_{k}$) and an absolutely continuous function 
$x:[0,T]\to \mathbb{R}^m$ such that

\begin{enumerate}
\item[(a)]  $(x_{k})_{k}$ converge uniformly to $x$

\item[(b)]  $(x_{k}^{\prime })_{k}$ converge uniformly to $x^{\prime }$

\item[(c)]  $(x_{k}^{\prime \prime })_{k}$ converge weakly in $L^{2}([0,T],%
\mathbb{R}^{m})$ to $x^{\prime \prime }$.
\end{enumerate}

By (H3) and Theorem 1.4.1 in \cite{a3} we get that 
\[
x''(t)\in \mathop{\rm co}F(x(t),x'(t))\subset \partial
V(x'(t)),a.e.\text{ on }[0,T],
\]
where co stands for the closed convex hull; hence, by Lemma 3.3 in \cite{b3}%
, we obtain that 
\[
\frac{d}{dt}V(x'(t))=\| x''(t)\| ^{2},a.e. \text{ on }[%
0,T];
\]
hence 
\begin{equation}
V(x'(T))-V(x'(0))=\int_0^{T}\| x''(t)\| ^{2}dt.
\label{16}
\end{equation}
On the other hand, since $x_{k}''(t)=u_{k}^{p-1}$, $\forall
t\in I_{k}^{p}$, by (iv), there exist $a_{k}^{p-1},b_{k}^{p-1}$, 
$z_{k}^{p-1}\in \frac{1}{k}B$, such that 
\begin{equation}
u_{k}^{p-1}-z_{k}^{p-1}\in
F(x_{k}^{p-1}-a_{k}^{p-1},y_{k}^{p-1}-b_{k}^{p-1})\subset \partial
V(y_{k}^{p-1}-b_{k}^{p-1}),\forall k\in \mathbb{N}^{\ast }  \label{17}
\end{equation}
and so the properties of the subdifferential of a convex function imply
that, for every $p<m(k)$, and for every $k\in \mathbb{N}^{\ast }$ we have 
\begin{align*}
V(x_{k}'&(t_{k}^{p})-b_{k}^{p})-V(x_{k}^{%
\prime}(t_{k}^{p-1})-b_{k}^{p-1})\geqslant \\
\geqslant &\langle
u_{k}^{p-1}-z_{k}^{p-1},x_{k}'(t_{k}^{p})-x_{k}^{%
\prime}(t_{k}^{p-1}) +b_{k}^{p-1}-b_{k}^{p}\rangle = \\
=&\langle
u_{k}^{p-1}-z_{k}^{p-1},\int_{t_{k}^{p-1}}^{t_{k}^{p}}x_{k}^{\prime%
\prime}(t)dt \rangle +\langle
u_{k}^{p-1}-z_{k}^{p-1},b_{k}^{p-1}-b_{k}^{p}\rangle = \\
=&\int_{t_{k}^{p-1}}^{t_{k}^{p}}\| x_{k}''(t)\| ^{2}dt-\langle
z_{k}^{p-1},\int_{t_{k}^{p-1}}^{t_{k}^{p}}x_{k}''(t)dt\rangle
+\langle u_{k}^{p-1}-z_{k}^{p-1},b_{k}^{p-1}-b_{k}^{p}\rangle ;
\end{align*}
hence 
\begin{multline}
V(x_{k}'(t_{k}^{p})-b_{k}^{p})-V(x_{k}^{%
\prime}(t_{k}^{p-1})-b_{k}^{p-1}) \\
\geqslant \int_{t_{k}^{p-1}}^{t_{k}^{p}}\| x_{k}''(t)\|
^{2}dt-\langle
z_{k}^{p-1},\int_{t_{k}^{p-1}}^{t_{k}^{p}}x_{k}''(t)dt\rangle
+\langle u_{k}^{p-1}-z_{k}^{p-1},b_{k}^{p-1}-b_{k}^{p}\rangle .  \label{18}
\end{multline}
Analogously if $T\in I_{k}^{m(k)}$, then by \eqref{11} we have 
\begin{equation}
\begin{aligned} V(x_{k}'&(T))-V(x_{k}'(t_{k}^{m(k)-1})-b_{k}^{m(k)-1})\\
\geqslant &\langle u_{k}^{m(k) -1}-z_{k}^{m(k)
-1},\int_{t_{k}^{m(k)-1}}^{T}x_{k}''(t)dt+b_{k}^{m(k)-1}\rangle \\
=&\int_{t_{k}^{m(k)-1}}^{T}\| x_{k}''(t)\| ^{2}dt -\langle z_{k}^{m(k)
-1},\int_{t_{k}^{m(k) -1}}^{T}x_{k}''(t)dt\rangle \\ &+\langle u_{k}^{m(k)
-1}-z_{k}^{m(k) -1},b_{k}^{m(k) -1}\rangle . \end{aligned}  \label{19}
\end{equation}
By adding the $m(k)-1$ inequalities from \eqref{18} and the inequality from %
\eqref{19}, we get 
\begin{equation}
V(x_{k}'(T))-V(y_0-b_{k}^{0})\geqslant \int_0^{T}\|
x_{k}''(t)\| ^{2}dt+\alpha (k),  \label{20}
\end{equation}
where 
\begin{align*}
\alpha (k)=&-\sum_{p=1}^{m(k)-1}\langle
z_{k}^{p-1},\int_{t_{k}^{p-1}}^{t_{k}^{p}}x_{k}''(t)dt\rangle
+\sum_{p=1}^{m(k)-1}\langle
u_{k}^{p-1}-z_{k}^{p-1},b_{k}^{p-1}-b_{k}^{p}\rangle \\
&-\langle z_{k}^{m(k) -1},\int_{t_{k}^{m(k)
-1}}^{T}x_{k}''(t)dt\rangle +\langle u_{k}^{m(k)
-1}-z_{k}^{m(k) -1},b_{k}^{m(k) -1}\rangle .
\end{align*}
Since 
\begin{align*}
| \alpha (k)| \leqslant &\sum_{p=1}^{m(k)-1}| \langle
z_{k}^{p-1},\int_{t_{k}^{p-1}}^{t_{k}^{p}}x_{k}''(t)dt\rangle |
+\sum_{p=1}^{m(k)-1}| \langle u_{k}^{p-1}-z_{k}^{p-1},b_{k}^{p-1}-b_{k}^{p}
\rangle| + \\
&+| \langle
z_{k}^{m(k)},\int_{t_{k}^{m(k)-1}}^{T}x_{k}''(t)dt\rangle | +|
\langle u_{k}^{m(k)-1}-z_{k}^{m(k) -1},b_{k}^{m(k) -1}\rangle | \\
\leqslant &\sum_{p=1}^{m(k)-1}\| z_{k}^{p-1}\| \|
\int_{t_{k}^{p-1}}^{t_{k}^{p}}x_{k}''(t)dt\|
+\sum_{p=1}^{m(k)-1}\| u_{k}^{p-1}-z_{k}^{p-1}\| \|b_{k}^{p-1}-b_{k}^{p}\| \\
&+\| z_{k}^{m(k)}\| \| \int_{t_{k}^{m(k)-1}}^{T}x_{k}''(t)dt\|
+\| u_{k}^{m(k) -1}-z_{k}^{m(k) -1}\| \| b_{k}^{m(k) -1}\| \\
\leqslant &\frac{(M+2) (3m(k)-1) }{k}
\end{align*}
it following that $\alpha (k)\to 0$ when $k\to \infty$;
hence, by \eqref{20}, we passing to the limit for $k\to \infty $, we
obtain 
\begin{equation}
V(x'(T))-V(y_0)\geqslant \limsup_{k\to \infty } \int_0^{T}\|
x_{k}''(t)\| ^{2}dt.  \label{21}
\end{equation}
Therefore, by \eqref{16} and \eqref{21}, 
\[
\int_0^{T}\| x''(t)\| ^{2}dt\geqslant \limsup_{k\to \infty }
\int_0^{T}\| x_{k}''(t)\| ^{2}dt
\]
and, since $(x'')_{k}$ converges weakly in $L^{2}([0,T],\mathbb{R}%
^m)$ to $x''$, by applying Proposition III.30 in \cite{b4}, we
obtain that $(x'')_{k}$ converge strongly in $L^{2}([0,T],\mathbb{R%
}^m)$ to $x''$, hence a subsequence again denoted by 
$(x'')_{k}$ converge poinwise a.e. to $x''$. Since
by \eqref{15} 
\[
\lim_{k\to \infty }d((x_{k}(t),x_{k}'(t),x_{k}''(t)),%
\mathop{\rm graph}(F))=0,
\]
and since by (H2) the graph of $F$ is closed (\cite{a3}, Proposition 1.1.2),
we have that 
\[
x''(t)\in F(x(t),x'(t))\quad a.e.\text{ on }[0,T].
\]

It remains to prove that $(x(t),x'(t))\in Q$, $\forall t\in [0,T]$.
Indeed, by \eqref{12}, \eqref{13}, and \eqref{14}, we have that 
\[
\| x_{k}(t)-x_{k}^{p}\| <\frac{\| y_0\| +M+2}{k}, \quad \|
x_{k}'(t)-y_{k}^{p}\| <\frac{M+1}{k},
\]
hence 
\[
\lim_{k\to \infty
}d((x_{k}(t),x_{k}'(t)),(x_{k}^{p},y_{k}^{p}))=0.
\]
Since, $(x_{k}^{p},y_{k}^{p})\in Q_0,\forall k\in \mathbb{N}^{\ast }$, by (a)
and (b) we have that 
\[
\lim_{k\to \infty }
d((x(t),x'(t)),(x_{k}(t),x_{k}'(t)))=0.
\]
On the other hand 
\begin{multline}
d((x(t),x'(t)),Q_0) \\
\leqslant d((x(t),x'(t)),(x_{k}(t),x_{k}'(t)))
+d((x_{k}(t),x_{k}'(t)),(x_{k}^{p},y_{k}^{p}))
+d((x_{k}^{p},y_{k}^{p}),Q_0);  \label{22}
\end{multline}
hence, by passing to the limit we obtain that 
\[
d((x(t),x'(t)),Q_0)=0,\quad \forall t\in [0,T].
\]
Since $Q_0$ is closed, we obtain that $(x(t),x'(t))\in Q_0$, for all 
$t\in \lbrack 0,T]$, which completes the proof. \hfill $\square$

\paragraph{Acknowledgements.}

This research was done while the author was visiting the Research unit
``Mathematics and Applications'' (Control Theory Group) of the Department of
Mathematics of Aveiro University, where the author was supported by a
post-doctoral fellowship. The author wishes to thank Vasile Staicu for
having introduced him to the subject of the paper; The author also wants to
thanks the referee for his/her remarks that improved the results and their
presentation.

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\noindent\textsc{Vasile Lupulescu}\newline
Universitatea ``Constantin Br\^{a}ncusi'' of T\^{a}rgu-Jiu\newline
Bulevardul Republicii, Nr.1\newline
1400 T\^{a}rgu-Jiu, Romania\newline
e-mail: vasile@utgjiu.ro

\end{document}