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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 110, 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-2005/110\hfil Second-order functional differential 
inclusions]
{Viable solutions for second order nonconvex functional differential
inclusions}
\author[V. Lupulescu\hfil EJDE-2005/110\hfilneg]{Vasile Lupulescu}
\address{Vasile Lupulescu \hfill \break 
``Constantin br\^{a}ncu\c{s}i'' 
University of T\^{a} rgu Jiu, 
bulevardul republicii, nr.1, 
210152 T\^{a}rgu Jiu, Romania}
\email{vasile@utgjiu.ro}

\date{}
\thanks{Submitted July 14, 2005. Published October 10, 2005.}
\subjclass[2000]{34A60}
\keywords{Functional differential inclusions; viability result}

\begin{abstract}
We prove the existence of viable solutions for an autonomous second-order
functional differential inclusions in the case when the multifunction that
define the inclusion is upper semicontinuous compact valued and contained in
the subdifferential of a proper lower semicontinuous convex function.
\end{abstract}

\maketitle

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

\section{Introduction}

Functional differential inclusions, well known as differential inclusions
with memory, express the fact that the velocity of the system depends not
only on the state of the system at given instant but depends upon the
history of the trajectory until this instant. The class of functional
differential inclusions contains a large variety of differential inclusions
and control systems. In particular, this class covers the differential
inclusions, the differential - difference inclusions and the Voltera
inclusions. For a detailed discussion on this topic we refer to \cite{ac}.

Let $\mathbb{R}^{m}$ be the $m$-dimensional Euclidean space with norm 
$\|\cdot\|$ and scalar product $\langle \cdot,\cdot\rangle $. Let $\sigma $
be a positive number and $\mathcal{C}_{\sigma }:=\mathcal{C}([-\sigma ,0], 
\mathbb{R}^{m})$ the Banach space of continuous functions from $[-\sigma ,0]$
to $\mathbb{R}^{m}$ with the norm $\|x(.)\|_{\sigma }:=\sup\{\|x(t)\|;t\in [
-\sigma ,0]\}$. For each $t\in [ 0,T]$, we define the operator $T(t)$ from 
$\mathcal{C}([-\sigma ,T],\mathbb{R}^{m})$ to $\mathcal{C}_{\sigma }$ as
follows: $(T(t)x)(s):=x(t+s)$, $s\in [-\sigma ,0]$. For a given nonempty
subset $K$ of $\mathbb{R}^{m}$ we introduce the set 
$\mathcal{K}_{0}=:\{\varphi \in \mathcal{C}_{\sigma };\varphi (0)\in K\}$.

The aim of this paper is to prove a viability result for the second order
functional differential inclusion 
\begin{equation}
x^{\prime\prime}\in F(T(t)x,x^{\prime}),\quad (T(0)x,x^{\prime}(0))=(\varphi
_{0},y_{0})\in \mathcal{K}_{0}\times \Omega  \label{e1.1}
\end{equation}
where $\Omega $ is an open set in $\mathbb{R}^{m}$ and $F:\mathcal{C}
_{\sigma }\times \Omega \to 2^{\mathbb{R}^{m}}$ is an upper semicontinuous,
compact valued multifunction such that $F(\psi ,y)\subset \partial V(y)$ for
every $(\psi ,y)\in \mathcal{K}_{0}\times \Omega $ and $V$ is a proper
convex and lower semicontinuous function.

Bressan, Cellina and Colombo \cite{bcc} prove the existence of local
solutions to the Cauchy problem $x^{\prime }\in F(x)$, $x(0)=x_{0}$, where 
$F $ is upper semicontinuous, cyclically monotone and compact valued
multifunction. While Rossi \cite{ro} prove a viability result for this
problem. The first viability result for the first order functional
differential inclusion was given by Haddad \cite{h1}, \cite{h2} in the case
when $F$ is upper semicontinuous with convex compact values. Lupulescu 
\cite{l2} has been proved the local existence of solutions for nonconvex
differential inclusion $x^{\prime }\in F(T(t)x)$, 
$T(0)x=\varphi _{0}\in \mathcal{C}_{\sigma }$, and the existence 
of viable solutions for this
problem has been studied by Cernea and Lupulescu \cite{cl} in the case when 
$F$ is upper semicontinuous compact values such that
 $F(\psi )\subset\partial V(\psi (0))$ for every 
 $\psi \in \mathcal{K}_{0}$.

The first viability result for second order differential inclusions 
\begin{equation}
x^{\prime\prime}\in F(x,x^{\prime}),\quad x(0)=x_{0},x^{\prime}(0)=y_{0}
\label{e1.2}
\end{equation}
were given by Cornet and Haddad \cite{ch} in the case in which $F$ is upper
semicontinuous and with convex compact values. The nonconvex case has been
studied by Lupulescu \cite{l1} and Cernea \cite{c1} in the finite
dimensional case. The nonconvex case in Hilbert spaces has been studied by
Ibrahim and Alkulaibi \cite{ia}. For other results, references and
applications in this framework we refer to the papers: Casting \cite{c1},
Auslender and Mechler \cite{am}, Aghezaaf and Sajid \cite{as}, Morchadi and
Sajid \cite{ms}, Marco and Murilio \cite{mm}, Syam \cite{s1} and book of
Motreanu and Pavel \cite{mp}. In the paper \cite{hm}, Duc Ha and Monteiro
Marques proved the several existence theorems for the nonconvex functional
differential inclusions governed by the sweeping process.

\section{Preliminaries and statement of the main result}

For $x\in \mathbb{R}^{m}$ and $r>0$ let 
$B(x,r):=\{y\in \mathbb{R}^{m};\|y-x\|<r\}$ be the open ball centered 
at $x$ with radius $r$, $\Omega $
and let $\overline{B}(x,r)$ be its closure. For 
$\varphi \in \mathcal{C}_{\sigma }$ let 
$B_{\sigma }(\varphi ,r):=\{\psi \in \mathcal{C}_{\sigma
};\|\psi -\varphi \|_{\sigma }<r\}$ and $\overline{B}_{\sigma }(\varphi
,r):=\{\psi \in \mathcal{C}_{\sigma };\|\psi -\varphi \|_{\sigma }\leq r\}$.
For $x\in \mathbb{R}^{m}$ and for a closed subset $A\subset \mathbb{R}^{m}$
we denote by $d(x,A)$ the distance from $x$ to $A$ given by 
$d(x,A):=\inf\{\|y-x\|;y\in A\}$.

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

We say that a multifunction 
$F:\mathcal{K}_{0}\times \Omega \to 2^{\mathbb{R}^{m}}$ 
is upper semicontinuous if for every
$(\varphi ,y)\in \mathcal{K}_{0}\times \Omega $ and for every $\varepsilon >0$
 there exists $\delta >0$
such that 
\begin{equation*}
F(\psi ,z)\subset F(\varphi ,y)+B(0,\varepsilon ),\text{ }(\forall )(\psi
,z)\in B_{\sigma }(\varphi ,\delta )\times B(y,\delta )\text{.}
\end{equation*}

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

For a multifunction $F:\mathcal{K}_{0}\times \Omega \to 2^{\mathbb{R} ^{m}}$
and for any $(\varphi ,y)\in \mathcal{K}_{0}\times \Omega $ we consider the
functional differential inclusion 
\begin{equation}
x^{\prime\prime}\in F(T(t)x,x^{\prime}),\quad T(0)x=\varphi
_{0},x^{\prime}(0)=y_{0}  \label{e2.1}
\end{equation}
under the following assumptions:

\begin{itemize}
\item[(H1)]  $K$ is a locally closed subset in $\mathbb{R}^{m}$, $\Omega $
is an open subset $\mathbb{R}^{m}$ and $F:\mathcal{K}_{0}\times \Omega
\rightarrow 2^{\mathbb{R}^{m}}$ is upper semicontinuous with compact values;

\item[(H2)]  There exists a proper convex and lower semicontinuous function 
$V:\mathbb{R}^{m}\rightarrow \mathbb{R}$ such that 
\begin{equation*}
F(\varphi ,y)\subset \partial V(y)\text{ for every }(\varphi ,y)\in \mathcal{K}_{0}
\times \Omega ;
\end{equation*}

\item[(H3)]  For every $(\varphi ,y)\in \mathcal{K}_{0}\times \Omega $ and
for every $z\in F(\varphi ,y)$ holds the following tangential condition: 
\begin{equation*}
\liminf_{h\downarrow 0}\frac{1}{h^{2}}d(\varphi (0)+hy+\frac{h^{2}}{2}z,K)=0.
\end{equation*}
\end{itemize}

\begin{remark} \label{rmk2.1}\rm
 A convex function $V:\mathbb{R}^{m}\to \mathbb{R}$ is continuous
 in the whole space $\mathbb{R}^{m}$ \cite[Corollary 10.1.1]{r1}
 and almost everywhere differentiable \cite[Theorem 25.5]{r1}.
Therefore, (H2) restricts strongly the multivaluedness of $F$.
\end{remark}

\begin{definition} \label{def2.1} \rm
 By a viable solution of the functional differential
inclusion \eqref{e2.1} we mean any continuous function $x:[-\sigma
,T]\to \mathbb{R}^{m}$, $T>0$, that is absolutely continuous on $
[0,T]$ with absolutely continuous derivative on $[0,T]$ such that $
T(0)x=\varphi _{0}$ on $[-\sigma ,T]$, $x'(0)=y_{0}$ and
\begin{gather*}
x''(t)\in F(T(t)x,x'(t)), \quad \text{a.e. on  }[0,T], \\
(x(t),x'(t))\in K\times \Omega , \quad\text{for every }t\in [ 0,T].
\end{gather*}
\end{definition}

Our main result is the following.

\begin{theorem} \label{thm3.1}
If Assumptions (H1)-(H3) are satisfied, then $K$ is a viable domain for
\eqref{e2.1}.
\end{theorem}

\section{Proof of the main result}

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

\begin{lemma} \label{lem3.1}
Assume that the hypotheses (H1) and (H3) are satisfied.
Then, for each $(\varphi ,y_{0})\in \mathcal{K}_{0}\times \Omega $
 there exist $r>0$ and $T>0$ such that
 $K\cap B(\varphi (0),r)$ is closed and for each
 $k\in N^{\ast }$ there exist
$m( k) \in N^{\ast },t_{k}^{p},x_{k}^{p},y_{k}^{p},z_{k}^{p}$
 and a continuous function $x_{k}:[ -\sigma ,T] \to R^{m}$ such
that for every $p\in \{0,1,\dots ,m(k)-1\}$ we have:
\begin{itemize}
\item[(i)] $h_{k}^{p}:=t_{k}^{p+1}-t_{k}^{p}<\frac{1}{k}$
and $t_{k}^{m(k)-1}\leq T<t_{k}^{m(k)}$,

\item[(ii)] $x_{k}( t) =\varphi ( t) $, for every $t\in [ -\sigma ,0]$,

\item[(iii)]  $x_{k}( t) =x_{k}^{p}+(t-t_{k}^{p})y_{k}^{p}+\frac{1}{2}
(t-t_{k}^{p})^{2}z_{k}^{p}$\textit{\ , \ for every }$t\in [
t_{k}^{p},t_{k}^{p+1}]$,

\item[(iv)]  $z_{k}^{p}\in F(T(t_{k}^{p})x_{k},y_{k}^{p})+\frac{1}{k}B$,

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

\item[(vi)]  $T( t_{k}^{p}) x_{k}\in \mathcal{K}_{0}\cap B_{\sigma}(\varphi ,r)$,
\end{itemize}
where $B:=B(0,1)$ and $Q_{0}:=(K\cap B(\varphi (0),r))\times
 \overline{B}(y_{0},r)$.
\end{lemma}

\begin{remark} \label{rmk3.2} \rm
 The following twp statements hold:
\begin{itemize}
\item[(i)] If $\alpha \in \mathcal{K}_{0}\cap B_{\sigma }(\varphi ,r)$
then $\alpha (0)\in K\cap B(\varphi (0),r)$,

\item[(ii)] If $K\cap B(\varphi (0),r)$ is closed in ${R}^{m}$ then
$\mathcal{K}_{0}\cap B_{\sigma }(\varphi ,r)$ is closed in
$C_{\sigma }$.
\end{itemize}
Indeed, the first statement is obvious. For the second statement, we assume
that $K\cap B(\varphi (0),r)$ is closed in ${R}^{m}$ and we consider a sequence
$(\alpha _{n})_{n}$ in $\mathcal{K}_{0}\cap B_{\sigma }(\varphi ,r)$
that is convergent(in the norm $\|\cdot\|_{\sigma }$) to
 $\alpha \in \mathcal{C}_{\sigma}$. Then follows that
$\alpha \in B_{\sigma }(\varphi ,r)$, $\alpha_{n}(0)\to \alpha (0)$
 and $\alpha _{n}(0)\in K\cap B(\varphi (0),r)$; therefore,
since $K\cap B(\varphi (0),r)$ is closed, we obtain that
$\alpha (0)\in K$ and thus
$\alpha \in \mathcal{K}_{0}\cap B_{\sigma}(\varphi ,r)$.
\end{remark}

\begin{proof}[Proof of Lemma \ref{thm3.1}]
Let $(\varphi ,y_{0})$ be arbitrary and fixed in
$\mathcal{K}_{0}\times \Omega $. Since $K$ is locally closed in
$\mathbb{R}^{m}$, there exists $r>0$ such that $K\cap B(\varphi (0),r)$ is
closed. Moreover, since $\Omega $ is open set in $\mathbb{R}^{m}$, we can
choose $r$ such that $\overline{B}(y_{0},r)\subset \Omega $.
By \cite[Proposition 1.1.3]{ac}, $F$ is locally bounded;
therefore, we cam assume that
there exists $M>0$ such that
\begin{equation}
\sup \{\|v\|;v\in F( \psi ) ,\psi \in \mathcal{K}_{0}\cap
B_{\sigma }( \varphi ,r) \}\leq M.  \label{e3.1}
\end{equation}
Since $\varphi $ is continuous on $[ -\sigma ,0] $ we can choose
$\eta >0$ small enough such that
\begin{equation}
\|\varphi ( t) -\varphi ( s) \|<\frac{r}{4},\text{ for
all }t,s\in [ -\sigma ,0] \text{ with }|t-s|<\eta .  \label{e3.2}
\end{equation}
Let
\begin{equation}
T:=\min \big\{ \eta ,\frac{r}{4(M+1)},\frac{r}{8(\|y_{0}\|+1)},\frac{1}{2}
\sqrt{\frac{r}{M+1}}\big\} .  \label{e3.3}
\end{equation}
Further on, for a fixed $k\in $\ $\mathbb{N}^{\ast }$, we put
$x_{k}(t)=\varphi ( t) $ for every $t\in [ -\sigma ,0] $
and for $p=0$ we take
$t_{k}^{0}=0,x_{k}^{0}=\varphi ( 0),y_{k}^{0}=y_{0}$
and we choose an arbitrary element
$z_{k}^{0}\in F(\varphi,y_{0})+\frac{1}{k}B$. Also,
we can define $t_{k}^{1},x_{k}^{1},y_{k}^{1},z_{k}^{1}$
and $x_{k}$ on $[ 0,t_{k}^{1}] $ in the same way that
in the next general case.

Suppose that, for a fixed $q\in \mathbb{N}^{\ast }$,
we have constructed $t_{k}^{p},x_{k}^{p},y_{k}^{p},z_{k}^{p}$
and $x_{k}$ on $[ 0,t_{k}^{p}] $ such that the conditions
 $(i)-(vi)$ are satisfied for each $p\in\{1,2,\dots ,q-1\}$.

To define the next step $h_{k}^{q}$, we denote by $H_{k}^{q}$ the
set of all $h\in (0,\frac{1}{k})$ for which the following conditions are
satisfied
\begin{itemize}
\item[(a)] $h\in (0,T-t_{k}^{q})$;

\item[(b)] there exists $u_{k}^{q}\in F(T(t_{k}^{q})x_{k},y_{k}^{q})$
such that $d(u_{k}^{q}+hy_{k}^{q}+\frac{h^{2}}{2}u_{k}^{q},K)
\leq \frac{h^{2}}{4k}$.
\end{itemize}

For a fixed $u\in F(T(t_{k}^{q})x_{k},y_{k}^{q})$, since
$(T(t_{k}^{q})x_{k})(0)=x( t_{k}^{q}) =x_{k}^{q}\in K$,
using tangential condition (H3) applied in
$(T( t_{k}^{q})x_{k},y_{k}^{q})\in \mathcal{K}_{0}\times \Omega $
we obtain that $H_{k}^{q}$ is nonempty and that. Since
 $H_{k}^{q}\cap [ \frac{d_{k}^{q}}{2},d_{k}^{q}]$ is also nonempty, let
us we chose
$h_{k}^{q}\in H_{k}^{q}\cap [ \frac{d_{k}^{q}}{2},d_{k}^{q}]$.
 We define $t_{k}^{q+1}:=t_{k}^{q}+h_{k}^{q}$ and so we have
$t_{k}^{q}<t_{k}^{q+1}$ and $\sum_{i=0}^{q}t_{k}^{i}<T$.
Since $h_{k}^{q}\in H_{k}^{q}$, it follows that the first condition
in $(i)$ is satisfies for $p=q$. Moreover, there exists
$u_{k}^{q}\in F(T(t_{k}^{q})x_{k},y_{k}^{q})$
such that
\begin{equation*}
d(x_{k}^{q}+h_{k}^{q}y_{k}^{q}+\frac{(h_{k}^{q})^{2}}{2}u_{k}^{q},K)\leq
\frac{(h_{k}^{q})^{2}}{4k}
\end{equation*}
and so there exists $v_{k}^{q}\in \mathbb{R}^{m}$ with
$\|v_{k}^{q}\|\leq \frac{1}{k}$ such that
\begin{equation}
x_{k}^{q+1}:=x_{k}^{q}+h_{k}^{q}y_{k}^{q}+\frac{(h_{k}^{q})^{2}}{2}
z_{k}^{q}\in K  \label{e3.4}
\end{equation}
where
\begin{equation*}
z_{k}^{q}:=u_{k}^{q}+v_{k}^{q}\in F(T(t_{k}^{q})x_{k},y_{k}^{q})
+\frac{1}{k} B,
\end{equation*}
Also, we remark that by \eqref{e3.1} we have
\begin{equation}
\|z_{k}^{q}\|\leq M+1  \label{e3.5}
\end{equation}
Hence, if we put
\begin{equation}
x_{k}(t)=x_{k}^{q}+(t-t_{k}^{q})y_{k}^{q}+\frac{1}{2}
(t-t_{k}^{q})^{2}z_{k}^{q}, \quad \text{for every }
t\in [t_{k}^{q},t_{k}^{q+1}],  \label{e3.6}
\end{equation}
it follows that the conditions $(iii)$ and $(iv)$ are also
satisfied for $p=q $.
Let
\begin{equation}
y_{k}^{q+1}:=y_{k}^{q}+h_{k}^{q}z_{k}^{p}.  \label{e3.7}
\end{equation}
By induction on $p$ (which is left to the reader) one verifies that
$x_{k}^{q+1}$ and $y_{k}^{q+1}$ defined above can be expressed as follows
\begin{equation}
x_{k}^{q+1}=\varphi (0)+(\sum_{j=0}^q h_{k}^{j})y_{0}+
\frac{1}{2}\sum_{j=0}^q (h_{k}^{j})^{2}z_{k}^{i}+
\sum_{i=0}^{q-1} \sum_{j=i+1}^q h_{k}^{i}h_{k}^{j}z_{k}^{i}  \label{e3.8}
\end{equation}
and
\begin{equation}
y_{k}^{q+1}=y_{0}+\sum_{j=0}^{q-1} h_{k}^{j}z_{k}^{j}.
\label{e3.9}
\end{equation}
Moreover, by \eqref{e3.8} and \eqref{e3.9}, from \eqref{e3.6} we obtain
\begin{equation}
x_{k}( t) =\varphi (0)+(t-t_{k}^{p})(y_{0}+\sum_{j=0}^{q-1}
h_{k}^{j}z_{k}^{j})+\frac{1}{2}(t-t_{k}^{p})^{2}z_{k}^{p}+w_{q},
\quad t\in [ 0,t_{k}^{q+1}]  \label{e3.10}
\end{equation}
where
\begin{equation*}
w_{q}=(\sum_{j=0}^{q-1} h_{k}^{j})y_{0}+\frac{1}{2}
\sum_{j=0}^{q-1} (h_{k}^{j})^{2}z_{k}^{i}
+\sum_{i=0}^{q-2} \sum_{j=i+1}^{q-1}h_{k}^{i}h_{k}^{j}z_{k}^{i}.
\end{equation*}
It easy to see that, using
$\sum_{j=0}^{q-1} h_{k}^{j}<T$, we obtain
\begin{equation}
\|w_{q}\|\leq T\|y_{0}\|+\frac{M+1}{2}T^{2}<\frac{r}{8}+\frac{r}{8}
=\frac{r}{4}.  \label{e3.11}
\end{equation}
Now, we check $(v)$ and $(vi)$ for $p=q$. To check condition  $(v)$,
we observe that by \eqref{e3.3}, \eqref{e3.5} and \eqref{e3.8} we have

\begin{equation} \label{e3.12}
\begin{aligned}
&\|x_{k}^{q+1}-\varphi (0)\| \\
&\leq \big(\sum_{j=0}^{q-1} h_{k}^{j}\big)\|y_{0}\|
+\frac{1}{2} \sum_{j=0}^{q-1} \big(h_{k}^{j}\big)^{2}(M+1)
+\sum_{i=0}^{q-2} \sum_{j=i+1}^{q-1} h_{k}^{i}h_{k}^{j}(M+1) \\
&\leq T\|y_{0}\|+\frac{M+1}{2}T^{2}<\frac{r}{8}+
\frac{r}{8}=\frac{r}{4}
\end{aligned}
\end{equation}
and by \eqref{e3.5} and \eqref{e3.9} we have
\begin{equation}
\|y_{k}^{q+1}-y_{0}\|\leq T(M+1)<\frac{r}{4}.  \label{e3.13}
\end{equation}
Therefore, by \eqref{e3.4} \eqref{e3.12} and \eqref{e3.13} we have
\begin{equation}
(x_{k}^{q+1},y_{k}^{q+1})\in (K\cap B(\varphi (0),r/4))\times
B(y_{0},r/4)\subset Q_{0}  \label{e3.14}
\end{equation}
and so the condition $(v)$ is checked for $p=q$.
Also, we observe that by \eqref{e3.3}, \eqref{e3.5}, \eqref{e3.10} and
\eqref{e3.11} we have
\begin{equation} \label{e3.15}
\begin{aligned}
\|x_{k}(t)-\varphi (0)\|
&\leq T[\|y_{0}\|+T(M+1)]+\frac{M+1}{2} T^{2}+\|w_{q}\|   \\
&\leq \frac{r}{8}+\frac{r}{8}+\frac{2r}{8}=\frac{r}{4},
\end{aligned}
\end{equation}
which means that $x_{k}(t)\in B(\varphi (0),r/4)$ for every
$t\in [0,t_{k}^{q+1}]$.

Furthermore, if $-\sigma \leq s\leq -t_{k}^{q}$, then, by the fact that 
$0<t_{k}^{q+1}<T$, $t_{k}^{q+1}+s\in [ -\sigma ,0] $ and by
\eqref{e3.2}, we have
\begin{equation*}
\|(T(t_{k}^{q+1})x_{k})(s)-\varphi (s)\|
=\|x_{k}(t_{k}^{q+1}+s)-\varphi(s)\|
=\|\varphi (t_{k}^{q+1}+s)-\varphi (s)\|<r/4.
\end{equation*}
If $-t_{k}^{q+1}\leq s\leq 0$, then $t_{k}^{q+1}+s\in [
0,t_{k}^{q+1}]. $ Hence, there exists $t_{k}^{p}<t_{k}^{q+1}$ such that $
t_{k}^{q+1}+s\in [ t_{k}^{p},t_{k}^{p+1}]$ and so, because $
|s|<t_{k}^{q+1}<T$, by \eqref{e3.2}, \eqref{e3.3} and \eqref{e3.15} we have
\begin{align*}
\|(T(t_{k}^{q+1})x_{k})(s)-\varphi (s)\|
&=\|x_{k}(t_{k}^{q+1}+s)-\varphi (s)\|\\
&\leq \|x_{k}(t_{k}^{q+1}+s)-\varphi (0)\|
+\|\varphi (s)-\varphi (0)\|\\
&<\frac{r}{4}+\frac{r}{4}=\frac{r}{2}.
\end{align*}
Hence we have that $T(t_{k}^{q+1})x_{k}\in B_{\sigma }(\varphi ,r)$ and
since $(T(t_{k}^{q+1})x_{k})(0)=x_{k}(t_{k}^{q+1})=x_{k}^{q+1}\in K$ we
deduce that
$T(t_{k}^{q+1})x_{k}\in \mathcal{K}_{0}\cap B_{\sigma }(\varphi,r)$
 and so the condition $(vi)$ is checked for $p=q$.

Thus the conditions $(i)-(vi)$ are satisfied for every
$p\in \{1,2\dots ,q\}$
and so the inductive procedure can be continued further.

In the following, we show that this iterative process if finite, i.e., there
exists $m(k)\in \mathbb{N}^{\ast }$ such that the second condition in $(i)$
is satisfied.

For this, we assume by contradiction that $t_{k}^{p}<T$ for every
$p\in \mathbb{N}$. Then the bounded increasing sequence
$(t_{k}^{p})_{p}$ converges to $t_{k}^{\ast }\leq T$.

Now, we show that $(x_{k}^{p})_{p}$ and $(y_{k}^{p})_{p}$ is a Cauchy
sequence. Indeed, for $q<p$, by \eqref{e3.5}, \eqref{e3.8} and \eqref{e3.9},
we have
\begin{gather*}
\|x_{k}^{p}-x_{k}^{q}\|\leq \|y_{0}\|(t_{k}^{p}-t_{k}^{q})+\frac{1}{2}
(M+1)(t_{k}^{p}-t_{k}^{q})^{2},\\
\|y_{k}^{p}-y_{k}^{q}\|\leq (M+1)(t_{k}^{p}-t_{k}^{q}).
\end{gather*}
Since $( t_{k}^{p}) _{p}$ is a Cauchy sequence, the sequences
$(x_{k}^{p})_{p}$ and $(y_{k}^{p})_{p}$ are also Cauchy. Therefore, there
exists $x_{k}^{\ast }=\lim_{p\to \infty }x_{k}^{p}$ and
$y_{k}^{\ast }=\lim_{p\to \infty }y_{k}^{p}$. Since
$K\cap B(\varphi (0),r)$ is closed and
$(x_{k}^{p},y_{k}^{p})\in (K\cap B(\varphi (0),r))\times B(y_{0},r)$
for every $p\in \mathbb{N}$, we have that
$(x_{k}^{\ast }$,
$y_{k}^{\ast })\in (K\cap B(\varphi (0),r))\times \overline{B}(y_{0},r)$.

Now, if we put $x_{k}( t_{k}^{\ast }) :=x_{k}^{\ast }$ then, for
any sequence $(s_{k}^{p})_{p}$ with
$t_{k}^{p}\leq s_{k}^{p}\leq t_{k}^{p+1}$
for every $p\in \mathbb{N}$, the  inequality
\begin{align*}
\|x_{k}(s_{k}^{p})-x_{k}^{\ast }\|
&\leq \|x_{k}(s_{k}^{p})-x_{k}^{p}\|+\|x_{k}^{p}-x_{k}^{\ast }\| \\
&\leq (s_{k}^{p}-t_{k}^{p})\|y_{0}\|+\frac{1}{2}
(M+1)(s_{k}^{p}-t_{k}^{p})^{2}+\|x_{k}^{p}-x_{k}^{\ast }\|
\end{align*}
implies $\|x_{k}(s_{k}^{p})-x_{k}^{\ast }\|\to 0$ as
$p\to \infty $.

Therefore, there exists $\lim_{t\to t_{k}^{\ast }}
x_{k}( t) =x_{k}^{\ast }=x_{k}( t_{k}^{\ast }) $.
Accordingly, $x_{k}$ is continuous on $[ -\sigma ,t_{k}^{\ast }] $
and hence $T(t_{k}^{p})x_{k}\to T(t_{k}^{\ast })x_{k}$ as
$p\to \infty $. Thus, since
$T(t_{k}^{p})x_{k}\in \mathcal{K}_{0}\cap B_{\sigma }(\varphi ,r)$
for every $p\in \mathbb{N}$ and
$\mathcal{K}_{0}\cap B_{\sigma }(\varphi ,r)$ is closed, we have
that $T(t_{k}^{\ast})x_{k}\in \mathcal{K}_{0}\cap B_{\sigma }(\varphi ,r)$.

Furthermore, by \eqref{e3.5} we deduce that there exists a subsequence
(again denote by) $(z_{k}^{p})_{p}$ such that
$z_{k}^{p}\to z_{k}^{\ast }$ as $p\to \infty $.

Also, since $u_{k}^{p}\in F(T(t_{k}^{p})x_{k},y_{k}^{p})$, by \eqref{e3.1}
we have $\|u_{k}^{p}\|\leq M$ for every $p\in \mathbb{N}$ and so there
exists a subsequence (again denote by) $(u_{k}^{p})_{p}$ such that
$u_{k}^{p}\to u_{k}^{\ast }$ as $p\to \infty $. Therefore,
since $(T(t_{k}^{p})x_{k},y_{k}^{p},u_{k}^{p})\in \mathop{\rm graph}
(F),y_{k}^{p}\to y_{k}^{\ast }$,
$u_{k}^{p}\to u_{k}^{\ast}$,
$T(t_{k}^{p})x_{k}\to T(t_{k}^{\ast })x_{k}$ as
$p\to\infty $ and $\mathop{\rm graph}(F)$ is closed we have
 that $u_{k}^{\ast}\in F(T(t_{k}^{\ast })x_{k},y_{k}^{\ast })$.

Since $(T(t_{k}^{p})x_{k})(0)=x_{k}(t_{k}^{\ast })=x_{k}^{\ast}\in K$,
 we can apply tangential condition (H3) in
($T(t_{k}^{\ast})x_{k},y_{k}^{\ast })$. Therefore, we can chose
$h\in (0,T-t_{k}^{\ast })$ such that
\begin{equation}
d(x_{k}^{\ast }+hy_{k}^{\ast }+\frac{h^{2}}{2}u_{k}^{\ast },K)\leq \frac{
h^{2}}{4k}.  \label{e3.16}
\end{equation}

We would like to prove that $h$ as defined above belongs to $H_{k}^{p}$ for
every $p$ sufficiently large.

Since $h\in (0,T-t_{k}^{\ast })$ implies that $h\in (0,T-t_{k}^{p})$
then the condition (a) of the definition $H_{k}^{p}$ is checked for
 every $p\in \mathbb{N}$. Since $(t_{k}^{p})_{p}$ is increasing to
$t_{k}^{\ast }$, there exists $p_{1}\in \mathbb{N}$ such that for
every $p\geq p_{1}$ we have $t_{k}^{\ast }-t_{k}^{p}<h$, and so
$t_{k}^{p}<t_{k}^{\ast}<t_{k}^{p}+h<t_{k}^{\ast }+h$ for every
$p\geq p_{1}$. Also, there exists $p_{2}\geq p_{1}$ such that
$p\geq p_{2}$ implies that
\begin{equation*}
\|x_{k}^{p}-x_{k}^{\ast }\|<\frac{h^{2}}{12k},\quad \|y_{k}^{p}-y_{k}^{
\ast }\|<\frac{h}{12k},\quad \|u_{k}^{p}-u_{k}^{\ast }\|<\frac{1}{6k}.
\end{equation*}
Therefore, for every $p\geq p_{2}$ we have
\begin{equation}
\begin{aligned}
\Delta _{k}&:=\|(x_{k}^{p}+hy_{k}^{p}+\frac{h^{2}}{2}u_{k}^{p})-(x_{k}^{\ast
}+hy_{k}^{\ast }+\frac{h^{2}}{2}u_{k}^{\ast })\| \\
&\leq \|y_{k}^{p}-x_{k}^{\ast }\|+h\|y_{k}^{p}-y_{k}^{\ast }\|
+\frac{h^{2}}{2}\|u_{k}^{p}-u_{k}^{\ast }\|\leq \frac{h^{2}}{4k}.
\end{aligned}\label{e3.17}
\end{equation}
Using the inequality
\begin{equation*}
d(x_{k}^{p}+hy_{k}^{p}+\frac{h^{2}}{2}u_{k}^{p},K)\leq d(x_{k}^{\ast
}+hy_{k}^{\ast }+\frac{h^{2}}{2}u_{k}^{\ast },K)+\Delta _{k},
\end{equation*}
by \eqref{e3.16} and \eqref{e3.17}, we obtain that $h$ and $u_{k}^{p}$
satisfy the second condition of the definition of $H_{k}^{p}$, for every
$p\geq p_{2}$.

Therefore, for $p\geq p_{2}$ we have that $h\in H_{k}^{p}$ and hence
$d_{k}^{p}:=\sup H_{k}^{p}\geq h$ for every $p\geq p_{2}$.
But $h_{k}^{p}\in[ \frac{d_{k}^{p}}{2},d_{k}^{p}]$, and so
$h_{k}^{p}\geq \frac{h}{2}>0$
for every $p\geq p_{1}$, which is in contradiction with
$h_{k}^{p}=t_{k}^{p+1}-t_{k}^{p}\to 0$ as $p\to \infty $.
This contradiction can be eliminated only of the iterative process is
finite, i.e.,if there exists $m( k) \in \mathbb{N}^{\ast }$ such
that $t_{k}^{m( k) -1}\leq T<t_{k}^{m( k) }$ and the
conditions $(i)-(vi)$ are satisfied for every
$p\in \{0,1,\dots ,m(k) -1\}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
Assume that hypotheses (H1)-(H3) are satisfied. Also, since
the multifunction $y\to \partial V(y)$ is locally bounded we can
choose $r>0$ and $M>0$ such that $V$ is Lipschitz
continuous with the constant $M$ in $\overline{B}(\varphi (0,r))$.

We prove that the sequence $x_{k}(\cdot )$, constructed by
Lemma \ref{lem3.1}, has a subsequence that converges to a solution 
of \eqref{e2.1}.
First, for every $k\geq 1$ we define the function
 $\theta_{k}:[0,T]\to [ 0,T]$ by
$\theta _{k}( t)=t_{k}^{p} $ for every
$t\in [ t_{k}^{p},t_{k}^{p+1}]$.

Since $|\theta _{k}( t) -t|\leq \frac{1}{k}$ for every
$t\in[ 0,T]$, then $\theta _{k}( t) \to t$ uniformly on
$[0,T]$.
By the fact that $x_{k}^{p}=x_{k}(\theta _{k}( t) )$ for every
$t\in [ t_{k}^{p},t_{k}^{p+1}]$ and for every $k\geq 1$ and by
$(v)$ and $(vi)$ we have
\begin{equation}
x_{k}(\theta _{k}( t) )\in K\cap B(\varphi (0),r),\text{ for every
}t\in [ 0,T]\text{ and for every }k\geq 1.  \label{e3.18}
\end{equation}
and
\begin{equation}
T(\theta _{k}( t) )x_{k}\in \mathcal{K}_{0}\cap B_{\sigma
}(\varphi ,r),\text{ for every }t\in [ 0,T]\text{ and for every }k\geq
1.  \label{e3.19}
\end{equation}
Also, by $(iii)$ and $(iv)$ we have
\begin{equation}
x_{k}''(t)\in F(T(\theta _{k}( t)
)x_{k},x_{k}'(\theta _{k}( t) ))+\frac{1}{k}B,\text{
a.e. on }[0,T]\text{ and for every }k\geq 1.  \label{e3.20}
\end{equation}
Moreover, by $(iii)$ and $(iv)$ we have
\begin{equation*}
x_{k}'(t)=y_{k}^{p}+(t-t_{k}^{p})z_{k}^{p}\text{ for every }t\in
[ t_{k}^{p},t_{k}^{p+1}]
\end{equation*}
and
\begin{equation*}
x_{k}''(t)=z_{k}^{p}\in F(T(t_{k}^{p})x_{k},y_{k}^{p})+\frac{1
}{k}B\text{ for every }t\in [ t_{k}^{p},t_{k}^{p+1}].
\end{equation*}
Hence, by \eqref{e3.3}, \eqref{e3.5}, and \eqref{e3.15} we obtain
\begin{gather*}
\|x_{k}''(t)\|=\|z_{k}^{p}\|\leq M+1,
\\
\|x_{k}'(t)\|\leq \|y_{k}^{p}\|+(t-_{k}^{p})\|z_{k}^{p}\|\leq
\|y_{k}^{p}-y_{0}\|+\|y_{0}\|+T(M+1)\leq \|y_{0}\|+2r,
\\
\|x_{k}(t)\|\leq \|x_{k}(t)-\varphi (0)\|+\|\varphi (0)\|\leq \|\varphi
(0)\|+r\,.
\end{gather*}
Therefore, $x_{k}''(\cdot )$ is bounded in $L^{2}([0,T],
\mathbb{R}^{m})$, $x_{k}(\cdot )$ $x_{k}'(\cdot )$ are bounded in
the space $C([0,T],\mathbb{R}^{m})$.
Moreover, for all $t',t''\in [ 0,T]$, we have
\begin{gather*}
\|x_{k}(t')-x_{k}(t'')\|=\|\int_{t'}^{t''}x_{k}'(t)dt\|
\leq \int_{t'}^{t''}\|x_{k}'(t)\|dt
\leq (\|y_{0}\|+2r)|t'-t''|,
 \\
\|x_{k}'(t')-x_{k}'(t'')\|=\|\int_{t'}^{t''}x_{k}''(t)dt\|
\leq \int_{t'}^{t''}\|x_{k}'(t)\|dt\leq (\|\varphi (0)\|+r)|t'-t''|,
\end{gather*}
i. e. the sequence $x_{k}(\cdot )$, is equi-lipschitzian and the
sequence $x_{k}'(\cdot )$ is equi-uniformly continuous.

Hence, by \cite[Theorem 0.3.4]{ac}, there exists a subsequence
(again denoted by) $x_{k}(\cdot )$ and an absolutely continuous
function $x:[0,T]\to \mathbb{R}^{m}$ such that
\begin{itemize}
\item[(a)]   $x_{k}(\cdot )$  converges uniformly to $x(\cdot )$,
\item[(b)]   $x_{k}'(\cdot )$  converges uniformly to $x'(\cdot )$,
\item[(c)]   $x_{k}''(\cdot )$  converges weakly in $L^{2}([0,T],
\mathbb{R}^{m})$ to $x''(\cdot )$.
\end{itemize}

Moreover, since all functions $x_{k}$ agree with $\varphi $ on $[-\sigma
,0], $ we can obviously say that $x_{k}\to x$ on $[-\sigma ,T]$, if
we extend $x$ in such a way that $x_{k}=\varphi $ on $[0,T]$. By $(a)$, $(b)$
and the uniformly converges of $\theta _{k}(\cdot )$ to $t$ on $[0,T]$ we
deduce that $x_{k}(\theta _{k}(t))\to x(t)$ uniformly on $[0,T]$ and
$x_{k}'(\theta _{k}(t))\to x'(t)$\ uniformly on $
[0,T]$. Also, it is clearly that $T(0)x=\varphi $ on $[-\sigma ,0]$.

Let us denote the modulus continuity of a function $\psi $ on
interval $I$ of $\mathbb{R}$ by
\begin{equation*}
\omega (\psi ,I,\varepsilon ):=\sup \{\|\psi ( t) -\psi (
s) \|;s,t\in I,|s-t|<\varepsilon \},\varepsilon >0.
\end{equation*}
Then we have
\begin{align*}
\|T(\theta _{k}(t))x_{k}-T(t)x_{k}\|_{\sigma }
&=\sup_{-\sigma \leq s\leq0} \|x_{k}(\theta _{k}(t)+s)-x_{k}(t+s)\| \\
&\leq \omega (x_{k},[ -\sigma ,T] ,\frac{1}{k})\leq \omega
(\varphi ,[ -\sigma ,0] ,\frac{1}{k})+\omega (x_{k},[ 0,T
] ,\frac{1}{k}) \\
&\leq \omega (\varphi ,[ -\sigma ,0] ,\frac{1}{k})+\frac{(\|y_{0}\|+2r)T}{k},
\end{align*}
hence
\begin{equation}
\|T(\theta _{k}(t)x_{k}-T(t)x_{k}\|_{\sigma }\leq \delta _{k}  \label{e3.21}
\end{equation}
for every $k\geq 1$, where
$\delta _{k}:=\omega (\varphi ,[ -\sigma ,0] ,\frac{1}{k})
+\frac{(\|y_{0}\|+2r)T}{k}$.
Thus, by continuity of $\varphi $, we have $\delta _{k}\to 0$ as
$k\to \infty $, hence
\begin{equation*}
\|T(\theta _{k}(t)x_{k}-T(t)x_{k}\|_{\infty }\to 0\quad
\text{as } k\to \infty ,
\end{equation*}
and so, since the uniform convergence of $x_{k}(\cdot )$ to $x(\cdot )$ on $
[ -\sigma ,T] $ implies
\begin{equation}
T(t)x_{k}\to T(t)x\quad \text{uniformly on }[ 0,T] ,
\label{e3.22}
\end{equation}
we deduce that
\begin{equation}
T(\theta _{k}(t)x_{k}\to T(t)x\quad \text{in }\mathcal{C}_{\sigma }.
\label{e3.23}
\end{equation}
Since $T(\theta _{k}(t))x_{k}\in \mathcal{K}_{0}\cap B_{\sigma }(
\varphi ,r) $ for every $t\in [ 0,T] $ and for every
$k\geq 1$, thus by \eqref{e3.19}, \eqref{e3.23} and by
Remark \ref{rmk3.2} we have
$T(t) x\in \mathcal{K}_{0}\cap B_{\sigma }( \varphi ,r) $.

Since $\|x_{k}'(t)-x_{k}'(\theta_{k}(t))\|\leq \frac{(M+1)T}{k}$,
 by \eqref{e3.20} and \eqref{e3.21}, we have
\begin{equation}
d(T(t)x_{k},x_{k}'(t),x_{k}''(t)),\mathop{\rm graph}
(F))\leq \delta _{k}+\frac{(M+1)T+1}{k}  \label{e3.24}
\end{equation}
for every $k\geq 1$.
By (H2), (b), (c), \eqref{e3.22} and  \cite[Theorem 1.4.1]{ac}, we
obtain
\begin{equation}
x''(t)\in \mathop{\rm co}F(T(t)x,x'( t) )\subset
\partial V( x'( t) ) \text{ a.e. on }[
0,T] ,  \label{e3.25}
\end{equation}
where $\mathop{\rm co}$ stands for the closed convex hull.
Since the functions $t\to x( t) $ and $t\to
V( x'( t) ) $ are absolutely continuous, we
obtain from  \cite[Lemma 3.3]{b1} and \eqref{e3.25} that
\begin{equation*}
\frac{d}{dt}V( x'( t) ) =\|x''( t) \|^{2}\quad \text{ a.e. on }[ 0,T]
\end{equation*}
hence
\begin{equation}
V(x'(t))-V(x'(0))=\int_{0}^{T}\|x''(t)\|^{2}dt  \label{e3.26}
\end{equation}
On the other hand, since $x_{k}''(t)=z_{k}^{p}$ for every $
t\in [ t_{k}^{p},t_{k}^{p+1}]$, by $(iv)$, there exists $w_{k}^{p}\in
\frac{1}{k}B$ such that
\begin{equation*}
z_{k}^{p}-w_{k}^{p}\in F(T(t_{k}^{p})x_{k},y_{k}^{p})\subset \partial
V(x_{k}'(t_{k}^{p})),\quad \forall k\in \mathbb{N}^{\ast }
\end{equation*}
and so the properties of subdifferential of a convex function imply that,
for every $p<m(k)-2$, and for every $k\in \mathbb{N}^{\ast }$ we have
\begin{align*}
V(x_{k}'(t_{k}^{p+1}))-V(x_{k}'(t_{k}^{p}))
&\geq \langle z_{k}^{p}-w_{k}^{p},x_{k}'(t_{k}^{p+1})-x_{k}'(t_{k}^{p})\rangle
\\
&=\langle z_{k}^{p},\int_{t_{k}^{p}}^{t_{k}^{p+1}}x_{k}''( t) dt\rangle
-\langle w_{k}^{p},\int_{t_{k}^{p}}^{t_{k}^{p+1}}x_{k}''( t) dt\rangle
\\
&=\int_{t_{k}^{p}}^{t_{k}^{p+1}}\|x_{k}''( t)
\|^{2}dt-\langle w_{k}^{p},\int_{t_{k}^{p}}^{t_{k}^{p+1}}x_{k}''( t) dt
\rangle ;
\end{align*}
hence
\begin{equation}
V(x_{k}'(t_{k}^{p+1})-V(x_{k}'(t_{k}^{p}))\geq
\int_{t_{k}^{p}}^{t_{k}^{p+1}}\|x''(t)\|^{2}dt-\langle
w_{k}^{p},\int_{t_{k}^{p}}^{t_{k}^{p+1}}x_{k}''(
t) dt\rangle .  \label{e3.27}
\end{equation}
Analogously, if $t\in [ t_{k}^{m(k)-1},T]$, then by $(i)$ we have
\begin{equation}
\begin{aligned}
&V(x_{k}'(T))-V(x_{k}'(t_{k}^{m(k)-1}))\\
&\geq \langle z_{k}^{m(k)-1}-w_{k}^{m(k)-1},
\int_{t_{k}^{m(k)-1}}^{T}x_{k}''( t) dt\rangle  \\
&=\int_{t_{k}^{m(k)-1}}^{T}\|x_{k}''( t)
\|^{2}dt-\langle w_{k}^{m(k)-1},\int_{t_{k}^{m(k)-1}}^{T}x_{k}''( t)
dt\rangle .
\end{aligned}
\label{e3.28}
\end{equation}
By adding the $m( k) -1$ inequalities from \eqref{e3.27} and the
inequality from \eqref{e3.28}, we get
\begin{equation}
V(x_{k}'(T))-V(x'(0))\geq \int_{0}^{T}\|x_{k}''(t)\|^{2}dt+\alpha (k),
 \label{e3.29}
\end{equation}
where
\begin{equation*}
\alpha (k)=-\sum_{p=0}^{m(k)-2} \langle
w_{k}^{p},\int_{t_{k}^{p}}^{t_{k}^{p+1}}x_{k}''(t)dt\rangle
-\langle w_{k}^{m(k)-1},\int_{t^{m(k)-1}}^{T}x_{k}''(t)dt\rangle .
\end{equation*}
Since
\begin{align*}
|\alpha (k)| &\leq \sum_{p=0}^{m(k)-2} |\langle
w_{k}^{p},\int_{t_{k}^{p}}^{t_{k}^{p+1}}x_{k}''(t)dt\rangle
|+|\langle w_{k}^{m(k)-1},\int_{t^{m(k)-1}}^{T}x_{k}'(t)dt\rangle |
\\
&\leq \sum_{p=0}^{m(k)-2} \|w_{k}^{p}\|\cdot\|
\int_{t_{k}^{p}}^{t_{k}^{p+1}}x_{k}''(t)dt\|+\|w_{k}^{m(k)-1}\|\cdot \|
\int_{t^{m(k)-1}}^{T}x_{k}''(t)dt\| \\
&\leq \frac{(M+1)(2m(k)-1)}{k}
\end{align*}
it follows that $\alpha (k)\to 0$ as $k\to \infty $;
hence, by \eqref{e3.29},  passing to the limit as $k\to \infty $,
we obtain
\begin{equation}
V(x'(t))-V(y_{0})\geq \limsup_{k\to \infty }
\int_{0}^{T}\|x_{k}''(t)\|^{2}dt.  \label{e3.30}
\end{equation}
Therefore, by \eqref{e3.26} and \eqref{e3.30} we have
\begin{equation*}
\int_{0}^{T}\|x''(t)\|^{2}dt\geq \limsup_{k\to\infty }
\int_{0}^{T}\|x_{k}''(t)\|^{2}dt
\end{equation*}
and, since $x_{k}''(\cdot )$ converges weakly in
$L^{2}([0,T], \mathbb{R}^{m})$ to $x''(\cdot )$, by the lower
semicontinuity of the norm in $L^{2}([0,T],\mathbb{R}^{m})$
(e.g. \cite[Proposition III 30]{b2}) we obtain that
\begin{equation*}
\lim_{k\to \infty }\int_{0}^{T}\|x_{k}''(t)\|^{2}dt
=\int_{0}^{T}\|x''(t)\|^{2}dt,
\end{equation*}
i. e. $x_{k}''(\cdot )$ converges strongly in $L^{2}([0,T],
\mathbb{R}^{m})$ to $x''(\cdot )$, hence a subsequence (again
denote by) $x_{k}''(\cdot )$ converges pointwise a.e. to $
x''(\cdot )$.

Since, by \eqref{e3.24}
\begin{equation*}
\lim_{k\to \infty } d((T(t)x_{k},x_{k}'(t),_{k}''(t)),
\mathop{\rm graph}(F))=0
\end{equation*}
and since, by (H1), the $\mathop{\rm graph}$ of $F$ is closed
(\cite[Proposition 1.1.2]{ac}), we have
\begin{equation*}
x''(t)\in F(T(t)x,x'(t))\quad \text{a.e. on }[0,T].
\end{equation*}

It remains to prove that
$(x( t) ,x'(t))\in K\times\Omega $ for every $t\in [ 0,T]$.
Indeed, since $\|x_{k}(t)-x_{k}^{p}\|\leq \frac{2\|y_{0}\|+3(M+1)T}{k}$,
$\|x_{k}'(t)-y_{k}^{p}\|\leq \frac{(M+1)T}{k}$ we have
\begin{equation*}
\lim_{k\to \infty } d((x_{k}(t),x_{k}'(t)),(x_{k}^{p},y_{k}^{p}))=0.
\end{equation*}
Since $(x_{k}^{p},y_{k}^{p})\in Q_{0}:=(K\cap B(\varphi (0),r))\times
\overline{B}(y_{0},r)\subset K\times \Omega $ for every
$k\in \mathbb{N}$, by $(a)$ and $(b)$ we have
\begin{equation*}
\lim_{k\to \infty } d((x(t),x'(t)),(x_{k}(t),x_{k}'(t)))=0.
\end{equation*}
On the other hand,
\begin{align*}
&d((x(t),x'(t)),Q_{0})\\
&\leq 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});
\end{align*}
hence, by passing to the limit as $k\to \infty $ we obtain
\begin{equation*}
d((x(t),x'(t)),Q_{0})=0,\quad \text{for every }t\in [ 0,T].
\end{equation*}
Since $Q_{0}$ is closed, we obtain that
$(x(t),x'(t))\in Q_{0}\subset K\times \Omega $ for all
$t\in [ 0,T]$, which completes the proof.
\end{proof}

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