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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 144, pp. 1--16.\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 2008 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2008/144\hfil Monotone solutions]
{Monotone solutions for a nonconvex functional differential
inclusions of second order}

\author[A. G. Ibrahim, F. A. Al-Adsani\hfil EJDE-2008/144\hfilneg]
{Ahmed G. Ibrahim, Feryal A. Al-Adsani} 

\address{Ahmed G. Ibrahim \newline
Department  of Mathematics, Faculty of Science, Cairo University,
Egypt}
\email{agamal2000@yahoo.com}

\address{Feryal A. Al-Adsani \newline
Department of Mathematics, Faculty of Educations for Women,
King Faisal University, Sudi Arabia}
\email{faladsani@hotmail.com}

\thanks{Submitted June 10, 2008. Published October 24, 2008.}
\subjclass[2000]{34A60, 49K24}
\keywords{Functional differential inclusion; monotone solution;
\hfill\break\indent second-order contingent cone}

\begin{abstract}
 We give sufficient conditions for the existence of a monotone
 solution for second-order functional differential inclusions.
 No convexity condition, on the values of the multifunction defining
 the inclusion, is involved in this construction.
\end{abstract}

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

\section{Introduction}

Let $K$ be a closed subset of $\mathbb{R}^n$, $\Omega $ an open subset
of $\mathbb{R}^n$, and $P$ a lower semicontinuous set-valued map
(multifunction) from $K$ to the family of all non-empty subsets of $K$, with
closed graph satisfying the following two conditions:
\begin{itemize}
\item[(i)] for all $x\in K$, $x\in P(x)$

\item[(ii)] for all $x,y\in K$, $y\in P(x)\Rightarrow P(y)\subseteq P(x)$.
\end{itemize}
Under these conditions, a preorder (reflexive and transitive relation) on $K$
 is defined as
\[
x\preceq y\Leftrightarrow y\in P(x)\,.
\]

Let $\sigma >0$ and $C([-\sigma,0],\mathbb{R}^n)$ be the space of
continuous functions from $[-\sigma,0]$ to $\mathbb{R}^n$ with the
uniform norm $\|x\|_{\sigma }=\sup \{\|x(t)\|:t\in [-\sigma,0]\}$.
For each $t\in [0,T];T>0$, we define the
operator $\tau (t)$ from $C([-\sigma,T],\mathbb{R}^n)$ to
$C([-\sigma,0],\mathbb{R}^n)$ as
\[
(\tau (t)x)(s)=x(t+s),\quad \text{for all }s\in [-\sigma,0].
\]

Here, $\tau (t)x$  represents the history of the state from
the time $t-\sigma $ to the present time $t$.

Let $K_{0}=\{\varphi \in C([-\sigma,T],\mathbb{R}^n):\varphi (0)\in K\}$
and $F$  be a set-valued map (multifunction) defined from
$K_{0}\times \Omega $ to the family of non-empty compact subsets
(not necessarily convex)
in $\mathbb{R}^n$ and $(\varphi _{0},y_{0})$ be a given element in
$K_{0}\times \Omega $. We consider the  second-order functional
differential inclusion
\begin{equation} \label{eP}
\begin{gathered}
x''(t)\in F(\tau (t)x,x'(t)),\quad \text{a.e. on }[0,T] \\
x(t)=\varphi _{0}(t),\quad \forall t\in [-\sigma,0] \\
x'(0)=y_{0} \\
x(t)\in P(x(t))\subset K,\quad \forall t\in [0,T] \\
x(s)\preceq x(t)\quad \text{whenever }0\leq s\leq t\leq T
\end{gathered}
\end{equation}

In the present work, we prove under reasonable conditions that there are a
positive real number $T$ and a continuous function
$x:[-\sigma,T]\to \mathbb{R}^n$ such that
\begin{enumerate}
\item the function $x$  is absolutely continuous on $[0,T]$ with
absolutely continuous derivative

\item $\tau (t)x\in K_{0}$, for all $t\in [0,T]$

\item $x'(t)\in \Omega $, a.e. on $[0,T]$

\item the functions $x,x',x''$ satisfy \eqref{eP}.
\end{enumerate}

Ibrahim and Alkulaibi \cite{i1} proved the existence of a
monotone solution for \eqref{eP} without delay.
They consider the problem
\begin{gather*}
x''(t)\in F(x(t),x'(t)),\quad \text{a.e. on }[0,T] \\
x(0)=x_{0},\quad x'(0)=y_{0} \\
x(t)\in K,\quad \forall t\in [0,T] \\
x(s)\preceq x(t), \quad \text{whenever }0\leq s\leq t\leq T\,.
\end{gather*}

Further, Lupulescu \cite{l4} proved the existence of a local solution, not
necessarily monotone, for \eqref{eP} in the particular case $P(x)=K$,
for all $x\in K$. Thus, the result, we are going to prove, generalizes the
results of Ibrahim and Alkulaibi \cite{i1} and Lupulescu \cite{l4}.

We mention, among others the works, \cite{c4,h1,h2,i1,m1}
for the proof of
existence of monotone solutions for differential inclusions or functional
differential inclusions and the works
\cite{a3,b1,c2,c3,c5,l1,l2,l3,l4,r1} for
solutions not necessarily monotone. Note that the case where
the solutions  are not necessarily monotone has been widely investigated
compared with that of monotone solutions which has been rarely investigated.

The present paper is organized as follows: In section 2,
some definitions and facts to be used later are introduced.
In section 3, the main result is proved.

\section{Preliminaries}

Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space with norm
$\|\cdot\|$ and scalar product $\langle \cdot,\cdot\rangle$.
For $x\in \mathbb{R}^n$ and $r>0$ let
$B(x,r)=\{y\in \mathbb{R}^n:\|y-x\|<r\}$ denote the open ball
centered at $x $ of radius $r$, and $\overline{B(x,r)}$ its closure.

For $\varphi \in C([-\sigma,0],\mathbb{R}^n)$ let
$B_{\sigma }(\varphi,r)=\{\psi \in C([-\sigma,0],\mathbb{R}^n):
\|\psi -\varphi \|_{\sigma }<r\}$ and
$\overline{B}_{\sigma }(\varphi,r)=\{\psi \in
C([-\sigma,0],\mathbb{R}^n):\|\psi -\varphi \|_{\sigma }\leq r\}$.

We also, denote by $d(x,A)=\inf \{\|x-y\|:y\in A\}$ the
distance from $x$ $\in \mathbb{R}^n$  to a closed subset
$A\subseteq \mathbb{R}^n$ .

A function $V:\mathbb{R}^n\to \mathbb{R}\cup \{\infty \}$ is said
to be proper if its effective domain
$D(V)=\{x\in \mathbb{R}^n:V(x)<\infty \}$ is non-empty.

The subdifferential of  a proper convex lower semicontinuous function
$V:\mathbb{R}^n\to \mathbb{R}$ at a point $x\in \mathbb{R}^n$ is
defined (in the sense of convex analysis)  by
\begin{equation*}
\partial V(x)=\big\{\xi \in \mathbb{R}^n:V(y)-V(x)\geq
\langle \xi,y-x\rangle,\;\forall y\in \mathbb{R}^n\big\}
\end{equation*}
The second-order contingent cone of a non-empty closed subset
$C\subset\mathbb{R}^n$ at a point $(x,y)\in C\times \mathbb{R}^n $
is defined by
\begin{equation*}
T_{C}^{2}(x,y)=\big\{z\in \mathbb{R}^n:\lim_{t\to 0^{+}}
\inf \frac{d(x+ty+\frac{t^{2}}{2}z,C)}{t^{2}}=0\big\}.
\end{equation*}
For the properties of the second-order contingent cone see
for example \cite{a2,a3,c2,l1}.

A multifunction $F:K_{0}\times \Omega \to 2^{\mathbb{R}^n}$ is
said to be upper semicontinuous at a point
$(\varphi,y)\in K_{0}\times\Omega $ if for every $\varepsilon >0$
there exists $\delta >0$, such that
\begin{equation*}
F(\psi,z)\subset F(\varphi,y)+B(0,\varepsilon ),
\end{equation*}
for all $(\psi,z)\in B_{\sigma }(\varphi,\delta )\times B(y,\delta )$.
For more information about the continuity properties for multifunctions we
refer the reader to \cite{a1,a2,c1}.

\section{Main Result}

\begin{lemma} \label{lem3.1}
Let $K$ be a non-empty closed subset of $\mathbb{R}^n$, $\Omega $ a
non-empty open subset of $\mathbb{R}^n$, $P$  a set-valued map from $K$
to the family of non-empty closed subsets of $K$ and
$K_{0}=\{\varphi \in C([-\sigma,0],\mathbb{R}^n),\varphi (0)\in K\}$.
 Let $F$ be an upper semicontinuous set-valued map from $K_{0}\times \Omega $
to the family of non-empty compact subsets of $\mathbb{R}^n$.
Assume also the following conditions:
\begin{itemize}
\item[(H1)] For all $x\in K$, $x\in P(x)$

\item[(H2)] There exists a proper convex lower semicontinuous function
$V:\mathbb{R}^n\to \mathbb{R}$ such that
$F(\varphi,y)\subseteq \partial V(y)$,
for every $(\varphi,y)\in K_{0}\times \Omega $

\item[(H3)] For $(\varphi,y)\in K_{0}\times \Omega $,
$F(\varphi,y)\subseteq T_{P(\varphi (0))}^{2}(\varphi (0),y)$,
where $T_{P(\varphi(0))}^{2}(\varphi (0),y)$ is the second order
contingent cone of the closed
subset $P(\varphi (0))$ at the point $(\varphi (0),y)$.
\end{itemize}

Let $(\varphi _{0},y_{0})$ be a fixed element in $K_{0}\times \Omega $. Then
there are two positive numbers $r$ and $T$ such that for each positive
integer $m$  there are:
\begin{enumerate}
\item A positive integer $\nu _{m}$.

\item A set of points
\begin{equation*}
P_{m}=\{t_{0}^{m}=0<t_{1}^{m}<\dots <t_{\nu _{m}-1}^{m}\leq T<t_{\nu
_{m}}^{m}\}
\end{equation*}

\item Three sets of elements in $\mathbb{R}^n$:
\begin{gather*}
X_{m} =\{x_{p}^{m}:p=0,1,\dots,\nu _{m}-1\}, \\
Y_{m} =\{y_{p}^{m}:p=0,1,\dots,\nu _{m}-1\}, \\
Z_{m}=\{z_{p}^{m}:p=0,1,\dots,\nu _{m}-1\},
\end{gather*}
with $x_{0}^{m}=\varphi _{0}(0)$ and $y_{0}^{m}=y_{0}$

\item A continuous function $x_{m}:[-\sigma,T]\to \mathbb{R}^n$
with $x_{m}(t)=\varphi _{0}(t)$, for all $t\in [-\sigma,0]$,
such that for each $p=0,1,\dots,\nu _{m}-1$, the following properties are
satisfied:
\begin{itemize}
\item[(i)] $h_{p+1}^{m}=t_{p+1}^{m}-t_{p}^{m}<\frac{1}{m}$

\item[(ii)] $z_{p}^{m}=u_{p}^{m}+w_{p}^{m}$ where $u_{p}^{m}\in F(\tau
(t_{p}^{m})x_{m},y_{p}^{m})$ and $w_{p}^{m}\in \frac{1}{m}B(0,1)$

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

\item[(iv)] $x_{p+1}^{m}=x_{p}^{m}+h_{p+1}^{m}y_{p}^{m}+\frac{1}{2}
(h_{p+1}^{m})^{2}\,z_{p}^{m}=x_{m}(t_{p+1}^{m})$

\item[(v)] $x_{p+1}^{m}\in P(x_{p}^{m})\cap B(\varphi _{0}(0),r)\subseteq K$
and \\
$y_{p+1}^{m}=y_{p}^{m}+h_{p+1}^{m}z_{p}^{m}\in \overline{B(y_{0},r)}
\subseteq \Omega $

\item[(vi)] $x_{m}(t)\in B(\varphi _{0}(0),r)$, for all
$t\in [t_{p}^{m},t_{p+1}^{m}]$.

\item[(vii)] $\tau (t_{p+1}^{m})x_{m}\in B_{\sigma }(\varphi _{0},r)
\cap K_{0}$.
\end{itemize}
\end{enumerate}
\end{lemma}

\begin{proof}
We follow the techniques developed in \cite{m1}.
From \cite[Prop. I.26]{c1}, for each
$y\in \mathbb{R}^n$, the subset $\partial V(y)$ is closed,
convex and bounded.
Moreover, by \cite[Thm. 0.7.2]{a1} the multifunction
 $y\to \partial V(y)$ is upper semicontinuous.
So, by \cite[Prop. 1.1.3]{a1}
there are two positive real numbers $r$ and $M$ such that
\begin{equation*}
\sup \{\|z\|:z\in \partial V(y)\}\leq M\,,
\end{equation*}
for all $y\in \overline{B(y_{0},r)}$.
Using condition (H2), we get
\begin{equation}
\sup \{\|z\|:z\in F(\psi,y)\}<M\,,  \label{e1}
\end{equation}
for all   $(\psi,y)\in (K_{0}\cap B_{\sigma }(\varphi
_{0},r))\times \overline{B(y_{0},r)}$.
Since $\Omega $ is open we can choose $r$ such that
$\overline{B(y_{0},r)}\subseteq \Omega $. It is obvious that
the closedness of $K$ implies the closedness of $K_{0}$ in
$C([-\sigma,0],\mathbb{R}^n)$. From the continuity of
$\varphi _{0}$ on $[-\sigma,0]$,
there is $\mu >0$ such that for all $t,s\in [-\sigma,0]$ we have
\begin{equation}
|t-s|<\mu \Longrightarrow \|\varphi _{0}(t)-\varphi _{0}(s)\|<\frac{r}{4}.
\label{e2}
\end{equation}
Put
\begin{equation}
T=\min \Big\{\mu,\frac{r}{4(M+1)},\frac{r}{8(\|y_{0}\|+1)},\sqrt{\frac{r}{4(M+1)
}}\Big\}  \label{e3}
\end{equation}
Thus the numbers $r$ and $T$ are well defined. Now let $m$ be a fixed
positive integer. We put $t_{0}^{m}=0$,
$x_{0}^{m}=\varphi _{0}(0)$ and $y_{0}^{m}=y_{0}$.
The sets $P_{m}$, $X_{m}$, $Y_{m}$ and $Z_{m}$ will be defined by induction.
We first define $x_{1}^{m}$, $t_{1}^{m}$, $y_{1}^{m}$, $z_{0}^{m}$ and
$x_{m}$ on $[0,t_{1}^{m}]$ such
that the properties (i)--(vii) are satisfied for $p=0$.

Using condition (H3), there is $u_{0}^{m}\in F(\varphi _{0},y_{0})$
such that
\begin{equation*}
\lim_{h\downarrow 0} \inf \frac{1}{h^{2}}d(\varphi_{0}(0)+h\,y_{0}^{m}
+\frac{h^{2}}{2}u_{0}^{m},P(\varphi _{0}(0)))=0.
\end{equation*}
So, a positive number $h_{1}^{m}$ is found such that
$h_{1}^{m}\leq \min \{\frac{1}{m},T\}$ and
\begin{equation*}
d(\varphi _{0}(0)+h_{1}^{m}\,y_{0}^{m}+\frac{(h_{1}^{m})^{2}}{2}
u_{0}^{m},P(\varphi _{0}(0)))\leq \frac{(h_{1}^{m})^{2}}{4m}.
\end{equation*}
Since $P(\varphi _{0}(0))$ is closed, there is
$x_{1}^{m}\in P(\varphi_{0}(0))$ with
\begin{equation*}
\|\varphi _{0}(0)+h_{1}^{m}y_{0}^{m}+\frac{(h_{1}^{m})^{2}}{2}
u_{0}^{m}-x_{1}^{m}\|\leq \frac{(h_{1}^{m})^{2}}{4m}.
\end{equation*}
Consequently there is $w_{0}^{m}\in \mathbb{R}^n$ such that
$\|w_{0}^{m}\| \leq \frac{1}{2m}$ and
\begin{equation*}
x_{1}^{m}=\varphi _{0}(0)+h_{1}^{m}y_{0}^{m}+\frac{(h_{1}^{m})^{2}}{2}
u_{0}^{m}+\frac{(h_{1}^{m})^{2}}{2}w_{0}^{m}.
\end{equation*}

Now we define $z_{0}^{m}=u_{0}^{m}+w_{0}^{m}$, therefore
$z_{0}^{m}\in F(\varphi _{0},y_{0})+\frac{1}{2m}B(0,1)$
and $x_{1}^{m}=\varphi _{0}(0)+h_{1}^{m}y_{0}^{m}
+\frac{(h_{1}^{m})^{2}}{2}z_{0}^{m}$.
We put $y_{1}^{m}=y_{0}^{m}+h_{1}^{m}z_{0}^{m}$ and
$t_{1}^{m}=t_{0}^{m}+h_{1}^{m}$ and for $t\in [t_{0}^{m},t_{1}^{m}]$
we define
\begin{center}
$x_{m}(t)=\varphi _{0}(0)+(t-t_{0}^{m})y_{0}^{m}+\frac{(t-t_{0}^{m})^{2}}{2}
z_{0}^{m}$.
\end{center}
Thus, the properties (i)--(iv) are clearly satisfied for $p=0$.

Since $\tau (t_{0}^{m})x_{m}=\varphi _{0}$, using relation \eqref{e1},
we obtain
\begin{equation*}
\sup \{\|v\|:v\in F(\tau (t_{0}^{m})x_{m},y_{0})\}\leq M.
\end{equation*}
Therefore, $\|z_{0}^{m}\| \leq M+\frac{1}{2m}<M+1$.
We get from the definition of $y_{1}^{m}$,
$\|y_{1}^{m}-y_{0}^{m}\| \leq h_{1}^{m}\|z_{0}^{m}\| \leq T(M+1)\leq r$.
Thus   $y_{1}^{m}\in \overline{B(y_{0},r)}$.
Since, $x_{1}^{m}\in P(x_{0}^{m})\subseteq K$ then to prove property (v)
for  $p=0$, it is sufficient to show that
\begin{equation*}
\|x_{1}^{m}-\varphi _{0}(0)\|<r.
\end{equation*}
We get using \eqref{e1}, \eqref{e3}
\begin{align*}
\|x_{1}^{m}-\varphi _{0}(0)\|
&=h_{1}^{m}\|y_{0}^{m}\|+\frac{(h_{1}^{m})^{2}
}{2}\|z_{0}^{m}\| \\
&\leq T\|y_{0}^{m}\|+\frac{T^{2}}{2}(M+1) \\
&\leq \frac{r}{8(\|y_{0}\|+1)}\|y_{0}\|+\frac{r}{8(M+1)}(M+1)
\\
&< \frac{r}{8}+\frac{r}{8}<r
\end{align*}
and hence (v) is satisfied for $p=0$.
To prove (vi) for $p=0$, we note that, for
$t\in [t_{0}^{m},t_{1}^{m}]$,
\begin{align*}
\|x_{m}(t)-\varphi _{0}(0)\|
&\leq (t-t_{0}^{m})\|y_{0}^{m}\|+\frac{(t-t_{0}^{m})^{2}}{2}\|z_{0}^{m}\| \\
&\leq h_{1}^{m}\|y_{0}^{m}\|+\frac{(h_{1}^{m})^{2}}{2}\|z_{0}^{m}\| \\
&\leq T\|y_{0}\|+\frac{T^{2}}{2}(M+1) \\
&\leq \frac{r}{8(\|y_{0}\|+1)}\|y_{0}\|+\frac{1}{2}\frac{r}{4(M+1)}(M+1) \\
&< \frac{r}{8}+\frac{r}{8}<r,
\end{align*}
which proves (vi) for $p=0$.

To prove property (vii) for $p=0$,
we note that if $-\sigma \leq s\leq-t_{1}^{m}$, then
$t_{1}^{m}+s\leq 0$ and by \eqref{e2}, \eqref{e3}, we get
\begin{equation*}
\|\tau (t_{1}^{m})x_{m}-\varphi \|_{\sigma }=\underset{-\sigma \leq s\leq 0}{
\sup }\|x_{m}(t_{1}^{m}+s)-\varphi _{0}(s)\|=\underset{-\sigma \leq s\leq 0}{
\sup }\|\varphi _{0}(t_{1}^{m}+s)-\varphi _{0}(s)\|<\frac{r}{4}.
\end{equation*}
while if $-t_{1}^{m}\leq s\leq 0$, then $0\leq t_{1}^{m}+s\leq t_{1}^{m}$
and hence by \eqref{e3} we get
\begin{align*}
\|x_{m}(t_{1}^{m}+s)-\varphi _{0}(s)\|
&\leq \|x_{m}(t_{1}^{m}+s)-\varphi
_{0}(0)\|+\|\varphi _{0}(0)-\varphi _{0}(s)\| \\
&\leq (h_{1}^{m}+s)\|y_{0}^{m}\|+\frac{(h_{1}^{m}+s)^{2}}{2}\|z_{0}^{m}\|+
\frac{r}{4} \\
&\leq T\|y_{0}^{m}\|+\frac{T^{2}}{2}\|z_{0}^{m}\|+\frac{r}{4} \\
&\leq \frac{r}{8(\|y_{0}\|+1)}\|y_{0}\|+\frac{r}{8(M+1)}(M+1)+
\frac{r}{4} \\
&< \frac{r}{8}+\frac{r}{8}+\frac{r}{4}=\frac{r}{2},
\end{align*}
which shows that $\tau (t_{1}^{m})x_{m}\in B_{\sigma }(\varphi _{0},r)$ and
hence (vii) is proved.

Now we suppose that $t_{p+1}^{m}$, $x_{p+1}^{m}$, $y_{p+1}^{m}$,
$z_{p}^{m}$ are well defined for $p=0,1,\dots,(q-1)$ and
$x_{m }$ is
defined on the interval $[-\sigma,t_{q}^{m}]$ such that all the properties
(i)--(vii) are satisfied for $p=0,1,\dots,(q-1)$.

We define $t_{q+1}^{m}$, $x_{q+1}^{m}$, $y_{q+1}^{m}$, $z_{q}^{m}$ and
$x_{m} $ on $[t_{q}^{m},t_{q+1}^{m}]$ such that the properties
(i)--(vii) are satisfied for $p=q$.
We denote by $H_{q}^{m}$ the set of all $h\in ] 0,\frac{1}{m}[ $
for which the following conditions are satisfied:
\begin{itemize}
\item[(a)] $h<T-t_{q}^{m}$.

\item[(b)] there exists $u_{q}^{m}\in F(\tau (t_{q}^{m})x_{m},y_{q}^{m})$
such that
\begin{equation*}
d(x_{q}^{m}+h y_{q}^{m}+\frac{h^{2}}{2}u_{q}^{m},p(x_{q}^{m}))\leq
\frac{h^{2}}{4m}.
\end{equation*}
\end{itemize}
From the fact that (v) and (vii) are true for $p=q-1$, we get
$y_{q}^{m}\in \Omega $ and $\tau (t_{q}^{m})x_{m}\in K_{0}$.
Moreover, since (iv) is true for $p=q-1$, then
$$
\tau (t_{q}^{m})x_{m}(0)=x_{m}(t_{q}^{m})=x_{q}^{m}.
$$
So, the condition (H3) gives:
$$
F(\tau (t_{q}^{m})x_{m},y_{q}^{m})\subseteq
T_{P(x_{q}^{m})}^{2}(x_{q}^{m},y_{q}^{m}).
$$
Therefore there is $u_{q}^{m}\in F(\tau (t_{q}^{m})x_{m},y_{q}^{m})$
such that
$$
\liminf_{h\downarrow 0} \frac{1}{h^{2}}d(x_{q}^{m}+h\,y_{q}^{m}+
\frac{h^{2}}{2}u_{q}^{m},P(x_{q}^{m}))=0,
$$
which shows that there is a positive number $h$ such that
$h<\min \{\frac{1}{m},T-t_{q}^{m}\}$ and
\begin{equation*}
d(x_{q}^{m}+h y_{q}^{m}+\frac{h^{2}}{2}u_{q}^{m},P(x_{q}^{m}))\leq
\frac{h^{2}}{4m}.
\end{equation*}
Hence $h\in H_{q}^{m}$. Since $H_{q}^{m}$ is bounded by the number $T$,
there is a number $d_{q}^{m}$ such that
$d_{q}^{m}=\sup \{\alpha :\alpha \in H_{q}^{m}\}$. Since
$H_{q}^{m}\cap [\frac{d_{q}^{m}}{2},d_{q}^{m}]\neq \phi $,
an element $h_{q+1}^{m}\in H_{q}^{m}\cap [\frac{d_{q}^{m}}{2},d_{q}^{m}]$
 is found such that
\begin{equation*}
d(x_{q}^{m}+h_{q+1}^{m} y_{q}^{m}+\frac{(h_{q+1}^{m})^{2}}{2}
u_{q}^{m}, P(x_{q}^{m}))\leq \frac{(h_{q+1}^{m})^{2}}{4m}.
\end{equation*}
From the closedness of $P(x_{q}^{m})$, there is
$x_{q+1}^{m}\in P(x_{q}^{m})\subseteq K$ with
$$
\|x_{q}^{m}+h_{q+1}^{m}y_{q}^{m}+\frac{(h_{q+1}^{m})^{2}}{2}
u_{q}^{m}-x_{q+1}^{m}\|\leq \frac{(h_{q+1}^{m})^{2}}{4m}.
$$
Consequently, there is $w_{q}^{m}\in \mathbb{R}^n$ with
$\|w_{q}^{m}\|\leq \frac{1}{2m}<\frac{1}{m}$ such that
\begin{align*}
x_{q+1}^{m}
&= x_{q}^{m}+h_{q+1}^{m}y_{q}^{m}+\frac{(h_{q+1}^{m})^{2}}{2}
u_{q}^{m}+\frac{(h_{q+1}^{m})^{2}}{2}w_{q}^{m} \\
&= x_{q}^{m}+h_{q+1}^{m}y_{q}^{m}+\frac{(h_{q+1}^{m})^{2}}{2}
(u_{q}^{m}+w_{q}^{m}).
\end{align*}
We define $z_{q}^{m}=u_{q}^{m}+w_{q}^{m}$. So that
\begin{gather*}
z_{q}^{m}\in F(\tau (t_{q}^{m})x_{m},y_{q}^{m})+\frac{1}{m}B(0,1),\\
x_{q+1}^{m}=x_{q}^{m}+h_{q+1}^{m}y_{q}^{m}+\frac{(h_{q+1}^{m})^{2}}{2}
z_{q}^{m}.
\end{gather*}
We put $y_{q+1}^{m}=y_{q}^{m}+h_{q+1}^{m}z_{q}^{m}$ and
$t_{q+1}^{m}=t_{q}^{m}+h_{q+1}^{m}$ and for
$t\in [t_{q}^{m},t_{q+1}^{m}]$, we define
\begin{equation*}
x_{m}(t)=x_{q}^{m}+(t-t_{q}^{m})y_{q}^{m}+\frac{(t-t_{q}^{m})^{2}}{2}
z_{q}^{m}.
\end{equation*}
Obviously the relations (i)--(iv) are satisfied for $p=q$.

Now we prove that (v) is true for $p=q$. Since (v) and (vii) are true
for $p=q-1$, then $\tau (t_{q}^{m})x_{m}\in B_{\sigma }(\varphi _{0},r)$ and
$y_{q}^{m}\in \overline{B(y_{0},r)}$, and hence by \eqref{e1}
 we get $\|z_{q}^{m}\|\leq M+1$.

Let us prove that $\|y_{q+1}^{m}-y_{0}\|<r$. We note that
$y_{q+1}^{m}=y_{q}^{m}+h_{q+1}^{m}z_{q}^{m}
=y_{q-1}^{m}+h_{q}^{m}z_{q-1}^{m}+h_{q+1}^{m}z_{q}^{m}$.
 By iterating we get
\begin{equation}
y_{q+1}^{m}=y_{0}^{m}+\sum_{s=0}^q h_{s+1}^{m}z_{s}^{m}.
\label{e4}
\end{equation}
Thus,
$$
\|y_{q+1}^{m}-y_{0}^{m}\|\leq \sum_{s=0}^q
h_{s+1}^{m}\|z_{s}^{m}\|\leq (M+1)\sum_{s=0}^q
h_{s+1}^{m}\leq (M+1)T<\frac{r}{4}<r.
$$
To prove that $x_{q+1}^{m}\in B_{\sigma }(\varphi _{0}(0),r) $
we first use the induction  technique to prove the  relation
\begin{equation}
x_{p+1}^{m}=\varphi _{0}(0)+\big(\sum_{j=0}^p h_{j+1}^{m}\big)y_{0}
+\frac{1}{2}\sum_{j=0}^p (h_{j+1}^{m})^{2}\,z_{j}^{m}
+ \sum_{i=0}^{p-1}
\sum_{j=i+1}^p h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m},
\label{e5}
\end{equation}
for $p=1,\dots,q$.
For $p=1$ we note that
\begin{align*}
x_{2}^{m}
&=x_{1}^{m}+h_{2}^{m}y_{1}^{m}+\frac{1}{2}(h_{2}^{m})^{2}z_{1}^{m}\\
&=x_{1}^{m}+h_{2}^{m}(y_{0}^{m}+h_{1}^{m}z_{0}^{m})
  +\frac{1}{2}(h_{2}^{m})^{2}z_{1}^{m} \\
&=(x_{0}^{m}+h_{1}^{m}y_{0}^{m}+\frac{1}{2}(h_{1}^{m})^{2}z_{0}^{m})
  +h_{2}^{m}(y_{0}^{m}+h_{1}^{m}z_{0}^{m})
  +\frac{1}{2}(h_{2}^{m})^{2}\,z_{1}^{m} \\
&=x_{0}^{m}+(h_{1}^{m}+h_{2}^{m})y_{0}^{m}
  +\frac{1}{2}((h_{1}^{m})^{2}z_{0}^{m}+(h_{2}^{m})^{2}z_{1}^{m})
  +h_{1}^{m}h_{2}^{m}z_{0}^{m} \\
&=\varphi_{0}(0)+(\sum_{j=0}^1 h_{j+1}^{m})y_{0}^{m}
 +\frac{1}{2}\sum_{j=0}^1 (h_{j+1}^{m})^{2}\,z_{j}^{m}
+\sum_{j=1}^1 h_{1}^{m}h_{j+1}^{m}z_{0}^{m}.
\end{align*}
Then  relation \eqref{e5} is true for $p=1$. Suppose that
 \eqref{e5} is true for $p=q-1$. This gives us
\begin{equation*}
x_{q}^{m}=\varphi _{0}(0)+\,\sum_{j=0}^{q-1} h_{j+1}^{m})y_{0}
+\frac{1}{2} \sum_{j=0}^{q-1} (h_{j+1}^{m})^{2}\,z_{j}^{m}
+\sum_{i=0}^{q-2} \sum_{j=i+1}^{q-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m}.
\end{equation*}
So, according to the definition of $x_{q+1}^{m}$ we have
\begin{align*}
x_{q+1}^{m}
&=x_{q}^{m}+h_{q+1}^{m}y_{q}^{m}+\frac{1}{2}(h_{q+1}^{m})^{2}z_{q}^{m} \\
&=\varphi _{0}(0)+(\sum_{j=0}^{q-1}h_{j+1}^{m})y_{0}^{m}
+\frac{1}{2}\sum_{j=0}^{q-1} (h_{j+1}^{m})^{2}\,z_{j}^{m}
+\sum_{i=0}^{q-2} \sum_{j=i+1}^{q-1}  h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m} \\
&\quad +h_{q+1}^{m}(y_{0}+\sum_{s=0}^{q-1} h_{s+1}^{m}z_{s}^{m})
+\frac{1}{2}(h_{q+1}^{m})z_{q}^{m} \\
&=\varphi _{0}(0)+(\sum_{j=0}^q h_{j+1}^{m})y_{0}^{m}
+\frac{1}{2} \sum_{j=0}^q  (h_{j+1}^{m})^{2}\,z_{j}^{m}
+\sum_{i=0}^{q-2} \sum_{j=i+1}^{q-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m} \\
&\quad +h_{q+1}^{m}(h_{1}^{m}z_{0}^{m}+h_{2}^{m}z_{1}^{m}
 +\dots +h_{q}^{m}z_{q-1}^{m}) \\
&=\varphi _{0}(0)+\sum_{j=0}^q  h_{j+1}^{m})y_{0}^{m}
+\frac{1}{2} \sum_{j=0}^q (h_{j+1}^{m})^{2}z_{j}^{m}
+\sum_{i=0}^{q-1} \sum_{j=i+1}^q h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m}.
\end{align*}
This implies that the relation \eqref{e5} is true for $p=q$.
Now, from the fact that $\|z_{p}^{m}\|\leq M+1$, for all $p=0,1,\dots,q$ we
get
\begin{align*}
&\|x_{q+1}^{m}-\varphi _{0}(0)\|\\
&\leq \|y_{0}\|(\sum_{j=0}^q h_{j+1}^{m})
+\frac{1}{2}\sum_{j=0}^q (h_{j+1}^{m})^{2}(M+1)
+(M+1) \sum_{i=0}^{q-1}
\sum_{j=i+1}^q h_{i+1}^{m}h_{j+1}^{m} \\
&\leq \|y_{0}\|T+\frac{1}{2}(M+1)T^{2}
+(M+1)\Big[h_{1}^{m}\sum_{j=1}^q h_{j+1}^{m}
+h_{2}^{m} \sum_{j=2}^q h_{j+1}^{m}+\dots\\
&\quad+h_{q}^{m} \sum_{j=q}^q h_{j+1}^{m}\Big] \\
&\leq \|y_{0}\|T+\frac{1}{2}(M+1)T^{2}+(M+1)T^{2} \\
&=\|y_{0}\|T+\frac{3}{2}(M+1)T^{2}\\
&<\frac{r}{8}+\frac{3r}{8} =\frac{r}{2}.
\end{align*}
Thus (v) is true for $p=q$.

Let us prove (vi) for $p=q$, namely
$\|x_{m}(t)-\varphi_{0}(0)\|<r$, for all $t\in [t_{q}^{m},t_{q+1}^{m}]$.
Let $t\in [t_{q}^{m},t_{q+1}^{m}]$. We have
\begin{align*}
x_{m}(t)&=x_{q}^{m}+(t-t_{q}^{m})y_{q}^{m}+\frac{1}{2}(t-t_{q}^{m})^{2}
\,z_{q}^{m} \\
&=\varphi _{0}(0)+(\sum_{j=0}^{q-1}  h_{j+1}^{m})y_{0}
+\frac{1}{2}\sum_{j=0}^{q-1}  (h_{j+1}^{m})^{2}\,z_{j}^{m}
+\sum_{i=0}^{q-2} \sum_{j=i+1}^{q-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m} \\
&\quad +(t-t_{q}^{m})(y_{0}
+\sum_{j=0}^{q-1}  h_{j+1}^{m}z_{j}^{m})
+\frac{1}{2} (t-t_{q}^{m})^{2}z_{q}^{m}.
\end{align*}
Thus,
\begin{align*}
&\|x_{m}(t)-\varphi _{0}(0)\|\\
&\leq \|y_{0}\|(\sum_{j=0}^{q-1}h_{j+1}^{m})
+\frac{1}{2}\sum_{j=0}^{q-1}(h_{j+1}^{m})^{2}\|z_{j}^{m}\|
\\
&\quad +\sum_{i=0}^{q-2}
\sum_{j=i+1}^{q-1} h_{i+1}^{m}h_{j+1}^{m}\|z_{i}^{m}\|
+h_{q+1}^{m}(y_{0}+\sum_{j=0}^{q-1}
h_{j+1}^{m}\|z_{j}^{m}\|)+\frac{1}{2}
(h_{q+1}^{m})^{2}\|z_{q}^{m}\| \\
&\leq \|y_{0}\|(\sum_{j=0}^q
h_{j+1}^{m})+\frac{1}{2}\sum_{j=0}^q
(h_{j+1}^{m})^{2}\|z_{j}^{m}\|
+\sum_{i=0}^{q-1}
\sum_{j=i+1}^q h_{i+1}^{m}h_{j+1}^{m}\|z_{i}^{m}\| \\
&\leq \|y_{0}\|T+\frac{1}{2}T^{2}(M+1)+T^{2}(M+1) \\
&\leq \|y_{0}\|T+\frac{3(M+1)}{2}T^{2}\\
&<\frac{r}{8}+\frac{r}{8}=\frac{r}{4}.
\end{align*}
We prove (vii) for $p=q$.
\begin{align*}
&\|\tau (t_{q+1}^{m})x_{m}-\varphi _{0}\|_{\sigma }\\
&=\sup_{-\sigma \leq s\leq 0}
 \|\tau (t_{q+1}^{m})x_{m}(s)-\varphi _{0}(s)\|. \\
&=\sup_{-\sigma \leq s\leq 0}
  \|x_{m}(t_{q+1}^{m}+s)-\varphi _{0}(s)\|. \\
&\leq \sup_{-\sigma \leq s\leq -t_{q+1}^{m}}
  \|\tau (t_{q+1}^{m})x_{m}(s)-\varphi _{0}(s)\|
 +\sup_{-t_{q+1}^{m}\leq s\leq 0 }
  \|\tau (t_{q+1}^{m})x_{m}(s)-\varphi _{0}(s)\|. \\
&\leq \sup_{-\sigma \leq s\leq -t_{q+1}^{m}}
 \|\varphi _{0}(t_{q+1}^{m}+s)-\varphi _{0}(s)\|
 +\sup_{-t_{q+1}^{m}\leq s\leq 0}
 \|x_{m}(t_{q+1}^{m}+s)-\varphi _{0}(s)\| \\
&\quad +\sup_{-t_{q+1}^{m}\leq s\leq 0}
 \|\varphi _{0}(0)-\varphi _{0}(s)\|. \\
&\leq \frac{r}{4}+\frac{r}{4}+\frac{r}{4}<r.
\end{align*}


It remains to show that there is a positive number $\nu _{m}$ such that
$t_{\nu _{m}-1}^{m}\leq T<t_{\nu _{m}}^{m}$. Therefore, we have to prove that
the iterative process is finite. For this purpose suppose that the iterative
process is not finite. So, for each non negative integer number $p$,
there are $t_{p}^{m}\in [0,T[$, $x_{p}^{m}$, $y_{p}^{m}$,
$z_{p}^{m}$ such that the relations (i)--(vii) are satisfied.
Since the sequence $\{t_{p}^{m}\}_{p\geq 1}$ is bounded and increasing,
there is $t_{\alpha }^{m}\in ]0,T]$ such that
$\lim_{p\to \infty } t_{p}^{m}=t_{\alpha }^{m}$. Let us show that
$\{x_{p}^{m}\}_{p\geq 1},\{y_{p}^{m}\}_{p\geq 1}$ are Cauchy sequences. Let
$p $ and $q$ be two positive integers such that $p>q$.
From the relation \eqref{e5} we have
\begin{align*}
&\|x_{p}^{m}-x_{q}^{m}\|\\
&=\|(\sum_{j=0}^{p-1} h_{j+1}^{m})y_{0}+\frac{1}{2}\sum_{j=0}^{p-1}
(h_{j+1}^{m})^{2}\,z_{j}^{m} +\sum_{i=0}^{p-2}
\sum_{j=i+1}^{p-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m}\, \\
&\quad -(\sum_{j=0}^{q-1}  h_{j+1}^{m})y_{0}
-\frac{1}{2}
\sum_{j=0}^{q-1}  (h_{j+1}^{m})^{2}\,z_{j}^{m}
-\sum_{i=0}^{q-2} \sum_{j=i+1}^{q-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m}\|. \\
&=\|(\sum_{j=q}^{p-1}
 h_{j+1}^{m})y_{0}+\frac{1}{2}
\sum_{j=q}^{p-1}(h_{j+1}^{m})^{2}z_{j}^{m}
+\sum_{i=0}^{q-1}
\sum_{j=q}^{p-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m}\|. \\
&\leq \sum_{j=q}^{p-1}h_{j+1}^{m})\|y_{0}\|
+\frac{1}{2}\sum_{j=q}^{p-1}
 (h_{j+1}^{m})^{2}\|z_{j}^{m}\|
+\sum_{i=q-1}^{p-2}\sum_{j=i+1}^{p-1}
h_{i+1}^{m}h_{j+1}^{m}\|z_{i}^{m}\|. \\
&\leq \|y_{0}\|(t_{p}^{m}-t_{q}^{m})+\frac{1}{2}
(M+1)(t_{p}^{m}-t_{q}^{m})^{2}+(M+1)
\sum_{j=q}^{p-1} h_{q}^{m}h_{j+1}^{m} \\
&\quad +(M+1)\sum_{j=q+1}^{p-1}h_{q+1}^{m}h_{j+1}^{m}
+\dots + (M+1)\sum_{j=p-2}^{p-1}
h_{p-2}^{m}h_{j+1}^{m}. \\
&=\|y_{0}\|(t_{p}^{m}-t_{q}^{m})+\frac{1}{2}
(M+1)(t_{p}^{m}-t_{q}^{m})^{2}+(M+1)h_{q}^{m}(t_{p}^{m}-t_{q}^{m}) \\
&\quad +(M+1)h_{q+1}^{m}(t_{p}^{m}-t_{q}^{m})+\dots
+(M+1)h_{p-2}^{m}(t_{p}^{m}-t_{q}^{m}).
\\
&=\|y_{0}\|(t_{p}^{m}-t_{q}^{m})+\frac{1}{2}(M+1)(t_{p}^{m}-t_{q}^{m})^{2}
\\
&\quad +(M+1)(t_{p}^{m}-t_{q}^{m})(h_{q}^{m}+h_{q+1}^{m}+\dots +h_{p-2}^{m}). \\
&=\|y_{0}\|(t_{p}^{m}-t_{q}^{m})+\frac{1}{2}
(M+1)(t_{p}^{m}-t_{q}^{m})^{2}+(M+1)(t_{p}^{m}-t_{q}^{m})^{2}.
\end{align*}
Since the sequence $\{t_{p}^{m}\}_{p\geq 1}$ is convergent,  the
sequence $\{x_{p}^{m}\}_{p\geq 1}$ is Cauchy. Then there
is $x_{\alpha}^{m}\in \mathbb{R}^n$ such that
$\lim_{p\to \infty }x_{p}^{m}=x_{\alpha }^{m}$. Also,
\begin{equation*}
\|y_{p}^{m}-y_{q}^{m}\|=\|\sum_{s=q}^{p-1}
h_{s+1}^{m}z_{s}^{m}\|\leq (M+1)(t_{p}^{m}-t_{q}^{m}).
\end{equation*}
Thus the sequence $\{y_{p}^{m}\}_{p\geq 1}$ is a Cauchy sequence in
$\mathbb{R}^n$. Hence there is $y_{\alpha }^{m}\in \mathbb{R}^n$ such that
$y_{\alpha }^{m}=\lim_{p\to \infty } y_{p}^{m}$.

From  property (v) we note that
\begin{equation}
x_{p}^{m}\in P(x_{p}^{m})\cap \overline{B(\varphi _{0}(0),r)}\subseteq K,
\label{e6}
\end{equation}
and
\begin{equation*}
y_{p}^{m}\in \overline{B(y_{0},r)}\subset \Omega .
\end{equation*}
Thus $x_{\alpha }^{m}\in K$ and $y_{\alpha }^{m}\in \overline{B(y_{0},r)}
\subset \Omega $.

Now we put $x_{m}(t_{\alpha }^{m})=x_{\alpha }^{m}$.
To show that $x_{m}$ is continuous at $t_{\alpha }^{m}$ let
$\{s_{p}^{m}:p\geq 1\}$ be a sequence in $[0,t_{\alpha }^{m}[$
 such that $\lim_{p\to \infty } s_{p}^{m}=t_{\alpha }^{m}$ and
$t_{p}^{m}\leq s_{p}^{m}\leq t_{p+1}^{m}$ for every $p\geq 1$.
We have
\begin{align*}
\|x_{m}(s_{p}^{m})-x_{m}(t_{\alpha }^{m})\|
& \leq \|x_{m}(s_{p}^{m})-x_{m}(t_{p}^{m})\|+\|x_{m}(t_{p}^{m})
-x_{\alpha }^{m}\|\\
&\leq (s_{p}^{m}-t_{p}^{m})\|y_{p}^{m}\|+
\frac{1}{2}(s_{p}^{m}-t_{p}^{m})^{2}(M+1)+\|x_{p}^{m}-x_{\alpha }^{m}\|.
\end{align*}
By taking the limit as $p\to \infty $, we obtain
\begin{equation*}
\lim_{p\to \infty } \|x_{m}(s_{p}^{m})-x_{m}(t_{\alpha}^{m})\|=0
\end{equation*}
which prove that $x_{m}$ is continuous at $t_{\alpha }^{m}$. Hence $x_{m}$
is continuous on $[-\sigma,t_{\alpha }^{m}]$. Consequently,
$$
\lim_{p\to \infty } \tau (t_{p}^{m})x_{m}=\tau (t_{\alpha}^{m})x_{m}.
$$
Note that from (vii),  $\tau (t_{p}^{m})x_{m}\in K_{0}\cap \overline{
B_{\sigma }(\varphi _{0},r)}$.
Since the subset $K_{0}\cap \overline{B_{\sigma }(\varphi _{0},r)}$,
is closed, we obtain
\begin{equation*}
\tau (t_{\alpha }^{m})x_{m}\in K_{0}\cap \overline{B_{\sigma }(\varphi
_{0},r)}\,.
\end{equation*}
Furthermore, by (ii) and the relation \eqref{e1}, the sequences
$\{z_{p}\}_{p\geq 1}$ and $\{u_{p}\}_{p\geq 1}$ are bounded in
$\mathbb{R}^n$. So, there are two convergent subsequences, denoted again by,
$\{z_{p}\}_{p\geq 1}$, $\{u_{p}\}_{p\geq 1}$. Thus there are two elements
$z_{\alpha }^{m}$, $u_{\alpha }^{m}$ of $\mathbb{R}^n$ such that
$\lim_{ p\to \infty } z_{p}^{m}=z_{\alpha }^{m}$,
$\lim_{p\to \infty }  u_{p}^{m}=u_{\alpha }^{m}$.

Now since $F$ is upper semicontinuous on $K_{0}\times \Omega $ with compact
values and since
$u_{p}^{m}\in F(\tau (t_{p}^{m})x_{m},y_{p}^{m})$, for all $p\geq 1$,
it follows that
$u_{\alpha }^{m}\in F(\tau (t_{\alpha }^{m})x_{m},y_{\alpha }^{m})$.
Applying condition (H3),
$$
\lim_{h\to 0^{+}} d(x_{m}(t_{\alpha }^{m})+hy_{\alpha
}^{m}+\frac{h^{2}}{2}u_{\alpha }^{m},P(x_{m}(t_{\alpha }^{m})))=0.
$$
Hence, there is $h\in ]0,T-t_{\alpha }^{m}[$ such that
\begin{equation}
d(x_{\alpha }^{m}+hy_{\alpha }^{m}+\frac{h^{2}}{2}u_{\alpha
}^{m},P(x_{\alpha }^{m}))\leq \frac{h^{2}}{16m}.  \label{e7}
\end{equation}
We prove that $h$ belongs to $H_{p}^{m}$ for every $p$ sufficient large.
Since $\{t_{p}^{m}\}_{p}$ is an increasing sequence to
$t_{\alpha }^{m}$ and since
$\lim_{p\to \infty }x_{p}^{m}=x_{\alpha}^{m}$,
$\lim_{p\to \infty } y_{p}^{m}=y_{\alpha }^{m}$
and $\lim_{p\to \infty } u_{p}^{m}=u_{\alpha }^{m}$.
Then we can find a natural number $p_{1}$ such that for every
$p>p_{1}$ we have
$t_{p}^{m}<t_{\alpha }^{m}<t_{p}^{m}+h<t_{\alpha }^{m}+h$,
\begin{gather}
\|x_{p}^{m}-x_{\alpha }^{m}\|\leq \frac{h^{2}}{24m}, \label{e8}\\
\|y_{p}^{m}-y_{\alpha }^{m}\|\leq \frac{h}{24m},  \label{e9}\\
\|u_{p}^{m}-u_{\alpha }^{m}\|\leq \frac{1}{12m} \,. \label{e10}
\end{gather}

From the lower semicontinuity of $P$ at $x_{p}^{m}$, there is a natural
number $p_{2}$ such that
$P(x_{\alpha }^{m})\subseteq P(x_{p}^{m})+\frac{h^{2}}{16m}B(0,1)$,
for all $p>p_{2}$. This gives that if $z\in \mathbb{R}^n$, then
\begin{equation}
d(z,P(x_{p}^{m}))\leq d(z,P(x_{\alpha }^{m}))+\frac{h^{2}}{16m},\forall
p>p_{2}.  \label{e11}
\end{equation}
Now let $p>\max (p_{1},p_{2})$. By \eqref{e7}--\eqref{e11},
we have
\begin{align*}
&d(x_{p}^{m}+hy_{p}^{m}+\frac{h^{2}}{2}u_{p}^{m},P(x_{p}^{m}))\\
&\leq d(x_{p}^{m}+hy_{p}^{m}+\frac{h^{2}}{2}u_{p}^{m},x_{\alpha }^{m}
+hy_{\alpha }^{m}+\frac{h^{2}}{2}u_{\alpha }^{m})\\
&\quad +d\big(x_{\alpha }^{m}+hy_{\alpha }^{m}+\frac{h^{2}}{2}u_{\alpha
}^{m},P(x_{\alpha }^{m})\big)+\frac{h^{2}}{16m} \\
&\leq \|x_{p}^{m}-x_{\alpha }^{m}\|+h\|y_{p}^{m}-y_{\alpha }^{m}\|+\frac{
h^{2}}{2}\|u_{p}^{m}-u_{\alpha }^{m}\|
+\frac{h^{2}}{8m}+\frac{h^{2}}{16m} \\
&\leq \frac{h^{2}}{24m}+\frac{h^{2}}{24m}+\frac{h^{2}}{24m}
+\frac{h^{2}}{16m}+\frac{h^{2}}{16m}\\
&=\frac{h^{2}}{8m}+\frac{h^{2}}{8m}=\frac{h^{2}}{4m}.
\end{align*}
Thus $h\in H_{p}^{m}$, for all $p\geq \max (p_{1},p_{2})$. From the
choice of $h_{p}^{m}$ we have
$$
\frac{1}{2}\sup H_{p}^{m}\leq h_{p}^{m}\leq \sup H_{p}^{m}.
$$
Hence, $h_{p}^{m}\geq \frac{h}{2}>\frac{h}{4}$ for all
$p\geq \max (p_{1},p_{2})$. This means that
$\lim_{p\to \infty }h_{p}^{m}=\lim_{p\to \infty }(t_{p+1}^{m}-t_{p}^{m})$
can not equal to zero, which contradicts with the fact that the sequence
$\{t_{p}^{m}\}_{p\geq 1}$ is convergent. So, the process must be finite.
\end{proof}

\begin{theorem} \label{maintheorem}
In addition to the assumptions of lemma \ref{lem3.1} we suppose that the graph of
$P$ is closed and the following condition is satisfied.
\begin{itemize}
\item[(H4)] for all $x\in K$ and all $y\in P(x)\,$we have $P(y)\subseteq P(x)$.
\end{itemize}
Then for all $(\varphi _{0},y_{0})\in K_{0}\times \Omega $ there exist
$T>0$ and an absolutely continuous function $x:[0,T]\to K$,
with absolutely continuous derivative such that
\begin{gather*}
x''(t)\in F(\tau (t)x,x'(t))\quad\text{a.e. on } [0,T]\\
x(t)=\varphi _{0}(t),\quad \forall t\in [-\sigma,0]\\
x'(0)=y_{0}.
\end{gather*}
Moreover, $x$ is monotone with respect to $P$ in the sense that for
all $t\in [0,T]$ and all $s\in [t,T]$ we have
$x(s)\in P(x(t))$; i.e., $0\leq t\leq s\leq T\Rightarrow x(t)\preceq x(s)$.
\end{theorem}

\begin{proof}
According to the definition of $x_{m}$, for all $m\geq 1$,
all $p=0,1,2,\dots,\nu _{m}-1$ and all $t\in [t_{p}^{m},t_{p+1}^{m}]$
we have
$$
x_{m}'(t)=y_{p}^{m}+(t-t_{p}^{m})z_{p}^{m},\quad
x_{m}''(t)=z_{p}^{m}\in F(\tau (t_{p}^{m})x_{m},y_{p}^{m})+\frac{1}{m}
B(0,1).
$$
Then from (ii) and (v) of lemma \ref{lem3.1} we get
\begin{equation}
\begin{aligned}
\|x_{m}'(t)\|
&\leq \|y_{p}^{m}\|+h_{p+1}^{m}\|z_{p}^{m}\|\leq \|y_{0}\|+r+T(M+1)  \\
&\leq \|y_{0}\|+r+\frac{r}{4},\quad \forall t\in [0,T]
\end{aligned} \label{e12}
\end{equation}
and
\begin{equation}
\|x_{m}''(t)\|\leq M+\frac{1}{m}\leq
M+1,\quad \forall t\in [0,T].  \label{e13}
\end{equation}
Then the sequences $(x_{m})$ and $(x_{m}')$ are equicontinuous in
$C([0,T],\mathbb{R}^n)$.
Applying Ascoli-Arzela theorem, there is a subsequence of $(x_{m}),$denoted
again by $(x_{m})$, and an absolutely continuous function
$x:[0,T]\to \mathbb{R}^n\,$with absolutely continuous derivative
$x'$ such that $(x_{m})$ converges uniformly to $x$ on $[0,T]$ and
$(x_{m}')$ converges uniformly to $x'$ on $[0,T]$ and
$(x_{m}'')$ converges weakly to $x''$ in
$L^{2}([0,T],\mathbb{R}^n)$. Furthermore, since all the functions $x_{m}$
equal $\varphi _{0}$ on $[-\sigma,0]$, we can say that $x_{m}$ converges
uniformly to $x$ on $[-\sigma,T]$ where $x=\varphi _{0}$ on $[-\sigma,0]$.

Now, for each $t\in [0,T]$ and each $m\geq 1$, let
$\delta_{m}(t)=t_{p}^{m}$, $\theta _{m}(t)=t_{p+1}^{m}$,
if $t\in ]t_{p}^{m},t_{p+1}^{m}]$ and $\delta _{m}(0)=\theta _{m}(0)=0$. For
$t\in ]t_{p}^{m},t_{p+1}^{m}]$ we get
\begin{align*}
x_{m}''(t) &= z_{p}^{m}\in F(\tau (t_{p}^{m})x_{m},y_{p}^{m})+
\frac{1}{m}B(0,1). \\
&= F(\tau (\delta _{m}(t))x_{m},x_{m}'(t_{p}^{m}))+\frac{1}{m}
B(0,1).
\end{align*}
Thus for all $m\geq 1$ and a.e. on $[0,T]$,
\begin{equation}
x_{m}''(t)\in F(\tau (\delta _{m}(t))x_{m},x_{m}'
(\delta _{m}(t)))+\frac{1}{m}B(0,1)  \label{e14}
\end{equation}
Also, for all $m\geq 1$ and all $t\in [0,T]$,
\begin{gather}
\tau (\theta _{m}(t))x_{m}\in B_{\sigma }(\varphi _{0},r)\cap K_{0} \label{e15}
\\
x_{m}(t)\in B(\varphi _{0}(0),r)  \label{e16}\\
x_{m}(\theta _{m}(t))\in P(x_{m}(\delta _{m}(t)))\subseteq K  \label{e17}
\end{gather}

\noindent\textbf{Claim:}
 For each $t\in [0,T],\lim_{m\to \infty } \tau (\theta _{m}(t))x_{m}=\tau (t)x$
in $C([-\sigma,0],\mathbb{R}^n)$.
Let $t\in [0,T]$. then
\begin{align*}
&\|\tau (\theta _{m}(t))x_{m}-\tau (t) x\|_{\sigma } \\
&\leq \|\tau (\theta _{m}(t))x_{m}-\tau (
t)x_{m}\|_{\sigma }+\|\tau (t)x_{m}-\tau (t)x\|_{\sigma } \\
&\leq \sup_{-\sigma \leq s\leq 0}\|x_{m}(\theta
_{m}(t)+s)-x_{m}(t+s)\|+\|\tau (t)x_{m}-\tau (t)x\|_{\sigma }
\\
&\leq \sup_{-\sigma \leq s_{1}\leq s_{2}\leq T\,,
|s_{2}-s_{1}|\leq \frac{1}{m}}
\|x_{m}(s_{2})-x_{m}(s_{1})\|+\|\tau (t)x_{m}-\tau (t)x\|_{\sigma }
\\
&\leq \sup_{-\sigma \leq s_{1}\leq s_{2}\leq 0,\,
|s_{2}-s_{1}|\leq \frac{1}{m}}
\|x_{m}(s_{2})-x_{m}(s_{1})\| \\
&\quad +\sup_{-\sigma \leq s_{1}\leq 0\leq s_{2}\leq T,\,
|s_{2}-s_{1}|\leq \frac{1}{m}}
\|x_{m}(s_{2})-x_{m}(s_{1})\| \\
&\quad +\sup_{0\leq s_{1}\leq s_{2}\leq T,\,
|s_{2}-s_{1}|\leq \frac{1}{m}}
\|x_{m}(s_{2})-x_{m}(s_{1})\|+\|\tau (t)x_{m}-\tau (t)x\|_{\sigma}
\\
&\leq \sup_{-\sigma \leq s_{1}\leq s_{2}\leq 0 \,,
|s_{2}-s_{1}|\leq \frac{1}{m}}
\|\varphi _{0}(s_{2})-\varphi _{0}(s_{1})\| \\
&\quad +\sup_{-\sigma \leq s_{1}\leq 0,\,
|s_{1}|\leq \frac{1}{m}}
\|x_{m}(0)-x_{m}(s_{1})\|
+\sup_{0\leq s_{2}\leq T,\,
|s_{2}|\leq \frac{1}{m}}
\|x_{m}(s_{2})-x_{m}(0)\| \\
&\quad+\sup_{0\leq s_{1}\leq s_{2}\leq T,\,
|s_{2}-s_{1}|\leq \frac{1}{m}}
\|x_{m}(s_{2})-x_{m}(s_{1})\|+\|\tau (t)x_{m}-\tau (t)x\|_{\sigma} \\
&\leq 2\sup_{-\sigma \leq s_{1}\leq s_{2}\leq 0,\,
|s_{2}-s_{1}|\leq \frac{1}{m}}
\|\varphi _{0}(s_{2})-\varphi _{0}(s_{1})\| \\
&\quad +2\sup_{0\leq s_{1}\leq s_{2}\leq T,\,
|s_{2}-s_{1}|\leq \frac{1}{m}}
\|x_{m}(s_{2})-x_{m}(s_{1})\|+\|\tau (t)x_{m}-\tau (t)x\|_{\sigma}\,.
\end{align*}
Using the continuity of $\varphi _{0}$, the fact that $(x_{m}')$ is
uniformly bounded, the uniform convergence of $(x_{m})$ towards $x$ and the
preceding estimate, we get
\begin{equation*}
\lim_{m\to \infty } \|\tau (\theta _{m}(t))x_{m}-\tau (t)x\|_{\sigma }=0.
\end{equation*}
Similarly, for each $t\in [0,T]$,
$\lim_{m\to \infty } \tau (\delta _{m}(t))x_{m}=\tau (t)x$ in
$C([-\sigma,0],\mathbb{R}^n)$. Also, since
$\lim_{m\to \infty }\delta _{m}(t)=t$
and $(x_{m}'')$ is uniformly bounded, then
\begin{equation}
\lim_{m\to \infty }x_{m}'(\delta _{m}(t))=x'(t)\quad
\forall t\in [0,T].
\label{e18}
\end{equation}
Thus by the upper semicontinuity of $F$, and by \eqref{e12}, we obtain
\begin{equation}
x''(t)\in \overline{Co}F(\tau (t)x,x'(t))\subseteq
\partial V(x'(t))\text{a.e. on }[0,T].  \label{e19}
\end{equation}

Our aim now is proving the  relation
\begin{equation}
x''(t)\in F(\tau (t)x,x'(t))\quad \text{a.e. on }[0,T].
\label{e20}
\end{equation}
Since $F$ is upper semicontinuous with closed values, then by
\cite[prop. 1.1.2]{a1}, the graph of $F$ is closed in
$[0,T]\times \mathbb{R}^n\times \mathbb{R}^n$.
So, if we prove that the sequence $(x_{m}'')$ has
a subsequence converges strongly point wise to $x''$ then
the relation \eqref{e14} assures that the relation \eqref{e20} is true.

In order to show that $(x_{m}'')$ has a subsequence
converges strongly point wise to $x''$, we note that the
condition (H2) and property (ii) of Lemma \ref{lem3.1} give
\begin{equation}
z_{p}^{m}-w_{p}^{m}\in F(\tau (t_{p}^{m})x_{m},y_{p}^{m})\subseteq \partial
V(y_{p}^{m})=\partial V(x_{m}'(t_{p}^{m})), \label{e21}
\end{equation}
for $p=0,1,2,\dots,\nu _{m}-2$.

From the definition of the subdifferential of  $V$, for
$p=0,1,2,\dots,\nu _{m}-2$, we have
\begin{align}
V(x_{m}'(t_{p+1}^{m}))-V(x_{m}'(t_{p}^{m}))
&\geq \langle z_{p}^{m}-w_{p}^{m},x_{m}'(t_{p+1}^{m})-x_{m}'(t_{p}^{m})\rangle
\notag \\
&= \langle z_{p}^{m}-w_{p}^{m},
\int_{t_{p}^{m}}^{t_{p+1}^{m}} x_{m}''(s)\,ds\rangle  \notag \\
&= \langle z_{p}^{m},z_{p}^{m}(t_{p+1}^{m}-t_{p}^{m})\rangle-
\langle w_{p}^{m}, \int_{t_{p}^{m}}^{t_{p+1}^{m}}  x_{m}''(s)\,ds\rangle \notag \\
&= h_{p+1}^{m}\|z_{p}^{m}\|^{2}
-\langle w_{p}^{m}, \int_{t_{p}^{m}}^{t_{p+1}^{m}} x_{m}''(s)ds\rangle \notag\\
&= \int_{t_{p}^{m}}^{t_{p+1}^{m}} \Vert x_{m}''(s)\Vert ^{2}ds
-\langle w_{p}^{m},\int_{t_{p}^{m}}^{t_{p+1}^{m}} x_{m}''(s)ds
\rangle  \label{e22}
\end{align} 
Analogously,
\begin{equation}
\begin{aligned}
V(x_{m}'(T))-V(x_{m}'(t_{\nu _{m}-1}^{m}))
&\geq \langle z_{\nu _{m}-1}^{m}-w_{\nu _{m}-1}^{m},
\int_{t_{\nu_{m}-1}^{m}}^T  x_{m}''(s)\,ds \rangle \\
&= \int_{t_{\nu _{m}-1}^{m}}^T \|x_{m}''(s)\|^{2}\,ds
-\langle w_{\nu _{m}-1}^{m},\int_{t_{\nu _{m}-1}}^T
x_{m}''(s)\,ds\rangle
\end{aligned} \label{e23}
\end{equation}
By adding the $\nu _{m}-1$ inequalities from \eqref{e22}
and the inequality \eqref{e23}, we get
\begin{equation}
\begin{aligned}
&V(x_{m}'(T))-V(x_{m}'(0)) \\
&= V(x_{m}'(T))-V(x_{m}'(t_{\nu _{m}-1}^{m}))
+V(x_{m}'(t_{\nu _{m}-1}^{m}))-V(x_{m}'(t_{\nu_{m}-2}^{m}))
+\dots \\
&\quad +V(x_{m}'(t_{1}^{m}))-V(x_{m}'(0))  \\
&\geq \int_0^T \|x_{m}''(s)\|^{2}\,ds
-\sum_{p=0}^{\nu _{m}-2} \langle w_{p}^{m},
\int_{t_{p}^{m}}^{t_{p+1}^{m}} x_{m}''(s)ds \rangle
- \langle  w_{\nu _{m}-1}^{m},\int_{t_{\nu _{_{m}}-1}^{m}}^T
x_{m}''(s)\,ds \rangle
\end{aligned} \label{e24}
\end{equation}
Now,
\begin{align*}
&\sum_{p=0}^{\nu _{m}-2}|\langle w_{p}^{m},
\int_{t_{p}^{m}}^{t_{p+1}^{m}} x_{m}''(s)\,ds\rangle|
+|\langle w_{\nu _{m}-1}^{m},\int_{t_{\nu _{_{m}}-1}^{m}}^T
x_{m}''(s)\,ds\rangle|\\
&\leq \sum_{p=0}^{\nu _{m}-2}\|w_{p}^{m}\|(M+1)
(t_{p+1}^{m}-t_{p}^{m})+\|w_{\nu _{m}-1}^{m}\|(M+1)(T-t_{\nu
_{m}-1}^{m})\\
&\leq \frac{T(M+1)}{m}
\end{align*}
Hence, by passing to the limit as $m\to \infty $ in \eqref{e24}
we obtain
\begin{equation}
V(x_{m}'(T))-V(y_{0})\geq \lim_{m\to \infty }
\sup \int_0^T \|x_{m}''(s) \|^{2}ds.  \label{e25}
\end{equation}
On the other hand from  relation \eqref{e19} and   \cite[Lemma 3.3]{b2},
we obtain
$$
\frac{d}{dt}V(x'(t))=\|x''(t)\|^{2},\quad\text{a.e.  on }[0,T]\,.
$$
Thus, $V(x'(T))-V(x'(0))=\int_0^T \|x''(s)\|^{2}ds$, which yields
directly that
\begin{equation}
V(x'(T))-V(y_{0})=\int_0^T \|x''(s)\|^{2}ds  \label{e26}
\end{equation}
Therefore, by \eqref{e25} and \eqref{e26}, we get
\begin{equation}
\int_0^T \Vert x''(s)\Vert
^{2}ds\geq \limsup_{m\to \infty } \int_0^T \|x_{m}''(s)\|^{2}\,ds  \label{e27}
\end{equation}
Since $(x_{m}'')$ converges weakly to $x''$ in
$L^{2}([0,T],\mathbb{R}^n)$. hence
\begin{equation}
\int_0^T \|x''(s)\|^{2}\,ds\leq
\liminf_{m\to \infty }\int_0^T \|x_{m}''(s)\|^{2}\,ds  \label{e28}
\end{equation}
By \eqref{e27} and \eqref{e28}, we obtain
$$
\lim_{m\to \infty } \int_0^T  \|x_{m}''(s)\|^{2}\,ds=
\int_0^T \|x''(s)\|^{2}\,ds,
$$
 this means that the sequences $(x_{m}'')$ converges strongly to
$x''$ in $L^{2}([0,T],\mathbb{R}^n)$. Consequently there is
a subsequence of $(x_{m}'')$, denoted again by
$(x_{m}'')$, converges point wise to $x''$.
From the facts that the graph of $F$ is closed, $\tau (t)x_{m}$ converges
uniformly to $\tau (t)x$, $(x_{m}')$ converges uniformly to
$x'$ and $(x_{m}'')$ converges point wise to
$x''$the relation$(18)$ is proved.

It remains to prove the following two properties:
\begin{enumerate}
\item $(x(t),x'(t))\in K\times \Omega $, for all $t\in [0,T]$.

\item $x(s)\in P(x(t))$ for all $t,s\in [0,T]$ and $t\leq s$.
\end{enumerate}
To prove the first property we note that the property (iii) of l
Lemma \ref{lem3.1} implies that
$x_{m}(\delta _{m}(t))\in \overline{B(\varphi _{0}(0),r)}\cap K$
and $x_{m}'(\delta _{m}(t))\in \overline{B(y_{0},r)}\cap \Omega $.
Since $\lim_{m\to \infty } x_{m}(\delta _{m}(t))=x(t)$ and
$\lim_{m\to \infty } x_{m}'(\delta_{m}(t))=x'(t)$ then
$x(t)\in \overline{B(\varphi _{0}(0),r)}\cap K$
and $x'(t)\in \overline{B(y_{0},r)}\cap \Omega $.

  To prove the second property, let $t,s\in [0,T]$
be such that $t\leq s$.  Then for $m$ large enough, we can find
$p,\,q\in \{0,1,2,\dots,\nu _{m}-2\}$ such that $p>q$,
$t\in [t_{q}^{m},t_{q+1}^{m}]$ and $s\in [t_{p}^{m},t_{p+1}^{m}]$.
Assume that $j=p-q$. Using property (v) of Lemma \ref{lem3.1} and condition
(H4) we get
$$
P(x_{m}(t_{p}^{m}))\subseteq P(x_{m}(t_{p-1}^{m}))\subseteq
P(x_{m}(t_{p-2}^{m}))\subseteq \dots \subseteq P(x_{m}(t_{q}^{m})).
$$
This implies
$P(x_{m}(\delta _{m}(s)))\subseteq P(x_{m}(\delta_{m}(t)))$.
Since $x_{m}(\delta _{m}(s))\in P(x_{m}(\delta _{m}(s)))$,
it follows that $P(x_{m}(\delta _{m}(t)))$ and hence the second
property is proved.
\end{proof}

AS an example, let $K=\mathbb{R}$ and $P(x)=[x,\infty )$.
Then $x\preceq y$ if and only if
$y\in P(x)$; i.e.,  if and only if $x\leq y$.
Then the solution obtained above is monotone in the usual sense.

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\end{document}