\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 $$\label{eP} \begin{gathered} x''(t)\in F(\tau (t)x,x'(t)),\quad \text{a.e. on }[0,T] \\ x(t)=\varphi _{0}(t),\quad \forall t\in [-\sigma,0] \\ x'(0)=y_{0} \\ x(t)\in P(x(t))\subset K,\quad \forall t\in [0,T] \\ x(s)\preceq x(t)\quad \text{whenever }0\leq s\leq t\leq T \end{gathered}$$ 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 $B(x,r)=\{y\in \mathbb{R}^n:\|y-x\|0$ 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}=00$such that for all$t,s\in [-\sigma,0]$we have $$|t-s|<\mu \Longrightarrow \|\varphi _{0}(t)-\varphi _{0}(s)\|<\frac{r}{4}. \label{e2}$$ Put $$T=\min \Big\{\mu,\frac{r}{4(M+1)},\frac{r}{8(\|y_{0}\|+1)},\sqrt{\frac{r}{4(M+1) }}\Big\} \label{e3}$$ 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,$\|z_{0}^{m}\| \leq M+\frac{1}{2m}q. From the relation \eqref{e5} we have \begin{align*} &\|x_{p}^{m}-x_{q}^{m}\|\\ &=\|(\sum_{j=0}^{p-1} h_{j+1}^{m})y_{0}+\frac{1}{2}\sum_{j=0}^{p-1} (h_{j+1}^{m})^{2}\,z_{j}^{m} +\sum_{i=0}^{p-2} \sum_{j=i+1}^{p-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m}\, \\ &\quad -(\sum_{j=0}^{q-1} h_{j+1}^{m})y_{0} -\frac{1}{2} \sum_{j=0}^{q-1} (h_{j+1}^{m})^{2}\,z_{j}^{m} -\sum_{i=0}^{q-2} \sum_{j=i+1}^{q-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m}\|. \\ &=\|(\sum_{j=q}^{p-1} h_{j+1}^{m})y_{0}+\frac{1}{2} \sum_{j=q}^{p-1}(h_{j+1}^{m})^{2}z_{j}^{m} +\sum_{i=0}^{q-1} \sum_{j=q}^{p-1} h_{i+1}^{m}h_{j+1}^{m}z_{i}^{m}\|. \\ &\leq \sum_{j=q}^{p-1}h_{j+1}^{m})\|y_{0}\| +\frac{1}{2}\sum_{j=q}^{p-1} (h_{j+1}^{m})^{2}\|z_{j}^{m}\| +\sum_{i=q-1}^{p-2}\sum_{j=i+1}^{p-1} h_{i+1}^{m}h_{j+1}^{m}\|z_{i}^{m}\|. \\ &\leq \|y_{0}\|(t_{p}^{m}-t_{q}^{m})+\frac{1}{2} (M+1)(t_{p}^{m}-t_{q}^{m})^{2}+(M+1) \sum_{j=q}^{p-1} h_{q}^{m}h_{j+1}^{m} \\ &\quad +(M+1)\sum_{j=q+1}^{p-1}h_{q+1}^{m}h_{j+1}^{m} +\dots + (M+1)\sum_{j=p-2}^{p-1} h_{p-2}^{m}h_{j+1}^{m}. \\ &=\|y_{0}\|(t_{p}^{m}-t_{q}^{m})+\frac{1}{2} (M+1)(t_{p}^{m}-t_{q}^{m})^{2}+(M+1)h_{q}^{m}(t_{p}^{m}-t_{q}^{m}) \\ &\quad +(M+1)h_{q+1}^{m}(t_{p}^{m}-t_{q}^{m})+\dots +(M+1)h_{p-2}^{m}(t_{p}^{m}-t_{q}^{m}). \\ &=\|y_{0}\|(t_{p}^{m}-t_{q}^{m})+\frac{1}{2}(M+1)(t_{p}^{m}-t_{q}^{m})^{2} \\ &\quad +(M+1)(t_{p}^{m}-t_{q}^{m})(h_{q}^{m}+h_{q+1}^{m}+\dots +h_{p-2}^{m}). \\ &=\|y_{0}\|(t_{p}^{m}-t_{q}^{m})+\frac{1}{2} (M+1)(t_{p}^{m}-t_{q}^{m})^{2}+(M+1)(t_{p}^{m}-t_{q}^{m})^{2}. \end{align*} Since the sequence\{t_{p}^{m}\}_{p\geq 1}$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 $$x_{p}^{m}\in P(x_{p}^{m})\cap \overline{B(\varphi _{0}(0),r)}\subseteq K, \label{e6}$$ 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 asp\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 $$d(x_{\alpha }^{m}+hy_{\alpha }^{m}+\frac{h^{2}}{2}u_{\alpha }^{m},P(x_{\alpha }^{m}))\leq \frac{h^{2}}{16m}. \label{e7}$$ 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$t_{p}^{m}p_{2}$. This gives that if$z\in \mathbb{R}^n$, then $$d(z,P(x_{p}^{m}))\leq d(z,P(x_{\alpha }^{m}))+\frac{h^{2}}{16m},\forall p>p_{2}. \label{e11}$$ Now let$p>\max (p_{1},p_{2}). By \eqref{e7}--\eqref{e11}, we have \begin{align*} &d(x_{p}^{m}+hy_{p}^{m}+\frac{h^{2}}{2}u_{p}^{m},P(x_{p}^{m}))\\ &\leq d(x_{p}^{m}+hy_{p}^{m}+\frac{h^{2}}{2}u_{p}^{m},x_{\alpha }^{m} +hy_{\alpha }^{m}+\frac{h^{2}}{2}u_{\alpha }^{m})\\ &\quad +d\big(x_{\alpha }^{m}+hy_{\alpha }^{m}+\frac{h^{2}}{2}u_{\alpha }^{m},P(x_{\alpha }^{m})\big)+\frac{h^{2}}{16m} \\ &\leq \|x_{p}^{m}-x_{\alpha }^{m}\|+h\|y_{p}^{m}-y_{\alpha }^{m}\|+\frac{ h^{2}}{2}\|u_{p}^{m}-u_{\alpha }^{m}\| +\frac{h^{2}}{8m}+\frac{h^{2}}{16m} \\ &\leq \frac{h^{2}}{24m}+\frac{h^{2}}{24m}+\frac{h^{2}}{24m} +\frac{h^{2}}{16m}+\frac{h^{2}}{16m}\\ &=\frac{h^{2}}{8m}+\frac{h^{2}}{8m}=\frac{h^{2}}{4m}. \end{align*} Thush\in H_{p}^{m}$, 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{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} and $$\|x_{m}''(t)\|\leq M+\frac{1}{m}\leq M+1,\quad \forall t\in [0,T]. \label{e13}$$ 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 allm\geq 1$and a.e. on$[0,T]$, $$x_{m}''(t)\in F(\tau (\delta _{m}(t))x_{m},x_{m}' (\delta _{m}(t)))+\frac{1}{m}B(0,1) \label{e14}$$ 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 $$\lim_{m\to \infty }x_{m}'(\delta _{m}(t))=x'(t)\quad \forall t\in [0,T]. \label{e18}$$ Thus by the upper semicontinuity of$F$, and by \eqref{e12}, we obtain $$x''(t)\in \overline{Co}F(\tau (t)x,x'(t))\subseteq \partial V(x'(t))\text{a.e. on }[0,T]. \label{e19}$$ Our aim now is proving the relation $$x''(t)\in F(\tau (t)x,x'(t))\quad \text{a.e. on }[0,T]. \label{e20}$$ 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 $$z_{p}^{m}-w_{p}^{m}\in F(\tau (t_{p}^{m})x_{m},y_{p}^{m})\subseteq \partial V(y_{p}^{m})=\partial V(x_{m}'(t_{p}^{m})), \label{e21}$$ 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{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} By adding the\nu _{m}-1inequalities from \eqref{e22} and the inequality \eqref{e23}, we get \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} 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 asm\to \infty $in \eqref{e24} we obtain $$V(x_{m}'(T))-V(y_{0})\geq \lim_{m\to \infty } \sup \int_0^T \|x_{m}''(s) \|^{2}ds. \label{e25}$$ 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 $$V(x'(T))-V(y_{0})=\int_0^T \|x''(s)\|^{2}ds \label{e26}$$ Therefore, by \eqref{e25} and \eqref{e26}, we get $$\int_0^T \Vert x''(s)\Vert ^{2}ds\geq \limsup_{m\to \infty } \int_0^T \|x_{m}''(s)\|^{2}\,ds \label{e27}$$ Since$(x_{m}'')$converges weakly to$x''$in$L^{2}([0,T],\mathbb{R}^n)$. hence $$\int_0^T \|x''(s)\|^{2}\,ds\leq \liminf_{m\to \infty }\int_0^T \|x_{m}''(s)\|^{2}\,ds \label{e28}$$ 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. \begin{thebibliography}{00} \bibitem{a1} J. P. Aubin and A. 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