\documentclass[twoside]{article} \usepackage{amssymb,amsmath} \pagestyle{myheadings} \markboth{\hfil A viability result \hfil EJDE--2002/76} {EJDE--2002/76\hfil Vasile Lupulescu \hfil} \begin{document} \title{\vspace{-1in} \parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 76, pp. 1--12. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A viability result for second-order differential inclusions % \thanks{\emph{Mathematics Subject Classifications:} 34G20, 47H20. \hfil\break\indent {\em Key words:} second-order contingent set, subdifferential, viable solution. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted December 9, 2001. Revised March 26, 2002. Published August 20, 2002.} } \date{} \author{Vasile Lupulescu} \maketitle \begin{abstract} We prove a viability result for the second-order differential inclusion $x''\in F(x,x'),\quad (x(0), x'(0))=(x_0,y_0)\in Q:=K\times \Omega,$ where $K$ is a closed and $\Omega$ is an open subsets of $\mathbb{R}^m$, and is an upper semicontinuous set-valued map with compact values, such that $F(x,y) \subset \partial V(y)$, for some convex proper lower semicontinuous function $V$. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} Bressan, Cellina and Colombo \cite{b2} proved the existence of local solutions to the Cauchy problem $x'\in F(x) ,\quad x(0) =\xi \in K,$ where $F$ is an upper semicontinuous, cyclically monotone, and compact valued multifunction. While Rossi \cite{r2} proved a viability result for this problem. On the other hand, for the second order differential inclusion $x''\in F(x,x') ,\quad x(0) =x_0,\quad x'(0)=y_0,$ existence results were obtained by many authors \cite{a1,a4,c1,c2,m1,s1}). In \cite{l1}, existence results are proven for the case when $F(.,.)$ is an upper semicontinuous set-valued map with compact values, such that $F(x,y) \subset \partial V(y)$ for some convex proper lower semicontinuous function $V$. The aim of this paper is to prove a viability result for the second-order differential inclusion $x''\in F(x,x') ,\quad (x(0) ,x'(0) )=(x_0,y_0)\in Q:=K\times \Omega ,$ where $K$ is a closed and $\Omega$ is an open subsets of $\mathbb{R}^m$, and $F:Q\subset \mathbb{R}^{2m}\to 2^{\mathbb{R}^m}$ is an upper semicontinuous set-valued map with compact values, such that $F(x,y) \subset \partial V(y)$, for some convex proper lower semicontinuous function $V$. \section{Preliminaries and statement of main result} Let $\mathbb{R}^m$ be the m-dimensional Euclidean space with scalar product $\langle .,.\rangle$ and norm $\| .\|$. For $x\in \mathbb{R}^m$ and $\varepsilon >0$ let $B_{\varepsilon }(x)=\{y\in \mathbb{R}^m:\| x-y\| <\varepsilon \}$ be the open ball, centered at $x$ with radius $\varepsilon$, and let $\overline{B}_{\varepsilon }(x)$ be its closure. Denote by $B$ the open unit ball $B=\{ x\in \mathbb{R}^m:\| x\| <1\}$. For $x\in \mathbb{R}^m$ and for a closed subsets $A\subset \mathbb{R}^m$ we denote by $d(x,A)$ the distance from $x$ to $A$ given by $d(x,A) =\inf \{ \| x-y\| :y\in A\} .$ Let $V:\mathbb{R}^m\to \mathbb{R}$ be a proper lower semicontinuous convex function. The multifunction $\partial V:\mathbb{R}^m\to 2^{\mathbb{R}^m}$ defined by $\partial V(x)=\{\xi \in \mathbb{R}^m:V(y)-V(x)\geqslant \langle \xi ,y-x\rangle ,\; \forall y\in \mathbb{R}^m\}$ is called subdifferential (in the sense of convex analysis) of the function $V$. We say that a multifunction $F:\mathbb{R}^m\to 2^{\mathbb{R}^m}$ is upper semicontinuous if for every $x\in \mathbb{R}^m$ and every $\varepsilon >0$ there exists $\delta >0$ such that $F(y) \subset F(x) +B_{\varepsilon }(0) , \quad \forall y\in B_{\delta }(x) .$ This definition of the upper semicontinuous multifunction is less restrictive than the usual (see Definition 1.1.1 in \cite{a3} or Definition 1.1 in \cite{d1}). Actually such a property is called ($\varepsilon$, $\delta$)-upper semicontinuity (see Definition 1.2 in \cite{d1}) and it is only equivalent to the upper semicontinuity for compact-valued multifunctions (see Proposition 1.1 in \cite{d1}). For $K\subset \mathbb{R}^m$ and $x\in K$ denote by $T_{K}(x)$ the Bouligand's contingent cone of $K$ at $x$, defined by $T_{K}(x)=\big\{v\in \mathbb{R}^m:\liminf_{h\to 0+} \frac{d(x+hv,K)}{h}=0\big\}.$ For $K\subset \mathbb{R}^m$ and $(x,y)\in K\times \mathbb{R}^m$ we denote by $T_{K}^{(2)}(x,y)$ the second-order contingent set of $K$ at $(x,y)$ introduced by Ben-Tal \cite{b1} and defined by $T_{K}^{(2)}(x,y)=\big\{v\in \mathbb{R}^m:\liminf_{h\to 0+} \frac{d(x+hy+% \frac{h^{2}}{2}v,K)}{h^{2}/2}=0\big\}.$ We remark that if $T_{K}^{(2)}(x,y)$ is non-empty then, necessarily, $y\in T_{K}(x)$. Moreover (see \cite{a4}, \cite{c2}, \cite{m1}), if $F$ is upper semicontinuous with compact convex values and if $x:[ 0,T] \to \mathbb{R% }^m$ is a solution of the Cauchy problem $x''\in F(x,x'),\quad x(0)=x_0,\quad x'(0)=y_0,$ such that $x(t)\in K$, $\forall t\in \lbrack 0,T]$, then $(x(t),x'(t))\in \mathop{\rm graph}(T_{K}),\quad \forall t\in [0,T),$ hence, in particular, $(x_0,y_0)\in \mathop{\rm graph}(T_{K})$. For a multifunction $F:Q:=K\times \Omega \subset \mathbb{R}^{2m} \to 2^{% \mathbb{R}^m}$ and for any $(x_0,y_0)\in \mathop{\rm graph}(T_{K})$ we consider the Cauchy problem $$x''\in F(x,x') ,\quad (x(0) ,x'(0) )=(x_0,y_0)\in Q \label{1}$$ under the following assumptions: \begin{enumerate} \item[(H1)] $K$ is a closed and $\Omega$ and open subset of $\mathbb{R}^{m}$, such that $Q:=K\times \Omega \subset \mathop{\rm graph}(T_{K})$ \item[(H2)] $F$ is an upper semicontinuous compact valued multifunction such that $F(x,y)\cap T_{K}^{(2)}(x,y)\neq \varnothing ,\quad \forall (x,y)\in Q;$ \item[(H3)] There exists a proper convex and lower semicontinuous function $V:$ $\mathbb{R}^{m}\to \mathbb{R}$ such that $F(x,y)\subset \partial V(y),\quad \forall (x,y)\in Q.$ \end{enumerate} \paragraph{Remark.} A convex function $V:\mathbb{R}^m\to \mathbb{R}$ is continuous in the whole space $\mathbb{R}^m$ (Corollary 10.1.1 in \cite{r1}) and almost everywhere differentiable (Theorem 25.5 in \cite{r1}). Therefore, (H3) strongly restricts the multivaluedness of $F$. \paragraph{Definition.} %1. By viable solution of the problem \eqref{1} we mean any absolutely continuous function $x:[0,T]\to \mathbb{R}^m$ with absolutely continuous derivative $x'$ such that $x(0)=x_0$, $x(0)=y_0$, \begin{gather*} x''(t)\in F(x(t),x'(t))\quad a.e.\text{ on }[0,T], \\ (x(t),x'(t))\in Q \quad \forall t\in [0,T]. \end{gather*} Our main result is the following:\medskip \begin{theorem} \label{thm1} If $F:Q\subset \mathbb{R}^{2m}\to 2^{\mathbb{R}^m}$ and $V:\mathbb{R}^m\to \mathbb{R}$ satisfy assumptions (H1)--(H3), then then for every $(x_0,y_0)\in Q$ there exist $T>0$ and $x:[0,T]\to \mathbb{R}^m$, a viable solution of the problem \eqref{1}. \end{theorem} \section{Proof of the main result} We start this section with the following technical result, which will be used to prove the main result. \begin{lemma} \label{lm2} Assume $Q=K\times \Omega \subset \mathbb{R}^{2m}$ satisfies (H1), $F:Q\to 2^{\mathbb{R}^m}$ satisfies (H2), $Q_0\subset Q$ is a compact subset and $(x_0,y_0)\in Q_0$. Then for every $k\in \mathbb{N}^{\ast }$ there exist $h_{k}^{0}\in (0,\frac{1}{k}]$ and $u_{k}^{0}\in \mathbb{R}^m$ such that \begin{gather*} x_0+h_{k}^{0}y_0+\frac{(h_{k}^{0}) ^{2}}{2}u_{k}^{0}\in K, (x_0,y_0,u_{k}^{0})\in \mathop{\rm graph}(F)+\frac{1}{k}(B\times B\times B). \end{gather*} \end{lemma} \paragraph{Proof.} Let $(x,y)\in Q$ be fixed. Since by (H2), $F(x,y)\cap T_{K}^{(2)}(x,y)\neq \emptyset$, there exists $v=v_{(x,y)}\in F(x,y)$ such that $\liminf_{h\to 0_+} \frac{d(x+hy+\frac{h^{2}}{2}v,K)}{h^{2}/2}=0.$ Hence, for every $k\in \mathbb{N}^{\ast }$ there exists $h_{k}=h_{k}(x,y)\in (0,% \frac{1}{k}]$ such that $$d(x+h_{k}y+\frac{h_{k}^{2}}{2}v,K)<\frac{h_{k}^{2}}{4k}. \label{2}$$ By the continuity of the map $(a,b)\to d(a+h_{k}b+\frac{h_{k}^{2}}{2}% v,K)$ it follows that $N(x,y)=\big\{(a,b):d(a+h_{k}b+\frac{h_{k}^{2}}{2}v,K)<\frac{h_{k}^{2}}{4k}% \big\}$ is an open set and, by \eqref{2}, it contains $(x,y)$. Then there exists $r:=r(x,y) \in (0,\frac{1}{k})$ such that $B_{r}(x,y)\subset N(x,y)$. Since $Q_0$ is compact there exists a finite subset $\{ (x_{j},y_{j})\in Q:1\leqslant j\leqslant m\}$ such that $Q_0\subset \bigcup_{j=1}^mB_{r_{j}}(x_{j},y_{j}).$ We set $h_0(k) :=\min \{ h_{k}(x_{j},y_{j}) :j\in \{ 1,\dots ,m\} \} .$ Since $(x_0,y_0)\in Q_0$, there exists $j_0\in \{1,2,\dots m\}$ such that $$(x_0,y_0)\in B_{r_{j_0}}(x_{j_0},y_{j_0})\subset N(x_{j_0},y_{j_0}). \label{3}$$ Denote by $h_{k}^{0}:=h_{k}(x_{j_0},y_{j_0})$ and remark that, by \eqref{2} and \eqref{3}, one has $h_{k}^{0}\in [ h_0(k) ,\frac{1}{k}]$ and there exists $z_0\in K$ such that we have that $\frac{d(x_0+h_{k}^{0}y_0+\frac{(h_{k}^{0}) ^{2}}{2}v_0,z_0)}{(h_{k}^{0}) ^{2}/2}\leqslant \frac{d(x_0+h_{k}y_0+\frac{(h_{k}^{0}) ^{2}}{2}v_0,K)}{% (h_{k}^{0}) ^{2}/2}+\frac{1}{2k}<\frac{1}{k},$ hence $$\| \frac{z_0-x_0-h_{k}^{0}y_0}{(h_{k}^{0}) ^{2}/2}-v_0\| <\frac{1}{k}. \label{4}$$ Let $u_{k}^{0}:=\frac{z_0-x_0-h_{k}^{0}y_0}{(h_{k}^{0}) ^{2}/2}.$ Then $x_0+h_{k}^{0}y_0+\frac{(h_{k}^{0}) ^{2}}{2}u_{k}^{0}\in K.$ By \eqref{4} and \eqref{3} we get successively: \begin{gather*} \| u_{k}-v_0\| <\frac{1}{k}, \\ d((x_0,y_0) ,(x_{j_0},y_{j_0})) \leqslant r_{j_0}<\frac{1}{k}, \end{gather*} hence $(x_0,y_0,u_{k}^{0})\in \mathop{\rm graph}(F) +\frac{1}{k}(B\times B\times B)$. \hfill$\square$ \paragraph{Proof of Theorem \ref{thm1}} Let $(x_0,y_0)\in Q\subset \mathop{\rm graph}(T_{K})$. Since $\Omega \subset \mathbb{R}^m$ is an open subset, there exist $r>0$ such that $\overline{B}% _{r}(y_0)\subset \Omega$. We set $Q_0:=\overline{B}_{r}(x_0,y_0)\cap (K\times \overline{B}_{r}(y_0)).$ Since $Q_0$ is a compact set, by the upper semicontinuity of $F$ and Proposition 1.1.3 in \cite{a3}, we have that $F(Q_0):=\bigcup_{(x,y)\in Q_0}F(x,y)$ is a compact set, hence there exists $M>0$ such that: $\sup \{ \| v\| :v\in F(x,y),\quad (x,y)\in Q_0\} \leqslant M.$ Let $$T=\min \big\{\frac{r}{2(M+1) },\sqrt{\frac{r}{M+1}}, \frac{r}{2(\| y_0\| +1)}% \big\}. \label{5}$$ We shall prove the existence of a viable solution of the problem \eqref{1} defined on the interval $[0,T]$. Since $(x_0,y_0)\in Q_0$ then, by Lemma \ref {lm2}, there exist $h_{k}^{0}\in [ h_0(k) ,\frac{1}{k}]$ and $u_{k}^{0}\in \mathbb{R}^m$ such that $x_0+h_{k}^{0}y_0+\frac{1}{2}(h_{k}^{0})^{2}u_{k}^{0}\in K$ and $(x_0,y_0,u_{k}^{0})\in \mathop{\rm graph}(F) +\frac{1}{k}(B\times B\times B)$. Define \begin{aligned} x_{k}^{1}:=&x_0+h_{k}^{0}y_0+\frac{1}{2}(h_{k}^{0})^{2}u_{k}^{0}; \\ y_{k}^{1}:=&y_0+h_{k}^{0}u_{k}^{0}. \end{aligned} \label{6} We remark that if \$h_{k}^{0}