Existence of Solutions of Monotone--Type Nonconvex \\ Differential Inclusions
The notion of weak cyclic monotonicity of set-valued maps generalizing the cyclyc monotonicity is introduced. The existence of solutions of differential inclusions with compact, upper semi-continuous, not necessarily convex right-hand sides in $\mathbb{R}^n$ is proved for weakly cyclic monotone right-hand sides. %%% !!!!! Under a strengthened monotonicity-type condition the
1307.2072.tex
\documentclass[11pt]{article}

\textheight 24cm \textwidth 17cm \oddsidemargin -.6cm \headheight
0pt \headsep -.6cm

\usepackage{amssymb}
\usepackage[centertags]{amsmath}
\usepackage{amsthm}
%\pagestyle{empty}

\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{example}{Example}
\newtheorem{definition}{Definition}
\newtheorem{remark}{Remark}

\newcommand{\Graph}{\operatorname{Graph}}
%\newcommand{\co}{\overline{co}\hbox{ }}
\newcommand{\co}{{\rm co}\hbox{ }}
\newcommand{\R}{\mathbb{R}}
\newcommand{\skal}[2]{\langle #1,#2 \rangle}
%%% !!! following new command
\newcommand{\sign}{\operatorname{sign}}

\def\dss{\displaystyle}


%%% !!! \title{Existence of Solutions of Monotone--Type Nonconvex \\ Differential Inclusions}
\title{Weak Cyclic Monotonicity and\\Existence of Solutions of Differential Inclusions }
\author{Elza Farkhi\thanks{School of Mathematical Sciences, Sackler Faculty of Exact
Sciences, Tel Aviv University, 69978 Tel Aviv, Israel, email: 
elza@post.tau.ac.il} 
%\quad Tzanko Donchev\thanks{Department of Mathematics, "Al. I.
%Cuza" University, Ia\c{s}i 700506, Romania, email: tzankodd@gmail.com}
 %\quad Robert Baier\thanks{University of Bayreuth, Department of Mathematics,
%Chair of Applied Mathematics, 95440 Bayreuth, Germany, email: 
%robert.baier@uni-bayreuth.de}  
}

\begin{document}

\maketitle

%%% !!! \centerline{Dedicated to the 70-th birthday of Acad. P. Kenderov}
%\centerline{Dedicated to the 70-th birthday of~Acad.~Petar~Kenderov}

\begin{abstract}
The notion of weak cyclic monotonicity of set-valued maps generalizing the cyclyc
monotonicity is introduced. The 
existence of solutions of differential inclusions with compact, upper semi-continuous,  
not necessarily convex right-hand sides 
in $\mathbb{R}^n$ is proved for weakly cyclic monotone 
right-hand sides.
%%% !!!!! Under a strengthened monotonicity-type condition the 
 
\end{abstract}
%
\begin{quote}
   \textbf{Key words:} differential inclusion, nonconvex right-hand side, existence of solutions,
      weakly monotone map, cyclic monotone map
\end{quote}
%
\begin{quote}
   \textbf{2010 Mathematics Subject Classification:} 34A60, 34A12, 47H05%, 34A36
%%% 34A60: Differential inclusions 
%%% 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions
%%% 26B35:  Functions of several variables: Special properties of functions of several variables, Hölder conditions, etc.
%%% only perhaps:
%%%   34A36: Discontinuous equations
%%%   26A16:  Functions of one variable: Lipschitz (Hölder) classes
%%%   47H05           Monotone operators and generalizations
%%% 
\end{quote}

\section{Introduction} 

We investigate the differential inclusion
\begin{equation}\label{1} 
 \dot x(t)\in F(x(t)),\quad x(0)= x_0\in \mathbb{R}^n,\quad t\in I=[0,T], 
\end{equation} 
where the set-valued mapping $F$ is upper semi-continuous, with non-empty compact, 
not necessarily convex values in $\mathbb{R}^n$. 

Among the relevant results on existence of solutions of such inclusions,
%%% !!!!! are not so many. 
we should first mention the case of upper semi-continuous maps with convex compact values
(see e.g. \cite[Chapter 2]{D}). 
%The convexity of $F(x)$ plays an essential role in the proof
%transfering the weak convergence into 
In particular, the classical existence theorem for right-hand side
which is the negation of a maximal monotone 
map  \cite[Sec.~3.2, Theorem~1]{AubCel:84}.
Maximal monotone set-valued maps are almost everywhere single-valued \cite{Zar:73,Kend:74},
 and it is easy to see that at the points where they are not single-valued, they
are upper semi-continuous with convex values.
Other important existence results for differential inclusions with non-convex right-hand sides
 are the results of Filippov \cite{FIL:zametki_71} in finite dimensions and De Blasi, Pianigiani
\cite{DB_P:91} in Banach spaces for  continuous %or ?Lipschitz 
$F$. The continuity may replace the monotonicity and convexity, as is shown there.
%, and of Hermes \cite{Her:71}, 
%who relaxed the Lipschitz continuity of $F$ to continuity with bounded variation.
%%% !!! The result of \cite{BreCelCol:89} is for upper-semi-continuous and cyclic monotone map $F$,

An important existence result for upper semi-continuous maps,
with not necessarily convex values, is established in \cite{BreCelCol:89},
where the condition of cyclic monotonicity is imposed. 
We remind that the values of a cyclic monotone map are subsets of the values of the subdifferential
of a proper convex function \cite{Rock:ConvAnal}.
Generally,  cyclic monotonicity is stronger than just monotonicity, but  
in $\R^1$ they are equivalent.

In \cite{KrRibTsa:07} the notion of colliding on the set of discontinuities of $F$ is investigated 
and geometric conditions to avoid it or to escape of this set are studied.
%In \cite{KrRibTsa:07} the phenomenon of
%``colliding'' on the set of discontinuities of $F$ is studied and conditions to avoid 
%or to escape from this set are investigated.

Existence of solutions is established in \cite{FarDonBai:14} 
uder  a weak  componentwise monotonicity condition and a standard growth condition.
This componentwise monotonicity condition is equivalent to 
 the strengthened one-sided Lipschitz (SOSL) condition \cite{LemVel:98} with constant zero
imposed on the negation of $F$, $-F(\cdot)$. 
In one dimension the latter condition is equivalent to
the  one-sided Lipschitz (OSL) property  in the sense of \cite{DonFar:98} of \ $-F$ with constant zero.
In the  proof a limiting procedure for the Euler polygons and their velocities is used. 
The method of Euler polygonal approximations is a widely exploited tool for deriving existence, 
approximation, well-posedness and stability properties of solutions of
differential equations and inclusions (see e.g. \cite{ADonFar:89,DonFar:98,Veliov:97}), 
as well as of variational and control problems \cite{DonLem:92, %ADontchevBook:83, 
Mord:93,Mord:bookII}. A typical proof of the existence of solutions
in the upper-semicontinuous case  uses the Arzela-Ascoli theorem applied
to the polygonal solutions and/or Mazur's weak closure theorem for the velocities. The
limit of a convergent sequence of Euler approximants is then a solution of the inclusion with convexified right-hand
side. In \cite{FarDonBai:14} the existence of solutions without convexity is achieved 
%applying the Helly's selection principle (see e.g. \cite[Chap.~10]{KF})  
%for the velocities of the Euler polygonal functions.
deriving monotonicity in the time of each coordinate of the velocities from the
weak  componentwise monotonicity condition. 
%which is the key for the existence proof. 
The Helly's selection principle (see e.g. \cite[Chap.~10]{KF}) is applied there instead of the Mazur's 
weak closure theorem to obtain strong compactness of the set of velocities.
%In fact, the weak componentwise monotonicity condition implies 


Here we formulate a condition called ``weak cyclic monotonicity'', which generalizes %both 
the cyclic monotonicity %and the weak componentwise monotonicity condition of \cite{FarDonBai:14},
and we prove the existence of solutions under this condition. 
In  the proof we construct a cyclic monotone subinclusion of the given inclusion
and apply the existence theorem of \cite{BreCelCol:89}.
Let us note that in general %in more than one dimension
the weak cyclic monotonicity condition is stronger than the weak monotonicity condition.
The question whether one can prove existence for weakly monotone maps
(maps having OSL negation with constant zero)
 in more than one dimension is still open.

% We give examples that show that our condition, although simple, does not imply monotonicity,
                                                                                                                                                                                                                                                                                             %hence does not imply cyclical monotonicity. 


\section{Weak cyclic monotonicity}

First we introduce some notation. For notions used in the paper, but not 
explicitly defined here we refer the reader to \cite{D, Rock:ConvAnal, AubCel:84} . 

Let $u,v\in \mathbb{R}^n$. We denote by $\skal{u}{v}$ the scalar product  of the vectors $u,v$ 
and by $|v|$ the Euclidean norm of $v$.
%and by $v_j$ its $j$--th coordinate, i.e.~$v= (v_1,v_2,\ldots,v_n)$. 
%Denote by $\mathbb{B}$ the unit ball in $\mathbb{R}^n$. 
For a bounded set $A\subset\R^n$,
we denote $\|A\|=sup\{|a|:a\in A\}$. The support function of the set $A$ in direction $v$
is denoted by $\delta^*(v,A)=\sup_{a\in A} \skal{v}{a}$.
%, the supporting face in direction $v$ is
%\[
%Y(v,A)=\{a\in A : \skal{v}{a}=\delta^*(v,A) \}.
%\]
The classical  convex subdifferential of a convex function $f$ at the point $x$ is denoted by $\partial f(x)$.

We use the fact that the set-valued map $F$ from $\mathbb{R}^n$ to a compact set in $\mathbb{R}^n$
is upper semicontinuous (USC) iff its graph is closed \cite[Section 1.1]{AubCel:84}.
%, i.e. if
% for every converging sequence $x_k\to x^\infty$ and every converging
%$v_k\in F(x_k)$, the limit of the velocities sequence satisfies $lim_{k\to\infty}v_k\in F(x^\infty)$.

We recall that the map $F:\R^n\rightrightarrows \mathbb{R}^n$ is monotone if for every $x,y\in\mathbb{R}^n,v_x\in F(x),v_y\in F(y)$,
\begin{equation}\label{mon} 
 \left<x- y, v_x- v_y\right> \geq 0 . 
\end{equation}
We call $F$  \textbf{\emph{weakly monotone}} if for every $x,y\in\mathbb{R}^n$ and every 
$v_x\in F(x)$ \emph{there exists} $v_y\in F(y)$ such that \eqref{mon} holds.

We also recall that the (monotone) set-valued function $F: \mathbb{R}^n\rightrightarrows \mathbb{R}^n$
is called 
\textbf{\emph{cyclic monotone}} if for every natural $m$, every $x_0,x_1,...,x_m\in\mathbb{R}^n$,
and every $v_i\in F(x_i), \ i=0,...m$,
\begin{equation}
\label{mc} 
 \skal{x_m- x_0}{v_m}\geq \sum_{i=1}^{m} \skal{x_i- x_{i-1}}{v_{i-1}}.
\end{equation}
It is well-known that monotone, hence all cyclyc monotone maps, are almost everywhere single-valued 
(in the sense of measure \cite{Zar:73} and in Baire category sense \cite{Kend:74}). 
The weaker property we define below 
may hold for everywhere non-single-valued maps.
%, as for constant maps $F(x)=A,\ x\in \mathbb{R}^n$, for instance.

In the next definitions and in all considerations below we fix the point $x_0$ to be the one of  \eqref{1}
and also fix an arbitrary $v_0\in F(x_0)$.
\begin{definition}
\label{Def_WECM_seq} 
%We call the finite sequence $\{(x_i,v_i)\}_{i=0}^k\subset Graph(F)$ %\mathbb{R}^n\times \mathbb{R}^n$ 
%an \textbf{\emph{Euler sequence}} for $F$ with stepsize $h$ %\in (0,\frac{T}{k}]$ if 
%\[
%x_{i+1}=x_{i}+hv_{i}, \ v_i\in F(x_i),\ i=0,...,k.
%\]
The finite %Euler 
sequence %for $F$ with stepsize $h>0$ , 
$\{(x_i,v_i)\}_{i=0}^k\subset Graph(F)$ is a \textbf{cyclic monotone sequence (CM sequence)}
%with stepsize $h$ (WCM(h))}} in the time interval of length $(k+1)h$ 
if \eqref{mc} holds for every $m=1,2,...k$. 
%In this case the Euler polygon (the piecewise-linear interpolant
%of the points $\{(ih,x_i)\}_{i=0}^{k+1}$ in $[0,(k+1)h]\times\mathbb{R}^n$)  is also called 
%weakly cyclic monotone with step $h$ (WCM(h)).
The one-term sequence $\{(x_0,v_0)\}\subset Graph(F)$ is  a CM sequence %(h)
 by definition. 
\end{definition}
It is clear from this definition that if one cuts several last terms
of the CM sequence $\{(x_i,v_i)\}_{i=0}^k$, the shorter
sequence $\{(x_i,v_i)\}_{i=0}^m$ for $m\le k$ is also CM.

Now we give a recursive definition of a weakly cyclic monotone map
based on cyclic monotone sequences on its graph.
\begin{definition}
\label{Def_WECM} 
The %dynamical system \eqref{1} 
set-valued map $F: \mathbb{R}^n\rightrightarrows \mathbb{R}^n$
is called \textbf{weakly cyclic monotone (WCM)} %in time $T$ 
if 
for every natural $m$, any CM %Euler 
sequence $\{(x_i,v_i)\}_{i=0}^{m-1}\subset Graph(F)$ %for $F$ with stepsize $h\in(0,\frac{T}{m-1})$,   
and  every $x_{m}\in\mathbb{R}^n$ %$x_{m}=x_{m-1}+h v_{m-1}$,
there exists $v_{m}\in F(x_{m})$ such that the sequence
$\{(x_i,v_i)\}_{i=0}^{m}$ is a CM %(h) %Euler 
sequence.
\end{definition}
In other words, $F$ is weakly cyclic monotone if every CM sequence  may be continued 
on the graph of $F$,  %can be continued (extended) 
preserving its cyclic monotonicity. 

Clearly, the weak cyclic monotonicity generalizes the cyclic monotonicity. 
%, but also the weak componentwise monotonicity condition, 
%for which is hard to find the common with the cyclic monotonicity.
%The WCM property is clearly weaker than the usual cyclic monotonicity of multifunctions in which
%the expression ``there exists'' in the above definition has to be replaced by ``for every''.
%Every constant non-single-valued map  is an example of weakly cyclic monotone map which is not
%monotone, hence not cyclic monotone.
On the other side, the weak cyclic monotonicity of set-valued maps is stronger than the weak monotonicity defined above.

\begin{remark}
\label{cond_WCM2}
The following condition %is equivalent to 
implies the WCM condition:
\begin{equation}
\label{WMC2}
\delta^*(x_m- x_0,F(x_m))\geq \sum_{i=1}^{m} \delta^*(x_i- x_{i-1},F(x_{i-1})),
\end{equation}
for every finite sequence $\{x_i\}_{i=0}^m$. Indeed,
%from Definition \ref{Def_WECM}  it is easy 
to see that 
the  condition \eqref{WMC2} implies weak cyclic monotonicity, 
we take an arbitrary CM %Euler 
sequence $\{(x_i,v_i)\}_{i=0}^{m-1}\subset Graph(F)$ %for $F$ with stepsize $h\in(0,\frac{T}{m-1})$,   
and   $x_{m}\in\mathbb{R}^n$ 
and chose $v_{m}\in F(x_{m})$ such that 
$\skal{x_m- x_0}{v_m}= \delta^*(x_m- x_0,F(x_m))$. Then by \eqref{WMC2}, 
$$\skal{x_m- x_0}{v_m}= \delta^*(x_m- x_0,F(x_m))\geq \sum_{i=1}^{m} \delta^*(x_i- x_{i-1},F(x_{i-1}))
\geq \sum_{i=1}^{m} \skal{x_i- x_{i-1}}{v_{i-1}},$$
which implies that the sequence
$\{(x_i,v_i)\}_{i=0}^{m}$ is a CM %(h) %Euler 
sequence.

On the other hand, 
the condition \eqref{WMC2} seems stronger than the WCM condition,
since in Definition \ref{Def_WECM} there exists $v_m$ satisfying \eqref{mc} 
only for a CM sequence $\{x_i,v_i\}_{i=1}^{m-1}$, while \eqref{WMC2}
means the existence of $v_m$ satisfying \eqref{mc} for every sequence 
$\{x_i,v_i\}_{i=1}^{m-1}$ in the graph of $F$.
\end{remark}

%\begin{lemma}
%??? If $F$ is weakly componentwise monotone, then it is WCM.
%\end{lemma}

%\begin{definition}
%\label{Def_WCM}
%%\textbf{\emph{Weak Cyclic Monotonicity}}: 
%The multifunction $F: \mathbb{R}^n\rightrightarrows \mathbb{R}^n$
%is \textbf{\emph{weakly cyclic monotone}} if for every natural $m$, every 
%$x_0,x_1,...,x_m\in\mathbb{R}^n$ and every $v_i\in F(x_i), \ i=0,...m-1$ 
%there exists $v_m\in F(x_m)$ such that 
%\eqref{mc} holds.
%\end{definition}
%
%An equivalent condition of the weak cyclic monotonicity is
%\begin{equation}
%\label{WMC2}
%\delta^*(x_m- x_0,F(x_m))\geq \sum_{i=1}^{m} \delta^*(x_i- x_{i-1},F(x_{i-1})).
%\end{equation}

%(?? for this may be in the definition for every $v_i$ in the support face should be for every $v_i in F(x_i)$ ??).

%and $Y(x_i-x_0,F(x_i))$ is replaced by the larger set $F(x_i)$.

%??/This definition is a generalization of the weak componentwise monotonicity?
\begin{remark}
\label{rem_wcm2}
Given a CM sequence $\{(x_i,v_i)\}_{i=0}^{m-1}$ and $x_m\in\R^n$, 
%with the next term $(x_m,v_m)$ on the graph of $F$, it is sufficient to 
if one can choose $v_m\in F(x_m)$ satisfying
\begin{equation}
\label{wcm2}
\skal{x_m- x_0}{v_m-v_{m-1}}\ge 0,
\end{equation}
then the sequence $\{(x_i,v_i)\}_{i=0}^{m}$ is CM.

Indeed, \eqref{wcm2} may be written as
\[
\skal{x_m- x_0}{v_m} \ge \skal{x_m- x_{m-1}}{v_{m-1}} + \skal{x_{m-1}- x_0}{v_{m-1}}.
\]
From this, one can prove 
 %together with the inequality \eqref{mc} taken with $m-1$ replacing $m$, implies
\eqref{mc} %, i.e. the weak cyclic monotonicity of $\{(x_i,v_i)\}_{i=0}^{m}$.
using a simple inductive argument.
\end{remark}
\begin{example}
Here we give a simple example of a WCM multifunction which is not cyclic monotone.
%\begin{enumerate}
%\item 
Given a compact set $A$, the constant map $F:\R^n\rightrightarrows\R^n$ defined by 
$F(x)\equiv A,\ x\in\R^n$, is WCM. Indeed, for a given CM %Euler 
sequence $\{(x_i,v_i)\}_{i=0}^{m-1}$ %for $F$ with stepsize $h\in(0,\frac{T}{m-1})$,   
and for given $x_m\in\mathbb{R}^n$ %$x_{m}=x_{m-1}+h v_{m-1}$,
one can choose $v_{m}=v_{m-1}\in A$ so as to fulfill \eqref{wcm2}.
If $A$ is not a singleton, then $F$ is not cyclic monotone since
cyclic monotone (and all monotone) maps are a.e. single-valued, as we have mentioned above.
%\item If $F_\alpha, \alpha\in A$ are WCM maps, then their union $F(x)=\cup_{\alpha\in A} F_\alpha(x)$
%is WCM. In particular, the union of cyclic monotone maps is weakly cyclic monotone.
%Indeed, to see this, 
%for the continuation of a CM sequence $\{(x_i,v_i)\}_{i=0}^{m-1}\subset Graph(F)$ with the next term
%$(x_m,v_m)$ on the graph of $F$, 
%if $v_{m-1}\in F_\alpha(x_{m-1})$ for some given $\alpha\in A$,
%. %to choose $v_m\in F(x_m)$ such that \eqref{wcm2} is satisfied,
%it is sufficient to choose $v_{m}\in F_\alpha(x_{m})$ satisfying \eqref{wcm2} - tova ne e osigureno!. % for $F_\alpha$.
%Thus, the CM sequence will remain on the same branch $F_\alpha$ of the graph of $F$.
%\item
%If $F$ satisfies the weak componentwise monotonicity
%\end{enumerate}
%Clearly, also the union of arbitrary many WCM maps is also WCM.
\end{example}


\section{The Main Result}

We impose the following assumptions in order to prove the existence of solutions: 
\vskip 0.5em 
\textbf{A1.} $F: \mathbb{R}^n\rightrightarrows \mathbb{R}^n$ has
compact, nonempty values %, is bounded on bounded sets 
and is upper semicontinuous.
%\vskip 0.5em 
%\textbf{A2.} \textbf{\emph{Linear growth condition}}
%There exist constants $A$ and $B$ such that $\| F(x)\|\leq A+ B|x|$
%for any $x\in \mathbb{R}^n$. 

%The following lemma is a corollary of the Gronwall inequality and
%{\bf A2}, and  its proof 
%is given in~\cite[Remark~3.1]{DonFar:2000} (see also \cite{DonFar:98}). 
%\begin{lemma} \label{l1}
% Under \textbf{A1, A2} there exist constants $L$ and $M$ such that $|x(t)|\leq L$ 
%and $|\dot x(t)|\leq M$ for every solution $x(\cdot)$ of 
%\[
%\dot x(t)\in \co F(x(t)+ \mathbb{B})+ \mathbb{B},\ x(0)= x_0. 
% \]
%\end{lemma}
\vskip 0.5em 
%\textbf{A3.} There exist constants $L$ and $M$ such that $|x(t)|\leq L$ 
%and $|\dot x(t)|\leq M$ for every solution $x(\cdot)$ of 
%\[
%\dot x(t)\in \co F(x(t)+ \mathbb{B})+ \mathbb{B},\ x(0)= x_0. 
% \]
%A sufficient condition for \textbf{A2} is the linear growth condition
%(see e.g. \cite{DonFar:2000,DonFar:98}). 

\textbf{A2.} The map $F$ is weakly cyclic monotone.
%
\begin{remark}
\label{rem_USC_cls_gph}
We recall that the map $F: \mathbb{R}^n\rightrightarrows \mathbb{R}^n$  is locally
bounded if for any $x_0\in\R^n$ there is $\varepsilon>0$ such that
$\{y\in F(x): |x-x_0|<\varepsilon\}$ is bounded. It is well-known (see e.g. \cite{Smirnov:02})
that a locally bounded map $F: \mathbb{R}^n\rightrightarrows \mathbb{R}^n$ is
upper semicontinuous if and only if it has closed graph. 
\end{remark}

The main result of this paper is:
\begin{theorem}\label{ThM}
 Under the conditions \textbf{A1, A2} %A3
there exists $T>0$ such that 
 the differential inclusion (\ref{1}) has a 
solution on $[0,T]$. 
\end{theorem} 
%\begin{proof}
 To proof the theorem, we use the following result proved in \cite{BreCelCol:89}.
\begin{theorem}
\label{th_BCC}
 If $F: \mathbb{R}^n\rightrightarrows \mathbb{R}^n$ has non-empty compact images
and is upper semicontinuous and cyclic monotone, then there exists $T>0$ such that 
 the differential inclusion (\ref{1}) has a 
solution on $[0,T]$. 
\end{theorem}

We reduce our considerations to the last theorem by the following claim.
\begin{theorem}\label{l1}
 If $F: \mathbb{R}^n\rightrightarrows \mathbb{R}^n$, 
satisfies {\bf A1,A2},  then for every fixed $x_0\in\mathbb{R}^n,v_0\in F(x_0)$ 
the following map is upper semicontinuous and cyclic monotone, with nonempty
compact images:
%\[
%G(x)=\bigcap_{N\in\mathbb{N}}\bigcap_{h<\frac{T}{N}} G_h(x),
%\]
%where
\begin{equation}
\label{def_G}
%G(x)=Y(x-\bar{x},F(x))
G(x)=\left\{ v\in F(x) : \skal{x-x_0}{v} \ge \sup\Bigl\{ \skal{x-x_m}{v_m}+ \sum_{i=1}^m \skal{x_i-x_{i-1}}{v_{i-1}} 
                                                          % : \{(x_i,v_i)\}_{i=0}^m \hbox{is WCM(h)}  
																													\Bigr\} \right\},
\end{equation}
where the supremum is taken on all natural $m$ and all CM %(?(h) ??Euler) 
sequences $\{(x_i,v_i)\}_{i=0}^m$. %(? satisfying \ $mh<T$).

%the  inclusion \eqref{1} has a solution.
\end{theorem}
\begin{proof} 
Let $x_0,v_0\in F(x_0)$ be fixed as above. 
 For a given finite CM sequence $S=\{(x_i,v_i)\}_{i=0}^m$
and for $x\in\R^n$ we denote 
\[
g(S,x)=\skal{x-x_m}{v_m}+\sum_{i=1}^m \skal{x_i-x_{i-1}}{v_{i-1}}.
\]
Define the function
\begin{equation}
\label{eq_def_g}
g(x)=\sup_{S}g(S,x),%\Bigl\{ \skal{x-x_m}{v_m}+\sum_{i=1}^m \skal{x_i-x_{i-1}}{v_{i-1}} \Bigr\}, 
%+\skal{x_m-x_{m-1}}{v_{m-1}}+...+\skal{x_1-x_{0}}{v_{0}} \},
\end{equation}
where the supremum is taken on all $m\in \mathbb{N}$ and all finite CM %(?(h) ??Euler) 
sequences $S=\{(x_i, v_i)\}_{i=0}^m \subset \Graph(G)$.
Clearly, $g$ is convex, as a supremum of affine functions. Further, 
\[
g(x_0)=\sup_{S}
       \Bigl\{ \skal{x_0-x_m}{v_m}+\sum_{i=1}^m \skal{x_i-x_{i-1}}{v_{i-1}} \Bigr\},
%\skal{x_m-x_{m-1}}{v_{m-1}}+...+\skal{x_1-x_{0}}{v_{0}} \},
\]
and %since by the definition of $G$, \ $\skal{x_0-x_m}{v_m}=-\delta^*(x_m-x_0,F(x_m))$,
using \eqref{mc} we get that $g(x_0)\le 0$. Taking $m=0$, we easily see
that $g(x_0)=0$, which implies that $g$ is proper.

Next, we prove that $G(x)$ is non-empty for every $x\in\mathbb{R}^n$.
Let us note that in the above notation
\[
G(x)=\left\{ v\in F(x) : \skal{x-x_0}{v} \ge  g(x)  \right\}.
\]
Given a finite CM sequence $S=\{(x_i,v_i)\}_{i=0}^m$ and 
setting $x_{m+1}=x$, by \textbf{A2} there is $v_{m+1}\in F(x)$, which
we denote by $v(S,x)$, such that
\[
\skal{x-x_0}{v(S,x)}\ge g(S,x). 
\]
Since $g(x)$ is a supremum,
we can choose a sequence $\{S^k\}_{k=1}^\infty$ such that 
\[
g(x)=\lim_{k\to\infty}g(S^k,x)\le \liminf_{k\to\infty}\skal{x-x_0}{v(S^k,x)}.
\]
Since $F(x)$ is compact, the limit of every convergent subsequence of $\{v(S^k,x)\}_{k=1}^\infty$ is in $F(x)$.
Denoting a limit of a converging subsequence of $\{v(S^k,x)\}_{k=1}^\infty$  by $v(x)\in F(x)$,
we get that $g(x)\le \skal{x-x_0}{v(x)}$, thus $v(x)\in G(x)$.
 Thus, \  $G(x)\neq\emptyset$ for any $x\in\mathbb{R}^n$.
Clearly, $G(x)$ is compact as a closed subset of the compact $F(x)$.

Now, by \cite[Proposition 1.1.3]{AubCel:84},
every upper semicontinuous map with compact values defined on a compact domain has a compact range,
therefore $F$ is locally bounded. Hence $G$ is locally bounded as a sub-mapping of $F$ and
according to Remark \ref{rem_USC_cls_gph}, in order to prove that $G$ is upper semicontinuous, 
it is enough to show that its graph is closed.
 %first show that $\co G(x)$ is upper semicontinuous. 
Take the sequences $x^k\to x^\infty$ and $v^k\to v^\infty$ with $v^k\in G(x^k)$.
Passing to the limit for $k\to\infty$ in the inequalities of \eqref{def_G} and taking in account that $F$ has closed graph,
we get that $v^\infty\in G(x^\infty)$, hence $G$ is USC.
%then $\lim\sup \delta^*(x_k-\bar{x},F(x_k))\le 
%\delta^*(x^*-\bar{x},F(x^*))$ since the support function is continuous with respect to 
%its first argument and $F$ is upper semicontinuous.  Then, since $G(x)=\co G(x)\cap F(x)$
%and the intersection of two USC maps is USC, we obtain that $G$ is USC.

To show that $G$ is cyclic monotone, we use the criterion that a map $G$ is cyclic monotone 
if and only if $G(x)$ is a subset of the subdifferential of a proper convex function \cite{Rock:ConvAnal}.
Our argument is similar to the one in Theorem 24.8 in \cite{Rock:ConvAnal}.

We now prove that $G(x)\subset \partial g(x)$ for any point $x\in \mathbb{R}^n$.
 For this, it is sufficient to show that for all $(x,v)\in\Graph(G)$ and for any $y\in\R^n$ and $\alpha<g(x)$,
\[
g(y)-\alpha >\skal{y-x}{v}.
\]
Indeed, since $g(x)$ is a supremum and $\alpha<g(x)$, there is
a finite CM sequence $S=\{(x_i,v_i)\}_{i=0}^k$,  %are points $(x_i,v_i)\in\Graph(G),i=1,...,k$, 
such that
\begin{equation}
\label{ineq_alpha}
\alpha < g(S,x)=\skal{x-x_k}{v_k}+\sum_{i=1}^k \skal{x_i-x_{i-1}}{v_{i-1}}.
%\skal{x-x_k}{v_k}+\skal{x_k-x_{k-1}}{v_{k-1}}+...+\skal{x_1-x_{0}}{v_{0}}.
\end{equation}
Now, adding the point $x_{k+1}=x,v_{k+1}=v\in G(x)$ to the sequence $S$, we obtain 
the CM sequence $\hat{S}=\{(x_i,v_i)\}_{i=0}^{k+1}$, and by the definition of $g$,
\[
g(y) \ge g(\hat{S},y)= \skal{y-x}{v}+\skal{x-x_k}{v_k}+\sum_{i=1}^k \skal{x_i-x_{i-1}}{v_{i-1}}.
\]
By \eqref{ineq_alpha}, 
\[
g(y) > \skal{y-x}{v}+ \alpha,
\]
which completes the proof.
\end{proof}
We obtain Theorem \ref{ThM} applying  Theorem \ref{th_BCC} for the inclusion
\[
\dot x(t) \in G(x(t)), \qquad x(0)=x_0, ,\quad t\in I=[0,T],
\]
with $G(x)$ defined in Theorem \ref{l1}.

\vspace{0.5em}
\begin{corollary}
Theorem \ref{ThM} holds if one replaces \textbf{A2} by \eqref{WMC2} 
for every finite sequence $\{x_i\}_{i=0}^m$.
\end{corollary}
%\textbf{Some open questions}: In one dimension, does WCM coincide with weak monotonicity?
%Is it weaker than the weak componentwise monotonicity?
\begin{remark}
As we mentioned above, in one dimension the weak monotonicity
coincides with the weak componentwise monotonicity for which the existence
of solutions is proved in \cite{FarDonBai:14}. We conjecture that there exist
solutions for weakly monotone right-hand sides also in higher dimensions.
\end{remark}

\vspace{0.5em}

\textbf{Acknowledgements.} 
The research  is 
partly supported by the Austrian Science Foundation (FWF) under grant
P 26640-N25 and by Minkowski Minerva 
Center for Geometry at Tel-Aviv University. 
The author is grateful to % V. Veliov,  and
%for the warm hospitality and fruitful discussions during
%her visit at the %%Research Unit "Operations Research and Control Systems" of the
%Institute of Mathematical Methods in Economics,
%Vienna University of Technology, and to 
R. Baier, T. Donchev and especially to an anonymous referee for their valuable %his/their 
remarks.

\begin{thebibliography}{l1}

\bibitem{AubCel:84}
Aubin J.-P., A.~Cellina.
\newblock {\em Differential Inclusions}, 
%%% volume 264 of {\em Grundlehren der mathe\-ma\-ti\-schen Wissenschaften}.
% \newblock Berlin--Heidelberg--New York--Tokyo, Springer-Verlag, 1984.
\newblock Berlin, Springer-Verlag, 1984.

\bibitem{BreCelCol:89}
Bressan A., A. Cellina, G. Colombo. 
 Upper semicontinuous differential inclusions without convexity. 
\emph{Proc.\ Amer.\ Math.\ Soc.}~106, 1989, no.~3, 771--775.

\bibitem{DB_P:91}
De Blasi, F. S., G.~Pianigiani. 
Non-convex-valued differential inclusions in Banach spaces. 
\emph{J. Math. Anal. Appl.}~ 157, 1991, no.~2, 469–-494.

\bibitem{D} 
Deimling K. 
{\em Multivalued Differential Equations},
%%% volume 1 of \emph{de Gruyter Series in Nonlinear Analysis and Applications}.
%%% Berlin--New York, De Gruyter, 1992.
Berlin, De Gruyter, 1992.

\bibitem{DonFar:98}
Donchev T., E.~Farkhi.
 Stability and {E}uler approximation of one-sided {L}ipschitz
 differential inclusions.
{\em SIAM J.~Control Optim.}~36, 1998, no.~2, 780--796.

\bibitem{DonFar:2000}
Donchev T., E.~Farkhi.
Approximations of one-sided Lipschitz differential inclusions
with discontinuous right-hand sides.
In: {\em Calculus of Variations and Differential Equations ({H}aifa, 1998)},
(eds.~A.~Ioffe, S.~Reich, I.~Shafrir),
Boca Raton, FL, Chapman \& Hall/CRC, 2000, 101--118.

\bibitem{ADonFar:89}
Dontchev A.~L., E.~M. Farkhi. Error Estimates for Discretized Differential Inclusions.
{\em Computing} 41 (1989), 349--358. 

\bibitem{DonLem:92}
Dontchev A. L., F.~Lempio.
 Difference methods for differential inclusions: {A} survey.
{\em SIAM Rev.}~34, 1992, no.~2, 263--294.

\bibitem{FarDonBai:14}
Farkhi E., Donchev T., Baier R..
Existence of Solutions for
Nonconvex Differential Inclusions of Monotone Type.
{\em Comptes rendus de l'Academie bulgare des Sciences},~67,  2014, no.~3, 323--330.

\bibitem{FIL:67}
Filippov A. F. 
 Classical solutions of differential equations with multi-valued right-hand side. 
\emph{SIAM J.~Control}~5, 1967, 609--621.

\bibitem{FIL:zametki_71}
 Filippov, A. F. 
The existence of solutions of multivalued differential equations. (Russian), 
{\em Mat. Zametki}~10, 1971, 307–-313.

\bibitem{Her:71}
Hermes H. 
 On continuous and measurable selections and the existence of solutions of 
 generalized differential equations. 
\emph{Proc.\ Amer.\ Math.\ Soc.}~29, 1971, 535--542.

\bibitem{Kend:74}
Kenderov P.~S. 
 The set-valued monotone mappings are almost everywhere single-valued.
{\em C.~R.\ Acad.\ Bulgare Sci.}~27, 1974, 1173--1175.

\bibitem{KrRibTsa:07}
Krastanov M. I., N. K. Ribarska., Ts. Y. Tsachev. 
 On the existence of solutions to differential inclusions with nonconvex right-hand sides. 
\emph{SIAM J.~Optim.}~18, 2007, no.~3, 733--751.

\bibitem{Lem:93}
Lempio F. 
 Modified {E}uler methods for differential inclusions.
In: {\em Set-Valued Analysis and Differential Inclusions. A Collection
of Papers resulting from a Workshop held in Pamporovo, Bulgaria, September
17--21, 1990} (eds.~A. B. Kurzhanski, V. M. Veliov), 
%%% volume~16 of {\em Progr.\ Systems Control Theory}, 
%%% Boston, MA--Basel--Berlin, 
Boston, MA, %%% --Basel--Berlin, 
Birkh\"auser, 1993, 131--148. 

\bibitem{LemVel:98}
Lempio F., V. Veliov. 
 Discrete approximations of differential inclusions. 
\emph{Bayreuth.\ Math.\ Schr.}~54, 1998, 149--232.

\bibitem{KF}  Kolmogorov A., S. Fomin. \emph{Introductory real analysis}. Dover Publ., 
New York, 1975.
%\bibitem{KF} 
%Kolmogorov A., S. Fomin. 
%\emph{Introductory real analysis}. 
%New York, Dover Publ., 1975.
%{\em Elements of Theory of Functions and Functional 
%Analysis}, FIZMATLIT, Moskow, 2004 (in Russian).

\bibitem{Mord:93}
 Mordukhovich B.~S. Discrete approximations and refined Euler–Lagrange conditions for differential inclusions.
\emph{SIAM J. Control Optim.} 33 (1995), 882–915.

\bibitem{Mord:bookII}
 Mordukhovich B.~S. \emph{Variational Analysis and Generalized Differentiation, II: Applications}. 
 {\em Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences)},  vol. 331. 
\newblock, Springer-Verlag, Berlin, 2006.

\bibitem{Rock:ConvAnal} 
 Rockafellar R. T. 
{\em Convex Analysis}, 
Princeton University Press, Princeton, 1970.

\bibitem{Smirnov:02}
Smirnov G.~V. {\em Introduction to the Theory of Differential Inclusions}, 
AMS, Providence, 2002.

\bibitem{Veliov:97}
 Veliov V. Stability-Like Properties of Differential Inclusions.
{\em Set-Valued Analysis} 5 (1997), 73-–88.

\bibitem{Zar:73}
Zarantonello E.~H. 
Dense single-valuedness of monotone operators.
{\em Israel J.~Math.}~15, 1973, 158--166.

\end{thebibliography}

\end{document}


1307.2072.extracted_bib
\begin{thebibliography}{l1}

\bibitem{AubCel:84}
Aubin J.-P., A.~Cellina.
\newblock {\em Differential Inclusions}, 
%%% volume 264 of {\em Grundlehren der mathe\-ma\-ti\-schen Wissenschaften}.
% \newblock Berlin--Heidelberg--New York--Tokyo, Springer-Verlag, 1984.
\newblock Berlin, Springer-Verlag, 1984.

\bibitem{BreCelCol:89}
Bressan A., A. Cellina, G. Colombo. 
 Upper semicontinuous differential inclusions without convexity. 
\emph{Proc.\ Amer.\ Math.\ Soc.}~106, 1989, no.~3, 771--775.

\bibitem{DB_P:91}
De Blasi, F. S., G.~Pianigiani. 
Non-convex-valued differential inclusions in Banach spaces. 
\emph{J. Math. Anal. Appl.}~ 157, 1991, no.~2, 469–-494.

\bibitem{D} 
Deimling K. 
{\em Multivalued Differential Equations},
%%% volume 1 of \emph{de Gruyter Series in Nonlinear Analysis and Applications}.
%%% Berlin--New York, De Gruyter, 1992.
Berlin, De Gruyter, 1992.

\bibitem{DonFar:98}
Donchev T., E.~Farkhi.
 Stability and {E}uler approximation of one-sided {L}ipschitz
 differential inclusions.
{\em SIAM J.~Control Optim.}~36, 1998, no.~2, 780--796.

\bibitem{DonFar:2000}
Donchev T., E.~Farkhi.
Approximations of one-sided Lipschitz differential inclusions
with discontinuous right-hand sides.
In: {\em Calculus of Variations and Differential Equations ({H}aifa, 1998)},
(eds.~A.~Ioffe, S.~Reich, I.~Shafrir),
Boca Raton, FL, Chapman \& Hall/CRC, 2000, 101--118.

\bibitem{ADonFar:89}
Dontchev A.~L., E.~M. Farkhi. Error Estimates for Discretized Differential Inclusions.
{\em Computing} 41 (1989), 349--358. 

\bibitem{DonLem:92}
Dontchev A. L., F.~Lempio.
 Difference methods for differential inclusions: {A} survey.
{\em SIAM Rev.}~34, 1992, no.~2, 263--294.

\bibitem{FarDonBai:14}
Farkhi E., Donchev T., Baier R..
Existence of Solutions for
Nonconvex Differential Inclusions of Monotone Type.
{\em Comptes rendus de l'Academie bulgare des Sciences},~67,  2014, no.~3, 323--330.

\bibitem{FIL:67}
Filippov A. F. 
 Classical solutions of differential equations with multi-valued right-hand side. 
\emph{SIAM J.~Control}~5, 1967, 609--621.

\bibitem{FIL:zametki_71}
 Filippov, A. F. 
The existence of solutions of multivalued differential equations. (Russian), 
{\em Mat. Zametki}~10, 1971, 307–-313.

\bibitem{Her:71}
Hermes H. 
 On continuous and measurable selections and the existence of solutions of 
 generalized differential equations. 
\emph{Proc.\ Amer.\ Math.\ Soc.}~29, 1971, 535--542.

\bibitem{Kend:74}
Kenderov P.~S. 
 The set-valued monotone mappings are almost everywhere single-valued.
{\em C.~R.\ Acad.\ Bulgare Sci.}~27, 1974, 1173--1175.

\bibitem{KrRibTsa:07}
Krastanov M. I., N. K. Ribarska., Ts. Y. Tsachev. 
 On the existence of solutions to differential inclusions with nonconvex right-hand sides. 
\emph{SIAM J.~Optim.}~18, 2007, no.~3, 733--751.

\bibitem{Lem:93}
Lempio F. 
 Modified {E}uler methods for differential inclusions.
In: {\em Set-Valued Analysis and Differential Inclusions. A Collection
of Papers resulting from a Workshop held in Pamporovo, Bulgaria, September
17--21, 1990} (eds.~A. B. Kurzhanski, V. M. Veliov), 
%%% volume~16 of {\em Progr.\ Systems Control Theory}, 
%%% Boston, MA--Basel--Berlin, 
Boston, MA, %%% --Basel--Berlin, 
Birkh\"auser, 1993, 131--148. 

\bibitem{LemVel:98}
Lempio F., V. Veliov. 
 Discrete approximations of differential inclusions. 
\emph{Bayreuth.\ Math.\ Schr.}~54, 1998, 149--232.

\bibitem{KF}  Kolmogorov A., S. Fomin. \emph{Introductory real analysis}. Dover Publ., 
New York, 1975.
%\bibitem{KF} 
%Kolmogorov A., S. Fomin. 
%\emph{Introductory real analysis}. 
%New York, Dover Publ., 1975.
%{\em Elements of Theory of Functions and Functional 
%Analysis}, FIZMATLIT, Moskow, 2004 (in Russian).

\bibitem{Mord:93}
 Mordukhovich B.~S. Discrete approximations and refined Euler–Lagrange conditions for differential inclusions.
\emph{SIAM J. Control Optim.} 33 (1995), 882–915.

\bibitem{Mord:bookII}
 Mordukhovich B.~S. \emph{Variational Analysis and Generalized Differentiation, II: Applications}. 
 {\em Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences)},  vol. 331. 
\newblock, Springer-Verlag, Berlin, 2006.

\bibitem{Rock:ConvAnal} 
 Rockafellar R. T. 
{\em Convex Analysis}, 
Princeton University Press, Princeton, 1970.

\bibitem{Smirnov:02}
Smirnov G.~V. {\em Introduction to the Theory of Differential Inclusions}, 
AMS, Providence, 2002.

\bibitem{Veliov:97}
 Veliov V. Stability-Like Properties of Differential Inclusions.
{\em Set-Valued Analysis} 5 (1997), 73-–88.

\bibitem{Zar:73}
Zarantonello E.~H. 
Dense single-valuedness of monotone operators.
{\em Israel J.~Math.}~15, 1973, 158--166.

\end{thebibliography}