\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 141, pp. 1--6.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/141\hfil Existence of solutions]
{Existence of solutions for nonconvex functional differential inclusions}

\author[Vasile Lupulescu\hfil EJDE-2004/141\hfilneg]
{Vasile Lupulescu} % in alphabetical order

\address{Vasile Lupulescu \hfill\break
 ``Constantin Br\^{a}ncu\c{s}i'' University of T\^{a}rgu-Jiu\\
 Bulevardul Republicii, nr. 1 \\
 1400 T\^{a}rgu-Jiu, Romania}
\email{vasile@utgjiu.ro}

\date{}
\thanks{Submitted September 24, 2004. Published November 29, 2004.}
\subjclass[2000]{34A60, 34K05, 34K25}
\keywords{Functional differential inclusions; existence result}

\begin{abstract}
 We prove the  existence of solutions for the functional differential
 inclusion  $x'\in F(T(t)x)$, where $F$ is upper semi-continuous,
 compact-valued  multifunction such that $F(T(t)x)\subset \partial V(x(t))$
 on $[0,T]$, $V$ is a proper convex and lower semicontinuous function, and
 $(T(t)x)(s)=x(t+s)$.
\end{abstract}

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


\section{Introduction}

Let $\mathbb{R}^{m}$ be the $m$-dimensional Euclidean space with the norm
$\|\cdot \|$ and the scalar product $\langle \cdot ,\cdot \rangle $.
When $I$ is a segment in $\mathbb{R}$, we denote by
$\mathcal{C}(I,\mathbb{R}^{m})$
the Banach space of continuous functions from $I$ to $\mathbb{R}^{m}$ with
the norm  $\|x(.)\|_{\infty }:=\sup \{\|x(t)\|;t\in I\}$. When $\sigma
$ is a positive number, we put
$\mathcal{C}_{\sigma }:=\mathcal{C}([-\sigma ,0],\mathbb{R}^{m})$ and for
any $t\in [ 0,T]$, $T>0$, we define the operator $T(t)$ from
$\mathcal{C}([-\sigma ,T],\mathbb{R}^{m})$ to $\mathcal{C}_{\sigma }$
as  $(T(t)x)(s):=x(t+s)$, $s\in [-\sigma ,0]$.

Let $\Omega $ be a nonempty subset in $\mathcal{C}_{\sigma }$. For a given
multifunction $F:\Omega \to 2^{\mathbb{R}^{m}}$ we consider the
following functional differential inclusion:
\begin{equation}
x'\in F(T(t)x).  \label{e1.1}
\end{equation}
We recall that  a continuous function $x(.):[-\sigma ,T]\to
\mathbb{R}^{m}$ is said to be a solution of \eqref{e1.1} if $x(.)$ is absolutely
continuous on $[0,T]$, $T(t)x\in \Omega $ for all $t\in [ 0,T]$ and
$x'(t)\in F(T(t)x)$ for almost all $t\in [0,T]$; see \cite{h1}.

The functional differential equation \eqref{e1.1} with $F$ 
single-valued, has been  studied by many authors; for results, references,
and applications, see for example \cite{h2,l1}.

The existence of solutions for the functional differential inclusion 
\eqref{e1.1} was proved by Haddad \cite{h1} when $F$ is upper semicontinuous
with convex compact values. The nonconvex case in Banach space has been
studied by Benchohra and Ntouyas \cite{b1}. The case when $F$ is lower
semicontinuous with compact value has been studied by Fryszkowski \cite{f1}.

In this paper we prove the existence of solutions for functional
differential inclusion \eqref{e1.1} when $F$ is upper
semicontinuous, compact valued multifunction such that
$F(\psi )\subset\partial V(\psi (0))$ for every $\psi \in \Omega $ and
$V$ is a proper convex and lower semicontinuous function.
Our existence result contains Peano's existence theorem as a particular case.
On the other hand, our result may be considered as an extension of the
previous result of Bressan, Cellina and Colombo \cite{b2}.

\section{Preliminaries and statement of the main result}

For $x\in \mathbb{R}^{m}$ and $r>0$ let $B(x,r):=\{y\in \mathbb{R}^{m};\|y-x\|<r\}$
be the open ball centered at $x$ with radius $r$, and let $\overline{B}(x,r)$
be its closure. For $\varphi \in \mathcal{C}_{\sigma }$
let $B_{\sigma }(\varphi ,r):=\{\psi \in \mathcal{C}_{\sigma };\|\psi
-\varphi \|_{\infty }<r\}$ and
$\overline{B}_{\sigma }(\varphi ,r):=\{\psi\in \mathcal{C}_{\sigma };\|\psi
-\varphi \|_{\infty }\leq r\}$. For $x\in \mathbb{R}^{m}$ and for a closed
subset $A\subset \mathbb{R}^{m}$ we denote
by $d(x,A)$ the distance from $x$ to $A$ given by
$d(x,A):=\inf\{\|y-x\|;y\in A\}$. Given a function
$V:\mathbb{R}^{m}\to \mathbb{R\cup \{+\infty \}}$ let
\begin{equation*}
D(V):=\{x\in \mathbb{R}^{m}:V(x)<\mathbb{+\infty }\}
\end{equation*}
be its effective domain. We say that $V$ is proper function if $D(V)$ is
nonempty.

Let $V:\mathbb{R}^{m}\to \mathbb{R}$ be a proper convex and lower
semicontinuous function. The multifunction
$\partial V:\mathbb{R}^{m}\to 2^{\mathbb{R}^{m}}$, defined by
\begin{equation}
\partial V(x):=\{\xi \in \mathbb{R}^{m};V(y)-V(x)\geq \left\langle \xi
,y-x\right\rangle ,\quad \forall y\in \mathbb{R}^{m}\},  \label{e2.1}
\end{equation}
is called subdifferential (in the sense of convex analysis) of the function $V$.

We say that a multifunction
$F:\Omega \subset \mathcal{C}_{\sigma}\to 2^{\mathbb{R}^{m}}$ is upper
semicontinuous if for every $\varphi \in \Omega $ and for every $\varepsilon >0$
there exists $\delta >0$ such that
\begin{equation*}
F(\psi )\subset F(\varphi )+B(0,\varepsilon ),\quad \forall \psi \in
\Omega \cap B_{\sigma }(\varphi ,\delta ).
\end{equation*}
The definition of the upper semicontinuous multifunctions is the same as
\cite[Definition 1.2]{d1}.

For a multifunction $F:\Omega \to 2^{\mathbb{R}^{m}}$ we consider
the functional differential inclusion \eqref{e1.1} under the following assumptions:
\begin{itemize}
\item[(H1)] $\Omega \subset \mathcal{C}_{\sigma }$ is an open set and
$F$ is upper semicontinuous with compact values;

\item[(H2)] There exists a a proper convex and lower semicontinuous
function $V:\mathbb{R}^{m}\to \mathbb{R}$ such that
\begin{equation}
F(\psi )\subset \partial V(\psi (0))\text{ for every }\psi \in \Omega .
  \label{e2.2}
\end{equation}
\end{itemize}

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

Our main result is the following:

\begin{theorem} \label{thm2.1}
 If $F:\Omega \to 2^{\mathbb{R}^{m}}$ and $V:\mathbb{R}^{m}\to \mathbb{R}$
satisfy assumptions (H1) and (H2) then for every $\varphi \in \Omega $
there exists $T>0 $ and $x(.):[-\sigma ,T]\to \mathbb{R}^{m}$ a solution of the
functional differential inclusion \eqref{e1.1} such that $T(0)x=\varphi $ on
$[-\sigma ,0]$.
\end{theorem}

\section{Proof of the main result}

Let $\varphi \in \Omega $ be arbitrarily fixed. Since the multifunction $%
x\to \partial V(x)$ is locally bounded \cite[Proposition 2.9]{b3},
there exists $r>0$ and $M>0$ such that $V$ is Lipschitz continuous with
constant $M$ on $B(\varphi (0),r)$. Since $\Omega $ is an open set we can
choose $r$ such that $\overline{B}_{\sigma }(\varphi ,r)\subset \Omega $.
Moreover, by \cite[Proposition 1.1.3]{a1}, $F$ is locally bounded; therefore, we
can assume that
\begin{equation}
\sup \{\|y\|: y\in F(\psi ),\,\psi \in B(\varphi ,r)\}\leq M\,. \label{e3.1}
\end{equation}
Since $\varphi $ is continuous on $[-\sigma ,0]$ we can choose $\eta >0$
such that
\begin{equation}
\|\varphi (t)-\varphi (s)\|<r/4\text{ for all }t,s\in [ -\sigma ,0]%
\text{ with }|t-s|<\eta .  \label{e3.2}
\end{equation}
Let $0<T\leq \min \{\eta ,r/4M\}$. We shall prove the existence of a
solution of \eqref{e1.1} defined on the interval $[-\sigma ,T]$. For this, we
define a family of approximate solutions and we prove that a subsequence
converges to a solution of \eqref{e1.1}.

First, for a fixed $n\in \mathbb{N}^{\ast }$, we set
\begin{equation}
x_{n}(t)=\varphi (t),\text{ }t\in [ -\sigma ,0].  \label{e3.3}
\end{equation}
Furthermore, we partition $[0,T]$ by points $t_{n}^{j}:=\frac{jT}{n}$,
$j=0,1,\dots ,n$, and, for every $t\in [ t_{n}^{j},t_{n}^{j+1}]$, we
define
\begin{equation}
x_{n}(t):=x_{n}^{j}+(t-t_{n}^{j})y_{n}^{j},  \label{e3.4}
\end{equation}
where $x_{n}^{0}=x_{n}(0):=\varphi (0)$ and
\begin{gather}
x_{n}^{j} = x_{n}^{j-1}+\frac{T}{n}y_{n}^{j-1},  \label{e3.5} \\
y_{n}^{j} \in F(T(t_{n}^{j})x_{n})  \label{e3.6}
\end{gather}
for every $j\in \{1,2,\dots ,n\}$.
It is easy to see that for every $j\in \{1,2,\dots ,n\}$ we have
\begin{equation}
x_{n}^{j}=\varphi (0)+\frac{T}{n}(y_{n}^{0}+y_{n}^{1}+\dots +y_{n}^{j-1}).
\label{e3.7}
\end{equation}
By \eqref{e3.1} and \eqref{e3.7} we infer
$\|x_{n}^{j}-\varphi (0)\|\leq \frac{jT}{n}M<r/4$, proving that
\begin{equation}
x_{n}(t_{n}^{j})=x_{n}^{j}\in B(\varphi (0),r/4)  \label{e3.8}
\end{equation}
for every $j\in \{1,2,\dots ,n\}$.

By \eqref{e3.1} and \eqref{e3.4} we have that
\begin{equation}
\|x_{n}(t)-x_{n}(t_{n}^{j})\|=\|x_{n}(t)-x_{n}^{j}\|\leq \frac{jT}{n}M<\frac{%
r}{4},  \label{e3.9}
\end{equation}
for every $j\in \{0,1,\dots ,n\}$.
Hence, from \eqref{e3.8} and \eqref{e3.9} we deduce that
\begin{equation*}
\|x_{n}(t)-\varphi (0)\|\leq
\|x_{n}(t)-x_{n}(t_{n}^{j})\|+\|x_{n}(t_{n}^{j})-\varphi (0)\|<\frac{r}{2}
\end{equation*}
and so
\begin{equation}
x_{n}(t)\in B(\varphi (0),\frac{r}{2})\text{, for every }t\in [ 0,T].
\label{e3.10}
\end{equation}
Moreover, by \eqref{e3.1}, \eqref{e3.4} and \eqref{e3.6}, we have
$\|x_{n}'(t)\|\leq M$ for every $t\in [ 0,T]$ and so the sequence $(x_{n}')$ is
bounded in $L^{2}([0,T],\mathbb{R}^{m})$.

For  $t,s\in [ 0,T]$, we have
\begin{equation*}
\|x_{n}(t)-x_{n}(s)\|\leq \big| \int_{s}^{t}\|x_{n}'(\tau )\|d\tau
\big| \leq M|t-s|
\end{equation*}
so that the sequence $(x_{n})$ is equiuniformly continuous.
Hence, by Theorem 0.3.4 in \cite{a1}, there exists a subsequence, still denoted by
$(x_{n})$, and an absolute continuous function $x:[0,T]\to \mathbb{R}%
^{m}$ such that:\begin{itemize}
\item[(i)] $\ (x_{n})$ converges uniformly on $[0,T]$ to $x$;

\item[(ii)] $(x_{n}')$ converges weakly in $L^{2}([0,T],\mathbb{R}^{m})$
to $x'$.
\end{itemize}
Moreover, since by \eqref{e3.3} all functions $x_{n}$ agree with $\varphi $ on
$[-\sigma ,0]$, we can obviously say that $x_{n}\to x$ on $[-\sigma,T]$,
if we extend $x$ in such a way that $x\equiv \varphi $ on $[-\sigma ,0] $.
Also, it is clearly that $T(0)x=\varphi $ on $[-\sigma ,0]$.

Further on, if we define $\theta _{n}(t)=t_{n}^{j}$ for all $t\in [
t_{n}^{j},t_{n}^{j+1}]$ then, by \eqref{e3.4} and \eqref{e3.6}, we have
\begin{equation}
x_{n}'(t)\in F(T(\theta _{n}(t))x_{n})\text{, a.e. on }[0,T].
\label{e3.11}
\end{equation}
and, by \eqref{e3.8},
\begin{equation}
x_{n}(\theta _{n}(t))\in B(\varphi (0),\frac{r}{4})\text{, for every }t\in
[ 0,T].  \label{e3.12}
\end{equation}
Also, since $|\theta _{n}(t)-t|\leq \frac{T}{n}$ for every $t\in [
0,T]$, then $\theta _{n}(t)\to t$ uniformly on $[0,T]$.
Moreover, by the uniformly converges of $(x_{n})$ and $(\theta _{n})$, we
deduce that $x_{n}(\theta _{n}(t))\to x(t)$ uniformly on $[0,T]$.

Now, we have to estimate $\|(T(\theta _{n}(t))x_{n})(s)-\varphi (s)\|$ for
each $s\in [ -\sigma ,0]$. If $-\theta _{n}(t)\leq s\leq 0$, then $%
\theta _{n}(t)+s\geq 0$ and so there exists $j\in \{0,1,\dots ,n-1\}$ such that
$\theta _{n}(t)+s\in [ t_{n}^{j},t_{n}^{j+1}]$.
Thus, by \eqref{e3.2}, \eqref{e3.10} and by the fact that
$|\theta _{n}(t)-t|\leq T$ and $|s|\leq T$,  we have
\begin{align*}
\|(T(\theta _{n}(t))x_{n})(s)-\varphi (s)\|
&=\|x_{n}(\theta_{n}(t)+s)-\varphi (s)\| \\
&\leq \|x_{n}(\theta _{n}(t)+s)-\varphi (0)\|+\|\varphi (s)-\varphi (0)\|\\
&<\frac{3r}{4}<r\,.
\end{align*}
If $-\sigma \leq s\leq -\theta _{n}(t)$ then $s+\theta _{n}(t)\leq 0$ and by
\eqref{e3.2} we have
\[
\|(T(\theta _{n}(t))x_{n})(s)-\varphi (s)\|
=\|\varphi (\theta_{n}(t)+s)-\varphi (s)\|\leq \frac{r}{4}<r.
\]
Therefore,
\begin{equation}
T(\theta _{n}(t))x_{n}\in B(\varphi ,r),\quad \text{ for every }t\in [ 0,T].
\label{e3.13}
\end{equation}
Let us denote the modulus continuity of a function $\psi $
defined on interval $I$ of $\mathbb{R}$ by
\begin{equation*}
\omega (\psi ,I,\varepsilon ):=\sup \{\|\psi (t)-\psi (s)\|;s,t\in
I,|s-t|<\varepsilon \},\text{ }\varepsilon >0.
\end{equation*}
Then we have:
\begin{align*}
\|T(\theta _{n}(t))x_{n}-T(t)x_{n}\|_{\infty }
&= \sup_{-\sigma\leq s\leq 0} \|x_{n}(\theta _{n}(t)+s)-x_{n}(t+s)\| \\
&\leq \omega (x_{n},[-\sigma ,T],\frac{T}{n}) \\
&\leq \omega (\varphi ,[-\sigma ,0],\frac{T}{n})+\omega (x_{n},[0,T],
\frac{T}{n}) \\
&\leq \omega (\varphi ,[-\sigma ,0],\frac{T}{n})+\frac{T}{n}M;
\end{align*}
hence
\begin{equation}
\|T(\theta _{n}(t))x_{n}-T(t)x_{n}\|_{\infty }\leq \delta _{n}\quad
\text{for every }t\in [ 0,T],  \label{e3.14}
\end{equation}
where $\delta _{n}:=\omega (\varphi ,[-\sigma ,0],\frac{T}{n})+\frac{T}{n}M$.
Thus, by continuity of $\varphi $, we have $\delta _{n}\to 0$ as
$n\to \infty $ and hence
\begin{equation*}
\|T(\theta _{n}(t))x_{n}-T(t)x_{n}\|_{\infty }\to 0\text{ as }
n\to \infty .
\end{equation*}
Therefore, since the uniform convergence of $x_{n}$ to $x$ on $[-\sigma ,T]$
implies
\begin{equation}
T(t)x_{n}\to T(t)x\quad \text{uniformly on }[-\sigma ,0],  \label{e3.15}
\end{equation}
we deduce that
\begin{equation}
T(\theta _{n}(t))x_{n}\to T(t)x\quad\text{in }\mathcal{C}_{\sigma }\,. \label{e3.16}
\end{equation}
Moreover, by \eqref{e3.13} and \eqref{e3.16}, we have that
$T(t)x\in \overline{B}_{\sigma }(\varphi ,r)\subset \Omega $.
Also, by \eqref{e3.11} and \eqref{e3.14}, we have
\begin{equation}
d((T(t)x_{n},x_{n}'(t)),\mathop{\rm graph}(F))\leq \delta _{n}\quad
\text{ for every }t\in [ 0.T].  \label{e3.17}
\end{equation}
By (H2), (ii), \eqref{e3.16} and \cite[Theorem 1.4.1]{a1}, we obtain
\begin{equation}
x'(t)\in coF(T(t)x)\subset \partial V(x(t))\quad \text{a.e. on }[0,T],
\label{e3.18}
\end{equation}
where $co$ stands for the closed convex hull.

Since the functions $t\to x(t)$ and $t\to V(x(t))$ are
absolutely continuous, we obtain from \cite[Lemma 3.3]{b3} and \eqref{e3.18} that
\begin{equation*}
\frac{d}{dt}V(x(t))=\|x'(t)\|^{2}\quad\text{a.e. on }[0,T];
\end{equation*}
therefore,
\begin{equation}
V(x(T))-V(x(0))=\int_{0}^{T}\|x'(t)\|^{2}dt.  \label{e3.19}
\end{equation}
On the other hand, since
\begin{equation*}
x_{n}'(t)=y_{n}^{j}\in F(T(t_{n}^{j})x_{n})\subset \partial
V(x_{n}(t_{n}^{j}))
\end{equation*}
for every $t\in [ t_{n}^{j},t_{n}^{j+1}]$ and for every
$j\in\{0,1,\dots ,n-1\}$, it follows that
\begin{align*}
V(x_{n}(t_{n}^{j+1}))-V(x_{n}(t_{n}^{j}))
&\geq \langle x_{n}'(t),x_{n}(t_{n}^{j+1})-x_{n}(t_{n}^{j})\rangle \\
&=\langle x_{n}'(t),\int_{t_{n}^{j}}^{t_{n}^{j+1}}x_{n}'(t)dt\rangle
=\int_{t_{n}^{j}}^{t_{n}^{j+1}}\|x'(t)\|^{2}dt.
\end{align*}
By adding the $n$ inequalities above, we obtain
\begin{equation*}
V(x_{n}(T))-V(x(0))\geq \int_{0}^{T}\|x_{n}'(t)\|^{2}dt
\end{equation*}
and passing to the limit as $n\to \infty $, we obtain
\begin{equation}
V(x(T))-V(x(0))\geq \underset{n\to \infty }{\lim \sup }
\int_{0}^{T}\|x_{n}'(t)\|^{2}dt.  \label{e3.20}
\end{equation}
Therefore, by b\eqref{e3.19} and \eqref{e3.20},
\begin{equation*}
\int_{0}^{T}\|x'(t)\|^{2}dt\geq \underset{n\to \infty }{%
\lim \sup }\int_{0}^{T}\|x_{n}'(t)\|^{2}dt
\end{equation*}
and, since $(x_{n}')$ converges weakly in $L^{2}([0,T],\mathbb{R}^{m})$ to $x'$,
by applying \cite[Proposition III.30]{b4}, we obtain
that $(x_{n}')$ converges strongly in $L^{2}([0,T],\mathbb{R}^{m})$%
. Hence there exists a subsequence, still denote by $(x_{n}')$,
which converges pointwiese a.e. to $x'$.

Since, by (H1), the graph of $F$ is closed \cite[Proposition 1.1.2]{a1},
by \eqref{e3.17},
\begin{equation*}
\lim_{n\to \infty } d((T(t)x_{n},x_{n}'(t)),\mathop{\rm graph}(F))=0,
\end{equation*}
we obtain
\begin{equation*}
x'(t)\in F(T(t)x)\quad \text{a.e. on }[0,T]\,.
\end{equation*}
Therefore, the functional differential inclusion \eqref{e1.1}  has solutions.

\begin{thebibliography}{99}

\bibitem{a1}  J. P. Aubin and A. Cellina; \emph{Differential inclusions}, Berlin,
Springer-Verlag, 1984.

\bibitem{b1}  M. Benchohra and S. K. Ntouyas; \emph{Existence results for
functional differential inclusions},  Electron. J. Diff. Eqns., Vol.
2001(2001), No. 41, pp. 1-8.

\bibitem{b2}  A. Bressan, A. Cellina and G. Colombo; \emph{Upper semicontinuous
differential inclusions without convexity}, Proc. Amer. Math. Soc.,
106(1984), 771-775.

\bibitem{b3}  H. Brezis; \emph{Operateurs Maximaux Monotones et Semigroups de
Contractions Dans Les Espaces de Hilbert}, Amsterdam, North-Holland, 1973.

\bibitem{b4}  H. Brezis, Analyse Functionelle, Th\'{e}orie et Applications,
Paris, Masson, 1983.

\bibitem{d1}  K. Deimling; \emph{Multivalued Differential Equations}, Walter de
Gruyter, Berlin, New York, 1992.

\bibitem{f1}  A. Fryszokwski; \emph{Existence of solutions of functional
differential inclusions in nonconvex case}, Ann. Pol. Math., 45(1985),
121-124.

\bibitem{h1}  G. Haddad; \emph{Topological Properties of Sets Solutions for
Functional Differential Inclusions}, Nonlinear Analysis, T.M.A, 5(12)(1981),
1349-1366.

\bibitem{h2}  J. K. Hale and S.M.V. Lumel; \emph{Introduction to functional
differential equations}, Springer-Verlag, 1993.

\bibitem{l1}  V. Lakshmikantham and S. Leela; \emph{Nonlinear differential
equations in abstract spaces}, Pergamon Press, Oxford, New York, 1981.

\bibitem{r1}  R. T. Rockafeller; \emph{Convex Analysis}, Princeton University Press,
1970.
\end{thebibliography}

\end{document}