\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 48, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2016/48\hfil Inclusions with primal lower nice functions]
{Existence of solutions to differential inclusions with primal lower nice functions}
\author[N. Fetouci, M. F. Yarou \hfil EJDE-2016/48\hfilneg]
{Nora Fetouci, Mustapha Fateh Yarou}
\address{Nora Fetouci \newline
LMPA Laboratory, Department of Mathematics, Jijel University, Algeria}
\email{norafetou2005@yahoo.fr}
\address{Mustapha Fateh Yarou \newline
LMPA Laboratory, Department of Mathematics, Jijel University, Algeria}
\email{mfyarou@yahoo.com}
\thanks{Submitted June 15, 2015. Published February 10, 2016.}
\subjclass[2010]{34A60, 49J52}
\keywords{Evolution problem; differential inclusion;
primal lower nice functions;
\hfill\break\indent Carath\'eodory perturbation}
\begin{abstract}
We prove the existence of absolutely continuous solutions to
the differential inclusion
\[
\dot{x}(t)\in F(x(t))+h(t,x(t)),
\]
where $F$ is an upper semi-continuous set-valued function with compact values
such that $F(x(t))\subset \partial f(x(t))$ on $[ 0,T]$, where $f$ is a
primal lower nice function, and $h$ a single valued Carath\'eodory perturbation.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}
In this article we study the first-order differential inclusion
\begin{equation}
\begin{gathered}
\dot{x}(t)\in F(x(t))+h(t,x(t)) \quad\text{a. e. } t \geq 0\\
x(0)=x_0
\end{gathered} \label{pr}
\end{equation}
where $F$ is an upper semi-continuous set-valued function with
nonconvex values and $h$ is a Carath\'eodory function. It is well
known that the problem admits a solution when $F$ has convex
values (see \cite{au}). When the values of $F$ fail to be convex
many results have been established in the case when
$F(x)\subset \partial f(x)$, for some proper convex lower semi-continuous
function $f$, see for instance \cite{br}. For some extensions of
these results, we refer to \cite{anc,cel,pap,sia}.
The case where the convexity of $f$ is dropped
has been studied in \cite{ben} by supposing $f$ is (Clarke) regular.
This class of function is of great importance in nonsmooth analysis
and optimization as it generalizes several
classes of functions such as convex proper lower semi-continuous
functions and uniformly regular functions. A more general problem
has been studied in the infinite dimensional setting in \cite{ya},
in which the author proved that for locally Lipschitz functions,
the class of convex functions, the class lower-$C^2$ functions and
the class of uniformly regular functions are strictly contained
within the class of regular functions. Another class of functions
which is of great interest in variational analysis and
optimization is the so called primal lower nice function (pln for
short). This class of functions covers all convex functions,
qualified convexly composite functions, and benefits from
remarkable features such the strong connection of pln functions
and their Moreau envelopes, in addition to the coincidence of
their proximal and Clarke subdifferential. The notion of pln
functions was introduced by Poliquin \cite{pol0} where a
subdifferential characterization of these functions was given in
the finite dimensional setting. Analogously to the convex case, it
was proved that pln functions are completely determined by their
subgradients, more precisely, if two lower semi-continuous
functions are pln at some point $x$ of their domain and have the
same proximal subgradient on a neighborhood of $x$, then the
difference between these functions is constant near $x$. Some
local regularity properties of the Moreau envelopes and the
related proximal mappings of prox regular functions and pln
functions in Hilbert space have been established, see \cite{ber}
and \cite{ma}. In this paper, we aim at showing that existence of
solution of (1.1) holds in the context of pln lower semi-continuous
functions $f$.
The paper is organized as follows:
In section 2 we recall some preliminary facts that we need in the sequel and
in section 3 we prove our existence result; first in
$\mathbb{R}^{n}$ and then in infinite dimensional Hilbert space.
\section{Preliminaries}
Throughout this article, $H$ stands for a real Hilbert space with scalar
product $\langle \cdot,\cdot\rangle $ and norm $\| \cdot\| $,
$\overline{B}(x,r)$ is the closed ball centered at $x$ with radius $r$.
\begin{definition} \label{def2.1} \rm
Let $X$ and $Y$ be topological spaces and $F$ a set-valued mapping defined
on $X$ with values in the space $\mathcal{P}(Y)$ of all nonempty subsets of $
Y$. We will say that $F:X\to \mathcal{P}(Y)$ is upper semi-continuous (usc) at $\bar{x}\in X$ if for every neighborhood $U$ of $F(
\bar{x})$ there exists a neighborhood $V$ of $\bar{x}$ such that $F(x)\subset U$ for all $x\in V$.
\end{definition}
\begin{definition} \label{def2.2} \rm
Let $f:H\to \mathbb{R\cup \{+\infty \}}$ be an extended real valued
lower semi-continuous function and
let $ \overline{x}\in \operatorname{dom}f$, that is $f(\overline{x})<+\infty $.
The proximal subdifferential of $f$ at $\overline{x}$ is the set
$\partial _{p}f(\overline{x})$ of all elements $v\in H$ for which
there exist $r>0$ and $\varepsilon >0$ such that
\[
\langle v,x-\overline{x}\rangle \leq f(x)-f(\overline{x}
)+ \varepsilon \| x-\overline{x}\| ^2 \quad \text{for all } x\in B(\overline{x},r ).
\]
The Fr\'echet subdifferential $\partial _{F}f(\overline{x})$ of $f$ at
$\overline{x}$ is defined by $v\in \partial _{F}f(\overline{x})$ provided
that for each $\varepsilon >0$, there exists some $\eta >0$ such that for all
$x\in B(\overline{x},\eta ),$
\[
\langle v,x-\overline{x}\rangle
\leq f(x)-f(\overline{x})+\varepsilon \| x-\overline{x}\| .
\]
The Clarke subdifferential of a lower semi-continuous function $f$ at
$\overline{x}$ is the set
\[
\partial _{C}f(\overline{x}) :=\{ v\in H:f^{\uparrow }(\overline{x};y)
\geq \langle v,y\rangle ,\text{ }\forall y\in H\}.
\]
where $f^{\uparrow }(\overline{x};y)$ is the generalized Rockafellar
derivative given by
\[
f^{\uparrow }(\overline{x};y)
=\limsup_{\substack{t\to 0^{+},\\ x\to ^f \overline{x}}}
\inf_{y'\to y} t^{-1}[f(x+ty')-f(x)]
\]
where $x\to^f \overline{x}$ means $x\to \overline{x}$ and
$ f(x) \to f(\overline{x})$. If $f$ is locally Lipschitz, the
generalized Rockafellar derivative $f^{\uparrow }(\overline{x};y)$
coincides with the Clarke directional derivative $f^0(\overline{x},y)$ defined by
\[
f^{0}(\overline{x},y)=\lim_{x\to \overline{x}}
\sup_{t \downarrow 0} \frac{f(x+t y)-V(x)}{t}
\]
If $x\notin \operatorname{dom}f$, $\partial _{C}f(x):=\emptyset $.
When $f$ is convex and lower semi-continuous, one has
\[
\partial _{p}f=\partial _{C}f=\partial _{F}f=\partial f.
\]
The operator $\partial f$ denotes the subdifferential in the sense
of convex analysis.
\end{definition}
Now, let us recall the definition of Primal Lower Nice functions
\cite{ma}.
\begin{definition} \label{def2.3} \rm
Let $f:H\to \mathbb{R\cup \{+\infty \}}$ be a proper function and consider
$x_0\in \operatorname{dom}f$. The function $f$ is said to be primal lower nice
(pln for short) at $x_0$, if there exist positive real numbers $s_0$,
$ c_0$, $Q_0$ such that for all $x\in \overline{B}(x_0,s_0)$, for all
$q\geq Q_0$ and all $v\in \partial _{p}f(x)$ with $\| v\| \leq c_0q$, one has
\[
f(y)\geq f(x)+\langle v,y-x\rangle -\frac{q}{2}\| y-x\| ^2\quad
\text{for each }y\in \overline{B}(x_0,s_0).
\]
\end{definition}
\begin{remark} \label{rmk2.4} \rm
(1)
Each extended real valued convex function is primal lower nice at each
point of its domain.
(2) Clearly, if $f$ is pln at $u_0$ with constants $s_0,c_0,Q_0$, one has
\[
\langle v_1-v_2, x_1-x_2\rangle \geq -q\|
x_1-x_2\| ^2
\]
for any $v_{i}\in \partial _{p}f(x_{i})$ with $\| v_{i}\| \leq c_0q$
whenever $q\geq Q_0$ and $x_{i}\in \overline{B}(u_0,s_0)$, $i=1,2$.
(3) If $f$ is pln at $u_0\in \operatorname{dom}f$ then for all $x$ in a
neighborhood of $u_0$, the proximal subdifferential of $f$ at $x$ coincides
with the Fr\'echet subdifferential and the Clarke subdifferential of $f$
at $x$, i.e, $\partial _{p}f(x)=\partial _{F}f(x)=\partial _{C}f(x)$.
In this case, we simply denote by $\partial f(x)$ the common subdifferential,
and by $\partial ^{0}f(x)$ its element of minimal norm for
$x\in \operatorname{dom}f$.
\end{remark}
The graph of the (proximal) subdifferential of a pln function enjoys the useful
closure property.
\begin{proposition} \label{prop1}
Let $f:H\to \mathbb{R\cup \{+\infty \}}$ be a proper lsc function which is
pln at $u_0\in \operatorname{dom}f$ with constants $s_0$, $c_0$,
$Q_0>0$, and let $T_0,T,v_0,\eta _0$ be positive real numbers such
that $T>T_0$ and $v_0+\eta _0=s_0$. Let $v(.)\in L^2( [
T_0,T] ,H) $ and $u(\cdot)$ be a mapping from $[ T_0,T ] $ into $H$.
Let $( u_n( \cdot) ) _n$ be a
sequence of mappings from $[ T_0,T] $ into $H$ and $(v_n( \cdot) ) _n$
be a sequence in $L^2( [T_0,T] ,H) $. Assume that:
\begin{enumerate}
\item $\{ u_n(t),n\in \mathbb{N}\} \subset \overline{B} ( u_0,\eta _0)
\cap \operatorname{dom}f$ for almost every $t\in [T_0,T] $,
\item $( u_n) _n$ converges almost everywhere to some mapping $u$ with
$u( t) \in \operatorname{dom}f$ for almost every $t\in [ T_0,T] $,
\item $v_n$ converges to $v$ with respect to the weak topology of
$L^2( [ T_0,T] ,H) $,
\item for each $n\geq 1$, $v_n( t) \in \partial f(u_n(t))$ for almost every
$t\in [ T_0,T] $.
\end{enumerate}
Then, for almost all $t\in [ T_0,T] $, $v( t) \in \partial f(u(t))$.
\end{proposition}
For the proof of the above proposition, we refer the reader to \cite{ma}.
\begin{proposition}[\cite{za}] \label{pr2}
Let $f:H{\to \mathbb{R}\cup }\{+\infty \} $ be a lower semi-continuous and
pln function at $\overline{x}\in \operatorname{dom}f$
with constants $\epsilon , c, T$ and $f$ is bounded from above on
$B(\overline{x},\epsilon )$. Assume also that $\partial f$ is
included in the Clarke subdifferential of $f$. Then the function $f$
is Lipschitz continuous and DC (difference of convex functions) near
$\overline{x}$. In fact, if $\epsilon >0$ is such that $f$ is also bounded
from below on $B(\overline{x},\epsilon )$, then for any $\alpha \in ] 0,1[ $
the function $f$ is Lipschitz continuous and DC on $B(\overline{x},\alpha \epsilon )$
\end{proposition}
For more details about pln functions we refer to \cite{ber} and \cite{pol0}.
\section{Main results}
Let us recall first the existence result for the subdifferential operator
of a pln function obtained in \cite{ma}.
\begin{theorem} \label{thma}
Let $f:H\to \mathbb{R}\cup \{ +\infty \} $ be a proper lsc function.
Consider $T_0\in [ 0,+\infty[ $ and let $x_0\in \operatorname{dom}f$ be such
that $f$ is pln at $x_0$ with constants $s_0$, $c_0$, $Q_0$. Then, there exist
a real number $T\in] T_0,+\infty [ $ and a unique absolutely continuous mapping
$x: [ T_0,T] \to B( x_0,s_0) $ which is a
solution of the problem
\begin{equation}
\begin{gathered}
0\in \dot{x}( t) +\partial f( x( t) ) \quad \text{for a.e }t\in [ T_0,T] , \\
x( T_0) =x_0.
\end{gathered} \label{P}
\end{equation}
Further, in the particular case when $x_0\in dom$ $\partial f$, the solution
$x(\cdot) $ above is actually Lipschitz continuous and it
also satisfies:
\begin{enumerate}
\item $f\circ x(\cdot) $ is Lipschitz continuous on $[ T_0,T] $.
\item For almost every $t\in [ T_0,T] $, the derivative
$(f\circ x) '(t)$ exists and
\[
( f\circ x) '(t)=-\| \dot{x}( t)\| ^2,
\]
and for any $T_0\leq s\leq t\leq T$:
\[
f( x(t)) -f(x(s))=-\int_{s}^{t}\| \dot{x}( \tau ) \| ^2d\tau .
\]
\end{enumerate}
\end{theorem}
Now we are able to establish existence result for the problem \eqref{pr}
in the context of finite dimensional space $\mathbb{R}^{n}$.
\begin{theorem}\label{thm1}
Under the following assumptions:
\begin{itemize}
\item[(H1)] $\Omega \subset \mathbb{R}^{n}$ is an open set and
$F:\Omega \rightharpoondown \mathbb{R}^{n}$ is an upper semi-continuous
compact valued multifunction.
\item[(H2)] $f:\Omega \to \mathbb{R\cup }\{+\infty \} $ is a lower
semi-continuous function, pln at $x_0\in \operatorname{dom}\partial f$
with constants $s_0$, $c_0$, $Q_0$ such that
\[
F(x)\subset \partial f(x)\quad \forall x\in \Omega .
\]
\item[(H3)] $h:\mathbb{R\times R}^{n}\to \mathbb{R}^{n}$
is a Carath\'eodory function, i. e. for every $x\in \mathbb{R}^{n}$,
$h(\cdot,x)$ is measurable, $h(t,\cdot)$ is continuous, and there
exists $m(\cdot)\in L^2(\mathbb{R}_{+}^{\ast })$ such that
\[
\| h(t,x)\| \leq ( 1+\| x\| )
m(t),\quad \forall x\in \mathbb{R}^{n},\text{ a.e. } t\in \mathbb{R}.
\]
\end{itemize}
Then, there exist $T>0$, and an absolutely continuous solution
$x:[0,T]\to \mathbb{R}^{n}$ of the differential inclusion
\begin{gather*}
\dot{x}(t)\in F(x(t))+h(t,x(t))\quad \text{a.e. }t\in [ 0,T] , \\
x(0)=x_0.
\end{gather*}
\end{theorem}
\begin{proof}
Since $\Omega $ is open, there exists $r>0$ such
that the compact set $K=\overline{B}(x_0,r)$ is contained in
$\Omega $. Moreover, by $( H_1) $ and \cite[Proposition 1.1.3]{au},
$F(K)=\underset{x\in K}{\cup }F(x)$ is
compact, hence there exists $M$ such that
\begin{equation}
\sup \{ \| u\| : u\in F(x), x\in K\} \leq M. \label{1}
\end{equation}
Since $f$ is pln at $x_0$, there exist positive real numbers $s_0$,
$ c_0$, $Q_0$ such that for all $ x\in \overline{B}(x_0,s_0)$,
$q\geq Q_0$ and $u\in \partial _{p}f(x)$, with $\| u\| \leq c_0q$ one
has
\[
f(y)\geq f(x)+\langle u,y-x\rangle -\frac{q}{2}\|y-x\| ^2
\]
for each $y\in \overline{B}(x_0,s_0)$.
Let us choose $T'>0$ such that
\[
\int_0^{T'}(M+\alpha m(t))dt<\frac{r}{2},
\]
where $\alpha =1+\| x_0\|
+\frac{r}{2}$. Set $r_0=\min (\frac{r}{2},s_0)$, take
$T=\min (\frac{r_0}{M},T')$ and let $I=[0,T]$. For each
integer $n\geq 1$ and for $0\leq i\leq n-1$, we set
$t_n^{i}=\frac{iT}{n}$;
$I_n^{i}=[t_n^{i},t_n^{i+1}[ $, and for every
$t\in I_n^{i}$, we define
\begin{equation}
x_n(t)=x_n^{i}+( t-t_n^{i}) u_n^{i}+\int_{i\frac{T}{n}
}^{t}h(s,x_n^{i})ds, \label{2}
\end{equation}
where $x_n(0)=x_n^{0}=x_0$ and
\begin{gather}
x_n(t_n^{i})=x_n^{i}=x_n^{i-1}+\frac{T}{n}u_n^{i-1} \quad
\text{for every } i\in \{1,2,\dots ,n\}, \label{3}
\\
u_n^{i}\in F(x_n^{i}), \quad \text{for every }
i\in \{0,1,2,\dots ,n\}.\label{4}
\end{gather}
$(x_n)$ is well defined on $[0,T]$. Clearly one has for every
$i\in \{1,2,\dots ,n\}$,
\begin{align*}
x_n^{i} - x_n^{0}
&=\frac{T}{n}(u_n^{0}+u_n^{1}+\dots +u_n^{i-1})\notag \\
&\leq \frac{T}{n}(\| u_n^{0}\| +\| u_n^{1}\| +\| u_n^2\|
+ \dots +\| u_n^{i-1}\| \notag \\
&\leq \frac{iTM}{n}\leq \frac{r}{2} ,
\end{align*}
proving that
\begin{equation}
x_n(t_n^{i})=x_n^{i}\in \overline{B}(x_0,\frac{r}{2}).\label{6}
\end{equation}
By \eqref{1} and \eqref{2} for all $t\in [ t_n^{i},t_n^{i+1}[$, we obtain
\begin{equation}
\begin{aligned}
\| x_n(t)-x_n(t_n^{i})\|
&\leq \int_{t_n^{i}}^{t}(M+( 1+\| x_0\| +\frac{r}{2}) m(\tau ))d\tau \\
&\leq \int_{t_n^{i}}^{t}(M+\alpha m(\tau ))d\tau <\frac{r}{2}.
\end{aligned} \label{7}
\end{equation}
From \eqref{6} and \eqref{7} one can deduce that
\begin{align*}
\| x_n(t)-x_0\|
&\leq \| x_n(t)-x_n(t_n^{i})\| +\| x_n(t_n^{i})-x_0\| \\
&\leq \frac{r}{2}+\frac{r}{2}=r,
\end{align*}
and so
\begin{equation}
x_n(t)\in \overline{B}(x_0,r),\quad \text{for each }t\in [0,T]. \label{9}
\end{equation}
By \eqref{2} we have
\begin{equation}
\dot{x}_n(t)=u_n^{i}+h(t,x_n(t))\quad \forall t\in ]t_n^{i},t_n^{i+1}[, \label{90}
\end{equation}
the last equality with \eqref{1} ensures that
\begin{equation}
\| \dot{x}_n(t)\| \leq M+\alpha m(t) \quad \text{for a.e } t\in [ 0,T], \label{10}
\end{equation}
then
\[
\int_0^{T}\| \dot{x}_n(t)\| ^2dt \leq \int_0^{T}( M+\alpha m(t)) ^2dt,
\]
and so the sequence $(\dot{x}_n)_n$ is bounded in $L^2([0,T],\mathbb{R}^{n})$.
Further, for all $t$, $s\in [ 0,T]$, $0\leq s0$ and an absolutely continuous function $ x:[0,T]\to H$
solution of the differential inclusion \eqref{pr}.
\end{theorem}
\begin{proof}
Since $\Omega $ is open, there exists $r>0$ such that $\overline{B}(x_0,r)$
is contained in $\Omega $. By the definition of $f$, for all
$x\in \overline{B}(x_0,s_0)$, for all $q\geq Q_0$ and all
$u\in \partial _{p}f(x)$, with $\| u\| \leq c_0q$, one has
\[
f(y)\geq f(x)+\langle u,y-x\rangle -\frac{q}{2}\| y-x\| ^2.
\]
Since $f$ is lower semi-continuous on $x_0$, taking $s_0$ smaller if necessary,
we may suppose that $f$ is bounded from bellow on $\overline{B}(x_0,s_0)$.
Let us fix $\beta \in ] 0,1[ $, Proposition \ref {pr2} implies that there is
$L>0$ such that $\partial _{p}f(x)\subset L\overline{B}$, whenever
$x\in \overline{B}(x_0,\beta s_0)$. By our assumption (H3') there is a positive
constant $m$ such that $h(t,x)\in K_1\subset mB$ for all
$(t,x)\in \mathbb{R}^{+} \mathbb{\times }\overline{B}(x_0,r)$.
Moreover, by (H2'), there exists a positive constant $m_1$ such that for any
$ x\in \overline{B}(x_0,r)$, $F( x) \subset ( 1+\|x_0\| +r) K\subset m_1B$.
Choose $T$ such that
\begin{equation}
0