\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 66, pp. 1--12.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/66\hfil Impulsive differential inclusions]
{Impulsive differential inclusions with constrains}
\author[T. Donchev\hfil EJDE-2006/66\hfilneg]
{Tzanko Donchev}
\address{Tzanko Donchev \hfill\break
Department of Mathematics, University of Architecture and Civil
Engineering, 1 ``Hr. Smirnenski'' str., 1046 Sofia, Bulgaria}
\email{tdd51us@yahoo.com}
\date{}
\thanks{Submitted March 27, 2006. Published May 26, 2006.}
\thanks{Supported by grant DP0346099 from the Australian Research Council}
\subjclass[2000]{34A60, 34A37, 34B15, 49K24}
\keywords{Impulsive differential inclusions; constrains; invariant solutions}
\begin{abstract}
In the paper, we study weak invariance of differential inclusions
with non-fixed time impulses under compactness type assumptions.
When the right-hand side is one sided Lipschitz an extension of the
well known relaxation theorem is proved. In this case also necessary
and sufficient condition for strong invariance of upper semi
continuous systems are obtained.
Some properties of the solution set of impulsive system (without
constrains) in appropriate topology are investigated.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\section{Preliminaries}
This paper is concerned with the impulsive differential inclusion
\begin{gather}
\dot x(t)\in F(t,x(t)),\quad x(0)=x_0\in D,\hbox{ a.e. } t\in
I= [0,1],\ t\neq \tau_i(x), \label{i3} \\
\Delta x|_{t=\tau_i(x)}= S_i(x),\quad i=1,\dots,p,\ x(t)
\in D, \label{i4}
\end{gather}
Here $D$ is a closed subset of a Banach space $E$ and $F: I\times
D\to E$ is multifunction with nonempty compact values. Every
absolutely continuous on $(\tau_i,\tau_{i+1})$ function for $
i=0,1,\dots,p,p+1$ ($\tau_0 =0$ and $\tau_{p+1}=1$) with
(possible) jumps $\Delta x|_{t=\tau_i(x)}= S_i(x(\tau_i(x)-0))$
called impulses, i.e. $ x(\tau_i(x)+0)= x(\tau_i(x)-0)+ S_i(x(
\tau_i(x)-0)$, is said to be a solution of \eqref{i3}--\eqref{i4}.
Further we assume that $x+S_i(x)\in D$ for every $x\in D$.
Differential inclusions without impulses have been studied
extensively; see \cite{A,AC,Bo,CLR,D,D2,HP,T} and references
therein. We refer to \cite{BS,FR2,KL,LBS,SP} for the theory of
impulsive differential equations. The existence of solutions of
impulsive differential inclusions in infinite dimensional spaces
is very comprehensively studied in \cite{BHN}, see also
\cite{BB3,BHNO,W}. In these works the authors use mainly fixed
point arguments. Method of averaging and some other qualitative
properties of impulsive differential inclusions are studied in
\cite{PIK,PIK1,PPV}. We refer to \cite{A1,Aol,CS} for impulsive
differential inclusions with constrains in finite dimensional
space.
Our first purpose is to present sufficient (and necessary)
conditions for the existence of solutions in arbitrary (not
necessarily separable) Banach space when the right-hand side
is almost USC. We also prove sufficient conditions when the
right-hand side is USC at some points and LSC in others. This
is done in the second section. Notice that our compactness
conditions are weaker that those in \cite{BHN,BB3,BHNO}.
In the present paper, we are not able to use fixed points approach.
We follow the method used in \cite{D,Bo} with some modifications.
Our second purpose is to describe the structure of solution set
(in appropriate metric).
We extend the well-known relaxation theorem. We use a modification
of the very short proof of this theorem presented in \cite{D1} for
systems without impulses. Namely we prove that the solution set of
\begin{gather}
\dot x(t)\in \overline{\mathop{\rm ext}} F(t,x(t)),\quad
x(0)=x_0 \hbox{ a.e. } t\in I= [0,1],\ t\neq \tau_i(x), \label{*} \\
\Delta x|_{t=\tau_i(x)}= S_i(x(\tau_i(x)-0)),\quad
i=1,\dots,p,\ x(t)\in D,
\label{**}
\end{gather}
is dense in the solution set of \eqref{i3}--\eqref{i4}. We do not
know any related results in this case (impulsive system with
non-fixed time of impulses).
For problems without constrains in finite dimension, i.e.
$D\equiv \mathbb{R}^n$, we show that the solution set of
\eqref{i3}--\eqref{i4} is $R_\delta$.
\subsection*{Notation and terminology}
The multifunction $G: E\to E$ with nonempty closed bounded values is said
to be upper semi-continuous (USC) at $x_0$, when for every
$\varepsilon>0$ there exists $\delta>0$ with $G(x_0)+
\varepsilon\mathbb{B} \supset G(x_0+\delta\mathbb{B})$. Here
$\mathbb{B}$ is the open unit ball. The multifunction $G(\cdot)$
is said to be lower semi-continuous (LSC) at $x_0$ when for every
$f\in G(x_0)$ and every sequence $ \{x_i\}_{i=1}^\infty$
converging to $x_0$ there exist $f_i\in G(x_i)$ such that $f_i\to
f_0$. When $G(\cdot)$ is USC (LSC) at every $x\in D$ it is called
USC (LSC). The multifunction $F(\cdot,\cdot)$ is said to be almost
USC when for every $\varepsilon>0$ there exists a compact set
$I_\varepsilon \subset I$ with Lebesgue measure
$\mathop{\rm meas}(I_\varepsilon) > 1- \varepsilon$ such that
$F(\cdot,\cdot)$ is USC on
$I_\varepsilon \times D$. The almost LSC maps are defined
analogously.
Given $M>0$ we define the cone $ \Gamma^M= \{(t,x)\in
\mathbb{R}^+\times E: \|x\|_E\le Mt\}$. If
$\Omega\subset \mathbb{R}\times E$ is nonempty and $f:\Omega\to E$
we say that
$f$ is $\Gamma^M$-continuous at the point $(t_0,x_0)$ when, given
$\varepsilon>0$ one can find $\delta>0$ such that $(t,x)\in
\Omega$, $t_00$ with $\langle \xi, x'-x\rangle\le
\alpha |x'-x|^2$, $\forall x'\in D$. The set of all such vectors is
a cone denoted by $N_D^P(x)$ and it is called proximal normal cone
to $D$ at $x$. If no such a vector exists we set $N_D^P(x)= 0$.
Denote $\mathbb{R}^+=[0,\infty)$. A Caratheodory function $w: I
\times\mathbb{R}^+\to\mathbb{R}^+$ is said to be Kamke function if
it is integrally bounded on the bounded sets, $w(t,0)\equiv 0$ and
the only solution of the differential equation $\dot s(t)=
w(t,s(t))$, $s(0)=0$ is $s(t)\equiv 0$.
In the next section we study the existence of solutions under
compactness type conditions. Notice that our assumptions are more
general that those in \cite{D,Bo}. It is also impossible to use
fixed point arguments (because the constrain sets are nonconvex). To
our knowledge there are no existence results in the existing
literature when the right-hand side changes its kind of
semicontinuity as Theorem \ref{mix} below.
In the last section we consider differential inclusion
\eqref{i3}--\eqref{i4} in $\mathbb{R}^n$. We prove that the
solution set is $R_\delta$ in appropriate metric.
The proof of the relaxation theorem is a modification of the author's
proof (\cite{D1}). It will be very difficult (if possible at all) to
prove such a theorem using the approach of Pianigiani and
Tolstonogov (\cite{P,T}). We also extend a very recent strong
invariance result of \cite{DRW}.
\section{Existence of solutions}
In this section we study the existence of solutions for the Cauchy
problem \eqref{i3}--\eqref{i4}.
First we need a result for a problem without impulses. We consider
$$
\dot x(t)\in F(t,x(t)),\quad x(0)=x_0\in D,\ t\in I. \label{DI}
$$
In this case we need the following hypotheses:
\begin{itemize}
\item[(H1)] There exists a Kamke function $\omega(\cdot,\cdot)$ such
that $\chi(F(t,A))\leq \omega(t,\chi(A))$ for every bounded
$A \subset D$ and a.e. $t\in I$. Here
\[
\chi(A)= \inf\{ r>0: A\hbox{ can be covered by finitely many balls
of radius }\leq r\}
\]
is the Hausdorff measure of non-compactness.
\item[(H2)]
There exists a null set $\mathcal{N}\subset I$ such that
$ F(t,x)\cap T_D(x)\neq \emptyset$ for every $t\in I\setminus
\mathcal{N}$ and every $x\in D$.
\end{itemize}
The following theorem has been proved under condition stronger
than (H1); see for example \cite{Bo,D}. We present a
complete proof because this theorem will be essential in this paper.
\begin{theorem} \label{noim}
Let $F(\cdot,\cdot)$ be almost USC with nonempty convex compact
values satisfying (H1). Assume there exist an
$L_1(I, \mathbb{R}^+)$ function $\lambda(\cdot)$ such that
$|F(t,x)|\leq \lambda(t) (1+|x|)$ (linear growth). The system \eqref{DI}
admits a solution defined on the whole interval $I$ for every
$x_0\in D$ if and only if (H2) holds.
\end{theorem}
\begin{proof}
As it is shown in \cite{D}, one can reduce the growth condition to
the case $|F(t,x)|\leq C$ for some constant $C>0$ without
destroying the other hypotheses.
For $\varepsilon>0$ we will prove that there exists $\varepsilon$
--solution $x(\cdot)$ on $[0,1]$, i.e.
\begin{enumerate}
\item $\mathop{\rm dist}(x(t),D)< \varepsilon$ for every $t\in I$,
\item $\dot x(t)\in F(t,x(t)+ \varepsilon \mathbb{B}\cap D)$ on a
set $I_\varepsilon$ with measure greater than $1-\varepsilon$
\item $\dot x(t)\in F(t,x(t))+ 2C\mathbb{B}$ otherwise.
\end{enumerate}
Fix $\varepsilon>0$. There exists a set $I_\varepsilon\subset I$
wit Lebesgue measure $\mathop{\rm meas}(I_\varepsilon)>1-\varepsilon$
such that $F(\cdot,\cdot)$ is USC on $I_\varepsilon\times E$ and
$F(t,x)\cap T_D(x)\neq \emptyset$ for every $x\in D$ and every
$t\in I_\varepsilon$. One can suppose also without loss of
generality that $\omega(\cdot,\cdot)$ is (uniformly) continuous on
$I_\varepsilon\times [0,2M]$.
Since $x_0\in D$, there exists a maximal number
$0\leq \tau\leq 1$ such that there exists a $\varepsilon$
--solution $x(\cdot)$ on $[0,\tau)$ and
$ x(\tau):=\lim_{t\uparrow \tau} x(\tau)\in D$ ($x(\tau)$ exists, because
$x(\cdot)$ is $C$--Lipschitz). We are done if $\tau=1$. Let
$\tau<1$. Two cases are possible:
Case a: $\tau\in I_\varepsilon$. Since $ F(\tau,x(\tau))\cap
T_D(x(\tau))\neq \emptyset$, one has that there exist $f\in
F(\tau,x(\tau))\cap T_D(x(\tau))$ and sequences $h_n\downarrow
0$ and $y_n\to 0$ such that $x(\tau)+ h_n(f-y_n)\in D$. Let
$\delta>0$ be such that be such that
$\omega(t,s)-\omega(\tau,\xi)|< \varepsilon$ and
$F(\tau,x(\tau))\supset F(t,y)$ when $|t-\tau|< \delta$,
$|s-\xi|\tau$ such
that $(\tau,T)\subset I\setminus I_\varepsilon$ and $T\in
I_\varepsilon$. We let $x(t)\equiv x(\tau)$ on $[\tau,T]$. Evidently
$x(\cdot)$ satisfies (1), (2) and (3) on $[0,T]$.
Applying Zorn' lemma one obtains that $x(\cdot)$ is extendable on
$[0,1]$.
Let $ \varepsilon_{i+1}= \frac{\varepsilon_i}{3}$
($\varepsilon_0=\varepsilon$) and let $ \{ x^i(\cdot)\}_{i=1
}^\infty$ be a sequence of $\varepsilon_i$-- solutions.
Their derivatives $\dot x_n(\cdot)$ are strongly measurable and
hence almost separable valued. Therefore there exist a separable
space $X_0\subset E$ and a null set $A$ such that which $\dot
x_n(t)\in X_0$ for every $n$ and every $t\in I\setminus A$. We can
assume without loss of generality that $x_n(t)$ are in $X_0$.
Define $ B(t)= \chi\big(\cup_{n=1}^\infty\{x_n(t) \}\big)$.
From \cite[Proposition 9.3]{D}, we know that
$$
\chi \Big(\big\{\int_t^{t+h}\dot x_k(t): k\geq 1\big\}\Big)\,dt
\leq \int_t^{t+h} \chi\big(\{\dot x_k(t):\ k\geq 1\}\big)\,dt.
$$
Taking into account the definition of $x_n(\cdot)$ one has that
$B(\cdot)$ is absolutely continuous and for every $\varepsilon>0$
there exists a compact set $I_\varepsilon$ with
$\mathop{\rm meas}(I_\varepsilon)>1-\varepsilon$ such that
$ \dot B(t)\leq \omega(t,B(t))+\varepsilon$ on $I_\varepsilon$ and
$ \dot B(t)\leq \omega(t,B(t))+2C$ on $I\setminus I_\varepsilon$. Since
$\varepsilon>0$ is arbitrary one has that $\dot B(t)\leq
\omega(t,B(t))$ for a.a. $t\in I$. However, $B(0)=0$ and hence
$B(t)\equiv 0$.
Due to Arzela's theorem the sequence $ \{ x_n(\cdot)\}_{n=
1}^\infty$ is $C(I,E)$ precompact. Hence passing to subsequences
$v_n(t)\to x(t)$ uniformly on $I$. The proof that $x(\cdot)$ is a
solution of \eqref{i3}--\eqref{i4} is standard.
\end{proof}
We will use the following assumptions in this article:
\begin{itemize}
\item[(A1)] $\tau_i(\cdot)$ are Lipschitz function with a constant $N$,
and $\tau_i(x)\geq \tau_i(x+S_i(x))$.
\item[(A2)] $\tau_i(x)< \tau_{i+1}(x)$ for every $x\in D$.
\item[(A3)] There exists a constant $C$ such that $|F(t,x)|\leq C$ for
every $x\in D$ and a.e. $t\in I$ and $NC<1$.
\end{itemize}
These assumptions prevent the beating phenomena (see the
following lemma, which proof follows \cite{PIK1,PPV}).
\begin{lemma}\label{l1}
Under (A1)--(A3), every solution of \eqref{i3}--\eqref{i4} (if
it exists) intersects every surface $t=\tau_i(x)$ at most once.
\end{lemma}
\begin{proof}
Suppose the contrary, i.e. there exists a solution $x(t)$ which
pass through the surface $t=\tau_i(x)$ at the time $t'=\tau_i(x')+0$
and on the time $t"$ the same surface $(t", x")$, $t"=\tau_i(x")$.
Due to (A2), $x(\cdot)$ is continuous on the interval
$(t',t")$. and $\dot x(t)\in F(t,x(t))$. Denote
$ h_i=\int_{t'}^{t"}\dot x(t)\,dt$. By (A2) and (A3), we have
\begin{align*}
t"-t'&= \tau_i(x")-\tau_i(x') \\
& = \tau_i(x'+S_i(x')+h_i)-
\tau_i(x'+S_i(x'))+ \tau_i(x'+S_i(x'))-\tau_i(x') \\
& \leq N\int_{t'}^{t"}|\dot x(t)|\,dt + \tau_i(x'+S_i(x'))-
\tau_i(x') \\
& \leq NC(t"-t')+ \tau_i(x'+S_i(x'))-\tau_i(x')
\end{align*}
i.e.
\[
(1-NC)(t"-t')\leq \tau_i(x'+S_i(x'))-\tau_i(x')\leq 0
\]
which is a contradiction.
\end{proof}
\begin{theorem} \label{nfth}
Let $F$ be almost USC with convex (and compact) values.
If (A1)--(A3), (H1)--(H2) hold, then the system
\eqref{i3}--\eqref{i4} has a solution.
\end{theorem}
\begin{remark} \label{rem1} \rm
Obviously the conclusion of Lemma \ref{l1} holds for the system
$\dot x\in G(t,x)$ when $|F(t,x)| \leq C$ is replaced by
$ G(t,x):=\overline{F(t,x)\cap C \mathbb{B}}\neq \emptyset$.
Moreover, the conclusion of Theorem \ref{nfth} holds when
(H2), (A1) are replaced by
\emph{
There exists a constant $C>0$ such that $NC<1$ and
$\overline{F(t,x)\cap C \mathbb{B}\cap T_D(x)}\neq\emptyset$
for every $x\in D$ and a.a. $t\in I$}.
\end{remark}
\begin{proof}[Proof of Theorem \ref{nfth}]
Note first that due to (A3) if
$G_\varepsilon(t,x)= \overline{\rm co\,} F(([t-\varepsilon,t+\varepsilon]\cap
I)\setminus A, x+ \varepsilon \mathbb{B} \cap D)$ then
$|G_\varepsilon(t,x)|\leq C$, where $A$ is a null set and
$\mathbb{B}$ is the unit ball in $E$.
Let $0$ be a point of impulse. Then we consider
\eqref{i3}--\eqref{i4} with an initial condition $x_0+S_1(0)\in
D$. Consequently one can suppose without loss of generality that
$0$ is not impulsive point. Due to Theorem \ref{noim} the problem
\eqref{DI} admits a nonempty $C(I,E)$ compact solution set. Therefore
there exists $s:= \max \{\tau>0:$ every solution of \eqref{DI} is
continuous on $(0,s)\}$. If $s=1$ then the proof is complete.
Otherwise $s<1$ is an impulsive point for some solution
$x(\cdot)$, i.e. $s= \tau_1(x(s))$. Since $x(s)\in D$, one has
that $x':= x(s)+ S_1(x(s-0))\in D$. We study the problem
\eqref{i3}--\eqref{i4} on $[s,1]$ with an initial condition $x(s)=
x'$. Applying Theorem \ref{noim} and Lemma \ref{l1} one can derive
the existence of solution $x(\cdot)$ on
$[s=\tau_1(x),T=\tau_2(x))$, where $T>s$. Since $x(\cdot)$ is
$C$-Lipschitz, one has that $ \lim_{t \uparrow T}x(t)$ exists. If
$T<1$, then we study \eqref{i3}--\eqref{i4} on $[T,1]$. One can
extend the solution $x(\cdot)$ on the whole interval $I$ using the
same method.
Due to (A1) and (A2), there exists an interval $[0,s]$ (with $s> 0$)
such that every $x^i(\cdot)$ is continuous on $[0,s]$.
\end{proof}
\begin{remark} \label{rmk2} \rm
It is possible to prove local existence of solutions, when (A3)
is replaced by the linear growth assumption as in Theorem
\ref{noim}. However, in this case it is possible the solution to
exists only on some neighborhood of $0$ (not on the whole $I$).
\end{remark}
\begin{corollary} \label{cimmp}
Under the conditions of Theorem \ref{nfth} there exists a constant
$\lambda>0$ such that for every solution $y(\cdot)$ of
\eqref{i3}--\eqref{i4} $\tau_{i+1}(y(t))-\tau_i(y(t))\geq \lambda$,
$i=1,2,\dots,p-1$.
\end{corollary}
\begin{proof}
Suppose the contrary, i.e. there exist a sequence $ \{y^k(\cdot)
\}_{k=1}^\infty$ such that
\begin{equation}\label{conv}
\min_i \left(\tau_{i+1}(y^k(t)) - \tau_i(y^k(t))\right)\to 0\quad
\hbox{as }k\to \infty.
\end{equation}
Denote by $\tau_i^k$ the $i$-th impulse of $y^k(\cdot)$. Passing
to subsequences if necessary, we may assume that
$ \lim_{k\to\infty}\tau_i^k= \tau_i$. Since $y^k(\cdot)$ are
$C$ Lipschitz on every $ [\tau_i^k,\tau_{i+1}^k]$ there exists
a subsequence converging to a solution $y(\cdot)$ of
\eqref{i3}--\eqref{i4} with impulsive points $\tau_i$ for
$i=1,2,\dots,p$. Due do (\ref{conv}) either
$\tau_i(x(\tau_i)+S_i(x(\tau_i)))= \tau_{i+1}$
for some $i$-contradiction with (A1), or
$\tau_i=\tau_{i+1}$-contradiction with (A2).
\end{proof}
Now we study the mixed semicontinuous case, namely we assume that
$F(\cdot,\cdot)$ is almost USC with convex compact values in some
points and it is almost LSC with compact values in others.
Note that the most papers consider the case when $F(\cdot,\cdot)$
is either almost USC or it is almost LSC. The tedious proofs of the
existence result in mixed semicontinuous case were simplified in
separable Banach spaces in the very recent papers \cite{D2} and
\cite{FG}. Here we extend the results presented in these papers to
the case of differential inclusions with impulses.
Assume that $E$ is a separable Banach space. Let
$\mathcal{A}\subset I\times D$ and let $\mathcal{A}\in
\mathcal{L}\otimes \mathcal{B}$, where $\mathcal{L}$ is the class
of Lebesgue measurable subsets of $I$ and $\mathcal{B}$ -- the
class of Borel subsets of $E$. We require also that for every $t$
with $(t,x)\in A$ the projections $\{z\in D:\ (t,z)\in
\mathcal{A}\}$ be relatively open (in $D$).
\begin{itemize}
\item[(A4)] $F(\cdot,\cdot)$ is almost LSC on $\mathcal{A}$.
$F(\cdot,x)$ is measurable for every $x\in D$, $F(t,\cdot)$ is USC
with convex values on $(I\times D)\setminus \mathcal{A}$.
Moreover, there exists a constant $C$ such that
$|F(t,x)|\leq C$ for every $x\in D$ and a.e. $t\in I$, and $NC<1$.
\item[(A5)] There exists a null set $\mathcal{N}\subset I$ such that
$ F(t,x)\subset T_D(x)$ for every $(t,x)\in \mathcal{A}$ when
$t\notin \mathcal{N}$.
\end{itemize}
The following lemma is used in the proof of Theorem
\ref{mix} below.
\begin{lemma}[{\cite[Theorem 2]{BC}}] \label{BrCo}
Let $X,Z$ be two Banach spaces, let $\Omega\subset I \times X$
be nonempty and
let $M>0$. Then any closed valued LSC multifunction from $\Omega$
into $Z$ admits a $\Gamma^M$-continuous selection.
\end{lemma}
\begin{theorem} \label{mix}
Under assumptions (A1)--(A5) and (H1)--(H2),
the system \eqref{i3}--\eqref{i4} has a solution.
\end{theorem}
\begin{proof} Since $F(\cdot,\cdot)$ is almost LSC, one has that
there exists a sequence $\{J_n\}_{n=1}^\infty$ of pairwise
disjoint compacts $J_n\subset I$ such that $F(\cdot,\cdot)$ is LSC
on $(J_n\times D)\cap \mathcal{A}$ for every $n$. Furthermore,
its union is of full measure (without loss of generality we ca
assume that it is equal to $I\setminus \mathcal{N}$).
Let $\Omega_n= ( J_n\times D)\cap \mathcal{A}$.
Then define
\[
G(t,x)= \begin{cases}
F(t,x) & (t,x)\in \big(J_n\times D)\setminus \Omega_n \\
G_n(t,x) & (t,x)\in \Omega_n, \ n=1,2,\dots \\
0 & t\in \mathcal{N}
\end{cases}
\]
Here $ G_n(t,x)=\cap_{\varepsilon>0}
\overline{\rm co\,} f_n(A_\varepsilon)$,
$A_\varepsilon=\big([t- \varepsilon,t+\varepsilon]\times(x+
\varepsilon U)\big)\cap\Omega_n$, where $f_n(\cdot,\cdot)$
are $\Gamma^{C+1}$ continuous selections of $F(\cdot,\cdot)$ on
$\Omega_n$. It is easy to see that due to (H1)
$G(\cdot,\cdot)$ is almost USC.
There exists a
measurable selection of $T_D(x)\cap G(\cdot,x)$. From
\cite[proposition 5.1]{D} we know that there exists an almost USC
$G_0(t,x)\subset G(t,x)$ with convex and compact values satisfying
the conditions of Theorem \ref{nfth} such that for every measurable
$u(\cdot),v(\cdot)$ with $v(t)\in G(t,(u(t))$ it follows that
$v(t)\in G_0(t,(u(t))$. Let $D'=\{ x_i\}_{i=1}^\infty$ be a dense
subset of $D$. Fix $\varepsilon>0$. Let $f_i(t)\in G(t,x_i)\cap
T_D(x_i)$ be measurable. Therefore there exists a compact
$I_\varepsilon\subset I$ with $\mathop{\rm meas}(I_\varepsilon)>1-
\varepsilon$ such that $f_i(\cdot)$ are continuous on
$I_\varepsilon$ for every $i$. Due to (H1) for every $t\in
I_\varepsilon$ every sequence $\{ f_i(t)\}_{i= 1}^\infty$ has a
density point say $f_t$. Let $x_i\to x$, therefore $f_t\in
G_0(t,x)$, because $G_0(t,\cdot)$ is USC. Thus $G_0(\cdot,\cdot)$
admits nonempty values and hence $G_0(t,x)\cap T_D(x)\neq
\emptyset$. Therefore $G_0$ satisfies all the conditions of Theorem
\ref{nfth}.
Hence the system \eqref{i3}--\eqref{i4} with $F$ replaced by $G_0$
has a solution. \end{proof}
\section{Properties of the solution set of impulsive differential
inclusion}
Let $\mathfrak{Im}_{k,L}$ be the set of all functions $x(\cdot)$
which are $L$-Lipschitz on $[t_i(x)+0,t_{i+1}(x)]$ and have no more
than $k$ jump points $t_1(x)0$ we consider $
I_\varepsilon= \cup_{j=1}^k (t_j- \frac{\varepsilon}{k}, t_j+
\frac{\varepsilon}{k})$. It is easy to see that $L^1$ limit of
this (sub)sequence on $I\setminus I_\varepsilon$ is $L$-Lipschitz
function. Since it is valid for every $\varepsilon>0$ one can
conclude that the limit function $x(\cdot)$ is $L$-Lipschitz on
every $(t_j,t_{j+1})$. The $L^1$ limit is unique and hence
$x(\cdot)\in \mathfrak{Im}_{k,L}$.
\end{proof}
We will also need the following assumption.
\begin{itemize}
\item[(A6)]
The functions $S_i: D\to D$ are Lipschitz continuous with
constant $\mu$ such that $C\mu< 1$.
\end{itemize}
Let $E$ be a Banach space with single valued duality map $J:E\to
E^*$.
Recall that the map $F:I\times D \to E$ is said to be
One Sided Lipschitz (OSL) when there exists a constant $L$ such that
$h_F(t,x,J(x-y))- h_F(t,y,J(x-y))\leq L|x-y|^2$ for every $x,y\in
D$, where $h_F: I\times E\times E^*\to \mathbb{R}$ is the lower
Hamiltonian defined as
$h_F(t,x,p)=\inf\{\langle p,v\rangle:v\in F(t,x)\}$.
We refer to \cite{D1,D2} and the
references therein for theory of OSL differential inclusions.
We will use the following lemma which is a particular case
of \cite[Lemma 2]{PIK}.
\begin{lemma} \label{2PK}
Let $a_1, a_2, b\geq 0$ and for $i=1,2,\dots,p$ let
\[
\delta_i^+\leq a_1\delta_i^-,\quad
\delta_i^-\leq a_2\delta_{i-1}^+ + b
\]
then $ \delta_i^-\leq b\sum_{j=0}^{i-1}(a_1a_2)^j + \delta_0
(a_1a_2)^i$, where $\delta_0^+\geq 0$.
\end{lemma}
The following result is the well known relaxation theorem.
However, to our knowledge this theorem has not been studied in
case of impulsive differential inclusions.
We follow the proof from \cite{D1} (given there for system without
impulses).
\begin{theorem} \label{relax}
Let $F(\cdot,\cdot)$ be almost continuous with nonempty convex
compact values. Further we assume that it is OSL and $|F(t,x)|\leq
C$. If $ \overline{\mathop{\rm ext}}\ F(t,x)\subset T_D(x)$ for every
$x\in D$ and a.a. $t\in I$ then under (A1)--(A6), (H1),
the solution set of \eqref{*}--\eqref{**} is dense in the solution
set of \eqref{i3}--\eqref{i4}.
\end{theorem}
\begin{proof} Let $x(\cdot)$ be a solution of
\eqref{i3}--\eqref{i4}. Denote
$ R(t,x)= \overline{\mathop{\rm ext}} F(t,x(t))$.
Since $F(\cdot,\cdot)$ is almost continuous, one has
that $R(\cdot,\cdot)$ is almost LSC \cite[Lemma 2.3.7]{T}).
Define:
\[
G_\varepsilon(t,y)=\overline{\big\{ v\in R(t,x(t)):\ \langle
J(x(t)-y), \dot x(t) - v\rangle< L|x(t)-y|^2+\varepsilon^2/2
\big\}}.
\]
From \cite[prop. 2.62 p. 55 vol. I]{HP} we know that
$G_\varepsilon(\cdot,\cdot)$ is almost LSC with nonempty compact
values.
One can prove as in the proof of Theorem \ref{mix}
that the system
\begin{equation} \label{est}
\begin{gathered}
\dot y(t)\in G_\varepsilon(t,y(t)),\quad
y(0)=x_0 \hbox{ a.e. } t\in I= [0,1],\ t\neq \tau_i(y), \\
\Delta y|_{t=\tau_i(y)}= S_i(y(\tau_i(y)-0)),\quad
i=1,\dots,p,\ x(t)\in D
\end{gathered}
\end{equation}
has a solution. Indeed, let $g_\varepsilon(t,x)\in G_\varepsilon
(t,x)$ be almost $\Gamma^{C+1}$ continuous, i.e.
$g_\varepsilon (\cdot,\cdot)$ is $\Gamma^{C+1}$ continuous
on $I_k\times D$ ($k=1,2,\dots$), where $ \cup_{k=1}^\infty I_k$
has full measure and $I_k$ are nonempty pairwise disjoint compact sets.
We let $ g_k(t,x):= \cap_{\delta>0} \overline{\rm co\,}
g_\varepsilon\left((t-\delta, t+\delta) \cap I_k,
x+\beta\mathbb{B}\right)$ for $t\in I_k$. Define
\[
g(t,x):=
\begin{cases}
g_k(t,x) & t\in I_k,\ k=1,2,\ldots \\
0 & \text{ otherwise}.
\end{cases}
\]
It is easy to see that $g(t,x)\subset F(t,x)$ is almost USC and it
satisfies all the assumptions of Theorem \ref{nfth}. Hence
\eqref{i3}--\eqref{i4} with $F$ replaced by $g$ admits a solution.
Let $y(\cdot)$ be a solution of (\ref{est}). On every common
interval of continuity of $x(\cdot)$ and $y(\cdot)$ one has $
\langle J(x(t)-y(t)),\dot x(t) -\dot y(t) \rangle\leq
L|x(t)-y(t)|^2+ \varepsilon^2/ 2$. That is
\[
\frac{d}{dt}|x(t)-y(t)|^2 \leq 2L|x(t)-y(t)|^2+ \varepsilon^2.
\]
Consequently, on every such interval $[\tau,\nu]$ one has
$|x(t)-y(t)|^2 \leq e^{2L(t-\tau)}\delta^2+ \varepsilon^2
\int_\tau^t e^{2L(t-s)}\,ds$, where $\delta= |x(\tau)-y(\tau)|$.
Hence
\begin{equation} \label{OSL}
|x(t)-y(t)|\leq e^{L(t-\tau)}\delta + f(t)\varepsilon,
\end{equation}
where $ f(t)= \max \{ 1, e^{L(t-\tau)}\}$. Let $ a_1=
\frac{1+N+C\mu}{1-C\mu}$, let $ a_2=\max \{e^L,1\}$ and let $
b=\varepsilon e^{q(L)}$, where
\[
q(L)= \begin{cases}
\frac{e^L-1}{L} & L\neq 0 \\
1 & L=0.
\end{cases}
\]
We are ready to apply Lemma \ref{2PK}.
Let $\delta_0^+= |x_0-y_0|$ (when $y(0)= y_0\neq x_0$ in
(\ref{est})).
We will show that for fixed $\delta>0$ there exists
$\varepsilon( \delta) >0$ such that:
For every $0<\varepsilon< \varepsilon(\delta)$ there exists a
solution $y(\cdot)$ of (\ref{est}) such that $|x(t)-y(t)|< \delta$
for $ t\in I\setminus \cup_{i=1}^p [\tau_i^-, \tau_i^+]$, where
$\tau_i^-= \max\{\tau_i^x, \tau_i^y\}$ and $\tau_i^+=
\min\{\tau_i^x, \tau_i^y\}$. Moreover,
\[
\sum_{i=1}^p |\tau_i^x- \tau_i^y|< \delta.
\]
Here $ \tau_i^x$ and $\tau_i^y$ are the jump points of $x(\cdot)$
and $y(\cdot)$. First we assume that $ \tau_i^x< \tau_{i+1}^y$
and $ \tau_i^y<\tau_{i+1}^x$ and afterward we will see that for
sufficiently small $\varepsilon$ it is the case.
The rest of the proof is very similar to the proof of Theorem 2 of
\cite{PIK} and will be given, for reader convenience.
Due to (\ref{OSL}) one has $|x(t)-y(t)|\leq a_1\delta_0^+ + b$ on
$ [0,\tau_1^-]$. Hence $ \delta_1^-= |x(\tau_1^-0)-
x(\tau_1^-0)|\leq a_1\delta_0^+ + b$, because obviously $f(t)< b$
and $ a_2> e^{L(t-\tau)}$ for any interval $[\tau,\nu]$. Evidently
denoting $ \delta_i^+= |x(\tau_i^+ +0)-y(\tau_i^+ +0)|$ one has
that $\delta_{i-1}^-\leq a_2\delta_i^+ +b$.
If $ \tau_1^y< \tau_1^x$ then $ |x(\tau_i^+-0)- y(\tau_i^- -0)|
\leq \delta_i^- + |x(\tau_i^+-0)-x(\tau_i^-)|\leq \delta_i^- +
C(\tau_i^+-\tau_i^-)$. Consequently,
$\tau_i^+-\tau_i^-=|\tau_i(x(\tau_i^+-0)- \tau_i(y(\tau_i^-0)|\leq \mu
|x(\tau_i^+-0)- y(\tau_i^- -0)|\leq \mu\left(\delta_i^-+
C(\tau_i^+-\tau_i^-) \right)$ and hence
\begin{equation} \label{tau}
\tau_i^+-\tau_i^-\leq \frac{\mu\delta_i^-}{1-C\mu}.
\end{equation}
Therefore, $ |x(\tau_i^+-0)- y(\tau_i^- -0)| \leq \delta_i^- +
C\frac{\mu\delta_i^-}{1-C\mu}$. For $ \delta_i^+$ we have
\begin{align*}
\delta_i^+
& \leq \delta_i^-+ |x(\tau_i^+)- x(\tau_i^--0)| +
|y(\tau_i^+)- y(\tau_i^--0)|\\
& \leq \delta_i^- +|S(x(\tau_i^+-0)-S(y(\tau_i^+-0)| + \left|
\int_{\tau_i^-}^{\tau_i^+} \dot x(t)- \dot y(t)\,dt\right| \\
& \leq \delta_i^- + 2C(\tau_i^+-\tau_i^-)+ N|x(\tau_i^+-0)-
y(\tau_i^- -0)| \\
& \leq \delta_i^- + \frac{2C\mu\delta_i^-}{1-C\mu} + N\big(
\delta_i^- + C\frac{\mu\delta_i^-}{1-C\mu}\big)\\
& = a_1\delta_i^-.
\end{align*}
Due the symmetry one can conclude that $ \delta_i^+\leq a_1
\delta_i^-$ also when $ \tau_1^x\leq \tau_1^y$. This is true for
$i=1,2,\dots,p$. It follows from Lemma \ref{2PK} that $
\delta_i^-\leq b\sum_{j=0}^{i-1}(a_1a_2)^j + \delta_0 (a_1a_2)^i$.
We have only to see that $ \tau_i^x< \tau_{i+1}^y$ and $ \tau_i^y<
\tau_{i+1}^x$ to complete the proof. For sufficiently small
$\varepsilon$ it follows that
\[
|\tau_i^x-\tau_i^y|< \min_{0*0$ there exists
$\varepsilon(\delta)$ such that for every
$0<\varepsilon<\varepsilon(\delta)$ there exists a solution
$y(\cdot)$ of (\ref{est}) with $ \sum_{i=1}^p |\tau_i^x-
\tau_i^y|< \delta$.
\end{proof}
Further, in this section we assume that $E\equiv \mathbb{R}^n$.
We will study \eqref{i3}--\eqref{i4} with the help of the assumption
\begin{itemize}
\item[(H3)] There exists a null set $\mathcal{N}\subset I$ with
$h_F(t,x,\zeta)\leq 0$, for all $\zeta\in N_D^P(x)$, for all
$x\in S$, for all $t\in I\setminus \mathcal{N}$.
\end{itemize}
\begin{remark} \label{R1} \rm
The condition (H3) is weaker than (H2) when $E$ is a Hilbert
space, however it is not applicable in more general spaces.
\end{remark}
We assume further that $F(\cdot,\cdot)$ is almost USC with nonempty
convex compact values. The following theorem is proved for
autonomous $F(\cdot)$ in \cite{CLR,CLSW} and extended to
non-autonomous case in \cite{DRW}.
\begin{theorem}\label {bl}
Under assumptions (A1)--(A3), the system \eqref{i3}--\eqref{i4}
has a solution if and only if (H3) holds.
\end{theorem}
\begin{proof} Assume that $t(x_0)$ is not impulsive point. Therefore,
there exists a neighborhood $x_0+ \varepsilon\mathbb{B}$ where
$t(x)$ is not a jump point for every $x\in x_0+
\varepsilon\mathbb{B}$. Consequently from Theorem 1 of \cite{DRW} we
know that there exists $t'>t$ such that the system
\eqref{i3}--\eqref{i4} has a solution on $[t,t']$. One can continue
as in the proof of Theorem \ref{nfth}. If $x_0$ is an impulsive
point then we consider the system \eqref{i3}--\eqref{i4} with $x_0$
replaced by $x(0)+ S_1(x_0)$ (the solution after impulse).
The proof of only if part is omitted, because it is the same as the proof
in case without impulses (cf. \cite{DRW}).
\end{proof}
When $F(\cdot,\cdot)$ is defined on the whole space the invariance
problem becomes simpler. We will call the solutions $x(\cdot)$ which
belong to $D$ \emph{viable}.
\begin{def} \label{dl} \rm
The system \eqref{i3}--\eqref{i4} is called weakly invariant when
there exists a viable solution $x(\cdot)$. The system
\eqref{i3}--\eqref{i4} is said to be (strongly) invariant when all the
solutions are viable.
We will say that the multivalued map $G(t,x)\subset F(t,x)$ is a
submultifunction (of $F$) when $G(\cdot,\cdot)$ is almost USC with
nonempty convex compact values.
\end{def}
The following lemma is an extension of \cite[Proposition 3 ]{DRW}.
\begin{lemma}[Invariance principle] \label{Ip}
Suppose that (A1)--(A3) hold. If $F(t,\cdot)$ is OSL, then
the system \eqref{i3}--\eqref{i4}
is invariant if and only if the system
\begin{gather}
\dot x(t)\in G(t,x(t)),\quad x(0)=x_0\in D,\quad\hbox{a.e. } t\in
I= [0,1],\; t\neq \tau_i(x), \label{i3g} \\
\Delta x|_{t=\tau_i(x)}= S_i(x),\quad i=1,\dots,p,\; x(t)
\in D, \label{i4g}
\end{gather}
is weakly invariant for every submultifunction $G$.
\end{lemma}
\begin{proof} Let \eqref{i3}--\eqref{i4} be strongly invariant.
Since the system (\ref{i3g})--(\ref{i4g}) has a solution (thanks to
Theorem \ref{bl}, one has that it is strongly (and hence weakly)
invariant.
Let (\ref{i3g})--(\ref{i4g}) be weakly invariant for every
submultifunction $G$. Let $x(\cdot)$ be a solution of
\eqref{i3}--\eqref{i4}. We define the multifunction
\[
G(t,y)= \left\{ v\in F(t,y):\ \langle x(t)-y, \dot x(t)- v \rangle
\leq L|x(t)-y|^2 \right\}.
\]
Since $F(t,\cdot)$ is OSL, one has that $G(\cdot,\cdot)$ is nonempty
valued. Let $u,v \in G(t,y)$ then
$\langle x(t)-y, \dot x(t)-\lambda u + (1-\lambda)v \rangle
\leq \left( \lambda + (1 -
\lambda)v\right) L|x(t)-y|^2$, i.e. $G$ is convex valued. Let $F$ be
USC on $\mathcal{A}\times \mathbb{R}^n$, let $(t,y)\in \mathcal{A}
\times \mathbb{R}^n$ and let $\dot x(\cdot)$ is continuous on
$\mathcal{A}$. If $\mathcal{A}\ni t_i\to t$, $y_i\to y$ and
$G(t_i,y_i)\ni v_i\to v$, then
\[
\lim_{i\to \infty}\langle x(t_i)-y_i, \dot x(t_i)- v_i \rangle =
\langle x(t)-y, \dot x(t)- v \rangle.
\]
Moreover, $ \lim_{i\to \infty} L|x(t_i)-y_i|^2 = L|x(t)-y|^2$. Thus
$G(\cdot,\cdot)$ is almost USC (because $G(t,x)\subset F(t,x)$).
Therefore, $G$ is a submultifunction of $F$. Let $y(\cdot)$ be
viable solution of (\ref{i3g})--(\ref{i4g}). If $[\mu,\nu]$ is an
interval without impulses of $x(\cdot)$ and $y(\cdot)$, then $
\langle x(t)-y(t), \dot x(t)- \dot y(t)\rangle \leq L|x(t)-y(t)
|^2$. Thus $ \frac{d}{dt}|x(t)-y(t)|\leq 2L|x(t)-y(t)|^2$. If
$x(\mu)=y(\mu)$, then $x(t)\equiv y(t)$ on $[\mu,\nu]$ thanks to
Gronwall inequality.
However, $x(0)=y(0)=x_0$. Consequently $x(t)\equiv y(t)$ on the
whole interval $I$ and hence $x(t)\in D$ for every $t\in I$. The
last implies that \eqref{i3}--\eqref{i4} is invariant.
\end{proof}
The following theorem is an immediate corollary of Theorem \ref{bl}
and Lemma \ref{Ip}. It extends \cite[Proposition 3]{DRW} to
impulsive systems.
\begin{theorem} \label{corl}
Let the conditions of Lemma \ref{Ip} hold. Then system
\eqref{i3}--\eqref{i4} is invariant if and only if for
every submultifunction $G$ there exists a null set $\mathcal{N}_G$
such that $ h_G(t,x,\zeta)\leq 0$, for all $\zeta\in N_D^P(x)$,
for all $x\in S$, for all $t\in I\setminus \mathcal{N}_G$.
\end{theorem}
Recall that $A$ is said to be absolute (metric)
retract \cite[p. 83]{D} if, given a metric space $\Omega$,
closed $B\subset\Omega$
and continuous $f : B\to A$, there exists a continuous
extension $\tilde f :\Omega\to A$ of $f$. $A$ is said to
be $R_\delta$ if $ A = \cap_{k\ge 1} A_k$ for decreasing
sequence of compact absolute retracts $A_k$.
The set $B$ is said to be contractible if there is $x_0\in B$ and
continuous $h:[0,1]\times B\to B$ such that $h(0,x)=x$ and
$h(1,x)=x_0$ on $B$.
It is well known that $A$ is $R_\delta$ if and only if
$A=\cap_{n\ge 1}B_n$ with decreasing sequence of closed
contractible sets (cf. \cite{H}).
\begin{theorem} \label{solset}
Let $D\equiv \mathbb{R}^n$ and let $F(\cdot,\cdot)$ be almost USC
with nonempty convex compact values. Under assumption (A1)--(A3) and
(A6) the solution set of differential inclusion
\eqref{i3}--\eqref{i4} is nonempty $R_\delta$ in
$\mathfrak{Im}_{p,L}$.
\end{theorem}
\begin{proof} Due to Lemma \ref{l1} the solution set of
\eqref{i3}--\eqref{i4} is in $\mathfrak{Im}_{p,L}$
(it is nonempty thanks to Theorem \ref{nfth}). Now we will use
the locally Lipschitz approximation of USC multifunctions.
Let $ I\setminus\mathcal{I} = \cup_{j=1}^\infty I_j$ be a
sequence of pairwise disjoint compacts such that $F$ is USC on
$I_j\times E$ and $\mathop{\rm meas}(\mathcal{I})=0$. Define
$F_n(t,x)= \sum_{\lambda\in \Lambda}\psi_\lambda(t,x)C_\lambda$ on
$I_j\times E$ with $ C_\lambda:= \overline{\rm co\,}
F(t,x+2r_n\mathbb{B})$, where
$ r_n=3^{-n}$. It is easy to see that
$F(t,x)\subset F_{n+1}(t,x)\subset F_n(t,x)\subset
\overline{\rm co\,} F(t,x+2r_n\mathbb{B})$.
We can take a strongly measurable selection $g_\lambda$ of
$F(\cdot,x_\lambda)$ and define $ f(t,x)= \sum_\lambda
\varphi_\lambda(x)g_\lambda(t)$. Therefore, $f(\cdot,x)$ is
strongly measurable and $f(t,\cdot)$ is locally Lipschitz
(cf. \cite[Lemma 2.2]{D}). Consequently, the system
\begin{gather}
\dot x(t)= f(t,x(t)),\hbox{ a.e. on } I,\ x(t)=y\
t\neq \tau_i(x), \label{e1} \\
\Delta x|_{t=\tau_i(x)}= S_i(x(\tau_i(x)-0)),\ i=1,\dots,p,
\label{e2}
\end{gather}
admits a unique solution, which depends continuously on $(t,y)$
(cf. \cite{BS,SP}). Thus the solution set of
\eqref{i3}--\eqref{i4} with $F$ replaced by $F_n$ has a nonempty
contractible solution set $Sol_n$ (cf. \cite[p. 82]{D} ). Furthermore
it is easy to see that the solution set $Sol$ of
\eqref{i3}--\eqref{i4} satisfies $ Sol= \cap_{n=1}^\infty
Sol_n$. Consequently $Sol$ is $R_\delta$ set.
\end{proof}
\begin{thebibliography}{00}
\bibitem{A1} Aubin J.-P., {\it Impulsive Differential Inclusions
and Hybrid Systems: A Viability Approach} Lecture Notes, Univ. Paris
2002.
\bibitem{A} Aubin J.-P., {\it Viability Theory}, Birkh\"auser,
Boston, 1991.
\bibitem{AC} Aubin J.-P., Cellina A., {\it Differential Inclusions},
Springer, Berlin, 1984.
\bibitem{Aol} Aubin J.-P., Lygeros J., Quincampoix M., Sastry S.,
Seube N., Impulse differential inclusions: a viability approach to
hybrid systems, {\it IEEE Tras. Aotomat. Control} {\bf 47(1)} (2002)
2-20.
\bibitem{BS} Bainov D., Simeonov P., {\it Systems with Impulsive
Effect. Stability, Theory and Applications}, Ellis Series in Maths
and Appl., Ellis Horwood, Chicester 1989.
\bibitem{BB3} Benchohra M., Boucherif A., On first order initial
value problems for impulsive differential inclusions in Banach
spaces, Dyn. Syst. Appl. {\bf 8} (1999) 119-126.
\bibitem{BHN} Benchohra M., Henderson J., Ntouyas S., {\it Impulsive
Differential Equations and Inclusions}, Hindawi Publishing Company,
New York (in press).
\bibitem{BHNO} Benchohra M., Henderson J., Ntouyas S., Ouahabi A.,
existence results for impulsive lower semicontinuous differential
inclusions, {\it Int. J. Pure Appl. Math.} {\bf 1} (2002) 431-443.
\bibitem{Bo} Bothe D., {\it Multivalued Differential Equations on
Graphs and Applications}, Ph.D. Thesis, Paderborn, 1992.
\bibitem{BC} Bressan A., Colombo G., Selections and representations
of multifunctions in paracompact spaces, {\it Studia Math.} {\bf
102} (1992) 209-216.
\bibitem{CS} Cardinali T., Servadei R., Periodic solutions of
nonlinear impulsive differential inclusions with constrains, {\it
Proc. Amer. Math. Soc.} {\bf 132} (2004) 2339-2349.
\bibitem{CLR} Clarke F., Ledyaev Yu., and Radulescu M., Approximate
invariance and differential inclusions in Hilbert spaces, {\it J.
Dynam. Control Syst.} {\bf 3} (1997) 493-518.
\bibitem{CLSW} Clarke F., Ledyaev Yu., Stern R. and Wolenski P., {\em
Nonsmooth Analysis and Control Theory,} Springer, New York, 1998.
\bibitem{D} Deimling K., {\it Multivalued Differential Equations},
De Gruyter Berlin, 1992.
\bibitem{D1} Donchev T., One Sided Lipschitz Multifunctions and
Applications, Proc. Louisiana Math. Control Conf., in {\it Optimal
Control, Stabilization and Nonsmooth Analysis}, de Querioz,
Malisoff, Wolenski (Eds.), {\it Lect. Notes Contr. Inf. Sci.} {\bf
301}, Springer, Berlin, 2004, pp. 333-342.
\bibitem{D2} Donchev T., Mixed type semicontinuous differential
inclusions in Banach spaces, {\it Ann. Polon. Math.} {\bf LXXVII.3}
(2001) 245-259.
\bibitem{DRW} Donchev T., Rios V., Wolenski P., Strong Invariance
and one-sided Lipschitz multifunctions, {\it Nonlinear Analysis
TMA} {\bf 60} (2005) 849-862
\bibitem{FR2} Frigon M., O'Regan D., First order impulsive initial
and periodic problems with variable moments, {\it J. Math. Anal.
Appl.} {\bf 233} (1999) 730-739.
\bibitem{FG} Fryszkowski A., Gornievicz L., Mixed Semicontinuous
Mappings and Their Applications to Differential Inclusions, {\it
Set-Valued Analysis} {\bf 8} (2000) 203-217.
\bibitem{HP} Hu S., Papageorgiou N., {\it Handbook of Multivalued
Analysis}, vol. I {\it Theory} 1997; vol. II {\it Applications}
2000, Kluwer Dodrecht.
\bibitem{H} Hyman D., On decreasing sequences of compact absolute
retracts, {\it Fundamenta Math.} {\bf 64} (1969) 91-97.
\bibitem{KL} Kaul S., Liu X., Vector Lyapunov functions for
impulsive differential systems with variable times, {\it Dynam.
Contin. Discrete Impuls. Systems} {\bf 6} (1999) 25-38.
\bibitem{LBS} Lakshmikantham V., Bainov D., Simeonov P., {\it Theory
of Impulsive Differential Equations}, World Scientific, Singapore,
1989.
\bibitem{P} Pianigiani G., On the fundamental theory of multivalued
differential equations, {\it J. Differential Equations} {\bf 25}
(1977) 30-38.
\bibitem{PIK} Plotnikov V., Kitanov N., On Continuous dependence
of solutions of impulsive differential inclusions and impulse
control problems, {\it Kybern. Syst. Analysis} (2002) {\bf No 5}
143 -154 (in Russian); English translation: Cybernetics System
Analysis {\bf 38} (2002) 749-758.
\bibitem{PIK1} Plotnikov V., Ivanov R., Kitanov N., Method of
averaging for impulsive differential inclusions, {\it Pliska. Stud.
Math. Bulgar.} {\bf 12} (1998) 43-55.
\bibitem{PPV} Plotnikov V., Plotnikov A., Vityuk A., {\it
Differential Equations with Multivalued Right-Hand Side. Asymptotic
Methods}, Astro Print Odessa, 1999. (Russian)
\bibitem{SP} Samoilenko A., Peresyuk N., {\it Differential Equations
with Impulsive Effects}, World Scientific, Singapore, 1995.
\bibitem{T} Tolstonogov A., {\it Differential Inclusions in a Banach
Space}, Kluwer, Dordrecht, 2000.
\bibitem{W} Watson P., Impulsive differential inclusions, {\it
Nonlin. World} {\bf 4} (1997) 395-402.
\end{thebibliography}
\end{document}
*