\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 53, pp. 1--7.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/53\hfil Filippov's theorem]
{Continuous version of Filippov's theorem for a
Sturm-Liouville type differential inclusion}
\author[A. Cernea\hfil EJDE-2008/53\hfilneg]
{Aurelian Cernea}
\address{Aurelian Cernea\newline
Faculty of Mathematics and Informatics,
University of Bucharest,
Academiei 14, 010014 Bucharest, Romania}
\email{acernea@math.math.unibuc.ro}
\thanks{Submitted January 22, 2008. Published April 10, 2008.}
\subjclass[2000]{34A60}
\keywords{Lower semicontinuous multifunction;
selection; solution set}
\begin{abstract}
Using Bressan-Colombo results, concerning the existence of
continuous selections of lower semicontinuous multifunctions with
decomposable values, we prove a continuous version of Filippov's
theorem for a Sturm-Liuoville differential inclusion. This result
allows to obtain a continuous selection of the solution set of the
problem considered.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{hypothesis}[theorem]{Hypothesis}
\section{Introduction}
In this paper we study second-order differential inclusions of the
form
\begin{equation}
(p(t)x'(t))'\in F(t,x(t))\quad \text{a. e. in }[0,T]),\quad x(0)=x_0,\quad
x'(0)=x_1,\label{e1.1}
\end{equation}
where $F:[0,T]\times X\to \mathcal{P}(X)$ is a set-valued map, $X$
is a separable Banach space, $x_0,x_1\in X$ and
$p:[0,T]\to (0,\infty )$ is continuous.
In some recent papers \cite{c4,l1} several existence results for
problem \eqref{e1.1} are obtained using fixed point techniques.
In \cite{c3} it is shown that Filippov's ideas \cite{f1}
can be suitably adapted in
order to prove the existence of mild solutions to problem \eqref{e1.1}.
The aim of this paper is to prove the existence of solutions
continuously depending on a parameter for the problem \eqref{e1.1}. Our
result may be interpreted as a continuous variant of the
celebrated Filippov's theorem \cite{f1} for problem \eqref{e1.1}. In
addition, as usual at a Filippov existence type theorem, our
result provides an estimate between the starting "quasi" solution
and the solution of the differential inclusion. At the same time
we obtain a continuous selection of the solution set of problem
\eqref{e1.1}
The key tool in the proof of our theorem is a result of Bressan
and Colombo \cite{b1} concerning the existence of continuous
selections of lower semicontinuous multifunctions with
decomposable values. The proof follows the general ideas as in
\cite{a1,c1,c2,c5,s1},
where similar results are obtained for other classes
of differential inclusions.
The paper is organized as follows: in Section 2 we present the
notations, definitions and the preliminary results to be used in
the sequel and in Section 3 we prove our results.
\section{Preliminaries}
Let $T>0$, $I:=[0,T]$ and denote by $\mathcal{L}(I)$ the $\sigma
$-algebra of all Lebesgue measurable subsets of $I$. Let $X$ be a
real separable Banach space with the norm $|\cdot|$. Denote by
$\mathcal{P}(X)$ the family of all nonempty subsets of X and by
$\mathcal{B}(X)$ the family of all Borel subsets of $X$. If
$A\subset I$ then $\chi _A:I\to \{0,1\}$ denotes the
characteristic function of $A$. For any subset $A\subset X$ we
denote by $\mathop{\rm cl}(A)$ the closure of $A$.
As usual, we denote by $C(I,X)$ the Banach space of all
continuous functions $x:I\to X$ endowed with the norm
$|x(.)|_C=\mbox{sup}_{t\in I}|x(t)|$ and by $L^1(I,X)$ the Banach
space of all (Bochner) integrable functions $x(.):I\to X$ endowed
with the norm $|x(.)|_1=\int_0^T|x(t)|dt$.
We recall first several preliminary results we shall use in the sequel.
\begin{lemma}[\cite{z1}] \label{lem2.1}
Let $u:I\to X$ be measurable and let
$G:I\to \mathcal{P}(X)$ be a measurable closed-valued
multifunction.
Then, for every measurable function $r:I\to (0, \infty )$, there
exists a measurable selection $g:I\to X$ of $G(\cdot )$ (i.e. such
that $g(t)\in G(t)$ a.e. (I)) such that
$$
|u(t)-g(t)| < d(u(t), G(t)) + r(t)\quad \text{a.e. in }(I),
$$
where the distance between a point $x\in X$ and a subset
$A\subset X$ is defined as usual by
$d(x, A) = \inf \{ |x-a| :a\in A\}$.
\end{lemma}
\begin{definition} \label{def2.2} \rm
A subset $D\subset L^1(I,X)$ is said to be
\emph{decomposable} if for any $u(\cdot ), v(\cdot )\in D$ and any
subset $A\in \mathcal{L}(I)$ one has $u\chi _{A} +v\chi _{B} \in
D$, where $B = I\backslash A$. We denote by $\mathcal{D}(I,X)$ the
family of all decomposable closed subsets of $L^1(I,X)$.
\end{definition}
Next $(S, d)$ is a separable metric space; we recall that a
multifunction $G(\cdot ) :S\to \mathcal{P}(X)$ is said to be lower
semicontinuous (l.s.c.) if for any closed subset $C\subset X$, the
subset $\{s\in S; G(s)\subset C \}$ is closed.
\begin{lemma}[\cite{b1}] \label{lem2.3}
Let $F^{*}:I\times S\to \mathcal{P}(X)$ be a closed-valued
$\mathcal{L}(I)\otimes \mathcal {B}(S)$-measurable multifunction
such that $F^{*}(t,.)$ is l.s.c. for any $t\in I$.
Then the multifunction $G:S\to \mathcal{D}(I,X)$ defined by
$$
G(s) = \{ v\in L^1(I,X):v(t)\in F^{*}(t, s)\; a.e.\; (I)
\}
$$
is l.s.c. with nonempty closed values if and only if there exists
a continuous mapping $q:S\to L^1(I,X)$ such that
$$
d(0, F^{*}(t, s))\leq q(s)(t)\quad \text{a.e. in } (I),\; \forall
s\in S.
$$
\end{lemma}
\begin{lemma}[\cite{b1}] \label{lem2.4}
Let $G(.):S\to \mathcal{D}(I,X)$ be a
l.s.c. multifunction with closed decomposable values and let
$\phi (.):S\to L^1(I,X)$, $\psi :S\to L^1(I,\mathbb{R})$ be
continuous such that the multifunction $H:S\to
\mathcal{D}(I,X)$ defined by
$$
H(s) = \mathop{\rm cl}\{v\in G(s): |v(t)-\phi (s)(t)| < \psi
(s)(t) \text{ a. e. } (I) \}
$$
has nonempty values.
Then $H(.)$ has a continuous selection, i.e. there exists a continuous
mapping $h:S\to L^1(I,X)$ such that
$$
h(s)\in H(s) \quad \forall s\in S.
$$
\end{lemma}
Consider $F:I\times X\to \mathcal{P}(X)$ a set-valued map,
$x_0,x_1\in X$ and $p:I\to (0,\infty )$ a continuous mapping
that defined the Cauchy problem \eqref{e1.1}.
A continuous mapping $x\in C(I,X)$ is called a solution of
problem \eqref{e1.1} if there exists a (Bochner) integrable function
$f\in L^1(I,X)$ such that:
\begin{gather}
f(t)\in F(t,x(t))\quad \text{a.e. } (I),\label{e2.1} \\
x(t)=x_0+p(0)x_1\int_0^t\frac{1}{p(s)}ds
+\int_0^t\frac{1}{p(s)}\int_0^sf(u)du\,ds\quad
\forall t\in I.\label{e2.2}
\end{gather}
Note that, if we denote $G(t):=\int_0^t\frac{1}{p(s)}$, $t\in I$,
then \eqref{e2.2} may be rewrite as
\begin{equation}
x(t)=x_0+p(0)x_1G(t)+\int_0^tG(t-u)f(u)du\quad \forall t\in
I,\label{e2.3}
\end{equation}
We shall call $(x(.),f(.))$ a \emph{trajectory-selection pair} of
\eqref{e1.1} if \eqref{e2.1} and \eqref{e2.2} are satisfied.
We shall use the following notation for the solution sets of
\eqref{e1.1}.
\begin{equation}
\mathcal{S}(x_0,x_1)=\{x: x\mbox{ is a solution of \eqref{e1.1}}\}.
\label{e2.4}
\end{equation}
\section{The main results}
To establish our continuous version of Filippov theorem
for problem \eqref{e1.1} we need the following hypotheses.
\begin{hypothesis} \label{hyp3.1} \rm
\begin{itemize}
\item[(i)] $F:I\times X\to \mathcal{P}(X)$
has nonempty closed values and is $\mathcal{L}(I)\otimes
\mathcal{B}(X)$ measurable.
\item[(ii)] There exists $L(.)\in L^1(I,\mathbb{R}_+)$ such that,
for almost all $t\in I, F(t,.)$ is $L(t)$-Lipschitz in the sense
that
$$
d_H(F(t,x),F(t,y))\leq L(t)|x-y| \quad \forall x, y\in X,
$$
where $d_H(.,.)$ is the Hausdorff distance
$$
d(A,B)=\max\{d^*(A,B), d^*(B,A)\},\quad d^*(A,B)=\sup\{d(a,B); a\in A\}
$$
\end{itemize}
\end{hypothesis}
\begin{hypothesis} \label{hyp3.2}
\begin{itemize}
\item[(i)] $S$ is a separable metric space and
$a,b:S\to X$, $c(.):S\to (0,\infty )$ are continuous
mappings.
\item[(ii)] There exists the continuous mappings
$g(.),q(.):S\to L^1(I,X)$, $y:S\to C(I,X)$ such that
\begin{gather*}
(p(t)(y(s))'(t))'=g(s)(t)\quad \forall s\in S,t\in I,\\
d(g(s)(t),F(t,y(s)(t))\leq q(s)(t)\quad \text{a.e. } (I),\; \forall \;
s\in S.
\end{gather*}
\end{itemize}
\end{hypothesis}
Let $M:=\sup_{t\in I}\frac{1}{p(t)}$. Note that $|G(t)|\leq Mt$
for all $t\in I$.
For the next result, we use the following notation:
$m(t)=\int_0^tL(u)du$ and
\begin{equation}
\begin{aligned}
\xi (s)(t)&=e^{MTm(t)}\Big(tMTc(s)+|a(s)-y(s)(0)|+
MTp(0)|b(s)-(y(s))'(0)|\Big)\\
&\quad +MT\int_0^tq(s)(u)e^{MT(m(t)-m(u))}du.
\end{aligned}\label{e3.1}
\end{equation}
\begin{theorem} \label{thm3.3}
Assume that Hypotheses 3.1 and 3.2 are satisfied.
Then there exist the continuous mappings $x:S\to C(I,X)$,
$f:S\to L^1(I,X)$ such that for any $s\in S$,
$(x(s)(.),f(s)(.))$ is a trajectory-selection pair of
$$
(p(t)x'(t))'\in F(t,x(t)),\quad x(0)=a(s),\quad x'(0)=b(s)
$$
and
\begin{gather}
|x(s)(t)-y(s)(t)|\leq \xi (s)( t)\quad \forall (t, s)\in I\times
S,\label{e3.2} \\
|f(s)(t)-g(s)(t)| \leq L(t)\xi (s)(t) + q(s)(t) + c(s)\quad \text{a.e. }
(I), \; \forall s\in S.\label{e3.3}
\end{gather}
\end{theorem}
\emph{Proof.} We make the following notations $\varepsilon
_n(s)=c(s)\frac{n+1}{n+2}$, $n=0,1,...$,
$d(s)=|a(s)-y(s)(0)|+MTp(0)|b(s)-(y(s))'(0)|$,
\begin{align*}
q_n(s)(t)&=(MT)^n\int_0^tq(s)(u)\frac {(m(t)-m(u))^{n-1}}{(n-1)!}du\\
&\quad + (MT)^{(n-1)}
\frac{(m(t))^{n-1}}{(n-1)!}(MTt\varepsilon _n(s)+d(s)),\quad
n\geq 1.
\end{align*}
Set also $x_0(s)(t)=y(s)(t)$, $\forall s\in S$.
We consider the multifunctions $G_0(.),H_0(.)$ defined,
respectively, by
\begin{gather*}
G_0(s)=\{v\in L^{1}(I,X):v(t)\in F(t,y(s)(t))\quad a.e.\,
(I)\}, \\
H_0(s)=\mathop{\rm cl}\{v\in G_0(s):|v(t)-g(s)(t)|<
q(s)(t)+\varepsilon _0(s)\}.
\end{gather*}
Since $d(g(s)(t),F(t,y(s)(t))\leq q(s)(t)