\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 $d(g(s)(t),F(t,y(s)(t))\leq q(s)(t)<q(s)(t)+\varepsilon
_0(s)$, according with Lemma \ref{lem2.1}, the set $H_0(s)$ is not empty.

Set $F_0^*(t,s)=F(t,y(s)(t))$ and note that
$$
d(0,F_0^*(t,s))\leq |g(s)(t)| + q(s)(t)=q^*(s)(t)
$$
and $q^*(.):S\to L^1(I,X)$ is continuous.

Applying now  Lemmas \ref{lem2.3} and \ref{lem2.4} we obtain the existence of a
continuous selection $f_0$ of $H_0$, i.e. such that
\begin{gather*}
f_0(s)(t)\in F(t,y(s)(t))\quad a.e.\,  (I),\; \forall s\in S, \\
|f_0(s)(t)-g(s)(t)|\leq q_0(s)(t)=q(s)(t)+\varepsilon _0(s)\quad
\forall s\in S,\; t\in I.
\end{gather*}

We define $x_1(s)(t)=a(s)+p(0)G(t)b(s)+\int_0^tG(t-u)f_0(s)(u)du$
and one has
\begin{align*}
&|x_1(s)(t)-x_0(s)(t)|\\
&\leq |a(s)-y(s)(0)|+MTp(0)|b(s)-(y(s))'(0)|
+MT\int_0^t|f_0(s)(u)- g(s)(u)|du\\
&\leq d(s)+MT\int_0^tq_0(s)(u)(u)du
+MTt\varepsilon _0(s)\leq q_1(s)(t).
\end{align*}

We shall construct, using the same idea as in \cite{c5}, two sequences
of approximations $f_n:S\to L^1(I,X)$, $x_n:S\to C(I,X)$
with the following properties
\begin{itemize}
\item[(a)] $f_n(.):S\to L^1(I,X)$, $x_n(.):S\to C(I,X)$ are continuous.

\item[(b)] $f_n(s)(t)\in F(t,x_n(s)(t))$, a.e. $(I)$, $s\in S$.

\item[(c)] $|f_n(s)(t)-f_{n-1}(s)(t)|\leq L(t)q_n(s)(t)$, a.e. $(I)$,
$s\in S$.

\item[(d)] $x_{n+1}(s)(t)=a(s)+p(0)G(t)b(s)+\int_0^tG(t-u)f_n(s)(u)du$,
$\forall t\in I, s\in S$.
\end{itemize}
Suppose we have already constructed $f_i(.),x_i(.)$ satisfying
(a)-(c) and define $x_{n+1}(.)$ as in (d). From (c) and (d) one has
\begin{equation}
\begin{aligned}
&|x_{n+1}(s)(t)-x_n(s)(t)| \\
&\leq MT\int_0^t|f_n(s)(u)-f_{n-1}(s)(u)|du\\
&\leq MT\int_0^tL(u)q_n(s)(u)du\\
&=(MT)^{n+1}\int_0^tq(s)(u)\frac {(m(t)-m(u))^n}{n)!}du
+ (MT)^n\frac{(m(t))^n}{n!}(MTt\varepsilon _n(s)+d(s))\\
&<q_{n+1}(s)(t).
\end{aligned}\label{e3.4}
\end{equation}
On the other hand,
\begin{equation}
d(f_n(s)(t),F(t,x_{n+1}(s)(t))\leq
L(t)|x_{n+1}(s)(t)-x_n(s)(t)|
<L(t)q_{n+1}(s)(t). \label{e3.5}
\end{equation}

Consider the following multifunctions for any $s\in S$
\begin{gather*}
G_{n+1}(s)=\{v\in L^{1}(I,X):v(t)\in F(t,x_{n+1}(s)(t))\quad
a.e.\, (I)\},
\\
H_{n+1}(s)=\mathop{\rm cl}\{v\in G_{n+1}(s):|v(t)-f_n(s)(t)|<
L(t)q_{n+1}(s)(t)\; a.e.\, (I)\}.
\end{gather*}
To prove that $H_{n+1}(s)$ is nonempty we note first that the real
function $t\to
r_n(s)(t)=c(s)\frac{(MT)^{n+1}tL(t)(m(t))^n}{(n+2)(n+3)n!}$ is
measurable and strictly positive for any $s$. Using \eqref{e3.5} we get
\begin{align*}
d(f_n(s)(t),F(t,x_{n+1}(s)(t))
&\leq L(t)|x_{n+1}(s)(t)-x_n(s)(t)|-r_n(s)(t)\\
&\leq L(t)q_{n+1}(s)(t)
\end{align*}
 and therefore according to Lemma \ref{lem2.1} there
exists $v\in L^1(I,X)$ such that $v(t)\in F(t,x_n(s)(t))$
a.e. $(I)$ and
$$
|v(t)-f_n(s)(t)|<d(f_n(s)(t),F(t,x_n(s)(t))+r_n(s)(t)
$$
and hence $H_{n+1}(s)$ is not empty.

Set $F_{n+1}^*(t,s)=F(t,x_{n+1}(s)(t))$ and note that we may write
\begin{align*}
d(0,F_{n+1}^*(t,s))
&\leq L(t)|x_{n+1}(s)(t)-x_n(s)(t)|\\
&\leq |f_n(s)(t)|+L(t)q_{n+1}(s)(t)\\
&=q_{n+1}^{*}(s)(t) \quad \text{a.e. } (I)
\end{align*}
and $p_{n+1}^*:S\to L^1(I,X)$ is continuous.

By Lemmas \ref{lem2.3} and \ref{lem2.4} there exists a continuous map
$f_{n+1}:S\to L^1(I,X)$ such that
\begin{gather*}
f_{n+1}(s)(t)\in F(t,x_{n+1}(s)(t))\quad \text{a.e. } (I),\forall s\in
S, \\
|f_{n+1}(s)(t)-f_n(s)(t)|\leq L(t)q_{n+1}(s)(t)\quad a.e. \,
(I),\forall s\in S.
\end{gather*}
From \eqref{e3.4} and (d) we obtain
\begin{equation}
\begin{aligned}
|x_{n+1}(s)(.)-x_n(s)(.)|_C
&\leq MT|f_{n+1}(s)(.)-f_n(s)(.)|_1\\
&\leq \frac {(MTm(T))^{n}}{n!}(MT|q(s)(.)|_1+MT^2c(s)+d(s)).
\end{aligned}\label{e3.6}
\end{equation}

Therefore $f_n(s)(.)$, $x_n(s)(.)$ are Cauchy sequences in the
Banach space $L^1(I,X)$ and $C(I,X)$, respectively. Let $f(.):S\to
L^1(I,X)$, $x(.):S\to C(I,X)$ be their limits. The function $s\to
MT|q(s)(.)|_1+MT^2c(s)+d(s)$ is continuous, hence locally bounded.
Therefore \eqref{e3.6} implies that for every $s'\in S$ the sequence
$f_n(s')(.)$ satisfies the Cauchy condition uniformly with respect
to $s'$ on some neighborhood of $s$. Hence, $s\to f(s)(.)$ is
continuous from $S$ into $L^1(I,X)$.

From \eqref{e3.6}, as before, $x_n(s)(.)$ is Cauchy in $C(I,X)$ locally
uniformly with respect to $s$. So, $s\to x(s)(.)$ is continuous
from $S$ into $C(I,X)$. On the other hand, since $x_n(s)(.)$
converges uniformly to $x(s)(.)$ and
$$
d(f_n(s)(t),F(t,x(s)(t))\leq L(t)|f_n(s)(t)-x(s)(t)|\quad a.e.\;
(I),\; \forall s\in S
$$
passing to the limit along a subsequence of $f_n(.)$ converging
pointwise to $f(.)$ we obtain
$$
f(s)(t)\in F(t,x(s)(t))\quad a.e.\; (I),\; \forall s\in S.
$$
Passing to the limit in d) we obtain
$$
x(s)(t)=a(s)+p(0)G(t)b(s)+\int_0^tG(t-u)f(s)(u)du.
$$
By adding inequalities (c) for all $n$ and using the fact that
$\sum _{i\geq 1}q_{i}(s)(t)\leq \xi (s)(t)$ we obtain
\begin{equation}
\begin{aligned}
|f_{n+1}(s)(t)-g(s)(t)|
&\leq \sum_{l=0}^n|f_{l+1}(s)(u)-f_l(s)(u)|+ |f_0(s)(t)-g(s)(t)|\\
&\leq\sum_{l=0}^nL(t)q_{l+1}(s)(t)+q(s)(t)+\varepsilon _0(s)\\
& \leq L(t)\xi (s)(t) + q(s)(t) + c(s).
\end{aligned}\label{e3.7}
\end{equation}
Similarly, by adding \eqref{e3.4} we get
\begin{equation}
|x_{n+1}(s)(t)-y(s)(t)|\leq \sum_{l=0}^nq_l(s)(t)\leq \xi
(s)(t).\label{e3.8}
\end{equation}

By passing to the limit in \eqref{e3.7} and \eqref{e3.8} we
obtain \eqref{e3.2} and
\eqref{e3.3}, respectively.

Theorem \ref{thm3.3} allows to obtain the next corollary which is a general
result concerning continuous selections of the solution set of
problem \eqref{e1.1}.

\begin{hypothesis} \label{hyp3.4} \rm
Hypothesis \ref{hyp3.1} is satisfied and there
exists $q_0\in L^1(I,\mathbb{R}_+)$ such that
$d(0,F(t,0))\leq q_0(t)$ a.e. $(I)$.
\end{hypothesis}

\begin{theorem} \label{thm3.5}
Assume that Hypothesis \ref{hyp3.4} are satisfied.
Then there exists a function $x:I\times X^2\to X$ such that
\begin{itemize}
\item[(a)] $x(.,(\xi ,\eta ))\in \mathcal{S}(\xi ,\eta )$, $\forall (\xi
,\eta )\in X^2$.

\item[(b)] $(\xi ,\eta ) \to x(.,(\xi ,\eta ) )$ is continuous from $X^2$
into $C(I,X)$.
\end{itemize}
\end{theorem}

\begin{proof} We take $S=X\times X$, $a(\xi ,\eta )=\xi $,
$b(\xi ,\eta )=\eta $ for all $(\xi ,\eta )\in X\times X$,
$c:X\times X\to (0,\infty )$ an arbitrary continuous function, $g(.)=0$,
$y=0$, $q(\xi ,\eta )(t)=q_0(t)$ $\forall (\xi ,\eta )\in
X\times X$, $t\in I$ and we apply Theorem \ref{thm3.3} in order to obtain
the conclusion of the theorem.
\end{proof}

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