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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 173, pp. 1--8.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/173\hfil Application of Pettis integration]
{Application of Pettis integration to differential inclusions
with three-point boundary conditions in Banach spaces}
\author[D. Azzam-Laouir, I. Boutana\hfil EJDE-2007/173\hfilneg]
{Dalila Azzam-Laouir, Imen Boutana} % in alphabetical order
\address{Dalila Azzam-Laouir \newline
Laboratoire de Math\'ematiques Pures
et Appliqu\'ees, Universit\'e de Jijel, Alg\'erie}
\email{azzam\_d@yahoo.com}
\address{Imen Boutana \newline
Laboratoire de Math\'ematiques Pures
et Appliqu\'ees, Universit\'e de Jijel, Alg\'erie}
\email{bou.imend@yahoo.fr}
\thanks{Submitted September 5, 2007. Published December 6, 2007.}
\subjclass[2000]{34A60, 28A25, 28C20}
\keywords{Differential inclusions; Pettis-integration; selections}
\begin{abstract}
This paper provide some applications of Pettis integration to differential
inclusions in Banach spaces with three point boundary conditions of the form
$$
\ddot{u}(t) \in F(t,u(t),\dot u(t))+H(t,u(t),\dot u(t)),\quad
\text{a.e. } t \in [0,1],
$$
where $F$ is a convex valued multifunction upper semicontinuous on
$E\times E$ and $H$ is a lower semicontinuous multifunction.
The existence of solutions is obtained under the non convexity condition
for the multifunction $\mathrm{H}$, and the assumption that
$\mathrm{F}(t,x,y)\subset \Gamma_{1}(t)$,
$\mathrm{H}(t,x,y)\subset \Gamma_{2}(t)$, where the multifunctions
$\Gamma_{1},\Gamma_{2}:[0,1]\rightrightarrows E$ are uniformly Pettis
integrable.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\numberwithin{equation}{section}
\section{Introduction}
In the theory of integration in infinite-dimensional spaces, Pettis
integrability is a more general concept than that of Bochner
integrability. Indeed, it is known that a Banach space $E$ is
infinite dimensional if and only if there exists a Pettis integrable
$E$-valued function, which is not Bochner integrable. There is a
rich literature dealing with the Pettis integral. For acquit
extensive account, we refer the reader to the monographe by Musial
\cite{M2}, where further references can be found. On the other hand,
the set-valued integration has shown to be useful tool for modeling
a lot of situations in several fields ranging from mathematical
economics to optimization and optimal control. Recently, special
attention has been paid to the Pettis integral of multifunctions.
For example, let us mention the recent contributions of Amrani and
Castaing \cite{A.C}, Amrani, Castaing and Valdier \cite{A.C.V}, and
Castaing \cite{C} which deal with the Pettis integral of bounded,
especially weakly compact, convex valued multifunctions. See also
\cite{A.H}, \cite{G}, \cite{H}, \cite{M1}, \cite{S} and the
references therein.
Existence of solutions for second order differential inclusions of
the form $\ddot{u}(t) \in F(t,u(t),\dot u(t))$ with three-point
boundary conditions, where $F:[0,1]\times E\times\ E
\rightrightarrows E$ is a convex compact valued multifunction,
Lebesgue-measurable on $[0,1]$, and upper semicontinuous on $E\times
E$, under the assumption that $F(t,x,y)\subset\Gamma(t)$ in the case
where $\Gamma$ is integrably bounded and the case where $\Gamma$ is
uniformly Pettis integrable, has been studied by Azzam-Laouir,
Castaing and Thibault \cite{A.C.T}.
Let $\theta$ be a given number in $]0,1[$; the aim of our article is
to provide existence results for the general problem of three point
boundary conditions associated with the differential inclusion\\
\begin{equation} \label{eP}
\begin{gathered}
\ddot{u}(t) \in F(t,u(t),\dot u(t))+ H(t,u(t),\dot
u(t)),\quad\text{a.e. }t\in [0,1],\\
u(0)=0;\quad u(\theta)=u(1).
\end{gathered}
\end{equation} We suppose that $F:[0,1]\times E\times E\rightrightarrows E$
is upper semicontinuous on $E\times E$ and measurable on $[0,1]$. We
take $H:[0,1]\times E\times E\rightrightarrows E$ as a measurable
multifunction lower semicontinuous on $E\times E$. Furthermore we
suppose that $F(t,x,y)\subset \Gamma_1(t)$, $H(t,x,y)\subset
\Gamma_2(t)$ for all $(t,x,y)\in [0,1]\times E\times E$ for some
convex $\Vert\cdot\Vert$-compact valued, and measurable
multifunctions $\Gamma_1,\Gamma_2:[0,1]\rightrightarrows E$ which
are uniformly Pettis integrable. Then we show that the differential
inclusion $(1.1)$ has at least a solution $u\in
\mathbf{W}^{2,1}_{P,E}([0,1])$.
\section{Notation and Preliminaries}
Throughout, $(E,\Vert\cdot\Vert)$ is a separable Banach space and
$E'$ is its Topological dual, $\overline{\mathbf{B}}_E$ is the
unit closed ball of E, $\mathcal{L}([0,1])$ is the
$\sigma$-algebra of Lebesgue-measurable sets of $[0,1]$,
$\lambda=dt$ is the Lebesgue measure on $[0,1]$, and
$\mathcal{B}(E)$ is the $\sigma$-algebra of Borel subsets of $E$.
By $\mathbf{L}^{1}_{E}([0,1])$ we denote the space of all
Lebesgue-Bochner integrable $E$ valued mappings defined on
$[0,1]$. We denote the topology of uniform convergence on weakly
compact convex sets by $\mathcal{T}^{w}_{co}$. Restricted to $E'$,
this is the Mackey topology, which is the strongest locally convex
topology on $E'$ and we denote it by $\mathcal{T}(E',E)$. We
recall some preliminary results. Let $f: [0,1]\to E$ be a scalarly
integrable mapping, that is, for every $x' \in E'$, the scalar
function $t\mapsto \langle x' ,f(t)\rangle$ is Lebesgue-integrable
on $[0,1]$. A scalarly integrable mapping $f: [0,1]\to E$ is
Pettis integrable if, for every Lebesgue measurable set $A$ in
$[0,1]$, the weak integral $\int_{A} f(t)dt$ defined by $\langle
x',\int_{A} f(t) dt\rangle=\int_{A} \langle x', f(t)\rangle dt$
for all $x'\in E'$, belongs to $E$. We denote by
$\mathbf{P}^1_E([0,1])$ the space of all Pettis-integrable
$E$-valued mappings defined on $[0,1]$. The Pettis norm of any
element $f\in\mathbf{P}^1_E([0,1])$ is defined by $\Vert
f\Vert_{Pe}= \sup_{x'\in\overline{\mathbf{B}}_{E'}}
\int_{[0,1]}|\langle x',f(t)\rangle|dt$. The space
$\mathbf{P}^1_E([0,1])$ endowed with $\Vert\cdot\Vert_{Pe}$ is a
normed space. A subset $\mathcal{K}\subset \mathbf{P}^1_E([0,1])$
is Pettis uniformly integrable (PUI for short) if, for every
$\varepsilon>0$, there exists $\delta>0$ such that
$$
\lambda(A)\leq\delta\Rightarrow \sup_{f\in \mathcal{K}}\Vert
\boldsymbol{1}_{A}f\Vert_{Pe}\leq\varepsilon,
$$
where $\boldsymbol{1}_{A}$ stands for the characteristic function
of $A$. If $f\in\mathbf{P}^1_E([0,1])$, the singleton $\{f\}$ is
PUI since the set $\{\langle x',f\rangle: \Vert x'\Vert\leq 1\}$
is uniformly integrable.
Let
$\mathbf{C}_{E}([0,1])$ be the Banach space of all continuous
mappings $u:[0,1]\to E$, endowed with the sup-norm, and let
$\mathbf{C}^{1}_{E}([0,1])$ be the Banach space of all continuous
mappings $u: [0,1]\to E$ with continuous derivative, equipped with
the norm
$$
\Vert u\Vert_{\mathbf{C}^{1}}=\max\{\max_{t\in[0,1]} \|u(t)\|,
\max_{t\in[0,1]}\|\dot{u}(t)\|\}.
$$
Recall that a mapping $v:[0,1]\to E$ is said to be scalarly
derivable when there exists some mapping $\dot{v}:[0,1]\to E$
(called the weak derivative of $v$) such that, for every $x' \in
E'$, the scalar function $\langle x',v(.)\rangle$ is a.e derivable
and its derivative is equal to $\langle x',\dot v(.)\rangle$. The
weak derivative $\ddot{v}$ of $\dot{v}$ when it exists is the weak
second derivative.
By $\mathbf{W}^{2,1}_{P,E} ([0,1])$ we denote the space of all
continuous mappings in $\mathbf{C}_{E}([0,1])$ such that their first weak
derivatives are continuous and their second weak derivatives belong to
$\mathbf{P}^{1}_{E}([0,1])$.
For closed subsets $A$
and $B$ of $E$, the excess of $A$ over $B$ is defined by
$$
e(A,B)=\sup_{a\in A}d(a,B)=\sup_{a\in A}(\inf_{b\in B}\Vert
a-b\Vert),
$$
and the support function
$\delta^*(\cdot,A)$ associated with $A$ is defined on $E'$ by
$$
\delta^*(x',A)=\sup_{a\in A}\langle x',a\rangle.
$$
Recall that we have
\begin{equation}
d(x,A)=\sup_{x'\in\overline{\mathbf{B}}_{E'}}[\langle
x',x\rangle-\delta^*(x',A)],\;\forall x\in E.\label{e*}
\end{equation}
For a set $A\subset E$,
$\overline{co}A$ is its closed convex hull.
Recall also that a set $K\subset \mathbf{P}^{1}_{E}([0,1])$ is
said to be decomposable if and only if for every $u,v\in K$ and
any $A\in \mathcal{L}([0,1])$ we have
$u.\boldsymbol{1}_{A}+v.(1-
\boldsymbol{1}_{A})\in K$.
\section{The main result}
We begin with a lemma which summarizes some properties of some Green
type function (see \cite{Az}, \cite{A.C.T} ). It will be used full in
the study of our boundary problems.
\begin{lemma} \label{lem3.1}
Let $E$ be a separable Banach space and let $G:[0,1]\times [0,1]
\to \mathbb{R}$ be the function defined by
\begin{equation}
G(t,s) =\begin{cases}
-s &\text{if } 0\leq s\leq t,\\
-t &\text{if } t