\documentclass[reqno]{amsart}
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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<s\leq \theta, \\
    t(s-1)/(1-\theta)&\text{if }\theta<s\leq 1,
\end{cases}\label{e3.1}
\end{equation}
if $0\leq t<\theta$, and
\begin{equation}
G(t,s) =\begin{cases}
-s &\text{if } 0\leq s< \theta,\\
    (\theta(s-t)+s(t-1))/(1-\theta)&\text{if } \theta\leq s\leq t, \\
    t(s-1)/(1-\theta)&\text{if }t<s\leq 1,
\end{cases}\label{e3.2}
\end{equation}
if $\theta\leq t\leq 1$.  Then the following assertions hold.

\noindent (1)  $G(.,s)$ is differentiable
 on $[0,1]$, for every $s\in[0,1]$, and its derivative is
\[
\frac{\partial G}{\partial t}(t,s) =
\begin{cases}
 0 &\text{if } 0\leq s\leq t,\\
-1 &\text{if } t<s\leq \theta, \\
(s-1)/(1-\theta)&\text{if }\theta<s\leq 1,
\end{cases} %(3.3)
\]
if $0\leq t< \theta$, and
\[
\frac{\partial G}{\partial t}(t,s) =
\begin{cases}
0 &\text{if } 0\leq s< \theta,\\
    (s-\theta)/(1-\theta)&\text{if } \theta\leq s\leq t, \\
    (s-1)/(1-\theta)&\text{if }t<s\leq 1,
\end{cases} %(3.4)
\]
if $\theta\leq t\leq 1$.

\noindent (2) $G(.,.)$ and $\frac{\partial G}{\partial t}(.,.)$  satisfies
$$
\sup_{t,s \in [0,1]} |G(t,s)|\leq 1,\quad
\sup_{t,s \in [0,1]} |\frac{\partial G}{\partial t}(t,s)|\leq 1. %(3.5)
$$

\noindent (3) For $f\in \mathbf{P}^1_E([0,1])$ and for the mapping
$u_f:[0,1]\to E$  defined by
$$
u_f(t)=\int_{0}^{1} G(t,s)f(s)ds,\quad \forall t\in [0,1], %(3.6)
$$
one has:
(3i) $u_{f}(0)=0$ and $u_{f}(\theta)=u_{f}(1)$.\\
(3ii) The mapping $t\mapsto u_{f}(t)$ is  continuous from [0,1]
into E,
i.e., $ u_{f} \in \mathbf{C}_{E}([ 0,1 ])$.\\
 (3iii) The mapping $u_{f}$ is scalarly derivable, that is, for
every $ x' \in E'$ ,
the scalar function $\langle x',u_{f}(.)\rangle$  is a.e derivable,
and its weak derivative $\dot{u}_{f}$ satisfies
\begin{align*}
\lim_{h\to 0}\langle x',\frac{u_{f}(t+h)-u_{f}(t)}{h}\rangle
&= \langle x' ,\dot{u_{f}}(t)\rangle\\
&= \int_{0}^{1}\frac{\partial G}{\partial t}(t,s)\langle
x',f(s)\rangle ds\\
&= \langle x',\int_{0}^{1}\frac{\partial G}{\partial t}(t,s),f(s)ds\rangle
\end{align*}
for all $t\in[0,1]$ and for all $x' \in E'$ . Consequently
$$
\dot u_{f}(t)=\int_{0}^{1}\frac{\partial G}{\partial t}(t,s)f(s)ds, \quad
\forall t\in [0,1], %(3.7)
$$
and $\dot{u}_{f}$ is a continuous mapping from $[0,1]$ into $E$.
(3vi) The mapping $\dot u_f$ is scalarly derivable, that is, there
exists a mapping $\ddot u_f:[0,1]\to E$ such that, for every $x'\in
E'$, the scalar function $\langle x',\dot u_f(\cdot)\rangle$ is a.e
derivable with $\frac{d}{dt}\langle x',\dot u_f(t)\rangle=\langle
x',\ddot u_f(t)\rangle$; furthermore
$$
\ddot u_f=f\quad\textmd{a.e. on } [0,1]. %(3.8)
$$
\end{lemma}

 Let us mention a useful consequence of Lemma \ref{lem3.1}.

\begin{proposition} \label{prop3.2}
Let $E$ be a separable Banach space and let $f:[0,1]\to E$ be a continuous
mapping (respectively a mapping in $\mathbf{P}^{1}_{E}([0,1]))$.
Then the mapping
$$
u_f(t)=\int_{0}^{1} G(t,s)f(s)ds,\quad \forall t\in [0,1],
$$
is the unique $\mathbf{C}^{2}_{E}([0,1])$-solution (respectively
$\mathbf{W}^{2,1}_{P,E}([0,1])$-solution) to the differential equation
\begin{gather*}
    \ddot u(t)=f(t)\quad \forall t\in [0,1],\\
    u(0)=0;\quad u(\theta)=u(1).
\end{gather*}
\end{proposition}


The following proposition is an analogous version of the continuous
selection theorem of Bressan and  Colombo \cite{B.C} and Fryszkowski
\cite{F}, in the case where the multifunction has values in
$\mathbf{P}^1_E([0,1])$. For the proof of this result we refer the
reader to \cite{A.B}.

\begin{proposition} \label{prop3.3}
Let $M : [0,1]\rightrightarrows \mathbf{P}^1_E([0,1]) $  be a lower
semicontinuous multifunction with closed  and decomposable values.
Then $M$ has a continuous selection.
\end{proposition}

For the proof of our Theorem, we  need the following Lemma due to
Grothendieck  \cite{GR}. See also \cite{C} for a more general result
concerning the Mackey topology for bounded sequences in
$\mathbf{L}^{\infty}_{E'}$.

 \begin{lemma} \label{lem3.4}
Let $(g_n)$ be a sequence of uniformly bounded mappings in
$\mathbf{L}^{\infty}_{\mathbb{R}}([0,T])$, which converges pointwise
to $0$. Then for all uniformly integrable subset $K$ of
$\mathbf{L}^{1}_{\mathbb{R}}([0,T])$, the sequence $(\langle g_n,
h\rangle)=(\int_{0}^{1}g_n(t) h(t) dt)$ converges uniformly to $0$,
for all $h\in K$.
\end{lemma}

Now we are able to give our main result.

\begin{theorem} \label{thm3.5}
Let $E$ be a separable Banach space and let $F:[0,1]\times E\times E
\rightrightarrows E$ be a convex compact valued multifunction, Lebesgue-measurable
on $[0,1]$, and upper semicontinuous on $E\times E$. Let $H:[0,1]\times E\times
E\rightrightarrows E$ be a multifunction with nonempty closed values such that $H$
is $\mathcal{L}([0,1])\otimes\mathcal{B}(E)\otimes\mathcal{B}(E)$-measurable and
lower semicontinuous on $E\times E$. Assume that for $i=1,2$ there are some convex
$\Vert\cdot\Vert$-compact valued, and measurable multifunctions
$\Gamma_{i}:[0,1]\rightrightarrows E$ which are Pettis uniformly integrable, such
that $F(t,x,y)\subset\Gamma_{1}(t)$ and $H(t,x,y)\subset\Gamma_{2}(t)$ for all
$(t,x,y)\in[0,1]\times E\times\ E$. Then the differential inclusion
\begin{gather*} %(\mathcal{P})
\ddot{u}(t) \in \mathrm{F}(t,u(t),\dot u(t))+\mathrm{H}(t,u(t),\dot u(t)),\quad
\text{a.e. } t \in [0,1],\\
u(0)=0;\quad  u(\theta)=u(1).
\end{gather*}
has at least one solution  $u\in \mathbf{W}^{2,1}_{P,E} ([0,1])$.
\end{theorem}


\begin{proof}
 \textit{Step 1}.  Taking $\overline{co}(\{0\}\cup \Gamma_i(t))$ if
necessary, we may suppose that $0\in \Gamma_{i}(t)$ for all
$t\in[0,1]$ and $i=1,2$.\\ For  $t\in[0,1]$, let
$\Gamma(t)=\Gamma_{1}(t)+\Gamma_{2}(t)$ and observe that the
multifunction $\mathrm{\Gamma}$ inherits all the properties of
$\mathrm{\Gamma}_{1}$ and $\mathrm{\Gamma}_{2}$. Let us consider the
differential inclusion
\begin{equation} \label{e3.9}
\begin{gathered}
\ddot{u}(t) \in \Gamma(t),\quad\text{a.e. } t \in [0,1],\\
u(0)=0; \quad  u(\theta)=u(1).
\end{gathered}
\end{equation}
We wish to show that the $\mathbf{W}^{2,1}_{P,E} ([0,1])$-solutions set
$\mathbf{X}_{\Gamma}$ of \eqref{e3.9} is nonempty and convex compact in
the Banach space $\mathbf{C}^{1}_{E}([0,1])$ endowed with the norm
$\|.\|_{\mathbf{C}^{1}}$.
Furthermore, if a sequence $(u_{n})$ of $\mathbf{X}_{\Gamma}$
$\|.\|_{\mathbf{C}^{1}}$-converges to $u$, then $(\dot{u}_{n})$ converges
pointwise to $\dot{u}$ and $(\ddot{u}_{n})$ converges
$\sigma(\mathbf{P}^{1}_{E},\mathbf{L}^{\infty}_E\otimes E')$ to $\ddot{u}$.
The proof of this last assertion is similar of the one
in \cite[Lemma 5]{A.C.T}; we
include it here for the convenience of the reader.

 Let us recall that the set
$\mathbf{S}_{\Gamma}^{Pe}$ of all Pettis integrable selections of $\Gamma$ is
nonempty and sequentially compact for the topology of pointwise convergence on
$\mathbf{L}^{\infty}\otimes E'$ and that the multivalued integral
$$
\int_{0}^{1} \Gamma(t) dt = \big\{\int_{0}^{1} f(t) dt;
f\in \mathbf{S}_{\Gamma}^{Pe}\big\}
$$
is convex and norm compact in $E$ (see \cite{A.C}, \cite{A.C.V}, \cite{C}).
 In view of Lemma \ref{lem3.1} and Proposition \ref{prop3.2}, the solutions set
$\mathbf{X}_{\Gamma}$ of \eqref{e3.9}
is characterized by
$$
\mathbf{X}_{\Gamma}= \{u_{f}:[0,1] \to E: u_{f}(t)=
\int_{0}^{1}G(t,s)f(s)ds,\forall t\in [0,1]; f\in
\mathbf{S}_{\Gamma}^{Pe}\}. %(3.10)
$$
Clearly $\mathbf{X}_{\Gamma}$ is convex. Furthermore, if $(t_{n})$ is a
sequence in $[0,1]$, which converges to $t \in [0,1]$ we have,
 by Lemma \ref{lem3.1},
\begin{equation}
\begin{aligned}
\|u_{f}(t_{n})-u_{f}(t)\|
&= \sup _{x' \in \overline{\mathbf{B}}_{E'}}\vert\langle
x',u_{f}(t_{n})-u_{f}(t)\rangle\vert\\
&= \sup _{x' \in \overline{\mathbf{B}}_{E'}}\vert\langle
x',\int_{0}^{1}G(t_n,s)f(s)ds-
\int_{0}^{1}G(t,s)f(s)ds\rangle\vert\\
&\leq \sup _{x' \in \overline{\mathbf{B}}_{E'}}\int_{0}^{1} \vert
G(t_{n},s)-G(t,s)\vert\vert\langle x',f(s)\rangle\vert ds\\
 &\leq \sup _{x'\in \overline{\mathbf{B}}_{E'}}\int_{0}^{1}
\vert G(t_{n},s)-G(t,s)\vert\vert\delta^{*}(x',\Gamma(s))\vert ds
\end{aligned} \label{e3.11}
\end{equation}
and
\begin{equation}
\Vert \dot{u}_{f}(t_{n})-\dot{u}_{f}(t)\Vert \leq \sup _{x'\in
\overline{\mathbf{B}}_{E'}}\int_{0}^{1} \vert \frac{\partial G}{\partial
t}(t_{n},s)-\frac{\partial G}{\partial
t}(t,s)\vert\vert\delta^{*}(x',\Gamma(s))\vert ds \label{e3.12}
\end{equation}
for all $f \in \mathbf{S}_{\Gamma}^{Pe}$. As the sequences
$(v_{n}(.)) :=(\vert G(t_{n},.)-G(t,.)\vert)$ and
$(w_{n}(.)) :=(\vert\frac{\partial G}{\partial t}(t_{n},.)
-\frac{\partial G}{\partial t}(t,.)\vert)$ are uniformly bounded and
converge pointwise to $0$ and as the set
$\{\vert\delta^{*}(x',\Gamma(.))\vert: x'
\in \overline{\mathbf{B}}_{E'}\}$ is uniformly integrable in
$\mathbf{L}_{\mathbb{R}}^{1}([0,1])$, by Lemma \ref{lem3.4} we conclude
that  $(v_{n}(.))$ and $(w_{n}(.))$ converge uniformly to $0$ on
this set in the duality $\langle
\mathbf{L}^{\infty}_{\mathbb{R}},\mathbf{L}^{1}_{\mathbb{R}}\rangle$.
Hence the second member of \eqref{e3.11} and \eqref{e3.12} tends to $0$.
This says that
$\mathbf{X}_{\Gamma}$ and $\{\dot{u}_{f}:u_f \in \mathbf{X}_{\Gamma}\}$ are
equicontinuous in $\mathbf{C}_{E}([0,1])$. Furthermore, the sets
$\mathbf{X}_{\Gamma}(t)=\{u_{f}(t): \;u_{f} \in \mathbf{X}_{\Gamma}\}$ and
$\{\dot{u}_{f}(t): \;u_{f} \in \mathbf{X}_{\Gamma}\}$ are relatively compact
in $E$ because they are included in the norm compact sets
$\int_{0}^{1}G(t,s)\Gamma (s) ds$
and $\int_{0}^{1}\frac{\partial G}{\partial t}(t,s)\Gamma (s) ds$ respectively.
 The Ascoli-Arzel\`a theorem yields that $\mathbf{X}_{\Gamma}$ is
relatively compact in
$\mathbf{C}^{1}_{E}([0,1])$ with respect to $\|.\|_{\mathbf{C}^{1}}$.
We claim that $\mathbf{X}_{\Gamma}$ is closed in
$(\mathbf{C}^{1}_{E}([0,1]),\|.\|_{\mathbf{C}^{1}})$. Let $(u_{f_{n}})$ be a
sequence in $\mathbf{X}_{\Gamma}$ converging to
$\xi \in \mathbf{C}^{1}_{E}([0,1])$
with respect to $\|.\|_{\mathbf{C}^{1}}$. As $\mathbf{S}_{\Gamma}^{Pe}$ is
sequentially compact for the topology of pointwize convergence on
$\mathbf{L}^{\infty}_E \otimes E'$, we extract from $(f_{n})$ a subsequence
that we do not relabel and which converges
$\sigma(\mathbf{P}^{1}_{E},\mathbf{L}^{\infty}_E\otimes {E}')$ to a mapping
$f \in \mathbf{S}_{\Gamma}^{Pe}$. In particular
\begin{equation}
\begin{aligned}
\lim_{n \to \infty} \langle x',\int_{0}^{1}G(t,s)f_{n}(s) ds
\rangle
&= \lim_{n \to \infty} \int_{0}^{1} \langle
G(t,s)x',f_{n}(s)  \rangle ds\\
&= \int_{0}^{1} \langle
G(t,s)x',f(s)  \rangle ds\\
&=  \langle x',\int_{0}^{1}G(t,s)f(s)ds \rangle.
\end{aligned} \label{e3.13}
\end{equation}
As the set valued integral $\int_{0}^{1}G(t,s)\Gamma (s) ds$
$(t\in [0,1])$ is
norm-compact, \eqref{e3.13} shows that the sequence
$(u_{f_{n}}(\cdot))=(\int_{0}^{1}G(.,s)f_{n}(s) ds)$ converges pointwise to
$u_{f}(.)$ for $E$ endowed with the strong topology.
Thus we get $\xi = u_{f}$. This
shows the compactness of $\mathbf{X}_{\Gamma}$ in
$\mathbf{C}^{1}_{E}([0,1])$.

\noindent \textit{Step 2}.   Let us observe that, for any
Lebesgue-measurable mappings  $v,w:[0,1]\to E$, there is a Pettis
integrable selection $s\in \mathbf{S}_{{\Gamma}_{1}}^{Pe}$ such that
$s(t)\in F(t,v(t),w(t))$ a.e. Indeed, there exist two sequences
$(v_{n})$ and $(w_{n})$ of simple $E$-valued mappings converging to
$v$ and $w$ respectively, for $E$ endowed with the norm topology.
Notice that the multifunctions $F(.,v_{n}(.),w_{n}(.))$ are
measurable. Let $s_{n}$ be a Lebesgue-measurable selection of
$F(.,v_{n}(.),w_{n}(.))$. As $s_{n}(t) \in
F(t,v_{n}(t),w_{n}(t))\subset{\Gamma}_{1}(t)$; for all $t\in[0,1]$
and $\mathbf{S}_{{\Gamma}_{1}}^{Pe}$ is sequentially
$\sigma(\mathbf{P}^{1}_{E},\mathbf{L}^{\infty}_E\otimes
E')$-compact, we may extract from $(s_{n})$ a subsequence $(s'_{n})$
which converges
$\sigma(\mathbf{P}^{1}_{E},\mathbf{L}^{\infty}_E\otimes E')$ to a
mapping $s\in \mathbf{S}_{{\Gamma}_{1}}^{Pe}$. Let
${(e_{k}^{*})}_{k\in \mathbb{N}}$ be a dense sequence for the Mackey
topology $\mathcal{T} (E',E)$. Let $k\in \mathbb{N}$ be fixed.
Applying the Mazur's trick to $(\langle e_{k}^{*}, s'_{n}(.)
\rangle)_{n}$ provides a sequence $(z_{n})$ with $z_{n}\in
co\{\langle e_{k}^{*}, s'_{m}(.) \rangle: m\geq n \}$ such that
$(z_{n})$ converges pointwise a.e. to $\langle e_{k}^{*}, s(.)
\rangle$. Using this fact and the pointwise convergence of the
sequences $(v_n)$ and $(w_n)$ and the upper semicontinuity of
$F(t,\cdot,\cdot)$, it is not difficult to check that $\langle
e_{k}^{*}, s(t) \rangle \leq \delta^{*}(e_{k}^{*}, F(t,v(t),w(t))$
a.e. Indeed, Let $A$ be a measurable set of $[0,1]$,
\begin{align*}
\int_{A} \langle e_{k}^{*}, s(t) \rangle dt&=
\lim_{n\to\infty}\int_{A}
\langle e_{k}^{*},s'_n(t) \rangle dt\\
&\leq \limsup_{n\to\infty} \int_{A}
\delta^{*}(e_{k}^{*}, F(t,v_n(t),w_n(t)))dt\\
&\leq \int_{A} \limsup_{n\to\infty}
\delta^{*}(e_{k}^{*},F(t,v_n(t),w_n(t))) dt\\
&= \int_{A} \delta^{*}(e_{k}^{*}, F(t,v(t),w(t))) dt.
\end{align*}
Then, for all $k\in \mathbb{N}$
\begin{equation}
\langle e^*_k, s(t)\rangle\leq \delta^{*}(e_{k}^{*}, F(t,v(t),w(t))),
\text{a.e. }t\in [0,1]. \label{e3.14}
\end{equation}
 On the other hand, in view of \cite[Lemma III.33 and Corollary I.15]{C.V},
we have
\begin{equation}
\sup_{x'\in E'}[\langle x', s(t)\rangle-\delta^{*}(x', F(t,v(t),w(t)))]=
\sup_{k\in \mathbf{N}}[\langle e^*_k, s(t)\rangle-\delta^{*}(e^*_k,
F(t,v(t),w(t)))]. \label{e3.15}
\end{equation}
Using relation \eqref{e*} given in Section 2 we get by \eqref{e3.14}
and \eqref{e3.15},
$$
d(s(t), F(t,v(t),w(t))\leq\sup_{k\in \mathbf{N}}[\langle e^*_k, s(t)\rangle
-\delta^{*}(e^*_k, F(t,v(t),w(t)))]\leq 0.
$$
Consequently $s(t)\in F(t,v(t),w(t))$ a.e. $t\in [0,1]$.

\noindent \textit{Step 3}. Let $\Phi:\mathbf{X}_{\Gamma}\rightrightarrows
\mathbf{P}^{1}_{E}([0,1])$  be the multifunction given by
$$
\Phi (u_{f})= \{v \in \mathbf{P}^{1}_{E}([0,1]):v(t)\in H(t,u_{f}(t),
\dot u_{f}(t)),\text{ a.e. on } [0,1]\}.
$$
We will prove that, for $\mathbf{X}_{\Gamma}$ endowed with the norm
$\Vert\cdot\Vert_{\mathbf{C}^{1}}$, the multifunction $\Phi$ admits
a continuous selection. It is clear that $\Phi$ has nonempty closed
decomposable values.
According to Proposition \ref{prop3.3}, it sufficient to prove that $\Phi$ is lower
semicontinuous. Let $u_{f_{0}}\in\;\mathbf{X}_{\Gamma}$,
$v_{0}\in\Phi (u_{f_{0}})$
and let $(u_{f_{n}})$ be a sequence in $\mathbf{X}_{\Gamma}$ converging to
$u_{f_{0}}$ in $(\mathbf{C}^{1}_{E}([0,1]),\Vert\cdot\Vert_{\mathbf{C}^{1}})$.
For any $n\in\mathbb{N}$, $H(.,u_{f_{n}}(.),\dot u_{f_{n}}(.))$
is measurable with nonempty closed values, so the multifunction
$\Lambda_{n}$ defined from $[0,1]$ into
$E$ by
$$
\Lambda _{n}(t)=\{w \in H(t,u_{f_{n}}(t),\dot u_{f_{n}}(t)):
\Vert w-v_{0}(t)\Vert=d(v_{0}(t),H(t,u_{f_{n}}(t),\dot
u_{f_{n}}(t)))\}
$$
is also measurable with nonempty closed values.
In view of the existence theorem of measurable selections (see
\cite{C.V}), there is a measurable mapping $v_{n} :[0,1] \to E$
such that $v_{n}(t) \in \Lambda _{n}(t)$, for all $t\in [0,1]$.
This yields $v_{n}(t) \in H(t,u_{f_{n}}(t),\dot u_{f_{n}}(t))$ and
\begin{align*}
\lim_{n\to\infty} \Vert v_{n}(t)- v_{0}(t)\Vert
&= \lim_{n\to \infty}d(v_{0}(t),H(t,u_{f_{n}}(t),\dot
u_{f_{n}}(t)))\\
&\leq \lim_{n\to \infty}e(H(t,u_{f_{0}}(t),\dot
u_{f_{0}}(t)),H(t,u_{f_{n}}(t),\dot u_{f_{n}}(t)))= 0.
\end{align*}
This says that $(v_{n})$ converges pointwise to $v_{0}$ and since
$H(t,x,y)\subset \Gamma_{2}(t)$ for all $(t,x,y) \in [0,1]\times E\times E$,
the convergence also
holds strongly in $\mathbf{P}_{E}^{1}([1,0])$. Indeed,
\begin{align*}
\lim_{n\to\infty} \Vert v_{n}- v_{0}\Vert_{Pe}
&= \lim_{n\to \infty} \sup_{x'\in \overline{\mathbf{B}}_{E'}}
\int_{0}^{1}\vert\langle x', v_{n}(t) -v_{0}(t)\rangle \vert dt\\
&= \lim_{n\to\infty} \sup_{x'\in \overline{\mathbf{B}}_{E'}}
\int_{0}^{1}\vert\langle x', v_{n}(t)\rangle-\langle x',
v_{0}(t)\rangle\vert dt.
\end{align*}
As $v_n(t)\in \Gamma_2(t)$ for all $n\in \mathbb{N}$ and as $\Gamma_2$
is scalarly uniformly integrable and hence the set
$\{\langle x', v_{n}(.)\rangle : \|x'\|\leq 1\}$ is uniformly
integrable in $\mathbf{L}_{E}^{1}([0,1])$, we get
$$
\lim_{n\to\infty}\Vert v_{n}- v_{0}\Vert_{Pe} = \lim_{n\to\infty}
\sup_{x'\in \overline{\mathbf{B}}_{E'}}
\int_{0}^{1}\lim_{n\to\infty} \vert\langle x',
v_{n}(t)\rangle-\langle x', v_{0}(t)\rangle\vert dt = 0.
$$
Therefore $\Phi$ is lower semicontinous. An application of
Proposition \ref{prop3.3} implies that, for $\mathbf{X}_{\Gamma}$ endowed
with the norm $\Vert\cdot\Vert_{\mathbf{C}^{1}}$, there exists a
continuous mapping $\varphi: \mathbf{X}_{\Gamma}\to
\mathbf{P}^{1}_{E}([0,1])$ such that $\varphi(u)\in \Phi(u)$ for
all $u\in \mathbf{X}_{\Gamma}$, or equivalently for each $u\in
\mathbf{X}_{\Gamma}$ the inclusion $\varphi(u)(t)\in H(t,u(t),\dot
u(t))$ holds for
a.e. $t \in [0,1]$.

\noindent\textit{Step 4}. For all $u\in \mathbf{X}_{\Gamma}$,
let us define the multifunction $\Psi$ by
$$
\Psi(u)=\{ v\in \mathbf{X}_{\Gamma}: \;\ddot{v}(t) \in F(t,u(t),\dot
u(t))+ \varphi(u)(t),\text{ a.e.}\}.
$$
In view of Step 2, and since $\varphi(u)\in \mathbf{S}_{\Gamma_{2}}^{Pe}$
for all $u\in \mathbf{X}_{\Gamma}$, for any measurable
selection $s$ of $F(.,u(.),\dot u(.))$ the mapping $ g:= s+\varphi(u)$ is in
$\mathbf{S}_{\Gamma}^{Pe}$ (because $g(t)= s(t)+\varphi(u)(t)\in
\Gamma_{1}(t)+\Gamma_{2}(t)= \Gamma(t))$ and the mapping $v$ defined by
$v(t)=\int_{0}^{1}G(t,s)g(s)ds$ is in $\Psi(u)$, and hence $\Psi(u)$
is a nonempty set. It clear that $\Psi(u)$ is a convex subset of
$\mathbf{X}_{\Gamma}$. We need to
check that $\Psi: \mathbf{X}_{\Gamma} \rightrightarrows \mathbf{X}_{\Gamma}$
is upper semicontinuous on the convex compact set $\mathbf{X}_{\Gamma}$.
Equivalently we need to check that the graph of $\Psi$, $\textmd{gph}(\Psi)=\{(u,v) \in
\mathbf{X}_{\Gamma} \times \mathbf{X}_{\Gamma}:v\in \Psi(u)\}$, is sequentially
closed in $\mathbf{X}_{\Gamma} \times \mathbf{X}_{\Gamma}$. Let $(u_{n},v_{n})$ be a
sequence in $\textmd{gph}(\Psi)$ converging to $(u,v) \in \mathbf{X}_{\Gamma}\times
\mathbf{X}_{\Gamma}$. By repeating the arguments given in Step 1, we obtain that
$(u_{n},v_{n})$ converges uniformly to $(u,v)$ in
$(\mathbf{C}^1_E([0,1]),\Vert\cdot\Vert_{\mathbf{C}^1})$, and that $(\ddot u_{n}$,
$\ddot v_{n})$ converges $\sigma(\mathbf{P}^{1}_{E},\mathbf{L}^{\infty}_E\otimes
E')$ to $(\ddot{u},\ddot{v})$. As $\ddot{v}_{n}(t)-\varphi(u_{n})(t) \in
F(t,u_{n}(t),\dot{u}_{n}(t))$, a.e., repeating the arguments given  at the end of
Step 2, we get $\ddot{v}(t)-\varphi(u)(t)\in F(t,u(t),\dot u(t))$, a.e.
This shows that
 $\textmd{gph}(\Psi)$ is closed in
$\mathbf{X}_{\Gamma} \times \mathbf{X}_{\Gamma}$ and
hence we get the upper semicontinuity of $\Psi$. An application of the Kakutani
fixed point theorem gives some $u\in\mathbf{X}_{\Gamma}$ such that $u \in \Psi(u)$.
This means $\ddot{u}(t) \in F(t,u(t),\dot u(t))+ \varphi(u)(t)$, a.e. Since
$\varphi(u)(t)\in H(t,u(t),\dot u(t))$, we get
\begin{gather*}
\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{gather*}
This completes the proof of the theorem.
\end{proof}


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\section*{Addendum posted by the editor on September 15, 2016}

A reader informed us that Proposition 3.3 and Theorem 3.5 are incorrect.
The same proposition (stated as Proposition 2.2) was also used in 

D. Azzam-Laouir, I. Boutana, A. Makhlouf;
Application of Pettis integration to delay second order 
differential inclusions, 
Electronic Journal of Qualitative Theory of Differential Equations 2012,
88 pp 1--15.

A reader pointed out the mistake and the authors posted a corrigendum
that says
\begin{quote}
In the above article, Proposition 2.2 is not true since the normed space
$P_E^1([0, 1])$ is not complete. Consequently, to correct Theorem 3.1 we
have to assume that $\Gamma_1$ is Pettis uniformly integrable and that 
$\Gamma_2$ is integrably bounded. 
Then in the proof we can use Proposition 2.2 with
$L_E^1([0, 1])$ instead of $P_E^1([0, 1])$
to conclude the result. This version of Proposition 2.2 can be found in

A. Fryszkowski, Continuous selections for a class of nonconvex mul-
tivalued maps, Studia Math., 76, (1983), pp. 163-174.
\end{quote}

We asked the authors to post a similar addendum to the EJDE article,
but the authors did not reply. 
So we attached this note.


\end{document}