\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 22, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/22\hfil Existence solutions]
{Existence of solutions for differential inclusions on closed moving
constraints in Banach spaces}

\author[A. M. Gomaa\hfil EJDE-2009/22\hfilneg]
{Adel Mahmoud Gomaa}

\address{Adel Mahmoud Gomaa\hfill\break
Mathematics Department, Faculty of Science,  Helwan University,
Cairo, Egypt}
\email{gomaa5@hotmail.com}

\thanks{Submitted December 1, 2008. Published January 27, 2009.}
\subjclass[2000]{32F27, 32C35, 35N15}
\keywords{Differential inclusions; moving constraints; existence solutions}

\begin{abstract}
In this paper, we prove the existence of solutions to
a multivalued differential equation with moving constraints.
We use a weak compactness type condition expressed in
terms of a strong measure of noncompactness.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

In this paper we study the existence of solutions to the
multivalued differential equation with moving constraints
\begin{equation} \label{eP}
\begin{gathered}
\dot x(t) \in F\big(t,x(t)\big) \quad \text{a.e. on }I, \\
 x(t) \in \Gamma(t) \quad \forall t \in [0,T], \\
x(0)=x_0 \in \Gamma(0).
\end{gathered}
\end{equation}
Where $F:[0,T]\times E\to P_{ck}(E)(P_{ck}$ is the family of
nonempty convex compact subsets of $E)$ and
$\Gamma: [0,T] \to P_f(E)$, $(P_f(E)$ is the family of closed subsets
of $E)$.
Problem \eqref{eP} has been studied by many authors;
see for example \cite{C.V, AC, 407.418, 43.50, 507.514} when $F$
is lower semicontinuous, and  \cite{23.30, C.V}  when $F$ is
upper semicontinuous with $\Gamma $ is independent of $t$.
For $\Gamma $ depending on $t$, we refer to \cite{AC, A.M.G, 1.48}.
In \cite{ont} we consider the differential inclusions
$\dot{x}(t)\in A(t)x(t)+F(t,x(t))$, $x(0)=x_{0}$ where
$\{A(t):0\le t\le T\}$ is a
family of densely defined closed linear operators generating a continuous
evolution operator ${{S}}(t,s)$ and $F$ is a multivalued function with
closed convex values in Banach spaces. there, we show how that
this results can be used in abstract control problems.
Also in \cite{wea} we consider the Cauchy problem
\begin{gather*}
\dot{x}(t)=f\big(t,x(t)\big), \quad t \in [0,T] \\
 x(0)=x_{0},
\end{gather*}
where $f:[0,T]\times E\to E$ and $E$ is a Banach space.
In \cite{China, Polon}, we study nonlinear differential
equations. In \cite{1172} we study  some differential
inclusions with delay and their topological properties.
Much work has been done in the study of topological properties
of solution for differential inclusions;
see \cite{59.65, 91.110, 255.263, 197.223, 363.379, onth, 155.168}.

In this paper we to prove the existence of solutions to
\eqref{eP} by using a measure of strong noncompactness, $\gamma$,
(see the next section).
Since the Kuratowksi measure of noncompactness and the ball measure
of noncomactness are measures of strong noncomactness and
we can construct many measures such $\gamma$ as in \cite{93.102},
in this paper Theorem \ref{thm3} is a generalization of results
for example Szufla \cite{507.514} and
Ibrahim-Gomaa \cite{A.M.G}. In Theorem \ref{thm4}, the assumption on
noncompactness is weaker than that of Benabdellah-Castaing
and Ibrahim \cite{1.48}.

\section{Preliminaries}

Let $E$ be a Banach space, $E^*$ its topological dual
space, $E_w$ the Banach space $E$ endowed with the weak topology, $B(0,1)$
unit ball of $E$, $I=[0,T]$, $(T>0)$, and $\lambda$ be the
Lebesgue measure on $I$ . Consider ${B}$ is the family of all bounded
subsets of $E$ and $C(I, E)$ is the space of all weakly continuous functions
from $I$ to $E$ endowed with the topology of weak uniform convergence.

\begin{definition} \label{def1.1} \rm
By a measure of strong noncompactness, $\gamma $, we will
understand a function $\gamma :{B}\to \mathbb{R}^{+}$ such
that, for all $U, V\in {{B}}$,
\begin{itemize}
\item[(M1)] $U\subset V\Longrightarrow \gamma (U)\le \gamma (V)$,
\item[(M2)] $\gamma (U\cup V)\le \max (\gamma (U),\gamma (V))$,
\item[(M3)] $\gamma (\overline{\rm conv}U)=\gamma (U)$,
\item[(M4)] $\gamma (U+V)\le \gamma (U)+\gamma (V)$,
\item[(M5)] $\gamma (cU)=|c|\gamma (U),\quad c\in \mathbb{R}$,
\item[(M6)] $\gamma (U)=0\Longleftrightarrow U$ is relatively
compact in $E$,
\item[(M7)] $\gamma (U\cup \{x\})=\gamma (U)$, $x\in E$.
\end{itemize}
\end{definition}

\begin{definition} \label{def1.2} \rm
For any nonempty bounded subset $U$ of $E$ the weak measure of
 noncompactness, $\beta$, and
the Kuratowski's measure of noncompactness, $\alpha $, is defined as:
\[
\alpha (U)=\inf \{\varepsilon >0: U
\text{ admits a finite number of sets
with diameter less than $\varepsilon$}.\}
\]
\end{definition}

For the properties of  $\beta$ and
$\alpha$ we refer the reader to \cite{BG, KD}.

\begin{definition} \label{def1.3} \rm
 By a Kamke function we mean a function $w:I\times {\mathbb{R}^{+}}
\to \mathbb{R}^{+}$ such that:
\begin{itemize}
\item[(i)] $w$ satisfies the Caratheodry conditions,
\item[(ii)] for all $t\in I$; $w(t,0)=0$,
\item[(iii)] for any $c\in (0,b]$, $u\equiv 0$ is the only absolutely
continuous function on $[0,c]$ which satisfies
$\dot{u}(t)\le w\big(t,u(t)\big)$ a.e.
on $[0,c]$ and such that $u(0)=0$.
\end{itemize}
\end{definition}

\begin{lemma}[\cite{JR, BG}] \label{lem1}
If $\gamma :{{B}\to \mathbb{R}^{+}}$ satisfies conditions
{\rm (M2), (M4), (M6)}, then, for
any nonempty $U\in {B}$,
\[
\gamma (U)\le \gamma (B(0,1))\alpha (U)
\]
\end{lemma}

\begin{lemma}[\cite{387.404, 607.614}] \label{lem2}
If $\gamma $ is a measure of weak (strong) noncompactness and
$A\subset C(I,E)$ be a family of strongly
equicontinuous functions, then
$\gamma (A(I))=\sup \{\gamma (A(t)): t\in I\}$.
\end{lemma}

\section{Main Results}

\begin{theorem}\label{thm3}
Let $\Gamma :I\to P_{f}(E)$ be a set-valued function
such that its graph, $G$, is left closed and
$F:I\times E\to P_{ck}(E)$ be a scalarly measurable set-valued
function such that for any $t\in I$, $F(t,.)$ is upper semicontinuous on $E$.
 Suppose that $F$ satisfies the following conditions:
\begin{itemize}
\item[(A1)] For each $\varepsilon >0$, there exists a closed subset
$I_{\varepsilon }$ of $I$ with $\lambda (I-I_{\varepsilon })<\varepsilon $
such that for any nonempty bounded subset $Z$ of $E$, one has
\[
\gamma (F(J\times Z))\le \sup_{t\in J}w(t,\gamma (Z))
\]
 for any compact subset $J$ of $I_{\varepsilon}$;

\item[(A2)] there is $\mu \in L^{1}(I,\mathbb{R}^{+})$, such that
$\Vert F(t,x)\Vert <\mu (t)(1+\Vert x\Vert)$, for all $(t,x)\in G$;

\item[(A3)] for each $(t,x)\in ([0,T[\times E)\cap G$ and $\varepsilon >0$
there is $(t_{\varepsilon },x_{\varepsilon })\in G$ such that
$0<t_{\varepsilon }-t<\varepsilon $ and that
\[
x_{\varepsilon }-x\in \int_{t}^{t_{\varepsilon }}F(s,x)ds+(t_{\varepsilon
}-t)\varepsilon B(0,1).
\]
\end{itemize}
Then, for any $x_{0}\in \Gamma (0)$, there is a solution for \eqref{eP}.
\end{theorem}

\begin{proof}
Let $(\varepsilon _{n})_{n\in \mathbb{N}}$ be a decreasing sequence
in $]0,1]$ with $\varepsilon _{n}=0$.
By  \cite[Proposition 6.1]{1.48}, there exist $m>1$,
a sequence $(\theta _{n})_{n\in \mathbb{N}}$ of right continuous functions
$\theta _{n}:I\to I$ such that $\theta _{n}(0)=0$,
$\theta _{n}(T)=T$ and $\theta _{n}(t)\in [t-\varepsilon _{n},t]$,
and a sequence $(x_{n})_{n\in \mathbb{N}}$ from $I$ to $E$ with
\begin{itemize}
\item[(i)] for all $t\in I$, $x_{n}(t)=x_{0}+\int_{0}^{t}\dot{x}_{n}(s)\,ds$,
where $\dot{x}_{n}\in L^{1}(I,E)$;

\item[(ii)] for all $t\in I,x_{n}(\theta _{n}(t))\in
\Gamma (\theta _{n}(t))$;

\item[(iii)] $\dot{x}_{n}(t)\in F(t,x_{n}(\theta _{n}(t))+\varepsilon
_{n}B(0,1)$ a.e on $I$;

\item[(iv)] $\Vert \dot{x}_{n}(t)\Vert \le m\mu (t)+1$, a.e on $I$.

\end{itemize}
From (iv) the sequence $(x_{n})$ is equicontinuous in $C(I,E)$. For each
$t\in I$, set
\[
A(t)=\{x_{n}(t):n\in \mathbb{N}\}\quad \text{and}\quad
\rho (t)=\gamma (A(t)).
\]
We claim that $(x_{n})_{n\in \mathbb{N}}$ is relatively compact
in the space $C(I,E)$. So we will show that $\rho \equiv 0$.
Since for each $(t,\tau )\in I\times I$, we have
\[
\gamma \{(x_{n})(\tau ):n\in \mathbb{N}\}\le \gamma \{(x_{n})(t):n\in
\mathbb{N}\}+\gamma \{(x_{n})(\tau )-(x_{n})(t):n\in \mathbb{N}\}
\]
and
\[
\gamma \{(x_{n})(t):n\in \mathbb{N}\}\le \gamma \{(x_{n})(\tau ):n\in
\mathbb{N}\}+\gamma \{(x_{n})(t)-(x_{n})(\tau ):n\in \mathbb{N}\},
\]
then, from Lemma \ref{lem1},
\[
|\rho (\tau )-\rho (t)|\le \gamma (B(0,1))\alpha (\{x_{n}(t)-x_{n}(\tau
): n\in \mathbb{N}\}),
\]
which implies
\[
|\rho (\tau )-\rho (t)|\le 2\gamma (B(0,1))|
 \int_{t}^{\tau }(m\mu(s)+1)\,ds|.
\]
It follows that $\rho $ is an absolutely continuous and hence differentiable
a.e. on $I$. Let $\varepsilon >0$. Since $\varepsilon _{n}\to 0$
as $n\to \infty $, then we can find $n_{0}\in \mathbb{N}$ such that
$T\varepsilon _{n}<\frac{\varepsilon }{\gamma (B(0,1))}$,
for all $n\ge n_{0}$.
Now let $(t,\tau )\in I\times I$ with $t\le \tau $. In view of Condition
(iii) and properties of $\gamma$ ((M4), (M7)), we have
\begin{align*}
\rho (\tau )-\rho (t)
&\leq \gamma (\int_{t}^{\tau }\dot{x}_{n}(s)\,ds:n\in N) \\
&\le \gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau
}F(s,\theta _{n}(s))ds)+\gamma (\{\varepsilon _{n}B(0,1)(\tau -t):n\in
\mathbb{N}\}) \\
&=\gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau }F(s,\theta
_{n}(s))ds)+\gamma (\{\varepsilon _{n}B(0,1)(\tau -t):n\ge n_{0}\}) \\
&\le \gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau
}F(s,\theta _{n}(s))ds)+\frac{\varepsilon }{\gamma (B(0,1))}\gamma (B(0,1))
\\
&\le \gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau
}F(s,\theta _{n}(s))ds)+\varepsilon .
\end{align*}
Thus,
\begin{equation}
\rho (\tau )-\rho (t)\le \gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau
}F(s,\theta _{n}(s))ds).  \label{a}
\end{equation}
Since $\rho $ is continuous and $w$ is Caratheodory we can find a closed
subset $I_{\varepsilon }$ of $I$, $\delta >0$, $\eta >0$ ($\eta <\delta $)
 and for $s_{1},s_{2}\in I_{\varepsilon }$; $r_{1},r_{2}\in [0,2T]$
such that if $|s_{1}-s_{2}|<\delta$, $|r_{1}-r_{2}|<\delta $, then
$|w(s_{1},r_{1})-w(s_{2},r_{2})|<\varepsilon $ and if $|s_{1}-s_{2}|<\eta $,
then $|\rho (s_{1})-\rho (s_{2})|<\frac{\delta }{2}$.
Consider the following  partition, of $[t,\tau ]$,
$t=t_{0}<t_{1}<\dots <t_{m}=\tau $ such that
$t_{i}-t_{i-1}<\eta $ for $i=1,\dots ,n$. From Condition (A1) we can
find a closed subset $J_{\varepsilon }$ of $I$ such that
$\lambda (I-J_{\varepsilon })<\varepsilon $ and that for any compact
subset ${K}$ of $J_{\varepsilon }$ and any bounded subset $Z$ of $E$,
$\gamma (f({K}\times Z))\le \sup_{s\in {K}}w(s,\gamma (Z))$.
Let $T_{i}=J_{\varepsilon }\cap [t_{i-1},t_{i}]\cap I_{\varepsilon}$,
$P=\cup_{i=1}^{m}T_{i}=[t,\tau ]\cap J_{\varepsilon }\cap I_{\varepsilon} $,
$Q=[t,\tau ]-P$ and $A_{i}=\{x_{n}(\theta _{n}(t)):n\in \mathbb{N},t\in
T_{i}\}$, $i=1,\dots ,m$. In view of the mean value theorem, properties
of $\gamma $ ((M3), (M5)) and Condition (A1), this implies
\begin{align*}
\gamma (\cup _{n\in \mathbb{N}}\int_{P}F(s,x_{n}(\theta _{n}(s)))ds)
&\leq \gamma (\sum_{i=1}^{m}\cup _{n\in \mathbb{N}}
\int_{T_{i}}F(s,x_{n}(\theta _{n}(s)))ds) \\
&\leq \gamma (\sum_{i=1}^{m}\lambda (T_{i})(\overline{\rm conv}
F(T_{i}\times A_{i}))) \\
&\leq \sum_{i=1}^{m}\lambda (T_{i})\gamma (\overline{\rm conv}
F(T_{i}\times A_{i})) \\
&\leq \sum_{i=1}^{m}\lambda (T_{i})\sup_{s_{i}\in T_{i}}w(s_{i},
\gamma (A_{i})).
\end{align*}
Now we have
\begin{align*}
\gamma (A_{i})
&= \gamma (\{x_{n}(\theta _{n}(s)):n\in \mathbb{N}, s\in T_{i}\})
\\
&\leq \gamma (\{x_{n}(s):n\in \mathbb{N},s\in T_{i}\})+\gamma
(\{x_{n}(\theta _{n}(s))-x_{n}(s):n\in \mathbb{N},s\in T_{i}\}) \\
&\leq \gamma (\{x_{n}(s):n\in \mathbb{N}, s\in T_{i}\})+\gamma
(\{\int_{s}^{\theta _{n}(s)}\dot{x}_{n}(r)dr:n\in \mathbb{N}, s\in T_{i}\}).
\end{align*}
 From Lemma \ref{lem1} we know that
\begin{align*}
&\gamma \Big(\Big\{\int_{s}^{\theta _{n}(s)}\dot{x}_{n}(r)dr:n\in
\mathbb{N},s\in T_{i}\Big\}\Big)\\
&\leq \gamma (B(0,1))\alpha \Big(\Big\{\int_{s}^{\theta
_{n}(s)}\dot{x}_{n}(r)dr:n\in \mathbb{N},s\in T_{i}\Big\}\Big).
\end{align*}
Also $\lim_{n\to \infty }|\theta _{n}(s)-s|=0$. So,
$\gamma (A_{i})=\gamma (\{x_{n}(s):n\in \mathbb{N},s\in T_{i}\})
+\frac{\delta }{2}$. Applying Lemma \ref{lem2}, we get
$\gamma (A_{i})=\sup_{\xi _{i}\in T_{i}}\rho (\xi _{i})
+\frac{\delta }{2}$. Since $w$ and $\rho $ are continuous on the
closed subsets of $T_{i}$, then
\begin{align*}
\gamma \Big(\cup _{n\in \mathbb{N}}\int_{P}F(s,x_{n}(\theta _{n}(s)))ds\Big)
&\leq \sum_{i=1}^{m}\lambda (T_{i})\sup_{s_{i}\in T_{i}}w(\big(s_{i},
 \sup_{\xi _{i}\in T_{i}}\rho (\xi _{i})+\frac{\delta }{2}\big) \\
&\leq \sum_{i=1}^{m}\lambda (T_{i})w\big(q_{i},\rho (\xi _{i})
 +\frac{\delta }{2}\big),
\end{align*}
where $q_{i}$ and $\xi _{i}$ are elements of $T_{i}$. Moreover,
for all $s\in T_{i}$, we have
$$
|\rho (s)-\rho (\xi _{i})+\frac{\delta }{2}|\le |\rho
(s)-\rho (\xi _{i})|+\frac{\delta }{2}<\frac{\delta }{2}+\frac{\delta }{2}
=\delta .
$$
This implies $|w(s,\rho (s))-w(q_{i},\rho (\xi_{i})
+\frac{\delta }{2})|<\varepsilon $ for all $s\in T_{i}$.
Consequently,
 $\lambda (T_{i})w(q_{i},\rho (\xi _{i})+\frac{\delta
}{2})\le \int_{T_{i}}w(s,\rho (s))\,ds
+\varepsilon \lambda (T_{i})$. So,
\begin{align*}
\gamma \Big(\cup _{n\in \mathbb{N}}\int_{P}F(s,x_{n}(\theta _{n}(s)))ds\Big)
&\leq \sum_{i=1}^{m}\Big(\int_{T_{i}}w(s,\rho (s))ds+\varepsilon
 \lambda (T_{i})\Big) \\
&= \int_{P}w(s,\rho (s))ds+\varepsilon  \lambda (P) \\
&\leq \int_{t}^{\tau }w(s,\rho (s))ds+\varepsilon (\tau -t).
\end{align*}
On the other hand,
\[
\gamma (\cup _{n\in \mathbb{N}}\int_{Q}F(s,x_{n}(\theta _{n}(s)))ds)\le 2m\gamma
(B(0,1)) \int_{Q}\mu (s)(1+\Vert x_{n}(\theta _{n}(s))\Vert )ds.
\]
As $\lambda (Q)<2\varepsilon $ and since $\varepsilon $ is arbitrary, then
\begin{equation}
\gamma (\cup _{n\in \mathbb{N}}\int_{t}^{\tau }F(s,x_{n}(\theta _{n}(s)))ds)
\le \int_{t}^{\tau }w(s,\rho (s))ds,  \label{b}
\end{equation}
Thus, from two relations (\ref{a}), (\ref{b}),
\[
\dot{\rho}(t)\le w(s,\rho (s))\quad\text{ a.e. on} I.
\]
$\rho (0)=0$ and $w$ is a Kamke function, then $\rho $ is identically equal
to zero. It follows that $(x_{n})$ is relatively compact in $C(I,E)$.
Since, for all $t\in I$,
\[
\gamma (\{x_{n}(\theta _{n}(t)) :n\in
\mathbb{N}\})\le \gamma (\{x_{n}(t):n\in \mathbb{N}\})
+\gamma (\int_{{\theta _{n}}(t)}^{t}(m\mu (s)+1)ds)B(0,1)
\]
and since $\lim_{n\to \infty }|{{\theta _{n}}(t)}-t|=0$,
the set $\tilde{A}(t):=\{x_{n}(\theta _{n}(t)):n\in
\mathbb{N}\}$ is relatively compact in $E$. By our assumption
$F(t,.)$ is upper semicontinuous, it follows that
$F(t,\overline{A}(t))$ is compact for all $t\in I$, Furthermore, we have
\[
\dot{x}_{n}(t)\in F(t,\overline{A(t)})+\varepsilon_n  B(0,1),\quad
\forall n\in \mathbb{N},\forall t\in I.
\]
Since $\dot{x}_{n}$ is uniformly integrable, by  \cite[Theorem 5.4]{1.48},
the sequence $\dot{x}_{n}$ is relatively
$\sigma (L^{1}(I, E), L^{\infty }(I, E))$ compact. Therefore there
are $x_{nk}\in {C(I,E)},g\in {L^{1}(I, E)}$
and a subsequence $(x_{nk})$ of $(x_{n})$ such that $(x_{nk})$ converges
to $x$ in ${C(I,E)}$ and $(\dot{x_{nk}})$ converges to $g$ in $L^{1}(I, E)$
for $\sigma (L^{1}(I, E), L^{\infty }(I, E))$, with
$$
x(t)=x_{0}+\int_{0}^{t}g(s)ds,  \forall t\in I.
$$
Thus $g=\dot{x}$. Clearly for all
$t\in I$, $\lim_{n\to \infty }x_{nk}(\theta _{n}(t))=x(t)$,
and $x(t)\in \Gamma (t)$, for all $t\in I$. Finally, in virtue
\cite[Theorem 5.6, Remark 6.3]{1.48} and the property (iii) we obtain
\[
\dot{x}(t)\in F(t,x(t))\quad \text{a.e.\thinspace on} I.
\]
\end{proof}

\begin{theorem}\label{thm4}
Let $\Gamma :I\to P_{f}(E)$ be a set-valued function
with closed  graph, $G$, and $F:G\to P_{ck}(E)$ be a
set-valued function such that for any $t\in I$, $F(t,.)$ is upper
semicontinuous on $E$. Assume that $F$ satisfies the following conditions:
\begin{itemize}

\item[(A1')] For each $\varepsilon >0$, there exists a closed
subset $I_{\varepsilon }$ of $I$ with
$\lambda (I-I_{\varepsilon })<\varepsilon $ such that for any
nonempty bounded subset $Z$ of $E$, one has
\[
\gamma (F(G\cap (I\times Z)))\le \sup_{t\in I}w(t,\gamma (Z)),
\]
for any compact subset $J$ of $I_{\varepsilon}$;

\item[(A2')] there is a positive number $c$ such that
\[
\Vert F(t,x)\Vert <c(1+\Vert x\Vert ),\forall (t,x)\in G;
\]

\item[(A3')]  for each $(t,x)\in ([0,T[\times E)\cap G$ and for any
$\varepsilon >0$ there is $(t_{\varepsilon },x_{\varepsilon })\in G$ such
that $0<t_{\varepsilon }-t<\varepsilon $ and
\[
\frac{x_{\varepsilon }-x}{t_{\varepsilon }-t}\in F(t,x)+\varepsilon
\overline{B(0,1)}.
\]
\end{itemize}
Then, for each $x_{0}\in \Gamma (0)$, there is a solution of \eqref{eP}.
\end{theorem}

\begin{proof}
Let $A_{\varepsilon }([0,\tau ])$ $(\varepsilon >0,\tau \in I)$ be the set
of all points $(x,\theta )$ where $\theta :[0,\tau ]\to [0,\tau] $ is an
increasing right continuous function with
 $\theta (0)=0,\theta (\tau )=\tau $ and for all $t\in ]0,\tau [$,
$\theta (t)\in [t-\varepsilon ,t]$ and $x:[0,\tau ]\to E$ is such that:
\begin{itemize}
\item[(i)] for all $t\in [0,\tau ]$, $x(t)=x_{0}+\int_{0}^{t}\dot{x}(s)\,ds$,
where $\dot{x}\in L^{1}(I, E)$;

\item[(ii)] for all $t\in [0,\tau ]$,
$x\big(\theta(t)\big)\in \Gamma (\theta(t))$;

\item[(iii)] for all $t\in [0,\tau ]$,
$\dot{x}(t)\in F\big(t,x(\theta(t)\big)
+\varepsilon \overline{B(0,1)}$, a.e.
\end{itemize}
Let $\varepsilon \in ]0,1]$ and
$(\theta ,x)\in A_{\varepsilon }([0,\tau ])$.
Then by (A2') and the fact that, for all
$t\in [0,\tau ]$, $\theta (t)\in [t-\varepsilon ,t]$, we have
\begin{align*}
\Vert x(\theta (t))\Vert
&\leq \Vert x_{0}\Vert +\int_{0}^{\theta (t)}\Vert
\dot{x}(s)\Vert \,ds\le \Vert x_{0}\Vert +\int_{0}^{t}\Vert \dot{x}(s)\Vert
\,ds \\
&\leq \Vert x_{0}\Vert +\varepsilon T+\int_{0}^{t}c(1+\Vert x(\theta
((s))\Vert )\,ds.
\end{align*}
By Gronwall's lemma, we obtain $\Vert x(\theta (t))\Vert \le (\Vert
x_{0}\Vert +T)e^{cT}$ which gives us
$$
\Vert x(\theta (t))\Vert +1\le (1+\Vert x_{0}\Vert +T)e^{cT}.
$$
Consequently we get for all $t\in [0,\tau ]$,
\begin{equation}
F(t,x(\theta (t)))\subseteq pc\overline{B(0,1)}  \label{**}
\end{equation}
where $p=(1+\Vert x_{0}\Vert +T)e^{cT}$. Let
$A_{\varepsilon }=\bigcup_{\tau \in I}A_{\varepsilon }([0,\tau ])$.
Obviously $A_{\varepsilon }\ne \emptyset $. Partially order
$A_{\varepsilon }$ such that for any
$(\theta _{i},x_{i})\in A_{\varepsilon }([0,\tau _{i}])
\subseteq A_{\varepsilon }$ $(i=1,2)$
$(\theta _{1},x_{1})\le (\theta _{2},x_{2})\Longleftrightarrow
\tau _{1}\le \tau _{2},\theta _{1}=\theta _{2}|_{[0,\tau _{1}]}$ and
$x_{1}=x_{2}|_{[0,\tau _{2}]}$. Let $C$ be a
subset of $A_{\varepsilon }$ such that each two elements of it are
comparable that is there exists a subset
$\mathbb{N}'\subseteq \mathbb{N}$
such that $C=\{(\theta _{j},x_{j}):j\in \mathbb{N}'\}\subseteq
A_{\varepsilon }$ and each $(\theta _{n},x_{n}), (\theta _{m},x_{n})\in C$
we have $(\theta _{n},x_{n})\le (\theta _{m},x_{m})$ or
$(\theta _{m},x_{m})\le (\theta _{n},x_{n})$. Now we prove that $C$
has an upper bound. Let $\tau =\sup_{j\in \mathbb{N}'}\tau _{j}$.
Also let $\theta :[0,\tau ]\to [0,\tau ]$ is such that, for each
$j\in \mathbb{N}',\theta |_{[0,\tau _{j}]}=\theta _{j}$ and
$x:[0,\tau [\to E$ with $x|_{[0,\tau _{j}]}=x_{j}$, for each
$j\in \mathbb{N}'$. Let $\{\tau _{k_{n}}\}$ be increasing sequence in
$\mathbb{N}'$ such that $\tau =\sup_{n\in \mathbb{N}}\tau _{k_{n}}$ and for
any $n,m\in \mathbb{N},\;m<n$ we have $\dot{x}_{k_{n}}=\dot{x}_{k_{m}}$
 a.e. on $[0,\tau _{k_{n}}]$. Now we can define $\dot{x}:[0,\tau [\to E$
by, for any $n\in \mathbb{N}$, $\dot{x}(t)=\dot{x}_{k_{n}}(t)$ a.e. on
$[0,\tau _{k_{n}}]$. From (\ref{**}) $\dot{x}$ is measurable and
$\Vert \dot{x} (t)\Vert \le pc+\varepsilon \le pc+1$. We claim that
$x, \dot{x}$ can be extend to $[0,\tau ]$.
Now for all $t\in [0,\tau [$,
$x(t)=x_{0}+\int_{]0,t]}\dot{x}(s)\,ds$, for all
$t\in [0,\tau [$, $\dot{x}(\theta (t))\in \Gamma (\theta (t))$ and
$\dot{x}(t)\in F(t,\dot{x}(\theta (t)))+\varepsilon
\overline{B(0,1)}$ a.e. on $[0,\tau [$. If $x'(t)=x_{0}+\int_{]0,t[}
\dot{x}(s)\,ds$ for all $t\in [0,\tau ]$ then, for
$(t,t')\in [0,\tau [\times [0,\tau [$, we have
$\Vert x'(t)-x'(t')\Vert \le \int_{[t,t'[}(ps+1)\,ds$. Then
$x^{*}:=\lim_{t\to \tau ^{-0}}(x_{0}+\int_{]0,t[}\dot{x}(s)\,ds)=\lim_{n\to
\infty }(x_{0}+\int_{]0,\tau _{k_{n}}[}\dot{x}(s)\,ds)$ exists. Since
$x'(\tau _{k_{n}})\in \Gamma (\tau _{k_{n}})$ and $G$ is left
closed, then $(\tau ,x^{*})\in G$ and hence the result. Let
$x^{*}=x(\tau )$ and $\dot{x}(\tau )=0$. Then
$x(\tau )=x_{0}+\int_{]0,\tau ]}\dot{x} (s)\,ds,\;x^{*}
=x(\tau )\in \Gamma (\tau )$ and
$\dot{x}(t)\in F(t,x(\theta (t)))+\varepsilon \overline{B(0,1)}$ a.e.
on $[0,\tau ]$. Consequently we
can extend $(\theta ,x)$ to $[0,\tau ]$ such that $(\theta ,x)$ belongs
to $A_{\varepsilon }([0,\tau ])$ and it is an upper bound for $C$.
By Zorn's lemma $(A_{\varepsilon },\le )$ has a maximal element
$(\theta _{\varepsilon },x_{\varepsilon })\in
A_{\varepsilon }([0,\tau _{\varepsilon }])$. We shall
prove that $\tau _{\varepsilon }=T$. Let $\tau _{\varepsilon }<T$.
If $\delta _{\varepsilon }>0$ such that
$\delta _{\varepsilon }<\inf(\varepsilon ,T-\tau _{\varepsilon })$.
Then by (A3') there exists $(\hat{t},\hat{x})\in G$ such that
$0<\hat{t}-\tau _{\varepsilon }\le \delta _{\varepsilon }$ and
\[
\frac{\hat{x}-x_{\varepsilon }}{\hat{t}-\tau _{\varepsilon }}\in F(\tau
_{\varepsilon },x_{\varepsilon }(\tau _{\varepsilon }))+\varepsilon
\overline{B(0,1)}.
\]
Let $\hat y\in F(\tau_{\varepsilon},x_{\varepsilon}(\tau_{\varepsilon}))+\varepsilon
\overline {B(0,1)}$ such that $\hat x-x_{\varepsilon}(\tau_{
\varepsilon})=(\hat t-\tau_{\varepsilon})\hat y$.
If $\hat{\theta}:[0,\hat{t}]\to [0,\hat{t}]$ and $\tilde{x}
:[0,\hat{t}]\to E$ are defined as:
\[
\hat{\theta}(t)=\begin{cases}
\theta _{\varepsilon } & \text{if }t\in [0,\tau_{\varepsilon}]  \\
\tau _{\varepsilon } & \text{if }t\in ]\tau_{\varepsilon},\hat t] \\
\hat{t} & \text{if } t=\hat t,
\end{cases}
\qquad
\tilde{x}(t)=\begin{cases}
x_{\varepsilon } & \text{if }t\in [0,\tau_{\varepsilon}] \\
\hat{x} & \text{if }t\in [\tau_{\varepsilon}, \hat t]
\end{cases}
\]
Then it is easy to check that \cite[p. 10.25]{1.48}
$(\hat{\theta},\tilde{x})\in A_{\varepsilon }([0,\hat{t}])$ and
$(\theta _{\varepsilon },x_{\varepsilon })<(\hat{\theta},\tilde{x})$. This
contradicts the fact that $(\theta _{\varepsilon },x_{\varepsilon })$ is
maximal. Now there exist $p>1$, (from (\ref{**})) a sequence
$(\theta _{n})_{n\in \mathbb{N}}$ of right continuous functions
$(\theta)_{n}:I\to I$ such that $\theta _{n}(0)=0$,
$\theta _{n}(T)=T$
and $\theta _{n}(t)\in [t-\varepsilon _{n},t]$, if we have decreasing
sequence $(\varepsilon _{n})$ such that
$0<\varepsilon _{n}\le 1\;\varepsilon _{n}\to 0$ as
$n\to \infty $ and
$T\varepsilon _{n}<\frac{\varepsilon }{\gamma (B(0,1))}$, for all
$n\ge n_{0}$
we can define a sequence $(x_{n})$ of approximated solutions as the follows:

$\forall t\in I,\quad  x_{n}(t)=x_{0}+\int_{0}^{t}\dot{x}_{n}(s)\,ds$,
 where $ \dot{x}_{n}\in L^{1}(I, E)$.
$(\theta _{n}(t),x_{n}(\theta _{n}(t)))\in G$.
$\dot{x}_{n}(t)\in F\big(t,x_{n}(\theta _{n}(t)\big) +\varepsilon
_{n}B(0,1)$, a.e on $I$.
$\Vert \dot{x}_{n}(t)\Vert \le pc+1$, a.e on $I$.

By the same arguments used in the proof of Theorem \ref{thm3} we can prove
that the sequence $(x_{n})$ converges to an absolutely continuous
function $x$ which is a solution for problem \eqref{eP}.
\end{proof}

\section{Conclusion}

Let us remark that, if we replace $\gamma $ in (A1') by $\alpha $,
the  condition
\begin{itemize}
\item[(A4)] For each $\varepsilon >0$, there exists a closed subset
$I_{\varepsilon }$ of $I$ with $\lambda (I-I_{\varepsilon })<\varepsilon $
such that for almost all $t\in I_{\varepsilon }$ and for any nonempty
bounded subset $Z$ of $E$, one has
\[
\inf_{\delta >0}\alpha (F(G\cap (([t-\delta ,t]\cap I)\times Z)))\le
w(t,\alpha (Z))
\]
\end{itemize}
implies Condition (A1') and the converse is not true. Indeed
Let $\varepsilon >0$. Since $w$ is Caratheodory function, we can find a
closed subset $I_{\varepsilon }$ of $I$ with $\lambda (I-I_{\varepsilon
})<\varepsilon $ such that $w$ is continuous on $I_{\varepsilon }$ and
Condition (A4) holds on $I_{\varepsilon }$. Let $Z$ be a nonempty
bounded subset of $E$. It follows from (A4) that, for any $\tau >0$ and
any $t\in I_{\varepsilon }$, there exists a $\delta _{\tau ,t}$ such that
$\alpha (F(G\cap (([t-\delta _{\tau ,t},t]\cap I)\times Z)))\le w(t,\alpha
(Z))+\tau $.
Let $\tau $ be arbitrary but fixed, $J$ be a compact subset of
$I_{\varepsilon }$. The collection
$\{(t-\frac{\delta _{t}}{2},t+\frac{\delta_{t}}{2}):t\in J\}$ is an
open cover for $J$. By compactness of $J$,
there exist $t_{1}',t_{2}'\dots ,t_{n}'$ such that
$ J\subseteq \cup _{i=1}^{n}(t_{i}'-\frac{\delta _{t_{i}'}}{2
},t_{i}'+\frac{\delta _{t_{i}'}}{2})\subseteq \cup
_{i=1}^{n}[t_{i}'-\frac{\delta _{t_{i}'}}{2},t_{i}'
+\frac{\delta _{t_{i}'}}{2}]$.
Now if $J_{i}=J\cap [t_{i}'-\frac{\delta _{t_{i}'}}{2},t_{i}'+\frac{\delta
_{t_{i}'}}{2}]$ and $t_{i}=\text{max}J_{i}, 1\le i\le n$, then
there exist $t_{1},t_{2}\dots t_{n}\in J$ such that $J_{i}\subseteq
[t_{i}-\delta _{t_{i}},t_{i}]$ and $J\subseteq \cup _{i=1}^{n}[t_{i}-\delta
_{t_{i}},t_{i}]$. This implies that,
\begin{align*}
\alpha (F(G\cap (J\times Z)))
&\leq \alpha (\cup _{i=1}^{n}F(G\cap
(([t_{i}-\delta _{t_{i}},t_{i}]\cap I)\times Z))) \\
&\leq \max_{1\le i\le n}\alpha (F(G\cap (([t_{i}-\delta _{t_{i}},t_{i}]\cap
I)\times Z))) \\
&\leq \max_{1\le i\le n}w(t_{i},\alpha (Z))+\tau \le \max_{t\in J}w(t,\alpha
(Z))+\tau
\end{align*}
Since $\tau $ is arbitrary, Condition (A1') holds. To show
that the converse is not true we give an example.
Let $f:[0,1]\times B(0,1)\to E$ be the single valued function
defined by $f(t,x)=k(t)x$, where $k:[0,1]\to \mathbb{R}$,
\[
k(t)=\begin{cases}
1 & \text{if $t$ is irrational} \\
1/t^2 & \text{if $t$ is rational}
\end{cases}
\]
Let also $w(t,s)=k(t)s$, for all $(t,s)\in I\times \mathbb{R}^{+}$.
Clearly, $w$ is a Kamke function. Let $\varepsilon >0$ and choose a
closed subset $I_{\varepsilon }$ of $I$ such that
$\lambda (I-I_{\varepsilon })<\varepsilon$ and $k$ is continuous on
$I_{\varepsilon }$. Then for any compact subset $J $ of $I_{\varepsilon }$
and any bounded subset $Z$ of $E$,
\begin{align*}
\alpha (f(G\cap (J\times Z)))\le \alpha (f(J\times Z))
&= \alpha \big(\cup _{t\in J,x\in Z}f\{(t,x)\}\big) \\
&= \alpha \big( \cup _{t\in J}k(t)Z\big) =\sup_{t\in J}k(t)\alpha (Z) \\
&= \sup_{t\in J}w\big(t,\alpha (Z)\big).
\end{align*}
Then Condition (A1') holds as the measure $\gamma $ replaced
by the measure $\alpha $. But for each $t\in (0,1)$ and each nonempty subset
$Z$ of $E$ we have
$\alpha \big(f([t-\delta ,t]\times Z)\big) =\alpha \big(\cup _{s\in
[t-\delta ,t]}k(s)Z\big)
=\alpha (Z)\cdot \big (\sup_{s\in [t-\delta ,t]}k(s)\big)
=\frac{\alpha (Z)}{(t-\delta )^{2}}$. %\end{align*}
Thus, 
$\inf_{\delta >0}\alpha (F(([t-\delta ,t]\cap I)\times Z))=\frac{\alpha (Z)}{
t^{2}}$.
So if t is irrational then
$\inf_{\delta >0}\alpha (F(([t-\delta ,t]\cap I)\times Z))=\frac{\alpha (Z)}{
t^{2}}>\alpha (Z)=k(t)\alpha (Z)=w(t,\alpha (Z))$.
Then (A4) does not hold and consequently Theorem \ref{thm4} is a
generalization of the following theorem.

\begin{theorem}[Benabdellah-Castaing and Ibrahim \cite{1.48}]
Let $F$ and $\Gamma $ be as
in Theorem \ref{thm4} except $F$ satisfies Condition {\rm (A4)} instead
of {\rm (A1')}. Then, for any $x_{0}\in \Gamma (0)$, there is a solution
for \eqref{eP}.
\end{theorem}


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\end{document}