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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 42, pp. 1--11.\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{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/42\hfil Existence of mild solutions]
{Existence of mild solutions of a semilinear evolution differential 
inclusions with nonlocal conditions}
\author[Reem A. Al-Omair, Ahmed G. Ibrahim\hfil EJDE-2009/42\hfilneg]
{Reem A. Al-Omair, Ahmed G. Ibrahim} % in alphabetical order

\address{Reem A. Al-Omair \newline
King Faisal University, Collage of Education in Jubail, Mathematics
Department, Jubail, Saudi Arabia}
\email{omair-reem@hotmail.com}

\address{Ahmed G. Ibrahim \newline
Cairo University, Faculty of Science, Mathematics Department, Egypt}
\email{agamal2000@yahoo.com}

\thanks{Submitted January 1, 2009. Published March 24, 2009.}
\subjclass[2000]{34A60}
\keywords{Evolution operator; generalized Cauchy operator;\hfill\break\indent
 measure of noncompactness; differential inclusions; nonlocal conditions;
mild solutions}

\begin{abstract}
 In this paper we prove the existence of a mild solution for a semilinear
 evolution differential inclusion with nonlocal condition and governed by a
 family of linear operators, not necessarily bounded or closed, in a Banach
 space. No compactness assumption is assumed on the evolution operator
 generated by the family operators. Also, we prove that the set of mild
 solutions is compact.
\end{abstract}

\maketitle

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

\section{Introduction}

The study of Cauchy problems with nonlocal conditions, which is a
generalization for the classical Cauchy problems with initial condition, was
motivated by physical problems (see \cite{b1,f2}). The pioneering work on
nonlocal conditions problems is due to Byszewski \cite{b5,b6,b7,b8}. In the
few past years, several papers have been devoted to study the existence of
solutions for differential equations or differential inclusions with
nonlocal conditions. Among others, we refer the reader to 
\cite{a1,b3,b4,c3,f1,x1}.

Let $E$ be a Banach space, $I=[0,b]$, $b>0$, $C(I,E)$ be the Banach space of
all continuous functions from $I$ to $E$ with the norm of uniform
convergence. Let $\mathcal{L}(E)$ be the space of bounded linear operators
on $E$ and $\{A(t):t\in I\} $ be a family of densely defined linear operator
(not necessarily bounded or closed) on $E$ and $T:\Delta=\{(t,s):0\leq
s\leq t\leq b\}\to\mathcal{L}(E)$ be the evolution operator generated by the
family $\{A(t):t\in I\}$.

Let $F$ be a Carath\`{e}odory type multifunction from $I\times E$ to the
collection of all nonempty convex compact subsets of $E$, and let $g:C(I,E)
\to E$ be a function.

Consider the following semilinear differential inclusion with nonlocal
condition: 
\begin{equation}  \label{ep}
\begin{gathered} x'(t)\in A(t)x(t)+F(t,x(t)),\quad t\in I \\ x(0)=g(x)
\end{gathered}
\end{equation}
By a mild solution of problem \eqref{ep} we mean a continuous function 
$x:I\to E$ such that 
\begin{equation*}
x(t)=T(t,0)g(x)+\int_0^t T(t,s)f(s)ds,\quad t\in I
\end{equation*}
where $f$ is a Bochner integrable function such that $f(s)\in F(s,x(s))$,
a.e. $t\in I$.

In the present paper we employ the methods of Kamenskii, Obukhowskii and
Zecca \cite{k1} and Cardinali and Rubbioni \cite{c1} to prove the existence
of mild solution for $(p)$ without a compactness assumption on the evolution
operator $T$ which is generated by the family $\{A(t):t\in I\}$. The price
that we pay to achieve this aim is that we assume that $F$ satisfies a
compactness condition involving the Hausdorff measure of noncompactness.

We would like to mention that in a recent paper Fan, Dong and Li \cite{f1}
proved the existence of mild solution for the semilinear differential
equations with nonlocal conditions in Banach space 
\begin{equation}  \label{eq}
\begin{gathered} 
x'(t)=A\big(x(t)\big) +f(t,x(t)) ,\quad t\in (0,b] \\
x(0)=g(x), 
\end{gathered}
\end{equation}
where $A$ is the infinitesimal generator of a strongly continuous semigroup
of bounded linear operators $\{U(t):t\in I\}$, $f:I\times E\to E$ is a Carath%
\`{e}odory type function and satisfies a compactness condition involving the
Hausdorff measure of noncompactness and $g$ is a continuous compact function.

Note that Corollary \ref{coro3.1} generalizes the result of Fan, Dong and Li 
\cite{f1} to the case when the function $f$ become a multifunction.
Moreover, since the nonlocal conditions Cauchy problems is a generalization
for the classical Cauchy problems with initial conditions, our work
generalizes many results in the literature, see for example \cite{c1,i1,k1}.
For differential inclusions with initial condition, we refer the reader to 
\cite{g1,i2}.

Our basic tools are the methods and results for semilinear differential
inclusions, the properties of non-compact measure and fixed point techniques.

\section{Preliminaries and notation}

Let $I$ $=[0,b]$, $b>0$ , $(E,\Vert \cdot \Vert )$ be a real Banach space, $%
C(I,E)$ the space of $E$-valued continuous functions on $I$ with the uniform
norm $\Vert x\Vert _{C}=\sup \{\Vert x(t)\Vert ,\;t\in I\}$, $L^{1}(I,E)$
the space of $E-$valued Bochner integrable functions on $I$ with the norm $%
\Vert f\Vert _{L^{1}}=\int_{0}^{b}\Vert f(t)\Vert dt$ and $P_{k}(E)$ $%
(P_{ck}(E))$ the collection of nonempty compact (convex compact) subsets of $%
E$

\begin{definition} \label{def2.1}\rm
Let $X$ and $Y$ be two topological spaces
and let $P(Y)$ the family of nonempty subsets of $Y$. A multifunction
$G:X\to P(Y)$ is said to be upper semicontinuous (u.s.c.) if
$G^{-1}(V)=\{x\in X:G(x)\subseteq V\}$ is an open subset of $X$
for every open $V\subseteq Y$. The multifunction $G$ is called closed if its
graph $\Gamma _{G}=\{(x,y) \in X\times Y:y\in G(x)\}$ is closed
subset of the topological space $X\times Y$. For details and
equivalent definitions see \cite{c2}.
\end{definition}

\begin{definition} \label{def2.2} \rm
Let $(\mathcal{A},\geq) $ be a partially ordered set. A function
$\beta :P(E)\to \mathcal{A}$ is called a measure of noncompactness
(MNC) in $E$  if
\begin{equation*}
\beta (\overline{\rm co}\Omega )=\beta (\Omega )
\end{equation*}
for every $\Omega\in P(E)$.
\end{definition}

\begin{definition} \label{def2.3}\rm
A measure of noncompactness $\beta $ is called:
\begin{itemize}
\item[(i)] monotone if $\Omega_{0},\Omega_{1}\in P(E)$,
$\Omega_{0}\subset\Omega_{1}$ implies
$\beta(\Omega_{0})\leq\beta(\Omega_{1})$

\item[(ii)] nonsingular if $\beta(\{a\}\cup\Omega)=\beta(\Omega)$
for every $a\in E$, $\Omega\in P(E)$;

\item[(iii)] regular if $\beta(\Omega)=0$ is equivalent to the relative
compactness of $\Omega$.

\end{itemize}
As an example of the measure of noncompactness possessing all these
properties is the Hausdorff of MNC which is defined by
\begin{equation*}
\chi (\Omega ) =\inf \{\varepsilon >0:\Omega \text{ has a finite
}\varepsilon -\text{net}\}.
\end{equation*}
For more information about the measure of noncompactness we refer
the reader to \cite{b2}.
\end{definition}

\begin{definition} \label{def2.4}\rm
A multifunction $G:E\to P_{k}(E)$ is said to be $\chi$-condensing
if for every bounded subset $\Omega \subseteq E$  the relation
\begin{equation*}
\chi (G (\Omega ) ) \geq \chi (\Omega )
\end{equation*}
 implies the relative compactness of $\Omega$.
\end{definition}

\begin{definition} \label{def2.5}\rm
A countable set $\{f_{n}:n\geq 1\}\subseteq L^{1}(I,E)$ is said to be
semicompact if
\begin{itemize}
\item[(i)] it is integrably bounded: $\| f_{n}(t)\| \leq\omega(t)$
for a.e. $t\in I$  and every $n\geq1$ where
$\omega\in L^{1}(I,\mathbb{R}^{+}) $

\item[(ii)] the set $\{f_{n}(t):n\geq 1\}$ is relatively compact in $E$
for a.e. $t\in I$.
\end{itemize}
Now, let for every $t\in I=I$ , $A(t):E\to E$ be a linear operator
such that
\begin{itemize}
\item[(i)] For all $t\in I$, $D(A(t))=D(A)\subseteq E$ is independent of $t$.

\item[(ii)] For each $s\in I$ and each $x\in E$ there is a unique solution
$v:[s,b]\to E$  for the evolution equation
\begin{equation} \label{east}
\begin{gathered}
v'(t)=A(t)v(t),\quad t\in [s,b] \\
v(s)=x.
\end{gathered}
\end{equation}
\end{itemize}
In this case an operator $T$ can be defined as
\begin{equation*}
T:\Delta =\{(t,s):0\leq s\leq t\leq b\}\to \mathcal{L}
(E),\quad T(t,s)(x)=v(t),
\end{equation*}
 where $v$ is the unique solution of \eqref{east} and $\mathcal{L}(E)$
is the family of linear bounded operators on $E$.
\end{definition}

\begin{definition} \label{def2.6}\rm
The operator $T$ is called the
evolution operator generated by the family $\{A(t):t\in I\}$.
It is known that (see \cite{p1})
\begin{enumerate}
\item  $T(s,s)=I_{E}$

\item $T(t,r)T(r,s)=T(t,s)$, for all $0\leq s\leq r\leq t\leq b$.

\item each operator $T(t,s)$ is strongly differentiable and
\[
 \frac{\partial T(t,s)}{\partial t} =A(t)T(t,s) \quad
 \frac{\partial T(t,s)}{\partial s} =-T(t,s)A(s)
\]
\end{enumerate}
\end{definition}

\begin{definition} \label{def2.7}\rm
The operator $G:L^{1}(I,X) \to C(I,X) $ defined by
\begin{equation}
Gf(t)=\int_0^t T(t,s)f(s)ds  \label{e1}
\end{equation}
is called the generalized Cauchy operator, where $T(.,.)$ is the
evolution operator generated by the family of operators
$\{A(t):t\in I\}$.
\end{definition}

In the sequel we will need the following known results.

\begin{lemma}[{cite[Proposition 4.2.1]{k1}}] \label{lem2.1}
Every semicompact
set is weakly compact in the space $L^{1}(I,E)$.
\end{lemma}

\begin{lemma}[{\cite[Theorem 2]{c1}}] \label{lem2.2}
The generalized Cauchy operator $G$ satisfies the properties
\begin{itemize}
\item[(G1)] there exists $\zeta \geq 0$ such that
\begin{equation*}
\| Gf(t)-G h(t)\| \leq \zeta \int_0^t \| f(s)-h(s)\| ds
\end{equation*}
for every $f,h\in L^{1}(I,E)$, $t\in I$.

\item[(G2)] for any compact $K\subseteq E$ and sequence
$(f_{n})_{n\geq 1}$, $f_{n}\in L^{1}(I,E) $ such that for all $n\geq 1$,
$f_{n}(t)\in K$, a. e. $t\in I$, the weak convergence
$f_{n}\rightharpoonup f_{0}$ in $L^{1}(I,E) $ implies the convergence
$Gf_{n}\to Gf_{0}$ in $C(I,E)$.
\end{itemize}
\end{lemma}

\begin{lemma}[{\cite[Theorem 5.1]{k1}}] \label{lem2.3}
 Let $S:L^{1}(I,E) \to C(I,E) $ be an operator satisfying
condition (G$_{2}$) and the  following Lipschits condition
(weaker than {\rm (G1)})
\begin{itemize}
\item[(G1')]
\[
\| Sf-Sg\| _{C(I,E) }\leq \zeta \| f-g\| _{L^{1}(I,E) }.
\]
\end{itemize}
\end{lemma}

Then for every semicompact set $\{f_{n}\}_{n=1}^{+\infty}\subset L^{1}(I,E) $
the set $\{Sf_{n}\}_{n=1}^{+\infty }$ is relatively compact in $C(I,E) $.
Moreover, if $(f_{n}) _{n\geq 1}$ converges weakly to $f_{0}$ in $L^{1}(I,E) 
$ then $Sf_{n}\to Sf_{0}$ in $C(I,E)$.

\begin{lemma}[{\cite[Theorem 4.2.2]{k1}}] \label{lem2.4}
Let $S:L^{1}(I,E) \to C(I,E) $ be an operator
satisfying conditions {\rm (G1), (G2)} and let the set
$\{f_{n}\}_{n=1}^{\infty }$ be integrably bounded with the property
$\chi (\{f_{n}(t):n\geq 1\}) \leq \eta (t)$, for a.e. $t\in I$, where
$\eta (.)\in L_{+}^{1}(I,\mathbb{R}^{+})$ and $\chi $ is
the Hausdorff MNC. Then
\begin{equation*}
\chi (\{Sf_{n}(t):n\geq 1\})\leq 2\zeta \int_0^t \eta (s)ds
\end{equation*}
for all $t\in I$, where $\zeta \geq 0$ is the constant in condition
{\rm (G1)}.
\end{lemma}

\begin{theorem}[{\cite[Corollary 3.3.1]{k1}}] \label{thm2.1}
 If $U$ is a closed convex subset of a Banach space and $R$ is  a
closed $\beta $-condensing multifunction from $U$ to the family
of nonempty convex compact subsets of $U$, where $\beta $ is a
nonsingular MNC defined on the subsets of $U$, then $R$ has a fixed point.
\end{theorem}

\begin{theorem}[{\cite[Proposition 3.5.1]{k1}}] \label{thm2.2}
 Let $W$ be a closed subset of a Banach space $E$ and
$R:W\to P_{ck}(E) $ be a closed multifunction which is $\beta $-condensing
 on every bounded subset of $W$, where $P_{ck}(E) $ is the family of
nonempty convex compact subsets of $E$ and $\beta $ is a monotone measure of
noncompactness. If the set of fixed points for $R$ is a bounded subset of $E$
then it is compact.
\end{theorem}

\section{Main Results}

In the following theorem we prove the existence of mild solution for %
\eqref{ep}.

\begin{theorem} \label{thm3.1}
Let $I=[0,b]$, $\{A(t):t\in I\}$ be
a family of linear (not necessarily bounded) operators, $A(t):D(A)\subset
E\to E$, $D(A)$ not depending on $t$ and dense subset of $E$ and
$T:\Delta =\{(t,s):0\leq s\leq t\leq b\}\to \mathcal{L}(E)$ be
the evolution operator generated by the family $\{A(t):t\in I\}$.

Let $F$ be a multifunction defined from $I\times E$ to the family of
nonempty closed convex subsets of $E$ such that
\begin{itemize}
\item[(H1)] for every $x$  the multifunction $t\to F(t,x)$ admits
a strongly measurable selection and for a.e. $t\in I$ the multifunction
$x\to F(t,x)$ is upper semicontinuous;

\item[(H2)] there exists a function $m\in L^{1}(I,\mathbb{R}^{+})$ such
that for every $x\in E$
\begin{equation*}
\| F(t,x)\| \leq m(t)(1+\| x\| )\quad \text{for a.e. } t\in I;
\end{equation*}

\item[(H3)] there exists a function $h\in L^{1}(I,\mathbb{R}^{+})$
such that for every bounded $D\subset E$:
\begin{equation*}
\chi(F(t,D))\leq h(t)\chi(D)\quad\text{for a.e. }t\in I,
\end{equation*}
where $\chi$ is the Hausdorff measure of noncompactness;

\item[(H4)] let $g:C(I,E) \to E$ be a continuous function
that maps every bounded set into relatively compact subset of $E$ and that
\begin{equation*}
\| g(x)\| \leq c\| x\| _{C}+d,\quad \forall x\in C(I,E) ,
\end{equation*}
for some positive constants $c$ and $d$.
\end{itemize}
Then the nonlocal condition Cauchy problem \eqref{ep}
has a mild solution provided that
\begin{equation}
M(c+\| m\| _{L^{1}}) \neq 1,  \label{e2}
\end{equation}
where $M=\sup_{(t,s)\in\Delta} \|T(t,s)\| _{\mathcal{L}(E)}$
\end{theorem}

\begin{proof}
From the strong continuity of the evolution operator $T$ on the compact set 
$\Delta $, a positive real number $M$ can be found such that 
\begin{equation*}
\Vert T(t,s)\Vert \leq M,\quad \forall (t,s)\in \Delta
\end{equation*}
Thus, the number $M$ mentioned in the statement of the theorem is well
defined. From assumptions (H1) and (H2), see \cite[Lemma 4]{c1}, it follows
that the super position multioperator 
\begin{equation*}
\mathop{\rm sel}{}_{F}:C(I,E)\rightarrow 2^{L^{1}(I,E)}
\end{equation*}
defied by 
\begin{equation*}
\mathop{\rm sel}{}_{F}(x)=\text{ }\mathcal{S}_{F(.,x(.))}^{1}=\{f\in
L^{1}(I,E):f(t)\in F(t,x(t)),\quad \text{a.e. }t\in I\}
\end{equation*}
is well defined and weakly closed in the following sense: if $(x_{n})_{n\geq
1},(f_{n})_{n\geq 1}$ are two sequences in $C(I,E)$ and $L^{1}(I,E)$
respectively such that $f_{n}\in \mathop{\rm sel}{}_{F}(x_{n})$, for all 
$n\geq 1$ and if $x_{n}\rightarrow x_{0}$ in $C(I,E)$ and $f_{n}$ converges
weakly to $f_{0}$ in $L^{1}(I,E)$ then $f_{0}\in \mathop{\rm sel}{}_{F}(x_{0})$.
 Therefore, the multifunction $R$ defined on $C(I,E)$ by 
\begin{equation}
R(x)=\{y\in C(I,E):y(t)=T(t,0)g(x)+\int_{0}^{t}T(t,s)f(s)ds,\;f\in 
\mathop{\rm sel}{}_{F}(x)\},  \label{e3}
\end{equation}
has nonempty values. It is easy to see that any fixed point for $R$ is a
mild solution for \eqref{ep}. So, our goal is to prove that $R$ satisfies
the conditions of Theorem \ref{thm2.1} in the preliminaries. The proof will
be given in steps.

\noindent\textbf{Step 1.} The values of $R$ are convex subsets in $C(I,E) $.
Let $x\in C(I,E) $, $y_{1},y_{2}\in R(x)$ and $0<\alpha <1$. From the
definition of $M$ we get 
\begin{align*}
(1-\alpha ) y_{1}+\alpha y_{2} & =(1-\alpha )T(t,0)g(x)+\alpha T(t,0)g(x) \\
&\quad +\int_0^t ((1-\alpha ) T(t,s)f(s)+\alpha T(t,s)h(s)) ds \\
& =T(t,0)g(x)+\int_0^t T(t,s)((1-\alpha ) f(s)+\alpha h(s))ds\in R(x),
\end{align*}
where $f,h\in \mathop{\rm sel}{}_{F}(x)$. Which means that $R(x)$ is convex
for each $x\in C(I,E)$.

\noindent \textbf{Step 2.} The values of $R$ are compact subsets in $C(I,E) $.
 Let $x\in C(I,E) $ and $(z_{n}) _{n\geq1}$ be a sequence in $R(x)$. To
prove that $R(x)$ is compact we have to show that $(z_{n}) _{n\geq1}$ has a
subsequence converging to a point $z$ in $R(x)$. By \eqref{e3} for each $%
n\geq1$, 
\begin{equation*}
z_{n}(t)=T(t,0)g(x)+\int_0^t T(t,s)f_{n}(s)ds, \quad \text{where }
f_{n}(s)\in F(s,x(s)) ,\text{ a.e. }t\in I.
\end{equation*}
By (H2) 
\begin{equation*}
\| f_{n}(t)\| \leq \| F(t,x(t)\| \leq m(t)(1+\| x\| _{C}),\quad \text{a.e. }
t\in I.
\end{equation*}
Thus the set $\{f_{n}:n\geq 1\}$ is integrably bounded in $L^{1}(I,E) $.
Further, the set $\{f_{n}(t):n\geq 1\}$ is relatively compact in $E$  for
a.e. $t\in I$ because by (H3) we have 
\begin{equation*}
\chi(\{f_{n}(t):n\geq1\}) \leq\chi(F(t,x(t))) \leq h(t)\chi(\{x(t)\})
=0,\quad \text{a.e. }t\in I.
\end{equation*}
Hence the set $\{f_{n}:n\geq 1\}$ is semicompact in $L^{1}(I,E) $. From
Lemma \ref{lem2.1}, it follows that $\{f_{n}:n\geq 1\}$ is weakly compact in 
$L^{1}(I,E) $. So, without loss of generality we can assume that the
sequence $(f_{n}) _{n\geq 1}$ converges weakly to a function $f$ in 
$L^{1}(I,E) $ with $f(t)\in F(t,x(t))$ a.e. $t\in I$. By applying Lemma 
\ref{lem2.3}, $Gf_{n}\to Gf$, where $G$ is the generalized Cauchy operator
defined by \eqref{e1}. Thus 
\begin{equation*}
\lim_{n\to \infty } z_{n}(t) =T(t,0)g(x)+G f(t) =T(t,0)g(x\text{)}+\int_0^t
T(t,s)f(s)ds.
\end{equation*}
So that $z\in R(x)$. Hence $R(x)$ is compact.

\noindent\textbf{Step 3.} $R$ is closed, i.e. its graph is closed. Let 
$(x_{n}) _{n\geq1}$ and $(y_{n}) _{n\geq1}$ be two sequences in $C(I,E) $
such that $\lim_{n\to \infty} x_{n}=x$, $\lim_{n\to\infty} y_{n}=y$ in 
$C(I,E) $ and $y_{n}\in R(x_{n})$ for all $n\geq1$. By means of the
definition of the generalized Cauchy operator, we can write 
\begin{equation*}
y_{n}=T(t,0)g(x_{n})+G f_{n}(t),\quad \forall n\geq1,\; \forall t\in I,
\end{equation*}
where $f_{n}\in \mathop{\rm sel}{}_{F}(x_{n})$. Since $g$ is continuous on 
$C(I,E) $, $\lim_{n\to\infty} g(x_{n})=g(x)$ in $E$. So, 
\begin{equation*}
\lim_{n\to\infty} T(t,0)g(x_{n})=T(t,0)g(x).
\end{equation*}

To apply Lemma \ref{lem2.3} we have to show that the set $\{f_{n}:n\geq 1\}$
is semicompact. Since the sequence $(x_{n})_{n\geq 1}$ converges uniformly
to $x$, we can find a positive real number $k$ such that $\Vert x_{n}\Vert
_{C}\leq k$, for all $n\geq 1$. By (H2), for every $n\geq 1$, 
\begin{align*}
\Vert f_{n}(t)\Vert & \leq \sup \{\Vert z\Vert :z\in F(t,,x_{n}(t))\} \\
& \leq (1+\Vert x_{n}\Vert _{C})m(t)\leq (1+k)m(t),\quad \text{a.e. }t\in I.
\end{align*}
This shows that the family $\{f_{n}:n\geq 1\}$ is integrably bounded.
Furthermore, by (H3) 
\begin{align*}
\chi (\{f_{n}(t):n\geq 1\})& \leq \chi (F(t,D_{t})),\quad
D_{t}=\{x_{n}(t):n\geq 1\} \\
& \leq h(t)\chi (D_{t}),\quad \text{for a.e. }t\in I.
\end{align*}

Since $(x_{n}) _{n\geq 1}$ converges to $x$ in $C(I,E) $, the set $D_{t}$ is
relatively compact for each $t\in I$ and consequently $\chi (D_{t}) =0$.
Thus for a.e. $t\in I$, the set $\{f_{n}(t):n\geq 1\}$ is relatively compact
in $E$. Then the set $\{f_{n}:n\geq 1\}$ is semicompact in $L^{1}(I,E) $.
Invoking Lemma \ref{lem2.1} this set is weakly compact in $L^{1}(I,E) $. So,
without loss of generality we can assume that $f_{n}$ converges weakly to a
function $f\in L^{1}(I,E)$. From \cite[Lemma 4]{c1}, we get $f(t)\in
F(t,x(t))$, a.e. $t\in I$. Furthermore, by Lemma \ref{lem2.3}, 
\begin{equation*}
\lim_{n\to \infty } Gf_{n}=Gf.
\end{equation*}
Thus 
\begin{align*}
y(t)& =\lim_{n\to \infty }y_{n}(t)=T(t,0)g(x)+G f \\
& =T(t,0)g(x)+\int_0^t T(t,s)f(s)ds,
\end{align*}
which yields to, $y\in R(x)$ and so $R$ has a closed graph.

\noindent\textbf{Step 4.} $R$ is condensing with respect to a nonsingular
measure of noncompactness defined on $C(I,E)$

Let us define a measure of noncompactness $\nu $ on $C(I,E)$ as foloows: for
each bounded subset $\Omega $ of $C(I,E)$ we put 
\begin{equation}
\nu (\Omega )=\max_{D\in \Delta (\Omega )}(\gamma (D),\mod_{C}(D))\in 
\mathbb{R}^{2},  \label{e4}
\end{equation}
where $\Delta (\Omega )$ is the collection of all the denumerable subsets of 
$\Omega $; 
\begin{equation}
\gamma (D)=\sup_{t\in I}e^{-Lt}\chi (\{x(t):x\in D\});  \label{e5}
\end{equation}
where $\mathop{\rm mod}{}_{C}(D)$ is the modulus of equicontinuity of the set of
functions $D$ given by the formula 
\begin{equation}
\mathop{\rm mod}{}_{C}(D)=\lim_{\delta \rightarrow 0}\sup_{x\in
D}\max_{|t_{1}-t_{2}|\leq \delta }\Vert x(t_{1})-x(t_{2})\Vert ;  \label{e6}
\end{equation}
and $L>0$ is a positive real number chosen so that 
\begin{equation}
q=2M\sup_{t\in I}\int_{0}^{t}e^{-L(t-s)}h(s)ds<1  \label{e7}
\end{equation}
where $M$ is defined above by $M=\sup_{(t,s)\in \Delta }\Vert T(t,s)\Vert $.

It is known that (see \cite[Example 2.1.4]{k1}) $\nu $ is monotone,
nonsingular and regular measure of noncompactness on the space $C(I,E)$.

To show that $R$ is $\nu $-condensing let $\Omega $ be a bounded subset of 
$C(I,E)$ such that 
\begin{equation*}
\nu (R(\Omega ))\geq \nu (\Omega ).
\end{equation*}%
We have to prove that $\Omega $ is relatively compact. Since $\nu $ is
regular it suffices to prove that $\nu (\Omega )=(0,0)$. For this purpose it
is enough to show that $\nu (R(\Omega ))=(0,0)$.

Let $D=\{y_{n}:n\geq1\}$ be a countable subset of $R(\Omega)$ which achieves
that maximum. Then there exists a countable set $\{x_{n}:n\geq1\}$ of 
$\Omega $ such that $y_{n}\in R(x_{n})$, $\forall n\geq1$. Hence for every 
$n\geq1$ and every $t\in I$ 
\begin{equation}
y_{n}(t) =T(t,0)g(x_{n})+\int_0^t T(t,s)f_{n}(s)ds =T(t,0)g(x_{n})+G
f_{n}(t),  \label{e8}
\end{equation}
where $f_{n}\in \mathop{\rm sel}{}_{F}(x_{n})$. From the assumption that 
$\nu(R(\Omega) ) \geq \nu(\Omega) $ and by \eqref{e4} we obtain 
\begin{align*}
(\gamma(\{y_{n}:n\geq1\}),\mathop{\rm mod}{}_{C}(\{y_{n}:n\geq1\}) ) &
=\nu(R(\Omega) )\geq\nu(\Omega) \\
& \geq(\gamma(\{x_{n}:n\geq1\}),\mathop{\rm mod}{}_{C}(\{x_{n}:n\geq1\}) ) .
\end{align*}
Thus 
\begin{equation}
\gamma(\{y_{n}:n\geq1\}) \geq\gamma(\{x_{n}:n\geq1\})  \label{e9}
\end{equation}
and 
\begin{equation}
\text{\ }\mathop{\rm mod}{}_{C}(\{y_{n}:n\geq1\}) \geq \mathop{\rm mod}{}
_{C}(\{x_{n}:n\geq1\})  \label{e10}
\end{equation}
Using (H3) and the relation \eqref{e5} we have, a.e. on $I$, 
\begin{align*}
\chi (\{f_{n}(s):n\geq 1\}) & \leq \chi ( F(s,\{x_{n}(s):n\geq 1\})) \\
& \leq \chi (\{z:z\in F(s,x_{n}(s)):n\geq 1\}) \\
& \leq h(s)\chi (\{x_{n}(s):n\geq 1\}) \\
& \leq h(s)e^{Ls}\sup_{t\in I} e^{-Lt}\chi (\{x_{n}(t):n\geq 1\}) \\
& \leq h(s)e^{Ls}\gamma (\{x_{n}:n\geq 1\}) .
\end{align*}
By Lemma \ref{lem2.4}, 
\begin{align*}
\chi (\{G f_{n}(s):n\geq 1\}) & \leq 2M\int_{0}^{s}\ h(t)e^{Lt}(\gamma
(\{x_{n}:n\geq 1\}) ) dt \\
& =2M\gamma (\{x_{n}:n\geq 1\}) \int_{0}^{s}\ h(t)e^{Lt}.
\end{align*}
Thus, by \eqref{e8} and \eqref{e9} we get 
\begin{equation}
\begin{aligned} \gamma (\{x_{n}:n\geq 1\}) & \leq \gamma (\{y_{n}:n\geq 1\})
\\ &=\sup_{t\in I} e^{-Lt}\chi (\{y_{n}(t):n\geq 1\}) \\ & \leq \sup_{t\in
I} e^{-Lt}\chi (\{T(t,0)g(x_{n}):n\geq 1\}) \\ &\quad +\sup_{t\in I}
e^{-Lt}(\chi (\{G f_{n}(t):n\geq 1\}) ). \end{aligned}  \label{e11}
\end{equation}
Since $\{x_{n}:n\geq 1\}$ is a bounded set, the condition (H4) assumes that
the set $\{g(x_{n}):n\geq 1\}$ is relatively compact in $E$. So, for each 
$t\in I$ the set $\{T(t,0)g(x_{n}):n\geq 1\}$ is relatively compact in $E$.
Thus 
\begin{equation*}
\chi (\{T(t,0)g(x_{n}):n\geq 1\}) =0,\quad \forall t\in I.
\end{equation*}
So, \eqref{e11} gives us 
\begin{align*}
\gamma(\{x_{n}:n\geq1\}) & \leq\gamma(\{y_{n}:n\geq1\}) \\
& \leq\sup_{t\in I} e^{-Lt}(\chi( \{G f_{n}(t):n\geq1\}) ) \\
& \leq\sup_{t\in I} e^{-Lt}2M \gamma( \{x_{n}:n\geq1\}) \int_0^t e^{Ls}h(s)ds
\\
& =2M \gamma(\{x_{n}:n\geq1\}) \sup_{t\in I}e^{-Lt}\int_0^t e^{Ls}h(s)ds \\
& =2M \gamma(\{x_{n}:n\geq1\}) \sup_{t\in I}\int_0^t e^{-L(t-s)}h(s)ds \\
& =\gamma(\{x_{n}:n\geq1\}) q.
\end{align*}
Because $q<1\ $we get $\gamma (\{x_{n}:n\geq 1\}) =0$ and consequently 
$\gamma (\{y_{n}:n\geq 1\}) =0$. Thus the set $\{y_{n}:n\geq 1\}$ is
relatively compact in $C(I,E) $. Hence 
\begin{equation*}
\lim_{\delta\to0} \sup_{n\geq1} \max_{| t_{1}-t_{2}| <\delta}
\|y_{n}(t_{1})-y_{n}(t_{2})\| =0.
\end{equation*}
Thus, by \eqref{e6}, $\mathop{\rm mod}{}_{C}(\{y_{n}:n\geq 1\}) =0$. So, 
$\nu (R(\Omega ) ) =(0,0)$ and then $\nu (\Omega) =(0,0)$.

\noindent\textbf{Step 5.} To apply the fixed point theorem 
(Theorem \ref{thm2.1}.), we consider the closed ball in $C(I,E)$: 
\begin{equation*}
\overline{B(0,r)}=\{x\in C(I,E) :\| x\|_{C}\leq r\},
\end{equation*}
where $r$ is a constant chosen so that 
\begin{equation}
r\geq\frac{Md+M\| m\| _{L^{1}}}{1-Mc-M\| m\| _{L^{1}}}  \label{e12}
\end{equation}
Note that from \eqref{e2} the number $r$ is well defined. Let us show that
the multifunction $R$ maps the closed ball $\overline{B(0,r)}$ into itself.
Let $x\in\overline{B(0,r)}$ and $y\in R(x)$. Then for every $t\in I$ 
\begin{equation*}
\| y(t)\| \leq\| T(t,0)g(x)\| +\int_0^t \| T(t,s)\| \|f(s)\| ds,
\end{equation*}
where $f(s)\in F(s,x(s))$, for a.e. $s\in I$. Using (H2), (H4) and the fact
that $M=\sup_{(t,s)\in\Delta} \| T(t,s)\| $, for any $t\in I$, we have 
\begin{align*}
\| y(t)\| & \leq M(c\| x\| _{C}+d) +M\int_0^t m(s)(\|x(s)\| +1) ds \\
& \leq Mcr+Md+M(r+1)\| m\| _{L^{1}} \\
& =r(Mc+M\| m\| _{L^{1}}) +Md+M\|m\| _{L^{1}} \\
\leq r.\quad \text{(from \eqref{e12})}
\end{align*}
It results that $\| y\| _{C}\leq r$. Then $y\in \overline{B(0,r)}$. Applying
Theorem \ref{thm2.1}, there is a $x\in C(I,E)$ such that $x\in R(x)$.
Clearly the function $x$ is a solution for \eqref{ep} This proves the
theorem.
\end{proof}

\begin{theorem} \label{thm3.2}
In addition of the assumptions in Theorem \ref{thm3.1}, suppose that
\begin{equation}
M(c+\| m\| _{L^{1}}) <1  \label{e13}
\end{equation}
where the numbers $M,c$ and the function $m$ are as in
Theorem \ref{thm3.1}. Then the set of mild solutions of \eqref{ep} is a compact
subset in $C(I,E)$.
\end{theorem}

\begin{proof}
We proved through Theorem \ref{thm3.1} that the values of the multifunction 
$R$ defined on $C(I,E) $ by 
\begin{equation*}
R(x)=\{y\in C(I,E) :y(t)=T(t,0)g(x)+\int_0^t T(t,s)f(s)ds, \quad f\in 
\mathop{\rm sel}{}_{F}(x)\}
\end{equation*}
are nonempty convex compact subsets in $C(I,E)$.

Since the set of mild solutions of $(p)$ is the set of fixed points of $R$,
then Theorem \ref{thm2.2} implies that the set of mild solutions of $(p)$
will be compact if we show that the set of fixed points of $R$ is a bounded
subset in $C(I,E) $. So, let $x$ be a fixed point for $R$. From conditions
(H2) and (H4), for all $t\in I$, 
\begin{align*}
\| x(t)\| & \leq \| T(t,0)g(x)\| +\big\| \int_0^t T(t,s)f(s)ds\big\| ,\quad
f\in \mathop{\rm sel}{}_{F}(x) \\
& \leq M(c\| x\| _{C}+d) +M(\|m\| _{L^{1}}(1+\| x\| _{C}) ) .
\end{align*}
Thus, 
\begin{equation*}
\| x\| _{C}(1-Mc-M\| m\|_{L^{1}}) \leq Md+M\| m\| _{L^{1}}.
\end{equation*}
So, 
\begin{equation*}
\| x\| _{C}\leq \frac{Md+M\| m\| _{L^{1}}}{ 1-M(c+\| m\| _{L^{1}}) }.
\end{equation*}
This means that the set of fixed points of the multifunction $R$ is bounded.
Thanks Theorem \ref{thm2.2}, the set of mild solutions for \eqref{ep} is
compact.
\end{proof}

In the following corollary we generalize the result of Fan, Dong and Li 
\cite{f1}. This generalization deals to prove that the set of mild solutions 
of \eqref{eq} is nonempty in the case when the function $f$ become a
multifunction.

\begin{corollary} \label{coro3.1}
Given a $C_{0}$-semigroup $Z=\{U(t):t\in I\}$, $A$ be
the infinitesimal generator of $Z$ and $F$ and $g$ be as in
Theorem \ref{thm3.1}.
Then there is a mild solution for the semilinear differential inclusion:
\[
x^{\cdot }(t) \in Ax(t)+F(t,x(t)), \quad
x(0) =g(x).
\]
\end{corollary}

\begin{proof}
Consider the family $\{A(t);t\in I\}$, where $A(t)=A$ for every $t\in I$ and
let $T:D\to\mathcal{L}(E)$ be an operator defined by $T(t,s)=U(t-s)$, $(t,s)
\in\Delta$. Then, see \cite{b5}, the operator $T$ is an evolution operator
generated by the family $\{A(t);t\in I\}$. By Theorem \ref{thm3.1} there is
a continuous function $x:I\to E$ such that 
\begin{align*}
x(t) & =T(t,0)g(x)+\int_0^t T(t,s)f(s)ds;\quad t\in I \\
& =U(t)g(x)+\int_0^t U(t-s)f(s)ds;\quad t\in I,
\end{align*}
where $f(s)\in F(s,x(s)) $, a.e. This means that $x$ is a mild solution for
the semilinear differential inclusion 
\begin{gather*}
x^{\cdot }(t) \in Ax(t)+F(t,x(t));\quad t\in I \\
x(0) =g(x).
\end{gather*}
\end{proof}

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\end{document}