\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 21, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2005/21\hfil Continuous selections of evolution inclusions]
{Continuous selections of set of mild solutions of evolution inclusions}
\author[A. Anguraj, C. Murugesan\hfil EJDE-2005/21\hfilneg]
{Annamalai Anguraj, Chinnagounder Murugesan} % in alphabetical order
\address{Annamalai Anguraj\hfill\break
Department of Mathematics\\
P.S.G. College of Arts \& Science\\
Coimbatore - 641 014, Tamilnadu, India}
\email{angurajpsg@yahoo.com}
\address{Chinnagounder Murugesan \hfill\break
Department of Mathematics\\
Gobi Arts \& Science College\\
Gobichettipalayam - 638 453, Tamilnadu, India}
\date{}
\thanks{Submitted November 3, 2004. Published February 11, 2005.}
\subjclass[2000]{34A60, 34G20}
\keywords{Mild solutions; differential inclusions; integrodifferential inclusions}
\begin{abstract}
We prove the existence of continuous selections of the set
valued map $\xi\to \mathcal{S}(\xi)$ where
$\mathcal{S}(\xi)$ is the set of all mild solutions of the
evolution inclusions of the form
\begin{gather*}
\dot{x}(t) \in A(t)x(t)+\int_0^tK(t,s)F(s,x(s))ds \\
x(0)=\xi ,\quad t\in I=[0,T],
\end{gather*}
where $F$ is a lower semi continuous set valued map Lipchitzean
with respect to $x$ in a separable Banach space $X$,
$A$ is the infinitesimal generator of a $C_0$-semi group of
bounded linear operators from $X$ to $X$, and $K(t,s)$ is a
continuous real valued function defined on $I\times I$ with
$t\geq s$ for all $ t,s\in I$ and $\xi \in X$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
Existence of solutions of differential inclusions and integrodifferential
equations has been studied by many authors \cite{a1,a2,b1}.
Existence of continuous selections of the solution sets of the Cauchy
problem
$\dot{x}(t)\in F(t,x(t)),\ x(0)=\xi$
was first proved by Cellina \cite{c1} for $F$ Lipchitzean with respect to
$x$ defined on an open subset of $R\times R^n$ and taking compact
uniformly bounded values. Cellina proved that the map that associates
the set of solutions $\mathcal{S}(\xi)$ of the above Cauchy problem
to the initial point $\xi$, admits a selection continuous from $R^n$ to
the space of absolutely continuous functions.
Extensions of Cellina's result to Lipchitzean maps with closed non empty
values in a separable Banach space has been obtained in \cite{b2} and \cite{c2}.
In \cite{s1} Staicu proved the existence of a continuous selection of the
set valued map $\xi\to \mathcal{S}(\xi)$ where $\mathcal{S}(\xi)$
is the set of all mild solutions of the Cauchy problem
$$
\dot{x}(t)\in Ax(t)+ F(t,x(t)),\ x(0)=\xi
$$
where $A$ is the infinitesimal generator of a $C_0$ - semi group
and $F$ is Lipchitzean with respect to $x$. Staicu also proved the
same result for the set of all weak solutions by considering that $-A$
is a maximal monotone map.
In this present work first we prove the existence of a continuous
selection of the set valued map $\xi\to \mathcal{S}(\xi)$ where
$\mathcal{S}(\xi)$ is the set of all mild solutions of the integrodifferential
inclusions of the form
\begin{equation}
\dot{x}(t) \in Ax(t)+\int_0^tK(t,s)F(s,x(s))ds,\quad
x(0)=\xi\ ,\quad t\in I=[0,T]\label{e1.1}
\end{equation}
where $F$ is a set valued map Lipchitzean with respect to $x$ in a
separable Banach space $X$,\ $A$ is the infinitesimal generator of a
$C_0$-semi group of bounded linear operators from $X$ to $X$ and $K(t,s)$
is a continuous real valued function defined on $I\times I$ with
$t\geq s$ for all $ t,s\in I$ and $\xi \in X$.
Then we extend our result for the evolution inclusions of the form
\begin{equation}
\dot{x}(t) \in A(t)x(t)+\int_0^tK(t,s)F(s,x(s))ds,\quad
x(0)=\xi\ ,\quad t\in I=[0,T]\,. \label{e1.2}
\end{equation}
\section{Preliminaries}
Let $T>0$, $I=[0,T]$ and denote by $\mathcal{L}$ the $\sigma$-algebra of
all Lebesgue measurable subsets of $I$. Let $X$ be a real separable
Banach space with norm $\| \cdot \|$. Let $2^X$ be the family of all
non empty subsets of $X$
and $\mathcal{B}(X)$ be the family of Borel subsets of $X$.
If $x\in X$ and $A$ is a subset of $X$, then we define
$$
d(x,A)=\inf \{\| x-y\|:y\in A\}.
$$
For any two closed and bounded non empty subsets $A$ and $B$ of $X$,
we define \emph{Housdorff distance} from $A$ and $B$ by
$$
h(A,B)=\max \{\sup \{d(x,B):x\in A\},\sup \{d(y,A):x\in B\}\}.
$$
Let $C(I,X)$ denote the Banach space of all continuous functions
$x:I\to X$ with norm
$$
\| x\|_\infty =\sup \{\| x(t)\| : t\in I\}.
$$
Let $L^1(I,X)$ denote the Banach space of all Bochner integrable
functions $x:I\to X$ with norm $\| x\|_1 =\int_0^T\| x(t)\| dt$.
Let $\mathcal{D}$ be the family of all decomposable closed non empty
subsets of $L^1(I,X)$.
A set valued map $\mathcal{G}:S\to 2^X$ is said to be
\emph{lower semi continuous (l.s.c)} if for every closed subset $C$
of $X$ the set $\{s\in S:\mathcal{G}(s)\subset C\}$ is closed in $S$.
A function $g:S\to X$ such that $g(s)\in \mathcal{G}(s)$ for all $s\in S$
is called a \emph{selection} of $\mathcal{G}(\cdot)$. Let $\{G(t):t\geq 0\}$
be a strongly continuous semi group of bounded linear operators from $X$
to $X$. Here $G(t)$ is a mapping (operator) of $X$ into itself for
every $t\geq 0$ with
\begin{enumerate}
\item $G(0)=I$ (the identity mapping of $X$ onto $X$)
\item $G(t+s)=G(t)G(s)$ for all $t,s\geq 0$.
\end{enumerate}
Now we assume the following:
\begin{enumerate}
\item[(H1)] $F:I\times X\to 2^X$ is a lower semi continuous set valued
map taking non empty closed bounded values.
\item[(H2)] $F$ is $\mathcal{L}\otimes\mathcal{B}(X)$ measurable.
\item[(H3)] There exists a $k\in L^1(I,R)$ such that the Hausdorff distance
satisfies $h(F(t,x(t)),F(t,y(t)))\leq k(t)\|x(t)-y(t)\|$ for all
$x,y\in X $ and a.e.\ $t\in I$
\item[(H4)] There exists a $\beta \in L^1(I,R)$ such that
$d(0,F(t,0))\leq \beta(t)$ a.e. $t\in I$
\item[(H5)] $K:D\to R$ is a real valued continuous function where
$D=\{(t,s)\in I\times I:t\geq s\}$ such that
$B=\sup\{\| K(t,s)\| :t\geq s\}$.
\end{enumerate}
To prove our theorem we need the following two lemmas.
\begin{lemma}[\cite{c1}] \label{lem2.1}
Let $F:I\times S\to 2^X,S\subseteq X$, be measurable with non empty closed
values, and let $F(t,\cdot )$ be lower semi continuous for each $t\in I$.
Then the map $\xi \to G_{F}(\xi)$ given by
$$
G_{F}(\xi)=\{v\in L^1(I,X):v(t)\in F(T,\xi)\quad \forall\ t\in I\}
$$
is lower semi continuous from $S$ into $\mathcal{D}$ if and only if there
exists a continuous function $\beta:S\to L^1(I,R)$ such that for all
$\xi\in S$, we have $d(0,F(t,\xi))\leq \beta(\xi)(t)$ a.e. $t\in I$.
\end{lemma}
\begin{lemma}[\cite{c1}] \label{lem2.2}
Let $\zeta: S\to \mathcal{D}$ be a lower semi continuous set valued map
and let $\varphi:S\to L^1(I,X) $ and $\psi :S\to L^1(I,X)$ be continuous maps. If for every $\xi \in S$ the set
$$
H(\xi)=\text{cl} \{ v \in \zeta (\xi) : \|v(t)-\varphi(\xi)(t)\|
< \psi (\xi)(t)\text{ a.e}\ t \in I\}
$$
is non empty, then the map $H:S\to \mathcal{D}$ defined above admits a
continuous selection.
\end{lemma}
\section{Integrodifferential inclusions}
\textbf{Definition.} % 3.1.
A function $x(\cdot,\xi):I\to X$ is called \emph{a mild solution} of
\eqref{e1.1} if there exists a function $f(\cdot,\xi)\in L^1(I,X)$ such that
\begin{enumerate}
\item[(i)] $f(t,\xi)\in F(t,x(t,\xi))\ \text{for almost all}\ t\in I$
\item[(ii)] $ x(t,\xi)= G(t)\xi +\int_0^t G(t-\tau)\int_0^\tau
K(\tau,s)f(s,\xi)dsd\tau $\ for each $t\in I.$
\end{enumerate}
\begin{theorem} \label{thm3.1}
Let $A$ be the infinitesimal generator of a $C_0$-semi group
$\{G(t):t\geq 0\}$ of bounded linear operators of $X$ into $X$ and the
hypotheses (H1)--(H5) be satisfied. Then there exists a function
$x(\cdot ,\cdot):I\times X\to X$ such that
\begin{enumerate}
\item[(i)] $x(\cdot ,\xi)\in\mathcal{S}(\xi)$ for every $\xi\in X$ and
\item[(ii)] $\xi \to x(\cdot ,\xi)$ is continuous from $X$ into $C(I,X)$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\epsilon >0$ be given. For $n\in N$ let
$\epsilon_n =\frac{1}{\epsilon^{n+1}}$. Let $M=\sup\{|G(t)|:t\in I\}$.
For every $\xi\in X$ define $x_0(\cdot,\xi):I\to X$ by
\begin{equation}
x_0(t,\xi)= G(t)\xi \label{e3.1}
\end{equation}
Now
\begin{equation*}
\| x_0(t,\xi_1)-x_0(t,\xi_2)\| =\|G(t)\xi_1-G(t)\xi_2 \|
=|G(t)|\ \|\xi_1-\xi_2\|
\leq M \|\xi_1-\xi_2\|
\end{equation*}
i.e.\ The map $\xi\to x_0(\cdot,\xi)$ is continuous from $X$ to $C(I,X)$.
For each $\xi\in X$ define $\alpha(\xi):I\to R$ by
\begin{equation}
\alpha(\xi)(t)=\beta(t)+k(t)\| x_0(t,\xi)\|\,. \label{e3.2}
\end{equation}
Now
$|\alpha(\xi_1)(t)-\alpha(\xi_2)(t)|