\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 $$\dot{x}(t) \in Ax(t)+\int_0^tK(t,s)F(s,x(s))ds,\quad x(0)=\xi\ ,\quad t\in I=[0,T]\label{e1.1}$$ 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 $$\dot{x}(t) \in A(t)x(t)+\int_0^tK(t,s)F(s,x(s))ds,\quad x(0)=\xi\ ,\quad t\in I=[0,T]\,. \label{e1.2}$$ \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 $$x_0(t,\xi)= G(t)\xi \label{e3.1}$$ 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 $$\alpha(\xi)(t)=\beta(t)+k(t)\| x_0(t,\xi)\|\,. \label{e3.2}$$ Now \$|\alpha(\xi_1)(t)-\alpha(\xi_2)(t)|