\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
$|\alpha(\xi_1)(t)-\alpha(\xi_2)(t)|<k(t)\| \xi_1-\xi_2\|$
i.e. $\alpha (\cdot)$ is continuous from $X$ to $L^1(I,R)$.
By (H4) and \eqref{e3.2} we have
\[
d(0,F(t,x_0(t,\xi))<\beta (t)+k(t)\| x_0(t,\xi)\|
\]
and so 
\begin{equation} \label{e3.3}
d(0,F(t,x_0(t,\xi))<\alpha(\xi)(t)\ \text{for a.e. } t\in I
\end{equation}
Define the set valued maps
$G_0:X\to  2^{L^1(I,X)}$ and $H_0:X\to  2^{L^1(I,X)}$ by
\begin{gather*}
G_0(\xi)=\{v\in L^1(I,X):v(t)\in F(t,x_0(t))\quad\mbox{a.e. } t\in I\},\\
H_0(\xi)= \mathop{\rm cl} \{v\in G_0(\xi):\| v(t)\| <\alpha(\xi)(t)+\epsilon
_0\}\,.
\end{gather*}
By Lemma \ref{lem2.2} and \eqref{e3.3} there exists a continuous selection 
$ h_0:X \to  L^1(I,X)$ of $H_0(\cdot)$.
Define $ m(t)=\int _0^t k(s) ds$
For $n\geq 1$ define $\beta _n(\xi)(t)$ by
\begin{equation}
\begin{aligned}
 \beta_n(\xi)(t)&=M^nB^nT^{n-1} \int_0^t \int _0^\tau \alpha (\xi)(s)
 \frac{[m(t)-m(s)]^{n-1}}{(n-1)}ds\, d\tau \\
&\quad +M^nB^nT^n\Big(\sum_{i=0}^n\epsilon _i\Big) 
\int _0^t \frac {[m(t)- m(\tau)]^{n-1}}{(n-1)} d
\end{aligned} \label{e3.4}
\end{equation}
Set $f_0(t,\xi)=h_0(\xi)(t)$. By the definition of $h_0$ we see that
 $f_0(t,\xi)\in F(t,x_0(t,\xi))$.
Define 
\begin{equation}
 x_1(t,\xi)=G(t)\xi +\int _0^t G(t-\tau)\int_0^\tau K(\tau,s)f_0(s,\xi)dsd\tau\
  \forall \ t\in I\backslash \{0\} \label{e3.5}
\end{equation}
By the definition of $H_0(\cdot)$ we see that
$\| f_0(t,\xi)\| <\alpha(\xi)(t)+\epsilon_0$ for all $t\in I\backslash \{0\}$.
From \eqref{e3.1} and \eqref{e3.5} we have
\begin{align*}
 \| x_1(t,\xi)-x_0(t,\xi)\| & \leq \int _0^t|G(t-\tau)|\int _0^\tau |K(\tau,s)|\| f_0(s,\xi)\| dsd\tau\\
& \leq M B\int_0^t\int_0^\tau \| f_0(s,\xi)\| dsd\tau \\
&\leq M B \int_0^t\int_0^\tau \{\alpha(\xi)(s)+\epsilon_0\}\\
&< M B \int_0^t\int_0^\tau \alpha(\xi)(s)dsd\tau+MBT\Big(\sum_{i=0}^1\epsilon _i\Big) \int _0^td\tau \\
&<\beta_1(\xi)(t).
\end{align*}
We claim that there are two sequences $\{f_n(\cdot,\xi)\}$ and 
$\{x_n(\cdot,\xi)\}$ such that for $n\geq 1$ the following properties 
are satisfied:
\begin{enumerate}
\item[(a)]  the map $\xi \to  f_n(\cdot,\xi)$ is continuous from $X$ into $L^1(I,X)$

\item[(b)] $\ f_n(t,\xi)\in F(t,x_n(t,\xi))$ for each $\xi\in X$ a.e.\  $t\in I$

\item[(c)] $\| f_n(t,\xi)-f_{n-1}(t,\xi)\| \leq k(t)\beta_n(\xi)(t)$ for a.e.\ 
$t\in I$ 

\item[(d)] $ x_{n+1}(t,\xi)=G(t)\xi +\int_0^tG(t-\tau)
\int_0^\tau K(\tau,s)f_n(s,\xi)dsd\tau$,  for all $t\in I$.
\end{enumerate}
We shall claim the above by induction on $n$.We assume that already there 
exist functions $f_1\dots f_n$ and $x_1\dots x_n$ satisfying (a)--(d). 
Define $x_{n+1}(\cdot,\xi):I\to  X$ by
$$
 x_{n+1}(t,\xi)=G(t)\xi +\int_0^tG(t-\tau)\int_0^\tau K(\tau,s)f_n(s,\xi)dsd\tau, 
 \quad  \forall t\in I.
$$
Then by (c) and (d), for $t\in I\backslash \{0\}$, we have
\begin{align}
&\| x_{n+1}(t,\xi)-x_n(t,\xi)\| \nonumber \\
&=\big\|\int_0^tG(t-\tau)\int_0^\tau K(\tau,s)\{f_n(s,\xi)-f_{n-1}(s,\xi)\}ds\,d
\tau\big\| \nonumber\\
& \leq MB\int_0^t \int_0^\tau \|f_n(s,\xi)-f_{n-1}\|ds\,d\tau
\label{e3.6}\\
&\leq MB \int_0^t \int_0^\tau k(s)\beta_n(\xi)(s)dsd\tau \nonumber\\
&\leq MB \int_0^t\int_0^\tau k(s)\Big\{M^nB^nT^{n-1} \int_0^s\int_0^u\alpha(\xi)(v)
\frac{[m(s)-m(v)]^{n-1}}{(n-1)}dv\,du \nonumber\\
&\quad +M^nB^nT^n\Big(\sum_{i=0}^n\epsilon _i\Big)
\int_0^s\frac{[m(s)-m(u)]^{n-1}}{(n-1)\ !}du\Big\}ds\,d\tau
\label{e3.7}\\
&<M^{n+1}B^{n+1}T^n \int_0^t \int _0^\tau \alpha
(\xi)(s)\frac{[m(t)-m(s)]^n}{n}ds \, d\tau \nonumber\\
&\quad +M^{n+1}B^{n+1}T^{n+1}\Big(\sum_{i=0}^{n+1}\epsilon _i\Big) \int _0^t 
\frac {[m(t)- m(\tau)]^n}{n}d\tau \label{e3.8} \\
&<\beta_n(\xi)(t).\nonumber
\end{align}
By (H3) we now have
\begin{equation}
d(f_n(t,\xi),F_n(t,x_{n+1}(t,\xi))\leq k(t)\|x_{n+1}(t,\xi)-x_n(t,\xi)\|
<k(t)\beta_{n+1}(\xi)(t) \label{e3.9}
\end{equation}
Define  a set valued map $G_{n+1}:X\to  2^{L^1(I,X)}$ by
$$
G_{n+1}(\xi)=\{v\in L^1(I,X):v(t)\in F(t,x_{n+1}(t,\xi))\quad \text{a.e.}\ t\in I\}
$$
By Lemma 3.1 and \eqref{e3.9}, $G_{n+1}$ is lower semi continuous from $X$ 
into $\mathcal{D}$.
Define a set valued map $H_{n+1}:X\to  2^{L^1(I,X)}$ by
\begin{equation}
H_{n+1}=\text{cl}\{v\in G_{n+1}(\xi):\|v(t)-f_n(t,\xi)\|<k(t)\beta_{n-1}(t)\quad
 \text{a.e. }  t\in I\} \label{e3.10}
\end{equation}
Therefore, $H_{n+1}(\xi)$ is non empty for each $\xi \in X$. By 
Lemma \ref{lem2.2} and \eqref{e3.10} we see that there exists a cotinuous 
selection
\begin{equation*}
h_{n+1}:X\to  L^1(I,X)\quad \text{of } H_{n+1}(\cdot).
\end{equation*}
Then $f_{n+1}(t,\xi)=h_{n+1}(\xi)(t)$ for each $\xi\in I$ and each $t\in I$
satisfies the properties (a)--(c) of our claim. By the property (c) and
\eqref{e3.6}--\eqref{e3.9} we have
\begin{align*}
\|x_{n+1}(\cdot,\xi)-x_n(\cdot,\xi)\|_\infty 
& \leq MB\|f_n(\cdot,\xi)-f_{n-1}(\cdot,\xi)\|_1\\
& \leq \frac{(MBT\|k\|_1)^n}{n!}\big\{MB\| \alpha(\xi)\|_1+MBT\epsilon\big\}
\end{align*}
Therefore, the sequence $\{f_n(\cdot,\xi)\}$ is a Cauchy sequence 
in $L^1(I,X)$ and the sequence $\{x_n(\cdot),\xi)\}$ is a Cauchy sequence 
in $C(I,X)$. Let $f(\cdot,\xi)\in L^1(I,X)$  be the limit of the Cauchy 
sequence $\{f_n(\cdot,\xi)\}$ and $x(\cdot,\xi)\in C(I,X)$ be the limit 
of the Cauchy sequuence $\{x_n(\cdot),\xi)\}$.

Now we can easily show that the map $\xi\to  f(\cdot,\xi)$ is continuous 
from $X$ into $L^1(I,X)$ and the map $\xi\to  x(\cdot,\xi)$ is continuous 
from $X$ into $C(I,X)$ and for all $\xi\in X$ and almost all 
$t\in I$, $f(t,\xi)\in F(t,x(t,\xi))$. Taking limit in (d) we obtain
$$
 x(t,\xi)=G(t)\xi+\int_0^t G(t-\tau)\int_0^\tau K(\tau,s)f(t,\xi)ds\,d\tau \quad 
 \forall \ t\in I.
$$
This completes the proof.
\end{proof}

\section{Evolution Inclusions}

Now we prove the existence of continuous selections of the set of mild 
solutions of evolution inclusions \eqref{e1.2}.

\noindent \textbf{Definition} % 4.1.
A function $x(\cdot,\xi):I\to  X$ is called \emph{a mild solution} of 
\eqref{e1.2} if there exists a function $f(\cdot,\xi)\in L^1(I,X)$ such that
\begin{enumerate}
\item[(i)] $f(t,\xi)\in F(t,x(t,\xi))$ for almost all $t\in I$
\item[(ii)] $ x(t,\xi)= G(t,0)\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}
We introduce the following norms, where $\omega $ is a constant,
\begin{gather*}
\|x\|_2=e^{-\omega t}\|x\|\,, \\
\|x\|_3=\sup \{\|x(t)\|_2,t\in I\} \,, \\
\|x\|_4=\int_0^t\|x\|_2dt\,.
\end{gather*}

\begin{theorem} \label{thm4.1} 
Let $A(t)$ be the infinitesimal generator of a $C_0$-semi group of a two 
parameter family  $\{G(t,\tau):t\geq 0, \tau\geq 0\}$ 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 $\|G(t,\tau)\|\leq Me^{\omega (t-\tau)}$ where $\omega$ is a constant. 
For every $\xi\in X$ define $x_0(\cdot,\xi):I\to  X$ by 
$$
x_0(t,\xi)= G(t,0)\xi 
$$
Now taking $\beta_n(\xi)(t)$ as in theorem 3.1, we can prove that there are 
two sequences $\{f_n(\cdot,\xi)\}$ and $\{x_n(\cdot,\xi)\}$ such that for 
$n\geq 1$ the following properties are satisfied:
\begin{enumerate}
\item[(a)]  The map $\xi \to  f_n(\cdot,\xi)$ is continuous from $X$ into $L^1(I,X)$.
\item[(b)] $f_n(t,\xi)\in F(t,x_n(t,\xi))$ for each $\xi\in X$ a.e.\ $t\in I$
\item[(c)] $\| f_n(t,\xi)-f_{n-1}(t,\xi)\| _2\leq k(t)\beta_n(\xi)(t)$ for a.e.\ 
$t\in I$
\item[(d)] $ x_{n+1}(t,\xi)=G(t,0)\xi +\int_0^tG(t,\tau)
\int_0^\tau K(\tau,s)f_n(s,\xi)dsd\tau$, for all $t\in I$.
\end{enumerate}
Now we have
\begin{align*}
\|x_{n+1}(\cdot,\xi)-x_n(\cdot,\xi)\|_3
& \leq MB\|f_n(\cdot,\xi)-f_{n-1}(\cdot,\xi)\|_4\\
& \leq \frac{(MBT\|k\|_4)^n}{n!}\big\{MB\| \alpha(\xi)\|_4+MBT\epsilon\big\}
\end{align*}
Then the Cauchy sequence $\{x_n(.,\xi)\}$ converges to a limit 
$x(.,\xi)\in C(I,X)$.
Now we can easily show that the map $\xi\to  f(\cdot,\xi)$ is continuous 
from $X$ into $L^1(I,X)$ and the map $\xi\to  x(\cdot,\xi)$ is continuous 
from $X$ into $C(I,X)$ and for all $\xi\in X$ and almost all 
$t\in I$, $f(t,\xi)\in F(t,x(t,\xi))$. Taking limit in (d) we obtain
$$
 x(t,\xi)=G(t,0)\xi+\int_0^t G(t,\tau)\int_0^\tau K(\tau,s)f(t,\xi)dsd\tau \quad 
 \forall  t\in I.
$$
This completes the proof.
\end{proof}

\subsection*{Acknowledgement}
The authors are grateful to Professor K. Balachandran for his valuable 
suggestions  for the improvement of this work.

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