\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 01, 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 (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/01\hfil Continuous selections of solution sets] {Continuous selections of solution sets to Volterra integral inclusions in Banach spaces} \author[S. Aizicovici, V. Staicu\hfil EJDE-2006/01\hfilneg] {Sergiu Aizicovici, Vasile Staicu} % in alphabetical order \address{Sergiu Aizicovici\hfill\break Department of Mathematics, Ohio University, Athens, OH 45701, USA} \email{aizicovi@math.ohiou.edu} \address{Vasile Staicu \hfill\break Department of Mathematics, Aveiro University, 3810-193 Aveiro, Portugal} \email{vasile@ua.pt} \date{} \thanks{Submitted July 26, 2005. Published January 6, 2006.} \subjclass[2000]{34G25, 45D05, 45N05, 47H06} \keywords{Volterra integral equation; m-accretive operator; \hfill\break\indent integral solution; multivalued map} \begin{abstract} We consider a nonlinear Volterra integral equation governed by an m-accretive operator and a multivalued perturbation in a separable Banach. The existence of a continuous selection for the corresponding solution map is proved. The case when the m-accretive operator in the integral inclusion depends on time is also discussed. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper we establish the existence of a continuous selection of the solution set to the nonlinear Volterra integral inclusion \begin{equation} u(t)+\int_{0}^{t}a(t-s)[Au(s) +F(s,u(s))]ds\ni\xi+g(t),\quad t\in I:=[0,T]\label{1.1} \end{equation} in a Banach space $X$. Here $A$ denotes an $m$-accretive operator in $X$, $F:I\times X\to2^{X}\backslash\{\emptyset\}$ is a multivalued perturbation, $a:I\to\mathbb{R}$, $\xi\in X$, $g:I\to X$, and the integral is taken in the sense of Bochner. The case when $A$ depends on time is also considered. Existence and continuous dependence results for Volterra equations of type \eqref{1.1} in infinite dimensional spaces were earlier proved in \cite{Aiz-Pa1,Aiz-Pa2,Aiz-Di-Pa}. Continuous selection theorems for semilinear abstract integrodifferential inclusions have recently been obtained in \cite{An-Mu}. As compared to \cite{Aiz-Pa1}, \cite{Aiz-Pa2}, we do not impose any compactness restriction on the semigroup generated by $-A$ (respectively, on the corresponding evolution operator, when $A$ is time-dependent), and allow $X$ to be a general (non-reflexive) Banach space. We note that in the special case when $a=1$ and $g=0$, equation \eqref{1.1} reduces to \begin{equation} u'(t)+Au(t)+F(t,u( t))\ni 0,\quad t\in I;u(0)=\xi. \label{1.2} \end{equation} The existence of continuous selections for the multivalued solution map associated with $(\ref{1.2})$ was proved by Staicu \cite{St} in a Hilbert space setting. The present work may be viewed as a direct attempt to extend the theory of \cite{St} to a broader class of nonlinear inclusions in a general Banach space. The plan of the paper is as follows. Section $2$ contains background material on multifunctions, $m$-accretive operators and evolution equations. The main results for equation \eqref{1.1} and its time dependent counterpart are stated in Section $3$. The proofs are carried out in Section $4$. Finally, in Section $5$ we present an example to which our abstract theory applies. \section{Preliminaries} For further background and details pertaining to this section we refer the reader to \cite{Au-Ce,Ba1,Cr-Paz,Hu-Pa1,Ka-Sh,Pav,Vr}. Throughout this paper, $X$ stands for a real separable Banach space with norm $\|\cdot \| $ and dual $(X^{\ast},\|\cdot\|_{\ast})$. If $\Omega$ is a subset of $X$, then the closure of $\Omega$ will be denoted by $\overline{\Omega}$, or alternatively by $cl(\Omega)$. Let $I=[0,T]$ and let $\mathcal{L}$ be the $\sigma-$algebra of all Lebesgue measurable subsets of $I$. By $C(I,X)$ (resp. $L^{1}(I,X)$) we denote the Banach space of all continuous (resp. Bochner integrable) functions $u:I\to X$ equipped with the standard norm $\|u\| _{\infty}=\sup_{t\in I}\| u(t)\| $ (resp. $\|u\| _{1}=\int _{0}^{T}\|u(t)\| dt$). $W^{1,1}( I,X)$ designates the space of all absolutely continuous functions $u:I\to X$ which can be written as \[ u(t)=u(0)+\int_{0}^{t}v(s) ds,\quad t\in I, \] for some $v\in$ $L^{1}(I,X)$. We will also use $L^{1}(I)$, $AC(I)$ and $BV(I)$ to indicate the space of all Lebesgue integrable functions, absolutely continuous functions, and respectively functions with bounded variation from $I$ to $\mathbb{R}$. A subset $K\subset$ $L^{1}(I,X)$ is called decomposable if for all $u$, $v\in K$ and $A\in\mathcal{L}$ , one has that $u\chi_{A} +v\chi_{I\backslash A}\in K$, where $\chi_{A}$ stands for the characteristic function of $A$. The family of all nonempty, closed and decomposable subsets of $L^{1}(I,X)$ is denoted by $\mathcal{D}$. The notation $2^{X}$ (resp. $\mathcal{P}(X))$ will designate the collection of all (resp. all nonempty closed) subsets of $X$. The Hausdorff distance on $\mathcal{P}(X)$ is defined by \[ h(A,B)=\max\big\{\sup_{a\in A}d(a,B) ,\sup_{b\in B}d(b,A)\big\},\quad \forall A,B\in\mathcal{P}( X), \] where $d(a,B)=\inf\{\|a-b\| :b\in B\}$. $\mathcal{B}(X)$ will denote the $\sigma-$algebra of Borel subsets of $X$ and $\mathcal{L\otimes B}(X)$ will stand for the $\sigma-$algebra on $I\times X$ generated by all sets of the form $A\times B$ with $A\in\mathcal{L}$ and $B\in\mathcal{B}(X) $. Let $S$ be a separable metric space and let $\mathcal{A}$ denote a $\sigma -$algebra of subsets of $S$. A multivalued map $G:S\to2^{X} \backslash\{\emptyset\}$ is said to be $\mathcal{A} $-\textit{measurable} if for each closed subset $C$ of $X$, the set $\{ s\in S:G(s)\cap C\neq\emptyset\}$ belongs to $\mathcal{A}$. A function $g:S\to X$ satisfying $g(s)\in G(s)$, for all $s\in S$, is called a selection of $G$. The multivalued map $G$ is said to be \textit{lower semicontinuous} (l. s. c.) if for every closed set $C$ of $X$, the set $\{s\in S:G(s)\subset C\}$ is closed in $S$. The following two results of \cite{Ce-Fr-Rz-St} will play a key role in the sequel. \begin{proposition} \label{prop1} Assume that $F^{\ast}:I\times S\to\mathcal{P}(X)$ is $\mathcal{L\otimes B}(S)$ measurable and that $F^{\ast}(t,.)$ is l.s.c. for each $t\in I$. Then the map $\xi\to G_{F}(\xi)$ given by \begin{equation} G_{F}(\xi)=\{v\in L^{1}(I,X):v( t)\in F^{\ast}(t,\xi),\text{ a.e. on }I\} \label{2.1} \end{equation} is l.s.c. from $S$ into $\mathcal{D}$ if and only if there exists a continuous map $\beta:S\to L^{1}(I)$, such that for every $\xi\in S$ \begin{equation} d(0,F^{\ast}(t,\xi))\leq\beta(\xi) (t),\quad \text{a.e. on }I. \label{2.2} \end{equation} \end{proposition} \begin{proposition} \label{prop2} Let $G:S\to\mathcal{D}$ be a l.s.c. multifunction, and let $\varphi:S\to L^{1}(I,X)$ and $\psi:S\to L^{1}(I)$ be continuous maps. Assume that for each $\xi\in S$, the set \begin{equation} H(\xi)=cl\{v\in G(\xi):\| v(t)-\varphi(\xi)(t)\| <\psi(\xi)(t),\text{ a.e. on }I\} \label{2.3} \end{equation} is nonempty. Then the map $\xi\to H(\xi)$ (with $H(\xi)$ given by \eqref{2.3}, from $S$ into $\mathcal{D}$, admits a continuous selection. \end{proposition} The remaining of this section is devoted to a brief discussion of accretive operators and related evolution equations. Let $A:X\to2^{X}$ be a set-valued operator in $X$ and let \[ D(A):=\{x\in X:Ax\neq\mathbb{\emptyset}\},\quad \mathcal{R}(A):=\bigcup_{x\in D(A)}Ax \] be the \textit{domain} and \textit{range} of $A$, respectively. We say that $A$ is an \textit{accretive operator} if \[ \|x_{1}-x_{2}\| \leq\|x_{1}-x_{2}+\lambda( y_{1}-y_{2})\| ,\quad \forall\lambda>0, \;\forall x_{i}\in D(A), \;\forall y_{i}\in Ax_{i}\;(i=1,2). \] If in addition $R(Id+\lambda A)=X$ for all (equivalently, some) $\lambda>0$, where $Id$ is the identity in $X$, then $A$ is said to be \textit{$m$-accretive.} The accretivity of $A$ is equivalent to the condition \[ \langle y_{1}-y_{2},x_{1}-x_{2}\rangle _{s}\geq 0,\quad \;\forall x_{i}\in D(A), \forall y_{i}\in Ax_{i}(i=1,2), \] with $\langle .,.\rangle _{s}$ given by $\langle y,x\rangle _{s}=\sup\{x^{\ast}(y):x^{\ast}\in J(x)\}$, where $J:X\to2^{X^{\ast}}$ is the duality map defined by \[ J(x)=\{x^{\ast}\in X^{\ast}:x^{\ast}(x)=\|x\| ^{2}=\|x^{\ast}\| _{\ast}^{2}\}. \] Consider now the initial value problem \begin{equation} u'(t)+Au(t)\ni f(t),\quad t\in I;u(0)=\xi, \label{2.4} \end{equation} where $A$ is m-accretive in $X$, $\xi\in\overline{D(A)}$ and $f\in L^{1}(I,X)$, whose solutions are meant in the sense of the following definition due to B\'{e}nilan \cite{Ben}. \begin{definition} \label{def3} \rm A function $u\in C(I,\overline{D(A)})$ is called an \textit{integral solution} of the problem \eqref{2.4} if $u(0)=\xi$ and the inequality \[ \|u(t)-x\| ^{2}\leq\|u(s)-x\| ^{2} +2\int_{s}^{t}\langle f(\tau)-y,u(\tau)-x\rangle _{s}d\tau \] holds for all $x$, $y\in X$, with $y\in Ax$, and all $0\leq s\leq t\leq T$. \end{definition} It is well-known that the problem \eqref{2.4} has a unique integral solution for each $f\in L^{1}(I,X)$ and each $\xi \in\overline{D(A)}$. The following property of integral solutions will be used in Section 4. \begin{proposition} \label{prop4} Let $u$ and $v$ be integral solutions of \eqref{2.4} that correspond to $(\xi,f)$ and $(\eta,g)$, respectively (where $\xi$, $\eta\in\overline{D(A)}$ and $f$, $g\in L^{1}(I,X))$. Then \begin{equation} \|u(t)-v(t)\| \leq\|\xi-\eta\| +\int_{0}^{t}\|f(\tau)-g( \tau)\| d\tau\label{2.5} \end{equation} for all $t\in I$. \end{proposition} We note that Benilan's original definition of an integral solution \cite{Ben} required the operator $A$ to be merely accretive. The accretivity alone doesn't generally guarantee the well-posedness of the problem \eqref{2.4} and the validity of the inequality \eqref{2.5}. If, however, $A$ is m-accretive, then problem \eqref{2.4} has a unique integral solution, and \eqref{2.5} holds. Now let $\{A(t):t\in I\}$ be a family of (possibly multivalued) operators on $X$, of domains $D(A(t)), $ with $\overline{D(A(t))}=\overline{D}$ (independent of $t)$ which satisfy the assumption \begin{enumerate} \item[(H1)] \begin{enumerate} \item[(i)] $R(Id+\lambda A(t))=X$, for all $\lambda>0$ and $t\in I$, \item[(ii)] There exists a continuous function $m_{1}:I\to X$ and a continuous nondecreasing function $m_{2}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ ($\mathbb{R}_{+}:=[0,\infty)$) such that \begin{align*} & \langle y_{1}-y_{2},x_{1}-x_{2}\rangle _{s}\\ &\geq -\|m_{1}(t)-m_{1}(s)\| \| x_{1}-x_{2}\| m_{2}(\max\{\|x_{1}\| ,\|x\| _{2}\}), \end{align*} for all $x_{1}\in D(A(t))$, $y_{1}\in A(t)x_{1}$, $x_{2}\in D(A(s))$, $y_{2}\in A(s)x_{2}$, $0\leq s\leq t\leq T$. \end{enumerate} \end{enumerate} We remark that (H1) implies that $A(t)$ is m-accretive for each $t\in I$. We consider the nonautonomous Cauchy problem \begin{equation} u'(t)+A(t)u(t)\ni f(t),\quad t\in I;u(0)=\xi, \label{2.6} \end{equation} where $A(t)$ satisfy (H1), $\xi\in\overline{D}$ and $f\in L^{1}(I,X)$. \begin{definition} \label{def5} \rm An integral solution of \eqref{2.6} is a function $u\in C(I,\overline{D})$ such that $u(0)=\xi$ and the inequality \begin{align*} \|u(t)-x\| ^{2} & \leq\|u(s)-x\| ^{2}+2\int_{s}^{t}[\langle f(\tau)-y,u(\tau)-x\rangle _{s}\\ & \quad +C\|u(\tau)-x\| \|m_{1}(\tau) -m_{1}(\theta)\| ]d\tau \end{align*} holds for all $0\leq s\leq t\leq T$, $\theta\in I$, $x\in D(A(\theta))$, $y\in A(\theta)x$, and $C=m_{2}(\max\{\|x\| ,\|u\|_{\infty}\})$, with $m_{1}$ and $m_{2}$ as in (H1)(ii). \end{definition} Recall (cf., e.g., \cite{Pav}) that \eqref{2.6} has a unique integral solution for each $\xi\in\overline{D}$ and $f\in L^{1}(I,X)$, provided that (H1) is satisfied. Moreover, the following analog of Proposition \ref{prop4} is true. \begin{proposition} \label{prop6} Let (H1) be satisfied and let $u$ and $v$ be integral solutions of \eqref{2.6} corresponding to $(\xi,f)$ and $(\eta,g)$, respectively (with $\xi$, $\eta\in\overline{D}$ and $f$, $g\in L^{1}(I,X))$. Then the inequality \eqref{2.5} holds for all $t\in I$. \end{proposition} \section{Main results} We consider the Volterra integral inclusion \eqref{1.1} under the following conditions: \begin{enumerate} \item[(H2)] $A$ is an m-accretive operator in $X$, with domain $D(A)$, and there exists an open subset $U$ of $X$ such that $U_{A}:=U\cap\overline{D(A)}$ is nonempty; \item[(H3)] $a\in AC(I)$ with $a'\in BV(I)$ and $a(0)=1;$ \item[(H4)] \begin{enumerate} \item[(i)] $F:I\times X\to\mathcal{P}(X)$ is $\mathcal{L\otimes B}(X)$ measurable, \item[(ii)] There exists $k\in L^{1}(I,(0,\infty))$ such that \[ h(F(t,x),F(t,y))\leq k( t)\|x-y\| ,\quad \forall x,\; y\in X,\quad\mbox{a.e. on } I\,, \] \item[(iii)] There exists $\beta\in L^{1}(I,\mathbb{R}_{\mathbb{+}})$ such that \[ d(0,F(t,0))\leq\beta(t),\quad \mbox{a.e. on } I\,; \] \end{enumerate} \item[(H5)] $g\in W^{1,1}(I,X)$ and $g(0)=0$. \end{enumerate} \begin{remark} \label{rmk7} \rm The restriction $a(0)=1$ in (H3) is only made for convenience. The essential condition is $a(0)>0$; see \cite[p. 317]{Cr-No}. \end{remark} For each $\xi\in U_{A}$, we reduce the study of \eqref{1.1} to that of the equivalent functional differential inclusion (cf. \cite{Aiz-Pa1,Cr-No}) \begin{equation} u'(t)+Au(t)+F(t,u(t))\ni\Gamma(u)(t),\quad t\in I;\quad u(0)=\xi, \label{3.1} \end{equation} where $\Gamma:C\big(I,\overline{D(A)}\big)\to L^{1}(I,X)$ is defined by \begin{gather} \Gamma(u)(t) =g'(t) +\int_{0}^{t}r(t-s)g'(s)ds-r(0)u(t)+r(t)\xi -\int_{0}^{t}u(t-s)dr(s), \label{3.2} \\ r(t)+\int_{0}^{t}a'(t-s)r( s)ds=-a'(t). \label{3.3} \end{gather} Note that by (H3), the function $r$ (as defined in \eqref{3.3}) is a function with bounded variation. \begin{definition} \label{def8} \rm A function $u\in C(I,\overline{D(A)})$ is said to be an integral solution of the equation \eqref{1.1} (equivalently, \eqref{3.1}) if there exists $\widehat{f}\in L^{1}(I,X)$ with $\widehat{f}(t)\in F(t,u(t))$, $a$. $e$. on $I$, such that $u$ is an integral solution, in the sense of Definition \ref{def3}, of the problem \eqref{2.4} with $\Gamma(u)(t) -\widehat{f}(t)$ in place of $f(t)$. \end{definition} In the following, $\mathcal{S}(\xi)$ denotes the set of all integral solutions of the equation \eqref{1.1}, which is viewed as a subset of $C(I,\overline{D(A)})$, for each $\xi\in U_{A}$. \begin{theorem} \label{thm9} Let assumptions (H2), (H3), (H4), (H5) be satisfied. Then there exists $u:I\times U_{A}\to\overline{D(A)}$ such that: \begin{gather} u(.,\xi)\in\mathcal{S}(\xi),\quad \forall \xi\in U_{A}, \label{3.4} \\ \xi\to u(.,\xi)\text{ is continuous from }U_{A}\text{ into }C(I,\overline{D(A)}). \label{3.5} \end{gather} \end{theorem} \begin{remark} \label{rmk10} \rm (i) In the case when $a=1$, $g=0$ and $X$ is a Hilbert space we recover \cite[Theorem 2.4]{St}. \noindent (ii) A similar result can be derived for the Volterra integral equation \[ u(t)+\int_{0}^{t}a(t-s)[Au( s)+F(s,u(s))]ds\ni g( \xi)+\int_{0}^{t}p(s)ds,\quad t\in I, \] where $g:U_{A}\to X$ is continuous, $p\in L^{1}(I,X)$, and conditions (H2), (H3) and (H4) are satisfied. For simplicity, we have restricted our attention to equations of the form \eqref{1.1}. \end{remark} Next, we are concerned with the time-dependent analog of $( \ref{1.1})$, namely \begin{equation} u(t)+\int_{0}^{t}a(t-s)[A( s)u(s)+F(s,u(s))] ds\ni\xi+g(t),\quad t\in I, \label{3.6} \end{equation} where $\{A(t):t\in I\}$ is a family of m-accretive operators in $X$ that satisfy assumption (H1), while $a$, $F$ and $g$ are subject to conditions (H3), (H4) and (H5), respectively, and $\xi\in\overline{D}$. As in \cite{Aiz-Di-Pa,Cr-No} we can replace \eqref{3.6} by an equivalent functional differential equation of the form $(\ref{3.1})$ (with $A(t)$ in place of $A$), where $\Gamma:C(I,\overline{D})\to L^{1}(I,X)$ is given by \eqref{3.2}. \begin{definition} \label{def11} \rm A function $u\in C(I,\overline{D})$ is called an integral solution of equation \eqref{3.6} if there exists $\widehat{f}\in L^{1}(I,X)$ with $\widehat{f}(t)\in F(t,u(t))$, $a$. $e$. on $I$, such that $u$ is an integral solution, in the sense of Definition \ref{def5}, of the problem \eqref{2.6} where $f(t)$ is replaced by $\Gamma(u)(t)-\widehat{f}(t)$. \end{definition} For each $\xi\in\overline{D}$, let $\mathcal{T}(\xi)$ denote the set of all integral solutions of the equation \eqref{3.6}), which is regarded as a subset of $C(I,\overline{D})$. The following counterpart of Theorem \ref{thm9} is valid. \begin{theorem} \label{thm12} Let conditions (H1), (H3), (H4), (H5) be satisfied. In addition assume that there exists an open subset $V$ of $X$ such that $V_{A}:=V\cap\overline{D}$ is nonempty. Then there exists $u:I\times V_{A}\to\overline{D}$ such that \begin{gather} u(.,\xi)\in\mathcal{T}(\xi),\quad \forall \xi\in V_{A}, \label{3.7} \\ \xi\to u(.,\xi)\text{ is continuous from }V_{A}\text{ into }C(I,\overline{D}). \label{3.8} \end{gather} \end{theorem} \section{Proofs} \begin{proof}[Proof of Theorem \ref{thm9}] We adapt the technique used in \cite{Ce-Fr-Rz-St,St} to handle \eqref{3.1}, which is the functional differential inclusion equivalent of the integral equation \eqref{1.1}. Fix $\varepsilon>0$ and set $\varepsilon_{n}:=\varepsilon/2^{n+1}$, $n\in\mathbb{N}$, where $\mathbb{N}$ denotes the set of all nonnegative integers. For $\xi\in U_{A}$, let $u_{0}(.,\xi):I\to\overline{D(A)}$ be the unique integral solution of \[ u'(t)+Au(t)\ni\Gamma(u) (t),\quad t\in I;\quad u(0)=\xi. \] The existence and uniqueness of $u_{0}(.,\xi)$ follows from \cite[Prop. 1 and Theorem 1]{Cr-No}, on account of (H2), (H4) and (H5). Set \begin{equation} \alpha(\xi)(t):=\beta(t)+k( t)\|u_{0}(t,\xi)\| ,\quad m( t):=\int_{0}^{t}k(s)ds,t\in I, \label{4.1} \end{equation} where $k(.)$ and $\beta(.)$ are as in (H4) (ii) and (iii), respectively. Also define \begin{equation} \gamma(t):=\left\vert r(0)\right\vert +var\{r:[0,t]\},\quad t\in I;\quad M:=e^{\int_{0}^{t}\gamma(s)ds}, \label{4.2} \end{equation} where the function $r(.)$ satisfies \eqref{3.3} and $var\{r:[0,t]\}$ indicates the total variation of $r$ over $[0,t]$. Let $f_{-1}(\xi)(t)\equiv0$. We will construct two sequences of functions $(u_{n}(.,\xi))_{n\in\mathbb{N}}\subset C(I,\overline{D(A)})$ and $(f_{n}(\xi))_{n\in\mathbb{N}}\subset L^{1}(I,X)$ satisfying the following conditions: \begin{enumerate} \item[(C1)] $u_{n}(.,\xi):I\to \overline{D(A)}$ is the unique integral solution of the problem \begin{equation} u'(t)+Au(t)\ni\Gamma(u) (t)-f_{n-1}(\xi)(t),\quad u(0)=\xi; \label{En} \end{equation} \item[(C2)] $\xi\to f_{n}(\xi)$ is continuous from $U_{A}$ into $L^{1}(I,X);$ \item[(C3)] $f_{n}(\xi)(t)\in F(t,u_{n}(t,\xi))$, for all $\xi\in U_{A}$, a.e. on $I$; \item[(C4)] $\|f_{n}(\xi)(t)-f_{n-1}(\xi)(t)\| \leq k(t)\beta_{n}(\xi)(t)$, for all $\xi\in U_{A}$, a.e. on $I$, \end{enumerate} where \[ \beta_{0}(\xi)(t):=(\alpha(\xi)(t)+\varepsilon_{0})(k(t))^{-1}, \] and, for $n\geq1$, \begin{equation} \beta_{n}(\xi)(t)=M^{n}\int_{0}^{t}\alpha( \xi)(s)\frac{[m(t)-m( s)]^{n-1}}{(n-1)!}ds+M^{n}T\frac{[ m(t)]^{n-1}}{(n-1)!}\sum_{i=0} ^{n}\varepsilon_{i}, \label{4.3} \end{equation} with $\alpha(\xi)(.)$, $m(.)$ and $M$ defined in \eqref{4.1} and \eqref{4.2}. We remark that $u_{0}(.,\xi)$ is the integral solution of \eqref{En} with $n=0$. We claim that the map $\xi\to u_{0}(.,\xi)$ is continuous from $U_{A}$ into $C(I,\overline{D(A)})$. Indeed, for $\xi_{1},\xi_{2}\in U_{A}$, we can invoke \eqref{2.5} to deduce, for $t\in I$, \begin{equation} \|u_{0}(t,\xi_{1})-u_{0}(t,\xi_{2})\| \leq\|\xi_{1}-\xi_{2}\| + \int_{0}^{t}\|\Gamma(u_{0}(.,\xi_{1}) )(s)-\Gamma(u_{0}(.,\xi_{2}) )(s)\| ds. \label{4.4} \end{equation} It is easily seen that the definition of $\Gamma$ (cf. \eqref{3.2}) implies \begin{equation} \begin{aligned} & \int_{0}^{t}\|\Gamma(u_{0}(.,\xi_{1}) )(s)-\Gamma(u_{0}(.,\xi_{2}) )(s)\| ds\\ & \leq r(t)\|\xi_{1}-\xi_{2}\| +\int_{0} ^{t}\gamma(s)\|u_{0}(.,\xi_{1}) -u_{0}(.,\xi_{2})\| _{\infty}(s)ds, \end{aligned} \label{4.5} \end{equation} where $\|u\| _{\infty}(s):=\sup_{\tau \in[0,s]}\| u(\tau)\| $ is the norm in $C([0,s],X)$ and $\gamma( .)$ is given by \eqref{4.2}. Since $r( .)\in BV(I)$, one has that $\|r\|_{\infty}:=\sup_{t\in I}\vert r(t)\vert <\infty$. Using \eqref{4.5} in \eqref{4.4} and applying Gronwall's lemma, we conclude that \[ \|u_{0}(.,\xi_{1})-u_{0}(.,\xi_{2}) \| _{\infty}\leq M(1+\|r\| _{\infty}) \| \xi_{1}-\xi_{2}\| . \] This yields the continuity of $\xi\to u_{0}(.,\xi)$ from $U_{A}$ into $C(I,\overline{D(A)})$, as claimed. Next, by (H4) (ii), (iii) and \eqref{4.1}, we have \begin{equation} d(0,F(t,u_{0}(t,\xi)))\leq \alpha(\xi)(t),\quad \mbox{a.e. on } I, \label{4.6} \end{equation} where it is to be noted that $\alpha(.)$ is continuous as a function from $U_{A}$ into $L^{1}(I)$. Define the multifunctions $G_{0}$, $H_{0}:U_{A}\to2^{L^{1}(I,X)}$ by \begin{gather} G_{0}(\xi):=\{v\in L^{1}(I,X):v( t)\in F(t,u_{0}(t,\xi)),\mbox{ a.e. on }I\}, \label{4.7} \\ H_{0}(\xi):=cl\{v\in G_{0}(\xi) :\|v(t)\| <\alpha(\xi)( t)+\varepsilon_{0},\mbox{ a.e. on }I\}. \label{4.8} \end{gather} Setting $F^{\ast}(t,\xi):=F(t,u_{0}(t,\xi))$ and invoking assumptions (H4) (i), (ii), \cite[Proposition 2.66, p.61]{Hu-Pa1}, the continuity of $\alpha(.)$ on $U_{A}$ and \eqref{4.6}, we conclude by applying Proposition \ref{prop1} that $G_{0}(.)$ is lower semicontinuous from $U_{A}$ into $\mathcal{D}$ and the set $H_{0}(\xi)$ is nonempty. Therefore, by Proposition \ref{prop2}, there exists $h_{0}\in C(U_{A},L^{1}(I,X))$ such that $h_{0}(\xi)\in H_{0}(\xi)$, $\forall\xi\in U_{A}$. Set \begin{equation} f_{0}(\xi)(t):=h_{0}(\xi)(t),\quad \forall\xi\in U_{A},t\in I, \label{4.9} \end{equation} and remark that, by virtue of \eqref{4.7}, \eqref{4.8}, \eqref{4.9} and the fact that $F$ is closed valued, $f_{0}(.)$ is continuous from $U_{A}$ into $L^{1}(I,X)$, $f_{0}(\xi)(t)\in F(t,u_{0}(t,\xi))$ and $\|f_{0}( \xi)(t)\| \leq k(t)\beta_{0}(\xi)(t)$, $a.e$. $on$ $I$. Recalling that $u_{0}(.,\xi)$ is the integral solution of \eqref{En} with $n=0$, we conclude that conditions $(C_{1})-( C_{4})$ hold for $n=0$. We now proceed inductively. Assume that the functions $\{u_{0},\quad u_{1},\dots ,u_{n}\}$ and $\{f_{0},\quad f_{1},\dots ,f_{n}\}$ have been constructed such that conditions $(C_{1})-(C_{4})$ are satisfied. For $\xi\in U_{A}$, let $u_{n+1}(.,\xi):I\to\overline{D(A)}$ be the unique integral solution of \eqref{En} with $n+1$ in place of $n$. (Taking into account that $f_{n}(\xi)\in L^{1}(I,X)$, we can again invoke \cite[Prop. 1 and Theorem 1]{Cr-No} to establish the existence and uniqueness of $u_{n+1}(.,\xi)$). Inasmuch as $u_{n}(.,\xi)$ and $u_{n+1}(.,\xi)$ satisfy \eqref{En}, and \eqref{En} with $n+1$ instead of $n$, respectively, we can apply Proposition \ref{prop4} to obtain, for $t\in I$, \begin{equation} \begin{aligned} \|u_{n+1}(t,\xi)-u_{n}(t,\xi) \| &\leq\int_{0}^{t}\|\Gamma(u_{n+1}( .,\xi))(s)-\Gamma(u_{n}( .,\xi))(s)\| ds\\ & \quad +\int_{0}^{t}\|f_{n}(\xi)(s) -f_{n-1}(\xi)(s)\| ds. \end{aligned}\label{4.10} \end{equation} From \eqref{3.2} it follows that \begin{equation} \begin{aligned} & \int_{0}^{t}\|\Gamma(u_{n+1}(.,\xi)) (s)-\Gamma(u_{n}(.,\xi))( s)\| ds\\ & \leq\int_{0}^{t}\gamma(s)\|u_{n+1}( .,\xi)-u_{n}(.,\xi)(s)\| _{\infty}(s)ds. \end{aligned} \label{4.11} \end{equation} Combining \eqref{4.10} with \eqref{4.11} and using Gronwall's lemma, we arrive at \begin{equation} \|u_{n+1}(t,\xi)-u_{n}(t,\xi) \| \leq M\int_{0}^{t}\|f_{n}(\xi)( s)-f_{n-1}(\xi)(s)\| ds,t\in I.\label{4.12} \end{equation} Employing property (C4) in $(\ref{4.12})$, we have \begin{equation} \|u_{n+1}(t,\xi)-u_{n}(t,\xi) \| \leq M\int_{0}^{t}k(s)\beta_{n}(\xi) (s)ds,t\in I.\label{4.13} \end{equation} If $n=0$, this implies, by virtue of \eqref{4.3}, \begin{equation} \|u_{1}(t,\xi)-u_{0}(t,\xi)\| \leq M\int_{0}^{t}\alpha(\xi)(s)ds <\beta_{1}(\xi)(t),\quad \mbox{a.e. on } I.\label{4.14} \end{equation} If $n>0$, then \eqref{4.13} and \eqref{4.3} lead to \begin{equation} \begin{aligned} \|u_{n+1}(t,\xi)-u_{n}(t,\xi)\| & \leq M^{n+1}\int_{0}^{t}k(s)\int_{0}^{s}\alpha( \xi)(\tau)\frac{[m(s)-m(\tau)]^{n-1}}{(n-1)!}d\tau ds\\ &\quad +M^{n+1}T[\sum_{i=0}^{n}\varepsilon_{i}]\int_{0}^{t}k( s)\frac{[m(s)]^{n-1}}{(n-1)!}ds, \end{aligned} \label{4.15} \end{equation} for $t\in I$. Recalling the definition of $m$ (cf. \eqref{4.1}), and interchanging the order of integration in the first term on the right-hand side of \eqref{4.15}, we get \begin{equation} \begin{aligned} & \|u_{n+1}(t,\xi)-u_{n}(t,\xi)\|\\ & \leq M^{n+1}\int_{0}^{t}\alpha(\xi)(\tau) \frac{[m(t)-m(\tau)]^{n}} {n!}d\tau+M^{n+1}T[\sum_{i=0}^{n}\varepsilon_{i}]\frac{[ m(t)]^{n}}{n!}\\ & <\beta_{n+1}(\xi)(t),\quad \mbox{a.e. on } I. \end{aligned} \label{4.16} \end{equation} By \eqref{4.14}, \eqref{4.16}, (C3) and (H4) (ii), it follows that \begin{equation} d(f_{n}(\xi)(t),F(t,u_{n+1}( t,\xi)))