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\title{On some evolution inclusions in non separable Banach spaces}
\author{Aurelian Cernea}
\address{University of Bucharest, \\ Faculty of Mathematics and Computer Sciences\\
14, Academiei Street,\\
010014 Bucharest,\\
Academy of Romanian Scientists,\\
Splaiul Independen\c{t}ei 54,\\
050094 Bucharest,\\
Romania}
\email{acernea@fmi.unibuc.ro}



\subjclass{34A60}
\keywords{Lusin measurable multifunctions, selection, mild solution.}
\begin{abstract}
We study a Cauchy problem of a class of nonconvex second-order integro-differential
inclusions and  a boundary value problem associated to a semilinear evolution inclusion defined by
nonlocal conditions in non-separable Banach spaces. The existence of mild solutions is established  under Filippov type assumptions.
\end{abstract}
\maketitle

\vskip 2em

\centerline{\it Dedicated to Professor Gheorghe Moro\c{s}anu on his $70^{th}$ anniversary}
\vskip 2em

\section{Introduction}

In this note we study two classes of evolution differential inclusions. First we consider the problem

$$
x''(t)\in A(t)x(t)+\int_0^tK(t,s)F(s,x(s))ds,\quad x(0)=x_0,x'(0)=y_0,\eqno
(1.1)
$$
where $F:[0,T]\times X\to \mathcal{P}(X)$ is a set-valued map
lipschitzian with respect to the second variable, $X$ is a
Banach space, $\{A(t)\}_{t\geq 0}$ is a family of linear
closed operators from $X$ into $X$ that genearates an evolution system of operators $\{G(t,s)\}_{t,s\in [0,T]}$,
$\Delta =\{(t,s)\in [0,T]\times [0,T];t\geq s\}$,
$K(.,.):\Delta \to \mathbb{R}$ is continuous and $x_0,y_0\in X$.
The general framework of evolution operators  $\{A(t)\}_{t\geq 0}$ that define problem (1.1) has been developed by Kozak (\cite{19}) and improved by Henriquez (\cite{17}).

Existence results and some qualitative properties of the mild solutions of
problem (1.1) may be found in \cite{14} in the case when $X$ is a separable Banach space.

De Blasi and Pianigiani (\cite{15}) obtained the existence of mild
solutions for semilinear differential inclusions on an arbitrary,
not necessarily separable, Banach space $X$. Even if Filippov's
ideas (\cite{16}) are still present, the approach in \cite{15} is fundamental
different: it consists in the construction of the measurable
selections of the multifunction. This construction does not use
classical selection theorems such as Kuratowski and
Ryll-Nardzewski's (\cite{20}) or Bressan and Colombo's (\cite{7}). 

The aim of this note is to obtain an existence result for problem
(1.1) similar to the one in \cite{15}. We will prove the existence of
solutions for problem (1.1) in an arbitrary space $X$ under
Filippov-type assumptions on $F$.

In several recent papers (\cite{2,3,6,12,13,17,18}) existence results and qualitative properties of mild solutions have been obtained for the following problem
$$
x''(t)\in A(t)x(t)+F(t,x(t)),\quad x(0)=x_0,x'(0)=y_0,\eqno (1.2)
$$
with $A(.)$ and $F(.,.)$ as above.  

On one hand, the result in the present paper extends to the integro-differential  framework (1.1) the result in \cite{12} obtained for problem (1.2) and, on the other hand, this paper extends to  second-order integro-differential inclusions a similar result in \cite{10} obtained for a class of first-order integro-differential inclusions.

The second class of evolution inclusions that we are considering is
$$
x'\in Ax+F(t,x)\quad a.e.\; ([0,T]),\eqno (1.3)
$$
$$
x(0)+\sum_{i=1}^ma _ix(t_i)=x_0,\eqno (1.4)
$$
where $X$ is a real separable Banach space, $a _i\in \mathbb{R}$, $a_i\neq 0$, $i=\overline{1,m}$,
$x_0\in X$, $0<t_1<t_2<...<t_m<T$, $F:[0,T]\times X\to \mathcal{P}(X)$ is a
set-valued map and $A$ is the infinitesimal generator of a linear semigroup $\{\mathcal{G}(t);\; t\geq 0\}$.

The nonlocal condition (1.4) was used by Byszewski (\cite{8,9}). If $a_i\neq 0$, $i=\overline{1,m}$
the results can be applied in kinematics to determine the evolution $t\to x(t)$ of the location
of a physical object for which the positions $x(0),x(t_1),...,x(t_m)$ are unknown but it is known
the condition (1.4). Consequently, to describe some physical phenomena the nonlocal condition may be
more useful than the standard initial condition $x(0)=x_0$. Obviously, when  $a_i=0$, $i=\overline{1,m}$,
one has the classical initial condition.

Existence of mild solutions of problem (1.3)-(1.4) has been obtained in \cite{4,5} for convex as well as nonconvex set-valued maps. All these results are based on some suitable theorems of fixed point theory. In our recent paper \cite{11}  it is shown that Filippov's ideas (\cite{1,16}) can be suitably adapted in order to prove the existence of solutions to problem (1.3)-(1.4) provided the Banach space $X$ is separable.

The result that we established in non separable Banach spaces for problem (1.3)-(1.4) may be interpreted as extension of the result in \cite{15} from Cauchy problems to  boundary value problems defined by nonlocal conditions and as an extension of the result in \cite{11} to non separable Banach spaces.

The paper is organized as follows: in Section 2 we present the notations, definitions and preliminary results to be used in the sequel and in Section 3 we prove the main results.


\section{Preliminaries}


Consider $X$, an arbitrary real Banach space with norm $|.|$ and
with the corresponding metric $d(.,.)$. Let $\mathcal{P}(X)$ be
the space of all bounded nonempty subsets of $X$ endowed with the
Hausdorff pseudometric
$$
\mbox{d}_H(A,B)=\max\{\mbox{d}^*(A,B),\mbox{d}^*(B,A)\},\quad
\mbox{d}^*(A,B)=\sup_{a\in A}\mbox{d}(a,B),
$$
where $\mbox{d}(x,A)=\inf_{a\in A}|x-a|$, $A\subset X,x\in X$.

Let $\mathcal{L}$ be the $\sigma $-algebra of the (Lebesgue)
measurable subsets of $R$ and, for $A\in \mathcal{L}$, let $\mu
(A)$ be the Lebesgue measure of $A$.

Let  $X$ be a Banach space and $Y$ be a metric space. An open
(resp., closed) ball in $Y$ with center $y$ and radius $r$ is
denoted by $B_Y(y,r)$ (resp., $\overline{B}_Y(y,r)$. In what
follows, $B=B_X(0,1)$.

A multifunction $F:Y\to \mathcal{P}(X)$ with closed bounded
nonempty values is said to be $d_H$-continuous at $y_0\in Y$ if
for every $\varepsilon >0$ there exists $\delta >0$ such that for
any $y\in B_Y(y_0,r)$ there is $\mbox{d}_H(F(y),F(y_0)) \leq
\varepsilon $. $F$ is called $\mbox{d}_H$-continuous if it is so
at each point $y_0\in Y$.

Let $A\in \mathcal{L}$, with $\mu (A)<\infty $. A multifunction
$F:Y\to \mathcal{P}(X)$ with closed bounded nonempty values is
said to be {\it Lusin measurable} if for every $\varepsilon >0$
there exists a compact set $K_{\varepsilon }\subset A$, with $\mu
(A\backslash K_{\varepsilon })<\varepsilon $ such that $F$
restricted to $K_{\varepsilon }$ is $\mbox{d}_H$-continuous.

It is clear that if $F,G:A\to \mathcal{P}(X)$ and $f:A\to X$ are
Lusin measurable, then so are $F$ restricted to $B$ ($B\subset A$
measurable), $F+G$ and $t\to \mbox{d}(f(t),F(t))$. Moreover, the
uniform limit of a sequence of Lusin measurable multifunctions is
Lusin measurable, too.

Let $I$ stand for the interval $[0,T]$, $T>0$, $C(I,X)$ is the Banach space of all continuous functions
from $I$ to $X$ with the norm $||x||_C=\sup_{t\in
I}|x(t)|$ and $L^1(I,X)$ is the Banach space of (Bochner)
integrable functions $u(.):I\to X$ endowed with the norm
$||u||_1=\int_0^T|u(t)|dt$.  Denote by $B(X)$
the Banach space of bounded linear operators from $X$ into $X$ with the norm
$||N||=\sup\{|N(y)|;\; |y|=1\}$.

In what follows $\{A(t)\}_{t\geq 0}$ is a family of linear
closed operators from $X$ into $X$ that genearates an evolution system of operators $\{G(t,s)\}_{t,s\in I}$.
By hypothesis the domain of $A(t)$, $D(A(t))$ is dense in $X$ and is independent of $t$.\vskip 0.5em

\begin{definition} (\cite{17,19}) A family of bounded linear operators $G(t,s):X\to X$,
$(t,s)\in \Delta:=\{(t,s)\in I\times I; s\leq t\}$ is called an evolution operator of the equation
$$
x''(t)=A(t)x(t)\eqno (2.1)
$$
if

\noindent i) For any $x\in X$, the map $(t,s)\to G(t,s)x$ is continuously differentiable and

a) $G(t,t)=0$, $t\in I$.

b) If $t\in I,x\in X$ then $\frac{\partial }{\partial t}G(t,s)x|_{t=s}=x$ and
$\frac{\partial }{\partial s}G(t,s)x|_{t=s}=-x$.

\noindent ii) If $(t,s)\in \Delta $, then $\frac{\partial }{\partial s}G(t,s)x\in D(A(t))$, the map $(t,s)\to G(t,s)x$
is of class $C^2$ and

a) $\frac{\partial ^2}{\partial t^2}G(t,s)x\equiv A(t)G(t,s)x$.

b) $\frac{\partial ^2}{\partial s^2}G(t,s)x\equiv G(t,s)A(t)x$.

c) $\frac{\partial ^2}{\partial s\partial t}G(t,s)x|_{t=s}=0$.

\noindent iii) If $(t,s)\in \Delta $, then there exist $\frac{\partial ^3}{\partial t^2\partial s}G(t,s)x$,
$\frac{\partial ^3}{\partial s^2\partial t}G(t,s)x$ and

a) $\frac{\partial ^3}{\partial t^2\partial s}G(t,s)x\equiv A(t)\frac{\partial }{\partial s}G(t,s)x$ and the map
$(t,s)\to A(t)\frac{\partial }{\partial s}G(t,s)x$ is continuous.

b) $\frac{\partial ^3}{\partial s^2\partial t}G(t,s)x\equiv \frac{\partial }{\partial t}G(t,s)A(s)x$.
\end{definition}

As an example for equation (2.1) one may consider the problem (e.g., \cite{19})
$$
\frac{\partial ^2z}{\partial t^2}(t,\tau )=\frac{\partial ^2z}{\partial \tau ^2}(t,\tau )+a(t)\frac{\partial z}{\partial t}(t,\tau ),\quad t\in [0,T],\tau \in [0,2\pi],
$$
$$
z(t,0)=z(t,\pi )=0,\quad \frac{\partial z}{\partial \tau}(t,0)= \frac{\partial z}{\partial \tau }(t,2\pi ),\; t\in [0,T],
$$
where $a(.):I\to \mathbb{R}$ is a continuous function.  This problem is modeled in the space $X=L^2(\mathbb{R},\mathbb{C})$ of $2\pi $-periodic 2-integrable functions from $\mathbb{R}$ to $\mathbb{C}$, $A_1z=\frac{d^2z(\tau )}{d\tau ^2}$ with domain
$H^2(\mathbb{R},\mathbb{C})$ the Sobolev space of $2\pi $-periodic functions whose derivatives belong to $L^2(\mathbb{R},\mathbb{C})$. It is well known that$A_1$ is the infinitesimal generator of strongly continuous cosine functions $C(t)$ on $X$. 
Moreover, $A_1$ has discrete spectrum; namely the spectrum of $A_1$ consists of eigenvalues $-n^2$, $n\in \mathbb{Z}$ with associated eigenvectors $z_n(\tau)=\frac{1}{\sqrt{2\pi }}e^{in\tau}$, $n\in \mathbb{N}$.
The set  $z_n, \; n\in \mathbb{N}$ is an orthonormal basis of $X$. In particular,
$A_1z=\sum_{n\in \mathbb{Z}}-n^2<z,z_n>z_n$, $z\in D(A_1)$. The cosine function is given by $C(t)z=\sum_{n\in \mathbf{Z}}\cos(nt)<z,z_n>z_n$ with the associated sine function $S(t)z=t<z,z_0>z_0+\sum_{n\in {\mathbf{Z}}^*}\frac{\sin(nt)}{n}<z,z_n>z_n$.

For $t\in I$ define the operator $A_2(t)z=a(t)\frac{dz(\tau )}{d\tau }$ with domain $D(A_2(t))=H^1(\mathbb{R},\mathbb{C})$. Set $A(t)=A_1+A_2(t)$. It has been proved in \cite{19} that this family generates an evolution operator as in Definition 2.1.

\begin{definition} A continuous mapping $x(.)\in C(I,X)$ is called a mild solution of
problem (1.1) if there exists a (Bochner) integrable function
$f(.)\in L^1(I,X)$ such that
$$
f(t)\in F(t,x(t))\quad a.e.\, (I),\eqno (2.2)
$$
$$
x(t)=-\frac{\partial }{\partial s}G(t,0)x_0+G(t,0)y_0+\int_0^tG(t,s)\int_0^sK(s,\tau )f(\tau )d\tau ,\; t\in I.\eqno (2.3)
$$
\end{definition}

We shall call $(x(.),f(.))$ a {\it trajectory-selection pair} of (1.1) if $f(.)$ verifies (2.2) and $x(.)$ is defined by (2.3).

We note that condition (2.3) can be rewritten as
$$
x(t)=-\frac{\partial }{\partial s}G(t,0)x_0+G(t,0)y_0+\int_0^tU(t,s)f(s)ds\quad \forall t\in I,\leqno
(2.4)
$$
where $U(t,s)=\int_s^tG(t,\tau )K(\tau ,s)d\tau$.
\vskip 0.5em

\noindent {\bf  Hypothesis H1.} i) There exists an evolution operator $\{G(t,s)\}_{t,s\in I}$ associated to the family
$\{A(t)\}_{t\geq 0}$.

ii) There exist $M,M_0\geq 0$ such that $|G(t,s)|_{B(X)}\leq M$, $|\frac{\partial }{\partial s}G(t,s)|\leq M_0$,
for all $(t,s)\in \Delta $.

iii) $K(.,.):\Delta \to \mathbb{R}$ is continuous.

\vskip 0.5em

\noindent {\bf Hypothesis H2.} i) $A$ is the infinitesimal generator of a 
strongly continuous and compact semigroup $\{\mathcal{G}(t);\; t\geq 0\}$ in $X$.

ii) There exists an operator $C:X\to X$ defined by
$$
C=[I+\sum_{i=1}^ma _i\mathcal{G}(t_i)]^{-1}.
$$


Let $m_0\geq 0$ be such that $|\mathcal{G}(t)|\leq m_0$ $\forall t\in I$.

According to \cite{4} if we assume that $\sum_{i=1}^m|a _i|<\frac{1}{m_0}$
then there exists $C$ as in Hypothesis H2 ii).

\begin{definition} A continuous mapping $x(.)\in C(I,X)$ is called a mild
solution of problem (1.3)-(1.4) if there exists a (Bochner) integrable
function $f(.)\in L^1(I,X)$ such that
$$
f(t)\in F(t,x(t))\quad a.e.\, (I)\eqno (2.5)
$$
$$
x(t)=\mathcal{G}(t)Cx_0-\sum_{i=1}^ma _i\mathcal{G}(t)C\int_0^{t_i}\mathcal{G}(t_i-u)f(u)du+\int_0^t\mathcal{G}(t-u)f(u)du,
t\in I.\eqno (2.6)
$$
\end{definition}

\begin{remark}  If we denote
$$
H(t,s)=\sum_{i=1}^ma _i\mathcal{G}(t)C\mathcal{G}(t_i-s)\chi _{[0,t_i]}(s)+\mathcal{G}(t-s)\chi _{[0,t]}(s),
$$
where $\chi _S(\cdot )$ is the characteristic function of the set $S$, then the solution $x(\cdot )$ in Definition
2.3 may be written as
$$
x(t)=\mathcal{G}(t)Cx_0-\int_0^TH(t,s)f(s)ds.\eqno (2.7)
$$
Obviously,
$$
|H(t,s)|\leq \sum_{i=1}^m|a _i|m_0^2||C||+m_0=:m\quad \forall \; t,s\in I.
$$
\end{remark}

In what follows $X$ is a real Banach space and we assume the
following hypotheses.\vskip 0.5em

\noindent {\bf Hypothesis H3.} i) {\it $F(.,.):I\times X\to \mathcal{P}(X)$
has nonempty closed bounded values and for any $x\in X$ $F(.,x)$
is Lusin measurable on $I$.}

ii) {\it There exists $l(.)\in L^1(I,(0,\infty ))$ such that, $\forall t\in I$}
$$
\mbox{d}_H(F(t,x_1),F(t,x_2))\leq l(t)|x_1-x_2|,\quad \forall \;
x_1,x_2\in X.
$$
iii) {\it There exists $q(.)\in L^1(I,(0,\infty ))$ such that $\forall t\in I$ we
have}
$$
F(t,0)\subset q(t)B.
$$

Denote $L=\int_0^Tl(s)ds$.

The technical results summarized in the following lemma are
essential in the proof of our results. For the proof, we refer the
reader to \cite{15}. 

\begin{lemma}
i) Let $F_i:I\to \mathcal{P}(X)$, i=1,2 be two Lusin measurable multifunctions and let $\varepsilon
_i>0$, i=1,2 be such that
$$
H_1(t):=(F_1(t)+\varepsilon _1B)\cap (F_2(t)+\varepsilon _2B)\neq
\emptyset ,\quad \forall t\in I.
$$

Then the multifunction $H_1:I\to \mathcal{P}(X)$ has a Lusin
measurable selection $h:I\to X$.

ii) Assume that Hypothesis  H3 is satisfied. Then for
any continuous $x(.):I\to X$, $u(.):I\to X$ measurable and any
$\varepsilon >0$ one has

a) the multifunction $t\to F(t,x(t))$ is Lusin measurable on $I$.

b) the multifunction $H_2:I\to \mathcal{P}(X)$ defined by
$$
H_2(t):=(F(t,x(t))+\varepsilon B)\cap
B_X(u(t),\mbox{d}(u(t),F(t,x(t)))+\varepsilon )
$$
has a Lusin measurable selection $g:I\to X$.
\end{lemma}


\section{The results}

Set $n(t)=\int_0^tl(u)\mbox{d}u$, $t\in I$, denote $K_0:=\sup_{(t,s)\in \Delta }|K(t,s)|$ and note that $|U(t,s)|\leq MK_0(t-s)\leq MK_0T$.

\begin{theorem}
We assume that Hypotheses H1 and H3 are
satisfied. Then, for every $x_0,y_0\in X$, Cauchy problem (1.1) has a mild solution $x(.)\in C(I,X)$.
\end{theorem}

\begin{proof} Let us first note that if $z(.):I\to X$ is
continuous, then every Lusin measurable selection $u:I\to X$ of
the multifunction $t\to F(t,z(t))+B$ is Bochner integrable on $I$.
More precisely, for any $t\in I$, there holds
$$
\begin{array}{l}
|u(t)|\leq d_H(F(t,z(t))+B,0)\leq d_H(F(t,z(t)),F(t,0))+d_H(F(t,0),0)+1\\ \leq l(t)|z(t)|+q(t)+1.
\end{array}
$$

Let $0<\varepsilon <1,\quad \varepsilon _n=\frac{\varepsilon
}{2^{n+2}}$.

Consider $f_0(.):I\to X$, an arbitrary Lusin measurable, Bochner
integrable function, and define
$$
x_0(t)=-\frac{\partial }{\partial s}G(t,0)x_0+G(t,0)y_0+\int_0^tU(t,s)f_0(s)ds,\quad t\in I.
$$

Since $x_0(.)$ is continuous, by Lemma 2.5 ii) there exists a
Lusin measurable function $f_1(.):I\to X$ which, for $t\in I$,
satisfies
$$
f_1(t)\in (F(t,x_0(t))+\varepsilon _1 B)\cap
B(f_0(t),d(f_0(t),F(t, x_0(t)))+\varepsilon _1)
$$
Obviously, $f_1(.)$ is Bochner integrable on $I$. Define
$x_1(.):I\to X$ by
$$
x_1(t)=-\frac{\partial }{\partial s}G(t,0)x_0+G(t,0)y_0+\int_0^tU(t,s)f_1(s)ds,\quad t\in I.
$$

By induction, we construct a sequence $x_n:I\to X,$ $n\geq 2$
given by
$$
x_n(t)=-\frac{\partial }{\partial s}G(t,0)x_0+G(t,0)y_0+\int_0^tU(t,s)f_n(s)ds,\quad t\in I,\eqno (3.1)
$$
where $f_n(.):I\to X$ is a Lusin measurable function which, for
$t\in I$, satisfies:
$$
f_n(t)\in (F(t,x_{n-1}(t))+\varepsilon _n B)\cap
B(f_{n-1}(t),d(f_{n-1}(t),F(t,x_{n-1}(t)))+\varepsilon _n).\eqno
(3.2)
$$

At the same time, as we saw at the begining of the proof, $f_n(.)$
is also Bochner integrable.

From (3.2), for $n\geq 2$ and $t\in I$, we obtain
$$
|f_n(t)-f_{n-1}(t)|\leq d(f_{n-1}(t),F(t,x_{n-1}(t)))+\varepsilon
_n\leq d(f_{n-1}(t),F(t,x_{n-2}(t)))+
$$
$$
d_H(F(t,x_{n-2}(t)),F(t,x_{n-1}(t)))+\varepsilon _n \leq
\varepsilon _{n-1}+l(t)|x_{n-1}(t)-x_{n-2}(t)| +\varepsilon _n.
$$
Since $\varepsilon _{n-1}+\varepsilon _n<\varepsilon _{n-2}$, for
$n\geq 2$, we deduce that
$$
|f_n(t)-f_{n-1}(t)|\leq \varepsilon
_{n-2}+l(t)|x_{n-1}(t)-x_{n-2}(t)|. \eqno (3.3)
$$

Denote $p_0(t):=d(f_0(t),F(t,x_0(t))),t\in I$. We next prove by
recurrence, that for $n\geq 2$ and $t\in I$
$$
\begin{array}{l}
|x_n(t)-x_{n-1}(t)|\leq \sum_{k=0}^{n-2}\int_0^t\varepsilon
_{n-2-k}\frac{(MK_0T)^{k+1}(n(t)-n(u))^k}{k!}du+\\
\varepsilon _0\int_0^t\frac{(MK_0T)^n(n(t)-n(u))^{n-1}}{(n-1)!}du+
\int_0^t\frac{(MK_0T)^n(n(t)-n(u))^{n-1}}{(n-1)!}p_0(u)du.
\end{array}
\eqno (3.4)
$$

We start with $n=2$. In view of (3.1), (3.2) and (3.3), for $t\in
I$, there is
$$
\begin{array}{l}
|x_2(t)-x_1(t)|\leq \int_0^t|U(t,s)|.|f_2(s)-f_1(s)|ds\leq \\
\int_0^tMK_0T[\varepsilon _0+l(s)|x_1(s)-x_0(s)|]ds\leq
\varepsilon _0MK_0Tt+\\
\int_0^t[MK_0Tl(s)\int_0^s|U(s,r)|.|f_1(r)-f_0(r)|dr]ds\leq
\varepsilon _0MK_0Tt+\\
\int_0^t[(MK_0T)^2l(s)\int_0^s(p_0(u)+\varepsilon _1)du]ds\leq
\varepsilon _0MK_0Tt+\int_0^t[(MK_0T)^2(p_0(u)+\\ \varepsilon _1)
\int_u^tl(s)ds]du=\varepsilon
_0MK_0Tt+\int_0^t(MK_0T)^2(n(t)-n(s))[p_0(s)+\varepsilon _0]ds,
\end{array}
$$
i.e, (3.4) is verified for $n=2$.

Using again (3.3) and (3.4), we conclude
$$
\begin{array}{l}
|x_{n+1}(t)-x_n(t)|\leq \int_0^t|U(t,s)|.|f_{n+1}(s)-f_n(s)|ds\leq \\
\leq \int_0^tMK_0T[\varepsilon _{n-1}+l(s)|x_n(s)-x_{n-1}(s)|]ds\leq \\
\leq \varepsilon
_{n-1}MK_0Tt+\int_0^tl(s)[\sum_{k=0}^{n-2}\int_0^s \varepsilon
_{n-2-k}\frac{(MK_0T)^{k+2}(n(s)-n(u))^k}{k!}du+\\
+\int_0^s\frac{(MK_0T)^{n+1}(n(s)-n(u))^{n-1}}{(n-1)!}(p_0(u)+\varepsilon
_0)du]ds=\\
\varepsilon _{n-1}MK_0Tt+\sum_{k=0}^{n-2}\varepsilon
_{n-2-k}\int_0^t[\int_0^s
\frac{(MK_0T)^{k+2}(n(s)-n(u))^k}{k!}l(s)du]ds+\\
+\int_0^tl(s)(\int_0^s
\frac{(MK_0T)^{n+1}(n(s)-n(u))^{n-1}}{(n-1)!}l(s)[p_0(u)+\varepsilon
_0]du)ds=\\
\varepsilon _{n-1}MK_0Tt+\sum_{k=0}^{n-2}\varepsilon
_{n-2-k}\int_0^t(\int_u^t
\frac{(MK_0T)^{k+2}(n(s)-n(u))^k}{k!}l(s)ds)du+\\
+\int_0^t(\int_u^t
\frac{(MK_0T)^{n+1}(n(s)-n(u))^{n-1}}{(n-1)!}l(s)ds)[p_0(u)+\varepsilon
_0]du=\\
\varepsilon _{n-1}MK_0Tt+\sum_{k=0}^{n-2}\varepsilon
_{n-2-k}\int_0^t \frac{(MK_0T)^{k+2}(n(s)-n(u))^{k+1}}{(k+1)!}du+\\
+\int_0^t\frac{(MK_0T)^{n+1}(n(s)-n(u))^n}{n!}[p_0(u)+\varepsilon
_0]du=\sum_{k=0}^{n-1}\varepsilon
_{n-1-k}\cdot \\\int_0^t\frac{(MK_0T)^{k+1}(n(s)-n(u))^k}{k!}du+
+\int_0^t\frac{(MK_0T)^{n+1}(n(s)-n(u))^n}{n!}[p_0(u)+\varepsilon
_0]du
\end{array}
$$
and statement (3.8) it is true for $n+1$.

From (3.8) it follows that for $n\geq 2$ and $t\in I$
$$
|x_n(t)-x_{n-1}(t)|\leq a_n,\eqno (3.5)
$$
where
$$
a_n=\sum_{k=0}^{n-2}\varepsilon
_{n-2-k}\frac{(MK_0T)^{k+1}n(T)^k}{k!}
+\frac{(MK_0T)^nn(T)^{n-1}}{(n-1)!}[\int_0^1p_0(u)du+\varepsilon
_0],
$$

Obviously, the series whose $n$-th term is $a_n$ converges. So,
from (3.5) we infer that $x_n(.)$ converges to a continuous
function, $x(.):I\to X$, uniformly on $I$.

On the other hand, in view of (3.3) there is
$$
|f_n(t)-f_{n-1}(t)|\leq \varepsilon _{n-2}+l(t)a_{n-1},\quad t\in
I, n\geq 3
$$
which implies that the sequence $f_n(.)$ converges to a Lusin
measurable function $f(.):I\to X$.

Since $x_n(.)$ is bounded and
$$
|f_n(t)| \leq l(t)|x_{n-1}(t)|+q(t)+1,
$$
we infer that $f(.)$ is also Bochner integrable.

Passing with $n\to \infty $ in (3.1) and using the Lebesgue
dominated convergence theorem, we obtain
$$
x(t)=-\frac{\partial }{\partial s}G(t,0)x_0+G(t,0)y_0+\int_0^tU(t,s)f(s)ds,\quad t\in I.
$$

On the other hand, from (3.2) we get
$$
f_n(t)\in F(t,x_n(t))+\varepsilon _nB,\quad t\in I,n\geq 1
$$
and letting $n\to \infty $ we obtain
$$
f(t)\in F(t,x(t)),\quad t\in I,
$$
which completes the proof. \end{proof}

\begin{theorem}
Assume that Hypotheses H2 and H3 are satisfied and $mL<1$.

Then, for every $x_0\in X$ problem (1.3)-(1.4) has a solution
$x(.):I\to X$.
\end{theorem}

\begin{proof} The proof follows the same pattern as in the proof of Theorem 3.1. This time
$$
x_n(t)=\mathcal{G}(t)Cx_0-\int_0^TH(t,s)f_n(s)\mbox{d}s,\quad \forall t\in I,
$$
with $f_n(.)$ as before and 
$$
|x_n(t)-x_{n-1}(t)|\leq \sum_{j=0}^{n-2}\varepsilon _{n-2-j}m^{j+1}L^jT +m^nL^{n-1}\int_0^T(p_0(s)+\varepsilon _0)ds
$$
 for $n\geq 2$ and $t\in I$. The estimate in (3.5) becames
$$
|x_n(t)-x_{n-1}(t)|\leq a_n,
$$
where
$$
a_n=\sum_{j=0}^{n-2}\varepsilon _{n-2-j}m^{j+1}L^jT +m^nL^{n-1}\int_0^T(p_0(s)+\varepsilon _0)ds
$$

Taking into account the fact that $mL<1$, we deduce that the series whose $n$-th term is $a_n$ is convergent.
\end{proof}


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