\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations}, 
Vol. 2005(2005), No. 73, pp. 1--17.\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-2005/73\hfil Existence results for a second-order]
{Existence results for a second-order abstract Cauchy problem
  with nonlocal conditions}
\author[E. Hern\'{a}ndez \& M. Pelicer\hfil EJDE-2005/73\hfilneg]
{Eduardo Hern\'{a}ndez M.,   Mauricio L. Pelicer}  % in alphabetical order

\address{Eduardo Hern\'{a}ndez M.  \hfill\break
Departamento de Matem\'atica \\
Instituto de Ci\^encias Matem\'aticas de S\~ao Carlos \\
Universidade de S\~ao Paulo \\
Caixa Postal 668 \\
13560-970 S\~ao Carlos, SP, Brazil}
\email{lalohm@icmc.sc.usp.br}

\address{Mauricio L. Pelicer \hfill\break
Departamento de Matem\'atica \\
Instituto de Ci\^encias Matem\'aticas de S\~ao Carlos \\
Universidade de S\~ao Paulo \\
Caixa Postal 668 \\
13560-970 S\~ao Carlos, SP, Brazil}
\email{mpelicer@icmc.sc.usp.br}

\date{}
\thanks{Submitted January 25, 2005. Published July 5, 2005.}
\subjclass[2000]{47D09, 47N20, 34G10}
\keywords{Abstract Cauchy problem; Cosine functions of operators}


\begin{abstract}
 In this paper we study the existence of mild and classical solutions
 for a  second-order abstract Cauchy problem with nonlocal
 conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}


\section{Introduction}


In this paper we  study the   existence of mild and classical
solutions for a class of second-order abstract   Cauchy problem
with nonlocal conditions described in the form
\begin{gather}\label{ne2}
 \frac{d}{dt}[x' (t)+g(t,x(t),x'(t))] =
Ax(t)+f(t,x(t),x'(t)), \quad t\in I=[0,a], \\
\label{ne21} x(0)= y_{0}+p(x,x'), \\
\label{ne22}  x'(0)=y_{1}+ q(x,x'),
\end{gather}
where $A$ is the infinitesimal generator of a strongly continuous
cosine function  of bounded linear operators
$(C(t))_{t\in\mathbb{R}}$ on a Banach space $X$ and
$ g,f:I\times X^{2}\to X$,
$p,q: C(I;X)\times C(I;X) \to X$ are appropriate functions.
\par  The system   \eqref{ne2}-\eqref{ne22} is  a
simultaneous generalization of the classical second order abstract
Cauchy problem studied by Travis and Weeb in \cite{TW2,TW3} and of
some recent developments for ordinary differential equations by
Stan\v ek in \cite{St1,St2,St3,St4}. This generalization and their
 applications to partial second order differential equations
 are the main  motivations of this paper.
\par Initial value problems with  nonlocal conditions
arises to deal specially with some situations in physics.  Motivated
for numerous applications, Byszewski studied in
  \cite{BY1}  the existence of mild, strong and
classical solutions for the semilinear abstract Cauchy problem with
nonlocal  conditions
\begin{gather*}
x'(t) = Ax(t)\;+\;f(t,x(t)), \quad t\in I=[0,a],  \\
x(0) = x_{0}\;+\; q(t_{1},t_{2},t_{3},\dots ,t_{n},x(\cdot))\in X.
\end{gather*}
In this system,  $A$ denotes the infinitesimal generator of a
strongly continuous semigroup of linear operators on $X$;
$0 <t_{1} < \dots <t_{n}\leq a$ are prefixed numbers;
 $f:[0,a]\times X \to X$, $q(t_{1},t_{2},t_{3},\dots ,t_{n},\cdot) :
C(I;X)\to X $ are appropriated functions  and the symbol $
q(t_{1},t_{2},t_{3},\dots ,t_{n},u(\cdot))$ is used  in the sense
that   $u(\cdot)$ can be evaluated only in the points $t_{i}$, for
instance $ q(t_{1},t_{2},t_{3},\dots ,t_{n},u(\cdot))=
\sum_{i=1}^{n}\alpha_{i}u(t_{i})$.

  The existence of mild solutions for  second order
abstract Cauchy problems with nonlocal conditions  is  studied in
Ntouyas \&  Tsamatos \cite{NT1,NT2}, Benchohra \&  Ntouyas
\cite{Be26,Be23,Be25,27}, Dauer \& Mahmudov
 \cite{Dauer1} and Hernández \cite{He1}.   The results in the first two paper are only
applicable to ordinary differential equations since the compactness
assumption assumed on the cosine function is valid if, only  if, the
underlying space is finite dimensional, see Travis \cite[p.
557]{TW2} for details. On the other hand, the results in
\cite{Be26,Be23,Be25,27} are proved using that  the cosine function
is continuous  in the uniform operator topology which implies that
their infinitesimal   generator is bounded,
  see \cite[p. 565]{TW2}.  We also observe that, in general,
    the nonlocal conditions
considered in these works   are described in the form
\[
x(0)=h(x)+x_{0}, \quad x'(0)=p(x)+\eta,
\]
where $h,p:C(I: X)\to X$ are   appropriate   functions and  $\eta
\in X$ is prefixed.  These restrictions are  an additional
 motivation for our paper.

 Concluding this introduction, we remark that the results in
this paper   can be applied in the study of second order partial
differential equations,    the operator  $A$ is assumed unbounded and
the system \eqref{ne2}-\eqref{ne22} can be considered  a generalization at
those studied in
  \cite{Be26,Be23,Be25,27,Dauer1,He1,St1,St2,St3,St4,TW2,TW3}.

 \section{ Preliminaries}

  Throughout this paper, $A$ is the
infinitesimal generator of a strongly continuous cosine family,
$(C(t))_{t\in \mathbb{R}}$, of bounded  linear operators  defined
on a Banach space $X$.
   We denote by   $(S(t))_{t\in
\mathbb{R}}$ the sine function associated to $(C(t))_{t\in
\mathbb{R}}$ which is defined by
$$
S(t) x := \int_{0}^{t} C(s) x ds,\quad x \in X, \;t \in \mathbb{R}.
$$
 Moreover,  $N$ and $\tilde{N}$ are   positive
 constants such that $\|C(t)\|
\leq N$ and $\|S(t)\| \leq \tilde{N}$ for every $t\in I$.

 In this paper,  $[D(A)]$ is the   space
 $D(A)=\{x\in X:C(\cdot) x\,\,\hbox{is of class $ C^{2}$  on } \mathbb{R}\}$,
 endowed with the norm $\| x\|_{A} = \| x\| + \| Ax \|$, $x \in D(A)$.
The notation
$E$ stands for the space  formed by the vectors $x \in X$ for
which  $C(\cdot)x$ is of class $C^{1}$ on $\mathbb{R}$. We know
from Kisi\'nsky \cite{Ki},  that $E$ endowed with the norm
\begin{eqnarray}\label{normE}
 \|x\|_{E} = \|x\|
\;+\; \sup_{0 \leq t \leq 1} \|A S(t) x\|, \quad x \in E,
\end{eqnarray}
 is a Banach space. The operator  valued function
 $g(t) = \begin{bmatrix}
{lr} C(t) & S(t) \\ A S(t) & C(t)
 \end{bmatrix}$
 is a strongly continuous group of linear operators  on the space   $E
\times X$ generated by the operator
$\mathcal{A} = \begin{bmatrix} 0 & I \\ A  & 0 \end{bmatrix}$
defined on $D(A) \times E$.  From this, it follows that $A S(t)
: E \to X$ is a bounded linear operator  and that
$AS(t) x \to 0$ as $t \to 0$, for each $x \in E$. Furthermore, if
$x :[0,\infty) \to X$ is  locally  integrable,  then
$y(t) =\int_{0}^{t} S(t -s) x(s)ds $ defines an $E$-valued continuous
function which  is a consequence of the fact that
\[
\int_{0}^{t} g(t -s) \begin{bmatrix} 0 \\ x(s) \end{bmatrix}
 ds =  \begin{bmatrix}
   \int_{0}^{t} S(t -s) x(s)\, ds \\[3pt]
 \int_{0}^{t} C(t -s) x(s)\, ds  \end{bmatrix}
  \]
defines an $E \times X$-valued continuous function.


  The existence of solutions  of the second-order
abstract Cauchy problem
\begin{gather}\label{aux}
 x''(t) =  A x(t) \;+\;h(t),\quad  t\in [0,a],\\
\label{aux1}  x(0) =  y_{0},\quad \\
\label{aux2} x'(0) = y_{1},
\end{gather}
where $h : [0, a] \to X$ is an integrable function  has been
discussed in \cite{TW2}. Similarly, the existence of solutions of
semilinear second order abstract Cauchy problem  has been treated
in \cite{TW3}. We only   mention here that the function
\begin{equation}\label{E1}
x(t) = C(t) y_{0} \;+\; S(t) y_{1} \;+\; \int_{0}^{t} S(t -s)
h(s)\, ds,\quad  t\in [0,a],
\end{equation}
is called   mild solution of (\ref{aux})-(\ref{aux2}) and that
when $y_{0} \in E$,  $x(\cdot)$ is continuously differentiable and
\begin{eqnarray}\label{38}
x'(t) = A S(t) y_{0} +C(t) y_{1} +\int_{0}^{t} C(t -s) h(s)\, ds.
\end{eqnarray}
 The regularity of mild solutions of (\ref{aux})-(\ref{aux2}) is studied
in   Travis $\&$ Weeb  \cite{TW3}. In our  work,
 we adopt the next concept  of classical solution  of (\ref{aux})-(\ref{aux2}).

\begin{definition}\label{classical} \rm
A function $u\in C(I;X)$ is a classical solution of
(\ref{aux})-(\ref{aux2}), if $u\in C^{2}(I;X)$ and
(\ref{aux})-(\ref{aux2}) are verified.
\end{definition}

\begin{remark}\label{rem1} \rm
As usual, we say that $u\in C^{1}([\sigma,\mu]:X)$ if $u'(\cdot)$
is continuous on $(\sigma,\mu)$ and the right and left lateral
derivatives of $u(\cdot)$ are  continuous functions on $
[\sigma,\mu)$ and $ (\sigma,\mu]$ respectively.
\end{remark}

  For additional  details concern to  cosine function theory,
we refer the reader to Fattorini \cite{Fa} and Travis \& Weeb
\cite{TW2,TW3}.


 The terminology  and notation  are those generally used in
functional analysis. In particular, if $(Z,\|\cdot\|_{Z})$ and
$(Y,\|\cdot\|_{Y})$ are Banach spaces, we indicate by
${\mathcal{L}}(Z;Y)$ the Banach space of bounded linear operators
from $Z$ into $Y$ and we abbreviate this notation to
${\mathcal{L}}(Z)$ whenever $Z=Y$.   In  this paper, $B_{r}(x;Z)$
denotes the closed ball with center at $x$ and radius $r>0$ in
$Z$. Additionally,  for a bounded function $\xi :I\to Z$ and $
t\in I $, we will employ the notation $\xi_{Z,\,t}$ for
\[
 \xi_{Z,t} =\sup\{\| \xi(s)\|_{Z} : s \in [0,t]\},
\]
 and we will write simply $\xi_{t}$ in the place of $\xi_{Z,\,t}$
 when no confusion arises.

  This  paper has  five  sections.  In section \ref{existence} we discuss the
existence of  mild solutions for  some abstract Cauchy problems
 similar to \eqref{ne2}-\eqref{ne22} and
in section \ref{regularity} we study the existence of classical
solutions for  \eqref{ne2}-\eqref{ne22}.   In section \ref{Examples}
some examples are considered.


\section{Existence of mild  solutions}\label{existence}

To begin this section we study the abstract Cauchy problem with
nonlocal conditions
\begin{gather}\label{mild1}
\frac{d}{dt}[x'(t)+g(t,x(t))]= Ax(t)+f(t,x(t)), \quad t\in I, \\
\label{P3}
 x(0)  =y_{0}+p(x),\\
 \label{P4}  x' (0) =y_{1}+ q(x) ,
 \end{gather}
 where   $f,g:I\times X\to X$ and  $ p,q:C(I;X)\to X$ are
appropriate functions.

If $u(\cdot)$ is  a  solution  of \eqref{mild1}-\eqref{P4} and
 the mapping $t\to g(t,u(t)) $ is enough smooth,  from  (\ref{E1})
and the relation  $ A\int_{r}^{s}S(\theta )x=C(s)x-C(r)x,\,x\in
X$, we obtain
\begin{align*}
 u(t)&=C(t) (y_{0}+p(u))+S(t)[y_{1}+q(u)+g(0, u(0))]-
 \int_{0}^{t}C(t-s)g(s,u(s))ds\\
 &\quad  +  \int_{0}^{t} S(t-s) f(s,u(s)) ds,\quad t\in I.
\end{align*}
 This  expression  is the  motivation of the following definition.

\begin{definition} \label{def3.1} \rm
A function $u\in C(I;X)$ is a mild solution of
\eqref{mild1}-\eqref{P4}, if   $u(0) =y_{0}+p(u)$ and
\begin{align*}
u(t) &= C(t)(y_{0} + p(u)) + S(t)(y_{1} + q(u) + g(0,u(0))) -
\int _0^t C(t-s)g(s,u(s))ds  \\
&\quad + \int_0^t S(t-s)f(s,u(s))ds, \quad t \in I.
\end{align*}
\end{definition}

Before establishing  our first result of existence,  we consider  the following
general lemma.


\begin{lemma}\label{lema4}
Let $(Z_{i},\|\cdot\|_{i})$, $i=1,2,3$, be Banach spaces,
   $L:I\times Z_{1}\to Z_{2}$ be a function,
 $\{ R(t):t\in I\}\subset \mathcal{L}(Z_{2},Z_{3}) $
   and assume that the next conditions hold.
  \begin{itemize}
    \item[(a)] The function $L(\cdot)$ satisfies  the following conditions.

    \begin{itemize}
    \item[(i)]    For every $r>0$, the set $ L(I\times B_{r}(0;Z_{1}))$
     is relatively compact    in $Z_{2}$.
        \item[(ii)] The function $L(t, \cdot):Z_{1} \to Z_{2}$
  is continuous $a.e.$ $t\in I$
  \item[(iii)] For each $z  \in Z_{1} $, the
  function  $L(\cdot, z) : I \to Z_{2}$  is strongly
  measurable.
        \item[(iv)] There exist an integrable function
          $m_{L}:I\to [0, \infty )$  and a continuous
  function $ W_{L}:[0,\infty)\to [0,\infty)$ such that
$$\| L(t,z )\|_{2} \leq m_{L}(t)W_{L}(\|
z\|_{1})\quad (t,z)\in I\times Z_{1}.$$
    \end{itemize}
  \item[(b)]  The operator family $(R(t))_{t\in I}$ is strongly continuous,
    this means that  $t\to R(t)z$ is continuous
on $I$ for every $z\in Z_{2}$.
      \end{itemize}
Then mapping $ \Gamma : C(I; Z_{1})\to C(I; Z_{3})$ defined by
  $$\Gamma u(t)=\int_0^t R(t-s)L(s,u(s)),
$$
 is completely  continuous.
\end{lemma}

\begin{proof} It is clear that $\Gamma(\cdot) $ is
well defined and  continuous. From  conditions (a) and (b), it
follows that the set $\{\,R(s)L(\theta,z):s,\theta\in I,z\in
B_{r}(0;Z_{1})\,\}$ is relatively compact in  $Z_{3}$. If $u\in
B_{r}(0;C(I;Z_{1}))$,\, from  the mean value Theorem for the
Bochner integral, see \cite[Lemma 2.1.3]{Ma},  we get
\begin{eqnarray}
\Gamma u(t) \in t \,{\overline{\mathop{\rm co}(\{\,R(s)L(\theta,z):s,\theta\in I,\,
z\in B_{r}(0;Z_{1})\})}}^{\,\,Z_{3}}
\end{eqnarray}
where  $\mathop{\rm co}(\cdot)$ denote  the convex hull. Thus,
$\{\Gamma u(t):u\in B_{r}(0;C(I;Z_{1})) \}$ is relatively compact
in $Z_{3}$ for every $t\in I.$

 Next,  we prove that
$\Gamma  (B_{r}(0;C(I;Z_{1}))=\{\Gamma u:u\in B_{r}(0;C(I;Z_{1}))
\}$ is equicontinuous on $I$. Let $ \varepsilon>0$ and $r>0$. From
the strong continuity of $(R(t))_{t\in I}$ and the compactness of
$L(I\times B_{r}(0;Z_{1}))$, we can choose  $\delta>0 $ such that
$$
\| R(t)L(s,z)-R(t')L(s,z)\|_{3}  \leq \varepsilon,\quad
  t',t,s\in I, \; z\in B_{r}(0;Z_{1}),
$$
when   $ | t-t'|\leq \delta $. Consequently, for    $u\in
B_{r}(0;C(I;Z_{1}))$, $t\in I$ and $| h| \leq \delta $ such
that $t+h\in I$, we get
\begin{align*}
\|  \Gamma u(t+h) - \Gamma u(t)\|_{3}
&\leq  \int_0^t \|  (R(t+h-s)-R(t-s)) L(s,u(s))\|_{3} ds\\
&\quad + \sup_{\theta \in I}\| R(\theta)\|_{\mathcal{L}(Z_{2};Z_{3}) }
 \int_t^{t+h} \| L(s,u(s))\|_{2} ds \\
 &\leq \varepsilon a+
 \sup_{\theta \in I}\| R(\theta)\|_{\mathcal{L}(Z_{2};Z_{3}) }
 W_{L}(r)\int_t^{t+h} m_{L}(s) ds,
\end{align*}
which shows the equicontinuity at $t\in I$ and so that $\Gamma
(B_{r}(0;C(I;Z_{1}))$ is equicontinuous on  $I$.
The assertion is now consequence of the Azcoli-Arzela
criterion. The proof is complete.
\end{proof}

For the rest of this article we use the following hypotheses:

\begin{itemize}
  \item[(H1)]  The functions
   $f,g :I \times X\to X$ satisfies the following conditions.
\begin{itemize}
  \item[(i)] The functions $f(t, \cdot):X \to X$, $g(t, \cdot):X \to X$
  are  continuous $a.e.$ $t\in I$;
 \item[(ii)] For each $x  \in X $, the functions
 $f(\cdot, x) : I \to X $,   $g(\cdot, x) : I \to X $ are
 strongly measurable.
 \end{itemize}
  \item[(H2)]  The functions $ p,q:C(I;X)\to X$ are
  continuous  and there are positive constants  $ l_p,l_q $ such that
\begin{gather*}
 \| p(u) - p(v)\| \leq l_p \| u-v\| _a ,
 \hspace{0.5cm} u,v \in C(I;X),\\
  \| q(u) - q(v)\| \leq l_q \| u-v\| _a ,
 \hspace{0.5cm} u,v \in C(I;X).
\end{gather*}
\end{itemize}

Now, we establish our first  result of existence.

\begin{theorem} \label{teo2}
Assume   (H1), (H2),   and  the following conditions:
\begin{itemize}
\item[(a)]  For every $r>0$, the set  $g(I\times B_{r}(0;X)
)$ is relatively compact in X and there exists a  constant
  $\alpha_r^g $ such that  $\| g(t,x)\| \leq \alpha_r^g$  for every
   $(t,x) \in I \times B_{r}(0;X)$.
\item[(b)] For  every $0< t'< t\leq a$ and every  $r>0$,
the set $$U(t,t',r)= \{ S(t')f(s,x): s \in [0,t],  x\in
B_{r}(0;X)\}$$ is relatively compact in $X$ and there exists a
positive  constant  $ \alpha_r^f$ such that $\| f(t,x)\| \leq
\alpha_r^f$ for every $(t,x) \in I \times B_{r}(0;X)$.
\end{itemize}
If
$$
 (N l_p + \tilde{N} l_q ) + \liminf_{r \to + \infty}
\frac{\tilde{N}\alpha_{r}^g + (N \alpha_r^g + \tilde{N}
\alpha_r^f)a}{r} <1 ,
$$
then there exists a mild solution of \eqref{mild1}-\eqref{P4}.
\end{theorem}

\begin{proof}
 On the  space $Y=C(I;X)$ endowed with the
norm of the uniform convergence,   we define the operator
$\Gamma: Y \to Y$ by
\begin{align*}
\Gamma u(t)& =  C(t)(y_{0} + p(u)) + S(t)(y_{1} +q(u) + g(0,u(0)))   \\
&\quad - \int_0^t C(t-s)g(s,u(s))ds + \int_0^t S(t-s)f(s,u(s))ds.
\end{align*}
We claim that there exists $r^{*}>0$ such that
$\Gamma(B_{r^{*}}(0,Y)) \subset B_{r^{*}}(0,Y)$. Assuming that
the claim is false, then for every $r>0$ there exists $x^r
\in B_{r}(0;Y)$  and $t^{r}\in I$ such that $\| \Gamma
x^r(t^{r})\| > r$. This yields
\begin{align*}
 r < \| x^r(t^{r})\|  & \leq  N( \| y_{0}\| + l_p r
+ \| p(0)\| )   + \tilde{N} ( \| y_{1}\| + l_q r+ \| q(0)\|
+  \alpha_{ r}^g  )\\
&\quad + N \int_0^a \alpha_r^g ds + \tilde{N} \int_0^a \alpha_r^f ds,
\end{align*}
and then
$$
1 \leq (N l_p + \tilde{N} l_q ) + \liminf_{r \to + \infty}
 \frac{\tilde{N}\alpha_{ r }^g +
 (N \alpha_r^g + \tilde{N} \alpha_r^f)a}{r} \,,
$$
which  contradicts   our  assumptions.

  Now, we prove that $\Gamma(\cdot)$ is a condensing operator
on  $B_{r^{*}} (0,Y)$. For  this purpose, we introduce the
decomposition $\Gamma = \sum_{i=1}^{3}\Gamma_i$, where
\begin{gather*}
\Gamma_{1} u(t) =  C(t)(y_{0} + p(u)) + S(t)(y_{1} +q(u)),\\
\Gamma_2 u (t)= S(t) g(0,u(0)) - \int_0^t C(t-s)g(s,u(s))ds,\\
\Gamma_3 u (t)=  \int_0^t S(t-s)f(s,u(s))ds.
 \end{gather*}
 From Lemma \ref{lema4}, condition (a) and the
  Lipschitz continuity of $t\to S(t)$    we infer
  that $\Gamma_{2}(\cdot)$ is completely continuous on $Y$ and from the
 estimate
\[
\| \Gamma_{1} u - \Gamma_{1} v\|_{a}  \leq  \left( Nl_p +
\tilde{N}  l_q \right) \| u-v\| _a,\quad u,v\in C(I;X),
\]
 that $ \Gamma_{1}(\cdot)$ is a contraction on $Y$.

 Next, by using the Ascoli-Arzela criterion,  we
prove  that $\Gamma_{3}(\cdot)$ is completely continuous on $Y$.
In the next steps $r$ is a positive number.

\noindent{\bf Step 1 } The set
 $\Gamma_{3} (B_{r}(0;Y))(t)= \{ \Gamma_{3} u(t) : u \in
B_{r}(0;Y) \}$ is relatively compact in $X$ for every  $t \in I$.
Let $t \in I, \varepsilon >0$ and
$0=s_1 < s_2 < \dots  < s_k =t$ be  numbers such that $|
s_{i}-s_{i+1}|\leq \varepsilon $ for every $i= 1, 2,\dots k-1$. If
$u\in B_{r}(0;Y)$, from  the mean value Theorem for Bochner
integral, see \cite[Lemma 2.1.3]{Ma},  we find that
 \begin{align*}
\Gamma_3 u(t) &= \sum_{i=1}^{k-1} \int_{s_i}^{s_{i+1}} S(s_i)
f(t-s,u(t-s))ds \\
&\quad + \sum_{i=1}^{k-1} \int_{s_i}^{s_{i+1}}(S(s) - S(s_i)) f(t-s,
u(t-s))ds  \\
&\in \sum_{i=1}^{k-1}( s_{i+1}- s_i ) \overline{\mathop{\rm co}(U(t,s_i,r))} +
\epsilon N\alpha_r^f a B_{1}(0,X) ,
\end{align*}
 where $co(\cdot)$  denote the convex hull.  Thus,
 $\Gamma_3  (B_{r}(0;Y))(t)$ is relatively compact in $X$.

\noindent{\bf Step 2. } The set $\Gamma_{3} (B_{r}(0;Y))$
  is uniformly  equicontinuous on $I$.
 For    $u \in B_{r}(0;Y)$, $ t \in I$ and $
h\in \mathbb{R}$   such that  $t+h \in I$, we get
\begin{align*}
&\|  \Gamma_{3} u(t+h) - \Gamma_{3} u(t)\| \\
&\leq  \int_0^t \| (S(t+h-s) - S(t-s)) f(s,u(s))\|  ds
+ \tilde{N}\int_t^{t+h} \| f(s,u(s))\|  ds  \\
& \leq  N\alpha_r^f ah  + \tilde{N} \alpha_{r}^f h,
\end{align*}
which implies   that $\Gamma_{3} (B_{r}(0;Y))$ is uniformly
equicontinuous on $I$.

 It follows from steps 1 and 2 that $\Gamma_3(\cdot) $
is completely continuous on $Y$. The previous remarks show that
$\Gamma(\cdot) $ is condensing from   $B_{r^{*}}(0,Y)$ into
$B_{r^{*}}(0,Y)$. The existence of a mild solution of system
\eqref{mild1}-\eqref{P4} is now a consequence of \cite[Corollary
4.3.2 ]{Ma}. The proof is completed.
\end{proof}

 Using arguments similar to the ones above, we can  prove the next result.

\begin{proposition}\label{prop1}
Let  assumptions  (H1), (H2) be satisfied.   Suppose,
furthermore, that condition $(a)$  of Theorem \ref{teo2} holds
and that there exists   $l_g \geq 0 $ such that
\[
\| g(t,x)-g(t,y)\| \leq l_g \| x-y\| , \quad t\in I,
x,y\in X.
\]
 If
\[
 (N l_p + \tilde{N} l_q ) + ( \tilde{N}+Na )l_{g} +
\tilde{N}a \liminf_{r \to + \infty} \frac{  \alpha_r^f }{r}<1 ,
\]
then there exists a mild solution of \eqref{mild1}-\eqref{P4}.
\end{proposition}

 Using the classical  principle of contraction,   we can prove  the
following  result.

\begin{theorem} \label{teo1}
Let  (H1), (H2)  be satisfied and assume that there exist
constants  $l_f$, $l_g$ such that
\begin{gather*}
\| g(t,x)-g(t,y)\|  \leq l_g \| x-y\| , \quad t\in I,
x,y\in X,\\
\| f(t,x)-f(t,y)\|  \leq l_f \| x-y\| , \quad t\in
I,x,y\in X.
\end{gather*}
If  $[N(l_p+al_g) + \tilde{N}(l_q+l_g + al_f)] < 1$, then there
exists a unique  mild solution of \eqref{mild1}-\eqref{P4}.
\end{theorem}

 Next, we study  the abstract Cauchy problem
\eqref{ne2}-\eqref{ne22}.

\begin{definition} \label{def3.6} \rm
A  function $u\in C(I;X)$ is called a mild solution of
\eqref{ne2}-\eqref{ne22} if $u\in C^{1}(I;X)$, conditions
(\ref{ne21}) and (\ref{ne22}) are satisfied and
\begin{align*}
u(t)& =  C(t)(y_{0} + p(u,u')) + S(t)( y_{1} + q(u, u') +
g(0,u(0),u'(0))) \\
&\quad - \int_0^t C(t-s)g(s,u(s),u'(s))ds + \int_0^t
S(t-s)f(s,u(s),u'(s))ds,\quad  t\in I.
\end{align*}
\end{definition}

To study the system  \eqref{ne2}-\eqref{ne22}
 we introduce the following conditions.
\begin{itemize}
 \item[(H3) ]  The function
   $f,g :I \times X\times X \to X$ satisfies the following conditions;
\begin{itemize}
  \item[(i)] The function $f(t, \cdot):X\times X  \to X$ is
    continuous $a.e.$ $t\in I$;
 \item[(ii)]  The function
 $f(\cdot, x,y) : I \to X $ is  strongly
 measurable for each $(x,y)  \in X\times X $.
  \item[(iii)] The function $g(\cdot)$ is $E$-valued and   $g:I \times  X\times X \to E$ is continuous.
\end{itemize}

\item[(H4)]  The function $ p,q:C(I;X)\times C(I;X) \to
X$ are   continuous, $p(\cdot)$ is  $E$-valued  and there exist
positive constants  $ l_p,l_q $ such that
\begin{gather*}
\| p(u_1,v_1) - p(u_2,v_2)\|_{E} \leq  l_p ( \| u_1 - u_2\|
_a + \| v_1 - v_2\| _a),\\
\| q(u_1,v_1) - q(u_2,v_2)\| \leq  l_q ( \| u_1 - u_2 \|_a +
\| v_1 - v_2\|_a ).
\end{gather*}
for every $u_i,v_i \in C(I;X)$.
 \end{itemize}

\begin{remark} \label{rmk3.7} \rm
In the rest of this paper, $ \rho=\sup_{\theta \in I}\|
AS(\theta)\|_{ {\mathcal{L}(E;X)}}$.
\end{remark}

\begin{theorem}\label{teo4}
Let $(y_{0},y_{1}) \in E \times X$ and assume
(H3), (H4) be satisfied. Suppose in addition that the following conditions
hold:
\begin{itemize}
\item[$\bf{(a)}$] For every  $r>0$, the set  $ f(I\times
B_{r}(0;X)\times B_{r}(0;X)) $ is relatively compact in $X $ and
there exists a constant  $\alpha_r^f $ such that $\| f(t,x,y)\|
\leq \alpha_r^f$  for every    $(t,x,y)\in I\times
B_{r}(0;X)\times B_{r}(0;X)$. \item[$\bf{(b)}$]  The function
$g(\cdot):I\times X\times X \to E$ is completely continuous and
for every  $r>0$ there exists a constant $\alpha_r^g $ such that
$\| g(t,x,y)\|_{E} \leq \alpha_r^g$  for every
   $(t,x,y)\in I\times  B_{r}(0;X)\times B_{r}(0;X)$.
    \item[$\bf{(c)}$]  For every $r>0$, the set
$\{t\to  g(t,u(t),v(t)):u,v\in B_{r}(0;C(  I;X)) \}$ is
   a equicontinuous subset of\, $C(I;X)$.
\end{itemize}
If $$
   (N+\rho)l_p + (N+
\tilde{N})l_q  + \liminf_{r \to \infty}  \frac{(N + \tilde{N})(
\alpha_{r }^g +a\alpha_r^f )+\alpha_{r }^g(1+ a (N+\rho) ) }{r}
<1, $$
  then there exists a mild solution of \eqref{ne2}-\eqref{ne22}.
\end{theorem}

\begin{proof} On the space  $Y=C(I;X)\times C(I;X)$
endowed with the   norm of the uniform convergence, $ \|
(u,v)\|_{a} = \| u\|_{a}+ \| v\|_{a} $, we define the
operator $\Gamma:Y \to Y$  by $ \Gamma (u, v)  =
(\Gamma_{1} (u, v),\Gamma_{2}(u, v))$ where
\begin{align*}
\Gamma_{1} (u, v)(t)&= C(t)(y_{0} + p(u,v)) +
 S(t)( y_{1} + q(u,v) + g(0,u(0),v(0))) \\
&\quad - \int_0^tC(t-s)g(s,u(s),v(s))ds + \int_0^t
S(t-s)f(s,u(s),v(s))ds, \\
\Gamma_{2}(u, v)(t)  &=  AS(t)( y_{0} + p(u,v)) + C(t)(y_{1} +
q(u,v) +
 g(0,u(0),v(0))) \\
& \quad - g(t,u(t),v(t)) - \int_0^t AS(t-s)g(s,u(s),v(s))ds   \\
&  \quad + \int_0^tC(t-s)f(s,u(s),v(s))ds.
\end{align*}
Using that $g(\cdot)$ and $p(\cdot)$ are $E$-valued continuous,
it's easy to prove  that  $ \Gamma(\cdot)$ is well defined and
continuous.

 Now, we show that there exists $r^{*}>0$ such that
$\Gamma(B_{r^{*}}(0,Y)) \subset B_{r^{*}}(0,Y)$. Assume that this
property is false. Then for every  $r>0$ there exists  $(u^r,
v^{r}) \in B_{r}(0;Y)$ such that $r< \| \Gamma (u^r,v ^{r})
\|_{a}$. This yields
\begin{align*}
&r<\|\Gamma^{1} (u, v)\|_{a}+ \|\Gamma^{2} (u, v)\|_{a} \\
&\leq N \left( \| y_{0}\| + l_p r + \| p(0,0)\|  \right) +
\tilde{N} ( \| y_{1}\|  + l_{q} r + \| q(0,0)\| + \alpha_{r}^g
)  \\
&\quad + a(N \alpha_{r}^g + \tilde{N} \alpha_r^f ) +\sup_{\theta
\in I}\| AS(\theta) \|_{{\mathcal{L}(E;X)}} \left( \| y_{0}\|_{E}
+ l_p r +
 \| p(0,0)\|_{E} \right) \\
&\quad + N ( \| y_{1}\|  + l_q r + \| q(0,0)\|  + \alpha_{r}^g )
+ \alpha_{r}^g   \\
&\quad  + \int_{0}^{a}\sup_{\theta \in I}\| AS(\theta) \|_{{\mathcal{L}(E;X)}}
\| g(s,u(s),v(s))\|_{E}ds + N \alpha_r^f  a \\
&\leq   (N+\rho) \left( \| y_{0}\|_{E} + l_p r + \| p(0,0)\|_{E}
\right) + \alpha_{r}^g \\
&\quad +  (N + \tilde{N}) \left( \|
y_{1}\| + l_q r + \| q(0,0)\| + \alpha_{r }^g \right)  +
a\left( \alpha_r^g (N+\rho)  + \alpha_r^f (N + \tilde{N}) \right)
\end{align*}
and hence
$$
1\leq (N+\rho)l_p + (N+ \tilde{N})l_q  + \liminf_{r
\to \infty}  \frac{(N + \tilde{N})( \alpha_{r }^g +a\alpha_r^f
)+\alpha_{r }^g(1+ a (N+\rho) ) }{r}, $$ which is  contrary to the
hypotheses.

 Next, we prove that $ \Gamma(\cdot)$ is  condensing  from
$B_{r^{*}}(0,Y)$ into $B_{r^{*}}(0,Y)$. Consider the decomposition
$ \Gamma=  \bar{\Gamma}_1 + \bar{\Gamma}_2$ where $
\bar{\Gamma}_{2} (u, v)  = (\bar{\Gamma}_{2}^{1} (u,
v),\bar{\Gamma}_{2}^{2} (u, v))$ and
\begin{align*}
\bar{\Gamma}_{2}^{1}(u, v)(t)&= S(t)g(0,u(0),v(0))- \int_0^t
C(t-s)g(s,u(s),v(s))ds \\
&\quad + \int_0^t S(t-s)f(s,u(s),v(s))ds, \\
\bar{\Gamma}_{2}^{2}(u, v)(t)
& = C(t)g(0,u(0),v(0)) -g(t,u(t),v(t)) \\
&\quad - \int_0^t AS(t-s)g(s,u(s),v(s))ds  +
\int_0^tC(t-s)f(s,u(s),v(s))ds.
\end{align*}
 Simple calculus  using
the properties of $p(\cdot)$ and $q(\cdot)$ proves that
\begin{eqnarray}
\| \bar{\Gamma}_{1}( u,v) - \bar{\Gamma}_{1} ( w,z)\|_{a}& \leq
& \left( (N+\rho)l_p + (N+\tilde{N})l_q \right) \| (
u,v)-(w,z)\|_{a},
\end{eqnarray}
and so that   $\bar{\Gamma}_{1}(\cdot) $ is a contraction on $Y.$

 On the other hand, from Lemma
\ref{lema4} and the properties of $f(\cdot)$ and $g(\cdot)$, it's
easy to infer that  $\bar{\Gamma}_{2}(\cdot)$ is completely
continuous on $Y$.
 From the previous remark, it follows that
  $\Gamma(\cdot) $ is a  condensing operator
 from $B_{r^{*}}(0,Y)$ into $B_{r^{*}}(0,Y)$. The
assertion is now a consequence of  \cite[Corollary 4.3.2 ]{Ma}.
\end{proof}

 Proceeding  as in the proof of Theorem \ref{teo4} we can
prove the next existence result.

\begin{proposition} \label{prop3.9}
Let $(y_{0},y_{1}) \in E \times X$ and conditions  (H3), (H4) be
satisfied.    Suppose that $f(\cdot)$ satisfies condition
${\bf(a)}$ of Theorem \ref{teo4} and that there exists a constant
$l_g \geq 0 $ such that
\begin{eqnarray}
\| g(t,x_1,z_1) - g(t,x_2,z_2)\|_{E} &\leq & l_g ( \| x_1 - x_2
\|  + \| z_1 - z_2\| ),
\end{eqnarray}
for every $ t\in I$ and every $ x_{i},z_{i}\in X$.
 If
$$
 (N+\rho)l_p + ( N+ \tilde{N})l_q+ l_g((N+\rho)
a+\tilde{N}+N+1)+ ( N+ \tilde{N})\liminf_{r \to \infty}
\frac{\alpha_r^f}{r})   <1, $$ then there exists a mild solution
of \eqref{ne2}-\eqref{ne22}.
\end{proposition}


\begin{theorem}\label{teo3}
Assume  (H3), (H4),  $(y_{0},y_{1}) \in E \times X$ and that there
exist constants $l_f,l_g$ such that
\begin{gather*}
\| f(t,x_1,z_1) - f(t,x_2,z_2)\|  \leq
l_f( \| x_1 - x_2 \|  + \| z_1 - z_2\|  ), \\
\| g(t,x_1,z_1) - g(t,x_2,z_2)\|_{E} \leq  l_g ( \| x_1 - x_2
\|  + \| z_1 - z_2\| ),
\end{gather*}
for every  $x_i,z_i \in X$. \par   If  $\max\{ N(l_p+al_g)+
\tilde{N}(l_q + l_g+a l_f),\,\,
   N( l_q + l_g+al_f ) + \rho  ( l_p +al_g)+l_g\} <1,$  then there
exists a unique mild solution of \eqref{ne2}-\eqref{ne22}.
\end{theorem}

\begin{proof} Let  $\Gamma(\cdot) $ be the map defined in the
proof of Theorem \ref{teo4}. It's clear that  $\Gamma(\cdot) $ is
well defined and continuous. Moreover,   for $u_i,v_i \in C(I;X)$
$$
\| \Gamma_1 (u_1,v_1) - \Gamma_1 (u_2,v_2) \|_{a} \leq
 [ N(l_p+al_g)+ \tilde{N}(l_q + l_g+a l_f)]\| (u_1,v_1) -
(u_2,v_2)\|_{a}
$$
and
\begin{align*}
&\| \Gamma_2 (u_1,v_1) - \Gamma_2 (u_2,v_2)\|_{a} \\
&\leq  \| AS(t)\| _{_{\mathcal{L}(E;X) }}\| p(u_1,v_1) -
p(u_2,v_2)\|_{E} \\
&\quad +  (N(l_q + l_g ) + l_g + a N l_f )\| (u_1,v_1) - (u_2,v_2)\|_{a}\\
&\quad +\int_0^t \| AS(t-s)\| _{_{\mathcal{L}(E;X) }}
 \| g(s,u_1(s),v_1(s)) - g(s,u_2(s),v_2(s)))\| _{E}ds \\
&\leq  \left( \rho l_p  + N( l_q + l_g ) + l_g +aN l_f
+a\rho l_g\right)\|  ( u_1,v_1) - (u_2,v_2)\|_{a}\\
&\leq  ( N( l_q + l_g+al_f ) + \rho  ( l_p +al_g)+l_g)
  \|  ( u_1,v_1) - (u_2,v_2)\|_{a},
\end{align*}
which  implies  that $\Gamma$ is a contraction. The statement of the
theorem is now a consequence of   the contraction mapping principle.
\end{proof}

\section{Classical Solutions}\label{regularity}

In this section we establish the existence of classical solutions for
 \eqref{ne2}-\eqref{ne22}.  First, we
introduce some definitions,   notation and preliminary results.

\begin{definition} \label{def4.1} \rm
 A  function $u\in C^{2}(I;X)$ is a classical
 solution  of  \eqref{ne2}-\eqref{ne22}, if
 the mapping  $ t\to u(t)+ \int_{0}^{t}g(s,u(s),u'(s))ds$ is in $C^{2}(I:X)$,
     $u(t)\in D(A) $ for every $t\in I$, and  \eqref{ne2}-\eqref{ne22}
     are satisfied.
\end{definition}

 In the next  pages, we use the assumption
\begin{itemize}
  \item[(H5)] The function $g(\cdot)$ is
 $[ D(A)]$-valued  and $g:I\times X\times X\to [D(A)] $ is continuous.
\end{itemize}

The remark below  is a consequence of  our preliminary results.

\begin{remark}\label{remark2} \rm
 If  $u(\cdot)$ is  a mild solution of \eqref{ne2}-\eqref{ne22},
$\varphi(0) \in E$ and  the function  $s\to g(s,u(s),u'(s))$ is
continuous from $I$ into  $E$, then $u \in C^{1}$ and
\begin{align*}
 u'(t)  &= AS(t)(y_{0} + p(u,u'))+ C(t)( y_{1} + q(u,
u')+ g(0,u(0),u'(0))) \\
& \quad - g(t,u(t), u'(t))
 -\int_{0}^{t} A S(t-s) g(s,u(s),u'(s)) ds \\
&\quad   + \int_{0}^{t} C(t-s) f(s,u(s),u'(s)) ds. \label{HH4}
\end{align*}
\end{remark}

\begin{lemma}\label{teo7}
Let $u(\cdot)$ be a mild  solution of \eqref{ne2}-\eqref{ne22} and
assume
  that (H5) holds. If  $y_{0}+ p(u,u')\in  D(A)$,
     $y_{1}+q(u,u')\in  E$,  $f(\cdot)$ is Lipschitz continuous on
     bounded subsets
   of $I\times X\times X$ and there exist  constants  $l_{g}^{1}>0$,
    $0<l_{g}^{2}<1$ such that
$$
\| g(t,x_1,y_1) - g(s,x_2,y_2)\|_{E} \leq l_{g}^{1} ( | t-s
|+ \| x_1 - x_2 \| )  + l_{g}^{2}\| y_1 - y_2\| \,,
$$
for every $ x_{i},y_{i}\in X$ and  every $t,s\in I$,  then
$u'(\cdot)$ is Lipschitz continuous on $I$.
\end{lemma}

\begin{proof} Let   $t\in I$ and $h\in \mathbb{R}$ be such
that $t+h\in I $. Using Remark \ref{remark2} and the Lipschitz
continuity of $ u(\cdot)$ on  $I$, we obtain
\begin{align*}
&\| u'(t+h)-u'(t) \|  \\
&\leq C_{1}h+ l_{g}^{2}\| u'(t+h)- u'(t)\|
 + \int_{t}^{t+h}\| S(t+h-s)Ag(s,u(s),u'(s))\| ds  \\
&\quad +  \int_{0}^{t}\| (S(t+h-s)- S(t-s))Ag(s,u(s),u'(s))\| ds \\
&\quad + \int_{0}^{h}\| C(t+h-s)f(s,u(s),u'(s))\| ds \\
&\quad  +N\int_{0}^{t}C_{2}[\,  h+\| u(s+h)- u(s)\| +
\| u'(s+h)-u'(s)\|  \, ]  ds \\
&\leq C_{3}h+l_{g}^{2} \| u'(t+h)- u'(t)\|+ NC_{2}\int_{0}^{t}
\int_{0}^{t}\| u'(s+h)- u'(s)\|  ds,
\end{align*}
where the constants   $C_{i}$ are  independent of $t$ and $h$.
Since  $l_{g}^{2}<1$, we can rewrite the last inequality in the
form
\[
\| u'(t+h)-u'(t) \|  \leq  C_{4}h+C_{5}\int_{0}^{t}\| u'(s+h)-
u'(s)\| ds,
\]
where  $C_{4},C_{5}$ are  independent of $t$ and $h$. This proves
that $u'(\cdot)$ is Lipschitz on $I$. The proof is complete
\end{proof}

 Let $(Z_{i},\|\cdot\|_{i})$,  $i=1,2,3$, be Banach spaces
  and $j(\cdot):I\times Z_{1}\times Z_{2}\to Z_{3}$ be a  differentiable
  function. We will  use the decomposition
\begin{align*}
&j(s,\bar{z}_{1},\bar{z}_{2})-j(t,z_{1},z_{2}) \\
&=(D_{1}j(t,z_{1},z_{2}),D_{2}j(t,z_{1},z_{2}),D_{3}j(t,z_{1},z_{2}))
  (s-t,\bar{z}_{1}-z_{1},\bar{z}_{2}-z_{2}) \\
&\quad + \| (s-t,\bar{z}_{1}-z_{1},\bar{z}_{2}-z_{2})\|_{ Z_{1},
Z_{2}} R_{Z_{1},Z_{2}}^{Z_{3}}(j(t,z_{1},z_{2}),s-t
,\bar{z}_{1}-z_{1},\bar{z}_{2}-z_{2}),
\end{align*}
where
$$ \| R_{Z_{1},Z_{2}}^{Z_{3}}(j(t,z_{1},z_{2}),h
,w_{1},w_{2})\| _{ Z_{3}} \to 0,
$$
when  $\|(h,w_{1},w_{2})\|_{ Z_{1}, Z_{2}}=  | h| +  \| w_{1}
\|_{ Z_{1}} +\| w_{2}\|_{ Z_{2}}  \to 0$. Moreover,  we will
write simply  $R_{Z_{1}}^{Z_{3}}$ and $\|(s,y,w)\|_{ Z_{1}}$
when $Z_{1}=Z_{2}$.

 The proof of the next Lemma will be omitted.

 \begin{lemma}\label{lema5}
Let $(Z_{i},\|\cdot\|_{Z_{i}} )$, $i=1,2,3$, be Banach
spaces, $\Omega_{1}\times \Omega_{2} \subset Z_{1}\times Z_{2}$
open,
  $K\subset \Omega_{1}\times \Omega_{2}$ compact and
   $j:I\times \Omega_{1}\times \Omega_{2}\to Z_{3}$ be a continuously
   differentiable  function.  Then, for  every $ \epsilon> 0$, there exists
 $\delta >0$ such that
 $$
\|  R_{Z_{1},Z_{2}}^{Z_{3}}(j(t,z_{1},z_{2}),s-t
,\bar{z}_{1}-z_{1},\bar{z}_{2}-z_{2})\| _{ Z_{3}} <\varepsilon,
\quad  t,s\in I,\; (z_{1},z_{2}), (\bar{z}_{1},\bar{z}_{2})\in K
$$
when
$ \| (s-t,\bar{z}_{1}-z_{1},\bar{z}_{2}-z_{2})\|_{ Z_{1},
Z_{2}}\leq \delta $.
\end{lemma}

\begin{theorem}\label{teo8}
 Let condition (H5) be satisfied and  $u(\cdot)$ be a  mild solution of
\eqref{ne2}-\eqref{ne22}. Assume that  the  functions
$f:I\times  X^{2}\to X $,  $g:I\times X^{2}\to E $ are  continuously
 differentiable,
 $ (y_{0}+p(u,u'), y_{1}+q(u,u'))\in D(A)\times E$  and that  there
exist  constants  $l_{g}^{1}>0$,  $0<l_{g}^{2}<1$ such that
$$
\| g(t,x_1,y_1) - g(s,x_2,y_2)\|_{E} \leq l_{g}^{1} ( | t-s
|+ \| x_1 - x_2 \| )  + l_{g}^{2}\| y_1 - y_2\| \,,
$$
for every $ x_{i},y_{i}\in X$ and  every $t,s\in I$.  If
 \begin{equation}\label{ine22}
\| D_{3}g(w)\|_{{\mathcal{L}}(X),\,a}+\int_{0}^{a}[\rho\|
D_{3}g(w(s))\|_{{\mathcal{L}}(X;E)}+ \|
D_{3}f(w(s))\|_{{\mathcal{L}}(X)}]ds <1,
 \end{equation}
  where $w(t)=(t,u(t),u'(t))$,  then $u(\cdot)$ is  a
classical  solution.
\end{theorem}

\begin{proof} First, we prove that $u(\cdot)$ is of class
$C^{2}$ on $I$ and for this purpose we introduce the integral
equation
\begin{equation}\label{auxequa}
\begin{aligned}
 v(t)&=P(t) -D_{3} g(w(t))(v(t))
-\int_{0}^{t}AS(t-s)D_{3}g(w(s))(v(s))ds \\
&\quad +\int_{0}^{t}C(t-s)D_{3}f(w(s))(
v(s))ds, \quad  t\in I,
\end{aligned}
\end{equation}
where
\begin{align*}
P(t)&=C(t)A u(0)+AS(t)u'(0) -D_{1}g(w(t))-D_{2}g(w(t))(u'(t)) \\
 &\quad -\int_{0}^{t}AS(t-s)[ D_{1}g(w(s)) +D_{2}g(w(s))(u'(s))]
 ds+C(t)\tilde{f}(0) \\
 &\quad + \int_{0}^{t}C(t-s)\left(D_{1}f(w(s))
 +D_{2}f(w(s))( u'(s))\right) ds.
\end{align*}
  The existence and uniqueness of solutions of the integral equation
(\ref{auxequa})  is  consequence of the contraction mapping principle
 and (\ref{ine22}), we omit  additional details.
  Let $v(\cdot)$ be the  solution
   (\ref{auxequa}) and let $t\in I$, $h\in\mathbb{R}$ be such that
 $t+h\in I$.  By using the relation
 $ A\int_{r}^{s}S(\theta)x=C(s)x-C(r)x$,  the notation
  $\zeta_{h}(t)=\partial_{h}u'(t)-v(t)$, $\tilde{f}=f(w(t))$,
  $\tilde{g}=g(w(t))$ and
\begin{gather*}
\Lambda g(t)= D_{1} g(w(t))+D_{2} g(w(t))( u'(t))+D_{3} g(w(t))(v(t)),\\
\Lambda f(t)= D_{1} f(w(t))+D_{2} f(w(t))( u'(t))+D_{3} f(w(t))( v(t)),
\end{gather*}
 we find that
\begin{align*}
&\| \zeta_{h}(t)\| \\
&\leq \xi_{1}(h,t) + \| [\partial_{h}C(t)]\tilde{g}(0)-
\frac{1}{h}\int_{0}^{h}AS(t+h-s)\tilde{g}(s) ds\|  +\| \Lambda
g(t)-\partial_{h}\tilde{g}(t)\|  \\
&\quad +\rho\int_{0}^{t}\| \Lambda
g(s)-\partial_{h}\tilde{g}(s)\|_{E}ds
 +    \| \frac{1}{h}\int_{0}^{h}C(t+h-s)\tilde{f}(s)ds
-C(t)\tilde{f}(0) \| \\
&\quad +  N\int_{0}^{t}\| \partial_{h}\tilde{f}(s)-\Lambda f(s)\| ds \\
&\leq \xi_{2}(h,t)+
\frac{1}{h}\int_{0}^{h}\| S(t+h-s)(A\tilde{g}(0)-A
\tilde{g}(s)\| ds +\| D_{3}g(w(t))\|_{{\mathcal{L}}(X)}\| \zeta_{h}(t)\|\\
&\quad + \| (1,\partial_{h} u(t),\partial_{h}u'(t) )\|_{ X}
\| R_{X}^{X}(\tilde{g}(t),h,h\partial_{h} u(t),h\partial_{h} u'(t))\|\\
&\quad + \int_{0}^{t}\left[ \rho\| D_{3}g(w(s))\|_{{\mathcal{L}}(X;E)}+
N\| D_{3}f(w(s))\|_{{\mathcal{L}}(X)}\right]\| \zeta_{h}(s)\| ds\\
&\quad +\rho\int_{0}^{t}\| (1,\partial_{h} u(s),\partial_{h} u'(s) )
\|_{ X} \| R_{X}^{E}(\tilde{g}(s),h,h\partial_{h} u(s),h
\partial_{h} u'(s) )\|_{E} ds\\
&\quad +N\int_{0}^{t}\| (1,\partial_{h} u(s),\partial_{h} u'(s)
)\|_{ X} \| R_{X}^{X}(\tilde{f}(s),h,h\partial_{h}
u(s),h\partial_{h} u'(s))\| ds,
\end{align*}
where $\xi_{i}(h,t)\to 0$, $i=1,2$,  as $h\to 0$. Since $\mu=1-\|
D_{3}g(w(\cdot))\|_{{\mathcal{L}}(X),a}>0$, we obtain
 \begin{align*}
 \| \zeta_{h}(t)\|
&\leq \xi_{3}(h,t)+ \frac{1}{\mu}\| (1,\partial_{h} u(t),
 \partial_{h}u'(t) )\|_{ X}
\| R_{X}^{X}(\tilde{g}(t),h,h\partial_{h} u(t),h\partial_{h} u'(t))\| \\
&\quad  + \frac{1}{\mu} \int_{0}^{t}\left[\rho \|
D_{3}g(w(s))\|_{{\mathcal{L}}(X;E)}+ N\|
D_{3}f(w(s))\|_{{\mathcal{L}}(X)}\right]\| \zeta_{h}(s)\| ds\\
&\quad  + \frac{\rho}{\mu}
 \int_{0}^{t}\| (1,\partial_{h} u(s),\partial_{h} u'(s) )
 \|_{ X} \| R_{X}^{E}(\tilde{g}(s),h,h\partial_{h}u(s),h\partial_{h}u'(s))\|_{E} ds\\
&\quad + \frac{N}{\mu} \int_{0}^{t}\| (1,\partial_{h}
u(s),\partial_{h} u'(s) )\|_{ X} \|
R_{X}^{X}(\tilde{f}(s),h,h\partial_{h}u(s),h\partial_{h}u'(s))\| ds
\end{align*}
where $\xi_{3}(h,t)\to 0$  as $h\to 0$. This inequality, jointly
with  the Lipschitz continuity of $u(\cdot)$ and  $u'(\cdot)$, see
Lemma \ref{teo7},  the Gronwall Bellman inequality and  Lemma
\ref{lema5},  permit to conclude that $u'' (\cdot)$ exists and
that $u''(\cdot)=v(\cdot)$ on $ I$.
\par From  \cite[Proposition 2.4]{TW3},
we know that the mild solution, $y(\cdot)$, of the abstract Cauchy
problem
\begin{equation}\label{last}
\begin{gathered}
 x''(t)  =  Ax(t) +f(t,u(t),u'(t))-  A\int_{0}^{t}g(s,u(s),u'(s))ds,
\quad t\in I, \\
 x(0)= y_{0}+p(u,u') \quad  x' (0) =  y_{1}+q(u,u')+g(0,u(0),u'(0)),
\end{gathered}
\end{equation}
is a classical solution (see Definition \ref{classical}). The
uniqueness of solution of (\ref{last}) and Remark   \ref{remark2},
permit to conclude that $y(t)=u(t)+\int_{0}^{t}g(s,u(s),u'(s))ds$
is a function of class $C^{2}$ on $I$  and that  $u(t)\in D(A)$
for every $t\in I$ since  $ g(\cdot)$ is $[D(A)]$-valued
continuous. This completes the  proof that $u(\cdot)$ is a
classical solution.
\end{proof}

\section{Applications}\label{Examples}

 In this section we apply  some of the results
established in this paper. First, we introduce the required
technical framework.
 On the space $X = L^{2}([0, \pi])$ we
consider the operator $ A f(\xi) = f''(\xi) $ with domain
$ D(A)= \{ f(\cdot) \in H^{2}(0,\pi) : f(0) = f(\pi) = 0 \}$.
 It is well known that $A$ is the infinitesimal generator of a strongly
continuous  cosine function, $(C(t))_{t\in\mathbb{R}}$, on $X$.
Furthermore, $A$ has discrete spectrum, the eigenvalues are
$-n^{2}$, $n \in \mathbb{N}$, with corresponding normalized
eigenvectors $z_{n} (\xi) := (\frac{2}{\pi})^{1/2} \sin(n \xi)$ and
\begin{itemize}

\item[(a)] $\{z_{n} : n \in \mathbb{N}\}$ is an
orthonormal basis of  $X$.

\item[(b)] If $\varphi \in D(A)$ then
$A \varphi =  - { \sum_{n=1}^{\infty} n^{2} \langle \varphi, z_{n}\rangle
z_{n}}$.

 \item[(c)] For $\varphi \in X$,
$C(t)\varphi = { \sum_{n=1}^{\infty} \cos(nt) \langle \varphi, z_{n}\rangle
z_{n}}$. It follows from this expression that   $S(t) \varphi
=\sum_{n=1}^{\infty} \frac{\sin(nt)}{n} \langle\varphi,z_{n}\rangle z_{n}$  for
every $\varphi \in \mathcal{B}$. Moreover,    $S(t)$ is a  compact
operator and $\|C(t)\| =\|S(t)\|= 1$ for every $ t \in \mathbb{R}$.

\item[(d)]  If $\Phi$ is the group of translations on $X$ defined
by $\Phi(t)x(\xi)=\tilde{x}(\xi+t)$, where $\tilde{x}(\cdot) $ is
the extension of $x(\cdot)$ with period $2\pi$, then
$C(t)=\frac{1}{2}(\Phi(t)+\Phi(-t))$ and    $A=B^{2}$, where $B$
is the infinitesimal generator of $\Phi$ and $ E=\{ x \in
H^{1}(0,\pi) : x(0) = x(\pi) = 0 \}$, see \cite{Fa} for details.
\end{itemize}

 First, we  consider the partial second-order differential equation
 with nonlocal conditions
\begin{gather}
\begin{gathered}
\frac{ \partial } {\partial t}[\frac{ \partial u( t,\xi)}
{\partial t} + G(t,\xi, u(t,\xi))] = \frac{\partial^{2}
u(t,\xi)} {\partial \xi^{2}} +F(t,\xi, u(t,\xi)),\\
\xi\in J=[0,\pi],\,\,t\in I=[0,a],
\end{gathered} \label{eqe1}\\
\label{eqe2}
u(t,0 )=u(t,\pi )=0,\quad  t\in I, \\
\label{eqe33} u(0,\xi )  =  y_{0}(\xi)+
\sum_{i=1}^{n}\alpha_{i}u(t_{i}, \xi ),\quad \xi \in J, \\
\label{eqe34}
\frac{ \partial u(0,\xi )}{\partial t} = y_{1}(\xi)+\sum_{i=1}^{k}
\beta_{i}u(s_{i},\xi), \quad  \xi \in J,
\end{gather}
where  $0<t_{i},s_{j}<a$, $\alpha_{i},\beta_{j}\in \mathbb{R}$ are
fixed numbers,  $y_{0},y_{1}\in X $ and  the functions
    $G, F:I\times J\times \mathbb{R} \to \mathbb{R}$
satisfy   the following conditions:
  \begin{itemize}
\item[(i)] $F(\cdot)$ is continuous and there exist functions
$\eta^{F}_{1},\eta^{F}_{2}\in C(I\times J:\mathbb{R}^{+})$ such
that
$$
| F(t,\xi,w)| \leq \eta^{F}_{1}(t, \xi )+
\eta^{F}_{2}(t,\xi)| w|, \quad t\in I,\xi\in J, w\in
\mathbb{R}.
$$
 \item[(ii)] $G(\cdot)$ is  continuous  and there
exists  $ \eta ^{G}\in C(I\times J;\mathbb{R}^{+})$ such that
\begin{align*}
\mid  G(t,\xi,x_{1})-G(t,\xi,x_{2})\mid  \leq & \eta^{G}(t,\xi)
\mid x_{1}- x_{2}\mid ,
\end{align*}
for every $(t,\xi)\in I\times J$ and every $x_{1},x_{2}\in \mathbb{R}$.
\end{itemize}

By defining  the functions $f,g: I\times X
\to X$ and $p,q: C(I;X) \to X$ by
$g(t,x)(\xi)=G(t,\xi, x(\xi) )$, $f(t,x)(\xi)=F(t,\xi,x(\xi))$, $
p(u)(\xi)= \sum_{i=1}^{n}\alpha_{i}u(t_{i},\xi )$ and $q(u)(\xi)=
\sum_{i=1}^{k}\beta_{i}u(s_{i},\xi)$, the system
\eqref{eqe1}-\eqref{eqe34} can be described as
  the abstract Cauchy problem with nonlocal conditions
  \eqref{mild1}-\eqref{P4}.
It is easy to see that $f(\cdot)$, $g(\cdot)$, $ p(\cdot)$,
$q(\cdot)$ satisfies the
    assumption of Proposition  \ref{prop1} and that
     $l_{g}=\sup_{(s,\xi)\in I\times J}\eta^{G}(s,\xi)$,
       $l_{p}=\sum_{i=1}^{n}| \alpha_{i}|  $,
    $l_{q}=\sum_{i=1}^{k}| \beta_{i}| $
     and
\[
\alpha^{f}_{r}=\sup\Big\{ \Big( \int_{0}^{\pi}
\eta^{F}_{1}(t,\xi)^{2} d \xi \Big)^{1/2}  +r
\eta^{F}_{2}(t,\cdot)_{\pi}\,:\,\,t\in I\Big\}.
 \]
The next result  is a consequence of Proposition \ref{prop1}.

\begin{theorem}\label{teo5}
Assume that (i) and  (ii) are  satisfied.  If
$$
\sum_{i=1}^{n}| \alpha_{i}| +\sum_{i=1}^{k}|
\beta_{i}| +(1+a)\sup_{(s,\xi)\in I\times J}\eta^{G}(s,\xi)+
a\sup_{s\in I}\eta^{F}_{2}(s,\cdot)_{\pi}ds
    <1,$$
     then there exists a mild solution of   \eqref{mild1}-\eqref{P4}.
\end{theorem}

Now, we consider briefly the partial differential equation
\begin{equation}\label{exemplo2}
\frac{ \partial } {\partial t}[\frac{ \partial u( t,\xi)}
{\partial t} + \int_{0}^{\pi} b(t,\eta,\xi)u(t,\eta)d \eta]
=\frac{\partial^{2} u(t,\xi)} {\partial \xi^{2}} +F(t,\xi,
u(t,\xi)),
\end{equation}
for $\xi\in J,\,t\in I$,  submitted to the conditions
(\ref{eqe2})-(\ref{eqe34}). To study this system we introduce the
next condition.
 \begin{itemize}
 \item[(iii)]  The functions   $  b(s,\eta, \xi)$,
 $\displaystyle { \frac{\partial^{i} b(s, \eta, \xi)}{\partial \xi^{i}}}$, $i=1,2$,
are continuous  on  $ \mathbb{R}^{3}$ and $b(\cdot, \pi) = b(\cdot, 0) = 0$
 on  $ I\times J$.
\end{itemize}
Let  $f(\cdot),p(\cdot),q(\cdot)$ defined as before and
$g(\cdot):I\times X\to X$ be the function defined by $g(t,x)(\xi)=
\int_{0}^{\pi} b(t,\eta,\xi) x(\eta)d\eta$. From the properties of
  $b(\cdot)$, we infer  that  $g(t,\cdot)$ is a $D(A)$-valued
  linear operator and that
$$
\sup\{\|
g(t,\cdot) \|,\|  g(t,\cdot) \|_{E},\| A g(t,\cdot)
\|_{{\mathcal{L}}(X)}:\, t\in I\}\leq
\alpha^{1/2},
$$
where
$$
\alpha :=\sup_{t\in [0,a]}\big\{\int_{0}^{\pi} \int_{0}^{\pi}
b(t,\eta, \xi)^{2}d \eta d\xi,\, \int_{0}^{\pi}
\int_{0}^{\pi}\big(\frac{\partial^{j}b(t,\eta,\xi)}{\partial
\xi^{j}}\big)^{2}d\eta d\xi :\,j=1,2 \big\}.
$$
Moreover, $ g(\cdot)$ is completely continuous since the inclusion
$i_{c}:[D(A)]\to X$ is compact.

 In the next result, the existence of a mild solution can be deduced
from  Theorem  \ref{teo2} or from Proposition \ref{prop1}.

\begin{theorem}\label{teo9}
Assume (i) and  (iii) be satisfied and that
$$
\sum_{i=1}^{n}| \alpha_{i}| +\sum_{i=1}^{k}| \beta_{i}|
+(1+a)\alpha^{\frac{1}{2}} + a\sup_{s\in
I}\eta^{F}_{2}(s,\cdot)_{\pi}ds<1.
$$
Then the partial  differential equation (\ref{exemplo2}) submitted to
      the conditions
(\ref{eqe2})-(\ref{eqe34}) has a mild solution.
\end{theorem}


 To finish this section,  we consider the  differential system
\begin{equation}\label{31}
\frac{ \partial } {\partial t}
\Big[\frac{ \partial u( t,\xi)}
{\partial t} + \int_{0}^{\pi} b(t,\eta,\xi)\frac{
\partial u( t,\eta)} {\partial t} d \eta\Big]
= \frac{\partial^{2}u(t,\xi)} {\partial \xi^{2}}
+F(t,u(t,\xi),\frac{\partial u} {\partial t}( t,\xi)),
\end{equation}
 for $\xi\in J$, $t\in I$,  subject to the conditions:
\begin{gather} \label{32}
u(t,0 )=u(t,\pi )=0,\quad  t\in I,\\
\label{33} u(0,\xi )=y_{0}(\xi)+\int_{0}^{a}P(u(s),\frac{
\partial u} {\partial t}(s))(\xi)d\mu(s),\\
\label{34}
 \frac{\partial u} {\partial t}(0,\xi )=
y_{1}(\xi)+\sum_{i=1}^{n}\alpha_{i}u(t_{i}, \xi )+
 \sum_{i=1}^{k}\beta_{i}\frac{\partial u} {\partial t}(s_{i},\xi),
\end{gather}
where $ \alpha_{i},\beta_{i}\in \mathbb{R}$,  $0<t_{i},s_{j}<a$
are prefixed numbers,  $\mu(\cdot)$ is a real function of bounded
variation on $I$ and  $ F:I\times J\times \mathbb{R}^{2} \to
\mathbb{R}$, $P:X\times X\to X $ satisfies the next conditions.
\begin{itemize}
\item[(iv)] $F(\cdot)$ is continuous and there exists  a constant
$L_{F} $ such that
$$
| F(t,x_{1}, w_{1})-F(s,x_{2}, w_{2})| \leq L_{F}\left(|
t-s|+| x_{1}- x_{2}|+ | w_{1}- w_{2}|\right),
$$
for  every $t,s\in I$ and every $x_{i},w_{i}\in \mathbb{R}$;

\item[(v)] $P$  is $E$-valued   and there
exist $l_{P}$ such that
$$
\| P(x_{1},w_{1})-P(x_{2},w_{2})\|_{E}\leq l_{P}(\|
x_{1}-x_{2}\|+\| w_{1}-w_{2}\| ), \quad
 x_{i},w_{i}\in X .
$$
(for examples of operators satisfying (v),  see \cite{Ma}).
\end{itemize}

 By defining the operators $f,g:I\times X\times X\to X$ and
$p,q:C(I;X)\times C(I;X) \to X$ by
\begin{gather*}
f(t,x,y)(\xi)= F(t,x(\xi),y(\xi)),\\
g(t,x,y)(\xi)=   \int_{0}^{\pi} b(t,\eta,\xi) y(\eta) d\eta,
  \quad x,y\in X, \\
p(u,v)(\xi)=\int_{0}^{\pi}P(u(s),v(s) )(\xi)d\mu (s),\quad u,v\in
C(I;X),\\
 q(u,v)(\xi)= \sum_{i=1}^{n}\alpha_{i}u(t_{i}, \xi )+
 \sum_{i=1}^{k}\beta_{i}v(s_{i},\xi),
 \quad u,v\in C(I;X),
\end{gather*}
we can model (\ref{31})-(\ref{34}) as
  the abstract Cauchy problem  \eqref{ne2}-\eqref{ne22}. As in the previous
  example,  $g(\cdot)$ is $[D(A)]$-valued continuous and
   $\| A g(t,\cdot) \|_{{\mathcal{L}}(X)}\leq
\alpha^{\frac{1}{2}}$ for every $t\in I$. Moreover,  the assumptions
of Theorem \ref{teo3} are satisfied with,  $l_{p}= l_{P}V(\mu) $,
where $V(\mu)$ is the variation of $\mu $,
 $ l_{q}= \sum_{i=1}^{n}|\alpha_{i}|+\sum_{i=1}^{k}|\beta_{i}|$,\,
  $l_{f}=L_{F}$,  $l_{g}=\alpha^{\frac{1}{2}}$  and $\rho=1$.
 The next result is a consequence of  Theorems
\ref{teo3}.

\begin{theorem}
 Assume conditions  (iii)-(v) are satisfied and
\[
 l_{P}V(\mu)+\sum_{i=1}^{n} |
\alpha_{i}|+\sum_{i=1}^{k}|
\beta_{i}|+3\alpha^{\frac{1}{2}}+L_{F}<1.
\]
 Then there exists a unique mild solution, $u(\cdot)$, of
(\ref{31})-(\ref{34}).
\end{theorem}

\subsection*{Acknowledgement} Mauricio L. Pelicer wishes to
acknowledge the  support of Capes Brazil, for this research.

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\end{document}