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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 28, 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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/28\hfil Existence results]
{Existence results for impulsive evolution differential
 equations with state-dependent delay}

\author[E. Hern\'{a}ndez,   R. Sakthivel, S.  Tanaka,\hfil EJDE-2008/28\hfilneg]
{Eduardo Hern\'{a}ndez,  Rathinasamy Sakthivel, Sueli Tanaka Aki}  % in alphabetical order

\address{Eduardo Hern\'{a}ndez, Sueli Tanaka Aki \newline
Departamento de Matem\'atica, I.C.M.C. Universidade de S\~ao Paulo, 
Caixa Postal 668, 13560-970, S\~ao Carlos SP, Brazil}
\email{lalohm@icmc.sc.usp.br}

\address{Rathinasamy Sakthivel \newline
Department of Mechanical Engineering\\
Pohang University of Science and Technology\\
Pohang- 790-784, South Korea}
\email{krsakthivel@yahoo.com}

\address{Sueli Tanaka Aki \newline
Departamento de Matem\'atica, I.C.M.C. Universidade de S\~ao Paulo, 
Caixa Postal 668, 13560-970, S\~ao Carlos SP, Brazil}
\email{smtanaka@icmc.sc.usp.br}


\thanks{Submitted November 26, 2007. Published February 28, 2008.}
\subjclass[2000]{35R10, 34K05}
\keywords{State-dependent delay; abstract Cauchy problem; \hfill\break\indent
 partial functional-differential equations; evolution operators}

\begin{abstract}
 We study  the existence of mild solution for impulsive evolution
 abstract differential equations with  state-dependent delay.
 A concrete application to partial delayed differential equations
 is considered.
\end{abstract}

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

\section{Introduction}

In this work we discuss the existence of mild solutions for impulsive
functional differential equations, with state-dependent delay, of the form
\begin{gather}\label{1}
 x' (t)= A(t)x(t)+f(t,x_{\rho (t,x_{t})}), \quad t\in I=[0,a],\\
\label{2} x_0 = \varphi \in \mathcal{B},\\
\Delta x(t_i) = I_i(x_{t_i}), \quad i=1,2,\dots,n, \label{3}
\end{gather}
where  $A(t): \mathcal{D}\subset X\to X$, $t\in I$, is a family of
closed linear operators defined on a common domain $\mathcal{D}$
which is  dense in a   Banach space $(X,\|\cdot\|)$;  the function
$x_{s}:(-\infty,0]\to X$, $x_{s}(\theta)=x(s+\theta)$,
belongs to some abstract phase space $\mathcal{B}$ described
axiomatically; $f:I\times \mathcal{B}\to X$, $\rho:I\times
\mathcal{B}\to (-\infty,a]$,   $ I_i:\mathcal{B} \to
 X$, $ i=1,2,\dots,n$, are appropriate functions;   $0<t_{1}<\dots.\, t_{n}<a $ are prefixed points
 and the symbol  $\Delta\xi(t)$ represents the jump of the function $
 \xi$ at $ t$, which is defined by $\Delta\xi(t)=\xi(t^+)-\xi(t^-)$.


   Various evolutionary processes from fields as diverse as physics,
 population dynamics, aeronautics, economics and engineering are
 characterized by the fact that they undergo abrupt changes of state at certain
 moments of time between intervals of continuous evolution. Because the
 duration of these changes are often negligible compared to the total
 duration of the process, such changes can be reasonably well-approximated
 as being instantaneous changes of state, or in the form of impulses.
 These process tend to more suitably modeled by impulsive differential
 equations, which allow for discontinuities in the evolution of the state.
 For more details on this theory and on its applications we refer to
  the monographs of  Lakshmikantham et al. \cite{l1}, and
 Samoilenko and Perestyuk \cite{s1} for the case of ordinary impulsive
 system and \cite{liu1, Rogo1, Rogo2, Hernandez1, Hernandez2} for
 partial differential and partial functional differential  equations  with impulses.

On the other hand, functional differential equations with
state-dependent delay appear frequently in applications as model of
equations and for this reason the study of this type of equations
has received great attention in the last years. There exists
 a extensive literature for
ordinary state-dependent delay equations, see among another works,
\cite{Arino1,Aiello1,Cao1,Domoshnitsky1,Hartung1,Hartung6,Hartung7}.
 The study of partial differential
equations with state dependent  delay have been initiated recently,
and
 concerning this matter we cite the pioneer works    Rezounenko et
 al.  \cite{wu1}, Hern\'{a}ndez el al. \cite{h3} and the papers
 \cite{h2,h4,h5,h1,Alex1}.


  To the best of our knowledge,  the study of
the existence of solutions for  systems described in the
abstract form \eqref{1}--\eqref{2} is a untreated problem, and this
fact,  is the main motivation of this  paper.


Throughout this paper,   $(X, \|\cdot\|) $  is a  Banach space,
 $\{A(t): t\in \mathbb{R} \}$ is a family of closed linear
operators  defined on a common domain $\mathcal{D}$ which  is
 dense in $X$, and we  assume  that the linear non-autonomous system
 \begin{equation} \label{eq2}
\begin{gathered}
  u'(t)=  A(t)u(t),\quad   s\leq t\leq a,   \\
  u(s) =x\in X,
 \end{gathered}
\end{equation}
has an associated evolution family of operators  $\{U(t,s): a\geq t\geq
s\geq 0\}$. In the next definition,   $\mathcal{L}(X)$ is the
space  of bounded linear operator from $X$ into $X$ endowed with the uniform
 convergence topology.

\begin{definition}  \label{def1.1} \rm
A family of linear operators   $\{U(t,s): a\geq t\geq
s\geq 0\} \subset \mathcal{L}(X)$   is called an evolution family of
operators for (\ref{eq2}) if the following conditions hold:
\begin{itemize}
\item[(a)] $U(t,s)U(s,r)=U(t,r)$  and   $U(r,r)x=x$ for every
 $ r \leq s \leq  t $ and all $x\in X$;
\item[(b)] For each $x \in X$ the function $(t,s)\to
U(t,s)x$ is continuous and  $U(t,s)\in \mathcal{L}(X)$ for every $t\geq
 s$; and
\item[(c)] For $ s \leq t\leq a  $,  the   function
$(s,t]\to \mathcal{L}(X) $,  $t\to  U(t,s)$
 is differentiable with $\frac{\partial }{\partial t}U(t,s)= A(t)U(t,s)$.
\end{itemize}
\end{definition}

In the sequel,   $\widetilde{M}$ is a  positive constant such that
 $\| U(t,s)\|\leq \widetilde{M}$ for every $t\geq s$, and
 we always assume that $U(t,s)$ is a compact operator for
 every $t>s$. We refer the reader to \cite{PA} for additional details
 on evolution  operator families.

 To consider the impulsive condition (1.3),  it is
convenient to introduce some additional concepts and notations. We
say that a function $u:[\sigma, \tau] \to X$  is a normalized piecewise
continuous function on $[\sigma, \tau]$ if $u$  is  piecewise
continuous and left continuous on $(\sigma, \tau]$. We denote by
${\mathcal{P}\mathcal{C}}([\sigma, \tau];X)$ the space formed by the
normalized piecewise continuous functions from $[\sigma, \tau]$ into
$X$. In particular, we  introduce the space
${\mathcal{P}\mathcal{C}}$ formed by all functions $u :[0, a] \to X$
such that $u$ is continuous at $t \neq t_{i}, u(t_{i}^{-})= u(t_{i})
$ and $u(t_{i}^{+}) $ exists, for all $i = 1, \dots, n$. In this
paper we always assume that ${\mathcal{P}\mathcal{C}}$ is endowed
with the norm $\|u\|_{{\mathcal{P}\mathcal{C}}} =\sup_{s \in I}
\|u(s)\|$.  It is clear that $({\mathcal{P}\mathcal{C}},\|\cdot
\|_{{\mathcal{P}\mathcal{C}}} ) $ is  a Banach space.

  To simplify the notations,   we  put $t_{0}
= 0,\; t_{n + 1} = a$ and for $u\in {\mathcal{P} \mathcal{C}} $  we
denote by  $\tilde{u}_{i} \in C([t_{i},t_{i+1}];X),\,i=0, 1, \dots,
n$, the function given by
\begin{equation} \label{ext1}
\widetilde{u}_{i}(t)= \begin{cases}
u(t), & \hbox{for } t\in  (t_{i},t_{i+1}],
\\u(t_{i}^{+}),& \hbox{for } t=t_{i}.
\end{cases}
 \end{equation}
  Moreover,  for  $B\subseteq {\mathcal{P}\mathcal{C}} $ we denote
 by  $\widetilde{B}_{i}$,  $i=0, 1,\dots,n$, the set
$ \widetilde{B}_{i}=\{ \tilde{u}_{i}: u\in B\}$.

\begin{lemma}\label{prop1}
  A set  $B\subseteq  {\mathcal{P}\mathcal{C}}$   is relatively compact
in   ${\mathcal{P}\mathcal{C}}$  if,
and  only if, the set  $\widetilde{B}_{i}$ is  relatively compact in
$C([t_{i},t_{i+1}];X)$,  for every  $i=0, 1,\dots,n$.
\end{lemma}

 In this work we will employ an axiomatic definition for
    the phase space $\mathcal{B}$ which is similar to those introduced  in
\cite{Hino}.  Specifically,   $\mathcal{B}$ will be a
linear space of  functions mapping $(-\infty,0]$ into $X$ endowed
with a seminorm $\| \cdot \|_{\mathcal{B}}$, and satisfies the
following conditions:
 \begin{itemize}
\item[(A)] If $x:(-\infty, \sigma + b]\to X$,
$b>0$, is such that $x|_{[\sigma, \sigma + b]} \in\mathcal{PC}([\sigma,
 \sigma +  b]:X) $ and $x_{\sigma}\in \mathcal{B}$, then for every $t\in [\sigma,
\sigma + b]$ the following  conditions hold:
\begin{itemize}
\item[(i)]$x_{t}$ is in $\mathcal{B}$, \item[(ii)]$\|
x(t)\| \leq H \| x_{t}\|_{\mathcal{B}}$,
\item[(iii)] $\| x_{t}\|_{\mathcal{B}} \leq K(t-\sigma)
\sup\{\| x(s)\|:\sigma\leq s\leq t\}+ M(t-\sigma)\|
x_{\sigma}\|_{\mathcal{B}}$,
\end{itemize}
 where $H>0$ is a constant; $ K,M:[0,\infty) \to [1,\infty)$,
$K$ is continuous, $M$ is locally bounded, and $H,K,M$ are
independent of $x(\cdot)$.

 \item[(B)] The space $\mathcal{B}$ is complete.
\end{itemize}

\begin{example}\rm
{\bf  Phase spaces $ \mathcal{P}{C_{h}}(X),\,
\mathcal{P}\mathcal{C}_{g}^{0}(X)$.}
 As usual, we say  that $ \psi:(-\infty,0]
\to X$ is normalized piecewise continuous, if $ \psi$ is
left continuous and the restriction of $ \psi$ to any interval $
[-r,0]$ is piecewise continuous.


  Let $g:(-\infty,0]\to [1,\infty)$ be a  continuous,
nondecreasing function with $g(0)=1$,  which satisfies the
conditions (g-1), (g-2) of \cite{Hino}.  This  means that
$\lim_{\theta \to -\infty} g(\theta) =\infty$ and that the function
${ \Lambda(t) := {\sup_{-\infty < \theta \leq -t}
\frac{g(t +\theta)}{g( \theta)}}}$ is locally bounded for $t \geq 0$.
 Next, we modify slightly the definition of the spaces $ C_{g},
C_{g}^{0} $ in \cite{Hino}. We denote by $
\mathcal{P}\mathcal{C}_{g}(X)$ the space formed by the normalized
piecewise continuous functions $ \psi$ such that
$\frac{ \psi}{g}  $\, is bounded on $(-\infty,0]$ and by $
\mathcal{P}\mathcal{C}_{g}^{0}(X)$ the subspace of $
\mathcal{P}\mathcal{C}_{g}(X)$ formed by the functions $ \psi$
such that $\frac{ \psi (\theta)}{g(\theta)}\to 0$ as
$\theta \to -\infty$. It is easy to see that $
\mathcal{P}\mathcal{C}_{g}(X)$ and $
\mathcal{P}\mathcal{C}_{g}^{0}(X)$ endowed with the norm $\|
\psi \|_{\mathcal{B}} : = \sup_{\theta \leq 0 }\frac{\|
 \psi (\theta )\|}{g(\theta )}, $ are phase spaces in the sense
considered in this work. Moreover,  in these cases $K\equiv 1$.
\end{example}


\begin{example}\rm {\bf Phase space   $\mathcal{P}\mathcal{C}_{r}
 \times L^{2}(g\,,X)$. }\label{example1}
  Let   $1\leq p<\infty$, $0\leq r<\infty$ and   $ g(\cdot)  $
  be a Borel nonnegative measurable function on $(-\infty ,r)$ which
satisfies the conditions (g-5)-(g-6) in the terminology of
\cite{Hino}.  Briefly, this means that $g(\cdot)$ is  locally integrable
on $(- \infty, -r)$ and that there exists a nonnegative and  locally
bounded function $ \Lambda$ on $(- \infty, 0]$ such that $ g(\xi
+\theta) \leq  \Lambda(\xi) g(\theta)  $
 for
all $ \xi \leq 0 $ and $ \theta \in (- \infty , -r) \setminus N_{\xi
}$, where $ N_{\xi} \subseteq (- \infty, -r)$ is a set with Lebesgue
measure $0$.

  Let  $\mathcal{B}:=\mathcal{P}\mathcal{C}_{r} \times L^{p}(g;X)$, $
r\geq 0,p>1$, be the space formed  of all classes of functions $
 \psi : (- \infty ,
0]\to X  $ such that  $ \psi|_{[-r,0]}\,\in
{\mathcal{P}\mathcal{C}}([-r,0],X) $,\,  $ \psi(\cdot)$ is
Lebesgue-measurable on $(-\infty,-r]$ and  $ g| \psi|^{p}$
is Lebesgue integrable on $ (- \infty ,-r]$. The semi-norm in $
\|\cdot\|_{\mathcal{B}} $ is defined by
$$
\|   \psi \|_{\mathcal{B}} : =
 \sup_{\theta\in [-r,0]}\|  \psi (\theta )\|
  +\Big( \int_{- \infty }^{-r} g(\theta ) \|
 \psi (\theta ) \|^{p}
d \theta \Big)^{1/p}.
$$


   Proceeding as in the proof of \cite[Theorem 1.3.8]{Hino}
it follows that $ \mathcal{B} $ is a phase space which
satisfies  the  axioms $\mathbf{A}$ and $\mathbf{B}$. Moreover,
 for  $r=0$ and $p=2$  this space coincides with $C_{0} \times
L^{2}(g,\,X)$, $H=1$; $M(t) =  \Lambda (-t)^{1/2}$ and
$K(t)=1+ \big(\int_{-t}^{0} g(\tau)d\tau \big)^{1/2}$ for
$t\geq 0$.
\end{example}

\begin{remark} \label{rmk1.5}\rm
 In retarded functional differential equations without impulses,
the axioms  of the abstract phase space $\mathcal{B}$ include the
continuity of the  function $t\to x_{t} $, see for instance
 \cite{Hino}.
  Due to the impulsive effect, this property is not
satisfied  in impulsive delay systems and,  for this reason,  has
been eliminated in our abstract description of $\mathcal{B}$.
\end{remark}


 The terminology  and notations  are those generally used in
functional analysis. In particular, for Banach a space
$(Z,\|\cdot\|_{Z} )$,  the notation  $ B_{r}(x,Z)$   stands   for the
closed ball with center at  $x$ and radius  $r>0$ in  $Z$.

 To prove  some of our  results,   we  use  a fixed point Theorem
which is referred  in the Literature as   Leray Schauder Alternative
Theorem,  see  \cite[Theorem 6.5.4]{GD}.

\begin{theorem}  \label{teo1}  Let $D$ be
  a convex subset of a Banach space
$X$ and assume that $0\in D$. Let $G:D\to D$ be a completely
continuous map. Then the map $G$ has a fixed point in $D$ or the set
$\{x\in D:x = \lambda G(x),\;0<\lambda<1\}$ is unbounded.
\end{theorem}

   In the next  section  we  study  the
 existence  of mild solutions for the abstract system
  \eqref{1}--\eqref{2}. In the  last  section  an application is discussed.

 \section{Existence Results} \label{existence}

  To prove our results on the  existence of mild solutions for the
abstract Cauchy problem \eqref{1}--\eqref{2}, we always assume that
$\rho:I\times\mathcal{B}\to (-\infty,a]  $ is continuous. In addition,
we introduce the following  conditions.

\begin{enumerate}

\item[(H0)] Let  $\mathcal{BPC}(\varphi)=\{u:(-\infty,a]\to X; u_0=\varphi,
 u|_{I}\in \mathcal{PC}\}$.  The function $t\to \varphi_t$ is continuous
 from
 $\mathcal{R}(\rho^{-})=\{\rho(s,x_{s}):\rho(s,x_{s})\leq 0,\, x\in \mathcal{BPC}(\varphi), s\in [0,a]\}$  into
 $\mathcal{B}$  and  there exists a continuous
and bounded function    $J^{\varphi}:\mathcal{R}(\rho^{-})
\to(0,\infty)$   such that $\|\varphi_t\|_{\mathcal{B}}\leq
J^{\varphi}(t)\|\varphi\|_{\mathcal{B}}$ for every
  $t\in\mathcal{R}(\rho^{-})$.

\item[(H1)]  The function
   $f:I \times  {\mathcal{B}}\to X$ satisfies the following properties.
\begin{enumerate}
\item The function $f(\cdot,\psi):I\to X$ is strongly measurable
for every  $\psi \in \mathcal{B}$.
 \item The function
 $f(t,\cdot) : \mathcal{B} \to X $  is
continuous for each $t \in I $.
 \item There exist an integrable
function  $m : I \to [0, \infty )$  and a continuous
nondecreasing function $ W:[0,\infty)\to (0,\infty)$ such
that $\| f(t,\psi )\| \leq  m(t)W(\|
\psi\|_{\mathcal{B}}), $ for every $(t,\psi)\in
 I\times
\mathcal{B}$.
\end{enumerate}

\item[(H2)] The maps $ I_i$ are completely continuous and
 there are positive constants $ c_i^j$, $ j=1,2$, such that $ \|I_i(\psi)\|\leq
 c_i^1\|\psi\|_{\mathcal{B}}+c_i^2$, $ i=1,2,\dots,n$, for every $ \psi\in
 \mathcal{B}$.

\item[(H3)] The function
 $I_i:\mathcal{B}\to X $ is continuous and there are positive constants $ L_i,
 $ $i=1,2,\dots,n$, such that $
\|I_i(\psi_1)-I_i(\psi_2)\|\leq L_i\|\psi_1-\psi_2\|_{\mathcal{B}}$,
 for every  $ \psi_j\in\mathcal{B}, $ $j=1,2$, $i=1,2,\dots,n. $

\end{enumerate}


\begin{remark}\label{remark1} \rm
The  condition   (H0),  is frequently verified by
functions   continuous and
bounded. If, for instance, the space $\mathcal{B}$ verifies axiom $C_{2}$
in the nomenclature of \cite{Hino}, then there exists a constant
${\mathrm{L}}>0$ such that $ \|\varphi \|_{\mathcal{B}} \leq
\mathrm{L}\sup_{ \theta \leq 0}\|\varphi ( \theta ) \|$ for every
$\varphi\in \mathcal{B}$ continuous and bounded, see
 \cite[Proposition 7.1.1]{Hino} for details.
Consequently,  $\|\varphi_{t}\|_{\mathcal{B}}\leq L \frac{\sup_{ \theta \leq
0}\|\varphi ( \theta ) \|}{\|\varphi\|_{\mathcal{B}}}\|\varphi\|_{\mathcal{B}}$
for every continuous and bounded function $\varphi \in \mathcal{B}\setminus
\{0 \}$ and every $t\leq 0$. We note  that the  spaces $ C_{r}
\times L^{p}(g;X)$, $ C_{g}^{0}(X) $ verify axiom $C_{2}$,
 see \cite[p.10]{Hino} and \cite[p.16]{Hino} for details.
\end{remark}

\begin{remark} \label{rmk2.2} \rm
 Let $\varphi\in \mathcal{B}$ and $t\leq 0$. The notation
$\varphi_{t}$ represents the function defined by
$\varphi_{t}(\theta)=\varphi(t+\theta)$. Consequently, if the function
$x(\cdot)$ in   axiom $\mathbf{A}$ is such that $x_{0}=\varphi$, then
$x_{t}=\varphi_{t}$.
  We also note  that, in general,  $\varphi_t\notin \mathcal{B}$. Consider
for example the characteristic function $\mathcal{X}_{[-r,0]}$, $
r>0 $, in the space ${\bf C_{r} \times L^{p}(g;X)}$.
\end{remark}

 In this paper, we  adopt the following
 concept of mild solution.

\begin{definition} \label{def2.3} \rm
A function $x:(-\infty,a]\to X$ is called a mild solution of
the abstract Cauchy problem  \eqref{1}--\eqref{2} if \, $x_{0}=\varphi$,
$x_{\rho(s,x_s)}\in\mathcal{B}$ for every $s\in I$ and
$$
 x(t)=U(t,0)\varphi(0)+
 \int_{0}^{t}U(t,s)f(s,x_{\rho(s,x_s)})ds+\sum_{0<t_i<t}U(t,t_i)I_{i}(x_{t_{i}}),\hspace{0.4cm}t\in I.
$$
\end{definition}

The next  result  is a consequence of  the phase space axioms.

 \begin{lemma}\label{lema1} If
$x:(-\infty,a]\to X$  is  a function such that $x_0=\varphi$
and $x|_{I} \in { \mathcal{P}C(I:X) } $, then
  $$
\|x_{s}\|_{\mathcal{B}}\leq
(M_{a}+J^{\varphi})\|\varphi\|_{\mathcal{B}}+{ K_{a}\sup\{\|x(\theta)
\|;\,  \theta\in [0,\,\max\{0,s\}]\} },\quad
s\in \mathcal{R}(\rho^{-}) \cup I,
$$
where $ J^{\varphi}=\sup_{t\in \mathcal{R}(\rho^{-})}
J^{\varphi}(t)$, $  M_a= \sup_{t \in I} M(t)  $ and
$ \ K_a= \sup_{t \in I} K(t)$.
\end{lemma}

 \begin{remark}\label{remark2} \rm
In the rest of this work,
$y:(-\infty,a]\to X$ is the function defined
 by $y_{0}=\varphi$ and $y(t)=U(t,0)\varphi(0)$ for $t\in I$.
 \end{remark}

  Now, we can prove our first existence result.

\begin{theorem}\label{teo2}
Let conditions {\rm (H0)--(H3)} be satisfied and assume
 that
 \begin{equation} \label{4} 1\,>\,K_a\widetilde{M}\Big(
 \liminf_{\xi\to\infty^{+}}\frac{W(\xi)}{\xi}\int_{0}^{a}m(s)ds+
  \sum_{i=1}^{n}L_{i} \Big) .
 \end{equation}
Then there exists a mild solution of \eqref{1}--\eqref{2}.
\end{theorem}

\begin{proof} On the space  $Y=\{u\in \mathcal{PC}:u(0)=\varphi(0)\}$
 endowed with the uniform convergence norm ($\|\cdot\|_{\infty}$), we
define the operator   $\Gamma:Y\to Y$   defined  by
$$
\Gamma  x(t)=U(t,0)\varphi(0)+\int_{0}^{t}U(t,s)
f(s,\bar{x}_{\rho(s,\bar{x}_s)})ds +\sum_{0<t_i<t}U(t,t_i)I_{i}
 (\bar{x}_{t_{i}})  ,
\quad t\in I,
$$
 where $\bar{x}:(-\infty,a]\to X$ is
such that  $\bar{x}_0=\varphi$ and $\bar{x}=x$ on $I$. From our
assumptions, it is  easy to see that  $\Gamma x(\cdot)\in Y$.

Let $\bar{\varphi}:(-\infty,a]\to X$ be the extension of $\varphi$
to $(-\infty,a] $ such that $\bar{\varphi}(\theta)=\varphi(0)$ on $I$
and $\widetilde{J}^{\varphi}=\sup\{ J^{\varphi}(s): s\in
\mathcal{R}(\rho^{-})\}$.  By using    Lemma \ref{lema1},  for $r>0$
 and $x^r\in
B_r(\bar{\varphi}|_{I},Y)$  we obtain
 \begin{align*}
&\|\Gamma x^r-\varphi(0)\| \\
&\leq
(\widetilde{M}+1)H\|\varphi\|_{\mathcal{B}}+\widetilde{M}\int_{0}^{a}m(s)
W(\|\overline{x^r}_{\rho(s,\overline{x^r_s})}\|_{\mathcal{B}})
ds\\
&\quad +\widetilde{M}\sum_{i=1}^{n}\left( L_{i}\|\overline{x}_{t_{i}}
\|_{\mathcal{B}}+\|I_{i}(0) \|\right)\\
&\leq (\widetilde{M}+1)H\|\varphi\|_{\mathcal{B}}+\widetilde{M}
\int_{0}^{a}m(s)W\Big((M_a+\widetilde{J}^{\varphi})\|\varphi\|_{\mathcal{B}}+K_a
 \sup_{\theta\in [0,a]}\|\overline{x^r}(\theta)\|\Big)ds\\
&\quad + \widetilde{M}\sum_{i=1}^{n}L_{i}\left(\|\overline{x}_{t_{i}} -\varphi\|_{\mathcal{B}}+\|\varphi\|_{\mathcal{B}} +\|I_{i}(0)\|\right)\\
&\leq (\widetilde{M}+1)H\|\varphi\|_{\mathcal{B}}+\widetilde{M}W
\left((M_a+\widetilde{J}^{\varphi})\|\varphi\|_{\mathcal{B}}+K_a
(r+\|\varphi(0)\|)\right)\int_{0}^{a}m(s)ds, \\
&\quad +\ \widetilde{M}\sum_{i=1}^{n}L_{i}\left( K_a
r+\|\varphi\|_{\mathcal{B}}+\|
I_{i}(0) \|\right)
\end{align*}
which from  (\ref{4}) implies that  $ \|\Gamma
x^r-\varphi(0)\|_{\infty}\leq r $ for $r$ large enough.


Let $r>0$ be  such that $\Gamma(B_r(\bar{\varphi}|_{I},Y))\subset
B_r(\bar{\varphi}|_{I},Y)$. Next, we will prove that $\Gamma(\cdot)$ is
completely continuous  from $B_r(\bar{\varphi}|_{I},Y)$ into
$B_r(\bar{\varphi}|_{I},Y)$.   To this end, we introduce the
decomposition
 $\Gamma=\Gamma_{1}+\Gamma_{2}$ where $(\Gamma_{1}x)_{0}=\varphi$,
 $(\Gamma_{2}x)_{0}=0$,  and
\begin{gather*}
  \Gamma_{1} x(t)=
 U(t,0)\varphi(0)+\int_{0}^{t}U(t,s)f(s,\bar{x}_{\rho(s,\bar{x}_s)})ds ,
\quad t\in I \\
\Gamma_{2} x(t)= \sum_{0<t_i<t}U(t,t_i)I_{i} (\bar{x}_{t_{i}}) , \quad
 t\in I.
\end{gather*}
    To begin, we  prove that the  set $\Gamma_{1}
 (B_r(\bar{\varphi}|_{I},Y))(t)=\{\Gamma_{1}
x(t):x\in B_r(\bar{\varphi}|_{I},Y)\}$ is relatively compact in $X$ for
every $t\in I$.

 The case $t=0$ is obvious. Let $0<\varepsilon<t\leq a$. If
$x\in B_r(\bar{\varphi}|_{I},Y)$, from  Lemma \ref{lema1} follows that
$\|\bar{x}_{\rho(t,\bar{x}_t)}\|_{\mathcal{B}}\leq
r^*:=(M_a+\widetilde{J}^{\varphi})\|\varphi\|_{\mathcal{B}}
+K_a(r+\|\varphi(0)\|)$
 which implies
\begin{equation} \label{des1}
 \big\|\int_{0}^{\tau}U(\tau,s)
f(s,\bar{x}_{\rho(s,\bar{x}_s)})ds\big\|\leq
r^{**}:=\widetilde{M}W(r^*)\int_{0}^{a}m(s)ds, \quad \tau\in I.
\end{equation}
 From the above inequality, we  find that
\begin{align*}
  \Gamma_{1}   x(t)&= U(t,0)\varphi(0) +
U(t,t-\varepsilon)\int_{0}^{t-\varepsilon}U(t-\varepsilon,s)f(s,\bar{x}_{\rho(s,\bar{x}_s)})ds
\\
 &\quad +\int_{t-\varepsilon}^{t}U(t,s)f(s,\bar{x}_{\rho(s,\bar{x}_s)})ds\\
&\in \{U(t,0)\varphi(0)\}+U(t,t-\varepsilon)B_{r^{**}}(0,X)+C_{\varepsilon},
\end{align*}
where $\mathop{\rm diam}(C_{\varepsilon})\leq 2\widetilde{M}W(r^*)
\int^{t}_{t-\varepsilon}m(s)ds\to 0$ as $\varepsilon\to 0$,  which
allows us to conclude that  $  \Gamma_{1}(B_r(\bar{\varphi}|_{I},Y))(t)$
is relatively  compact in $X$.

Now, we prove that  $  \Gamma_{1} (B_r(\bar{\varphi}|_{I},Y))$ is
equicontinuous on $I$. Let $0<t< a$ and $\varepsilon >0$. Since the
set
 $  \Gamma_{1}(B_r(\bar{\varphi}|_{I},Y))(t)$ is relatively compact  compact in
 $X$, from the   properties of the evolution family $U(t,s)$, there
 exists   $0<\delta\leq a-t $ such that $\|U(t+h,t)x-x\| <\varepsilon$,
 for every $ x\in  \Gamma_{1} (B_r(\bar{\varphi}|_{I},Y))(t)$
 and  all   $0<h<\delta$. Under these conditions, for   $x\in
B_r(\bar{\varphi}|_{I},Y)$ and $0<h<\delta$ we obtain
\begin{align*}
  \|\Gamma_{1} x(t+h)-  \Gamma_{1} x(t)\|
&\leq \|U(t+h,0)\varphi(0)-U(t,0)\varphi(0)\| \\
&\quad +\|(U(t+h,t)-I)\int_{0}^{t}U(t,s)
f(s,\bar{x}_{\rho(s,\bar{x}_s)})ds\|\\
&\quad +\widetilde{M}\int_{t}^{t+h}m(s)W(r^*)ds\\
&\leq  2\varepsilon  +\widetilde{M}W(r^*)\int_{t}^{t+h}m(s)ds,
\end{align*}
which proves   that  $  \Gamma_{1} (B_r(\bar{\varphi}|_{I},Y))$ is right
equicontinuous at $t\in (0,a)$.  A  similar procedure shows that
$  \Gamma_{1} ( B_r(\bar{\varphi}|_{I},Y))$ is right equicontinuous  at zero
and  left equicontinuous at $t\in (0,a]$. Thus,  the set $  \Gamma_{1}
(B_r(\bar{\varphi}|_{I},Y))$ is equicontinuos on $I$.

 Using the same arguments  as  in  \cite[Theorem 2.2]{h3},
 it follows that $  \Gamma_{1}$ is a continuous map, which complete the
 proof  that $\Gamma_{1}$ is completely continuous.  On the other hand,
from  the assumptions and the phase space axioms it follows that
 \begin{align*}
\|\Gamma_{2} x-\Gamma_{2} y\|_{\infty}  &\leq
 K_a\widetilde{M}\sum_{i=1}^{n}L_{i}\|x-y\|_{ \infty}
 \end{align*}
which proves that $\Gamma_{2}$  is a contraction on
$ B_r(\bar{\varphi}|_{I},Y)$ and that $\Gamma$ is a condensing map on
$ B_r(\bar{\varphi}|_{I},Y)$.

Finally, the existence  of a mild solutions is a consequence of
 \cite[Theorem 4.3.2]{Ma}.  The proof is complete.
\end{proof}

In the next result, $\mathcal{BPC}(\varphi)$ is the set introduced
in assumption (H0).


\begin{theorem}\label{teo3}
Let  {\rm (H0)--(H2)}  be satisfied. If
 $\rho(t,x_{t})\leq t$
for every $(t,x)\in I\times\mathcal{BPC}(\varphi)$,
  $ \mu=1- K_a\widetilde{M}\sum_{i=1}^{n}c_{i}>0$ and
$$
K_a\widetilde{M}\int_{0}^{a}m(s)ds<\int_{C}^{\infty}\frac{ds}{W(s)},
$$
where
$$
C=(M_a+J^{\varphi}+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}
 + \frac{\widetilde{M}K_a}{\mu }\sum_{i=1}^{n}
\Big[ c_{i}^1(M_a+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+
 c_{i}^2\Big]
$$
 then there  exists a mild solution of \eqref{1}--\eqref{2}.
\end{theorem}

\begin{proof}
 On the space $\mathcal{BPC}=\{u:(-\infty,a]\to X; u_0=0,
 u|_{I}\in \mathcal{PC}\}$ provided with the sup-norm   $
 \|\cdot\|_{\infty}$,   we define the operator $ \Gamma:
 \mathcal{BPC}\to\mathcal{BPC}$ by $(\Gamma u)_{0}=0 $ and
\[
\Gamma x(t)=  \int_0^t{U(t,s)f(s,\bar{x}_{\rho(s,\bar{x}_s)})}ds
  +\sum_{0<t_i<t}U(t,t_i)I_i(\bar{x}_{t_i}), \quad t\in I,
\]
where $ \bar{x}=x+y$ on  $(-\infty,a]$ and  $y(\cdot)$ is the
function  defined in Remark \ref{remark2}. To use Theorem \ref{teo1}, we
 establish \emph{a priori} \ estimates for the solutions of the integral
 equation $ z=\lambda \Gamma z, \lambda\in (0,1)$. Let $x^{\lambda}$ be a
 solution of $z=\lambda\Gamma z, \lambda\in(0,1)$. By using  Lemma \ref{lema1}, the
 notation
$\alpha^{\lambda}(s)=\sup_{\theta\in[0,s]}\|x^{\lambda}(\theta)\|$,
and the fact that $ \rho(s,\overline{(x^{\lambda})}_s)\leq s$,
 for each $ s\in I $, we find that
\begin{align*}
\|x^{\lambda}(t)\|&\leq
\widetilde{M}\int_0^t{m(s)}W\Big((M_a+J^{\varphi}+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+K_a\alpha^{\lambda}(s)
\Big)ds\\&\quad +\widetilde{M}\sum_{0<t_i\leq
 t}c_{i}^1\Big[(M_a+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+
K_a\alpha^{\lambda}(t)\Big]+\widetilde{M}\sum_{i=1}^n c_{i}^2,
\end{align*}
and so,
\begin{align*}
 \alpha^{\lambda}(t)
&\leq \widetilde{M}\sum_{i=1}^{n} \left[
 c_{i}^1(M_a+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+ c_{i}^2\right] +
 K_a\widetilde{M}\sum_{0<t_i\leq t}c_{i}^1\alpha^{\lambda}(t) \\
&\quad + \widetilde{M}\int_0^t{m(s)}W\Big((M_a+J^{\varphi}
 +\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+K_a\alpha^{\lambda}(s)
\Big)ds,
\end{align*}
which implies
\begin{align*}
\alpha^{\lambda}(t)&\leq \frac{\widetilde{M}}{\mu }\sum_{i=1}^{n}
\big[
 c_{i}^1(M_a+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+
 c_{i}^2\big]\\
 &\quad +\frac{\widetilde{M} }{\mu}\int_0^t{m(s)}W((M_a+J^{\varphi}
+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+K_a\alpha^{\lambda}(s))ds,
\end{align*}
for every $t\in [0,a]$. By defining $\xi^{\lambda}(t)=(M_a+J^{\varphi}
+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+K_a\alpha^{\lambda}(t) $,
we find that
\begin{align*}
\xi^{\lambda}(t) &\leq  (M_a+J^{\varphi}+\widetilde{M}HK_a)
\|\varphi\|_{\mathcal{B}} + \frac{\widetilde{M}K_a}{\mu }\sum_{i=1}^{n}
\big[
 c_{i}^1(M_a+\widetilde{M}HK_a)\|\varphi\|_{\mathcal{B}}+
 c_{i}^2\big]\\
&\quad +\frac{\widetilde{M}K_a }{\mu}\int_0^t{m(s)}W(\xi^{\lambda}(s))ds.
\end{align*}
Denoting by $ \beta_{\lambda}(t)$ the right hand side of the last
 inequality, if follows that
\[
\beta'_{\lambda}(t)\leq
 \frac{\widetilde{M}K_a}{\mu}m(t)W(\beta_{\lambda}(t))
\]
and hence
\[
\int_{\beta_{\lambda}(0)=C}^{\beta_{\lambda}(t)}\frac{ds}{W(s)}\leq
 \frac{\widetilde{M}K_a}{\mu}\int_0^a{m(s)}ds
<\int_{C}^\infty{\frac{ds}{W(s)}},
\]
which implies that the set of functions $ \{\beta_{\lambda}(\cdot):
 \lambda\in (0,1)\} $ is bounded in   $ C(I,\mathbb{R})$.  This show that the set   $ \{
 x^{\lambda}(\cdot): \lambda\in (0,1)\}$ is bounded in $ \mathcal{BPC}$.

To prove that the map $\Gamma $ is completely continuous, we
consider
 the  decomposition $ \Gamma =\Gamma_1 +\Gamma_2 $ where   $ (\Gamma_i
 x)_{0}=0, i=1,2$, and
\begin{align*}
\Gamma_1 x(t)=
\int_{0}^{t}U(t,s)f(s,\bar{x}_{\rho(s,\bar{x}_s)})ds,
 \quad t\in I,\\
\Gamma_2 x(t)=  \sum_{0<t_i<t}U(t,t_i)I_i(\bar{x}_{t_i}),\quad t\in
I.
\end{align*}
Proceeding as in the proof of  Theorem \ref{teo2} we can prove that
 $\Gamma_{1}$ is completely continuous. The continuity of $\Gamma_2$ can be
 proven using the phase space axioms.  To prove that $\Gamma_{2}$ is
 also completely  continuous, we use  Lemma \ref{prop1}.  For   $r>0$,
  $t\in [t_{i},t_{i+1}]\cap
(0,a]$, $i\geq 1$,  and $u\in
B_r=B_r(0,{\mathcal{B}\mathcal{P}\mathcal{C}})$  we find that
\[
\widetilde{\Gamma_{2}u}(t)\in
\begin{cases}
 \sum_{j=1}^{i}U(t,t_{j})I_{j}(B_{ r^{*}}(0,
 X)),& t\in ( t_{i},t_{i+1} ),\\[3pt]
 \sum_{j=0}^{i}U(t_{i+1},t_{j})I_{j}( B_{ r^{*}}(0,
X)),& t=t_{i+1},
\\[3pt]
\sum_{j=1}^{i-1}U(t_{i},t_{j})I_{j}(B_{ r^{*}}(0, X))+ I_{i}(
B_{ r^{*}}(0; X)),&  t=t_{i},\,
\end{cases}
\]
where  $r^*:=(M_a+\widetilde{M}HK_{a})\|\varphi\|_{\mathcal{B}} + K_a r$,
which proves that $[\widetilde{\Gamma_{2} (B_r)}]_{i}(t)$ is relatively
compact in $X$   for every $t\in [t_{i},t_{i+1}]$, since the maps $I_{j}$
are completely continuous.
  Moreover, using  the compactness of the
operators $I_{i}$ and  properties  of the   evolution  family   $
 U(\cdot)$,   we can prove that   $\widetilde{\,[\Gamma_{2}(B_r)]_{i}}(t)$  is
equicontinuous at $t$,  for every $t\in [t_{i},t_{i+1}] $ and each
$
 i=1,2,\dots,n$, which   complete  the proof that $\Gamma_{2}$ is
 completely continuous.

The existence of a mild solution is now  a consequence of  Theorem
 \ref{teo1}.   The proof is complete.
\end{proof}


\section{Applications}

In this section we consider an application of our abstract results.
  Consider the partial differential equation
\begin{equation} \label{eq1}
\begin{aligned}
\frac{\partial u(t,\xi)}{\partial t}
&=  \frac{\partial^{2}u(t,\xi)}
{\partial \xi^{2}} + a_{0}(t, \xi) u(t,\xi)\\
&\quad  + \int_{-\infty}^{t} a_{1}(s - t) u(s-\rho_{1}
(t)\rho_{2}(\int_{0}^{\pi}a_{2}(\theta )| u(t,\theta)|^{2}d\theta ),
\xi) ds
\end{aligned}
\end{equation}
for $t\in I=[0,a]$, $\xi\in [0,\pi]$.
The above equation is subject to the conditions
\begin{gather}
u(t, 0)   =   u(t, \pi)  =  0, \quad t \geq 0, \label{eqq2} \\
u(\tau, \xi)  =  \varphi(\tau, \xi),\quad \tau \leq
0,\; 0 \leq \xi \leq \pi. \label{eq3}\\
\Delta u(t_j,\xi) = \int_{-\infty}^{t_j}{\gamma_{j}(s-t_j)u(s,\xi)}ds,
 \quad  j=1,2,\dots ,n.
\end{gather}

To study this system, we consider the space  $X = L^{2}([0, \pi])$
and the operator $A:D(A)\subset X\to X$  given by
 $Ax=x'' $
with $ D(A) := \{x \in X : x'' \in X, \;x(0) = x(\pi) = 0 \}$. It is
well known that $A$ is the infinitesimal generator of an analytic
semigroup  $(T(t))_{t\geq 0}$ on $ X$.
Furthermore, $A$ has  discrete spectrum with eigenvalues $- n^{2}$,
$n  \in \mathbb{N}$,
and corresponding normalized eigenfunctions given by $z_{n} (\xi) =
(\frac{2}{\pi})^{1/2} \sin (n \xi)$.  In addition,  $\{z_{n} : n \in
\mathbb{N} \}$ is an orthonormal basis of $ X$ and  $
T(t)x = \sum_{n=1}^{\infty} e^{-n^{2}t}  \langle x, z_{n} \rangle
 z_{n}$  for $x\in X$ and $t\geq 0$.
It follows  from this representation  that $T(t)$ is compact for
every  $t>0$ and that  $\|T(t)\|\leq e^{-t}$ for every $t\geq 0$.

 On the domain $D(A)$, we define the operators
 $A(t):D(A)\subset X\to X$   by
$ A(t)x(\xi)=Ax(\xi) + a_0(t,\xi)x(\xi) $.
 By assuming that $a_0(\cdot)$ is continuous and that
$a_0(t,\xi) \leq - \delta_0$  ($\delta_0 > 0$) for every
$t \in \mathbb{R}, \xi  \in [0, \pi]$, it  follows that   the system
\begin{gather*}
  u'(t)=  A(t)u(t) \quad t\geq s, \\
  u(s) =x\in X,
 \end{gather*}
has an associated evolution family    given by
  $U(t,s)x(\xi)=[T(t-s)e^{\int_{s}^{t}a_0(\tau,\xi)d\tau}x](\xi)$.
 From this expression, it follows that $U(t,s)$ is a compact linear
operator   and that $\|U(t,s)\| \leq e^{-(1 + \delta_0)(t-s)} $  for
every $t,s\in I$ with  $t>s$.

\begin{proposition} \label{prop3.1}
Let  $\mathcal{B} =\mathcal{P}\mathcal{C}_{0}
 \times L^{2}(g,X)$ and  $\varphi \in \mathcal{B}$. Assume  that condition {\rm (H0)}
holds,  $\rho_i: [0,\infty )\to [0,\infty )$, $i=1,2$, are
continuous  and that the following conditions are
 verified.
\begin{itemize}
\item[(a)] The functions $ a_i: \mathbb{R}\to\mathbb{R}$
  are   continuous and
$ L_{f}=( \int^{0}_{-\infty}   \frac{(a_{1} (s))^{2}}{ g(s)}ds)^{1/2}$ is finite.

\item[(b)] The functions $\gamma_{i}:\mathbb{R}\to \mathbb{R}$,
 $i=1,2,\dots,n$, are continuous, bounded
 and  $ L_{i}:=\Big( \int_{-\infty}^{0}\frac{(\gamma_{i}(s))^{2}}{g
 (s)}ds\Big)^{1/2} <\infty$ for every $i=1,2,\dots,n$.
\end{itemize}
Then there exists a mild solution of \eqref{eq1}--\eqref{eq3}.
\end{proposition}

\begin{proof}
 From the assumptions, we have that
\begin{gather*}
f(t,\psi)(\xi)= \int^0_{-\infty}a_1(s)\psi(s,\xi) ds,\\
\rho(s,\psi)= s-\rho_1(s)\rho_2 \Big(\int^{\pi}_{0}a_2(\theta)|
\psi(0, \xi)|^{2} d\theta\Big),\\
  I_{i}(\psi)(\xi)= \int_{-\infty}^{0} \gamma_{i}(s)\psi(s,
\xi)ds,\quad
 i = 1, 2, \dots, n,
\end{gather*}
 are well defined functions, which permit to transform system
 \eqref{eq1}--\eqref{eq3} into  the abstract
system  \eqref{1}--\eqref{2}. Moreover, the functions $f$, $I_{i}$
are  bounded  linear operator, $\|f \|\leq L_1$ and
 $\|I_{i} \|\leq L_i$ for   every $i=1,2,\dots n$.
Now, the existence of a mild
 solutions can be deduced from a direct application of
 Theorem  \ref{teo3}.
  The proof is complete.
\end{proof}

 From Remark  \ref{remark1} we have the following result..

\begin{corollary}
   Let $\varphi\in\mathcal{B}$ be continuous and bounded.  Then
there exists a mild solution of {\rm \eqref{eq1}--\eqref{eq3}} on $I$.
\end{corollary}

\subsection*{Acknowledgements} The authors are grateful to the
anonymous  referees for their comments and suggestions.

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\end{document}