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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 84, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/84\hfil Quasilinear impulsive equations]
{Existence of mild solutions for quasilinear integrodifferential
equations with \\ impulsive conditions}

\author[K. Balachandran, F. P. Samuel\hfil EJDE-2009/84\hfilneg]
{Krishnan Balachandran, Francis Paul Samuel}  % in alphabetical order

\address{Krishnan Balachandran \newline
Department of Mathematics,
Bharathiar University,
Coimbatore 641 046, India}
\email{balachandran\_k@lycos.com}

\address{Francis Paul Samuel \newline
Department of Mathematics,
Bharathiar University,
Coimbatore 641 046, India}
\email{paulsamuel\_f@yahoo.com}

\thanks{Submitted February 24, 2009. Published July 10, 2009.}
\subjclass[2000]{34A37, 34G60, 34G20, 47H10}
\keywords{Semigroup; mild solution; impulsive conditions}

\begin{abstract}
 We prove the existence and uniqueness of mild solutions of
 quasilinear integrodifferential equations with nonlocal
 and impulsive conditions in Banach spaces. The results are obtained
 by using a fixed point technique and semigroup theory.
 Examples are provided to illustrate the theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

Many evolution process are characterized by the fact that at
certain moments of time they experience a change of state abruptly.
 These processes are subject to short-term perturbations whose duration
is negligible in comparison with the duration of the process.
Consequently, it is natural to assume that these perturbations act
instantaneously, that is, in the form of impulses. It is known, for
example, that many biological phenomena involving thresholds, bursting
rhythm models in medicine and biology, optimal control model in economics,
pharmacokinetics and frequency modulated systems, do exhibit impulsive
effects. Thus differential equations involving impulsive effects appear
as a natural description of observed evolution phenomena of several real
world problems.

Existence of solutions of impulsive differential equation of the form
\begin{gather}
u'(t) = Au(t)+f(t,u(t)), \quad t \in (0,a]\label{1e1}\\
u(0)+g(u)= u_0,\label{1e2}\\
\Delta u(t_i)=I_i(u(t_i)), \quad i=1,2,3,\dots , p,\;
 0<t_1<t_2<,\dots t_p<a \label{1e3}
\end{gather}
has been studied by Liang et al \cite{L2}. The impulsive condition
is the combination of traditional initial value problem and
short-term perturbations whose duration can be negligible in
comparison with the duration of process. They have advantages over
traditional initial value problem because they can be used to
model phenomena that cannot be modelled by traditional initial
value problem. Recently, the study of the impulsive differential
equations has attracted a great deal of attention. The theory of
impulsive differential equations is an important branch of
differential equations \cite{L1,P0,S1,Y1,Z1}.

Several authors  have studied the existence of solutions
of abstract quasilinear evolution equations in Banach space
\cite{B1,B2,B3,C1,D1,S2}.  Bahuguna \cite{B1}, Oka \cite{O1}
and Oka and Tanaka \cite{O2}
discussed the existence
of solutions of quasilinear integrodifferential equations in
Banach spaces. Kato \cite{K1} studied the nonhomogeneous evolution
equations where as Chandrasekaran \cite{C1} proved the existence of mild
solutions of the nonlocal Cauchy problem for a nonlinear
integrodifferential equation. An equation of this type occurs in a
nonlinear conversation law with memory
 \begin{gather}
u_t(t,x)+\Psi(u(t,x))_x = \int_0^t b(t-s)\Psi(u(t,x))_x\ ds +
f(t,x),\quad  t \in [0,T], \label{1e4}\\
u(0,x)= \phi(x),\quad  x \in \mathbb{R}.\label{1e5}
 \end{gather}
It is clear that if nonlocal condition (\ref{1e2}) is introduced
to (\ref{1e4}), then it will also have better effect than the
classical condition $u(0,x)= \phi(x)$.

 The aim of this paper is to prove the existence and uniqueness
of mild solutions of quasilinear impulsive evolution
integrodifferential equation of the form
 \begin{gather}
u'(t)+A(t,u)u(t)= f(t,u(t))+\int_0^t g(t,s,u(s))ds, \label{1e6}\\
u(0)+h(u)= u_0,\label{1e7}\\
\Delta u(t_i)=I_i(u(t_i)), \quad i=1,2,3,\dots , m,\;
 0 <t_1<t_2<,\dots t_m<T. \label{1e8}
 \end{gather}

 Let $A(t,u)$ be the infinitesimal generator of a $C_0$-semigroup in a
Banach space $X$. Let $PC([0,T];X)$ consist of functions $u$
from $[0,T]$ into $X$, such that $u(t)$ is continuous at $t\neq t_i$
and left continuous at $t=t_i$, and the right limit $u(t_i^+)$ exists for
$i=1,2,3,\dots m$. Evidently $PC([0,T],X)$ is a Banach space with the norm
 $$
 \|u\|_{PC}=\sup_{t\in [0,T]}\|u(t)\|.
$$
 Let $u_0 \in X$,  $f : [0,T] \times X   \to X$,
$g : \Omega \times X \to X$, $h: PC([0,T]:X)\to X$
and $\Delta u(t_i) = u(t_i^+)-u(t_i^-)$ constitutes an impulsive condition.
Here $[0,T]=J$ and $\Omega  = \{(t,s) : 0 \leq s \leq t \leq T\}$.
The results obtained in this paper are generalizations of the results given by
Balachandran and Uchiyama \cite{B3} and Pazy \cite{P1}.

\section{Preliminaries}

  Let $X$ and $Y$ be two Banach spaces such that $Y$ is densely and
continuously embedded in $X$. For any Banach spaces $Z$ the norm
of $Z$ is denoted by $\|\cdot\|$ or $\|\cdot \|_{Z}$. The space of
all bounded linear operators from $X$ to $Y$ is denoted by
$B(X,Y)$ and $B(X,X)$ is written as $B(X)$. We recall some
definitions and known facts from Pazy \cite{P1}.

\begin{definition} \label{def2.1} \rm
   Let $S$ be a linear operator in $X$ and let $Y$ be a subspace of $X$.
The operator $\tilde{S}$ defined by
$D(\tilde{S})= \{x\in D(S)\cap Y : Sx \in Y\}$ and $\tilde{S}x = Sx$
for $x \in D(\tilde{S})$ is called the part of $S$ in $Y$.
\end{definition}

\begin{definition} \label{def2.2} \rm
 Let $B$ be a subset of $X$ and for every $0 \leq t \leq T$ and
$b\in B$, let $A(t, b)$ be the infinitesimal generator of a $C_0$
semigroup $S_t,_b(s), s\geq 0$, on $X$. The family of operators
$\{A(t, b)\}, (t, b)\in [0,T] \times  B$, is stable if there are constants
$M\geq 1$ and $\omega$ such that
\begin{gather*}
\rho(A(t, b)) \supset (\omega, \infty) \quad\text{for }
 (t, b) \in [0,T] \times  B, \\
\|\prod_{j =1}^k R(\lambda : A(t_j, b_j))\|  \leq  M(\lambda - \omega)^{-k}
\end{gather*}
for $\lambda > \omega$ every finite sequences
$0 \leq t_1 \leq t_2 \leq \dots  \leq t_k\leq T$,
$b_j \in B,\  1 \leq j\leq k$.
 The stability of $\{A(t, b)\}, (t, b) \in [0,T] \times B$ implies
(see \cite{P1})
 that
 \[
 \|\prod_{j =1}^k S_{t_j,b_j}(s_j)\| \leq M
\exp \big\{\omega\sum_{j=1}^k s_j\big\},\quad  s_j \geq 0
 \]
 and any finite sequences $0 \leq t_1 \leq t_2 \leq \dots  \leq t_k\leq T$,
$b_j \in B$, $1 \leq j\leq k$.  $k = 1,2,\dots $
\end{definition}

\begin{definition} \label{def2.3} \rm
 Let $S_{t,b}(s), s\geq 0$ be the $C_0$-semigroup generatated by
$A(t, b)$, $(t ,b) \in J \times B$. A subspace $Y$ of $X$ is called
$A(t, b)$-admissible if $Y$ is invariant subspace of $S_{t,b}(s)$
and the restriction of $S_{t,b}(s)$ to $Y$ is a $C_0$-semigroup in $Y$.
\end{definition}

     Let $B\subset X$ be a subset of $X$ such that for every
$(t, b) \in [0,T] \times  B$, $A(t, b)$ is the infinitesimal generator
 of a $C_0$-semigroup $S_{t,b}(s), s \geq0$ on $X$.
We make the following assumptions:
\begin{itemize}
\item[(H1)] The family $\{A(t, b)\},(t, b) \in [0,T] \times B$ is stable.
\item[(H2)] $Y$ is $A(t, b)$-admissible for $(t, b)\in [0,T] \times B$
     and the family $\{\tilde{A}(t, b)\}, (t, b) \in [0,T] \times  B$
     of parts $\tilde{A}(t, b)$ of $A(t, b)$ in $Y$, is stable in $Y$.
\item[(H3)] For $(t, b)\in [0,T] \times  B$, $D(A(t, b)) \supset Y$, $A(t, b)$
     is a bounded linear operator from $Y$ to $X$ and $t \to A(t, b)$
     is continuous in the $B(Y, X)$ norm $\|.\|$ for every $b\in B$.
\item[(H4)] There is a constant $L > 0$ such that
\[
\|A(t, b_1) - A(t, b_2)\|_Y{_\to}_ X\ \leq L \|b_1 - b_2\|_X
\]
holds for every $b_1, b_2 \in B$ and  $0\leq t \leq T$.
\end{itemize}
Let $B$ be a subset of $X$ and $\{A(t, b)\}, (t, b)\in [0,T] \times B$
be a family of operators satisfying the conditions (H1)--(H4).
If $u \in PC([0,T] : X)$ has values in $B$ then there is a unique evolution
system $U(t,s;u), 0 \leq s  \leq t  \leq  T$, in $X$ satisfying,
(see \cite[Theorem 5.3.1 and Lemma 6.4.2, pp. 135, 201-202]{P1}
\begin{itemize}
\item[(i)]  $\|U(t,s;u)\| \leq M e^{\omega(t-s)}$   for
  $0 \leq s  \leq t  \leq T$.
where $M$ and $\omega$ are stability constants.
\item[(ii)]  $\frac{\partial^+}{\partial t} U(t,s;u)y =A(s, u(s))U(t,s;u)y$
    for $y\in Y$,  for   $0 \leq s  \leq t  \leq T$.
\item[(iii)] $\frac{\partial}{\partial s}  U(t,s;u)y = -U(t,s;u)A(s, u(s))y$
  for $y\in Y$,  for $0 \leq s\leq t \leq T$.
\end{itemize}
Further we assume that
\begin{itemize}
\item[(H5)] For every $u \in PC([0,T] : X)$ satisfying $u(t)\in B$ for
  $0  \leq t  \leq T$, we have
\[
U(t,s;u)Y\subset Y,\quad 0 \leq s\leq t \leq T
\]
and $U(t,s;u)$ is strongly continuous in $Y$ for $0 \leq s\leq t \leq T$.
\item[(H6)] Closed bounded convex subsets of $Y$ are closed in $X$.
\item[(H7)] For every $(t,b) \in J \times B$,
 $f(t, b) \in Y$ and $((t,s),b) \in \Omega \times B$,
 $g(t,s, b) \in Y$.
\item[(H8)] $h: PC([0,T]: B) \to Y$ is Lipschitz continuous in $X$ and
bounded in $Y$, that is, there exist constant $H  >  0$ such that
\begin{gather*}
\|h(u) - h(v)\|_Y \leq H\|u - v\|_{PC},\ \ u,v\in PC([0,T];X).
\end{gather*}
For the conditions (H9) and (H10) let $Z$ be taken as both $X$ and $Y$.
\item[(H9)] $g:\Omega \times Z \to Z$ is continuous and there
exist constants $G > 0$ and $G_1>0$ such that
\begin{gather*}
\int_0^t\|g(t,s,u)- g(t,s,v)\|_Z ds
\leq G \|u - v\|_Z),\ \ u,v \in X, \\
  G_1= \max\{\int_0^t\|g(t,s,0)\|_Z \ ds :(t,s)\in \Omega \}.
\end{gather*}
\item[(H10)]     $f : [0,T] \times Z  \to Z$ is continuous and
there exist constants $F > 0$ and $F_1 > 0$ such that
\begin{gather*}
      \|f(t,u) - f(t,v)\|_Z
 \leq F  \|u - v\|_Z , \ u,v\in X, \\
     F_1 = \max_{t\in [0,T]} \|f(t,0)\|_Z.
\end{gather*}
\end{itemize}
     Let us take
$M_0 = \max \{\|U(t,s;u)\|_{B(Z)}, 0 \leq s\leq t \leq T, \ u\in B\}$.
\begin{itemize}
\item[(H11)]$I_i: X  \to X$ is continuous and there exist constant $l_i > 0,\\
 i=1,2,3,\dots,m$ such that
\begin{eqnarray*}
\|I_i(u) - I_i(v)\| &\leq& l_i\|u - v\|, \ \ u,v\in X.
\end{eqnarray*}
\item[(H12)] There exist a positive constant $r>0$ such that
\begin{gather*}
 M_0\Big[\|u_0\|_Y +Hr+\|h(0)\|+T[r(F+G)+F_1+G_1]+\sum_{i = 1}^m (l_ir+\|I_i(0)\|)\Big]
 \leq r \ \ \mbox{and}\\
\begin{aligned}
 q &=\Big\{KT\Big[\|u_0\|_Y +Hr+\|h(0)\|+T[r(F+G)+F_1+G_1]+\sum_{i = 1}^m (l_ir+\|I_i(0)\|)\Big]\\
 &\quad+M_0\Big[H+T(F+G)+\sum_{i = 1}^m l_i\Big]<1.
\end{aligned}
\end{gather*}
\end{itemize}

\begin{definition} \label{def2.4} \rm
        A function $u\in PC([0,T] : X)$ is a mild solution of equations
\eqref{1e6}--\eqref{1e8} if it satisfies
\begin{equation}
\begin{aligned}
u(t)&=U(t,0;u)u_0 - U(t,0;u)h(u)+\int_0^tU(t,s;u)\Big[f(s,u(s))\\
&\quad+\int_0^s g(s,\tau,u(\tau))d\tau \Big]ds+\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i)), \ \ 0\leq t\leq T
\end{aligned} \label{2e1}
\end{equation}
\end{definition}
\begin{definition} \label{def2.5} \rm
A function $u\in PC([0,T] : X)$ such that $u(t)\in D(A(t,u(t)) \ $ for $t\in (0,T], u\in C^1((0,T]\backslash\{t_1,t_2,\dots,t_m\}:X)$ and satisfies \eqref{1e6}--\eqref{1e8} in X is called a classical solution of  \eqref{1e6}--\eqref{1e8} on $[0,T]$,
\end{definition}

Further there exists a constant $K  >  0$ such that for every
$u, v \in PC([0,T] : X)$ and every $y \in Y$ we have
\begin{equation}\label{2e2}
 \|U(t,s;u)y - U(t,s;v)y\| \leq K T\|y\|_Y \|u - v\|_{PC}.
\end{equation}

\section{Existence Result}

\begin{theorem} \label{thm3.1}
        Let $u_0 \in Y$ and let $B = \{u\in X: \|u\|_X \leq r\}$, $r > 0$.
If the assumptions {\rm (H1)--(H12)} are satisfied, then
\eqref{1e6}--\eqref{1e8} has a unique mild solution
$u\in PC([0,T] : Y)$.
\end{theorem}

 \begin{proof}
Let $S$ be a nonempty closed subset of $PC([0,T]:X)$ defined by
$$ S = \{u:u\in PC([0, T]: X),\|u(t)\|_{PC} \leq r \mbox{ for}\  0\leq t\leq T\}.$$

 Consider a mapping $\Phi$ on $S$ defined by
\begin{equation}
\begin{aligned}
 (\Phi u)(t) &= U(t,0;u)u_0 - U(t,0;u)h(u)+\int_0^t U(t,s;u)\Big[f(s,u(s))\\
&\quad +\int_0^sg(s,\tau, u(\tau))d\tau\Big]ds+\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i)).
\end{aligned} \label{3e1}
\end{equation}
We claim that $\Phi$ maps $S$ into $S$. For $u\in S$, we have
\begin{align*}
 & \|\Phi u(t)\|_Y \\
&\leq  \|U(t,0;u)u_0\|+\|U(t,0;u)h(u)\|\\
    &\quad + \int_0^t\ \|U(t,s;u)\|\Big[\|f(s,u(s))- f(s,0)\|
    + \|f(s,0)\|\\
    &\quad + \|\int_0^s [g(s,\tau,u(\tau))- g(s,\tau,0)]
    d\tau\|+ \|\int_0^s g(s,\tau,0)d\tau\|\Big]ds\\
    &\quad+\sum_{0 < t_i < t}\|U(t,t_i;u)I_i(u(t_i))\|\\
&\leq  M_0 \|u_0\|_Y+M_0\Big[H\|u\|+\|h(0)\|\Big]+M_0\Big[\int_0^tF\|u(s)\|ds+F_1T \\
&\quad +  \int_0^tG\|u(s)\|ds+G_1T\Big]
  +M_0\sum_{i = 1}^m\Big(l_i\|u\|+\|I_i(0)\|\Big) \\
&\leq  M_0\Big[\|u_0\|_Y +Hr+\|h(0)\|+T\Big[r(F+G)+F_1+G_1\Big]+\sum_{i = 1}^m\Big(l_ir+\|I_i(0)\|\Big).
\end{align*}
 From  assumption (H12), one gets
    $\|\Phi u(t)\|_Y     \leq r$.
Therefore $\Phi$ maps $S$ into itself. Moreover, if $u,v \in S$, then
\begin{align*}
&\|\Phi u(t)- \Phi v(t)\|\\
&\leq \|U(t,0;u)u_0-U(t,0;v)u_0\|+\|U(t,0;u)h(u)-U(t,0;v)h(v)\|\\
&\quad +  \int_0^t\|U(t,s;u)\Big[f(s,u(s))
  + \int_0^s g(s,\tau,u(\tau))d\tau\Big]+\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i))\\
&\quad - U(t,s;v)\Big[f(s,v(s))
  + \int_0^s g(s,\tau,v(\tau))d\tau\Big]-\sum_{0 < t_i < t}U(t,t_i;v) I_i(v(t_i))\|ds
\end{align*}
Using  assumptions (H8)-(H12), one can get
\begin{align*}
&\|\Phi u(t)- \Phi v(t)\|\\
&\leq KT\|u_0\|_Y\|u - v\|_{PC}
 +KT\Big[H\|u\|+\|h(0)\|\Big]\|u - v\|_{PC}
\end{align*}
\begin{align*}
&\quad + M_0H\|u - v\|_{PC}
  + KT\|u - v\|_{PC}\Big[\int_0^t(F\|u(s)\|+F_1)ds\\
&\quad+\int_0^t(G\|u(s)\|+G_1)ds\Big]
  + KT\|u - v\|_{PC}\sum_{i = 1}^m\Big(l_ir+\|I_i(0)\|\Big)\\
&\quad + M_0\Big[\int_0^tF\|u(s)-v(s)\|ds
  +\int_0^tG\|u(s)- v(s)\|ds\Big]+M_0\sum_{i = 1}^ml_i\|u - v\|_{PC}\\
&\leq \Big\{KT\Big[\|u_0\|_Y+Hr+\|h(0)\|+T[r(F+G)+F_1+G_1]\\
&\quad + \sum_{i = 1}^m(l_ir+\|I_i(0)\|)\Big]
  +M_0\Big[H+T(F+G)+\sum_{i = 1}^ml_i\Big]\Big\}\|u - v\|_{PC}\\
&= q \|u - v\|_{PC}, \ \ \ u,v\in PC([0,T];X)
\end{align*}
where $0 < q < 1$.  From this inequality it follows that for any $t\in [0,T]$,
\[
 \|\Phi u(t)- \Phi v(t)\|\leq q \|u - v\|_{PC},
\]
 so that $\Phi$ is a contraction on $S$. From the contraction mapping
theorem it follows that $\Phi$ has a unique fixed point $u\in S$ which is
the mild solution of \eqref{1e6}--\eqref{1e8} on $[0,T]$.
Note that $u(t)$ is in $PC([0,T] : Y)$ by (H6) see
\cite[pp. 135, 201-202 lemma 7.4]{P1}. In fact, $u(t)$ is weakly
continuous as a $Y$-valued  function. This implies that $u(t)$ is
separably valued in $Y$, hence
it is strongly measurable. Then $\|u(t)\|_ {PC}$ is bounded and measurable
function in $t$. Using the relation $u(t)= \Phi u(t)$,
we conclude that $u(t)$ is in $PC([0,T] :Y)$.
\end{proof}

\noindent\textbf{Remark.} %3.1
Using the additional assumption $A(t,b)u_0, b\in B$ is bounded in $Y$
one can establish a unique local
classical solution for the equations \eqref{1e6}--\eqref{1e8}.

\section{Quasilinear Delay Integrodifferential Equation}

Next we consider the following quasilinear delay integrodifferential
equation with impulsive nonlocal conditions \eqref{1e7} and \eqref{1e8}
 \begin{gather}
u'(t)+A(t,u)u(t)= f(t,u(\alpha(t)))+\int_0^t g(t,s,u(\beta(s)))ds, \quad
 t\in [0,T],\label{4e1}
 \end{gather}
where $A, f$ and $h$ are as before. Assume the following additional
conditions:
\begin{itemize}
\item[(H13)] $\alpha,\beta$: $[0,T] \to [0,T]$ are absolutely
continuous and there exists constants $\delta_1, \delta_2 > 0$ and such that
$\alpha^\prime(t) \geq \delta_1$ and $\ \beta^\prime(t) \geq \delta_2$ for $0 < t \leq T$.
\item[(H14)] There exist a positive constant $k>0$ such that
\begin{gather*}
 M_0\Big[\|u_0\|_Y +Hk+\|h(0)\|+T\Big[k/\delta_1\delta_2(F\delta_2+G\delta_1)+F_1+G_1\Big]\\
 +\sum_{i = 1}^m\Big(l_ik+\|I_i(0)\|\Big) \leq k \\
\begin{aligned}\mbox{and}\ \ \
 p &=\{KT\Big[\|u_0\|_Y+Hk+\|h(0)\|+T[k/\delta_1\delta_2(F\delta_2+G\delta_1)+F_1+G_1]\\
&\quad + \sum_{i = 1}^m(l_ik+\|I_i(0)\|)\Big]
  +M_0\Big[H+T/\delta_1\delta_2(F\delta_2+G\delta_1)+\sum_{i = 1}^ml_i\Big]\Big\}<1.
\end{aligned}
\end{gather*}
\end{itemize}
For a mild solution of the equation \eqref{4e1} and \eqref{1e7}-\eqref{1e8}
we mean a function $u\in PC([0,T]:X)$ and $u_0 \in X$ satisfying
the integral equation
\begin{equation}
\begin{aligned}
u(t)&=U(t,0;u)u_0 - U(t,0;u)h(u)+\int_0^tU(t,s;u)\Big[f(s,u(\alpha(s)))\\
&\quad+\int_0^s g(s,\tau,u(\beta(\tau)))d\tau \Big]ds
  +\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i)), \quad 0\leq t\leq T.
\end{aligned} \label{4e4}
\end{equation}

\begin{theorem} \label{thm4.1}
  If the assumptions {\rm (H1)--(H11)} and {\rm (H13)--(H14)} are
satisfied, then the equation \eqref{4e1} with nonlocal and impulsive
conditions \eqref{1e7}-\eqref{1e8} has a unique mild solution
$u\in PC([0,T] : Y)$.
\end{theorem}

\begin{proof}
Let $S$ be a nonempty closed subset of $PC([0,T]:X)$ defined by
$ S = \{u:u\in PC([0, T]: X), \|u(t)\|_{PC} \leq k\text{ for }
0\leq t\leq T\}$.

 Consider a mapping $\Psi$ on $S$ defined by
\begin{align*}
 (\Psi u)(t) &= U(t,0;u)u_0 - U(t,0;u)h(u)+\int_0^t U(t,s;u)\Big[f(s,u(\alpha(s)))\\
&\quad +\int_0^sg(s,\tau, u(\beta(\tau)))d\tau\Big]ds+\sum_{0 < t_i < t}U(t,t_i;u) I_i(u(t_i)).
\end{align*}
         Obviously $\Psi$ maps $S$ into $S$, by (H14) and
\[
 \|\Psi u(t)- \Psi v(t)\|\leq p \|u - v\|_{PC}.
\]
 Since $p < 1$, $\Psi$ is a contraction on $S$ and so $\Psi$ has a
unique fixed point $u\in S$ which is
the mild solution of the problem \eqref{4e1} and
 \eqref{1e7}-\eqref{1e8} on $[0,T]$.\\
\end{proof}

\noindent\textbf{Remark.} %4.1
 Using the additional assumption $A(t,b)u_0, b\in B$ is bounded in $Y$
 a unique local classical solution for the equations
\eqref{4e1}, \eqref{1e7}, \eqref{1e8} can be established.

\section{Examples}

In this section we shall give two examples to illustrate the theorems.

\begin{example} \label{exa1} \rm
 Consider the  nonlinear partial integrodifferential equation
\begin{gather}
\begin{aligned}
&\frac{\partial}{\partial t}z(t,y)+
\frac{\partial^3}{\partial y^3}z(t,y)+z(t,y)
 \frac{\partial}{\partial y}z(t,y) \\
&=k_0(y)\sin z(t,y)+k_1\int_0^t e^{-z(s,y)}ds,
\end{aligned} \label{5e1}\\
z(0,y)+\sum_{i=1}^mc_iz(t_i,y)=z_0(y),\quad  y \in \mathbb{R},\label{5e2}\\
\Delta z|_{t=t_i}=I_i(z(y))=(\alpha_i|z(y)|+ t_i)^{-1}, \quad
 1\leq i\leq m  \label{5e3}
\end{gather}
where the constants $c_i$ and $\alpha_i$ are small and $k_0(y)$
is continuous on $\mathbb{R}$, and $k_1>0$.
\end{example}

Let $H^s$ be the Hilbert space introduced in \cite{P1}.
Take $X= L^2(R)=H^0(R)$ and $Y=H^s(R)$, $s\geq 3$.
Define an operator $A_0$ by $D(A_0)=H^3(R)$ and $A_0z=D^3z$ for
$z\in D(A_0)$ where $D=d/dy$.
Then $A_0$ is the infinitesimal generator of a $C_0$-group of
isometries on $X$. Next we define for every $v\in Y$ an operator
$A_1(v)$ by $D(A_1(v))=H^1(R)$ and $z\in D(A_1(v)), A_1(v)z=vDz$.
Then we have for every $v\in Y$ the operator $A(v)=A_0+A_1(v)$
is the infinitesimal generator of $C_0$ semigroup $U(t,0;v)$ on
$X$ satisfying $\|U(t,0;v)\|\leq e^{\beta t}$ for every
 $\beta\geq c_0\|v\|_s$ where $c_0$ is a constant independent
of $v\in Y$. Let $B_r$ be the ball of radius $r>0$ in $Y$ and it
is proved that the family of operators $A(v),v\in B_r$ satisfies
the conditions (H1)--(H7) (see \cite {P1}).

Put $u(t)=z(t,\cdot)$,\ \ 
$h(u)=\sum_{i=1}^mc_iz(t_i,\cdot)$  and
$$
f(t,u)=k_0(\cdot)\sin z(t,\cdot),\quad g(t,s,u)=k_1e^{-z(s,\cdot)}.
$$
With this choice of $A(u)$, $I_i$, $f$, $g$, $h$
we see that the equation \eqref{5e1}--\eqref{5e3} is an abstract
formulation of \eqref{1e6}--\eqref{1e8}.

Further other conditions (H8)--(H11) are obviously satisfied and
it is possible to choose $c_i$, $\alpha_i$, $k_0, k_1$ in such a
way that the constant $q<1$.
Hence by Theorem \ref{thm3.1} the equation \eqref{5e1}--\eqref{5e3} has a
unique mild solution on $J $.

\begin{example} \label{exa2} \rm
Consider the delay partial integrodifferential equation
\begin{equation}
\begin{aligned}
&\frac{\partial}{\partial t}z(t,y)+
\frac{\partial^3}{\partial y^3}z(t,y)+z(t,y)
 \frac{\partial}{\partial y}z(t,y) \\
&=k_0(y)\arctan z(\sin t,y)+k_1\int_0^t e^{-z(\sin s,y)}ds,
\end{aligned}\label{5e4}
\end{equation}
with the same impulsive and nonlocal conditions as in Example \ref{exa1}.
Here $ f(t,u)=k_0(\cdot)\arctan z(\sin t,\cdot)$ and
$\alpha(t)=\beta(t)=\sin t$.
With the same $A(u), I_i, g, h$ we see that the equations \eqref{5e4}
with \eqref{5e2}--\eqref{5e3} is an abstract formulation
of \eqref{4e1} with \eqref{1e7}--\eqref{1e8}. Note that (H1)--(H11)
are already satisfied and it is possible to choose the constants so
that the conditions (H13) and (H14) are also satisfied.
Now by Theorem \ref{thm4.1} the equation \eqref{5e4} has a unique mild
solution on $J$.
\end{example}

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\end{document}