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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 18, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/18\hfil Existence of solutions]
{Existence of solutions for nonlinear impulsive neutral integro-differential
equations of Sobolev type with nonlocal conditions in \\
Banach spaces}

\author[B. Radhakrishnan, A. Mohanraj, V. Vinoba \hfil EJDE-2013/18\hfilneg]
{Bheeman Radhakrishnan, Aruchamy Mohanraj, Velu Vinoba}  

\address{Bheeman Radhakrishnan \newline
Department of Applied Mathematics \& Computational Sciences,
PSG College of Technology, 
Coimbatore - 641 004, TamilNadu, India}
\email{radhakrishnanb1985@gmail.com}

\address{Aruchamy Mohanraj \newline
Department of Mathematics and Computer Sciences,
SVS College of Engineering,
Coimbatore - 642 109, TamilNadu, India}
\email{mohanraj.mat@gmail.com}

\address{Velu Vinoba \newline
Department of Mathematics,
K. N. Govt. Arts College for Women,
Thanjavur,  TamilNadu, India}
\email{vinoba2012@gmail.com}

\thanks{Submitted September 6, 2012. Published January 21, 2013.}
\subjclass[2000]{34A37, 47D06, 47H10, 74H20, 34K40}
\keywords{Existence; neutral differential equation; fixed point theorem;
\hfill\break\indent impulsive differential equation}

\begin{abstract}
 In this article, we prove the existence of mild and strong
 solutions for  nonlinear impulsive integro-differential equations
 of Sobolev type with nonlocal initial conditions.
 The results are obtained by using semigroup theory and the Schauder
 fixed point theorem. An example is provided to illustrate the theory.
\end{abstract}

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


\section{Introduction}

 Differential equations arise in many areas of science and technology,
specifically whenever a deterministic relation involving some continuously
varying quantities (modeled by functions) and their rates of change in
space and/or time are known or postulated. This is illustrated
in classical mechanics where the motion of a body is described by
its position and velocity as the time varies. It is well known that
the systems described by partial differential equations can be
expressed as abstract differential equations \cite{p1}.
These equations occur in various fields of study and each system
can be represented by different forms of differential or integro-differential
equations in Banach spaces. Using the method of semigroups, various solutions
of nonlinear and semilinear evolution equations have been discussed by Pazy \cite{p1}.
 The study of abstract nonlocal semilinear initial value problems was
initiated by Byszewski \cite{by1,by2,by3}. Because it is demonstrated
that the nonlocal problems have better effects in applications than the
classical Cauchy problems. Such problems with nonlocal conditions have
been extensively studied in the literature \cite{kb1,kb2,kb3,kb4,kb5,x1}.
Showalter \cite{S2} established the existence
of solutions of semilinear evolution equations of Sobolev type in Banach spaces.
This type of equations arise in various applications such as in the flow of
fluid through fissured rocks, thermodynamics, and shear in second-order fluids.
For more details, we refer the reader to \cite{B1, La1, Li}.

 Neutral differential equations arise in many areas of applied mathematics
and for this reason these equations have received much attention during
the last few decades. There are also a number of applications in which
the delayed argument occurs in the derivative of the state variable
as well as in the independent variable, as in the so called neutral
differential difference equations. A neutral functional differential
equation is one in which the derivatives of the past history or
derivatives of functionals of the past history are involved as well
as the present state of the system. A good guide to the literature
for neutral functional differential equations is the book by Hale
and Verduyn Lunel \cite{Ha3} and the references therein.
Hernandez \cite{H1} established the existence results for partial
 neutral functional differential equations with nonlocal conditions modeled as
\begin{equation} \label{e1.1}
\begin{gathered}
\frac{d}{dt}\big[u(t)+F(t,u(t))\big]=Au(t)+G(t,u(t)) \\
u_{\sigma}= \varphi+ q(x_{t_1},x_{t_2},\dots, x_{t_n}) \quad \text{in } \Omega,
\end{gathered}
\end{equation}
where $A$ is the infinitesimal generator of an analytic semigroup $T(t)$
on a Banach space. He made use of fixed point theorems and the results
 mentioned in Pazy \cite{p1}. For results on neutral partial differential
equations with nonlocal and classical conditions, we refer to the
papers of Hernandez and Henryquez \cite{H2}, Fu and Ezzinbi \cite{F1},
and references therein. Controllability of functional differential
systems of Sobolev type in Banach spaces has been first studied by
 Balachandran and Dauer \cite{kb}.

 Differential equations arise in many real world problems such as physics,
population dynamics, ecology,biological systems, biotechnology, optimal
control and so forth. Much has been done the assumption that the state
variables and systems parameters change continuously.
However, one may easily visualize that abrupt changes such as shock,
harvesting and disasters may occur in nature. These phenomena are short
time perturbations whose duration is negligible in comparison with the
duration of the whole evolution process. Consequently, it is natural to assume,
in modeling these problems, that these perturbations act instantaneously,
that is in the form of impulses. The theory of impulsive differential
equation \cite{L1, S1} is much richer than the corresponding theory of
differential equations without impulsive effects. The impulsive condition
 $$
\Delta u(t_i) = u(t_i^{+}) - u(t_i^{-} )= I_i(u(t_i^{-} )),\quad
 i = 1,\ 2, \dots,m,
$$
is a combination of traditional initial value problems and short-term
perturbations whose duration is negligible in
comparison with the duration of the process. Lin and Liu \cite{L2} discussed
the iterative methods for the solution of impulsive functional
 differential systems.

Motivated by the above approach, the goal of this paper is to use the
fixed point theorem to obtain the mild solution of the nonlinear
 impulsive neutral integro-differential equation of Sobolev type
with nonlocal conditions.

\section{Preliminaries}

Consider the nonlinear impulsive neutral integrodifferential equation
of Sobolev type with nonlocal conditions of the form
\begin{gather} \label{e2.1}
\begin{gathered}
  \frac {d}{dt}\big[Bu(t)+e(t,u(t))\big]+Au(t)
=   f(t,u(t))+\int^t_0  k(t,s,u(s))ds,\\\
t\in(0, a],\; t\neq t_k,
\end{gathered}\\
\label{e2.2}
  u(0)+\sum^n_{i=1}c_iu(t_i)=u_0 \\
\label{e2.3}
  \Delta u(t_k)=I_k(u_{t_k}),\quad k=1,2,\dots, m,
\end{gather}
where $0\leq t_1<t_2<\dots <t_p\leq a$, $B$ and $A$ are linear operators
with domains contained in a Banach space $X$ and ranges contained
in a Banach space $Y$ and the nonlinear operators
$f:I\times X \to Y$, $k:I^2\times X \to Y$,
$e:I\times X \to Y$ and $I_k:X\to Y$ are appropriate functions and
the symbol $\Delta u(t_k)$ represent the jump of the function $u$
at $t$, which is defined by $\Delta u(t_k)=u(t^+)-u(t^-)$.
Here $I =[0,a]$. In this paper, we establish the existence of a nonlinear
impulsive neutral integro-differential equation of Sobolev type with nonlocal
conditions using Schauder fixed point theorem.

To prove our main theorem we assume certain conditions on the operators $A$ and
$B$. Let $X$ and $Y$ be Banach spaces with norm $|\cdot|$ and $\|\cdot\|$
respectively. The operators $A:\mathcal{D}(A)\subset X \to Y$ and
 $B:\mathcal{D}(B)\subset X \to Y$ satisfy the following hypothesis:
\begin{itemize}
\item[(M1)] $A$ and $B$ are closed linear operators,
\item[(M2)] $\mathcal{D}(B)\subset \mathcal{D}(A)$ and $B$ is bijective,
\item[(M3)] $B^{-1} : Y \to  \mathcal{D}(B)$ is continuous.
\end{itemize}
 The hypothesis (M1)--(M3) and the closed graph theorem imply
the boundedness of the linear operator $AB^{-1} : Y \to Y$ and $-AB^{-1}$
 generates a uniformly continuous semigroup $S(t), t\geq 0$,
of bounded linear operators from $Y$ into $Y$ and so
$  \max_{t \in I} \|S(t)\|$ is finite. We denote
$  M=\max_{t\in I}\|S(t)\|$, $R=\|B^{-1}\|$.
 Let  $\mathbb B_r=\{x \in X:|x| \leq r \}$ and
$c= \sum^{p}_{i=1}|c_i|$.

 In this article, we assume that there exists an operator
 $E$ on $ \mathcal{D}(E)=X$ given by the formula
\[
 E=\Big[I+\sum^n_{i=1}c_iB^{-1}S(t_i)B\Big]^{-1}\quad \text{and}\quad
 Eu_0\in  \mathcal{D}(B),
\]
with
\begin{align*}
 &E\Big\{B^{-1}e(t,u(t))-B^{-1}S(t_i)e(0,u(0))
 +\int^{t_i}_0AS(t_i-s)B^{-1}e(s,u(s))ds \\
 &+ \int^{t_i}_0    B^{-1}S(t_i-s)[f(s,u(s))
 +  \int^s_0  k(s,\tau,u(\tau))d\tau]ds\\
 &-\sum_{0<t_k<t_i}  B^{-1}S(t_i-t_k)I_ku(t_k) \Big\}\in  \mathcal{D}(B),
\end{align*}
for $ i=1,2,\dots ,p$.

 The existence of $E$ can be observed from the following fact 
(see \cite{by1}).
Suppose that $\{S(t)\}$ is a $C_0$ semigroup of operators on $X$
such that $\|B^{-1}S(t_i)B\|\leq Ce^{-\delta t_i}(i=1,2,\dots ,n)$
 where $\delta$ is a positive constant and $C\leq1$.
If $ \sum^p_{i=1}|c_i|e^{-\delta t_i}<1/C$ then
 $\| \sum^p_{i=1}c_iB^{-1}S(t_i)B\|<1$. So such an operator $E$ exists on $X$.

\begin{definition} \label{def2.1} \rm
 A continuous solution $u$ of the integral equation
\begin{equation} \label{e2.4}
\begin{aligned}
u(t)
 &= B^{-1}S(t)BEu_0 +\sum^n_{i=1}c_iB^{-1}S(t)BE\\
&\quad\times\Big\{B^{-1}e(t,u(t))-B^{-1}S(t_i)e(0,u(0))
 -  \sum_{0<t_i<t}   B^{-1}S(t_i-t_k)I_ku(t_k)\Big\}\\
&\quad +\sum^n_{i=1}c_iB^{-1}S(t)BE
 \Big\{\int^{t_i}_0 B^{-1}S(t_i-s)
\Big[Ae(s,u(s))+f(s,u(s))\\
&\quad +\int^s_0 k(s,\tau,u(\tau))d\tau\Big]ds\Big\}
 +B^{-1}S(t)e(0,u(0))-B^{-1}e(t,u(t))\\
&\quad +\int^t_0  S(t-s)B^{-1}
 \Big[Ae(s,u(s))+f(s,u(s)) + \int^s_0  k(s,\tau,u(\tau))d\tau\Big]ds\\
&\quad +  \sum_{0<t_i<t}  B^{-1}S(t-t_k)I_ku(t_k)
\end{aligned}
\end{equation}
is said to be a mild solution of problem \eqref{e2.1}-\eqref{e2.3} on $I$.
\end{definition}

\begin{definition} \label{def2.2} \rm
A function $u$ is said to be a strong solution of  \eqref{e2.1}-\eqref{e2.3}
 on $I$ if $u$ is differentiable almost everywhere on $I$,
$u'\in L^1(I, X)$, $u(0)+ \sum^n_{i=1}c_iu(t_i)=u_0$ and
\begin{gather*}
\frac {d}{dt}\big[(Bu(t)+e(t,u(t))\big]+Au(t)
=   f(t,u(t))+\int^t_0 k(t,s,u(s))ds,\quad t\in (0, a],\; t\neq t_k\\
  \Delta u(t_k) =  I_k(u_{t_k}),\quad k=1,2,,\dots, m
\end{gather*}
almost everywhere on  $I$.
\end{definition}

\begin{remark} \label{rmk2.1} \rm
 A mild solution of the neutral integro-differential
 \eqref{e2.1}-\eqref{e2.3} satisfies
the condition \eqref{e2.2}, for \eqref{e2.4}
\begin{align*}
  u(0)
&= Eu_0+\sum^n_{i=1}c_iE\Big\{B^{-1}e(t,u(t))
-B^{-1}S(t_i)e(0,u(0))\\
&\quad -\sum_{0<t_i<t}B^{-1}S(t_i-t_k)I_ku(t_k)\Big\}\\
&\quad +\sum^n_{i=1}c_iE\Big\{\int^{t_i}_0S(t_i-s)B^{-1}
\Big[Ae(s,u(s))+f(s,u(s))\\
&\quad +\int^s_0k(s,\tau,u(\tau))d\tau \Big]ds\Big\}
\end{align*}
and
\begin{align*}
  u(t_j)
&= B^{-1}S(t_j)BEu_0
 +\sum^n_{i=1}c_iB^{-1}S(t_j)BE\Big\{B^{-1}e(t,u(t))
-B^{-1}S(t_i)e(0,u(0))\\
&\quad -\sum_{0<t_i<t}B^{-1}S(t_i-t_k)I_ku(t_k)\Big\}
\\
&\quad +\sum^n_{i=1}c_iB^{-1}S(t_j)BE\Big\{\int^{t_i}_0S(t_i-s)B^{-1}
[Ae(s,u(s))+f(s,u(s))\\
&\quad +\int^s_0k(s,\tau,u(\tau))d\tau]ds\Big\}
 +B^{-1}S(t_j)e(0,u(0))-B^{-1}e(t_j,u(t_j))\\
&\quad +\int^{t_j}_0 S(t_j-s)B^{-1}\Big[Ae(s,u(s))+f(s,u(s))
 +\int^s_0k(s,\tau,u(\tau))d\tau\Big]ds \\
&\quad +\sum_{0<t_i<t}B^{-1}S(t_j-t_k)I_ku(t_k).
\end{align*}
Therefore,
\begin{align*}
&  u(0)+\sum^n_{j=1}c_ju(t_j) \\
&= \Big[I+\sum^n_{j=1}c_jB^{-1}S(t_j)B\big]Eu_0
 +\Big[I+\sum^n_{j=1}c_jB^{-1}S(t_j)B\Big]
\sum^n_{i=1}c_iE\Big\{B^{-1}e(t,u(t))\\
&\quad -B^{-1}S(t_i)e(0,u(0))
 -\sum_{0<t_i<t}B^{-1}S(t_i-t_k)I_ku(t_k)\Big\}
\\
&\quad+\Big[I+\sum^n_{j=1}c_jB^{-1}S(t_j)B\Big]
\sum^n_{i=1}c_iE\Big\{\int^{t_i}_0S(t_i-s)B^{-1}[Ae(s,u(s))+f(s,u(s))\\
&\quad  +  \int^s_0k(s,\tau,u(\tau))d\tau]ds\Big\}
+\sum^n_{j=1}c_j\Big[ B^{-1}S(t_j)e(0,u(0))-B^{-1}e(t_j,u(t_j))\\
&\quad +\int^{t_j}_0S(t_j-s)B^{-1}\Big[Ae(s,u(s))
+f(s,u(s))+\int^s_0k(s,\tau,u(\tau))d\tau \Big]ds \\
&\quad+\sum_{0<t_i<t}B^{-1}S(t_j-t_k)I_ku(t_k)\Big]\\
&= u_0
\end{align*}
\end{remark}

To prove the existence result, we use the following hypotheses:
\begin{itemize}
\item[(M4)] The function $f:I\times X\to Y$ is continuous in $t$
and there exists a constant $L_f>0$ such that
$$\|f(t,u)\|\leq L_f,\ \text{for}\ t\in I\ \text{and}\ u \in X.
$$

\item[(M5)] The function $k:I^2\times X \to Y$ is continuous in $t$
and there exists a constant $L_k>0$ such that
$$\|k(t,s,u)\|\leq L_k,\ \text{for}\ s,t\in I\ \text{and}\ u \in X.
$$

\item[(M6)] The function $e:I\times X \to Y$ is continuous in $t$ and
there exist constants $L_e>0,\ L_0 >0$ and $L_1>0$ such that
\begin{gather*}
\|e(t,u(t))\|  \leq L_e,\quad \text{for $t\in I$ and $u \in X$}\\
\|e(0,u(0))\|  \leq L_0,\quad \text{for $t\in I$ and $u \in X$}\\
\|Ae(t,u(t))\| \leq L_1,\quad \text{for $t\in I$ and $u \in X$.}
\end{gather*}

\item[(M7)] The maps $I_k:X \to Y$ are continuous and there exists 
a constant $\mathcal I>0$ such that
$$
\|I_k(u)\|\leq \mathcal I, \quad \text{for $k\in \mathbb{N}$ and $y \in X$.}
$$

\item[(M8)]
\begin{align*}
&R\|BEu_o\|M+cR^2\|BE\|M[L_e+M\mathcal I+ML_0+aM(L_1+L_f+L_ka)]\\
&+RM[L_0+L_1+L_f+L_ka+a(L_1+G_1)+\mathcal I+RK_1]\leq r.
\end{align*}
\end{itemize}

\section{Main Results}

\begin{theorem} \label{thm3.1}
If  assumptions {\rm (M1)-(M7)} hold, then Problem \eqref{e2.1}-\eqref{e2.3}
 has a mild solution on $I$.
\end{theorem}

\begin{proof}
Let $E = \mathcal{C}(I,Y)$ and
$\mathcal{Y}_0=\{u \in Y : u(t)\in \mathbb B_r,\ t\in I \}$.
 Clearly, $\mathcal{Y}_0$ is a bounded closed convex subset of $Y$.
We define a mapping $F :\mathcal{Y}_0 \to \mathcal{Y}_0$ by
\begin{align*}
  (Fu)(t)
&= B^{-1}S(t)BEu_0
 +\sum^n_{i=1}c_iB^{-1}S(t)BE \Big\{B^{-1}e(t,u(t))\\
&\quad -B^{-1}S(t_i)e(0,u(0))
-\sum_{0<t_i<t}B^{-1}S(t_i-t_k)I_ku(t_k)\Big\}
\\
&\quad+\sum^n_{i=1}c_iB^{-1}S(t)BE\Big\{\int^{t_i}_0S(t_i-s)B^{-1}
[Ae(s,u(s))+f(s,u(s))\\
&\quad +\int^s_0k(s,\tau,u(\tau))d\tau]ds\Big\}
 +B^{-1}S(t)e(0,u(0))-B^{-1}e(t,u(t))\\
&\quad+\int^t_0S(t-s)B^{-1}
\Big[Ae(s,u(s))+f(s,u(s))+\int^s_0k(s,\tau,u(\tau))d\tau\Big]ds\\
&\quad +\sum_{0<t_i<t}B^{-1}S(t-t_k)I_ku(t_k)
\end{align*}
 Now we shown that $F:\mathcal{Y}_0\to \mathcal{Y}_0$ is continuous.
Let $\{u_n\}^{\infty}_0\subset \mathcal{Y}_0$ with $u_n\to u$ in $\mathcal{Y}_0$.
Then there is an integer $r$ such that $\|u_n(t)\|\leq r$, for all $n$
 and $t\in I$, so $u_n\in \mathbb B_r$ and $u\in \mathbb B_r$.
From the assumptions $(M_1)-(M_7)$, we have
\begin{itemize}
\item[(a)] $I_k$, $k=1,2,\dots ,p$ is continuous.

\item[(b)] $e(t,u_n(t))\to e(t, u(t))$, for $t\in I$ and since
\[
\|e(t,u_n(t))- e(t,u(t))\|<2[L_{e}+L_0].
\]

\item[(c)] $Ae(t,u_n(t))\to Ae(t, u(t))$, for $t\in I$ and since
\[
\|Ae(t,u_n(t))- Ae(t,u(t))\|<2[L_{1}+L_3].
\]

\item[(d)] $f(t,u_n(t))\to f(t,u(t))$, for $t\in I$ and since
\[
\|f(t,u_n(t))- f(t, u(t))\|<2 [L_f+F_0].
\]

\item[(e)] $k(t,s,u_n(s))\to k(t,s,u(s))$, for $t,\ s\in I$ and since
\[
\|k(t,s,u_n(s))- k(t,s,u(s))\|<2 [L_k+K_0].
\]
\end{itemize}
By the dominated convergence theorem, we have
\begin{align*}
 \|Fu_n-Fu\|
&\leq R^2Mc\|BE\|\{\|e(t,u_n(t))- e(t,u(t))\|\}\\
&\quad +R^2Mc\|BE\| \int^{t_i}_0 S(t_i-s)\Big[\{\|Ae(s,u_n(s))- Ae(s,u(s))\|\}\\
&\quad +\{\|f(s,u_n(s))- f(s,u(s))\|\}\\
&\quad +\int^s_0\{\|k(s,\tau,u_n(\tau))- k(s,\tau,u(\tau))\|\}d\tau\Big]ds\\
&\quad +R^2Mc\|BE\|\sum_{0<t_i<t}S(t_i-t_k)\{\|I_k(u_n(t_k))-I_k(u(t_k))\|\}\\
&\quad +R\{\|e(t,u_n(t))- e(t,u(t))\|\}\\
&\quad +RM\int^t_0\Big[\{\|Ae(s,u_n(s))- Ae(s,u(s))\|\}\\
&\quad +\{\|f(s,u_n(s))- f(s,u(s))\|\}\\
&\quad +\int^s_0\{\|k(s,\tau,u_n(\tau))- k(s,\tau,u(\tau))\|\}d\tau\Big] ds\\
&\quad +RM\sum_{0<t_i<t}\{\|I_k(u_n(t_k))-I_k(u(t_k))\|\}
\to 0 \quad \text{as } n\to \infty.
\end{align*}
Thus $F$ is continuous.
 Moreover, $F$ maps $\mathcal{Y}_0$ into a precompact subset of
 $\mathcal{Y}_0$. We prove that the set
 $\mathcal{Y}_0(t) = \{(Fu)(t) : u\in {\mathcal{Y}_0}\}$ is precompact
in $X$ for every fixed $t\in I$. We shall show that
$F(\mathcal{Y}_0) =\mathcal  Z = \{Fu : u\in \mathcal{Y}_0\}$ is
an equicontinuous family of functions.

 For $0 < s < t$, we have
\begin{align*}
&  \|(Fu)(t)-(Fu)(s)\| \\
&\leq \|B^{-1}(S(t)-S(s))B Eu_0\|\\
&\quad +\sum^n_{i=1}c_i\|B^{-1}(S(t)-S(s))BE\|\Big\{\|B^{-1}e(t,u(t))
-B^{-1}S(t_i)e(0,u(0))\\
&\quad -\sum_{0<t_i<t}B^{-1}S(t_i-t_k)I_ku(t_k)\|\Big\}
 +\sum^n_{i=1}c_i\|B^{-1}(S(t)-S(s))BE\|\\
&\quad \times\{\int^{t_i}_0\|S(t_i-s)B^{-1}
 \Big[Ae(s,u(s))+f(s,u(s))+\int^s_0k(s,\tau,u(\tau))d\tau\Big]\|ds\}
 \\
&\quad +\|B^{-1}(S(t)-S(s))e(0,u(0))\|+\|B^{-1}(e(t,u(t))-e(s,u(s)))\|\\
&\quad +\int^t_0  \|(S(t-\theta)-S(s-\theta)B^{-1}
 \big\{Ae(\theta,u(\theta))+f(\theta,u(\theta)) 
 +  \int^{\theta}_0  k(s,\tau,u(\tau)) d\tau\big\} \|d\theta\\
&\quad +\int^t_{\theta}\|S(t-\theta)B^{-1}
 \big[Ae(\theta,u(\theta))+f(\theta,u(\theta))+\int^{\theta}_0
 k(\theta,\tau,u(\tau))d\tau\big]\|d\theta\\
&\quad +\sum_{0<t_i<t}\|B^{-1}(S(t-s))I_ku(t_k)\|\\
&\leq  \Big\{R\|BEu_0\|+R^2\|BE\|[L_e+M\mathcal I+ML_0]c \\
&\quad +R^2Ma\|BE\|[L_1+L_f+L_ka]c+RL_0\Big\} \|s(t)-S(s)\|\\
&\quad +\{RL_0+RM[L_1+L_f+L_ka]\}|t-s|\\
&\quad +R(L_e+L_f+L_ka)\int^t_0\|S(t-\theta)-S(s-\theta)\|d\theta.
\end{align*}
The right hand side of the above inequality is independent of
$u\in \mathcal{Y}_0$ and tends to zero as $s\to t$ as a consequence
of the continuity of $S(t)$ in the uniform operator topology for
 $t > 0$ which follows from the compactness of $S(t)$, $t > 0$.
It is also clear that $Z$ is bounded in $Y$.
Thus by Arzela-Ascoli's theorem, $Z$ is precompact.
Hence by the Schauder fixed point theorem, $F$ has a fixed
point in $Y_0$ and any fixed point of $F$ is a mild solution of
 \eqref{e2.1}-\eqref{e2.3} on $I$ such that $u(t)\in X$, for $t\in I$.
\end{proof}

 Next we prove that the problem \eqref{e2.1}-\eqref{e2.3}
 has a strong solution.

\begin{theorem} \label{thm3.2}
 Assume that
\begin{itemize}
\item[(i)] Conditions {\rm (M1)--(M8)} hold.
\item[(ii)] $Y$ is a reflexive Banach space with norm $\|\cdot\|$.
\item[(iii)] $f : I\times X\to Y$ is continuous in $t$ on $I$ and
there exists  a constant $G_1 > 0$ such that
\[
\|f(t,u)-f(s,v)\|\leq G_1[|t-s|+\|u-v\|],\]
 for $t, s \in I$ and $u, v \in X$.

\item[(iv)] $k : I^2\times X \to Y$ is continuous in $t$ and there
 exists a constant $K_1 > 0$ such that
\[
\|k(t,\tau, u)- k(s,\tau, u)\|\leq K_1[|t-s|],
\]
 for  $\tau,s,t\in I$, $u \in X$,

\item[(v)] $e:I\times X \to Y$ is continuous and there exist 
constants $K >0$ and $K_1 >0$ such that
\begin{gather*}
\|Ae(t,u(t)-Ae(s,u(s))\|\leq L_2[|t-s|],\quad\text{for } s,t\in I,\; u \in X,\\
\|e(t,u(t)-e(s,u(s))\|\leq L[|t-s|],\quad\text{for } s,t\in I,\; u \in X.
\end{gather*}
\item[(vi)] $Eu_0\in \mathcal{D}(B)$,
\begin{align*}
& E\Big\{B^{-1}e(t,u(t))-B^{-1}S(t_i)e(0,u(0))\\
&\quad +\int^{t_i}_0B^{-1}S(t_i-s) 
\Big[Ae(s,u(s))+f(s,u(s))+\int^s_0k(s,\tau,u(\tau))d\tau\Big]\,ds\\
&-\sum_{0<t_i<t} B^{-1}S(t_i-t_k)I_ku(t_k) \Big\}\in D(B),
\end{align*}
for $i=1,2,\dots ,p$.
\end{itemize}
Then $u$ is a strong solution of problem \eqref{e2.1}--\eqref{e2.3} on $I$.
\end{theorem}

\begin{proof} Since all the assumptions of Theorem 3.1
are satisfied, then \eqref{e2.1}-\eqref{e2.3} has a mild solution belonging
to $\mathcal{C}(I,X)$. Now we shall show that $u$ is a strong solution
of \eqref{e2.1}-\eqref{e2.3} on $I$.
For any $t\in I$, we have
\begin{align*}
&\|u(t+h)-u(t)\|\\
&\leq \|B^{-1}[T(t+h)-T(t)]BEu_0
\\
&\quad +\sum^n_{i=1}c_i \|B^{-1}(S(t+h)-S(t))BE\|\Big\{\|B^{-1}e(t,u(t))
-B^{-1}S(t_i)e(0,u(0))
\\
&\quad -\sum_{0<t_i<t} B^{-1}S(t_i-t_k)I_ku(t_k)\|\Big\}
 +\sum^n_{i=1}c_i\|B^{-1}(S(t+h)-S(t))BE\|
\\ %first times
&\quad \times\Big\{\int^{t_i}_0\|S(t_i-s)B^{-1}[Ae(s,u(s))+f(s,u(s))
+\int^s_0k(s,\tau,u(\tau))d\tau]\|ds\Big\}\\
&\quad +\|B^{-1}(S(t+h)-S(t))e(0,u(0))\|+\|B^{-1}(e(t+h,u)-e(t,u))\|\\
&\quad +\int^h_0\|S(t+h-s)B^{-1}
\big[Ae(s,u(s))+f(s,u(s)+\int^{s}_0k(s,\tau,u(\tau)) d\tau\big]\|ds\\
&\quad +\int^{t+h}_h\|(S(t+h-s)B^{-1}\big[Ae(s,u(s))
 +f(s,u(s))+\int^{s}_0k(s,\tau,u(\tau)) d\tau\big]\|ds\\
&\quad +\int^t_0\|S(t-s)B^{-1}\big[Ae(s,u(s))+f(s,u(s))
 +\int^{s}_0k(s,\tau,u(\tau)) d\tau\big]\|ds\\
&\quad +\sum_{0<t_i<t}\|B^{-1}(S(t+h-t_k)-S(t-t_k))I_ku(t_k)\|\\
&\leq \|B^{-1}S(t)[S(h)-I]BEu_0 \|\\
&\quad +\sum^n_{i=1}c_i\|B^{-1}S(t)(S(h)-I)BE\|\Big\{\|B^{-1}e(t,u(t))\|
+\|B^{-1}S(t_i)e(0,u(0))\|
\\
&\quad +\sum_{0<t_i<t}\|B^{-1}S(t_i-t_k)I_ku(t_k)\|\Big\}
 +\sum^n_{i=1}c_i\|B^{-1}S(t)(S(h)-I)BE\|
\\ %second times
&\quad \times\Big\{\int^{t_i}_0\|S(t_i-s)B^{-1}\|
 \Big[\|Ae(s,u(s))\|+\|f(s,u(s))\| \\
&\quad +\int^s_0\|k(s,\tau,u(\tau))\|d\tau\Big]ds\Big\}
 +\|B^{-1}S(t)(S(h)-I)e(0,u(0))\|\\
&\quad +\|B^{-1}(e(t+h,u)-e(t,u))\|
 +\int^h_0\|(S(t+h-s)B^{-1}\|
\Big[\|Ae(s,u(s))\|\\
&\quad +\|f(s,u(s))\| +\int^{s}_0\|k(s,\tau,u(\tau))\| d\tau\Big]ds\\
&\quad +\int^{t+h}_h\|(S(t+h-s)B^{-1}\|
\Big[\|Ae(s,u(s))\|+\|f(s,u(s))\|\\
&\quad +\int^{s}_0\|k(s,\tau,u(\tau))\| d\tau\Big]ds
 +\int^t_0\|S(t-s)B^{-1}\|\Big[\|Ae(s,u(s))\| +\|f(s,u(s))\|\\
&\quad  +\int^{s}_0\|k(s,\tau,u(\tau))\| d\tau\Big]ds
 +\sum_{0<t_i<t}\|B^{-1}S(t-t_k)(S(h)-I)I_ku(t_k)\|
\\
&\leq \|B^{-1}S(t)[S(h)-I]BEu_0 \\
&\quad +\sum^n_{i=1}c_i\|B^{-1}S(t)(S(h)-I)BE\|\{\|B^{-1}e(t,u(t))\|
+\|B^{-1}S(t_i)e(0,u(0))\|
\\
&\quad +\sum_{0<t_i<t}\|B^{-1}S(t_i-t_k)I_ku(t_k)\|\}
 +\sum^n_{i=1}c_i\|B^{-1}S(t)(S(h)-I)BE\|
\\ % third times
&\quad \times\Big\{\int^{t_i}_0\|S(t_i-s)B^{-1}\|
\Big[\|Ae(s,u(s))\|+\|f(s,u(s))\|\\
&\quad  + \int^s_0\|k(s,\tau,u(\tau))\|d\tau\Big]ds\Big\}
+\|B^{-1}S(t)(S(h)-I)e(0,u(0))\|\\
&\quad +\|B^{-1}(e(t+h,u)-e(t,u))\|\\
&\quad +\int^h_0\|S(t+h-s)B^{-1}\|
\Big[\|Ae(s,u(s))\|+\|f(s,u(s))\|+\int^{s}_0\|k(s,\tau,u(\tau))\| d\tau\Big]ds\\
&\quad +\int^t_0\|S(t-s)B^{-1}\|\Big[\|Ae(s+h,u(s+h))-Ae(s,u(s))\|\\
&\quad +\|f(s+h,u(s+h))-f(s,u(s))\|
+\int^{s}_0\|k(s+h,\tau,u(\tau))-k(s,\tau,u(\tau))\| d\tau\Big]ds\\
&\quad +\sum_{0<t_i<t}\|B^{-1}S(t-t_k)(S(h)-I)I_ku(t_k)\|
\end{align*}
using our assumptions we observe that
\begin{align*}
&\|u(t+h)-u(t)\|\\
&\leq    R\|BEu_0\|Mh\|AB^{-1}\|
 +R^2Mhc\|BE\|[L_e+M\mathcal I+ML_0]\|AB^{-1}\|\\
&\quad +R^2Mhc\|BE\|[Ma(L_1+L_e+L_ka)]\|AB^{-1}\|\\
&\quad +RMhL_0+RLh+RMh(L_1+L_e+K_1a)+RMh\mathcal I\\
&\quad +RM\int^t_0\{L_2[h+\|u(s+h)-u(s)]+G_1[h+\|u(s+h)-u(s)]\}ds\\
&\quad +RM \int^t_0
\Big\{\int^s_0 \|k(s+h,\tau,u(\tau))-k(s,\tau,u(\tau))\|d\tau\\
&\quad + \int^{s+h}_s   \|k(s+h,\tau,u(\tau))\|d\tau\Big\}ds
\\
&\leq h\{R\|BEu_0\|M\|AB^{-1}\|\\
&\quad +R^2Mc\|BE\|[L_e+M\mathcal I+ML_0+Ma(L_1+L_e+L_ka)]\|AB^{-1}\|\\
&\quad +RM[L_0+L_1+L_e+L_ka+a(L_2+G_1+K_1+K_1a)+\mathcal I]+RL_k\}\\
&\quad +RM(L_2+G_1)\int^t_0\|u(s+h)-u(s)\|ds\\
&\leq Ph+\mathcal{Q}\int^t_0\|u(s+h)-u(s)\|ds
\end{align*}
where
\begin{gather*}
\begin{aligned}
\mathcal{P}&= R\|BEu_0\|M\|AB^{-1}+R^2Mc\|BE\|
 \Big[L_e+M\mathcal I+ML_0\\
&\quad +Ma(L_1+L_e+L_ka)\Big]\|AB^{-1}\| + RM\big[L_0+L_1\\
&\quad +L_e+L_ka+a(L_2+G_1+K_1+K_1a)+\mathcal I\big] +RL_k,
\end{aligned}\\
\mathcal{Q}= RM(L_2+G_1).
\end{gather*}
Hence by Gronwall's inequality
\[
\|u(t+h)-u(t)\|\leq  \mathcal{P} he^{\mathcal{Q}}, \quad\text{for } t\in I.
\]
Therefore, $u$ is Lipschitz continuous on $I$.
The Lipschitz continuity of $u$ on $I$ combined with (iii)--(v) implies that
\[
t\to f(t,u(t)),\quad
t\to e(t,u(t)), \quad
t\to\int^t_0k(t,s,u(s))ds.
\]
Hence, $u$ is strong solution of the problem \eqref{e2.1}-\eqref{e2.3}
 on $(0,a]$.
\end{proof}

\section{Example}
 Consider the  partial integro-differential equation of neutral type
\begin{equation} \label{e4.1}
\begin{gathered}
\begin{aligned}
& \frac{\partial}{\partial t}\Big[z(t,x)-z_{xx}(t,x)
 +\int^t_{-\infty} a_1(s-t)z_t(s,x)ds\Big]-z_{xx}(t,x) \\
&= \rho(t,z(t,x))+\int^t_0 a(t,s)z(s,x)ds,\quad x \in [0,\pi],\; t\in I,
\end{aligned}\\
z(t,0)= z(t,\pi)=0,\quad t\in I, \\
z(0,x)+\sum^p_{k=1}z(t_k,x)= z_0(x)\quad 0<t_1<t_2<\dots <t_p<b; \;
 x\in[0,a]
\\
\Delta z|_{t=t_i}= I_i(z(x))=(\gamma_i(z(x))+t_i)^{-1}, \quad z\in X,  \;
 1\leq i\leq p,
\end{gathered}
\end{equation}
where $a(t, s)$ is continuous such that $\|a(t,s)\|\leq L_1$ and the
constant $\gamma_i $ is small.

Let us take $X=Y={\mathcal L}^2[0,\pi]$ to be endowed with the usual
norm $\| \cdot \|_{\mathcal L_2}$. and let
\begin{gather*}
e(t,z) = \int^t_{-\infty}a_1(s-t)z_t(s,x)ds\\
f(t,z) = \rho(t,z(t,x))\\
\int^t_0k(t,s,z)ds = \int^t_0 a(t,s)z(s,x)ds\\
I_i(z(x)) = (\gamma_i(z(x))+t_i)^{-1}.
\end{gather*}
Define the operator $A:\mathcal{D}(A)\subset X \to Y$ and
$B:\mathcal{D}(B)\subset X \to Y$ by
\[
  Az=-z_{xx}, \quad  Bz=z-z_{xx},
\]
where each domain $\mathcal{D}(A)$ and $\mathcal{D}(B)$ is given by
\[
  \{z\in X: z,z_x \text{ are absolutely continuous, }
z_{xx}\in X, z(0)=z(\pi)=0\}.
\]
Then the above problem can be formulated abstractly as
\begin{gather*}
  \frac {d}{dt}\big[Bu(t)+e(t,u(t))\big]+Au(t)
=   f(t,u(t))+\int^t_0  k(t,s,u(s))ds,\\\
t\in(0, a],\; t\neq t_k,\\
  u(0)+\sum^n_{i=1}c_iu(t_i)=u_0 \\
  \Delta u(t_k)=I_k(u_{t_k}),\quad k=1,2,\dots, m,
\end{gather*}
Then $A$ and $B$ can be written, respectively, as
\begin{gather*}
  Az = \sum^{\infty}_{n=1}n^2\langle z,z_n\rangle z_n,  \; z\in \mathcal{D}(A)\\
Bz = \sum^{\infty}_{n=1}(1+n^2)\langle z,z_n\rangle z_n, \; z\in \mathcal{D}(B),
\end{gather*}
where $z_n(x)=\sqrt{2/\pi}\sin (nx)$, $n=1,2,\dots $, is the orthogonal
set of vectors of $A$. Furthermore for $z\in X$, we have
\begin{gather*}
  B^{-1}z = \sum^{\infty}_{n=1}\frac {1}{1+n^2}\langle z,z_n\rangle z_n,\\
-AB^{-1}z = \sum^{\infty}_{n=1}\frac {-n^2}{1+n^2}\langle z,z_n\rangle z_n, \\
S(t)z =  \sum^{\infty}_{n=1}\exp\Big(\frac {-n^2 t}{1+n^2}\Big)\langle z,z_n
\rangle z_n.
\end{gather*}
It is easy to see that $AB^{-1}$ generates a strongly continuous semigroup
$S(t)$ on $Y$ and $S(t)$ is compact such that $|S(t)|\leq e^{-t}$
for each $t>0$. For this $S(t),  B,  B^{-1}$ we assume that the operator
 $E$ exists. So all the conditions of the Theorem 3.1 are satisfied.
Hence the equation \eqref{e4.1} has a mild solution.

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\end{document}