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

\AtBeginDocument{{\noindent\small {\em
Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 111, pp. 1--8.\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-2005/111\hfil Existence and uniqueness solutions]
{Existence and uniqueness of mild and classical solutions of
impulsive evolution equations}
\author[A. Anguraj, M. M. Arjunan\hfil EJDE-2005/111\hfilneg]
{Annamalai Anguraj, Mani Mallika Arjunan}

\address{Annamalai Anguraj \hfill\break
Department of Mathematics\\
P.S.G. College of Arts and Science\\
Coimbatore - 641 014, Tamilnadu, India}
\email{angurajpsg@yahoo.com}

\address{Mani Mallika Arjunan \hfill\break
Department of Mathematics\\
P.S.G. College of Arts and Science\\
Coimbatore- 641 014, Tamilnadu, India}
\email{arjunphd07@yahoo.co.in}

\date{}
\thanks{Submitted June 15, 2005. Published October 17, 2005.}
\subjclass[2000]{34A37, 34G60, 34G20}
 \keywords{Semigroups; evolution equations; impulsive conditions}

\begin{abstract}
 We consider the non-linear impulsive evolution equation
 \begin{gather*}
 u'(t)=Au(t)+f(t,u(t),Tu(t),Su(t)),  \quad  0<t<T_0, \;  t\neq t_i,\\
 u(0) =u_0,\\
 \Delta u(t_i) =I_i(u(t_i)),\quad  i=1,2,3,\dots,p.
 \end{gather*}
 in a Banach space $ X$, where $ A $ is the infinitesimal generator
 of a $C_0 $ semigroup. We study the existence
 and uniqueness of the mild solutions of the evolution equation
 by using semigroup theory and then show that the mild solutions
 give rise to a classical solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

 The theory of impulsive differential equations is an important
branch of differential equations, which has an extensive physical
background. Impulsive differential equations arise frequently in the
modelling many physical systems whose states are subjects to sudden
change at certain moments, for example, in population biology,
the diffusion of chemicals, the spread of heat, the radiation of
electromagnetic waves and etc.,(see \cite{g1,l1,s1}).

 Existence of solutions of impulsive differential equation of the form
\begin{gather}
u'(t)= f(t,u(t),Tu(t),Su(t)),  \quad  0<t<T_0,  \;  t\neq t_i, \label{e1} \\
u(0) =u_0,   \label{e2}  \\
\Delta u(t_i) =I_i(u(t_i)),\quad  i=1,2,3,\dots,p. \label{e3}
\end{gather}
has been studied by many authors \cite{g2,g3,z1}. In the special case
where $f$ is uniformly continuous, Guo and Liu \cite{g2} established
existence theorems of maximal and minimal solutions for \eqref{e1}--\eqref{e3}
with strong conditions. Guo and Liu \cite{l4}, Liu \cite{l5}
 obtained the same
conclusion applying the monotone iterative technique when $ f $ does not
contain integral operator $S $ in \eqref{e1}.
But they did not obtain a unique solution for \eqref{e1}--\eqref{e3}.
 Recently, in the special case where \eqref{e1}--\eqref{e3} has no
impulses, Liu \cite{l5} obtained a unique solution
by the monotone iterative technique with coupled upper and lower
 quasi-solutions when $ f=f(t,u,u,Tu,Su)$.
A similar conclusion was obtained by Liu \cite{l3}. However, one of the
require assumptions in \cite{g2,l4,l5} is that $ f $ satisfies some compactness-type
conditions, which as we know is difficult and inconvenient to verify
in abstract spaces. Rogovchenko \cite{r1}, studied the existence and
uniqueness of the classical solutions by the successive approximations
for the evolution equation with an unbounded operator $ A $; i.e, equations
of the form
$$
u'(t) =Au(t)+f(t,u(t)),\quad t>0,\; t\neq t_i,
$$
with impulsive condition in \eqref{e2}-\eqref{e3}, where $A $ is sectorial
operator with some conditions given on the fractional operators
$ A^\alpha,\alpha \geq 0 $ .

Liu \cite{l2} studied the existence of mild solutions of the impulsive
evolution equation
$$
u'(t) =Au(t)+f(t,u(t)),  \quad   0<t<T_0,\;  t\neq t_i,
$$
where $ A $ is the infinitesimal generator of $ C_0 $ semigroup with
the impulsive condition in \eqref{e2}-\eqref{e3} by using semigroup theory.

In this paper we study the existence and uniqueness of mild solutions
for the nonlinear impulsive evolution equation
\begin{gather*}
u'(t) =Au(t)+f(t,u(t),Tu(t),Su(t)), \quad  0<t<T_0,\;  t\neq t_i,\\
u(0) =u_0,\\
\Delta u(t_i) =I_i(u(t_i)), \quad i=1,2,3,\dots,p.
\end{gather*}
in a Banach space $ X $, where $ A $ is the infinitesimal generator
of $ C_0 $ semigroup $ \{G(t)| t \geq 0\} $. Then we prove
 that the existence and uniqueness of mild solutions give rise
to the existence and uniqueness of classical solutions if $ f $ which
is continuously differentiable.

\section{Preliminaries and Hypotheses}

Let $ X $ be a Banach space. Let
$ PC([0,T_0],X)$ consist of functions $u$ that are a map from $ [0,T_0] $
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 $i=1,2,3,\dots p$. Evidently $ PC([0,T_0],X) $ is a Banach space
with the norm
\begin{equation} \label{e4}
\|u \|_{PC} = \sup_{t\in [0,T_0]}\| u(t)\|\,.
\end{equation}
Consider the impulsive evolution equation of the form
\begin{gather}
u'(t) =Au(t)+f(t,u(t),Tu(t),Su(t)), \quad  0<t<T_0,\; t \neq t_i, \label{e5}\\
u(0) =u_0, \label{e6}\\
\Delta u(t_i) =I_i(u(t_i)),\quad  i=1,2,3,\dots,p. \label{e7}
\end{gather}
in a Banach space $ X,$ where
$ f \in C([0,T_0] \times X \times X \times X,X)$,
\begin{gather}
Tu(t) = \int_0^t{K(t,s)u(s)}\mathrm{d}s,\   K \in C[D,R^+], \label{e8}\\
Su(t) = \int_0^{T_0}{H(t,s)u(s)}\mathrm{d}s,\ H \in C[D_0,R^+], \label{e9}
\end{gather}
where
$ D=\{(t,s)\in R^2 : 0\leq s\leq t\leq T_0 \}$,
$ D_0=\{(t,s)\in R^2 : 0\leq t,s\leq T_0 \}$
and  $ 0<t_1<t_2<t_3<\dots <t_i<\dots <t_p<T_0$,
$\Delta u(t_i)= u(t_i^+)-u(t_i^-)$.

We assume the following hypotheses:
\begin{itemize}
\item[(H1)] $f : [0,T_0] \times X \times X \times X \to X $, and
$ I_i : X \to X $,  $i =1,2,\dots,p$. are continuous and there exists
constants $ L_1,L_2,L_3 >0$, $h_i >0$, $i =1,2,3,\dots,p$. such that
\begin{gather}
\begin{aligned}
&\|f(t,u_1,u_2,u_3)-f(t,v_1,v_2,v_3)\| \\
& \leq L_1\|u_1-v_1\|+L_2\|u_2-v_2\|
 + L_3\|u_3-v_3\|, \quad t\in [0,T_0],\; u,v \in X;
\end{aligned}\label{e10}\\
\|I_i(u)-I_i(v)\|   \leq h_i\|u-v\|, \quad u,v \in X.\label{e11}
\end{gather}
\end{itemize}
 Let $ G(\cdot) $ be the $ C_0 $ semigroup generated by
the unbounded operator $ A $. Let $ B(X) $ be the Banach space
of all linear and bounded operators on $ X $.
Let
\begin{gather*}
M =\max_{t\in [0,T_0]} \|G(t)\|_{B(X)}, \quad
L = \max\{L_1,L_2,L_3\}\,.
\\
K^* =\sup_{t\in [0,T_0]} \int_0^t{|K(s,t)|}\mathrm{d}t <\infty,\quad
H^* = \sup_{t\in [0,T_0]} \int_0^{T_0}{|H(s,t)|}\mathrm{d}t
<\infty
\end{gather*}
\begin{itemize}
\item[(H2)] The constants $ L,L_1,L_2,L_3,K^*,H^* $ satisfy the inequality
$$
M \Big[LT_0(1+K^*+H^*)+ \sum_{i=1}^p h_i\Big]<1
$$
\end{itemize}

\section{Existence Theorems}

\subsection{Mild solution}
A function  $ u(\cdot)\in PC([0,T_0],X) $ is a mild solution of equations
\eqref{e5}--\eqref{e7} if it satisfies
\begin{equation}
\begin{aligned}
u(t)& =G(t)u_0+\int_0^t{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s\\
&\quad + \sum_{0<t_i<t}{G(t-t_i)I_i(u(t_i))}, \  0\leq t\leq T_0.
\end{aligned}\label{e12}
\end{equation}

\begin{theorem} \label{thm3.1}
 Assume that (H1)-(H2) are satisfied. Then for every $u_0\in X$,
 for  $ t\in [0,T_0] $ the equation
\begin{equation}
u(t) =G(t)u_0+\int_0^t{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s
+ \sum_{0<t_i<t}{G(t-t_i)I_i(u(t_i))},\label{e13}
\end{equation}
has a unique solution.
\end{theorem}

\begin{proof}
 Let $ u_0\in X $ be fixed. Define an operator $ F $ on $ PC([0,T_0],X) $ by
\begin{equation}
\begin{aligned}
(Fu)(t)& = G(t)u_0+\int_0^t{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s\\
&\quad +\sum_{0<t_i<t}{G(t-t_i)I_i(u(t_i))},\ 0\leq t\leq T_0
\end{aligned} \label{e14}
\end{equation}
Then it is clear that $ F :PC([0,T_0],X) \to  PC([0,T_0],X) $.
Now we show that $ F $ is contraction.
For any $ u,v \in PC([0,T_0],X)$,
\begin{equation}
\begin{aligned}
\|(Fu)(t)-(Fv)(t)\|
& \leq \int_0^t \|G(t-s)\|_{B(X)}\|f(s,u(s),Tu(s),Su(s))\\
&\quad -f(s,v(s),Tv(s),Sv(s))\|\mathrm{d}s\\
&\quad + \sum_{0<t_i<t}{\|G(t-t_i)\|_{B(X)}\|I_i(u(t_i))-I_i(v(t_i))\|}
\end{aligned}\label{e15}
\end{equation}
Using Assumption (H1) and definition of $M$, we have
\begin{equation}
\begin{aligned}
&\|(Fu)(t)-(Fv)(t)\|\\
& \leq  M \int_0^t \|f(s,u(s),Tu(s),Su(s)
 -f(s,v(s),Tv(s),Sv(s)\|\mathrm{d}s\\
&\quad +M \sum_{0<t_i<t}{\|I_i(u(t_i))-I_i(v(t_i))\|} \\
&\leq M\Big[\int_0^t L_1\|u-v\|+L_2\|Tu-Tv\|
 +L_3\|Su-Sv\|\Big]\mathrm{d}s\\
&\quad +M \|u-v\|_{PC}\sum_{i=1}^p h_i
\end{aligned} \label{e16}
\end{equation}
Now,
\begin{equation}
\begin{aligned}
\int_0^tL_2\|Tu-Tv\|\mathrm{d}s
& \leq L_2 \int_0^t \int_0^s{\|K(s,\tau)\|\|u(\tau)-v(\tau)\|}
\mathrm{d}\tau \mathrm{d}s\\
& \leq L_2 \int_0^t{\|u(s)-v(s)\|}\int_0^s{\|K(s,\tau)\|}
\mathrm{d}\tau \mathrm{d}s\\
& \leq L_2\|u(t)-v(t)\|\int_0^t{K^*}\mathrm{d}s\\
&  \leq L_2\|u-v\|_{PC}K^*T_0
\end{aligned}\label{e17}
\end{equation}
Similarly,
\begin{equation}
\int_0^t{L_3\|Su-Sv\|}\mathrm{d}s\leq L_3\|u-v\|_{PC}H^*T_0\label{e18}
\end{equation}
Substitute \eqref{e17}, \eqref{e18} in  \eqref{e16}, we have
\begin{equation}
\begin{aligned}
&\|(Fu)(t)-(Fv)(t)\| \\
& \leq M\Big[L_1T_0\|u-v\|_{PC}+L_2T_0\|u-v\|_{PC}K^*
 +L_3T_0\|u-v\|_{PC}H^*\\
&\quad +M \|u-v\|_{PC} \sum_{i=1}^p h_i\Big]\\
& \leq M\Big[L_1T_0+L_2T_0K^*+L_3T_0H^*
+\sum_{i=1}^p h_i\Big]\|u-v\|_{PC}\,.
\end{aligned}\label{e19}
\end{equation}
Using  the definition of $L$, we have
\begin{equation}
\begin{aligned}
\|Fu-Fv\|_{PC}&=\max_{t\in[0,T_0]}\|Fu(t)-Fv(t)\|\\
&\leq M\Big[LT_0(1+K^*+H^*)+\sum_{i=1}^ph_i\Big]
\|u-v\| _{PC}.
\end{aligned} \label{e20}
\end{equation}
Now from  Assumption (H2), we have
\begin{equation}
\|(Fu)(t)-(Fv)(t)\|\leq \|u-v\|_{PC},\quad u,v\in PC([0,T_0],X).\label{e21}
\end{equation}
Therefore, $ F $ is a contraction operator on $ PC([0,T_0],X)$.
This completes the proof.
\end{proof}

Next we study the classical solutions.
First we give the definition.

\subsection{Classical solutions}
A classical solution of Equations \eqref{e5}--\eqref{e7}
is a function $ u(\cdot)$ in
$PC ([0,T_0],X)\cap C^1 ((0,T_0)\backslash \{t_1,t_2,\dots,t_p\},X)$,
$u(t)\in D(A)$ for
 $ t\in (0,T_0)\backslash \{t_1,t_2,\dots,t_p \}$,
which satisfies equations \eqref{e5}--\eqref{e7} on $ [0,T_0] $.

 To prove the main theorem we need the following two Lemmas.

\begin{lemma} \label{lem3.2.1}
 Consider the evolution equation
\begin{gather}
u'(t) =Au(t)+f(t,u(t),Tu(t),Su(t)), \quad t_0<t<T_0,\label{e22}\\
u(0) =u_0,\label{e23}
\end{gather}
If $ u_0\in D(A)$, and $ f \in C^1((0,T_0)\times X\times X\times X,X)$,
then it has a unique classical solution, which satisfies
\begin{equation}
u(t)= G(t-t_0)u_0+\int_{t_0}^t{G(t-s)f(s,u(s),Tu(s),Su(s)}\mathrm{d}s,\quad
 t\in [t_0,T_0).\label{e24}
\end{equation}
\end{lemma}
The above lemma can be proved easily using the \cite[Theorem 6.1.5]{p1}.

\begin{lemma} \label{lem3.2.2}
Assume hypotheses (H1)-(H2) are satisfied. Then for the unique
classical solution $ u(\cdot)=u(\cdot,u_0)$ on $ [0,t_1)$ of
equations \eqref{e5}-\eqref{e6} without the impulsive conditions
(guaranteed by Lemma \ref{lem3.2.1}), one can define $u(t_1) $ in such a way
that $ u(\cdot)$ is left continuous at $ t_1 $ and $ u(t_1)\in D(A)$.
\end{lemma}

\begin{proof} Consider the following evolution equation without
the impulsive condition on $ (0,T_0)$,
\begin{gather*}
w'(t) =Aw(t)+f(t,w(t),Tw(t),Sw(t)),\qquad  0<t<T_0, \label{e25}\\
w(0) = w_0, \label{e26}
\end{gather*}
 From Lemma \ref{lem3.2.1}, there is a classical solution given by
\begin{equation}
w(t)=G(t)u_0+\int_0^{t}{G(t-s)f(s,w(s),Tw(s),Sw(s))}\mathrm{d}s,
\quad t\in [0,T_0).\label{e27}
\end{equation}
and $ w(t)\in D(A) $ for $ t\in [0,T_0)$.
Next, applying Lemma \ref{lem3.2.1} to the function  $u(\cdot) $, one has,
for $ t\in [0,t_1)\subset [0,T_0)$,
\begin{equation}
u(t)=G(t)u_0+\int_0^{t}{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s,
\quad t\in [0,t_1),\label{e28}
\end{equation}
Now, we can define
\begin{equation}
u(t_1)=G(t_1)u_0+\int_0^{t_1}{G(t_1-s)f(s,u(s),Tu(s),Su(s))}
\mathrm{d}s,\label{e29}
\end{equation}
So that $ u(\cdot) $ is left continuous at $ t_1$.
 Then apply Lemma \ref{lem3.2.1} on $ [0,t_1]$ to get
\begin{equation}
u(t)=w(t), \quad t\in [0,t_1]. \label{e30}
\end{equation}
Thus we have,
$u(t_1)=w(t_1)\in D(A)$ %,\label{e31}
which completes the proof.
\end{proof}
Before proving the main theorem,  we prove the following theorem.

\begin{theorem} \label{thm3.2.1}
 Assume that $ u_0\in D(A)$, $ q_i\in D(A)$, $i=1,2,\dots,p$. and that
$ f \in C^1((0,T_0)\times X\times X\times X,X) $. Then the impulsive equation
\begin{gather}
u'(t) =Au(t)+f(t,u(t),Tu(t),Su(t)), \quad  0<t<T_0,\; t \neq t_i,\label{e32}\\
u(0) =u_0, \label{e33}\\
\Delta u(t_i) =q_i, \quad  i=1,2,3,\dots,p.\label{e34}
\end{gather}
has a unique classical solution $ u(\cdot) $ which,
for $ t\in[0,T_0)$, satisfies
\begin{equation}
u(t) =G(t)u_0+\int_0^t{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s
 +\sum_{0<t_i<t}{G(t-t_i)q_i}.\label{e35}
\end{equation}
\end{theorem}

\begin{proof}
Consider the interval on $ J_1=[0,t_1)$, apply Lemma \ref{lem3.2.1}
 to the equation
\begin{gather}
u'(t) =Au(t)+f(t,u(t),Tu(t),Su(t)),
 \quad 0<t<t_1, \label{e36} \\
u(0) =u_0,\label{e37}
\end{gather}
has a unique classical solution $ u_1(\cdot) $ which satisfies
\begin{equation}
u_1(t)=G(t)u_0+\int_0^t{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s,
 \quad t\in [0,t_1),\label{e38}
\end{equation}
Now, define
\begin{equation}
u_1(t_1)=G(t_1)u_0+\int_0^{t_1}{G(t_1-s)f(s,u(s),Tu(s),Su(s))}
\mathrm{d}s ,\label{e39}
\end{equation}
Applying Lemma \ref{lem3.2.2}, we see that $ u_1(\cdot) $ is left continuous at
$ t_1$, and $ u_1(t_1)\in D(A)$. Next on $ J_2=[t_1,t_2)$,  consider
the equation
\begin{gather}
u'(t) =Au(t)+f(t,u(t),Tu(t),Su(t)), \qquad t_1<t<t_2,\label{e40}\\
u(t_1) =u_1(t_1)+q_1,\label{e41}
\end{gather}
Since $ u_1(t_1)+q_1\in D(A), $ \ once again we can use Lemma \ref{lem3.2.1} again
 to get a unique classical solution $ u_2(\cdot) $ which satisfies
\begin{equation}
u_2(t)=G(t-t_1)[u(t_1)]+\int_ {t_1}^{t}{G(t-s)f(s,u(s),Tu(s),Su(s))}
\mathrm{d}s.\label{e42}
\end{equation}
Now, define
\begin{equation}
u_2(t_2)=G(t_2-t_1)[u_1(t_1)+q_1]
 +\int_ {t_1}^{t_2}{G(t_2-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s.\label{e43}
\end{equation}
Therefore, $ u_2(\cdot)$ is left continuous at $ t_2 $ and
$ u_2(t_2)\in D(A) $ using Lemma \ref{lem3.2.2}.
Continuous in this process on $ J_k=[t_{k-1},t_k)$, $(k=3,4,5,\dots,p+1)$
to get the classical solutions
\begin{equation}
\begin{aligned}
u_k(t)& =G(t-t_{k-1})[u_{k-1}(t_{k-1})+q_{k-1}]\\
&\quad +\int_ {t_{k-1}}^{t}{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s.
\end{aligned}\label{e44}
\end{equation}
for $ t\in [t_{k-1},t_k)$, with $ u_i(\cdot)$ left continuous at $ t_i $
 and $  u_i(t_i)\in D(A)$,   $ i=1,2,\dots,p$.
Now, define
\begin{equation}
u(t)=
\begin{cases}
u_1(t) & 0\leq t\leq t_1, \\
u_k(t) & t_k-1<t\leq t_k, k=2,3,\dots,p.\\
u_{k+1}(t) & t_p<t<t_{p+1}=T_0. \\
\end{cases} \label{e45}
\end{equation}
Therefore, $ u(\cdot) $ is the unique classical solution of
equations \eqref{e32}--\eqref{e34}. Using induction method we show
that \eqref{e35} is satisfied on $ [0,T_0)$.
First \eqref{e35} is satisfied on $ [0,t_1]$. If \eqref{e35} is satisfied
on $ (t_{k-1},t_k]$, then for $ t\in(t_k,t_{k+1}]$,
\begin{align*}
&u(t)\\
& =u_{k+1}(t) =G(t-{t_k})[u_k(t_k)+q_k]\\
&\quad +\int_{t_k}^t{G(t-s)f(s,u_{k+1}(s),Tu_{k+1}(s),Su_{k+1}(s))}
\mathrm{d}s \\
&= G(t-t_k)[G(t_k)u_0+\int_0^{t_k}{G(t_k-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s\\
&\quad +\sum_{0<t_i<t_k}{G(t_k-t_i)q_i+q_k}]\\
&\quad +\int_{t_k}^t{G(t-s)f(s,u_{k+1}(s),Tu_{k
+1}(s),Su_{k+1}(s))} \mathrm{d}s \\
&= G(t-t_k)G(t_k)u_0
 +\int_0^{t_k}{G(t-s)f(s,u(s),Tu(s),Su(s))} \mathrm{d}s\\
&\quad + \sum_{0<t_i<t_k}{G(t-t_i)q_i}+G(t-t_k)q_k+\int_{t_k}^t
{G(t-s)f(s,u(s),Tu(s),Su(s))} \mathrm{d}s \\
&=G(t)u_0+\int_0^t{G(t-s)f(s,u(s),Tu(s).Su(s))}\mathrm{d}s
+\sum_{0<t_i<t}{G(t-t_i)q_i}. %\label{e46}
\end{align*}
Thus  \eqref{e35} is also true on $(t_k,t_{k+1}]$.
Therefore,  \eqref{e35} is true on $ [0,T_0)$, which completes the proof.
\end{proof}
Next Theorem gives the proof of the main theorem.

\begin{theorem} \label{thm3.2.2}
 Assume the hypotheses (H1)-(H2) are satisfied.
Let $ u(\cdot)= u(\cdot,u_0)$ be the unique mild solution of
\eqref{e5}--\eqref{e7} obtained in  Theorem \ref{thm3.1}.
Assume that $ u_0\in D(A)$, $ I_i(u(t_i))\in D(A)$, $i=1,2,\dots,p$,
and that $ f \in C^1((0,T_0)\times X\times X\times X,X) $. Then
$u(\cdot)$ gives rise to a unique classical solution of
\eqref{e5}--\eqref{e7}.
\end{theorem}

\begin{proof}. Let $ u(\cdot) $ be the mild solution. Let
$ q_i=I_i(u(t_i))$, $i=1,2,\dots,p$. Then from Theorem \ref{thm3.2.1},
equation \eqref{e32}-\eqref{e34} has a unique classical solution
$ w(\cdot) $ which satisfies for $ t\in [0,T_0)$
\begin{align*}
w(t)& =G(t)u_0+\int_0^t{G(t-s)f(s,w(s),Tw(s),Sw(s))} \mathrm{d}s\notag\\
&\quad +\sum_{0<t_i<t}{G(t-t_i)I_i(w(t_i))} %\label{e47}
\end{align*}
Since $ u(\cdot)$ is the mild solution of  \eqref{e5}--\eqref{e7},
for $ t\in [0,T_0]$,
\begin{align*}
u(t)& =G(t)u_0+\int_0^t{G(t-s)f(s,u(s),Tu(s),Su(s))}\mathrm{d}s\notag\\
&\quad +\sum_{0<t_i<t}{G(t-t_i)I_i(u(t_i))} %\label{e48}
\end{align*}
Suppose  $u(t) $ and $w(t) $ are two mild solutions, then
\[
w(t)-u(t) =\int_0^t G(t-s)[f(s,w(s),Tw(s),Sw(s))
 -f(s,u(s),Tu(s),Su(s))] \mathrm{d}s. %\label{e49}
\]
Since by Theorem \ref{thm3.1}, the mild solutions is unique, it follows that
 $ w(t)-u(t)=0 $. This implies that $ w(\cdot)=u(\cdot) $.
This shows that $u(\cdot)$ is also a classical solution.
This completes the proof.
\end{proof}

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\end{document}