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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 149, 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 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/149\hfil Monotone iterative method]
{Monotone iterative method for  semilinear impulsive
evolution equations of mixed type in Banach spaces}

\author[P. Chen, J. Mu\hfil EJDE-2010/149\hfilneg]
{Pengyu Chen, Jia Mu}  % in alphabetical order

\address{Pengyu Chen \newline
Department of Mathematics, Northwest Normal University, 
Lanzhou 730070, China}
\email{chpengyu123@163.com}

\address{Jia Mu \newline
Department of Mathematics, Northwest Normal University,
Lanzhou 730070, China}
\email{mujia05@lzu.cn}

\thanks{Submitted August 4, 2010. Published October 21, 2010.}
\subjclass[2000]{34K30, 34K45, 35F25}
\keywords{Initial value problem; lower and upper solution;
\hfill\break\indent impulsive integro-differential evolution equation;
 $C_0$-semigroup; cone}

\begin{abstract}
 We use a monotone iterative method in the presence of lower
 and upper solutions to discuss the existence and uniqueness
 of mild solutions for the initial value problem
 \begin{gather*}
    u'(t)+Au(t)= f(t,u(t),Tu(t)),\quad t\in J,\; t\neq t_k,\\
   \Delta u |_{t=t_k}=I_k(u(t_k)) ,\quad k=1,2,\dots ,m,\\
   u(0)=x_0,
 \end{gather*}
 where $A:D(A)\subset E\to E$ is a closed linear operator
 and $-A$ generates a strongly continuous semigroup
 $T(t)(t\geq 0)$ in $E$.
 Under wide monotonicity conditions and the non-compactness measure
 condition of the nonlinearity $f$, we obtain the
 existence of extremal mild solutions and a unique mild solution
 between lower and upper solutions requiring only that $-A$ generate a
 strongly continuous semigroup.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}

The theory of impulsive differential equations is a new and
important branch of differential equation theory, which has an
extensive physical, chemical, biological, and engineering background;
hence it has emerged as an important area of research in the previous
decades, see for example \cite{l1}.
Consequently, some basic results on impulsive differential equations
have been obtained and applications to different areas have been
considered by many authors;  see \cite{b1,g1,l2,l5} and their
references.

In this article, we use a monotone iterative method in the presence
of lower and upper solutions to discuss the existence of mild
solutions to the initial value problem (IVP) of first order
semilinear impulsive integro-differential evolution equations of
Volterra type in an ordered Banach space $E$
\begin{equation}
\begin{gathered}
    u'(t)+Au(t)= f(t,u(t),Tu(t)),\quad t\in J,\; t\neq t_k,\\
   \Delta u |_{t=t_k}=I_k(u(t_k)) ,\quad k=1,2,\dots ,m,\\
   u(0)=x_0,
 \end{gathered} \label{e1}
\end{equation}
where $A:D(A)\subset E\to E$ be a closed linear operator and $-A$
generates a strongly continuous semigroup ($C_0$-semigroup, in
short) $T(t)(t\geq 0)$ in $E$; $f\in C(J\times E\times E, E)$,
$J=[0,a]$, $a>0$ is a constant, $0<t_1<t_2<\dots<t_m<a$; $I_k\in
C(E,E)$ is an impulsive function, $k=1,2,\dots,m$; $x_0\in E$; and
\begin{equation}
Tu(t)=\int_{0}^tK(t,s)u(s)ds \label{e2}
\end{equation}
is a Volterra integral operator with integral kernel $K\in
C(\Delta,\mathbb{R}^{+})$, $\Delta=\{(t,s): 0\leq s\leq t\leq a\}$;
$\Delta u|_{t=t_k}$ denotes the jump of $u(t)$ at $t=t_k$; i.e.,
$\Delta u|_{t=t_k}=u(t_k^+)-u(t_k^-)$, where $u(t_k^+)$ and
$u(t_k^-)$ represent the right and left limits of $u(t)$ at $t=t_k$
respectively. Let $PC(J,E)=\{u:J\to E , u(t)$ is continuous at
$t\neq t_k$, and left continuous at $t=t_k$, and $u(t_k^+)$ exists,
$k=1,2,\dots ,m\}$. Evidently, $PC(J,E)$ is a Banach space with the
norm $ \| u\|_{PC}=\sup_{t\in J}\| u(t)\|$. Denote $E_1$ by the norm
$\|\cdot\|_1=\|\cdot\|+\| A\cdot\|$. Let
$J'=J\backslash\{t_1,t_2,\dots,t_m\}$. An abstract function $u\in
PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$ is called a solution of IVP
\eqref{e1} if $u(t)$ satisfies all the equalities in \eqref{e1}.

The monotone iterative technique in the presence of lower and upper
solutions is an important method for seeking solutions of
differential equations in abstract spaces. Recently, Du and
Lakshmikantham \cite{d2}, Sun and Zhao \cite{s1} investigated the
existence of minimal and maximal solutions to initial value problem
of ordinary differential equation without impulse by using the
method of upper and lower solutions and the monotone iterative
technique. Guo and Liu \cite{g1} developed the monotone iterative
method for impulsive integro-differential equations, and built a
monotone iterative method for impulsive ordinary
integro-differential equation for the IVP in $E$
\begin{equation}
\begin{gathered}
    u'(t)= f(t,u(t),Tu(t)),\quad t\in J,\; t\neq t_k,\\
   \Delta u |_{t=t_k}=I_k(u(t_k)) ,\quad k=1,2,\dots ,m,\\
   u(0)=x_0.
 \end{gathered} \label{e3}
\end{equation}
They proved that if \eqref{e3} has a lower solution $v_0$ and an
upper solution $w_0$ with $v_0\leq w_0$, and the nonlinear
term $f$ satisfies the monotonicity condition
\begin{equation}
\begin{gathered}
f(t,x_2,y_2)-f(t,x_1,y_1)\geq -M(x_2-x_1)-M_1(y_2-y_1), \\
 v_0(t)\leq x_1\leq x_2\leq w_0(t), \quad
Tv_0(t)\leq y_1\leq y_2\leq Tw_0(t), \quad \forall t\in J.
\end{gathered} \label{e4}
\end{equation}
They also required that the nonlinear term $f$ and the impulsive
function $I_k$ satisfy the noncompactness measure condition
\begin{gather}
\alpha(f(t,U,V))\leq L_1\alpha(U)+L_2\alpha(V),\label{e5}\\
\alpha(I_k(D))\leq M_k\alpha(D), \quad k=1,2,\dots,m,\label{e6}
\end{gather}
where $U, V, D\subset E$ are arbitrarily bounded sets, $L_1, L_2$
and $M_k$ are positive constants and satisfy
\begin{equation}
2a(M+L_1+aK_0L_2)+\sum_{k=1}^{m}M_k<1,\label{e7}
\end{equation}
where $K_0=\max_{(t,s)\in \Delta}K(t,s)$, $\alpha(\cdot)$ denotes
the Kuratowski measure of noncompactness in $E$. Then IVP\eqref{e3}
has minimal and maximal solutions between $v_0$ and $w_0$, which can
be obtained by a monotone iterative procedure starting from $v_0$
and $w_0$ respectively. Latter, Li and Liu \cite{l2} expanded the
results in \cite{g1}, they obtained the existence of the extremal
solutions to the initial value problem for impulsive ordinary
integro-differential equation  \eqref{e3}, but did not require the
noncompactness measure condition  \eqref{e6} for impulsive function
$I_k$ and the restriction condition \eqref{e7}.

On the other hand, some authors consider the initial
value problem of evolution equations, see \cite{b1,l3,l4,l5,s2,z1}
 and the
reference therein. But they all require the semigroup
$T(t)(t\geq 0)$ generated by $-A$ be equicontinuous semigroup,
this is a very strong assumption. In this paper, we will study
the initial value problem of impulsive integro-differential
evolution equation  \eqref{e1} not requiring the equicontinuity
of the semigroup $T(t)(t\geq 0)$ generated by $-A$.
We obtain the existence of extremal mild
solutions and a unique mild solution between lower and upper
solutions only requiring the semigroup $T(t)(t\geq 0)$ generated
by $-A$ is a  $C_0$-semigroup in $E$.

\section{Preliminaries}

Let $E$ be an ordered Banach space with the norm $\|\cdot\|$ and
partial order $\leq$, whose positive cone $P=\{x\in E: x\geq 0\}$ is
normal with normal constant $N$. Let $C(J,E)$ denote the Banach
space of all continuous $E$-value functions on interval $J$ with the
norm $\|u\|_{C}=\max_{t\in J}\|u(t)\|$. Evidently, $C(J,E)$ is also
an ordered Banach space reduced by the convex cone $P'=\{u\in
Y|u(t)\geq 0, t\in J\}$, and $P'$ is also a normal cone. Let
$\alpha(\cdot)$ denote the Kuratowski measure of noncompactness of
the bounded set. For the details of the definition and properties of
the measure of noncompactness,  see \cite{d1}. For any  $B\subset
C(J,E)$ and $t\in J$, set $B(t)=\{u(t) : u\in B\}\subset E$. If $B$
is bounded in $C(J,E),$ then $B(t)$ is bounded in $E$, and
$\alpha(B(t))\leq \alpha(B)$.

We first give the following lemmas  to be used in proving our main
results.

\begin{lemma}[\cite{h1}] \label{lem1}
 Let $B=\{u_n\}\subset PC(J,E)$ be a
bounded and countable set. Then  $\alpha(B(t))$ is Lebesgue integral
on $J$, and
$$
\alpha\Big(\Big\{\int_J u_n(t)dt: n\in \mathbb{N}\Big\}\Big)\leq
2\int_J \alpha(B(t))dt.
$$
\end{lemma}

Let $A:D(A)\subset E\to E$ be a closed linear operator
and $-A$ generates a $C_0$-semigroup $T(t)(t\geq 0)$ in $E$. Then
there exist constants $C>0$ and $\delta\in \mathbb{R}$ such that
$$
\| T(t)\|\leq Ce^{\delta t},\quad t\geq 0.
$$
Let $I=[t_0,T](t_0\geq 0)$, $T>t_0$ be a constant. It is well-known
\cite[Chapter 4, Theorem 2.9]{p1} that for any $x_0\in D(A)$ and
$h\in C^1(I,E)$, the initial value problem of the linear evolution
equation
\begin{equation}
\begin{gathered}
    u'(t)+Au(t)=h(t),\quad t\in I,\\
    u(t_0)=x_0,
 \end{gathered} \label{e8}
\end{equation}
has a unique classical solution $u\in C^1(I,E)\cap C(I,E_1)$ given
by
\begin{equation}
u(t)=T(t-t_0)x_0+\int_{t_0}^tT(t-s)h(s)ds, \quad t\in I.\label{e9}
\end{equation}
If $x_0\in E$ and $h\in C(I,E)$, the function $u$ given by
\eqref{e9} belongs to $C(I,E)$, we call it a mild solution \cite{p1}
of IVP\eqref{e8}.

To prove our main results, for any $h\in PC(J,E)$, we consider the
initial value problem (IVP) of linear impulsive evolution equation
in $E$
\begin{equation}
\begin{gathered}
    u'(t)+Au(t)=h(t),\quad t\in J',\\
   \Delta u |_{t=t_k}=y_k,\quad k=1,2,\dots ,m,\\
   u(0)=x_0,
 \end{gathered} \label{e10}
\end{equation}
where $y_k\in E$, $k=1,2,\dots,m$, $x_0\in E$.

\begin{lemma} \label{lem2}
Let $T(t)(t\geq 0)$ be a $C_0$-semigroup
in $E$ generated by $-A$, for any $h\in PC(J,E)$,
$x_0\in E$ and $y_k\in E$, $k=1,2,\dots,m$, then the linear IVP
\eqref{e10} has a unique mild solution $u\in PC(J,E)$ given by
\begin{equation}
u(t)=T(t)x_0+\int_0^t
T(t-s)h(s)ds + \sum _{0<t_k<t}T(t-t_k)y_k , \quad t\in
J.\label{e11}
\end{equation}
\end{lemma}

\begin{proof}
 Let $y_0=\theta$, $J_k=[t_{k-1},t_k]$, $k=1,2,\dots,m+1$,
where $t_0=0$ and $t_{m+1}=a$. If $u\in PC(J,E)$ is a mild solution
of IVP\eqref{e10}, then the restriction of $u$ on $J_k$ satisfies
the initial value problem of linear evolution equation without
impulse
\begin{gather*}
    u'(t)+Au(t)=h(t),\quad t_{k-1}<t\leq t_k,\\
    u(t_{k-1}^+)=u(t_{k-1})+y_{k-1}.
 \end{gather*}
Hence, on $(t_{k-1},t_k], u(t)$ can be expressed by
$$
u(t)=T(t-t_{k-1})(u(t_{k-1})+y_{k-1})+\int_{t_{k-1}}^t
T(t-s)h(s)ds.
$$
Iterating successively in the above equality
with $u(t_j)$, $j=k-1,k-2,\dots,1$, we see that $u$ satisfies
\eqref{e11}.

Inversely, we can verify directly that the function $u\in PC(J,E)$
defined by \eqref{e11} is a mild solution of IVP\eqref{e10}. Hence
IVP\eqref{e10} has a unique mild solution $u\in PC(J,E)$ given by
\eqref{e11}.
\end{proof}

\begin{definition} \label{def1}\rm
If a function $v_0\in PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$
satisfies
\begin{equation}
\begin{gathered}
v_0'(t)+Av_0(t)\leq f(t,v_0(t),Tv_0(t))\quad t\in J',\\
\Delta v_0 |_{t=t_k}\leq I_k(v_0(t_k)),\quad k=1,2,\dots ,m,\\
v_0(0)\leq x_0,
\end{gathered} \label{e12}
\end{equation}
we call it a lower solution of  IVP\eqref{e1}; if all the
inequalities in \eqref{e12} are reversed, we call it an upper
solution of IVP\eqref{e1}.
\end{definition}

\begin{definition} \label{def2}\rm
A $C_0$-semigroup  $T(t)(t\geq 0)$ in $E$ is called to be positive,
if order inequality $T(t)x\geq \theta$ holds for each
$x\geq \theta, x\in E$ and $t\geq 0$.
\end{definition}

It is easy to see that for any $M\geq 0$, $-(A+MI)$ also generates
a $C_0$-semigroup $S(t)=e^{-Mt}T(t)(t\geq 0)$ in $E$.
And $S(t)(t\geq 0)$ is a positive $C_0$-semigroup
if $T(t)(t\geq 0)$ is a positive $C_0$-semigroup (about the
positive $C_0$-semigroup,  see \cite{l3}).

Evidently, $PC(J,E)$ is also an ordered Banach space with the
partial order $\leq$ induced by the positive cone $K_{PC}=\{u\in
PC(J,E) : u(t)\geq 0, t\in J\}$. $K_{PC}$ is also normal with the
same normal constant $N$. For $v,w\in PC(J,E)$ with $v\leq w$, we
use $[v,w]$ to denote the order interval $\{u\in PC(J,E): v\leq
u\leq w\}$ in $PC(J,E)$, and $ [v(t),w(t)]$ to denote the order
interval $\{u\in E: v(t)\leq u(t)\leq w(t), t\in J\}$ in $E$.

\section{Main results}

\begin{theorem} \label{thm1}
Let $E$ be an ordered Banach space, whose positive cone $P$ is
normal, $A:D(A)\subset E\to E$ be a closed linear operator, the
positive $C_0$-semigroup $T(t)(t\geq 0)$ generated by $-A$ is
compact in $E, f\in C(J\times E\times E, E)$ and $I_k\in C(E,E)$,
$k=1,2,\dots,m$. Assume that  IVP\eqref{e1} has a lower solution
$v_0\in PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$  and an upper solution
$w_0\in PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$ with $v_0\leq w_0$.
Suppose also that the following conditions are satisfied:
\begin{itemize}
\item[(H1)] There exist a constant $M>0$ such that
$$
f(t,u_2,v_2)-f(t,u_1,v_1)\geq-M(u_2-u_1),
$$
for any $t\in J$, and $ v_0(t)\leq u_1\leq u_2\leq w_0(t)$,
$Tv_0(t)\leq v_1\leq v_2\leq Tw_0(t)$.

\item[(H2)] $I_k(u)$ is increasing on order
interval $[v_0(t),w_0(t)]$ for $t\in J, k=1,2,\dots,m$.
\end{itemize}
Then the IVP\eqref{e1} has minimal and maximal mild solutions
$\underline{u}$ and $\overline{u}$ between $v_0$ and $w_0$.
\end{theorem}

\begin{proof}
 Let $\overline{M}=\sup_{t\in J}\| S(t)\|$, we define the
mapping $Q:[v_0,w_0]\to PC(J,E)$ by
\begin{equation}
\begin{aligned}
Qu(t)&=S(t)x_0+\int_0^t S(t-s)(f(s,u(s),Tu(s))+Mu(s))ds\\
&\quad +\sum_{0<t_k<t}S(t-t_k)I_k(u(t_k)).
\end{aligned} \label{e13}
\end{equation}
Clearly, $Q:[v_0,w_0]\to PC(J,E) $ is continuous. By Lemma
\ref{lem2}, the mild solution of IVP\eqref{e1} is equivalent to the
fixed point of the operator $Q$. Since $S(t)(t\geq 0)$ is a positive
$C_0$-semigroup, combine this with the assumptions (H1) and (H2),
$Q$ is increasing in $[v_0,w_0]$.

We first show $v_0\leq Qv_0$,
$Qw_0\leq w_0$. Let $h(t)=v_0'(t)+Av_0(t)+Mv_0(t)$, by
\eqref{e12},
$h\in PC(J,E)$ and $h(t)\leq f(t,v_0(t),Tv_0(t))+Mv_0(t), t\in J'$.
By Lemma \ref{lem2}, we have
\begin{align*}
v_0(t)&=S(t)v_0(0)+\int_0^t S(t-s)h(s)ds + \sum
_{0<t_k<t}S(t-t_k)\Delta v_0 |_{t=t_k}\\
&\leq S(t)x_0+\int_0^t S(t-s)(f(s,v_0(s),Tv_0(s))+Mv_0(s))ds\\
&\quad +\sum_{0<t_k<t}S(t-t_k)I_k(v_0(t_k))\\
&=Qv_0(t),\quad t\in J,
\end{align*}
namely, $v_0\leq Qv_0$. Similarly, it can be show that
$ Qw_0\leq w_0$. So $Q:[v_0,w_0]\to[v_0,w_0]$ is a continuously
increasing operator.

Next, we show that $Q:[v_0,w_0]\to[v_0,w_0]$ is completely
continuous. Let
\begin{equation}
\begin{gathered}
(Wu)(t)=\int_0^t S(t-s)(f(s,u(s),Tu(s))+Mu(s))ds,\\
(Vu)(t)=\sum_{0<t_k<t}S(t-t_k)I_k(u(t_k)),\quad
u\in[v_0,w_0].
\end{gathered}\label{e14}
\end{equation}
On the one hand, we prove that for any $0<t\leq a$,
$Y(t)= \{(Wu)(t):u\in[v_0,w_0]\}$ is precompact in $E$. For
 $0<\epsilon<t$ and $u\in[v_0,w_0]$,
\begin{equation}
\begin{aligned}
(W_\epsilon u)(t)
&=\int_0^{t-\epsilon} S(t-s)(f(s,u(s),Tu(s))+Mu(s))ds \\
& =S(\epsilon)\int_0^{t-\epsilon}
S(t-s-\epsilon)(f(s,u(s),Tu(s))+Mu(s))ds.
\end{aligned}\label{e15}
\end{equation}
For any $u\in[v_0,w_0]$, by Assumption (H1), we have
\begin{align*}
f(t,v_0(t),Tv_0(t))+Mv_0(t)
&\leq f(t,u(t),Tu(t))+Mu(t)\\
& \leq f(t,w_0(t),Tw_0(t))+Mw_0(t).
\end{align*}
By the normality of the cone $P$, there exists $\overline{M_1}>0$
such that
$$
\| f(t,u(t),Tu(t))+Mu(t)\|\leq \overline{M_1},\quad u\in[v_0,w_0].
$$
By the compactness
of $S(\epsilon)$, $Y_\epsilon(t)=\{(W_\epsilon
u)(t):u\in[v_0,w_0]\}$ is precompact in $E$. Since
\begin{align*}
\| (Wu)(t)-(W_\epsilon u)(t)\|
& \leq \int_{t-\epsilon}^t\| S(t-s)\|\cdot\|
  f(s,u(s),Tu(s))+Mu(s)\| ds \\
&\leq \overline{M}~\overline{M_1}\epsilon,
\end{align*}
the set $Y(t)$ is totally bounded  in $E$. Furthermore, $Y(t)$ is
precompact in $E$.

On the other hand, for any
$0\leq t_1\leq t_2\leq a$, we have
\begin{equation}
\begin{aligned}
&\| (Wu)(t_2)-(Wu)(t_1)\|\\
&=\|\int_0^{t_1} (S(t_2-s)-S(t_1-s))(f(s,u(s),Tu(s))+Mu(s))ds \\
&\quad +\int_{t_1}^{t_2} S(t_2-s)(f(s,u(s),Tu(s))+Mu(s))ds\| \\
&\leq \overline{M_1}\int_0^{t_1}
\| S(t_2-s)-S(t_1-s)\| ds+\overline{M}~\overline{M_1}(t_2-t_1) \\
& \leq \overline{M_1}\int_0^a\| S(t_2-t_1+s)-S(s)\| ds
 +\overline{M}~\overline{M_1}(t_2-t_1).
\end{aligned} \label{e16}
\end{equation}
The right side of \eqref{e16} depends on $t_2-t_1$, but is
independen of $u$. As $T(\cdot)$ is compact, $S(\cdot)$ is also
compact and therefore $S(t)$ is continuous in the uniform operator
topology for $t>0$. So, the right side of \eqref{e16} tends to zero
as  $t_2-t_1\to 0$. Hence $W([v_0,w_0])$ is
equicontinuous function of cluster in $C(J,E)$.

The same idea can be used to prove the compactness of $V$.

For  $0\leq t\leq a$, since
$\{Qu(t):u\in[v_0,w_0]\}=\{S(t)x_0+(Wu)(t)+(Vu)(t):u\in[v_0,w_0]\}$,
and $Qu(0)=x_0$ is precompact in $E$. Hence, $Q([v_0,w_0])$ is
precompact in $C(J,E)$ by the Arzela-Ascoli theorem. So
$Q:[v_0,w_0]\to[v_0,w_0]$ is completely continuous. Hence, $Q$ has
minimal and maximal fixed points $\underline{u}$ and $\overline{u}$
in $[v_0,w_0]$, and therefore, they are the minimal and maximal mild
solutions of the IVP\eqref{e1} in $[v_0,w_0]$, respectively.
\end{proof}

\begin{theorem} \label{thm2}
Let $E$ be an ordered Banach space, whose positive cone $P$ is
normal, $A:D(A)\subset E\to E$ be a closed linear operator and $-A$
generates a positive $C_0$-semigroup $T(t)(t\geq 0)$ in $E$, $f\in
C(J\times E\times E, E)$ and $I_k\in C(E,E)$, $k=1,2,\dots,m$. If
the IVP\eqref{e1} has a lower solution $v_0\in PC(J,E)\cap
C^1(J',E)\cap C(J',E_1)$  and an upper solution $w_0\in PC(J,E)\cap
C^1(J',E)\cap C(J',E_1)$ with $v_0\leq w_0$, conditions {\rm (H1)}
and {\rm (H2)} hold, and satisfy
\begin{itemize}
\item[(H3)] There exist a constant $L>0$ such that for all $t\in J$,
$$
\alpha(\{f(t,u_n,v_n)\})\leq L(\alpha(\{u_n\})
+\alpha(\{v_n\})),
$$
and increasing
or decreasing sequences $\{u_n\}\subset[v_0(t),w_0(t)]$ and
 $\{v_n\}\subset[v_0(t),w_0(t)]$.
\end{itemize}
Then the IVP\eqref{e1} has minimal and maximal mild solutions
between $v_0$ and $w_0$, which can be obtained by a monotone
iterative procedure starting from $v_0$ and $w_0$ respectively.
\end{theorem}

\begin{proof} From  Theorem \ref{thm1}, we know
that $Q:[v_0,w_0]\to[v_0,w_0]$ is a continuously increasing
operator. Now, we define two
sequences $\{v_n\}$ and $\{w_n\}$ in $[v_0,w_0]$ by the iterative
scheme
\begin{equation}
v_n=Qv_{n-1},\quad
w_n=Qw_{n-1},\quad n=1,2,\dots.\label{e17}
\end{equation}
Then from the monotonicity of $Q$, it follows that
\begin{equation}
v_0\leq v_1\leq v_2\leq\dots\leq
v_n\leq \dots \leq w_n\leq \dots\leq w_2\leq w_1\leq
w_0.\label{e18}
\end{equation}
We prove that $\{v_n\}$ and $\{w_n\}$ are convergent
in $J$.
For convenience, let$ B=\{v_n: n\in \mathbb{N}\}$ and
$ B_0=\{v_{n-1}: n\in \mathbb{N}\}$.
Then $B=Q(B_0)$. Let $J_1'=[0,t_1]$, $J_k'=(t_{k-1},t_k]$,
$k=2,3,\dots m+1$.
 From $B_{0}=B\bigcup\{v_0\}$ it follows
that $ \alpha(B_{0}(t))=\alpha(B(t))$ for $t\in J$.
Let $\varphi(t):=\alpha(B(t)), t\in J$, Going
from $J_1'$ to $J_{m+1}' $interval by interval we show
that $\varphi(t)\equiv 0$ in $J$.

For $t\in J$, there exists a $J_k'$ such that
$t\in J_k'$. By \eqref{e2} and Lemma \ref{lem1}, we have that
\begin{align*}
\alpha(T(B_0)(t))
&=\alpha\Big(\Big\{\int_0^tK(t,s)v_{n-1}(s)ds: n\in \mathbb{N}\Big\}
 \Big) \\
&\leq \sum _{j=1}^{k-1}\alpha\Big(\Big\{\int_{t_{j-1}}^{t_j}K(t,s)v_{n-1}(s)ds:
  n\in \mathbb{N}\Big\}\Big)\\
&\quad +\alpha\Big(\Big\{\int_{t_{k-1}}^tK(t,s)v_{n-1}(s)ds:
  n\in \mathbb{N}\Big\}\Big)\\
&\leq 2K_0\sum _{j=1}^{k-1}\int_{t_{j-1}}^{t_j}\alpha(B_0(s))ds
 +2K_0\int_{t_{k-1}}^t\alpha(B_0(s))ds  \\
&=2K_0\sum_{j=1}^{k-1}\int_{t_{j-1}}^{t_j}\varphi(s)ds
 +2K_0\int_{t_{k-1}}^t\varphi(s)ds \\
&=2K_0\int_0^t\varphi(s)ds,
\end{align*}
and therefore,
\begin{equation}
\int_0^t\alpha(T(B_0)(s))ds\leq 2aK_0\int_0^t\varphi(s)ds.\label{e19}
\end{equation}
For $t\in J_1'$, from \eqref{e13}, using Lemma \ref{lem1},
assumption (H3) and \eqref{e19}, we have
\begin{align*}
\varphi(t)&=\alpha(B(t))=\alpha(Q(B_0)(t))\\
&=\alpha\Big(\Big\{S(t)x_0+\int_0^t
S(t-s)(f(s,v_{n-1}(s),Tv_{n-1}(s))+Mv_{n-1}(s))ds\Big\}\Big)\\
&\leq2\overline{M}\int_0^t\alpha(\{f(s,v_{n-1}(s),Tv_{n-1}(s))
 +Mv_{n-1}(s)\})ds  \\
&\leq 2\overline{M}\int_0^t(L(\alpha(B_{0}(s))+\alpha(Q(B_{0})(s)))
 +M\alpha(B_{0}(s)))ds  \\
&\leq2\overline{M}(L+M+2aLK_0)\int_0^t\varphi(s)ds.
\end{align*}
Hence by the Bellman inequality, $\varphi(t)\equiv 0$ in $J_1'$. In
particular, $\alpha(B(t_1))=\alpha(B_{0}(t_1))=\varphi(t_1)=0$, this
implies that $B(t_1)$ and $B_{0}(t_1)$ are precompact
in $E$. Thus $I_1(B_{0}(t_1))$  is precompact
in $E$, and $\alpha(I_1(B_{0}(t_1)))=0$.

Now, for $t\in J_2'$, by \eqref{e13} and the above argument for
$t\in J_1'$, we have
\begin{align*}
\varphi(t)&=\alpha(B(t))=\alpha(Q(B_0)(t))     \\
&=\alpha\Big(\Big\{S(t)x_0+\int_0^t S(t-s)(f(s,v_{n-1}(s),Tv_{n-1}(s))
 +Mv_{n-1}(s))ds\\
&\quad +S(t-t_1)I_1(v_{n-1}(t_1))\Big\}\Big) \\
&\leq2\overline{M}(L+M+2aLK_0)\int_0^t\varphi(s)ds\\
&=2\overline{M}(L+M+2aLK_0)\int_{t_1}^t\varphi(s)ds.
\end{align*}
Again by Bellman inequality, $\varphi(t)\equiv 0$ in $J_2'$, from
which we obtain that $\alpha(B_{0}(t_2))=0$ and
$\alpha(I_2(B_{0}(t_2)))=0$.

Continuing such a process interval by interval up to $J_{m+1}'$, we
can prove that $\varphi(t)\equiv 0$ in every $J_k',
k=1,2,\dots,m+1$. Hence, for any $t\in J, \{v_n(t)\}$ is precompact,
and $\{v_n(t)\}$ has a convergent subsequence. Combing this with the
monotonicity \eqref{e18}, we easily prove that $\{v_n(t)\}$ itself
is convergent, i.e., $\lim_{n\to \infty}v_n(t)=\underline{u}(t)$,
$t\in J$. Similarly$, \lim_{n\to \infty}w_n(t)=\overline{u}(t)$,
$t\in J$.

Evidently $\{v_n(t)\}\in PC(J,E)$, so $\underline{u}(t)$ is bounded
integrable in every $J_k$, $k=1,2,\dots,{m+1}$. Since for any
$t\in J_k$,
\begin{align*}
v_n(t)&=Qv_{n-1}(t)\\
&=S(t)x_0+\int_0^t
S(t-s)(f(s,v_{n-1}(s),Tv_{n-1}(s))+Mv_{n-1}(s))ds\\
&\quad +\sum_{0<t_i<t}S(t-t_i)I_i(v_{n-1}(t_i)),
\end{align*}
letting $n\to\infty$, by the Lebesgue dominated convergence theorem,
for all $t\in J_k$, $k=1,2,\dots,{m+1}$, we have
$$
\underline{u}(t)=S(t)x_0+\int_0^t
S(t-s)(f(s,\underline{u}(s),T\underline{u}(s))+M\underline{u}(s))ds+
\sum_{0<t_i<t}S(t-t_i)I_i(\underline{u}(t_i)),
$$
and $\underline{u}(t)\in C(J_k,E)$, $k=1,2,\dots,{m+1}$.
So, for $t\in J$, we have
$$
\underline{u}(t)=S(t)x_0+\int_0^t
S(t-s)(f(s,\underline{u}(s),T\underline{u}(s))+M\underline{u}(s))ds+
\sum_{0<t_k<t}S(t-t_k)I_k(\underline{u}(t_k)).
$$
Therefore, $\underline{u}(t)\in PC(J,E)$, and
$\underline{u}=Q\underline{u}$. Similarly, $\overline{u}(t)\in
PC(J,E)$, and $\overline{u}=Q\overline{u}$. Combing this with
monotonicity \eqref{e18}, we see that $v_0\leq \underline{u}\leq
\overline{u}\leq w_0$. By the monotonicity of $Q$, it is easy to see
that $\underline{u}$ and $\overline{u}$ are the minimal and maximal
fixed points of $Q$ in $[v_0,w_0]$. Therefore, $\underline{u}$ and
$\overline{u}$ are the minimal and maximal mild solutions of the
IVP\eqref{e1} in $[v_0,w_0]$, respectively.
\end{proof}

\begin{corollary} \label{coro1}
Let $E$ be an ordered Banach space, whose positive cone $P$ is
regular, $A:D(A)\subset E\to E$ be a closed linear operator and $-A$
generates a positive $C_0$-semigroup $T(t)(t\geq 0)$ in $E$, $f\in
C(J\times E\times E, E)$ and $I_k\in C(E,E)$, $k=1,2,\dots,m$. If
the IVP\eqref{e1} has a lower solution $v_0\in PC(J,E)\cap
C^1(J',E)\cap C(J',E_1)$  and an upper solution $w_0\in PC(J,E)\cap
C^1(J',E)\cap C(J',E_1)$ with $v_0\leq w_0$, and conditions {\rm
(H1)} and {\rm (H2)} are satisfied, then  the IVP\eqref{e1} has
minimal and maximal mild solutions between $v_0$ and $w_0$, which
can be obtained by a monotone iterative procedure starting from
$v_0$ and $w_0$ respectively.
\end{corollary}

Now we discuss the uniqueness of the mild solution to
 IVP\eqref{e1} in $[v_0,w_0]$. If we replace the assumption
(H3) by the assumption:
\begin{itemize}
\item[(H4)] There exist positive
constants $\overline{C}$ and $\overline{L}$ such that
$$
f(t,u_2,v_2)-f(t,u_1,v_1)\leq
 \overline{C}(u_2-u_1)+\overline{L}(v_2-v_1),
$$
for any $t\in J$, and $ v_0(t)\leq u_1\leq u_2\leq w_0(t)$,
$Tv_0(t)\leq v_1\leq v_2\leq Tw_0(t)$,
\end{itemize}
We have the following unique existence result.

\begin{theorem} \label{thm3}
 Let $E$ be an ordered Banach space, whose
positive cone $P$ is normal, $A:D(A)\subset E\to E$ be a closed
linear operator and $-A$ generates a positive $C_0$-semigroup
$T(t)(t\geq 0)$ in $E$, $f\in C(J\times E\times E, E)$ and $I_k\in
C(E,E)$, $k=1,2,\dots,m$. If  the IVP\eqref{e1} has a lower solution
$v_0\in PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$  and an upper solution
$w_0\in PC(J,E)\cap C^1(J',E)\cap C(J',E_1)$ with $v_0\leq w_0$,
such that conditions {\rm (H1), (H2), (H4)} hold, then the
IVP\eqref{e1} has a unique mild solution between $v_0$ and $w_0$,
which can be obtained by a monotone iterative procedure starting
from $v_0$ or $w_0$.
\end{theorem}

\begin{proof} We firstly prove
that (H1) and (H4) imply (H3).
For $t\in J$, let $\{u_n\}\subset [v_0(t),w_0(t)]$ and
$\{v_n\}\subset [ Tv_0(t),Tw_0(t)]$ be two increasing sequences.
For $m,n\in \mathbb{N}$ with $m>n$, by (H1) and (H4),
\begin{align*}
\theta&\leq f(t,u_m,v_m)-f(t,u_n,v_n)+M(u_m-u_n) \\
&\leq (M+\overline{C})(u_m-u_n)+\overline{L}(v_n-v_m).
\end{align*}
By this and the normality of cone $P$, we have
\begin{align*}
&\| f(t,u_m,v_m)-f(t,u_n,v_n)\|\\
&\leq N\| (M+\overline{C})(u_m-u_n)+\overline{L}(v_n-v_m)\|
+M\| u_m-u_n\| \\
&\leq (N(M+\overline{C})+M)\| u_m-u_n\|+N\overline{L}\| v_n-v_m\|.
\end{align*}
From this inequality and the definition of the measure of
noncompactness, it follows that
\begin{align*}
\alpha(\{f(t,u_n,v_n)\})
&\leq (N(M+\overline{C})+M)\alpha(\{u_n\})
 +N\overline{L}\alpha(\{v_n\})\\
&\leq L_1(\alpha(\{u_n\})+\alpha(\{v_n\})),
\end{align*}
where $L_1=\max\{(N(M+\overline{C})+M),N\overline{L}\}$. If
$\{u_n\}$ and $\{v_n\}$ are two decreasing sequences, the above
inequality is also valid. Hence (H3) holds. Therefore, by Theorem
\ref{thm2}, the IVP\eqref{e1} has minimal and maximal mild solutions
$\underline{u}$ and$ \overline{u}$ between $v_0$ and $w_0$. By the
proof of Theorem \ref{thm2}, \eqref{e17} and \eqref{e18} are valid.
Going from $J_1'$ to $J_{m+1}'$ interval by interval we show that
$\underline{u}(t)\equiv\overline{u}(t)$ in every $J_k'$.

For $t\in J_1'$, by \eqref{e13} and assumption (H4), we have
\begin{align*}
\theta &\leq \overline{u}(t)-\underline{u}(t)
 =Q\overline{u}(t)-Q\underline{u}(t)  \\
&=\int_0^t S(t-s)\big[f(s,\overline{u}(s),T\overline{u}(s))
 -f(s,\underline{u}(s),T\underline{u}(s))
 +M(\overline{u}(s)-\underline{u}(s))\big]ds \\
&\leq \overline{M}(M+\overline{C}
 +a\overline{L}K_0)\int_0^t(\overline{u}(s)-\underline{u}(s))ds.
\end{align*}
 From this and the normality of cone $P$ it follows that
$$
\| \overline{u}(t)-\underline{u}(t)\|
\leq N\overline{M}(M+\overline{C}+a\overline{L}K_0)
\int_0^t\|\overline{u}(s)-\underline{u}(s)\| ds.
$$
By this and Bellman inequality, we obtained
that $\underline{u}(t)\equiv\overline{u}(t)$ in $J_1'$.

For $t\in J_2'$, since $I_1(\overline{u}(t_1))=I_1(\underline{u}(t_1))$,
using \eqref{e13} and completely the same argument as above
for $t\in J_1'$, we can prove that
\begin{align*}
\| \overline{u}(t)-\underline{u}(t)\|
&\leq N\overline{M}(M+\overline{C}+a\overline{L}K_0)
 \int_0^t\|\overline{u}(s)-\underline{u}(s)\| ds \\
& =N\overline{M}(M+\overline{C}+a\overline{L}K_0)
 \int_{t_1}^t\|\overline{u}(s)-\underline{u}(s)\| ds.
\end{align*}
Again, by the Bellman inequality, we obtain
that $\underline{u}(t)\equiv\overline{u}(t)$ in $J_2'$.

Continuing such a process interval by interval up to $J_{m+1}'$, we
see that $\underline{u}(t)\equiv\overline{u}(t)$ over the whole of
$J$. Hence, $\widetilde{u}:=\underline{u}=\overline{u}$ is the
unique mild solution of the IVP\eqref{e1} in $[v_0,w_0]$, which can
be obtained by the monotone iterative procedure \eqref{e18} starting
from $v_0$ or $w_0$.
\end{proof}

If lower solution and upper solutions for the IVP\eqref{e1} do not
exist, then we have the following result.

\begin{theorem} \label{thm4}
 Let $E$ be an ordered Banach space, whose
positive cone $P$ is normal, $A:D(A)\subset E\to E$ be a
closed linear operator and $-A$ generates a
positive $C_0$-semigroup $T(t)(t\geq 0)$ in $E$,
$f\in C(J\times E\times E, E)$ and $I_k\in C(E,E)$, $k=1,2,\dots,m$.
If there exist $a>0$, $x_0\in D(A)$, $x_0\geq \theta$, $y_k\in D(A)$,
$y_k\geq \theta$, $k=1,2,\dots,m$, $h\in PC(J,E)\cap C^1(J',E)$ and
$h(t)\geq \theta$, such that
\begin{gather*}
f(t,x,Tx)\leq ax+h(t), \quad
I_k(x)\leq y_k, x\geq \theta; \\
f(t,x,Tx)\geq ax-h(t), \quad I_k(x)\geq-y_k, x\leq \theta.
\end{gather*}
Then we have:
\begin{itemize}
\item[(i)] If the $C_0$-semigroup $T(t)(t\geq 0)$ generated
by $-A$ is compact in $E$, and conditions {\rm (H1)} and {\rm (H2)}
are satisfied, then  the IVP\eqref{e1} has minimal and maximal mild
solutions.

\item[(ii)] If conditions {\rm (H1), (H2), (H3)} are satisfied,
 then the IVP\eqref{e1} has minimal and maximal mild solutions.

\item[(iii)] If the positive cone $P$ is regular, and
conditions {\rm (H1)} and {\rm (H2)} are satisfied, then
 the IVP\eqref{e1} has minimal and maximal mild solutions.

\item[(iv)] If conditions {\rm (H1), (H2), (H4)} are satisfied,
then the IVP\eqref{e1} has a unique mild solution.
\end{itemize}
\end{theorem}

\begin{proof} Firstly, we consider the IVP of linear
impulsive evolution equation in $E$
\begin{equation}
\begin{gathered}
    u'(t)+Au(t)-au(t)=h(t),\quad t\in J',\\
   \Delta u |_{t=t_k}=y_k,\quad k=1,2,\dots ,m,\\
   u(0)=x_0.
 \end{gathered} \label{e20}
\end{equation}
Since $-A+aI$ generates a positive $C_0$-semigroup
$S(t)=e^{at}T(t)(t\geq 0)$ in $E$. So, by  \cite[Chapter 4, Theorem
2.9]{p1} and Lemma \ref{lem2}, we know that  IVP\eqref{e20} has a
unique positive classical solution $ \widetilde{u}\in PC(J,E)\cap
C^1(J',E)\cap C(J',E_1)$. Let $v_0=-\widetilde{u}$,
$w_0=\widetilde{u}$, it is easy to see that $v_0$ and $w_0$ are
lower solution and upper solution of
 the IVP\eqref{e1} respectively. So, our
conclusions (i), (ii), (iii) and (iv) follow from the
Theorem \ref{thm1}, Theorem \ref{thm2}, Corollary \ref{coro1} and
Theorem \ref{thm3} respectively.
\end{proof}

\section{Applications}

Consider the IVP of impulsive parabolic partial differential
equation
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}+A(x,D)u(t)
= f(x,t,u(t),Tu(t)),\quad x\in\Omega, t\in J, t\neq t_k,\\
   \Delta u |_{t=t_k}=I_k(u(x,t_k)) ,\quad x\in\Omega, k=1,2,\dots ,m,\\
   Bu=0,\quad (x,t)\in \partial\Omega\times J,\\
   u(x,0)=\varphi(x),\quad x\in\Omega,
 \end{gathered} \label{e21}
\end{equation}
where $J=[0,a]$, $0<t_1<t_2<\dots<t_m<a$, integer
$N\geq 1$, $\Omega\subset\mathbb{R}^N$ is a bounded domain with a
sufficiently smooth boundary $\partial\Omega$,
$$
A(x,D)=-\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}(x)
 \frac{\partial^2}{\partial x_i \partial
y_j}+\sum _{i=1}^{N}a_i(x)\frac{\partial}{\partial
x_i}+a_0(x)
$$
is a strongly elliptic operator of second
order, coefficient functions $a_{ij}(x), a_i(x)$ and $a_0(x)$ are
H\"{o}lder continuous in $\Omega, Bu=b_0(x)u+\delta \frac{\partial
u}{\partial n}$ is a regular boundary operator
on $\partial\Omega, f:\overline{\Omega}\times
J\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is
continuous, $I_k:\mathbb{R}\to\mathbb{R}$ are also
continuous, $k=1,2,\dots ,m$.

Let $E=L^p(\Omega)$ with $p>N+2$, $P=\{u\in L^p(\Omega): u(x)\geq 0,
a.e.~x\in \Omega\}$, and define the operator $A$ as follows:
$$
D(A)=\{u\in W^{2,p}(\Omega): Bu=0\}, \quad
 Au=A(x,D)u.
$$
Then $E$ is a Banach space, $P$ is a regular cone of $E$, and $-A$
generates a positive and analytic $C_0$-semi-group $T(t)(t\geq 0)$
in $E$ (see \cite{l3,l4,p1}). So, the problem \eqref{e21} can be
transformed into the IVP \eqref{e1}. To solve the IVP\eqref{e21}, we
also need following assumptions:

(a) Let $f(x,t,0,0)\geq 0$, $I_k(0)\geq 0$,
$\varphi(x)\geq 0$, $x\in\Omega$, and there exists a function
$w=w(x,t)\in PC(J,E)\cap C^{2,1}(\overline{\Omega}\times J)$,
such that
\begin{gather*}
    \frac{\partial w}{\partial t}+A(x,D)w\geq f(x,t,w,Tw),\quad
(x,t)\in\Omega\times J, t\neq t_k,\\
   \Delta w|_{t=t_k}\geq I_k(w(x,t_k)) ,\quad x\in\Omega,\;
  k=1,2,\dots ,m,\\
   Bw=0,\quad (x,t)\in\partial\Omega\times J,\\
   w(x,0)\geq\varphi(x),\quad x\in\Omega.
 \end{gather*}

(b) There exists a constant $M>0$ such that
$$
f(x,t,x_2,y_2)-f(x,t,x_1,y_1)
\geq-M(x_2-x_1),$$for any $t\in J$, and
$ 0\leq x_1\leq x_2\leq w(x,t), 0\leq y_1\leq y_2\leq Tw(x,t)$.

(c) For any $u_1, u_2\in [0,w(x,t)]$ with $u_1\leq u_2$, we have
$$
I_k(u_1(x,t_k))\leq I_k(u_2(x,t_k)),\quad x\in\Omega,\;
 k=1,2,\dots ,m.
$$

Assumption (a) implies that $v_0\equiv 0$ and $w_0\equiv w(x,t)$ are
lower and upper solutions of the IVP\eqref{e1} respectively, and
from (b) and (c), it is easy to verify that conditions (H1) and (H2)
are satisfied. So, from Corollary \ref{coro1}, we have the following
result.

\begin{theorem} \label{thm5}
 If the assumptions (a), (b) and (c) are
satisfied, then the IVP\eqref{e21} has minimal and maximal mild
solutions between $0$ and $w(x,t)$, which can be obtained by a
monotone iterative procedure starting from $0$ and $w(x,t)$
respectively.
\end{theorem}

\subsection*{Acknowledgements}
 The author is deeply indebted to the
anonymous referee for his/her valuable suggestions which improve
the presentation of this paper.
This work is supported by grants 10871160 from the  NNSF of
China, 0710RJZA103 from the the NSF of Gansu Province,
 and by Project of NWNU-KJCXGC-3-47.


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\end{document}