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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 241, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/241\hfil Existence of mild solutions]
{Existence of mild solutions to partial differential equations
with non-instantaneous impulses}

\author[P. Chen, X. Zhang, Y. Li \hfil EJDE-2016/241\hfilneg]
{Pengyu Chen, Xuping Zhang, Yongxiang Li}

\address{Pengyu Chen (corresponding author)\newline
Department of Mathematics,
Northwest Normal University,
Lanzhou 730070, China}
\email{chpengyu123@163.com, Phone +86 0931 7971111}

\address{Xuping Zhang \newline
Department of Mathematics,
Northwest Normal University,
Lanzhou 730070, China}
\email{lanyu9986@126.com}

\address{Yongxiang Li \newline
Department of Mathematics,
Northwest Normal University,
Lanzhou 730070, China}
\email{liyx@nwnu.edu.cn}

\thanks{Submitted March 7, 2016. Published September 1, 2016.}
\subjclass[2010]{34K45, 47D06}
\keywords{Evolution equation; initial value problem; non-instantaneous impulse;
\hfill\break\indent 
   mild solution; equicontinuous semigroup; existence of solutions}

\begin{abstract}
 In this article, we  study the existence of piecewise-continuous mild solutions
 for the initial value problems for a class of semilinear evolution equations.
 These equations have non-instantaneous impulses in Banach spaces 
 and the corresponding solution semigroup is noncompact.  We assume that the 
 nonlinear term  satisfies certain local growth condition and a noncompactness 
 measure condition.  Also we assume the non-instantaneous impulsive  
 functions satisfy some Lipschitz conditions. An example is given to illustrate
 our results.
\end{abstract}

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

\section{Introduction}

In this article, we study the existence of piecewise-continuous mild solutions 
(PC-mild solutions) for the initial value problem (IVP)  of
semi-linear evolution equations with non-instantaneous impulses in Banach space $E$,
\begin{equation}
\begin{gathered}
    u'(t)+Au(t)= f(t,u(t)),\quad t\in\cup_{k=0}^{m}(s_k,t_{k+1}],\\
   u(t)=\gamma_k(t,u(t)),\quad t\in\cup_{k=1}^{m}(t_k,s_{k}],\\
   u(0)=u_0,
 \end{gathered} \label{e1.1}
\end{equation}
where $A:\mathcal {D}(A)\subset E\to E$ is a closed linear operator,
$-A$ is the infinitesimal generator of a strongly continuous semigroup
 ($C_0$-semigroup) $T(t)(t\geq 0)$ in $E$, $0<t_1<t_2<\dots<t_m<t_{m+1}:=a$,
$a>0$ is a constant, $s_0:=0$ and $s_k\in (t_k,t_{k+1})$ for each
$k=1,2,\dots,m$, $f:[0,a]\times E\to E$ is a given nonlinear function
satisfying some assumptions, $\gamma_k:(t_k,s_{k}]\times E\to E$
is non-instantaneous impulsive function for all $k=1,2,\dots,m$, $u_0\in E$.

The theory of instantaneous impulsive differential equations describes 
processes which experience a sudden change in their states at certain moments. 
Processes with such a character arise naturally and often, especially
 in phenomena studied in physics, chemistry, biology,
population and dynamics, engineering and economics, see the monographs by
 Benchohra et al \cite{bhn06}, Lakshmikantham et al \cite{lbs89} and the 
papers of Guo \cite{g08}, Li and Liu \cite{ll07} for more comments and citations. 
Particularly, the theory of instantaneous impulsive evolution equations has 
become more important in resent years because of its
wide applicability in control, mechanics, electrical engineering, biological 
and medical fields. The theory of instantaneous impulsive evolution equations 
in Banach spaces has been emerging as an important area of investigation 
in the last few decades. For more details on this theory and its applications, 
we refer to the  the references 
\cite{abh09,a06,cl11,chenl13,fl10,llx09,llx11},  where numerous properties 
of their solutions are studied and detailed bibliographies are given.


The most important feature of instantaneous impulsive ordinary and partial 
differential equations is this class of equations is linked to
their utility in simulating processes and phenomena subject to short time 
perturbations during their evolution, and the perturbations are performed
 discretely and their duration is negligible in
comparison with the total duration of the processes and phenomena when construct 
mathematical models.
In short, these ordinary and partial differential equations with instantaneous 
impulses consider basically problems for which the impulses are abrupt and 
instantaneous. However, one can see that the models with instantaneous
impulses could not explain the certain dynamics of evolution processes in
 pharmacotherapy. Just as  pointed out by Hern\'{a}ndez and O'Regan in \cite{ho13}, 
when we consider the simplified situation concerning the hemodynamic equilibrium 
of a person, the introduction of the drugs in the bloodstream and the consequent 
absorption for the body are gradual and continuous process. 
Therefore, one can interpret this situation as an impulsive action
which starts abruptly and stays active on a finite time interval. 
We call such phenomenon non-instantaneous impulses during construct mathematical 
models.
It is reported that many models arising from realistic models can be described 
as partial differential equations with non-instantaneous impulses.

In the past two years, nonlinear differential equations with non-instantaneous 
impulses have been studied by several authors and some interesting results 
have been obtained, see
 \cite{ab15,cmx16,gd15,ho13,por13,wl14,wzl14,xw15}. In 2013,
 Hern\'{a}ndez and O'Regan \cite{ho13} firstly studied the initial value problem 
for a new class of abstract evolution equations with non-instantaneous 
impulses in Banach spaces. In the same year, Pierri et al \cite{por13} obtained
the existence of mild solutions for a class of semi-linear abstract differential 
equations with non-instantaneous impulses by using the theory of analytic semigroup.
 Gautam and Dabas \cite{gd15} studied the existence, uniqueness and continuous 
dependence results of mild solution for fractional functional integro-differential
 equations with non-instantaneous impulses by using
the theory of analytic $\alpha$-resolvent family and fixed point theorems. Colao
et al \cite{cmx16} obtained the existence of solutions for a second-order 
differential equation with non-instantaneous impulses and delay on an unbounded 
interval by establish a compactness criterion
in a certain class of functions. Yu and Wang \cite{xw15} investigated the 
existence of solutions to periodic boundary value problems for nonlinear
evolution equations with non-instantaneous impulses on Banach spaces by
 using the theory of semigroup and fixed point methods. Wang and Li \cite{wl14}
 obtained the existence of solutions for periodic boundary value problem of 
nonlinear ordinary differential equations with non-instantaneous impulses.  
In addition, fractional ordinary and partial differential equations with
 non-instantaneous impulses have also been studied in \cite{ab15,wzl14}.

The motivation of this article is as follows: 
to the best of the authors knowledge, all the existing articles (see, for
example \cite{cmx16,gd15,ho13,por13,xw15}) used various fixed point theorems 
to study abstract evolution equations with non-instantaneous impulses when
the corresponding semigroup $T(t)(t\geq 0)$ is compact, this is very convenient 
to the equations with compact resolvent.
But for the case that the corresponding semigroup $T(t)(t\geq 0)$ is noncompact, 
we have not seen the relevant papers to study abstract evolution equations with 
non-instantaneous impulses. Therefore, inspired by the previous works, we  
study the existence of PC-mild solution for \eqref{e1.1} under the assumption
 that the corresponding solution semigroup is noncompact. 
By using the properties of Kuratowski measure of noncompactness, 
$k$-set-contraction mapping fixed point theorem (see Lemma \ref{lem2.5}) 
and a new estimation technique for the measure of noncompactness 
(see Lemma \ref{lem2.6}), we obtained the existence of PC-mild solution for \eqref{e1.1}.
Our work can be considered as a supplement for the case that the 
corresponding solution semigroup is compact.

In the following section we first introduce some notations and
preliminary lemmas which will be used in this paper, at the same time the
definition of PC-mild solution for \eqref{e1.1} has been given. 
In section 3 we state and prove the existence of PC-mild solution 
for \eqref{e1.1}  under the appropriate assumptions. 
In the last paragraph we give an example to illustrate the feasibility
 of our abstract results.

\section{Preliminaries}

We begin by giving some notation. Let $E$ be a Banach space with the norm 
$\|\cdot\|$. We use $\theta$ to present the zero element in $E$. 
For any constant $a>0$, denote $J=[0,a]$. Let $C(J,E)$ be the 
Banach space of all continuous functions from $J$ into $E$ endowed
with the supremum-norm
$\|u\|_C=\sup_{t\in J}\|u(t)\|$ for every  $u\in C(J,E)$.
From the associate literature, we consider the following space of piecewise 
continuous functions,
\begin{align*}
PC(J,E)=\big\{&u:J\to E: u \text{ is continuous for }t\neq t_k,\\
&\text{left continuous at $t =t_k$ and $u(t_k^+ )$  exists for }k=1,2,\dots,m
\big\}.
\end{align*}
It is easy to see that $PC(J,E)$ is a Banach space endowed with the PC-norm
$$
\|u\|_{PC}=\max\Big\{\sup_{t\in J}\|u(t^+)\|,\;
\sup_{t\in J}\|u(t^-)\|\Big\},\quad  u\in PC(J,E),
$$
where $u(t^+)$ and $u(t^-)$ represent respectively the right and left 
limits of $u(t)$ at $t\in J$.

For each finite constant $r>0$, let
$$
\Omega_r=\{u\in PC(J,E): \|u(t)\|\leq r,\; t\in J\},
$$
then $\Omega_r$ is a bounded closed and convex set in $PC(J,E)$.

Let $\mathcal {L}(E)$ be the Banach space of all linear and bounded operators on $E$. 
Since the semigroup $T(t)(t\geq 0)$ generated by $-A$ is a $C_0$-semigroup in $E$, 
denote
\begin{equation}
M:= \sup_{t\in J}\|T(t)\|_{\mathcal {L}(E)},\label{e2.1}
\end{equation}
then $M\geq 1$ is a finite number.

\begin{definition} \label{def2.1} \rm
A $C_0$-semigroup $T(t)(t\geq 0)$ in $E$ is said to be equicontinuous if 
$T(t)$ is continuous by operator norm for every $t>0$.
\end{definition}

Now we introduce some basic definitions and properties about
 Kuratowski measure of noncompactness that will be used in the proof 
of our main results.

\begin{definition}[\cite{bg80,d85}] \label{def2.2} 
The Kuratowski measure of noncompactness $\alpha(\cdot)$ defined on bounded 
set $S$ of Banach space $E$ is
$$
\alpha(S):=\inf\{\delta>0:S=\cup_{i=1}^{m}S_i \text{ with }
 \operatorname{diam}(S_i)\leq \delta\text{ for }i=1,2,\dots,m\}.
$$
\end{definition}

The following properties about the Kuratowski measure of noncompactness 
are well known.

  \begin{lemma}[\cite{bg80,d85}] \label{lem2.3}
 Let $E$ be a Banach space and $S$, $U\subset E$ be bounded. 
The following properties are satisfied:
\begin{itemize}
\item[(i)] $\alpha(S)=0$ if and only if $\overline{S}$ is compact,
 where $\overline{S}$ means the closure hull of $S$;

\item[(ii)] $\alpha(S)=\alpha(\overline{S})=\alpha(\operatorname{conv} S)$,
 where $\operatorname{conv}S$ means the convex hull of $S$;

\item[(iii)] $\alpha(\lambda S)=|\lambda|\alpha(S)$ for any 
$\lambda\in \mathbb{R}$;

\item[(iv)] $S\subset U$ implies $\alpha(S)\leq\alpha(U)$;

\item[(v)] $\alpha(S\cup U)=\max\{\alpha(S),\alpha(U)\}$;

\item[(vi)] $\alpha(S+U)\leq\alpha(S)+\alpha(U)$, where 
$S+U=\{x\mid x=y+z, y\in S, z\in U\}$;

\item[(vii)] If the map $Q:\mathcal {D}(Q)\subset E\to X$ is
 Lipschitz continuous with constant $k$, then 
$\alpha(Q(V))\leq k\alpha(V)$ for any bounded subset 
$V\subset \mathcal {D}(Q)$, where $X$ is another Banach space.
\end{itemize}
\end{lemma}

In this article, we denote by $\alpha(\cdot)$, $\alpha_C(\cdot)$ and 
$\alpha_{PC}(\cdot)$ the Kuratowski measure of noncompactness on the bounded 
set of $E$, $C(J,E)$ and $PC(J,E)$, respectively. 
For any $D\subset C(J,E)$ and $t\in J$, set $D(t)=\{u(t)\mid u\in D\}$ then 
$D(t)\subset E$. If $D\subset C(J,E)$ is bounded,
then $D(t)$ is bounded in $E$ and $\alpha(D(t))\leq \alpha_C(D)$. 
For more details about the properties of the Kuratowski measure of
 noncompactness, we refer to the monographs
\cite{bg80,d85}.

  \begin{definition}[\cite{d85}] \label{def2.4} \rm
Let $E$ be a Banach space, and let $S$ be a nonempty subset of $E$.
 A continuous mapping $Q:S\to E$ is called to be $k$-set-contractive 
if there exists a constant $k\in[0,1)$
such that, for every bounded set $\Omega \subset S$,
$$
\alpha(Q(\Omega))\leq k\alpha(\Omega).
$$
\end{definition}

  \begin{lemma}[\cite{d85}] \label{lem2.5} 
 Let $E$ be a Banach space. Assume that $\Omega \subset E$ is a bounded closed and 
convex set on $E$, the operator $Q:\Omega \to\Omega$ is
$k$-set-contractive. Then $Q$ has at least one fixed point in $\Omega$.
\end{lemma}

\begin{lemma}[\cite{cl13,li05}] \label{lem2.6}
Let $E$ be a Banach space,
and let $D\subset E$ be bounded. Then there exists a countable set
$D_0\subset D$, such that $\alpha(D)\leq 2\alpha(D_0)$.
\end{lemma}

  \begin{lemma}[\cite{h83}] \label{lem2.7} 
 Let $E$ be a Banach space, and
let $D=\{u_n\}\subset PC([b_1,b_2],E)$ be a bounded and countable set for 
constants $-\infty<b_1< b_2<+\infty$. Then
$\alpha(D(t))$ is Lebesgue integral on $[b_1,b_2]$, and
$$
\alpha\Big(\Big\{\int_{b_1}^{b_2} u_n(t)dt: n\in \mathbb{N}\Big\}\Big)
\leq 2\int_{b_1}^{b_2}\alpha(D(t))dt.
$$
\end{lemma}

\begin{lemma}[\cite{bg80}] \label{lem2.8}
 Let $E$ be a Banach space, and let $D\subset C([b_1,b_2],E)$ be bounded and
 equicontinuous.  Then $\alpha(D(t))$ is continuous on $[b_1,b_2]$, and
\[
 \alpha_C(D)=\max_{t\in [b_1,b_2]}\alpha(D(t)).
\]
\end{lemma}

At last, we give the definition of mild solution for  \eqref{e1.1} 
according to the developments of Hern\'{a}ndez and O'Regan \cite{ho13}.

  \begin{definition} \label{def2.9} \rm
 A function $u\in PC(J,E)$ is called a mild solution of \eqref{e1.1} 
if $u$ satisfies
\begin{gather*}
u(t)=T(t)u_0+\int_0^t T(t-s)f(s,u(s))ds,\quad t\in[0,t_1];\\
u(t)=\gamma_k(t,u(t)),\quad  t\in(t_k,s_k],\; k=1,2,\dots,m;\\
u(t)=T(t-s_k)\gamma_k(s_k,u(s_k)) 
+\int_{s_k}^t T(t-s)f(s,u(s))ds,\\
\text{for } t\in(s_k,t_{k+1}],\;k=1,2,\dots,m.
\end{gather*}
\end{definition}

\section{Main results}

To obtain the existence of PC-mild solution for \eqref{e1.1}, we introduce 
the following hypotheses:
\begin{itemize}
\item [(H1)]  The nonlinear function $f:J \times E\to E$ is continuous, 
for some $r>0$, there exist a constant $\rho>0$, Lebesgue integrable 
function $\varphi:J\to [0,+\infty)$ and a nondecreasing continuous function 
$\Psi:[0,+\infty)\to (0,+\infty)$ such that for all $t\in J$ and 
$u\in E$ satisfying $\|u\|\leq r$,
    $$
    \|f(t,u)\|\leq \varphi(t) \Psi(\|u\|)\quad \text{and}\quad
    \liminf_{r\to +\infty}\frac{\Psi(r)}{r}=\rho<+\infty.
    $$

\item [(H2)]  The impulsive function $\gamma_k:[t_k,s_k]\times E\to E$ is continuous,
 and there exists a constant $K_{\gamma_k}>0$, $k=1,2,\dots,m$, such that for all 
$u$, $v\in E$
    $$
    \|\gamma_k(t,u)-\gamma_k(t,v)\|\leq K_{\gamma_k}\|u-v\|,\quad \forall\;t\in (t_k,s_k].
    $$

\item [(H3)]  There exist positive constant
$L_k$ ($k=0,1,\dots,m$) such that for any countable set
$D\subset E$,
$$
\alpha(f(t,D))\leq L_k\alpha(D),\quad t\in (s_k,t_{k+1}],\;k=0,1,\dots,m.
$$
\end{itemize}
For brevity of notation, we denote
\begin{gather}
K:=\max_{k=1,2,\dots,m}K_{\gamma_k},\quad 
\Lambda:=\max_{k=0,1,\dots,m}\|\varphi\|_{L[s_{k},t_{k+1}]}, \label{e3.1} \\
L:=\max_{k=0,1,\dots,m}L_k(t_{k+1}-s_k). \label{e3.2}
\end{gather}

 \begin{theorem} \label{thm3.1}
Assume that the semigroup $T(t)(t\geq 0)$ generated by $-A$ is equicontinuous, 
the function $\gamma_k(\cdot,\theta)$ is bounded for $k=1,2,\dots,m$. 
If the conditions {\rm (H1)--(H3)} are satisfied, then \eqref{e1.1} has
 at least one PC-mild solution $u\in PC(J,E)$ provided that
\begin{equation}
M\max\{\rho\Lambda+K,K+4L\}<1.\label{e3.3}
\end{equation}
\end{theorem}

\begin{proof} Define the operator $\mathcal {F}$ on $PC(J,E)$ by
\begin{equation}
(\mathcal {F}u)(t)=(\mathcal {F}_1u)(t)+(\mathcal {F}_2u)(t),\label{e3.4}
\end{equation}
where
\begin{gather}
(\mathcal {F}_1u)(t)=\begin{cases}
T(t)u_0 & t\in[0,t_1]; \\
\gamma_k(t,u(t)), &t\in(t_k,s_k],\; k=1,2,\dots,m;\\
T(t-s_k)\gamma_k(s_k,u(s_k)), & t\in(s_k,t_{k+1}],\;k=1,2,\dots,m,
 \end{cases} \label{e3.5}
\\
(\mathcal {F}_2u)(t)=\begin{cases}
\int_{s_k}^t T(t-s)f(s,u(s))ds,& t\in(s_k,t_{k+1}],\; k=0,1,\dots,m; \\
0,& \text{otherwise}.
 \end{cases} \label{e3.6}
\end{gather}
By direct calculations, it is easy to see that the operator $\mathcal {F}$
is well defined on $PC(J,E)$. From Definition \ref{def2.9}, one can easily see
that the PC-mild solution of \eqref{e1.1} is equivalent to the fixed
 point of operator $\mathcal {F}$ defined by \eqref{e3.4}.
Next, we will prove that the operator $\mathcal {F}$ has at least
 one fixed point.

Firstly, we show that $\mathcal {F}u\in PC(J,E)$ for all $u\in PC(J,E)$. 
For $0\leq \tau<t\leq t_1$, by the strongly continuity of the  semigroup 
$T(t)(t\geq 0)$, \eqref{e2.1} and \eqref{e3.4}, we know that
\begin{equation} \label{e3.7}
\begin{aligned}
&\|(\mathcal {F}u)(t)-(\mathcal {F}u)(\tau)\|\\
&\leq \|T(t)u_0-T(\tau)u_0\|+\big\|\int_\tau^tT(t-s)f(s,u(s))ds\big\|\\
&\quad + \big\|\int_0^\tau
[T(t-s)-T(\tau-s)]f(s,u(s))ds\big\|\\
&\leq M\|T(t-\tau)u_0-u_0\|+M\int_\tau^t \|f(s,u(s))\|ds\\
&\quad +\int_0^\tau
\|T(t-\tau)T(\tau-s)f(s,u(s))-T(\tau-s)f(s,u(s))\|ds\\
&\to 0\quad \text{as } t\to \tau.
\end{aligned}
\end{equation}
From this inequality it follows  that $\mathcal {F}u\in C([0,t_1],E)$.

From \eqref{e3.4} and the continuity of the non-instantaneous impulsive functions 
$\gamma_k(t,u(t))$, $k=1,2,\dots,m$, it is easy to know that 
$\mathcal {F}u\in C((t_k,s_k],E)$ for every $k=1,2,\dots,m$. 
Completely similar with the proof for the continuity of $(\mathcal {F}u)(t)$ 
with respect to $t$ on $[0,t_1]$, we can prove that 
$\mathcal {F}u\in C((s_k,t_{k+1}],E)$ for $k=1,2,\dots,m$. 
Therefore, we have proved that $\mathcal {F}u\in PC(J,E)$ for $u\in PC(J,E)$, 
namely, $\mathcal {F}$ maps $PC(J,E)$ to $PC(J,E)$.

Next, we prove that there exists a constant $R>0$, such that 
$\mathcal {F}(\Omega_R)\subset \Omega_R$.
If this is not true, then for each $r>0$, there would exist 
$u_r\in \Omega_r$ and $t_r \in J$ such that $\|(\mathcal {F}u_r)(t_r)\|>r$. 
If $t_r\in[0,t_1]$, then by \eqref{e2.1}, \eqref{e3.4} and the assumption 
(H1), we know that
\begin{equation}
\begin{aligned}
\|(\mathcal {F}u_r)(t_r)\|
&\leq M \|u_0\|+M\int_0^{t_r}\|f(s,u_r(s))\|ds\\
&\leq M \|u_0\|+M\int_0^{t_r}\Psi(\|u_r\|)\varphi(s)ds\\
&\leq M \|u_0\|+M\Psi(r)\|\varphi\|_{L[0,t_1]}.
\end{aligned}\label{e3.8}
\end{equation}
 If $t_r\in(t_k,s_k]$, $k=1,2,\dots,m$, then by \eqref{e2.1}, \eqref{e3.4}
and  assumption (H2), we obtain
\begin{equation}
\|(\mathcal {F}u_r)(t_r)\|=\|\gamma_k(t_r,u_r(t_r))\|
\leq K_{\gamma_k}\|u_r(t_r)\|+\|\gamma_k(t_r,\theta)\|
 \leq K_{\gamma_k}r+N,\label{e3.9}
\end{equation}
where
$$
N=\max_{k=1,2,\dots,m}\sup_{t\in J}\|\gamma_k(t,\theta)\|.
$$
 If $t_r\in(s_k,t_{k+1}]$, $k=1,2,\dots,m$, then by \eqref{e2.1}, \eqref{e3.4}
and the assumptions (H1) and (H2), we obtain
\begin{equation}
\begin{aligned}
\|(\mathcal {F}u_r)(t_r)\|
&\leq M \|\gamma_k(s_k,u_r(s_k))\|+M\int_{s_k}^{t_r} \|f(s,u_r(s))\|ds\\
&\leq M (K_{\gamma_k}\|u_r(s_k)\|+\|\gamma_k(s_k,\theta)\|)
 +M\Psi(r)\int_{s_k}^{t_r}\varphi(s)ds\\
&\leq M (K_{\gamma_k}r+N)+M\Psi(r)\|\varphi\|_{L[s_{k},t_{k+1}]}.
\end{aligned} \label{e3.10}
\end{equation}
Combining \eqref{e2.1}, \eqref{e3.1}, \eqref{e3.4}, \eqref{e3.8}-\eqref{e3.10}
with the fact $r<\|(\mathcal {F}u_r)(t_r)\|$, we obtain
\begin{equation}
r<\|(\mathcal {F}u_r)(t_r)\|
\leq M\Big(\|u_0\|+\Psi(r) \Lambda+Kr+N\Big).\label{e3.11}
\end{equation}
Dividing both side of \eqref{e3.11} by $r$ and taking the lower limit
as $r\to +\infty$, we have
$$
1\leq M(\rho\Lambda+K),
$$
which contradicts  \eqref{e3.3}.

Next, we prove that the operator $\mathcal {F}_1:\Omega_R\to \Omega_R$ 
is Lipschitz continuous. For $t\in(t_k,s_k]$, $k=1,2,\dots,m$ and
 $u$, $v\in \Omega_R$, by \eqref{e3.5} and the assumption (H2), we obtain
\begin{equation}
\|(\mathcal {F}_1u)(t)-(\mathcal {F}_1v)(t)\|
\leq K_{\gamma_k}\|u(t)-v(t)\|\leq K_{\gamma_k}\|u-v\|_{PC}.\label{e3.12}
\end{equation}
For $t\in(s_k,t_{k+1}]$, $k=1,2,\dots,m$ and $u$, $v\in \Omega_R$,
by \eqref{e3.5} and the assumption (H2), we know that
\begin{equation}
\|(\mathcal {F}_1u)(t)-(\mathcal {F}_1v)(t)\|
\leq MK_{\gamma_k}\|u(s_k)-v(s_k)\|\leq MK_{\gamma_k}\|u-v\|_{PC}.\label{e3.13}
\end{equation}
From \eqref{e3.12}, \eqref{e3.13}, \eqref{e2.1} and \eqref{e3.1}, we obtain
\begin{equation}
\|\mathcal {F}_1u-\mathcal {F}_1v\|_{PC}\leq MK \|u-v\|_{PC}.\label{e3.14}
\end{equation}

In the following, we prove that $\mathcal {F}_2$ is continuous in $\Omega_R$.
To this end, let $u_n\in \Omega_R$ be a sequence such that 
$\lim_{n\to +\infty}u_n=u$ in $\Omega_R$. By the
continuity of nonlinear term $f$ with respect to the second variable, for 
each $s\in J$ we have
\begin{equation}
\lim_{n\to +\infty}f(s,u_n(s))=f(s,u(s)).\label{e3.15}
\end{equation}
By  assumption (H1), we obtain that for every $s\in J$,
\begin{equation}
\|f(s,u_n(s))-f(s,u(s))\|\leq 2\varphi(s) \Psi(R).\label{e3.16}
\end{equation}
Using the fact that the function $s\to 2\varphi(s) \Psi(R)$ is Lebesgue integrable 
for $s\in [s_k,t]$ and $t\in (s_k,t_{k+1}]$, $k=0,1,\dots,m$, by 
\eqref{e2.1}, \eqref{e3.6}, \eqref{e3.15}, \eqref{e3.16} and the Lebesgue 
dominated convergence theorem, we know that
\begin{equation}
\begin{aligned}
\|(\mathcal {F}_2u_n)(t)-(\mathcal {F}_2u)(t)\|
&\leq M \int_{s_k}^t
\|f(s,u_n(s))-f(s,u(s))\|ds\\
&\to 0\quad \text{as } n\to +\infty.
\end{aligned} \label{e3.17}
\end{equation}
Then we infer that
$$
\|\mathcal {F}_2u_n-\mathcal {F}_2u\|_{PC}\to0\quad \text{as } n\to +\infty,
$$
which means that $\mathcal {F}_2$ defined by \eqref{e3.6} is continuous
in $\Omega_R$.

Now, we demonstrate that the operator $\mathcal {F}_2:\Omega_R\to \Omega_R$ 
is equicontinuous. For any $u\in \Omega_R$ and $s_k< t' < t''\leq t_{k+1}$ 
for $k=0,1,\dots,m$, we obtain that
\begin{align*}
&\|(\mathcal {F}_2u)(t'')-(\mathcal {F}_2u)(t')\| \\
&\leq \big\|\int_{t'}^{t''} T(t''-s)f(s,u(s))ds\big\|
+\big\|\int_{s_k}^{t'}\big[T(t''-s)-T(t'-s)\big]f(s,u(s))ds\big\|\\
&:= I_1+I_2,
\end{align*}
where
\begin{gather*}
I_1=\big\|\int_{t'}^{t''} T(t''-s)f(s,u(s))ds\big\|,\\
I_2=\big\|\int_{s_k}^{t'}\big[T(t''-s)-T(t'-s)\big]f(s,u(s))ds\big\|.
\end{gather*}
Therefore, we only need to check $I_1$ and $I_2$ tend to $0$ independently 
of $u\in \Omega_R$ when $t''-t'\to 0$.
For $I_1$, by \eqref{e2.1} and the assumption (H1), we can easily see that
$$
I_1\leq M\Psi(R)\int_{t'}^{t''}\varphi(s)ds\to 0\quad \text{as } t''-t'\to 0.
$$
For $\epsilon>0$ small enough, by \eqref{e2.1}, the assumption
(H1), equicontinuity of the $C_0$-semigroup $T (t)(t\geq 0)$ and the 
Lebesgue dominated convergence theorem, we obtain that
\begin{align*}
I_2&\leq \big\|\int_{s_k}^{t'-\epsilon}\big[
T(t''-s)-T(t'-s)\big]f(s,u(s))ds\big\|\\
&\quad +\big\|\int_{t'-\epsilon}^{t'}\big[
T(t''-s)-T(t'-s)\big]f(s,u(s))ds\big\|\\
&\leq \Psi(R)\int_{\epsilon}^{t'-s_k}\|
T(t''-t'+s)-T(s)\|\varphi(t'-s)ds
 +2M \Psi(R)\int_{t'-\epsilon}^{t'}\varphi(s)ds\\
&\to 0\quad \text{as } t''-t'\to 0 \text{ and } \epsilon\to0.
\end{align*}
As a result, $\|(\mathcal {F}_2u)(t'')-(\mathcal {F}_2u)(t')\|$ 
tends to $0$  independently of $u\in \Omega_R$ as $t''-t'\to 0$, 
which means that $\mathcal {F}_2:\Omega_R\to \Omega_R$ is
equicontinuous.

For any bounded $D\subset \Omega_R$,
by Lemma \ref{lem2.6}, we know that there exists a countable set $D_0=\{u_n\}\subset D$,
such that
\begin{equation}
\alpha(\mathcal {F}_2(D))_{PC}\leq 2\alpha(\mathcal {F}_2(D_0))_{PC}.\label{e3.18}
\end{equation}
Since $\mathcal {F}_2(D_0)\subset \mathcal {F}_2(\Omega_R)$ is bounded and
equicontinuous, we know from Lemma \ref{lem2.8} that
\begin{equation}
\alpha(\mathcal {F}_2(D_0))_{PC}=\max_{t\in [s_k, t_{k+1}],\,k=0,1,\dots,m}
\alpha(\mathcal {F}_2(D_0)(t)).\label{e3.19}
\end{equation}
For every $t\in [s_k,t_{k+1}]$, $k=0,1,\dots,m$, by Lemma \ref{lem2.7}, the assumption
(H3) and \eqref{e3.6}, we have
\begin{equation}
\begin{aligned}
\alpha(\mathcal {F}_2(D_0)(t))
&= \alpha\Big(\Big\{\int_{s_k}^t
T(t-s)f(s,u_n(s))ds\Big\}\Big)\\
&\leq  2M\int_{s_k}^t
\alpha(\{f(s,u_n(s))\})ds\\
&\leq 2M\int_{s_k}^t L_k\alpha(D_0(s))ds\\
&\leq 2ML_k(t_{k+1}-s_k)\alpha(D)_{PC}.
\end{aligned} \label{e3.20}
\end{equation}
Therefore, from \eqref{e3.3}, \eqref{e3.18}, \eqref{e3.19} and \eqref{e3.20}
we know that
\begin{equation}
\alpha(\mathcal {F}_2(D))_{PC}\leq 4ML\alpha(D)_{PC}.\label{e3.21}
\end{equation}
From \eqref{e3.14} and Lemma \ref{lem2.3} (vii), we know that for any bounded
$D\subset \Omega_R$,
\begin{equation}
\alpha(\mathcal {F}_1(D))_{PC}\leq MK\alpha(D)_{PC}.\label{e3.22}
\end{equation}
By \eqref{e3.21}, \eqref{e3.22} and Lemma \ref{lem2.3} (vi), we have
\begin{equation}
\alpha(\mathcal {F}(D))_{PC}\leq\alpha(\mathcal {F}_1(D))_{PC}
+\alpha(\mathcal {F}_2(D))_{PC}\leq M(K+4L)\alpha(D)_{PC}.\label{e3.23}
\end{equation}
\eqref{e3.23} combining with \eqref{e3.3} and Definition \ref{def2.4}
we know that the operator $\mathcal {F}:\Omega_R\to \Omega_R$ is a
$k$-set-contractive. It follows from
Lemma \ref{lem2.5} that $\mathcal {F}$ has at least one fixed point
$u\in \Omega_R$, which is just a PC-mild solution of \eqref{e1.1}.
This completes the proof
\end{proof}

\begin{remark} \label{rmk3.2} \rm
The analytic semigroup and differentiable semigroup are equicontinuous
semigroup \cite{p83}. In the application of partial differential
equations, such as parabolic and strongly damped wave equations, the
corresponding solution semigroup are analytic semigroup. 
Therefore, Theorem \ref{thm3.1} has a broad applicability.
\end{remark}

\begin{remark} \label{rmk3.3} \rm
 Theorem \ref{thm3.1} complements results in \cite{ho13} 
and \cite{por13}. Theorem \ref{thm3.1} can be applied to a class of partial 
differential equations of evolution type for which the corresponding solution 
semigroup  are not compact.
\end{remark}

\section{An example}

To illustrate the applicability of our main results, we consider the
parabolic partial differential equation with non-instantaneous impulses,
\begin{equation}
\begin{gathered}
\frac{\partial}{\partial t}u(x,t)+\mathcal {A}u(x,t)
=\frac{e^{-t}}{2+|u(x,t)|},\quad x\in \Omega,\;
 t\in [0,\frac{1}{3})\cup (\frac{2}{3},1], \\
\mathcal {B}u(x,t)=0,\quad x\in \partial\Omega,\quad t\in [0,1],\\
u(x,t)=\frac{e^{-(t-\frac{1}{3})}}{4}\frac{|u(x,t)|}{1+|u(x,t)|},\quad
x\in \Omega,\; t\in [\frac{1}{3},\frac{2}{3}), \\
u(x,0)=\varphi(x),\quad x\in \Omega,
 \end{gathered} \label{e4.1}
\end{equation}
where $J=[0,1]$, integer $N\geq 1$, $\Omega\subset\mathbb{R}^N$ is a
bounded domain, whose boundary $\partial\Omega$ is an $(N-1)$-dimensional
$C^{2+\mu}$-manifold for some $0 <\mu < 1$,
$$
\mathcal {A}u:=-\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\partial}{\partial x_i
}\Big(a_{ij}(x)
 \frac{\partial u}{\partial x_j}\Big)+a_0(x)u
$$
is a uniformly elliptic differential operator on
$\overline{\Omega}$ with the coefficients $a_{ij}\in C^{1+\mu}(\overline{\Omega})$
$(i,j=1,2,\dots,N)$ and $a_0\in C^{\mu}(\overline{\Omega})$ for some
$\mu\in(0,1)$, $a_0(x)\geq 0$ on $\overline{\Omega}$.
That is, $[a_{ij}(x)]_{N\times N}$ is a positive
definite symmetric matrix for every $x\in \overline{\Omega}$ and
there exists a constant $\mu_0>0$ such that
\begin{gather*}
\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}(x)\eta_i\eta_j\geq \mu_0\mid\eta\mid^2,\quad
\forall\;\eta=(\eta_1,\eta_2,\dots,\eta_N)\in \mathbb{R}^N,\;
x\in \overline{\Omega}; \\
\mathcal {B}u:=\delta\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}(x)
 \cos(\nu,x_i)\frac{\partial u}{\partial x_j}+(1-\delta)u
\end{gather*}
is a boundary operator on $\partial\Omega$, where $\nu$ is an outer unit
normal on $\partial\Omega$, $\delta=0$ or $1$; $\varphi\in L^p(\Omega)$
with $p\geq 2$.

Let $E=L^p(\Omega)$ with $p\geq2$. Then $E$ is a Banach space equipped
with the $L^p$-norm $\|\cdot\|_p$. Consider the
operator $A:D(A)\subset E\to E$ defined by
\begin{equation}
D(A)=\big\{u\in W^{2,p}(\Omega): \mathcal {B}u=0\Big\},\quad
Au=\mathcal {A}u.\label{e4.2}
\end{equation}
It is well known from \cite{a88} and \cite{p83} that $-A$ generates
an analytic $C_0$-semigroup $T(t)$ $(t\geq 0)$ on $E$, and
\begin{equation}
\| T(t)\|\leq 1, \quad  \text{for any } t\geq0,\label{e4.3}
\end{equation}
which means that the $C_0$-semigroup
$T(t)$ $(t\geq 0)$ is contraction on $E$.


Let $a=t_2=1$, $t_0=s_0=0$, $t_1=\frac{1}{3}$, $s_1=\frac{2}{3}$, 
$u(t)=u(\cdot,t)$, $f(t,u(t))=\frac{e^{-t}}{2+|u(\cdot,t)|}$,
$\gamma_1(t,u(t))=\frac{e^{-(t-\frac{1}{3})}}{4}\frac{|u(\cdot,t)|}{1+|u(\cdot,t)|}$, 
$u_0=\varphi(\cdot)$, then the
parabolic partial differential equation \eqref{e4.1} can be rewritten 
into the abstract form of \eqref{e1.1} in $L^p(\Omega)$ for $m=1$.

From the definition of nonlinear term $f$ and non-instantaneous impulsive 
function $\gamma_1$, we can easily to verify that the assumptions (H1)-(H3) 
and the condition \eqref{e3.3} hold with
$$
M=1,\quad \varphi(t)=\frac{|\Omega|}{2}e^{-t},\quad 
\Psi\equiv 1,\quad \Lambda= \frac{|\Omega|}{6},\quad 
K=K_{\gamma_1}=\frac{1}{4},\quad L=\frac{1}{12}.
$$
Therefore, by Theorem \ref{thm3.1}, 
the parabolic partial differential equation \eqref{e4.1} 
has at least one PC-mild solution.

\subsection*{Acknowledgments}
This research was supported by NNSFs of China (11501455, 11261053),
by the Science Research Project for Colleges and Universities
of Gansu Province (2015A-003, 2015A-213) and by
Key Project of Gansu Provincial National Science Foundation (1606RJZA015).


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\end{document}