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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 194, pp. 1--10.\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-2013/194\hfil Controllability of impulsive systems]
{Controllability of impulsive functional differential systems
 with nonlocal conditions}

\author[Y. Liu, D. O'Regan \hfil EJDE-2013/194\hfilneg]
{Yansheng Liu, Donal O'Regan}  % in alphabetical order

\address{Yansheng Liu \newline
Department of Mathematics,
Shandong Normal University, Jinan, 250014, China}
\email{yanshliu@gmail.com}

\address{Donal O'Regan \newline
Department of Mathematics, National University of Ireland,
Galway, Ireland}
\email{donal.oregan@nuigalway.ie}

\thanks{Submitted  April 12, 2012. Published August 30, 2013.}
\subjclass[2000]{34K10, 34K21, 34K35}
\keywords{Controllability; fixed point theorem; nonlocal conditions;
\hfill\break\indent  impulsive functional differential equations}

\begin{abstract}
 In this article, we study the controllability of impulsive functional
 differential equations with nonlocal conditions. We establish
 sufficient conditions for controllability, via the measure of
 noncompactness and M\"{o}nch fixed point theorem.
\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}

 Consider the impulsive functional differential equation
\begin{equation} \label{e1.1}
\begin{gathered}
x'(t)=A(t)x(t) + f(t, x(t), x_t) + Bu(t),\quad\text{a.e. } t\in [0, a];\\
\Delta x\big|_{t=t_i}=I_i(x(t_i)),\quad i=1, 2, \dots k;\\
x(t)=\phi(t),\quad t\in [-\tau,  0);\\
x(0)+M(x)=x_0,
\end{gathered}
\end{equation}
where  $\Delta x |_{t=t_i}=x(t_i+0)- x(t_i-0)$, $A(t)$ is a family of linear
operators which generates an evolution operator
$$
U: \Delta =\{(t,s)\in J\times J: 0\le s\le t\le a\}\to L(X),
$$
$X$ is a Banach space, $J=[0,a]$, $L(X)$ is the space of all bounded linear
operators in $X$, $M: PC(J,X)\to X$, $B$ is a bounded linear
operator from a Banach space $V$ to $X$ and the control function
$u(\cdot )$ is given in $L^2(J,V)$,
$0=t_0<t_1<t_2<\dots< t_k<t_{k+1}=a$, $I_i: X\to X, i = 1,\dots, k$ are impulsive functions, $f: J\times X \times L([-\tau, 0], X)\to X$
is a given function satisfying some assumptions that will be specified later,
$\phi\in L([-\tau, 0], X)$ and $L([-\tau, 0], X)$ is  the space of
X-valued Bochner integrable functions on $[-\tau, 0]$ with the norm
$\|\phi\|_{L[-\tau, 0]}=\int_{-\tau}^0\|\phi(t)\|dt$.

Abstract differential systems in infinite-dimensional spaces appear
in many bran\-ches of science and engineering, such as heat flow in
materials with memory, viscoelasticity, and other physical
phenomena. Systems with short-term perturbations are often naturally
described by impulsive differential equations  \cite{Lakshmi,Samo}.
Impulsive interruptions are observed in mechanics, radio
engineering, communication security, control theory, optimal
control, biology,  medicine, bio-technologies,
electronics, neural networks and economics (see for example
\cite{cgr, d, gcn, lh, tc,yzn}). We also refer the reader to recent
results in impulse theory \cite{fql, fyl, rt,zs}.  The semilinear nonlocal
initial problem was first discussed by Byszewski \cite{Bysz, Bysza}.
It was studied extensively under various conditions on $A$ (or
$A(t)$) and $f$ by several authors (see \cite{cgr,jlw} and the
references therein). Recently, Ji et al \cite{jlw} studied
the impulsive differential equation
\begin{equation} \label{e1.2}
\begin{gathered}
x'(t)=A(t)x(t) + f(t, x(t)) + Bu(t),\quad\text{a.e. } t\in [0, a];\\
\Delta x\big|_{t=t_i}=I_i(x(t_i)),\quad i=1, 2, \dots k;\\
x(0)+M(x)=x_0.
\end{gathered}
\end{equation}
Time delays are often encountered
unavoidably in many practical systems such as automatic control
systems, population models, inferred grinding models, the AIDS
epidemic, and neural networks; see \cite{gukh,hale,kolm,hayk,nicu}
and the references therein. They describe phenomenon present in real
systems where the rate of change of the state depends on not only
the current state of the system but also its state at some time in
history. Therefore, it is natural and necessary to study
\eqref{e1.2} with time delay, i.e. the \eqref{e1.1}.

To the best of our knowledge there is  no  paper studying such
systems. The purpose of the present paper is to fill this  gap. In
this paper some sufficient conditions for controllability are
established by using the measure of noncompactness and M\"{o}nch's
fixed point theorem. The main features in the present paper are as
 follows. First, the \eqref{e1.1} considers the effect of time
 delay.  Also we relax the assumptions on the functions $f$, $M$, and $I_i$
 in \cite{jlw}.

The organization of this article is as follows. We shall introduce
some preliminaries and some lemmas in Section 2. The main results
and their proof are given in Section 3.

\section{Preliminaries}


For the sake of simplicity, we put $J_0 = [0, t_1]$ and
$J_i = (t_i, t_{i+1}]$, $i = 1,\dots, k$.
 Let $PC(J, X)=\{x: x$ is a
map from $J$ into $X$ such that $x(t)$ is continuous at $t\neq t_i$,
and left continuous at $t=t_i$, and the right limit $x(t_i^+)$
exists for $i=1, 2, \dots, k \}$.
Evidently, $PC(J, X)$ is a Banach space with the norm
$$
\|x\|_{PC}=\sup_{t\in J}\{\|x(t)\|\},\quad \forall x\in PC(J, X).
$$

Notice that the interaction of time delay and impulse  give rise to
discontinuity. Therefore, we introduce the special complete space
$L([-\tau, 0], X)$ to overcome the difficulty arising from time
delay.
For any function $y\in PC(J, X)$
and any $t\in J$, we denote a function $y_t$ by
\begin{equation} \label{e2.1}
y_t(\theta)=\begin{cases}
y(t+\theta), &   t+\theta\ge 0;\\
\phi(t+\theta), &   t+\theta< 0
\end{cases}
 \end{equation}
for $\theta\in [-\tau,  0]$,  where $\phi(t)$ is the same as in \eqref{e1.1}.
Then it is easy to see  $y_t\in L([-\tau, 0],X)$.
Moreover, we have the following Lemma.

\begin{lemma} \label{l2.0}
 Suppose $y_n, y_0\in PC(J, X)$ with $\|y_n-
y_0\|_{PC}\to 0$ as $n\to +\infty$. Then for each $t\in J$, we have
$$
\|y_{nt}- y_{0t}\|_{L[-\tau, 0]}\to 0,\quad \text{as } n\to +\infty,
$$
where $y_{nt}(\theta)$ and $ y_{0t}(\theta)$ are defined by
\eqref{e2.1}.
\end{lemma}

\begin{proof}
From \eqref{e2.1}, it follows that
$$
\|y_{nt}- y_{0t}\|_{L[-\tau, 0]}=\begin{cases}
\int_0^t|y_n(s)-y_0(s)|ds, &   t\le \tau;\\
\int_{t-\tau}^t|y_n(s)-y_0(s)|ds, &   t\ge \tau.
\end{cases}
$$
The conclusion follows.
\end{proof}

 The basic space to study \eqref{e1.1} in this paper is
 $PC(J,X)$. For a bounded subset $\Omega$ of Banach space $X$, let $\beta(\Omega)$
be the Hausdorff  noncompactness measure of $\Omega$, which is defined by
$\beta(\Omega)=\inf\{ \varepsilon >0: \Omega\text{ has a finite $\varepsilon$-net in }X\}$
(see \cite{bg,koz}). In this paper,
 the Hausdorff measure of noncompactness of a bounded set in $X$,
 $PC(J, X)$, and
 $L([-\tau, 0], X)$  are denoted by $\beta(\cdot)$, $\beta_{PC}(\cdot)$,
and $\beta_{\tau}(\cdot)$, respectively.
As in \cite{h}, we have the following result on the Hausdorff
noncompactness measure.

\begin{lemma} \label{l2.1}
 Suppose $E$ is a Banach space. Let $H$ be a countable
set of strongly  measurable function $x: J\to E$ such that there
exists a $\mu\in  L[J, R^+]$ with $\|x(t)\|\le \mu(t)$ a.e.
 $t\in J$ for all   $x\in H$. Then $\beta (H(t))\in L[J, R^+]$ and
  $$
  \beta\big(\big\{\int_J x(t)dt: x\in H\Big\}\Big)\le 2\int_J\beta
  (H(t))dt,
  $$
where $\beta(\cdot)$ denotes the Hausdorff  noncompactness measure,
$J=[0,a]$.
\end{lemma}

\begin{lemma}[M\"{o}nch fixed point theorem \cite{m}] \label{l2.2}
Suppose $E$ is a Banach space. Let $D$ be a closed and convex subset of
$E$ and $u\in D$. Assume that the continuous operator $A: D\to D$ has
the following property:
$C\subset D$ countable, $ C \subset \overline{co}(\{u\}\cup   A(C))$
implies $C$ is relatively compact.
Then $A$ has a fixed point in $D$.
\end{lemma}

\begin{definition} \label{d2.1} \rm
A function $x\in PC(J;X)$ is said to be a mild solution
of \eqref{e1.1} if $x(0) + M(x) =  x_0$ and
$$
x(t)=U(t, 0)x(0)+\int_0^tU(t, s)\big((f(s, x(s), x_s)+Bu(s)\big)ds
+ \sum_{0<t_i<t} U(t,t_i)I_i(x(t_i)),
$$
for all $t \in J$, where $x_s$ is defined by \eqref{e2.1}.
\end{definition}

\begin{definition} \label{d2.2} \rm
Equation \eqref{e1.1} is said  to be nonlocally controllable on $J$ if,
for every $x_0, x_1\in X$, there exists a control
$u \in  L^2(J,V)$ such that the mild solution $x$ of \eqref{e1.1} satisfies
$x(b) + M(x) = x_1$.
\end{definition}

A two parameter family of bounded linear operators $U(t, s)$,
$0 \le s \le t \le a$ on $X$ is called an evolution system if the following
two conditions are satisfied:
\begin{itemize}
\item[(i)] $U(s, s) = I$,
$U(t, r)U(r, s) = U(t, s)$ for $0  \le s \le t \le a$;

\item[(ii)] $(t, s)\to U(t, s)$ is strongly continuous for
$0  \le s \le t \le a$.
\end{itemize}
Since the evolution system $U(t, s)$ is strongly continuous on the
compact set $J \times J$, then there exists $L_U > 0$ such that
$\|U(t, s)\| \le L_U$ for any $(t, s)\in J \times J$.
More details about evolution systems can be found in \cite{p}.

\section{Main results}

We will use the following hypotheses:
\begin{itemize}
\item[(S1)]  $A(t)$ is a family of linear operators,
$A(t):\mathscr{D}(A)\to X$, $\mathscr{D}(A)$ not depending on $t$ is a
dense subset of $X$, generating an equicontinuous evolution system
$\{U(t, s) : (t, s)\in J\times J\}$, i.e., $(t, s)\to \{U(t, s)x :
x\in\Omega\}$ is equicontinuous for $t > 0$ and for all bounded
subsets $\Omega$.

\item[(S2)]  $f: J\times X\times L([-\tau, 0], X)\to X$ satisfies:
\begin{itemize}
\item[(i)] $t\to f(t,x, y)$ is strongly measurable for each
  $x\in X, y\in L([-\tau, 0], X)$;
  $(x, y)\to f(t, x, y)$ is continuous for almost all $t\in J$;

\item[(ii)]  there exist  functions $a_1, b_1, \mu_1\in L(J;R^+)$ such
that 
$$\|f(t, x, y)\|\le a_1(t)\|x\| + b_1(t)\|y\|_{L[-\tau, 0]} +
\mu_1(t),
$$ 
for all $t\in J$, $x\in X$, $y\in L([-\tau, 0], X)$;

\item[(iii)] there exist $l_1, l_2\in  L^1(J;R^+)$ such that for any
bounded subsets $B_1\subset X, B_2\subset L([-\tau, 0], X)$,
$$
\beta(f(t,B_1, B_2))\le l_1(t)\beta(B_1)+l_2(t)\beta_{\tau}(B_2);
$$
\end{itemize}

\item[(S3)] $M: PC(J, X) \to X$ is a continuous operator and there exist
nonnegative numbers $a_2, b_2, l_3$  such that
\begin{gather*}
\|M(y)\|\le a_2\|y\| +b_2,\quad \forall y\in PC(J, X);\\
\beta(M(B_1))\le l_3\beta_{PC}(B_1),\quad \text{for any bounded } B_1\subset PC(J, X);
\end{gather*}

\item[(S4)] the linear operator $W: L^2(J,V)\to X$  defined by
$$
Wu=\int_0^aU(a, s)Bu(s)ds
$$
is such that:
\begin{itemize}
\item[(i)] $W$ has an invertible operator $W^{-1}$ which take
values in $L^2(J,V)/kerW$ and there exist positive constants $L_B$
and $L_W$ such that $\|B\|\le L_B$ and $\|W^{-1}\|\le L_W$;

\item[(ii)] there is $K_W\in L^1(J,R^+)$ such that, for any bounded set $Q
\subset X$, $$\beta_V((W^{-1}Q)(t))\le K_W(t)\beta(Q).$$
\end{itemize}

\item[(S5)] $I_i: X\to X  (i = 1, \dots, k)$  is a  continuous operator
and there exist nonnegative numbers $c_i, d_i, k_i\ (i=1, 2,
\dots, k)$ such that:
\begin{gather*}
\|I_i(x)\|\le c_i\|x\|+d_i,\quad \forall x\in X,\;i=1, 2, \dots, k;\\
\beta(I_i(B_1))\le k_i\beta(B_1),\quad
\text{for any bounded } B_1\subset X,\; i=1, 2, \dots, k.
\end{gather*}
\end{itemize}


\begin{theorem}\label{t2}
Assume that {\rm (S1)--(S5)} are satisfied.
In addition, assume that
\begin{gather} \label{e3.1}
\begin{aligned}
c&:=L_U\Big[(1+L_BL_Wa^{1/2})\Big(a_2+\int_0^a\big(a_1(s)+\tau
 b_1(s)\big)ds +\sum_{i=1}^kc_i\Big)\\
&\quad +L_UL_BL_Wa_2a^{1/2}\Big]<1,
\end{aligned}\\
\label{ed}
\begin{aligned}
d&:=L_U\Big[\big(l_3 + 2\int_0^a(l_1(s)+ \tau l_2(s))ds +
\sum_{i=1}^kk_i \big)\big(1+2L_BL_U\int_0^aK_W(s)ds\big)\\
&\quad +2l_3L_B\int_0^aK_W(s)ds \Big]< 1.
\end{aligned}
\end{gather}
Then the impulsive functional
differential system \eqref{e1.1} is nonlocally controllable on
$J$.
\end{theorem}


\begin{proof}  From (S4)(i), one can define the  control:
\begin{equation} \label{e3.2}
\begin{aligned}
 u_x(t)&= W^{-1}[x_1-M(x) -U(a, 0)(x_0-M(x))\\
&\quad -\int_0^aU(a, s)f(s, x(s), x_s)ds
 - \sum_{i=1}^k U(a, t_i)I_i(x(t_i)) ](t),
\end{aligned}
\end{equation}
for all $x\in PC(J, X)$.
Using this control, define the following operator on $PC(J, X)$ by
\begin{equation} \label{e3.3}
\begin{aligned}
 (Gx)(t)=& U(t, 0)(x_0-M(x)) + \int_0^tU(t, s)\big(f(s, x(s), x_s)+Bu_x(s)\big)ds\\
&\quad +  \sum_{0< t_i< t} U(t, t_i)I_i(x(t_i)),\quad \forall  x\in
 PC(J, X).
\end{aligned}
\end{equation}
 Obviously, $Gx\in PC(J, X)$. We shall show that $G$ has a fixed point,
 which is then a solution of \eqref{e1.1}. Clearly, if $x$ is  a fixed
point of $G$, then $x_1=M(x) + G(x)(a)$, which implies that the system
\eqref{e1.1} is controllable.

First we show that $G$ is continuous. To do this, suppose
$x_n, x \in PC(J, X)$ and $x_n\to x$ as $n\to +\infty$.
Then by (S3) and (S5) we know that
 \begin{equation} \label{e3.4}
\begin{aligned}
&\|Gx_n - Gx\|_{PC} \\ &\leq  L_U\Big(\|M(x_n)-M(x)\|+\int_0^a\|
f(s, x_n(s), x_{ns})- f(s, x(s), x_s)\|ds \\
&\quad + L_B\int_0^a\|u_{x_n}(s)- u_x(s)\|ds + \sum_{i=1}^k
\|I_i(x_n(t_i))- I_i(x(t_i))\|\Big)\\
&\le L_U\Big(\|M(x_n)-M(x)\|+\int_0^a\| f(s, x_n(s), x_{ns})
 - f(s, x(s), x_s)\|ds \\
&\quad  + L_B a^{1/2}\|u_{x_n}- u_x\|_{L^2}+
\sum_{i=1}^k \|I_i(x_n(t_i)- I_i(x(t_i))\|\Big). 
\end{aligned}
\end{equation}
Notice  that
\begin{equation} \label{e3.5}
\|x_{ns}-x_s\|_{L[-\tau, 0]}\leq \tau\|x_n-x\|_{PC}.
\end{equation}
From  \eqref{e3.2}, we have
\begin{equation} \label{e3.6}
\begin{aligned}
&\|u_{x_n}- u_x\|_{L^2}\\
&\le L_W\|M(x_n)-M(x)\|+L_WL_U \Big[\|M(x_n)-M(x)\|\\
&\quad +\int_0^a\|
f(s, x_n(s), x_{ns})- f(s, x(s), x_s)\|ds + \sum_{i=1}^k
\|I_i(x_n(t_i)- I_i(x(t_i))\|\Big].
\end{aligned}
\end{equation}
Then by \eqref{e3.4}--\eqref{e3.6}, (S2)--(S5), and the Lebesgue
dominated convergence theorem, we obtain
$$
\|Gx_n - Gx\|_{PC} \to 0\quad {\rm as} \quad n\to +\infty,
$$
so $G$ is continuous.

Next, choose a positive number $r$ satisfying
\begin{equation} \label{er}
\begin{aligned}
r&>\frac{L_U}{1-c}\Big[(1+L_UL_BL_Wa^{1/2})\Big(\|x_0\|+b_2
+\int_0^ab_1(s)ds\cdot \|\phi\|_{L[-\tau, 0]}\\
&\quad+ \int_0^a\mu_1(s)ds
 +\sum_{i=1}^kd_i\Big)+L_BL_Wa^{1/2}(\|x_1\|+b_2)\Big].
\end{aligned}
\end{equation}
We now show that
\begin{equation}
\label{e3.7} G: B(0, r)\to B(0, r),
\end{equation}
where $B(0, r)=\{x\in PC(J, X): \|x\|_{PC}\le r\}$.
In fact, for each $x\in PC(J, X)$, by \eqref{e3.2}, we have
\begin{align*}
\|u_x\|_{L^2}
&=  \Big(\int_0^a\|u_x(s)\|^2ds\Big)^{1/2}
\\
&\leq L_W(\|x_1\|+ a_2\|x\|_{PC}+b_2) + L_WL_U \Big[\|x_0\|+
a_2\|x\|_{PC}+b_2
\\
&\quad + \int_0^a\big(a_1(s)\|x(s)\|+b_1(s)\|x_s\|_{L[-\tau,
0]}+\mu_1(s)\big)ds + \sum_{i=1}^k(c_i\|x(t_i)\|+d_i)\Big]
\\
&\leq L_W(\|x_1\|+ a_2\|x\|_{PC}+b_2) + L_WL_U \Big[\|x_0\|+
a_2\|x\|_{PC}+b_2\\
&\quad + \int_0^a\big(a_1(s)\|x\|_{PC}+b_1(s)(\tau\|x\|_{PC}
+\|\phi\|_{L[-\tau, 0]}) +\mu_1(s)\big)ds\\
&\quad  +\sum_{i=1}^k(c_i\|x\|_{PC}+d_i)\Big].
\end{align*}
This together with \eqref{e3.3} guarantees that
\begin{align*}
&\|Gx\|_{PC} \\
&\leq  L_U\Big[\|x_0\|+ \|M(x)\| + \int_0^a
\|f(s, x(s), x_s)+Bu_x(s)\|ds + \sum_{i=1}^k \|I_i(x(t_i))\|\Big]
\\
&\leq  L_U\Big[\|x_0\|+ a_2\|x\|_{PC}+b_2 +
\int_0^a\big(a_1(s)\|x(s)\|+b_1(s)\|x_s\|_{L[-\tau, 0]}+\mu_1(s)\big)ds \\
&\quad+ L_B\int_0^a\|u_x(s)\|ds+
\sum_{i=1}^k(c_i\|x(t_i)\|+d_i)\Big]\\ &\leq  L_U\Big[\|x_0\|+ a_2\|x\|_{PC}+b_2 +
\int_0^a\big(a_1(s)\|x\|_{PC}+b_1(s)(\tau\|x\|_{PC}+\|\phi\|_{L[-\tau,
0]})\\
&\quad +\mu_1(s)\big)ds + L_Ba^{1/2}\|u_x\|_{L^2}+
\sum_{i=1}^k(c_i\|x\|_{PC}+d_i)\Big] \\ &\leq
c\|x\|_{PC}+ L_U\Big[(1+L_UL_BL_Wa^{1/2})\Big(\|x_0\|+b_2
+\int_0^ab_1(s)ds\cdot \|\phi\|_{L[-\tau, 0]}\\
&\quad +\int_0^a\mu_1(s)ds+
\sum_{i=1}^kd_i\Big)+L_BL_Wa^{1/2}(\|x_1\|+b_2)\Big].
 \end{align*}
 From  \eqref{er} we have  $\|Gx\|_{PC}\le r$ if
$\|x\|_{PC}\le r$; that is,
 \eqref{e3.7} holds.

Next we prove that if   $D\subset B(0, r)$ is countable and
\begin{equation} \label{e3.9}
D \subset \overline{co}(\{u_0\}\cup   G(D)),
\end{equation}
where $u_0\in B(0, r)$, then $D$ is relatively compact.
Without loss of generality,  suppose that $D =\{x_n\}_{n=1}^\infty$.
First we show $\{Gx_n\}_{n=1}^\infty$ is equicontinuous on each $J_i$,
$i = 0, \dots, k$. If this is true then
$\overline{co}(\{u_0\}\cup  G(D)) $ is also equicontinuous on  each $J_i$.
To this end, notice   that for each $x\in D$, $t', t''\in J_i$, we have
\begin{equation} \label{e3.10}
\begin{aligned}
&\|(Gx)(t'')- (Gx)(t')\|\\
&=  \|[U(t'',0)-U(t', 0)](x_0-M(x))\|
 +\|\sum_{j=1}^i\big(U(t'',t_j)-U(t', t_j)\big)I_j(x(t_j))\|\\
&\quad +\|\int_0^{t''}U(t'',s)\big(f(s, x(s), x_s)+Bu_x(s)\big)ds\\
&\quad -\int_0^{t'}U(t', s)\big(f(s,x(s), x_s)
 +Bu_x(s)\big)ds\|\\
&\leq  \|[U(t'', 0)-U(t',0)](x_0-M(x))\|
 +\sum_{j=1}^i\|\big(U(t'', t_j)-U(t',t_j)\big)I_j(x(t_j))\|\\
&\quad +\int_0^{t'}\|U(t'', s)-U(t',s)\big(f(s, x(s), x_s)+Bu_x(s)\big)\|ds\\
&\quad +\int_{t'}^{t''}\|U(t'', s)\|\cdot\|f(s, x(s), x_s)+Bu_x(s)\|ds
\end{aligned}
\end{equation}

From the equicontinuity property of $U(\cdot, s)$ and the absolute
continuity of the Lebesgue integral, we see that the right-hand side
of the inequality \eqref{e3.10} tends to zero independent of $x\in
D$ as $|t''- t'|\to 0, t'', t'\in J_i$. Therefore, $G(D)$ is
equicontinuous on every $J_i$.

Next notice that
$$
\|x_{ns}-x_{ms}\|_{L[-\tau, 0]}\le \tau\|x_n-x_m\|_{PC}, \quad
\|x_{n}(s)-x_{m}(s)\|\le \|x_n-x_m\|_{PC}, s\in J,
$$
which implies
$$
\beta_{\tau}(\{x_{ns}\}_{n=1}^\infty )\le
\tau\beta_{PC}(\{x_n\}_{n=1}^\infty),\quad
 \beta(\{x_n(s) \}_{n=1}^\infty )\le
\beta_{PC}(\{x_n\}_{n=1}^\infty), s\in J.
$$
Then from (S2), (S3), (S4) and (S5),  for each $t\in J$, we have
\begin{align*}
&\beta_V(\{u_{x_n}(t)\}_{n=1}^\infty )\\
&\leq  K_W(t)\beta\Big(\{M(x_n)
 +U(a, 0)(x_0-M(x_n))+\int_0^aU(a, s)f(s, x_n(s), x_{ns})ds\\
&\quad + \sum_{i=1}^k U(a, t_i)I_i(x_n(t_i)) \}_{n=1}^\infty\Big)\\
&\leq  K_W(t)\Big( l_3(1+L_U)\beta_{PC}(\{x_n\}_{n=1}^\infty)
  + 2L_U\int_0^a\big[l_1(s)\beta(\{x_n(s)\}_{n=1}^\infty )\\
&\quad + l_2(s)\beta_{\tau}(\{x_{ns}\}_{n=1}^\infty)\big]ds
 +L_U\sum_{i=1}^kk_i\beta(\{x_n(t_i)\}_{n=1}^\infty) \Big)\\
&\leq  K_W(t)\Big( l_3(1+L_U) + 2L_U\int_0^a\big[l_1(s)+ \tau l_2(s)\big]ds
 +L_U\sum_{i=1}^kk_i \Big)\beta_{PC}(\{x_n\}_{n=1}^\infty),
\end{align*}
and
 \begin{equation}\label{e3.11}
\begin{aligned}
&\beta(\{(Gx_n)(t)\}_{n=1}^\infty)\\
&\leq \beta\Big(\{U(t, 0)(x_0-M(x_n)) \}_{n=1}^\infty \Big)\\
&\quad +\beta\Big(\{\int_0^tU(t, s)\big(f(s, x_n(s),
x_{ns})+Bu_{x_n}(s)\big)ds  \}_{n=1}^\infty \Big)\\
&\quad + \beta\Big(\{\sum_{0< t_i< t} U(t, t_i)I_i(x_n(t_i)) \}_{n=1}^\infty \Big)\\
&\leq  L_U l_3\beta_{PC}(\{x_n\}_{n=1}^\infty)
 + 2L_U\int_0^a\big[l_1(s)\beta(\{x_n(s)\}_{n=1}^\infty
)+l_2(s)\beta_{\tau}(\{x_{ns}\}_{n=1}^\infty )\big]ds\\
&\quad +2L_UL_B\int_0^a\beta_V(\{u_{x_n}(s)\}_{n=1}^\infty
)ds+L_U\sum_{i=1}^kk_i\beta(\{x_n(t_i)\}_{n=1}^\infty)\\
&\leq  L_U\Big[l_3 + 2\int_0^a\big[l_1(s)+ \tau l_2(s)\big]ds
 + 2L_B\Big( l_3(1+L_U) + 2L_U\int_0^a\big[l_1(s)+\tau l_2(s)\big]ds\\
&\quad +L_U\sum_{i=1}^kk_i \Big)\int_0^aK_W(s)ds +
\sum_{i=1}^kk_i \Big]\beta_{PC}(\{x_n\}_{n=1}^\infty)\\
&\leq  L_U\Big[\big(l_3 + 2\int_0^a(l_1(s)+ \tau l_2(s))ds +
\sum_{i=1}^kk_i \big)\big(1+2L_BL_U\int_0^aK_W(s)ds\big)\\
&\quad +2l_3L_B\int_0^aK_W(s)ds \Big]\beta_{PC}(\{x_n\}_{n=1}^\infty)\\
&=  d\cdot\beta_{PC}(\{x_n\}_{n=1}^\infty).
\end{aligned}
\end{equation}

Note since  $\{Gx_n\}_{n=1}^\infty$ is equicontinuous on each $J_i$,
$i = 0, \dots, k$ we have (from a well known result on measures of
noncompactness)
$$
\beta_{PC}(\{Gx_n\}_{n=1}^\infty)= \sup_{ 0 \leq i \leq k}\,\sup_{t\in
J_i}\beta(\{(Gx_n)(t)\}_{n=1}^\infty).
$$
This together with \eqref{ed},
\eqref{e3.9} and \eqref{e3.11} guarantees that
$$
\beta_{PC}(\{x_n\}_{n=1}^\infty)
\le \beta_{PC}(\{Gx_n\}_{n=1}^\infty)
\le d \cdot\beta_{PC}(\{x_n\}_{n=1}^\infty),
$$
which implies that $D =\{x_n\}_{n=1}^\infty$  is relatively compact.

From M\"{o}nch's fixed point theorem, $G$ has a fixed point in
$B(0, r)$ and immediately the system \eqref{e1.1} is nonlocally
controllable on $J$.
\end{proof}

\begin{remark}\label{rmk3} \rm
Note that \eqref{e1.1}  with no effect of time
delay was considered in \cite{jlw}. The assumptions on $f$, $M$, and $I_i$
in \cite{jlw} are relaxed in this paper.
For example  $M$ is not necessarily compact here,
 and the the assumptions (S2),  (S3), and (S5) in our paper are weaker
than assumptions (H2),  (H3), and (H5) in \cite{jlw}.
\end{remark}

\subsection*{Acknowledgements} The authors wish to thank the anonymous
 referees for their valuable suggestions.

This research was supported by grants 11171192 from the NNSF of
China, and BS2010SF025 from the Promotive Research Fund for Excellent 
Young and Middle-Aged Scientists of Shandong Province.

\begin{thebibliography}{00}

\bibitem{bg} J. Banas, K. Goebel;
\emph{Measure of Noncompactness in Banach Spaces},
Marcel Dekker, New York, 1980.

\bibitem{Bysz} L. Byszewski;
\emph{Theorems about the existence and uniqueness of
solutions of a semilinear evolution nonlocal Cauchy problem}, J.
Math. Anal. Appl., 162 (1991) 494-505.

\bibitem{Bysza} L. Byszewski, H. Akca;
\emph{Existence of solutions of a semilinear
functional-differential evolution nonlocal problem}, Nonlinear Anal.,
34 (1998) 65-72.

\bibitem{cgr} M. Choisy, J. F. Guegan, P. Rohani;
\emph{Dynamics of infectious diseases
and pulse vaccination: Teasing apart the embedded resonance effects},
Physica D., 22 (2006) 26-35.

\bibitem{d} A. d'Onofrio;
\emph{On pulse vaccination strategy in the SIR epidemic
model with vertical transmission}, Appl. Math. Lett., 18 (2005)
729-32.

\bibitem{fql} X. Fu, J. Qi, Y. Liu;
\emph{General comparison principle for impulsive variable time differential
equations with application}, Nonlinear Anal., 42 (2000) 1421-1429.

\bibitem{fyl} X. Fu, B. Yan, Y. Liu;
\emph{Introduction to Impulsive Differential System}, China Science Publisher,
Beijing, 2005 (in Chinese).

\bibitem{gcn} S. Gao, L. Chen, J. J. Nieto, A. Torres;
\emph{Analysis of a delayed epidemic model with pulse vaccination
and saturation incidence},
Vaccine., 24 (2006) 6037-6045.

\bibitem{gukh} K. Gu, V. Kharitonov, J. Chen;
\emph{Stability of Time-Delay Systems}, Birkh\"auser, Boston,
Massachusetts, 2003.

\bibitem{hale} J. Hale, S. Verduyn Lunel;
\emph{Introduction to Functional Differential
Equations}, Springer-Verlag, New York, 1993.

\bibitem{hayk} S. Haykin;
\emph{Neural Networks}, Prentice Hall, New Jersey, 1999.

\bibitem{heard} M. L. Heard;
\emph{A quasilinear hyperbolic integrodifferential equation related
to a nonlinear string}, Trans. American Math. Soc., 285 (1984) 805-823.

\bibitem{h}  H. P. Heinz;
\emph{On the behaviour of measures of noncompactness with respect to
differentiation and integration of vector-valued
functions},  Nonlinear Anal., 7 (1983) 1351-1371

\bibitem{ho} E. Hernandez, D. O'Regan;
\emph{Controllability of Volterra-Fredholm type systems in Banach space},
J. Franklin Inst., 346 (2009) 95-101.

\bibitem{jlw} S. Ji, G. Li, M. Wang;
\emph{Controllability of impulsive differential systems with nonlocal
conditions}, Applied Mathematics and Computation, 217 (2011)
6981-6989.

\bibitem{koz} M. Kamenskii, P. Obukhovskii, P. Zecca;
\emph{Condensing Multivalued Maps and Semilinear Differential
Inclusions in Banach Spaces}, De Gruyter, 2001.

\bibitem{kolm} V. Kolmanovskii, A. Myshkis;
\emph{Applied Theory of Functional Differential Equations},
 Kluwer Academic Publishers, Netherlands, 1992.

\bibitem{Lakshmi} V. Lakshmikantham, D. D. Bainov, P. S. Simeonov;
\emph{Theory of Impulsive Differential Equations}, World Scientific,
Singapore, 1989.

\bibitem{lh} S. K. Ntouyas, D. O'Regan;
\emph{Some remarks on controllability of evolution equations in Banach spaces},
Elect. J Diff. Eqns., Vol. 2009 (2009), No. 79, 1-6.

\bibitem{m} H. M\"{o}nch;
\emph{Boundary value problems for nonlinear ordinary differential equations
of second order in Banach spaces}, Nonlinear Anal.,
 4 (1980) 985-99.

\bibitem{nara} R. Narasimha;
\emph{Nonlinear vibration of an elastic string},
J. Sound Vibration, 8 (1968) 134-146.

\bibitem{nicu} S. Niculescu;
\emph{Delay Effects on Stability: A Robust
Control Approach}, Springer-Verlag, New York, 2001.

\bibitem{p} A. Pazy;
\emph{Semigroups of Linear Operators and Applications
to Partial Differential Equations}, Springer-Verlag, New York, 1983.

\bibitem{rt} I. Rachunkov, M. Tvrdy;
\emph{Non-ordered lower and upper functions in
second-order impulsive periodic problems}, Dyn. Contin. Discrete
Impuls. Syst. Ser. A Math. Anal., 12 (2005) 397-415.

\bibitem{Samo} A. M. Samoilenko, N.A. Perestyuk;
\emph{Impulsive Differential Equations}, World Scientific, Singapore, 1995.

\bibitem{tc} S. Tang, L. Chen;
\emph{Density-dependent birth rate, birth pulses and
their population dynamic consequences}, J. Math. Biol., 44 (2002)
185-199.

\bibitem{yzn} J. Yan, A. Zhao, J. J. Nieto;
\emph{Existence and global attractivity of
positive periodic solution of periodic single-species impulsive
Lotka-Volterra systems}, Math. Comput. Modelling., 40 (2004) 509-518.

\bibitem{zs} S. T. Zavalishchin, A. N. Sesekin;
\emph{Dynamic Impulse Systems: Theory and
Applications}, Kluwer Academic Publishers Group, Dordrecht, 1997.


\end{thebibliography}

\end{document}