\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations}, 
Vol. 2004(2004), No. 31, pp. 1--14.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}



\begin{document}

\title[\hfilneg EJDE-2004/31\hfil 
 N-th order impulsive integro-differential equations] 
{N-th order impulsive integro-differential equations in Banach spaces}

\author[Manfeng Hu \& Jiang Zhu\hfil EJDE-2004/31\hfilneg]
{Manfeng Hu \&  Jiang Zhu} % in alphabetical order

\address{Manfeng Hu \hfill\break
Department of Science, Jiangnan University \\
Wuxi 214000, China, and Department of Mathematics, 
Xuzhou Normal University\\
Xuzhou 221116,  China}

\address{Jiang Zhu \hfill\break
Department of Mathematics, Xuzhou Normal University\\
Xuzhou 221116,  China} \email{jzhuccy@yahoo.com.cn}

\date{}
\thanks{Submitted August 14, 2003. Published March 3, 2004.}
\subjclass[2000]{45J05, 47H07, 34A12} 
\keywords{Integro-differential equations in Banach spaces, cone, 
\hfill\break\indent 
partial ordering, monotone iterative technique}

\begin{abstract}
 We investigate the maximal and minimal solutions of initial value
 problem for N-th order nonlinear impulsive integro-differential
 equation in Banach space by establishing a comparison result and
 using the upper and lower solutions methods.
\end{abstract}

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

\section{Introduction}

The theory of impulsive differential equations in Banach spaces
has become an important area of investigation in recent years. In
\cite{Guo2}, the existence of solution of initial value problem
for second order nonlinear impulsive integro-differential equation
in Banach space was studied by establishing a comparison result
and using the upper and lower solutions methods. Now, in this
paper, we shall investigate the existence of solution of
initial-value problem (IVP) for N-th order nonlinear impulsive
integro-differential equation in Banach space by establishing a
new comparison result and using the upper and lower solutions
methods. Consider the IVP for impulsive integro-differential
equation in a Banach space $E$:
\begin{equation}
\begin{gathered}
u^{(n)}=f(t,u(t),u'(t),\dots ,u^{(n-1)}(t),(Tu)(t)),\quad\forall
t\in
J,t\neq t_{i} \\
\Delta u|_{t=t_{i}}=L_{i}^{0}u^{(n-1)}(t_{i}) \\
\Delta u'|_{t=t_{i}}=L_{i}^{1}u^{(n-1)}(t_{i}) \\
\dots \\
\Delta u^{(n-2)}|_{t=t_{i}}=L_{i}^{n-2}u^{(n-1)}(t_{i}) \\
\Delta u^{(n-1)}|_{t=t_{i}}=-L_{i}^{n-1}u^{(n-1)}(t_{i})\\
u(0)=u_{0},\; u'(0)=u_{1},\dots ,u^{(n-1)}(0)=u_{n-1}
\end{gathered} \label{1.1}
\end{equation}
where  $i=1,2,\dots ,m$, $J=[0,a](a>0)$, $u_{j}\in E(j=0,1,2,\dots
,n-1)$, $f\in C[J\times E\times E\times \dots \times E,E]$,
$0<t_{1}<\dots <t_{i}<\dots <t_{m}<a$, $L_{i}^{j}(i=1,2,\dots ,m$;
$j=0,1,\dots ,n-1)$ are constants, and
\begin{equation}
(Tu)(t)=\int_{0}^{t}k(t,s)u(s)ds,\quad  \forall t\in J \,,
\label{2}
\end{equation}
$k\in C[D,R_{+}]$, $D=\{(t,s)\in J\times J:t\geq s\}$, $R_{+}$ is
the set of all nonnegative real numbers, and $k_{0}=\max
\{k(t,s):(t,s)\in D\}$. $\Delta u^{(j)}|_{t=t_{i}}$ denotes the
jump of $u^{(j)}(t)$ at $t=t_{i}$, i.e.
\[
\Delta
u^{(j)}\big|_{t=t_{i}}=u^{(j)}(t_{i}^{+})-u^{(j)}(t_{i}^{-}),
\]
where $u^{(j)}(t_{i}^{+})$ and $u^{(j)}(t_{i}^{-})$ represent the
right-hand limit and left-hand limit of $u^{(j)}(t)$ at
${t=t_{i}}$ respectively. In \eqref{1.1} and the following,
$u^{(n-1)}(t_{i})$ is understood as $u^{(n-1)}(t_{i}^{-})$.

Let $PC[J,E]=\{u:u$ is a map from $J$ into $E$ such that $u(t)$ is
continuous at $t\neq t_{i}$, left continuous at $t=t_{i}$ and
$u(t_{i}^{+})$ exist for $i=1,2,\dots ,m\}$, $PC^{j}[J,E]=\{u\in
PC^{j-1}[J,E]$: $ u^{(j)}(t)$ is continuous at $t\neq t_{i}$, left
continuous at $t=t_{i}$ and $u^{(j)}(t_{i}^{+})$ exist for
$i=1,2,\dots ,m\}(j=1,2,\dots ,n-2)$ and $ PC^{n-1}[J,E]=\{u\in
PC^{n-2}[J,E]:u^{(n-1)}(t)$ is continuous at $t\neq t_{i}$, and
$u^{(n-1)}(t_{i}^{+})$,$u^{(n-1)}(t_{i}^{-})$ exist for $
i=1,2,\dots ,m\}$. Evidently, $PC[J,E]$ is a Banach space with
norm
\[
\Vert u\Vert _{pc}=\sup_{t\in J}\Vert u(t)\Vert .
\]
It is clear that $PC^{j}[J,E]$ is a Banach space with norm
\[
\Vert u\Vert _{j}=\max \{\Vert u\Vert _{pc},\Vert u'\Vert
_{pc},\dots ,\Vert u^{(j)}\Vert _{pc}\},\quad (j=1,2,\dots ,n-1)
\]
Let $J'=J\backslash \{t_1,\dots ,t_{m}\}$, $\tau
=\max\{t_{i}-t_{i-1}:i=1,2,\dots ,m+1\}$, (where
$t_{0}=0,t_{m+1}=a$), $J_{0}=[0,t_{1}]$,
$J_{1}=(t_{1},t_{2}],\dots ,J_{m-1}=(t_{m-1},t_{m}]$,
$J_{m}=(t_{m},a]$. A map $u\in PC^{n-1}[J,E]\bigcap C^{n}[J',E]$
is called a solution of \eqref{1.1} if it satisfies \eqref{1.1}.

\section{Comparison Result}

Let $E$ be partially ordered by a cone $P$ of $E$, i.e. $x\leq y$
if and only if $y-x\in P$. $P$ is said to be normal if there
exists a positive\ constant $N$ such that $\theta \leq x\leq y$
implies $\Vert x\Vert \leq N\leq \Vert y\Vert $, where $\theta $
denotes the zero element of $E$, and $ P $ is said to be regular
if $x_{1}\leq x_{2}\leq \dots \leq x_{n}\leq \dots \leq y$ implies
$\Vert x_{n}-x\Vert \to 0$ as $n\to \infty $ for some $x\in E$. It
is well known that the regularity of $P$ implies the normality of
$P$. For details on cone theory, see \cite{Guo1}.

\begin{lemma}[Comparison result] \label{lm2.1}
Assume that $p\in PC^{n-1}[J,E]\bigcap C^{n}[J',E]$ satisfies
\begin{equation}
\begin{gathered}
p^{(n)}(t)\leq -M_{0}p-M_{1}p'-M_{2}p''-\dots
-M_{n-1}p^{(n-1)}-NTp,\quad \forall t\in J,t\neq t_{i} \\
\Delta p|_{t=t_{i}}=L_{i}^{0}p^{(n-1)}(t_{i}) \\
\Delta p'|_{t=t_{i}}=L_{i}^{1}p^{(n-1)}(t_{i}) \\
\dots  \\
\Delta p^{(n-2)}|_{t=t_{i}}=L_{i}^{n-2}p^{(n-1)}(t_{i}) \\
\Delta p^{(n-1)}|_{t=t_{i}}\leq -L_{i}^{n-1}p^{(n-1)}(t_{i}),\quad
(i=1,2,\dots ,m) \\
p^{(n-1)}(0)\leq p^{(j)}(0)\leq  \theta ,\quad j=0,1,\cdots ,n-2,
\end{gathered} \label{2.1}
\end{equation}
where $M_{j}\geq 0$, $L_{i}^{j}\geq 0$ $(j=0,1,\dots
,n-1;i=1,2,\dots ,m)$ are constants and
\begin{equation}
\sum_{i=1}^{m}L_{i}^{n-1}+(m+1)M_{0}\tau \leq 1  \label{2.2}
\end{equation}
where
\begin{equation}
M_{0}=M_{n-1}+\sum_{i=0}^{n-2}c_{i}^{\star }+k_{1}^{\star
}a+\sum_{j=0}^{n-2}(M_{j}\sum_{i=1}^{m}L_{i}^{j})+\sum
_{j=0}^{n-2}(d_{j}^{\star }\sum_{i=1}^{m}L_{i}^{j})a \label{2.3}
\end{equation}
\begin{gather*}
c_{i}^{\ast }=\sum_{j\leq
i}\frac{a^{i-j}}{(i-j)!}M_{j}+\frac{a^{i+1}
}{(i+1)!}Nk_{0},i=0,1,\dots ,n-2\,, \\
k_{1}^{\ast
}=\frac{a^{n-2}}{(n-2)!}M_{0}+\frac{a^{n-3}}{(n-3)!}M_{1}+\dots
+M_{n-2}+\frac{k_{0}Na^{n-1}}{(n-1)!},\\
d_{0}^{\ast }=Nk_{0},\\
d_{1}^{\star }=Nk_{0}a+M_{0},\\
d_{j}^{\ast
}=\frac{Nk_{0}a^{j}}{j!}+\frac{M_{0}a^{j-1}}{(j-1)!}+\dots
+M_{j-2}a+M_{j-1},j=2,3,\dots ,n-2,
\end{gather*}
then $p^{(j)}(t)\leq \theta ,\forall t\in J,(j=0,1,\dots ,n-1)$,
where $p^{(0)}(t)=p(t)$.
\end{lemma}

\begin{proof}
 Let $p_{1}(t)=p^{(n-1)}(t),t\in J$. Then $p_{1}\in
PC[J,E]\bigcap C^{1}[J',E]$ and
\[
p^{(n-2)}(t)=p^{(n-2)}(0)+\int_{0}^{t}p_{1}(s)ds+\sum
_{0<t_{i}<t}[p^{(n-2)}(t_{i}^{+})-p^{(n-2)}(t_{i}^{-})],
\]
\begin{align*}
p^{(n-3)}(t) & =  p^{(n-3)}(0)+tp^{(n-2)}(0)+\int_{0}^{t}ds_{1}
\int_{0}^{s_{1}}p_{1}(s_{2})ds_{2} \\
& \quad+\int_{0}^{t}\sum
_{0<t_{i}<s}[p^{(n-2)}(t_{i}^{+})-p^{(n-2)}(t_{i}^{-})]ds \\
& \quad
+\sum_{0<t_{i}<t}[p^{(n-3)}(t_{i}^{+})-p^{(n-3)}(t_{i}^{-})]
\end{align*}
\dots
\begin{align*}
p'(t) & =  p'(0)+tp''(0)+\dots +\frac{
t^{n-3}}{(n-3)!}p^{(n-2)}(0) \\
& \quad +\int_{0}^{t}ds_{1}\int_{0}^{s_{1}}ds_{2}\dots
\int_{0}^{s_{n-3}}p_{1}(s_{n-2})ds_{n-2} \\
&\quad +\int_{0}^{t}ds_{1}\int_{0}^{s_{1}}ds_{2}\dots
\int_{0}^{s_{n-4}}\sum_{0<t_{i}<s_{n-3}}[p^{(n-2)}(t_{i}^{+})-p^{(n-2)}
(t_{i}^{-})]ds_{n-3}
\\
& \quad +\dots
 +\int_{0}^{t}\sum_{0<t_{i}<s}[p''(t_{i}^{+})-p''(t_{i}^{-})]ds
 +\sum_{0<t_{i}<t}[p'(t_{i}^{+})-p'(t_{i}^{-})]
\end{align*}
\begin{align*}
p(t) & =  p(0)+tp'(0)+\dots +\frac{t^{n-2}}{(n-2)!}p^{(n-2)}(0)\\
& \quad +\int_{0}^{t}ds_{1}\int_{0}^{s_{1}}ds_{2}\dots
\int_{0}^{s_{n-2}}p_{1}(s_{n-1})ds_{n-1} \\
& \quad +\int_{0}^{t}ds_{1}\int_{0}^{s_{1}}ds_{2}\dots
\int_{0}^{s_{n-3}}\sum
_{0<t_{i}<s_{n-2}}[p^{(n-2)}(t_{i}^{+})-p^{(n-2)}(t_{i}^{-})]ds_{n-2}
\\
&\quad +\dots
+\int_{0}^{t}\sum_{0<t_{i}<s}[p'(t_{i}^{+})-p'(t_{i}^{-})]ds
+\sum_{0<t_{i}<t}[p(t_{i}^{+})-p(t_{i}^{-})]\,.
\end{align*}
It is easy to see by induction that for $m=1,2,\dots ,n-1$,
\[
\int_{0}^{t}ds_{1}\int_{0}^{s_{1}}ds_{2}\dots
\int_{0}^{s_{m-1}}A(s_{m})ds_{m}=\frac{1}{(m-1)!}
\int_{0}^{t}(t-s)^{m-1}A(s)ds,
\]
where $A$ denotes an integrable function. So, we have
\begin{equation}
\begin{gathered}
p^{(n-1)}(t)  =  p_{1}(t) \\
p^{(n-2)}(t)  =  p^{(n-2)}(0)+\int_{0}^{t}p_{1}(s)ds+\sum
_{0<t_{i}<t}[p^{(n-2)}(t_{i}^{+})-p^{(n-2)}(t_{i}^{-})] \\
\begin{aligned}
p^{(n-3)}(t) & =
p^{(n-3)}(0)+tp^{(n-2)}(0)+\int_{0}^{t}(t-s)p_{1}(s)ds \\
&\quad +\int_{0}^{t}\sum
_{0<t_{i}<s}[p^{(n-2)}(t_{i}^{+})-p^{(n-2)}(t_{i}^{-})]ds \\
&\quad
+\sum_{0<t_{i}<t}[p^{(n-3)}(t_{i}^{+})-p^{(n-3)}(t_{i}^{-})]
\end{aligned}\\
\dots      \\
\begin{aligned}
p'(t) & =  p'(0)+tp''(0)+\dots +\frac{
t^{n-3}}{(n-3)!}p^{(n-2)}(0)+\frac{1}{(n-3)!}
\int_{0}^{t}(t-s)^{n-3}p_{1}(s)ds \\
& \quad+\frac{1}{(n-4)!}\int_{0}^{t}(t-s)^{n-4}\sum
_{0<t_{i}<s}[p^{(n-2)}(t_{i}^{+})-p^{(n-2)}(t_{i}^{-})]ds \\
&\quad
+\cdots+\int_{0}^{t}\sum_{0<t_{i}<s}[p''(t_{i}^{+})-p''(t_{i}^{-})]ds
+\sum_{0<t_{i}<t}[p'(t_{i}^{+})-p'(t_{i}^{-})]
\end{aligned}\\
\begin{aligned}
p(t) & =  p(0)+tp'(0)+\dots +\frac{t^{n-2}}{(n-2)!}p^{(n-2)}(0)+
\frac{1}{(n-2)!}\int_{0}^{t}(t-s)^{n-2}p_{1}(s)ds \\
&\quad +\frac{1}{(n-3)!}\int_{0}^{t}(t-s)^{n-3}\sum
_{0<t_{i}<s}[p^{(n-2)}(t_{i}^{+})-p^{(n-2)}(t_{i}^{-})]ds \\
&\quad
+\cdots+\int_{0}^{t}\sum_{0<t_{i}<s}[p'(t_{i}^{+})-p'(t_{i}^{-})]ds
+\sum_{0<t_{i}<t}[p(t_{i}^{+})-p(t_{i}^{-})]
\end{aligned}
\end{gathered}  \label{2.7}
\end{equation}
Substituting (\ref{2.7}) into (\ref{2.1}), we get
\begin{equation}
\begin{aligned}
p_{1}'(t) & \leq
-M_{n-1}p_{1}(t)-c_{0}(t)p(0)-c_{1}(t)p'(0)-\dots
-c_{n-2}(t)p^{(n-2)}(0) \\
& \quad
-\int_{0}^{t}k_{1}(t,s)p_{1}(s)ds-\sum_{j=0}^{n-2}M_{j}\sum
_{0<t_{i}<t}[p^{(j)}(t_{i}^{+})-p^{(j)}(t_{i})] \\
& \quad -\sum_{j=0}^{n-2}\int_{0}^{t}d_{j}(t,s)\sum
_{0<t_{i}<s}[p^{(j)}(t_{i}^{+})-p^{(j)}(t_{i})]ds,\quad \forall
t\in J
\end{aligned} \label{2.8}
\end{equation}
Where
\begin{gather*}
c_{0}(t)=M_{0}+N\int_{0}^{t}k(t,s)ds, \\
c_{1}(t)=tM_{0}+M_{1}+N\int_{0}^{t}sk(t,s)ds, \\
\dots  \\
c_{n-2}(t)=\frac{t^{n-2}}{(n-2)!}M_{0}+\frac{t^{n-3}}{(n-3)!}M_{1}+\dots
+M_{n-2}+\frac{N}{(n-2)!}\int_{0}^{t}s^{n-2}k(t,s)ds, \\
\begin{aligned}
k_{1}(t,s)&=\frac{(t-s)^{n-2}}{(n-2)!}M_{0}+\frac{(t-s)^{n-3}}{(n-3)!}M_{1}\\
&\quad +\dots
+M_{n-2}+\frac{N}{(n-2)!}\int_{s}^{t}(r-s)^{n-2}k(t,r)dr,
\\
\end{aligned}\\
d_{0}(t,s)=Nk(t,s),\\
d_{1}(t,s)=N\int_{s}^{t}k(t,r)dr+M_{0}, \\
d_{2}(t,s)=\frac{N}{1!}\int_{s}^{t}k(t,r)(r-s)dr+M_{0}(t-s)+M_{1},\\
\dots  \\
\begin{aligned}
d_{n-2}(t,s)&=\frac{N\
}{(n-3)!}\int_{s}^{t}k(t,r)(r-s)^{n-3}dr+M_{0}
\frac{(t-s)^{n-3}}{(n-3)!}\\
&\quad +\dots +M_{n-4}(t-s)+M_{n-3}\,.
\end{aligned}
\end{gather*}
For $g\in P^{\ast }$, the dual cone of $P$, let
$v(t)=g(p_{1}(t))$, then $v\in PC[J,R]\bigcap C'[J',R]$. By
(\ref{2.8}) and (\ref{2.1}), we have
\begin{equation}
\begin{gathered}
\begin{aligned}
v'(t) & \leq  -M_{n-1}v(t)-\sum_{j=0}^{n-2}c_{j}(t)g(p^{(j)}(0))
-\int_{0}^{t}k_{1}(t,s)v(s)ds \\
&\quad
-\sum_{j=0}^{n-2}M_{j}\sum_{0<t_{i}<t}g(p^{(j)}(t_{i}^{+})-p^{(j)}(t_{i})) \\
& \quad -\sum_{j=0}^{n-2}\int_{0}^{t}d_{j}(t,s)\sum
_{0<t_{i}<s}g(p^{(j)}(t_{i}^{+})-p^{(j)}(t_{i}))ds,\quad \forall
t\in J
\end{aligned} \\
g(p^{(j)}(t_{i}^{+})-p^{(j)}(t_{i}))=L_{i}^{j}v(t_{i})\quad
(j=0,1,2,\dots ,n-2;i=1,2,\dots ,m) \\
\Delta v|_{t=t_{i}}\leq -L_{i}^{n-1}v(t_{i})\quad (i=1,2,\dots ,m) \\
v(0)\leq g(p^{(j)}(0))\leq 0,\quad j=0,1,\cdots ,n-2,
\end{gathered} \label{2.9}
\end{equation}
We now show that
\begin{equation}
v(t)\leq 0,\quad  \forall t\in J.  \label{2.10}
\end{equation}
Assume that (\ref{2.10}) is not true, i.e. there exists a
$0<t^{\ast }\leq a$
 such that $v(t^{\ast })>0$. Let $t^{\ast }\in J_{j}=(t_{j},t_{j+1}]$
and $\inf_{0\leq t\leq t^{\ast }}v(t)=-\lambda $. We have $\lambda
\geq 0$. Assume that there exist a $J_{k}=(t_{k},t_{k+1}](k\leq
j)$  such that $v(t_{\ast })=-\lambda $ hold for some $t_{\ast
}\in J_{k}$ or $ v(t_{k}^{+})=-\lambda $. We may assume that
$v(t_{\ast })=-\lambda$, since for the  case
$v(t_{k}^{+})=-\lambda $ the proof is similar. By (\ref{2.9}), we
have
\begin{equation}
\begin{aligned}
v'(t) & \leq   \lambda \lbrack
M_{n-1}+\sum_{j=0}^{n-2}c_{j}(t)+\int_{0}^{t}k_{1}(t,s)ds+\sum
_{j=0}^{n-2}(M_{j}\sum_{i=1}^{m}L_{i}^{j}) \\
&\quad +\sum_{j=0}^{n-2}\int_{0}^{t}(d_{j}(t,s)\sum
_{0<t_{i}<s}L_{i}^{j}ds] \\
& \leq \lambda M_{0},\quad \forall t\in [ 0,t^{\ast }],
\end{aligned} \label{2.11}
\end{equation}
where $M_{0}$ is given by (\ref{2.3}),
\begin{equation}
\begin{gathered}
\Delta v|_{t=t_{i}}\leq -L_{i}^{n-1}v(t_{i})\leq \lambda
L_{i}^{n-1}\quad
(i=1,2,\dots ,m)\\
v(0)\leq 0
\end{gathered} \label{2.12}
\end{equation}
Now, the mean value theorem implies
\begin{equation}
\begin{gathered}
v(t^{\ast })-v(t_{j}^{+})=v'(\xi _{j})(t^{\ast }-t_{j}),\quad
t_{j}<\xi _{j}<t^{\ast }; \\
v(t_{j})-v(t_{j-1}^{+})=v'(\xi _{j-1})(t_{j}-t_{j-1}),\quad
t_{j-1}<\xi _{j-1}<t_{j}; \\
\dots \\
v(t_{k+2})-v(t_{k+1}^{+})=v'(\xi _{k+1})(t_{k+2}-t_{k+1}),\quad
t_{k+1}<\xi _{k+1}<t^{k+2}; \\
v(t_{k+1})-v(t_{\ast })=v'(\xi _{k})(t_{k+1}-t_{\ast }),\quad
t_{\ast }<\xi _{k}<t_{k+1}.
\end{gathered} \label{2.13}
\end{equation}
By (\ref{2.12}) we have
\begin{equation}
v(t_{i}^{+})=v(t_{i})+\Delta v|_{{t=t_{i}}}\leq v(t_{i})+\lambda
L_{i}^{n-1},\quad \forall t_{i}\leq t^{\ast }.  \label{2.14}
\end{equation}
By (\ref{2.11}), (\ref{2.13}), (\ref{2.14}), we obtain
\begin{equation}
\begin{gathered}
v(t^{\ast })-v(t_{j})-\lambda L_{j}^{n-1}\leq \lambda M_{0}\tau  \\
v(t_{j})-v(t_{j-1})-\lambda L_{j-1}^{n-1}\leq \lambda M_{0}\tau  \\
\dots \\
v(t_{k+2})-v(t_{k+1})-\lambda L_{k+1}^{n-1}\leq \lambda M_{0}\tau  \\
v(t_{k+1})+\lambda \leq \lambda M_{0}\tau\,.
\end{gathered}  \label{2.15}
\end{equation}
Adding these inequalities,  we obtain
\[
v(t^{\ast })+\lambda -\lambda \sum_{i=k+1}^{j}L_{i}^{n-1}\leq
(j-k+1)\lambda M_{0}\tau
\]
and so
\[
0<v(t^{\ast })\leq -\lambda +\lambda
\sum_{i=1}^{m}L_{i}^{n-1}+(m+1)\lambda M_{0}\tau
\]
Evidently $\lambda \neq 0$, so, $\lambda >0$, then, we have
\[
1<\sum_{i=1}^{m}L_{i}^{n-1}+(m+1)M_{0}\tau ,
\]
which contradicts (\ref{2.2}), hence $v(t)\leq 0,\forall t\in J$.
Since $g\in P^{\ast }$ is arbitrary, it implies that
$p^{(n-1)}(t)\leq \theta$ for all $t\in J$. By $p^{(n-2)}(0)\leq
\theta $, $\Delta
p^{(n-2)}|_{t=t_{i}}=L_{i}^{n-2}p^{(n-1)}(t_{i})\leq \theta $;
this implies $p^{(n-2)}(t)\leq \theta$ for all $t\in J$.
Continuing in this manner, $p^{(i)}(t)\leq \theta$ for all $t\in
J$, $i=0,1,\dots ,n-3$. The proof is complete.
\end{proof}

\begin{lemma} \label{lm2.2}
 Assume $\sigma \in PC[J,E]$, and $M_{j}$, $N$,
$L_{i}^{j}$ $(j=0,1,2,\dots ,n-1;i=1,2,\dots ,m)$ are constants,
then $u\in PC^{n-1}[J,E]\bigcap C^{n}[J',E]$ is a solution of the
linear IVP
\begin{equation}
\begin{gathered}
u^{(n)}=-\sum_{j=0}^{n-1}M_{j}u^{(j)}-NTu+\sigma (t),\quad \forall
t\in J,\;t\neq t_{i} \\
\Delta u^{(j)}|_{t=t_{i}}=L_{i}^{j}u^{(n-1)}(t_{i}),\quad
(j=0,1,\dots ,n-2) \\
\Delta u^{(n-1)}|_{t=t_{i}}=-L_{i}^{n-1}u^{(n-1)}(t_{i}),\quad
(i=1,2,\dots ,m) \\
u(0)=u_{0},\; u'(0)=u_{1},\dots ,u^{(n-1)}(0)=u_{n-1}.
\end{gathered}\label{2.16}
\end{equation}
if and only if $u\in PC^{n-1}[J,E]$ is a solution of the linear
impulsive integral equation
\begin{equation}
\begin{aligned}
u(t)&= u_{0}+tu_{1}+\frac{t^{2}}{2!}u_{2}+\dots
+\frac{t^{n-1}}{(n-1)!}
u_{n-1} \\
&\quad +\frac{1}{(n-1)!}\int_{0}^{t}(t-s)^{n-1}[-\sum
_{j=0}^{n-1}M_{j}u^{(j)}(s)-N(Tu)(s)+\sigma (s)]ds \\
&\quad
+\sum_{0<t_{i}<t}[-\frac{(t-t_{i})^{n-1}}{(n-1)!}L_{i}^{n-1}+\frac{
(t-t_{i})^{n-2}}{(n-2)!}L_{i}^{n-2}\\
&\quad +\dots +(t-t_{i})L_{i}^{1}
 +L_{i}^{0}]u^{(n-1)}(t_{i})\quad \forall t\in J\,.
\end{aligned}\label{2.17}
\end{equation}
\end{lemma}

The proof of this lemma is similar to the proof of Lemma 3 in
\cite{Guo2}; therefore, we omit it.

\begin{lemma} \label{lm2.3}
Let $\sigma \in PC[J,E]$, $M_{j}\geq 0$, $N\geq 0$, $L_{i}^{j}\geq
0$ $(j=0,1,2,\dots ,n-1;i=1,2,\dots ,m)$ be constants. Assume
\begin{equation}
\begin{gathered}
\begin{aligned}
\beta _{j}&=\frac{\sum_{i=0}^{n-1}M_{i}+Nk_{0}a}{(n-j)!}
a^{n-j}+\sum_{i=1}^{m}[\frac{(a-t_{i})^{n-j-1}}{(n-j-1)!}
L_{i}^{n-1}\\
&\quad +\dots +(a-t_{i})L_{i}^{j+1}+L_{i}^{j}]<1
\end{aligned}\\
\beta =\max_{j}\{\beta _{j}\}
\end{gathered}\label{2.18}
\end{equation}
where $j=0,1,\dots ,n-1)$. Then the impulsive integral equation
(\ref{2.17}) has a unique solution in $PC^{n-1}[J,E]$.
\end{lemma}

\begin{proof} Define operator $F$ by
\begin{align*}
(Fu)(t)= & u_{0}+tu_{1}+\frac{t^{2}}{2!}u_{2}+\dots
+\frac{t^{n-1}}{(n-1)!}
u_{n-1} \\
&\quad +\frac{1}{(n-1)!}\int_{0}^{t}(t-s)^{n-1}[-\sum
_{j=0}^{n-1}M_{j}u^{(j)}(s)-N(Tu)(s)+\sigma (s)]ds \\
&\quad
+\sum_{0<t_{i}<t}[-\frac{(t-t_{i})^{n-1}}{(n-1)!}L_{i}^{n-1}+\frac{
(t-t_{i})^{n-2}}{(n-2)!}L_{i}^{n-2} \\
&\quad +\dots +(t-t_{i})L_{i}^{1}+L_{i}^{0}]u^{(n-1)}(t_{i})\quad
(\forall t\in J).
\end{align*}
Then for all $t\in J',j=1,2,\dots ,n-1$,
\begin{align*}
(Fu)^{(j)}(t)&=  u_{j}+tu_{j+1}+\dots
+\frac{t^{n-j-1}}{(n-j-1)!}u_{n-1} \\
&\quad +\frac{1}{(n-j-1)!}\int_{0}^{t}(t-s)^{n-j-1}[-\sum
_{j=0}^{n-1}M_{j}u^{(j)}(s)-N(Tu)(s)+\sigma (s)]ds \\
&\quad +\sum_{0<t_{i}<t}[-\frac{(t-t_{i})^{n-j-1}}{(n-j-1)!}
L_{i}^{n-1}+\dots +(t-t_{i})L_{i}^{j+1}+L_{i}^{j}]u^{(n-1)}(t_{i})
\end{align*}
and $F:PC^{n-1}[J,E]\to PC^{n-1}[J,E]$. For $u,v\in
PC^{n-1}[J,E]$, by (\ref{2.15}) we have
\begin{align*}
&\Vert (Fu)^{(j)}(t)-(Fv)^{(j)}(t)\Vert  \\
& \leq  \frac{\sum_{i=0}^{n-1}M_{i}+Nk_{0}a}{(n-j-1)!} \Vert
u-v\Vert _{n-1}\int_{0}^{t}(t-s)^{n-j-1}
+\sum_{i=1}^{m}[\frac{(t-t_{i})^{n-j-1}}{(n-j-1)!}L_{i}^{n-1} \\
&\quad +\frac{(t-t_{i})^{n-j-2}}{(n-j-2)!}L_{i}^{n-2}+\dots
+(t-t_{i})L_{i}^{j+1}+L_{i}^{j}]\Vert u-v\Vert _{n-1} \\
& \leq  \beta _{j}\Vert u-v\Vert _{n-1} \quad
 (\forall t\in J,j=0,1,\dots ,n-1)
\end{align*}
and
\begin{equation}
\Vert Fu-Fv\Vert _{n-1}\leq \beta \Vert u-v\Vert _{n-1},\quad
\forall u,v\in PC^{n-1}[J,E]  \label{2.22}
\end{equation}
where $\beta _{j},\beta $ is defined by (\ref{2.18}), The Banach
fixed point implies that $F$ has a unique fixed point in
$PC^{n-1}[J,E]$, and the lemma is proved.
\end{proof}

\section{Main Theorem}

Let us list some conditions used for stating the main result.
\begin{itemize}
\item[(H1)]
 There exist $v_{0}$,$w_{0}\in PC^{n-1}[J,E]\bigcap C^{n}[J',E]$
with $v_{0}(t)\leq w_0(t)(t\in J)$ such that
\begin{gather*}
v_{0}^{(n)}\leq f(t,v_{0},v_{0}',\dots
,v_{0}^{(n-1)},Tv_{0}),\quad
\forall t\in J,t\neq t_{i} \\
\Delta v_{0}^{(j)}|_{t=t_{i}}=L_{i}^{j}v_{0}^{(n-1)}(t_{i}),\quad
(j=0,1,\dots ,n-2;i=1,2,\dots ,m) \\
\Delta v_{0}^{(n-1)}|_{t=t_{i}}\leq -L_{i}^{n-1}v_{0}^{(n-1)}(t_{i}) \\
 v_{0}^{(j)}(0)\leq u_{j},v_{0}^{(n-1)}(0)-v_{0}^{(j)}(0)\leq
u_{n-1}-u_{j}\quad (j=0,1,2,\dots ,n-1),
\end{gather*}
and
\begin{gather*}
w_{0}^{(n)}\geq f(t,w_{0},w_{0}',\dots
,w_{0}^{(n-1)},Tw_{0}),\quad
 \forall t\in J,t\neq t_{i} \\
\Delta w_{0}^{(j)}|_{t=t_{i}}=L_{i}^{j}w_{0}^{(n-1)}(t_{i}),\quad
(j=0,1,\dots ,n-2;i=1,2,\dots ,m) \\
\Delta w_{0}^{(n-1)}|_{t=t_{i}}\geq -L_{i}^{n-1}w_{0}^{(n-1)}(t_{i}) \\
w_{0}^{(j)}(0)\geq u_{j},\quad w_{0}^{(n-1)}(0)-\
w_{0}^{(j)}(0)\geq u_{n-1}-u_{j},\quad (j=0,1,2,\dots ,n-1),
\end{gather*}
where $L_{i}^{j}\geq 0$, $(i=1,2,\dots ,m;j=0,1,\dots ,n-1)$,
$v_{0}$ and $w_{0}$ are lower and upper solution of \eqref{1.1}
respectively.

\item[(H2)] There exist constants $M_{i}\geq 0$ $(i=0,1,\dots ,n-1)$
and $N\geq 0$ such that
\begin{gather*}
f(t,u_{0},u_{1},u_{2},\dots
,u_{n-1},v)-f(t,\bar{u}_{0},\bar{u}_{1},\bar{u}
_{2},\dots ,\bar{u}_{n-1},\bar{v}) \\
\geq -\sum_{j=0}^{n-1}M_{j}(u_{j}-\bar{u}_{j})-N(v-\bar{v}),\quad
 \forall t\in J \\
v_{0}^{(j)}\leq \bar{u}_{j}\leq u_{j}\leq w_{0}^{(j)},\quad
(j=0,1,\dots ,n-1)\\
Tv_{0}\leq \bar{v}\leq v\leq Tw_{0}
\end{gather*}
\end{itemize}
Let $[v_{0},w_{0}]=\{u\in PC^{n-1}[J,E]:v_{0}^{(j)}(t)\leq
u^{(j)}(t)\leq w_{0}^{(j)}(t),t\in J,j=0,1,\dots ,n-1\}$

\begin{theorem} \label{thm3.1}
 Let cone $P$ be regular and $f$ be uniformly
continuous on $J\times B_{r}\times B_{r}\times \dots \times B_{r}$
for any $r>0$, where $B_{r}=\{x\in E:\Vert x\Vert \leq r\}$.
Suppose that conditions (H1) and (H2) are satisfied,
$L_{i}^{j}\geq 0(j=0,1,\dots ,n-1;i=1,2,\dots ,m)$ and
inequalities (\ref{2.2}), (\ref{2.18}) hold. Then \eqref{1.1} has
minimal and maximal solutions $\bar{u}$ and $u^{\ast }$ in
$[v_{0},w_{0}]$; Moreover, there exist monotone sequences
$\{v_{k}(t)\}$ and $\{w_{k}(t)\}$ such that
$\{v_{k}^{(j)}(t)\}$,$\{w_{k}^{(j)}(t)\}(j=0,1,2,\dots ,n-1)$
converge uniformly on $J_{j}(j=0,1,\dots ,m)$ to the
$\bar{u}^{(j)}(t)$ and $(u^{\ast })^{(j)}(t)(j=0,1,2,\dots ,n-1)$
respectively, and
\begin{equation}
\begin{gathered}
v_{0}^{(j)}(t)\leq v_{1}^{(j)}(t)\leq \dots \leq
v_{k}^{(j)}(t)\leq \dots
\leq \bar{u}^{(j)}(t) \\
\leq u^{(j)}(t)\leq (u^{\ast })^{(j)}(t)\leq \dots \leq
w_{k}^{(j)}(t)\leq \dots \leq w_{1}^{(j)}(t)\leq w_{0}^{(j)}(t)
\end{gathered} \label{3.1}
\end{equation}
for all $t\in J$, $j=0,1,\dots ,n-1$, where $u(t)$ is any solution
of \eqref{1.1} in $[v_{0},w_{0}]$.
\end{theorem}

\begin{proof} For  $\eta \in [v_{0},\omega _{0}]$, consider the
linear problem (\ref{2.16}) with
\begin{equation}
\sigma (t)=f(t,\eta (t),\eta '(t),\dots ,\eta ^{(n-1)}(t),(T\eta
)(t))+\sum_{j=0}^{n-1}M_{j}\eta ^{(j)}(t)+N(T\eta )(t)
\label{3.2}
\end{equation}
By Lemma \ref{lm2.3}, (\ref{2.16}) has a unique solution $u\in
PC^{n-1}[J,E]$. Let $u=A\eta $. Then $A:[v_{0},w_{0}]\to
PC^{n-1}[J,E]\bigcap C^{n}[J',E]\subset PC[J,E]$, we now show that
\begin{itemize}
\item[(a)] $v_{0}^{(j)}(t)\leq (Av_{0})^{(j)}(t),(Aw_{0})^{(j)}(t)\leq
(w_{0})^{(i)}(t)$, $t\in J,j=0,1,2,\dots ,n-1$

\item[(b)] $\eta _{1},\eta _{2}\in [v_{0},w_{0}]$,
$\eta _{1}^{(j)}\leq \eta_{2}^{(j)}$ implies $(A\eta
_{1})^{(j)}\leq (A\eta _{2})^{(j)}$, $t\in J$, $j=0,1,2,\dots
,n-1$.
\end{itemize}
To prove (a), we set $v_{1}=Av_{0}$ and $p=v_{0}-v_{1}$. From
(\ref{2.16}) and (\ref{3.2}), we have
\begin{gather*}
\begin{aligned}
v_{1}^{(n)}&=f(t,v_{0},v_{0}',\dots
,v_{0}^{(n-1)},Tv_{0})+\sum_{j=0}^{n-1}M_{j}v_{0}^{(j)}+N(Tv_{0}) \\
&\quad -\sum_{j=0}^{n-1}M_{j}v_{1}^{(j)}-N(Tv_{1}),\quad \forall
t\in J,t\neq t_{i}
\end{aligned}\\
\Delta v_{1}^{(j)}|_{t=t_{i}}=L_{i}^{j}v_{1}^{(n-1)}(t_{i}),\quad
 (j=0,1,\dots ,n-2) \\
\Delta
v_{1}^{(n-1)}|_{t=t_{i}}=-L_{i}^{n-1}v_{1}^{(n-1)}(t_{i}),\quad
 (i=1,2,\dots ,m) \\
v_{1}(0)=u_{0},\; v_{1}'(0)=u_{1},\dots ,v_{1}^{(n-1)}(0)=u_{n-1}
\end{gather*}
so, by (H1),
\begin{gather*}
p^{(n)}(t)\leq -\sum_{j=0}^{n-1}M_{j}p^{(j)}(t)-N(Tp)(t),\quad
\forall t\in J,t\neq t_{i}, \\
\Delta p^{(j)}|_{t=t_{i}}=L_{i}^{j}p^{(n-1)}(t_{i}),\quad
(j=0,1,\dots ,n-2;i=1,2,\dots ,m), \\
\Delta p^{(n-1)}|_{t=t_{i}}\leq -L_{i}^{n-1}p^{(n-1)}(t_{i}),\quad
(i=1,2,\dots ,m), \\
p^{(n-1)}(0)\leq p^{(j)}(0)\leq \theta ,\quad (j=0,1,2,\dots
,n-2),
\end{gather*}
which implies by virtue of Lemma \ref{lm2.1} that $p^{(j)}(t)\leq
\theta$ $(j=0,1,\dots ,n-1)$ for $t\in J$, i.e.
$v_{0}^{(j)}(t)\leq (Av_{0})^{(j)}(t)$, for all $t\in J$,
$j=0,1,2,\dots ,n-1$. Similarly, we can show that
$(Aw_{0})^{(j)}(t)\leq (w_{0})^{(j)}(t)$ for all $t\in J$,
$j=0,1,2,\dots,n-1$.

To prove (b), let $\eta _{1},\eta _{2}\in [v_{0},w_{0}]$, such
that $\eta _{1}^{(j)}\leq \eta _{2}^{(j)}$ and $p=A\eta _{1}-A\eta
_{2}$. Then, from \eqref{2.16} and (H2), we have
\begin{gather*}
p^{(n)}(t)\leq -\sum_{j=0}^{n-1}M_{j}p^{(j)}(t)-N(Tp)(t),\quad
\forall t\in J,t\neq t_{i}, \\
\Delta p^{(j)}|_{t=t_{i}}=L_{i}^{j}p^{(n-1)}(t_{i}),\quad
(j=0,1,\dots ,n-2;i=1,2,\dots ,m), \\
\Delta p^{(n-1)}|_{t=t_{i}}=-L_{i}^{n-1}p^{(n-1)}(t_{i}),\quad
(i=1,2,\dots ,m), \\
p^{(j)}(0)=\theta ,\quad (j=0,1,2,\dots ,n-1).
\end{gather*}
So, Lemma \ref{lm2.1} implies (b). Let
\begin{equation}
v_{k}=Av_{k-1},\quad w_{k}=Aw_{k-1},\quad k=1,2,\dots ,
\label{3.3}
\end{equation}
By (a) and (b) above, we have
\begin{equation}
v_{0}^{(j)}(t)\leq v_{1}^{(j)}(t)\leq \dots \leq
v_{k}^{(j)}(t)\leq \dots \leq w_{k}^{(j)}(t)\leq \dots \leq
w_{1}^{(j)}(t)\leq w_{0}^{(j)}(t), \label{3.4}
\end{equation}
for all $t\in J,j=0,1,2,\dots ,n-1$. On account of the definition
of $v_{k}$, we have
\begin{equation}
\begin{aligned}
v_{k}(t)&= u_{0}+tu_{1}+\frac{t^{2}}{2!}u_{2}+\dots
+\frac{t^{n-1}}{(n-1)!}
u_{n-1} \\
&\quad +\frac{1}{(n-1)!}\int_{0}^{t}(t-s)^{n-1}[-\sum
_{j=0}^{n-1}M_{j}v_{k}^{(j)}(s)-N(Tv_{k})(s)+\sigma _{k-1}(s)]ds \\
&\quad
+\sum_{0<t_{i}<t}[-\frac{(t-t_{i})^{n-1}}{(n-1)!}L_{i}^{n-1}
+\frac{(t-t_{i})^{n-2}}{(n-2)!}L_{i}^{n-2} \\
&\quad +\dots
+(t-t_{i})L_{i}^{1}+L_{i}^{0}]v_{k}^{(n-1)}(t_{i}),\quad (\forall
t\in J,k=1,2,3,\dots )
\end{aligned}\label{3.5}
\end{equation}
where
\begin{equation}
\begin{aligned}
\sigma _{k-1}(t)&=f(t,v_{k-1}(t),v_{k-1}'(t),\dots
,v_{k-1}^{(n-1)}(t),(Tv_{k-1})(t)) \\
&\quad +\sum_{j=0}^{n-1}M_{j}v_{k-1}^{(j)}(t)+N(Tv_{k-1})(t),
\quad \forall t\in J,\quad k=1,2,3,\dots
\end{aligned} \label{3.6}
\end{equation}
so,
\begin{equation}
\begin{aligned}
&v_{k}^{(j)}(t)\\
&=  u_{j}+tu_{j+1}+\dots +\frac{t^{n-j-1}}{(n-j-1)!}u_{n-1}\\
&\quad+\frac{1}{(n-j-1)!}\int_{0}^{t}(t-s)^{n-j-1}
[-\sum_{j=0}^{n-1}M_{j}v_{k}^{(j)}(s)-N(Tv_{k})(s)+\sigma
_{k-1}(s)]ds
\\
&\quad+\sum_{0<t_{i}<t}[-\frac{(t-t_{i})^{n-j-1}}{(n-j-1)!}L_{i}^{n-1}+
\frac{(t-t_{i})^{n-j-2}}{(n-j-2)!}L_{i}^{n-2} \\
&\quad+\dots +(t-t_{i})L_{i}^{j+1}+L_{i}^{j}]v_{k}^{(n-1)}(t_{i}),
\end{aligned} \label{3.7}
\end{equation}
for all $t\in J'$, $j=1,2,\dots ,n-1$, $k=1,2,3,\dots$. Similar to
the (\ref{2.22}), for $k,i=1,2,\dots$, we can obtain
\[
\Vert v_{k+i}-v_{k}\Vert _{n-1}\leq \beta \Vert v_{k+1}-v_{k}\Vert
_{n-1}+\beta ^{\ast }\Vert \sigma _{k+i-1}-\sigma
_{k-1}\Vert_{pc},
\]
where $\beta $ is defined by (\ref{2.18}) and
\begin{equation}
\beta ^{\ast }=\max
\{\frac{a^{n}}{n!},\frac{a^{n-1}}{(n-1)!},\dots ,\frac{
a^{2}}{2},a\}\,.  \label{3.8}
\end{equation}
Hence, for $k,i=1,2,\dots$,
\begin{equation}
\Vert v_{k+i}-v_{k}\Vert _{n-1}\leq \frac{\beta ^{\ast }}{1-\beta
}\Vert \sigma _{k+i-1}-\sigma _{k-1}\Vert _{pc}\,.  \label{3.9}
\end{equation}
Since the regularity of $P$ implies the normality of $P$, we see
from (\ref{3.4}) that $V_{j}=\{v_{k}^{(j)}:k=0,1,2,\dots \}$
$(j=0,1,\dots ,n-1)$ is a bounded set in $PC^{j}[J,E]$. It is easy
to show that the uniform continuity of $f$ on $J\times B_{r}\times
B_{r}\times \dots \times B_{r}$ implies the boundedness of $f$ on
$J\times B_{r}\times B_{r}\times \dots \times B_{r}$, so by
(\ref{3.6}),there is a constant $b>0$ such that
\[
\Vert \sigma _{k-1}\Vert _{pc}\leq b\quad (k=1,2,\dots )
\]
and therefore, from (\ref{3.7}) we know that functions
$\{v_{k}^{(j)}(t)\}$ $(j=0,1,\dots ,n-2)$ are equicontinuous on
each $J_{i}$ $(i=0,1,\dots ,m)$. From (\ref{3.4}) and the
regularity of $P$, we can infer that $\{v_{k}^{(j)}(t)\}$
converges uniformly to $\bar{u}^{(j)}(t)\in PC[J,E]$ in $J$; i.e.,
\begin{equation}
\Vert v_{k}^{(j)}-\bar{u}^{(j)}\Vert _{pc}\to 0\quad (k\to \infty
) \label{3.10}
\end{equation}
 From (\ref{3.6}),(\ref{3.10}) and the uniform continuity of $f$ on
$J\times B_{r}\times B_{r}\times \dots \times B_{r}$, we get
\[
\Vert \sigma _{k-1}-\bar{\sigma}\Vert _{pc}\to 0\quad (k\to \infty
)
\]
where
\[
\bar{\sigma}(t)=f(t,\bar{u}(t),\bar{u}'(t),\dots
,\bar{u}^{(n-1)}(t),(T\bar{u})(t))
+\sum_{j=0}^{n-1}M_{i}\bar{u}^{(j)}(t)+N(T\bar{u})(t),
\]
for all $t\in J$. Taking limits in (\ref{3.5}),we obtain
\begin{align*}
\bar{u}(t)&= u_{0}+tu_{1}+\frac{t^{2}}{2!}u_{2}+\dots
+\frac{t^{n-1}}{(n-1)!}u_{n-1} \\
&\quad
+\frac{1}{(n-1)!}\int_{0}^{t}(t-s)^{n-1}[-\sum_{j=0}^{n-1}M_{j}\bar{
u}^{(j)}(s)-N(T\bar{u})(s)+\sigma _{k-1}(s)]ds \\
&\quad
+\sum_{0<t_{i}<t}[-\frac{(t-t_{i})^{n-1}}{(n-1)!}L_{i}^{n-1}+\frac{
(t-t_{i})^{n-2}}{(n-2)!}L_{i}^{n-2} \\
&\quad +\dots +(t-t_{i})L_{i}^{1}+L_{i}^{0}]\bar{u}^{(n-1)}(t_{i})
\end{align*}
which by Lemma \ref{lm2.2} implies $\bar{u}\in
PC^{n-1}[J,E]\bigcap C^{n}[J',E]$ and $\bar{u}(t)$ is a solution
of \eqref{1.1}.

In the same way, we can show that $\| w_{k}-u^{\ast }\|_{n-1}\to
0(k\to \infty )$ for some $u^{\ast }\in PC^{n-1}[J,E]\bigcap
C^{n}[J',E]$ and $u^{\ast }(t)$ is a solution of \eqref{1.1}.

Finally, let $u\in PC^{n-1}[J,E]\bigcap C^{n}[J',E]$ be any
solution of \eqref{1.1} in $[v_{0},w_{0}]$ satisfying
$v_{0}^{(j)}(t)\leq u^{(j)}(t)\leq w_{0}^{(j)}(t)$, for all $t\in
J$, $j=0,1,\dots ,n-1$. Assume that $v_{m-1}^{(j)}(t)\leq
u^{(j)}(t)\leq w_{m-1}^{(j)}(t)$ for all $t\in J$, $j=0,1,\dots
,n-1$. Then by (H2) and Lemma \ref{lm2.1}, we can infer that
$v_{m}^{(j)}(t)\leq u^{(j)}(t)\leq w_{m}^{(j)}(t)$ for all $t\in
J$, $j=0,1,\dots,n-1$. Hence, by induction, $v_{k}^{(j)}(t)\leq
u^{(j)}(t)\leq w_{k}^{(j)}(t)$ for all $t\in J$, $j=0,1,\dots
,n-1$, $k=1,2,\dots $, which implies that $\bar{u}^{(j)}(t)\leq
u^{(j)}(t)\leq (u^{\ast})^{(j)}(t)$ for all $t\in J$, $j=0,1,\dots
,n-1$. Hence \eqref{3.1} follows from \eqref{3.4}. The proof is
complete.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
 The condition that $P$ is regular will be satisfied if $E $ is weakly
complete (reflexive, in particular) and $P$ is normal (see
\cite[theorem 2]{Du}).
\end{remark}

\section{An Example}

Consider the initial-value problem infinite system for scalar
third order integro-differential equations
\begin{equation}
\begin{gathered}
\begin{aligned}
u_{n}^{(3)}&=\frac{1}{100n^{2}}[(t-u_{n})^{2}+t^{2}u_{n+1}
+(u_{2n}')^{2}+(t-u_{n}'')^{2}] \\
&\quad +\frac{t}{800n^{3}}(t-\int_{0}^{t}e^{-ts}u_{n}(s)ds)^{2},
\quad 0\leq t\leq 2,t\neq 1
\end{aligned}\\
\Delta u_{n}|_{t=1}=\frac{1}{3}u_{n}''(1) \\
\Delta u_{n}'|_{t=1}=\frac{1}{2}u_{n}''(1) \\
\Delta u_{n}''|_{t=1}=-\frac{1}{60}u_{n}''(1)\\
u_n(0)=u'_n(0)=u''_n(0)=0,\quad n=1,2,\dots
\end{gathered} \label{4.1}
\end{equation}

\noindent\textbf{Claim:} The system \eqref{4.1} admits minimal and
maximal solutions which are continuously differentiable on
$[0,1)\cup (1,2]$ and satisfy
\[
0\leq u_{n}(t)\leq
\begin{cases}
t^3/n^2, &{if } 0\leq t\leq 1 \\
(t^{3}+t^{2}+t)/n^2, &{if } 1<t\leq 2
\end{cases}
\]
\[
0\leq u_{n}'(t)\leq
\begin{cases}
3t^{2}/n^{2}, &{if } 0\leq t\leq 1 \\
(3t^{2}+2t+1)/n^{2}, &{if } 1<t\leq 2
\end{cases}
\]
\[
0\leq u_{n}''(t)\leq
\begin{cases}
6t/n^{2}, &{if } 0\leq t\leq 1 \\
(6t+2)/n^{2}, &{if } 1<t\leq 2
\end{cases}
\]
where $n=1,2,3,\dots $.

\begin{proof}
Let $E=l^{1}=\{u=(u_{1},u_{2},\dots ,u_{n},\dots
):\sum_{n=1}^{\infty }|u_{n}|<\infty \}$ with the norm $\Vert
u\Vert=\sum_{n=1}^{\infty }|u_{n}|$ and let
$P=\{u=(u_{1},u_{2},\dots ,u_{n},\dots )\in l^{1}:u_{n}\geq
0,n=1,2,\dots \}$. Then $P$ is a normal cone in $E$. Since $l^{1}$
is weakly complete, from remark 3.1 we know that $P$ is regular.
System \eqref{4.1} can be regarded as an IVP of form \eqref{1.1},
where
\begin{gather*}
a=2,k(t,s)=e^{-ts}, \quad
u=(u_{1},u_{2},\dots ,u_{n},\dots ),\\
v=(v_{1},v_{2},\dots ,v_{n},\dots),\quad
w=(w_{1},w_{2},\dots ,w_{n},\dots ), \\
z=(z_{1},z_{2},\dots ,z_{n},\dots ),\quad f=(f_{1},f_{2},\dots
,f_{n},\dots ),
\end{gather*}
in which
\begin{equation}
f_{n}(t,u,v,w,z)=\frac{1}{100n^{2}}
[(t-u_{n})^{2}+t^{2}u_{n+1}+(v_{2n})^{2}+(t-w_{n})^{2}]
+\frac{t}{800n^{3}}(t-z_{n})^{2} \label{4.2}
\end{equation}
and $m=1$, $t_{1}=1$, $L_{1}^{0}=\frac{1}{3}$,
$L_{1}^{1}=\frac{1}{2}$, $L_{1}^{2}=\frac{1}{60}$,
$u_{0}=u_{1}=u_{2}=(0,0,\dots ,0,\dots )$.

Evidently, $f\in C[J\times E\times E\times E\times E,E]$
$(J=[0,2])$. Let $v_{0}(t)=(0,0,\dots ,0,\dots )$, for $0\leq
t\leq 2$ and
\[
w_{0}(t)=\begin{cases}
(t^{3},\dots ,t^{3}/n^{2},\dots ), &\quad{if } 0\leq t\leq 1; \\
(t^{3}+t^{2}+t,\dots ,\frac{t^{3}+t^{2}+t}{n^{2}},\dots ),
&\quad{if }1<t\leq 2.
\end{cases}
\]
We have $v_{0}\in C^{3}[J,E]$, $w_{0}\in PC^{2}[J,E]\cap
C^{3}[J',E]$, where $J'=J\backslash
\{1\}=[0,1)\cup(1,2],v_{0}(t)\leq w_{0}(t)$ $(t\in J)$ and
\begin{gather*}
w_{0}'(t)=\begin{cases} (3t^{2},\dots ,\frac{3t^{2}}{n^{2}},\dots
), &\quad{if } 0\leq t\leq 1
\\
(3t^{2}+2t+1,\dots ,\frac{3t^{2}+2t+1}{n^{2}},\dots ), &\quad{if }
1<t\leq 2
\end{cases} \\
w_{0}''(t)=\begin{cases}
(6t,\dots ,\frac{6t}{n^{2}},\dots ), &\quad{if } 0\leq t\leq 1 \\
(6t+2,\dots ,\frac{6t+2}{n^{2}},\dots ), &\quad{if }1<t \leq 2
\end{cases}\\
w_{0}^{(3)}=(6,\dots ,\frac{6}{n^{2}},\dots ),\quad  \forall 0\leq
t\leq 2.
\end{gather*}
It is clear that
\begin{gather*}
v_{0}'(t)\leq w_{0}'(t),v_{0}''(t)\leq
w_{0}''(t),\quad  \forall t\in J \\
v_{0}(0)=w_{0}(0)=(0,0,\dots ,0,\dots )=u_{0}, \\
v_{0}'(0)-v_{0}(0)=w_{0}'(0)-w_{0}(0)=u_{1}-u_{0}=(0,0,\dots
,0,\dots )
\\
v_{0}''(0)-v_{0}'(0)=w_{0}''(0)-w_{0}'(0)=u_{2}-u_{1}=(0,0,\dots
,0,\dots ) \\
v_{0}^{(3)}(t)=(0,0,\dots ,0,\dots ),\quad \forall t\in J \\
\Delta v_{0}|_{t=1}=(0,0,\dots ,0,\dots)=\frac{1}{3}v_{0}''(1) \\
\Delta v_{0}'|_{t=1}=(0,0,\dots ,0,\dots )=\frac{1}{2} v_{0}''(1) \\
\Delta v_{0}''|_{t=1}=(0,0,\dots ,0,\dots )=-\frac{1}{60}v_{0}''(1) \\
\Delta w_{0}|_{t=1}=(2,\dots ,\frac{2}{n^{2}},\dots
)=\frac{1}{3}w_{0}''(1) \\
\Delta w_{0}'|_{t=1}=(3,\dots ,\frac{3}{n^{2}},\dots
)=\frac{1}{2}w_{0}''(1) \\
\Delta w_{0}''|_{t=1}=(2,\dots ,\frac{2}{n^{2}},\dots
)>-\frac{1}{60}w_{0}''(1) \\
f_{n}(t,v_{0}(t),v_{0}'(t),v_{0}''(t),(Tv_{0})(t))=
\frac{2t^{2}}{100n^{2}}+\frac{t^{3}}{800n^{3}}\geq
0=v_{0}^{(3)}(t),\quad \forall t\in J.
\end{gather*}
When $0\leq t\leq 1$,we have
\begin{align*}
&f_{n}(t,w_{0}(t),w_{0}'(t),w_{0}''(t),(Tw_{0})(t)) \\
&=
\frac{1}{100n^{2}}\big[(t-\frac{t^{3}}{n^{2}})^{2}+t^{2}\frac{t^{3}}{(n+1)^{2}}
 +(\frac{3t^{2}}{(2n)^{2}})^{2}+(t-\frac{6t}{n^{2}})^{2}\big]\\
&\quad
+\frac{t}{800n^{3}}(t-\int_{0}^{t}e^{-ts}\frac{s^{3}}{n^{2}}ds)^{2}\\
& \leq  \frac{1}{100n^{2}}(t^{2}+\frac{t^{5}}{(n+1)^{2}}
+\frac{9t^{4}}{4n^{2}}+t^{2})+\frac{t^{3}}{800n^{3}}
 \leq  \frac{6}{n^{2}}.
\end{align*}
When $0<t\leq 2$, we have
\begin{align*}
&f_{n}(t,w_{0}(t),w_{0}'(t),w_{0}''(t),(Tw_{0})(t)) \\
&= \frac{1}{100n^{2}}\big[(t-\frac{t^{3}+t^{2}+t}{n^{2}})^{2}
+t^{2}\frac{t^{3}+t^{2}+t}{(n+1)^{2}}
 +(\frac{3t^{2}+2t+1}{(2n)^{2}})^{2}+(t-\frac{6t+2}{n^{2}})^{2}\big]\\
&\quad
+\frac{t}{800n^{3}}(t-\int_{0}^{t}e^{-ts}\frac{s^{3}+s^{2}+s}{n^{2}}ds)^{2} \\
& \leq
\frac{1}{100n^{2}}(t^{2}+\frac{t^{5}+t^{4}+t^{3}}{(n+1)^{2}}+\frac{
(t^{2}+2t+1)^{2}}{4n^{2}}+t^{2}) +\frac{t^{3}}{800n^{3}}
 \leq  \frac{6}{n^{2}}.
\end{align*}
Hence $v_{0},w_{0}$ satisfy (H1). On the other hand, for $t\in J$
\begin{gather*}
v_{0}(t)\leq \overline{u}\leq u\leq w_{0}(t), \quad
v_{0}'(t)\leq \overline{v}\leq v\leq w_{0}'(t), \\
v_{0}''(t)\leq \overline{w}\leq w\leq w_{0}'(t),\quad
(Tv_{0})(t)\leq \overline{z}\leq u\leq (Tw_{0})(t),
\end{gather*}
we have
\begin{gather*}
0\leq  \overline{u}_{n}\leq u_{n}\leq \frac{t^{3}+t^{2}+t}{n^{2}},
\quad
0\leq \overline{v}_{n}\leq v_{n}\leq \frac{3t^{2}+2t+1}{n^{2}}, \\
0\leq \overline{w}_{n}\leq w_{n}\leq \frac{6t+2}{n^{2}}, \quad
0\leq \ \overline{z}_{n}\leq z_{n}\leq
\frac{3t^{4}+4t^{3}+6t^{2}}{12},
\end{gather*}
$n=1,2,\dots$. Therefore, by (\ref{4.2}),
\begin{align*}
&f_{n}(t,u,v,w,z)-f_{n}(t,\overline{u},\overline{v},\overline{w},
\overline{z}) \\
& \geq
\frac{1}{100n^{2}}\big[(t-u_{n})^{2}-(t-\overline{u}_{n})^{2}+(t-w_{n})^{2}
  -(t-\overline{w}_{n})^{2}\big]\\
&\quad
+\frac{t}{800n^{3}}\big[(t-z_{n})^{2}-(t-\overline{z}_{n})^{2}\big] \\
& \geq
-\frac{1}{100n^{2}}[2t(u_{n}-\overline{u}_{n})+2t(w_{n}-\overline{w}_{n}]
  -\frac{2t^{2}}{800n^{3}}(z_{n}-\ \overline{z}_{n})
  -\frac{1}{25}(u_{n}-\ \overline{u}_{n})\\
&\quad -\frac{1}{25}(w_{n}-\overline{w}_{n})
  -\frac{1}{100}(z_{n}- \overline{z}_{n}).
\end{align*}
Consequently, (H2) is satisfied for $M_{0}=1/25=M_{2}$, $M_{1}=0$,
$N=1/100$. It is clear that $k_{0}=1$ and $\tau =1$, and it is
easy to verify that inequalities (\ref{2.2}) and (\ref{2.18})
hold. Hence, our conclusion follows from Theorem \ref{thm3.1}.
\end{proof}

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\end{document}