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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 103, pp. 1--11.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/103\hfil Volterra integral equations]
{On non-absolute functional Volterra integral equations
and impulsive differential equations in ordered Banach spaces}

\author[S. Heikkil\"a, S. Seikkala\hfil EJDE-2008/103\hfilneg]
{Seppo Heikkil\"a, Seppo Seikkala}  % in alphabetical order

\address{Seppo Heikkil\"a \newline
Department of Mathematical Sciences,
University of Oulu, Box 3000\\
FIN-90014 University of Oulu, Finland}
\email{seppo.heikkila@oulu.fi}

\address{Seppo Seikkala \newline
Division of Mathematics,
Department of Electrical Engineering,
University of Oulu, 90570 Oulu, Finland}
\email{seppo.seikkala@ee.oulu.fi}

\thanks{Submitted April 24, 2008. Published August 6, 2008.}
\subjclass[2000]{26A39, 28B15, 34G20, 34K45, 45N05, 46E40, 47H07}
\keywords{HL integrability; Bochner integrability; ordered Banach space;
\hfill\break\indent
 dominated convergence; monotone convergence; integral equation;
 boundary value problem}

\begin{abstract}
 In this article we derive existence and comparison results for
 discontinuous non-absolute functional integral equations of
 Volterra type in an ordered Banach space which has a regular order cone.
 The obtained results are then applied to first-order impulsive
 differential equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remarks}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}
\newcommand{\kint}{\;\rlap{${}^K$}\;}

\section{Introduction}\label{s1}

In \cite{HKS09} a  theory for HL integrable functions with values
in  ordered Banach spaces was developed, and applied to Fredholm
integral equations and concrete boundary value problems of second
order ordinary differential equations. In this paper we apply that
theory and a fixed point result in abstract spaces to prove
existence and comparison results for non-absolute functional
Volterra integral equations in an ordered Banach space $E$, and
give applications to first-order impulsive  initial value problems
involving discontinuities and functional dependencies.

The main features of this paper are:
\newline
-- The $E$-valued functions in considered equations are discontinuous
and depend functionally on the unknown function. Thus integro-differential
 equations are included.
\newline
-- Integrals in  integral equations are  non-absolute integrals, and
differential equations of  impulsive problems may be singular.
\newline
-- Impulses are allowed to occur in  well-ordered sets, in particular,
in finite sets or in increasing sequences.


The main tools are:\newline
-- Fixed point results in partially
ordered sets, proved in \cite{HeiLak94} by generalized iteration
methods.
\newline
-- Dominated and monotone convergence theorems for  HL integrable
mappings and results on the existence of supremum and infimum of
chains of locally HL integrable mappings from a  real interval to
$E$, proved in \cite{HKS09}.

\section{Preliminaries}\label{s2}

In this section we study properties of HL integrable, a.e.
differentiable  and locally HL integrable functions from a real
interval to a Banach space $E$.

A \emph{K-partition of} a compact real interval $I$ is formed by  a
finite  collection of closed  subintervals $[t_{i-1},t_i]$ of $I$
whose union is $I$, and  tags $\xi_i\in [t_{i-1},t_i]$. A function
$u:I \to E$ is \emph{HL integrable} if there is a function $F:I\to
E$, called a \emph{primitive of} $u$, which  has the following
property: If $\epsilon > 0$, there is such a function $\delta:I\to
(0,\infty)$ that
$$
\sum_i \big\|u(\xi_i)(t_i-t_{i-1})-(F(t_i)-F(t_{i-1}))\big\|<\epsilon
$$
%
for every K-partition $\{(\xi_i,[t_{i-1},t_i])\}$ of $I$ with $[t_{i-1},t_i]\subset (\xi_i-\delta(\xi_i),\xi_i+\delta(\xi_i))$ for all $i$.
If $u$ is HL integrable on $I$, it is HL integrable on every closed subinterval $J=[a,b]$ of $I$, and
$F(b)-F(a)$ is  the {\it Henstock-Kurzweil} integral of $u$ over $J$, i.e.,
%
\begin{equation}\label{E2.0}
F(b)-F(a)=\kint \int_{J}u(s)\,ds=\kint \int_a^bu(s)\,ds.
\end{equation}
The proofs for the results of the next Lemma can be found,
e.g. in \cite{Sye05}.

\begin{lemma}\label{L6.0} \begin{itemize}
\item[(a)] The Henstock-Kurzweil integrals of a.e. equal HL
integrable functions
 are equal.

\item[(b)] Every HL integrable function is strongly measurable.

\item[(c)] A Bochner integrable function $u:I\to E$ is HL integrable,
and $\int_Ju(s)\,ds =\kint \int_{J}u(s)\,ds$ whenever $J$ is a closed
subinterval of $I$.
\end{itemize}
\end{lemma}

The set $H(I,E)$ of all HL integrable functions  $u: I\to E$ is a vector
space with respect to the usual addition and scalar multiplication of
functions. Identifying   a.e. equal functions
it follows that the space $L^1(I,E)$ of all Bochner integrable functions
 $u: I\to E$ is a subset of $H(I,E)$.

A function $u: I\to E$ is called \emph{absolutely continuous
($AC$)} on $I$ if for each $\epsilon > 0$ there corresponds such a
 $\delta > 0$, that for any sequence $[a_j,b_j]$, $j = 1,\dots,n$ of
disjoint subintervals of $I$ with $\sum_{j=1}^n (b_j-a_j) <
\delta$ we have $\sum_{j=1}^n \| u(b_j) - u(a_j)\| < \epsilon$.

We say that a function $u:I\to E$ is \emph{generalized absolutely continuous in
the restricted sense ($ACG^*$)} on $I$ if $I$ can be
expressed as such a countable union of its  subsets $B_n$,
$n\in\mathbb N$, that for all $\epsilon > 0$ and $n\in\mathbb N$
there exists such a $\delta_n> 0$ that
$$
\sum_i\sup\{\|u(d)-u(c)\|: [c,d]\subseteq [c_i,d_i]\} < \epsilon
$$
whenever $\{[c_i,d_i]\}$ is a finite sequence of non-overlapping
intervals which have endpoints in $B_n$ and satisfy
$\sum_i(d_i-c_i)< \delta_n$.  If $u$ is $AC$ on $I$, it is
continuous  and $ACG^*$ on $I$.

A function $v: I\to E$ is said to be \emph{a.e. (strongly)
differentiable}, if the strong derivative $ v'(t) = {\lim}_{h\to
0}\frac {v(t+h) - v(t)}h $ exists for  a.e. $t\in I.$

As for the proof of the following result, see, e.g.,
\cite[subsection 7.4.1]{Sye05}.

\begin{theorem}\label{T601.102}
Given $u,\,v: I\to E$ and $(t_0,x_0)\in I\times E$, then
the following conditions are equivalent.
\begin{itemize}
\item[(a)] $u$ is continuous and $ACG^*$ on $I$, $u'(t) = v(t)$ for
a.e. $t\in I$ and $u(t_0) = x_0$.

\item[(b)] $v$ is HL-integrable and $u(t) = x_0 +
\kint \int_{t_0}^tv(s)ds$  for all $t\in I$.
\end{itemize}
\end{theorem}

If $u: I\to E$ is a.e. differentiable, define $u'(t) = 0$ at those points
 $t\in I$ where $u$ is not differentiable.

The next result is a consequence of Theorem \ref{T601.102}.

\begin{corollary}\label{C6.1}
If $u: I\to E$ is a.e. differentiable, then $u$ is continuous and
$ACG^*$ on $I$ if and only if $u'$ is HL-integrable, and
$$
u(t)-u(t_0) = \kint \int_{t_0}^t u'(s)ds \quad\text{for all }
t_0,\,t\in I.
$$
\end{corollary}

The following result is needed in section 4.

\begin{lemma}\label{L6.4}
If $u: I\to \mathbb R$ is absolutely continuous, and
$v: I\to E$ is continuous, $ACG^*$  and a.e. differentiable, then
$$
u(t)v(t) - u(t_0)v(t_0) = \kint \int_{t_0}^t (u(s)v'(s) + u'(s)v(s))ds
\quad\text{for all } t_0,\,t\in I.
$$
\end{lemma}

\begin{proof} Let  $t,\,t+h\in I$, $h\ne 0$ be given.
Since $u$ and $v$ are continuous on a compact interval $I$, they
are also bounded, whence
$$
u(t+h)v(t+h) - u(t)v(t) = (u(t+h) - u(t))v(t+h) + u(t)(v(t+h) - v(t)).
$$
implies when   $M = \max\{\|v(t)\|:t\in I\}$ and
$m = \max\{|u(t)|:t\in I\}$
that
$$
\| u(t+h)v(t+h) - u(t)v(t)\|\le M\,|u(t+h) - u(t)| + m\,\|v(t+h) - v(t)\|.
$$
Because $u$ is an absolutely continuous real-valued function,
it is $ACG^*$ on  $I$. It then follows from the above inequality
that $u\cdot v$ is continuous and $ACG^*$ on $I$.
Moreover, $u$ and $v$ are a.e. differentiable, whence $u\cdot v$ is a.e.
differentiable and
$$
(u\cdot v)'(t) = u(t)v'(t) + u'(t)v(t) \quad\text{for a.e. } t\in
I.
$$
The assertion follows then from Corollary \ref{C6.1}.
\end{proof}

The following result is adapted from \cite{PiMa02}.

\begin{proposition}\label{P6.1}
If $v:I\to E$ is HL-integrable and $u:I\to \mathbb R$ is of bounded
variation, then $u\cdot v$ is HL-integrable.
\end{proposition}


Given an interval $J$ of $\mathbb R$, not necessarily closed or
bounded, denote by $H_{loc}(J,E)$ the space of all strongly
measurable functions $u:J\to E$  which are HL integrable on each
compact subinterval of $J$. We assume that $H_{\rm loc}(J,E)$ is
ordered a.e. pointwise; i.e.,
\begin{equation}\label{E6.1}
 u\le v \quad\text{if and only if $u(s)\le v(s)$  for a.e. } s\in
J.
\end{equation}
The  results of the next Lemma follow from  \cite[Proposition 2.1
and Lemma 2.5]{HKS09}.

\begin{lemma}\label{L6.1}
Given an ordered Banach space, let $u,\,v:J\to E$ be strongly measurable,
$u_\pm\in H_{\rm loc}(J,E)$, and assume that $u_-(s)\le u(s)\le
v(s)\le u_+(s)$ for a.e. $s\in J$. Then  $u\in H_{\rm loc}(J,E)$.
Moreover,
$$
\kint \int_a^tu(s)\,ds\le\kint \int_a^tv(s)\,ds \quad \text{for all }
a,\,t\in J, \ a\le t.
$$
\end{lemma}


Next we present Dominated and  Monotone Convergence Theorems
for locally HL-integrable functions, which are needed in applications.

\begin{theorem}\label{T6.1}
Given a real interval $J$ and a Banach space $E$ ordered by a
normal order cone, let $(u_n)_{n=1}^\infty$ be a sequence of
strongly measurable functions from $J$ to $E$, let $u_\pm\in
H_{\rm loc}(J,E)$, and assume that $u_-\le u_n\le u_+$ for each
$n=1,2,\dots$, and that $u_n(s)\to u(s)$ for a.e. $s\in J$. Then
$u,\,u_n\in  H_{\rm loc}(J,E)$, $n=1,2,\dots$, and
$\kint \int_{a}^tu_n(s)ds\to\kint \int_{a}^tu(s)ds$ for all $a,\,t\in J$,
$a< t$.
\end{theorem}

\begin{proof}
The given hypotheses imply by Lemma \ref{L6.1} that $u_n\in H_{\rm
loc}(J,E)$, $n=1,2,\dots$. If  $a,\,t\in J$, $a < t$, are fixed,
then $u_{\pm}\in  H([a,t],E)$ and $u_n\in H([a,t],E)$,
$n=1,2,\dots$, and  $u_n(s)\to u(s)$ for a.e. $s\in [a,t]$. Thus
$u\in H([a,t],E)$ and $\kint \int_{a}^tu_n(s)ds\to\kint \int_{a}^tu(s)ds$ by
\cite[Theorem 3.1]{HKS09}.
\end{proof}

As an easy consequence of Theorem \ref{T6.1} we obtain the
following result.

\begin{theorem}\label{T6.2}
Given a real interval $J$ and a Banach space $E$ ordered by a
regular order cone, let $(u_n)_{n=1}^\infty$ be a monotone
sequence of strongly measurable functions from a real interval $J$
to $E$. Assume that $u_\pm\in H_{\rm loc}(J,E)$, and that $u_-\le
u_n\le u_+$ for each $n=1,2,\dots$. Then there exists a function
$u\in H_{\rm loc}(J,E)$ such that $u(t)=\lim_nu_n(t)$ for a.e.
$t\in J$, and $\kint \int_{a}^tu_n(s)ds\to\kint \int_{a}^tu(s)ds$ for all
$a,\,t\in J$, $a < t$.
\end{theorem}

\begin{proof}
 Since $(u_n(s))$ is monotone and $u_-(s)\le u_n(s)\le u_+(s)$
for a.e. $s\in [a,b)$, and since the order cone of $E$ is regular,
then $(u_n)$ converges  a.e. pointwise to a function $u:J\to E$.
The  conclusions follow then from Theorem \ref{T6.1}.
\end{proof}

In our study of Volterra integral  equations  we need the
following result, which is proved in \cite[Proposition 3.2]{HKS09}.

\begin{lemma}\label{L6.2}
Assume that $W$ is a nonempty set  in an order interval
 $[w_-,w_+]$ of  $H_{\rm loc}(J,E)$, where $J$ is a real interval $J$
 and $E$ a Banach space ordered by a regular order cone.
\begin{itemize}
\item[(a)] If $W$ is well-ordered, it contains an increasing sequence
which converges a.e. pointwise to $\sup W$.

\item[(b)] If $W$ is inversely well-ordered, it contains a decreasing
sequence which converges a.e. pointwise to $\inf W$.
\end{itemize}
\end{lemma}

Since each increasing sequence of $H_{\rm loc}(J,E)$ is
well-ordered and each decreasing sequence of $H_{\rm loc}(J,E)$ is
inversely well-ordered, we obtain as a consequence of Lemma
\ref{L6.2} and
\cite[Proposition 1.1.3, Corollary 1.1.3]{HeiLak94},
the following results.

\begin{corollary}\label{C6.2}
Given a real interval $J$ and a Banach space $E$ ordered by a normal
order cone,
assume that $(u_n)$ is a sequence  of $H_{\rm loc}(J,E)$, and that
there exist  functions $w_\pm\in H_{\rm loc}(J,E)$ such that
$u_n\in [w_-,w_+]$ for each $n$.
\begin{itemize}
\item[(a)] If $(u_n)$ is increasing, it converges a.e.\ pointwise to
$u_*=\sup_nu_n$ in the space $H_{\rm loc}(J,E)$, and $u_*$ belongs to
$[w_-,w_+]$.

\item[(b)] If $(u_n)$ is decreasing, it converges a.e.\ pointwise to
$u^*=\inf_nu_n$ in the space $H_{\rm loc}(J,E)$, and $u^*$ belongs to
$[w_-,w_+]$.
\end{itemize}
\end{corollary}

The following fixed point result is a consequence of
\cite[Theorem A.2.1]{CarHei00},
\cite[Theorem 1.2.1 and Proposition 1.2.1]{HeiLak94}.


\begin{lemma}\label{L6.5}
Given a partially ordered set $P=(P,\le)$ and its
order interval $[w_-,w_+]=\{w\in P\mid w_-\le u\le w_+\}$,
assume that $G:P\to [w_-,w_+]$ is increasing, i.e., $Gu\le Gv$
whenever $u\le v$ in $P$, and that each well-ordered chain of the range
$G[P]$ of $G$ has a supremum in $P$ and each inversely
well-ordered chain of $G[P]$ has an infimum in $P$.  Then $G$ has
least and  greatest fixed points, and they are increasing
with respect to $G$.
\end{lemma}


\section{Existence and comparison results for a functional Volterra integral
equation}\label{S6.3}

 Throughout this section $E = (E,\le,\|\cdot\|)$ is an ordered Banach
space with a regular order cone, which means by
 \cite[Lemma 1.3.3]{HeiLak94}, that all order bounded and monotone
sequences of $E$ converge.

In this section we study  the  functional Volterra integral equation
 \begin{equation}\label{E6.2}
u(t) =q(t,u)+ \kint \int_a^t k(t,s)f(s,u(s),u)\,ds,  \quad   t\in
J=[a,b),\end{equation}
where $q:J\times H_{\rm loc}((a,b),E)\to E$, $f:J\times E\times
H_{\rm loc}((a,b),E)\to E$ and $k:\Lambda\to\mathbb R_+$, where
$\Lambda=\{(t,s)\in J\times J: s\le t\}$ and
$-\infty < a < b\le\infty$.

Assuming that $H_{\rm loc}((a,b),E)$ is equipped with a.e.
pointwise ordering (\ref{E6.1}), we impose the following
hypotheses on the functions $q$, $f$ and $k$.
\begin{itemize}
\item[(q0)] $q(t,\cdot)$ is increasing for a.e. $t\in J$, $q(\cdot,u)$ is
strongly measurable for all $u\in H_{\rm loc}((a,b),E)$, and there
exist $\alpha_\pm\in H_{\rm loc}((a,b),E)$ such that $\alpha_-\le
q(\cdot,u) \le\alpha_+$ for all $u\in H_{\rm loc}((a,b) ,E)$.
\item[(f0)] There exist functions $u_\pm\in H_{\rm loc}((a,b),E)$ such that $u_-\le f(\cdot,x,u)\le u_+$ for all $x\in  E$ and $u\in H_{\rm loc}((a,b) ,E)$.
\item[(f1)] The mapping $f(\cdot,u(\cdot),u)$ is strongly measurable for each  $u\in H_{\rm loc}((a,b),E)$.
\item[(f2)] $f(s,z,u)$ is increasing with respect to $z$ and $u$ for a.e. $s\in J$.
\item[(k0)] $k$ is continuous and the mappings $s\mapsto k(t,s)u_\pm(s)$ belong to $H_{\rm loc}(J,E)$ for each $t\in J$.
\end{itemize}
Our main existence and comparison result for the integral equation
\eqref{E6.2} reads as follows.

\begin{theorem}\label{T6.3}
Assume that the hypotheses {\rm (q0), (f0), (f1), (f2), (k0)} are
satisfied. Then the equation \eqref{E6.2} has least and greatest
solutions in $H_{\rm loc}((a,b),E)$. Moreover, these solutions
$u_*$ and $u^*$ are increasing with respect to $q$ and $f$.
\end{theorem}

\begin{proof}
The hypotheses (q0), (k0) and (f0) ensure that the equations
\begin{equation}\label{E6.3}
w_\pm(t)=\alpha_\pm(t) + \,\kint \int_a^t k(t,s)u_\pm(s)\,ds,
 \quad t\in J,
\end{equation}
define functions $w_\pm:J\to E$. Noticing that the integral on
the right-hand side of (\ref{E6.3}) is continuous in its upper
limit $t$, and that the integrand  is continuous in $t$ for fixed $s$,
one can show by applying also Theorem \ref{T6.1}, that the second
term on the right-hand side of  (\ref{E6.3}) is continuous in $t$.
Thus the functions $w_\pm$ belong to the set
 $P:= H_{\rm loc}((a,b),E)$. By using the hypotheses (q0), (k0), (f0)--(f2),
Lemmas \ref{L6.0} and \ref{L6.1} and Theorem \ref{T6.1} it can be shown
that the equation
\begin{equation}\label{E6.4}
Gu(t)=q(t,u) + \,\kint \int_a^t k(t,s)f(s,u(s),u)\,ds,\qquad t\in J,
\end{equation}
defines an increasing  mapping $G:P\to [w_-,w_+]$.
Since $G[P]\subset[w_-,w_+]$, it follows from Lemma \ref{L6.2}
that each well-ordered chain of
$G[P]$ has a supremum in $P$ and each inversely well-ordered chain
of $G[P]$ has an infimum in $P$.

The above proof shows that all the hypotheses of Lemma \ref{L6.5} are
valid for the operator $G$ defined by \eqref{E6.4}. Thus $G$ has least
and greatest fixed points $u_*$ and $u^*$.
Noticing that fixed points of $G$ defined by \eqref{E6.4} are solutions
of \eqref{E6.2} and vice versa, then $u_*$ and $u^*$ are least and
greatest solutions of \eqref{E6.2}.
It follows from \eqref{E6.4}, by Lemma \ref{L6.1}, that $G$ is
increasing with respect to $q$ and $f$, whence the last assertion of
Theorem  follows from the last assertion of Lemma \ref{L6.5}.
\end{proof}

Next we consider  a case when
the extremal solutions of the integral equation \eqref{E6.2} can be
obtained by ordinary iterations.

\begin{proposition}\label{P6.2}
Assume that the hypotheses {\rm (q0), (f0), (f1), (f2),
(k0)} hold, and let $G$ be defined by \eqref{E6.4}.
\begin{itemize}
\item[(a)] The sequence $(u_n)_{n=0}^\infty:=(G^nw_-)_{n=0}^\infty$ is
increasing and converges  a.e. pointwise  to a function $u_*\in
H_{\rm loc}((a,b),E)$. Moreover, $u_*$ is the least solution of
\eqref{E6.2} if
 $q(t,u_n)\to q(t,u_*)$ for a.e. $t\in J$ and
$f(s,u_n(s),u_n)\to f(s,u_*(s),u_*)$ for all $t\in J$ and for a.e.  $s\in[a,t]$;

\item[(b)] The sequence $(v_n)_{n=0}^\infty :=(G^nw_+)_{n=0}^\infty$ is
decreasing and converges  a.e. pointwise  to a function $u^*\in
H_{\rm loc}((a,b),E)$. Moreover, $u^*$ is the greatest solution of
\eqref{E6.2} if
 $q(t,v_n)\to q(t,u^*)$ for a.e. $t\in J$ and
$f(s,v_n(s),v_n)\to f(s,u^*(s),u^*)$ for  a.e.  $s\in J$.
\end{itemize}
\end{proposition}

\begin{proof} (a)  The sequence $(u_n):=(G^nw_-)$ is
increasing and contained in the order interval $[w_-,w_+]$. Hence
the asserted a.e. pointwise  limit $u_*\in H_{\rm loc}((a,b),E)$
exists  by Corollary \ref{C6.2} (a). Moreover, $(u_n)$ equals to
the sequence of successive approximations $u_n:J\to E$  defined by
\begin{equation}\label{E6.5}
u_{n+1}(t) = q(t,u_n)+\,\kint \int_a^t k(t,s)f(s,u_n(s),u_n)\,ds,\quad
u_0(t)=w_-(t),  \;  t\in J, \ n\in\mathbb N.
\end{equation}
%
In view of these results, the hypotheses of (a) and Theorem \ref{T6.2},
it  follows from (\ref{E6.5}) as $n\to\infty$ that
$u_*$ is a solution of \eqref{E6.2}.

If $u$ is any solution of \eqref{E6.2}, then $u=Gu\in [w_-,w_+]$.
 By induction one can show that $u_n=G^nw_-\in [w_-,u]$ for each $n$.
Thus $u_*=\sup_nu_n\le u$, which proves that $u_*$ is the least
solution of \eqref{E6.2}.

The proof of part (b) is similar to that of (a) and is omitted.
\end{proof}

\begin{example}\label{Ex6.1} \rm
Let $E$ be the  space $c_0$ of all sequences $(c_n)_{n=1}^\infty$ of real numbers converging to zero,
ordered componentwise and equipped with the sup-norm.
Define $h_n,\, \alpha_n:[0,\infty)\to \mathbb R$ and
$k:\Lambda\to\mathbb R_+$ by equations
\begin{equation}\label{E6.51}
\begin{gathered}
h_n(t)=\frac 2{\sqrt{n}}\cos\bigl(\frac 1{t^2}\bigr)
+\frac 2{\sqrt{n}t^2}\sin\bigl(\frac 1{t^2}\bigr), \quad t > 0, \ h_n(0)=0,\\
\alpha_n(t)=\frac{1}{\sqrt{n}t}H\Bigl(t-\frac{2n-1}{2n}\Bigr),\quad
 n=1,2,\dots, \\
 k(t,s)=\frac st, \quad t > 0, \quad \alpha_n(0)=k(0,\cdot)=0,
\end{gathered}
\end{equation}
 The solutions of the infinite system of integral equations
\begin{equation}\label{E6.52}
w_n(t) = \pm\alpha_n(t) + \kint \int_{0}^tk(t,s)\Big(h_n(s)
\pm \frac 1{\sqrt{n}}\Big)\,ds,
\quad  n=1,2,\dots,
\end{equation}
in $H_{\rm loc}((0,\infty),c_0)$ are
\begin{equation}\label{E6.53}
w_\pm(t) =\left(w_{n\pm}(t)\right)_{n=1}^\infty
= \left(\pm \frac {1}{\sqrt{n}t}H\Bigl(t-\frac{2n-1}{2n}\Bigr)
+\frac t{\sqrt{n}}\cos\bigl(\frac 1{t^2}\bigr)
\pm \frac{t}{2\sqrt{n}}\right)_{n=1}^\infty.
\end{equation}
In particular, Theorem \ref{T6.3} can be applied to show that the
infinite system of integral equations
\begin{equation}\label{E6.54}
u_n(t) = q_n(u)\alpha_n(t) + \kint \int_{0}^t k(t,s)
\Big(h_n(s)+\frac 1{\sqrt{n}} g_n(u)\Big)\,ds, \quad n=1,2,\dots,
\end{equation}
where $u=(u_n)_{n=1}^\infty$ has least and greatest solutions
$u_*=(u_{*n})_{n=1}^\infty$ and $u^*=(u^*_n)_{n=1}^\infty$ in
$H_{\rm loc}((0,\infty),c_0)$, if all the functions
$q_n,\,g_n:H_{\rm loc}([0,\infty),c_0)\to \mathbb R$ are
increasing, and if $-1\le g_n(u),\,q_n(u)\le 1$ for all $u\in
H_{\rm loc}((0,\infty),c_0)$ and $n=1,2,\dots$. Moreover, both
$u_*$ and $u^*$ belong to the order interval  $[w_-,w_+]$ of
$H_{\rm loc}(0,\infty),c_0)$, where the functions $w_\pm$ are
given by (\ref{E6.53}).
\end{example}

\begin{remark}\label{R6.10} \rm
The functions $h_n$ in Example 3.1 do not
belong to $H([0,t_1],\mathbb R)$ for any $t_1>0$. However,
$k(t,s)=\frac st$ is continuous and the functions $k(t,\cdot)h_n$
belong to $H_{\rm loc}([0,\infty),\mathbb R)$, whence the
hypothesis (k0) is valid.

Continuity of $k$ and Theorem \ref{T6.1} ensure that the integral
on the right-hand side of equation \eqref{E6.2} is continuous in $t$.
If also the function $q$ is continuous in $t$ in that equation,
then its solutions  are continuous.
\end{remark}

\section{An application to an impulsive IVP}\label{S6.4}
\setcounter{equation}{0}

Let $E$ be a Banach space ordered by a regular order cone.
The result of Theorem \ref{T6.3} will now be applied to
the following impulsive initial
value problem (IIVP)
\begin{equation}\label{E6.6}
\begin{gathered}
 u'(t)+p(t)u(t)=f(t,u(t),u) \quad\text{a.e. on }  J=[a,b), \\
u(a)=x_0, \quad \Delta u(\lambda)=D(\lambda,u), \quad
 \lambda\in W,
\end{gathered}
 \end{equation}
where $p\in L^1(J,\mathbb R)$, $f:J\times E\times H_{\rm
loc}(J,E)\to E$, $x_0\in E$, $\Delta
u(\lambda)=u(\lambda+0)-u(\lambda)$, $D:W\times H_{\rm
loc}(J,E)\to E$, and $W$ is a well-ordered (and hence countable)
subset of $(a,b)$.

Denoting $W^{<t}=\{\lambda\in W\mid \lambda < t\}$, $t\in J$, and
by $ACG^*_{\rm loc}(J,E)$ the set of all  continuous functions
from $J$ to $E$ which are  $ACG^*$ on every compact subinterval of
$J$, we say that $u:J\to E$ is a solution of the IIVP \eqref{E6.6}
if it satisfies the equations of \eqref{E6.6}, and if it belongs
to the set
\begin{align*}
V=\{&u:J\to E\mid \sum_{\lambda\in W}\|\Delta u(\lambda)\|<\infty
\quad\text{and} \\
& t\mapsto u(t)-\sum_{\lambda\in W^{<t}}\Delta u(\lambda)
 \in ACG^*_{\rm loc}(J,E)\}.
\end{align*}
It is easy to verify that $V$ is a subset of $H_{\rm loc}(J,E)$.

The following result, which is a generalization to
 \cite[Lemma 3.1]{CaHe00}, allows
us to convert the IIVP \eqref{E6.6} to an improper Volterra integral
equation.

\begin{lemma}\label{L6.6}
If $p\in L^1(J,\mathbb R)$, $g\in H_{\rm loc}(J,E)$,
$x_0\in E$ and $c:W\to E$, and if
$\sum_{\lambda\in W}\|c(\lambda)\|<\infty$, then the problem
\begin{equation}\label{E6.7}
\begin{gathered}
 u'(t)+p(t)u(t)=g(t) \quad\text{a.e. on }  J, \\
u(a)=x_0, \quad \Delta u(\lambda)=c(\lambda), \quad \lambda\in W,
\end{gathered}
 \end{equation}
has a unique solution $u$. This solution can be represented as
\begin{equation}\label{E6.8}
u(t)= e^{-\int_{a}^tp(s)ds}x_0+
\sum_{\lambda\in W^{<t}}e^{-\int_{\lambda}^tp(s)ds}c(\lambda)
+ \kint \int_a^te^{-\int_s^tp(\tau)d\tau}g(s)ds
\end{equation}
for $t\in J$. Moreover, $u$ is increasing with respect to $g$, $c$
and $x_0$.
\end{lemma}

\begin{proof} Let $u:J\to E$ be defined by (\ref{E6.8}).
Given a compact subinterval $I=[a,t_1]$ of $J$,
define a mapping $\Gamma:I\to I$ by
$$
\Gamma(s) = \min\{t\in W\cup\{t_1\}\mid s < t\}, \quad
s\in[a,t_1), \quad \Gamma(t_1)=t_1.
$$
Denote by $C$ the well-ordered chain of $\Gamma$-iterations of $a$,
i.e. (cf. \cite[Theorem 1.1.1]{HeiLak94}) $C$ is the only well-ordered
subset of $J$
with the following properties:
$a=\min C$, and if $s > a$, then $s\in C$ if and only if
 $s=\sup\Gamma\{t\in C|t < s\}$.

It follows from \cite[Corollary 1.1.1]{HeiLak94} that $W\subset C$,
and $I$ is a disjoint union of $C$ and open
intervals $(s,\Gamma(s))$, $s\in C$. Moreover, $C$ is countable as a
well-ordered set of real numbers. Hence, rewriting (\ref{E6.8}) as
$$
u(t)= e^{-\int_{a}^tp(s)ds}\Big[x_0+
\sum_{\alpha\in W^{<t}}e^{-\int_{\alpha}^{a}p(s)ds}c(\alpha)
+ \kint \int_{a}^te^{-\int_s^{a}p(\tau)d\tau}g(s)ds\Big],
$$
it is easy to verify that
\begin{equation}\label{E6.9}
u'(t)+p(t)u(t)=g(t) \quad\text{for a.e. }  t\in I, \quad
u(a)=x_0.\end{equation}
For each $\alpha\in W$ the open interval $(\alpha,\Gamma(\alpha))$  does
not contain any point of $W$, so that
\begin{equation}\label{E6.10}
\Delta u(\alpha)=u(\alpha+0)-u(\alpha) = \lim_{t\to
\alpha+0}e^{-\int_{\alpha}^tp(s)ds}c(\alpha)=c(\alpha), \quad
 \alpha\in W.
\end{equation}
It follows from (\ref{E6.8}) and (\ref{E6.10}) that
\begin{equation}\label{E6.11}
u(t)-\sum_{\alpha\in W^{<t}}\Delta u(\alpha) =
u(t)-\sum_{\alpha\in W^{<t}}c(\alpha) = v(t) +w(t),
\end{equation}
where
\begin{gather*}
v(t)= e^{-\int_{a}^tp(s)ds}x_0
+ \kint \int_{a}^te^{-\int_s^tp(\tau)d\tau}g(s)ds, \quad t\in I,\\
w(t)=\sum_{\alpha\in W^{<t}}(e^{-\int_{\alpha}^tp(s)ds}-1)c(\alpha),
 \quad t\in I.
\end{gather*}
Thus, for $a\le \bar t < t\le t_1$ we obtain
\begin{align*}
&w(t)-w(\bar t)\\
&=\sum_{\alpha\in W\cap(a,\bar t)}(e^{-\int_{\alpha}^tp(s)ds}
 -e^{-\int_{\alpha}^{\bar t}p(s)ds})c(\alpha)
 + \sum_{\alpha\in W\cap[\bar t,t)}(e^{-\int_{\alpha}^tp(s)ds}-1)
 c(\alpha)\\
&=\sum_{\alpha\in W\cap(a,\bar t)}
\int_{\bar t}^t-p(s)e^{-\int_{\alpha}^sp(\tau)d\tau}ds\,c(\alpha)
 + \sum_{\alpha\in W\cap[\bar t,t)}
\int_{\alpha}^t-p(s)e^{-\int_{\alpha}^sp(\tau)d\tau}ds\,c(\alpha).
\end{align*}
Applying this representation and denoting
$M=e^{\int_{a}^{t_1}|p(s)|ds}\sum_{\alpha\in W}\|c(\alpha)\|$, it
follows that
$$
\|w(t)-w(\bar t)\|\le M\int_{\bar t}^t|p(s)|ds \quad \text{ for }
 a\le \bar t < t\le t_1.
$$
This implies that $w$ is absolutely continuous. Obviously, $w$ is a.e.
differentiable and the function $v$ is continuous and belongs to
$ACG^*(I,E)$ by Theorem \ref{T601.102} and Proposition \ref{P6.1}.

The above result holds for every $t_1\in (a,b)$, so that $u\in V$ by
(\ref{E6.11}). This, (\ref{E6.9}) and (\ref{E6.10}) imply that $u$
is a solution of problem (\ref{E6.7}).

If $v\in V$ is a solution of (\ref{E6.7}), then $w=u-v$ is a
function of $V$ and $\Delta w(\alpha)=0$ for each $\alpha\in W$,
whence $w\in ACG^*_{\rm loc}(J,E)$ and $w$ is a solution of the
initial value of problem
\begin{equation}\label{E6.111}
 w'(t)+p(t)w(t)=0 \quad\text{a.e. on } J, \quad w(a)=0.
\end{equation}
For every fixed $t\in J$ the function
$$
h(s)=e^{\int_{a}^{s}p(\tau)d\tau}, \quad s\in I=[a,t],
$$
is absolutely continuous on $I$ and real-valued.
It then follows from Lemma \ref{L6.4} that
$$
h(t)w(t)-h(a)w(a)=\kint \int_a^t(h'(s)w(s)+h(s)w'(s))\,ds, \quad t\in J,
$$
or equivalently,
$$
h(t)w(t)-h(a)w(a)=\kint \int_a^t(e^{\int_{a}^{s}p(\tau)d\tau}(p(s)w(s)+w'(s))\,ds,
\quad t\in J.
$$
This equation and (\ref{E6.111}) imply that $h(t)w(t)\equiv 0$,
so that $w(t)\equiv 0$, whence $u=v$.

The last assertion of Lemma is a direct consequence from the representation
(\ref{E6.8}) and Lemma \ref{L6.1}.
\end{proof}

We shall impose the following hypotheses on the function $D$.
\begin{itemize}
\item[(D0)] $D(\lambda,\cdot)$ is increasing for all $\lambda\in  W$, and
there exist $c_\pm:W\to E$ such that
$c_-(\lambda)\le D(\lambda,u)\le c_+(\lambda)$ for all $\lambda\in
W$ and $u\in H_{\rm loc}(J ,E)$, and that
$\sum_{\lambda\in W}\|c_\pm(\lambda)\|< \infty$.
\end{itemize}

As an application of Theorem \ref{T6.3} we get the following existence
and comparison result for the IIVP \eqref{E6.6}.

\begin{theorem}\label{T6.4}
Let the functions $f$ and $D$  in \eqref{E6.6} satisfy the hypotheses
{\rm (f0)--(f2), (D0)}.
If $p\in L^1(J,\mathbb R)$, and if the improper integrals
$\kint \int_a^te^{\kint \int_{a}^sp(\tau)d\tau}h_\pm(s)ds$ exist for some $t\in J$,
then the IIVP \eqref{E6.6} has for each $x_0\in E$
least and greatest solutions $u_*$ and $u^*$ in $V$.
Moreover, these solutions are increasing with respect to
$x_0$, $D$ and $f$.
 \end{theorem}

\begin{proof}
 The  hypotheses given for $D$ and $p$ ensure that for each
$x_0\in E$ the relations
\begin{equation}\label{E6.12}
\begin{gathered}
q(t,u)= e^{-\int_{a}^tp(s)ds}x_0+
\sum_{\lambda\in W^{<t}}e^{-\int_{\lambda}^tp(s)ds}D(\lambda,u),\\
\text{for } t\in J,\,u\in H_{\rm loc}(J,E);\\
k(t,s)=e^{-\int_s^tp(\tau)d\tau}, \quad
(t,s)\in \Lambda=\{(t,s)\in J\times J\,|\, s\le t\},
\end{gathered}
\end{equation}
define mappings $q:J\times H_{\rm loc}(J,E)\to E$ and
$k:\Lambda\to\mathbb R_+$ which satisfy the hypotheses (q0), and
(k0) of Theorem \ref{T6.3}. Then the integral equation
\eqref{E6.6}, which by (\ref{E6.12}) can be rewritten as a fixed
point equation
\begin{equation}\label{E6.13}
\begin{aligned}
u(t)=Gu(t)
&:= e^{-\int_{a}^tp(s)ds}x_0
 +\sum_{\lambda\in W^{<t}}e^{-\int_{\lambda}^tp(s)ds}D(\lambda,u) \\
&\quad + \kint \int_a^te^{-\int_s^tp(\tau)d\tau}f(s,u(s),u)ds,
\end{aligned}
\end{equation}
has by Theorem \ref{T6.3} least and greatest solutions $u_*$ and
$u^*$, and they are increasing with respect to $q$ and $f$.
Because by Lemma \ref{L6.6} the solutions of the IIVP \eqref{E6.6}
are the same as the solutions of the integral equation
(\ref{E6.13}), then $u_*$ and $u^*$ are least and greatest
solutions of the (IIVP) \eqref{E6.6}, and they are increasing with
respect to $x_0$, $D$ and $q$.
\end{proof}

The next result is a consequence of Proposition \ref{P6.2}.

\begin{proposition}\label{P6.3}
Assume that the hypotheses of Theorem 4.1 hold, and let $G$ be
defined by (\ref{E6.13}).
\begin{itemize}
\item[(a)] The sequence $(u_n)_{n=0}^\infty=(G^nw_-)_{n=0}^\infty$ is
increasing and converges  a.e. pointwise  to a function $u_*\in
H_{\rm loc}(J,E)$. Moreover, $u_*$ is the least solution of
\eqref{E6.6} if
 $D(\lambda,u_n)\to D(\lambda,u_*)$ for each $\lambda\in W$ and
$f(s,u_n(s),u_n)\to f(s,u_*(s),u_*)$ for a.e.  $s\in J$;

\item[(b)] The sequence $(v_n)_{n=0}^\infty =(G^nw_+)_{n=0}^\infty$ is
decreasing and converges  a.e. pointwise  to a function $u^*\in
H_{\rm loc}(J,E)$. Moreover, $u^*$ is the greatest solution of
\eqref{E6.6} if
 $D(\lambda,v_n)\to D(\lambda,u^*)$ for each $\lambda\in W$ and
$f(s,v_n(s),v_n)\to f(s,u^*(s),u^*)$ for  a.e.  $s\in J$.
\end{itemize}
\end{proposition}

\begin{example}\label{Ex6.2} \rm
Let $E$ be, as in Example \ref{Ex6.1}, the  space $c_0$ of the
sequences of real numbers converging to zero, ordered
componentwise and equipped with the sup-norm. The solutions of the
infinite system of IIVP's
\begin{equation}\label{E6.14}
\begin{gathered}
w_n'(t)+\frac 1{1+t}w_n(t)
= \frac 2{\sqrt{n}(1+t)}\Bigl(\cos\bigl(\frac 1{t^2}\bigr)
 +\frac 2{t}\sin\bigl(\frac 1{t^2}\bigr)\Bigr) \pm \frac 1{\sqrt{n}},\\
w_n(0+)=0, \quad \Delta w_n\Bigl(t-\frac{2n-1}{2n}\Bigr)
=\pm\frac 1{\sqrt{n}},\quad  n=1,2,\dots,
\end{gathered}
\end{equation}
in $H_{\rm loc}((0,2),c_0)$ are
\begin{equation}\label{E6.15}
\begin{aligned}
&\left(w_{n\pm}(t)\right)_{n=1}^\infty\\
&= \left(\frac 1{2\sqrt{n}(1+t)}\left(\pm\frac{4n-1}{n} H
\Bigl(t-\frac{2n-1}{2n}\Bigr)+2t^2\cos\bigl(\frac 1{t^2}\bigr)
\pm 2t\pm t^2\right)\right)_{n=1}^\infty.
\end{aligned}
\end{equation}
Thus  Theorem \ref{T6.4} can be applied to show that least and
greatest solutions $u_*=(u_{*n})_{n=1}^\infty$ and
$u^*=(u^*_n)_{n=1}^\infty$ of infinite system of IIVP's
\begin{equation}\label{E6.16}
\begin{gathered}
u_n'(t)+\frac 1{1+t}u_n(t)=  \frac 1{\sqrt{n}(1+t)}
 \Bigl(\cos\bigl(\frac 1{t^2}\bigr)+\frac 2{t}
 \sin\bigl(\frac 1{t^2}\bigr)\Bigr) +\frac 1{\sqrt{n}} g_n(u),\\
w_n(0+)=0, \quad \Delta w_n\Bigl(t-\frac{2n-1}{2n}\Bigr)
=\frac 1{\sqrt{n}} D_n(u) ,\quad  n=1, 2,\dots,
\end{gathered}
 \end{equation}
exist  in $H_{\rm loc}((0,2),c_0)$ and belong to its order
interval  $[w_-,w_+]$, if we assume that  all the functions
$D_n,\,g_n:H_{\rm loc}([0,2),c_0)\to \mathbb R$, are increasing,
and if $-1\le D_n(u),\,g_n(u)\le 1$ for all $u\in H_{\rm
loc}((0,2),c_0)$ and $n=1,2,\dots$.
\end{example}



\begin{remark}\label{R6.3} \rm
The functional dependence on the last argument $u$ of $q$, $f$ and $D$  can
be formed, e.g., by bounded, linear and positive operators, such as integral
operators of Volterra and/or Fredholm type with nonnegative kernels.  Thus the
results derived in this paper can be applied also to integro-differential
equations.

If $a > -\infty$, then $H_{\rm loc}([a,b),E)$ contains those
functions $u:[a,b)\to E$ which are HL integrable on every compact
subinterval of $(a,b)$ and for which the improper integral
$$
\kint \int_{a+}^tu(s)\,ds=\lim_{c\to a+}\kint \int_c^tu(s)\,ds
$$
exists for some $t\in (a,b)$ (cf. \cite[Theorem 2.1]{Fdt02}).
Noticing also that Bochner integrable functions are HL integrable,
the results of Sections 3 and 4 generalize the corresponding
results of \cite{HKS08} in the case when $a > -\infty$.

As for other papers dealing with functional Volterra integral
equations and differential equations via non-absolute integrals;
see, e.g. \cite{Fdt02,FdScw06,{Stc08},SN02,SN07}.
\end{remark}

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\end{document}