\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 $W^{ a$, 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
$$
u(t)= e^{-\int_{a}^tp(s)ds}\Big[x_0+
\sum_{\alpha\in W^{ -\infty$, then $H_{\rm loc}([a,b),E)$ contains those
functions $u:[a,b)\to E$ which are HL integrable on every compact
subinterval of $(a,b)$ and for which the improper integral
$$
\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}