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\journal{Journal of Computational \& Applied Mathematics}

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\begin{document}
\begin{frontmatter}
%% Title, authors and addresses
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 \title{{The Title of the Demonstration Paper}\tnoteref{label00}}
 \tnotetext[label00]{Demonstration of elsarticle.cls}
 \author{C.T.H.Baker\corref{cor1}\fnref{label01}}
 \ead{cthbaker@na-net.ornl.gov}
 \ead[url]{{http://www.chester.ac.uk/maths/baker.html} \
 {http://www.maths.man.ac.uk/}\name} 

   \address{Dept.\ of Mathematics, University of Chester, CH1 4BJ \&
 University of Manchester, M13 9PL}
   \cortext[cor1]{Corresponding  author [copyright reserved]} 
   \fntext[label01]{Emeritus Professor of Mathematics in the University of
 Manchester}

\author{A.N. Author, jnr.\fnref{label1}}
\ead{s.author@notmail.com}
\address{University of the South Pole, Antarctica \fnref{label1}}
  \fntext[label1]{Frozen professor}

%\author{}%\address{}
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\begin{abstract}
 This paper is a demonstration of the use of {\tt
elsarticle.cls}.
\end{abstract}

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%% \linenumbers
%% main text
\section{Introduction}
\label{sect:Introduction}
Consider the following results (that appear in, \eg \cite{bib:1}), but
are given here only for completeness of the presentation and to make up the page.
\begin{lem}\label{lem:1}
If 
\begin{subequations}
\begin{equation}
 A : X \stackrel{1:1\ onto}{\to} 
Y
\end{equation}
\begin{equation}
 B : Y  \stackrel{1:1\ onto}{\to}  Z
\end{equation}
then $ BA : X  \stackrel{1:1\ onto}{\to}  Z$.
\end{subequations}
\end{lem}
\begin{thm} \label{thm1}
If, in Lemma 1, $X_{\|\cdot\|}$, $Y_{\|\cdot\|}$, $Z_{\|\cdot\|}$, are B-spaces  and
\begin{subequations}
\begin{equation}
 A : X \to Y \hbox{ with } \|A\| < 1,
\end{equation}
\begin{equation}
 B : Y \to Z \hbox{ with } \|B\| < 1,
\end{equation}
\end{subequations}
then $\| B A\| < 1$.
\end{thm}
The assumptions are stronger than necessary\footnote{Added in proof:
This paper was discovered to contain a fundamental error but the
second author could not be contacted to agree to issuing a correction.}.  We defer the
proofs to the Appendices. The \emph{Appendices} start with the command
\verb@\appendix@; appendix sections are then type-set as normal
sections: \appendix
 \section{B-spaces}
%%
 \label{App-sec:B-spaces}
Recall that a  B-space (Banach space) is a normed linear space that is complete.


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\bibitem{bib:1} Anonymous, A., {Sample paper}, 
{J.\ Sample Papers}, {1} (2222), pp.~1--1500.

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%%
\begin{document}
\begin{frontmatter}
\hfill {\tt Desktop  : jcam-1-cb-ys-iwanasp.tex}\\[-5mm]
\title{Periodic Solutions of  Non-linear Discrete
 Volterra Equations with Finite Memory}
%\\[-15mm]}
\author[CB]{Christopher T H Baker\corref{cor1}}
\ead{cthbaker@na-net.ornl.gov}
\author[YS]{Yihong Song\corref{cor2}\fnref{fn1}}
\ead{yihongsong@hotmail.com}

\cortext[cor1]{Corresponding author  [copyright reserved]}

\fntext[fn1]{ Supported by JGSF of Jiangsu (No.\ 07KJB110104)}

\address[CB]{Department of Mathematics, University of Chester, CH1 4BJ,\\
Emeritus professor, School of Mathematics, University of Manchester, UK}
\address[YS]{Department of Mathematics, Suzhou University, Jiangsu 215006, P. R. China}
\pagestyle{plain}
\setcounter{page}{1}
\begin{abstract}
 In this paper we discuss the existence of periodic solutions of
 \emph{discrete} (and \emph{discretized}) non-linear Volterra equations with finite
 memory. The literature contains a number of results on periodic
 solutions of non-linear Volterra \emph{integral equations} with finite memory,
of a type that arises in biomathematics.  The ``summation'' equations studied here
can  arise as discrete models in their own right but
 are (as we demonstrate) of a type that arise from the discretization of such
integral equations.
Our main results are in two parts: ($i$) results for discrete equations and
($ii$) consequences for quadrature methods applied to integral equations. The
first set of results are  obtained using a variety of fixed point theorems. The second set of results
 address the preservation of properties of integral equations on discretizing them.
An expository style is adopted and
examples are given to illustrate the discussion. \\[-.25\baselineskip]
\end{abstract}
  %\vspace{-.5\baselineskip}
  \begin{keyword}
\parbox{\textwidth}%{Key words:}
 {Periodic solutions, discrete equations, finite memory,
fixed point theorems, quadrature, simulation.
} \\
{\it AMS Subject Classification:} 65Q05 \sep 37M05 \sep 39A12\sep 47B34 \sep 47B60
  \end{keyword}
\end{frontmatter}
%\vspace{-\baselineskip}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
The equation
\begin{equation}\label{eq-contin-2}
\x(t) = \int_{t-\tau}^t \ksf(t,s) \f(s,\x(s)) {\rm d}s,
 \quad
\hbox{for $ t\in \Rset$ with } \tau >0,\  \x(t)\in \Rset,
\end{equation}
is an integral equation of Volterra type, studied
\cite{cook} as a model of certain epidemic problems.
  We
study periodic solutions of an
analogous \emph{discrete} system
\begin{equation}\label{eq-nvd-1}
x(n)=\sum^n_{j-N}k(n,j)f(j,x(j)),
 \quad N \in  \Nset, \quad x(n)\in \Rset,
\end{equation}
 given $N$, under certain conditions on $\{k(n,j)\}$ and
$\{f(n,x(n))\}$ (see below). {Here, $\Rset=(-\infty,\infty)$, $\Rset_+=[0,\infty)$, we write
$\Zset$ to denote the integers, $\Zset_+$ the set of non-negative integers, $\Nset$ the positive
integers $\Zset_+\!\setminus\! \{0\}$. The set of  quotients of positive integers
we denote by $\Qset_+$.}
\begin{remark}{\small
If $k(t,s) \equiv 1$, then (\ref{eq-contin-2}) reduces to the integral equation
%%% % \vspace{-.5\baselineskip}
\begin{equation}\label{eq-contin-1}
\x(t) = \int_{t-\tau}^t\f(s,\x(s)) {\rm d}s.
\end{equation}
%%% % \vspace{-.75\baselineskip}
 Examples studied in the literature also include those of the form
\bsq
 % % \vspace{-.25\baselineskip}
\bq
\label{ex-inteq-1}
\x(t) = \int^t_{t-\tau} \ksf(t,s)\g(s) \{\x(s)\}^\gamma \ds
%% % \vspace{-\baselineskip}
\eq
and, more generally,
\bq\label{ex-inteq-2}
\x(t) = \int^t_{t-\tau} \ksf(t,s)\g(s) [\{\x(s)\}^\alpha +  \{\x(s)\}^\beta] \ds
\eq
\esq
subject to certain conditions \cite{aga01}.
 Eq.\  (\ref{eq-contin-1}) was discussed,
in \cite{cook}, as a model for the spread of certain infectious diseases with
periodic contact rate that varies seasonally,
 where $\tau>0$ is the length of time an individual remains infectious
and it is assumed that there exists $\varpi>0$ with
$\f(t+\varpi,u)=\f(t,u)$ for all $t\in \Rset$
and $u\in \Rset_+$ (see \cite{cook} for details).
Using Krasnosel${}^\prime\!$ski\u\i 's fixed
point theorem in a cone \cite[p.137]{bk:Kras},
Cooke and Kaplan \cite{cook} established the
existence, if $\tau$ is sufficiently large, of a
nontrivial periodic non-negative solution to
(\ref{eq-contin-1}) with period $\varpi$. Later, Leggett and
Williams \cite{legg} generalized the
results in \cite{cook}.
We shall give some results for discrete versions of (\ref{ex-inteq-1})
and (\ref{ex-inteq-2}) in the present paper.
 }\end{remark}
%% % \vspace{-\baselineskip}
\subsection{ Discrete equations}
%% % \vspace{-\baselineskip}
 Eq.\  (\ref{eq-nvd-1}) is a discrete analogue  of (\ref{eq-contin-2}),
 and (\ref{eq-nvd-1}) arises in  the numerical analysis of (\ref{eq-contin-2})
(see below, and \cite{aga00,baker,brunner}), as well as in discrete models.
It appears that little work has been done on questions of periodic
solutions of the {\em implicit}, \emph{non-linear},
\emph{finite-memory} discrete system\footnote{ For periodic solutions
of {\em explicit} discrete systems, see, \eg
\cite{elay0,fur}. Periodic solutions of (\ref{eq-nvd-2}) with
$f(j,x(j))=x(j)$ were discussed in \cite{elay94}.  For periodic
solutions of {\em implicit} discrete systems with \emph{unbounded
memory}, see \cite{song2,song4}.  } (\ref{eq-nvd-1}).  This motivates
us to investigate periodic solutions of (\ref{eq-nvd-1}), and to
parallel some results for (\ref{eq-contin-2}) {stated in
\cite{aga01}}. We establish existence results for periodic solutions
of (\ref{eq-nvd-1}), via a variety of fixed point theorems (\cf
\cite{aga01-cup}).  In \S \ref{sec-prelims-fpt}, we give some basic
results and recall several fixed point theorems, and give our main
results in \S \ref{sec-MR}. In \S \ref{sec-numerics}, we discuss
quadrature methods for (\ref{eq-contin-2}) and demonstrate the
application of the general results of \S \ref{sec-MR}.


 Given the terminology ``integral equation'' for (\ref{eq-contin-2}), it
seems appropriate to term (\ref{eq-nvd-1})  a ``summation equation'' (of
Volterra type, with a finite memory). Eq.\ (\ref{eq-nvd-1}) has also been
called a recurrence relation or a difference
equation.
For each integer $n$, eqn (\ref{eq-nvd-1}) is (if $k(n,n) \neq 0$) is an
{\em implicit} equation for $x(n)$.
 One may consider (\ref{eq-nvd-1}) for
$n \in \Zset$, for $n \geq n_0$ ($n \in \Zset$), or for $n \in \Zset_{+}$, for example.
  A particular solution $\{x(\varphi; n)\}_{n\geq 0}$ of (\ref{eq-contin-2})
   for $n\in \Zset_+$
corresponds to a choice
 \bq\label{x-varphi} x(\varphi; n) := \varphi(n) \hbox{ for } n \in
\{-N,\ldots, -1\};  \eq
  $x(\varphi;n)$ is an analogue of a solution
$\x(\varrphi;t) $ of  (\ref{eq-contin-2}) for $t \geq 0$, wherein $\x(\varrphi;t) = \varrphi(t)$
for $t \in [-\tau,0]$. There is no reason, without imposing assumptions on $k$ and $f$, to suppose
that  eqn (\ref{eq-nvd-1}) has a solution $x(n)$ given $\{x(j)\}_{j < n}$.
\begin{example}\label{theta-numerics}
Suppose that  $\tau > 0$ and $\hfr > 0$ is chosen with
${N}_\tau \in \Nset$ and $ \hfr = \tau / {N}_\tau$,
 and suppose that $\vartheta \in [0,1]$. Then the
``\emph{composite}'' or ``\emph{repeated}'' version (\S \ref{sec-Quad})
of the  $\vartheta$-rule applied to (\ref{eq-contin-2}) yields the form
(\ref{eq-nvd-1}) with \bsq \label{theta-eq}
 % % \vspace{-.5\baselineskip}
 \begin{equation}\label{theta-eq-a}
\hbox{$k(n,j) = \hfr \ksf (n \hfr, j \hfr)$ for $j \not\in
\{n-{N}_\tau, n\}$,}
\eq
\bq
\nonumber
\hbox{$k(n,n-{N}_\tau) =
 (1-\vartheta) \hfr \times
\ksf(n \hfr, (n-{N}_\tau) \hfr)$, \quad $k(n,n) = \vartheta \hfr \times \ksf (n \hfr, n \hfr)$}
 \end{equation}
 and
  \begin{equation}
 f(n,v) = \f(n\, \hfr,v).
 \end{equation}
 \esq
By the opening assumptions,  $\hfr \in {\frH}_\tau$ where
    we define    (for any $\sigma \in \Rset_+$)
     \bq
\label{H-sub-tau}
  {\frH}_{\sigma} := \left\{ {\sigma}/{{N}_\sigma} \ \hbox{ for some } {N}_\sigma \in
\Nset \right\}.  \eq
 In the case $\vartheta = 0$, we obtain equations $x(n)=
\sum^{n-1}_{j-{N}_\tau}\hfr \ksf (n \hfr,j \hfr)\f(j \hfr,x(j))$. These
relations are \emph{explicit} and are special cases (for $N= {N}_\tau$) of the form
\begin{equation}\label{eq-nvd-2}
x(n)=\sum^{n-1}_{j-N}k(n,j)f(j,x(j)),
 \end{equation}
 However, the cases
$\vartheta =\half$ and $\vartheta = 1$ yield implicit recurrence relations
 and it is known that these relations display, in many cases,
better stability properties than $\vartheta = 0$.
 Under Lipschitz conditions, the convergence rates of
$x(\varphi; n)$ to $\x(\varrphi;
t)$, as $\hfr \searrow
  0$, where $\varphi(j) = \varrphi(j \hfr)$ for $j \in [-\tau, 0]$ (and
$n \hfr = t$, $n \to \infty$), are optimized by the choice $\vartheta = \half$.
\end{example}
 { Our main
results are in two parts: ($i$) results for discrete equations
(\emph{per se}) and ($ii$) consequences for quadrature methods applied
to integral equations. One aim of the paper is to demonstrate that it is possible
by picking appropriate families of quadrature rules to preserve
appropriate periodicity displayed in the integral equation model.}
{  One might suggest that seeking to preserve
periodicity displays some parallels with the
search for symplectic numerical methods for certain
classes of differential equations, a topic that is seen of some significance in
numerical analysis.}
\begin{remark}{\small
{ Consider the perspective of mathematical modelling in
biomathematics, involving (say) models of population growth. It is
widely recognized that \emph{when population sizes are small},
models based on a continuous or differentiable function defined on
$\Rset$ may be less suited than models based on discretely defined
functions (where values may be observable only at discrete times).
\emph{The construction of discrete models in (for example)
biomathematics is governed by an attempt to model the scientific
understanding.} Periodicity of solutions (or almost periodicity) is
a matter of scientific interest. Where continuous models are to be
preferred (on grounds of realism) over discrete models, the
scientifically faithful forms of such continuous models rarely have
closed-form solutions. Where practically useful insight into
solutions are sought, one may turn to numerical methods, applied to
realistic models, to provide approximate values. In this situation,
one seeks `appropriate' numerical formulae. }{
 The discretized  equations obtained when one discretizes
integral equations reflect (1) \emph{the modelling} (based on our
scientific understanding of the processes modelled) \emph{implicit
in the construction of the integral equation} and (2) \emph{the
independent choice of a numerical method} (which is not necessarily
governed by the underlying science). }}
\end{remark}
%% % \vspace{-\baselineskip}
\section{Preliminaries to the fixed point analysis}\label{sec-prelims-fpt}
%% % \vspace{-\baselineskip}
 For basic functional analysis see, \eg  \cite{L-Sobolev}. Suppose that $X$
is a linear space; if $\| \cdot \|$ is a norm on $X$ we denote by $X_{\|\cdot\|}$ the corresponding
normed linear space and if $X_{\|\cdot\|}$ is a Banach space we denote it by ${\mathcal X}$ or $
\{X; \| \cdot \|\}$.  The closure of a set $\setS$ is denoted $\overline{\setS^{\,}}$, and the
boundary $\partial \setS$ of an open set $\setS$ is   $\partial \setS :=
\overline{\setS^{\,}}\!\setminus\! \setS$. For the basic properties of \emph{convex sets} and
\emph{cones}\footnote{
 ${\Conel}\subset X$ is a cone  in  ${X_{\|\cdot\|}}$ if it
is a  closed  convex set and
($i$) for any non-zero $v \in \Conel$ and any
 $\lambda \geq 0$, we have $ \lambda v
\in \Conel$, and ($ii$) if $ v \in X$ is non-zero then at least one of
the pair $\{ v, -v\}$ does
not lie in $\Conel$.} (needed below) see \cite{guo,bk:Kras}.
 Denote by $ \ell(\Zset) $ the linear space
whose elements are sequences with real $x(n)$:
 \begin{equation}
 \ell(\Zset) := \{ x
\ |x = \{x(n)\}_{n\in \Zset}, \ x(n) \in {\Rset} \} \end{equation}
 ($x$ or $\{x(n)\}_{n \in \Zset} $ denotes $\{ \ldots,x(-2), x(-1),
x(0),x(1),x(2), \ldots \}$, and we have the obvious definitions of
addition and scalar multiplication).
\begin{defn}\label{def:periodic} Let $\omega\in \Nset$ be a given positive
integer.  Then $x=\{x(n)\}_{n\in \Zset}$ is an \emph{$\omega$-periodic sequence} if $
x(n+\omega)=x(n)$ for all $n\in \Zset$. $A_\omega=A_\omega(\Zset)$ denotes the (finite-dimensional)
subspace of $\ell(\Zset)$ consisting of  all $\omega$-periodic sequences.  We define the norm
$|\cdot|_\omega$ on $A_\omega(\Zset)$ by setting $|x|_\omega=\sup_{n\in \Zset}|x(n)|=\max_{1\le
n\le \omega}|x(n)|$ for $x=\{x(n)\}_{n\in \Zset}\in A_\omega(\Zset)$. A solution $x$ of
(\ref{eq-nvd-1}) is called periodic if it satisfies (\ref{eq-nvd-1}) for $n \in \Zset$ and $x \in
A_\omega$ for some integer $\omega \in \Nset$.
\end{defn}
Obviously, an $\omega$-periodic sequence is $\omega_\star$-periodic where
$\omega_\star/ \omega \in \Nset$. We next state, as lemmata, two basic results
(applicable for given $\omega \in \Nset$).
\begin{lem}\label{lem-A-omega}
 ${\calA}_\omega=\{A_\omega(\Zset); | \cdot |_\omega\}$
is an $\omega$-dimensional Banach
space.
\end{lem}
% % \vspace{-.5\baselineskip}
\begin{lem}\label{lem-cpct-map}
 Let $\setD \subseteq {\calA}_\omega$. If an operator $T:\setD \to
 {\calA}_\omega$ is continuous and maps bounded sets into bounded sets, then $T$ is completely
 continuous (compact).
\end{lem}
% %\vspace{-.5\baselineskip}
For Lemma \ref{lem-cpct-map},  recall that $T:\setD \to {\calA}_\omega$
is \emph{completely continuous}, or \emph{compact}, if, and only if, for any bounded sequence
$\{x_n\}$ in $\setD$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that
$\{Tx_{n_k}\}$ is convergent in ${\calA}_\omega$.  By the general Heine-Borel theorem, every
bounded set in ${\calA}_\omega$ contains a convergent sequence.

The existence of one or more periodic solutions of (\ref{eq-nvd-1})
will be established via fixed point theories. We associate with
(\ref{eq-nvd-1}) an operator $T$
 on $\ell(\Zset)$ (or a subspace), with $(Tx)(n) :=\sum^n_{j-N}k(n,j)f(j,x(j))$,
and we identify a solution of (\ref{eq-nvd-1}) as a fixed point of
$T$. We will use (citing, and paraphrasing, \cite{aga01})
either \emph{Krasnosel${}^\prime$ski\u\i's fixed point theorem}
(stated here as Theorem \ref{krans-fixed} -- see \cite[Theorem 2.1.1]{aga01}), the
\emph{``non-linear alternative''} (Theorem \ref{non-linear-alternative}-- see \cite[Theorem
1.2.1]{aga01}) or the \emph{Leggett-Williams fixed point theorem}
(Theorem \ref{thm-LW}-- see
\cite[Theorem 4.3.1]{aga01}).  These theorems were used in
 \cite{aga01} to discuss (\ref{eq-contin-2}).
\begin{thm}\label{krans-fixed} {\bf (Krasnosel${}^\prime$ski\u\i)}
Let $\calX \equiv \{X; \|\cdot\|\}$ be a Banach space  and
let $\Cone\subset \calX$ be a cone in $\calX$. Assume
$\Omega_{1,2}$ are bounded open subsets of $\calX$ with $0\in \Omega_1$,
$\overline{\Omega}_1 {\subseteq} \Omega_2$, and let
$
T: \Cone\cap (\overline{\Omega}_2\!\setminus\!\Omega_1)\to \Cone
$
be a completely continuous operator such that either
(i) $\|Tu\|\le \|u\|$, $u\in \Cone\cap \partial \Omega_1$ and
$\|Tu\|\ge \|u\|, u\in \Cone\cap \partial \Omega_2$  or (ii) $
\|Tu\|\ge \|u\|$, $u\in \Cone\cap \partial \Omega_1$ and $\|Tu\|\le
\|u\|, u\in \Cone\cap \partial \Omega_2$ is true. Then $T$ has a fixed
point in $\Cone\cap(\overline{\Omega}_2
  \!\setminus\!\Omega_1)$.
\end{thm}
\begin{thm}\label{non-linear-alternative} {\bf (Non-linear  alternative)}
{Suppose that $\convexC$ is a convex subset of a normed
linear space $ {X}_{\|\cdot\|}$, and let
$\setU$ be an open subset of $\convexC$, with $p^\star\in \setU$.}
Then every compact, continuous
map $T_\natural: \overline{\setU^{\,}}\to \convexC$ has at
least one of the following two
properties: (i) $T_\natural$ has a fixed point in $\overline{\setU^{\,}}$;
 (ii) there is an
$x_\lambda\in
\partial \setU$ with $x_\lambda=(1-\lambda)p^\star+\lambda T_\natural x_\lambda$ for some $0<\lambda<1$.
\end{thm}
Note that   (i) and (ii) are not (as sometimes implied by the description \emph{alternative})
mutually exclusive, but if conclusion (ii) is shown to be false then conclusion (i) holds. In
applications, $\convexC$ can be a cone.
\begin{thm}\label{thm-LW} {\bf (Leggett-Williams) }
Let ${X}_{\|\cdot\|}$ define a Banach space ${\calX}$, $\Cone\subset {X}$ a cone in ${\calX}$,
$r_1, r_2 \in (0, \infty)$, $r_1\ne r_2$ with $R=\max\{r_1,r_2\}$ and $r=\min\{r_1,r_2\}$. Define $
 {C}_\eta=\{x\in \Cone: \|x\|
 < \eta\}$ (${\partial C}_\eta=\{x\in \Cone \, \big| \, \|x\|=\eta\}$,
 $\overline{C}_\eta=\{x\in C \, \big| \, \|x\| \le \eta\}$).
 Let $T:\overline{C}_R \to \Cone$ be a continuous,
compact map such that (i) there exists $u_0\in \Cone\!\setminus\!\{0\}$ with
$T u\not \leqslant  u$ for $u\in
{\partial C}_{r}\cap C(u_0)$ where
$C(u_0)=\{u\in C \, \big| \, \hbox{$ \exists \lambda>0$ with $u\geqslant \lambda u_0$}\}$;
and (ii) $\|Tu\|\le \|u\|$ for $u\in {\partial C}_{R}$.
Then $T$ has at least one fixed point $x\in \Cone$ with $r\le \|x\|\le
R$.
\end{thm}
%% % \vspace{-1.3\baselineskip}
\section{Main results on discrete equations}\label{sec-MR}

The authors' main results will be stated as {\sc Propositions}.
Associated with  (\ref{eq-nvd-1}), we define (for suitable $k, f$)
 an operator $T$ on $\ell(\Zset)$ by
\begin{equation}\label{eq-operator}
(Tx)(n)=\sum^n_{j-N}k(n,j)f(j,x(j)),\quad n\in \Zset.
\end{equation}
A fixed point of $T$ is a solution of (\ref{eq-nvd-1}). We shall rely
 on assumptions that parallel
 assumptions made in discussions \cite{aga01} of the continuous case
(\ref{eq-contin-2}). We shall also make additional varying  assumptions  from amongst the following
together with additional conditions.  We have
 \bq
% % \vspace{-.5\baselineskip}
\omega \in \Nset.
  \eq
Throughout, one may
replace $\omega$ by $\omega_\star =\widehat \jmath \times \omega$ with given
$\widehat\jmath \in \Nset$.

 % % \vspace{-.5\baselineskip}
\begin{assump}\label{A1}
\begin{equation}\label{eq-for-k}
\hbox{ $k:\Zset \times \Zset \to \Rset_+$ and $
k(n+\omega,j+\omega)=k(n,j) $ for every $n,j\in \Zset$}.
\end{equation}
\end{assump} %\vspace{-\baselineskip}
 \begin{assump} {($a$)} \label{A2}  % % \vspace{-\baselineskip}
\bsq\label{eq-for-f1-all}
 \bq \label{eq-for-f1} f:\Zset \times \Rset_+ \to {\Rset_+}
\eq
\vspace{-\baselineskip}
\noindent\hbox{\!{\assume\,  \ref{A2}} {($b$)}}
  %\vspace{-\baselineskip}
   \begin{equation}
 \label{eq-for-f1-a+}
 \ \hbox{  $\qquad f(n,\cdot)$ is continuous on $\Rset_+$ for each $n\in \Zset$}.\!\!\!
 \end{equation}
\vspace{-\baselineskip}
\begin{equation}
\label{eq-for-f1-b} \! \hbox{{\assume\, \ref{A2}} {($c$)}\quad } f(n+\omega,u)=f(n,u) \hbox{
for all $n\in \Zset$ and $u\in \Rset_+$}.
\end{equation}
\esq
 \vspace{-.5\baselineskip}
 \end{assump}
 \noindent
\begin{assump}{($a$)}\label{A3}
% % \vspace{-.5\baselineskip}
{The function $\psi$ is a non-decreasing continuous map}
\bsq\label{eq-for-f3}
  \begin{equation}
 \label{eq-for-f3-a}
 \psi:\Rset_+\to \Rset_+;\
\psi(u_1) \geq \psi(u_2) \hbox{ if } u_1 \geq u_2 \in \Rset_+.
 \end{equation}
{\assume\,\ref{A3}} {($b$)} \ The functions $f$, $\psi$  in (\ref{eq-for-f3-a}), and $q\in
{\calA}_\omega$ satisfy
\begin{equation} \label{eq-for-f3-b}
\hbox{  $ f(n,u)\le q(n)\psi(u)$ for all $u\in \Rset_+$ and $n\in \Zset$.}
\end{equation}
{\assume\,\ref{A3}} {($c$)}
 There exists a constant ${a}_0 \in (0,1)$, such that the
functions $f$, $\psi$ in (\ref{eq-for-f3-a}), and $q\in {\calA}_\omega$ satisfy
   \begin{equation}
\label{eq-for-f3-c}
 \hbox{  $ {a}_0q(n)\psi(u)\le f(n,u)$ for all $u\in
\Rset_+$ and $n\in \Zset$}.
\end{equation}
  \esq
{\assume\,\ref{A3}} {($d$)} There exists a function {${ {\blue\xi}}:(0,1)\to
\Rset_+\!\setminus\! \{0\}$} such that, for $\psi$ in (\ref{eq-for-f3-a}),
\begin{equation}\label{eq-for-f4}
 \psi(\mu \upsilon)\ge { {\blue\xi}}(\mu)\psi(\upsilon), \hbox{ for any
   $0<\mu<1$,   $\upsilon \ge 0$}.
 \end{equation}
\end{assump}
\begin{assump} With given $f$, $\psi$, $q$, there exists a continuous function
 ${ {\chi}} : \Rset_+ \to \Rset_+$ with
\bq \label{eq-LW-1}
 { {\chi}}(u)q(n)\le f(n,u)\le q(n)\psi(u) \hbox{
for all $u\in \Rset_+$ and $n\in \Zset$}. \eq
\end{assump}
(The second inequality in (\ref{eq-LW-1}) arises in (\ref{eq-for-f3-b}).) In  Proposition
\ref{propn:thm-from-LW} we require (\ref{eq-LW-1}) with ${ {\chi}}(u)/u$
 nonincreasing for $u$ in an
interval $(0,r)$. Some further assumptions are stated, later, in terms of
 $\kappa_{\max}$, $\kappa_{\min}$:
\begin{defn}\label{def:kappa-1-2} ${\kappa_{\max, \min}(q_\natural)}$ are defined
for any $q_\natural \in A_\omega$ by
 \begin{equation}\label{eq-for-f5-6}\!
{\kappa}_{\max}(q_\natural) :=\max_{1\le n\le
\omega}\!\sum^n_{j-N} \! k(n,j)q_\natural(j), \
{\kappa}_{\min}(q_\natural) := \min_{1\le n\le
\omega}\!\sum^n_{j-N} \! k(n,j)q_\natural(j).
\end{equation}
\end{defn}
% % \vspace{-\baselineskip}
\begin{remark}\label{kern-cond}{\small
The following assumptions relate to the integral equation (\ref{eq-contin-2}).
{ It is natural to assume the continuity of $\ksf$ in the light of
familiar models arising in biomathematics, but we note that results in the paper
can be modified (see below) for a class of weakly-singular, Abel-type, kernels
and modified discretization methods.}}
\end{remark}
\begin{assump}\label{cts-case-k+f}
We have $\varpi \in \Rset_+$ and
 \bsq \label{ie-cond-k+f}
 \bq \label{ie-cond-k}\ksf\in C(\Rset \times \Rset \to
\Rset_+) \hbox{ and,  for every } t,s\in \Rset,
  \ksf(t+\varpi, s+\varpi)=\ksf(t,s) .
 \eq
  Further,
  \bq\label{ie-cond-f}
  \hbox{   $\f: C(\Rset \times \Rset_+ \to \Rset_+)$, and
$\f(t+\varpi,u)=\f(t,u)$} \eq
 \esq
\end{assump}
\begin{proposition}\label{propn:Ex-quad-discrete}
Suppose that {\assume} \ref{cts-case-k+f} applies. When $\hfr >0$,
when
 \bq \hfr W_\ell \in \Rset_+, \ \hbox{ for } \ell
\in \{ n-N, n-N+1, \ldots, n\},\ n \in \Zset \eq and  $k$ and when
$\ksf$, $f$  and $\f$ are related by
 \bq k(n,j) = \hfr W_{n-j} \ksf(n\hfr, j\hfr), \ j \in \{ n-N, n-N+1, \ldots,
n\}, \ n \in \Zset \eq \bq f(n,u) = \f(n\hfr, u), \ n \in \Zset, \ u
\in \Rset, \eq
   then {\assumes} \ref{A1} and  \ref{A2}
    are satisfied with $\varpi = \hfr \times \omega $ where
$ \omega \in \Nset$.
Suppose, additionally, that
 \bsq\label{collected} \bq \label{ie-cond-q} \hbox{$\qsf \in C( \Rset
\to \Rset$)
 and   $\qsf(t)
= \qsf(t+\varpi)$ for $t \in \Rset$}.
\eq
 Then if $q(n) = \qsf(nh)$ (for $n \in \Zset$) and  $\varpi = \hfr \times \omega $, it follows
that $q \in \calA_\omega$.  Relations (\ref{eq-for-f3-b})  or
(\ref{eq-for-f3-c})  follow if, respectively,
 \begin{equation} \label{int-eq-d}
\hbox{  $ \f(t,u)\le q(t)\psi(u)$ for all $u\in \Rset_+$ and $t\in \Rset$,}
\end{equation}
or
 \begin{equation} \label{int-eq-e}
 \hbox{  $ {a}_0\qsf(t)\psi(u)\le \f(t,u)$ for all $u\in
\Rset_+$ and $t\in \Rset$}.
\end{equation}
\esq
\end{proposition}
\begin{remark}{\small \label{H-sub-varpi}
Conditions on $\ksf$, $\f$ and on $\qsf$, in the literature
 on (\ref{eq-contin-2}),
include the assumptions on $\ksf$, $\f$ and on $\qsf$ of Proposition \ref{propn:Ex-quad-discrete}.
 Now, for $\varpi \in \Rset_+$, $ {\frH}_{\varpi}  \subset \Rset$ is defined as $\{
{ \varpi}/{ {N}_\varpi } \ |\,  {N}_\varpi \in \Nset\}$.
  In view of  Proposition \ref{propn:Ex-quad-discrete}, we
later suppose that {$h \in {\frH}_\varpi$}. Referring to
Example \ref{theta-numerics}, where $h = \tau/{N}_{\tau}$, we
also ask that {$h \in {\frH}_\tau$}. This restricts $\tau$ and
$\varpi$ (we need $ \tau = \{{N}_\tau/{
N}_\varpi\}\varpi$). This restriction will be overcome; see Proposition
\ref{propn:replace-tau} {\it et seq}.
}\end{remark}
 % % \vspace{-\baselineskip}
\subsection{Positive periodic solutions via Krasnosel${}^\prime\!$ski\u\i 's fixed point theorem}
%% % \vspace{-\baselineskip}
First, we use {Krasnosel${}^\prime\!$ski\u\i}'s fixed point
theorem to establish a
result for (\ref{eq-nvd-1}).
 In  Krasnosel${}^\prime\!$ski\u\i's
theorem, take $\mathcal X$ to be  ${\calA}_\omega \equiv \{ {A}_\omega, |\cdot|_\omega\}$, and $
\Cone=\{x \in {\calA}_\omega  \ | \ x(n)\geq
 M|x|_\omega   \hbox{ for } n\in\Zset\}$ with $M$ from
 (\ref{eq-for-f6}):  $x\in \Cone$ when $
x(n)\ge M|x|_\omega$  for  $ 1\le n \le \omega$.
 It is readily shown
that $\Cone$ is a cone in ${\calA}_\omega$. Let $T: \Cone
 \to \ell(\Zset)$ be
defined by (\ref{eq-operator}).
 Then, $T:\Cone \to {\calA}_\omega$. Indeed, for any
$x  \in \Cone\subset {\calA}_\omega$, it follows from
(\ref{eq-for-k}) and (\ref{eq-for-f1}) that
%\vspace{-\baselineskip}
\bq \nonumber
(Tx)(n+\omega)=\sum^{n+\omega}_{j+\omega-N}k(n+\omega,j)f(j,x(j))
=\eq
% & \vspace{-\baselineskip}
\bq \nonumber
\sum^{n}_{j-N}k(n+\omega,j+\omega)f(j+\omega,x(j+\omega))
=\sum^{n}_{j-N}k(n,j)f(j,x(j))=(Tx)(n), \eq so $Tx\in
{\calA}_\omega$. Next we require that
  $T:\Cone \to {\calA}_\omega$ is {continuous} and compact.


Continuity is readily established; consider the compactness of $T$.  Let $\Omega$ be a bounded set
in $C$ (there exists $r>0$ with $|x|_\omega\le r$ for
 all $x=\{x(n)\}_{n\in
\Zset}\in\Omega$). Since $f(j,\cdot)$ is continuous on $\Rset_+$ for each $j\in \{1, \ldots, \omega\}$, there exists $M_r>0$ such that $|f(j,u)|\le M_r$ for all $u\in [0,r]$ and
$j\in \{1, \ldots,  \omega\}$. For $n\in \{1, \ldots, \omega\}$ and $x=\{x(n)\}_{n\in
\Zset}\in \Omega$, we have
\[
|(Tx)(n)|\le \sum^n_{j-N}k(n,j)|f(j,x(j))| \le M_r\max_{n\in \{1,
\ldots, \omega\}}\sum^\omega_{j=1-N}k(n,j),
\]
  so $T\Omega$ is a bounded subset of ${\calA}_\omega$,
 and $\overline{T\Omega}$ is compact. Thus, $T$ is compact.\hfill $\Box$
\begin{proposition}\label{propn:thm-from-Kras}
Suppose (from the listed {\sc Assumptions} on $k$, $f$, $\psi$, ${ {\blue\xi}}$) that
(\ref{eq-for-k}),  (\ref{eq-for-f1}), (\ref{eq-for-f1-a+}),  (\ref{eq-for-f1-b})
 (\ref{eq-for-f3-a}),  (\ref{eq-for-f3-b}),  (\ref{eq-for-f3-c}).
  (\ref{eq-for-f4}) hold and, given (\ref{eq-for-f5-6}), suppose (where $q$ is
the function in (\ref{eq-for-f3-b}))
\bsq\label{subeq-for-Kras}
\bq \label{eq-for-f5}
{\kappa}_{\min}(q) > 0;
\eq
 there exists $M \in (0,1)$ with
\begin{equation}\label{eq-for-f6}
{M}/{{ {\blue\xi}}(M)}\le {a}_0 {{\kappa}_{\min}(q)}/{{
\kappa}_{\max}(q)};
\end{equation}
 there exists $\alpha >0$ with
\begin{equation}\label{eq-for-f7}
\alpha>{\kappa}_{\max}(q)\psi(\alpha);
\end{equation}
and  there exists $\beta>0$, $\beta\ne \alpha$, with
\begin{equation}\label{eq-for-f8}
\beta<{a}_0{\kappa}_{\min}(q)\psi(M\beta).
\end{equation}
\esq Then, (\ref{eq-nvd-1}) has at least one positive periodic solution $x  \in
{\calA}_\omega(\Zset)$, with \bq \label{A-or-B}
 0<\min\{\alpha, \beta\}<|x|_\omega<\max\{\alpha,\beta\};
\,
x(n)\ge M\min\{\alpha,\beta\} \hbox{ for } n\in \Zset.
\eq
\end{proposition}
{\bf Proof}. For $a \in \Rset$ let  $\Omega_a=\{x\in {\calA}_\omega:|x|_\omega<a\}$.
 To apply Krasnosel${}^\prime\!$ski\u\i 's fixed
point theorem, we shall show that the following conditions
hold:
%% % \vspace{-\baselineskip}
\begin{quote}
 (a)\quad $T:\Cone\to \Cone$,\\
(b)\quad $|Tx|_\omega\le |x|_\omega$ for $x\in
\Cone\cap\partial \Omega_\alpha={\setS}_\alpha$ and\\
 (c) \quad $|Tx|_\omega\ge |x|_\omega$ for $x\in \Cone\cap\partial
\Omega_\beta={\setS}_\beta$.
\end{quote}
(a) Let $x \in \Cone$. Then (\ref{eq-for-f3}) implies that, for $n\in \{1, \ldots, \omega\}$,
% % \vspace{-\baselineskip}
\begin{eqnarray}\label{eq-Tx}% % \vspace{-.5\baselineskip}
|(Tx)(n)|&\le& \psi(|x|_\omega)\max_{n\in\{1, \ldots,
J\}}\sum^n_{j-N}k(n,j)q(j) ={
\kappa}_{\max}(q)\psi(|x|_\omega).
\end{eqnarray}
On the other hand, since $x\in \Cone$, we have $x(n)\ge M|x|_\omega$ for $n\in \Zset$, and
therefore (\ref{eq-for-f3}), (\ref{eq-for-f4}), (\ref{eq-Tx}) and (\ref{eq-for-f6}) give for $n\in
\{1, \ldots, \omega\}$,
\begin{eqnarray*}
(Tx)(n)&\ge & {a}_0\sum^n_{j-N}k(n,j)q(j)\psi(x(j))\ge
{a}_0\psi(M|x|_\omega)\sum^n_{j-N}k(n,j)q(j)\\ &\ge& {a}_0{
{\blue\xi}}(M)\psi(|x|_\omega)\sum^n_{j-N}k(n,j)q(j)\ge
 {\kappa}_{\min}({q}){a}_0{ {\blue\xi}}(M)\psi(|x|_\omega)\\
&\ge&\{{\kappa}_{\min}({q})/{\kappa}_{\max}({q})\}\,
 {a}_0{ {\blue\xi}}(M)|Tx|_\omega\ge M|Tx|_\omega;
\end{eqnarray*}
thus, $Tx\in \Cone$ and (a) holds.

To establish (b), let $x\in \Cone\cap\partial \Omega_\alpha={\setS}_\alpha$. In this case,
$|x|_\omega=\alpha$ and $x(n)\ge M\alpha$ for all $n\in \Zset$. Now for $n\in \{1, \ldots,
\omega\}$, we have
\[
|(Tx)(n)|\le \psi(|x|_\omega)\sum^n_{j-N}k(n,j)q(j)\le
\psi(\alpha) {\kappa}_{\max}({q}).
\]
This, with (\ref{eq-for-f7}), yields $ |Tx|_\omega \le
\psi(\alpha){\kappa}_{\max}({q})<\alpha =|x|_\omega$;
 thus (b) is satisfied.

To establish (c), let $x\in \Cone\cap\partial \Omega_\beta=
{\setS}_\beta$. Then $|x|_\beta=\beta$ and $M\beta\le x(n)\le \beta$
for all $n\in \Zset$. Now, for $n\in \{1, \ldots, \omega\}$, it
follows from (\ref{eq-for-f3})  that \bq |(Tx)(n)|\ge {a}_0
\sum^n_{j-N}k(n,j)q(j)\psi(x(j)) \geq
{a}_0{\kappa}_{\min}({q})\psi(M\beta), \eq which, together with
(\ref{eq-for-f8}), yields
\[
\hbox{ $(Tx)(n)\ge {a}_0{\kappa}_{\min}({q})\psi(M\beta)>\beta=|x|_\omega $ for $n\in
\{1, \ldots, \omega\}$},
\]
and thus $|Tx|_\omega>|x|_\omega$; that is, (c) holds.

Applying Krasnosel${}^\prime\!$ski\u\i 's Theorem \ref{krans-fixed}, we conclude that
(\ref{eq-nvd-1}) has a solution $x\in {\calA}_\omega$  with $x\in
\Cone\cap(\overline{\Omega_\alpha} \!\setminus\! \Omega_\beta)$ if $\beta<\alpha$, or $x\in
\Cone\cap(\overline{\Omega_\beta}\!\setminus\!\Omega_\alpha)$ if $\alpha<\beta$. Finally, we note
that $|x|_\omega\ne \alpha$ and $|x|_\omega\ne \beta$. In fact, if $|x|_\omega=\alpha$, then from
$x=Tx$ we have $ \alpha=|x|_\omega=|Tx|_\omega\le \psi(\alpha){\kappa}_{\max}({
q})<\alpha=|x|_\omega$ (which is a contradiction). A similar argument shows that $|x|_\omega\ne
\beta$. This completes the proof.\hfill $\Box$
\begin{example}  Consider (compare (\ref{ex-inteq-1})) the non-linear system
\begin{equation}\label{eq-exap}
% % \vspace{-.5\baselineskip}
x(n)=\sum^n_{j-N}k(n,j){g}(j)[x(j)]^\gamma \quad \hbox{ for $n\in
\Zset$},
\end{equation}
where $0<\gamma <1$.  Assume that $k$ satisfies
(\ref{eq-for-k}) and, for $\omega \in \Nset$,
 that
\begin{equation}\label{eq-for-exap-1}
{g}  \in {\calA}_\omega,\
  {g}(n)\ge 0 ,  \hbox{  and  }\kappa_{\min}(g) >0
\end{equation}
(with ${\kappa}_{\min}(g)$,  $ {\kappa}_{\max}(g)$ defined by
(\ref{eq-for-f5-6})). Then (\ref{eq-exap}) has at least one positive periodic solution $x  \in
{\calA}_\omega$, where with
 $ M={\half}\left({\kappa}_{\min}({g})/{
\kappa}_{\max}({g})\right)^{\frac{1} {1-\gamma}} $,
$
\beta={\half}M^{\frac{\gamma}{1-\gamma}}({\kappa}_{\min}({g}))^{\frac{1}{1-\gamma}} $
$<|x|_\omega<2({\kappa}_{\max}({g}))^{\frac{1}{1-\gamma}}$ $ =$ $ \alpha$
 and $ x(n)\ge M\beta $ for $n\in \Zset$.

To see that the above result is true, we apply Proposition
\ref{propn:thm-from-Kras} with  $q=g$,
 $f(n,u)=q(n)u^\gamma$, $\psi(u)=u^\gamma$, ${a}_0=1$ and ${ {\blue\xi}}(u)=u^\gamma$.
Now, the continuity and periodicity conditions on $k$, $f$ in  Proposition
\ref{propn:thm-from-Kras} and continuity and monotonicity
 properties of $\psi$ are clearly satisfied;
we verify (\ref{subeq-for-Kras}).
Now (\ref{eq-for-f5}) is satisfied by assumption.
 To establish (\ref{eq-for-f6}) for ${ {\blue\xi}}$, we note that
\[
\frac{M}{{ {\blue\xi}}(M)}=M^{1-\gamma}=\left({\half}\right)^{1-\gamma} \frac{{
\kappa}_{\min}({g})} {{\kappa}_{\max}({g})}
\le
\frac{{\kappa}_{\min}({g})}{{\kappa}_{\max}({g})} ={a}_0\frac{{
\kappa}_{\min}({g})}{{\kappa}_{\max}({g})}.
\]
Also (\ref{eq-for-f7}) holds since $\alpha/(\psi(\alpha))=\alpha^{1-\gamma}=2^{1-\gamma}{
\kappa}_{\max}({g})
>{\kappa}_{\max}({g})$.
 Since
${\beta}/{\psi(M\beta)}=\{{1}/{M^\gamma}\}\beta^{1-\gamma}
=\{{1}/{M^\gamma}\}\left({\half}\right)^{1-\gamma}M^\gamma {
\kappa}_{\min}({g})=\left({\half}\right)^{1-\gamma}{\kappa}_{\min}({g})<
 {a}_0{\kappa}_{\min}({g})$,
 (\ref{eq-for-f8}) is also true. Now apply
{Proposition} \ref{propn:thm-from-Kras}.
\end{example}
With additional conditions on $k$ and $f$ in (\ref{eq-nvd-1}),
applications  of    {Proposition} \ref{propn:thm-from-Kras} will yield additional
positive periodic solutions of (\ref{eq-nvd-1}). For completeness
we provide one  result on multiple solutions.

\begin{proposition}\label{propn:thm-multiple}
Suppose (from the listed {\sc Assumptions} on $k$, $f$, $\psi$, ${
{\blue\xi}}$) that (\ref{eq-for-k}), (\ref{eq-for-f1}),
(\ref{eq-for-f1-a+}), (\ref{eq-for-f1-b}) (\ref{eq-for-f3-a}),
(\ref{eq-for-f3-b}), (\ref{eq-for-f3-c}), (\ref{eq-for-f4}) hold,
and also that (\ref{eq-for-f6}) is satisfied.  Also, given
(\ref{eq-for-f5-6}), suppose (where $q \in {\mathcal A}_\omega$ is
the function in (\ref{eq-for-f3-b})) that there are constants
$0<\gamma_0<\gamma_1<\gamma_2$ with (i)
$\gamma_0<{a}_0{\kappa}_{\min}({q})\psi(M\gamma_0)$, with (ii)
$\gamma_1>{\kappa}_{\max}({q})\psi(\gamma_1)$ and with (iii)
$\gamma_2<{a}_0{\kappa}_{\min}({q})$ $\psi(M\gamma_2)$. Then
(\ref{eq-nvd-1}) has at least two positive periodic solutions
$x_1=\{x_1(n)\}_{n\in \Zset}$, and $ x_2=\{x_2(n)\}_{n\in \Zset}
\in {\calA}_\omega$ with
$0<\gamma_0<|x_1|_\omega<\gamma_1<|x_2|_\omega<\gamma_2$,
$x_1(n)\ge M\gamma_0$ and $x_2(n)\ge M\gamma_1$ for $n\in \Zset$.
\end{proposition}
{\bf Proof}. {The existence of $x_1$ follows from  Proposition
\ref{propn:thm-from-Kras}  with $\alpha=\gamma_1$ and
$\beta=\gamma_0$, and the existence of $x_2$ follows from  Proposition
\ref{propn:thm-from-Kras} with $\alpha=\gamma_1$ and
$\beta=\gamma_2$.} \hfill $\Box$
%% % \vspace{-\baselineskip}
\subsection{Non-negative periodic solutions via the non-linear alternative}
%% % \vspace{-\baselineskip}
We use the non-linear alternative to obtain an existence result for
(\ref{eq-nvd-1}).
\begin{proposition}\label{propn:thm-nonli-alterna}
Suppose (from the listed {\sc Assumptions} on $k$,  $f$,
 $\psi$, and  $q \in {\mathcal A}_\omega$)  that (\ref{eq-for-k}),
 (\ref{eq-for-f1}), (\ref{eq-for-f1-a+}),  (\ref{eq-for-f1-b}),
  (\ref{eq-for-f3-a}),  (\ref{eq-for-f3-b})
are satisfied, and
in addition assume that, with
 $\kappa_{\max}({q})$ as in (\ref{eq-for-f5-6}),
there exists ${\alpha > 0}$  with
\begin{equation}\label{eq-na-4}
\alpha>{\kappa}_{\max}({q})\psi(\alpha).
 \end{equation}Then (\ref{eq-nvd-1}) has a non-negative solution $x
 \in {\calA}_\omega$ with $|x|_\omega <\alpha$.
\end{proposition}
{ \bf Proof}. Any non-negative solution (that is, with $x(n) \geq 0$ for $n \in \Zset$) of
(\ref{eq-nvd-1}) is a solution of
\begin{equation}\label{eq-nvd-1-natural}
x(n)=\sum^n_{j-N}k(n,j)f_\natural(j,x(j)),
 \quad N \in  \Nset, \quad x(n)\in \Rset.
\end{equation}
where
 \bq f_\natural (n,u) = f(n, |u|).
 \eq
  By construction, and the assumptions on $f$,
 $f_\natural(n,\cdot)$ is continuous on $\Rset$ for each $n\in \Zset$
 and
 \begin{equation}\label{eq-na-1}
 \hbox{$f_\natural(n+\omega,u)=f_\natural(n,u)\ge 0$ for all $n\in \Zset$ and $u\in \Rset$}.
 \end{equation}
 We apply the alternative theorem to (\ref{eq-nvd-1-natural}),  setting
  \bq
 T_\natural x(n) =\sum^n_{j-N}k(n,j)f_\natural(j,x(j))\eq and taking ${X}_{\|\cdot \|}$ to be
 ${A}_\omega$ with norm $|\cdot |_\omega$  and
$\setU=\{x\in {A}_\omega\ \, \big| \, \ |x|_\omega<\alpha\}$. It is
easy to see that the operator $T_\natural$ on ${A}_\omega$ (where
$(T_\natural x)(n)=\sum^n_{j-N}k(n,j)f_\natural(j,x(j))$ for $n\in
\Zset$) maps ${A}_\omega$ to ${A}_\omega$ by conditions
(\ref{eq-for-k}) and by the assumed properties of $f$. In addition,
 these
properties guarantee that $T_\natural:{A}_\omega \to
{A}_\omega$ is continuous and compact. Let $x_\lambda
 \in {A}_\omega$ be any solution of
\[
x_\lambda(n)=\lambda
\{\sum^n_{j-N}k(n,j)f_\natural(j,x_\lambda(j)) \}, \quad n\in
\Zset,
\]
for $0<\lambda<1$. Notice that (\ref{eq-for-k}) and (\ref{eq-for-f1-all})
  imply $x_\lambda(n)\ge 0$ for all $n\in \Zset$. Now for
$n\in \{1, \ldots, \omega\}$, we have $ |x_\lambda(n)|\le
\sum^n_{j-N}k(n,j)q(j)\psi(x_\lambda(j))\le
\psi(|x_\lambda|_\omega)\sum^n_{j-N}k(n,j)q(j)\le
{\kappa}_{\max}({q})\psi(|x_\lambda|_\omega) $ and
therefore
\begin{equation}\label{eq-na-5}
|x_\lambda|_\omega \le {\kappa}_{\max}({q}) \psi(|x_\lambda|_\omega).
\end{equation}
In addition, (\ref{eq-na-4}) and (\ref{eq-na-5}) implies that $|x_\lambda|_\omega\ne \alpha$. Apply
the non-linear alternative: since we have shown that option (ii) (in the statement of that theorem)
cannot occur, we deduce that (\ref{eq-nvd-1}) has a solution $x=\{x(n)\}_{n\in \Zset}\in
{\calA}_\omega$ with $x(n)\ge 0$ for all $n\in \Zset$.
   Further, $|x|_\omega<\alpha$. (We have $|x|_\omega \le \alpha$ by the non-linear alternative and
$|x|_\omega\ne \alpha$ by an argument similar to that used to show that $|x_\lambda|_\omega\ne
\alpha$.) \hfill $\Box$
\begin{remark}{\small
If there is $n_0\in \{1, \ldots, \omega\}$ such
that $k(n_0,n_0)f(n_0,0)>0$, then the \emph{null}
periodic sequence,
 $\{y(n) = 0\}_{n\in \Zset}$, is not a solution of
(\ref{eq-nvd-1}). Then, the non-negative solution
$x$ with $|x|_\omega<\alpha$, in
Proposition \ref{propn:thm-nonli-alterna}, satisfies $|x|_\omega>0$.
}\end{remark}
% % \vspace{-.5\baselineskip}
\subsection{Non-negative periodic solutions via the Leggett-Williams theorem}
%% % \vspace{-.5\baselineskip}
It is possible to use the Leggett-Williams fixed point theorem to
establish the existence of non-negative periodic
solutions of (\ref{eq-nvd-1}).
\begin{proposition}\label{propn:thm-from-LW}
Suppose (from the listed {\sc Assumptions} on $k$, $f$, $\psi$,
and  $q \in {\mathcal A}_\omega$)  that {(\ref{eq-for-k}),
(\ref{eq-for-f1}), (\ref{eq-for-f1-a+}),  (\ref{eq-for-f1-b}), (\ref{eq-for-f3})
 hold;
suppose, further, that ${ {\chi}}$ satisfies (\ref{eq-LW-1}). In addition,
 assume  that
\begin{equation}\label{eq-LW-2}
{\kappa}_{\min}({q})
 >0;
\end{equation}
that there exists $R > r>0$ such that $\displaystyle \frac{{ {\chi}}(u)}{u}$ is nonincreasing on
$(0,r)$ and \bsq
\begin{equation}\label{eq-LW-3}
 r<{\kappa}_{\min}({q}){ {\chi}}(r),
\end{equation}
\begin{equation}\label{eq-LW-5}
R>\psi(R){\kappa}_{\max}({q}).
\end{equation}
\esq}Then (\ref{eq-nvd-1}) has a non-negative solution $x \in {\calA}_\omega $ with $r\le
|x|_\omega <R$.
\end{proposition}
  %%error corrected%%
  {\bf Proof}. In  { Theorem \ref{thm-LW} (due to Leggett and Williams)}, take
 ${\mathcal X}= {\calA}_\omega$ and
 $\Cone=\{x\in {\calA}_\omega \, \big| \, x(n)\ge 0\}$ for
$n\in \{1, \ldots, \omega\}$, \ie $\{x\in {\calA}_\omega \, \big| \, x(n)\ge 0$ for $n\in \Zset\}$.
Clearly, $\Cone$ is a cone in ${\calA}_\omega$. Let $u_0 =\{u_0(n)\}_{n\in \Zset}$ with $u_0(n)= 1$
for all $n\in \Zset$ and note that
 $\Cone(u_0)=\{x\in \Cone:$ there exists $\lambda>0$
with $x(n)\ge \lambda$ for $n\in \{1, \ldots, \omega\}$. From
(\ref{eq-for-k}) and (\ref{eq-for-f1}), it follows that $T:\Cone\to
\Cone$   (where $(Tx)(n)= \sum^n_{j-N}k(n,j)f(j,x(j))$ for $n\in
\Zset$ and $x \in \Cone$) is continuous and compact.

To apply { Theorem} \ref{thm-LW}, we first show
\begin{equation}\label{eq-LW-6}
\hbox{ $|Tx|_\omega\le |x|_\omega$ for $x\in S_R=\{x\in \Cone
\, \big| \, |x|_\omega=R\}$.}
\end{equation}
If $x\in S_R$, then $|x|_\omega=R$ and  for $n\in \{1, \ldots,
\omega\}$, since $$\DSS \sum^n_{j-N}k(n,j)q(j)\psi(x(j)) \le
\psi(|x|_\omega)\sum^n_{j-N}k(n,j)q(j),$$  we have
\bq\label{eq-LW-6-1} (Tx)(n)\le  \psi(R){\kappa}_{\max}({q}). \eq
This, together with (\ref{eq-LW-5}), gives
\begin{equation}\label{eq-LW-7}
|Tx|_\omega \le \psi(R){\kappa}_{\max}({q})<R=|x|_\omega,
\end{equation}
which implies that (\ref{eq-LW-6}) is true. Next we show
\begin{equation}\label{eq-LW-8}
Tx\not \leqslant  x \quad \hbox{ for $ x\in {\setS}_r\cap \Cone(u_0)$,
where $\setS_r=\{x\in \Cone \, \big| \, |x|_\omega=r\}$}.
\end{equation}
To show this, let $x=\{x(n)\}_{n\in \Zset}\in \setS_r\cap C(u_0)$, hence $|x|_\omega=r$ and $r\ge
x(n)>0$ for $n\in \{1, \ldots, \omega\}$. Now, for $n\in \{1, \ldots, \omega\}$, we have
\begin{eqnarray*}
(Tx)(n)\ge \sum^n_{j-N}k(n,j)q(j)\frac{{
{\blue\xi}}(x(j))}{x(j)}x(j) \ge \frac{{
{\blue\xi}}(r)}{r}\sum^n_{j-N}k(n,j)q(j)x(j).
\end{eqnarray*}
Let $n_0\in \{1, \ldots, \omega\}$ be such that $\min_{n\in \{1, \ldots, \omega\}}x(n)=x(n_0)$ and
this together with the previous inequality yields, for $n\in \{1, \ldots, \omega\}$,
\[
(Tx)(n)\ge \frac{{ {\blue\xi}}(r)}{r}x(n_0)\sum^n_{j-N}k(n,j)q(j)\ge
\left(\frac{{ {\blue\xi}}(r)}{r}{\kappa}_{\min}({q})\right)x(n_0).
\]
By (\ref{eq-LW-3}) we obtain $(Tx)(n)>x(n_0)$ for $n\in \{1, \ldots, \omega\}$ and
$(Tx)(n_0)>x(n_0)$, which means that (\ref{eq-LW-8}) is true.

Applying  { Theorem} \ref{thm-LW}, we conclude that (\ref{eq-nvd-1})
has a non-negative periodic solution $x=\{x(n)\}_{n\in \Zset}\in \Cone$
with $r\le |x|_\omega \le R$. Note that $|x|_\omega\ne R$ since if
$|x|_\omega = R$, then from $x=Tx$, (\ref{eq-LW-6-1}) and
(\ref{eq-LW-7}), we have $ R=|x|_\omega=|Tx|_\omega \le
\psi(R){\kappa}_{\max}({q})<R=|x|_\omega$, a contradiction.
 \hfill $\Box$
\begin{example}  Consider the following discrete non-linear system
\begin{equation}\label{eq-examp-2}
x(n)=\sum^n_{j-N}k(n,j){g}(j)([x(j)]^\alpha+[x(j)]^\beta), \quad
n\in \Zset,
\end{equation}
with $0<\alpha<1$, $\beta\ge 1$ and (\ref{eq-for-k}) satisfied. In
addition assume
\begin{equation}\label{eq-examp-2-1}
\hbox{ ${g}\in A_w$ with ${g}(n+\omega)={g}(n)\ge 0$ for all $n\in \Zset$},
\end{equation}
\begin{equation}\label{eq-examp-2-2}
\kappa_{\min}({g})
>0
\quad \hbox{ and }\quad
 \kappa_{\max}({g}) <{\half}.
\end{equation}
 Then (\ref{eq-examp-2}) has a non-negative solution
$x\in {\calA}_\omega$ with
\[ \big(\half \kappa_{\min}(g)\big)^{\frac{1}{1-\alpha}} \equiv
\left({\half}\min_{n\in \{1, \ldots,
\omega\}}\sum^n_{j-N}k(n,j){g}(j)\right)^{\frac{1}{1-\alpha}}\le
|x|_\omega<1.
\]

To establish this, let $f(n,u)={g}(n)[u^\alpha+u^\beta]$, $\psi(u)=[u^\alpha+u^\beta]$, ${
{\blue\xi}}(u)=u^\alpha$, and $q(n)={g}(n)$ with $ r= \big(\half
\kappa_{\min}(g)\big)^{\frac{1}{1-\alpha}}$ and $R=1$. Obviously, (\ref{eq-for-f1}),
(\ref{eq-LW-1}) and (\ref{eq-LW-2}) hold. To establish (\ref{eq-LW-3}), notice that $
\frac{r}{{ {\blue\xi}}(r)}=r^{1-\alpha}={\half} \kappa_{\min}({g})
< \kappa_{\min}({g})$.
Also, ${{ {\blue\xi}}(u)}/{u}={1}/{u^{1-\alpha}}$ is non-increasing on $[0,r]$ as $0<\alpha<1$, and
finally (\ref{eq-LW-5}) holds with $R=1$. We can now
 apply  Proposition  \ref{propn:thm-from-LW}.

{
Consider (\ref{eq-examp-2-2}) for varying $\alpha \in (0,1)$, $\beta\geq 1$.
We can denote $x(n)$ by $x_{\alpha,\beta}(n)$, to
indicate dependence on the values $\alpha$, $\beta$.
 The
complexity of (\ref{eq-examp-2}) as an equation for $x_{\alpha,\beta}(n)$ when $\{
x_{\alpha,\beta}(n-\ell)\}_{\ell=1}^N$ are known increases with increasing $\beta$:
for example, if $\alpha = \half$ and $1< \beta \in \Nset$ then
(\ref{eq-examp-2}) can be viewed as a polynomial equation of degree
$2\beta$ for $\sqrt{x(n)}=
\sqrt{ x_{\alpha,\beta}(n)}$.
 As  our
 discussion is confined to the unit ball $|x|<1$, the periodic
solution $x_{\alpha,\beta}(n)$ that is addressed in the result
is located in this unit ball and satisfies $0<r\leq |x_{\alpha,\beta}|_{\omega}<1$
 for any  $\alpha \in (0,1)$, $\beta>1$.}
\end{example}
    % % \vspace{-1.25\baselineskip}
   \section{Discretization}
\label{sec-numerics}
%\vspace{-\baselineskip}
We develop further the discussion of the
discretization of (\ref{eq-contin-2}).
We first pause to
indicate the general nature of results found concerning
(\ref{eq-contin-2}), in the literature and refer to the application of
 Krasnosel${}^\prime\!$ski\u\i 's fixed
point theorem in  {\cite[p.~139]{aga01}}. We shall require
 continuity of $\ksf$ and $\f$. With such modifications
 we have the following version of  {\cite[Theorem 4.4.1]{aga01}}:
\begin{thm}\label{agar} Given (\ref{eq-contin-2}) with
$0 < \tau \in \Rset_+$, suppose (i) that $\ksf$ satisfies
 (\ref{ie-cond-k}) of {\assume} \ref{cts-case-k+f} (ii) $\f$ satisfies
 (\ref{ie-cond-f}) of {\assume} \ref{cts-case-k+f} and (iii) $\psi$,
 $\qsf$, $m$, $\ksf$ and $\f$ collectively satisfy
 (\ref{collected}). Suppose (iv) that (\ref{subeq-for-Kras}) hold when
 $\kappa_{\max, \min}({q})$ are replaced by $\widehat \kappa_{\max,
 \min}({\q}) \equiv \widehat \kappa^\tau_{\max\min}({\q})$ with
% % \vspace{-.5\baselineskip}
 \bq
\label{kappa-cond}
 \widehat \kappa^\tau_{\max}({\q}) = \inf_{t \in [0,\varpi]}\int_{t-\tau}^t
 \ksf(t,s) \q(s) \ds,\
\widehat \kappa^\tau_{\min}({\q}) =  \sup_{t \in [0,\varpi]}\int_{t-\tau}^t
 \ksf(t,s) \q(s) \ds.
\eq
Then there exists at least one $\varpi$-periodic solution
$\x$ of (\ref{eq-contin-2}) such that
\bq \label{int_eq-A-or-B}
 0< \!
\min\{\alpha, \beta\}\! < \!
\sup_t|\x(t)|\! < \!
\max\{\alpha,\beta\};
\,
\x(t)\ge M\min\{\alpha,\beta\} \hbox{ for } t\in \Rset.
\eq
\end{thm}% % \vspace{-\baselineskip}
A  $\varpi$-periodic solution
$\x$  is a solution      that satisfies $\x(t+\varpi) = \x(t)$ for all $t\in \Rset $.
\begin{proposition}\label{propn:replace-tau} Suppose that $0< \tau \in \Rset_+$ and that
 $\widetilde\tau \equiv \widetilde\tau(\varepsilon)$
with $ 0< \tau -\varepsilon < \widetilde\tau \leq \tau$.
 If the conditions  of Theorem \ref{agar} apply, they apply also when $\tau$ is
replaced by $\tilde \tau$ provided that $\varepsilon$ is sufficiently small --- and the conclusion
of Theorem \ref{agar} applies to a $\varpi$-periodic solution \bq \label{perturbed-ie} \tilde\x(t)
= \int_{t-\tilde \tau}^t \ksf(t,s)\f(s,\tilde\x(s))\ds. \eq Further,  provided that $\varepsilon$
is sufficiently small, if the conditions of   Theorem \ref{agar} apply to (\ref{perturbed-ie})
 they establish the existence of a $\varpi$-periodic solution
$\x$ of (\ref{eq-contin-2}).
\end{proposition}
 {\bf Proof:} We are concerned only
with the effect of replacing
 $\kappa_{\max,
\min}({q})$  in (\ref{subeq-for-Kras}) by  $\widehat
\kappa^{\wt\tau}_{\max, \min}({q})$ instead of
 by $\widehat \kappa^\tau_{\max, \min}({q})$.
 Under the assumptions, the integrals $\int_{t'}^t
 \ksf(t,s) \q(s) \ds$ depend continuously on $t' \in [\tau,
 \tau+\varepsilon]$, uniformly in $t$.  The integral with lower limit
 $t-\tau$ and that with lower limit $t-\widetilde\tau$ are therefore
 arbitrarily close (uniformly in $t$) for correspondingly small
 $\varepsilon$.  Indeed,
$|\int_{t-\tau}^t
 \ksf(t,s) \q(s) \ds - \int_{t-\tilde\tau}^t
 \ksf(t,s) \q(s) \ds| \leq$
$ \varepsilon \sup_{t \in \Rset} \sup_{s\in[{t-\tau},
{t-\tilde\tau}]} |\ksf(t,s)|$ $ \sup_{s\in[{t-\tau},
{t-\tilde\tau}]} |\g(s)| $ and hence is bounded by $ \varepsilon
\sup_{t \in [0,\varpi]} \sup_{s\in[{t-\tau}, {t-\tilde\tau}]}
|\ksf(t,s)|$ $ \sup_{s\in[{t-\tau}, {t-\tilde\tau}]} |\g(s)|$.
Thus the pair ($\widehat \kappa^\tau_{\min}({q})$,
   $\widehat \kappa^{\wt\tau}_{\min}({q})$) and the pair
 ($\widehat \kappa^\tau_{\max}({q})$,
  $\widehat \kappa^{\wt\tau}_{\max}({q})$) are in each case
arbitrarily close for $\tau-\wt \tau$
sufficiently close. \hfill $\Box$


We shall exploit the preceding result, which indicates (in broad
terms) that one can consider $\tau > 0$ to be replaced by a nearby
$\tilde \tau$. Assume that
\bq\label{request}
\tau - \varepsilon < \tilde\tau \leq \tau, \hbox{ and } \tilde\tau = {N}_{\tilde\tau} \hfr
\hbox{ where }  {N}_{\tilde\tau} \in \Nset,
\eq
 for sufficiently small $\varepsilon > 0$ that  Proposition   \ref{propn:replace-tau} applies.
The process for ensuring (\ref{request}) is discussed later (\S \ref{sec:Approximate}).
  We consider discretization of
\begin{equation}\label{eq-contin-2-bis}
\x(t) = \int_{t-\tilde\tau}^t \ksf(t,s) \f(s,\x(s)) {\rm d}s,
 \quad \hbox{with } \tilde\tau >0,\  \x(t)\in \Rset,
\end{equation}
(which is (\ref{eq-contin-2}), with $\tau$ replaced by $\tilde\tau$)
using quadrature.
 When we discretize (\ref{eq-contin-2-bis}), we
 seek (for $ {N}_{\tilde\tau} \in \Zset_+$,
$\hfr = \tilde\tau/{N}_{\tilde\tau}$) a discrete system of the
type
\begin{equation}\label{eq-nvd-X}
\wt x(n)=\sum^n_{j-{N}_{\tilde\tau}}\hfr W_{n-j}
\ksf(n\hfr,j\hfr)\f(j\hfr, \wt x(j)),
  \quad \wt x(n)\in \Rset,
\end{equation}
with $ \wt x(j) \approx \x(j\hfr)$. Our aim is to examine conditions
on the integral equation
 (\ref{eq-contin-2}) which allow the analysis, as in \cite{aga00},
of periodic solutions, and to discuss whether one can apply, to
(\ref{eq-nvd-X}), the discrete analysis developed earlier.

 We assume
that $\ksf$ satisfies conditions (for example, those in Theorem
\ref{agar})  that guarantee the existence of a $\varpi$-periodic
solution $\x$ of (\ref{eq-contin-1}), on the assumption that
$\varepsilon$ is sufficiently small that the conditions continue to
be satisfied when $\tau$ is replaced by $\tilde\tau$ in
(\ref{request}); see  Proposition  \ref{propn:replace-tau}. To simulate
Theorem \ref{agar} (and   similar results) using discrete equations,
we present in \S \ref{sec-Quad} {(see also \cite{TR2007-1})} various
quadratures that can be used to discretize (\ref{eq-contin-2}),
via (\ref{eq-contin-2-bis}),
while ensuring the existence of periodic solutions of (\ref{eq-nvd-X}).
\begin{remark}{\small \label{rem:general}{\small
 From a purely computational perspective, one might discretize
(\ref{eq-contin-2}) using the given $\tau$,  perhaps by
implementing an error control mechanism, but without any
attempt to conserve periodic equations.
It is to be expected that the analysis carried out here will yield insight into
approximations obtained by such means.}
}\end{remark}
 % % \vspace{-\baselineskip}
\subsection{Quadrature rules: basic properties and examples}
\label{sec-Quad}
 % % % \vspace{-1\baselineskip}
 We restrict attention to  simple approximations
associated with sampling the integrand at \emph{equally-spaced abscissae}.
Though we need to discretize
integrals over $[t-\tau,t]$ (first replacing the integral by that
over  $[t-\tilde\tau,t]$), our quadrature rule is defined by  the approximation
for an integral over $[0,1]$, of the type
  \bq \label{basic-rule}
   \int_0^1 \psi(s) \ds \approx
\sum_{\ell=0}^N w_\ell\, \psi(\ell \, h)  =:
{\mathcal Q}_{1/N}(\psi) , \quad h = 1/N,
 \eq
with the convention that an affine change of variable is used to
secure,  for arbitrary finite $a < b$, the related, or \emph{induced}, approximation
% % \vspace{-.75\baselineskip}
 \bq \label{induced}
{\mathcal Q}^{[a,b]}_{1/N}( \psi)  :=
\sum_{j=0}^N \{(b-a) w_j\}
 \psi(a +j {h_\natural})  \approx \int_a^b \psi(s) \ds, \quad h_\natural= (b-a)/N
\eq ($ {\mathcal Q}_{1/N}(\psi)$  is  ${\mathcal
Q}^{[0,1]}_{1/N}(\psi)$).
The sum ${\mathcal Q}^{[a,b]}_{(b-a)/N}(\psi)$ in (\ref{induced}) is
the \emph{quadrature ``approximation''} to $\int_a^b\psi(s) \ds$.  We
define the \emph{quadrature ``rule''} ${\mathcal Q}$ as the map that,
with $N \in \Nset$, assigns to finite $a,b \in \Rset$ and $ \psi \in
R[a,b]$ the value (\ref{induced}).  We restrict the \emph{permitted} quadrature
(\ref{basic-rule}) to formulae satisfying
  \bq \label{basic-cond}
\sum_{\ell=0}^N w_\ell = 1, \hbox{ and }
 w_j \geq 0 \hbox{ for } j \in \{0,1, \ldots , N\}.
  \eq
Certain Newton-Cotes rules,  the 2-point Radau rule, the composite versions, and
 classical Romberg and certain Gregory rules  provide examples
 (see \cite{baker,Nature_of,TR2007-1}).



 Both (\ref{basic-rule}) and (\ref{induced})  define the quadrature
 rule ${\mathcal Q}$, but  a rule is defined uniquely by the \emph{``weights''}
$\{w_\ell\}_0^N$ and the abscissae  in (\ref{basic-rule}).
Since some weights $w_j$ may vanish, the values of $h$ and $N$ are
 fixed by requiring that (\ref{basic-rule}) with $h\in (0,1]$ cannot be rewritten
using a larger value  $h^{*} > h$  (smaller value $ N^{*} < N  \in \Nset$
where $h^{*} = 1/N^{*}$),
as an approximation $ \sum_{\ell=0}^{N^{*}} w^{*}_\ell\, \psi(\ell \, h^{*})$.
A quadrature \emph{family} $\mathfrak Q$ (a collection of rules) is
defined by (\ref{basic-rule}) for $h \in \setH^{\mathfrak Q} \subset (0,1]$.
 We assume
\bq \label{Indexing}
\setH^{\mathfrak Q} := \{h_1, h_2, h_3, \ldots \},
\quad \setN^{\mathfrak Q} := \{N^{[1]}, N^{[2]}, N^{[3]},  \ldots \} \
(h_\ell  = 1/N^{[\ell]}),
 \eq
are defined by a monotonically decreasing sequence of values
$h_\ell$ with $N^{[\ell]} \in \setN^{\mathfrak Q} \subseteq \Nset$.
The choice of $h \in \setH^{\mathfrak Q}$ (equivalently  $N \in
\setN^{\mathfrak Q}$) is assumed to define  uniquely a particular
rule in ${\mathfrak Q}$. The selection of a particular type of
quadrature rule restricts $\setN^{\mathfrak Q}$ (the possible
$N^{[\ell]}$): for $m$-times repeated rules  $N^{[\ell]}$ has the
form $m \times N^{[0]} $ while for  classical Romberg rules
{(see \cite{baker,Nature_of,TR2007-1})} $N^{[\ell]}$ has the
form $2^m \times N^{[0]} $ ($\ell,m \in \Nset$).
 Subsequently, we  employ approximations
\bq \label{convol-quad} \int_{(n-N)\hfr}^{n\hfr} \psi(s)
\ds \approx \sum_{j-N}^n
 \hfr W_{n-j} \psi (j \hfr) \quad
(\hbox{with
$n \in \Nset$, } \sum_{j=0}^N W_{j} = N),
 \eq
  derived as particular
cases of (\ref{induced}) on setting $a = (n-N)\hfr$ and $b= n \hfr$.

 We now describe some additional families of quadrature.
 A \emph{composite}  or
\emph{$m$-times repeated} quadrature formulae ($m \in \Nset$) is
 based on summation over $\ell$ of the
contributions ${\mathcal Q}^{[\ell h,(\ell + 1) h ]}( \psi)$  (\cf (\ref{induced}))
to give the approximation
 % % \vspace{-.5\baselineskip}
\[
 \int_0^1  \psi(s) \ds =  \sum_{\ell=0}^{m-1}\int_{\ell/m}^{(\ell+1)/m} \psi(s) \ds
\approx  \] % % \vspace{-\baselineskip}
 \bq \label{composite}
 \sum_{\ell=0}^{m-1} {\mathcal Q}_{1/mN}^{[\ell/m,(\ell+1)/m]}( \psi) =:  (m
\times {\mathcal Q}_{1/N})(\psi) \quad (\hbox{for } m\in \Nset).
 \eq
Approximation (\ref{composite}) is of the generic form
(\ref{basic-rule}); $(m \times {\mathcal Q}_{1/N})
(\psi)$ could be written
${\mathrm Q}_{1/mN}(\psi)$.
 As (\ref{induced}) followed from
(\ref{basic-rule}), so one obtains,  from (\ref{composite}),
\bq
 \int_a^b  \psi(s) \ds \approx  (m
\times {\mathcal Q}_{(b-a)/N})^{[a,b]}( \psi) =
{\mathrm Q}_{(b-a)/mN}^{[a,b]}( \psi) \quad (\hbox{for } m\in \Nset).
\eq
 For Newton-Cotes rules,  $N$ is limited by (\ref{basic-cond}), so $h=1/N$ is
bounded away from zero, but the use of composite versions with increasing $m$
overcomes this.

 We denote by $R[0,1]$ the space of \emph{bounded Riemann-integrable
functions} on $[0,1]$. If the points $\boldsymbol{\eta}= \{
\eta_\ell \}_0^{M+1}$
  and  $\boldsymbol{\zeta}= \{ \zeta_\ell \}_0^{M}$
    together satisfy $
0=  \eta_0 \leq \zeta_1 \leq \eta_1
\ldots \leq \eta_{M} \leq \zeta_M \leq \eta_{M+1} = 1 $
with $\zeta_\ell \neq \zeta_{\ell+1}, \ \ell \in \{0,1, \ldots, M\}$.
If $\psi \in R[0,1]$, $\Sigma(\boldsymbol{\eta}, \boldsymbol{\zeta}
; \psi) = \sum_{\ell=0}^M \{ \eta_{\ell+1}- \eta_{\ell}\} \psi(\zeta_\ell)$
is a \emph{Riemann sum}.
Below,  Proposition \ref{propn:CvgtQ} exploits  a link between
certain quadratures and Riemann sums.
If $M\! \to\! \infty$ so that $\sup\{ \eta_{\ell+1}\!
-\!
\eta_{\ell}\} \searrow 0$ then $ \Sigma(\boldsymbol{\eta}, \boldsymbol{\zeta}
; \psi)\! \to \!\int_0^1\psi(s) \ds$.

 % \vspace{-1\baselineskip}
 \subsection{Convergence of quadrature rules as $h \to 0$}
% \vspace{-1\baselineskip}
The  types of permitted quadrature rules discussed above provide generic sums,  denoted
 $\{ {\mathcal Q}_{h}(\psi)\}$, that approximate the integral $\int_0^1 \psi(s) \ds$.
Thus, the expression
\bq
{\mathcal Q}_{h}^{T}(\psi) = \half h \psi(0) + h \psi(h) + \cdots +
 h \psi(1-h) + \half h \psi(1),
\quad h = 1/m, \ N \in \Nset,
\eq  %{ (\emph{cf}. (\ref{repeat-trap}))}
  can be computed for $h = 1, h= \half, h =
\frac{1}{4}, h = \frac{1}{8}, \ldots$, or for $h = 1,
\frac{1}{5}, h = \frac{1}{25},  h = \frac{1}{125},
\ldots$ and in general for $h= h_0, h= h_1, h= h_2, \ldots$
 where $h_\ell = 1/N_\ell$, $N_\ell \in \Nset$, and $N_{\ell+1}
  > N_{\ell}$.   The elements of the  corresponding sequence
$\{{\mathcal Q}_{h_\ell}^{T}(\psi)\}$ converge (as $h_\ell \to 0$) to a limit that
coincides with the integral if $\psi \in R[0,1]$.
In general, convergence can be arbitrarily slow, or as
${\mathcal O}(h_\ell^\rho)$ for arbitrary $\rho \in \Nset$ --- depending
on the integrand. Similar convergence issues arise for a  general family
of quadratures and a  sequence of positive values $h \in \setH^{\mathfrak Q}
 =\{ h_1, h_2,  h_3, \ldots\}$ wherein $\lim_{\ell \to \infty} h_\ell = 0$.
 \begin{defn}\label{def-conv} The family  $\mathfrak Q =
 \{ {\mathcal Q}_h\}_{h \in \setH^{\mathfrak Q}}$  is termed
 \emph{convergent}  if, for any $\psi \in R[0,1 ]$,
 $\DSS \lim_{h  \searrow 0,\, h \in \setH^{\mathfrak Q}}
  {\mathcal Q}^{[0,1]}_h(\psi) = \int_0^1 \psi(s) \ds $.
\end{defn}
\begin{assump}\label{A-Quad} The family  $\mathfrak Q$ is convergent.
\end{assump}
\begin{proposition}
\label{propn:CvgtQ}
 Repeated quadrature rules, classical Romberg rules,
and Gregory rules having
a fixed number of correction terms, using abscissae at step  ${h}\in \setH^{\mathfrak Q}$,
  define families that satisfy Assumption \ref{A-Quad}.
\end{proposition}
\noindent {\bf Proof}. { The associated quadrature approximations
are either  Riemann sums, weighted sum of a finite set of Riemann sums,
or differ by ${\mathcal O}({h})\, \|\psi\|_\infty$ from a Riemann sum.
 The Riemann sums referred to here converge to
the required integral as ${h} \searrow 0$. (See  \cite[p.~123]{baker},
\cite[Theorem~1]{Nature_of}, \cite{TR2007-1}.)}\hfill $\Box$
\begin{proposition}\label{propn:Uset}
Suppose that $\setfrakU$ is a set of functions defined on $[0,1]$ that are
uniformly bounded and equi-continuous, and Assumption \ref{A-Quad} is valid. Then
$ \DSS
\lim_{{h} \searrow 0}\sup_{u \in \setfrakU}|{\mathcal Q}_{h}(u)
 - \int_0^1 u(s)\ds | = 0$, where ${h} \searrow 0$ with
 ${h} \in {\setH}$.
\end{proposition}
\noindent {\bf Proof} { If  $u \in C[0,1]$, \emph{a fortiori}  $ u \in R[0,1]$} so
$\lim_{h \searrow 0}|{\mathcal Q}_h(u) -\int_0^1 u(s) \ds | = 0$. This
convergence is uniform on compact sets, and $\setfrakU$ is compact
in $C[0,1]$% by the Arzela-Ascoli theorem
 \hfill $\Box$

{ The proof of the next Proposition is also straightforward \cite{TR2007-1}.}
\begin{proposition}\label{propn:equicont}
Suppose that $\ksf(t,s)$, $f(t,u)$ satisfy Assumption \ref{cts-case-k+f} and
$\q \in \calA_\varpi$.
Then the family of integrands $\ksf(t,t+(\sigma-1)\tau)
\f(t+(\sigma-1)\tau, \q(t+(\sigma-1)\tau))$ for $\sigma \in [0,1]$
is uniformly bounded and equicontinuous for $t \in \Rset$.
\end{proposition}
% % \vspace{-\baselineskip}
\subsection{Approximate integration on $[t-\tau, t]$}
\label{sec:Approximate}
%% % \vspace{-\baselineskip}
For $h \in {\mathfrak H}^ {\mathfrak Q}$,  ${\mathcal Q}_h \in {\mathfrak Q}$  induces
(with finite $ a< b \in \Rset$) a corresponding
formula ${\mathcal Q}^{[a,b]}_\hfr(\psi)$ ($\hfr = (b-a)h$, $\psi \in R[a,b]$),
with $\DSS \lim_{\hfr  \searrow 0,\, \hfr \in (b-a)\setH^{\mathfrak Q}}
  {\mathcal Q}^{[a,b]}_h(\psi) = \int_a^b \psi(s) \ds $.
(The notation $ (b-a)\setH^{\mathfrak Q}$ denotes
$ \{(b-a) h_1, (b-a)h_2, (b-a)h_3, \ldots\}$.)
Henceforth, we are mainly concerned with integrands $\psi(s)$ of the form
  $\ksf(t,s) \f(s,\x(s))$ with $t \in \Rset$ (integrated for
$s \in [t-\tau, t]$).
Given arbitrary $\tau > 0$ and $t \in \Rset$, a convergent family of
quadrature rules $\mathfrak Q$ induces corresponding  $ \{{\mathcal
Q}_\hfr^{[t-\tau, t]}\}$ that generate  discrete equations obtained
from (\ref{eq-contin-2}). {For the specific $\tau$ in
(\ref{eq-contin-2}), we  require  quadrature ({\it cf}.
(\ref{convol-quad}))  to approximate integrals over  $[t-\tau, t]$,
in which $t = n \hfr$, $\hfr \in \frH_{\varpi} \cap \frH_{\tau}$. If
$\tau/\varpi \not\in \Qset_+$, this cannot be achieved, so we
replace  $\tau$ by an approximation $\tilde\tau$ with
$\tilde\tau/\varpi \in \Qset_+$, and $\hfr \in \frH_{\varpi} \cap
\frH_{\tilde\tau}$. The approximation $\tilde\tau \approx \tau$
(with  $\tilde\tau \leq \tau$) can be made arbitrarily close by
taking suitable sufficiently small $\hfr \in \frH_\varpi$. To use
rules from the family $\mathfrak Q$ we also require ${\mathfrak h}
\in \wt \tau {\mathfrak H}^{\mathfrak Q}$; this is a refinement of
(\ref{request}) requiring $N_{\wt\tau} \in \Nset$; now, $N_{\wt\tau}
\in {\mathfrak N}^{\mathfrak Q}$. In commonplace families  of
quadrature,
 the members $N^{[\ell]}$ in the sequence $ {\mathfrak N}^{\mathfrak Q}$ form either
a increasing arithmetic progression or an increasing geometric progression.
\begin{assump}\label{revised-request}
 We are given positive, $\tau, \varpi$, and a family of quadrature rules
$\{\mathfrak Q_{h}\}_{h\in {\mathfrak H}^{\mathfrak Q}}$ associated with
  ${\mathfrak N}^{\mathfrak Q}$ in (\ref{Indexing}). We assume  that, given arbitrary
$\varepsilon > 0$, there exists a corresponding $\hfr^\star >0$ such that,
whenever  $\hfr \leq \hfr_\star$  and
$\hfr \in
  {\mathfrak H}_{\varpi}$,  there exists
 $\wt \tau \in [\tau - \varepsilon, \tau]$
with $\wt \tau = N_{\wt \tau} \hfr$, where
$ N_{\wt \tau} \in  {\mathfrak N}^{\mathfrak Q}$.
\end{assump}
\begin{remark}{\small
 In Assumption \ref{revised-request},
 $ \hfr \in {\mathfrak H}_{\wt\tau}
\cap {\mathfrak H}_{\varpi}\cap{\wt\tau}{\mathfrak H}_{\mathfrak Q}$
(and this condition is to be satisfied by $\hfr^\star$, in the sense
that
 $ \hfr^\star \in {\mathfrak H}_{\wt\tau^\star}
\cap {\mathfrak H}_{\varpi}\cap{\wt\tau^\star}{\mathfrak
H}_{\mathfrak Q}$). Suppose $\tau/ \varpi = \varrho \in \Rset$; if
$\wt \tau/ \varpi \in \Qset_+$ then  $\wt \tau/ \varpi =
\mu_1/\mu_2$ with $\mu_{1,2} \in \Nset$. If $\hfr \in  \wt
\tau{\mathfrak H}^{\mathfrak Q}$ then $\hfr = \wt \tau
 /\wt N$ where $\wt N \in {\mathfrak N}^{\mathfrak Q}$ (then $\wt N = \wt \tau/ \hfr$).
If  $\hfr \in \frH_\varpi$; then ${ N}_\varpi := \varpi/\hfr \in
\Nset$. Hence, $\mu_1/\mu_2 = \wt N / N_\varpi$ where $\wt N \in
{\mathfrak N}^{\mathfrak Q}$ and we need to be able to approximate
$\varrho$ arbitrarily closely by fractions of the form $ \wt N /
N_\varpi$. }
\end{remark}
 Given $\varpi$, $\hfr$, with $\varpi = N_\varpi \hfr$, identify $\omega=\omega(\hfr)$ with $ {{N}_\varpi}$, and write
 $\omega^\star =
\omega(\hfr^\star)$ for $ {{N}^\star_\varpi}$; if $\hfr <
\hfr^\star$ then $ {{N}_\varpi} >  {{N}^\star_\varpi}$.
 Select
 ${N}_{\tilde\tau}$ so that ${N}_{\tilde\tau} =\tilde\tau / \hfr
\in   {\mathfrak N}^{\mathfrak Q}$
 where
\bq \nonumber
 \tilde\tau =
 \frac{{N}_{\tilde\tau}}{{N}_\varpi} \varpi =
\max_{n \in  {\mathfrak N}^{\mathfrak Q}}\big\{
 \frac{n }{{N}_\varpi} \varpi < \tau \big\} \
\hbox{and }
\tilde\tau^\star  =
 \frac{{N}_{\tilde\tau^\star}}{{N}^\star_\varpi} \varpi
 =
\max_{n^\star\in  {\mathfrak N}^{\mathfrak Q}}\big\{
 \frac{n^\star }{{N}^\star_\varpi} \varpi < \tau \big\}.
\eq
To ensure that $0 \leq \tau - \wt\tau
<\tau - \wt\tau^\star $ when $\hfr < \hfr^\star$, we
require $ {N}_{\tilde\tau}  > {N}_{\tilde\tau^\star} \in
{\mathfrak N}^{\mathfrak Q}$. With $ \varrho = \tau/ \varpi$,
 ${N}_{\tilde\tau}$ must be the largest integer
in ${\mathfrak N}^{\mathfrak Q}$ such that ${N}_{\tilde\tau}
< \varrho  N_\varpi$.
 The commonplace ${\mathfrak N}^{\mathfrak Q}$ give
 $N^{[\ell]} =  \mu + N^{[\ell-1]}$ with $0< \mu  \in \Nset$
or $N^{[\ell]} =  \nu \times N^{[\ell-1]}$ with $1 < \nu  \in
\Nset$, and, for simplicity and  convenience, we take the latter
case with $N^{[\ell]} =\nu^{\ell}$. Then, ${N}_{\tilde\tau} =
\nu^{\tilde \lambda}$ with ${\tilde \lambda}=
 \lfloor \log_{\nu} (\varrho  N_\varpi) \rfloor$ (and
  ${N}_{\tilde\tau}> {N}_{\tilde\tau^\star}$).

 Our approximate integration proceeds, with a choice of
$\hfr \in \frH_\varpi\subset (0, \tau]$, to
determination of $\wt \tau$, to application of a quadrature rule.
 When
\bq \label{1st}
\DSS |\int_{t-\tau}^t \psi(s) \ds -
 \int_{t-\tilde\tau}^t \psi(s) \ds| \leq \varepsilon \sup_{s
 \in [t-\tau,t-\tilde\tau]} |\psi(s)| ,
\eq
 the quadrature involved in (\ref{basic-rule}) induces,
 for $t= n\hfr$,  and
on setting $N$ equal to ${N}_{\tilde\tau}$, the approximation
\bq \label{2nd}
 \int_{t-\tilde\tau}^t \psi(s) \ds \approx
\sum_{\ell=0}^{{N}_{\tilde\tau}} \tilde\tau w_\ell\, \psi(t-\tilde\tau +\ell \, \hfr)
 =:  {\mathcal Q}^{[t-\tilde\tau, t]}_{\hfr}(\psi),
\quad (\hfr = \tilde\tau/{N}_{\tilde\tau}),
\eq
so the sum in (\ref{2nd}) approximates the first integral in (\ref{1st})
with a sum of the type in (\ref{convol-quad}).}

Indeed, for  the integral in (\ref{eq-contin-2-bis}) over
$[n\hfr-\tilde\tau,n\hfr]$,  the $m$-times repeated quadrature
corresponding to (\ref{basic-rule}) induces the approximation
 \bsq
 \bq \nonumber
\int_{t-\tilde\tau}^t \psi(s)
    {\ds} \approx \sum^n_{j-{N}_{\tilde\tau}}\hfr W^{{\scriptsize
(m\times{\mathcal Q})}}_{n-j}\, \psi(j\hfr)
 \ (t= n\hfr,  \hfr =\tilde\tau/{N}_{\tilde\tau}, \ {N}_{\tilde\tau}= mN \in \Nset),
 \eq
with $W^{(m\times{\mathcal Q})}_{j} =
w_s$ { if } $j\equiv{s\!\!\mod(m)}$
and $ s \neq 0$,
 $W^{(m\times{\mathcal Q})}_{0}=
w_N$,
$W^{(m\times{\mathcal Q})}_{r m} =
 w_0 +w_N $ if
 $rm\notin \{0, {N}_{\tilde\tau}\}$,
$ W^{(m\times{\mathcal
Q})}_{{N}_{\tilde\tau}} = w_0$.   By assumption,
 $w_j \geq 0$ for $j \in \{0,1, \ldots, N\}$; hence,
 $W^{(m\times{\mathcal Q})}_j $ is non-negative
for $j \in \{0,1, \ldots, N_{\wt \tau}\}$.
 \esq
Likewise,  a given rule $R$ from amongst the classical Romberg rules gives an
approximation expressible as $\int_0^1\psi(s) \ds \approx
\sum_{j=0}^{2^m} w^{[R]}_j \psi(j/\{2^m\})$, where $ m \in \Nset$, and
where $ w^{[R]}_j = w^{[R]}_{2^m-j} > 0$.  We thus obtain,  for $t = n\hfr$,
formulae of the type
  \bq \int_{t-\tilde\tau}^t \psi(s) \ds \approx \sum^{n}_{j-2^m}
\hfr W^{[R]}_{n-j} \psi({t-\tilde\tau}+j \hfr), \ \hfr = \tilde\tau/{2^m},
\hbox{ with ${N}_{\tilde\tau} = 2^m$.}
\eq
(Note the restricted form of $\hfr$.) Further, the
$r$-th Gregory rule $G_r^{{N}_{\tilde\tau}}$
(using ${N}_{\tilde\tau}+1$ abscissae, and
with $r \leq {N}_{\tilde\tau}$) gives for $t=
n\hfr$ an approximation
 \bq\label{Greg} \int_{t-\tilde\tau}^t \psi(s) \ds
\approx\
\sum^n_{j-{N}_{\tilde\tau}} \hfr W^{[G]}_{n-j}\, \psi(j\hfr)
 \quad (t = n \hfr, \tilde\tau = {N}_{\tilde\tau} \hfr,\, G \equiv
G_r^{{N}_{\tilde\tau}})
 \eq
in which $ W^{[G]}_{n-j}=W^{[G]}_{j}$, for $
j\in \{0,1, \ldots, {N}_{\tilde\tau}\}$.
These rules include the Newton-Cotes case (for $r = {N}_{\tilde\tau}$), so
the weights need not be positive, as we require.
% %\vspace{-2\baselineskip}
\subsection{ Dependence on $\tau$}
\begin{remark}{\small 
\label{sec-dep-tau} Prior to discretization, we replaced
 $\tau$ by a nearby value ${\tilde\tau}$ in the integral equation.  To
analyze this further, we can appeal to variations of parameters
formulae and to inequalities similar to that of Gronwall.

Consider a solution $\x(t) \equiv \x(\tau;t)$)  of
 $\x(\tau;t) = \int_{t-\tau}^t \ksf(t,s) \f(s,\x(\tau; s)) {\rm d}s,
 $ (that is, of (\ref{eq-contin-2})).
One can  see that if
$ \delta\x(\tau; t)  = \x(\tilde\tau; t) -\x(\tau; t)$ then
\bq
\delta \x(\tau; t)  = \int_{t-\tau}^t
 \ksf(t,s) \{\f(s,\x(\widetilde\tau; s)) -\f(s,\x(\tau; s))\}
{\rm d}s
 + \int_{t-\tau}^{t-\tau}
\ksf(t,s) \f(s,\x(\widetilde\tau; s)) {\rm d}s.
\eq
Further, if it exists, the first-order sensitivity,
 can be expressed as
$\frac{\partial}{\partial\tau}\x(\tau;t)$.
Assuming the existence of the derivatives,
\bq
 \frac{\partial}{\partial\tau}\x(\tau;t) =  \int_{t-\tau}^t\{
 \ksf(t,s) \f_2(s,\x(\tau; s))
\frac{\partial}{\partial\tau}\x(\tau;s)\} {\rm d}s
 -\ksf(t,{t-\tau}) \f({t-\tau},\x(\tau; {t-\tau})).
\eq
The two displayed equations permit further investigation;
 under mild additional conditions and assumptions, not pursued here,
 $\delta\x(\tau; t)$ can be related to the size of $\tau-\tilde\tau$.
}
\end{remark} % % \vspace{-\baselineskip}
\subsection{Numerics of integral equations} \label{sec-nume-cont}
%% % \vspace{-\baselineskip}
 We can now address the issue of guaranteeing
 conditions for periodic solutions when we discretize
an  integral equation (\ref{eq-contin-2}) for which a result like Theorem
\ref{agar} holds. Given a
$\varpi$-periodic solution $\x(t)$ of  (\ref{eq-contin-2}),
the sequence $\{ x(n) = \x(n\hfr)\}_{n\in
\Nset}$ is $\omega$-periodic where $\omega = \varpi/\hfr$ and we
seek conditions ensuring a $\omega$-periodic solution of our
 discretized equations. Obviously, $\omega$ is a function of $\hfr$, $\omega =
\omega(\hfr)$.
 We now consider an equation of the type (\ref{eq-nvd-X}) but in the form
\begin{equation}\label{eq-nvd-X-bis}
\wt x(n)=\sum^n_{j-{N}_{\tilde\tau}(\hfr)}\hfr W_{n-j}
\ksf(n\hfr,j\hfr)\f(j\hfr, \wt x(j)),
 \quad \hfr \in
\frH_\varpi, \quad \wt x(n)\in \Rset.
\end{equation}
This is of the form (\ref{eq-nvd-1}) with $k(n,j) := \hfr W_{n-j}
\ksf(n\hfr,j\hfr)$, $f(j, u) := \f(j\hfr, u)$.
We define (\ref{eq-nvd-X-bis}) to be the ${\mathcal Q}_\hfr$-based
discretization of (\ref{eq-contin-2}) and
 we examine the assumptions made in Propositions
\ref{propn:Ex-quad-discrete}--\ref{propn:thm-from-LW}
when applied to (\ref{eq-nvd-X-bis}). Now if  $\varpi =
\omega \hfr$ for $\omega \in\Nset$,  then
 $k(n,j) =
k(n+\omega, j+\omega)$
 when
$\ksf(n\hfr+\varpi,j\hfr+\varpi ) =\ksf(n\hfr,jh)$, and
 $f(n,u) =f(n+\omega, u)$ when
$\f(n\hfr+\varpi,u) =\f(n\hfr,u)$. The following is typical of the results we seek.
\begin{proposition}
Given $\q \in \calA_\varpi$, and $\f$ satisfying (\ref{ie-cond-f}),
suppose that $\varpi = \omega \hfr$ and define
$q(n) = \q(n\hfr)$, $f(n,u) = \f(n\hfr, u)$.
(a) Suppose that $\psi$ satisfies (\ref{eq-for-f3-a})
 and the
functions $\f$, $\psi$  and $\q$ satisfy
 $\f(t,u)\le \q(n)\psi(u)$ for all $u\in \Rset_+$ and $t\in \Rset$,
Then (\ref{eq-for-f3-b})
is satisfied. (b) If there exists a constant ${a}_0 \in (0,1)$,
such that $\f$, $\psi$,
and $\q$ satisfy   $\, {a}_0\q(t)\psi(u)\le \f(t,u)$ for all $u\in \Rset_+$
and $t\in \Rset$, then  (\ref{eq-for-f3-c}) is satisfied.
\end{proposition}
Other assumptions in the discrete case depend on
$\kappa_{\min,\max}(q)$ ($q \in {\mathcal A}_\omega$, {\it cf}.
(\ref{eq-for-f5-6})), which,  with  $q(n) = \q(n\hfr)$ compare
with $\widehat\kappa^{\wt \tau}_{\min,\max}(\q)$ in (\ref{kappa-cond}).
Since
  % % \vspace{-\baselineskip}
\begin{eqnarray}\nonumber \int_{t-\tau}^t \ksf(t,s) \q(s) \ds  =
\DSS \int_0^1 \ksf(t,t+(\sigma-1)\tau) \q(t+(\sigma-1)\tau)
{\dsigma}
  &\\ \nonumber
  \approx \sum_j w_j  \ksf(t,t+(j/N_\tau-1)\tau)
\q(t+(j/N_\tau-1)\tau), &\end{eqnarray}
  Proposition \ref{propn:Uset} and Proposition \ref{propn:equicont} have obvious
corollaries that combine (on setting $f(t,u) = u$) to give us
Proposition \ref{propn:Cvgt} below. (Recall that  the
family ${\mathfrak Q}$ of quadrature rules inducing  $\{{\mathcal Q}_\hfr\}
$  is  assumed to be convergent.)
\begin{proposition}
\label{propn:Cvgt} Suppose that $\ksf(t,s)$  satisfies (\ref{ie-cond-k}) in Assumption
\ref{cts-case-k+f} and $\q \in \calA_\varpi$ is continuous.  Then,
$\sum^n_{j-{N}_{\tilde\tau}(\hfr)} \hfr W_{n-j}\ksf(t,jh)
\q(jh) -
 \int_{t-\tau}^t \ksf(t,s)  \q(s) \ds$
 is arbitrarily small (uniformly for $ t \hfr$ in  $\Rset$)
 and $\widehat\kappa^{\wt \tau}_{\min,\max}(\q)$
differ from $\kappa_{\min,\max}(q)$ by  arbitrarily small amounts
for sufficiently small $\hfr$.
\end{proposition}


It now follows that our discrete theory is applicable
to the equations obtained by application of convergent quadrature
of the type discussed above,
given sufficient conditions on the integral equation. The following result is typical:
\begin{proposition}
Suppose  the conditions of Theorem \ref{agar} hold, and consider
the  ${\mathcal Q}_\hfr$-discretization (\ref{eq-nvd-X-bis}). Then the conditions of
Proposition \ref{propn:thm-from-Kras}
apply to  (\ref{eq-nvd-X-bis}) for all sufficiently small $\hfr \in \setH$.
\end{proposition}
 % % \vspace{-2\baselineskip}
\section{Questions not addressed in the paper}
 %\vspace{-\baselineskip}
{
Remarks \ref{rem:general} and \ref{sec-dep-tau} indicated  routes
for further research, usually requiring added conditions on $\f$ or $f$.
 The following list provides an  indication of subjects for further study
that have not already been mentioned in the text.
\begin{itemize}
\item As a first step, we demonstrated here the existence of periodic solutions.
A subsequent issue is how the dynamics of a solution $x(\varphi;t)$
of (\ref{eq-nvd-1}) for $n \geq 0$ (given $x(n)=\varphi(n)$ for $n \in
\{-N, 1-N \cdots, -1\}$) relate to the existence of periodic solutions
of (\ref{eq-nvd-1}) for $t \in \Rset$.  Does this emulate the way the
dynamics of a solution $\x(\varrphi; t)$ of (\ref{eq-contin-2}) for $t
\geq 0$, given $\x(t) = \varrphi(t)$ for $t \in [-\tau, 0]$, relate to
the existence of periodic solutions $\x$ of (\ref{eq-contin-2}),
considered for all $ t \in \Rset$?
\item
What can be said about $\x(\varrphi;n\hfr) - \widetilde x(\varphi;n)$ for small $\hfr$
when $\varphi(n) = \varrphi(n\hfr)$? (This is easily investigated when $\f(t,u)$
satisfies a uniform Lipschitz condition in $u$.)
\item  Where a solution $\{x(n)\}$ of  (\ref{eq-nvd-1}) exists, and we set
 $y(n) = f(n, x(n))$,  we have $ y(n) = f(n, \sum_{j-N}^n k(n,j) y(j))$
(for $n \in \Zset$).
These equations yield a viewpoint that may be pursued further.
\item
When $c= \{c(n)\}$ is suitably periodic, we might consider the
equation
$$ x(n)=\sum^n_{j-N}k(n,j)f(j,x(j)) +c(n), $$
 by a study of  $\hat x(n) = x(n) - c(n)$ where
$ \hat x(n)=\sum^n_{j-N}k(n,j)
  f\big(j, \hat x(j) - c(j)\big)$.
\end{itemize}}

\noindent
\appendix
\normalsize \centerline{\sc Appendices}
\renewcommand{\baselinestretch}{.875}
% % \vspace{-1\baselineskip}
\section{Cones and compression operators}
%\vspace{-\baselineskip}
\begin{defn} Let ${\mathcal X} = \{X, \| \cdot \|\}$ be a Banach
space (a complete, normed, linear space). (i) A closed, convex, set
$C$ in ${\mathcal X}$ is a \emph{(positive) cone} when:
% % \vspace{-\baselineskip}
\begin{eqnarray}
(a)\quad  \hbox{if } u \in C \hbox{ then }& \lambda u \in C \hbox{ for } \lambda \geq 0\\
(b)\quad\  \hbox{if } u \in C  \hbox{ and }& -u \in C  \hbox{ then }
u =0.
\end{eqnarray}
(ii) A cone $C$ in $\mathcal X$ induces a \emph{partial ordering}
$\leqslant$ in $\mathcal X$ by the definition $u \leqslant v$ if and
only if $v-u \in C$. For $u, v$ in $C$ such that $v-u \notin C$ we
write $u \not\leqslant v$. A Banach space with a partial ordering
induced by a cone is a \emph{partially ordered Banach space}. (iii)
An operator $T$ is a \emph{compression} of a cone $C$ in an ordered
Banach space if (a)  $ T(0) = 0$; (b)
 there exist $r, R$ with $0 < r < R$ such that
$T(u) \not \leqslant u \hbox{ if } u \in C,\ \| u \| \leq r$, { and
} $u \neq 0$;
 and also, (c)
for all $\varepsilon > 0$, $ (1 +\varepsilon) u \not \leqslant T(u)\
\hbox{ if } u \in C,\ \| u \| \geq R$.
\end{defn}

  The ``compression of the cone" theorem \cite[p.~137]{bk:Kras} due to
Krasnosel${}^\prime\!$ski\u\i\, (1920--1997) reads as follows.
\begin{thm}\label{Kras:compression}
Let the positive completely continuous operator $T$ be a compression
of the cone $C$. Then $T$ has at least one non-zero fixed point on
$C$.
\end{thm}
The proof of this theorem relies upon a result due to the Polish
mathematician Julius Schauder (1899-1943): \emph{A completely
continuous operator that transforms a bounded convex and closed set
into itself has at least one fixed point in the given set.} Theorem
\ref{Kras:compression} was refined by Leggett and Williams
\cite{legg} who proved the following result, in which the conditions
associated with a compression are relaxed.
\begin{thm}
Given a cone $C \in {\mathcal X}$, define
 $C_\rho : = \{ v \in C | \| v \| \leq \rho\}$ ($ \rho \in (0, \infty)$)
and $C_\infty = C$. Suppose $ u  \in C\!\setminus\!\{0\}$  and
 $C[u] :=
\{ v \in C | \alpha v \geqslant u \hbox{ for some } \alpha > 0\}$.
 For some $R > 0$,
suppose that  $T: C_R \to C$ is completely continuous with $T(0)
=0$, and there exists $r$ with $0 < r < R$ such that $T(u) \not
\leqslant u$ if $ u \in C,\ \| u \| = r$
 and  for all $\varepsilon > 0$, $
(1 +\varepsilon) u \not \leqslant T(u)\ \hbox{ if } u \in C,\ \| u
\| = R$. Then $T$ has a fixed point $x \in C$ with $r \leq \| x\|
\leq R$
\end{thm}
% % \vspace{-\baselineskip}
\section{ Modifications for weakly singular equations and their discretization}
%\vspace{-\baselineskip}
  The discussion that we have provided can be adapted to the treatment
  of a typical class of weakly singular (Abel-type) kernels. We
  indicate one approach.

Consider
\begin{equation}\label{eq-contin-2-repeat}
\x(t) = \int_{t-\tau}^t \ksf^\sharp(t,s) \f(s,\x(s)) {\rm d}s,
 \quad
\hbox{for $ t\in \Rset$ with } \tau >0,\  \x(t)\in \Rset.
\end{equation}
and suppose:\\
{\assume} \ref{A1}${\!\,}^\natural$
 The functions $\ksf$ and $\f$ satisfy {\sc Assumption} \ref{A1}
and the possibly unbounded kernel $\ksf^{\natural}$ satisfies \bq
\ksf^{\natural}(t,s) = \ksf(t,s)/ |t-s|^\nu \quad \hbox{with }\quad
\nu \in (0,1). \eq \newline With this Assumption,
% \bq \label{ie-cond-k-natural}
$  \ksf^{\natural}(t+\varpi, s+\varpi)=\ksf^{\natural}(t,s)
 \hbox{ for } \varpi \in \Rset_+ \hbox{ for every }
 t,s\in \Rset$.
For every  $u \in C(\Rset)$ we use the notation $ T^\sharp u( t) =
\displaystyle  \int_{t_\tau}^t \ksf^\sharp (t,s) f(s, u(s)) \ds $.
Without ambiguity, we  employ the notation $T^\sharp$ for the
operator defined on given $\setD \subseteq {\calA}_\omega$, the
latter being equipped with the uniform norm. \emph{This operator
$T^\sharp$ is a compact map  from $\setD \in {\calA}_\omega$ to $
{\calA}_\omega$}, and the abstract functional analysis can be
applied to the integral equation case.

Preparatory to the numerics, if we define $ A(t) = \displaystyle
\int_{t_\tau}^t (t-s)^{-\nu} \ds =\displaystyle
\tau^{1-\nu}/\{1-\nu\} $, we can write
 \bq T^\sharp
u(t) =\displaystyle  \int_{t_\tau}^t \frac{\ksf (t,s) f(s, u(s))-
\ksf (t,t) f(t, u(t))}{(t-s)^\nu} \ds + A(t)\ksf (t,t) f(t, u(t)).
  \eq
   To obtain a
discretized equation we can now apply quadrature to approximate the
integral term (the integrand is continuous).
 As in the case of continuous kernels,
we can apply the general discrete theory if we discretize the
integral appropriately.

The use of quadrature as indicated may give low-order accuracy. More
generally, we can adapt other numerical techniques found in the
literature for an Abel-type equation
 $%\bq
u(t) = \displaystyle \int_0^t \ksf^\sharp (t,s) f(s, u(s)) \ds
 $; % \eq
 this equation
differs in having a lower limit $0$ rather than { ${t-\tau}$}.
% % \vspace{-\baselineskip}
\begin{thebibliography}{9}

\itemsep=-0.0mm
\bibitem{aga00} R.P. Agarwal, {\em Difference Equations and
Inequalities}, 2nd Ed., Dekker, 2000.

\end{thebibliography}

\end{document} 