\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 40, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/40\hfil Averaging for differential equations]
{Time averaging for ordinary differential equations and retarded
functional differential equations}
\author[M. Lakrib, T. Sari\hfil EJDE-2010/40\hfilneg]
{Mustapha Lakrib, Tewfik Sari} % in alphabetical order
\address{Mustapha Lakrib \newline
Laboratoire de Math\'ematiques, Universit\'e Djillali Liab\`es,
B.P. 89, 22000 Sidi Bel Abb\`es, Alg\'erie}
\email{m.lakrib@univ-sba.dz}
\address{Tewfik Sari \newline
Laboratoire de Math\'ematiques, Informatique et Applications,
Universite de Haute Alsace, 4 rue des fr\`eres Lumi\`ere, 68093
Mulhouse
and EPI MERE (INRIA-INRA), UMR MISTEA, INRA 2,
pl. Viala, 34060 Montpellier, France}
\email{Tewfik.Sari@uha.fr}
\thanks{Submitted September 15, 2009. Published March 19, 2010.}
\subjclass[2000]{34C29, 34C15, 34K25, 34E10, 34E18}
\keywords{Averaging; ordinary differential equations; stroboscopic method;
\hfill\break\indent retarded functional differential equations;
nonstandard analysis}
\begin{abstract}
We prove averaging theorems for non-autonomous ordinary
differential equations and retarded functional differential equations
in the case where the vector fields are continuous in the spatial
variable uniformly with respect to the time and the solution of
the averaged system exists on some given interval.
Our assumptions are weaker than those required in the results of
the existing literature. Usually, we require that the non-autonomous
differential equation and the autonomous averaged equation
are locally Lipschitz and that the solutions of both equations
exist on some given interval. Our results are formulated in classical
mathematics. Their proofs use the stroboscopic method which is
a tool of the nonstandard asymptotic theory of differential equations.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
Averaging is an important method for analysis of
nonlinear oscillation equations containing a small parameter. This
method is well-known for ordinary differential equations (ODEs) and fundamental averaging results (see, for instance,
\cite{Bogolyubov-Mitropolsky,Guckenheimer-Holmes,Hale2,Sanders-Verhulst,Sari1}
and references therein) assert that the solutions of a
non-autonomous equation in \emph{normal form}
\begin{equation}
\label{eq11}
x'(\tau)=\varepsilon f(\tau,x(\tau)),
\end{equation}
where $\varepsilon$ is a small positive parameter, are
approximated by the solutions of the autonomous averaged equation
\begin{equation}
\label{eq12}
y'(\tau)=\varepsilon F(y(\tau)).
\end{equation}
The approximation holds on time intervals of order $1/\varepsilon$
when $\varepsilon$ is sufficiently small. In \eqref{eq12},
the function $F$ is the average of the function $f$ in
\eqref{eq11} defined by
\begin{equation}\label{average}
F(x)=\lim_{T\to\infty}\frac{1}{T}\int_0^Tf(t,x)dt.
\end{equation}
The method of averaging for ODEs is known also as the
Krylov-Bogolyubov-Mitropolsky (KBM) method \cite{Bogolyubov-Mitropolsky}.
The method of averaging was extended by Hale \cite{Hale1} (see
also of \cite[Section 2.1]{Lehman}) to the case of retarded
functional differential equations (RFDEs) containing a
small parameter when the equations are considered in normal form
\begin{equation}
\label{eq13}
x'(\tau)=\varepsilon f(\tau,x_\tau),
\end{equation}
where, for $\theta\in[-r,0]$, $x_\tau(\theta)=x(\tau+\theta)$.
Equations of the form \eqref{eq13}
cover a wide class of differential equations including those with
point-wise delay for which a method of averaging was developed by Halanay
\cite{Halanay}, Medvedev \cite{Medvedev} and Volosov et
al. \cite{Volosov-Medvedev-Morgunov}. Note that the
averaged equation corresponding to \eqref{eq13} is the ODE
\begin{equation}
\label{eq14}
y'(\tau)=\varepsilon F(\tilde{y}^\tau),
\end{equation}
where, for $\tau$ fixed and $\theta\in[-r,0]$,
$\tilde{y}^\tau(\theta)=y(\tau)$
and the average function $F$ is defined by (\ref{average}).
Recently, Lehman and Weibel
\cite{Lehman-Weibel} proposed to retain the delay in the averaged
equation and proved that equation
\eqref{eq13} is approximated by the averaged RFDE
\begin{equation}\label{lwa}
y'(\tau)=\varepsilon F(y_\tau).
\end{equation}
They observed, using numerical simulations, that equation
\eqref{eq13} is better approximated by the RFDE (\ref{lwa})
than by the ODE \eqref{eq14}, see Remark~\ref{rem3}.
The change from the fast time scale $\tau$ to the slow time scale
$t=\varepsilon\tau$ transforms equations \eqref{eq11} and
\eqref{eq12}, respectively, into
\begin{equation}
\label{eq15}
\dot{x}(t)=f(t/\varepsilon ,x(t))
\end{equation}
and
\begin{equation}
\label{eq16}
\dot{y}(t)=F(y(t)).
\end{equation}
Thus a method of averaging can be developed for
\eqref{eq15}, that is, if $\varepsilon$ is sufficiently
small, the difference between the solution $x$ of
\eqref{eq15} and the solution $y$ of \eqref{eq16},
with the same initial condition, is small on finite time
intervals.
The analog of equation \eqref{eq15} for RFDEs is
\begin{equation}
\label{eq17}
\dot{x}(t)=f\left(t/\varepsilon,x_t\right).
\end{equation}
The averaged equation corresponding to \eqref{eq17} is the
RFDE
\[
\dot y(t)=F(y_t),
\]
where the average function $F$ is defined by (\ref{average}).
Notice that the RFDEs \eqref{eq13} and \eqref{eq17} are not
equivalent under the change of time $t=\varepsilon\tau$,
as it was the case
for the ODEs \eqref{eq11} and \eqref{eq15}. Indeed, by rescaling
$\tau$ as $t=\varepsilon\tau$ equation \eqref{eq13} becomes
\begin{equation}
\label{eq18}
\dot{x}(t)= f\left(t/\varepsilon,x_{t,\varepsilon}\right),
\end{equation}
where, for $\theta\in[-r,0]$,
$x_{t,\varepsilon}(\theta)=x(t+\varepsilon\theta)$. Equation
\eqref{eq18} is different from~\eqref{eq17}, so that
the results obtained for \eqref{eq18} cannot be applied to
\eqref{eq17}. This last equation deserves a special
attention. It was considered by Hale and Verduyn Lunel in
\cite{Hale-Lunel1} where a method of averaging is developed for
infinite dimensional evolutionary equations which include RFDEs
such \eqref{eq17} as a particular case (see also
Section~12.8 of Hale and Verduyn Lunel's book \cite{Hale-Lunel2}
and Section~2.3 of \cite{Lehman}).
Following our previous works
\cite{Lakrib0,Lakrib1,Lakrib2,LakibThesis,Lakrib-Sari,Lakrib-Sari1,Sari1,Sari2,Sari3}
we consider in this paper all equations \eqref{eq15},
\eqref{eq17} and \eqref{eq18}. Our aim is to give
theorems of averaging under weaker conditions than those of the
literature. We want to emphasize that our main contribution is the
weakening of the regularity conditions on the equation under which
the averaging method is justified in the existing literature.
Indeed, usually classical averaging theorems require that the
vector field $f$ in \eqref{eq15}, \eqref{eq17} and
\eqref{eq18} is at least locally Lipschitz with respect to
the second variable uniformly with respect to the first one (see
Remarks \ref{rem1}, \ref{rem2} and \ref{rem4} below). In our
results this condition is weakened and it is only assumed that
$f$ is continuous in the second variable uniformly with respect to
the first one. Also, it is often assumed that the solutions $x$
and $y$ exist on the same finite interval of time. In this paper
we assume only that the solution $y$ of the averaged equation
exists on some finite interval and we give conditions on the
vector field $f$ so that, for $\varepsilon$ sufficiently small,
the solution $x$ of \eqref{eq15}, \eqref{eq17} or
\eqref{eq18} will be defined at least on the same interval.
The {\em uniform quasi-boundedness} of the vector field $f$ is
thus introduced for this purpose. Recall that the property of
quasi-boundedness is strongly related to results on continuation
of solutions of RFDEs. It should be noticed that the existing
literature \cite{Hale1,Hale-Lunel1,Lehman} proposed also important
results on the infinite time interval $[0,\infty)$, provided that
more hypotheses are made on the non-autonomous system and its
averaged system. For example, to a hyperbolic equilibrium point of
the averaged system there corresponds a periodic solution of the
original equation if $\varepsilon$ is small. Of course, for such
results, stronger assumptions on the regularity of the vector
field $f$ are required.
In this work our averaging results are formulated in classical
mathematics. We prove them within \emph{Internal Set Theory} (IST) \cite{Nelson} which is an axiomatic approach to
\emph{Nonstandard Analysis} (NSA) \cite{Robinson}.
The idea to use NSA in perturbation theory of differential
equations goes back to the 1970s with the Reebian school. Relative
to this use, among many works we refer the interested reader, for
instance, to
\cite{b2,vdb,DD,Lobry-Sari,Lobry-SariIJC,Lobry-Sari-Touhami,LST,Lutz,Lutz-Sari,Sari-Yadi,y}
and the references therein. It has become today a well-established tool
in asymptotic theory of differential equations.
Among the famous discoveries of the nonstandard asymptotic theory of differential equations
we can cite the \emph{canards}
which appear in slow-fast vector fields and are closely
related to the stability loss delay phenomenon in dynamical bifurcations
\cite{bcdd,b1,bfsw,bs0,bs,DD,fr,ga,W,Wech}.
The structure of the paper is as follows. In Section
\ref{Notations and Main Results} we introduce the notations and
present our main results : Theorems \ref{theorem21},
\ref{theorem24} and~\ref{theorem28}. We discuss also both periodic
and almost periodic special cases. In Section~\ref{The
Stroboscopic Method} we start with a short tutorial to NSA and
then state our main (nonstandard) tool, the so-called
\emph{stroboscopic method}. In Section \ref{Proofs of the
Results}, we give the proofs of Theorems \ref{theorem21},
\ref{theorem24} and \ref{theorem28}. Some of the auxiliary results
can be found, for instance, in \cite{LakibThesis,Sari1,Sari2}. They are
included here to keep the paper self-contained for the benefit of
the reader. In Section \ref{discussion} we say exactly what are
the results that are already proved in our previous articles and
we discuss the differences with the previous works.
Let us notice that our proofs do not need to be translated
into classical mathematics, because IST is a
conservative extension of ZF, that is, any
classical statement which is a theorem of IST is also a
theorem of ZFC.
\section{Notation and Main Results}
\label{Notations and Main Results}
In this section we will present our main results on averaging for
fast oscillating ODEs \eqref{eq15},
RFDEs in normal form \eqref{eq18} and
fast oscillating RFDEs \eqref{eq17}. First we introduce some
necessary notations.
We assume that $r\geq 0$ is a fixed real
number and denote by
$\mathcal{C}=\mathcal{C}([-r,0],\mathbb{R}^d)$ the Banach space of
continuous functions from $[-r,0]$ into $\mathbb{R}^d$ with the
norm $\|\phi\|=\sup\{|\phi(\theta)|: \theta\in[-r,0]\}$, where
$|\cdot|$ is a norm of $\mathbb{R}^d$. Let $L\geq 0$. If
$x:[-r,L]\to\mathbb{R}^d$ is a continuous function then,
for each $t\in[0,L]$, we define $x_t\in \mathcal{C}$ by setting
$x_t(\theta)=x(t+\theta)$ for all $\theta\in [-r,0]$. Note that
when $r=0$ the Banach space $\mathcal{C}$ can be identified with
$\mathbb{R}^d$ and $x_t$ with $x(t)$ for each $t\in[0,L]$.
\subsection{Averaging for ODEs}
\label{Averaging for ODEs}
Let $U$ be an open subset of $\mathbb{R}^d$ and let $f:\mathbb{R}_+\times U\to\mathbb{R}^d$,
$(t,x)\mapsto f(t,x)$, be a continuous function. Let
$x_0\in U$ be an initial condition. We consider the initial
value problem
\begin{equation}
\label{eq21}
\dot{x}(t)=f\left(t/\varepsilon,x(t)\right),\quad x(0)=x_0,
\end{equation}
where $\varepsilon>0$ is a small parameter.
We state the precise assumptions on this problem in the
following definition.
\begin{definition}\label{definition0}\rm
A vector field $f:\mathbb{R}_+\times U\to\mathbb{R}^d$ is said to be a KBM-vector field
if it is continuous and satisfies the following conditions
\begin{itemize}
\item[(C1)]
The function $f$ is continuous in the second variable uniformly
with respect to the first one.
\item[(C2)]
For all $x\in U$, there exists a limit
$
F(x):=\lim_{T\to\infty}\frac{1}{T}\int_0^Tf(t,x)dt$.
\item[(C3)]
The initial value problem
\begin{equation}
\label{eq22}
\dot y(t)=F(y(t)),\quad y(0)=x_0
\end{equation}
has a unique solution.
\end{itemize}
\end{definition}
Notice that from conditions (C1) and (C2) we deduce that the
average of the function $f$, that is, the function
$F:U\to\mathbb{R}^d$ in \text{(C2)}, is
continuous (see Lemma~\ref{lemma41}). So, the averaged
initial value problem \eqref{eq22} is well defined.
The main theorem of this section is on averaging for fast
oscillating ODEs. It
establishes nearness of solutions of \eqref{eq21} and
\eqref{eq22} on finite time intervals, and reads as follows.
%%%%%%%%%%%%%%%
\begin{theorem} \label{theorem21}
Let $f:\mathbb{R}_+\times U\to\mathbb{R}^d$ be a
KBM-vector field. Let $x_0\in U$. Let $y$ be the solution of
\eqref{eq22} and let $L\in J$, where $J$ is the positive
interval of definition of $y$. Then, for every $\delta>0$, there
exists $\varepsilon_0=\varepsilon_0(L,\delta)>0$ such that, for
all $\varepsilon\in(0,\varepsilon_0]$, every solution $x$ of
\eqref{eq21} is defined at least on $[0,L]$ and satisfies
$|x(t)-y(t)|<\delta$ for all $t\in [0,L]$.
\end{theorem}
Let us discuss now the result above when the function $f$ is
periodic or more generally almost periodic in the first variable.
We will see that some of the conditions in Theorem \ref{theorem21}
can be removed. Indeed, in the case where $f$ is periodic in~$t$,
from continuity plus periodicity properties one can easily deduce
condition (C1). Periodicity also implies condition (C2) in an
obvious way. The average of $f$ is then given, for every
$x\in U$, by
\begin{equation}\label{periodic}
F(x)=\frac{1}{T}\int_0^Tf(t,x)dt,
\end{equation}
where $T$ is the period.
In the case where $f$ is almost periodic in $t$ it is well-known that for
all $x\in U$, the limit
\begin{equation}\label{almostperiodic}
F(x)=\lim_{T\to\infty}\frac{1}{T}\int_s^{s+T}f(t,x)dt
\end{equation}
exists uniformly with respect to $s\in\mathbb{R}$. So, condition
(C2) is satisfied when \text{$s=0$}. We point out also that in a
number of cases encountered in applications the function $f$ is a
finite sum of periodic functions in $t$. As in the periodic case
above, condition~(C1) is satisfied. Hence we have the following
result.
\begin{corollary}[Periodic and Almost periodic cases]
The conclusion of Theorem~\ref{theorem21} holds when
$f:\mathbb{R}_+\times U\to\mathbb{R}^d$ is a
continuous function which is periodic (or a sum of periodic
functions) in the first variable and satisfies condition (C3). It
holds also when $f$ is continuous, almost periodic in the first
variable and satisfies conditions (C1) and (C3).
\end{corollary}
\begin{remark}\label{rem1} \rm
In the results of the classical literature,
for instance \cite[Theorem 1, p. 202]{Lehman},
it is assumed that $f$ has bounded partial
derivatives with respect to the second variable.
\end{remark}
\subsection{Averaging for RFDEs in normal form}
\label{Averaging for RFDEs in normal form}
This section concerns the use of the method of averaging to
approximate initial value problems of the form
\begin{equation}
\label{eq23}
\dot{x}(t)= f(t/\varepsilon,x_{t,\varepsilon}),
\quad x(t)=\phi(t/\varepsilon), t\in[-\varepsilon r,0].
\end{equation}
Here $f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$,
$(t,x)\mapsto f(t,x)$, is a continuous function, $\Omega=\mathcal{C}([-r,0],U)$, $r>0$, where $U$ is an open subset of $\mathbb{R}^d$, $\phi\in
\Omega$ is an initial condition and $\varepsilon>0$ is a
small parameter. For each $t\geq 0$, $x_{t,\varepsilon}$ denotes
the element of $\mathcal{C}$ given by
$x_{t,\varepsilon}(\theta)=x(t+\varepsilon\theta)$ for all
$\theta\in[-r,0]$.
We recall that the change of time scale $t=\varepsilon\tau$ transforms
\eqref{eq23} into the following initial
value problem, associated to a RFDE in normal form :
\[
x'(\tau)=\varepsilon f(\tau,x_\tau), \quad x_0=\phi.
\]
\begin{definition}\label{KBMvf} \rm
A vector field $f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ is said to be a KBM-vector field
if it is continuous and satisfies the following conditions.
\begin{itemize}
\item[(H1)]
The function $f$ is continuous in the second variable uniformly
with respect to the first one.
\item[(H2)]
The function $f$ is quasi-bounded in the second variable
uniformly with respect to the first one, that is, for every
compact subset $W\subset U$, $f$ is
bounded on $\mathbb{R}_+\times \Lambda$, where
$\Lambda=\mathcal{C}([-r,0],W)$.
\item[(H3)]
For all $x\in \Omega$, the limit $
F(x):=\lim_{T\to\infty}\frac{1}{T}\int_0^Tf(t,x)dt$
exists.
\item[(H4)]
The initial value problem
\begin{equation}
\label{eq24}
\dot y(t)=G(y(t)),\quad y(0)=\phi(0)
\end{equation}
has a unique solution. Here $G:U\to\mathbb{R}^d$ is defined by
$G(x)=F(\tilde{x})$ where, for each
$x\in U$, $\tilde{x}\in\Omega$ is given by
$\tilde{x}(\theta)=x$, $\theta\in[-r,0]$.
\end{itemize}
\end{definition}
As we will see later, condition (H2) is used essentially to prove
continuability of solutions of \eqref{eq23} at least on
every finite interval of time on which the solution of
\eqref{eq24} is defined. For more details and a complete
discussion about quasi-boundedness and its crucial role in the
continuability of solutions of RFDEs, we refer the reader to
Sections 2.3 and 3.1 of \cite{Hale-Lunel2}.
In assumption (H4) we anticipate the existence of solutions of
\eqref{eq24}. This will be justified a posteriori by Lemma
\ref{lemma41} where we show that the function
$F:\Omega\to\mathbb{R}^d$ in (H3), which is the
average of the function $f$, is continuous. This implies the
continuity of $G:U\to\mathbb{R}^d$ in
\eqref{eq24} and then guaranties the existence of solutions.
The result below is our main theorem on averaging for RFDEs in
normal form. It states closeness of solutions of
\eqref{eq23} and \eqref{eq24} on finite time
intervals.
\begin{theorem} \label{theorem24}
Let $f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ be a
KBM-vector field. Let $\phi\in\Omega$. Let $y$ be the
solution of \eqref{eq24} and
let $L\in J$, where $J$ is the positive interval of definition of
$y$. Then, for every $\delta>0$, there exists
$\varepsilon_0=\varepsilon_0(L,\delta)>0$ such that, for all
$\varepsilon\in(0,\varepsilon_0]$, every solution $x$ of
\eqref{eq23} is defined at least on $[-\varepsilon r,L]$ and
satisfies $|x(t)-y(t)|<\delta$ for all $t\in [0,L]$.
\end{theorem}
As in Section \ref{Averaging for ODEs}, we discuss now both periodic
and almost periodic special cases. In
each one, some of the conditions in Theorem~\ref{theorem24} can be
either removed or weakened. Let us consider the following (weak) condition
which will be used hereafter instead of condition (H2):
\begin{itemize}
\item[(H5)]
The function $f$ is quasi-bounded, that is, for every compact
interval $I$ of $\mathbb{R}_+$ and every compact subset $W\subset U$, $f$ is bounded on $I\times
\Lambda$, where $\Lambda=\mathcal{C}([-r,0],W)$.
\end{itemize}
When $f$ is periodic it is easy to see that condition (H1) derives from the continuity
and the periodicity properties of $f$. On the other
hand, by periodicity and condition (H5), condition
(H2) is also satisfied. The average $F$ in condition (H3) exists and is now given by formula
\eqref{periodic} where $T$ is the period.
When $f$ is almost periodic, condition (H5) imply condition (H2) and the average $F$ is given by formula
\eqref{almostperiodic}.
Quite often the function $f$ is a finite sum of periodic functions
so that condition (H1) is satisfied. Hence we have the following
result.
%%%%%%%%%%%%%%%
\begin{corollary}[Periodic and Almost periodic cases]
\label{theorem25}
The conclusion of Theorem \ref{theorem24} holds
when $f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ is a
continuous function which is periodic (or a sum of periodic
functions) in the first variable and satisfies condition (H4) and
(H5). It holds also when $f$ is continuous, almost periodic in the
first variable and satisfies conditions (H1), (H4) and (H5).
\end{corollary}
Consider now the special case of equations with point-wise
delay of the form
\[
\dot{x}(t)=f(t/\varepsilon,x(t), x(t-\varepsilon r))
\]
which is obtained, by letting $\tau=t/\varepsilon$, from equation
\[
x'(\tau)=\varepsilon f(\tau,x(\tau), x(\tau-r)).
\]
In this case,
for both periodic and almost
periodic functions, condition
(H5) follows from the continuity property and then may
be removed in Corollary \ref{theorem25}.
\begin{remark}\label{rem2} \rm
In the results of the
literature, for instance \cite[Theorem 3, p. 206]{Lehman},
$f$ is assumed to be locally Lipschitz with
respect to the second variable.
Note that local Lipschitz condition with respect to the second
variable implies condition (H1). It also assures the local
existence for the solution of \eqref{eq23}. But, in
opposition to the case of ODEs, it is well known (see Sections 2.3
and 3.1 of \cite{Hale-Lunel2} that without condition
(H5) one cannot extend the solution $x$ to finite time
intervals where the solution $y$ is defined in spite of the
closeness of $x$ and $y$. So, in the existing literature it is
assumed that the solutions $x$ and $y$ are both defined
at least on the same interval $[0,L]$.
\end{remark}
\begin{remark}\label{rem3} \rm
In the introduction, we noticed that
Lehman and Weibel \cite{Lehman-Weibel} proposed to retain the delay in the averaged
equation (\ref{lwa}). At time scale
$t=\varepsilon\tau$, their observation is that equation
\eqref{eq23} is better approximated by the averaged RFDE
\begin{equation}\label{lwat}
\dot{y}(t)=F(y_{t,\varepsilon})
\end{equation}
than by the averaged ODE \eqref{eq24}. It should be noticed that
the averaged RFDE \eqref{lwat} depends on the small parameter $\varepsilon$, which is not
the case of the averaged equation \eqref{eq24}.
\end{remark}
\subsection{Averaging for fast oscillating RFDEs}
The aim here is to approximate the solutions of the initial value problem
\begin{equation}
\label{eq25}
\dot{x}(t)=
f\left(t/\varepsilon,x_t\right),\quad x_0=\phi,
\end{equation}
where $f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$,
$(t,x)\mapsto f(t,x)$, is a continuous function, $\Omega=\mathcal{C}([-r,0],U)$, $r>0$, where $U$ is an open subset of $\mathbb{R}^d$,
$\phi\in\Omega$ is an initial condition and $\varepsilon>0$ is a
small parameter.
\begin{definition} \rm
A vector field
$f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ is said to be a KBM-vector field
if it is continuous and satisfies conditions (H1), (H2), (H3) in Definition \ref{KBMvf} and the following condition
\begin{itemize}
\item[(H6)]
The initial value problem
\begin{equation}
\label{eq26}
\dot y(t)=F(y_t),\quad y_0=\phi
\end{equation}
has a unique solution.
\end{itemize}
\end{definition}
The averaged initial value problem \eqref{eq26} associated to
\eqref{eq25} is well defined since conditions (H1) and (H3) imply the
continuity of the function $F:\Omega\to\mathbb{R}^d$ in (H3).
We may state our main result on averaging for fast oscillating RFDEs.
It shows that the solution of \eqref{eq26}
is an approximation of solutions of \eqref{eq25}
on finite time intervals.
\begin{theorem} \label{theorem28}
Let $f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ be a
KBM-vector field. Let $\phi\in\Omega$. Let $y$
be the solution of \eqref{eq26} and let $L\in J$ be
positive, where $J$ is the interval of definition of $y$. Then,
for every $\delta>0$, there exists
$\varepsilon_0=\varepsilon_0(L,\delta)>0$ such that, for all
$\varepsilon\in(0,\varepsilon_0]$, every solution $x$ of
\eqref{eq25} is defined at least on $[-r,L]$ and satisfies
$|x(t)-y(t)|<\delta$ for all $t\in [0,L]$.
\end{theorem}
In the same manner as in Section \ref{Averaging for RFDEs in
normal form} we have the following
result corresponding to the periodic and almost periodic special
cases.
\begin{corollary}[Periodic and Almost periodic cases]
\label{theorem29}
The conclusion of Theorem \ref{theorem28} holds
when $f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ is a
continuous function which is periodic (or a sum of periodic
functions) in the first variable and satisfies condition (H5) and
(H6). It holds also when $f$ is continuous, almost periodic in the
first variable and satisfies conditions (H1), (H5) and (H6).
\end{corollary}
For fast oscillating equations with
point-wise delay of the form
\[
\dot{x}(t)=f(t/\varepsilon,x(t), x(t-r)),
\]
in the periodic case as well as in the almost
periodic one, condition (H5) derives from the
continuity property and then can be removed in
Corollary \ref{theorem29}.
\begin{remark}\label{rem4} \rm
In the results of the classical literature, for instance, \cite[Theorem 4, p. 210]{Lehman},
it is assumed that $f$ is locally Lipschitz with
respect to the second variable and the existence of the solutions
$x$ and $y$ on the same interval $[0,L]$ is required.
\end{remark}
\section{The Stroboscopic Method}
\label{The Stroboscopic Method}
\subsection{Internal Set Theory}
\label{IST:A short tutorial}
In this section we give a short tutorial of NSA.
Additional information can be found in
\cite{vdb,DD,Nelson,Robinson}.
\emph{Internal Set Theory} (IST) is a theory extending ordinary
mathematics, say ZFC (Zermelo-Fraenkel set theory with the axiom
of choice), that axiomatizes (Robinson's) \emph{nonstandard
analysis} (NSA). We adjoin a new undefined unary predicate
\emph{standard} (st) to ZFC. In addition to the usual axioms of
ZFC, we introduce three others for handling the new predicate in a
relatively \emph{consistent} way. Hence \emph{all theorems} of ZFC
\emph{remain valid} in IST. What is new in IST is an addition, not
a change. In the external formulas, we use the following abbreviations \cite{Nelson}~:
$$\forall^{\rm st} A \text{ for }\forall x({\rm st} x \Rightarrow A)
\quad\text{and}\quad \exists^{\rm st} A\text{ for }\exists x
({\rm st}x~\& A).$$
A real number $x$ is said to be \emph{infinitesimal} if $|x|0~|x|\leq a$.
By idealization, there exists a standard and finite set $a'$ such that
$\forall x\in A~\exists a\in a'~|x|\leq a$.
Let $a=\max(a')$. Then $\forall x\in A~|x|\leq a$. Hence $A\subset F=\{x\in \mathbb{R}^d:|x|\leq a\}$, from where we deduce that
${}^oA\subset{}^oF=F$. This proves that ${}^oA$ is bounded.
Finally, we conclude that ${}^oA$ is compact in $\mathbb{R}^d$ since it is closed and bounded.
\emph{Step 2.} \emph{We show that ${}^oA\subset U$}.
Let $x$ be standard in ${}^oA$. Let $y\in A$ such that $y\simeq x$. Since $A$ is near-standard in $U$ and $x$ is the standard part of $y$, we have $x\in U$. By transfer we deduce that every $x\in{}^oA$ belongs to $U$. Thus ${}^oA\subset U$.
\emph{Step 3.} \emph{We show that there exists a standard and compact set $W$ such that
$A\subset W\subset U$}. Let $W$ be the standard and compact neighborhood, around the standard and compact set ${}^oA$,
given by
$W=\{y\in\mathbb{R}^d/\ \exists x\in {}^oA: |x-y|\leq\rho\}$,
for some standard $\rho>0$ chosen such that $W\subset U$.
Let $x\in A$ and $x_0\in U$, $x_0$ standard, such that $x\simeq x_0$. Thus $x_0\in {}^oA$.
Hence $x\in W$, since $W$ is a standard neighborhood of $x_0$, which proves that $A\subset W$.
\end{proof}
Let $I\subset \mathbb{R}$ be some interval and
$f:I\to\mathbb{R}^d$ be a function, with $d$ standard. We
say that $f$ is \emph{S-continuous at} a standard point $x\in I$ if,
for all $y\in I$, $y\simeq x$ implies $f(y)\simeq f(x)$, $f$ is
\emph{S-continuous on} $I$ if $f$ is S-continuous at each standard
point of $I$ and $f$ is \emph{S-uniformly-continuous on} $I$
if, for all $x,y\in I$, $x\simeq y$ implies $f(x)\simeq f(y)$.
If $I$ is standard and compact, S-continuity on $I$ and S-uniform-continuity
on $I$ are the same. When $f$ (and then
$I$) is standard, the first definition is the same as
saying that $f$ is continuous at a standard point $x$,
the second definition corresponds to the continuity of $f$ on $I$
and the last one to the uniform continuity
of $f$ on $I$.
We need the following result on S-uniformly-continuous functions on compact intervals of $\mathbb{R}$.
This result is a particular case of the
so-called ``Continuous Shadow Theorem" \cite{DD}.
\begin{theorem} \label{theorem31}
Let $I$ be a standard and compact interval of
$\mathbb{R}$, $D$ be a standard subset of $ \mathbb{R}^d$ (with
$d$ standard) and $x:I\to D$ be a function. If
$x$ is S-uniformly-continuous on $I$ and for each $t\in I$,
$x(t)$ is near-standard in $D$ then there exists
a standard and continuous function $y:I\to D$ such that,
for all $t\in I$, $x(t)\simeq y(t)$.
\end{theorem}
The function $y$ in Theorem \ref{theorem31} is unique. It is
defined as the unique standard function $y$ which, for $t$
standard in $I$, is given by $y(t)={}^ox(t)$. The function $y$ is
called the \emph{standard part} or \emph{shadow} of the function
$x$ and denoted by $y={}^ox$.
\subsection{The Stroboscopic Method for ODEs}
Let $U$ be a standard open subset of $\mathbb{R}^d$.
Let $x_0\in U$ be standard and let $F:\mathbb{R}_+\times
U\to\mathbb{R}^d$ be a standard and continuous
function. Let $I$ be some subset of $\mathbb{R}$ and let
$x:I\to U$ be a function such that $0\in I$ and
$x(0)\simeq x_0$.
\begin{definition}[\emph{F}-Stroboscopic property]
\label{definition1} \rm
A real number $t\geq 0$ is said to be an
\emph{instant of observation} if $t$ is limited, $[0,t]\subset I$
and $x(s)$ is near standard in $U$ for all $s\in [0,t]$.
The function $x$ is said to satisfy the
\emph{F}-Stroboscopic property on $I$ if there exists $\mu>0$ such that,
for all instant of observation $t\in I$, there exists $t'\in I$ such
that $\mu0$ be limited such that $[0,L]\subset I$.
Suppose that
\begin{itemize}
\item[(i)] For all $t\in[0,L]$, $x(t)$ is near-standard in $U$.
\item[(ii)]
There exist some positive integer $N$ and some infinitesimal
partition $\{t_n:n=0,\ldots,N+1\}$ of $[0,L]$ such that $t_0=0$,
$t_N\leq L0$ be standard such that $[0,L]\subset I$.
Suppose that conditions (i) and (ii) in Lemma \ref{lemma31} hold.
Then the shadow $y={}^ox$ of the function $x$ is a solution
of \eqref{eq31}.
Moreover, the approximation $x(t)\simeq y(t)$ holds for all
$t\in[0,L]$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma31} the function $x$ is S-uniformly-continuous on
$[0,L]$. From hypothesis (i) and Theorem \ref{theorem31} we deduce that
$y$ is continuous on $[0,L]$ and $y(t)\simeq x(t)$ for all $t\in[0,L]$.
Let us show now that the function $y$ is a solution of \eqref{eq31}, that is, for all $t\in[0,L]$
it satisfies
\[
y(t)=x_0+\int_0^t F(s,y(s))ds.
\]
Let $t\in [0,L]$ be standard and let $n\in\{0,\ldots,N\}$ be such that
$t\in[t_n,t_{n+1}]$. Then
\begin{equation}\label{eq33}
\begin{aligned}
y(t)-x_0 \simeq x(t_n)-x(0)
&= \sum_{k=0}^{n-1}(x(t_{k+1})-x(t_k))\nonumber\\
& = \sum_{k=0}^{n-1}(t_{k+1}-t_k)[F(t_k,x(t_k))+\eta_k],
\end{aligned}
\end{equation}
where $\eta_k\simeq 0$ for all $k\in\{0,\ldots,n-1\}$. As $F$ is
standard and continuous, and $x(t_k)\simeq y(t_k)$
with $x(t_k)$ near-standard in $U$, we have $F(t_k,x(t_k))=F(t_k,y(t_k))+\beta_k$
where $\beta_k\simeq 0$ for all $k\in\{0,\ldots,n-1\}$.
Hence \eqref{eq33} gives
\[
y(t)-x_0 \simeq
\sum_{k=0}^{n-1}(t_{k+1}-t_k)[F(t_k,y(t_k))+\beta_k+\eta_k]
\simeq \int_0^t F(s,y(s))ds.
\]
Thus the
approximation
\begin{equation}
\label{eq34}
y(t)\simeq x_0+ \int_0^t F(s,y(s))ds
\end{equation}
holds for all standard $t\in[0,L]$. Actually \eqref{eq34} is
an equality since both sides of which are standard. We have thus,
for all standard $t\in[0,L]$,
\begin{equation}
\label{eq35}
y(t)=x_0+\int_0^t F(s,y(s))ds
\end{equation}
and by transfer \eqref{eq35} holds for all $t\in[0,L]$. The
proof is complete.
\end{proof}
The following statement is a consequence of Lemma \ref{lemma32}.
%%%%%%%%%%%%%
\begin{lemma}
\label{lemma33}
Let $L>0$ be standard such that $[0,L]\subset I$.
Suppose that
\begin{itemize}
\item[(i)]
For all $t\in[0,L]$, $x(t)$ is near-standard in $U$.
\item[(ii)] The function $x$ satisfies the
F-stroboscopic property on $[0,L]$ (Definition \ref{definition1}).
\end{itemize}
Then the function $x$ is S-uniformly-continuous on $[0,L]$ and
its shadow is a solution $y$ of \eqref{eq31}. So, we have
$x(t)\simeq y(t)$ for all $t\in[0,L]$
\end{lemma}
\begin{proof}
First of all we have $\lambda\in A_\mu$ for all standard real
number $\lambda>0$, where $A_\mu$ is the subset of $\mathbb{R}$
defined by
$ A_\mu=\{\lambda\in\mathbb{R}\ /\
\forall t\in[0,L]\ \exists t'\in I:
\mathcal{P}_\mu(t,t',\lambda)\}
$
and $\mathcal{P}_\mu(t,t',\lambda)$ is the property
$$
\mu\mu$} for all $t\in[0,L]$, there are a
positive integer $N$ and an infinitesimal partition $\{t_n:
n=0,\ldots,N+1\}$ of $[0,L]$ such that $t_0=0$, $t_N\leq
L0$ be standard in $J$.
Fix $\rho>0$ to be standard such that the (standard)
neighborhood around
$\Gamma=\{y(t):t\in[0,L]\}$ given by
$W=\{z\in\mathbb{R}^d\ /\ \exists t\in[0,L]:
|z-y(t)|\leq\rho\}$ is included in $U$.
Let $A$ be the subset of $[0,L]$ defined by
\[
A=\{L_1\in[0,L]\ /\ [0,L_1]\subset I\text{ and } \{x(t):
t\in[0,L_1]\}\subset W\}.
\]
The set $A$ is nonempty ($0\in A$) and bounded above by $L$. Let
$L_0$ be the upper bound of $A$ and let $L_1\in A$ be such that
$L_0-\muL_1+\mu$, $[L_1,L_1']\subset I$ and $x(t)\simeq \ y(t)$
for all $t\in[L_1,L_1']$. By \eqref{eq37} we have
$x(t)\simeq y(t)$ for all $t\in[0,L_1']$.
It remains to verify that $L\leq L_1'$. If this is not true, then
$[0,L_1']\subset I$ and $\{x(t): t\in[0,L_1']\}\subset W$ imply
$L_1'\in A$. This contradicts the fact that $L_1'>L_0$. So the
proof is complete.
\subsection{The Stroboscopic Method for RFDEs}\label{section3.3}
Let $r\geq 0$ be standard. Let $\Omega=\mathcal{C}([-r,0],U)$, where $U$ is a standard open subset of $\mathbb{R}^d$ and $\phi\in\Omega$ be standard. Let $F:\mathbb{R}_+\times
\Omega\to\mathbb{R}^d$ be a standard and continuous
function. Let $I$ be some subset of $\mathbb{R}$ and let
$x:I\to U$ be a function such that
$[-r,0]\subset I$, $x_0\simeq\phi$ and, for each $t\in I$, $t\geq 0$, $x_t\in\Omega$.
%%%%%%%%%%%%%%%%%%
\begin{definition}[\emph{F}-Stroboscopic property]
\label{definition2} \rm
A real number $t\geq0$ is said to be an instant of observation if
$t$ is limited, $[0,t]\subset I$ and for all $s\in[0,t]$,
$x(s)$ is near-standard in $U$ and $F(s,x_s)$ is limited.
The function $x$ is said to satisfy the
\emph{F}-Stroboscopic property on $I$ if there exists $\mu>0$ such that,
for all instant of observation $t$, there
exists $t'\in I$ such that $\mu0$ be limited such that $[0,L]\subset I$.
Suppose that
\begin{itemize}
\item[(i)]
For all $t\in[0,L]$,
$x(t)$ is near-standard in $U$ and $F(t,x_t)$ is limited.
\item[(ii)]
There exist some positive integer $N$ and some infinitesimal
partition $\{t_n:n=0,\ldots,N+1\}$ of $[0,L]$ such that $t_0=0$,
$t_N\leq L0$ be standard such that $[0,L]\subset I$.
Suppose that conditions (i) and (ii) in Lemma \ref{lemma34} are
satisfied. Then the shadow
$y={}^ox$, of the function $x$ is a solution of \eqref{eq39} and
satisfies
\begin{equation}
\label{eq311}
x(t)\simeq y(t),\quad \forall t\in[0,L].
\end{equation}
\end{lemma}
\begin{proof}
The proof is the same as the proof of Lemma \ref{lemma32}.
Notice that by \eqref{eq311} we obtain that for all $t\in[0,L]$,
$x_t$ is near-standard
in $\Omega$ with $x_t\simeq y_t$. The details are omitted.
\end{proof}
From Lemma \ref{lemma35} we deduce the result below.
\begin{lemma} \label{lemma36}
Let $L>0$ be standard such that $[0,L]\subset I$.
Suppose that
\begin{itemize}
\item[(i)]
For all $t\in[0,L]$, $x(t)$ is near-standard in $U$ and
$F(t,x_t)$ is limited.
\item[(ii)] The function $x$ satisfies the
F-stroboscopic property on $[0,L]$ (Definition \ref{definition2}).
\end{itemize}
Then the function $x$ is S-uniformly-continuous on $[0,L]$ and
its shadow is a solution of \eqref{eq39} and satisfies
approximation \eqref{eq311}.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{lemma33}, we obtain a function
$c:[0,L]\to I$ satisfying, for all $t\in[0,L]$,
$$\mu0$ be standard in $J$ and let $W_0\subset U$
be the standard neighborhood around
$\Gamma_0=\{y(t):t\in[0,L]\}$ defined by
$W_0=\{z\in\mathbb{R}^d\ /\ \exists t\in[0,L]: |z-y(t)|\leq\rho_0\}$,
where $\rho_0>0$ is a given standard real number.
Now, since
$F$ is standard and continuous, and $[0,L]\times\Gamma$ is a
standard compact subset of $\mathbb{R}_+\times\Omega$,
where $\Gamma=\{y_t:t\in[0,L]\}$, there
exists ${\rho}>0$ and standard such that $F$ is limited on
$[0,L]\times W$, where ${W}\subset \Omega$
is the standard neighborhood
around $\Gamma$ given by
$
{W}=\{z\in\mathcal{C}\ /\ \exists t\in[0,L]: |z-y_t|\leq{\rho}\}.
$
Consider the set
\[
A=\{L_1\in[0,L]\,/\, [0,L_1]\subset I, \, \{x(t):
t\in[0,L_1]\}\subset W_0 \text{ and } \{x_t:
t\in[0,L_1]\}\subset {W}\}.
\]
The set $A$ is nonempty ($0\in A$) and bounded
above by $L$. Let $L_1\in A$ such that $L_0-\muL_1+\mu$, $[L_1,L_1']\subset I$ and
$x(t)\simeq \ y(t)$, for all
$t\in[L_1,L_1']$.
Combining with \eqref{eq312} yields
\begin{equation}
\label{eq314} x(t)\simeq y(t),\quad \forall t\in[0,L_1'].
\end{equation}
Now, taking into account that $x_0\simeq\phi=y_0$, from
\eqref{eq314} we deduce that $x_t\simeq y_t$ for all
$t\in[0,L_1']$.\\
It remains to verify that $L\leq L'_1$. Assume that $L_1'\leq L$. Then $[0,L_1']\subset
I$, $\{x(t): t\in[0,L_1']\}\subset W_0$ and $\{x_t:
t\in[0,L_1']\}\subset W$. This implies $L_1'\in A$, which is
absurd since $L_1'>L_0$. Thus $L_1'>L$. Finally, for any standard
$L\in J$ we have shown that $x(t)\simeq y(t)$ for all
$t\in[0,L]\subset[0,L_1']$. This completes the proof of the
theorem.
\section{Proofs of the Results}
\label{Proofs of the Results}
We prove Theorems \ref{theorem21}, \ref{theorem24} and
\ref{theorem28} within IST. By {\rm transfer} it suffices to prove
those results for {\em standard data} $f$, $x_0$ and $\phi$. We
will do this by applying Stroboscopic Lemma for ODEs (Theorem
\ref{theorem32}) in both cases of Theorems \ref{theorem21} and
\ref{theorem24}, and Stroboscopic Lemma for RFDEs
(Theorem~\ref{theorem33}) in case of Theorem \ref{theorem28}. For
this purpose we need first to translate all conditions (C1) and
(C2) in Section \ref{Averaging for ODEs}, and (H1), (H2) and (H3)
in Section~\ref{Averaging for RFDEs in normal form} into their
external forms and then prove some technical lemmas.
Let $U$ be a standard open subset of
$\mathbb{R}^d$ and let $\Omega=\mathcal{C}([-r,0],U)$, where
$r\geq 0$ is standard. Let $f:\mathbb{R}_+\times
U\to\mathbb{R}^d$ or
$f:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ be a standard
and continuous function.
The external formulations of conditions (C1) and
(C2) are:
\begin{itemize}
\item[(C1')]
$\forall^{\rm st} x\in U$ $\forall x'\in U$
$\forall t\in\mathbb{R}_+$ $\big(x'\simeq x \Rightarrow
f(t,x')\simeq f(t,x)\big).$
\item[(C2')]
$\exists^{\rm st}F: U\to\mathbb{R}^d$ $ \forall^{\rm st}
x\in U$ $\forall R\simeq +\infty$
$F(x)\simeq\frac{1}{R}\int_0^Rf(t,x)dt$.
\end{itemize}
The external formulation of conditions (H1), (H2) and (H3) are,
respectively:
\begin{itemize}
\item[(H1')]
$\forall^{\rm st} x\in\Omega$ $\forall x'\in\Omega$ $\forall
t\in\mathbb{R}_+$ $\big(x'\simeq x \Rightarrow f(t,x')\simeq
f(t,x)\big).$
\item[(H2')]
$\forall^{\rm st} W$ compact, $W\subset U$, $\forall t\in\mathbb{R}_+$, $\forall x\in\Lambda=\mathcal{C}([-r,0],W)$,
$f(t,x)$ is limited.
\item[(H3')]
$\exists^{\rm st}F:\Omega\to\mathbb{R}^d$ $ \forall^{\rm st}
x\in\Omega$ $\forall R\simeq +\infty$
$F(x)\simeq\frac{1}{R}\int_0^Rf(t,x)dt$.
\end{itemize}
\subsection{Technical Lemmas}
In Lemmas \ref{lemma41} and \ref{lemma42} below we formulate some
properties of the average $F$ of the function $f$ defined in (C2) and~(H3).
\begin{lemma}
\label{lemma41} Suppose that the function $f$ satisfies conditions
(C1) and (C2) when $r=0$ and conditions (H1) and (H3) when $r>0$.
Then the function $F$ in (C2) or (H3) is continuous and satisfies
\[
F(x)\simeq\frac{1}{R}\int_0^R f(t,x)dt
\]
for all $x\in U$ or $x\in\Omega$, $x$ near-standard in $U$ or in $\Omega$ and all $R\simeq +\infty$.
\end{lemma}
\begin{proof} The proof is the same in both cases $r=0$ and $r>0$. So,
there is no restriction to suppose that $r=0$. Let $x,
{}^ox\in U$ be such that ${}^ox$ is standard and
$x\simeq {}^ox$. Fix $\delta>0$ to be infinitesimal. By condition
(C2) there exists $T_0>0$ such that
\[
\Big|F(x)-\frac{1}{T}\int_0^Tf(t,x)dt\Big|<\delta,\quad \forall
T>T_0.
\]
Hence there exists $T\simeq+\infty$ such
that
\[
F(x)\simeq\frac{1}{T}\int_0^T f(t,x)dt.
\]
By condition (C1') we have $f(t,x)\simeq f(t,{}^ox)$ for all
$t\in\mathbb{R}_+$. Therefore
\[
F(x)\simeq\frac{1}{T}\int_0^T f(t,{}^ox)dt.
\]
By condition (C2') we deduce that $F(x)\simeq F({}^ox)$. Thus $F$
is continuous. Moreover, for all $T\simeq+\infty$, we have
\[
F(x)\simeq F({}^ox)\simeq\frac{1}{T}\int_0^T f(t,{}^ox)dt\simeq\frac{1}{T}\int_0^T f(t,x)dt.
\]
So, the proof is complete.
\end{proof}
\begin{lemma}
\label{lemma42} Suppose that the function $f$ satisfies
conditions (C1) and (C2) when $r=0$ and conditions (H1) and (H3)
when $r>0$. Let $F$ be as in (C2) or (H3). Let $\varepsilon>0$ be
infinitesimal. Then, for all limited $t\in\mathbb{R}_+$ and all
$x\in U$ or $x\in\Omega$, $x$~near-standard in $U$ or in
$\Omega$, there exists $\alpha=\alpha(\varepsilon,t,x)$ such that
$0<\alpha\simeq 0$, $\varepsilon/\alpha\simeq 0$ and
\[
\frac{\varepsilon}{\alpha}\int_{t/\varepsilon}^{t/\varepsilon+T\alpha/\varepsilon}
f(\tau,x)d\tau\simeq TF(x), \quad \forall T\in [0,1].
\]
\end{lemma}
\begin{proof}
The proof is the same in both cases $r=0$ and $r>0$. Let $t$ be
limited in $\mathbb{R}_+$ and let $x$ be near-standard in
$\Omega$. We denote for short $g(r)= f(r,x)$. Let
$T\in[0,1]$. We consider the following two cases.
Case 1: $t/\varepsilon$ is limited. Let $\alpha>0$ be such that
$\varepsilon/\alpha\simeq 0$. If $T\alpha/\varepsilon$ is limited
then we have $T\simeq0$ and
\[
\frac{\varepsilon}{\alpha}\int_{t/\varepsilon}^{t/\varepsilon+T\alpha/\varepsilon}g(r)dr\simeq0\simeq
TF(x).
\]
If $T\alpha/\varepsilon\simeq+\infty$ we write
\[
\frac{\varepsilon}{\alpha}\int_{t/\varepsilon}^{t/\varepsilon+T\alpha/\varepsilon}g(r)dr=
\big(T+\frac{t}{\alpha}\big)
\frac{1}{t/\varepsilon+T\alpha/\varepsilon}\int_0^{t/\varepsilon+T\alpha/\varepsilon}g(r)dr-
\frac{\varepsilon}{\alpha}\int_0^{t/\varepsilon}g(r)dr.
\]
By Lemma \ref{lemma41} we have
\[
\frac{1}{t/\varepsilon+T\alpha/\varepsilon}\int_0^{t/\varepsilon+T\alpha/\varepsilon}g(r)dr\simeq
F(x).
\]
Since
$\frac{\varepsilon}{\alpha}\int_0^{t/\varepsilon}g(r)dr\simeq0$
and $t/\alpha\simeq 0$, we have
\[
\frac{\varepsilon}{\alpha}\int_{t/\varepsilon}^{t/\varepsilon+T\alpha/\varepsilon}g(r)dr\simeq
TF(x).
\]
This approximation is satisfied for all $\alpha>0$ such that
$\varepsilon/\alpha\simeq 0$. Choosing then $\alpha$ such that
$0<\alpha\simeq 0$ and $\varepsilon/\alpha\simeq 0$ gives the
desired result.
Case 2: $t/\varepsilon$ is unlimited. Let $\alpha>0$. We have
\begin{equation}
\label{eq41}
\frac{\varepsilon}{\alpha}\int_{t/\varepsilon}^{t/\varepsilon+T\alpha/\varepsilon}g(r)dr
= T\eta(\alpha)
+\frac{t}{\alpha}\left[\eta(\alpha)-\eta(0)\right],
\end{equation}
where
\[
\eta(\alpha)=\frac{1}{t/\varepsilon+T\alpha/\varepsilon}\int_0^{t/\varepsilon+T\alpha/\varepsilon}g(r)dr.
\]
By Lemma \ref{lemma41} we have $\eta(\alpha)\simeq F(x)$ for all
$\alpha\geq 0$.
Return to \eqref{eq41} and assume that $\alpha$ is not
infinitesimal. Then
\begin{equation}
\label{eq42}
\frac{\varepsilon}{\alpha}
\int_{t/\varepsilon}^{t/\varepsilon+T\alpha/\varepsilon}g(r)dr\simeq TF(x),
\end{equation}
By overspill \eqref{eq42} holds for some
$\alpha\simeq 0$ which can be chosen such that
$\varepsilon/\alpha\simeq 0$.
\end{proof}
\subsection{Proof of Theorem \ref{theorem21}}
\label{Proof1}
We need first to prove the following result which discuss some
properties of solutions of a certain ODE needed in the sequel.
\begin{lemma} \label{lemma43}
Let $\omega\in \mathbb{R}_+$, $\omega\simeq+\infty$.
Let $g:\mathbb{R}_+\times B(0,\omega)\to\mathbb{R}^d$ and
$h:\mathbb{R}_+\to\mathbb{R}^d$ be continuous functions, where
$B(0,\omega)\subset\mathbb{R}^d$ is the ball of center $0$
and radius $\omega$.
Let $x_0$ be limited in $\mathbb{R}^d$. Suppose that
\begin{itemize}
\item[(i)]
$g(t,x)\simeq h(t)$ holds for all $t\in[0,1]$ and all
$x\in B(0,\omega)$.
\item[(ii)]
$\int_0^t h(s)ds$ is limited for all $t\in[0,1]$.
\end{itemize}
Then any solution $x$ of the initial value problem
\[
\dot x=g(t,x),\ t\in[0,1];\quad x(0)=x_0
\]
is defined and limited on $[0,1]$ and satisfies
\[
x(t)\simeq x_0+\int_0^t h(s)ds,\quad \forall t\in[0,1].
\]
\end{lemma}
\begin{proof}
Assume that there exists $t\in[0,1]$ such that $x(t)\in B(0,\omega)$
and $x(t)\simeq\infty$. Then we have
\[
x(t)=x_0+\int_0^{t}
g(s,x(s))ds\simeq x_0+\int_0^{t} h(s)ds
\]
whence, in view of hypothesis (ii), $x(t)$ is limited; this is a
contradiction. Therefore $x(t)$ is defined and limited for all
$t\in[0,1]$.
\end{proof}
Let us now prove Theorem \ref{theorem21}. Assume that $x_0$ and
$L$ are standard. To prove Theorem \ref{theorem21} is equivalent
to show that, for every infinitesimal $\varepsilon>0$, every
solution $x$ of \eqref{eq21} is defined at least on $[0,L]$
and satisfies $x(t)\simeq y(t)$ for all $t\in [0,L]$. Fix
$\varepsilon>0$ to be infinitesimal and let $x:I\to U$ be
a maximal solution of~\eqref{eq21}. We claim that $x$
satisfies the \emph{F}-stroboscopic property on $I$ (Definition \ref{definition1}).
To see this, let $t_0\geq 0$ be an instant of observation of the stroboscopic
method for ODEs; that is $t_0$ is limited, $t_0\in I$ and $x(t)$
is near-standard in $U$ for all $t\in[0,t_0]$. By
Lemma~\ref{lemma42} there exists
$\alpha=\alpha(\varepsilon,t_0,x(t_0))$ such that $0<\alpha\simeq
0$, $\varepsilon/\alpha\simeq 0$ and
\begin{equation}
\label{eq43}
\frac{\varepsilon}{\alpha}\int_{t_0/\varepsilon}^{t_0/\varepsilon+T\alpha/\varepsilon}
f(t,x(t_0))dt\simeq TF(x(t_0)),\quad \forall T\in[0,1].
\end{equation}
Introduce the function
\[
X(T)=\frac{x(t_0+\alpha T)-x(t_0)}{\alpha},\quad T\in[0,1].
\]
Differentiating and substituting the above into \eqref{eq21}
gives, for $T\in[0,1]$,
\begin{equation}
\label{eq44}
\frac{dX}{dT}(T) =f\Big(\frac{t_0}{\varepsilon}+
\frac{\alpha}{\varepsilon}T,x(t_0)+\alpha X(T)\Big).
\end{equation}
By (C1') and Lemma \ref{lemma43} the function $X$, as a solution
of \eqref{eq44}, is defined and limited on $[0,1]$ and, for
$T\in[0,1]$,
\[
X(T) \simeq
\int_0^T
f\big(\frac{t_0}{\varepsilon}+\frac{\alpha}{\varepsilon}t,x(t_0)\big)dt
= \frac{\varepsilon}{\alpha}
\int_{t_0/\varepsilon}^{t_0/\varepsilon+T\alpha/\varepsilon} f(t,x(t_0))dt.
\]
Using now \eqref{eq43} this leads to the approximation
\[
X(T)\simeq TF(x(t_0)), \quad\forall T\in[0,1].
\]
Define $t_1=t_0+\alpha$ and set $\mu=\varepsilon$. Then
$\mu<\alpha=t_1-t_0\simeq 0$, $[t_0,t_1]\subset I$ and
$x(t_0+\alpha T)=x(t_0)+\alpha X(T)\simeq x(t_0)$ for all
$T\in[0,1]$, that is, $x(t)\simeq x(t_0)$ for all $t\in[t_0,t_1]$,
whereas
\[
\frac{x(t_1)-x(t_0)}{t_1-t_0}=X(1)\simeq
F(x(t_0)),
\]
which shows the claim. Finally, by (C3) and Theorem
\ref{theorem32}, the solution $x$ is defined at least on $[0,L]$
and satisfies $x(t)\simeq y(t)$ for all $t\in [0,L]$. The proof of
Theorem~\ref{theorem21} is complete.
%%%%%%%
\subsection{Proof of Theorem \ref{theorem24}}
\label{section4.3}
We start by showing the following auxiliary lemma,
which is needed to prove Lemmas~\ref{lemma44} and \ref{lemma45}.
Lemma \ref{lemma44} is used in the proof of Theorem \ref{theorem24} and
Lemma \ref{lemma45} is used in the proof of Theorem \ref{theorem25}.
\begin{lemma}
\label{lemma440} Let $U$ be a standard open subset of $\mathbb{R}^d$
and $\Omega=\mathcal{C}([-r,0],U)$, where $r\geq 0$ is standard. Let
$g:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ be a
continuous function. Suppose that
\begin{itemize}
\item[(A)]
for all standard and compact subset $W\subset U$,
all $t\in \mathbb{R}_+$ and all $x\in
\Lambda=\mathcal{C}([-r,0]),W)$, $g(t,x)$ is limited.
\end{itemize}
Let $\phi\in\Omega$ be standard. Let
$x:I=[-\varepsilon r,b)\to U$, with $b>0$, be a maximal solution of the initial
value problem
\begin{equation}
\label{eq450} \dot{x}(t) = g(t,x_{t,\varepsilon}),\quad
x(t)=\phi(t/\varepsilon),\ t\in[-\varepsilon r,0],
\end{equation}
or $x:I=[-r,b)\to U$, with $b>0$, be a maximal solution
of the initial value problem
\begin{equation}
\label{eq451} \dot{x}(t) = g(t,x_{t}),\quad
x_0=\phi.
\end{equation}
Let $t_0\in[0,b)$ be limited such that $x(t)$ is near-standard
in $U$ for all $t\in[0,t_0]$.
If $b\simeq t_0$ then $x(t')$ is not near-standard in $U$ for some $t'\in [t_0,b)$.
\end{lemma}
\begin{proof} The proof is the same for the solution $x$ of the initial value problem
\eqref{eq450} or the solution
$x$ of the initial value problem
\eqref{eq451}. We give the details in the first case.
Assume by contradiction that $x(t)$ is near-standard in $U$ for all $t\in[t_0,b)$.
\emph{Claim 1:} \emph{$\sup_{t\in[t_0,b)}|g(t,x_{t,\varepsilon})|$
is limited.} Since $x([-\varepsilon r,0])=\phi([-r,0])$,
$x(\theta)$ is near-standard in $U$ for all
$\theta\in[-\varepsilon r,0]$. Thus $x(t)$ is near-standard in $U$
for all $t\in[-\varepsilon r,b)$. By Lemma \ref{lemma4400}, there
exists a standard and compact set $W$ such that $x([-\varepsilon
r,b))\subset W\subset U$. We have $x_{t,\varepsilon}\in
\Lambda=\mathcal{C}([-r,0],W)$ for all $t\in[t_0,b)$. By
assumption (A), $\sup_{t\in[t_0,b)}|g(t,x_{t,\varepsilon})|$ is
limited.
\emph{Claim 2:} \emph{$\lim_{t\to b}x(t)$ exists and is in $U$.} Let $(\tau_n)_n$ be a sequence in $[t_0,b)$ which converges to $b$.
For $n,m\in \mathbb{N}$, we have
\[
|x(\tau_m)-x(\tau_n)|
=\big|\int_{\tau_n}^{\tau_m}g(t,x_{t,\varepsilon})dt\big| \leq
|\tau_m-\tau_n|\sup_{t\in[t_0,b)}|g(t,x_{t,\varepsilon})|.
\]
By Claim 1, the sequence $(x(\tau_n))_n$ is a Cauchy sequence, and hence, it converges to some $\xi\in \mathbb{R}^n$.
Let $t\in[t_0,b)$ and $n\in \mathbb{N}$ such that $\tau_n\geq t$. By
$$
|x(\tau_n)-x(t)|\leq \int_{t}^{\tau_n}|g(s,x_{s,\varepsilon})|ds \leq
(\tau_n-t)\sup_{s\in[t_0,b)}|g(s,x_{s,\varepsilon})|,
$$
we conclude that $\lim_{t\to b}x(t)=\xi$.
By Claim 1, for each $t\in[t_0,b)$, we have
\[
|x(t)-x(t_0)|\leq \int_{t_0}^{t}|g(s,x_{s,\varepsilon})|ds \leq
(t-t_0)\sup_{s\in[t_0,t]}|g(s,x_{s,\varepsilon})|\simeq 0.
\]
Since $x(t_0)$ is near-standard in $U$ and $x(t)\simeq x(t_0)$,
we have $\xi\in U$.
Now, one can extend $x$ to a continuous function on $[-\varepsilon r,b]$
by
setting $x(b)=\xi$. Consequently,
$x_{b,\varepsilon}\in\Omega$ and then one can find a solution
of \eqref{eq450} through the point $(b,x_{b,\varepsilon})$ to
the right of $b$, which contradicts the noncontinuability
hypothesis on~$x$. So the proof is complete.
\end{proof}
\begin{lemma}
\label{lemma44} Let $U$ be a standard open subset of
$\mathbb{R}^d$ and $\Omega=\mathcal{C}([-r,0],U)$, where $r\geq 0$
is standard. Let
$g:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ be a
continuous function. Suppose that condition (A) in
Lemma~\ref{lemma440} holds. Let $\phi\in\Omega$ be standard and
let $x:I=[-\varepsilon r,b)\to U$, with $0**t_0$,
$t'\simeq t_0$ such that $x(t')$ is not near-standard in $U$. Now,
by the continuity of $x$, there exists $t_1\in[t_0,t']$ such that
$x(t)$ is near-standard in $U$ for all $t\in[t_0,t_1]$ and
$x(t_1)\not\simeq x(t_0)$. By property (i) of the lemma, $x$ is
S-uniformly-continuous on $[0,t_1]$. Thus $x(t_1)\simeq x(t_0)$,
which is a contradiction.
\end{proof}
The proof of Theorem \ref{theorem24} is as follows.
Assume that $\phi$ and $L$ are standard. To prove Theorem \ref{theorem24} is equivalent to
prove that, when $\varepsilon>0$ is infinitesimal, every solution $x$ of
\eqref{eq23} is defined at least on $[-\varepsilon r,L]$ and
satisfies $x(t)\simeq y(t)$ for all $t\in [0,L]$.
Let $\varepsilon>0$ be infinitesimal. Let $x$ be a maximal solution of
\eqref{eq23} defined on $I$, an interval of $\mathbb{R}$.
Let $t_0\geq 0$ be an instant of observation of the stroboscopic method for ODEs; that is
$t_0$ is limited, $t_0\in I$ and $x(t)$ is
near-standard in $U$ for all $t\in[0,t_0]$. Since $x(t_0)$ is near-standard so is $\tilde{x}^{t_0}$ where
$\tilde{x}^{t_0}\in\Omega$ is defined by
$\tilde{x}^{t_0}(\theta)=x(t_0)$ for all $\theta\in[-r,0]$. Now we
apply Lemma \ref{lemma42} to obtain some constant
$\alpha=\alpha(\varepsilon,t_0,\tilde{x}^{t_0})$ such that
$0<\alpha\simeq 0$, $\varepsilon/\alpha\simeq 0$ and
\begin{equation}
\label{eq46}
\frac{\varepsilon}{\alpha}\int_{t_0/\varepsilon}^{t_0/\varepsilon+T\alpha/\varepsilon}
f(t,\tilde{x}^{t_0})dt\simeq TF(\tilde{x}^{t_0})=TG(x(t_0)),\quad \forall
T\in[0,1].
\end{equation}
By Lemma \ref{lemma44} $x(t)$ is defined and near-standard in $U$ for all $t\geq
t_0$ and $t\simeq t_0$. Hence one can consider the function
\[
X(\theta,T)=\frac{x(t_0+\alpha
T+\varepsilon\theta)-x(t_0)}{\alpha},\quad \theta\in[-r,0], \
T\in[0,1].
\]
We have, for $T\in[0,1]$,
\[
X(0,T)=\frac{x(t_0+\alpha T)-x(t_0)}{\alpha},\quad x_{t_0+\alpha
T,\varepsilon}=\tilde{x}^{t_0}+\alpha X(\cdot,T).
\]
Note that, since $\tilde{x}^{t_0}(\theta)+\alpha X(\theta,T)$ is near-standard in $U$ for all $\theta\in[-r,0]$ and all $T\in[0,T]$, by Lemma \ref{lemma4400} there exists a standard and compact set $W$ such that $\{\tilde{x}^{t_0}(\theta)+\alpha X(\theta,T): \theta\in[-r,0],\ T\in[0,T]\}\subset W\subset U$.
From this we deduce that
$\tilde{x}^{t_0}+\alpha X(\cdot,T)\in \Lambda=\mathcal{C}([-r,0],W)$ for all $T\in[0,1]$.
Differentiate now $X(0,\cdot)$ to obtain
\[
\frac{\partial X}{\partial T}(0,T)
=f\Big(\frac{t_0}{\varepsilon}+
\frac{\alpha}{\varepsilon}T,\tilde{x}^{t_0}+\alpha
X(\cdot,T)\Big),\quad T\in[0,T].
\]
Integration between $0$ and $T$, for
$T\in[0,1]$, yields
\begin{equation}
\label{eq47}
X(0,T)=\int_0^Tf\Big(\frac{t_0}{\varepsilon}+\frac{\alpha}{\varepsilon}t,
\tilde{x}^{t_0}+\alpha X(\cdot,t))\Big)dt.
\end{equation}
Here after we will consider the following two cases:
Case 1: $T\in[0,\varepsilon r/\alpha]$. Using
(H2') and taking into account that $\varepsilon r/\alpha\simeq 0$,
\eqref{eq47} leads to the approximation
\begin{equation}
\label{eq48}
X(0,T)\simeq 0.
\end{equation}
Case 2: $T\in[\varepsilon r/\alpha,1]$. By Lemma \ref{lemma44}
the restriction of $x$ to interval $[0,t_0+\alpha]$ is S-uniformly-continuous
so that, for $\theta\in[-r,0]$,
\[
\alpha X(\theta,T)=x(t_0+\alpha T+\varepsilon\theta)-x(t_0) \simeq
0,
\]
since $t_0+\alpha
T+\varepsilon\theta\in[t_0,t_0+\alpha]\subset[0,t_0+\alpha]$ and
$t_0+\alpha T+\varepsilon\theta\simeq t_0$.
Return now to \eqref{eq47}. For $T\in[0,1]$, we write
\[
X(0,T) = \Big(\int_0^{\varepsilon r/\alpha}
+\int_{\varepsilon r/\alpha}^T\Big)
f\big(\frac{t_0}{\varepsilon}+\frac{\alpha}{\varepsilon}t,
\tilde{x}^{t_0}+\alpha X(\cdot,t)\big)dt.
\]
Using \eqref{eq48}, (H1'), (H2') and \eqref{eq46}, we thus
get, for \text{$T\in[0,1]$},
\begin{align*}
X(0,T) &\simeq \int_{\varepsilon
r/\alpha}^Tf\Big(\frac{t_0}{\varepsilon}+\frac{\alpha}{\varepsilon}t,
\tilde{x}^{t_0}\Big)dt
\simeq \int_0^T
f\Big(\frac{t_0}{\varepsilon}+\frac{\alpha}{\varepsilon}t,
\tilde{x}^{t_0}\Big)dt\hfill\\
& = \frac{\varepsilon}{\alpha}
\int_{t_0/\varepsilon}^{t_0/\varepsilon+T\alpha/\varepsilon} f(t, \tilde{x}^{t_0})dt \simeq TG(x(t_0)).\hfill
\end{align*}
Defining $t_1=t_0+\alpha$
and setting $\mu=\varepsilon$, the following properties are true:
$\mu<\alpha=t_1-t_0\simeq 0$, $[t_0,t_1]\subset I$, $x(t_0+\alpha
T)=x(t_0)+\alpha X(0,T)\simeq x(t_0)$ for all $T\in[0,1]$, that
is, $x(t)\simeq x(t_0)$ for all $t\in[t_0,t_1]$, and
\[
\frac{x(t_1)-x(t_0)}{t_1-t_0}=X(0,1)\simeq
G(x(t_0)).
\]
This proves that $x$ satisfies the
\emph{F}-stroboscopic property on $I$ (Definition \ref{definition1}). Taking (H4) into account, we
finally apply Theorem \ref{theorem32} (Stroboscopic Lemma for
ODEs) to obtain the desired result, that is, the solution $x$ is
defined at least on $[-\varepsilon r,L]$ and satisfies $x(t)\simeq
y(t)$ for all $t\in [0,L]$.
The theorem is proved.
\subsection{Proof of Theorem \ref{theorem28}}
\label{section4.4}
We first prove the following result.
\begin{lemma} \label{lemma45}
Let $U$ be a standard open subset of $\mathbb{R}^d$ and $\Omega=\mathcal{C}([-r,0],U)$, where $r\geq 0$ is standard.
Let $g:\mathbb{R}_+\times\Omega\to\mathbb{R}^d$ be a
continuous function. Suppose that condition (A) in Lemma
\ref{lemma440} holds. Let $\phi\in\Omega$ be standard and let
$x:I=[-r,b)\to U$, with $0****t_0$,
$t'\simeq t_0$ such that $x(t')$ is not near-standard in $U$.
By the continuity of $x$ there exists
$t_1\in[t_0,t']$ such that $x(t)$ is near-standard in $U$ for all $t\in[t_0,t_1]$ and
$x(t_1)\not\simeq x(t_0)$.
By property (i) of the lemma $x$ is S-uniformly-continuous on $[-r,t]$.
Since $t\simeq t_0$, it follows that $x(t)\simeq
x(t_0)$, which is absurd. This proves that $x(t)$ is defined and near-standard in $U$ for
all $t\simeq t_0$.
\end{proof}
Let us prove Theorem \ref{theorem28}. Let $\phi$ and $L$ be standard. To prove Theorem
\ref{theorem28} is equivalent to show that for every infinitesimal
$\varepsilon>0$, every solution $x$ of \eqref{eq25} is
defined at least on $[-r,L]$ and $x(t)\simeq y(t)$ holds for all
$t\in [0,L]$.
We fix $\varepsilon>0$ to
be infinitesimal and we let $x:I\to U$ to be a
maximal solution of \eqref{eq25}. We show that
$x$ satisfies the \emph{F}-stroboscopic property on $I$ (Definition \ref{definition2}). Let
$t_0\geq 0$ be an instant of observation of the stroboscopic method for RFDEs; that is
$t_0$ is limited, $t_0\in I$ and $x(t)$ is near-standard in $U$ and $F(x_t)$ is
limited for all $t\in[0,t_0]$. According to (H2') Lemma~\ref{lemma45} applies. Thus $x_t$ is near-standard in $\Omega$ for all
$t\in[0,t_0]$.
Now, applied to $t_0$ and $x_{t_0}$, Lemma \ref{lemma42} gives
\begin{equation}
\label{eq49}
\frac{\varepsilon}{\alpha}\int_{t_0/\varepsilon}^{t_0/\varepsilon+T\alpha/\varepsilon}
f(t,x_{t_0})dt\simeq TF(x_{t_0}),\quad \forall T\in[0,1]
\end{equation}
for some $\alpha=\alpha(\varepsilon,t_0,x_{t_0})$ such that
$0<\alpha\simeq 0$ and $\varepsilon/\alpha\simeq 0$.
Let $X:[-r,0]\times[0,1]\to\mathbb{R}^d$ be the function
given by
\[
X(\theta,T)=\frac{x(t_0+\alpha
T+\theta)-x(t_0+\theta)}{\alpha},\quad \theta\in[-r,0], \
T\in[0,1].
\]
By Lemma \ref{lemma45} the function $X$ is well defined. It
satisfies, for $T\in[0,1]$,
\[
X(0,T)=\frac{x(t_0+\alpha T)-x(t_0)}{\alpha},\quad x_{t_0+\alpha
T}=x_{t_0}+\alpha X(\cdot,T).
\]
Hence, for $T\in[0,1]$,
\[
\frac{\partial X}{\partial T}(0,T)
=f\Big(\frac{t_0}{\varepsilon}+
\frac{\alpha}{\varepsilon}T,x_{t_0}+\alpha X(\cdot,T)\Big).
\]
Solving this equation gives, for $T\in[0,1]$,
\begin{equation}
\label{eq410}
X(0,T)=\int_0^Tf\Big(\frac{t_0}{\varepsilon}+\frac{\alpha}{\varepsilon}t,
x_{t_0}+\alpha X(\cdot,t)\Big)dt.
\end{equation}
According to Lemma \ref{lemma45} the solution $x$ is
S-uniformly-continuous on \text{$[-r,t_0+\alpha]$}. Therefore, for
$\theta\in[-r,0]$ and $T\in[0,1]$, $X(\theta,T)$ satisfies, since
$t_0+\alpha T+\theta\simeq t_0+\theta$,
\begin{equation}
\label{eq411}
\alpha X(\theta,T)=x(t_0+\alpha T+\theta)-x(t_0+\theta) \simeq 0.
\end{equation}
By (H1'), \eqref{eq411},
\eqref{eq410} and \eqref{eq49}, we have for all $T\in[0,1]$ the approximation
\[
X(0,T) \simeq
\int_0^T
f\Big(\frac{t_0}{\varepsilon}+\frac{\alpha}{\varepsilon}t,x_{t_0}\Big)dt
= \frac{\varepsilon}{\alpha}
\int_{t_0/\varepsilon}^{t_0/\varepsilon+T\alpha/\varepsilon} f(t,x_{t_0})dt
\simeq TF(x_{t_0}).
\]
Let $t_1=t_0+\alpha$ and set $\mu=\varepsilon$. The instant $t_1$
and the constant $\mu$ are such that: $\mu<\alpha=t_1-t_0\simeq
0$, $[t_0,t_1]\subset I$, $x(t_0+\alpha T)=x(t_0)+\alpha
X(0,T)\simeq x(t_0)$ for all $T\in[0,1]$, that is, $x(t)\simeq
x(t_0)$ for all $t\in[t_0,t_1]$ and
\[
\frac{x(t_1)-x(t_0)}{t_1-t_0}=X(0,1)\simeq F(x_{t_0}),
\]
which is the \emph{F}-stroboscopic property on $I$. Finally, using (H6) we
get, by means of Theorem \ref{theorem33} (Stroboscopic Lemma for
RFDEs), that the solution $x$ is defined at least on $[-r,L]$ and
satisfies \text{$x(t)\simeq y(t)$} for all $t\in [0,L]$. So the
proof is complete.
\section{Discussion}
\label{discussion} In this paper we presented averaging results
for ODEs and RFDEs. The results are proved in an unified manner
for both ODEs and RFDEs, by using the Stroboscopic method which is
a nonstandard tool in the asymptotic theory of differential
equations. It should be noticed that the usual approaches for
averaging make use of different tools for ODEs
\cite{Bogolyubov-Mitropolsky,Guckenheimer-Holmes,Sanders-Verhulst}
and for RFDEs
\cite{Halanay,Hale1,Hale2,Hale-Lunel1,Hale-Lunel2,Lehman,Lehman-Weibel,Medvedev,Volosov-Medvedev-Morgunov}.
The results on RFDEs presented in this paper were obtained
in \cite{LakibThesis}, in which the Stroboscopic method for
RFDEs was stated for the first time (see also \cite{Lakrib-Sari1,Sari3}).
We recall that the stroboscopic method was initially proposed
for ODEs. In this paper, we presented a slightly
modified version of this method (see Theorem \ref{theorem32})
and then extended it (see Theorem \ref{theorem33}) in the context
of RFDEs.
Here, the stroboscopic method is slightly extended since
the time $t'$ in Definition \ref{definition1} is assumed to exist
only for those limited values of $t$ for which $x(s)$ is
near-standard in $U$ for all $s\in[0,t]$ (see also Definition 6 in
\cite{Sari3}). In the previous papers the time $t'$ was assumed to
exist for those limited values of $t$ for which $x(t)$ is
near-standard in $U$, without any assumption on $x(s)$ for
$s\in[0,t]$ (see Theorem~1 in \cite{Sari2} or Definition 5 in
\cite{Sari3}). In the stroboscopic method for RFDEs,
the main assumption is that the time $t'$
in Definition \ref{definition2} is assumed to exist
for those limited values of $t$ for which $x(s)$ is
near-standard in $U$ and $F(s,x_s)$ is limited for all $s\in[0,t]$.
The Stroboscopic method for ODEs was first obtained by Callot and
Reeb \cite{Callot-Sari,Reeb}. For more information on the
discovery of the stroboscopic method, and its use in averaging and
asymptotic analysis the reader can consult
\cite{Lutz,SariStrob,Sari2}. Lemma \ref{lemma32} of the present
paper is simply the Stroboscopic Theorem of Callot (see Theorem 1
in \cite{Callot-Sari} or Lemma 1 in \cite{Sari2}). Theorem
\ref{theorem32} is similar to Theorem 1 in \cite{Sari2}.
Lemma~\ref{lemma33} is similar to Lemma~2 in \cite{Sari2}. In the
case $r=0$, Lemma \ref{lemma41} is Lemma 4 in \cite{Sari2}; in the
case $r>0$ it is Lemma 4.3.6 in \cite{LakibThesis}. In the case
$r=0$, Lemma \ref{lemma42} is Lemma 2 in \cite{Sari1} or Lemma 5
in \cite{Sari2}; in the case $r>0$ it is Lemma 4.3.7 in
\cite{LakibThesis}. Lemma \ref{lemma43} is Lemma 1 in \cite{Sari1}
or Theorem 2 in \cite{Sari2}. The results of Sections
\ref{section3.3}, \ref{section4.3} and \ref{section4.4} are
extensions of some of the results in \cite{LakibThesis}. In our
previous papers \cite{Lakrib0,Lakrib1,Lakrib2,Lakrib-Sari}, the
stroboscopic method for RFDEs was not yet established and the
results of averaging were obtained directly through evaluations of
integrals.
The KBM theorem of averaging on ODEs obtained previously (see
Theorems 1 in \cite{Sari1} or Theorem 6 in \cite{Sari2}) concerned
nonstandard differential equations of the form $\dot
x=g(t/\varepsilon,x)$ where $g(t,x)$ is a perturbation of a
standard KBM vector field $f(t,x)$ satisfying the conditions in
Definition \ref{definition0}. From a classical point of view this
theorem includes the case of deformations of the form
$$
\dot x(t)=f(t/\varepsilon,x(t),\varepsilon),
$$
where the vector field
$f(t,x,\varepsilon)$ depends also on $\varepsilon$. More precisely
the theorem concerns all initial value problems in a neighborhood
of a KBM vector field in a suitable topology (see Theorem 7 in \cite{Sari2}).
The KBM theorems of averaging on ODEs and RFDEs obtained in this paper
does not concern deformations of the above form in the case of
ODEs or of the form
$$\dot x(t)=f(t/\varepsilon,x_t,\varepsilon)
\quad
\mbox{ or }
\quad
\dot x(t)=f(t/\varepsilon,x_{t,\varepsilon},\varepsilon),$$
in the case of RFDEs. However we think that our approach will permit
also to consider these deformations. We leave this problem for
future investigations. An other adapted version of the
stroboscopic method for the case of ordinary differential
inclusions has been given in~\cite{Lakrib3} and used there to
prove an averaging result.
\subsection*{Acknowledgements}
The authors would like to thank the anonymous referee very much
for his/her valuable comments and suggestions.
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\end{document}
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