\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 46, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2005/46\hfil Rectifiability of solutions]
{Rectifiability of solutions\\ of the one-dimensional
p-Laplacian}
\author[M. Pa\v{s}i\'{c}\hfil EJDE-2005/46\hfilneg]
{Mervan Pa\v{s}i\'{c}}
\address{Mervan Pa\v{s}i\'{c} \hfill\break
Department of Mathematics\\
Faculty of Electrical Engineering and Computing\\
University of Zagreb\\
Unska 3, 10000 Zagreb, Croatia}
\email{mervan.pasic@fer.hr}
\date{}
\thanks{Submitted November 11, 2004. Published April 24, 2005.}
\subjclass[2000]{35J60, 34B15, 28A75}
\keywords{Nonlinear p-Laplacian; bounded continuous solutions; graph; \hfill\break\indent
qualitative properties; length; rectifiability}
\begin{abstract}
In the recent papers \cite{Pasic03} and \cite{PZ04} a class of
Carath\'{e}odory functions $f(t,\eta ,\xi )$ rapidly sign-changing
near the boundary point $t=a$, has been constructed so that the
equation $-(|y'|^{p-2}y')'=f(t,y,y')$ in $(a,b)$ admits continuous
bounded solutions $y$ whose graphs $G(y)$ do not possess a finite
length. In this paper, the same class of functions $f(t,\eta ,\xi )$
will be given, but with slightly different input data compared to
those from the previous papers, such that the graph $G(y)$ of each
solution $y$ is a rectifiable curve in $\mathbb{R}^{2}$. Moreover,
there is a positive constant which does not depend on $y$ so that
$\mathop{\rm length}(G(y))\leq c<\infty $.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
Let $-\infty 0$.
In the author's paper \cite{Pasic03}, a class of Carath\'{e}odory functions
$f(t,\eta ,\xi )$ rapidly sign-changing and singular near the boundary
point $t=a$ has been constructed in such a way that all solutions of the
problem \eqref{Eq} admit very singular boundary behaviour.
It has been verified by means of the upper Minkowski-Bouligand dimension
of the graph $G(y)$ of each solution $y$ as well as by the order of growth
for singular behaviour of the $L^{p}$ norm of the derivative of solutions.
This can also be sharpened by the use of notions of the upper Minkowski
content and the $s$ - dimensional density of the graph $G(y)$ of each
solution $y$, as in the paper \cite{PZ04}.
Taking a closer look at \cite[Section 7]{Pasic03}, see also \cite[Section 5]{PZ04}, one can find that there are two hypotheses involved to the nonlinear term $f(t,\eta ,\xi )$ which can reduce the singular behaviour of the graph $G(y)$ of each solution $y$ of the problem \eqref{Eq}. In the first one, it is supposed that there is a continuous positive function $\tilde{\omega} (t)$ such that:
\begin{equation}
\begin{gathered}
f(t,\eta ,\xi )<0,\quad t\in (a,b),\quad\eta >\tilde{\omega} (t)\
\mbox{ and }\ \xi \in \mathbb{R}, \\
f(t,\eta ,\xi )>0,\quad t\in (a,b),\quad\eta <-\tilde{\omega} (t)\
\mbox{ and }\ \xi \in \mathbb{R}.
\end{gathered}\label{first}
\end{equation}
In the second one, it is supposed that there is a decreasing sequence $a_{k}$ of real numbers
from interval $(a,b)$ such that $a_{k}\searrow a$ and:
\begin{equation}
\begin{gathered}
f(t,\eta ,\xi )>0,\quad t\in (a_{2k},a_{2k-1}),\quad \eta \in (-\tilde{\omega}_{0},\tilde{\omega} (t))\ \mbox{ and }\xi \in \mathbb{R}, \\
f(t,\eta ,\xi )<0,\quad t\in (a_{2k+1},a_{2k}),\quad \eta \in (-\tilde{\omega} (t),\tilde{\omega}_{0})\ \mbox{ and }\xi \in \mathbb{R},
\end{gathered}\label{second}
\end{equation}
where $\tilde{\omega}_{0}$ is an arbitrarily given real number such that $\tilde{\omega} (t)\leq \tilde{\omega}_{0}$ for all
$t\in (a,b)$. Because of \eqref{second}, we say that the nonlinear term $f(t,\eta ,\xi )$ is rapidly sign-changing
near $t=a$.
In \cite[Corollary 7.2]{Pasic03}, see also \cite[Corollary 5.9]{PZ04}, including the hypotheses \eqref{first} and \eqref{second},
a class of the functions $\tilde{\omega} (t)$ and a class of the sequences $a_{k}$ were given in such a way that a type of the singular boundary behaviour of solutions of the problem \eqref{Eq} is valid.
In particular, one can conclude that the graph $G(y)$ of each solution $y$ of \eqref{Eq}
is not a rectifiable curve in $\mathbb{R}^{2}$, that is to say, the graph $G(y)$
does not possess a finite length.
This can be summarized this way.
\begin{theorem}\label{theo1}
Let a Carath\'{e}odory function $f(t,\eta ,\xi )$ satisfy the hypotheses \eqref{first} and \eqref{second}
where the function $\tilde{\omega} (t)$ and the sequence $a_{k}$ are given by:
\begin{equation}
\begin{gathered}
\tilde{\omega} (t)=c(t-a)^{q}\quad \mbox{in $[a,b],$ where $c>0$ and $q\in (0,1)$},\\
a_{k}=a+(b-a)(\frac{1}{k})^{1/\beta },\quad
\mbox{$\forall k\in \mathbb{N}$, where $\beta >q$}.
\end{gathered}\label{omegaak}
\end{equation}
Then problem \eqref{Eq} admits the solutions $y\in C^{2}(a,b)$ such that the graph $G(y)$ is not a rectifiable curve in $\mathbb{R}^{2}$,
that is to say, $G(y)$ does not possess a finite length.
\end{theorem}
In this paper, under the hypotheses \eqref{first} and \eqref{second},
one gives a possibility to avoid the nonrectifiability of $G(y)$ stated in
Theorem \ref{theo1}.
It will be done by choosing a class of sequences $a_{k}$ from the interval
$(a,b)$ which decays exponentially to $a$, that is
to say, $a_{k}-a\approx ce^{-kT}$ as $k\approx \infty $, for any $c>0$
and $T>0$. Such a class of sequences $a_{k}$ is quite different from
the previous one given in \eqref{omegaak}.
It concerns the following main result.
\begin{theorem}\label{theo2}
Let a Carath\'{e}odory function $f(t,\eta ,\xi )$ satisfy the
hypotheses \eqref{first} and \eqref{second},
where the function $\tilde{\omega} (t)$ is determined as in \eqref{omegaak}
and the sequence $a_{k}$ is given by
\begin{equation}
a_{k}=a+(b-a)e^{-(k-1)T},\quad \mbox{$\forall k\in \mathbb{N},$}\quad
\mbox{where $T>0$}.\label{ak}
\end{equation}
Then the graph $G(y)$ of each solution $y\in C^{2}(a,b)$ of problem \eqref{Eq}
is a rectifiable curve in $\mathbb{R}^{2}$.
Moreover, there is a positive constant $c$ which does not depend on $y$ so that:
\begin{equation}
\mathop{\rm length}(G(y))\leq c<\infty . \label{rect}
\end{equation}
\end{theorem}
Thus, under such a class of Carath\'{e}odory functions $f(t,\eta ,\xi )$,
the rectifiability of the graph $G(y)$ of solutions of the problem \eqref{Eq}
depends only on the appropriate choice of the sequence $a_{k}$ appearing
in \eqref{second}.
The main statement \eqref{rect} will be proved in Section 3, by the use of
the upper Minkowski content $M^{1}(G(y))$ of the graph $G(y)$ defined in
Section 2. The first step is to give a condition such that
$\mathop{\rm length}(G(y))\leq M^{1}(G(y))$. The second step is to derive an
upper bound for $M^{1}(G(y))$ which depends only on the behaviour of the
function $\tilde{\omega} (t)$ and of the sequence $a_{k}$ both appearing
inside the main hypotheses \eqref{first} and \eqref{second}.
Finally, at the end of Section 3, an example of such a class of
Carath\'{e}odory functions $f(t,\eta ,\xi )$ that satisfies the
hypotheses \eqref{first} and \eqref{second} will be given.
\subsection*{Further Remarks}
A. The rectifiability of the graph $G(y)$ considered in this paper contributes
to the completion of some results presented in the papers
\cite[Section 7]{Pasic03} and \cite[Section 5]{PZ04}.
\noindent B. Including the hypotheses \eqref{first} and \eqref{second}, the existence of at least one solution of the problem \eqref{Eq}
was considered in \cite[Appendix]{Pasic03}.
\noindent C. The main result of the paper can be applied to the study of rectifiable oscillations
of the second order differential equations. Such oscillations appears in the famous linear Euler's equation
as well as in the $p$ - Laplace equation, see \cite{Pasic05}.
\noindent D. Although the nonlinear term $f(t,\eta ,\xi )$ could have very singular behaviour near $t=a$,
the rectifiability of $G(y)$ can be understood as a regular property of solutions of the problem \eqref{Eq}.
As for some known results on the regularity of solutions for $p$ - Laplace equations,
see for instance \cite{KPZ01}, \cite{Rak90}, \cite{RT90}, and \cite{RZ90},
and references therein.
\section{How to measure the length of $G(y)$?}
Let $y$ be a real continuous function defined on the interval $[a,b]$ and let
the length of the graph $G(y)$, denoted by $\mathop{\rm length}(G(y))$, be defined as in \eqref{length}.
Since we are going to work with the solutions $y$ of a nonlinear differential equation,
from computational point of view, the formula \eqref{length}
does not seem to be convenient. For the purpose of proving the rectifiability of the graph $G(y)$
and in order to prove the main statement \eqref{rect},
we will calculate the length of the graph $G(y)$
by means of the upper Minkowski content $M^{1}(G(y))$ as follows.
Let $G_{\varepsilon }(y)$ denote the $\varepsilon -$ neighborhood of the graph $G(y)$
and let $|G_{\varepsilon }(y)|$ denote the Lebesgue measure of $G_{\varepsilon }(y)$. The upper Minkowski content of $G(y)$,
denoted by $M^{1}(G(y))$, is defined by
\[
M^{1}(G(y))=\limsup_{\varepsilon \to 0}(2\varepsilon )^{-1}
|G_{\varepsilon }(y)|.
\]
Referring to \cite[Chapter 5.2]{Falc99}, \cite[Chapter 5.5]{Matt95}
and \cite[Chapter 9]{Trico95}, it holds that
\begin{equation}
\mathop{\rm length}(G(y))=M^{1}(G(y)), \label{2length2}
\end{equation}
provided the graph $G(y)$ is a rectifiable curve in $\mathbb{R}^{2}$.
However, this result cannot be immediately used here because the condition
of rectifiability of the graph $G(y)$
is not yet presumed but it must be proved. Hence, we will
use the following complementary claim.
\begin{lemma}\label{2lemma1}
Let $y$ be a real continuous function defined on the interval $[a,b]$
and let $y|_{[a+\varepsilon ,b]}$ denote the function restriction of $y$
on the interval $[a+\varepsilon ,b]$.
We assume that the graph $G(y|_{[a+\varepsilon ,b]})$ is a rectifiable
curve in $\mathbb{R}^{2}$ for all $\varepsilon >0$. If
$M^{1}(G(y))<\infty $ then the graph $G(y)$ is also a rectifiable curve
in $\mathbb{R}^{2}$. Moreover, there holds true
\[
\lim_{\varepsilon \rightarrow 0}\mathop{\rm length}(G(y|_{[a+\varepsilon ,b]}))
=\mathop{\rm length}(G(y))\leq M^{1}(G(y)).
\]
\end{lemma}
\begin{proof}
By means of the rectifiability of the graphs $G(y|_{[a+\varepsilon ,b]})$, $\varepsilon >0$, and by the use of
\eqref{2length2} we observe that
\[
\mathop{\rm length}(G(y|_{[a+\varepsilon ,b]}))=M^{1}(G(y|_{[a+\varepsilon
,b]}))\leq M^{1}(G(y))<\infty , \quad \forall \varepsilon >0.
\]
Since the sequence $\mathop{\rm length}(G(y|_{[a+\varepsilon ,b]}))$
is increasing, while $\varepsilon $ is decreasing, it implies the existence of
its limit which satisfies
\begin{equation}
\lim_{\varepsilon \rightarrow 0}\mathop{\rm length}(G(y|_{[a+\varepsilon ,b]}))
\leq M^{1}(G(y))<\infty \label{2var1}.
\end{equation}
Next, it easy to check that:
\begin{equation}
\mathop{\rm length}(G(y))\leq\lim_{\varepsilon \rightarrow 0}
\mathop{\rm length}(G(y|_{[a+\varepsilon ,b]})) \label{2var2}.
\end{equation}
Indeed, for any given dissection $a=t_{0}0$, and combining this inequality with \eqref{2var1}
and \eqref{2var2}.
\end{proof}
With the help of Lemma \ref{2lemma1} applied to solutions of the
problem \eqref{Eq} we obtain the following consequence.
\begin{corollary}\label{2cor1}
For all solutions $y$ of the problem \eqref{Eq} such that $M^{1}(G(y))<\infty $
we have
\begin{equation}
\mathop{\rm length}(G(y))= M^{1}(G(y)).\label{2main}
\end{equation}
\end{corollary}
\begin{proof}
Let $y$ be a solution of \eqref{Eq}. Let us mention that
$y\in W_{loc}^{1,p}((a,b])$, that is to say, $y\in W^{1,p}(a+\varepsilon ,b)$
for all $\varepsilon >0$. It implies that $y$ is an absolutely continuous
function (of bounded variation) on $[a+\varepsilon ,b]$.
Therefore, the graph $G(y|_{[a+\varepsilon ,b]})$ is a rectifiable curve
in $\mathbb{R}^{2}$ for all $\varepsilon >0$.
Since $M^{1}(G(y))<\infty $, we may apply Lemma \ref{2lemma1} to obtain
$\mathop{\rm length}(G(y))\leq M^{1}(G(y))<\infty $. Now, it is
possible to use \eqref{2length2} and thus, \eqref{2main} is shown.
\end{proof}
Thus, according to Corollary \ref{2cor1}, in order to prove the rectifiability
of $G(y)$ it is enough to show that $M^{1}(G(y))<\infty $. For this we need
the following general result.
\begin{lemma}\label{2lemma2}
Let $\tilde{\omega} (t)$ be a continuously increasing and concave function
such that $\tilde{\omega} (a)=0$. Next,
let $a_{1}=b$ and let $a_{k}$ be a decreasing sequence of real numbers
from interval $(a,b)$ satisfying:
\begin{equation}
\begin{aligned}
&\mbox{$a_{k}\searrow a$ and there is an $\varepsilon _{1}>0$ such that for all $\varepsilon \in (0,\varepsilon _{1})$} \\
&\mbox{there is an $m(\varepsilon )\in \mathbb{N}$ such that
$a_{j-1}-a_{j}>4\varepsilon$, for each $j\leq m(\varepsilon )$,}\\
&\mbox{and $m(\varepsilon )\rightarrow \infty $ as $\varepsilon \rightarrow 0.$}
\end{aligned} \label{2ak}
\end{equation}
Let a Carath\'{e}odory function $f(t,\eta ,\xi )$ satisfy
hypotheses \eqref{first} and \eqref{second}.
Then the graph $G(y)$ of each solution $y\in C^{2}(a,b)$ of problem \eqref{Eq}
satisfies
\begin{equation}
\begin{aligned}
M^{1}(G(y))
&\leq (\pi +6)(b-a)+ (\pi +6)\sum_{j=1}^{\infty}\tilde{\omega} (a_{j}) \\
&\quad +\limsup_{\varepsilon \to 0}[\frac{1}{\varepsilon }
(a_{m(\varepsilon )}-a)\tilde{\omega} (a_{m(\varepsilon )})+(\pi +4)\varepsilon
\,m(\varepsilon )].
\end{aligned} \label{2M}
\end{equation}
\end{lemma}
\begin{proof}
With the help of hypothesis \eqref{first} and \cite[Lemma 3.1]{Pasic03},
we have that each solution $y\in C^{2}(a,b)$ of problem \eqref{Eq}
satisfies
\begin{equation}
-\tilde{\omega} (t)\leq y(t)\leq \tilde{\omega} (t),
\quad \mbox{for all $t\in [a,b]$}.\label{2comp}
\end{equation}
Next, taking into account hypothesis \eqref{second} and
\cite[Remark A in Appendix]{Pasic03}, we have also that
\begin{equation}
\mbox{$y$ is concave in $(a_{2k},a_{2k-1})$\quad and\quad $y$
is convex in $(a_{2k+1},a_{2k})$,} \label{2convex}
\end{equation}
for each $k\geq 1$.
Now, according to assertions \eqref{2ak}, \eqref{2comp} and \eqref{2convex}
it is clear the all assumptions of \cite[Definition 5.1]{PZ04} and
\cite[Lemma 5.3]{PZ04} are fulfiled,
where $\tilde{\tilde{\omega}}(t)=\tilde{\omega} (t)$ and
$\tilde{\theta}(t)=-\tilde{\omega} (t)$. So, by means of \cite[Lemma 5.3]{PZ04},
we obtain
\[
\begin{aligned}
|G_{\varepsilon }(y|_{[a,a_{1}]})|
&\leq 2(a_{m(\varepsilon)}-a+2\varepsilon)(\tilde{\omega} (a_{m(\varepsilon )})+\varepsilon ) \\
&\quad+2(\pi +6)\varepsilon \sum_{j=2}^{m(\varepsilon )}[\tilde{\omega} (a_{j-1})+(a_{j-1}- a_{j})]+2(\pi +4)\varepsilon ^{2}m(\varepsilon ),\\
\end{aligned}
\]
where $a_{1}=b$. Since $m(\varepsilon )\rightarrow \infty $,
$a_{m(\varepsilon )}\rightarrow a$, and $\tilde{\omega} (0)=0$, the right
hand side in the previous inequality can be simplified and so,
the statement \eqref{2M} is proved.
\end{proof}
As a consequence of the preceding lemma we have the following statement.
\begin{corollary}\label{2cor2}
Let the hypotheses of Lemma \ref{2lemma2} be still fulfilled.
Let $\tilde{\omega} (t)$ and $a_{k}$ be such that there are three positive
constants $c_{1}$, $c_{2}$ and $c_{3}$ satisfying
\begin{equation}
\sum_{j=1}^{\infty}\tilde{\omega} (a_{j})\leq c_{1}, \quad
(a_{m(\varepsilon )}-a)\tilde{\omega} (a_{m(\varepsilon )})
\leq c_{2}\varepsilon ,\quad \mbox{and}\quad
\varepsilon \,m(\varepsilon )\leq c_{3},\label{2extra}
\end{equation}
for sufficiently small $\varepsilon $. Then there is a positive constant
$c$ such that each solution $y\in C^{2}(a,b)$ of the problem \eqref{Eq}
satisfies $M^{1}(G(y))\leq c<\infty $.
\end{corollary}
Thus, according to Corollary \ref{2cor1}, in order to prove \eqref{rect},
we need to ensure that a class of the functions $\tilde{\omega} (t)$ given
in \eqref{omegaak} and a class of the sequences $a_{k}$ given in \eqref{ak}
satisfy all the assumptions of Corollary \ref{2cor2}.
\section{The proofs of the main results and an example}
\begin{proof}[Sketch of the proof of Theorem \ref{theo1}]
Including some additional assumptions on the Caratheodory function $f(t,\eta ,\xi )$ which are completely balanced
with the main hypotheses \eqref{first} and \eqref{second}, one can show (see in \cite[Corollary 7.2]{Pasic03} and
\cite[Corollary 5.9]{PZ04}) that the graph $G(y)$ of each solution $y\in C^{2}(a,b)$ of the problem \eqref{Eq} satisfies:
\begin{equation}
\dim _{M}G(y)=s\quad\mbox{and}\quad 01$ and combining \eqref{3dim} with \eqref{3rel} we obtain
that $M^{1}(G(y))=\infty $.
With the help of \eqref{2length2}, it follows that the graph $G(y)$ does not
possess a finite length.
\end{proof}
Next, we give the proof of the main result of the paper.
\begin{proof}[Proof of Theorem \ref{theo2}]
Let us mention that the function $\tilde{\omega} (t)$ and the sequence $a_{k}$
are given by
\begin{equation}
\begin{aligned}
&\tilde{\omega} (t)=c(t-a)^{q}\quad \mbox{in $[a,b],$}\quad
\mbox{where $c>0$ and $q\in (0,1)$},\\
&a_{k}=a+(b-a)e^{-(k-1)T},\quad \mbox{$\forall k\in \mathbb{N},$}\quad
\mbox{where $T>0$}.
\end{aligned}\label{3omegaak}
\end{equation}
According to Corollary \ref{2cor2}, it is sufficient to show that such
$\tilde{\omega} (t)$ and $a_{k}$
satisfy the all conditions of Lemma \ref{2lemma2} as well as
condition \eqref{2extra}.
It is clear that the function $\tilde{\omega} (t)$ is continuous and concave
in $(a,b)$, and that $\tilde{\omega} (0)=0$. Also, the sequence $a_{k}$ is
decreasing, $a_{1}=b$ and $a_{k}\searrow a$.
Let $m(\varepsilon )$ be a sequence of integers determined by the following
inequality:
\begin{equation}
\frac{1}{T}\ln \frac{(e^{T}-1)(b-a)}{4\varepsilon }
\leq m(\varepsilon )\leq \frac{1}{T}\ln \frac{(e^{2T}-e^{T})(b-a)}{4\varepsilon },
\quad \mbox{$\varepsilon >0$}.\label{3meps}
\end{equation}
With the help of \eqref{3omegaak} and \eqref{3meps} it is easy to check
the following calculations. Firstly, we derive that
\begin{equation}
\frac{1}{T}\ln \frac{(e^{2T}-e^{T})(b-a)}{4\varepsilon }-\frac{1}{T}\ln \frac{(e^{T}-1)(b-a)}{4\varepsilon }=\frac{1}{T}\ln e^{T}=1.
\label{3proof1}
\end{equation}
Also, for each $j\leq m(\varepsilon )$, we obtain
\begin{equation}
\begin{aligned}
a_{j-1}-a_{j}&=(b-a)(e^{2T}-e^{T})e^{-jT}\geq (b-a)(e^{2T}-e^{T})e^{-m(\varepsilon )T}\\
& \geq (b-a)(e^{2T}-e^{T})e^{-\frac{1}{T}T\ln \frac{(e^{2T}-e^{T})(b-a)}{4\varepsilon }}=4\varepsilon .
\end{aligned}\label{3proof2}
\end{equation}
Next, there is a positive constant $c_{1}$ such that for all $\varepsilon >0$ we obtain:
\begin{equation}
\sum_{j=1}^{\infty}\tilde{\omega} (a_{j})=c\sum_{j=1}^{\infty}(a_{j}-a)^{q}=c(b-a)^{q}e^{qT}\sum_{j=1}^{\infty}e^{-jqT}
\leq c_{1}.
\label{3proof3}
\end{equation}
Furthermore, there is a positive constant $c_{2}$ such that for all
$\varepsilon >0$ we derive
\begin{equation}
\begin{aligned}
(a_{m(\varepsilon )}-a)\tilde{\omega} (a_{m(\varepsilon )})
&= c[(b-a)e^{T}]^{q+1}e^{-(q+1)Tm(\varepsilon )}\\
&\leq c[(b-a)e^{T}]^{q+1}e^{-Tm(\varepsilon )}\\
&\leq c[(b-a)e^{T}]^{q+1}e^{-T\frac{1}{T}\ln \frac{(e^{T}-1)(b-a)}{4\varepsilon }} =c_{2}\varepsilon .
\end{aligned}\label{3proof4}
\end{equation}
Since $\lim_{\varepsilon \rightarrow 0} [\varepsilon \ln(1/\varepsilon )]=0$, there is a positive constant $c_{3}$ such that for all $\varepsilon >0$ we have:
\begin{equation}
\varepsilon \,m(\varepsilon )\leq \frac{\varepsilon }{T}\ln \frac{(e^{2T}-e^{T})(b-a)}{4\varepsilon }\leq c_{3}.
\label{3proof5}
\end{equation}
We now conclude with the help of \eqref{3proof1} that it is possible to choose a $m(\varepsilon )\in \mathbb{N}$
in the statement \eqref{3meps}. Also,
according to \eqref{3proof2} we obtain that the sequence $a_{k}$ satisfies
the condition \eqref{2ak}, where $\varepsilon_{1}$
can be taken arbitrarily.
Next, by means of \eqref{3proof3}, \eqref{3proof4} and \eqref{3proof5}
we obtain that the function $\tilde{\omega} (t)$ and the sequence $a_{k}$
satisfy all the conditions from \eqref{2extra}. Therefore,
we may apply Corollary \ref{2cor2} which immediately gives the desired
statement \eqref{rect}.
\end{proof}
\begin{example}\label{3exampl}\rm
Let $g_{k}(t)$ be an arbitrarily given sequence of measurable functions
such that $g_{k}(t)>0$ in $(a,b)$, for each $k=0,1,2,\dots $.
Let $f(t,\eta ,\xi )$ be a nonlinear function defined by:
\begin{align*}
f(t,\eta ,\xi )
=& -g_{0}(t)(\eta - \tilde{\omega} (t))^{+}+ g_{1}(t)(\eta
+ \tilde{\omega} (t))^{-} \\
&+\sum_{k=1}^{\infty }\bigl[ g_{2k}(t)(\eta - \tilde{\omega} (t))^{-}
\sin (\frac{\pi }{a_{2k-1}-a_{2k}}
(t-a_{2k}))K_{[a_{2k},a_{2k-1}]}(t) \\
&-g_{2k+1}(t)(\eta + \tilde{\omega} (t))^{+}
\sin (\frac{\pi }{a_{2k}-a_{2k+1}}
(t-a_{2k+1}))K_{[a_{2k+1},a_{2k}]}(t)\bigl],
\end{align*}
where $t\in [a,b]$, $\eta\in\mathbb{R}$ and $\xi\in\mathbb{R}$.
Here $K_{A}(t)$ denotes, as usual, the characteristic function of a set
$A$ and also, $\eta ^{-}=\max \{0,-\eta \}$ and
$\eta ^{+}=\max \{0,\eta \}$. It is easy to check that
such a function $f(t,\eta ,\xi )$ is of Carath\'{e}odory type and that
it satisfies the desired hypotheses \eqref{first} and \eqref{second}.
\end{example}
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