\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 60, pp. 1--25.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/60\hfil Some metric-singular properties]
{Some metric-singular properties of the graph of solutions of the
one-dimensional p-Laplacian}

\author[Mervan Pa\v{s}i\'{c} \& Vesna \v{Z}upanovi\'{c}\hfil EJDE-2004/60\hfilneg]
{Mervan Pa\v{s}i\'{c} \& Vesna \v{Z}upanovi\'{c}} % in alphabetical order

\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}

\address{Vesna \v{Z}upanovi\'{c}\hfill\break
Department of Mathematics\\
Faculty of Electrical Engineering and Computing\\
University of Zagreb\\
Unska 3, 10000 Zagreb, Croatia}
\email{vesna.zupanovic@fer.hr}

\date{}
\thanks{Submitted October 28, 2003. Published April 19, 2004.}
\subjclass[2000]{35J60, 34B15, 28A75}
\keywords{Nonlinear p-Laplacian, bounded solutions, qualitative properties,
\hfill\break\indent
 graph, singularity, Minkowski content, $s$-dimensional density}

\begin{abstract}
  We study the asymptotic behaviour of $\varepsilon$-neighbourhood
  of the graph of a type of rapidly oscillating continuous functions.
  Next, we estate necessary and sufficient conditions for rapid
  oscillations of solutions of the main equation. This enables us
  to verify some new singular properties of bounded   continuous
  solutions of a class of nonlinear $p$-Laplacian by  calculating
  lower and upper bounds for the Minkowski content and the
  $s$-dimensional density of the graph of each solution and
  its derivative.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

Let $-\infty <a<b<\infty $. Let $y$ be a real function defined on $[a,b]$
and let $y'$ denote the derivative of $y$ in the classical sense.
The main subjects of the paper are the graph $G(y)$ of a function $y$ and
its $\varepsilon -$ neighbourhood $G_{\varepsilon }(y)$, that is
\begin{gather*}
G(y)=\{(t,y(t)):a\leq t\leq b\} \\
G_{\varepsilon }(y)=\{(t_{1},t_{2})\in \mathbb{R}^{2}:d((t_{1},t_{2}),G(y))\leq
\varepsilon \}.
\end{gather*}
Here $\varepsilon >0$ and $d((t_{1},t_{2}),G(y))$ denotes the distance
between $(t_{1},t_{2})$ and $G(y)$.

In the author's paper \cite{Pasic03} for arbitrarily given $s\in (1,2)$ it
is constructed a class of Caratheodory functions $f(t,\eta ,\xi )$ such that
the graph $G(y)$ of each smooth enough solution $y$ of the main equation
\begin{equation}
\begin{gathered}
-(|y'|^{p-2}y')'=f(t,y,y')\quad\mbox{in } (a,b),   \\
y(a)=y(b)=0,   \\
y\in W_{\rm loc}^{1,p}((a,b])\cap C([a,b]),
\end{gathered}  \label{1eq}
\end{equation}
satisfies
\begin{equation}
\begin{gathered}
\dim _{M}G(y)=s\quad\mbox{and}\quad\dim _{M}G(y')>1,   \\
\dim _{\rm Mloc}(G(y);a)=s\quad\quad\mbox{and}\quad \dim _{\rm Mloc}(G(y);t)=1\mbox{ for each }
t\in (a,b].
\end{gathered}  \label{1dims}
\end{equation}
Here $\dim _{M}G(y)$ denotes the upper Minkowski-Bouligand (box-counting)
dimension of the graph $G(y)$ and $\dim _{\rm Mloc}(G(y);t)$ denotes the locally
upper Minkowski-Bouligand dimension of $G(y)$ at a point $t\in [a,b]$,
defined by
\begin{gather*}
\dim _{M}G(y)=\limsup_{\varepsilon \to 0}\big(
2-\frac{\log |G_{\varepsilon }(y)|}{\log \varepsilon }\big) , \\
\dim _{\rm Mloc}(G(y);t)=\limsup_{\varepsilon \to 0}\dim_{M}(G(y)\cap B_{\varepsilon }(t,y(t))).
\end{gather*}
Here $|G_{\varepsilon }(y)|$ denotes the Lebesgue measure of
$G_{\varepsilon}(y)$ and $B_{\varepsilon }(t_{1},t_{2})$ denotes a ball in
$\mathbb{R}^{2}$ centered at the point $(t_{1},t_{2})$ with radius
$\varepsilon >0$.

The orders of growth for the asymptotic behaviour of $|G_{\varepsilon }(y)|$
and $|G_{\varepsilon }(y')|$ are given in \eqref{1dims}. That is,
when $\varepsilon \approx 0$ we have
\begin{equation}
|G_{\varepsilon }(y)|\approx \varepsilon ^{2-s}\quad\mbox{and}\quad
|G_{\varepsilon}(y')|\approx \varepsilon ^{2-q}\quad \mbox{for some }q>1,
\label{1Geapr}
\end{equation}
where $y$ is any smooth enough solution of \eqref{1eq}. In particular from
\eqref{1dims} we also have
\begin{equation}
\begin{gathered}
y\notin W^{1,p}(a,b)\quad\mbox{and}\quad\mathop{\rm length}(G(y))
=\mathop{\rm length}(G(y'))=\infty ,  \\
y\in W^{1,p}(a+\varepsilon ,b)\quad\mbox{and}\quad
\mathop{\rm length}(G(y|_{[a+\varepsilon,b]}))<\infty \mbox{ for any }\varepsilon >0.
\end{gathered}  \label{1length}
\end{equation}
In the present paper, we derive some new singular properties of the graphs
$G(y)$ and $G(y')$,  improving \eqref{1dims},  \eqref{1Geapr} and
\eqref{1length}. In this purpose, we need an equivalent way to define
box-counting dimension
\begin{equation}
\dim _{M}G(y) =\inf \{\tau \geq 0:M^{\tau }(G(y))=0\}
 =\sup \{\tau \geq 0:M^{\tau }(G(y))=\infty \}, \label{1Mink}
\end{equation}
where $M^{\tau }(G(y))$ denotes the $\tau$-dimensional upper Minkowski
content of the graph $G(y)$ defined by
\[
M^{\tau }(G(y))=\limsup_{\varepsilon \to 0}(2\varepsilon )^{\tau-2}
|G_{\varepsilon }(y)|.
\]
According to \eqref{1dims} and \eqref{1Mink} we may conclude that
\[
M^{\tau }(G(y))=0\mbox{ for all }\tau >s\quad\mbox{and}\quad M^{\tau }(G(y))=\infty
\mbox{ for all }\tau <s.
\]
Thus, the following three cases are possible
\begin{itemize}
\item[(i)] $M^{s}(G(y))=0$

\item[(ii)]  $M^{s}(G(y))=\infty $

\item[(iii)] $0<M^{s}(G(y))<\infty $.
\end{itemize}
In Section 2 and Section 5, for arbitrarily given $s\in (1,2)$ and under
related assumptions on the nonlinearity $f(t,\eta ,\xi )$ as in \cite
{Pasic03}, we will prove that each solution $y$ of \eqref{1eq} satisfies
$0<M^{s}(G(y))<\infty $. That is, the graph $G(y)$ can be called as an
$s$-set in respect to Minkowski content. Moreover, we find lower and upper
bounds for $M^{s}(G(y))$ such that
\begin{equation}
0<\frac{1}{2^{7}}(b-a)^{s}\leq M^{s}(G(y))\leq m_{s}(b-a)^{s}<\infty ,
\label{1Mlu}
\end{equation}
where the positive constant $m_{s}$ only depends on $s$. It is interesting
in \eqref{1Mlu} that the $s$-dimensional ''length'' of $G(y)$ depends on
the $s-$ power of the length of interval $[a,b]$, which improves
$\mathop{\rm length}(G(y))=\infty $ appearing in \eqref{1length}. Furthermore, we will
give the existence of an $\varepsilon _{0}>0$ such that
\begin{equation}
0<\frac{1}{2^{6}}(b-a)^{s}\varepsilon ^{2-s}\leq |G_{\varepsilon }(y)|
\quad \mbox{for each }\varepsilon \in (0,\varepsilon _{0}),  \label{1Gel}
\end{equation}
where $y$ is any solution of \eqref{1eq}. This statement gives us a lower
bound for asymptotic behaviour of $|G_{\varepsilon }(y)|$ as $\varepsilon
\approx 0$. It is more precise than corresponding one in \eqref{1Geapr}.

As the second, for arbitrarily given $s\in (1,2)$ and under the same
assumptions on the nonlinearity $f(t,\eta ,\xi )$ as in getting of
\eqref{1Mlu}--\eqref{1Gel},  we will prove in Section 3 that each smooth enough
solution $y$ of \eqref{1eq} satisfies
\begin{equation}
0<\frac{1}{2^{4}}(b-a)^{s/2}\leq M^{1+s/2}(G(y')).  \label{1Mderl}
\end{equation}
Because of \eqref{1Mink},  the inequality \eqref{1Mderl} improves
corresponding result from \eqref{1dims}, where $\dim _{M}G(y')>1.$
Here, we have shown that $\dim _{M}G(y')\geq 1+s/2$. Furthermore,
we will show the existence of an $\varepsilon _{0}>0$ such that
\begin{equation}
0<\frac{\sqrt{2}}{2^{4}}(b-a)^{s/2}\varepsilon ^{1-s/2}\leq |G_{\varepsilon
}(y')|\quad \mbox{for each }\varepsilon \in (0,\varepsilon _{0}).
\label{1Gederl}
\end{equation}
It gives us a lower bound for the asymptotic behaviour of $|G_{\varepsilon
}(y')|$ as $\varepsilon \approx 0$.

Next, by means of \eqref{1length} we have that the graph $G(y)$ of each
solution $y$ of \eqref{1eq} is high concentrated (in some sense) at the
boundary point $t=a$. How much of the graph $G(y)$ is concentrated near $t=a$
in the sense of Minkowski content, we will consider in Section 6. In this
purpose, we define the $s$-dimensional upper (Minkowski) density of $G(y)$
at a point $t\in [a,b]$ as follows
\[
D^{s}(G(y);t)=\limsup_{r\to 0}\frac{M^{s}(G(y)\cap B_{r}(t,y(t)))}{(2r)^{s}}.
\]
Let us remark that by means of $y\in W^{1,p}(a+\varepsilon ,b)$,  where
$\varepsilon >0$, it is clear that
\[
D^{s}(G(y);t)=0\quad\mbox{for any }t\in (a,b]\quad\mbox{and}\quad s\in (1,2).
\]
In Section 6, for arbitrarily given $s\in (1,2)$ and under the same
assumptions on the nonlinearity $f(t,\eta ,\xi )$ as in getting of
\eqref{1Mlu}--\eqref{1Gel}, we will find a constant $d_{s}$ such that each
solution $y$ of \eqref{1eq} satisfies
\begin{equation}
0<d_{s}\leq D^{s}(G(y);t=a).  \label{1Dsl}
\end{equation}
Moreover, we will prove that
\begin{equation}
d_{s}(2r)^{s}\leq M^{s}(G(y)\cap B_{r}(a,y(a)))\quad\mbox{for each }r\in (0,b-a),
\label{1Mslr}
\end{equation}
where $y$ is any smooth enough solution of \eqref{1eq}. This inequality
complete the statement \eqref{1Mlu}.

To derive the statements \eqref{1Mlu}--\eqref{1Gederl}, in Section
2, Section 3 and Section 5, we consider some metric properties of two types
of rapid oscillations of real continuous functions: the first one is a kind
of oscillations where the function is rapidly jumping over given obstacles,
and the second one is a kind of oscillations where the convex and concave
properties of the function are rapidly changing. In Section 4, some
necessary conditions on the nonlinearity $f(t,\eta ,\xi )$ will be given
such that each solution of \eqref{1eq} is rapidly oscillating. These
conditions on $f(t,\eta ,\xi )$ will be very close to corresponding
sufficient conditions used in Section 2. In appendix of the paper, we will
give some technical results which play an important role in the proofs of
the main results.

\subsection*{Further Remarks}
A. It seems there is no known article dealing with this kind of problems.
But, in the proofs of the main results we use some methods recently
introduced in the author's paper \cite{Pasic03}.

\noindent B. The existence result for the equation \eqref{1eq} where the
Caratheodory function $f(t,\eta ,\xi )$ satisfies the assumptions as in the
paper, was considered in the appendix of \cite{Pasic03}. It was based on
some known results about the existence of continuous solutions for the
equations with singular nonlinearity, explored via the sub- and
super-solutions technique. See O'Regan's book \cite[Chapter 14]{ORegan}.

\noindent C. Even the Minkowski content $M^{s}$ is not a measure in the axiomatic
sense, see for instance \cite{Falc99} and \cite{Matt95}, the main results of
the paper give some additional informations about the singular-metric
boundary behaviour of the graph of any sufficiently smooth bounded solutions
of the equation \eqref{1eq}.

\noindent D. To make a comparison between singularity and regularity of bounded
solutions, we refer to \cite{Rak90} and references therein, where regularity
of solutions of quasilinear elliptic equations associated with $p$-Laplacian
are studied.

\noindent E. About the properties and calculations of fractal dimensions and Minkowski
content of several types of sets in $\mathbb{R}^{n}$,  we refer to
\cite{Bou25, Falc95, Falc99, Lem90, Matt95, Peit92, Trico82, Trico95}.

\noindent F. Our main results can be generalized to the case of nonlinear variational
inequalities and quasilinear elliptic systems associated with the
one-dimensional $p$-Laplacian. See \cite{Pasic04}.

\section{Rapid oscillations and lower bounds for $|G_{\varepsilon }(y)|$ and
$M^{s}(G(y))$}


For a function $y:[a,b]\to \mathbb{R}$ let us introduce a type of
very rapid oscillations of $y$ near the boundary point $t=a$. A classical
example for such type of oscillations is the function
$y(t)=t^{\alpha }\cos 1/t^{\beta }$ near $t=0$,  where $0<\alpha <\beta$.

\begin{definition} \rm \label{def2.1}
Let $a_{k}$ be a decreasing sequence of real numbers
from interval $(a,b)$ satisfying
\begin{equation}
\begin{gathered}
\mbox{$a_{k}\searrow a$ and there is an $\varepsilon _{0}>0$
such that for each $\varepsilon \in (0,\varepsilon _{0})$}  \\
\mbox{there is a $k(\varepsilon )\in \mathbb{N}$ such that
$a_{j-1}-a_{j}\leq \varepsilon /2$ for each $j\geq k(\varepsilon )$.}
\end{gathered}  \label{2ak}
\end{equation}
Let $\theta $ and $\omega $ be two measurable and bounded functions, both
defined on $[a,b]$,  such that $\theta (t)\leq \omega (t)$ for each $t\in
[a,b]$. We say that a function $y$ defined on $[a,b]$ is
$(\theta ,\omega,a_{k})$-\emph{rapidly oscillating} if there is a sequence
$\sigma _{k}\in(a_{k},a_{k-1})$,  $k>1$ such that
\begin{equation}
y(\sigma _{2k})\geq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega \quad
\mbox{and}\quad y(\sigma _{2k+1})\leq \mathop{\rm ess\,inf}
_{_{(a_{2k+1},a_{2k})}}\theta ,\ k\geq 1.  \label{2osc}
\end{equation}
\end{definition}

For the record, the number $k(\varepsilon )$ could be called as the index of
$\varepsilon -$ density of a sequence $a_{k}$. It is interesting to show how
to calculate the number $k(\varepsilon )$ for a given sequence $a_{k}$ which
satisfies \eqref{2ak}.

\begin{example} \label{ex2.2} \rm
 Let $a_{k}$ be a sequence of real numbers from interval
$(a,b)$ given by
\begin{equation}
a_{k}=a+\frac{b-a}{2}(\frac{1}{k})^{1/\beta },\quad k\geq 1\mbox{ and } 0<\beta
<\infty .  \label{2akb}
\end{equation}
Such a type of the sequence $a_{k}$ is appearing in oscillations of the
function $y(t)=t^{\alpha }\sin 1/t^{\beta }$ near $t=0$,  where $a=0$,  $b=1$
and $0<\alpha <\beta $. One can take for $k(\varepsilon )$ to be any natural
number which satisfies
\begin{equation}
2(\frac{\beta \varepsilon }{b-a})^{-\frac{\beta }{\beta +1}}\leq
k(\varepsilon )\quad \mbox{for each }\varepsilon \in (0,\varepsilon _{0})\mbox{ and for any }
\varepsilon _{0}>0.  \label{2kepsl}
\end{equation}
Indeed, using an elementary inequality
\[
\big( \frac{1}{j-1}\big) ^{1/\beta }-\big( \frac{1}{j}\big) ^{1/\beta
}\leq \frac{2^{1+1/\beta }}{\beta }\big( \frac{1}{j}\big) ^{1+1/\beta}\,,
\]
where $\beta >0$ and $j\geq 2$,  it is easy to show that the sequence $a_{k}$
defined in \eqref{2akb} satisfies the statement \eqref{2ak} in respect
to $k(\varepsilon )$ determined in \eqref{2kepsl}.
\end{example}

Now, we consider the following lemma which is a modification of
\cite[Lemma 2.1, p. 271]{Pasic03}. It gives us an elementary and useful
metric property of the $(\theta ,\omega ,a_{k})$-rapidly oscillating
functions.

\begin{lemma} \label{lm2.3}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{2ak}. Let $\theta (t)$ and
$\omega (t)$ be two measurable and bounded real functions on $[a,b]$,
$\theta (t)\leq \omega (t)$, $t\in [a,b]$, such that
\begin{equation}
\mathop{\rm ess\,inf}_{(a_{2k+2},a_{2k+1})}\theta \geq
\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta,\quad
\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\omega \leq
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega ,\quad k\geq 1.  \label{2essto}
\end{equation}
Let $y$ be $(\theta ,\omega ,a_{k})$-rapidly oscillating function on
$[a,b]$ and let $y\in C((a,b])$. Then we have
\begin{equation}
|G_{\varepsilon }(y)|\geq \int_{a}^{a_{k(\varepsilon )}}(\omega (t)-\theta
(t))dt\quad\mbox{for each }\varepsilon \in (0,\varepsilon _{0}),
\label{2Gey}
\end{equation}
where $k(\varepsilon )$ and $\varepsilon _{0}$ are appearing in
\eqref{2ak}.
\end{lemma}

Let us remark that the condition \eqref{2essto} can be easy verified if,
for instance, $\theta $ is decreasing and $\omega $ is increasing on $[a,b]$.


\begin{proof}[Proof of Lemma \ref{lm2.3}]
Let $\varepsilon $ be a fixed real number such
that $\varepsilon \in (0,\varepsilon _{0})$,  where $\varepsilon _{0}$ is
from \eqref{2ak}. We use the notation
\begin{gather*}
A({\varepsilon },\theta ,\omega )=\{(t,y)\in \mathbb{R}^{2}:t\in
(a,a_{k(\varepsilon )}],\;y\in [\theta (t),\omega (t)]\},   \\
B_{2k}=[a_{2k},a_{2k-1}]\times [\mathop{\rm ess\,inf}_{(a_{2k-1},a_{2k-2})}\theta ,
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega ],   \\
B_{2k+1}=[a_{2k+1},a_{2k}]\times [\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta ,
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega ].
\end{gather*}
Let $y$ be a $(\theta ,\omega ,a_{k})$-rapidly oscillating function on $[a,b]$
such that $y\in C((a,b])$. By Definition \ref{def2.1}, it implies the existence of a
sequence $\sigma _{k}\in (a_{k},a_{k-1})$ such that
\begin{equation}
y(\sigma _{2k})\geq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega \quad
\mbox{and}\quad y(\sigma _{2k+1})\leq \mathop{\rm ess\,inf}
_{_{(a_{2k+1},a_{2k})}}\theta ,\quad k\geq 1.  \label{2jump}
\end{equation}
Let $k$ be a fixed natural number such that $k\geq k(\varepsilon )+1$. Let
$(t_{0},y_{0})$ be an arbitrarily given point of $B_{k}$. Let us remark that
from \eqref{2jump} we get:
\begin{gather*}
\mbox{if }(t_{0},y_{0})\in B_{2j}\mbox{ then }y(\sigma _{2j-1})\leq
y_{0}\leq y(\sigma _{2j}), \\
\mbox{if }(t_{0},y_{0})\in B_{2j+1}\mbox{ then }y(\sigma _{2j+1})\leq
y_{0}\leq y(\sigma _{2j}),\ j\geq 1.
\end{gather*}
In particular, it implies that
\[
\big\{(t,y_{0}):t\in (\sigma _{k},\sigma _{k-1})\big\}\cap G(y|_{[\sigma _{k},\sigma
_{k-1}]})\neq \emptyset ,
\]
where $G(y|_{[\sigma _{k},\sigma _{k-1}]})$ denotes the graph of the
function $y|_{[\sigma _{k},\sigma _{k-1}]}$ (here $y|_{I}$ denotes the
function-restriction of $y$ on interval $I$). Hence, there is a point $s\in
(\sigma _{k},\sigma _{k-1})$ such that $(s,y_{0})\in G(y|_{[\sigma
_{k},\sigma _{k-1}]})$. Now, by the help of \eqref{2ak} we get
\begin{align*}
d((t_{0},y_{0}),G(y|_{[\sigma _{k},\sigma _{k-1}]}))
&\leq d((t_{0},y_{0}),(s,y_{0}))
\leq d((a_{k},y_{0}),(\sigma _{k-1},y_{0})) \\
&=\sigma _{k-1}-a_{k}\leq a_{k-2}-a_{k}\\
&=a_{k-1}-a_{k}+a_{k-2}-a_{k-1}\leq
\frac{\varepsilon }{2}+\frac{\varepsilon }{2}=\varepsilon .
\end{align*}
Thus, we have proved that
\begin{equation}
B_{k}\subseteq G_{\varepsilon }(y|_{[\sigma _{k},\sigma _{k-1}]})
\quad \mbox{ for any }k\geq k(\varepsilon )+1\quad\mbox{and}\quad\varepsilon \in (0,\varepsilon _{0}),
\label{2Bk}
\end{equation}
where $G_{\varepsilon }(y|_{[\sigma _{k},\sigma _{k-1}]})$ denotes the
$\varepsilon$-neighbourhood of the graph
$G(y|_{[\sigma _{k},\sigma_{k-1}]})$. Let us remark that from
 \eqref{2essto} we may easy conclude that
\begin{equation}
A({\varepsilon },\theta ,\omega )\subseteq \cup _{k=k(\varepsilon
)+1}^{\infty }B_{k},
\label{2Aeto}
\end{equation}
 where $\varepsilon \in (0,\varepsilon _{0})$.
  According to \eqref{2Bk} and \eqref{2Aeto} we obtain:
\[
A({\varepsilon },\theta ,\omega )\subseteq \cup _{k=k(\varepsilon
)+1}^{\infty }G_{\varepsilon }(y|_{[\sigma _{k},\sigma _{k-1}]})\subseteq
G_{\varepsilon }\bigl(\cup _{k=k(\varepsilon )+1}^{\infty }y|_{[\sigma
_{k},\sigma _{k-1}]}\bigl)\subseteq G_{\varepsilon }(y),\
\]
where $\varepsilon \in (0,\varepsilon _{0})$. Taking the Lebesgue measure on
the both sides in the previous statement, it proves the inequality
\eqref{2Gey}. \end{proof}

How to calculate the right hand side in \eqref{2Gey} it is shown in the
following example.

\begin{example} \label{ex2.4} \rm
 Let $y$ be a real continuous function on $[a,b]$ and
let $y$ be $(\theta ,\omega ,a_{k})$-rapidly oscillating, where $\theta $,
$\omega $ and $a_{k}$ are given by:
\begin{equation}
\begin{gathered}
a_{k}=a+\frac{b-a}{2}(\frac{1}{k})^{1/\beta },\quad k\geq 1,  \\
\theta (t)=-(t-a)^{\alpha }\quad\mbox{and}\quad \omega (t)=(t-a)^{\alpha },\quad
t\in [a,b], \\
\mbox{where $\alpha$ and $\beta$ satisfy  $0<\alpha <\beta <\infty$.}
\end{gathered}  \label{2akto}
\end{equation}
We can take for $k(\varepsilon )$ to be any number that satisfies
\begin{equation}
\begin{gathered}
c_{0}\big( \frac{1}{\varepsilon }\big) ^{\frac{\beta }{\beta +1}}\leq
k(\varepsilon )\leq 2c_{0}\big( \frac{1}{\varepsilon }\big) ^{\frac{\beta
}{\beta +1}}\quad\mbox{for each }\varepsilon \in (0,\varepsilon _{0}),  \\
\mbox{where }c_{0}=2\big( \frac{b-a}{\beta }\big) ^{\frac{\beta }{\beta +1
}}\mbox{ and } \varepsilon _{0}=\frac{b-a}{\beta }.
\end{gathered}  \label{2kepslu}
\end{equation}
It is clear that \eqref{2kepslu} implies \eqref{2kepsl} and so the main
conclusion of Example \ref{ex2.2} is still valid. That is, the sequence $a_{k}$
given in \eqref{2akto} satisfies the condition \eqref{2ak}, where
$k(\varepsilon )$ and $\varepsilon _{0}$ are taken to be as in
\eqref{2kepslu}.

In contrast to \eqref{2kepsl},  where $\varepsilon _{0}$ is an arbitrary
positive number, we need in \eqref{2kepslu} an
$\varepsilon_{0}=(b-a)/\beta $. This condition on $\varepsilon _{0}$
is essentialy to ensure that $k(\varepsilon )\in \mathbb{N}$ for all
$\varepsilon \in (0,\varepsilon _{0})$.

Next, since $\theta (t)=-(t-a)^{\alpha }$ and $\omega (t)=(t-a)^{\alpha }$
are decreasing and increasing on $[a,b]$,  respectively, it is clear that
$\theta $ and $\omega $ satisfy the condition \eqref{2essto}.

Thus, the $(\theta ,\omega ,a_{k})$ defined by \eqref{2akto} satisfies the
hypotheses of Lemma \ref{lm2.3}. Hence, the inequality \eqref{2Gey} can be applied
to our situation here. It gives us
\begin{align*}
|G_{\varepsilon }(y)| &\geq \int_{a}^{a_{k(\varepsilon )}}(\omega
(t)-\theta (t))dt=2\int_{a}^{a_{k(\varepsilon )}}(t-a)^{\alpha }dt \\
&=\frac{2}{\alpha +1}(a_{k(\varepsilon )}-a)^{\alpha +1}=\frac{2^{-\alpha
}(b-a)^{\alpha +1}}{\alpha +1}(\frac{1}{k(\varepsilon )})^{\frac{\alpha +1}{
\beta }}\quad\mbox{for each }\varepsilon \in (0,\varepsilon _{0}).
\end{align*}
 From the right inequality in \eqref{2kepslu} we have in particular
\[
\frac{1}{k(\varepsilon )}\geq \frac{1}{4}(\frac{\beta }{b-a})^{\frac{\beta }{
\beta +1}}\varepsilon ^{\frac{\beta }{\beta +1}}\quad
\mbox{for each }\varepsilon \in (0,\varepsilon _{0}=\frac{b-a}{\beta }).
\]
These inequalities give us
\[
|G_{\varepsilon }(y)|\geq c\varepsilon ^{\frac{\alpha +1}{\beta +1}}\quad
\mbox{ for each }\varepsilon \in (0,\varepsilon _{0}),
\]
where
\[
\varepsilon _{0}=\frac{b-a}{\beta }\quad\mbox{and}\quad
c=\frac{2^{-\alpha }}{\alpha
+1}(\frac{1}{4})^{\frac{\alpha +1}{\beta }}\beta ^{\frac{\alpha +1}{\beta +1}
}(b-a)^{\frac{(\alpha +1)\beta }{\beta +1}}.
\]
\end{example}

To derive a lower bound for $M^{s}(G(y))$ where $y$ is any solution
of \eqref{1eq} we need to impose on the function $f(t,\eta ,\xi )$ some
sufficient conditions such that each solution of \eqref{1eq} is
$(\theta,\omega ,a_{k})$-rapidly oscillating. It is the subject of the following
lemma which is a modification of \cite[Lemma 4.1, p. 280]{Pasic03} and
\cite[Lemma 4.2, p. 281]{Pasic03}.

\begin{lemma} \label{lm2.5}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{2ak}. Let $\tilde{\theta}_{0}$
and $\tilde{\omega}_{0}$ be two arbitrarily given real numbers, and
let $\theta (t)$ and $\omega (t)$ be two measurable and bounded real
functions on $[a,b]$, $\theta (t)\leq \omega (t)$, $t\in [a,b]$, which
satisfy \eqref{2essto} and
\begin{equation}
\tilde{\theta}_{0}\leq \mathop{\rm ess\,inf}_{(a,b)}\theta <
\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta \,, \quad
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega
<\mathop{\rm ess\,sup}_{(a,b)}\omega \leq \tilde{\omega}_{0},\quad k\geq 1.
  \label{2ttot}
\end{equation}
Let the sets $J_{k}$ be defined by
\[
J_{2k}=(\tilde{\theta}_{0},\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}
\omega )\quad \mbox{and}\quad J_{2k+1}=(\mathop{\rm ess\,inf}
_{(a_{2k+1},a_{2k})}\theta ,\tilde{\omega}_{0}),\quad k\geq 1.
\]
Next, let for each $k\geq 1$ the Caratheodory function $f(t,\eta ,\xi) $
satisfy
\begin{gather}
f(t,\eta ,\xi )\geq 0,\quad t\in (a_{2k},a_{2k-1}),\ \eta \in J_{2k},\ \xi
\in \mathbb{R},  \label{2fp} \\
\int_{A_{2k}} \mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2k}\times
R}f(t,\eta ,\xi )dt
>\frac{c(p)}{(a_{2k-1}-a_{2k})^{p-1}}\frac{(\tilde{\omega}_{0}-\tilde{
\theta}_{0})^{p}}{(\tilde{\omega}_{0}-\mathop{\rm ess\,sup}
_{(a_{2k},a_{2k-1})}\omega )},  \label{2fsp}
\\
f(t,\eta ,\xi )\leq 0,\quad t\in (a_{2k+1},a_{2k}),\ \eta \in J_{2k+1},\ \xi
\in \mathbb{R},  \label{2fn} \\
\int_{A_{2k+1}} \mathop{\rm ess\,sup}_{(\eta ,\xi )\in J_{2k+1}\times
R}f(t,\eta ,\xi )dt
<-\frac{c(p)}{(a_{2k}-a_{2k+1})^{p-1}}\frac{(\tilde{\omega}_{0}-\tilde{
\theta}_{0})^{p}}{(\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta -
\tilde{\theta}_{0})},  \label{2fsn}
\end{gather}
where $c(p)=2[4(p-1)]^{p-1}$ and $A_{k}$ is a family of sets
\[
A_{k}=[a_{k}+\frac{1}{4}(a_{k-1}-a_{k}),a_{k-1}-\frac{1}{4}
(a_{k-1}-a_{k})],\quad k\geq 1.
\]
Then each solution $y$ of the equation \eqref{1eq} such that
\begin{equation}
\tilde{\theta}_{0}\leq y(t)\leq \tilde{\omega}_{0},\quad t\in (a,b)
\label{2comp}
\end{equation}
is $(\theta ,\omega ,a_{k})$-rapidly oscillating on $[a,b]$.
\end{lemma}

The proof of this lemma will be sketched in Appendix of this paper. It is
worth to mention that the condition \eqref{2comp} will be easy achieved in
Theorem \ref{thm2.7} below.
An example for such a class of Caratheodory functions $f(t,\eta ,\xi )$
which satisfies the hypotheses \eqref{2fp}--\eqref{2fsn} is given as
follows.

\begin{example} \rm \label{ex2.6}
To simplify the notation, let $\tilde{\theta}
_{0}$, $\tilde{\omega}_{0}$,  $\theta _{2k+1}$,  and $\omega _{2k}$ be defined
as follows:
\begin{gather*}
\tilde{\theta}_{0}=\mathop{\rm ess\,inf}_{(a,b)}\theta \quad\mbox{and}\quad
\tilde{\omega}_{0}=\mathop{\rm ess\,sup}_{(a,b)}\omega , \\
\theta _{2k+1}=\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta
\quad\mbox{and}\quad \omega _{2k}=\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega .
\end{gather*}
Let $g=g(t,\eta ,\xi )$ be a Caratheodory function defined by
\begin{align*}
g&=\frac{\pi c(p)(\tilde{\omega}_{0}-\tilde{\theta}_{0})^{p}}{\sin \frac{\pi
}{4}} \sum_{k=1}^{\infty }\bigl[\frac{(\eta -\tilde{\omega}_{0})^{-}}{(
\tilde{\omega}_{0}-\omega _{2k})^{2}}\frac{\sin (\frac{\pi }{a_{2k-1}-a_{2k}}
(t-a_{2k}))}{(a_{2k-1}-a_{2k})^{p}}K_{[a_{2k},a_{2k-1}]}(t) \\
&-\frac{(\eta -\tilde{\theta}_{0})^{+}}{(\theta _{2k+1}-\tilde{\theta}
_{0})^{2}}\frac{\sin (\frac{\pi }{a_{2k}-a_{2k+1}}(t-a_{2k+1}))}{
(a_{2k}-a_{2k+1})^{p}}K_{[a_{2k+1},a_{2k}]}(t)\bigl],
\end{align*}
where $c(p)$ is appearing in \eqref{2fsp} and \eqref{2fsn}, and where
$K_{A}(t)$ denotes, as usual, the characteristic function of a set $A$. Also,
$\eta ^{-}=\max \{0,-\eta \}$ and $\eta ^{+}=\max \{0,\eta \}$.

Even $K_{A}(t)$ is not continuous in $t$, it is not dificult to check that
$g(t,\eta ,\xi )$ is continuous in all its variables. Next, for any fixed
$k\in \mathbf{N,}$ it is clear that
\begin{gather*}
(\eta -\tilde{\omega}_{0})^{-}\sin (\frac{\pi }{a_{2k-1}-a_{2k}}
(t-a_{2k}))\geq 0\quad \mbox{for any }t\in [a_{2k},a_{2k-1}],\ \eta \in J_{2k}, \\
\frac{(\eta -\tilde{\omega}_{0})^{-}}{\tilde{\omega}_{0}-\omega _{2k}}\geq
1\quad \mbox{for any }\eta \in J_{2k},\mbox{ where }J_{2k}=(\tilde{\theta}
_{0},\omega _{2k}).
\end{gather*}
Hence, for any $k\in \mathbb{N}$, $t\in [a_{2k},a_{2k-1}]$,  $\eta \in
J_{2k}$, and $\xi \in \mathbb{R}$ we have
\begin{gather*}
g(t,\eta ,\xi )=\frac{\pi c(p)(\tilde{\omega}_{0}-\tilde{\theta}_{0})^{p}}{
\sin \frac{\pi }{4}}\frac{(\eta -\tilde{\omega}_{0})^{-}}{(\tilde{\omega}
_{0}-\omega _{2k})^{2}}\frac{\sin (\frac{\pi }{a_{2k-1}-a_{2k}}(t-a_{2k}))}{
(a_{2k-1}-a_{2k})^{p}}\geq 0,\  \\
\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2k}\times R}g(t,\eta ,\xi
)\geq \frac{\pi c(p)}{(\sin \frac{\pi }{4})}\frac{(\tilde{\omega}_{0}-\tilde{
\theta}_{0})^{p}}{(\tilde{\omega}_{0}-\omega _{2k})}\frac{\sin (\frac{\pi }{
a_{2k-1}-a_{2k}}(t-a_{2k}))}{(a_{2k-1}-a_{2k})^{p}}.
\end{gather*}
 From the first equality, it follows that $g(t,\eta ,\xi )$ satisfies
\eqref{2fp}. Also, integrating the second inequality over the set
\[
\lbrack a_{2k}+\frac{1}{4}(a_{2k-1}-a_{2k}),a_{2k-1}-\frac{1}{4}
(a_{2k-1}-a_{2k})],
\]
it immediately shows that $g(t,\eta ,\xi )$ satisfies \eqref{2fsp} too. On
the same way the dual hypotheses \eqref{2fn} and \eqref{2fsn} can be
verified. Thus, the function $f(t,\eta ,\xi )=g(t,\eta ,\xi )$ satisfies the
assumptions of Lemma \ref{lm2.5}.
\end{example}

Now, we are able to formulate the first main result of the paper.

\begin{theorem} \label{thm2.7}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{2ak}. Let $\theta (t)$ and
$\omega (t)$ be two measurable and bounded real functions on $[a,b]$,
$\theta (t)\leq \omega (t)$, $t\in [a,b]$, which satisfy \eqref{2essto},
\eqref{2ttot} and
\begin{equation}
\mathop{\rm ess\,inf}_{(a,b)}\theta <0<\mathop{\rm ess\,sup}_{(a,b)}\omega .
\label{2nula}
\end{equation}
Next, let for each $k\geq 1$ the Caratheodory function $f(t,\eta ,\xi) $
satisfy \eqref{2fp}--\eqref{2fsn}, and
\begin{equation}
\begin{gathered}
f(t,\eta ,\xi )<0,\quad t\in (a,b),\ \eta >\tilde{\omega}_{0} \mbox{ and }
\xi \in \mathbb{R},   \\
f(t,\eta ,\xi )>0,\quad t\in (a,b),\ \eta <\tilde{\theta}_{0} \mbox{ and }
\xi \in \mathbb{R},
\end{gathered}  \label{2fcomp}
\end{equation}
where $\tilde{\theta}_{0}$ and $\tilde{\omega}_{0}$ are two
arbitrarily given real numbers satisfying \eqref{2ttot}. Then for any
$s\in (1,2)$ and for each solution $y$ of  \eqref{1eq} there
holds true
\begin{equation}
\begin{gathered}
|G_{\varepsilon }(y)|\geq \int_{a}^{a_{k(\varepsilon )}}(\omega (t)-\theta
(t))dt\quad\mbox{for each } \varepsilon \in (0,\varepsilon _{0}),\\
M^{s}(G(y))\geq 2^{s-2}\limsup_{\varepsilon \to
0}\varepsilon ^{s-2}\int_{a}^{a_{k(\varepsilon )}}(\omega (t)-\theta (t))dt,
\end{gathered}  \label{2GeMs}
\end{equation}
where $\varepsilon _{0}$ is appearing in \eqref{2ak}.
\end{theorem}

\begin{proof}
 Let $y$ be a solution of the equation \eqref{1eq}. Because
of \cite[Lemma 3.2, p. 279]{Pasic03}, from \eqref{2ttot}, \eqref{2nula}
and \eqref{2fcomp} we get $\tilde{\theta}_{0}\leq y(t)\leq \tilde{\omega}
_{0}$ for each $t\in [a,b]$. Therefore the assumptions of Lemma \ref{lm2.5} are
satisfied and so $y$ is $(\theta ,\omega ,a_{k})$-rapidly oscillating
function on $[a,b]$. Now, by means of Lemma \ref{lm2.3} and the definition of
$M^{s}(G(y))$,  the desired statement \eqref{2GeMs} easy follows.
\end{proof}

The following result is the main consequence of \eqref{2GeMs}. We can
derive also lower bounds appearing in \eqref{1Mlu} and \eqref{1Gel}.

\begin{corollary} \label{coro2.8}
For arbitrarily given real number $s\in (1,2)$,
let $a_{k}$, $\theta $ and $\omega $ be given by
\begin{equation}
\begin{gathered}
a_{k}=a+\frac{b-a}{2}(\frac{1}{k})^{1/\beta },\quad k\geq 1,   \\
\theta (t)=-(t-a)\quad\mbox{and}\quad \omega (t)=t-a,\quad t\in (a,b), \\
\mbox{ where $\beta $ satisfies $1<\beta <\infty$ and
$\frac{2\beta }{\beta +1}=s$.}
\end{gathered}  \label{2aktos}
\end{equation}
Let the Caratheodory function $f(t,\eta ,\xi )$ satisfy
\eqref{2fp}--\eqref{2fsn}, and \eqref{2fcomp} in respect to such given
$(\theta ,\omega,a_{k})$,  where $\tilde{\theta}_{0}$ and
 $\tilde{\omega}_{0}$ are two
arbitrarily given real numbers satisfying \eqref{2ttot}. Then each
solution $y$ of the equation \eqref{1eq} satisfies
\begin{equation}
\begin{gathered}
|G_{\varepsilon }(y)|\geq \frac{1}{2^{6}}(b-a)^{s}\varepsilon ^{2-s}\ \
\emph{for\ each}\ \varepsilon \in (0,\varepsilon _{0}=\frac{b-a}{\beta }),\\
M^{s}(G(y))\geq \frac{1}{2^{7}}(b-a)^{s}.
\end{gathered}  \label{2GeMsy}
\end{equation}
\end{corollary}

\begin{proof}
First, we know, by Example \ref{ex2.4}, that the sequence
$a_{k}$ defined in \eqref{2aktos} satisfies the condition \eqref{2ak} in
respect to $k(\varepsilon )$ determined by \eqref{2kepslu}. Next, since
$\theta (t)=-(t-a)$ is decreasing and $\omega (t)=t-a$ is increasing, it is
clear that the functions $\theta $ and $\omega $ defined in \eqref{2aktos}
satisfy the conditions \eqref{2essto},  \eqref{2ttot} and \eqref{2nula}.

Thus, the hypotheses of Theorem \ref{thm2.7} are satisfied and so the statement
\eqref{2GeMs} may be used here. Putting given data from \eqref{2aktos} into
\eqref{2GeMs} we can calculate
\begin{equation}
|G_{\varepsilon }(y)|\geq \int_{a}^{a_{k(\varepsilon )}}2(t-a)dt
=\big(\frac{b-a}{2}\big) ^{2}\big( \frac{1}{k(\varepsilon )}\big) ^{2/\beta}.  \label{2start}
\end{equation}
Since $\beta >1$ and $2\beta /(\beta +1)=s$,  by the help of the right
inequality in \eqref{2kepslu} we get
\[
\big( \frac{b-a}{2}\big) ^{2}\big( \frac{1}{k(\varepsilon )}\big)
^{2/\beta}\geq \frac{(b-a)^{2}}{4}\big( \frac{1}{4}\big)
^{2/\beta}\frac{\beta ^{\frac{2}{\beta +1}}}{(b-a)^{\frac{2}{\beta +1}}}
\varepsilon ^{\frac{2}{\beta +1}}\geq \frac{1}{2^{6}}(b-a)^{s}\varepsilon
^{2-s}.
\]
Putting this inequality into \eqref{2start} we have proved \eqref{2GeMsy}.
\end{proof}

At the end of this section we give an example for such a class of
Caratheodory functions $f(t,\eta ,\xi )$ that satisfies the assumptions of
Theorem \ref{thm2.7}.

\begin{example} \rm \label{ex2.9}
Let $h_{1}(t)$ and $h_{2}(t)$ be two measurable
functions on $(a,b)$ such that $h_{1}(t)>0$ and $h_{2}(t)>0$. Let
$f(t,\eta,\xi )$ be a function defined by
\[
f(t,\eta ,\xi )=\mathbf{-}h_{1}(t)(\eta -\mathop{\rm ess\,sup}
_{(a,b)}\omega )^{+}+h_{2}(t)(\eta -\mathop{\rm ess\,inf}_{(a,b)}\theta
)^{-}+g(t,\eta ,\xi ),
\]
where the function $g(t,\eta ,\xi )$ is constructed in Example \ref{ex2.6} above.
Since $g(t,\eta ,\xi )$ satisfies the assumptions of Lemma \ref{lm2.5}, it is clear
that such given $f(t,\eta ,\xi )$ satisfies the assumptions of
Theorem \ref{thm2.7}.
\end{example}

\section{Lower bounds for $|G_{\varepsilon }(y')|$ and  $M^{s}(G(y'))$}

We proceed with some observations from the previous section but
they are pointed to the derivative $y'$ of any smooth enough real
function $y $. It is started with a relation between the $(\theta
,\omega ,a_{k})- $rapid oscillations of a function $y$ and the
asymptotic behaviour of $|G_{\varepsilon }(y')|$ as $\varepsilon
\approx 0$.

\begin{lemma} \label{lm3.1}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{2ak}. Let $\theta (t)$ and
$\omega (t)$ be two measurable and bounded real functions on $[a,b]$,
$\theta (t)\leq \omega (t)$, $t\in [a,b]$, which satisfy \eqref{2essto}.
Next, let $y$ be $(\theta ,\omega ,a_{k})$-rapidly oscillating on $[a,b]$ and
let $y\in C((a,b])\cap C^{1}(a,b)$ . Then we have
\begin{equation}
|G_{\varepsilon }(y')|\geq \sum_{k=k(\varepsilon /2)}^{\infty }(
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega
-\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta )\quad\mbox{ for each }
\varepsilon \in (0,\varepsilon _{0}),  \label{3Gedy}
\end{equation}
where $k(\varepsilon )$ and $\varepsilon _{0}$ are defined in
\eqref{2ak}.
\end{lemma}

\begin{proof} Let $y$ be a $(\theta ,\omega ,a_{k})$-rapidly oscillating
function on $[a,b]$. By means of Definition \ref{def2.1}, there is a sequence $\sigma
_{k}\in (a_{k},a_{k-1})$,  $k>1$ such that
\begin{equation}
y(\sigma _{2k})\geq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega \quad
\mbox{and}\quad  y(\sigma _{2k+1})\leq \mathop{\rm ess\,inf}
_{_{(a_{2k+1},a_{2k})}}\theta ,\quad k\geq 1.  \label{3jump}
\end{equation}
Let $k(\varepsilon )$ and $\varepsilon _{0}$ be taken from \eqref{2ak}.
So, we have that $a_{j-1}-a_{j}\leq \varepsilon /4\ $for each
$j\geq k(\frac{\varepsilon }{2})$. Hence, the sequence $\sigma _{k}$
also satisfies
\begin{equation}
\mbox{$\sigma _{k}\searrow a$ and $\sigma _{j-1}-\sigma _{j}\leq
\varepsilon /2$ for each $j\geq k(\frac{\varepsilon }{2})$
and $\varepsilon \in (0,\varepsilon _{0})$.}  \label{3sigma}
\end{equation}
Applying Lagrange's mean value theorem on $(\sigma _{k},\sigma _{k-1})$ we
get a sequence $s_{k}\in (\sigma _{k},\sigma _{k-1})$,  $k>1$ such that
\begin{equation}
y'(s_{2k+1})=\frac{y(\sigma _{2k})-y(\sigma _{2k+1})}{\sigma
_{2k}-\sigma _{2k+1}}\quad\mbox{and}\quad
 y'(s_{2k})=\frac{y(\sigma_{2k-1})-y(\sigma _{2k})}{\sigma _{2k-1}-\sigma _{2k}}.
\label{3Lagr}
\end{equation}
Let us use the following notation
\begin{gather*}
z(t)=y'(t),\ t\in (a,b), \\
\delta _{2k+1}=\frac{\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega -
\mathop{\rm ess\,inf}_{_{(a_{2k+1},a_{2k})}}\theta }{\sigma
_{2k}-\sigma _{2k+1}}, \\
\delta _{2k}=\frac{\mathop{\rm ess\,inf}_{_{(a_{2k-1},a_{2k-2})}}
\theta -\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega }
{\sigma_{2k-1}-\sigma _{2k}}.
\end{gather*}
Hence, from \eqref{2essto}, \eqref{3jump}, and \eqref{3Lagr} follows
\begin{equation}
z(s_{2k+1})\geq \delta _{2k+1}\geq 0\quad\mbox{and}\quad z(s_{2k})\leq \delta
_{2k}\leq 0,\quad k\geq 1,  \label{3zjump}
\end{equation}
where $s_{k}\in (\sigma _{k},\sigma _{k-1})$. To prove \eqref{3Gedy} we
need a modification of Lemma \ref{lm2.3}.
\end{proof}

\begin{lemma}[A version of Lemma \ref{lm2.3}]
Let $\sigma _{k}$ be a decreasing sequence of real numbers from interval $(a,b)$
satisfying \eqref{3sigma},
where $k(\varepsilon )$ and $\varepsilon _{0}$ are taken from \eqref{2ak}.
Let $\delta _{k}$ be a sequence of real numbers such that $\delta
_{2k+1}\geq 0$ and $\delta _{2k}\leq 0$,  $k\geq 1$. Let $z(t)$ be a
continuous function in $(a,b]$ such that there is a sequence $s_{k}\in
(\sigma _{k},\sigma _{k-1})$ satisfying \eqref{3zjump}. Then there holds
\begin{equation}
|G_{\varepsilon }(z)|\geq \sum_{k=k(\varepsilon /2)}^{\infty }\delta
_{2k+1}(\sigma _{2k}-\sigma _{2k+1})\quad \mbox{for each }\varepsilon
\in (0,\varepsilon _{0}).  \label{3Gedelta}
\end{equation}
\end{lemma}
To prove \eqref{3Gedelta}, we suggest reader to use the same
argumentation from the proof of Lemma \ref{lm2.3} to prove that
$B_{2k+1}\subseteq G_{\varepsilon }(z|_{[\sigma _{2k+1},\sigma _{2k}]})$
for each $k\geq k(\varepsilon )+1$ and $\varepsilon \in (0,\varepsilon _{0})$,
where $B_{2k+1}=[\sigma _{2k+1},\sigma _{2k}]\times [0,\delta _{2k+1}]$.

Using the above notation for $\delta _{k}$ and $z(t)$,  from \eqref{3Gedelta}
follows
\begin{align*}
|G_{\varepsilon }(y')|
&\geq \sum_{k=k(\varepsilon /2)}^{\infty }
\frac{(\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega -
\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta )}{\sigma _{2k}-\sigma
_{2k+1}}(\sigma _{2k}-\sigma _{2k+1}) \\
&=\sum_{k=k(\varepsilon /2)}^{\infty }(\mathop{\rm ess\,sup}
_{(a_{2k},a_{2k-1})}\omega -\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}
\theta).
\end{align*}
Thus, Lemma \ref{lm3.1} is verified.

Next, we consider an easy example for the calculation of the right hand side
in \eqref{3Gedy}.

\begin{example} \rm \label{ex3.2}
Let $y$ be a real function smooth enough in $(a,b)$
and let $y$ be $(\theta ,\omega ,a_{k})$-rapidly oscillating, where the data
$\theta $,  $\omega $ and $a_{k}$ are given in \eqref{2akto} above.
Regarding to Example \ref{ex2.4}, we know such given $\theta $,  $\omega $
and $a_{k}$
satisfies the condition \eqref{2ak} and \eqref{2essto}, where
$k(\varepsilon )$ and $\varepsilon _{0}$ are given in \eqref{2kepslu}.
Hence, we may calculate the right hand side in \eqref{3Gedy} for such
function $y$. First, we have
\begin{equation}
\begin{aligned}
&\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega
-\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta
=\omega (a_{2k-1})-\theta (a_{2k})  \\
&=(\frac{b-a}{2})^{\alpha }\big[\big( \frac{1}{2k-1}\big) ^{\alpha
/\beta }+\big( \frac{1}{2k}\big) ^{a/\beta }\big]\geq \frac{(b-a)^{\alpha
}}{2^{\alpha -1}}\left( \frac{1}{2k}\right) ^{\alpha /\beta }.
\end{aligned}\label{3essto}
\end{equation}
In particular, from \eqref{2kepslu} we have
\begin{equation}
\frac{1}{k(\varepsilon /2)}\geq \frac{1}{4}
\big( \frac{\beta }{2(b-a)}\big) ^{\frac{\beta }{\beta +1}}
\varepsilon ^{\frac{\beta }{\beta +1}}\quad
\mbox{for each }\varepsilon \in (0,\varepsilon _{0}=\frac{b-a}{\beta }).
\label{3kepspol}
\end{equation}
According to \eqref{3essto} and \eqref{3kepspol},  from \eqref{3Gedy}
follows
\begin{align*}
|G_{\varepsilon }(y')| &\geq \sum_{k=k(\varepsilon /2)}^{\infty }
\frac{(b-a)^{\alpha }}{2^{\alpha -1}}\Big( \frac{1}{2k}\Big) ^{\alpha
/\beta }\geq \frac{(b-a)^{\alpha }}{2^{\alpha -1}}\big( \frac{1}{
2k(\varepsilon /2)}\big) ^{\alpha /\beta } \\
&\geq \frac{(b-a)^{\alpha }}{2^{\alpha -1}}\frac{1}{8^{\alpha /\beta }}
\big( \frac{\beta }{2(b-a)}\big) ^{\frac{\alpha }{\beta +1}}\varepsilon ^{
\frac{\alpha }{\beta +1}},
\end{align*}
that is
$|G_{\varepsilon }(y')|\geq c\varepsilon ^{\alpha /(\beta +1)}$
for each $\varepsilon \in (0,\varepsilon _{0})$,
where
\[
\varepsilon _{0}=\frac{b-a}{\beta }\quad\mbox{and}\quad
c=\frac{(b-a)^{\frac{\alpha
\beta }{\beta +1}}}{2^{\alpha -1}8^{\alpha /\beta }}\big( \frac{\beta }{2}
\big) ^{\frac{\alpha }{\beta +1}}.
\]
\end{example}

The main result of this section is the following.

\begin{theorem} \label{thm3.3}
Let the hypotheses of Theorem \ref{thm2.7} be still valid,
that is: let the sequence $a_{k}$ satisfy \eqref{2ak}, let the functions
$\theta (t)$ and $\omega (t)$,  $\theta (t)\leq \omega (t)$, $t\in [a,b]$,
satisfy \eqref{2essto},  \eqref{2ttot} and \eqref{2nula}, and
let the function $f(t,\eta ,\xi )$ satisfy the assumptions
\eqref{2fp}--\eqref{2fsn} and \eqref{2fcomp}. Then each solution $y$ of
\eqref{1eq} such that $y\in C^{1}(a,b)$ satisfies
\begin{equation}
\begin{gathered}
|G_{\varepsilon }(y')|\geq h(\varepsilon )\quad
\mbox{for each $\varepsilon \in (0,\varepsilon _{0})$,} \\
M^{s}(G(y'))\geq 2^{s-2}\limsup_{\varepsilon
\to 0}\varepsilon ^{s-2}h(\varepsilon )\quad \mbox{for any } s\in (1,2),
\end{gathered}  \label{3GeMs}
\end{equation}
where $\varepsilon _{0}$ is appearing in \eqref{2ak} and
\[
h(\varepsilon )=\sum_{k=k(\varepsilon /2)}^{\infty }(\mathop{\mathrm{ess\ sup
}}_{(a_{2k},a_{2k-1})}\omega -\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}
\theta ).
\]
\end{theorem}

\begin{proof}
As a consequence of the assumptions \eqref{2fp}--\eqref{2fsn} and
\eqref{2nula}--\eqref{2fcomp} we may use Lemma \ref{lm2.5}.
Hence, each solution $y$ of \eqref{1eq} is $(\theta ,\omega ,a_{k})$-rapidly
oscillating and so we may use Lemma \ref{lm3.1} too. It gives us the
inequality \eqref{3Gedy} from which
immediately follows \eqref{3GeMs}.
\end{proof}

Now, we are able to prove the inequalities \eqref{1Mderl} and \eqref{1Gederl}.

\begin{corollary} \label{coro3.4}
Let the hypotheses of Corollary \ref{coro2.8} be still
valid, that is: for arbitrarily given real number $s\in (1,2)$, let $a_{k}$,
$\theta $ and $\omega $ be given by \eqref{2aktos}, and let the function
$f(t,\eta ,\xi )$ satisfies \eqref{2fp}--\eqref{2fsn}, and \eqref{2fcomp}
in respect to such given $(\theta ,\omega ,a_{k})$. Then each solution $y$
of \eqref{1eq} such that $y\in C^{1}(a,b)$ satisfies
\begin{equation}
\begin{gathered}
|G_{\varepsilon }(y')|\geq \frac{\sqrt{2}}{2^{4}}
(b-a)^{s/2}\varepsilon ^{1-s/2}\quad
\mbox{for each $\varepsilon \in (0,\varepsilon _{0}=\frac{b-a}{\beta })$},\\
M^{1+s/2}(G(y'))\geq \frac{1}{2^{4}}(b-a)^{s/2}.
\end{gathered}  \label{3GeMsdy}
\end{equation}
\end{corollary}

\begin{proof}
 From the proof of Corollary \ref{coro2.8}, we know that the sequence
$a_{k}$ given in \eqref{2aktos} satisfies the condition \eqref{2ak},
where $k(\varepsilon )$ and $\varepsilon _{0}$ are determined in
\eqref{2kepslu}. Also, the functions $\theta $ and $\omega $ given in
\eqref{2aktos} satisfy the condition \eqref{2essto},  \eqref{2ttot} and
\eqref{2nula}. Therefore, we may apply Theorem \ref{thm3.3} and to
calculate the right
hand side in \eqref{3GeMs},  where $\theta $,  $\omega $ and $a_{k}$ are
given in \eqref{2aktos}. In this direction, by the help of \eqref{3essto}
for $\alpha =1$ and \eqref{3kepspol} we obtain
\begin{align*}
\sum_{k=k(\varepsilon /2)}^{\infty }(\mathop{\rm ess\,sup}
_{(a_{2k},a_{2k-1})}\omega -\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta )
&\geq (b-a)\sum_{k=k(\varepsilon /2)}^{\infty }\big( \frac{1}{2k}
\big) ^{1/\beta } \\
&\geq (b-a)\big( \frac{1}{2k(\varepsilon /2)}\big) ^{1/\beta }\\
&\geq\big( \frac{\beta }{2}\big) ^{\frac{1}{\beta +1}}
\big( \frac{1}{8}\big) ^{\frac{1}{\beta }}(b-a)^{\frac{\beta }{\beta +1}}
\varepsilon ^{\frac{1}{\beta +1}}\\
&\geq \frac{\sqrt{2}}{2^{4}}(b-a)^{\frac{s}{2}}\varepsilon ^{1-\frac{s}{2}},
\end{align*}
where \eqref{2aktos} is used, that is, $\beta >1$ and $2\beta /(\beta
+1)=s$. Now from \eqref{3GeMs} immediately follows
\[
|G_{\varepsilon }(y')|\geq \frac{\sqrt{2}}{2^{4}}
(b-a)^{s/2}\varepsilon ^{1-s/2} \quad\mbox{for each}\ \varepsilon \in
(0,\varepsilon _{0}=\frac{b-a}{\beta }).
\]
Also, we have
\begin{align*}
M^{1+s/2}(G(y')) &\geq 2^{(1+s/2)-2}\limsup
_{\varepsilon \to 0}\big[\varepsilon ^{(1+s/2)-2}\frac{\sqrt{2}}{
2^{4}}(b-a)^{s/2}\varepsilon ^{1-s/2}\big] \\
&=\frac{2^{s/2+1/2}}{2^{5}}(b-a)^{s/2}\limsup
_{\varepsilon \to 0}(\varepsilon ^{s/2-1}\varepsilon ^{1-s/2})\geq
\frac{1}{2^{4}}(b-a)^{s/2}.
\end{align*}
This proves the desired statement \eqref{3GeMsdy}.
\end{proof}

\section{Necessary conditions for rapid oscillations}

Let us mention that the inequality \eqref{2Gey},  stated in Lemma \ref{lm2.3}, was
useful in the calculation of a lower bound of $|G_{\varepsilon }(y)|$,  where
$y$ is a $(\theta ,\omega ,a_{k})$-rapidly oscillating function. Next, in
Lemma \ref{lm2.5} some (sufficient) conditions on the nonlinearity $f(t,\eta ,\xi )$
were given such that each solution $y$ of \eqref{1eq} is $(\theta ,\omega
,a_{k})$-rapidly oscillating. In this section, we consider an inverse of
Lemma \ref{lm2.5}. Precisely, supposing that there is at least one $(\theta ,\omega
,a_{k})$-rapidly oscillating and smooth enough solution of \eqref{1eq} it
is shown what type of (necessary) conditions on the nonlinearity $f(t,\eta
,\xi )$ must be satisfied. Let us remark that in both cases we are working
with solutions $y$ of \eqref{1eq} which satisfy a basic condition
\begin{equation}
\begin{gathered}
\tilde{\theta}_{0}\leq y(t)\leq \tilde{\omega}_{0}\quad\mbox{ for each }t\in
[a,b],\quad \mbox{where }   \\
\tilde{\theta}_{0}\leq \mathop{\rm ess\,inf}_{(a,b)}\theta \quad\mbox{and}\quad
\tilde{\omega}_{0}\geq \mathop{\rm ess\,sup}_{(a,b)}\omega .
\end{gathered}  \label{4comp}
\end{equation}
As we have seen in Theorem \ref{thm2.7}, the condition \eqref{4comp} can be easily
verified if the assumption \eqref{2fcomp} is imposed on the nonlinear term
$f(t,\eta ,\xi )$.

\begin{theorem} \label{thm4.1}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{2ak}. Let $\theta (t)$ and
$\omega (t)$ be two measurable and bounded real functions on $[a,b]$,
$\theta (t)\leq \omega (t)$, $t\in [a,b]$, which satisfy \eqref{2essto} and
\[
\mathop{\rm ess\,sup}_{(a,b)}\omega >\mathop{\rm ess\,inf}
_{(a_{2k},a_{2k-1})}\theta, \quad \mathop{\rm ess\,sup}
_{(a_{2k+1},a_{2k})}\omega >\mathop{\rm ess\,inf}_{(a,b)}\theta ,\quad
k\geq 1.
\]
If there is at least one solution $y\in C^{2}(a,b)$ of \eqref{1eq}
which is $(\theta ,\omega ,a_{k})$-rapidly oscillating and satisfies
\eqref{4comp},  then for each $k\geq 1$, the Caratheodory function
$f(t,\eta ,\xi)$ needs to satisfy the following inequalities:
\begin{equation}
\begin{aligned}
&\int_{a_{2k+1}}^{a_{2k-2}}\mathop{\rm ess\,sup}_{(\eta ,\xi )\in
I\times R}f^{+}(t,\eta ,\xi )dt  \\
&\geq \frac{1}{(a_{2k-2}-a_{2k+1})^{p-1}}\frac{(\mathop{\rm ess\,sup}
_{(a_{2k},a_{2k-1})}\omega -\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}
\theta )^{p}}{\mathop{\rm ess\,sup}_{(a,b)}\omega -\mathop{\mathrm{ess\
inf}}_{(a_{2k},a_{2k-1})}\theta },
\end{aligned} \label{4fp}
 \end{equation}
and
\begin{equation}
\begin{aligned}
&\int_{a_{2k+2}}^{a_{2k-1}}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in
I\times R}f_{\_}(t,\eta ,\xi )dt  \\
&\leq -\frac{1}{(a_{2k-1}-a_{2k+2})^{p-1}}\frac{(\mathop{\rm ess\,sup}
_{(a_{2k+1},a_{2k})}\omega -\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}
\theta )^{p}}{\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\omega -
\mathop{\rm ess\,inf}_{(a,b)}\theta },
\end{aligned} \label{4fn}
\end{equation}
where $I=(\mathop{\rm ess\,inf}_{(a,b)}\theta $,
$\mathop{\rm ess\,sup}_{(a,b)}\omega )$, and $\ f_{\_}=\min \{f,0\}\leq
0$,  and $f^{+}=\max \{f,0\}\geq 0$.
\end{theorem}

Taking in \eqref{2ttot} that $\tilde{\theta}_{0}=\mathop{\rm ess\,inf}
_{(a,b)}\theta $ and $\mathop{\rm ess\,sup}_{(a,b)}\omega =\tilde{\omega
}_{0}$ we  see that both types of conditions, the sufficient conditions
from Lemma \ref{lm2.5} and the necessary conditions from Theorem \ref{thm4.1}
are similar each other, especially in the variable $t$.

\begin{proof} Let $y$ be a solution of the equation \eqref{1eq} which
is $(\theta ,\omega ,a_{k})$-rapidly oscillating and satisfies the condition
\eqref{4comp}. According to \eqref{2osc} and \eqref{2essto} it is easy
to check that there are two numbers $s^{*}\in (a_{2k+1},a_{2k-1})$ and
$t^{*}\in (a_{2k},a_{2k-2})$,  $s^{*}<$ $t^{*}$,  such that
\begin{gather}
y'(s^{*})\geq 0\quad\mbox{and}\quad y(s^{*})\geq
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega \geq \mathop{\rm ess\,inf}
_{(a_{2k},a_{2k-1})}\theta ,  \label{4szv} \\
y(t^{*})=\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\theta
\quad\mbox{and}\quad
y(t)\geq \mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\theta ,\quad t\in
(s^{*},t^{*}),  \label{4tzv}
\end{gather}
where $k$ is a fixed natural number. It yields
\begin{align*}
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega
-\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\theta
&\leq y(s^{*})-y(t^{*})\leq \int_{s^{*}}^{t^{*}}|y'|dt \\
&\leq (t^{*}-s^{*})^{1/p'}(\int_{s^{*}}^{t^{*}}|y'|^{p}dt)^{1/p}\\
&\leq (a_{2k-2}-a_{2k+1})^{1/p'}(\int_{s^{*}}^{t^{*}}|y'|^{p}dt)^{1/p},
\end{align*}
where $1/p+1/p'=1$. Thus, we have
\begin{equation}
\frac{(\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega -
\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\theta )^{p}}{
(a_{2k-2}-a_{2k+1})^{p-1}}\leq \int_{s^{*}}^{t^{*}}|y'|^{p}dt.
\label{4start}
\end{equation}
On the other hand, by means of \eqref{4szv} and \eqref{4tzv} and
multiplying the equation \eqref{1eq} by the test function
$\varphi =y-\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\theta $ and integrating both
sides over $[s^{*},t^{*}]$ we obtain
\begin{align*}
&\int_{s^{*}}^{t^{*}}|y'|^{p}dt \\
&=\big(|y'|^{p-2}y'(y-\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\theta )\big)
|_{s^{*}}^{t^{*}}
+\int_{s^{*}}^{t^{*}}f(t,y,y')(y-\mathop{\rm ess\,inf}
_{(a_{2k},a_{2k-1})}\theta )dt\\
&\leq \int_{s^{*}}^{t^{*}}f(t,y,y')(y-\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}
\theta )dt \\
&\leq (\mathop{\rm ess\,sup}_{(a,b)}\omega -\mathop{\rm ess\,inf}
_{(a_{2k},a_{2k-1})}\theta )\int_{a_{2k+1}}^{a_{2k-2}}
\mathop{\rm ess\,sup}_{(\eta ,\xi )\in I\times R}f^{+}(t,\eta ,\xi )dt,
\end{align*}
where $I=(\mathop{\rm ess\,inf}_{(a,b)}\theta $,
$\mathop{\rm ess\,sup}_{(a,b)}\omega )$ and $f^{+}=\max \{f,0\}\geq 0$.
Now, combining previous inequality with \eqref{4start} we immediately
derive the inequality \eqref{4fp}.

Similar to \eqref{4szv} and \eqref{4tzv} and by the help of
\eqref{2osc} and \eqref{2essto} we get two numbers $s^{*}\in (a_{2k+2},a_{2k})$
and $t^{*}\in (a_{2k+1},a_{2k-1})$,  $s^{*}<$ $t^{*}$,  such that
\begin{gather}
y'(s^{*})\leq 0\quad\mbox{and}\quad y(s^{*})\leq
\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta
\leq \mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\omega ,  \label{4anszv} \\
y(t^{*})=\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\omega \quad\mbox{and}\quad
 y(t)\leq \mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\omega ,\quad t\in
(s^{*},t^{*}).  \label{4antzv}
\end{gather}
Using the same observation as in the proof of \eqref{4fp}, the
inequalities \eqref{4anszv} and \eqref{4antzv} verify \eqref{4fn}.
\end{proof}

\section{Upper bounds for $|G_{\varepsilon }(y)|$ and $M^{s}(G(y))$}

In this section, we will derive an upper bound for the behaviour of
$|G_{\varepsilon }(y)|$ as $\varepsilon \approx 0$, where $y$ is any smooth
enough solution of \eqref{1eq}. In this direction, we introduce the second
type of rapid oscillations near the boundary point $t=a$. For the record,
the function $y(t)=t^{\alpha }\cos (1/t^{\beta }),$where $0<\alpha <\beta
<\infty $ has such type of rapid oscillations near $t=0$.

\begin{definition} \rm \label{def5.1}
Let $a_{k}$ be a decreasing sequence of real numbers
from interval $(a,b)$ satisfying
\begin{equation}
\begin{gathered}
\mbox{$a_{k}\searrow a$  and there is an $\varepsilon _{1}>0$
such that for each $\varepsilon \in (0,\varepsilon _{1})$}  \\
\mbox{there is an $m(\varepsilon )\in \mathbb{N}$ such that
$a_{j-1}-a_{j}>4\varepsilon$, for each $j\leq m(\varepsilon )$.}
\end{gathered}  \label{5ak}
\end{equation}
Let $\tilde{\theta}(t)$ and $\tilde{\omega}(t)$ be two measurable and
bounded functions, both defined on $[a,b]$,  such that $\tilde{\theta}(t)\leq
\tilde{\omega}(t)$, for each $t\in [a,b]$. We say that a real function $y$
defined on $[a,b]$ has \emph{convex-concave rapid oscillations} in respect
to $(\tilde{\theta},\tilde{\omega},a_{k})$ if there hold true:
\begin{equation}
\mbox{$y$ is concave in $(a_{2k},a_{2k-1})$ and $y$
is convex in $(a_{2k+1},a_{2k})$,}  \label{5cc}
\end{equation}
for each $k\geq 1$ and
\begin{equation}
\tilde{\theta}(t)\leq y(t)\leq \tilde{\omega}(t)\quad\mbox{for each }
 t\in [a,b].  \label{5comp}
\end{equation}
\end{definition}

In contrast to \eqref{2ak} where the number $k(\varepsilon )$ was
appearing like an index of $\varepsilon $-density of $a_{k}$,  here in
\eqref{5ak} the number $m(\varepsilon )$ could be taken as an index of
$\varepsilon $-separation of the most finite numbers of $a_{k}$. Let us
remark that there is a sequence $a_{k}$ which satisfies both conditions
\eqref{2ak} and \eqref{5ak}. It will be the case of the following example,
in which we show a calculation of the number $m(\varepsilon )$.

\begin{example} \rm \label{ex5.2}
Let $a_{k}$ be a sequence of real numbers defined as in
Example \ref{ex2.2} above, that is
\begin{equation}
a_{k}=a+\frac{b-a}{2}(\frac{1}{k})^{1/\beta },\quad k\geq 1,\ 0<\beta <\infty .
\label{5akb}
\end{equation}
Let us take for $m(\varepsilon )$ any natural number which satisfies
\begin{equation}
m(\varepsilon )\leq 2\big( \frac{b-a}{\beta 2^{4+2/\beta }}\big)
^{\frac{\beta }{\beta +1}}\varepsilon ^{-\frac{\beta }{\beta +1}}\quad
\quad\mbox{ for each $\varepsilon \in (0,\varepsilon _{1})$ and $\varepsilon _{1}>0$.}
\label{5mepsl}
\end{equation}
Using an elementary inequality
\[
\frac{1}{\beta }\big( \frac{1}{j}\big) ^{1+1/\beta }
\leq \big( \frac{1}{j-1}\big) ^{1/\beta }-\big( \frac{1}{j}\big) ^{1/\beta },
\]
where $\beta >0$ and $j\geq 2$,  it is easy to show that the sequence $a_{k}$
defined in \eqref{5akb} satisfies the condition \eqref{5ak} in respect
to $m(\varepsilon )$ determined in \eqref{5mepsl}.
\end{example}

It is clear now that by the help of Example \ref{ex2.2} and
Example \ref{ex5.2} we have in
\eqref{5akb} a sequence of real numbers which satisfies both conditions
\eqref{2ak} and \eqref{5ak}.

Now, we are interested to relate the convex-concave rapid oscillation of a
function $y$ with the asymptotic behaviour of $|G_{\varepsilon }(y)|$ as
$\varepsilon \approx 0$.

\begin{lemma} \label{lm5.3}
Let $a_{k}$ be a decreasing sequence of real
numbers from $(a,b)$ satisfying \eqref{5ak}. Let $\tilde{\theta}(t)$ and
$\tilde{\omega}(t)$ be two continuous functions on $[a,b]$ satisfying
$\tilde{\theta}(a)=\tilde{\omega}(a)=0$ and
\begin{equation}
\mbox{$\tilde{\theta}$ is decreasing, and $\tilde{\omega}$
is increasing in $[a,b]$.}  \label{5incdec}
\end{equation}
Let $y\in C^{2}((a,b])\cap C([a,b])$ have convex-concave rapid
oscillations in respect to $(\tilde{\theta},\tilde{\omega},a_{k})$. Then for
each $\varepsilon \in (0,\varepsilon _{1})$ we have
\begin{equation}
\begin{aligned}
|G_{\varepsilon }(y|_{[a,a_{1}]})|
&\leq (a_{m(\varepsilon)}-a+2\varepsilon )
\big(\tilde{\omega}(a_{m(\varepsilon )})-\tilde{\theta}
(a_{m(\varepsilon )})+2\varepsilon \big)  \\
&\quad +\varepsilon \sum_{j=2}^{m(\varepsilon )}\big[(6+\pi )\big(\tilde{\omega}
(a_{j-1})-\tilde{\theta}(a_{j-1})\big)+2(a_{j-1}-a_{j})\big]  \\
&\quad+ 2(\pi +4)\varepsilon ^{2}m(\varepsilon ),
\end{aligned} \label{5Geu}
\end{equation}
where $\varepsilon _{1}$ and $m(\varepsilon )$ are defined in the
assumption \eqref{5ak} and where $a_{1}$ is the first member of $a_{k}$.
\end{lemma}

The proof of this lemma is omitted because it is very similar to the proof of
\cite[Lemma 2.2, p. 273-277]{Pasic03}

Next, we give an example for the calculation of the right hand side in
\eqref{5Geu}.

\begin{example} \rm \label{ex5.4}
Let $y\in $\emph{$C^{2}((0,1])\cap C([0,1])$} be a real
continuous function on $[0,1]$ and let $y$ have convex-concave rapid
oscillations in respect to \emph{$(\tilde{\theta},\tilde{\omega},a_{k}),$}
where $\tilde{\theta}$, $\tilde{\omega}$ and $a_{k}$ are given by
\begin{equation}
\begin{gathered}
a_{k}=\frac{1}{2}(\frac{1}{k})^{1/\beta },\quad k\geq 1,  \\
\tilde{\theta}(t)=-2t^{\alpha }\quad\mbox{and }\quad
 \tilde{\omega}(t)=2t^{\alpha },\quad t\in [0,1], \\
\mbox{where $\alpha$ and $\beta$  satisfy
$0<\alpha <\beta<\infty$.}
\end{gathered}  \label{5akttot}
\end{equation}
We take for $m(\varepsilon )$ any number which satisfies
\begin{equation}
(\beta 2^{4+2/\beta }\varepsilon )^{-\frac{\beta }{\beta +1}}\leq
m(\varepsilon )\leq 2(\beta 2^{4+2/\beta }\varepsilon )^{-\frac{\beta }{
\beta +1}}\quad\mbox{for each }\varepsilon \in (0,\varepsilon _{1}),
\label{5mepslu}
\end{equation}
where $\varepsilon _{1}=1/(\beta 2^{4+2/\beta })$. It is easy to check that
$m(\varepsilon )\in \mathbb{N}$ for each $\varepsilon \in (0,\varepsilon
_{1})$. Since from \eqref{5mepslu} follows \eqref{5mepsl} we know that
the sequence $a_{k}$ defined in \eqref{5akttot} satisfies the condition
\eqref{5ak}, where $m(\varepsilon )$ is determined by \eqref{5mepslu}.
Also, it is clear that the functions $\tilde{\theta}$ and $\tilde{\omega}$
defined in \eqref{5akttot} satisfy the condition \eqref{5incdec}. Thus,
the data $\tilde{\theta}$, $\tilde{\omega}$ and $a_{k}$ from \eqref{5akttot}
 satisfy the assumptions of Lemma \ref{lm5.3} and therefore we may calculate
\eqref{5Geu},  where $a=0$.

Let us remark that, in particular, from \eqref{5mepslu} we have
\begin{equation}
\frac{1}{m(\varepsilon )}\leq (\beta 2^{4+2/\beta }\varepsilon )^{\frac{
\beta }{\beta +1}}\quad\mbox{for each }\varepsilon \in (0,\varepsilon _{1}).
\label{5jmeps}
\end{equation}
 From \eqref{5akttot},  and using the inequality \eqref{5jmeps},  for each
$\varepsilon \in (0,\varepsilon _{1})$ we get
\begin{equation}
(a_{m(\varepsilon )}+2\varepsilon )\big(\tilde{\omega}(a_{m(\varepsilon )})-
\tilde{\theta}(a_{m(\varepsilon )})+2\varepsilon \big)
\leq c_{1}\varepsilon ^{\frac{\alpha +1}{\beta +1}}+c_{2}\varepsilon ^{\frac{
\beta +\alpha +1}{\beta +1}}
+c_{3}\varepsilon ^{\frac{\beta +2}{\beta +1}}+4\varepsilon ^{2},
\label{5prva}
\end{equation}
where
\[
c_{1}=2^{1-\alpha +(4+\frac{2}{\beta })\frac{\alpha +1}{\beta +1}}\beta ^{
\frac{\alpha +1}{\beta +1}},\quad
 c_{2}=2^{3-\alpha +(4+\frac{2}{\beta })\frac{
\alpha }{\beta +1}}\beta ^{\frac{\alpha }{\beta +1}},\quad
 c_{3}=2^{(4+\frac{2}{
\beta })\frac{1}{\beta +1}}\beta ^{\frac{1}{\beta +1}}.
\]
 From \eqref{5akttot}, \eqref{5mepslu},  and using the inequality
\begin{equation}
\sum_{j=1}^{n}\big(\frac{1}{j}\big)^{H}\leq \frac{2}{1-H}n^{1-H} \label{5nej}
\end{equation}
for each $n\in \mathbb{N}$ and $H\in (0,1)$,
we obtain
\begin{equation}
(6+\pi )\varepsilon \sum_{j=2}^{m(\varepsilon )}\big(\tilde{\omega}(a_{j-1})-
\tilde{\theta}(a_{j-1})\big)\leq c_{4}\varepsilon ^{\frac{\alpha +1}{\beta +1
}}\label{5druga}
\end{equation}
for each $\varepsilon \in (0,\varepsilon _{1})$,
where
\[
c_{4}=\frac{6+\pi }{\beta -\alpha }2^{4-\alpha -\frac{\alpha }{\beta }-(4+
\frac{2}{\beta })\frac{\beta -\alpha }{\beta +1}}\beta ^{\frac{\alpha +1}{
\beta +1}}.
\]
 From \eqref{5akttot},  and using the inequality
\begin{equation}
\big( \frac{1}{j-1}\big) ^{1/\beta }-\big( \frac{1}{j}\big) ^{1/\beta
}\leq \frac{1}{\beta }\big( \frac{1}{j-1}\big) ^{1+1/\beta }  \label{5jb}
\end{equation}
for each $j\in \mathbb{N}$ and $\beta >0$,
we get
\begin{equation}
2\varepsilon \sum_{j=2}^{m(\varepsilon )}(a_{j-1}-a_{j})\leq
c_{5}\varepsilon  \quad\mbox{for each }\varepsilon \in (0,\varepsilon _{1}),
\label{5treca}
\end{equation}
where
\[
c_{5}=\frac{1}{\beta }\sum_{j=1}^{\infty }\big( \frac{1}{j}\big)
^{1+1/\beta }.
\]
 From \eqref{5akttot},  and using the right inequality from \eqref{5mepslu}
we have
\begin{equation}
2(\pi +4)\varepsilon ^{2}m(\varepsilon )\leq c_{6}\varepsilon ^{\frac{\beta
+2}{\beta +1}} \quad\mbox{for each }\varepsilon \in (0,\varepsilon _{1}),
\label{5cetvrta}
\end{equation}
where
\[
c_{6}=(\pi +4)2^{2-(4+\frac{2}{\beta })\frac{\beta }{\beta +1}}\beta ^{-
\frac{\beta }{\beta +1}}.
\]
Putting the inequalities \eqref{5prva}, \eqref{5druga}, \eqref{5treca},
and \eqref{5cetvrta} into \eqref{5Geu} we obtain that for each
$\varepsilon \in (0,\varepsilon _{1})$ there holds
\[
|G_{\varepsilon }(y|_{[0,a_{1}]})|\leq (c_{1}+c_{4})\varepsilon ^{\frac{
\alpha +1}{\beta +1}}+c_{5}\varepsilon +(c_{3}+c_{6})\varepsilon ^{\frac{
\beta +2}{\beta +1}}+c_{2}\varepsilon ^{\frac{\beta +\alpha +1}{\beta +1}
}+4\varepsilon ^{2},
\]
where the constants $c_{1}$,  $c_{2}$,  $c_{3}$,  $c_{4}$,  $c_{5}$, $c_{6}$
are determined in the process above.


Next, we give some sufficient conditions on the nonlinearity
$f(t,\eta ,\xi )$ such that each smooth enough solution can have
convex-concave rapid oscillations in the sense of Definition \ref{def5.1}.
\end{example}

\begin{lemma} \label{lm5.5}
Let $a_{k}$ be a decreasing sequence of real
numbers from $(a,b)$ which satisfies \eqref{5ak}. Let $\tilde{\theta}(t)$
and $\tilde{\omega}(t)$ be two continuous functions on $[a,b]$ satisfying
$\tilde{\theta}(a)=\tilde{\omega}(a)=0$ and
\begin{equation}
\mbox{$\tilde{\theta}$ is decreasing and convex on $[a,b]$,
$\tilde{\omega}$ is increasing and concave on $[a,b]$}.
 \label{5dcic}
\end{equation}
Let the Caratheodory function $f(t,\eta ,\xi )$ satisfy
\begin{equation}
\begin{gathered}
f(t,\eta ,\xi )<0,\quad t\in (a,b),\ \eta >\tilde{\omega}(t),\
\xi \in \mathbb{R}, \\
f(t,\eta ,\xi )>0,\quad t\in (a,b),\ \eta <\tilde{\theta}(t),\
\xi \in \mathbb{R},
\end{gathered}  \label{5fcomp}
\end{equation}
and let for each $k\in \mathbb{N}$,
\begin{equation}
\begin{gathered}
f(t,\eta ,\xi )>0,\quad t\in (a_{2k},a_{2k-1}),\ \eta \in (\tilde{\theta}
_{0},\tilde{\omega}(t)),\ \xi \in \mathbb{R},   \\
f(t,\eta ,\xi )<0,\quad  t\in (a_{2k+1},a_{2k}),\ \eta \in (\tilde{\theta}(t),
\tilde{\omega}_{0}),\ \xi \in \mathbb{R},
\end{gathered}  \label{5fpn}
\end{equation}
where $\tilde{\theta}_{0}$ and $\tilde{\omega}_{0}$ be two arbitrary
given real numbers such that $\tilde{\theta}_{0}\leq \tilde{\theta}(b)<0$
and $0<\tilde{\omega}(b)\leq \tilde{\omega}_{0}$. Then each solution $y$ of
\eqref{1eq} such that $y\in C^{2}((a,b])\cap C([a,b])$ has convex-concave
rapid oscillations in respect to $(\tilde{\theta},\tilde{\omega},a_{k})$.
\end{lemma}

The assumption \eqref{5dcic} can be avoided in a particular case as
follows.

\begin{lemma} \label{lm5.6}
Let $a_{k}$ be a decreasing sequence of real
numbers from $(a,b)$ which satisfies \eqref{5ak}. Let $\tilde{\theta}(t)$
and $\tilde{\omega}(t)$ be two real functions on $[a,b]$ defined by
\begin{equation}
\tilde{\theta}(t)=-2(t-a)\quad\mbox{and}\quad
 \tilde{\omega}(t)=2(t-a)\quad \mbox{for each}\ t\in [a,b].  \label{5ttot}
\end{equation}
Let the Caratheodory function $f(t,\eta ,\xi )$ satisfy the hypotheses
\eqref{5fcomp} and \eqref{5fpn} in respect to $\tilde{\theta}$ and
$\tilde{\omega}$ defined in \eqref{5ttot}. Then each solution $y$ of
\eqref{1eq} such that $y\in C^{2}((a,b])\cap C([a,b])$ has convex-concave rapid
oscillations in respect to $(\tilde{\theta},\tilde{\omega},a_{k})$.
\end{lemma}

\begin{proof}[Sketch of the proofs of Lemmas \ref{lm5.5} and \ref{lm5.6}]
 It is an elementary fact
that the hypothesis \eqref{5fpn} implies the condition \eqref{5cc}, that
is, the convex-concave property of smooth solutions of \eqref{1eq} is an
easy consequence of \eqref{5fpn} (see a discussion in
\cite[Appendix,p. 302]{Pasic03}). But, it is not so easy to establish how
the hypotheses \eqref{5dcic} and \eqref{5fcomp} (or \eqref{5fcomp} and
\eqref{5ttot}) give us the condition \eqref{5comp} for any solution $y$
of \eqref{1eq}. A way to prove it, we suggest the argumentation presented
in the proof of \cite[Lemma 3.1, p. 278-279]{Pasic03}. See also Lemma
\ref{lm6.4}
below.

Thus, the hypotheses \eqref{5dcic}--\eqref{5ttot} guarantee us that each
solution $y$ of \eqref{1eq} such that \emph{$y\in C^{2}((a,b])\cap
C([a,b]) $ }satisfies the conditions \eqref{5cc} and \eqref{5comp}, that
is to say, the solutions $y$ of \eqref{1eq} have convex-concave rapid
oscillations in respect to $(\tilde{\theta},\tilde{\omega},a_{k})$.
\end{proof}

Now, we can state the main result of this section.

\begin{theorem} \label{thm5.7}
Let $a_{k}$ be a decreasing sequence of real
numbers from $(a,b)$ which satisfies \eqref{5ak}. Let $\tilde{\theta}(t)$
and $\tilde{\omega}(t)$ be two continuous functions on $[a,b]$, $\tilde{
\theta}(a)=\tilde{\omega}(a)=0$ which satisfy \eqref{5incdec} and either
\eqref{5dcic} or \eqref{5ttot}. Next, let the Caratheodory function
$f(t,\eta ,\xi )$ satisfy the hypotheses \eqref{5fcomp} and \eqref{5fpn}.
Then each solution $y$ of \eqref{1eq} such that $y\in C^{2}((a,b])\cap
C([a,b])$ satisfies
\begin{equation}
\begin{aligned}
M^{s}(G(y)) &\leq 2^{s-2}\limsup_{\varepsilon
\to 0}\varepsilon ^{s-2}\big\{(a_{m(\varepsilon )}-a)\big(\tilde{
\omega}(a_{m(\varepsilon )})-\tilde{\theta}(a_{m(\varepsilon )})\big) \\
&\quad+\varepsilon \sum_{j=2}^{m(\varepsilon )}\big[(6+\pi )\big(\tilde{\omega}
(a_{j-1})-\tilde{\theta}(a_{j-1})\big)+2(a_{j-1}-a_{j})\big] \\
&\quad +2(\pi +4)\varepsilon ^{2}m(\varepsilon )\big\},
\end{aligned} \label{5Msu}
\end{equation}
where $s\in (1,2)$.
\end{theorem}

\begin{proof}
>From Lemma \ref{lm5.5} or Lemma \ref{lm5.6} we have that each solution $y$ of
\eqref{1eq} such that $y\in C^{2}((a,b])\cap C([a,b])$ has convex-concave rapid
oscillations in respect to given $(\tilde{\theta},\tilde{\omega},a_{k}).$
Now, from Lemma \ref{lm5.3} we have that such solutions $y$ of \eqref{1eq} satisfy
the inequality \eqref{5Geu}. Multiplying both side in \eqref{5Geu} by
$(2\varepsilon )^{s-2}$ and taking $\limsup$ as
$\varepsilon \to 0$ we immediately obtain the inequality \eqref{5Msu}.
Note that it is not necessary to consider the part
$y|_{[a_{1},b]}, $ because $y\in W^{1,p}(a_{1},b)$.
\end{proof}

At the end of this section, we are able to derive the constant $m_{s}$
appearing in \eqref{1Mlu}.

\begin{corollary} \label{coro5.8}
For arbitrarily given $s\in (1,2)$ let $a_{k}$, $\tilde{\theta}(t)$, and
$\tilde{\omega}(t)$ be defined by
\begin{equation}
\begin{gathered}
a_{k}=a+\frac{b-a}{2}(\frac{1}{k})^{1/\beta },\quad k\geq 1,  \\
\tilde{\theta}(t)=-\tilde{\omega}(t)\quad \mbox{and}\quad
 \tilde{\omega}(t)=2(t-a),\quad t\in (a,b),  \\
\mbox{where $1<\beta <\infty$ and $\frac{2\beta }{\beta +1}=s$.}
\end{gathered}  \label{5aktos}
\end{equation}
Let the Caratheodory function $f(t,\eta ,\xi )$ satisfy the hypotheses
\eqref{5fcomp} and \eqref{5fpn} in respect to such given
$(\tilde{\theta},\tilde{\omega},a_{k})$. Then each solution $y$ of \eqref{1eq}
such that $y\in C^{2}(a,b)$ satisfies
\begin{equation}
M^{s}(G(y))\leq m_{s}(b-a)^{s},  \label{5Mss}
\end{equation}
where
\begin{equation}
m_{s}=2^{6}+2^{2}(6+\pi )\frac{2-s}{s-1}.  \label{5ms}
\end{equation}
\end{corollary}

\begin{proof} As in Example \ref{ex5.4}, we take for $m(\varepsilon )$ any
number that satisfies
\begin{equation}
\big( \frac{b-a}{\beta 2^{4+2/\beta }}\big) ^{\frac{\beta }{\beta +1}
}\varepsilon ^{-\frac{\beta }{\beta +1}}
\leq m(\varepsilon )
\leq 2\big(\frac{b-a}{\beta 2^{4+2/\beta }}\big) ^{\frac{\beta }{\beta +1}
}\varepsilon ^{-\frac{\beta }{\beta +1}},  \label{5mepsab}
\end{equation}
where $\varepsilon \in (0,\varepsilon _{1}=\frac{b-a}{\beta 2^{4+2/\beta }})$.
 From \eqref{5aktos} and \eqref{5mepsab} we have
\begin{align*}
&(a_{m(\varepsilon )}-a)\big(\tilde{\omega}(a_{m(\varepsilon )})-\tilde{
\theta}(a_{m(\varepsilon )})\big) \\
&=4(\frac{b-a}{2})^{2}(\frac{1}{m(\varepsilon )})^{2/\beta } \\
&\leq 2^{(4+\frac{2}{\beta })\frac{2}{\beta +1}}\beta ^{\frac{2}{\beta +1}
}(b-a)^{\frac{2\beta }{\beta +1}}\varepsilon ^{\frac{2}{\beta +1}}\leq
2^{6}(b-a)^{s}\varepsilon ^{2-s}
\quad\mbox{for each }\varepsilon \in (0,\varepsilon _{1}),
\end{align*}
where $s\in (1,2)$. This implies
\begin{equation}
2^{s-2}\limsup_{\varepsilon \to 0}\varepsilon
^{s-2}\big[(a_{m(\varepsilon )}-a)\big(\tilde{\omega}(a_{m(\varepsilon )})-
\tilde{\theta}(a_{m(\varepsilon )})\big)\big]\leq 2^{6}(b-a)^{s}.
\label{5lim1}
\end{equation}
Using inequality \eqref{5nej}, from \eqref{5aktos} and
\eqref{5mepsab} we have
\begin{align*}
&\varepsilon \sum_{j=2}^{m(\varepsilon )}(6+\pi )\big(\tilde{\omega}
(a_{j-1})-\tilde{\theta}(a_{j-1})\big) \\
&=\frac{b-a}{2}(6+\pi )4\varepsilon
\sum_{j=2}^{m(\varepsilon )}(\frac{1}{j-1})^{1/\beta } \\
&\leq 2(6+\pi )(b-a)\frac{2}{1-1/\beta }\varepsilon (m(\varepsilon
))^{1-1/\beta } \\
&\leq (6+\pi )2^{3-\frac{1}{\beta }-(4+\frac{2}{\beta })\frac{\beta -1}{
\beta +1}}\frac{\beta ^{\frac{2}{\beta +1}}}{\beta -1}(b-a)^{\frac{2\beta }{
\beta +1}}\varepsilon ^{\frac{2}{\beta +1}} \\
&\leq 2^{2}(6+\pi )\frac{2-s}{s-1}(b-a)^{s}\varepsilon ^{2-s}\quad
\mbox{for each }\varepsilon \in (0,\varepsilon _{1}).
\end{align*}
For any $s\in (1,2)$, this implies
\begin{equation}
2^{s-2}\limsup_{\varepsilon \to 0}\varepsilon
^{s-2}\big[\varepsilon \sum_{j=2}^{m(\varepsilon )}(6+\pi )
\big(\tilde{\omega}(a_{j-1})-\tilde{\theta}(a_{j-1})\big)\big]
\leq \frac{2^{2}(6+\pi )(2-s)}{s-1}(b-a)^{s}.  \label{5lim2}
\end{equation}
Using  inequality \eqref{5jb}, from \eqref{5aktos} and \eqref{5mepsab}
 we have
\begin{align*}
2\varepsilon \sum_{j=2}^{m(\varepsilon )}(a_{j-1}-a_{j})
&=(b-a)\varepsilon
\sum_{j=2}^{m(\varepsilon )}\big((\frac{1}{j-1})^{1/\beta }-(\frac{1}{j}
)^{1/\beta }\big) \\
&\leq \varepsilon \frac{b-a}{\beta }\sum_{j=2}^{\infty }(\frac{1}{j-1}
)^{1+1/\beta }\quad\mbox{for each }\varepsilon \in (0,\varepsilon _{1}).
\end{align*}
For any $s\in (1,2)$, this implies
\begin{equation}
2^{s-2}\limsup_{\varepsilon \to 0}\varepsilon
^{s-2}\big[2\varepsilon \sum_{j=2}^{m(\varepsilon )}(a_{j-1}-a_{j})\big]=0.
\label{5lim3}
\end{equation}
Finally, from \eqref{5aktos} and \eqref{5mepsab}, we have
\[
2(\pi +4)\varepsilon ^{2}m(\varepsilon )\leq 4(\pi +4)
\big( \frac{b-a}{\beta 2^{4+2/\beta }}\big) ^{\frac{\beta }{\beta +1}}
\varepsilon ^{2-\frac{\beta }{\beta +1}}.
\]
For any $s\in (1,2)$,  this implies
\begin{equation}
2^{s-2}\limsup_{\varepsilon \to 0}\varepsilon
^{s-2}\big[2(\pi +4)\varepsilon ^{2}m(\varepsilon )\big]=0.  \label{5lim4}
\end{equation}
Putting \eqref{5lim1}, \eqref{5lim2},  \eqref{5lim3},  and \eqref{5lim4}
 into \eqref{5Msu} we obtain \eqref{5Mss} and \eqref{5ms}.
\end{proof}

At the end of this section we consider both inequalities in \eqref{1Mlu}.
In this direction, let us remark that both types of hypotheses on the
nonlinear term $f(t,\eta ,\xi )$ appearing in Corollary \ref{coro2.8} and
Corollary \ref{coro5.8} are completely harmonized.
It is because the data $\theta $, $\omega $,
$\tilde{\theta}$,  and $\tilde{\omega}$ defined in \eqref{2aktos} and
\eqref{5aktos} satisfy
\begin{equation}
\begin{gathered}
\tilde{\theta}(t)<\theta (t)<0<\omega (t)<\tilde{\omega}(t), \\
\mathop{\rm ess\,inf}_{(a_{2k+1},a_{2k})}\theta \geq
\mathop{\rm ess\,sup}_{(a_{2k+1},a_{2k})}\tilde{\theta},   \\
\mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega \leq
\mathop{\rm ess\,inf}_{(a_{2k},a_{2k-1})}\tilde{\omega}.
\end{gathered}  \label{5nula}
\end{equation}
Therefore, we may combine Corollary \ref{coro2.8} and Corollary \ref{coro5.8}
 to obtain the following consequence.

\begin{corollary} \label{coro5.9}
For arbitrarily $s\in (1,2)$ let $a_{k}$, $\theta $,  $\omega $,
$\tilde{\theta}(t)$, and $\tilde{\omega}(t)$ be
given by \eqref{2aktos} and \eqref{5aktos}. Let the Caratheodory
function $f(t,\eta ,\xi )$ satisfy the hypotheses \eqref{2fp}--\eqref{2fsn},
and \eqref{2fcomp}, and \eqref{5fcomp}--\eqref{5fpn} in respect to such
given $(\theta ,\omega ,\tilde{\theta},\tilde{\omega},a_{k})$. Then each
solution $y$ of \eqref{1eq} such that $y\in C^{2}(a,b)$ satisfies
\begin{equation}
\frac{1}{2^{7}}(b-a)^{s}\leq M^{s}(G(y))\leq m_{s}(b-a)^{s},  \label{5kraj}
\end{equation}
where the constant $m_{s}$ is defined in \eqref{5ms}.
\end{corollary}

Thus, because of \eqref{5kraj}, we have proved the desired statement
\eqref{1Mlu}.

\section{Lower bound for the $s$-dimensional upper density}

In this section, we derive the inequalities \eqref{1Dsl} and \eqref{1Mslr}.
In this direction, we need some preliminaries. The first one is a
version of Lemma \ref{lm2.3} above.

\begin{lemma} \label{lm6.1}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{2ak}. Let $\theta (t)$ and
$\omega (t)$ be two measurable and bounded real functions on $[a,b]$,
$\theta (t)\leq \omega (t)$, $t\in [a,b]$, which satisfy \eqref{2essto}.
Let $y$ be $(\theta ,\omega ,a_{k})$-rapidly oscillating function on $[a,b]$
and let $y\in C((a,b])$. Let $k(\varepsilon )$ and $\varepsilon _{0}$ be
from \eqref{2ak}. Then for any $c\in (a,b)$ such that
\begin{equation}
\mbox{there exists $\varepsilon _{c}\in (0,\varepsilon _{0})$
satisfying $a_{k(\varepsilon )-1}\in (a,c)$ for each
$\varepsilon \in (0,\varepsilon _{c})$},  \label{6epsc}
\end{equation}
we have
\begin{equation}
|G_{\varepsilon }(y|_{[a,c]})|\geq \int_{a}^{a_{k(\varepsilon )}}(\omega
(t)-\theta (t))dt \quad\mbox{for each }\varepsilon \in
(0,\varepsilon_{c}).  \label{6Geyac}
\end{equation}
\end{lemma}

\begin{proof}
 Regarding to the last line in the proof of Lemma \ref{lm2.3} above and
using the same notations, we have already proved that
\begin{equation}
A({\varepsilon },\theta ,\omega )\subseteq G_{\varepsilon }\bigl(\cup
_{k=k(\varepsilon )+1}^{\infty }y|_{[\sigma _{k},\sigma _{k-1}]}\bigl)\quad
\mbox{for each } \varepsilon \in (0,\varepsilon _{0}),  \label{6Aeto}
\end{equation}
where $k(\varepsilon )$ and $\varepsilon _{0}$ are appearing in \eqref{2ak}.
Let $c\in (a,b)$ be a real number which satisfies \eqref{6epsc}. It is
clear that
\begin{equation}
G(y|_{(a,a_{k(\varepsilon )-1}]})\subseteq G(y|_{(a,c]})
 \quad\mbox{for each } \varepsilon \in (0,\varepsilon _{c}).  \label{6Gy}
\end{equation}
Since $\varepsilon _{c}\in (0,\varepsilon _{0})$ and $\sigma _{k}\in
(a_{k},a_{k-1})$,  combining \eqref{6Aeto} and \eqref{6Gy} we obtain:
\begin{align*}
A({\varepsilon },\theta ,\omega )
&\subseteq G_{\varepsilon }\bigl(\cup_{k=k(\varepsilon )+1}
^{\infty }y|_{[\sigma _{k},\sigma _{k-1}]}\bigl) \\
&=G_{\varepsilon }(y|_{(a,\sigma (k(\varepsilon ))]})\subseteq
G_{\varepsilon }(y|_{(a,a_{k(\varepsilon )-1}]})
\subseteq G_{\varepsilon}(y|_{(a,c]});
\end{align*}
that is,
\begin{equation}
A({\varepsilon },\theta ,\omega )\subseteq G_{\varepsilon }(y|_{(a,c]})\quad
\mbox{for each }\varepsilon \in (0,\varepsilon _{c}).  \label{6AGy}
\end{equation}
Taking the Lebesgue measure of the both sides in \eqref{6AGy} we get
\eqref{6Geyac}.
\end{proof}

A choice for the number $\varepsilon _{c}$ appearing in \eqref{6epsc} and
\eqref{6Geyac} will be given in Corollary \ref{coro6.3} below.
Analogously to Theorem \ref{thm2.7} we can state the following result.

\begin{theorem} \label{thm6.2}
Let $a_{k}$ be a decreasing sequence of real
numbers from interval $(a,b)$ satisfying \eqref{2ak}. Let $\theta (t)$ and
$\omega (t)$ be two measurable and bounded real functions on $[a,b]$,
$\theta (t)\leq \omega (t)$, $t\in [a,b]$, which satisfy \eqref{2essto},
\eqref{2ttot} and \eqref{2nula}. Next, let the Caratheodory function
$f(t,\eta ,\xi )$ satisfy \eqref{2fp}--\eqref{2fsn}, and \eqref{2fcomp}.
Let $c\in (a,b)$ and $\varepsilon _{c}$ be numbers satisfying \eqref{6epsc}.
Then each solution $y$ of the equation \eqref{1eq} satisfies
\begin{gather*}
|G_{\varepsilon }(y|_{[a,c]})|\geq \int_{a}^{a_{k(\varepsilon )}}(\omega
(t)-\theta (t))dt\quad\mbox{for each }\varepsilon \in
(0,\varepsilon_{c}),  \\
M^{s}(G(y|_{[a,c]}))\geq 2^{s-2}\limsup_{\varepsilon
\to 0}\varepsilon ^{s-2}\int_{a}^{a_{k(\varepsilon )}}(\omega
(t)-\theta (t))dt,
\end{gather*}
for any $s\in (1,2)$.
\end{theorem}

The proof of this theorem is the same as the proof of Theorem \ref{thm2.7},
but using Lemma \ref{lm6.1} instead of Lemma \ref{lm2.3}.
Now, from Theorem \ref{thm6.2} we obtain the following resutls.

\begin{corollary} \label{coro6.3}
For an arbitrarily real number $s\in (1,2)$,
let $a_{k}$, $\theta $ and $\omega $ be given by \eqref{2aktos}. Let the
Caratheodory function $f(t,\eta ,\xi )$ satisfy \eqref{2fp}--\eqref{2fsn},
and \eqref{2fcomp} in respect to such given $(\theta ,\omega ,a_{k})$.
Then for each $c\in (a,b)$ and for each solution $y$ of the equation
\eqref{1eq} there holds
\begin{equation}
\begin{gathered}
|G_{\varepsilon }(y|_{[a,c]})|\geq \frac{1}{2^{6}}(c-a)^{s}\varepsilon
^{2-s}\quad \mbox{for each } \varepsilon \in (0,\varepsilon _{c}), \\
M^{s}(G(y|_{[a,c]}))\geq \frac{1}{2^{7}}(c-a)^{s},
\end{gathered}  \label{6GeMss}
\end{equation}
where
\begin{equation}
\varepsilon _{c}=\min \{\varepsilon _{0},\frac{1}{\beta }\frac{(c-a)^{\beta
+1}}{(b-a)^{\beta }}\}\quad\mbox{and }\varepsilon _{0}=\frac{b-a}{\beta }.
\label{6eps0c}
\end{equation}
\end{corollary}

\begin{proof} It is simple to check that every $c\in (a,b)$ satisfies the
condition \eqref{6epsc} in respect to $a_{k}$,  $k(\varepsilon )$ and
$\varepsilon _{c}$ given in \eqref{2aktos}, \eqref{2kepslu} and
\eqref{6eps0c},  respectively. Next, using the same calculation as in the proof
of Corollary \ref{coro2.8}, the statement \eqref{6GeMss} immediately follows from
Theorem \ref{thm6.2}.
\end{proof}

To derive the inequalities \eqref{1Dsl} and \eqref{1Mslr} we
need a comparison result for solutions of \eqref{1eq} which is a
modification of \cite[Lemma 3.1, p. 278]{Pasic03}.

\begin{lemma} \label{lm6.4}
Let $\tilde{\theta}(t)=-2(t-a)$ and $\tilde{\omega}(t)=2(t-a)$.
Let the Caratheodory function $f(t,\eta ,\xi )$ satisfy
\begin{gather*}
f(t,\eta ,\xi )<0,\quad t\in (a,b),\ \eta >\tilde{\omega}(t),\
\xi \in \mathbb{R},   \\
f(t,\eta ,\xi )>0,\quad  t\in (a,b),\ \eta <\tilde{\theta}(t),\ \xi \in \mathbb{R}.
\end{gather*}
Then each solution $y$ of \eqref{1eq} such that $y\in C^{2}(a,b)$
satisfies
\begin{equation}
\tilde{\theta}(t)\leq y(t)\leq \tilde{\omega}(t)\quad\mbox{ for each }
t\in [a,b].  \label{6comp}
\end{equation}
\end{lemma}

The proof of this lemma is omitted because it is the same as the proof of
\cite[Lemma 3.1, p. 278]{Pasic03}.
Finally, according to Corollary \ref{coro6.3} and Lemma \ref{lm6.4} we are
able to state the main result of this section.

\begin{corollary} \label{coro6.5}
For an arbitrarily  $s\in (1,2)$,  let the
hypotheses of Corollary \ref{coro6.3} and Lemma \ref{lm6.4} be still valid.
Then each solution $y$ of \eqref{1eq} such that $y\in C^{2}(a,b)$ satisfies
\begin{equation}
\begin{gathered}
M^{s}(G(y)\cap B_{r}(a,0))\geq \frac{1}{2^{7}}(\frac{r}{\sqrt{5}})^{s}\quad
\mbox{for each } r\in (0,b-a), \\
D^{s}(G(y);t=a)\geq \frac{1}{2^{7}}(\frac{1}{2\sqrt{5}})^{s};
\end{gathered}  \label{6Dsl}
\end{equation}
that is, the constant $d_{s}$ appearing in \eqref{1Dsl} and
\eqref{1Mslr} satisfies
\[
d_{s}=\frac{1}{2^{7}}(\frac{1}{2\sqrt{5}})^{s}.
\]
\end{corollary}

\begin{proof} Let us remark that because of \eqref{5nula} the assumptions
of Corollary \ref{coro6.3} and Lemma \ref{lm6.4} are completely harmonized,
where $a_{k}$, $\theta $ and $\omega $ be given by \eqref{2aktos} and where
$\tilde{\theta}(t)=-2(t-a)$ and $\tilde{\omega}(t)=2(t-a)$. Therefore,
the main conclusions of Corollary \ref{coro6.3} and Lemma \ref{lm6.4} may be
used together.

By \eqref{6comp} and making intersections of $\tilde{\theta}(t) $ and
$\tilde{\omega}(t)$ with $B_{r}(a,0)$,  it is clear that for any
$r\in (0,\sqrt{5}(b-a))$ we have
\[
G(y|_{[a,a+\frac{r}{\sqrt{5}}]})\subseteq G(y)\cap B_{r}(a,0),
\]
where $y$ is any smooth enough solution of \eqref{1eq}. Since $M^{s}$ is a
monotone set function, it yields
\begin{equation}
M^{s}(G(y|_{[a,a+\frac{r}{\sqrt{5}}]}))\leq M^{s}(G(y)\cap B_{r}(a,0)).
\label{6Msr}
\end{equation}
Next, we apply Corollary \ref{coro6.3} to $y|_{[a,a+\frac{r}{\sqrt{5}}]}$ and from
\eqref{6GeMss} we derive
\begin{equation}
M^{s}(G(y|_{[a,a+\frac{r}{\sqrt{5}}]}))\geq \frac{1}{2^{7}}(\frac{r}{\sqrt{5}
})^{s}\quad\mbox{for any }r\in (0,b-a).  \label{6Msa}
\end{equation}
Combining \eqref{6Msr} and \eqref{6Msa} we conclude that
\[
M^{s}(G(y)\cap B_{r}(a,0))\geq \frac{1}{2^{7}}(\frac{r}{\sqrt{5}})^{s}\
\mbox{for any }r\in (0,b-a).
\]
Multiplying both inequalities by $1/(2r)^{s}$ and taking
$\limsup$ as $r\to 0$ we immediately derive the
desired statement \eqref{6Dsl}.
\end{proof}

\section{Appendix}

In this appendix, we sketch the proof of Lemma \ref{lm2.5}. Under the
assumptions of Lemma \ref{lm2.5}, we verify that each solution $y$ of the equation
\eqref{1eq} which satisfies \eqref{2comp} is $(\theta ,\omega ,a_{k})$-rapidly
oscillating on $[a,b]$. In this proof, a simple method of the
localisation of integration in \eqref{1eq} is exploited, often used in
analysis of local regular properties of solutions of PDE's (see for instance
\cite{KPZ01, Lio69,Rak90,RT90}).

Regarding  Definition \ref{def2.1}, it is sufficient to prove that for any
fixed $k\in \mathbb{N}$ the hypotheses \eqref{2fp}, \eqref{2fsp} and \eqref{2comp}
verify that
\begin{equation}
\exists \sigma _{2k}\in (a_{2k},a_{2k-1})\mbox{ such that }y(\sigma
_{2k})\geq \mathop{\rm ess\,sup}_{(a_{2k},a_{2k-1})}\omega ,
\label{ADjumpg}
\end{equation}
and on the other hand, that the hypotheses \eqref{2fn}, \eqref{2fsn} and
\eqref{2comp} verify that
\begin{equation}
\exists \sigma _{2k+1}\in (a_{2k+1},a_{2k})\mbox{ such that }y(\sigma
_{2k+1})\leq \mathop{\rm ess\,inf}_{_{(a_{2k+1},a_{2k})}}\theta .
\label{ADjumpd}
\end{equation}
Since \eqref{ADjumpd} is the dual statement of \eqref{ADjumpg},  in order
to simplify the proof, we will prove only the statement \eqref{ADjumpg}
for a fixed $k\in \mathbf{N.}$ In this direction, let $\sigma $ and $r$ be
two real numbers defined by
\[
\sigma =\frac{a_{2k}+a_{2k-1}}{2}\quad\mbox{and}\quad
r=\frac{1}{4}(a_{2k-1}-a_{2k}).
\]
Let $B_{r}=B_{r}(\sigma )$ denote a ball with radius $r>0$ centered at the
point $\sigma $. Then we have:
\begin{gather*}
B_{2r} =B_{2r}(\sigma )=(a_{2k},a_{2k-1}), \\
B_{r} =B_{r}(\sigma )=(a_{2k}+\frac{1}{4}(a_{2k-1}-a_{2k}),a_{2k-1}
-\frac{1}{4}(a_{2k-1}-a_{2k})).
\end{gather*}
Also, let $\tilde{\theta}_{0}=\mathop{\rm ess\,inf}_{_{(a,b)}}\theta $,
and $\tilde{\omega}_{0}=\mathop{\rm ess\,sup}_{(a,b)}\omega $,  and
$\omega _{2r}=\mathop{\rm ess\,sup}_{B_{2r}}\omega $,  and
$J_{2r}=(\tilde{\theta}_{0},\omega _{2r})$. Because of \eqref{2essto} and
\eqref{2ttot} we have that $\tilde{\theta}_{0}<\omega _{2r}<\tilde{\omega}_{0}$.
Using the preceding notation, we can rewrite the main assumptions
\eqref{2fp} and \eqref{2fsp} in the form
\begin{gather}
f(t,\eta ,\xi )\geq 0,\quad t\in B_{2r},\ \eta \in J_{2r},\ \xi \in \mathbb{R},
\label{ADfp} \\
\begin{aligned}
\int_{B_{r}}\mathop{\rm ess\,inf}_{(\eta ,\xi )\in J_{2r}\times
R}f(t,\eta ,\xi )dt
&>\frac{c(p)}{4^{p-1}}\frac{1}{r^{p-1}}\frac{(\tilde{\omega}_{0}
-\tilde{\theta}_{0})^{p}}{\tilde{\omega}_{0}-\omega _{2r}} \\
&=(p-1)^{p-1}\frac{(\tilde{\omega}_{0}-\tilde{\theta}_{0})^{p}}{\tilde{
\omega}_{0}-\omega _{2r}}\frac{|B_{r}|}{r^{p}},
\end{aligned}  \label{ADfsp}
\end{gather}
where we have used $|B_{r}|=2r$ and $c(p)=2[4(p-1)]^{p-1}$. Next, let $y$ be
a solution of \eqref{1eq} which satisfies \eqref{2comp}. Let us suppose
the contrary statement to \eqref{ADjumpg},  that is
\begin{equation}
y(t)<\omega _{2r}=\mathop{\rm ess\,sup}_{B_{2r}}\omega \quad
\mbox{for each }t\in B_{2r}.  \label{ADpom}
\end{equation}
According to \eqref{2comp} and \eqref{ADfp}--\eqref{ADpom} we have
\begin{gather}
f(t,y,y')\geq 0 \quad\mbox{in }B_{2r},  \label{ADfyp} \\
\int_{B_{r}}f(t,y,y')dt>(p-1)^{p-1}\frac{(\tilde{\omega}_{0}-\tilde{
\theta}_{0})^{p}}{\tilde{\omega}_{0}-\omega _{2r}}\frac{|B_{r}|}{r^{p}}.
\label{ADfysp}
\end{gather}
Now we can repeat a similar argument as in \cite[Theorem 5, p. 256]{KPZ01}
or \cite[Lemma 4.1, p. 280]{Pasic03}. It is known that for any
$c_{0}>1$ there exists a function $\Phi \in C_{0}^{\infty }(\mathbb{R})$,
$0\leq \Phi \leq 1$ in $\mathbb{R}$ such that the following properties are
fulfilled, see \cite[Lemma 5, pp. 267]{KPZ01},
\begin{equation}
\begin{gathered}
\Phi (t)=1\quad \mbox{for }t\in B_{r}\quad\mbox{and}\quad
\Phi (t)=0\quad \mbox{for }t\in \mathbb{R}\setminus B_{2r},  \\
\Phi (t)>0\quad \mbox{for }t\in B_{2r}\quad\mbox{and}\quad
|\Phi '(t)|\leq \frac{c_{0}}{r},\quad t\in \mathbb{R}.
\end{gathered}  \label{ADPhi}
\end{equation}
For any $c_{0}>1$,  we take a test function $\varphi $ defined by
\[
\varphi (t)=\begin{cases}
(y(t)-\tilde{\omega}_{0})\Phi ^{p}(t)  &\mbox{if } t\in B_{2r}, \\
0 & \mbox{otherwise}.
\end{cases}
\]
It is clear that $\varphi \in W_{0}^{1,p}(B_{2r})\cap L^{\infty }(B_{2r})$.
Putting in \eqref{1eq} this test function we obtain
\begin{equation}
\begin{aligned}
&\int_{B_{2r}}|y'|^{p}\Phi ^{p}dt  \\
&\leq p\int_{B_{2r}}|y'|^{p-1}\Phi ^{p-1}(\tilde{\omega}
_{0}-y(t))|\Phi '|dt-\int_{B_{2r}}f(t,y,y')(\tilde{\omega}
_{0}-y(t))\Phi ^{p}dt. \end{aligned} \label{ADstart}
\end{equation}
Next, using that
$\tilde{\omega}_{0}-y(t)\leq \tilde{\omega}_{0}-\tilde{\theta}_{0}$,  and
$(p-1)p'=p$,  and
$\delta _{1}(p\delta _{2})\leq\delta _{1}^{p'}+(\frac{p}{p'})^{p-1}\delta _{2}^{p}$
in particular for
\[
\delta _{1}=|y'|^{p-1}\Phi ^{p-1}\quad\mbox{and}\quad
\delta _{2}=(\tilde{\omega}_{0}-y(t))|\Phi '|,
\]
 from \eqref{ADstart} we obtain
\begin{align*}
0&=\bigl[1-\frac{p'}{p'}\bigl]\int_{B_{2r}}|y'|^{p}\Phi ^{p}dt \\
&\leq \bigl(\frac{p}{p'}\bigl)^{p-1}(\tilde{\omega}_{0}
-\tilde{\theta}_{0})^{p}\int_{B_{2r}}|\Phi '|^{p}dt
-\int_{B_{2r}}f(t,y,y')(\tilde{\omega}_{0}-y(t))\Phi ^{p}dt.
\end{align*}
Now, by  \eqref{ADpom},  \eqref{ADfyp} and \eqref{ADPhi}, we
have
\[
0\leq \bigl(\frac{p}{p'}\bigl)^{p-1}(\tilde{\omega}_{0}
-\tilde{\theta}_{0})^{p}|B_{2r}\setminus B_{r}|
\bigl(\frac{c_{0}}{r}\bigl)^{p}-(\tilde{\omega}_{0}-\omega _{2r})
\int_{B_{r}}f(t,y,y')dt.
\]
Since $|B_{2r}\setminus B_{r}|=|B_{r}|$ and passing to the limit as
$c_{0}\to 1$ we obtain
\[
\int_{B_{r}}f(t,y,y')dt\leq (p-1)^{p-1}\frac{|B_{r}|}{r^{p}}\frac{(
\tilde{\omega}_{0}-\tilde{\theta}_{0})^{p}}{\tilde{\omega}_{0}-\omega _{2r}}.
\]
This inequality contradicts \eqref{ADfysp} and so the assumption \eqref{ADpom}
is not possible. Thus, the statement \eqref{ADjumpg} is proved.

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\end{document}