\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 220, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/220\hfil 
Oscillation for modified Euler type equations]
{Oscillation constant for modified Euler type half-linear equations}

\author[P. Hasil, M. Vesel\'y \hfil EJDE-2015/220\hfilneg]
{Petr Hasil, Michal Vesel\'y}

\address{Petr Hasil \newline
 Department of Mathematics and Statistics \newline
 Faculty of Science, Masaryk University \newline
 Kotl\'a\v{r}sk\'a 2, CZ 611 37 Brno, Czech Republic}
\email{hasil@mail.muni.cz}

\address{Michal Vesel\'y  (corresponding author)  \newline
 Department of Mathematics and Statistics \newline
 Faculty of Science, Masaryk University \newline
 Kotl\'a\v{r}sk\'a 2, CZ 611~37 Brno, Czech Republic}
\email{michal.vesely@mail.muni.cz}

\thanks{Submitted March 16, 2015. Published August 24, 2015.}
\subjclass[2010]{34C10, 34C15}
\keywords{Half-linear equations; Pr\"{u}fer angle; oscillation theory;
\hfil\break\indent conditional oscillation; oscillation constant}

\begin{abstract}
 Applying the modified half-linear Pr\"ufer angle, we study oscillation
 properties of the half-linear differential equation
 $$
 [ r(t) t^{p-1} \Phi(x')]' + \frac{s(t)}{t \log^pt} \Phi(x) = 0, \quad
 \Phi(x)=|x|^{p-1}\operatorname{sgn} x.
 $$
 We show that this equation is conditionally oscillatory in a very general case.
 Moreover, we identify the critical oscillation constant
 (the borderline depending on the functions $r$ and $s$ which separates
 the oscillatory and non-oscillatory equations).
 Note that the used method is different from the standard  method based 
 on the half-linear Pr\"ufer angle.
\end{abstract}

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

\section{Introduction}

For any $p > 1$, we analyze the oscillation behavior of the
second-order half-linear differential equation
\begin{equation} \label{introRS}
[R(t) \Phi(x')]' + S(t) \Phi(x) = 0, \quad \Phi(x)=|x|^{p-1}\operatorname{sgn} x,
\end{equation}
where $R$ is a positive continuous function and $S$ is a continuous function.
In this paper, we focus on the so-called conditional oscillation. 
To recall the notion of the conditional oscillation in a~simple form, we 
consider  \eqref{introRS} with
$S(t) = \gamma C(t)$, where $C$ is a continuous function
and $\gamma \in \mathbb{R}$, i.e.,
\begin{equation} \label{introRgC}
[R(t) \Phi(x')]' + \gamma C(t) \Phi(x) = 0.
\end{equation}
We say that  \eqref{introRgC} is conditionally oscillatory if there exists
a positive constant $\Gamma$ such that  \eqref{introRgC} is oscillatory
for $\gamma > \Gamma$ and non-oscillatory for $\gamma < \Gamma$.
Such a constant $\Gamma$ is called the critical oscillation constant
of  \eqref{introRgC}. Of course,
once we know its value, we are able to formulate oscillation results directly
for  \eqref{introRS}. For this reason, some results of this kind are 
presented in different forms in the literature (see the below given results and 
references).

Now we collect the most relevant references of the treated topic.
 The reader can find the fundamental theory overview on half-linear equations 
in books \cite{HasVes-agarwal2002,DV-DoslyRehak2005}. The conditional oscillation
of half-linear differential equations is studied, e.g.,
in papers \cite{DV-DoslyHasil2011,DV-Elbert1984,hasil2008} 
(see also \cite{dosly-funk,E-Sch,DV-AAA2013-14,HasVes-2015}).
Of course, the identification of oscillation constants is a subject
of research also in the field of difference
equations (see \cite{DV-AAA2013}) and in the field of dynamic
equations on time scales (see \cite{DV-hasvit}).
In the linear case, we refer to \cite{DV-gesztesy,DV-AAA2012,schmidt-1,vitovec} 
(and also \cite{DV-Kruger2,DV-Kruger3,DV-Kruger4}).

Very general results about the conditional oscillation   can be found
in~\cite{DV-ADE2013,DV-JJMV,DV-2014}. In \cite{DV-ADE2013,DV-2014}, 
there are treated equations with asymptotically almost periodic coefficients 
and coefficients having mean values (for the definitions of the asymptotic 
almost periodicity and the mean value, see Definitions \ref{d1} 
and \ref{d2v3} below). In addition, the results of \cite{DV-JJMV} 
cover equations whose coefficients are unbounded.
The main results of \cite{DV-JJMV,DV-2014} are reformulated in 
Theorems \ref{intro-TH1} and \ref{intro-TH2} below. Note that the main result 
of \cite{DV-2014} is superior to the results of \cite{DV-ADE2013}. 
To mention the basic motivation explicitly in Theorems \ref{intro-TH1} and
\ref{intro-TH2}, we use
the standard symbols $\mathbb{R}^+ :=(0,\infty)$ and
$\mathbb{R}_a := [a, \infty)$ for any $a > 0$  in the rest of this article.

\begin{theorem}[\cite{DV-2014}] \label{intro-TH1}
Let $r: \mathbb{R}_1 \to \mathbb{R}^+$ be a continuous function,
for which mean value $M({r^{1/(1-p)}})$ exists and for which
it holds
$$
0 < \inf_{t \in \mathbb{R}_1} \, r (t) \le \sup_{t \in \mathbb{R}_1}
 \, r (t) < \infty, 
$$
and let $s: \mathbb{R}_1 \to \mathbb{R}$ be a continuous function having
mean value $M(s)$. Let
$$
\widetilde{\Gamma }:= \Big(\frac{p-1}{p}\Big)^p \big[ M \big({r^{\frac{1}{1-p}}} 
\big)\Big]^{1-p}. 
$$
Consider the equation
\begin{equation}
[r(t) \Phi(x')]' +\frac{s(t)}{t^{p}} \Phi(x) = 0. \label{hl2}
\end{equation}
Equation \eqref{hl2} is oscillatory if $M(s) > \widetilde{\Gamma }$,
 and non-oscillatory if
$M(s) < \widetilde{\Gamma }$.
\end{theorem}

\begin{theorem}[\cite{DV-JJMV}] \label{intro-TH2}
 Let $r: \mathbb{R}_1 \to \mathbb{R}^+$ and
$s: \mathbb{R}_1 \to \mathbb{R}$ be continuous functions such that there exist
$\delta > 1$ and $\varepsilon \in (0, 1/2)$ for which
\begin{equation} \label{h9h}
 \int_t^{t+1} r (\tau) \,\mathrm{d}\tau  < \delta t^\varepsilon, \quad 
\int_t^{t+1} |  s (\tau) | \,\mathrm{d}\tau  < \delta t^\varepsilon, \quad
 t \in \mathbb{R}_1.
 \end{equation}
Consider the equation
\begin{equation} \label{01}
 [r^{1-p}(t)\, \Phi (x')]' +  \frac{s(t)}{t^p}  \Phi(x)  = 0.
\end{equation}
\begin{itemize}
\item[(i)] If  there exist $\alpha, R, S \in \mathbb{R}^+$
satisfying  
$$
R^{p - 1}  S > \Big(\frac{p-1}{p}\Big)^p
$$ 
and
$$ 
\frac{1}{\alpha} \int_{t}^{t + \alpha} r (\tau) \,\mathrm{d}\tau \ge R, \quad
\frac{1}{\alpha} \int_{t}^{t + \alpha} s (\tau) \,\mathrm{d}\tau \ge S
$$
for all sufficiently large $t$, then  \eqref{01} is oscillatory.

\item[(ii)] If there exist $\alpha, R, S \in \mathbb{R}^+ $
satisfying  
\begin{gather*}
R^{p - 1}  S < \Big(\frac{p-1}{p}\Big)^p , \\
\frac{1}{\alpha} \int_{t}^{t + \alpha} r (\tau) \,\mathrm{d}\tau \le R, \quad
\frac{1}{\alpha} \int_{t}^{t + \alpha} s (\tau) \,\mathrm{d}\tau \le S
\end{gather*}
for all sufficiently large $t$,
then  \eqref{01} is non-oscillatory.
\end{itemize}
\end{theorem}

In this paper, motivated by Theorems \ref{intro-TH1} and \ref{intro-TH2}, 
our aim is to extend the family of conditionally oscillatory
equations. More precisely, we identify the critical oscillation constant
for the Euler type equations in the form
\begin{equation} \label{introRtS}
[ r(t) t^{p-1} \Phi(x')]' + \frac{s(t)}{t \log^pt} \Phi(x) = 0,
\end{equation}
where $r: \mathbb{R}_{\rm e} \to \mathbb{R}^+$ and 
$s: \mathbb{R}_{\rm e} \to \mathbb{R}$ are continuous functions.
We introduce a new modification of the half-linear Pr\"{u}fer angle in combination
with the Riccati transformation.
We remark that the results proven in this paper are stronger than the ones
known for  \eqref{01} (compare \eqref{opr} and \eqref{opr2} with \eqref{h9h}).
To the best of our knowledge, the presented results are new even for
 \eqref{introRtS} with periodic coefficients $r, s$.

The article is organized as follows.
The notion of the Pr\"{u}fer angle is mentioned in the next section.
Auxiliary results are collected in Section~\ref{Sec3}.
The content of Sections~\ref{Sec2} and~\ref{Sec3} gives
the description of our method which is used in the proofs
of the main results in Section~\ref{Sec4}.
Section~\ref{Sec4} is finished by corollaries and examples.

\section{Equation for Pr\"{u}fer angle}
\label{Sec2}

In this section, we derive the equation for the modified half-linear Pr\"{u}fer 
angle which will be fundamental for our investigation. To do this, we have 
to begin with the well-known notion of the Riccati equation
corresponding to the general half-linear equation
\begin{equation} \label{start}
[R(t) \Phi(x')]' + S(t) \Phi(x) = 0, \quad \Phi(x)=|x|^{p-1}\operatorname{sgn} x,
\end{equation}
where $p > 1$ and $R: \mathbb{R}_{\rm e} \to \mathbb{R}^+$
and $S: \mathbb{R}_{\rm e} \to \mathbb{R}$ are continuous functions.
The Riccati equation
\begin{equation} \label{ok}
 w'(t) +S (t)+(p-1)R^{\frac{1}{1-p}} (t)|w(t)|^{\frac{p}{p-1}}=0
\end{equation}
can be obtained applying the transformation
\begin{equation} \label{75}
w(t)=R (t)\Phi\Big(\frac{x'(t)}{x(t)}\Big)
\end{equation}
to  \eqref{start}, where $x$ is a non-trivial solution of  \eqref{start}. 
Of course, the transformation is valid whenever $x(t)\neq 0$.
For more details including the used calculations, 
see \cite[Section~1.1.4]{DV-DoslyRehak2005}.

To introduce the modified half-linear Pr\"{u}fer transformation, we should 
shortly recall the half-linear trigonometric functions. 
The half-linear sine and  cosine functions,
denoted by $\sin_p$ and $\cos_p$, are defined via the initial problem
\begin{equation} \label{54h}
 [\Phi (x')]'+(p-1)\Phi(x)=0, \quad x(0)=0, \quad x'(0)=1.
\end{equation}
More precisely, $\sin_p$ is given as the odd extension
of the solution of \eqref{54h} with period
$$
2 \pi_p := \frac{4\pi}{p\sin\frac{\pi}{p}}.
$$
Then, $\cos_p$ is introduced as the derivative of $\sin_p$.
The functions $\sin_p$ and $\cos_p$ have many properties similar to the common
sine and cosine functions. In this paper, we will use only the Pythagorean 
identity, whose half-linear version is
\begin{equation} \label{pyth}
 |\sin_p x|^p+|\cos_p x|^p=1, \quad x \in \mathbb{R}.
\end{equation}
For more details concerning the half-linear trigonometric functions,
we refer to \cite[Section 1.1.2]{DV-DoslyRehak2005}.

We introduce the modified Pr\"ufer transformation in the form
\begin{equation} \label{6k}
 x(t)=\rho(t)\sin_p \varphi(t),\quad 
R^{q-1}(t) x'(t)= \frac{\rho(t)}{\log t}\cos_p\varphi(t)
 \end{equation}
together with the substitution
\begin{equation} \label{76}
v(t)=(\log t)^{p/q}w(t), \quad t \in \mathbb{R}_{\rm e},
\end{equation}
where $\log$ stands for the natural logarithm and $q$ is the number 
conjugated with $p$, i.e.,
\begin{equation}\label{cxb}
p+q=pq.
\end{equation}
Note that substitutions similar to \eqref{76} can be used also in the
Riccati equation \eqref{ok}. This approach leads to the so-called adapted 
(or weighted) Riccati equation.
Nevertheless, we use this process only partially (see below) and we have 
to take into consideration the modified Pr\"ufer transformation \eqref{6k} 
as well.

Using \eqref{76}, \eqref{75}, \eqref{6k}, and \eqref{cxb} successively, we obtain
\begin{equation}\label{tf}
\begin{aligned}
 v (t) &=(\log t)^{p/q}w(t)
  = (\log t)^{p/q} R (t)\Phi\Big(\frac{x'(t)}{x(t)}\Big) \\
&=(\log t)^{p/q} R (t)
\frac{\Phi(R^{1-q}(t) \cos_p\varphi(t))}{\Phi(\log t\;\sin_p\varphi(t))}\\
&=\Phi\Big(\frac{\cos_p\varphi (t)}{\sin_p\varphi (t)}\Big).
\end{aligned}
\end{equation}
One can easily verify that
\begin{equation} \label{9663sd}
\hat{v}(t) := \Phi\Big(\frac{\cos_pt}{\sin_pt}\Big) 
\end{equation}
solves the equation
\begin{equation} \label{reqsin}
 \hat{v}'(t)+p-1+(p-1)|\hat{v}(t)|^q = 0.
\end{equation}
Equation \eqref{reqsin} is the Riccati equation associated to the equation 
in \eqref{54h}.
Hence, due to \eqref{pyth}, \eqref{cxb}, \eqref{9663sd}, and \eqref{reqsin}, 
we have
\begin{equation}                        \label{v1b}
\begin{aligned}
v'(t) &= [\hat{v}(\varphi(t))]'=[-p+1-(p-1)|\hat{v}(\varphi(t))|^q] \varphi'(t) \\
 &=(1-p) \Big[1+ \Big|\frac{\cos_p\varphi (t)}{\sin_p\varphi (t)}
  \Big|^{p}\Big]\varphi' (t) \\
&=  \frac{1-p}{|{\sin_p\varphi (t)}|^{p} }\, \varphi' (t).
\end{aligned}
\end{equation}

On the other hand, considering  \eqref{76} together with \eqref{ok}, we have
\begin{equation}
 \label{v2b} \begin{aligned}
v'(t) 
&= \frac{p}{q} (\log t)^{\frac{p}{q}-1} \frac{w(t)}{t} + (\log t)^{p/q} w'(t) \\
&= \frac{p}{q} \cdot \frac{v(t)}{t \log t} + (\log t)^{p/q}
  [-S(t) - (p-1)R^{1-q}(t)|w(t)|^q ] \\
&= \frac{p}{q} \cdot \frac{v(t)}{t \log t} - (\log t)^{p/q} S(t)
  -(p-1)R^{1-q}(t)\,\frac{|v(t)|^q}{\log t}.
\end{aligned}
\end{equation}
We combine \eqref{v1b} and \eqref{v2b}. This leads to
$$
\frac{1-p}{|{\sin_p\varphi (t)}|^{p} }\, \varphi' (t)
= \frac{p}{q} \cdot \frac{v(t)}{t \log t} - (\log t)^{p/q} S(t)
  -(p-1)R^{1-q}(t) \frac{|v(t)|^q}{\log t}.
$$
Taking into account \eqref{tf}, we obtain
\begin{align*}
 (1-p) \varphi' (t)
&= \frac{p}{q} \cdot \frac{1}{t \log t} \, \Phi(\cos_p\varphi(t))\sin_p\varphi(t) \\
&\quad- (\log t)^{p/q} S(t) |{\sin_p\varphi (t)}|^{p}
  -(p-1)R^{1-q}(t) \frac{|\cos_p\varphi (t)|^p}{\log t}
\end{align*}
which gives us the desired equation for the Pr\"{u}fer angle as
\begin{equation} \label{6gd}
\begin{aligned}
 \varphi'(t) 
&= R^{1-q}(t)\,\frac{|\cos_p\varphi (t)|^p}{\log t} \\
&\quad -\frac{1}{t \log t}\, \Phi(\cos_p\varphi(t))\sin_p\varphi(t)
   + \frac{(\log t)^{p/q}}{p-1} \, S(t) |{\sin_p\varphi (t)} |^{p}.
\end{aligned}
 \end{equation}

In this article, we apply  \eqref{6gd} to the study of the half-linear equations 
with the coefficients in the form
\begin{equation} \label{RSrs}
 R(t) = r(t) t^{p-1}, \quad 
S(t)= \frac{s(t)}{t \log^pt}, \quad t \in \mathbb{R}_{\rm e}.
\end{equation}
Thus, we will analyze the equations
\begin{equation} \label{start0}
[r(t) t^{p-1}  \Phi(x')]' + \frac{s(t)}{t \log^pt} \Phi(x) = 0,
\end{equation}
where $r: \mathbb{R}_{\rm e} \to \mathbb{R}^+$ and
 $s: \mathbb{R}_{\rm e} \to \mathbb{R}$ are continuous
functions. We point out that the form of  \eqref{6gd} for the coefficients in
\eqref{RSrs} is
\begin{equation} \label{ysd}
\begin{aligned}
\varphi'(t)
 &= \frac{1}{t\log t} \Big[r^{1-q}(t)|\cos_p\varphi (t)|^p\\
 &\quad  -\Phi(\cos_p\varphi(t))\sin_p\varphi(t)  
   + s(t) \frac{|{\sin_p\varphi (t)} |^{p}}{p-1} \Big],
\end{aligned}
\end{equation}
which can be simply verified.

\section{Pr\"{u}fer angle of average function}\label{Sec3}

In this section, we assume that the coefficients 
$r: \mathbb{R}_{\rm e} \to \mathbb{R}^+$ and 
$s: \mathbb{R}_{\rm e} \to \mathbb{R}$ in  \eqref{start0} are such that
\begin{gather} \label{opr}
\lim_{t \to \infty} 
\frac{\int_t^{t + \alpha} r^{1-q} (\tau) \,\mathrm{d}\tau}{\sqrt{t \log t}} = 0,
\\
 \label{opr2}
\lim_{t \to \infty} \frac{\int_t^{t + \alpha} |s (\tau)| \,\mathrm{d}\tau}
{\sqrt{t \log t}} = 0
\end{gather}
hold for some $\alpha \in \mathbb{R}^+$. For this number $\alpha$, 
we define the function $\psi$ which
determines the average value of an arbitrarily given solution $\varphi$ 
of  \eqref{ysd} over  intervals of the length $\alpha$; i.e., we put
$$
\psi (t) := \frac{1}{\alpha}\int_t^{t + \alpha} \varphi (\tau)\, \,\mathrm{d}\tau,
\quad t \in \mathbb{R}_{\rm e}, 
$$
where $\varphi$ is a solution of \eqref{ysd}
on $\mathbb{R}_{\rm e}$. We formulate and prove auxiliary results concerning 
the function $\psi$.

\begin{lemma} \label{L1} 
It holds 
\begin{equation} \label{1a}
\lim_{t \to \infty} \sqrt{t \log t} |\varphi(s )- \psi (t)| = 0
 \end{equation}
uniformly with respect to $s \in [t, t + \alpha]$.
\end{lemma}

\begin{proof} 
For $s \in [t, t + \alpha]$, it is seen that
\begin{align*}
&\limsup_{t \to \infty}  \sqrt{t \log t} |\varphi(s)- \psi (t)| \\
&\leq  \limsup_{t \to \infty}  \sqrt{t \log t} 
\int_t^{t+ \alpha}|\varphi'(\tau)|\,\mathrm{d}\tau \\
&= \limsup_{t \to \infty}  \sqrt{t \log t}  \int_t^{t+ \alpha} 
 \Big|\frac{1}{\tau \log \tau }\Big[r^{1-q}(\tau)|\cos_p\varphi (\tau)|^p 
\\ 
&\quad - \Phi(\cos_p\varphi (\tau) )\sin_p\varphi (\tau)     +   s(\tau)
\frac{|\sin_p\varphi (\tau)|^p}{p-1}\Big] \Big| \,\mathrm{d}\tau \\
&\leq \limsup_{t \to \infty}  \sqrt{t \log t}  \int_t^{t+ \alpha} 
 \frac{1}{t \log t }\Big[r^{1-q}(\tau)|\cos_p\varphi (\tau)|^p
\\ 
&\quad + | \Phi(\cos_p\varphi (\tau) )\sin_p\varphi (\tau) |   +    | s(\tau)  |
\frac{|\sin_p\varphi (\tau)|^p}{p-1}\Big]  \,\mathrm{d}\tau .
\end{align*}
Since
(see directly \eqref{pyth})
\begin{equation}
\label{nb} |\sin_p x|^p\le 1, \quad |\cos_p x|^p \le 1, \quad x \in \mathbb{R},
\end{equation}
and, consequently,
\begin{equation}
\label{nb2} |\Phi(\cos_p x )\sin_p x| 
= |\cos_p x|^{p-1} |\sin_p x| \le 1, \quad x \in \mathbb{R},
\end{equation}
we have
\begin{align*} 
0 &\le \limsup_{t \to \infty}  \sqrt{t \log t} |\varphi(s)- \psi (t)| \\
 &\leq \limsup_{t \to \infty}  \frac{1}{\sqrt{t \log t} } 
  \int_t^{t+ \alpha}   r^{1-q}(\tau) + 1 +
\frac{ | s(\tau)  |}{p-1}   \,\mathrm{d}\tau, \quad s \in [t, t + \alpha].
\end{align*}
Using \eqref{opr} and \eqref{opr2}, we obtain
$$ 
0 \le \liminf_{t \to \infty}  \sqrt{t \log t} |\varphi(s)- \psi (t)|
 \le \limsup_{t \to \infty}  \sqrt{t \log t} |\varphi(s)- \psi (t)| = 0 
$$
uniformly with respect to $s \in [t, t + \alpha]$.
\end{proof}

\begin{lemma} \label{L3b} It holds
\begin{align*}
\psi' (t) &= \frac{1}{t \log t} \Big[\frac{|\cos_p \psi (t)|^p}{\alpha} 
\int_t^{t+\alpha}  r^{1-q}(\tau)\,\mathrm{d}\tau  - \Phi(\cos_p \psi (t))\sin_p \psi (t)  \\ 
&\quad + \frac{|\sin_p \psi (t)|^p}{(p-1) \alpha} \int_t^{t+ \alpha} s(\tau) 
 \,\mathrm{d}\tau  + \Psi (t) \Big]
\end{align*}
for all $t > \mathrm{e}$, where $\Psi: \mathbb{R}_{\rm e} \to \mathbb{R}$ is a continuous 
function such that $\lim_{t \to \infty} \Psi (t) = 0$.
\end{lemma}

\begin{proof} For any $t > \\mathrm{e}$, we have
\begin{align*} 
\psi' (t)  &= \frac{1}{\alpha}  \int_{t}^{t+\alpha} 
 \frac{1}{\tau \log \tau} \Big[ r^{1-q}(\tau) |\cos_p \varphi (\tau)|^p \\
 &\quad - \Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau)+ s(\tau) 
 \frac{|\sin_p\varphi (\tau)|^p}{p-1} \Big] \,\mathrm{d}\tau.
\end{align*}
We can replace $\psi' (t)$ by
\[
\frac{1}{\alpha} \int_{t}^{t+\alpha} \frac{1}{t \log t} 
\Big[ r^{1-q}(\tau) |\cos_p \varphi (\tau)|^p 
 - \Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau)+ s(\tau) 
\frac{|\sin_p\varphi (\tau)|^p}{p-1}  \Big] \,\mathrm{d}\tau,
\]
because we can easily estimate (see \eqref{opr}, \eqref{opr2}, 
\eqref{nb}, \eqref{nb2})
\begin{align*}   
&\limsup_{t \to \infty}  \frac{t \log t}{\alpha} 
\Big|  \int_{t}^{t+ \alpha} \frac{1}{\tau \log \tau} 
 \Big[ r^{1-q}(\tau) |\cos_p \varphi (\tau)|^p 
  -  \Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau)\\
& + s(\tau) \frac{|\sin_p\varphi (\tau)|^p}{p-1} \Big] \,\mathrm{d}\tau
-\int_{t}^{t+ \alpha} \frac{1}{t \log t} 
\Big[ r^{1-q}(\tau) |\cos_p \varphi (\tau)|^p 
-\Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau)\\
&+ s(\tau) \frac{|\sin_p\varphi (\tau)|^p}{p-1} \Big] \,\mathrm{d}\tau \Big|
\\ 
&\le \limsup_{t \to \infty}  \frac{t \log t}{\alpha} 
 \int_{t}^{t+ \alpha} \Big[\frac{1}{t \log t} - \frac{1}{\tau \log \tau} \Big] 
\Big[ r^{1-q}(\tau) +1  +| s(\tau)| \frac{1}{p-1} \Big] \,\mathrm{d}\tau
\\
&\le \limsup_{t \to \infty}  \frac{t \log t}{\alpha} 
 \int_{t}^{t+ \alpha} 
\Big[\frac{1}{t \log t} - \frac{1}{(t + \alpha) \log (t + \alpha)} \Big] \\
&\quad \times \Big[ r^{1-q}(\tau) +1  +| s(\tau)| \frac{1}{p-1} \Big] \,\mathrm{d}\tau
\\
&\le \limsup_{t \to \infty}  \frac{t \log t}{\alpha} \cdot 
\frac{\alpha}{t (t + \alpha) \log t } \int_{t}^{t+ \alpha} 
\Big[ r^{1-q}(\tau) +1  +| s(\tau)| \frac{1}{p-1} \Big] \,\mathrm{d}\tau
 \\
&\le \limsup_{t \to \infty}  \frac{1}{t + \alpha} 
\int_{t}^{t+ \alpha} \Big[ r^{1-q}(\tau) +1  +| s(\tau)| \frac{1}{p-1} \Big] \,\mathrm{d}\tau
= 0.
\end{align*}

Applying the uniform continuity of the half-linear trigonometric functions and 
\eqref{1a} in Lemma~\ref{L1}, we see that
\begin{equation} \label{vb} 
\lim_{t \to \infty} 
\Big( \Phi(\cos_p \psi  (t))\sin_p \psi  (t)  
- \frac{1}{\alpha} \int_t^{t+\alpha} \Phi(\cos_p\varphi(\tau))
 \sin_p\varphi (\tau) \,\mathrm{d}\tau \Big) = 0.
\end{equation}

In addition, the half-linear trigonometric functions are continuously 
differentiable and periodic (see, e.g., \cite[Section 1.1.2]{DV-DoslyRehak2005}). 
Hence, they have the Lipschitz property on $\mathbb{R}$ and, consequently, 
there exists a constant $L\in \mathbb{R}^+$ for which
\begin{gather} \label{54}
 \big| |\cos_p x|^p-|\cos_p y|^p\big|
 \leq L |x - y|, \quad x, y \in \mathbb{R},
\\
\label{55}
\big| |\sin_p x|^p-|\sin_p y|^p\big|
 \leq L |x - y|, \quad x, y \in \mathbb{R}. 
\end{gather}
Therefore, from \eqref{opr}, Lemma \ref{L1}, and \eqref{54}, it follows
\begin{equation} \label{va}
 \begin{aligned}
&\limsup_{t \to \infty} \Big|\frac{|\cos_p \psi (t)|^p}{\alpha} 
\int_t^{t+\alpha} r^{1-q}(\tau)\,\mathrm{d}\tau
 - \frac{1}{\alpha} \int_t^{t+\alpha} r^{1-q}(\tau) 
 |\cos_p \varphi (\tau)|^p \,\mathrm{d}\tau \Big| \\
&\le \frac{1}{\alpha} \limsup_{t \to \infty} \int_t^{t+\alpha} r^{1-q}(\tau) \big| 
 |\cos_p \psi (t)|^p - |\cos_p \varphi (\tau)|^p  \big| \,\mathrm{d}\tau \\
&\le  \frac{1}{\alpha} \limsup_{t \to \infty} 
 \frac{\sqrt{t \log t}}{\sqrt{t \log t}} \int_t^{t+\alpha } r^{1-q}(\tau) L  
 | \psi (t) - \varphi (\tau)| \,\mathrm{d}\tau = 0.
\end{aligned}
\end{equation}
Analogously, from \eqref{opr2}, Lemma \ref{L1}, and \eqref{55}, we have
\begin{equation} \label{vc}
\begin{aligned}
&\limsup_{t \to \infty} \Big| \frac{|\sin_p \psi (t)|^p}{(p-1) \alpha} 
\int_t^{t+ \alpha} s(\tau) \,\mathrm{d}\tau
 - \frac{1}{\alpha} \int_t^{t+\alpha} s(\tau)\frac{|\sin_p\varphi (\tau)|^p}{p-1}
 \,\mathrm{d}\tau \Big| \\
&\le \frac{1}{(p-1)\alpha} \limsup_{t \to \infty} 
\int_t^{t+\alpha} | s(\tau)| \cdot \big|  |\sin_p \psi (t)|^p 
- |\sin_p \varphi (\tau)|^p  \big| \,\mathrm{d}\tau \\
&\le \frac{1}{(p-1)\alpha} \limsup_{t \to \infty} 
\frac{\sqrt{t \log t}}{\sqrt{t \log t}} 
\int_t^{t+\alpha} | s(\tau)| L | \psi (t) - \varphi (\tau)| \,\mathrm{d}\tau = 0.
\end{aligned}
\end{equation}
Finally, the statement of the lemma directly comes from the combination
of \eqref{vb}, \eqref{va}, and \eqref{vc}. The continuity of $\Psi$ is obvious.
\end{proof}

\section{Oscillation constant}\label{Sec4}

At first, we recall known results concerning the studied equations with 
constant coefficients.

\begin{theorem} \label{v1} 
If $A, B \in \mathbb{R}^+$ satisfy $B/A > q^{-p}$, then the equation
$$ 
[A t^{p - 1}  \Phi(x')]' + \frac{B}{ t \log^p t} \Phi(x) = 0 
$$
is oscillatory.
\end{theorem}

\begin{proof} 
See, e.g., \cite[Theorem 1.4.4]{DV-DoslyRehak2005} (or directly \cite{E} 
and \cite{DV-Elbert1984}).
\end{proof}

\begin{theorem} \label{v2} If $C, D \in \mathbb{R}^+$ satisfy
$D/C < q^{-p}$, then the equation
$$ [C t^{p - 1}  \Phi(x')]' + \frac{D}{ t \log^p t} \Phi(x) = 0 $$
is non-oscillatory.
\end{theorem}

\begin{proof} See \cite[Theorem 1.4.4]{DV-DoslyRehak2005} 
(or \cite{E,DV-Elbert1984}).
\end{proof}

Now we can prove the announced results which identify the critical 
oscillation constant for the analyzed equations with very general coefficients.

\begin{theorem} \label{TH}
 Let $\alpha, R, S \in \mathbb{R}^+$ be such that \eqref{opr2} is valid and 
$R^{p-1} S > q^{-p}$. If there exists $T > {\rm e}$ with the property that
$$  
\frac{1}{\alpha} \int_t^{t + \alpha} r^{1-q} (\tau) \,\mathrm{d}\tau \ge R, \quad
\frac{1}{\alpha} \int_t^{t + \alpha} s (\tau) \,\mathrm{d}\tau \ge S, \quad
 t \in \mathbb{R}_T,
 $$
then  \eqref{start0} is oscillatory.
\end{theorem}

\begin{proof} 
We study the equation for the Pr\"{u}fer angle corresponding to the average 
function $\psi$. It is well-known that the non-oscillation of solutions of
 \eqref{start0} is equivalent to the boundedness from above of
the Pr\"{u}fer angle $\varphi$ given by  \eqref{ysd}. We refer, 
e.g., to \cite{dosly-funk,DV-DoslyHasil2011,DV-Schmidt2000,DV-AAA2013-14}. 
It also suffices to consider directly \eqref{6k} and  \eqref{ysd} when 
$\sin_p \varphi (t) = 0$. Based on Lemma~\ref{L1}, we know that the boundedness 
(from above) of $\varphi$ is equivalent to the boundedness (from above) 
of $\psi$. Hence, we will show that $\psi$ is unbounded from above.
At first, we assume that \eqref{opr} is true.

Taking into account Lemma \ref{L3b}, we have
\begin{align*}
\psi' (t) &= \frac{1}{t \log t} 
\Big[\frac{|\cos_p \psi (t)|^p}{\alpha} 
 \int_t^{t+\alpha}  r^{1-q}(\tau)\,\mathrm{d}\tau  - \Phi(\cos_p \psi (t))\sin_p \psi (t)  \\
 &\quad + \frac{|\sin_p \psi (t)|^p}{(p-1) \alpha} \int_t^{t+ \alpha} s(\tau) 
 \,\mathrm{d}\tau  + \Psi (t) \Big] \\
&\ge \frac{1}{t \log t} \Big[R |\cos_p \psi (t)|^p  
  - \Phi(\cos_p \psi (t))\sin_p \psi (t)  \\ 
&\quad + S\frac{|\sin_p \psi (t)|^p}{p-1}  + \Psi (t) \Big]
\end{align*}
for all $t \in \mathbb{R}_T$ and for some continuous function 
$\Psi: \mathbb{R}_T \to \mathbb{R}$ satisfying
\begin{equation} \label{hhh}
\lim_{t \to \infty} \Psi (t) = 0.
\end{equation}

Let $\varepsilon > 0$ be arbitrary. From
\eqref{pyth} and \eqref{hhh}, we have
\begin{equation} \label{8zz}
\varepsilon\Big( |\cos_p x|^p  + \frac{|\sin_p x|^p}{p-1} \Big) > |\Psi (t)|
\end{equation}
for all $x \in \mathbb{R}$ and for all large $t \in \mathbb{R}_T$. Thus,
\begin{equation} \label{hg}
\begin{aligned}
\psi' (t) & > \frac{1}{t \log t} \Big[(R-\varepsilon) |\cos_p \psi (t)|^p  
  - \Phi(\cos_p \psi (t))\sin_p \psi (t)  \\ 
& \quad + (S-\varepsilon) \frac{|\sin_p \psi (t)|^p}{p-1} \Big]
\end{aligned}
\end{equation}
for all large $t \in \mathbb{R}_T$. Let $\varepsilon$ be so small that 
$(R-\varepsilon)^{p-1} (S-\varepsilon) > q^{-p}$ and $R-\varepsilon > 0$. 
Using Theorem \ref{v1} for $A = (R-\varepsilon)^{\frac{1}{1-q}}$
and $B=S-\varepsilon$, we know that the equation
$$
 [(R-\varepsilon)^{\frac{1}{1-q}} t^{p - 1}  \Phi(x')]' 
+ \frac{S-\varepsilon}{ t \log^p t} \Phi(x) = 0 
$$
is oscillatory; i.e., any solution $\hat{\varphi}: \mathbb{R}_T \to \mathbb{R}$ 
of the equation
\begin{equation} \label{thg} 
\begin{aligned}
 \hat{\varphi}' (t)
&=\frac{1}{t \log t}\Big[(R-\varepsilon) |\cos_p \hat{\varphi} (t)|^p -
\Phi(\cos_p \hat{\varphi} (t))\sin_p \hat{\varphi} (t) \\
&\quad + (S -\varepsilon)
\frac{|\sin_p \hat{\varphi} (t)|^p}{p-1}\Big]
\end{aligned}
\end{equation}
has the property that $\limsup_{t \to \infty} \hat{\varphi} (t) = \infty$. 
Indeed, one can simply compute that
$$
B/A = (R-\varepsilon)^{p-1} (S-\varepsilon) > q^{-p}.
$$
Considering \eqref{hg} with \eqref{thg} and  the $2\pi_p$-periodicity of 
$\sin_p$ and $\cos_p$, we see that
$\limsup_{t \to \infty} \hat{\varphi} (t) = \infty$ implies
$\limsup_{t \to \infty} \psi (t) = \infty$.

To finish the proof, we have to consider the case when \eqref{opr} is not valid. 
Evidently, there exists a continuous function 
$\tilde{r}: \mathbb{R}_{\rm e} \to \mathbb{R}^+$ with the properties
$$ 
{r}^{1-q} (t) \ge \tilde{r}^{1-q} (t) ,\quad  
\frac{1}{\alpha} \int_t^{t + \alpha} \tilde{r}^{1-q} (\tau) \,\mathrm{d}\tau \ge R,
\quad  t \in \mathbb{R}_T, 
$$
and
\begin{equation} \label{6tyd}
  \lim_{t \to \infty} 
\frac{\int_t^{t + \alpha} \tilde{r}^{1-q} (\tau) \,\mathrm{d}\tau}{\sqrt{t \log t}} = 0.
\end{equation}
We actually know that the equation
$$ 
[\tilde{r}(t) t^{p-1} \Phi(x')]' + \frac{s(t)}{t \log^pt} \Phi(x) = 0
 $$
is oscillatory (cf. \eqref{opr} and \eqref{6tyd}). 
Since ${r} (t) \le \tilde{r} (t)$ for all
$t \in \mathbb{R}_T$, the Sturmian half-linear comparison theorem 
(see, e.g., \cite[Theorem 1.2.4]{DV-DoslyRehak2005}) gives the oscillation 
of  \eqref{start0}.  
\end{proof}

\begin{theorem} \label{TH2} 
Let $\alpha, R, S \in \mathbb{R}^+$ be such that  \eqref{opr2} is valid and 
$R^{p-1} S < q^{-p}$. If there exists $T > {\rm e}$ with the property that
\begin{equation} \label{x}
  \frac{1}{\alpha} \int_t^{t + \alpha} r^{1-q} (\tau) \,\mathrm{d}\tau \le R, \quad
\frac{1}{\alpha} \int_t^{t + \alpha} s (\tau) \,\mathrm{d}\tau \le S, \quad
t \in \mathbb{R}_T,  
\end{equation}
then  \eqref{start0} is non-oscillatory.
\end{theorem}

\begin{proof} 
We  proceed analogously as in the proof of Theorem \ref{TH}. 
In this proof, we  show that $\psi$ is bounded from above. 
Note that \eqref{opr} is valid (see the first inequality in \eqref{x}).
From Lemma~\ref{L3b}, we obtain
\[
\psi' (t)
\le \frac{1}{t \log t} \Big[R |\cos_p \psi (t)|^p  
 - \Phi(\cos_p \psi (t))\sin_p \psi (t)  
 + S\frac{|\sin_p \psi (t)|^p}{p-1}  + \Psi (t) \Big]
\]
for any $t \in \mathbb{R}_T$ and for a continuous function 
$\Psi: \mathbb{R}_T \to \mathbb{R}$ satisfying \eqref{hhh}.
In addition, using \eqref{8zz}, we obtain
\begin{equation} \label{hg2}
\begin{aligned}
\psi' (t) 
&< \frac{1}{t \log t} \Big[(R +\varepsilon) |\cos_p \psi (t)|^p  
 - \Phi(\cos_p \psi (t))\sin_p \psi (t)  \\ 
& \quad + (S + \varepsilon) \frac{|\sin_p \psi (t)|^p}{p-1} \Big]
\end{aligned}
\end{equation}
for any $\varepsilon > 0$ and all sufficiently large $t \in \mathbb{R}_T$.
 We choose $\varepsilon$ in such a way that  
$(R+\varepsilon)^{p-1} (S + \varepsilon) < q^{-p}$.
We put $C = (R + \varepsilon)^{\frac{1}{1-q}}$
and $D=S + \varepsilon$
in Theorem \ref{v2}. Since 
$$
D/C = (R+\varepsilon)^{p-1} (S+\varepsilon) < q^{-p}, 
$$ 
we know that the equation
$$ 
[(R + \varepsilon)^{\frac{1}{1-q}} t^{p - 1}  \Phi(x')]' 
+ \frac{S + \varepsilon}{ t \log^p t} \Phi(x) = 0 
$$
is non-oscillatory. This fact means that any solution 
$\check{\varphi}: \mathbb{R}_T \to \mathbb{R}$ of the equation
\begin{equation} \label{thg2} 
\begin{aligned}
 \check{\varphi}' (t)
&=\frac{1}{t \log t}\Big[(R+\varepsilon) |\cos_p \check{\varphi} (t)|^p -
\Phi(\cos_p \check{\varphi} (t))\sin_p \check{\varphi} (t) \\
& \quad + (S + \varepsilon)
\frac{|\sin_p \check{\varphi} (t)|^p}{p-1}\Big]
\end{aligned}
\end{equation}
has the property that $\limsup_{t \to \infty} \check{\varphi} (t) < \infty$.
Finally, considering \eqref{hg2} together with \eqref{thg2} and considering
 the $2\pi_p$-periodicity of the generalized trigonometric functions, we have
the inequality $\limsup_{t \to \infty} \psi (t) < \infty$. Therefore,
 the statement of the theorem is proven. 
\end{proof}

Now we mention definitions which enable us to  formulate the below given 
Corollaries \ref{c1} and \ref{c2} (motivated by the results of
 \cite{DV-ADE2013,DV-2014} recalled in Introduction).

\begin{definition} \label{d1} \rm
 Let $f: \mathbb R_{\rm e} \to \mathbb R $ be a continuous function.
The mean value $M(f)$ of  $f$ is defined as
$$ 
M(f) := \lim_{t \to \infty} \frac{1}{t} \int_{a}^{a + t} f (\tau ) \,\mathrm{d}\tau
$$
if the limit is finite and if it exists uniformly with respect to 
$a \in \mathbb{R}_{\rm e}$.
\end{definition}

\begin{definition} \label{def2} \rm
A {continuous} function $f: \mathbb R \to \mathbb R $
is called {almost periodic} if, for all $\varepsilon > 0$,
there exists $l(\varepsilon)> 0$ such that any interval of length 
$l(\varepsilon)$ of the real line contains at least one point $s$ for which
$$ 
| f ({t + s}) - f (t) | < \varepsilon, \quad t \in \mathbb{R}. 
$$
\end{definition}

As a direct generalization of the almost periodicity, we consider the notion 
of the so-called asymptotic almost periodicity.

\begin{definition} \label{d2v3} \rm
We say that a  {continuous} function $f: \mathbb R_{\rm e} \to \mathbb R $
is {asymptotically almost periodic} if $f$ can be expressed in the form
$ f (t) = f_1 (t) + f_2 (t)$, $t \in \mathbb R_{\rm e}$, where $f_1$ 
is almost periodic and $f_2$ has the property that 
$\lim_{t \to \infty} f_2 (t) = 0$. 
\end{definition}

Concerning coefficients with mean values, we obtain a new result which reads 
as follows.

\begin{corollary} \label{c1}
 Let continuous functions $r: \mathbb{R}_{\rm e} \to \mathbb{R}^+$ and 
$s: \mathbb{R}_{\rm e} \to \mathbb{R}$ be such that the mean values 
$M(r^{1-q}) \in \mathbb{R}^+$, $M(s) \in \mathbb{R}$ exist and let 
\eqref{opr2} be valid for some $\alpha \in \mathbb{R}^+$. 
Let $[M(r^{1-q})]^{p-1} M(s) \ne q^{-p}$. Then, \eqref{start0} is oscillatory 
if and only if $[M(r^{1-q})]^{p-1} M(s) > q^{-p}$.
\end{corollary}

\begin{proof} 
The corollary follows from Theorems \ref{TH} and \ref{TH2}. 
Let $\varepsilon > 0$ be arbitrary. The existence of $M(r^{1-q})$ and $M(s)$ 
implies the existence of $n \in \mathbb{N}$ such that
$$
\big|  \frac{1}{n\alpha} \int_t^{t + n\alpha} r^{1-q} (\tau) \,\mathrm{d}\tau
- M(r^{1-q})\big| < \varepsilon, \quad 
\big|  \frac{1}{n\alpha} \int_t^{t + n\alpha} s (\tau) \,\mathrm{d}\tau - M(s)\big|
< \varepsilon 
$$
for all $t \in \mathbb{R}_{\rm e}$. If $[M(r^{1-q})]^{p-1} M(s) > q^{-p}$, 
then it suffices to choose $\varepsilon$ so that 
$$
M(r^{1-q}) - \varepsilon > 0, \quad [M(r^{1-q}) - \varepsilon ]^{p-1}
 [ M(s) - \varepsilon ] > q^{-p}, 
$$ 
to put $R = M(r^{1-q}) - \varepsilon$ and $S = M(s) - \varepsilon$,  and to 
replace $\alpha$ by $n \alpha$
in Theorem \ref{TH}. Obviously, \eqref{opr2} is true also for $n \alpha$. 
If $[M(r^{1-q})]^{p-1} M(s) < q^{-p}$,
 then we choose $\varepsilon$ so that 
$$
M(s) + \varepsilon > 0, \quad [M(r^{1-q}) + \varepsilon ]^{p-1} 
[ M(s) + \varepsilon ] < q^{-p},
$$ 
we consider $R = M(r^{1-q}) + \varepsilon$ and 
$S = M(s) + \varepsilon$, and we replace $\alpha$ by $n \alpha$
in Theorem \ref{TH2}.
\end{proof}

\begin{remark} \rm
We point out that the requirement about the validity of \eqref{opr2} for 
some $\alpha \in \mathbb{R}^+$ cannot be omitted in the statement of 
Corollary \ref{c1}. Indeed, the existence of $M(s)$ does not imply \eqref{opr2}. 
We remark that, from the same reason, Theorem \ref{intro-TH1} does not 
follow from Theorem \ref{intro-TH2}.  
\end{remark}

For asymptotically almost periodic coefficients, we get a new result as well. 
Again, we formulate it explicitly.

\begin{corollary} \label{c2} 
Let functions $r^{1-q}: \mathbb{R}_{\rm e} \to \mathbb{R}^+$ and 
$s: \mathbb{R}_{\rm e} \to \mathbb{R}$ be asymptotically almost periodic 
and let $[M(r^{1-q})]^{p-1} M(s) \in \mathbb{R}^+ \setminus \{ q^{-p}\}$. 
Then,  \eqref{start0} is oscillatory if and only if 
$[M(r^{1-q})]^{p-1} M(s) > q^{-p}$.
\end{corollary}

\begin{proof} 
Since any asymptotically almost periodic function has mean value and it is
 bounded (see, e.g., \cite{HasVes-711c9,HasVes-798fi}), this corollary is a 
consequence of Corollary \ref{c1}.
\end{proof}

\begin{remark} \rm
Let us pay our attention to  \eqref{hl2}. We repeat that the result 
about  \eqref{hl2}, which corresponds to Corollary~\ref{c1}, is proven 
in \cite{DV-2014} and that the one, which corresponds to Corollary~\ref{c2}, 
is proven in \cite{DV-ADE2013}.  
\end{remark}

\begin{remark} \rm
In Corollaries \ref{c1} and \ref{c2}, the
case $[M(r^{1-q})]^{p-1} M(s) = q^{-p}$
cannot be solved as oscillatory or non-oscillatory
for general coefficients (which have mean values or which are asymptotically 
almost periodic). We conjecture that this case is not possible to solve 
even for general almost periodic coefficients. Our conjecture is based 
on constructions of almost periodic functions mentioned in \cite{ves000p} 
(see also \cite{vestgyuk,ves000pp}).
\end{remark}

At the end, we give some examples to illustrate the proven results.

\begin{example}  \rm
Let us consider constants $u> 0$, $v\in\mathbb{R}$, and $w\ne 0$ and 
the function $h : \mathbb{R}_1 \to \mathbb{R}$ given by the formula
\begin{equation*}
h (t) := \begin{cases}
v + nw 2^n(t - n), & t \in [n, n + \frac{1}{2^{n}}), \, n \in \mathbb{N}; \\
v + nw 2^n(n + \frac{2}{2^{n}} - t ),
 &  t \in [n + \frac{1}{2^{n}}, n + \frac{2}{2^{n}}), \, n \in \mathbb{N};\\
v, & t \in [n + \frac{2}{2^{n}}, n + 1), \, n \in \mathbb{N}.
\end{cases}
\end{equation*}
We analyze the equation
\begin{equation} \label{eqex01}
\Big[\frac{( tx')^3}{u}\Big]' + \frac{h(t)}{t \log^4 t} \, x^3 = 0.
\end{equation}
Hence, we deal with \eqref{start0}, where
$p=4$ and
$$
r(t) = \frac{1}{u}, \quad s(t)= h(t), \quad t \in \mathbb{R}_{\rm e}.
$$
One can verify that  
$$ 
M(h) = v, \quad M(r^{1-q}) = M(r^{-\frac{1}{3}}) 
= M(u^{\frac{1}{3}}) = u^{ \frac{1}{3}} > 0.
$$
Therefore, applying Corollary~\ref{c1} (condition \eqref{opr2} is trivially 
valid for all $\alpha \in \mathbb{R}^+$), we know that  \eqref{eqex01} 
is oscillatory when $4^4 u v > 3^4$
and non-oscillatory when $4^4 u v < 3^4$. Note that the second coefficient 
$h$ has mean value, but it is not asymptotically
almost periodic (it suffices to consider that 
$\limsup_{t\to\infty}| h(t)|=\infty$).
\end{example}


\begin{example} \rm
For $a, b, c > 0$, we consider the equation
\begin{equation} \label{eqex02} 
\Big[\big(\frac{t^2 x'}{1 + t [1+\sin t \cos t ]}\big)^5\Big]' \\
+ \frac{|\sin(at)|+\arctan[\sin(b t) + \cos (b t)]}{c (t + \sqrt{t}\,){\log^6t}}
 x^5 = 0
\end{equation}
which is in the form of  \eqref{start0} with
\begin{gather*}
 r(t) = \Big(\frac{t}{1 + t [1 +\sin t \cos t ]}\Big)^5, \quad t \in \mathbb{R}_{\rm e},\\ 
s(t) = \frac{t (|\sin(at)|+\arctan[\sin(b t) + \cos (b t)] )}{c (t + \sqrt{t}\,) },
 \quad t \in \mathbb{R}_{\rm e}, 
\end{gather*}
and $p=6$.
Since $r^{1-q}$ and $s$ are asymptotically almost periodic functions 
(see, e.g., \cite{HasVes-711c9,HasVes-798fi}),
we can use directly Corollary \ref{c2}.
We have
\begin{gather*}
M(r^{1-q}) = M(r^{-1/5}) 
= M\Big(\frac{1}{t}+ 1 +\sin t \cos t\Big) = 1, \\
c M(s) = M (|\sin(at)|+\arctan[\sin(b t) + \cos (b t)] ) 
= M (|\sin(at)| ) = \frac{2}{\pi}.
\end{gather*}
Therefore, \eqref{eqex02} is oscillatory for 
$$
c < \overline{\Gamma} := \Big(\frac{6}{5}\Big)^6 {\frac{2}{\pi}}
$$
and non-oscillatory for $c > \overline{\Gamma}$.
\end{example}

\begin{example} \rm
Let $p = 3/2$ and let
 $f, g : [-1, 1]\to \mathbb{R}^+$ be continuous functions. 
We find the oscillation constant for the equation
\begin{equation} \label{eqex03}
 [\sqrt{t f(\sin t)} \, \Phi(x')]'
  + \frac{g(\sin t)}{t\log^p t }\, \Phi(x) = 0 .
\end{equation}
Evidently, the functions $r(t) = \sqrt{f(\sin t)}$ and
$s(t) = g(\sin t)$ are periodic and
$$
M(r^{-2}) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{\,\mathrm{d}t}{f(\sin  t )}, \quad
M(s) = \frac{1}{2\pi} \int_{-\pi}^{\pi} g(\sin t) \,\mathrm{d}t.
$$
We can apply, e.g., Corollary \ref{c2}. If
$$
\widehat{\Gamma} := \frac{1}{2\pi} \int_{-\pi}^{\pi} g(\sin t) \,\mathrm{d}t
\sqrt{\frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{\,\mathrm{d}t}{f(\sin  t )}}
> 3^{-3/2},
$$
then  \eqref{eqex03} is oscillatory. If $\widehat{\Gamma}
< 3^{-3/2}$, then
 \eqref{eqex03} is non-oscillatory. \end{example}


\subsection*{Acknowledgements}
The first author is supported by Grant P201/10/1032 of the Czech Science Foundation.
The second author is supported by the project ``Employment of Best Young Scientists
for International Cooperation Empowerment'' (CZ.1.07/2.3.00/30.0037)
co-financed from European Social Fund and the state budget of the Czech Republic.

\begin{thebibliography}{99}

\bibitem{HasVes-agarwal2002} R. P. Agarwal, A. R. Grace, D. O'Regan;
\emph{Oscillation theory for second order linear, half-linear,
    superlinear and sublinear dynamic equations}.
Kluwer Academic Publishers, Dordrecht, 2002.

\bibitem{HasVes-711c9} C. Corduneanu;
\emph{Almost periodic oscillations and waves}.
Springer, New York, 2009.

\bibitem{dosly-funk} O. Do\v{s}l\'{y}, H. Funkov\'{a};
\emph{Euler type half-linear differential equation with
periodic coefficients}.
 Abstract Appl. Anal. \textbf{2013} (2013), art. ID 714263, pp. 1--6.

\bibitem{DV-DoslyHasil2011} O. Do\v{s}l\'y, P. Hasil;
\emph{Critical oscillation constant for half-linear differential equations with
periodic coefficients}.
Ann. Mat. Pura Appl. {\bf 190} (2011), no. 3, pp. 395--408.

\bibitem{DV-DoslyRehak2005}
O. Do\v{s}l\'y, P. \v{R}eh\'ak;
\emph{Half-linear differential equations}.
Elsevier, Amsterdam, 2005.

\bibitem{E} \'A. Elbert;
\emph{Asymptotic behaviour of autonomous half-linear differential systems
on the plane}.
Studia Sci. Math. Hungar. {\bf 19} (1984), no. 2-4, pp. 447--464.

\bibitem{DV-Elbert1984} \'{A}. Elbert;
\emph{Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations}.  Ordinary and partial differential equations (Dundee, 1982), pp. 187--212,
Lecture Notes in Math., vol. 964, Springer, Berlin, 1982.

\bibitem{E-Sch}
\'A. Elbert, A. Schneider;
\emph{Perturbations of half-linear Euler differential equation.}
Results Math.
{\bf 37} (2000), no. 1-2, pp. 56--83.

\bibitem{HasVes-798fi} A. M. Fink;
\emph{Almost periodic differential equations}.
Springer-Verlag, Berlin, 1974.

\bibitem{DV-gesztesy}
F. Gesztesy, M. \"{U}nal;
\emph{Perturbative oscillation criteria and Hardy-type inequalities}.
Math. Nachr. \textbf{189} (1998), pp. 121--144.

\bibitem{hasil2008} P. Hasil;
\emph{Conditional oscillation of half-linear differential
equations with periodic coefficients}.
Arch. Math. {\bf 44} (2008), no. 2, pp. 119--131.

\bibitem{DV-AAA2012} P. Hasil, M. Vesel\'{y};
\emph{Critical oscillation constant for difference equations
with almost periodic coefficients}.
Abstract Appl. Anal. \textbf{2012} (2012), art. ID 471435, pp. 1--19.

\bibitem{DV-ADE2013} P. Hasil, M. Vesel\'{y};
\emph{Oscillation of half-linear differential equations with asymptotically
almost periodic coefficients}.
Adv. Differ. Equ. \textbf{2013} (2013), no. 122, pp. 1--15.

\bibitem{DV-hasvit} P. Hasil, J. V\'{i}tovec;
\emph{Conditional oscillation of half-linear Euler-type
dynamic equations on time scales}. Electron. J. Qual. Theory Differ. Equ. \textbf{2015} (2015), no.~6, pp. 1--24.

\bibitem{DV-JJMV} J. Jaro\v{s}, M. Vesel\'{y};
\emph{Conditional oscillation of Euler type half-linear differential equations 
with unbounded coefficients}.
Submitted to Studia Sci. Math. Hungar.

\bibitem{DV-Kruger2} H. Kr\"{u}ger, G. Teschl;
\emph{Effective Pr\"{u}fer angles and relative oscillation criteria}.
J.~Differ. Equ. \textbf{245} (2008), no. 12, pp. 3823--3848.

\bibitem{DV-Kruger3} H. Kr\"{u}ger, G. Teschl;
 \emph{Relative oscillation theory for Sturm--Liouville operators extended}.
J. Funct. Anal. \textbf{254} (2008), no. 6, pp. 1702--1720.

\bibitem{DV-Kruger4} H. Kr\"{u}ger, G. Teschl;
\emph{Relative oscillation theory, weighted zeros of the Wronskian, and the spectral shift function}.
Commun Math. Phys. \textbf{287} (2009), no. 2, pp. 613--640.

\bibitem{DV-Schmidt2000} K. M. Schmidt;
\emph{Critical coupling constant and eigenvalue asymptotics of perturbed
periodic Sturm-Liouville operators}.
Commun Math. Phys. {\bf 211} (2000), pp. 465--485.

\bibitem{schmidt-1} K. M. Schmidt;
\emph{Oscillation of perturbed Hill equation and lower spectrum of radially
periodic Schr\"odinger operators in the plane}.
Proc. Amer. Math. Soc. {\bf 127} (1999), pp. 2367--2374.

\bibitem{vestgyuk} M. Vesel\'{y};
\emph{Almost periodic homogeneous linear difference systems without almost periodic solutions}.
J. Differ. Equ. Appl. \textbf{18} (2012), no. 10, pp. 1623--1647.

\bibitem{ves000p} M. Vesel\'{y};
\emph{Construction of almost periodic functions with given properties}.
Electron. J. Differ. Equ. \textbf{2011} (2011), no. 29, pp. 1--25.

\bibitem{ves000pp} M. Vesel\'{y};
\emph{Construction of almost periodic sequences with given properties}.
Electron. J. Differ. Equ. \textbf{2008} (2008), no. 126, pp. 1--22.

\bibitem{DV-AAA2013-14} M. Vesel\'{y}, P. Hasil;
\emph{Conditional oscillation of Riemann--Weber
half-linear differential equations with asymptotically
 almost periodic coefficients}. Studia Sci. Math. Hungar. 
\textbf{51} (2014), no. 3, pp. 303--321.

\bibitem{HasVes-2015} M. Vesel\'{y}, P. Hasil;
\emph{Non-oscillation of half-linear differential equations
with periodic coefficients}.
Electron. J. Qual. Theory Differ. Equ. \textbf{2015} (2015), no.~1, pp. 1--21.

\bibitem{DV-AAA2013} M. Vesel\'{y}, P. Hasil;
\emph{Oscillation and non-oscillation of asymptotically almost periodic
half-linear difference equations}.
Abstract Appl. Anal. \textbf{2013} (2013), art. ID 432936, pp. 1--12.

\bibitem{DV-2014} M. Vesel\'{y}, P. Hasil, R. Ma\v{r}\'ik;
\emph{Conditional oscillation of half-linear differential equations 
with coefficients having mean values}.
 Abstract Appl. Anal. \textbf{2014} (2014), art. ID 258159, pp. 1--14.

\bibitem{vitovec} J. V\'{i}tovec;
\emph{Critical oscillation constant for Euler-type dynamic equations on time scales}. 
Appl. Math. Comput. \textbf{243} (2014), pp. 838--848.

\end{thebibliography}

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