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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 120, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/120\hfil Non-oscillation of periodic half-linear equations]
{Non-oscillation of periodic half-linear equations in the critical case}

\author[P. Hasil, M. Vesel\'y \hfil EJDE-2016/120\hfilneg]
{Petr Hasil, Michal Vesel\'y}

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

\thanks{Submitted September 3, 2015. Published May 13 2016.}
\subjclass[2010]{34C10, 34C15}
\keywords{Half-linear equations; Pr\"{u}fer angle; oscillation theory;
\hfill\break\indent conditional oscillation; oscillation constant}

\begin{abstract}
 Recently, it was shown that the Euler type half-linear differential equations
 \[
 [ r (t)  t^{p-1}\Phi(x')]' + \frac{s (t)}{ t \log^p t} \Phi(x) = 0
 \]
 with periodic coefficients $r, s$ are conditionally oscillatory and the
 critical oscillation constant was found.
 Nevertheless, the critical case remains unsolved.
 The objective of this article is to study the critical case. Thus, we consider
 the critical value of the coefficients and we prove that any considered
 equation is non-oscillatory. Moreover, we analyze the situation when the
 periods of coefficients $r, s$ do not need to coincide.
\end{abstract}

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

\section{Introduction}

In this article, we study the oscillation behaviour of the equation
\begin{equation} \label{introeq}
[ r^{-p/q}(t)  t^{p-1}\Phi(x')]' + \frac{s (t)}{ t \log^p t} \Phi(x) = 0,
\quad \Phi(x)=|x|^{p-1}\operatorname{sgn} x,
\end{equation}
where $p>1$, $\log$ is the natural logarithm, $r > 0$ and $s$ are continuous
functions, and $q$ is the number conjugated with $p$, i.e., $q = p/(p-1)$.
The main motivation of the presented research comes from \cite{naseEJDE},
where the equation
\begin{equation} \label{introeq2}
[ r(t)  t^{p-1}\Phi(x')]' + \frac{s (t)}{ t \log^p t} \Phi(x) = 0
\end{equation}
is proved to be conditionally oscillatory.
It means that there exists the so-called critical oscillation constant,
which is a positive value given by coefficients $r$ and $s$ with the
following property:
\begin{itemize}
\item[(1)] If the coefficients indicate a value greater than the critical one,
then  \eqref{introeq2} \textit{is oscillatory};
\item[(2)] If the coefficients indicate a value less than the critical one,
then  \eqref{introeq2} \textit{is non-oscillatory}.
\end{itemize}
We point out that for the equations studied here, 
all solutions are oscillatory if and only if 
a non-trivial solutions is oscillatory.

Note that, in \cite{naseEJDE}, Equation \eqref{introeq2} is considered
without the power $-p/q$ in the first term. Nevertheless, since  function
 $r$ is positive, it does not have any impact.
We consider \eqref{introeq2} in the presented form only because of technical reasons,
i.e., the technical parts of our processes are more transparent.
The described result from \cite{naseEJDE} rewritten for \eqref{introeq} is
explicitly mentioned in Theorem~\ref{s1} below.

Since the case when the coefficients indicate exactly the critical
value is open, the aim of this article is to fill this gap. We will consider
\eqref{introeq} with periodic continuous coefficients.
We will not require any common period for the coefficients $r$ and $s$.

Now, let us give a short overview of the literature.
The fundamental theory concerning half-linear
differential equations can be found in books
\cite{HasVes-agarwal2002,DV-DoslyRehak2005}.
As basic papers about half-linear equations, we refer to \cite{E,DV-Elbert1984}.
For the analyzed conditional oscillation
of half-linear differential equations, we mention, e.g.,
papers \cite{DV-DoslyHasil2011,hasil2008,HasVes-2015ADE,HasVes-2015,DV-2014}
and the paper \cite{naseEJDE} which we have already mentioned as the primary
motivation.
The corresponding results dealing with difference equations
and with dynamic equations on time scales are also present in the literature,
but they are still behind the continuous case.
See \cite{DV-AAA2013,HasVes-2015ADE2} for the discrete equations
and \cite{DV-hasvit} for the dynamic equations on time scales.
In the linear case, there are many relevant results. We mention at least
the most relevant papers \cite{DV-gesztesy,DV-AAA2012,schmidt-1,DV-vitoce}.

This article is organized as follows. In the next section, we give
only necessary preliminaries including the half-linear trigonometric functions
and the equation for the Pr\"{u}fer angle, which will allow us to investigate
the (non-)oscillation of \eqref{introeq}.
In Section \ref{Sec3}, we prove auxiliary results and we mention the later
used known results.
Finally, in Section \ref{Sec4}, we formulate, prove, and illustrate by examples
the main result. To the best of our knowledge, the presented result
 is new in the linear case as well (see Corollary \ref{vn2} below).

\section{Preliminaries}

In this section, we describe the equation for the modified
half-linear Pr\"{u}fer angle given by the studied type of equations.
At first, we briefly recall the notion of half-linear
trigonometric functions.

The half-linear sine function denoted by $\sin_p$ is introduced
as the odd $2\pi_p$-periodic extension of the solution of the initial
problem
\begin{equation*}
[\Phi (x')]'+(p-1)\Phi(x)=0, \quad x(0)=0, \quad x'(0)=1
\end{equation*}
on $[0, \pi_p]$, where
\begin{equation*}
\pi_p:=\frac{2\pi}{p\sin(\pi/p)}.
\end{equation*}
We denote the derivative of the half-linear sine function as $\cos_p$
and we call it the half-linear cosine function. It holds
\begin{equation} \label{was}
|\cos_p a| 	\leq 1, \quad
|\sin_p a| 	\leq 1, \quad a \in \mathbb{R}.
\end{equation}
For more details about $\sin_p$ and $\cos_p$, we refer to
\cite[Section 1.1.2]{DV-DoslyRehak2005}.

Now, let us turn our attention to the half-linear equation
\begin{equation} \label{hl}
[ r^{-p/q}(t) \, t^{p-1}\Phi(x')]' + \frac{s (t)}{ t \log^p t} \,\Phi(x) = 0
 \end{equation}
and the corresponding equation for the Pr\"ufer angle
\begin{equation} \label{ysd}
\varphi'(t)
= \frac{1}{t\log t} \Big[r(t)|\cos_p\varphi (t)|^p
-\Phi\left(\cos_p\varphi(t)\right)\sin_p\varphi(t)
+ s(t) \, \frac{|{\sin_p\varphi (t)}|^{p}}{p-1} \Big],
\end{equation}
where $r: \mathbb{R} \to \mathbb{R}$ is a continuous, positive,
 and $\alpha$-periodic function and $s: \mathbb{R} \to \mathbb{R}$
is a continuous and $\beta$-periodic function.

We use the Riccati type transformation
\begin{equation*}
w(t)=r^{-p/q}(t)  t^{p-1}\Phi\Big(\frac{x'(t)}{x(t)}\Big)
\end{equation*}
to \eqref{hl}. This leads to the equation
\begin{equation} \label{ok}
 w'(t) +\frac{s (t)}{ t \log^p t}+(p-1)
[r^{-p/q}(t) \, t^{p-1}]^{\frac{1}{1-p}}
	|w(t)|^{\frac{p}{p-1}}=0.
\end{equation}
Then, using the substitution
\begin{equation*}
v(t)=(\log t)^{\frac{p}{q}}w(t), \quad t \in ({\mathrm{e}}, \infty),
\end{equation*}
in \eqref{ok} and taking into account the
modified Pr\"ufer transformation
\begin{equation*}
 x(t)=\rho(t)\sin_p \varphi(t),\quad
	[r^{-p/q}(t) \, t^{p-1}]^{q-1} x'(t)=
	\frac{\rho(t)}{\log t}\cos_p\varphi(t),
 \end{equation*}
we easily obtain \eqref{ysd}. The more comprehensive description
of the derivation of \eqref{ysd}
is given in our previous paper \cite{naseEJDE}.

Further, let us mention the definition of the mean value of an arbitrary
periodic function which is essential for our results.

\begin{definition} \label{def1} \rm
The mean value $M(f)$ of a periodic function $f: \mathbb{R} \to \mathbb{R}$
with period $P > 0$ is defined as
$$
M(f) := \frac{1}{P} \int_{0}^P f (\tau) \,d  \tau.
$$
\end{definition}

Finally, for the upcoming use, we put
\begin{equation} \label{war}
 \tilde{r} := \sup \{r (t): t> \mathrm{e}\}, \quad
\tilde{s} := \sup  \{|s (t)|: t> \mathrm{e}\}
 \end{equation}
and we denote $2\varrho := \min \,\{p-1, 1\}$.


\section{Auxiliary results} \label{Sec3}

Let $\vartheta > 0 $ be arbitrary. We define
	\begin{equation}                    \label{th}
\psi (t) := \frac{1}{\sqrt{t}}\int_t^{t + \sqrt{t}} \varphi (\tau)\, \,d  \tau,
\quad t \ge {\mathrm{e}} + \vartheta, 	
\end{equation}
where $\varphi$ is a solution of \eqref{ysd}
on $[ {\mathrm{e}} + \vartheta ,\infty) $. Now, we formulate and prove auxiliary
results concerning this function $\psi$.

\begin{lemma} \label{L1}
	If $\varphi$ is a solution of \eqref{ysd} on $[{\mathrm{e}} + \vartheta, \infty)$, then
the function $\psi: [{\mathrm{e}} + \vartheta, \infty) \to \mathbb{R}$ defined by \eqref{th}
	satisfies
	\begin{equation} \label{jsusa}
|\varphi(\tau)-\psi (t)| \leq \frac{C}{\sqrt{t} \log t}, \quad
t \geq {\mathrm{e}} + \vartheta,\; \tau \in [t, t + \sqrt{t}],
	\end{equation}
	for some constant $C > 0$.
\end{lemma}

\begin{proof}
The continuity of  $\varphi$ implies that, for any $t \geq {\mathrm{e}} + \vartheta$,
there exists $\tilde{t} \in [t,t+\sqrt{t}]$ such that
$\psi(t)=\varphi(\tilde{t})$.
Hence, for all $t \geq {\mathrm{e}} + \vartheta$, $\tau \in [t,t + \sqrt{t}]$,
we obtain
\begin{align*}
|\varphi(\tau)-  \psi(t)|
& =|\varphi(\tau)-\varphi (\tilde{t})|\\
&\leq \int_t^{t+\sqrt{t}}|\varphi'(\tau)|\,\,d  \tau \\
&\leq \frac{1}{t \log t} \Big[  \int_t^{t+\sqrt{t}} r(\tau) |\cos_p\varphi(\tau)|^p
	+  |\Phi(\cos_p\varphi (\tau))\sin_p\varphi (\tau) | \, \,d  \tau \\
&\quad + \int_t^{t+\sqrt{t}} \frac{|\sin_p\varphi(\tau)|^p}{p-1}
 |s(\tau)| \, \,d  \tau \Big],
\end{align*}	
i.e., we obtain (see \eqref{was}, \eqref{war})	
\[
|\varphi(\tau)-  \psi(t)|
\leq \frac{1}{t \log t} \int_t^{t+\sqrt{t}}	\big( \tilde{r} + 1
	+ \frac{\tilde{s}}{p-1} \big) \,d  \tau
\leq \frac{C}{\sqrt{t} \log t},
\]
where
\begin{equation} \label{dpos}
C := \tilde{r} + 1 	+ \frac{\tilde{s}}{p-1}.
\end{equation}
\end{proof}

\begin{lemma} \label{l2}
The inequality
\begin{align*}
&\Big| \psi' (t) - \frac{1}{t \log t} \Big[\frac{|\cos_p \psi (t)|^p}{\sqrt{t}}
\int_t^{t+\sqrt{t}} r (\tau)\,d \tau  \\
& - \Phi(\cos_p \psi (t))\sin_p \psi (t)
+ \frac{|\sin_p \psi (t)|^p}{(p-1) \sqrt{t}}
 \int_t^{t+ \sqrt{t}} s(\tau) \,d  \tau  \Big] \Big| \\
&< \frac{D}{t^{1 + \varrho} \log t}
\end{align*}
holds for some $D > 0$ and for all $t > {\mathrm{e}} + \vartheta$.
\end{lemma}

\begin{proof}
For all $t > {\mathrm{e}} + \vartheta$, we have
\begin{align*}
	\psi'(t)
&= \big(1 + \frac{1}{2 \sqrt{t}}\big)
\frac{\varphi(t +\sqrt{t})}{\sqrt{t}} - \frac{\varphi(t)}{\sqrt{t}}
- \frac{1}{2 \sqrt{t^3}} \int_t^{t + \sqrt{t}} \varphi (\tau) \,d  \tau \\
&=  \frac{1}{ \sqrt{t}} \int_{t}^{t+\sqrt{t}} {\varphi' (\tau)} \,d  \tau
+ \frac{1}{2 t}   \varphi(t +\sqrt{t}) - \frac{1}{2 \sqrt{t^3}}
 \int_t^{t + \sqrt{t}} \varphi (\tau) \,d  \tau	 \\
&=  \frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}} \frac{1}{\tau\log \tau}
\Big[r(\tau)|\cos_p\varphi (\tau)|^p
		-\Phi\big(\cos_p\varphi(\tau)\big)\sin_p\varphi(\tau)  \\
&\quad + s(\tau)  \frac{|{\sin_p\varphi (\tau)}|^{p}}{p-1} \Big] \,d  \tau
+    \frac{1}{2 \sqrt{t^3}} \int_t^{t + \sqrt{t}}
\big[\varphi(t +\sqrt{t}) - \varphi (\tau) \big] \,d  \tau.
\end{align*}
Since (see also \eqref{was}, \eqref{war}, and \eqref{dpos})
\begin{align*}
 &\Big| \frac{1}{2 \sqrt{t^3}} \int_t^{t + \sqrt{t}}
[\varphi(t +\sqrt{t}) - \varphi (\tau) ] \,d  \tau \Big| \\
&\le  \frac{1}{2 \sqrt{t^3}} \int_t^{t + \sqrt{t}}
 \int_\tau^{t + \sqrt{t}} | \varphi'(\sigma) | \,d  \sigma \,d  \tau \\
&\le \frac{1}{2 \sqrt{t^3}}  \int_t^{t + \sqrt{t}}
\int_\tau^{t + \sqrt{t}} \frac{1}{ \sigma \log \sigma}
\big|r(\sigma)|\cos_p\varphi (\sigma)|^p
		-\Phi\left(\cos_p\varphi(\sigma)\right)\sin_p\varphi(\sigma)  \\
&\quad + s(\sigma) \, \frac{|{\sin_p\varphi (\sigma)}|^{p}}{p-1} \Big|
\,d  \sigma \,d  \tau \\
&\le \frac{1}{2 \sqrt{t^5} \log t}
\int_t^{t + \sqrt{t}} \int_t^{t + \sqrt{t}} [\tilde{r}
		+ 1 + \frac{\tilde{s}}{p-1} ] \,d  \sigma \,d  \tau \\
&\le \frac{C}{2 \sqrt{t^3} \log t}  ,
		\end{align*}
it suffices to consider
\[
\frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}} \frac{1}{\tau\log \tau}
\Big[r(\tau)|\cos_p\varphi (\tau)|^p
		-\Phi\big(\cos_p\varphi(\tau)\big)\sin_p\varphi(\tau)
+ s(\tau) \, \frac{|{\sin_p\varphi (\tau)}|^{p}}{p-1} \Big] \,d  \tau .
\]
In fact, we will consider
\begin{equation} \label{yf}
\begin{aligned}
 &\frac{1}{ \sqrt{t^3} \log t} \int_t^{t + \sqrt{t}}
\Big[r(\tau)|\cos_p\varphi (\tau)|^p
		-\Phi\left(\cos_p\varphi(\tau)\right)\sin_p\varphi(\tau)  \\
&\quad + s(\tau)  \frac{|{\sin_p\varphi (\tau)}|^{p}}{p-1} \Big] \,d  \tau,
\end{aligned}
\end{equation}
because
\begin{align*}
 &\Big|\int_{t}^{t+\sqrt{t}} \frac{1}{\tau \log \tau}
		[ r(\tau) |\cos_p\varphi (\tau)|^p-
\Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau)]\,d  \tau\\
& +\int_{t}^{t+\sqrt{t}} \frac{1}{\tau \log \tau}
 \frac{|\sin_p\varphi(\tau)|^p}{p-1}\, s(\tau) \,d  \tau	\\
&  - \int_{t}^{t+\sqrt{t}} \frac{1}{t \log t}
		[ r(\tau) |\cos_p\varphi (\tau)|^p-
\Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau)]\,d  \tau \\
&  - \int_{t}^{t+\sqrt{t}} \frac{1}{t \log t}
 \frac{|\sin_p\varphi(\tau)|^p}{p-1} s(\tau) \,d  \tau \Big| \\
& \le  \int_t^{t + \sqrt{t}} [\tilde{r} + 1 + \frac{\tilde{s} }{p - 1} ]
[ \frac{1}{t \log t} - \frac{1}{\tau \log \tau}  ] \,d  \tau \\
 &\le C \sqrt{t}  \frac{(t + \sqrt{t}) \log (t + \sqrt{t})
- t \log t}{t (t + \sqrt{t})  \log (t + \sqrt{t}) \log t } \\
&\le \frac{K C}{t \log t}
\end{align*}
for all $ t \ge {\mathrm{e}} + \vartheta$, where $K > 0$ is such a constant that
$$
\frac{(t + \sqrt{t}) \log (t + \sqrt{t}) - t \log t}
{\log (t + \sqrt{t})} \le K \sqrt{t}, \quad t \ge {\mathrm{e}} + \vartheta.
$$
Considering the form of \eqref{yf}, to finish the proof,
it suffices to prove the following inequalities
\begin{gather}
\Big| \frac{|\cos_p\psi (t)|^p}{\sqrt{t}} \int_{t}^{t+\sqrt{t}}  r(\tau) \,d  \tau
- \frac{1}{\sqrt{t}} \int_{t}^{t+\sqrt{t}}  r(\tau) |\cos_p\varphi (\tau)|^p
\,d  \tau \Big|
\le \frac{E_1}{\sqrt{t} \log t}, \label{55}
\\
\begin{aligned}
&\Big| \frac{1}{\sqrt{t}}\int_t^{t + \sqrt{t}} \Phi(\cos_p\psi(t))\sin_p\psi (t) \,d  \tau
 - \frac{1}{\sqrt{t}}\int_t^{t + \sqrt{t}} \Phi(\cos_p\varphi(\tau))
\sin_p\varphi (\tau)  \,d  \tau \Big| \\
&\le \frac{E_2}{t^\varrho \log^{2\varrho} t},
\end{aligned}  \label{56}
\\
\Big| \frac{ |\sin_p\psi (t)|^p}{\sqrt{t}}
\int_{t}^{t+\sqrt{t}}  s(\tau)\,d  \tau
 - \frac{1}{\sqrt{t}} \int_{t}^{t+\sqrt{t}}  s(\tau) |\sin_p\varphi (\tau)|^p
\,d  \tau \Big|
\le \frac{E_3}{\sqrt{t} \log t} \label{57}
\end{gather}
for some constants $E_1, E_2, E_3 > 0$ and for all $t \ge {\mathrm{e}} + \vartheta$.

From \cite[pp. 4-5]{DV-DoslyRehak2005}, we know that
there exists $A > 0$ for which
\begin{gather}
	\big| |\cos_p a|^p-|\cos_p b|^p\big|
	\leq A |a - b |, \quad a, b \in \mathbb{R}, \label{8a}
\\
	\big| |\sin_p a|^p-|\sin_p b|^p\big|
	\leq A |a - b|, \quad a, b \in \mathbb{R},  \label{8c}
\\
	\big|\sin_p a - \sin_p b \big|
	\leq A |a - b|, \quad a, b \in \mathbb{R}. \label{8d}
\end{gather}
In addition, directly from the definition of $\Phi$ and $\cos_p$, it follows the
existence of $B > 0$ such that
\begin{equation} \label{nb} 	
|\Phi(\cos_p a) -\Phi(\cos_p b) |
	\leq  [{B} |a - b|]^{\min\{1, p-1\}}, \quad a, b \in \mathbb{R}.
	\end{equation}

At first, we consider inequality \eqref{55} which follows from
(see also \eqref{war}, \eqref{jsusa}, and \eqref{8a})
\begin{align*} %\label{ju}
&\Big|  \frac{1}{\sqrt{t}} \int_{t}^{t+\sqrt{t}} r(\tau)
\left( |\cos_p\psi (t)|^p - |\cos_p\varphi (\tau)|^p \right) \,d  \tau \Big| \\
&\le \frac{1}{\sqrt{t}} \int_{t}^{t+\sqrt{t}} r(\tau) A |\psi (t)
 - \varphi (\tau)| \,d  \tau \\
&\le \frac{\tilde{r} A C}{\sqrt{t} \log t}, \quad
t \ge {\mathrm{e}} + \vartheta.
\end{align*}
Similarly, we can obtain \eqref{57} from (see \eqref{war}, \eqref{jsusa}, and
\eqref{8c})
\begin{align*}
&\Big| \frac{1}{\sqrt{t}}  \int_{t}^{t+\sqrt{t}}  s(\tau)
\left(|\sin_p\psi (t)|^p- |\sin_p\varphi (\tau)|^p \right)\,d  \tau \Big| \\
&\le \frac{1}{\sqrt{t}} \int_{t}^{t+\sqrt{t}} |s(\tau)| A |\psi (t)
- \varphi (\tau)| \,d  \tau \\
&\le \frac{\tilde{s} A C }{\sqrt{t} \log t}, \quad t \ge {\mathrm{e}} + \vartheta.
\end{align*}
It remains to show \eqref{56}. We have (see \eqref{was})
\begin{align*}
&\Big|\frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}}
[\Phi(\cos_p\psi(t))\sin_p\psi (t)  -
  \Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau)]  \,d  \tau \Big| \\
& \le  \frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}}  \big|
\Phi(\cos_p\psi(t))\sin_p\psi (t)  -
\Phi(\cos_p\psi(t))\sin_p\varphi (\tau) \big| \,d  \tau \\
&\quad +  \frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}}
\big| \Phi(\cos_p\psi(t))\sin_p\varphi (\tau)  -
\Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau) \big| \,d  \tau
\\
& \le  \frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}}
| \sin_p\psi (t)  -  \sin_p\varphi (\tau) | \,d  \tau \\
&\quad +  \frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}}
|\Phi(\cos_p\psi(t))   - \Phi(\cos_p\varphi(\tau)) | \,d  \tau
\end{align*}
for all $t \ge {\mathrm{e}} + \vartheta$ and, using 
\eqref{jsusa}, \eqref{8d},
and \eqref{nb}, we obtain
\begin{align*}
&\Big|\frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}} [
\Phi(\cos_p\psi(t))\sin_p\psi (t)  -
 \Phi(\cos_p\varphi(\tau))\sin_p\varphi (\tau)  ]\,d  \tau \Big|
\\
& \le  \frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}}
 A | \psi (t)  -  \varphi (\tau) | \,d  \tau  \\
&\quad +  \frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}} 
 [  B | \psi(t)    -  \varphi(\tau) | ]^{\min \{1, p-1\}} \,d  \tau
\\
&\le \frac{1}{\sqrt{t}} \int_t^{t + \sqrt{t}}
 \frac{A C  }{\sqrt{t}  \log t} 
+\Big[\frac{BC}{\sqrt{t} \log t}\Big]^{\min \{1, p-1\}}\,d\tau \\
&\le \frac{A C  }{\sqrt{t}  \log t}
+ \big(\frac{BC }{\sqrt{t}  \log t} \big)^{\min \{1, p-1\}}
\end{align*}
for all $t \ge {\mathrm{e}} + \vartheta$, i.e., \eqref{56} is valid for
$$
E_2 := AC + [B C]^{\min \{1, p-1\}}.
 $$
The proof is complete.
\end{proof}

Now we recall a known result and we provide its direct consequence
which we will use in the proof of Theorem \ref{n1} in the next section.

\begin{theorem} \label{vp}
If $M, N > 0$ are such that  $M^{p-1} N  = q^{-p}$, then the equation
\begin{equation} \label{pt}
 \Big[\big(M + \frac{1}{t } \big)^{-p/q}\Phi(x')\Big]'
+ \frac{1 }{ t^p} \big(N + \frac{1}{t }\big) \Phi(x) = 0
\end{equation}
is non-oscillatory.
\end{theorem}

For a proof of the above theorem, see \cite{dosly-funk}.

\begin{corollary} \label{vn}
If $M, N > 0$ are such that  $M^{p-1} N  = q^{-p}$, then the equation
\begin{equation}\label{fv}
\Big[\big(M + \frac{1}{\log t} \big)^{-p/q} t^{p-1}\Phi(x')\Big]'
+ \frac{N + \frac{1}{\log t} }{ t \log^p t} \Phi(x) = 0
 \end{equation}
is non-oscillatory.
\end{corollary}

\begin{proof}
Let us consider \eqref{fv}, where $x=x(t)$ and $(\cdot)' = \frac{\,d }{\,d  t}$.
Using the transformation of the independent variable
$s = \log t$ when $x(t)=y(s)$,
we have
$$
\frac{1}{t}\frac{\,d }{\,d  s}
\Big[\big(M +\frac{1}{s}\big)^{-p/q} t^{p-1}
	\Phi\Big(\frac{1}{t}\frac{\,d  y}{\,d  s}\Big)\Big]'
	+\frac{1}{t s^p}\big(N+\frac{1}{
	s}\big)\Phi(y)=0.
$$
This equation can be easily simplified into the form
\begin{equation} \label{pt2}
\Big[\big(M+\frac{1}{s}\big)^{-p/q} \Phi(y')\Big]'
	+\frac{1}{s^p}\big(N+\frac{1}{s}\big)\Phi(y)=0.
\end{equation}
Hence (cf. \eqref{pt} and \eqref{pt2}), it suffices to apply
Theorem~\ref{vp}.
\end{proof}

\section{Results} \label{Sec4}

Applying Lemma \ref{l2} and Corollary \ref{vn}, we prove the following theorem.

\begin{theorem} \label{n1}
Let $\alpha, \beta > 0$. If
$r : \mathbb{R} \to \mathbb{R}$ is $\alpha$-periodic
and $s : \mathbb{R} \to \mathbb{R}$ is $\beta$-periodic
such that
\begin{equation} \label{pp}
 \Big[\frac{1}{\alpha} \int_0^\alpha r (\tau) \,d  \tau \Big]^{p-1}
\frac{1}{\beta} \int_0^\beta s (\tau) \,d  \tau
=  [M(r)]^{p-1} M(s) = q^{-p},
\end{equation}
then \eqref{hl} is non-oscillatory.
\end{theorem}

\begin{proof}
In this proof, we consider the equation for the Pr\"{u}fer angle
 $\varphi$ and the corresponding equation for $\psi$.
The used method is based on the fact that the non-oscillation of solutions of
\eqref{hl} is equivalent to the boundedness from above of a solution
$\varphi$ of \eqref{ysd}. We can refer to \cite{naseEJDE} or also to
the papers \cite{dosly-funk,DV-DoslyHasil2011,Kruger,DV-Schmidt2000,DV-AAA2013-14}.
In addition, Lemma \ref{L1} implies that a solution
$\varphi: [{\mathrm{e}} + \vartheta, \infty) \to \mathbb{R}$ of \eqref{ysd}
is bounded from above if and only if $\psi$ given by \eqref{th} is bounded
from above.

From Lemma \ref{l2}, we have
\begin{align*}
\psi' (t)
&< \frac{1}{t \log t} \Big[\frac{|\cos_p \psi (t)|^p}{\sqrt{t}}
\int_t^{t+ \sqrt{t}}  r (\tau)\,d \tau
 - \Phi\left(\cos_p \psi (t)\right) \sin_p \psi (t)  \\
&\quad + \frac{|\sin_p \psi (t)|^p}{(p-1) \sqrt{t}}
 \int_t^{t+ \sqrt{t}} s(\tau) \,d  \tau  + \frac{D}{t^{\varrho} } \Big]
\end{align*}
for all $t > {\mathrm{e}} + \vartheta$ and for some $D$.
Especially,
\begin{equation} \label{d1}
\begin{aligned}
\psi' (t)
& < \frac{1}{t \log t} \Big[\frac{|\cos_p \psi (t)|^p}{\sqrt{t}}
\int_t^{t+ \sqrt{t}}  r (\tau)\,d \tau
  - \Phi\left(\cos_p \psi (t)\right)\sin_p \psi (t)  \\
 & \quad +  \frac{|\sin_p \psi (t)|^p}{(p-1) \sqrt{t}}
  \int_t^{t+ \sqrt{t}} s(\tau) \,d  \tau  + \frac{D}{\log^{2} t} \Big]
\end{aligned}
\end{equation}
for all $t > {\mathrm{e}} + \vartheta$. Then, using the periodicity of coefficients
$r, s$, we obtain (see \eqref{war} and \eqref{d1})
\begin{equation} \label{d2}
\begin{aligned}
\psi' (t)
&< \frac{1}{t \log t} \Big[ {|\cos_p \psi (t)|^p}
\Big(M(r) + \frac{\tilde{r} \alpha}{\sqrt{t}} \Big)
 - \Phi\big(\cos_p \psi (t)\big)\sin_p \psi (t)  \\
&\quad+ \frac{|\sin_p \psi (t)|^p}{p-1}
\Big(M(s) + \frac{\tilde{s} \beta}{\sqrt{t}} \Big)  + \frac{D}{\log^{2} t} \Big]
\end{aligned}
\end{equation}
for all $t > {\mathrm{e}} + \vartheta$. Indeed, for any periodic continuous function
$f$ with period $P > 0$ and positive mean value $M(f)$, we have
\begin{align*}
 &\frac{1}{\sqrt{t}} \int_t^{t+ \sqrt{t}}  f (\tau)\,d \tau
 = \frac{1}{\sqrt{t}} \Big( \int_t^{t+ Pn}  f (\tau)\,d \tau
+ \int_{t+ Pn}^{t + \sqrt{t}}  f (\tau)\,d \tau \Big) \\
&\le   \frac{1}{Pn}\int_t^{t+ Pn}  f (\tau)\,d \tau
+ \frac{1}{\sqrt{t}} \int_{t+ Pn}^{t + P (n+1)}  |f (\tau)| \,d \tau
\le M(f) + \frac{\tilde{f} P}{\sqrt{t}},
 \end{align*}
where $\tilde{f} := \max \{|f (t)|: t \in [0, P]\} $  and
$n \in \mathbb{N} \cup \{0\}$ is such that $Pn \le \sqrt{t}$ and that
$P(n + 1) > \sqrt{t}$.

For $R := \max  \{1 , p-1\}$, the well-known Pythagorean identity
(see, e.g., \cite[Section 1.1.2]{DV-DoslyRehak2005}) gives
\begin{equation} \label{zz}
R \Big( |\cos_p a|^p  + \frac{|\sin_p a|^p}{p-1} \Big)
\ge 1, \quad a \in \mathbb{R}.
\end{equation}
Considering \eqref{d2} and \eqref{zz}, we have
\begin{align*} %\label{d3}
\psi' (t) &< \frac{1}{t \log t} \Big[ {|\cos_p \psi (t)|^p}
 \Big(M(r) + \frac{\tilde{r} \alpha}{\sqrt{t}}+  \frac{RD}{\log^{2} t} \Big)  \\
& \quad  - \Phi\left(\cos_p \psi (t)\right)\sin_p \psi (t)
 + \frac{|\sin_p \psi (t)|^p}{p-1}
\Big(M(s) + \frac{\tilde{s} \beta}{\sqrt{t}} +  \frac{RD}{\log^{2} t} \Big) \Big]
\end{align*}
for all $t > {\mathrm{e}} + \vartheta$ and, consequently, we have
\begin{equation} \label{d4}
\begin{aligned}
\psi' (t)
&< \frac{1}{t \log t} \Big[ {|\cos_p \psi (t)|^p}
\Big(M(r) + \frac{1}{\log {t}} \Big)  \\
& \quad- \Phi\left(\cos_p \psi (t)\right)\sin_p \psi (t)
 + \frac{|\sin_p \psi (t)|^p}{p-1 }   \Big(M(s) +  \frac{1}{\log {t}} \Big) \Big]
\end{aligned}
\end{equation}
for all large $t$.

The equation
\begin{equation} \label{d5}
\begin{aligned}
\varphi' (t)
&= \frac{1}{t \log t} \Big[ {|\cos_p \varphi (t)|^p}
\Big(M(r) + \frac{1}{\log {t}} \Big)
- \Phi\left(\cos_p \varphi (t)\right)\sin_p \varphi (t)  \\
&\quad  + \frac{|\sin_p \varphi (t)|^p}{p-1 }
\Big(M(s) +  \frac{1}{\log {t}} \Big) \Big]
\end{aligned}
\end{equation}
has the form of the equation for the Pr\"ufer angle $\varphi$ which
corresponds to \eqref{fv}, where $M = M(r)$ and $N= M(s)$.
Therefore (see \eqref{pp}), Corollary \ref{vn} guarantees that any solution
$\varphi: [{\mathrm{e}} + \vartheta, \infty) \to \mathbb{R}$ of \eqref{d5} is
bounded from above.
Comparing \eqref{d4} with \eqref{d5} and considering the
 $2\pi_p$-pe\-riodicity of the half-linear trigonometric functions,
we know that the considered function $\psi$ is bounded from above.
This means that any non-zero solution of \eqref{hl}
is non-oscillatory.
\end{proof}

Now we explicitly mention a result which is the basic motivation for our
current research.


\begin{theorem} \label{s1}
Let $r, s : \mathbb{R} \to \mathbb{R}$ be periodic.
\begin{itemize}
\item[(i)] If $ [M(r)]^{p-1} M(s) > q^{-p}$, then  \eqref{hl} is oscillatory.
\item[(ii)] If $ [M(r)]^{p-1} M(s) < q^{-p}$, then  \eqref{hl} is non-oscillatory.
\end{itemize}
\end{theorem}


The statements of the above theorem can be obtained immediately from the
main results of \cite{naseEJDE}.
Using Theorem~\ref{s1}, we can generalize
Theorem~\ref{n1} as follows.

\begin{theorem} \label{n2} Let
$r, s : \mathbb{R} \to \mathbb{R}$ be periodic. Equation
\eqref{hl} is oscillatory if and only if $[M(r)]^{p-1} M(s) > q^{-p}$.
\end{theorem}

We get a new result even for linear equations. Thus, we formulate the
corollary below.

\begin{corollary} \label{vn2}
Let $r: \mathbb{R} \to \mathbb{R}$ be continuous, positive, and periodic
function and let $s: \mathbb{R} \to \mathbb{R}$ be continuous and periodic function.
The equation
\begin{equation}  \label{ex}
\Big[ \frac{t}{r (t)}  x' \Big]' + \frac{s (t)}{ t \log^2 t}  x  = 0
 \end{equation}
is oscillatory if and only if $ 4 M(r)  M(s) > 1$.
\end{corollary}

To illustrate the presented results, we give some examples of equations
whose oscillation properties do not follow from previously known oscillation
criteria. First, we mention an example to illustrate Theorem \ref{n1}.

\begin{example} \rm
For any $p > 1$, the equation
\begin{equation} \label{ex01}
 \Big[ \Big(\frac{2+\sin(\sqrt{q}t)}{2q}\Big)^{-p/q} t^{p-1} \Phi (x')\Big]'
	+ \frac{p-1+\cos(pt)}{p t \log^pt}\Phi (x) = 0
\end{equation}
is in the critical case because
$$
M(r)=M\Big(\frac{2+\sin(\sqrt{q}t)}{2q}\Big)
 =\frac{1}{q}
 =M\Big(\frac{p-1+\cos\left(pt\right)}{p}\Big)=M(s).
$$
Hence, $[M(r)]^{p-1} M(s) = q^{-p}$ and \eqref{ex01} is non-oscillatory
due to Theorem \ref{n1}.
\end{example}

Of course, the oscillation behaviour of \eqref{ex01} is solvable in many 
slightly modified situations as well.
For example, its coefficients may involve parameters. 
Thus, we can apply Theorem \ref{n2} as follows.

\begin{example} \rm
Let $a>1$ and $b, c, d \ne 0$ be real parameters.
We consider the equation
\begin{equation} \label{ex02}
 \Big[ \Big(\frac{a+\sin(ct)}{q}\Big)^{-p/q} t^{p-1} 
\Phi (x')\Big]'
	+ \frac{p-1+\cos(dt)}{b t \log^pt}\Phi (x) = 0
\end{equation}
with
\begin{gather*}
M(r)=M\Big(\frac{a+\sin(ct)}{q}\Big)=\frac{a}{q},\\
M(s)=M\Big(\frac{p-1+\cos(dt)}{b  }\Big)=\frac{p-1}{b}.
\end{gather*}
Therefore, by Theorem \ref{n2}, Equation \eqref{ex02} is oscillatory 
for $a^{p-1} p/b > 1$ and non-oscillatory for $a^{p-1} p/b \leq 1$.
\end{example}

Finally, we mention the following simple example of linear equations 
whose oscillation properties are solvable by Corollary \ref{vn2}.

\begin{example} \rm
Consider the equation
\begin{equation} \label{ex03} 
\begin{aligned}
&\Big[\frac{t}{a_1+b_1\sin(c_1t)+d_1\cos(c_1t)}\,x'\big]'\\ 
& +\frac{a_2+b_2\sin(c_2t)\cos(c_2t) + d_2\arcsin [\cos(c_2t)]}{t\log^2t} x=0,
\end{aligned}
\end{equation}
where $a_i,b_i,c_i,d_i \in \mathbb{R}$, $c_i \ne 0$, $i\in\{1,2\}$, 
$a_1> |b_1|+|d_1|$. It is seen that $M(r)=a_1$ and $M(s)=a_2$ 
(cf. \eqref{ex} and \eqref{ex03}). Hence, \eqref{ex03} is oscillatory
for $a_1a_2>1/4$ and non-oscillatory for $a_1a_2\leq1/4$. We emphasize that
this conclusion remains valid even for, e.g., $c_1=1$ and $c_2=\pi$
or $c_2=\sqrt{2}$, when $r$ and $s$
do not possess any common period.
\end{example}

As a final remark, we consider again the critical case.
In this paper, we deal with the critical case of equations with periodic 
coefficients. It is not possible to categorize as oscillatory and 
non-oscillatory equations in the critical case for ``too general'' coefficients.
 We can illustrate this fact by the Euler type half-linear equations
\begin{equation*}
[r(t)\Phi(x')]'+\frac{s(t)}{t^p}\Phi(x)=0.
\end{equation*}
We refer to \cite{dosly-funk,DV-DoslyVesel,DV-ADE2013,DV-AAA2013-14}. 
Concerning equations of the form given by \eqref{hl}, we conjecture that 
the critical case is not generally solvable even for almost periodic 
functions $r, s$ (for the definition of almost periodicity, see, 
e.g., \cite{HasVes-711,HasVes-798fi}).
This conjecture is based on constructions in \cite{ves000p} 
(see also \cite{vestgyuk,ves000pp}).

\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.

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\end{document}