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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 225, pp. 1--13.\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-2015/225\hfil Monotocity properties of oscillatory solutions]
{Monotocity properties of oscillatory solutions of two-dimensional
systems of differential equations}

\author[M. Bartu\v{s}ek \hfil EJDE-2015/225\hfilneg]
{Miroslav Bartu\v{s}ek}

\address{Miroslav  Bartu\v{s}ek \newline
Faculty of Science, Masaryk University Brno,
Kotl\'{a}\v{r}sk\'{a} 2, 611 37 Brno,  Czech Republic}
\email{bartusek@math.muni.cz}

\thanks{Submitted May 5, 2015. Published August 31, 2015.}
\subjclass[2010]{34C10, 34C15, 34D05}
\keywords{Monotonicity; oscillatory solutions; two-dimensional systems}

\begin{abstract}
 Sufficient conditions for the monotonicity of the sequences of the absolute
 values of all local extrema of components of a two-dimensional systems
 are obtained.
\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}
\allowdisplaybreaks


\section{Introduction}\label{sec1}

In this article, we study the system of differential equations
\begin{equation}\label{e1}
\begin{gathered}
y_1' = f_1 (t, y_1, y_2)\\
y_2' = f_2 (t, y_1, y_2)\,,
\end{gathered}
\end{equation}
where $f_1$ and $f_2$ are continuous on 
$D= \{(t,u,v) : t\in \mathbb{R}_+=[0, \infty), 
u, v\in \mathbb{R}\}$ $\mathbb{R}=(-\infty, \infty)$, and
\begin{gather}
f_1 (t,u,v) v >0\quad  \text{on }  D, \; v\ne 0\,, \label{e2}
\\
f_2 (t,u,v) u<0 \quad  \text{on } D, \; u\ne 0\,. \label{e3}
\end{gather}

\begin{definition}\label{de1} \rm
A function $y=(y_1, y_2): I= [t_y, \bar t_y)\subset \mathbb{R}_+ \to \mathbb{R}^2$
is called a solution of \eqref{e1} if $y_i\in C^1(I)$,
$i=1,2$, and \eqref{e1} holds on $I$. A solution $y$ is oscillatory
on $I$ if there exist two sequences of zeros of $y_1$ and $y_2$
tending to $\bar t_y$,  and $y_1$, $y_1$ are
nontrivial in any left neighbourhood of $\bar t_y$.
\end{definition}

\begin{remark}\label{rem1} \rm
The definition of an oscillatory solution $y$ of \eqref{e1} is not
restrictive. If $y_1$ has a sequence of zeros tending to 
$\bar t_y$ and $y_1$ is nontrivial in any left neighbourhood of  
$\bar t_y$, then according to \eqref{e2} and \eqref{e3} the same is
valid for $y_2$.
\end{remark}


Sometimes, solutions are studied on finite intervals since \eqref{e1} 
may have solutions that cannot be defined in a neighbourhood of $\infty$ 
(so called noncontinuable solutions, singular solutions of the 2-nd kind, 
see e.g.\ \cite{8, 4, 5}).

The prototype of \eqref{e1} is the second-order equation with
$p$-Laplacian
\begin{equation}\label{e4}
\big(y^{[1]}\big)' + f \big(t, y, y^{[1]}\big)=0
\end{equation}
where
\begin{gather}\label{e44}
y^{[1]}(t) = a(t) \big| y' (t) \big|^p \operatorname{sgn}y'
(t)\\
p>0\,,\quad a\in C^0(\mathbb{R}_+)\,, \quad a>0 \quad \text{on }
\mathbb{R}_+\,, \quad f\in C^0(\mathbb{R}_+\times \mathbb{R}^2),\nonumber\\
f(t,u,v)u>0 \quad \text{on } D\,, \; u\ne 0\,;\label{e401}
\end{gather}
Equation \eqref{e4} is equivalent to the system
\begin{equation}\label{e5}
\begin{gathered}
y_1' = a^{-1/p} (t) |y_2|^{1/p} \operatorname{sgn}y_2\,, \\
y_2' =-f(t, y_1, y_2)
\end{gathered}
\end{equation}
with the relation between solutions of \eqref{e4} and \eqref{e5}
given by
$$
y_1=y\,, \quad y_2=y^{[1]}\,.
$$
Note, that  \eqref{e4} is a special case of \eqref{e1}--\eqref{e3}
with
\begin{equation}\label{e71}
f_1(t,u,v)= a^{-1/p} (t) |v|^{1/p} \operatorname{sgn}v\,, \quad
f_2(t,u,v)=-f(t,u,v)\,.
\end{equation}

The study of oscillatory solutions of \eqref{e1} or \eqref{e4} is of 
interest to  many authors  at the present time; see e.g. \cite{12, 5}.

Let $y$ be an oscillatory solution of \eqref{e4} defined on 
$I=[t_y, \bar t_y) \subset \mathbb{R}_+$ such that
it has no accumulation point of zeros on $I$. 
Then a left  neighbourhood of $\bar t_y$ exists such that all zeros of 
$y$ and $y^{[1]}$ in it can be  described  by two increasing sequences 
$\{t_k\}_{k=1}^\infty$ and $\{\tau_k\}_{k=1}^\infty$, respectively.
Note, that  by virtue of \eqref{e4} and \eqref{e401}, $y(\tau_k)$ 
and $y^{[1]}(t_k)$, $k=1,2,\dots$ are local extrems of $y$ and $y^{[1]}$,
 respectively (see \cite{9}).
Then, the following problem
 for \eqref{e4} has a long history.
\smallskip
 
\noindent\textbf{Problem.}
  Find sufficient conditions for the sequence 
$\big\{|y(\tau_k)|\big\}_{k=1}^\infty$  
$\big(|y^{[1]}(t_k)|_{k=1}^\infty \big)$ of the absolute values of the 
local extrema of $y$ (of $y^{[1]}$) to be monotone.

This problem was initiated by Milloux \cite{13}  and then it was considered 
by many authors for linear (e.g.\ \cite{8}) and special types of nonlinear 
equations of the form
 \begin{equation}\label{e50}
 y'' + f(t, y, y') =0
 \end{equation}
(the first results are given by Bihari \cite{6}); 
the history of this problem is described more precisely in monograph
 \cite{2} and paper \cite{7}.

 Concerning   equation \eqref{e4}, some results of \cite{9} are summed 
up in the following theorems.
Let $y $ be an oscillatory solution of \eqref{e4} and let 
$\{t_k\}_{k=1}^\infty$ and $\{\tau_k\}_{k=1}^\infty$ be given as above.


\begin{theorem}[\cite{9}]\label{thA}
Let $|f(t, u,v)|$ be non-decreasing with respect to $t$ in $D$ and
  $a$\quad  be  non-decreasing on $\mathbb{R}_+$.

(i)  Let   $f(t, -u,v) =-f(t,u,v)$  on  $D$,
$f(t,u,v)$ be  non-increasing with respect to $v$ on 
$D\cap \{v\geq 0, u\geq 0\}$, and  be non-decreasing with respect 
to $v$ on $D\cap \{v\leq 0, u\geq 0\}$.
Then $\big\{|y(\tau_k)|\big\}_{k=1}^\infty$
is non-increasing.


(ii)  Let   $f(t, u,-v) =f(t,u,v)$ on $D$,
$|f(t,u,v)|$ be  non-decreasing with respect to $v$ on $D\cap \{v\geq 0\}$.
Then
$\big\{|y^{[1]}(t_k)|\big\}_{k=1}^\infty$ is non-decreasing.
\end{theorem}

For a special case  of \eqref{e4},  the results in Theorem~\ref{thA} 
are proved under weaker assumptions,
the monotonicity with respect to $t$ of 
$a^{1/p}(t)|f(t,u,v)|$ is supposed instead of the monotonicities of $a$ and 
 $|f(t,u,v)|$.


\begin{theorem}[\cite{9}] \label{thB}
 Let  $f(t, u,v) \equiv r(t)  h(u)$, where  
$r\in C^0 (\mathbb{R}_+)$ and  $r>0$, let $ h\in \mathbb{R}$ be an odd function with
$h(u)>0$ for $u>0$,  and  let $a^{1/p} r \in C^1(\mathbb{R}_+)$ be non-increasing.
Then $\big\{|y(\tau_k)|\big\}_{k=1}^\infty$ is non-decreasing and 
$\big\{|y^{[1]}(t_k)|\big\}_{k=1}^\infty$  is non-increasing.
\end{theorem}


\begin{remark}\label{re1} \rm
If we change ``non-decreasing'' to ``non-increasing'',
and ``non-in\-creasing''  to ``non-decreasing'',
then Theorems~\ref{thA} and \ref{thB} still hold.
\end{remark}


The same Problem is studied for \eqref{e1} in \cite{10}.
Our goal is to generalize the results of \cite{10} and
  of Theorems~\ref{thA} and \ref{thB} to equation \eqref{e1}.
  We prove them under weaker assumptions and under different ones as well.
 We will remove the assumption that the oscillatory solution is defined 
on the interval without accumulation points of zeros of $y_1$; it will be 
shown that oscillatory solutions of \eqref{e1} have no such points on 
their definition intervals under the  assumptions in our theorems.

In Theorem~\ref{thA},  some kind of monotonicity of $f$ with respect to $v$ 
is used; we show that this assumption is not needed. We are also  
able to weaken the assumption concerning to the monotonicity with respect to $t$.

The basics of the method of the proofs are used in \cite{7, 2} for  \eqref{e50}
 and in \cite{10} for \eqref{e1}. We study a solution $y$ of \eqref{e1} 
locally on two consecutive quarter-waves using the inverse functions to 
$y_1$ on each of them.

The structure of zeros of a solution of \eqref{e1} can be
complicated (see  \cite{3} for equation \eqref{e50}). So, we
introduce the following definition (see \cite{1} for \eqref{e1}).

\begin{definition}\label{de2} \rm
Let $y$ be a solution of \eqref{e1} defined on $[t_y, \bar t_y)$.
A number $c\in [t_y, \bar t_y)$ is called an $H$-point of $y$ if
there are sequences $\{\tau_k\}_{k=1}^\infty$ and 
$\{\bar \tau_k\}_{k=1}^\infty$ of numbers from $[t_y, \bar t_y)$ tending
to $c$ such that
$$
y_1 (\tau_k) =0\,,\quad y_1(\bar \tau_k)\ne 0\,,\quad 
(\tau_k- c) (\bar\tau_k-c)>0\,,\quad k=1,2,\dots
$$
\end{definition}

In Definition \ref{de2}, it is sufficient to work only with $y_1$, as
according to \eqref{e1}--\eqref{e3}, $y_1$ has a sequence of zeros
tending to $c$ from the left (right) side and $y_1$ is nontrivial in
any left (right) neighbourhood of $c$ if and only if the same
properties hold for $y_2$. Moreover, if $c$ is an $H$-point of $y$,
then
\begin{equation}\label{e6}
y_1(c) = y_2(c) =0\,.
\end{equation}
Conditions for the nonexistence of $H$-points of a solution
\eqref{e1} are given in \cite{1}; for equation \eqref{e50}, see also e.g.\
\cite{4}. On the other side, there exists an equation of the form
\eqref{e50}  with a solution with infinitely many $H$-points
tending to $\infty$, see \cite{3}.

\begin{definition}\label{de3} \rm
Let $i\in \{1,2\}$ and $y$  be a solution of \eqref{e1}  defined on 
$[t_y, \bar t_y)$. Then $y_i$ has a local extreme at $t=T$ if a neighbourhood $I$ 
(a right  neighbourhood $I$) of $T$ exists such that either  
$y_i(t) \geq y_i(T)$  or $y_i (t)\leq y_i(T)$ for $t\in I$ in case $T> t_y$ 
(and $y_i'=0$ in case $T=t_y$).
\end{definition}

\section{Preliminary results}\label{sec2}

At first, we give some auxiliary results concerning  zeros of a
solution of \eqref{e1}.

\begin{lemma}\label{le1}
Let $y$ be a solution of \eqref{e1} defined on $I$, let  $c\in I$ be such that
\begin{equation}\label{e7}
y_1(c)=y_2(c)=0\,,
\end{equation}
and let  $y$ be nontrivial in any right (left) neighbourhood of $c$.
Then there is a sequence $\{t_k\}_{k=1}^\infty$ of zeros of $y_1$
such that $t_k>c$ ($t_k<c$) for $k=1,2,\dots$ and
$\lim_{k\to\infty} t_k=c$. Hence, $c$ is $H$-point of $y$.
\end{lemma}

\begin{proof}
Suppose $y$ is a nontrivial solution in any right neighbourhood of
$c$ and \eqref{e7} holds. Let, contrarily, a right neighbourhood
$I$ of $c$ exist such that
\begin{equation}\label{e8}
y_1(t)>0\quad \text{for } t\in I\,.
\end{equation}
Then \eqref{e1} ($i=2$) and \eqref{e3} imply $y_2$ is decreasing
on $I$, and due to \eqref{e7}, we have $y_2(t)<0$ on $I$. From
this, from \eqref{e1} ($i=1$), and \eqref{e2}, we have $f_1\big(t,
y_1(t), y_2(t)\big)<0$ on $I$ or $y_1$ is decreasing on $I$. As
$y_1(c)=0$, we can conclude $y_1(t)<0$ on $I$. The contradiction
with \eqref{e8} proves the statement.

The case $y_1(t)<0$ for $t\in I$ can be studied similarly.
\end{proof}

\begin{lemma}\label{le2}
Let  a solution $y$ of \eqref{e1} be oscillatory on $I =[t_y, \bar
t_y)\subset \mathbb{R}_+$ without $H$-points. Then all zeros of
either $y_1$ or $y_2$ are simple and isolated, and  sequences
 $\{t_k\}_{k=1}^\infty$ and $\{\tau_k\}_{k\in {\mathcal{N}}_0}$
 exist such that  either $\mathcal{N}_0=\{1,2,\dots\}$ or
 $\mathcal{N}_0=\{0,1,2,\dots\}$,
\begin{gather}\label{e9}
t_y\leq t_k<\tau_k<t_{k+1}< \bar t_y\,,\quad
 k=1,2,\dots\,, \;\lim_{k\to\infty} t_k=\bar t_y\,,
\\
y_1(t_k)=0\,, \quad  y_1(t)\ne 0 \quad \text{for } t\ne
t_k\,,\; t\in I\,,\; k\in \{1,2,\dots\}\,, \nonumber\\
 y_2(\tau_k)=0\,, \quad y_2(t) \ne 0  \quad \text{for } t\ne \tau_k\,, \; t\in I\,,
\; k\in \mathcal{N}_0\,. \nonumber
\end{gather}
Moreover,
\begin{equation}\label{e10}
\begin{gathered}
y_1(t)\,y_2(t) >0 \quad \text{on } (t_k, \tau_k)\,,\\
y_1(t)\,y_2(t) <0 \quad \text{on } (\tau_k, t_{k+1})\,, \;
k=1,2,\dots
\end{gathered}
\end{equation}
\end{lemma}


\begin{proof}
Note that $y$ is not trivial on $I$ due to $y$  being oscillatory.
As $y$ has no $H$-point, Lemma~\ref{le1} implies  \eqref{e7} is
not valid for any $c\in I$,  and according to \eqref{e1} and \eqref{e2} 
any zero of $y_1$ is simple and isolated. Let
$y_2(c)=0=y_2'(c)$. Then, according  to \eqref{e1} and
\eqref{e3}, $y_1(c)=0$, which  contradicts the proved  part. Hence,
any zero of $y_2$ is simple and isolated.

Let $T_1<T_2$ be successive zeros of $y_1$ and let, for the sake of
simplicity, $y_1(t)>0$ on $(T_1, T_2)$. Then \eqref{e1} and
\eqref{e3} imply $y_2$ is decreasing on $(T_1, T_2)$. By
 Rolle's Theorem $y_1'$ has a zero $T_3\in (T_1, T_2)$, so
we obtain from \eqref{e1} and \eqref{e3} that $y_2 (T_3) =0$ and
$$
y_2(t)>0\quad \text{on } [T_1, T_3)\,,\quad 
y_2(t)<0\quad \text{on } (T_3, T_2]\,.
$$
Inequalities \eqref{e9} and \eqref{e10} follow from this. 
If $y_1(t)<0$ on $(T_1, T_2)$, the proof is similar.
\end{proof}


\begin{lemma}\label{le33} 
Let the assumptions of Lemma~\ref{le2} hold.  
Then  $\{ y_1(\tau_k)\}_{k\in {\mathcal{N}}_0}$
$($resp. $\{y_2(t_k)\}_{k=1}^\infty)$ is the sequence of all local 
extrema of $y_1$ (of $y_2$) on $I$.
\end{lemma}

\begin{proof}
By Lemma~\ref{le2}, $t_k$ are simple zeros of $y_1$, and  $y_1$ changes 
its sign when $t$  is going through $t_k$; hence, using \eqref{e1} and 
\eqref{e3}, a neighbourhood of $t_k$ exists  such that $y'_2(t) \, y_1(t) <0$ in it. 
Thus, $y_2$ has a local extreme at $t=t_k$.
Similarly, it can be proved that $y_1$ has a local extrum at 
$t=\tau_k$ using \eqref{e1} and \eqref{e2}.
\end{proof}

\begin{lemma}\label{le3}
Let $y$ be a solution of \eqref{e1} defined on $[t_y, \bar t_y)$
without $H$-points,
\begin{equation}\label{e111}
\frac{\big|f_2(t,u,v)\big|}{\big|f_1(t,u,v)\big|} \quad \text{be
non-decreasing with respect to $t$}
\end{equation}
on $D$ with  $u v <0$. For any integer $m$, let there be  a continuous 
function $g_m: (0,m]\to (0, \infty)$  such that
\begin{equation}\label{e29}
\frac{g(|v|)\big|f_2(t,u,v)\big|}{f_1(t,u,v)}\quad
\text{is non-decreasing}
\end{equation}
with respect to $v$ for $|v|\in \big[\frac{1}{m}, m\big]$,  $t\in [0, m]$, 
and  $|u|\leq m$ with $uv<0$.

(i) Then the sequence of all positive (the absolute
values of all negative) local extrema of $y_1$ is non-increasing.

(ii) Let, moreover,
\begin{equation}\label{e11}
f_1(t, -u, v) = f_1(t, u, v)\,,\quad  f_2(t, -u,v)=-f_2(t,u,v)\quad 
\text{on } D\,.
\end{equation}
Then the sequence of the absolute values of all local extrema of
$y_1$ is non-increasing, i.e.,
$\big\{y_1(\tau_k)\big\}_{k\in {\mathcal{N}}_0}$ is non-increasing, where
$\{\tau_k\}_{k\in {\mathcal{N}}_0}$ is given by Lemma \ref{le2}.
\end{lemma}

\begin{proof}
Let $y$ be defined on $[t_y, \bar t_y)$ without $H$-points. We use
the notation from Lemma~\ref{le2}, and first we prove case (ii). Let
$n-1\in \mathcal{N}_0$ be fixed  and put
$$
T_0=\tau_{n-1}\,,\quad T_1=t_n\,,\quad T_2=\tau_n\,,\quad
J_0=[T_0, T_1]\,,\quad J_1=[T_1, T_2]\,.
$$
Suppose, without the loss of generality, $y_2(t)>0$ on 
$(T_0, T_2)$ (if $y_2<0$ the proof is similar). For convenience, we
describe the situation more precisely using \eqref{e1}--\eqref{e3}
and \eqref{e10}. We have
\begin{equation}\label{e12}
\begin{gathered}
y_1<0  \text{ is increasing,} \quad y_2>0 
 \text{ is increasing,}\\
f_1\big(t,y_1(t), y_2(t)\big) >0\,,\quad 
f_2\big(t, y_1(t), y_2(t)\big) >0 \quad \text{on }(T_0,T_1),\\
y_1>0  \text{ is increasing,} \quad  y_2>0 \quad \text{is
decreasing,}\\
f_1\big(t,y_1(t), y_2(t)\big) >0\,,\quad
f_2\big(t, y_1(t), y_2(t)\big) <0 \quad \text{on }(T_1, T_2).
\end{gathered}
\end{equation}
Define $s_0(z)$, $z\in \big[0, |y_1(T_0)|\big]$ ($s_1(z)$,
 $z\in [0, y_1(T_2)$) as the inverse function to $|y_1|$ (to $y_1$) on
$J_0$ (on $J_1$). Let $\bar z =\min \big(|y_1(T_0)|,
y_1(T_2)\big)$. We prove that
\begin{equation}\label{e13}
y_2 \big(s_0(z)\big) \geq y_2 \big(s_1(z)\big)\quad
\text{for } z\in [0, \bar z]\,.
\end{equation}
Note that $y_2(s_0(0)) = y_2(s_1(0))>0$.
Assume, to the contrary, that there exists $\tilde z\in (0, \bar z)$  such that
\begin{equation}\label{e14}
y_2 \big(s_0(\tilde z)\big) < y_2 \big(s_1(\tilde z)\big)\,.
\end{equation}
Thus,  an integer $m$ exists such that
\begin{equation}\label{e15}
T_2 \leq m\,, \quad 0<\tilde z\leq m\,,\quad 
y_2 \big(s_i(z)\big) \in \big[\tfrac{1}{m}, m\big]
\end{equation}
for $i=1,2$, $z\in [0, \tilde z]$,  and the function
\begin{equation}\label{e16}
\frac{g_m(v)\big|f_2(t,u,v)\big|}{f_1(t,u,v)}\quad 
\text{is non-decreasing with respect to $v$}
\end{equation}
for $t\in [0,m]$, $0< |u| \leq m$ and $v\in \big[\frac{1}{m}, m\big]$. Put
$$
G(v) =\int_0^v g_m (\sigma)\, d\sigma\,, \quad 
H(z) =G\big(y_2(s_0(z))\big) -G\big(y_2(s_1(z))\big)\,.
$$
Note that for $z\in (0, \tilde z]$,
\begin{equation}\label{e17}
y_2\big(s_0(z)\big) < y_2\big(s_1(z)\big) \Leftrightarrow 
H(z) <0\,.
\end{equation}
Furthermore, using \eqref{e111}, \eqref{e11}, \eqref{e12}, we have
\begin{equation}
\begin{aligned}
\frac{d}{dz} H(z) 
&= -\frac{ g_m(y_2(s_0)) f_2 (s_0, -z,
y_2(s_0))}{f_1(s_0, -z, y_2(s_0))} - \frac{ g_m(y_2(s_1)) f_2
(s_1, z, y_2(s_1))}{f_1(s_1, z, y_2(s_1))} \\[4pt]
& \geq -\frac{ g_m(y_2(s_0)) f_2 (s_1, -z, y_2(s_0))}{f_1(s_1, -z,
y_2(s_0))}+ \frac{ g_m(y_2(s_1)) f_2 (s_1,- z, y_2(s_1))}{f_1(s_1,
-z, y_2(s_1))}
\end{aligned} \label{e18}
\end{equation}
for $z\in (0, \tilde z]$,  $s_0=s_0(z)$,  and $s_1=s_1(z)$. Then
\eqref{e15}, \eqref{e16}, \eqref{e17} and \eqref{e18} imply
\begin{equation}\label{e19}
z\in (0, \tilde z]\,, \quad\text{and} \quad  
H(z) <0 \Rightarrow \frac{d}{dz} H(z) \geq
0\,.
\end{equation}
As \eqref{e14} and \eqref{e17} imply $H(\tilde z) <0$, 
we have from \eqref{e19} that
$$
H(z) \leq H(\tilde z) <0 \quad \text{for } z\in(0, \tilde z]\,,
$$
which contradicts $H(0)= G\big(y_2(T_1)\big) - G\big(y_2(T_1)\big)=0$. 
Hence, \eqref{e13} holds. Furthermore, we prove that
\begin{equation}\label{e191}
|y_1(T_0)| \geq y_1(T_2)\,.
\end{equation}
Assume, to the contrary, that
\begin{equation}\label{e20}
|y_1(T_0)| < y_1(T_2)\,.
\end{equation}
Then $\bar z=|y_1(T_0)|$ and \eqref{e13} imply
$$
0=y_2 (T_0)=y_2\big(s_0(\bar z)\big) \geq y_2 \big(s_1(\bar z)\big)\,.
$$
From this and from \eqref{e12}, $s_1(\bar z)=T_2$.
As \eqref{e20} implies
$|y_1(T_0)|=\bar z = y_1 (s_1(\bar z)) <y_1(T_2)$, 
we have  from \eqref{e12} that $s_1(\bar z) <T_2$. This contradiction 
proves\eqref{e191}.  Hence, as $n$ was arbitrary,  (ii) holds.
\smallskip

\noindent\textbf{Case (i).} The proof  is similar to case (i). 
We study the solution on intervals $[\tau_{n-1}, t_n]$ and $[\tau_{n+1},
t_{n+2}]$ instead of  on $J_0$ and $J_1$. Note, that
$y_1(\tau_{n-1})$ and $y_1(\tau_{n+1})$ are two consecutive local
extrema with the same signs. As $y_2(t)\, y_2(s) >0$ for $t\in
[\tau_{n-1}, t_n)$ and $s\in [\tau_{n+1}, t_{n+2})$, condition
\eqref{e11} is not necessary  (see \eqref{e18}).
\end{proof}


\begin{remark}\label{re2}  \rm
If ``non-decreasing'' and ``non-increasing'' is replaced by
``non-in\-creasing''  and ``non-decreasing'', respectively, 
with the exception of \eqref{e29},  then Lemma~\ref{le3} holds, too.
It is important to note that \eqref{e29} must have the given form.
\end{remark}

\begin{lemma}\label{le4} 
Let $y$ be a solution of \eqref{e1} defined on $[t_y, \bar t_y)$
without $H$-points,
\begin{equation}\label{e21}
\Big|\frac{f_2(t,u,v)}{f_1(t,u,v)}\Big| \text{ be
non-decreasing with respect to $t$ on $D$, $uv>0$.}
\end{equation}
For any integer $m$, assume there is a continuous function 
$g_m: (0,m] \to (0, \infty)$  such that
\begin{equation}\label{e291}
\frac{g(|v|)\big|f_2(t,u,v)\big|}{f_1(t,u,v)} \text{ is non-increasing}
\end{equation}
with respect to $v$ for $v\in (0, m]$, and for $v\in [-m,0)$, and
for  any $t\in [0, m]$, $\frac{1}{m} \leq |u|\leq m$, $uv>0$.
\smallskip

(i) Then the sequence of all positive (the absolute values of
 all negative) local extrema of $y_2$ is non-decreasing.

(ii) If, moreover,
 \begin{equation}\label{e211}
 f_1(t,u,-v) =-f_1(t,u,v)\,,\quad f_2(t, u, -v)=f_2(t,u,v)\,,
\end{equation}
 then the sequence of the  absolute values of all local extrema of
 $y_2$ is non-decreasing, i.e.\ $\big\{ |y_2(t_k)|\big\}_{k=1}^\infty$ 
is non-decreasing, where
 $\{t_k\}_{k=1}^\infty$ is given by Lemma~\ref{le2}.
 \end{lemma}

\begin{proof}
Let $y$ be defined on $[t_y, \bar t_y)$ without $H$-points. We use
the notation in Lemma~\ref{le2}. First we prove case (ii). Let
$n\in \{1,2,\dots\}$ be fixed. Put $T_1=t_n$, $T_2=\tau_n$, $T_3=
t_{n+1}$, $J_1=[T_1, T_2]$ and $J_2=[T_2, T_3]$. 
Suppose, without loss of the generality, that $y_1(t)>0$ on $J_1\cup J_2$, 
the proof is similar in case $y_1<0$.  For convenience, we describe the
situation more precisely using \eqref{e1}--\eqref{e3} and
\eqref{e10}. We have
\begin{equation}\label{e22}
\begin{gathered}
y_1>0  \text{ is increasing },\quad
y_2>0  \text{ is decreasing}\\
f_1\big(t, y_1(t), y_2(t)\big) >0\,, \quad
f_2\big(t, y_1(t), y_2(t)\big)<0 \quad \text{on $(T_1, T_2)$}, \\
y_1>0   \text{ is decreasing}, \quad
y_2<0 \quad \text{is decreasing}\\
f_1\big(t, y_1(t), y_2(t)\big) <0\,, \quad
f_2\big(t, y_1(t), y_2(t)\big)<0 \quad \text{on $(T_2, T_3)$}\,.
\end{gathered}
\end{equation}
Let $z\in[0, y_1(T_2)]$. Define $s_1(z)$ and $s_2(z)$ as  the
inverse functions to $y_1$ on $J_1$ and $J_2$, respectively. We
prove that
\begin{equation}\label{e23}
y_2\big(s_1(z)\big) \leq \big|y_2\big(s_2(z)\big)\big|\quad
\text{for}\quad z\in \big[0, y_1(T_2)\big]\,.
\end{equation}
Assume to the contrary that there exists $\bar z\in \big(0, y_1(T_2)\big)$
such that
\begin{equation}\label{e24}
y_2\big(s_1(\bar z)\big) > \big|y_2\big(s_2(\bar z)\big)\big|\,.
\end{equation}
Then there exist an integer $m$  such that
 \begin{gather}
T_3\leq m\,,\quad \big[\bar z, y_1(T_2)\big]\subset \big[ \tfrac{1}{m}, m\big]\,,\\
\max\big\{y_2\big(s_1(\bar z)\big), \big|y_2\big(s_2(\bar z )\big)\big|\big\}
\leq m , \label{e25}\\
\frac{g_m(v)|f_2(t,u,v)|}{f_1(t,u,v)} \quad 
\text{ is non-increasing in $v$} \nonumber\\
\text{for $t\in [0,m]$, $u\in \big[\tfrac{1}{m}, m\big]$, and $v\in (0,m)$.}
\label{e26}
\end{gather}
Put
$$
G(v) =\int_0^v g_m(\sigma)\, d\sigma\,, \quad 
H(z) = G\big(y_2(s_1(z))\big) -G\big(|y_2(s_2(z))|\big)\,.
$$
Note, that
\begin{equation}\label{e27}
y_2 \big(s_1(z))\big) -|y_2\big(s_2(z)\big)| >0 \Leftrightarrow
 H(z)>0 \text{ for } z\in [0, y_1(T_2)]\,.
\end{equation}
Furthermore, using \eqref{e21}, \eqref{e211} and \eqref{e22},
\begin{equation}
\begin{aligned}
\frac{d}{dz}H(z)
&= \frac{g_m(y_2(s_1)) f_2(s_1, z, y_2(s_1))}{f_1(s_1, z, y_2(s_1))}
+  \frac{g_m(y(s_2)) f_2(s_2, z, y_2(s_2))}{f_1(s_2, z, y_2(s_2))}\nonumber\\
&\geq \frac{g_m(y_2(s_1)) f_2(s_2, z, y_2(s_1))}{f_1(s_2, z, y_2(s_1))} -
\frac{g_m(y(s_2)) f_2(s_2, z, |y_2(s_2)|)}{f_1(s_2, z,| y_2(s_2)|)}
\end{aligned}\label{e28}
\end{equation}
for $z\in \big[ \bar z, y_1(T_2)\big)$,  $s_1 = s_1(z)$, and  $s_2=s_2(z)$. As
 $t=s_i(z)$, $u=z$, $v=|y_j(s_i)|$ satisfies \eqref{e26} for
$z\in \big[\bar z, y_1(T_2)\big]$, $i=1,2$ and $j=1,2$,
\eqref{e25}, \eqref{e27}, \eqref{e28} imply
$$
z\in \big(\bar z, y_1(T_2)\big)\,, \quad
H(z)>0 \Rightarrow \frac{d}{dz} \, H(z) \geq 0\,.
$$
By \eqref{e24} and \eqref{e27}, $H(\bar z)>0$ and we can  conclude
$$
H(z)\geq H(\bar z) >0\,, \quad z\in \big[\bar z, y_1(T_2)\big]
$$
which  contradicts $H\big(y_1(T_2)\big) =0$. Hence, \eqref{e23} holds and
$$
y_2(t_n) = y_2(s_1(0))\leq \big| y_2(s_2(0))\big| = |y_2(t_{n+1})|\,.
$$
As $n$ was arbitrary, the conclusion holds. Case (i) can be proved from
case (ii)  as in the proof of Lemma~\ref{le3}.
\end{proof}


\begin{remark}\label{re20} \rm
If ``non-decreasing'' and ``non-increasing'' is replaced by ``non-in\-creasing'' 
and ``non-decreasing'', respectively, with the exception of \eqref{e291}, 
then Lemma~\ref{le4} holds, too. Again \eqref{e291} must have  the given form.
\end{remark}

\begin{remark}\label{re10}
The results of Lemmas \ref{le3} (ii) and \ref{le4} (ii) are proved 
in \cite{10} under stronger assumptions concerning  the monotonicity with 
respect to $t$.
\end{remark}

In Lemmas \ref{le3} and \ref{le4} no assumptions are made on functions 
$f_1$ and $f_2$ with respect to the second variable. The following results 
are obtained without assumptions on $f_1$ and $f_2$ with respect to the 
third variable.

\begin{lemma}\label{le8}
Let $y$ be a solution of \eqref{e1} defined on $[t_y, \bar t_y)$
without $H$-points,
\begin{equation*}
\frac{\big|f_2(t,u,v)\big|}{\big|f_1(t,u,v)\big|} \text{ be
non-decreasing with respect to $t$ on $D$, $u v \ne 0$.}
\end{equation*}

(i)  For any integer $m$, assume  there exists a continuous function
$g_m: (0,m]\to (0, \infty)$  such that
\begin{equation*}
\frac{g(|u|)\big|f_1(t,u,v)\big|}{f_2(t,u,v)}
\text{ is non-increasing}
\end{equation*}
with respect to $u$ for $|u|\in [\frac{1}{m}, m]$,
for any  $t\in [0, m]$, and  $|v|\in \big(0,m\big]$.
Then the results of Lemma~\ref{le4} hold.

(ii) For any integer $m$, assume there exists a continuous function
$\bar g_m: (0,m]\to (0, \infty) $  such that
\begin{equation}\label{e381}
\frac{\bar g(|u|)\big|f_1(t,u,v)\big|}{f_2(t,u,v)}\quad
\text{is non-decreasing}
\end{equation}
with respect to $u$ for $u\in (0, m]$,  and  for $u\in [-m, 0)$,  
for any $t \in [0, m]$, and $|v|\in \big[\frac{1}{m},m\big]$. 
Then the results of Lemma~\ref{le3} hold.
\end{lemma}

\begin{proof}
By the transformation
\begin{equation} \label{e34}
z_1(t) = -y_2(t)\,,\quad z_2(t)=y_1(t)\,,
\end{equation}
system \eqref{e1} is equivalent to
\begin{equation}
z_i'  =F_i(t, z_1, z_2)\,, \quad i=1,2, \label{e35}
\end{equation}
where $F_1 (t, z_1, z_2)= - f_2(t,z_2,-z_1)$, $F_2(t, z_1,z_2)=f_1(t, z_2, -z_1)$
in $D$. From  \eqref{e2} and \eqref{e3},
\begin{gather*}
 F_1 (t, z_1, z_2)  z_2  =  -f_2 (t, z_2, -z_1)  z_2 >0 \quad \text{for }
 z_2\ne 0\,,\\
 F_2(t, z_1, z_2)  z_1 = f_1 (t, z_2, -z_1)  z_1 <0 \quad \text{for }  z_1\ne 0\,,
\end{gather*}
Remarks \ref{re2} and \ref{re20} can be applied to \eqref{e35}.
If we use the back transformation \eqref{e34}, we can obtain the results
of the lemma. Note that case (i) \big((ii)\big) follows from Remark~\ref{re2}
(Remark~\ref{re20}).
\end{proof}

The following lemmas give  sufficient conditions for the validity of 
either  \eqref{e291} or \eqref{e381}.


\begin{lemma}\label{le5} 
Let $m$ be an integer and let 
$\frac{\partial}{\partial v} \frac{f_2(t,u,v)}{f_1(t,u,v)}$ be continuous on
$D$ for $uv \ne 0$.
Then there is a function $g_m: (0,m] \to \mathbb{R}_+$  such that
$$
J(v) =\frac{g_m(|v|) |f_2(t,u,v)|}{f_1(t,u,v)}
$$
 is non-increasing in $v$ for $v\in (0, m]$  and  for $v\in [-m, 0)$,  
 for any $t\in [0,m]$, and $|u|\in \big[\frac{1}{m}, m\big]$.
\end{lemma}

\begin{proof}
 Put $\bar D =\big\{ (t,u): t\in [0,m], |u| \in \big [ \frac{1}{m}, m\big]\big\}$,
\[
g(z) = \exp \big\{ -\int_z^m \min \big(A_1(\sigma), A_2(-\sigma)\big)\, d\sigma\big\}\,, 
\quad  z\in (0, m]\,,
\]
with
\begin{gather*}
B(t,u,v)= -\frac{d}{dv}\Big(\frac{|f_2(t,u,v)|}{f_1(t,u,v)}\Big) 
\frac{|f_1(t,u,v)|}{|f_2(t,u,v)|}\,,\\
A_1(z) =\min_{(t,u)\in \bar D}   B(t,u,z)\,, \quad 
A_2(-z) =\min_{(t,u)\in \bar D} B(t,u,-z)\,.
\end{gather*}
Let $v\in (0, m]$ and $(t, u) \in \bar D$. Then \eqref{e2} implies $f_1(t,u,v)>0$,
\[
\frac{g'(v)} {g(v)} = \min \big(A_1(v), A_2(-v)\big) \leq B(t,u,v)
\]
or
\[
 g'(v) \frac{|f_2(t,u,v)|}{f_1(t,u,v)} 
\leq - g(v) \frac{d}{dv} \frac{|f_2(t,u,v)|}{f_1(t,u,v)}
\]
and, hence $ J'(v) \leq 0$.

Let $v\in [-m, 0)$. Then \eqref{e2} implies $f_1(t,u,v)<0$,
\begin{align*}
\frac{ g'(-v)} {g(-v)} 
&=- \min \big(A_1(-v), A_2(v)\big) \geq -B(t,u,v)\\
&=-\frac{d}{dv} \Big( \frac{|f_2(t,u,v)|}{f_1(t,u,v)}\Big) 
 \frac{f_1(t,u,v)}{|f_2(t,u,v)|}\,,
\end{align*}
or
\[
g'(|v|) \frac{|f_2(t,u,v)|}{f_1(t,u,v)} 
\leq - g(|v|) \frac{d}{dv} \frac{|f_2(t,u,v)|}{f_1(t,u,v)}\,,
\]
and so  $J'(v) \leq 0$.
\end{proof}

The following lemma can be proved similarly as Lemma~\ref{le5}.

\begin{lemma}\label{le7}
Let $m$ be an integer and let 
$\frac{\partial}{\partial u} \frac{f_1(t,u,v)}{f_2(t,u,v)}$ be continuous on
$D$ for $uv \ne 0$.
Then there is  a function $g_m: (0,m] \to \mathbb{R}_+$  such that
$$
J(u) =\frac{g_m(|u|) |f_1(t,u,v)|}{f_2(t,u,v)}
$$
 is non-decreasing  in $u$ for $u\in (0, m]$   and for  $u\in [-m, 0)$,  
for any $t\in [0,m]$, and $|v|\in \big[\frac{1}{m}, m\big]$.
\end{lemma}

\section{Main results}\label{sec3}

\begin{theorem}\label{th1}
Suppose
\[
\Big| \frac{f_2(t,u,v)}{f_1(t,u,v)}\Big| 
\text{ is non-decreasing (non-increasing) on $D$ for $uv\ne 0$}
\]
and either


(i)  $\dfrac{\partial}{\partial u} \dfrac{f_2(t,u,v)}{f_1(t,u,v)}$ 
 is continuous  on $D$, $uv\ne 0$,  or

(ii) for any integer $m$,  there is a  continuous function 
$g_m: (0,m] \to (0, \infty)$   such that
\[
 \frac{g_m (|u|) |f_1(t,u,v)|}{f_2(t,u,v)}  \text{ is non-decreasing}
\]
with respect to $u$ for $u\in (0, m]$
and with respect to $u$ for $u\in[-m, 0)$,
 and  for any $t\in[0,m]$ and $|v|\in\big[\frac{1}{m},m\big]$; or

(iii) for any integer $m$, there is  a continuous function 
$\bar g_m: (0,m] \to (0, \infty)$   such that
\begin{equation*}%\label{e37}
 \frac{\bar g_m (|v|) |f_2(t,u,v)|}{f_1(t,u,v)} \text{ is non-decreasing}
\end{equation*}
with respect to $v$ for $|v|\in \big[\frac{1}{m},m\big]	$ and  for any 
$t\in [0,m]$ and $|u|\in (0, m]$.

Let $y$  be an oscillatory solution of \eqref{e1} defined on 
$[t_y, \bar t_y]\subset R_+$. Then
\begin{enumerate}
\item There exists  no $H$-point of $y$, $y$ can not be defined at 
$t= \bar t_y$, and all zeros of  $y_1$ can be described by  the increasing sequence
$\{\tau_k\}_{k=1}^\infty$.

\item The sequence of all positive local extrema of $y_1$  is non-increasing
 (is non-decreasing).

\item The sequence of the absolute values of all negative local extrema
 of $y_1$  is non-increasing (is non-decreasing).


\item If, moreover,
\[
f_1(t,-u,v) = f_1(t,u,v)\,,\quad f_2(t,-u,v) = -f_2(t,u,v)
\]
on $D$, then the sequence $\big\{|y_1(\tau_k)|\big\}_{k=1}^\infty$  
of the absolute values of all local extrema    of $y_1$  is non-increasing 
(is non-decreasing).
\end{enumerate}
\end{theorem}

\begin{proof}
As $y$ is oscillatory, $T_y\in [t_y,\bar t_y)$ exists such that $y_1(T_y) \ne 0$. 
Let $[T_y, \bar T_y)\subset [t_y, \bar t_y)$ be the maximal interval to the 
right on which $y$ has no $H$-points. We prove that
\begin{equation}\label{e31}
\bar T_y = \bar t_y\,.
\end{equation}
Assume, to the  contrary, that  $\bar T_y < \bar t_y$. 
Then $\bar T_y$ is $H$-point of $y$, and according to \eqref{e6},
\begin{equation}\label{e32}
y_1(\bar T_y) = y_2 (\bar T_y)=0\,.
\end{equation}
 From this and from Lemma~\ref{le1}, $y$ is oscillatory on $[T_y, \bar T_y)$.
Moreover, Lemmas~\ref{le3}, \ref{le8} (ii) and  \ref{le7} applied to $y$ and  
the interval $[T_y, \bar T_y)$ imply the validity of (1)--(4) in all cases (i)--(iii).
Note, that case (i) follows from Lemmas \ref{le8} (ii) and \ref{le7}, case 
(ii) from Lemma \ref{le8} (ii), and case (iii) from Lemma \ref{le3}.
But, according to Lemmas~\ref{le3} and \ref{le4},
the sequences of the absolute values of all local extrema of $y_1$ and $y_2$ 
are monotone and they have  the opposite kind of monotonicity. 
Hence, the only case where \eqref{e32} holds is $y_1(t) \equiv y_2(t) \equiv 0$ 
in a left neighbourhood of $\bar T_y$. But that contradicts $y$  being 
oscillatory; thus \eqref{e31} holds and $y$ has no $H$-points on  
$[T_y, \bar t_y)$.

If either $t_y=T_y$ or if $y$ has no $H$-points on $[t_y, T_y)$, then the 
statement follows  from Lemmas~\ref{le3}, \ref{le8} (ii) and  \ref{le7}.
Let $c\in[t_y, T_y)$  be the maximal $H$-point of $y$.
 Then \eqref{e6} implies
 \begin{equation}\label{e33}
 y_1(c) = y_2(c) =0\,,
 \end{equation}
and according to Lemma~\ref{le1}, a decreasing sequence 
$\{\bar t_k\}_{k=1}^\infty$ exists such that 
$\bar t_k\in (c, T_y]$, $y_1(\bar t_k) =0$, $k=1, 2, 3, \dots$ and  
$\lim_{k\to \infty}\bar t_k=c$. From this and from \eqref{e1}--\eqref{e3},
 a sequence $\{\bar \tau_k\}_{k=1}^\infty$, of zeros of $y_2$ exists such that 
$\bar t_k>\bar \tau_k> \bar t_{k+1}$ and $\lim_{k\to \infty}\bar \tau_k=c$. 
As the intervals $J_k=[\bar t_k, T_y)$ are  without $H$-points, we can apply 
Lemmas \ref{le3}, \ref{le8} (ii) and  \ref{le7} on $J_k$. 
If, for simplicity, $y_1 (T_y)>0$, then the sequence of all local maxima of 
$y_1$ on $J_k$ is non-increasing and greater
or equal to $y_1(T_y)$. Hence, if $k\to \infty$, 
$\{y_1(\bar \tau_y)\}_{k=1}^\infty$ is non-decreasing and
 $y_1(c) = \lim_{k\to \infty}y_1(\bar \tau_k) \geq y_1(T_y)>0$. 
This contradicts \eqref{e33} and proves that $H$-points do not exist 
on $[t_y, \bar t_y)$,  which  is impossible.
\end{proof}

The following result can be proved similarly as in Theorem \ref{th1}.

\begin{theorem}\label{th3}
Suppose
\[
\Big|\frac{f_2(t,u,v)}{f_1(t,u,v)} \Big|\text{ is non-decreasing (non-increasing) 
with respect to $t$}
\]
on $D$, $uv\ne 0$, and either

(i) 
\[
 \frac{\partial}{\partial v} \frac{f_2(t,u,v)}{f_1(t,u,v)}
\text{ is continuous on $D$, $uv\ne 0$},
\]
 or

(ii) for any integer $m$ there is a continuous function 
$g_m: (0,m]\to (0,\infty)$  such that
\[
\frac{g_m(|u|)|f_1(t,u,v)|}{f_2(t,u,v)}  \text{ is non-increasing }
\]
with respect to $u$ for $|u|\in \big[\frac{1}{m}, m\big]$,  for any 
$t\in [0, m]$, and $|v|\in (0, m]$, or

 (iii) for any integer $m$ there is a continuous function 
$\bar g_m: (0,m]\to (0,\infty)$  such that
\begin{equation*}
\frac{\bar g_m(|v|)|f_2(t,u,v)|}{f_1(t,u,v)}  \text{ is non-increasing }
\end{equation*}
with respect to $v$ for $v\in (0, m]$ and $v\in [-m, 0)$,  
for any $t\in [0, m]$, and $|u|\in \big[\frac{1}{m}, m\big]$.

Let $y$  be an oscillatory solution of \eqref{e1} defined on 
$[t_y, \bar t_y]\subset R_+$. Then:
\begin{enumerate}
\item There exists  no $H$-point of $y$, $y$ can not be defined at 
$t= \bar t_y$ and all zeros of  $y_2$ can be described by  increasing sequence
$\{t_k\}_{k=1}^\infty$.

\item The sequence of all positive local extrema of $y_2$  is non-decreasing 
(is non-increasing).

\item The sequence of the absolute values of all negative local extrema 
of $y_2$  is non-decreasing (is non-increasing).

\item If, moreover,
\begin{equation*}
f_1(t,u,-v) =- f_1(t,u,v)\,,\quad 
f_2(t,u,-v) = f_2(t,u,v) 
\end{equation*}
on $D$, then the sequence $\big\{|y_2(t_k)|\big\}_{k=1}^\infty$  
of the absolute values of all local extrema  of $y_2$  is non-decreasing 
(is non-increasing).
\end{enumerate}
\end{theorem}

\section{Applications}

We apply  our results to equation \eqref{e4}.

\begin{theorem}\label{th5}
Suppose $f(t,-u,v) = -f(t,u,v)$ on $D$ and
$$
a^{1/p}(t) |f(t,u,v)| \quad \text{is non-decreasing (non-increasing)}
$$
with respect to $t$ on $D$. Let $y$ be  an oscillatory solution of \eqref{e4} 
defined on $[t_y, \bar t_y)$ and  $\{\tau_k\}_{k=1}^\infty$ be the 
increasing sequence of all zeros of  $y^{[1]}$ on $[t_y, \bar t_y)$.
Let either

 (i)   $\frac{\partial}{\partial u} f(t,u,v)$ be continuous
on $D$, $uv\ne 0$, or

 (ii) For any integer $m$ there is a positive function $g_m\in C^0(0, m]$  
such that
$$
g_m (u)  f(t,u,v) \text{ is non-decreasing with respect to $u$}
$$
for $u\in (0, m]$, $t\in [0,m]$ and $|v|\in \big[\frac{1}{m},m\big]$, or

(iii) for any integer $m$ there is a positive function $\bar g_m\in C^0(0,m]$  
such that
$$
\bar g_m (|v|)  f(t,u,v)\operatorname{sgn}v \text{ is non-decreasing with respect to $v$}
$$
for $|v|\in \big[\frac{1}{m},m\big]$, $t\in [0,m]$ and $u\in (0, m]$.

Then  $\big\{ |y(\tau_k)|\big\}_{k=1}^\infty$ is non-increasing  (non-decreasing).
\end{theorem}

\begin{proof}
The result follows from Theorem \ref{th1} since
  $f_1(t,u,v)=a^{-1/p}(t) |v|^{1/p}\operatorname{sgn}v$ 
and $f_2(t,u,v)=-f_1(t,u,v)$. If we  denote the function $g_m$ from 
Theorem \ref{th1} (ii) by $\tilde g_m$, then 
$\tilde g_m(z)=1/g_m(z)$. Similarly, if we denote $\bar g_m$ 
from Theorem~\ref{th1} (iii) by  ${\bar g}_m$, then
${\bar g}_m(z)= \bar g_m(z) z^{1/p}$.
\end{proof}

\begin{theorem}\label{th44}
Suppose $f(t, u, -v)= f(t,u,v)$ on  $D$ and
$a^{1/p}(t) |f(t,u,v,)|$  is  non-decreasing (non-increasing)
with respect to $t$ on $D$. Let $y$ be an oscillatory solution 
of \eqref{e4} defined on $[t_y, \bar t_y)$ and $\{ t_k\}_{k=1}^\infty$ 
be the increasing sequence of all zeros of $y$ on $[t_y, \bar t_y)$. 
Let either

(i)
$\frac{\partial}{\partial v}  f(t,u,v)$  be continuous on $D$, $uv \ne 0$, or


(ii) for any integer $m$ there is a positive function $g_m\in C^0(0,m]$  such that
$ g_m (|u|)  f(t,u,v)$ is non-increasing with respect to $u$
for $|u|\in \big[\frac{1}{m},m\big]$, $t\in [0,m]$ and $v\in (0, m]$, or

(iii)  for any integer $m$ there is a positive function $\bar g_m\in C^0(0, m]$  
such that
$\bar g_m (v)  |f(t,u,v)|$ is non-increasing with respect to $v$
for $v\in \in (0, m]$,  $t\in [0,m]$ and $|u|\in  \big[\frac{1}{m},m\big]$.

 Then $\big\{ |y^{[1]}(t_k)|\big\}_{k=1}^\infty$ is non-decreasing (non-increasing).
\end{theorem}


The proof of the above theorem is similar to that of Theorem \ref{th5},
 using Theorem \ref{th3}.

\begin{remark}\label{re11}
Theorem \ref{thA} (i) is a special case of Theorem \ref{th5} (iii) 
and Theorem \ref{thA}  (ii) follows from Theorem \ref{th44} (iii).
\end{remark}


Finally, we formulate our results for the equation
\begin{equation}\label{e402}
y^{[1]} + r(t) \, \bar f(y)\, h(y^{[1]}) =0
\end{equation}
where $p>0$, $y^{[1]}$ is given by \eqref{e44}, $\bar f\in C^0(\mathbb{R})$,
$h\in C^0(\mathbb{R})$,  $\bar f(u) u>0$ for $u\ne 0$, and $h(v)>0$ on $\mathbb{R}$.

\begin{corollary}\label{co1}
Suppose $a^{1/p} r$ is  non-decreasing (non-increasing) on $\mathbb{R}_+$.
Let $y$ be  an oscillatory solution of \eqref{e402} defined on 
$[t_y, \bar t_y)$ and $\{ t_k\}_{k=1}^\infty$ and $\{\tau_k\}_{k=1}^\infty$ 
be the increasing sequences of all zeros of $y$ and $y^{[1]}$ 
on $[t_y, \bar t_y)$, respectively.

(i)  If  $\bar f(-u) =-\bar f(u)$ on $\mathbb{R}$, then the sequence
$\big\{ |y(\tau_k)|\big\}_{k=1}^\infty$
is non-increasing (non-decreasing).


(ii) If $h(-v)=h(v)$ on  $\mathbb{R}$, then the sequence
$\big\{ |y^{[1]} (t_k)|\big\}_{k=1}^\infty$
is non-decreasing (non-increasing).
\end{corollary}

\begin{proof}
Suppose $a^{-1/p} r$ is non-decreasing.
Put $f(t,u,v)= r(t)\, \bar f(u) \, h(v)$.
Case (i) follows from Theorem \ref{th5} (ii) 
with $g_m(u) =(\bar f (u))^{-1}$. 
Case (ii) follows from Theorem \ref{th44} (iii) with  
$\bar g_m(v)=\frac{1}{h(v)}$.
\end{proof}

\begin{remark}\label{re6} \rm
Note that \cite[Theorems 3.6 and 3.7]{9} and Theorem \ref{thB}  are  
special cases of Corollary \ref{co1} for $h\equiv 1$. 
Some results in \cite[Theorems 4.2 and 4.6]{11} are special cases of
 Corollary \ref{co1} (with $p=1$, $h\equiv 1$).
\end{remark}


\subsection*{Acknowledgements}
This work was supported by  grant GAP 201/11/0768 from the Grant Agency 
of the Czech Republic.


\begin{thebibliography}{00}

\bibitem{7}  Bartu\v{s}ek, M.;
\emph{Monotonicity theorems for second order non-linear differential equations}, 
Arch. Math. (Brno) XVI, No. 3, 1980, 127--136.

\bibitem{10}  Bartu\v{s}ek, M.;
\emph{On properties of oscillatory solutions of two-dimensional differential systems},
 Trudy Instituta Prikladnoj Matematiky im. Vekua, 8, 1980, 5--11. (In Russian.)

\bibitem{3}  Bartu\v{s}ek, M.;
\emph{On Properties of Oscillatory Solutions of Nonlinear Differential Equations 
of $n$-th Order}, In: Diff. Equat. and Their Appl., Equadiff 6, 
Lecture Notes in Math., Vol. 1192, Berlin 1985, pp. 107--113.

\bibitem{1}  Bartu\v{s}ek, M.;
\emph{On oscillatory solutions of differential inequalities}, 
Czechoslovak Math. J. {\bf 42} (117) (1992).

\bibitem{2}  Bartu\v{s}ek, M.;
\emph{Asymptotic Properties of  Oscillatory Solutions of Differential Equation 
of the $n$-th Order}, Folia Fac. Sci. Natur. Univ. Masaryk. Brun., Math. 3,
 Masaryk University, 1992.

\bibitem{9}  Bartu\v{s}ek, M.; Kokologiannaki, Ch. G.;
\emph{Monotonicity properties of oscillatory solutions of differential 
equation $(a(t) |y'|^{p-1} y')'+ f(t, y, y')=0$}, 
Arch. Math. (Brno) 49, 2013, 121--129.

\bibitem{6} Bihari, I.;
\emph{Oscillation and monotonicity theorems concerning non-linear differential 
equations of the second order}, Acta Acad. Sci. Hung., IX, No. 1--2, 1958, 83--104.

\bibitem{12}  Do\v{s}l\'{y}, O.; \v{R}eh\'{a}k, P.;
\emph{Half-linear Differential Equations}, Elsevier, Amsterdam, 2005.

\bibitem{8} Hartman, Ph.;
\emph{Ordinary Differential Equations}, 
John Willey \& Sons, New York - London - Sydney, 1964.


\bibitem{4}  Kiguradze, I. T.; Chanturia, T. A.;
\emph{Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential 
Equations}, Kluwer, Dordrecht 1993.

\bibitem{13}  Milloux, H.;
\emph{Sur l'\'{e}quation diff\'{e}rentielle $x''+A(t) x =0$}, 
Prace Math., 41 (1934), 39--53.

\bibitem{5}  Mirzov, J. D.;
\emph{Asymptotic  Properties of Solutions of Systems of Nonlinear 
Nonautonomous Ordinary Differential Equations}, 
Folia Fac. Sci. Natur. Univ. Masaryk. Brun., Math. 14, Masaryk University, 2004.

\bibitem{11} Rohleder, M.;
\emph{On the existence of oscillatory solutions of the second  order nonlinear ODE},
 Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 51, No. 2, 2012, 107--127.

 \end{thebibliography}

 \end{document}










\begin{thebibliography}{WI}
\itemsep=5pt



\bibitem{1}  Bartu\v{s}ek M.,  On Oscillatory Solutions of Differential Inequalities, Czechoslovak Math. J. {\bf 42} (117) 1992,  45--52.


\bibitem{2}  Bartu\v{s}ek M., Asymptotic properties of  oscillatory solutions of differential equation of the $n$-th order, Folia Fac. Sci. Natur. Univ. Masaryk. Brun., Math. 3, Masaryk University, 1992.

\bibitem{3}  Bartu\v{s}ek M.,
On Properties of Oscillatory Solutions of Nonlinear Differential Equations of $n$-th Order, In: Diff. Equat. and Their Appl., Equadiff 6, Lecture Notes in Math., Vol. 1192, Berlin 1985, pp. 107--113.


\bibitem{4}
 Kiguradze I. T. and Chanturia T. A.,  Asymptotic properties of solutions of nonautonomous ordinary differential equations, Kluwer, Dordrecht 1993.

\bibitem{5}
 Mirzov J. D.,  Asymptotic  properties of solutions of systems of nonlinear nonautonomous ordinary differential equations, Folia Fac. Sci. Natur. Univ. Masaryk. Brun., Math. 14, Masaryk University, 2004.


\bibitem{6}
Bihari I., Oscillation and Monotonicity Theorems Concerning Non-linear Differential Equations of the Second Order, Acta Acad. Sci. Hung., IX, No. 1--2, 1958, 83--104.



\bibitem{7}
 Bartu\v{s}ek M.,
Monotonicity Theorems for Second Order Non-linear Differential Equations, Arch. Math. (Brno) XVI, No. 3, 1980, 127--136.


\bibitem{8} Hartman Ph., Ordinary differential equations, John Willey \& Sons, New York - London - Sydney, 1964.

\bibitem{9}
 Bartu\v{s}ek M., Kokologiannaki Ch. G., Monotonicity Properties of Oscillatory Solutions of Differential Equation $(a(t) |y'|^{p-1} y')'+ f(t, y, y')=0$, Arch. Math. (Brno) 49, 2013, 121--129.

 \bibitem{10}
 Bartu\v{s}ek M.,
 On Properties of Oscillatory Solutions of Two-dimensional Differential Systems, Trudy Instituta Prikladnoj Matematiky im. Vekua, 8, 1980, 5--11. (In Russian.)

 \bibitem{11}
 Rohleder M., On the Existence of Oscillatory Solutions of the Second  Order Nonlinear ODE,    Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 51, No. 2, 2012, 107--127.

 \bibitem{12}
 Do\v{s}l\'{y} O., \v{R}eh\'{a}k P., Half-linear differential equations, Elsevier, Amsterdam, 2005.

 \bibitem{13}
 Milloux H., Sur l'\'{e}quation diff\'{e}rentielle $x''+A(t) x =0$, Prace Math., 41, 1934, 39--53.

 \end{thebibliography}
 \end{document}