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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 162, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/162\hfil Iterative oscillation results]
{Iterative oscillation results for second-order differential
 equations with advanced argument}

\author[Irena Jadlovsk\'a \hfil EJDE-2017/162\hfilneg]
{Irena Jadlovsk\'a}

\address{Irena Jadlovsk\'a \newline
Department of Mathematics and Theoretical Informatics,
Faculty of Electrical Engineering and Informatics,
Technical University of Ko\v{s}ice,
B.~N\v{e}mcovej 32, 042 00 Ko\v{s}ice, Slovakia}
\email{irena.jadlovska@tuke.sk}

\thanks{Submitted April 28, 2017. Published July 4, 2017.}
\subjclass[2010]{34C10, 34K11}
\keywords{Linear differential equation; advanced argument; second-order;
\hfill\break\indent  oscillation}

\begin{abstract}
 This article concerns the oscillation of solutions to a linear second-order
 differential equation with advanced argument.
 Sufficient oscillation conditions involving limit inferior are given which
 essentially improve  known results. We base our technique  on the iterative
 construction of solution estimates and some of the recent ideas developed
 for first-order advanced differential equations.  We demonstrate the advantage
 of our results on Euler-type advanced equation. Using MATLAB software,
 a comparison of the effectiveness of  newly obtained criteria as well
 as the necessary iteration length in  particular cases are discussed.
\end{abstract}

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

\section{Introduction}

We consider the  linear second-order advanced differential equation 
 \begin{equation}\label{e}
y''(t)+q(t)y(\sigma(t)) = 0, \quad t\ge t_0>0,
 \end{equation}
where $q\in \mathcal{C}([t_0,\infty))$ and $\sigma\in \mathcal{C}^1([t_0,\infty))$ 
are such that $q(t)>0$,  $\sigma(t)\ge t$ and $\sigma'(t)\ge 0$.

By a solution of  \eqref{e},  we understand  a nontrivial function
$y\in \mathcal{C}^2([t_0,\infty))$,  which satisfies \eqref{e} on $[t_0,\infty)$.
 We restrict our attention to those solutions $y$ of \eqref{e}  which 
satisfy  $\sup\{  |y(t)|: t\ge T\}>0$, for all  $T\ge t_0$.  
We recall that a solution of \eqref{e} is said to be oscillatory if it 
has arbitrarily large zeros, and otherwise  it is said to be nonoscillatory.
Equation \eqref{e} is called oscillatory if all of its solutions are 
oscillatory as well.

Differential equations with deviating argument are deemed to be adequate in 
modeling of the countless processes in all areas of science.
As is well known,  a distinguishing feature of \emph{delay differential equations}
under consideration is  the dependence of the  evolution rate of the 
processes described by such equations on the past history.
 This consequently results in predicting the future in a more reliable and 
efficient way, explaining at the same time many qualitative phenomena 
such as periodicity, oscillation or instability. The  concept of the delay 
incorporation into systems  plays an essential role in modeling  
to represent time taken to complete some hidden processes, see \cite{el,a4}.

Contrariwise, \emph{advanced differential equations} can find use in many
 applied problems  whose  evolution rate depends not only on the present, 
but also on the future. Therefore, an advance could be introduced into 
the equation to highlight the influence of potential future actions, 
which are available at the presence and should be beneficial in the process 
of decision making.  For instance, population dynamics, economical problems 
or mechanical control engineering are typical fields where such phenomena 
is believed to occur (see  \cite{el} for details).

The first oscillation results for  differential equations with deviating
 argument were obtained in the classical paper by Fite  \cite{fite} in $1921$. 
Since then, a great deal of the effort has been made by many researchers 
in order to advance the knowledge further (for the summary of most 
essential contributions on the subject, see,  e.g.,  
monographs \cite{a1,a2,a4,erbe,Lad} and the references cited therein).

Most of the literature, however, has been devoted to the investigation 
of differential equations with  delay argument,  and very little is known up 
to now about those with advanced arguments.  In particular, two main approaches  
for the investigation of \eqref{e} have appeared (see  \cite[Chapter 2]{a2}, 
\cite{dz,Kop,KN}). Taking Kusano's and Naito's comparison 
theorem \cite[Theorem 1]{KN} into account, the oscillatory behavior of \eqref{e} 
can be  treated as that of the ordinary differential equation
\begin{equation}\label{b}
y''(t)+q(t)y(t) = 0.
\end{equation}
It seems obvious that in such a case, all  impact of the advanced argument is 
completely neglected.  On the other hand, an another  approach has been 
based on the comparison with the first-order advanced differential equation
\begin{equation}\label{a}
y'(t)- \Big(\int_{t}^{\infty}q(s)\mathrm{d} s\Big)y(\sigma(t)) = 0,
 í\end{equation}
in the sense that oscillation of  \eqref{e} is inherited from that of 
\eqref{a} (see \cite[Theorem 2.1.12]{a2}).  Here, the advance may generate
 oscillations.  In particular, by applying the famous Hille's result \cite{hille} 
and  the well-known oscillation criterion due to Ladas \cite{ladas} 
to  \eqref{b} and \eqref{a}, respectively,  one can immediately get the 
following couple of oscillation criteria for \eqref{e}:
\begin{gather}\label{hille0}
\liminf_{t\to \infty}t\int_{t}^{\infty}q(s)\mathrm{d} s>\frac{1}{4}, \\
\label{ladde}
\liminf _{t\to \infty}\int_{t}^{\sigma(t)}\int_{u}^{\infty}q(s)\mathrm{d} s\,\mathrm{d} u
 >\frac{1}{\mathrm{e}}.
\end{gather}
The  question naturally arises: 
\begin{quote}
Is it possible to establish an effective 
oscillation result of Hille type which simultaneously takes  into account
 the presence of the advance and the second order nature of the equation 
studied as well?
\end{quote}
 The purpose of this article is to give an affirmative answer to this quastion, 
i.e.,  to propose an approach for investigation the  \eqref{e} when both 
above-mentioned conditions \eqref{hille0} and \eqref{ladde} fail.  
The use is made of some of the recent results developed for first-order 
delay/advanced differential equations which have been based on the iterative 
application of the Gr\"{o}nwall's inequality (see \cite{braverman,chat}). 
This technique enables one to obtain sufficient conditions for oscillation 
of \eqref{e} involving $\liminf$, which essentially use  value of the advanced 
argument. Our method of the proof that is quite different from the very 
recent  study \cite{bac} is essentially new.

Finally, we demonstrate the advantage of our results on Euler-type advanced 
equations. Using MATLAB software,  a comparison of the effectiveness of  
newly obtained criteria is provided as well as the necessary  iteration 
length in  particular cases.

 \section{Main results}

 In this section, we establish a number of new  oscillation criteria for \eqref{e}.

In the sequel, all functional inequalities are assumed to hold eventually, 
that is, they are satisfied for all $t$ large enough.
 

 \begin{remark} \label{rmk2} \rm
  As $-y(t)$ is also a solution of \eqref{e},
  we may restrict ourselves only to the case where $y(t)$ is eventually positive.
 \end{remark}

\begin{remark} \label{rmk3} \rm
 In view of the well-known Leighton's criterion \cite{lei} and the comparison
 theorem \cite[Theorem 1]{KN},   equation  \eqref{e} is oscillatory if
 $\int^{\infty}q(s)\mathrm{d} s = \infty$.
 Therefore, we assume throughout the paper that 
$\int^{\infty}q(s)\mathrm{d} s<\infty$.
\end{remark}

We define
   $$ 
\tilde{q}(t)=q(t)\Big(1+ \int_{t}^{\sigma(t)}\int_{u}^{\infty}q(s)\mathrm{d} s\, \mathrm{d} u
 \Big).
$$


\begin{theorem} \label{thm1}
 Assume that the second-order differential equation
 \begin{equation}\label{sec}
 y''(t)+\tilde{q}(t)y(t) = 0
 \end{equation}
 is oscillatory. Then  \eqref{e} is oscillatory.
\end{theorem} 

\begin{proof}
 Suppose to the contrary that $y$ is a positive solution of \eqref{e} on 
$[t_0,\infty)$. Obviously,  there exists $t_1\ge t_0$ such that
 \begin{equation}\label{struct}
 y(t)>0, \quad y'(t)>0, \quad y''(t)\le 0, \quad \text{for}\quad t\ge t_1.
 \end{equation}
 An integration of \eqref{e} from $t$ to $\infty$ in view of \eqref{struct} leads
 to
 \begin{align}
 y'(t)
&\ge  \int_{t}^{\infty}q(s)y(\sigma(s))\mathrm{d} s \label{eqa1}  \\
&\ge y(\sigma(t))\Big(\int_{t}^{\infty}q(s)\mathrm{d} s\Big)\label{eqa2}.
 \end{align}
Integrating \eqref{eqa2} from $t$ to $\sigma(t)$, we have
 \begin{equation}\label{ee}
 y(\sigma(t)) \ge  y(t)+ \int_{t}^{\sigma(t)}y(\sigma(u))\int_{u}^{\infty}q(s)\mathrm{d} s\,\mathrm{d} u.
 \end{equation}
 Using that  $y(\sigma(t))\ge y(t)$ in \eqref{ee}, one obtains
\begin{align*}
y(\sigma(t)) 
& \ge y(t)+y(\sigma(t))\int_{t}^{\sigma(t)}\int_{u}^{\infty}q(s)\mathrm{d} s\, \mathrm{d} u \\
& \ge y(t)\Big(1+ \int_{t}^{\sigma(t)}\int_{u}^{\infty}q(s)\mathrm{d} s\, \mathrm{d} u \Big).
\end{align*}
Combining the last inequality and \eqref{e} yields
 \begin{equation}\label{keyy}
 y''(t)+ \tilde{q}(t)y(t)\le 0.
 \end{equation}
 Define $w(t) = y'(t)/y(t)$ to  see that $w(t)$ satisfies  
the first-order Riccati inequality
 \begin{equation*}
 w'(t) - \tilde{q}(t)-w^2(t)\le 0,
 \end{equation*}
 which in turn implies (see \cite[Lemma 2.2.1]{a1}) that the
 equation \eqref{sec} has a positive solution; a   contradiction.
    The proof is complete.
\end{proof}

\begin{corollary} \label{coro1}
 If
 \begin{equation}\label{hille}
 \liminf _{t\to \infty} t\int_{t}^{\infty}\tilde{q}(s)\mathrm{d} s>\frac{1}{4},
 \end{equation}
 then \eqref{e} is oscillatory.
\end{corollary}

\begin{remark} \label{rmk4} \rm
 The criterion \eqref{hille} of Hille type takes the presence of the 
advanced argument into account and thus can be applied even if the 
corresponding known one \eqref{hille0} fails.
\end{remark}

The lemma below is a slight modification of \cite[Lemma 1]{jaros}
 originally given for the first-order equation with delayed argument. 
For the sake of clarity, we also include its complete proof.

\begin{lemma} \label{lem1}
 Let $y(t)$ be an eventually positive solution of \eqref{e}. Then 
 \begin{gather}\label{rho}
  \rho: =\liminf _{t\to \infty} \int_{t}^{\sigma(t)}\int_{u}^{\infty}q(s)\mathrm{d} s 
\mathrm{d} u\le \frac{1}{\mathrm{e}}, \\
\label{esti}
 \liminf _{t\to \infty} \frac{y(\sigma(t))}{y(t)} \ge \lambda,
\end{gather}
 where $ \lambda$ is the smaller root of the transcendental equation
 $\lambda = \mathrm{e}^{\rho \lambda}$.
\end{lemma}
 
\begin{proof}
Let 
$$
\alpha = \liminf _{t\to \infty} \frac{y(\sigma(t))}{y(t)}.
$$
Dividing  \eqref{eqa2} by $y(t)$ and integrating from $t$ to $\sigma(t)$, we have
\begin{equation*}
\ln \Big(\frac{y(\sigma(t))}{y(t)} \Big)
\ge \int_{t}^{\sigma(t)}\frac{y(\sigma(u))}{y(u)}\int_{u}^{\infty}q(s)
\mathrm{d} s\,\mathrm{d} u,
\end{equation*}
or
\begin{equation*}
\frac{y(\sigma(t))}{y(t)}
\ge \exp\Big(\int_{t}^{\sigma(t)}\frac{y(\sigma(u))}{y(u)}\int_{u}^{\infty}q(s)
\mathrm{d} s\,\mathrm{d} u \Big),
\end{equation*}
which clearly implies
\begin{equation}\label{aa}
\alpha \ge \mathrm{e}^{ \rho \alpha}.
\end{equation}
Note that \eqref{aa} is impossible when $\rho>1/\mathrm{e}$, since 
$\lambda <\exp \rho \lambda$ for all $\lambda>0$ and so \eqref{e} has no 
positive solutions. If $\rho\le 1/\mathrm{e}$, then the equation 
$\lambda = \exp \rho \lambda$ has roots $\lambda\le \tilde{\lambda}$, with 
$\lambda = \tilde{\lambda} = \mathrm{e}$ if and only if $\rho =1/\mathrm{e}$ and \eqref{aa} holds
 if and only if $\lambda\le \alpha\le \tilde{\lambda}$.
\end{proof}

As an  immediate consequence of Lemma \ref{lem3}, we have the following result,
which applies when \eqref{ladde} fails.

 \begin{theorem} \label{thm2}
 Let  \eqref{rho} hold and $\lambda$ be as in Lemma \ref{lem1}. Assume that  
the second-order differential equation
 \begin{equation}\label{ab}
 y''(t)+k\lambda q(t)y(t)= 0
 \end{equation}
 is oscillatory for some $k\in (0,1)$. Then \eqref{e} is oscillatory.
\end{theorem}

\begin{proof}
 Suppose to the contrary that $y$ is a positive solution of \eqref{e} on 
$[t_0,\infty)$.
Then it follows from Lemma \ref{lem1} that there exists 
$t_1\in [t_0,\infty)$ such that, for every $k\in (0,1)$,
\begin{equation}\label{key1}
\frac{y(\sigma(t))}{y(t)}\ge k \lambda \quad \text{on } [t_1,\infty).
\end{equation}
Using \eqref{key1} in \eqref{e}, it is easy to see that $y$ is a positive
 solution of the inequality
\begin{equation*}
y''(t)+ k\lambda y(t)\le 0.
\end{equation*}
The same as in the proof of Theorem \ref{thm1}, we can conclude that the
 corresponding equation  \eqref{ab} also has a positive solution, a contradiction. 
The proof is complete.
\end{proof}


\begin{corollary}\label{coro2}
 Let \eqref{rho} hold and $\lambda$ be as in Lemma \ref{lem1}. If
 \begin{equation}\label{hille2}
 \liminf _{t\to \infty} t\int_{t}^{\infty} q(s)\mathrm{d} s>\frac{1}{4\lambda},
 \end{equation}
 then \eqref{e} is oscillatory.
\end{corollary}


In the next lemma, we derive some useful estimates which are based on the 
iterative application of the Gr\"{o}nwall  inequality and permit us to 
improve all the previous results.

\begin{lemma} \label{lem3}
 Let $y(t)$ be an eventually positive solution of \eqref{e}.  Define
 \begin{gather*}%\label{a_1}
 a_1(s,t) = \exp\Big(\int_{t}^{s}\int_{u}^{\infty}q(x)\mathrm{d} x\, \mathrm{d} u \Big), \\
 \label{a_n}
 a_{n+1}(s,t)  
=  \exp\Big(\int_{t}^{s}\int_{u}^{\infty}q(x)a_{n}(\sigma(x),u)\mathrm{d} x\, \mathrm{d} u 
\Big),\quad  n \in \mathbb{N}.
 \end{gather*}
 Then
 \begin{equation}\label{est}
 y(s)\ge y(t)a_n(s,t), \quad s\ge t,
 \end{equation}
 for $t$ large enough.
\end{lemma}

\begin{proof}
 We will prove  Lemma \ref{lem3} by mathematical induction. 
Since  $y$ is an eventually positive solution of \eqref{e}, there exists 
$t_1\ge t_0$ such that $y$ satisfies \eqref{struct} on $[t_1,\infty)$. 
Thus  $y(\sigma(t))\ge y(t)$ and by virtue of \eqref{eqa2}, we have
 \begin{equation*}
 y'(t)\ge y(t)\int_{t}^{\infty}q(s)\mathrm{d} s.
 \end{equation*}
Applying the Gr\"{o}nwall inequality,  we obtain
 \begin{equation}\label{eqd}
 \begin{split}
 y(s)&\ge y(t)\exp\Big(\int_{t}^{s}\int_{u}^{\infty}q(x)\mathrm{d} x\,\mathrm{d} u \Big), \quad 
s\ge t\ge t_1,
\end{split}
 \end{equation}
that is,  the estimate \eqref{est} is valid for $n = 1$.

 Next, we assume  that \eqref{est} holds for some $n>1$. Then
 \begin{equation}\label{est1}
y(\sigma(s))\ge y(t)a_n(\sigma(s),t), \quad \sigma(s)\ge t.
 \end{equation}
 Substituting \eqref{est1} into \eqref{eqa1} yields
 \[
 y'(t)\ge \int_{t}^{\infty}q(s)y(\sigma(s))\mathrm{d} s
 \ge  y(t)\int_{t}^{\infty}q(s)a_n(\sigma(s),t)\mathrm{d} s.
\]
 Again, applying the Gr\"{o}nwall inequality, we have
 \begin{equation}\label{est2}
 y(s)\ge y(t)\exp\Big(\int_{t}^{s}\int_{u}^{\infty}q(x)
a_n(\sigma(x),u)\mathrm{d} x\,\mathrm{d} u \Big),
 \end{equation}
 i.e.,
 \begin{equation*}
 y(s)\ge y(t)a_{n+1}(s,t).
 \end{equation*}
This established the induction step and completes the proof.
\end{proof}

 \begin{theorem} \label{thm3}
Let $a_n(t,s)$ be as in Lemma \ref{lem3}. Assume that  the first-order 
advanced differential equation
 \begin{equation}\label{frst}
 y'(t) - \Big(\int_{t}^{\infty}q(s)a_n(\sigma(s),\sigma(t))\mathrm{d} s\Big)
y(\sigma(t))  = 0
 \end{equation}
is oscillatory for some $n\in \mathbb{N}$. Then  \eqref{e} is oscillatory.
\end{theorem}

\begin{proof}
 Suppose to the contrary that $y$ is a positive solution of \eqref{e} on
 $[t_0,\infty)$.
Then  there exists $t_1\ge t_0$ such that $y$ satisfies \eqref{struct} on 
$[t_1,\infty)$. It follows from Lemma \ref{lem3} that
\begin{equation}\label{key}
y(\sigma(s))\ge y(\sigma(t))a_n(\sigma(s),\sigma(t)), \quad s\ge t,
\end{equation}
 for some $n\in \mathbb{N}$ and $t$ large enough. Integrating \eqref{e} from $t$ 
to $\infty$ and using \eqref{key}, we are led to
 \begin{equation}\label{eq6}
 y'(t)\ge \int_{t}^{\infty}q(s)y(\sigma(s))\mathrm{d} s
  \ge y(\sigma(t))\int_{t}^{\infty}q(s)a_n(\sigma(s),\sigma(t))\mathrm{d} s,
 \end{equation}
 which means that  $y$ is a positive solution of the first-order advanced 
differential inequality
 \begin{equation*}\label{frstt}
 y'(t)- \Big(\int_{t}^{\infty}q(s)a_n(\sigma(s),\sigma(t))\mathrm{d} s\Big)
y(\sigma(t))\ge 0.
 \end{equation*}
In view of  \cite[Theorem 1]{philos}, the equation \eqref{frst} also has a 
positive solution, a contradiction. The proof is complete.
\end{proof}

\begin{corollary}\label{coro3}
 Let $a_n(t,s)$ be as in Lemma \ref{lem3}. If
 \begin{equation}\label{c3}
  \liminf _{t\to \infty}\int_{t}^{\sigma(t)}\int_{u}^{\infty}q(s)
a_{n}(\sigma(s),\sigma(u))\mathrm{d} s\,\mathrm{d} u>\frac{1}{\mathrm{e}},
 \end{equation}
 for some $n\in \mathbb{N}$, then \eqref{e} is oscillatory.
\end{corollary}

\begin{remark} \label{rmk5} \rm
 The above theorem permits us to deduce oscillation of \eqref{e} from that 
of the first-order advanced differential equation  \eqref{frst}.  
One can see that, even for $n=1$,  the criterion \eqref{c3} is sharper 
than \eqref{ladde} and thus provides a better result.
\end{remark}

\begin{theorem} \label{thm4}
 Assume that the second-order differential equation
 \begin{equation}\label{ac}
 y''(t)+q(t)a_n(\sigma(t),t)y(t) = 0
 \end{equation}
 is oscillatory for some $n\in N$.  Then \eqref{e} is oscillatory.
\end{theorem}

\begin{proof} 
Suppose to the contrary that $y$ is a positive solution of \eqref{e} on 
$[t_0,\infty)$.
Then  there exists $t_1\ge t_0$ such that $y$ satisfies \eqref{struct} on 
$[t_1,\infty)$. It follows from Lemma \ref{lem3} that
\begin{equation}\label{keyyy}
y(\sigma(t))\ge y(t)a_n(\sigma(t),t)
\end{equation}
for some $n\in \mathbb{N}$ and $t$ large enough.
Using \eqref{keyyy} in \eqref{e}, we see that $y$ is a positive solution of
\begin{equation*}
y''(t)+ q(t)a_n(\sigma(t),t)y(t)\le 0.
\end{equation*}
As in the proof of Theorem \ref{thm1}, we can see that the 
corresponding equation \eqref{ac} also has a positive solution, a contradiction. 
The proof is complete.
\end{proof}

\begin{corollary}\label{coro34}
 If
 \begin{equation}
 \liminf _{t\to \infty} t\int_{t}^{\infty}q(s)a_n( \sigma(s),s)\mathrm{d} s>\frac{1}{4}
 \end{equation}
 for some $n\in\mathbb{N}$,  then \eqref{e} is oscillatory.
\end{corollary}

We define
$$ 
\tilde{q}_n(t)=q(t)\Big(1+ \int_{t}^{\sigma(t)}\int_{u}^{\infty}q(s)
 a_n(\sigma(s),t)\mathrm{d} s\, \mathrm{d} u \Big), \quad n\in \mathbb{N},
$$
where $a_n(s,t)$ is as in Lemma \ref{lem3}.

\begin{theorem} \label{thm5}
 Assume that the second-order differential equation
 \begin{equation}\label{secn}
 y''(t)+\tilde{q}_n(t)y(t) = 0
 \end{equation}
 is oscillatory for some $n\in\mathbb{N}$. Then \eqref{e} is oscillatory.
\end{theorem}

\begin{proof}
Suppose to the contrary that $y$ is a positive solution of \eqref{e} on 
$[t_0,\infty)$. Then  there exists $t_1\ge t_0$ such that $y$ 
satisfies \eqref{struct} on $[t_1,\infty)$.  As in the proof of
 Theorem \ref{thm3}, we obtain \eqref{eq6}, that is,
 \begin{equation}\label{eq7}
 y'(t) \ge y(\sigma(t))\int_{t}^{\infty}q(s)a_n(\sigma(s),\sigma(t))\mathrm{d} s.
 \end{equation}
 Integrating \eqref{eq7} from $t$ to $\sigma(t)$ and using \eqref{est}, i.e.,
 $$
y(\sigma(u))\ge  y(t)a_n(\sigma(u),t), \quad  \sigma(u)\ge t,
$$
 we obtain
 \begin{align*}
  y(\sigma(t)) 
&\ge  y(t)+ \int_{t}^{\sigma(t)}y(\sigma(u))\int_{u}^{\infty}q(s)
 a_n(\sigma(s),\sigma(u))\mathrm{d} s\,\mathrm{d} u \\
& \ge y(t)\Big(1+ \int_{t}^{\sigma(t)}a_n(\sigma(u),t)\int_{u}^{\infty}q(s)
 a_n(\sigma(s),\sigma(u))\mathrm{d} s\, \mathrm{d} u \Big).
 \end{align*}
 The rest of the proof is similar to that of Theorem \ref{thm1} and so we omit it.
\end{proof}

\begin{corollary}\label{coro4}
 If
 \begin{equation}
 \liminf _{t\to \infty} t\int_{t}^{\infty}\tilde{q}_n(s)\mathrm{d} s>\frac{1}{4}
 \end{equation}
for some $n\in\mathbb{N}$,  then \eqref{e} is oscillatory.
\end{corollary}

\begin{lemma} \label{lem4}
 Let $y(t)$ be an eventually positive solution of \eqref{e}. Then
\begin{equation}\label{rhon}
 \rho_n: = \liminf _{t\to \infty}\int_{t}^{\sigma(t)}\int_{u}^{\infty}
q(s)a_{n}(\sigma(s),\sigma(u))\mathrm{d} s\,\mathrm{d} u\le\frac{1}{\mathrm{e}}, \\
\end{equation}
 and
 $$
\liminf _{t\to \infty} \frac{y(\sigma(t))}{y(t)} \ge \lambda_n,  
$$
 where
 $a_n(t,s)$ is as in Lemma \ref{lem3} and $\lambda_n$ is the smaller root of 
the equation
 $$ 
\lambda_n = \mathrm{e}^{\rho_n \lambda_n}.
$$
\end{lemma}

\begin{proof}
We proceed as in the proof of Theorem \ref{thm3} to  obtain that $y$ satisfies
 \eqref{eq6}.   The next arguments are the same as in the proof of 
Lemma \ref{lem1} so we can omit them.
\end{proof}

\begin{theorem} \label{thm6}
 Let \eqref{rhon} hold and $\lambda_n$ be as in Lemma \ref{lem4}. 
Assume that  the second-order differential equation
 \begin{equation}
 y''(t)+k\lambda_n q(t)y(t)= 0
 \end{equation}
 is oscillatory for some $n\in \mathbb{N}$ and  $k\in (0,1)$. Then \eqref{e} is oscillatory.
\end{theorem}

\begin{corollary}\label{coro5}
 Let \eqref{rhon} hold and  $\lambda_n$ be as in Lemma \ref{lem4}.  If
 \begin{equation}
 \liminf _{t\to \infty} t\int_{t}^{\infty}q(s)\mathrm{d} s>\frac{1}{4\lambda_n},
 \end{equation}
for some $n\in \mathbb{N}$,  then \eqref{e} is oscillatory.
\end{corollary}

Finally, we discuss the efficiency of  newly obtained criteria on Euler-type 
differential equations.

\begin{example} \label{examp1} \rm
 Consider the second-order advanced Euler differential equation
 \begin{equation}\label{ex}
 y''(t)+\frac{a}{t^2}y(ct) = 0, \quad c\ge 1, \quad a>0, \quad t\ge 1.
 \end{equation}
Known oscillation criteria  \eqref{hille0} and \eqref{ladde} give
 \begin{equation}\label{c2}
 a>\frac{1}{4}
 \end{equation}
 and
\begin{equation}\label{c1}
  a\ln c>\frac{1}{\mathrm{e}},
\end{equation}
 respectively.

The recent result  \cite[Corollary 1]{bac} gives
 \begin{equation}\label{bb}
 a\Big(\frac{c^\beta - 1}{\beta}+\frac{1}{1-a}+\frac{c^\beta}{1-\beta}\Big)>1,
 \end{equation}
where $\beta  = \frac{1-\sqrt{1-4a}}{2}$ and $a\le 1/4$.
From Corollary \ref{coro1}, we have that \eqref{ex} is oscillatory if
\begin{equation}\label{cc3}
 a(1+a\ln c)>\frac{1}{4}.
\end{equation}
To apply Corollary \ref{coro2},  we set $\rho: = a\ln c\le 1/\mathrm{e}$.   
Then the smaller root of the equation
 $\lambda = \mathrm{e}^{ \rho \lambda}$
 is 
 $$
\lambda = -\frac{W(-\ln \mathrm{e}^{\rho}) }{\ln \mathrm{e}^{\rho}} = -\frac{W(-\rho)}{\rho},
$$
 where $W(\cdot)$  denotes the principal branch of the Lambert function,
 see \cite{lambert} for details. Consequently, the oscillation
 criterion \eqref{hille2} becomes
 $$
-a \frac{W(-\rho)}{\rho}>\frac{1}{4}, 
$$
 that is,
\begin{equation}\label{c4}
 -\frac{W(-a\ln c)}{\ln c}>\frac{1}{4}.
\end{equation}

Now, we set $n = 1$.  After  simple calculations,  the following conditions 
for oscillation of \eqref{ex}, i.e.,
\begin{gather}\label{c5}
\frac{a}{1-a}\ln c >\frac{1}{e}, \\
\label{c56}
ac^a>\frac{1}{4}, \\
\label{c6}
a\Big(1+\frac{c^{a}(c^a-1)}{1-a}\Big)>\frac{1}{4},\\
\label{c7}
\frac{(a-1)W(\frac{a}{a-1}\ln c)}{\ln c}>\frac{1}{4}, \quad \text{where }
 \frac{a}{a-1}\ln c\le 1/e,
\end{gather}
result from Corollaries \ref{coro3}, \ref{coro34}, \ref{coro4} and \ref{coro5}, 
respectively.  A comparison of the effectiveness of the above-mentioned criteria
in terms of the required value $c$ for a given coefficient $a = 0.23$ is 
shown in the Table \ref{tab1}.

\begin{table}[htb]
 \caption{Comparison of the strength of criteria \eqref{c2}--\eqref{c7} for a given $a= 0.23$ }
 \label{tab1}
\begin{center}
 \begin{tabular}{|c c|}
 \hline
   criterion  &  required  $c$   \\
 \hline
 \eqref{c2} & inapplicable\\
 \eqref{c1}&  $4.950436$  \\
 \eqref{bb}& $2.274700$ \\
 \eqref{cc3}& $1.459467$\\
 \eqref{c4}& $1.395881$\\
 \eqref{c5}& $3.426695$\\
 \eqref{c56} & $1.436966$\\
 \eqref{c6}& $1.304194$\\
 \eqref{c7}& $1.292806$\\
\hline
 \end{tabular}
\end{center}
\end{table}

On the other hand, if we set $a = 0.19$ and $ c= 2$ in \eqref{ex}, then it 
is easy to verify that all criteria \eqref{c1}$-$\eqref{c7} fail.
In such a case, it is interesting to compare the length of the iteration 
process in  particular cases corresponding to Corollaries  
\ref{coro3}-\ref{coro5}. As can be seen from Table \ref{tab2}, $13$ iteration 
steps are necessary when applying Corollary \ref{coro3}, 
Corollary \ref{coro34} requires $7$ steps,  while Corollaries \ref{coro4} 
and \ref{coro5} ensure the oscillation of \eqref{ex} after the same number 
of iterations ($6$ steps).
\end{example}

\begin{table}[htb]
 \caption{Comparison of iterative processes for \eqref{ex} resulting from
 Corollaries \ref{coro3}, \ref{coro34}, \ref{coro4}, \ref{coro5},  respectively.}
 \label{tab2}
\begin{center}
 \begin{tabular}{|lc|}
\hline
$n$&  crit.~val.~$1/\mathrm{e}$ (Cor.~\ref{coro3}) \\
\hline
$1$&  $0.162590$  \\
$2$ & $0.179813$\\
$3$& $0.191500$\\
$4$& $0.200467$\\
$5$& $0.208003$\\
$6$& $0.214830$\\
$7$& $0.221447$\\
 $8$& $0.235846$\\
 $9$& $0.244837$\\
 $10$& $0.256514$\\
 $11$& $0.273525$\\
 $12$& $0.302947$\\
 $13$& $0.372771$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|lc|}
 \hline
 $n$&  crit.~val.~$1/4$  (Cor.~\ref{coro34})  \\
 \hline
 $1$&  $0.216745$  \\
 $2$ & $0.228721$\\
 $3$& $ 0.235918$\\
 $4$& $ 0.241002$\\
 $5$& $0.245011$\\
 $6$& $ 0.248452$\\
 $7$& $ 0.251627$\\
\hline
\end{tabular}
\\[5pt] 
 \begin{tabular}{|lc|}
 \hline
 $n$&  crit.~val.~$1/4$  (Cor.~\ref{coro4}) \\
 \hline
 $1$&  $0.231658$  \\
 $2$ & $ 0.237998$\\
 $3$& $ 0.24264$\\
 $4$& $ 0.246414$\\
 $5$& $0.249743$\\
 $6$& $ 0.252893$\\
\hline
\end{tabular}
\quad 
 \begin{tabular}{|lc|}
 \hline
 $n$&  crit.~val.~$1/4$  (Cor.~\ref{coro5})\\
 \hline
 $1$&  $0.227666$  \\
 $2$ & $0.235188$\\
 $3$& $ 0.240441$\\
 $4$& $ 0.244541$\\
 $5$& $0.248028$\\
 $6$& $ 0.251219$\\
 $7$& $ \text{undefined}$\\
\hline
\end{tabular}
\end{center}
\end{table}


\subsection*{Acknowledgements}
 This research was supported by the internal grant project no.~FEI-2015-22.

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\end{document}