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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 143, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/143\hfil Oscillation results?]
{Oscillation results for even-order quasilinear neutral
functional differential equations}

\author[B. Bacul\'ikov\'a, J. D\v{z}urina, T. Li \hfil EJDE-2011/143\hfilneg]
{Blanka Bacul\'ikov\'a, Jozef D\v{z}urina, Tongxing Li}
 % in alphabetical order

\address{Blanka Bacul\'ikov\'a \newline
Department of Mathematics,
Faculty of Electrical Engineering and Informatics,
Technical University of Ko\v{s}ice,
Letn\'a 9, 042\,00~Ko\v{s}ice, Slovakia}
\email{blanka.baculikova@tuke.sk}

\address{Jozef D\v{z}urina \newline
Department of Mathematics,
Faculty of Electrical Engineering and Informatics,
Technical University of Ko\v{s}ice,
Letn\'a 9, 042\,00~Ko\v{s}ice, Slovakia}
\email{jozef.dzurina@tuke.sk}

\address{Tongxing Li \newline
School of Control Science and Engineering,
Shandong University, Jinan,  Shandong 250061, China}
\email{litongx2007@163.com}

\thanks{Submitted April 28, 2011. Published November 1, 2011.}
\subjclass[2000]{34K11, 34C10}
\keywords{Oscillation; Neutral differential equation; even-order}

\begin{abstract}
 In this article, we use the Riccati transformation technique
 and some inequalities, to establish  oscillation theorems
 for all solutions to even-order quasilinear neutral differential
 equation
 $$
 \Big(\big[\big(x(t)+p(t)x(\tau(t))\big)^{(n-1)}\big]^\gamma\Big)'
 +q(t)x^\gamma\big(\sigma(t)\big)=0,\quad t\geq t_0.
 $$
 Our main results are illustrated with examples.
\end{abstract}

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

\section{Introduction}

Neutral differential equations find numerous applications in natural
science and technology; see Hale \cite{hale}. Recently, there has
been much research activity concerning the oscillation and
non-oscillation of solutions of various types of neutral functional
differential equations; see for example
\cite{rmw,gsl,erbz,dzurina,baculikova,han,han1,hasanbulli}
 and the references cited therein.

In this article, we consider the oscillatory behavior of solutions
to the even-order neutral differential equation
\begin{equation}\label{lh1.1}
\Big(\big[\big(x(t)+p(t)x(\tau(t))\big)^{(n-1)}\big]^\gamma\Big)'
+q(t)x^\gamma\big(\sigma(t)\big)=0,\quad t\geq t_0.
\end{equation}
We will use the following assumptions:
\begin{itemize}
\item[(A1)] $n\geq2$ is even and $\gamma\geq1$ is the ratio of odd
positive integers;

\item[(A2)] $p\in C([t_0,\infty),[0,a])$, where $a$ is a constant;

\item[(A3)] $q\in C([t_0,\infty),[0,\infty))$, and $q$
is not eventually zero on any half line $[t_*,\infty)$;

\item[(A4)] $\tau, \sigma\in C([t_0,\infty),\mathbb{R})$,
$\lim_{t\to \infty}\tau(t)=\lim_{t\to\infty}\sigma(t)=\infty$,
$\sigma^{-1}$ exists and $\sigma^{-1}$ is continuously
differentiable.
\end{itemize}

We consider only those solutions $x$ of  \eqref{lh1.1} for which
 $\sup\{|x(t)|:t\geq T\}>0$ for all $T\geq t_0$.
We assume that  \eqref{lh1.1} possesses such a solution. As usual, a
solution of  \eqref{lh1.1} is called oscillatory if it has
arbitrarily large zeros on $[t_0,\infty)$; otherwise, it is called
non-oscillatory. Equation \eqref{lh1.1} is said to be oscillatory if all
its solutions are oscillatory.

For the oscillation of even-order neutral differential equations,
Zafer \cite{zafer}, Karpuz et al. \cite{bk}, Zhang et al.
\cite{zhang} and Li et al. \cite{li} considered the oscillation of
even-order neutral equation
\begin{equation}\label{ghj}
\big(x(t)+p(t)x(\tau(t))\big)^{(n)}+q(t)x(\sigma(t))=0,\quad t\geq t_0
\end{equation}
by using the results given in \cite{chg}.
Meng and Xu \cite{meng} studied the oscillation property of the
even-order quasi-linear neutral equation
$$
\big[r(t)|(z(t))^{(n-1)}|^{\alpha-1}(z(t))^{(n-1)}\big]'+
q(t)|x(\sigma(t))|^{\alpha-1}x(\sigma(t))=0,\quad t\geq t_0,
$$
with $z(t)=x(t)+p(t)x(\tau(t))$.
To the best of our knowledge, there are no results on the
oscillation of  \eqref{lh1.1} when $p(t)>1$ and $\gamma>1$. The
purpose of this paper is to establish some oscillation results for
 \eqref{lh1.1}. The organization of this article is as follows:
In Section 2, we give some oscillation criteria for
\eqref{lh1.1}. In Section 3, we give several examples to illustrate
our main results.

Below, when we write a functional inequality without specifying its
domain of validity we assume that it holds for all sufficiently
large $t$.

\section{Main results}

In this section, we  establish some oscillation criteria for
\eqref{lh1.1}.
Let  $f^{-1}$ denote the inverse function of $f$, and
for the sake of convenience, we let
\begin{gather*}
z(t):=x(t)+p(t)x(\tau(t)), \quad
Q(t):=\min\{q(\sigma^{-1}(t)),q(\sigma^{-1}(\tau(t)))\},\\
(\rho'(t))_+:=\max\{0,\rho'(t)\}.
\end{gather*}
To prove our main results, we use the following lemmas.

\begin{lemma}[{\cite[Lemma 2.2.1]{rmw}}] \label{le1}
Let $u(t)$ be a positive and $n$-times differentiable function on an
interval $[T,\infty)$ with its $n$-th derivative $u^{(n)}(t)$
non-positive on $[T,\infty)$ and not identically zero on any interval
 $[T_1,\infty)$, $T_1\geq T$. Then there exists an
integer $l$, $0\leq l\leq n-1$, with $n+l$ odd, such that, for some
large $T_2\geq T_1$,
\begin{gather*}
(-1)^{l+j}u^{(j)}(t)>0\quad  \text{on } [T_2,\infty)\; (j=l,
l+1,\dots,n-1) \\
u^{(i)}(t)>0\quad  \text{on } [T_2,\infty)\; (i=1, 2,\dots,l-1)\
\text{when}\ l>1.
\end{gather*}
\end{lemma}

\begin{lemma}[{\cite[P. 169]{rmw}}] \label{le2}
Let $u$ be as in Lemma \ref{le1}. If
$\lim_{t\to\infty}u(t)\neq0$, then, for every $\lambda$,
$0<\lambda<1$, there is $T_\lambda\geq t_0$ such that,
for all $t\geq T_\lambda$,
$$
u(t)\geq\frac{\lambda}{(n-1)!}t^{n-1}u^{(n-1)}(t).
$$
\end{lemma}


\begin{lemma}[\cite{chg}] \label{le12}
Let $u$ be as in Lemma \ref{le1} and $u^{(n-1)}(t)u^{(n)}(t)\leq 0$
for $t\geq t_*$. Then for every constant $\theta$, $0<\theta<1$,
there exists a constant $M_\theta>0$ such that
$$
u'(\theta t)\geq M_\theta t^{n-2}u^{(n-1)}(t).
$$
\end{lemma}

\begin{lemma}\label{le3}
Assume that $x$ is an eventually positive solution of
\eqref{lh1.1}, and $n$ is even.
Then there exists  $t_1\geq t_0$ such that, for
$t\geq t_1$,
$$
z(t)>0,\quad z'(t)>0,\quad z^{(n-1)}(t)>0,\quad  z^{(n)}(t)\leq 0,
$$
and $z^{(n)}$ is not identically zero on any interval $[a,\infty)$.
\end{lemma}

 The proof of the above lemma is similar to that of
\cite[Lemma 2.3]{meng}, with $\gamma$ being the ratio of odd
integers. We omit it.


\begin{lemma}\label{lle2.1}
Assume that $\gamma\geq1$, $x_1$, $x_2\in \mathbb{R}$. If
$x_1\geq0$ and  $x_2\geq0$, then
\begin{equation}\label{lh2.1}
{x_1}^\gamma+{x_2}^\gamma\geq\frac{1}{2^{\gamma-1}}(x_1+x_2)^\gamma.
\end{equation}
\end{lemma}


\begin{proof}
(i) Suppose that $x_1=0$ or $x_2=0$. Then we have
\eqref{lh2.1}.
(ii) Suppose that $x_1>0$ and $x_2>0$. Define  $f$
by $f(x)=x^\gamma$, $x\in (0,\infty)$. Clearly,
$f''(x)=\gamma(\gamma-1)x^{\gamma-2}\geq0$ for $x>0$. Thus, $f$ is a
convex function. By the definition of convex function, for $x_1$,
$x_2\in(0,\infty)$, we have
$$
f\big(\frac{x_1+x_2}{2}\big)\leq\frac{f(x_1)+f(x_2)}{2}.
$$
That is,
$$
{x_1}^\gamma+{x_2}^\gamma\geq\frac{1}{2^{\gamma-1}}(x_1+x_2)^\gamma.
$$
This completes the proof.
\end{proof}

First, we establish the following comparison theorems.

\begin{theorem}\label{lth3.1}
Assume that $(\sigma^{-1}(t))'\geq\sigma_0>0$ and
$\tau'(t)\geq\tau_0>0$. Further, assume that there exists a constant
$\lambda$, $0<\lambda<1$, such that
\begin{equation}\label{mh1}
\big[\frac{y(\sigma^{-1}(t))}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}
y(\sigma^{-1}(\tau(t)))\big]'
+\frac{1}{2^{\gamma-1}}\Big(\frac{\lambda}{(n-1)!}t^{n-1}\Big)^\gamma
Q(t)y(t)\leq 0
\end{equation}
has no eventually positive solution. Then  \eqref{lh1.1} is
oscillatory.
\end{theorem}

\begin{proof}
Let $x$ be a non-oscillatory solution of
\eqref{lh1.1}. Without loss of generality, we assume that there
exists $t_1\geq t_0$ such that $x(t)>0$, $x(\tau(t))>0$ and
$x(\sigma(t))>0$ for all $t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$.
From \eqref{lh1.1}, we obtain
$$
\big((z^{(n-1)}(t))^\gamma\big)'=-q(t)x^\gamma(\sigma(t))\leq0,\quad
 t\geq t_1.
$$

By Lemma \ref{le3} with $n$ even, there exists $t_2\geq t_1$
such that $z^{(n)}(t)\leq 0$ for $t\geq t_2$.
Thus, from Lemma \ref{le1},
there exist $t_3\geq t_2$ and an odd integer $l\leq n-1$ such that,
for some large $t_4\geq t_3$,
\begin{equation}\label{01}
(-1)^{l+j}z^{(j)}(t)>0, \quad j=l, l+1,\dots,n-1,\; t\geq t_4
\end{equation}
and
\begin{equation}\label{02}
z^{(i)}(t)>0, \quad i=1, 2,\dots,l-1, \; t\geq t_4.
\end{equation}
Hence, in view of \eqref{01} and \eqref{02}, we obtain $z'(t)>0$ and
$z^{(n-1)}(t)>0$. Therefore, $\lim_{t\to\infty}z(t)\neq 0$.
Then, by Lemma \ref{le2}, for every $\lambda$, $0<\lambda<1$, there
exists $T_\lambda$ such that, for all $t\geq T_\lambda$,
\begin{equation}\label{10.10}
z(t)\geq\frac{\lambda}{(n-1)!}t^{n-1}z^{(n-1)}(t).
\end{equation}
It follows from \eqref{lh1.1} that
\begin{equation}\label{d1}
\frac{((z^{(n-1)}(\sigma^{-1}(t)))^\gamma)'}{(\sigma^{-1}(t))'}
+q(\sigma^{-1}(t))x^\gamma(t)=0.
\end{equation}
The above inequality at times
$\sigma^{-1}(t)$ and $\sigma^{-1}(\tau(t))$, yields
\begin{equation}
\begin{split}
&\frac{((z^{(n-1)}(\sigma^{-1}(t)))^\gamma)'}{(\sigma^{-1}(t))'}
 +a^\gamma\frac{((z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma)'}{(\sigma^{-1}(\tau(t)))'} \\
&+q(\sigma^{-1}(t))x^\gamma(t)+a^\gamma
q(\sigma^{-1}(\tau(t)))x^\gamma(\tau(t))=0.
\end{split} \label{d11}
\end{equation}
By \eqref{lh2.1} and the definition of $z$,
\begin{equation}
\begin{aligned}
q(\sigma^{-1}(t))x^\gamma(t)+a^\gamma
q(\sigma^{-1}(\tau(t)))x^\gamma(\tau(t))
&\geq Q(t)[x^\gamma(t)+a^\gamma x^\gamma(\tau(t))] \\
&\geq  \frac{1}{2^{\gamma-1}}Q(t)[x(t)+ax(\tau(t))]^\gamma\\
&\geq  \frac{1}{2^{\gamma-1}}Q(t)z^\gamma(t)
\end{aligned} \label{d111}
\end{equation}
 It follows from
\eqref{d11} and \eqref{d111} that
\begin{equation}\label{xjl2}
\frac{((z^{(n-1)}(\sigma^{-1}(t)))^\gamma)'}{(\sigma^{-1}(t))'}+a^\gamma\frac{((z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma)'}{(\sigma^{-1}(\tau(t)))'}+\frac{1}{2^{\gamma-1}}Q(t)z^\gamma(t)
\leq0.
\end{equation}
From this inequality,  $(\sigma^{-1}(t))'\geq\sigma_0>0$ and
$\tau'(t)\geq\tau_0>0$, we obtain
\begin{equation}\label{xxjl2}
\frac{((z^{(n-1)}(\sigma^{-1}(t)))^\gamma)'}{\sigma_0}
+a^\gamma\frac{((z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma)'}{\sigma_0\tau_0}
+\frac{1}{2^{\gamma-1}}Q(t)z^\gamma(t) \leq 0.
\end{equation}
Set $y(t)=(z^{(n-1)}(t))^\gamma>0$. From
\eqref{10.10} and \eqref{xjl2}, we see that $y$ is an eventually
positive solution of
$$
\big[\frac{y(\sigma^{-1}(t))}{\sigma_0}
+\frac{a^\gamma}{\sigma_0\tau_0} y(\sigma^{-1}(\tau(t)))\big]'
+\frac{1}{2^{\gamma-1}}\Big(\frac{\lambda}{(n-1)!}t^{n-1}\Big)^\gamma
Q(t)y(t)\leq 0.
$$
The proof is complete.
\end{proof}


\begin{theorem}\label{lth3.2}
Let $\tau^{-1}$ exist. Assume that $\tau(t)\leq t$,
$(\sigma^{-1}(t))'\geq\sigma_0>0$ and $\tau'(t)\geq\tau_0>0$.
Moreover, assume that there exists a constant $\lambda$,
$0<\lambda<1$, such that
\begin{equation}\label{ssmh1}
u'(t)+\frac{1}{2^{\gamma-1}
\big(\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\big)}
\Big(\frac{\lambda}{(n-1)!}t^{n-1}\Big)^\gamma
Q(t)u(\tau^{-1}(\sigma(t)))\leq 0
\end{equation}
has no eventually positive solution. Then   \eqref{lh1.1} is
oscillatory.
\end{theorem}

\begin{proof}
Let $x$ be a non-oscillatory solution of
\eqref{lh1.1}. Without loss of generality, we assume that there
exists $t_1\geq t_0$ such that $x(t)>0$, $x(\tau(t))>0$ and
$x(\sigma(t))>0$ for all $t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$.
Proceeding as in the proof of Theorem \ref{lth3.1}, we obtain that
$y(t)=(z^{(n-1)}(t))^\gamma>0$ is non-increasing and satisfies
inequality \eqref{mh1}. Define
$$
u(t)=\frac{y(\sigma^{-1}(t))}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}
y(\sigma^{-1}(\tau(t))).
$$
Then, from $\tau(t)\leq t$, and $\sigma^{-1}$ begin increasing, we have
$$
u(t)\leq \big(\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\big)
y(\sigma^{-1}(\tau(t))).
$$
Substituting the above formulas into \eqref{mh1}, we find $u$ is an
eventually positive solution of
\begin{equation} \label{eu1}
u'(t)+\frac{1}{2^{\gamma-1}\big(\frac{1}{\sigma_0}
+\frac{a^\gamma}{\sigma_0\tau_0}\big)}
\Big(\frac{\lambda}{(n-1)!}t^{n-1}\Big)^\gamma
Q(t)u(\tau^{-1}(\sigma(t)))\leq 0.
\end{equation}
The proof  is complete.
\end{proof}

From Theorem \ref{lth3.2} and \cite[Theorem 2.1.1]{gsl}, we
establish the following corollary.

\begin{corollary}\label{xlth3.2}
Let $\tau^{-1}$ exist. Assume that $\tau(t)\leq t$,
$(\sigma^{-1}(t))'\geq\sigma_0>0$, $\tau'(t)\geq\tau_0>0$,
$\tau^{-1}(\sigma(t))<t$ and
\begin{equation}\label{81mh1}
\liminf_{t\to\infty}\int_{\tau^{-1}(\sigma(t))}^tQ(s)
(s^{n-1})^\gamma{\rm
d}s>\frac{2^{\gamma-1}
\big(\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\big)}{e}
\big((n-1)!\big)^\gamma.
\end{equation}
Then   \eqref{lh1.1} is oscillatory.
\end{corollary}

\begin{proof}  From \eqref{81mh1}, one can choose a positive
constant $0<\lambda<1$ such that
$$
\liminf_{t\to\infty}\lambda
^\gamma\int_{\tau^{-1}(\sigma(t))}^tQ(s) (s^{n-1})^\gamma{\rm
d}s>\frac{2^{\gamma-1}\big(\frac{1}{\sigma_0}
+\frac{a^\gamma}{\sigma_0\tau_0}\big)}{e}((n-1)!)^\gamma.
$$
Applying \cite[Theorem 2.1.1]{gsl} to \eqref{eu1}, with
$\tau^{-1}(\sigma(t))<t$, we complete the proof.
\end{proof}


\begin{theorem}\label{jlth3.2}
Assume that $(\sigma^{-1}(t))'\geq\sigma_0>0$,
$\tau'(t)\geq\tau_0>0$ and $\tau(t)\geq t$. Furthermore, assume that
there exists a constant $\lambda$, $0<\lambda<1$, such that
\begin{equation}\label{jssmh1}
u'(t)+\frac{1}{2^{\gamma-1}
\big(\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\big)}
\Big(\frac{\lambda}{(n-1)!}t^{n-1}\Big)^\gamma
Q(t)u(\sigma(t))\leq 0
\end{equation}
has no eventually positive solution. Then  \eqref{lh1.1} is
oscillatory.
\end{theorem}

\begin{proof} Let $x$ be a non-oscillatory solution of
\eqref{lh1.1}. Without loss of generality, we assume that there
exists $t_1\geq t_0$ such that $x(t)>0$, $x(\tau(t))>0$ and
$x(\sigma(t))>0$ for all $t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$.
Proceeding as in the proof of Theorem \ref{lth3.1}, we obtain that
$y(t)=(z^{(n-1)}(t))^\gamma>0$ is nonincreasing and satisfies
inequality \eqref{mh1}. Define
$$
u(t)=\frac{1}{\sigma_0}y(\sigma^{-1}(t))+\frac{a^\gamma}{\sigma_0\tau_0}
y(\sigma^{-1}(\tau(t))).
$$
Then, from $\tau(t)\geq t$, we have
$$
u(t)\leq
\Big(\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\Big)
y(\sigma^{-1}(t)).
$$
Substituting the above formulas into \eqref{mh1}, we find $u$ is an
eventually positive solution of
\begin{equation} \label{eu2}
u'(t)+\frac{1}{2^{\gamma-1}
\big(\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\big)}
\Big(\frac{\lambda}{(n-1)!}t^{n-1}\Big)^\gamma
Q(t)u(\sigma(t))\leq 0.
\end{equation}
The proof is complete.
\end{proof}

From Theorem \ref{jlth3.2} and \cite[Theorem 2.1.1]{gsl}, we
establish the following corollary.

\begin{corollary}\label{jxlth3.2}
Assume that $(\sigma^{-1}(t))'\geq\sigma_0>0$,
$\tau'(t)\geq\tau_0>0$, $\tau(t)\geq t$, $\sigma(t)<t$ and
\begin{equation}\label{j81mh1}
\liminf_{t\to\infty}\int_{\sigma(t)}^tQ(s)
(s^{n-1})^\gamma{\rm d}s
>\frac{2^{\gamma-1}\big(\frac{1}{\sigma_0}
+\frac{a^\gamma}{\sigma_0\tau_0}\big)}{e}\big((n-1)!\big)^\gamma.
\end{equation}
Then   \eqref{lh1.1} is oscillatory.
\end{corollary}

\begin{proof}  From \eqref{j81mh1}, one can choose a positive
constant $0<\lambda<1$ such that
$$
\liminf_{t\to\infty}\lambda ^\gamma\int_{\sigma(t)}^tQ(s)
(s^{n-1})^\gamma{\rm
d}s>\frac{2^{\gamma-1}\big(\frac{1}{\sigma_0}
+\frac{a^\gamma}{\sigma_0\tau_0}\big)}{e}((n-1)!)^\gamma.
$$
Applying \cite[Theorem 2.1.1]{gsl} to \eqref{eu2}, with
$\sigma(t)<t$,
we complete proof.
\end{proof}

By employing Riccati transformation, we obtain the following
criteria.


\begin{theorem}\label{lth3.3}
Let $(\sigma^{-1}(t))'\geq\sigma_0>0$, $\sigma^{-1}(t)\geq t$,
$\sigma^{-1}(\tau(t))\geq t$ and $\tau'(t)\geq\tau_0>0$. Assume that
there exists $\rho\in C^1([t_0,\infty),(0,\infty))$ such that
\begin{equation}\label{gxq}
\limsup_{t\to\infty}\int_{t_0}^t
\Big[\frac{1}{2^{\gamma-1}}\rho(s)Q(s)
-\frac{\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}}
{(\gamma+1)^{\gamma+1}}\frac{((\rho'(s))_+)^{\gamma+1}}{(\theta M
s^{n-2})^\gamma\rho^\gamma(s)}\Big]{\rm d}s=\infty
\end{equation}
holds for some constant $\theta$, $0<\theta<1$ and for all constants
$M>0$. Then  \eqref{lh1.1} is oscillatory.
\end{theorem}

\begin{proof} Let $x$ be a non-oscillatory solution of
\eqref{lh1.1}. Without loss of generality, we assume that there
exists $t_1\geq t_0$ such that $x(t)>0$, $x(\tau(t))>0$ and
$x(\sigma(t))>0$ for all $t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$.
Proceeding as in the proof of Theorem \ref{lth3.1},  there exists
$t_2\geq t_1$ such that \eqref{01}, \eqref{02} and \eqref{xxjl2}
hold for $t\geq t_2$.

Using the Riccati transformation
\begin{equation}\label{8.191}
\omega(t)=\rho(t)\frac{(z^{(n-1)}(\sigma^{-1}(t)))^\gamma}
{z^\gamma(\theta t)},\quad t\geq t_2.
\end{equation}
Then $\omega(t)>0$ for $t\geq t_2$. Differentiating \eqref{8.191},
we obtain
\begin{equation}\label{lq1}
\begin{split}
\omega'(t)&=\rho'(t)\frac{(z^{(n-1)}(\sigma^{-1}(t)))^\gamma}
{z^\gamma(\theta t)}
+\rho(t)\frac{((z^{(n-1)}(\sigma^{-1}(t)))^\gamma)'}{z^\gamma(\theta
t)}\\
&\quad -\gamma\theta\rho(t)\frac{(z^{(n-1)}(\sigma^{-1}(t)))^\gamma
z'(\theta t)}{z^{\gamma+1}(\theta t)}.
\end{split}
\end{equation}
By Lemma \ref{le12} and Lemma \ref{le3}, we have
$$
z'(\theta t)\geq M t^{n-2}z^{(n-1)}(t)
 \geq M t^{n-2}z^{(n-1)}(\sigma^{-1}(t)),
$$
for every $\theta$, $0<\theta<1$ and for some $M>0$. Thus, from
\eqref{8.191} and \eqref{lq1}, we obtain
\begin{equation}\label{litong1}
\omega'(t)\leq\frac{(\rho'(t))_+}{\rho(t)}\omega(t)
+\rho(t)\frac{((z^{(n-1)}(\sigma^{-1}(t)))^\gamma)'}{z^\gamma(\theta t)}
 -\gamma\theta M t^{n-2}\frac{(\omega(t))^{(\gamma+1)/\gamma}}
{\rho^{1/\gamma}(t)}.
\end{equation}
Next, define function
\begin{equation}\label{8.192}
\psi(t)=\rho(t)\frac{(z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma}{z^\gamma(\theta t)},\quad t\geq t_2.
\end{equation}
Then $\psi(t)>0$ for $t\geq t_2$. Differentiating \eqref{8.192}, we
see that
\begin{equation}\label{lq2}
\begin{split}
\psi'(t)&=\rho'(t)\frac{(z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma}{z^\gamma(\theta t)}
+\rho(t)\frac{((z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma)'}{z^\gamma(\theta t)} \\
&\quad -\gamma\theta\rho(t)\frac{(z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma
 z'(\theta t)}{z^{\gamma+1}(\theta t)}.
\end{split}
\end{equation}
In view of Lemmas \ref{le12} and \ref{le3}, we have
\[
z'(\theta t)\geq M t^{n-2}z^{(n-1)}(t)\geq M
t^{n-2}z^{(n-1)}(\sigma^{-1}(\tau(t))),
\]
 for every $\theta$,
$0<\theta<1$ and for some $M>0$. Hence, by \eqref{8.192} and
\eqref{lq2}, we obtain
\begin{equation}\label{litong2}
\begin{split}
\psi'(t)
&\leq\frac{(\rho'(t))_+}{\rho(t)}\psi(t)+\rho(t)\frac{((z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma)'}{z^\gamma(\theta t)} \\
&\quad -\gamma\theta
M t^{n-2}\frac{(\psi(t))^{(\gamma+1)/\gamma}}{\rho^{1/\gamma}(t)}.
\end{split}
\end{equation}
Therefore,  from \eqref{litong1} and \eqref{litong2} it follows that
\begin{equation}
\begin{split}
&\frac{\omega'(t)}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\psi'(t)\\
&\leq \rho(t)
\big[\frac{\frac{((z^{(n-1)}(\sigma^{-1}(t)))^\gamma)'}{\sigma_0}
+\frac{a^\gamma}{\sigma_0\tau_0}((z^{(n-1)}
(\sigma^{-1}(\tau(t))))^\gamma)'} {z^\gamma(\theta t)}\big] \\
&\quad+ \frac{1}{\sigma_0}\big[\frac{(\rho'(t))_+}{\rho(t)}\omega(t)
-\gamma\theta M
t^{n-2}\frac{(\omega(t))^{(\gamma+1)/\gamma}}{\rho^{1/\gamma}(t)}\big]
\\
&\quad +\frac{a^\gamma}{\sigma_0\tau_0}
\big[\frac{(\rho'(t))_+}{\rho(t)}\psi(t)
-\gamma\theta M t^{n-2}\frac{(\psi(t))^{(\gamma+1)
/\gamma}}{\rho^{1/\gamma}(t)}\big].
\end{split}\label{tongxing4}
\end{equation}
Thus, from the above inequality and \eqref{xxjl2}, we have
\begin{equation}
\begin{split}
&\frac{\omega'(t)}{\sigma_0}
+\frac{a^\gamma}{\sigma_0\tau_0}\psi'(t)\\
&\leq-\frac{1}{2^{\gamma-1}}\rho(t) Q(t)
 +\frac{1}{\sigma_0}\big[\frac{(\rho'(t))_+}{\rho(t)}\omega(t)
-\gamma\theta M
t^{n-2}\frac{(\omega(t))^{(\gamma+1)/\gamma}}{\rho^{1/\gamma}(t)}\big] \\
&\quad +\frac{a^\gamma}{\sigma_0\tau_0}
\big[\frac{(\rho'(t))_+}{\rho(t)}\psi(t)
-\gamma\theta M t^{n-2}\frac{(\psi(t))^{(\gamma+1)
/\gamma}}{\rho^{1/\gamma}(t)}\big].
\end{split}\label{tongxing5}
\end{equation}
Set
$$
A:=\frac{(\rho'(t))_+}{\rho(t)},\quad
B:=\frac{\gamma\theta M t^{n-2}}{\rho^{1/\gamma}(t)},\quad
v:=\omega(t), \psi(t).
$$
Then, using \eqref{tongxing5} and the inequality
\begin{equation}\label{sz1}
Av-Bv^{(\gamma+1)/\gamma}\leq\frac{\gamma^\gamma}
{(\gamma+1)^{\gamma+1}}\frac{A^{\gamma+1}}{B^\gamma},\quad
B>0,
\end{equation}
we have
$$
\frac{\omega'(t)}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\psi'(t)
\leq-\frac{1}{2^{\gamma-1}}\rho(t)Q(t)
+\frac{\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}}
{(\gamma+1)^{\gamma+1}}\frac{((\rho'(t))_+)^{\gamma+1}}{(\theta M
t^{n-2})^\gamma\rho^\gamma(t)}.
$$
Integrating the above inequality from $t_2$ to $t$, we obtain
\[
\int_{t_2}^t\Big[\frac{1}{2^{\gamma-1}}\rho(s)Q(s)
-\frac{\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}}
{(\gamma+1)^{\gamma+1}}\frac{((\rho'(s))_+)^{\gamma+1}}{(\theta M
s^{n-2})^\gamma\rho^\gamma(s)}\Big]{\rm d}s
\leq
\frac{\omega(t_2)}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}\psi(t_2),
\]
which contradicts \eqref{gxq}. The proof  is complete.
\end{proof}


\begin{remark} \label{rmk2.1} \rm
From \eqref{tongxing5}, define a Philos-type function $H(t,s)$, and
obtain some oscillation criteria for \eqref{lh1.1},
the details are left to the reader.
\end{remark}

\begin{theorem}\label{jklth3.3}
Let $n=2$, $(\sigma^{-1}(t))'\geq\sigma_0>0$, $\sigma^{-1}(t)\geq
t$, $\sigma^{-1}(\tau(t))\geq t$ and $\tau'(t)\geq\tau_0>0$. Assume
that there exists $\rho\in C^1([t_0,\infty),(0,\infty))$ such that
\begin{equation}\label{jkgxq}
\limsup_{t\to\infty}\int_{t_0}^t
\Big[\frac{1}{2^{\gamma-1}}\rho(s)Q(s)
-\frac{\frac{1}{\sigma_0}+\frac{a^\gamma}{\sigma_0\tau_0}}
{(\gamma+1)^{\gamma+1}}\frac{((\rho'(s))_+)^{\gamma+1}}{\rho^\gamma(s)}
\Big]{\rm d}s=\infty.
\end{equation}
Then  \eqref{lh1.1} is oscillatory.
\end{theorem}

\begin{proof} Define
$$
\omega(t)=\rho(t)\frac{(z'(\sigma^{-1}(t)))^\gamma}{z^\gamma(t)},
\quad
\psi(t)=\rho(t)\frac{(z'
(\sigma^{-1}(\tau(t))))^\gamma}{z^\gamma(t)}.
$$
The remainder of the proof is similar to that of Theorem
\ref{lth3.3}.
\end{proof}

\section{Applications}

Han et al. \cite{han, han1} considered the oscillation of
solutions to the second-order neutral equation
$$
(x(t)+p(t)x(\tau(t)))''+q(t)x(\sigma(t))=0,\quad t\geq t_0,
$$
where
\begin{equation}\label{adx}
0\leq p(t)\leq p_0<\infty,\quad \tau'(t)\geq\tau_0>0, \quad
\tau\circ\sigma=\sigma\circ\tau.
\end{equation}
Li et al. \cite{li} investigated the oscillation of
\eqref{ghj} when \eqref{adx} holds.
It is easy to see that our results
weaken the restrictions in
\cite{han, han1,li}, since we do not assume
$\tau\circ\sigma=\sigma\circ\tau$; instead we assume
$\tau^{-1}(\sigma(t))<t$, and bounds on
$\sigma'$, $(\sigma{^-1})'$ and $\tau^{-1}$.
Below, we give three examples that illustrate our results.

\begin{example} \label{examp3.1} \rm
Consider the even-order equation
\begin{equation}\label{66}
\Big(\big[\big(x(t)+ax(t-3)\big)^{(n-1)}\big]^\gamma\Big)'
+\frac{\beta}{(t^{n-1})^\gamma}x^\gamma\left(t-6\right)=0,\quad
t\geq 1,
\end{equation}
where $\gamma>1$ is the quotient of odd positive integers, $a>0$ and
$\beta>0$ are constants.
Let $\tau(t)=t-3$, $p(t)=a$, $q(t)=\beta/(t^{n-1})^\gamma$ and
$\sigma(t)=t-6$. Then $\tau^{-1}(t)=t+3$,
$\tau^{-1}(\sigma(t))=t-3$, $\sigma^{-1}(t)=t+6$,
$\sigma^{-1}(\tau(t))=t+3$ and $Q(t)=\beta/((t+6)^{n-1})^\gamma$.
Since
$$
\liminf_{t\to\infty}\int_{\tau^{-1}(\sigma(t))}^tQ(s)
(s^{n-1})^\gamma{\rm
d}s>\frac{\beta}{2^{\gamma(n-1)}}\liminf_{t\to\infty}\int_{t-3}^t{\rm
d}s=\frac{3\beta}{2^{\gamma(n-1)}},
$$
by applying Corollary \ref{xlth3.2}, Equation \eqref{66} is
oscillatory when
$$
\frac{3\beta}{2^{\gamma(n-1)}}
\geq\frac{2^{\gamma-1}(1+a^\gamma)((n-1)!)!}{e}.
$$
\end{example}

\begin{example} \label{examp3.2} \rm
 Consider the even-order equation
\begin{equation}\label{x66}
\Big(\big[\big(x(t)+ax(t+3)\big)^{(n-1)}\big]^\gamma\Big)'
+\frac{\beta}{(t^{n-1})^\gamma}
x^\gamma\big(\frac{t}{2}\big)=0,\quad t\geq 1,
\end{equation}
where $\gamma>1$ is the quotient of odd positive integers, $a>0$ and
$\beta>0$ are constants.
Let $\tau(t)=t+3$, $p(t)=a$, $q(t)=\beta/(t^{n-1})^\gamma$ and
$\sigma(t)=t/2$. Then $\sigma^{-1}(t)=2t$,
$\sigma^{-1}(\tau(t))=2(t+3)$ and
$Q(t)=\beta/((2t+6)^{n-1})^\gamma$.
Since
$$
\liminf_{t\to\infty}\int_{\sigma(t)}^tQ(s)
(s^{n-1})^\gamma{\rm d}s=\infty,
$$
by applying Corollary \ref{jxlth3.2}, Equation \eqref{x66} is
oscillatory.
\end{example}


\begin{example} \label{examp3.3}\rm
 Consider the even-order equation
\begin{equation}\label{96}
\Big(\big[\big(x(t)+ax\left(2t\right)\big)^{(n-1)}\big]^\gamma\Big)'
+\frac{\beta}{t}x^\gamma\big(\frac{t}{3}+1\big)=0,\quad t\geq1,
\end{equation}
where $\gamma>1$ is the quotient of odd positive integers, $a>0$ and
$\beta>0$ are constants.
Let $\tau(t)=2t$, $p(t)=a$, $q(t)=\beta/t$ and $\sigma(t)=(t/3)+1$.
Then  $\sigma^{-1}(t)=3(t-1)$, $\sigma^{-1}(\tau(t))=3(2t-1)$ and
$Q(t)=\beta/(6t-3)$.
Set $\rho(t)=1$. Then, by Theorem \ref{lth3.3},
every solution of \eqref{96} is oscillatory.
\end{example}

Note that the known results in the literature are not applicable to
Equations \eqref{66}, \eqref{x66} and \eqref{96}.

\subsection*{Acknowledgments}
This work is the result of the project implementation: Development of
the Center of Information and Communication Technologies for Knowledge
Systems (ITMS project code: 26220120030) supported by the Research \&
Development Operational Program funded by the ERDF.

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\end{document}