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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 29, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/29\hfil Oscillation of solutions]
{Oscillation of solutions to third-order half-linear neutral 
 differential equations}

\author[J. D\v zurina, E. Thandapani, S. Tamilvanan \hfil EJDE-2012/29\hfilneg]
{Jozef D\v zurina, Ethiraju Thandapani, Sivaraj Tamilvanan}  % in alphabetical order

\address{Jozef  D\v zurina \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{Ethiraju Thandapani \newline
Ramanujan Institute for Advanced Study in Mathematics,
University of Madras, Chennai, 600 005, India}
\email{ethandapani@yahoo.co.in}

\address{Sivaraj Tamilvanan \newline
Ramanujan Institute for Advanced Study in Mathematics,
University of Madras, Chennai, 600 005, India}
\email{saitamilvanan@yahoo.in}

\thanks{Submitted November 12, 2011. Published February 21, 2012.}
\subjclass[2000]{34K11, 34C10}
\keywords{Third-order neutral differential equation; Riccati transformation;
 \hfill\break\indent oscillation of solutions}

\begin{abstract}
 In this article, we study the oscillation of solutions to the
 third-order neutral differential equations
 $$
 \Big(a(t)\big([x(t)\pm p(t)x(\delta(t))]''\big)^\alpha\Big)' +
 q(t)x^\alpha(\tau(t)) = 0.
 $$
 Sufficient conditions are established so that every
 solution is either oscillatory or converges to zero.
 In particular, we extend the results obtain in \cite{bac} for $a(t)$
 non-decreasing,  to the non-increasing case.
\end{abstract}

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

 \section{Introduction}
In recent years, there has been great interest in studying the oscillatory
behavior of differential equations;
see for example \cite{erbe,gre,gyori,kig,lad} and the references cited therein.
Compared to first and second order, third-order neutral differential
equations have received less attention, even though such equations arise
in many physical problems.
Motivated by this observation, we study the oscillation of solutions to
the third-order half-linear neutral differential equations
\begin{equation} \label{eE+}
 \Big(a(t)\big([x(t)+ p(t)x(\delta(t))]''\big)^\alpha\Big)' +
 q(t)x^\alpha(\tau(t)) = 0,\quad t\geq t_0,
\end{equation}
 and
\begin{equation} \label{eE-}
 \Big(a(t)\big([x(t)- p(t)x(\delta(t))]''\big)^\alpha\Big)' +
  q(t)x^\alpha(\tau(t)) = 0, \quad t\geq t_0\,.
\end{equation}

We assume the following conditions:
\begin{itemize}
\item[(H1)] $a(t)$, $p(t)$, $q(t)$, $\tau(t)$, $\delta(t)$ are in $C([0, \infty))$;
 $a(t)$,  $q(t)$, $\tau(t)$, $\delta(t)$ are positive functions;
 $\alpha$ is the quotient of two  odd positive integers.

\item[(H2)] There is constant $p$ such that $0\leq p(t)\leq p<1$; the delay arguments
satisfy   $\tau(t)\leq t$, $\delta(t)\leq t$, $\lim_{t\to \infty} \tau(t)
   = \lim_{t\to \infty} \delta(t) = \infty$.

\item[(H3)] $a(t)$ is positive and non-increasing;
 $A(t):=\int_{t_0}^t a^{-1/\alpha}(s)\, d{s} \to \infty$ as
 $t\to\infty$.
\end{itemize}

By a solution to \eqref{eE+}, we mean a function
$x(t)$ in $\mathcal{C}^2[T_x,\infty)$ for which
$ a(t)(z''(t))^{\alpha}$ is in $\mathcal{C}^1[T_x,\infty)$
and \eqref{eE+} is satisfied on some interval $[T_x,\infty)$,
where $T_x\geq t_0$, and  $z(t)= x(t)+p(t)x(\delta(t))$.
The same concept of a solution applies to \eqref{eE-}.

Dzurina \cite{bac} obtained sufficient conditions for the oscillation
of solutions to \eqref{eE+} and to \eqref{eE-}, under the assumption that $a(t)$
is non-decreasing.
Here we establish similar results when $a(t)$ is non-increasing.
We follow the same strategy as in \cite {bac}, but with new estimates
in Lemmas \ref{lem23}, \ref{lem24}, \ref{lem25}.


We consider only  solutions $x(t)$  for which
$\sup \{|x(t)|: t\geq T\} >0$ for all $T\geq T_x$.
We say that a solution is  oscillatory if it has arbitrarily
large zeros, and non-oscillatory otherwise.
All functional inequalities are assumed to hold eventually;
that is, for all $t$ large enough.
Note that if $x(t)$ is a solution so is $-x(t)$; so our proofs are done
only for positive solutions.

 In Section 2, we present oscillation results for \eqref{eE+},
while in Section 3 we present similar results for \eqref{eE-}.
In both section we give examples to illustrate our results.

\section{Oscillation results for \eqref{eE+}}

For a solution $x(t)$ of \eqref{eE+}, we define the corresponding function
\begin{equation} \label{e21}
  z(t) = x(t)+p(t)x(\delta(t)).
\end{equation}
To obtain sufficient conditions for the oscillation of solutions to \eqref{eE+},
we need the the following lemmas.


\begin{lemma}[{\cite[Lemma 1]{bac}}] \label{lem21}
Let $x(t)$ be a positive solution of \eqref{eE+}. Then there are only
two possible cases:
\begin{itemize}
\item[(I)] $z(t) > 0$, $z'(t)>0$, $z''(t)>0$,
$(a(t)(z''(t))^\alpha)' < 0$;

\item[(II)] $z(t) > 0$, $z'(t)<0$, $z''(t)>0$,
$(a(t)(z''(t))^\alpha)' < 0$.
\end{itemize}
\end{lemma}


\begin{lemma}[{\cite[Lemma 2]{bac}}] \label{lem22}
Let $x(t) $ be a positive solution of \eqref{eE+}, and let the
corresponding function $z(t)$ satisfy Case (II) of Lemma \ref{lem21}.
If
\begin{equation}\label{e22}
  \int_{t_0}^{\infty}\int_{v}^{\infty}
  \Big[\frac{1}{a(u)}\int_{u}^{\infty}q(s)ds\Big]^{1/\alpha}du \,dv = \infty,
\end{equation}
then $\lim_{t \to \infty} x(t) = \lim_{t\to \infty} z(t) = 0$.
\end{lemma}


\begin{lemma}\label{lem23}
 Assume that $u(t)>0$, $u'(t)>0$, $(a(t)(u'(t))^\alpha)' \leq 0 $
on $[t_0,\infty)$.
Then for each $\ell\in(0,1) $ there exists $T_\ell\geq t_0 $ such that
$$
 \frac{u(\tau(t))}{A(\tau(t))}\geq \ell\frac{u(t)}{A(t)}\quad \text{for }
 t\geq T_\ell.
$$
\end{lemma}

\begin{proof}
Since  $a(t)(u'(t))^\alpha$ is non-increasing, so is
$a^{1/\alpha}(t)(u'(t))$.
Then by the  definition of $A(t)$, we have
\begin{equation}\label{e23}
\begin{split}
u(t) - u(\tau(t)) 
&=\int_{\tau(t)}^t a^{1/\alpha}(s)(u'(s))\frac{1}{a^{1/\alpha}(s)}\,ds\\
&\leq a^{1/\alpha}(\tau(t))u'(\tau(t))\big(A(t) - A(\tau(t))\big).
\end{split}
\end{equation}
Also
$$
u(\tau(t)) \geq u(\tau(t)) -u(t_0) \geq a^{1/\alpha}(\tau(t))u'(\tau(t))
\big( A(\tau(t)) -A(t_0) \big).
$$
Since $\lim_{t\to \infty} \frac{A(\tau)-A(t_0)}{A(\tau)}=1$,
for each $\ell \in (0,1)$ there exists  $T_\ell\geq t_0 $ such
that $\big( A(\tau(t)) -A(t_0) \big) > \ell  A(\tau(t))$ for $t\geq T_\ell$.
From the above inequality,
\begin{equation} \label{e24}
\frac{u(\tau(t))}{u'(\tau(t))}\geq \ell
a^{1/\alpha}(\tau(t))A(\tau(t)),\quad t\geq T_\ell.
\end{equation}
Combining \eqref{e23} and \eqref{e24}, we obtain
$$
\frac{u(t)}{u(\tau(t))}\leq 1+ \frac{A(t)-A(\tau(t))}{\ell A(\tau(t))}
\leq \frac{A(t)}{\ell A(\tau(t))},
$$
which completes the proof.
\end{proof}

\begin{lemma}\label{lem24}
Assume that $z(t) > 0$, $z'(t)>0$, $z''(t)>0$,
$\big(a(t)(z''(t))^\alpha\big)' \leq 0 $ on $(T_\ell,\infty)$.
 Then
$$
\frac{z(t)}{z'(t)} \geq \frac{a^{1/\alpha}(t)A(t)}{2} \quad\text{for } t\geq T_\ell.
$$
\end{lemma}

\begin{proof}
Since  $a(t)(z''(t))^\alpha$ is positive and non-increasing, so is
$a^{1/\alpha}(t) z''(t)$. From $z'(t)>0$, $a(t)>0$, we have
\begin{equation} \label{e24i}
z'(t) \geq z'(t)- z'(\tau(t))
\geq \int_{T_\ell}^{t}\frac{a^{1/\alpha}(s)z''(s)}{a^{1/\alpha}(s)} ds
 \geq a^{1/\alpha}(t)A(t) z''(t).
\end{equation}
Since $A'(t)=a^{-1/\alpha}(t)$,
\begin{equation} \label{e24b}
 A'(t) z'(t) \geq A(t)z''(t),\quad t\geq T_\ell.
\end{equation}
 Integrating both sides of the above inequality, and using that
 $A(T_\ell)z'(T_\ell)>0$, we obtain
 $$
\int_{T_\ell}^{t}A'(s) z'(s)ds \geq A(t) z'(t)- \int_{T_\ell}^{t}A'(s) z'(s)ds.
$$
Therefore,
 \begin{equation}\label{e25}
 \int_{T_\ell}^{t}A' (s) z'(s)ds  \geq \frac{1}{2}A(t)z'(t).
 \end{equation}
Since $a(t) $ is non-increasing, we have $A(t)>0$, $A'(t)>0$, $A''(t)\geq 0$.
and
 \begin{equation}\label{e26}
 ( A'(t) z(t))' =  A'(t) z'(t)  +  A''(t) z(t) \geq  A'(t) z'(t).
\end{equation}
Integrating on both sides of the above equality, then using that
$A'(T_\ell)z(T_\ell)>0$ and \eqref{e25}, we obtain
$$
A'(t)z(t)  \geq \frac{1}{2}A(t)z'(t),\quad t\geq T_\ell,
$$
 which implies the desired result.
\end{proof}


The next lemma follows from \eqref{e24b}.

\begin{lemma} \label{lem25}
Assume that $ z'(t)>0$, $z''(t)>0$, $\big(a(t)(z''(t))^\alpha\big)' \leq 0 $
on $(T_\ell,\infty)$.
Then
$$
\frac{A(t)z''(t)}{A'(t)z'(t)} \leq 1, \quad\text{for } \quad t\geq T_\ell.
$$
\end{lemma}

For simplicity of notation, we introduce
$$
P_\ell(t) = \ell^\alpha(1-p)^\alpha q(t) a(\tau(t))
\Big(\frac{A(\tau(t))}{A(t)}\Big)^\alpha\Big(\frac{A(\tau(t))}{2}\Big)^\alpha 
$$
with $\ell \in (0,1)$ and $t\geq T_\ell$;
\begin{equation}\label{e27}
P= \liminf_{t\to \infty}A^\alpha(t)\int_{t}^{\infty}P_\ell(s)ds, \quad
Q= \limsup_{t\to \infty}\frac{1}{A(t)} \int_{t_0}^{t} A^{\alpha+1}(s)P_\ell(s)ds.
\end{equation}
Further, for $z(t)$ satisfying Case (I) of Lemma \ref{lem21},
we define
\begin{gather} \label{e28}
 w(t)= a(t)\Big(\frac{z''(t)}{z'(t)}\Big)^\alpha,\\
\label{e29} r= \liminf_{t\to \infty} A^\alpha(t)w(t),\quad
 R = \limsup _{t\to \infty}  A^\alpha(t) w(t).
\end{gather}

\begin{lemma}\label{lem26}
Let $x(t)$ be a positive solution of \eqref{eE+}.
\begin{itemize}
\item[(a)] Let $P<\infty$, $Q< \infty$ and $z(t)$ satisfy Case (I)  of Lemma \ref{lem21}.
Then $P \leq r -  r^{1+\frac{1}{\alpha}}$ and $P+Q \leq 1$.

\item[(b)] If $P=\infty $ or $Q= \infty$, then $z(t)$ does not satisfy
 Case (I) of Lemma \ref{lem21}.
\end{itemize}
\end{lemma}

\begin{proof}
Part (a).  Assume that $x(t) $ is a positive solution of \eqref{eE+} and the
corresponding function $z(t) $ satisfies Case(I) of Lemma \ref{lem21}.
From the definition of $z(t)$, we have
$$
x(t) = z(t)- p(t)x(\delta(t)) > z(t) - p(t)z(\delta(t)) \geq (1-p)z(t).
$$
Using this inequality in \eqref{eE+}, we obtain
\begin{equation}\label{e211}
(a(t)(z''(t))^\alpha)' \leq - (1-p)^\alpha q(t)z^\alpha(\tau(t)) \leq 0.
\end{equation}
Then from its  definition, $w(t)$ is positive and satisfies
\begin{equation}\label{e212}
\begin{split}
w'(t) &= \frac{1}{(z'(t))^\alpha} \big(a(t) (z''(t))^\alpha\big)'
 - \alpha a(t)\Big(\frac{z''(t)}{z'(t)}\Big)^{\alpha+1}\\
&\leq -q(t)(1-p)^\alpha  \frac{z^\alpha(\tau(t))}{(z'(t))^\alpha}
  - \frac{\alpha}{a^{1/\alpha} (t)} w^{1+\frac{1}{\alpha}}(t).
\end{split}
\end{equation}
From Lemma \ref{lem23} with $u(t) = z'(t)$, we have
$$
\frac{1}{z'(t)} \geq \ell \frac{A(\tau(t))}{A(t)}\frac{1}{z'(\tau(t))}, \quad
t\geq T_\ell,
$$
where $\ell $ is the same as in $P_\ell$. Now \eqref{e212} becomes
$$
w'(t) \leq - \ell^\alpha q(t)(1-p)^\alpha \Big(\frac{A(\tau(t))}{A(t)}\Big)^\alpha
 \frac{z^\alpha(\tau(t))}{(z'(\tau(t)))^\alpha}
 - \frac{\alpha}{a^{1/\alpha}(t)} w^{1+\frac{1}{\alpha}}(t).
$$
From Lemma \ref{lem24},  we have $z(t) \geq \frac{a^{1/\alpha}(t)A(t)}{2}z'(t)$, so that
 \begin{equation}\label{e213}
 w'(t)+ P_\ell(t)+ \frac{\alpha}{a^{1/\alpha}(t)} w^{1+\frac{1}{\alpha}}(t) \leq 0.
 \end{equation}
 Since $P_\ell (t)>0 $ and $w(t)>0 $ for $t\geq T_\ell$.
It follows  that $w'(t)\leq 0 $ and
$-w'(t)\geq \alpha w^{1+(1/\alpha)}(t)/a^{1/\alpha}(t)$;
 thus
 $$
 \Big(\frac{1}{w^{1/\alpha}(t)}\Big)' > \frac{1}{a^{1/\alpha}(t)}.
$$
 Integrating the above inequality from $T_\ell $ to $t$, and using that
$w^{-1/\alpha}(T_\ell)>0$,  we obtain
 $$
w(t)< \frac{1}{\big(\int_{T_\ell}^t a^{-1/\alpha}(s)\,ds\big)^\alpha}\,,
$$
which in view of (H3) implies that $\lim_{t\to \infty}w(t) = 0$.

On the other hand, from the definition of $w(t) $ and
  Lemma \ref{lem25},
$$
A^\alpha(t)w(t)=a(t)\Big(\frac{A(t)z''(t)}{z'(t)}\Big)^\alpha
=\Big(\frac{A(t)z''(t)}{A'(t)z'(t)}\Big)^\alpha \leq 1^\alpha.
$$
Then
\begin{equation}\label{e214}
  0\leq r \leq R\leq 1.
\end{equation}
Next we prove the first inequality in (a).
Let $\epsilon > 0$. Then from the definition of $P$ and $r$,
we can choose $t_2 \geq T_\ell $, sufficiently large such that
$$
A^\alpha(t)\int_{t}^{\infty}P_\ell(s)ds \geq P-\epsilon\quad \text{and}\quad
A^\alpha(t)w(t) \geq r-\epsilon \quad\text{ for } t\geq t_2.
$$
Integrating \eqref{e213} from $t$ to $\infty $ and using that
$\lim_{t\to \infty}w(t) = 0$, we have
\begin{equation}\label{e215}
  w(t) \geq \int_{t}^{\infty}P_\ell(s)ds
+ \alpha \int_{t}^{\infty} \frac{ w^{1+\frac{1}{\alpha}}(s)}
  {a^{1/\alpha}(s)}ds \quad\text{ for } t\geq t_2.
\end{equation}
Multiplying the above inequality by $A^\alpha(t) $ and simplifying, we obtain
\begin{align*}
     A^\alpha(t)w(t)
&\geq   A^\alpha(t) \int_{t}^{\infty}P_\ell(s)ds +
     \alpha A^\alpha(t)\int_{t}^{\infty} \frac{A^{\alpha+1}(s) w^{1+\frac{1}{\alpha}}(s)}
  {A^{\alpha+1 }(s)a^{1/\alpha}(s)}ds \\
&\geq   (P-\epsilon) + (r-\epsilon)^{1+\frac{1}{\alpha}} A^\alpha(t)\int_{t}^{\infty}
\frac{\alpha A'(s)}{A^{\alpha +1}(s)}ds,
\end{align*}
and so
$$
A^{\alpha}(t)w(t)\geq (P-\epsilon) + (r-\epsilon)^{1+\frac{1}{\alpha}}.
$$
Taking the limit inferior on both sides as $t\to \infty$, we obtain
$$
r\geq (P-\epsilon) + (r-\epsilon)^{1+\frac{1}{\alpha}}.
$$
Since $\epsilon > 0 $ is arbitrary, we obtain the desired result
$$
P\leq r - r^{1+\frac{1}{\alpha}}.
$$
Next, we prove the second inequality in  (a).
 Multiplying \eqref{e213} by $A^{\alpha +1}(t) $ and integrating it from $t_2$ to $t$,
we obtain
$$
\int_{t_2}^{t}A^{\alpha +1}(s)w'(s)ds
\leq  - \int_{t_2}^{t} A^{\alpha +1}(s) P_\ell(s)ds
-\alpha \int_{t_2}^{t} \frac{\left(A^{\alpha}(s)  w(s)\right)
^{(\alpha+1)/\alpha}}{a^{1/\alpha}(s)}ds.
$$
Integrating by parts,
\begin{align*}
A^{\alpha + 1}(t)w(t)
&\leq A^{\alpha + 1}(t_2)w(t_2) - \int_{t_2}^{t} A^{\alpha +1}(s)P_\ell(s)ds\\
&\quad - \alpha\int_{t_2}^{t}  \frac{\left(A^{\alpha}(s)  w(s)
 \right)^{(\alpha+1)/\alpha}}{  a^{1/\alpha}(s)}ds
 +\int_{t_2}^{t}w(s)\left( A^{\alpha +1}(s)\right)' ds.
\end{align*}
Hence,
\begin{align*}
A^{\alpha + 1}(t)w(t)
&\leq  A^{\alpha + 1}(t_2)w(t_2) - \int_{t_2}^{t} A^{\alpha +1}(s)P_\ell(s)ds\\
&\quad +\int_{t_2}^{t}\big[\frac{(\alpha+1)A^\alpha(s)w(s)}{
a^{1/\alpha}(s)}
 -\frac{\alpha (A^\alpha(s)w(s))^{(\alpha +1)/\alpha}}{  a^{1/\alpha}(s)} \big] ds.
\end{align*}
 Using the inequality
\begin{equation} \label{eBD}
 Bu - Du^{(\alpha +1)/\alpha}
\leq \frac{\alpha^\alpha}{(\alpha+1)^{\alpha+1}}\frac{B^{\alpha+1}}{D^\alpha}
\end{equation}
 with $ u = A^\alpha(t)w(t)$,
$D = \frac{\alpha}{a^{1/\alpha}(t)}$, and $B = \frac{\alpha+1}{a^{1/\alpha}(t)}$,
  we obtain
$$
A^{\alpha + 1}(t)w(t) \leq A^{\alpha + 1}(t_2)w(t_2)
- \int_{t_2}^{t} A^{\alpha +1}(s)P_\ell(s)ds + A(t)-A(t_2).
$$
It follows that
$$
A^{\alpha }(t)w(t) \leq \frac{1}{A(t)}A^{\alpha + 1}(t_2)w(t_2)
- \frac{1}{A(t)}\int_{t_2}^{t} A^{\alpha +1}(s)P_\ell(s)ds
  + 1-\frac{A(t_2)}{A(t)}.
$$
  Taking the limit superior on both sides as $t\to \infty$, we obtain
  \begin{equation}\label{e216}
  R\leq -Q + 1.
  \end{equation}
Combining this inequality with \eqref{e214}, we have
  $$
P\leq r-r^{1+\frac{1}{\alpha}} \leq r \leq R\leq -Q + 1,
$$
which completes the proof of Part (a).
\smallskip

 Part (b).  Assume that $x(t) $ is a positive solution of \eqref{eE+}.
We shall show that $z(t) $  can not satisfy  Case (I) of Lemma \ref{lem21}.
On the contrary, first, we assume that $P=\infty$.
Then \eqref{e215},
$$
A^{\alpha}(t)w(t)\geq A^\alpha(t)\int_{t}^{\infty}P_\ell(s)ds.
$$
Note that by \eqref{e214}, the left-hand side is bounded above by $1$.
Also note that limit inferior of the right-hand side is  $P=\infty$.
This leads to a contradiction.

Now, we assume that $Q=\infty$. Then by \eqref{e216},  $R=-\infty$,
which contradicts $0\leq R\leq 1$ in \eqref{e214}.
The proof is complete.
\end{proof}

Now we present oscillation results whose proofs
follow the steps in \cite[Theorems 1 and 2]{bac}.

\begin{theorem} \label{thm21}
Assume that \eqref{e22} holds, and let $x(t) $ be a solution of \eqref{eE+}.
If
\begin{equation}\label{e217}
P := \liminf_{t\to \infty}A^{\alpha }(t)\int_{t}^{\infty}P_\ell(s)ds >
   \frac{\alpha^\alpha}{(\alpha+1)^{\alpha+1}},
\end{equation}
then $x(t)$ is either oscillatory or $\lim_{t\to \infty}x(t) = 0$.
   \end{theorem}

\begin{proof}
Suppose $x$ is a non-oscillatory solution of \eqref{eE+}.
Since $-x$ is also a solution, we can assume without loss of generality
that $x$ is positive.
If $P=+\infty$, then by Lemma \ref{lem26}, $z(t)$ does not have property (I).
That is, $z(t)$ satisfies Case (II) of Lemma \ref{lem21}.
Therefore, n from Lemma \ref{lem22}, we
have $\lim_{t\to\infty}x(t)=0$.

Now assume that $z(t)$ satisfies Case (I) of Lemma \ref{lem21}.
Let $w(t)$ and $r$ be defined by
\eqref{e28} and \eqref{e29}, respectively. Then from Lemma \ref{lem26},
we have $P\leq r -r^{(\alpha+1)/\alpha}$.
Using  \eqref{eBD} with $B=D=1$, we have
$$
P\leq \frac{\alpha^\alpha}{(\alpha+1)^{\alpha +1}},
$$
which contradicts \eqref{e217}. The proof is complete.
\end{proof}

\begin{theorem} \label{thm22}
Assume that  \eqref{e22} holds, and let $x(t)$ be a solution of \eqref{eE+}.
If
   \begin{equation}\label{e218}
   P+Q>1,
   \end{equation}
then $x(t)$ is either oscillatory or $\lim_{t\to \infty}x(t) =0$.
\end{theorem}

\begin{proof}
Suppose $x$ is a non-oscillatory solution of \eqref{eE+}.
Since $-x$ is also a solution, we can assume without loss of generality
that $x$ is positive.
If $P$ or $Q$ equal $infty$, then by Lemma \ref{lem26}, $z(t)$ does not satisfy
Case (I), and $z(t)$ must satisfy Case (II). Then from Lemma \ref{lem22},
$\lim_{t\to\infty}x(t)=0$.

Now assume that Case (I) holds. Let $w(t)$ and $r$ be defined as above.
Then from Lemma \ref{lem26}, $P+Q\leq 1$. which contradicts \eqref{e218}.
The proof is complete.
\end{proof}
 
As a consequence of Theorem \ref{thm22}, we have the following results.

\begin{corollary}\label{coro23}
Assume that  \eqref{e22} holds. If
$$
\lim_{t\to \infty} \inf A^{\alpha }(t) \int_{t}^{\infty} q(s) a(\tau(s))
 \frac{(A(\tau(s)))^{2\alpha}}{A^\alpha} ds
> \frac{(2\alpha)^\alpha}{\ell^\alpha(1-p)^\alpha}
 \frac{\alpha^\alpha}{(\alpha+1)^{\alpha+1}},
$$
then every solution $x(t) $ of  \eqref{eE+} is either
 oscillatory or $ \lim_{t\to \infty}x(t) = 0$.
\end{corollary}

\begin{corollary}\label{coro24}
Assume that  \eqref{e22} holds. If
 $$
Q = \lim_{t\to \infty} \sup \frac{1}{A(t)} \int_{t_0}^{t} A^{\alpha +1}(s)P_\ell(s)ds > 1,
$$
 then $x(t) $ is either oscillatory or  $\lim_{t\to \infty}x(t) = 0$.
\end{corollary}

We conclude this section with an example.
Consider the third-order neutral differential equation
\begin{equation}\label{e219}
\big[\frac{1}{t^3}([x(t)+ \frac{1}{3} x(\frac{t}{2})]'')^3\big]' +
\frac{\lambda}{t^{10}}x^3(\frac{t}{2}) = 0,\quad \lambda >0, \quad t\geq 1.
\end{equation}
Here $a(t)=1/t^3$, $p=1/3$, $\alpha = 3$, $\tau(t) = \delta(t)= t/2$,
$q(t)= \lambda/t^{10}$.
It is easy to see that \eqref{e22} holds. Hence by Corollary \ref{coro23},
every non-oscillatory solution of \eqref{e219} converges to zero provided
that $\lambda > 3^6\times 4^5$.

\section{Oscillation results for \eqref{eE-}}

 For each solution $x(t)$ of \eqref{eE-},
 we define the associated function
\begin{equation}\label{e31}
z(t) = x(t) - p(t)x(\tau(t)).
\end{equation}

\begin{lemma}[{\cite[Lemma 7]{bac}}] \label{lem31}
Let $x(t)$ be a positive solution of equation\eqref{eE-}.
Then there are the following four cases for $z(t)$:
\begin{itemize}
\item[(I)]  $z(t) > 0$, $z'(t)>0$, $z''(t)>0$,
$(a(t)(z''(t))^\alpha)' < 0$;

\item[(II)] $z(t) > 0$, $z'(t)<0$, $z''(t)>0$,
$(a(t)(z''(t))^\alpha)' < 0$;

\item[(III)] $z(t) < 0$, $z'(t)<0$, $z''(t)>0$,
$(a(t)(z''(t))^\alpha)' < 0$;

\item[(IV)] $z(t) <0$, $z'(t)<0$, $z''(t)<0$,
 $(a(t)(z''(t))^\alpha)' < 0$.
\end{itemize}
\end{lemma}

\begin{lemma}[{\cite[Lemma 8]{bac}}] \label{lem32}
Let $x(t)$ be a positive solution of \eqref{eE-} and
$z(t)$ satisfy Case (II) of Lemma \eqref{lem31}. If \eqref{e22} holds, then
$\lim_{t\to \infty}x(t) = \lim_{t\to \infty}z(t) = 0$.
\end{lemma}

For simplicity of notation,  we introduce
$$
\overline{P_\ell}(t) = \ell^\alpha q(t)a(\tau(t))
\Big(\frac{A(\tau(t))}{A(t)}\Big)^\alpha \Big(\frac{A(\tau(t))}{2}\Big)^\alpha
$$
with $\ell \in (0,1) $;
$$
\overline{P} = \liminf_{t\to \infty}A^{\alpha}(t)
\int_{t}^{\infty}\overline{P_\ell}(s)ds,\quad
\overline{Q} = \limsup_{t\to \infty}\frac{1}{A(t)}
 \int_{t_0}^{t}A^{\alpha +1}(s)\overline{P_\ell}(s) ds.
$$
Also $w(t), r, R$ are defined as in \eqref{e28} and \eqref{e29}.

\begin{lemma}\label{lem33}
 Let $x(t)$ be a positive solution of \eqref{eE-}.
\begin{itemize}
\item[(a)] Let $\overline{P}<\infty$ and $\overline{Q}< \infty$.
 Assume that  $z(t)$ satisfies Case (I)  of Lemma \ref{lem31}.
Then $\overline{P} \leq r -  r^{1+\frac{1}{\alpha}}$ and
$\overline{P} + \overline{Q} \leq 1$.

\item[(b)] If $\overline{P}=\infty $ or $\overline{Q}= \infty$,
then $z(t)$ can not satisfy  Case (I) of Lemma \ref{lem31}.
\end{itemize}
 \end{lemma}

 \begin{proof}
 Assume that $x(t)$ is a positive solution of \eqref{eE-} and the associated
function $z(t) $ satisfies Case (I) of Lemma \ref{lem31}.
Since $0<z(t)<x(t)$, equation \eqref{eE-} can be written as
 $$
\big(a(t)(z''(t))^\alpha\big)' < - q(t)z^\alpha(\tau(t)) < 0.
$$
 The rest of the proof is similar to that of Lemma \ref{lem26} and hence
it is omitted.
 \end{proof}

The following theorem presents an oscillation criterion for equation \eqref{eE-}.

 \begin{theorem} \label{thm34}
Assume that  \eqref{e22} holds. If
 \begin{equation}\label{e32}
 \liminf_{t\to \infty} A^{\alpha}(t)\int_{t}^{\infty}\overline{P_\ell}(s)ds >
 \frac{\alpha^\alpha}{(\alpha+1)^{\alpha+1}},
 \end{equation}
 then every solution $x(t) $ of \eqref{eE-} is either oscillatory or
 $\lim_{t\to \infty} x(t)= 0$.
 \end{theorem}

The proof of the above theorem is similar to that of \cite[Theorem 3]{bac};
hence it is omitted.
From the above theorem we have a simplified criterion  as follows.

\begin{corollary}\label{coro32}
 Assume that \eqref{e22} holds. If
\begin{equation}\label{e33}
\liminf_{t\to \infty} A^{\alpha}(t)\int_{t}^{\infty}q(s)a(\tau(s))
\frac{(A(\tau(s)))^{2\alpha}}{A^\alpha(s)}ds >
\frac{(2\alpha)^\alpha}{(\alpha+1)^{\alpha+1}},
\end{equation}
then every solution $x(t) $ of \eqref{eE-} is either oscillatory or
$\lim_{t\to \infty} x(t) = 0$.
\end{corollary}

\begin{theorem}\label{thm33}
Assume that \eqref{e22} holds. Let $x(t)$ be a solution of \eqref{eE-}.
If 
\begin{equation}\label{e34}
\overline{P} + \overline{Q} > 1,
\end{equation}
then every solution of \eqref{eE-} is either oscillatory or $\lim_{t\to \infty} x(t) =
0$.
\end{theorem}

The proof of the above theorem is similar to that of Theorem \ref{thm22};
hence it is omitted.

\begin{corollary}\label{coro34}
Assume that \eqref{e22} holds. If 
\begin{equation}\label{e35}
\limsup_{t\to \infty} \frac{1}{A(t)}\int_{t_0}^{t}A^{\alpha +1}(s)
\overline{P_\ell}(s)ds > 1,
\end{equation}
then every solution $x(t) $ of \eqref{eE-} is either oscillatory or
$\lim_{t\to \infty} x(t) = 0$.
\end{corollary}

As an example, consider the third-order neutral
differential equation 
\begin{equation}\label{e36}
\Big(\frac{1}{t^3}([x(t)- \frac{1}{3}x(\frac{t}{2})]'')^3\Big)' +
\frac{\lambda}{t^{10}}x^3(\frac{t}{2}) = 0,\quad \lambda >0, \quad
 t\geq 1.
\end{equation}
Corollary \ref{coro32} implies that every solution of \eqref{e36}
is either oscillatory or approaches zero as $t\to\infty$,
 provided $\lambda > 6^4 \times 2^7$.

We conclude this article  with by remarking that
when $a(t)$ is constant, our results coincide with the results in \cite{bac}.


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\end{document}