\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 27, 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  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/27\hfil Kamenev-type oscillation criteria]
{Kamenev-type oscillation criteria for second-order
quasilinear differential equations}
\author[Z. Xu, Y. Xia \hfil EJDE-2005/27\hfilneg]
{Zhiting Xu, Yong Xia}

\address{Zhiting  Xu \hfill\break
 School of  Mathematical Sciences, South China Normal University \\
 Guangzhou, 510090, China}
\email{xztxhyyj@pub.guangzhou.gd.cn}

\address{Yong Xia \hfill\break
Department of Mathematics, Dongguan Institute of Technology\\
Dongguan, 511700, China}
\email{xiay@dgut.edu.cn}


\date{}
\thanks{Submitted August 11, 2004. Published March 6, 2005.}
\subjclass[2000]{34C10, 34C15}
\keywords{Oscillation;  second order quasilinear
 differential equation; \hfill\break\indent integral operator}

\begin{abstract}
 We obtain  Kamenev-type oscillation criteria for
 the second-order quasilinear  differential equation
 $$
 (r(t)|y'(t)|^{\alpha-1}y'(t))'+ p(t)|y(t)|^{\beta-1}y(t)=0 \,.
 $$
 The criteria obtained extend the integral averaging technique and
 include earlier results due to Kamenev, Philos and Wong.
\end{abstract}

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

\section{Introduction}

This paper concerns the oscillation of solutions to the second order
quasilinear differential equation
\begin{equation}
(r(t)|y'(t)|^{\alpha-1}y'(t))'+
p(t)|y(t)|^{\beta-1}y(t)=0, \quad t\geq t_0>0,
\label{e1.1}
\end{equation}
where  $r\in C^1([t_0, \infty), \mathbb{R}^+)$,
$p \in C([t_0, \infty), \mathbb{R})$,
and $\alpha$, $\beta>0$, $(\alpha\neq \beta)$, are constants.

In this paper we shall assume that the following conditions hold:
\begin{itemize}
\item[(A1)]  $ R(t):=\int^t_{t_0}r^{-1/\alpha}(s)ds\to \infty$, as
$t \to \infty$,

\item[(A2)]  $ \liminf_{t \to\infty}\int^t_{t_0}p(s)ds=-M_0>-\infty$.
\end{itemize}

By a solution to \eqref{e1.1}, we mean a function $y\in C^1( [T_y,
\infty), \mathbb{R})$, $T_y\geq t_0$, which has the property
$r(t)|y'(t)|^{\alpha-1}y'(t) \in C^1([ T_y, \infty), \mathbb{R})$
and satisfies \eqref{e1.1}. We restrict our attention only to the
nontrivial solutions of \eqref{e1.1}, i.e., to the solution $y(t)$
such that $\sup \{|y(t)|: t\geq T \}>0$ for all $T\geq T_y$. A
nontrivial solution of \eqref{e1.1} is called oscillatory if it
has arbitrary large zeros, otherwise, it is called nonoscillatory.
Equation \eqref{e1.1} is called oscillatory if all its solutions
are oscillatory.

When $\alpha=\beta$, Equation \eqref{e1.1} reduces to second order half-linear
differential equation
\begin{equation}
(r(t)|y'(t)|^{\alpha-1}y'(t))'+
p(t)|y(t)|^{\alpha-1}y(t)=0.
\label{e1.2}
\end{equation}

Oscillatory and nonoscilltory property  of \eqref{e1.2} have been
widely discussed in the literatures (see, for example,
\cite{a1,d1,e1,e2,h1,h2,k2,l1,l2,l3,l4,l5,l6,m1,w2,w3,y1} and the
reference therein). However, relatively less attention \cite{t1}
has been given to oscillation of \eqref{e1.1}. Some of the
oscillation criteria \cite{a1,l4,l5,m1,w2} for \eqref{e1.2} have
been obtained by using the averaging technique from the papers of
Kamenev \cite{k1} and Philos \cite{p1}
 for linear differential equation
\begin{equation}
(r(t)y'(t))'+p(t)y(t)=0.
\label{e1.3}
\end{equation}
It is, therefore, natural to ask if it is possible to establish oscillation
criteria for \eqref{e1.1}. Motivated by the idea of Wong \cite{w1}, in this paper we
extend the results of Kamenev \cite{k1}, Philos \cite{p1}, Wong \cite{w1}
 to \eqref{e1.1}
by general means given in \cite{p1,w1}.
Our methodology is somewhat different from that of
previous authors. We believe that our approach is simple and also provides a more
unified account of the study of Kamenev-type oscillation theorems. We will also
show that do not need any restriction on the sign of the function $p$.

\section{Main results}

First, we introduce the concept of general means \cite{p1,w1} and present some
properties, which will be used in the proof of our results.

Let $D=\{ (t,s): t \geq s\geq t_0\}$ and $D_0=\{(t,s): t > s\geq t_0\}$. We will
say that the function $H \in C(D, \mathbb{R})$ belongs to a class $\Im$
if
\begin{itemize}
\item[(H1)]   $H(t,t)=0$ for $t \geq t_0$, $H(t,s) >0$ on $D_0$

\item[(H2)] $H$ has a continuous and non-positive partial derivative
in $D_0$ with respect to the second variable

\item[(H3)]  There exist functions $\rho \in C^1([t_0, \infty), \mathbb{R}^+ )$
and $h \in C(D, \mathbb{R})$ such that
$$
\frac{\partial}{\partial s}[H(t,s)\rho(s)]=-H(t,s)h(t,s),
\quad (t,s)\in D_0.
$$
\end{itemize}
Let $\rho \in C^1([t_0, \infty), \mathbb{R}^+ )$ and $H \in \Im$. We take the
integral operator $A$, which is defined in \cite{w1}, in terms of $H(t,s)$ and
$\rho(s)$ as
\begin{equation}
A_T (\phi; t):=\int^t_TH(t,s)\phi(s)\rho(s)ds, \quad t \geq T\geq t_0,
\label{e2.1}
\end{equation}
where $\phi \in C([t_0, \infty), \mathbb{R})$. It is easily seen that the integral
operator $A$ satisfies the following properties:
\begin{gather}
A_T(\alpha_1 h_1+\alpha_2h_2; t)
=\alpha_1A_T(h_1,t)+\alpha_2A_T(h_2, t), \label{e2.2} \\
A_T (h_3, t) \geq 0 \quad \forall\, h_3\geq0, \label{e2.3} \\
A_T(h'_4; t)=-H(t,T)h_4(T)\rho(T)
+A_T(\rho^{-1} h_4 h;t). \label{e2.4}
\end{gather}
Here $h_1, h_2, h_3\in C([t_0, \infty),\mathbb{R})$,
$h_4\in C^1([t_0, \infty),\mathbb{R})$, and $\alpha_1$, $\alpha_2$ are real
numbers.

For an arbitrary function $\xi \in C([ t_0, \infty), \mathbb{R}^+)$, define the
kernel
\begin{equation}
H(t,s):=\Big(\int^t_s\frac{du}{\xi(u)}\Big)^m, \quad m>1, \label{e2.5}
\end{equation}
with $\int^{\infty}_a 1/\xi (\tau)d{\tau}=\infty$. An important
particular case is $\xi(\tau)=\tau^n$, where $n \leq 1$ is real. When
$\xi(\tau)=1$ we have $H(t,s)=(t-s)^m$, and when $\xi(\tau)=\tau$ we have
$H(t,s)=(\ln t/\ln s)^m$. It is easily verified that the kernel  \eqref{e2.5}
satisfies (H1)--(H3).


We are now able to state and show the main results.


\begin{theorem} \label{thm2.1}
Suppose that there exist functions
$\rho \in C^1([t_0, \infty), \mathbb{R}^+)$, $H, h \in C(D, \mathbb{R})$
with $H \in \Im$ and for any $M>0$ such that
\begin{equation}
\limsup_{t \to \infty}\frac{1}{H(t,t_0)}A_{t_0}
(p-\theta g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1};t)=\infty,
\label{e2.6}
\end{equation}
where
$$
\theta=(\alpha+1)^{-(\alpha+1)}, \quad
g(t)= \frac{\beta M}{\alpha}r^{-1/\alpha}(t)R^{-1}(t).
$$
Then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
 Let $y(t)$ be a nonoscillatory solution of \eqref{e1.1}. Without loss
of generality, we assume that $y(t)\neq 0$ for all $t\geq t_0$. Furthermore, we
suppose that $y(t)>0$ for $t\geq t_0$, since the substitution $u=-y$ when $y(t)<0$
transforms \eqref{e1.1} into an equation of the same form  to the assumptions
of the theorem. Now, we put
\begin{equation}
W(t)=\frac{r(t)|y'(t)|^{\alpha-1}y'(t)}{|y(t)|^{\beta-1}y(t)}.
\label{e2.7}
\end{equation}
Then it follows from \eqref{e1.1} that
\begin{equation}
\begin{aligned}
W'(t)&=  -p(t)
-\beta r(t)\frac{|y'(t)|^{\alpha+1}}{|y(t)|^{\beta+1}}\\
&= -p(t)-\beta r^{-1/\alpha}(t) |y(t)|^{(\beta-\alpha)/\alpha}
|W(t)|^{(\alpha+1)/\alpha}, \quad \mbox{for }  t\geq t_0,
\end{aligned} \label{e2.8}
\end{equation}
and consequently,
\begin{equation}
\frac{r(t)|y'(t)|^{\alpha-1}y'(t)}{|y(t)|^{\beta}}
=C-\int^t_{t_0}p(s)ds-\beta \int^t_{t_0}r(s)
\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds,
\label{e2.9}
\end{equation}
where $C=r(t_0)|y'(t_0)|^{\alpha-1}y'(t_0)/|y(t_0)|^{\beta}$.


Next, we consider the following three cases for the behavior of $y'(t)$:


\noindent\emph{Case 1.} $y'(t)$ is oscillatory.
Then there exists a sequence
$\{t_m\}$, $ ( m=1,2 \dots)$, in $[t_0, \infty)$ with
$\lim_{m \to\infty}t_m=\infty$ and such
that $y'(t_m)=0$, $(m=1,2,\dots)$. Thus, \eqref{e2.9} gives
$$
\beta \int^{t_m}_{t_0}r(s)
\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds
=C-\int^{t_m}_{t_0}p(s)ds, \quad m=1,2, \dots,
$$
and hence, by (A2), we conclude that
\begin{equation}
\int^{\infty}_{t_0}r(s)
\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds<\infty.
\label{e2.10}
\end{equation}
So, for some positive constant $N$, we have
$$
\int^t_{t_0}r(s)
\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds\leq N^{\alpha+1},
\quad \mbox{for }   t\geq t_0.
$$
By the H\"{o}lder inequality
\begin{align*}
\big|\int^t_{t_0}\frac{y'(s)}
{[y(s)]^{(\beta+1)/(\alpha+1)}}ds\big|
&\leq  \big[ \int^t_{t_0}r(s)\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds\big]
^{1/(\alpha+1)}\big[ \int^t_{t_0}r^{-1/\alpha}(s)ds\big]^{\alpha/(\alpha+1)}\\
&\leq  NR^{\alpha/(\alpha+1)}(t).
\end{align*}
Hence
$$
\big|[y(t)]^{(\alpha-\beta)/(\alpha+1)}
-[y(t_0)]^{(\alpha-\beta)/(\alpha+1)}\big|\leq
\frac{N(\alpha+1)}{|\alpha-\beta|}R^{\alpha/(\alpha+1)}(t).
$$
So, there exist a $t_1\geq t_0$ and a constant $M>0$ so that for $t\geq t_1$
$$
|y(t)|^{(\alpha-\beta)/(\alpha+1)}
\leq M^{-\alpha/(\alpha+1)}R^{\alpha/(\alpha+1)}(t),
$$
or
\begin{equation}
|y(t)|^{(\beta-\alpha)/\alpha}
\geq MR^{-1}(t).
\label{e2.11}
\end{equation}
Substituting from \eqref{e2.11} into \eqref{e2.8}, we have
\begin{equation}
\begin{aligned}
W'(t)&\leq  -p(t)-\beta Mr^{-1/\alpha}(t)R^{-1}(t)
|W(t)|^{(\alpha+1)/\alpha}\\
&=  -p(t)-\alpha g(t)|W(t)|^{(\alpha+1)/\alpha}.
\end{aligned} \label{e2.12}
\end{equation}
Applying the operator $A_T$, $(T\geq t_0)$, to \eqref{e2.12}, and
using \eqref{e2.4}, we have
\begin{equation}
A_T(p  ; t)\leq H(t,T)\rho(T)W(T)+A_T(\rho^{-1}|h||W|;t)
-\alpha A_T(g|W|^{(\alpha+1)/\alpha};t).
\label{e2.13}
\end{equation}
The Young inequality gives
$$
\rho^{-1}|h||W|\leq \alpha g|W|^{(\alpha+1)/\alpha}
+\theta g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1}.
$$
Substitute the above inequality into \eqref{e2.13}, we get
\begin{equation}
A_T(p ;t)\leq H(t, T)\rho(T)W(T)
+\theta A_T(g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1};t).
\label{e2.14}
\end{equation}
Set $T=t_0$ and divide \eqref{e2.14} through by $H(t, t_0)$, so
\begin{equation}
\frac{1}{H(t,t_0)}A_{t_0}(p
-\theta g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1};t)\leq \rho(t_0)W(t_0).
\label{e2.15}
\end{equation}
Taking limsup in \eqref{e2.15} as $t \to \infty$, condition \eqref{e2.6}
gives a desired contradiction. \smallskip


\noindent\emph{Case 2.} $y'(t)>0$ on $[T, \infty)$ for some $T\geq
t_0$. In this case, from \eqref{e2.9} it follows that
\eqref{e2.10} holds for $t\geq T$. Once again, we can complete the
proof by the procedure of the proof of Case 1. \smallskip


\noindent\emph{Case 3.} $y'(t)<0$ on $[ T, \infty)$ for some $T\geq t_0$.
If \eqref{e2.10} holds, then we can arrive at a contradiction by the
procedure of Case 1.
So we suppose that
$$
\int^{\infty}_{t_0}r(s) \frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds=\infty.
$$
Using \eqref{e2.9}, we have, for $t\geq T$
\begin{equation}
-\frac{r(t)|y'(t)|^{\alpha-1}y'(t)}{|y(t)|^{\beta}}\geq -(C+M_0)
+\beta \int^t_T r(s)\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds.
\label{e2.16}
\end{equation}
By the assumption, we can choose $T_1\geq T$ such that
$$
\beta  \int^{T_1}_Tr(s)\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds
=1+C+M_0,
$$
and then for any $t\geq T_1$, we get
$$
\frac{-\frac{r(t)|y'(t)|^{\alpha-1}y'(t)}{|y(t)|^{\beta}}
\big(-\beta\frac{y'(t)}{y(t)}\big)}{-(C+M_0)+\beta\int^t_Tr(s)
\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds}
\geq -\beta \frac{y'(t)}{y(t)}.
$$
Integrate the above inequality from $T_1$ to $t$ to obtain
$$
\ln \big[-(C+M_0)+\beta\int^t_Tr(s)
\frac{|y'(s)|^{\alpha+1}}{|y(s)|^{\beta+1}}ds\big]\geq
\ln \big[\frac{y(T_1)}{y(t)}\big]^{\beta},
$$
which together with \eqref{e2.16} yields
$$
-\frac{r(t)|y'(t)|^{\alpha-1}y'(t)}{|y(t)|^{\beta-1}y(t)}
\geq \big(\frac{y(T_1)}{y(t)}\big)^{\beta},
$$
from which it follows that
$$
y'(t)\leq -y^{\beta/\alpha}(T_1)r^{-1/\alpha}(t), \quad \mbox{for }
t\geq T_1,
$$
then, by (A1),
$$ y(t)\leq
y(T_1)-y^{\beta/\alpha}(T_1)\int^t_{T_1}r^{-1/\alpha}(s)ds \to
-\infty, \quad \mbox{as }   t\to \infty,
$$
contradicting the assumption that $y(t)>0$. This completes the
proof.
\end{proof}


\begin{corollary} \label{coro2.1}
Replace condition \eqref{e2.6} in Theorem 2.1  by
\begin{equation}
\limsup_{t \to \infty}\frac{1}{H(t,t_0)}A_{t_0}(p ;t)=\infty,
\label{e2.17}
\end{equation}
and assume that
\begin{equation}
\limsup_{t \to \infty}\frac{1}{H(t,t_0)}
A_{t_0}(g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1} ;t)<\infty.
%\label{e2.18}
\end{equation}
Then conclusion of Theorem 2.1 holds.
\end{corollary}

It is clear that \eqref{e2.17} is a necessary condition for \eqref{e2.6} to hold.
In case \eqref{e2.6} fails to satisfied, then the following theorem may
be applicable.


\begin{theorem} \label{thm2.2}
Let $\rho$, $H$ and $h$ be as in Theorem 2.1. Suppose
that there exists $\phi_1, \phi_2\in C([t_0, \infty), \mathbb{R})$ and
for all $T\geq t_0$, $M>0$ such that
\begin{equation}
\limsup_{t \to \infty}\frac{1}{H(t,T)}A_T(p ;t)\geq \phi_1(T)
\label{e2.19}
\end{equation}
and
\begin{equation}
\limsup_{t \to \infty}\frac{1}{H(t,T)}
A_T(g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1} ;t)\leq \phi_2(T),
\label{e2.20}
\end{equation}
where $\phi_1$ and $\phi_2$ satisfy
\begin{equation}
\liminf_{t \to \infty}\frac{1}{H(t, T)}A_T(g\rho^{-(\alpha+1)/\alpha}
(\phi_1-\theta\phi_2)_+^{(\alpha+1)/\alpha};t)=\infty,
\label{e2.21}
\end{equation}
where $\theta$ and $g$ are the same as in Theorem 2.1.
Then \eqref{e1.1} is oscillatory.
\end{theorem}


\begin{proof}
Let $y(t)$ be a non-oscillatory solution of \eqref{e1.1}, say
$y(t)>0$ for $t\geq t_0$, and let $W(t)$ be as defined in the proof of
Theorem 2.1 for all $t\geq t_0$, we get \eqref{e2.8}.
As in the proof of Theorem 2.1, we consider three cases of the
behavior of $y'(t)$. \smallskip

\noindent\emph{Case 1.}  $y'(t)$ is oscillatory.
Proceeding as the proof of Theorem 2.1 (Case 1),
 \eqref{e2.13} and \eqref{e2.14} hold. Then by \eqref{e2.14}, we have,
for all $T\geq t_0$,
$$
\frac{1}{H(t,T)}A_T(p ; t)-\frac{\theta}{H(t,T)}
A_T(g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1}; t)\leq \rho(T)W(T).
$$
Taking limsup in above inequality as $t \to \infty$ and applying \eqref{e2.19}
and \eqref{e2.20}, we obtain
$$
\phi_1(T)-\theta\phi_2(T)\leq \rho(T)W(T),
$$
from which it follows that
\begin{equation}
\frac{1}{H(t,T)}A_T\big(g\rho^{-(\alpha+1)/\alpha}(\phi_1-\theta\phi_2)_+
^{(\alpha+1)/\alpha}; t\big)\leq
\frac{1}{H(t,T)}A_T\big(g|W|^{(\alpha+1)/\alpha};t\big).
\label{e2.22}
\end{equation}
On the other hand, by \eqref{e2.13}, we have
\begin{align*}
 &\frac{\alpha}{H(t,T)}A_T(g|W|^{(\alpha+1)/\alpha}; t)
-\frac{1}{H(t,T)}A_T(\rho^{-1}|h||W|; t)\\
&\leq  \rho(T)W(T)-\frac{1}{H(t,T)}A_T(p ; t).
\end{align*}
Thus, by \eqref{e2.19},
\begin{equation}
\begin{aligned}
 &\liminf_{t \to\infty}\left\{\frac{\alpha}{H(t,T)}A_T
(g|W|^{(\alpha+1)/\alpha}; t)
-\frac{1}{H(t,T)}A_T(\rho^{-1}|h||W|; t)\right\}\\
&\leq  \rho(T)W(T)-\phi_1(T)\leq C_0.
\end{aligned}\label{e2.23}
\end{equation}
where $C_0$ is a constant.
Now, we claim that
\begin{equation}
 \liminf_{t \to\infty}\frac{1}{H(t,T)}A_T
\big(g|W|^{(\alpha+1)/\alpha}; t\big)<\infty.
\label{e2.24}
\end{equation}
If this inequality does not hold, then there exists a sequence
$\{t_j\}_{j=1}^{\infty} \in [t_0, \infty)$ with $\lim_{j \to \infty}t_j=\infty$
such that
\begin{equation}
\liminf_{j \to\infty}\frac{1}{H(t_j,T)}A_T
\big(g|W|^{(\alpha+1)/\alpha}; t_j\big)=\infty.
\label{e2.25}
\end{equation}
Hence, by \eqref{e2.23}, for $j$ large enough, we have
$$
\frac{\alpha}{H(t_j,T)}A_T\big(g|W|^{(\alpha+1)/\alpha}; t_j\big)
-\frac{1}{H(t_j,T)}A_T(\rho^{-1}|h||W|; t_j)\leq  C_0+1.
$$
This and \eqref{e2.25} give, for $j$ large enough,
$$
\frac{A_T(\rho^{-1}|h||W|; t_j)}{A_T(g|W|^{(\alpha+1)/\alpha};
t_j)}-\alpha\geq-\frac{\alpha}{2},
$$
that is
\begin{equation}
A_T(\rho^{-1}|h||W|; t_j)\geq\frac{\alpha}{2}A_T
(g|W|^{(\alpha+1)/\alpha}; t_j), \quad
\mbox{for all large }   j.
\label{e2.26}
\end{equation}
By the H\"{o}lder inequality
\begin{equation}
\begin{aligned}
 &A_T(\rho^{-1}|h||W|; t_j)\\
&\leq \big[A_T(g|W|^{(\alpha+1)/\alpha}; t_j)\big]^{\alpha/(\alpha+1)}
\big[A_T(g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1};
t_j)\big]^{1/(\alpha+1)}.
\end{aligned} \label{e2.27}
\end{equation}
 From \eqref{e2.26} and \eqref{e2.27}, we obtain
\begin{equation}
\frac{1}{H(t_j, T)}A_T\big(g^{-\alpha}\rho^{-(\alpha+1)}|h|^{\alpha+1};
t_j\big)\geq \big(\frac{\alpha}{2}\big)^{\alpha+1}\frac{1}{H(t_j,T)}
A_T(g|W|^{(\alpha+1)/\alpha}; t_j).
\label{e2.28}
\end{equation}
By \eqref{e2.20}, the left-hand side of \eqref{e2.28} is bounded, which contradicts \eqref{e2.25}.
Therefore, \eqref{e2.24} holds. Hence by \eqref{e2.22},
\begin{align*}
& \liminf_{t \to \infty}\frac{1}{H(t,T)}
A_T(g\rho^{-(\alpha+1)/\alpha}(\phi_1
-\theta\phi_2)_+^{(\alpha+1)/\alpha};t)\\
&\leq  \liminf_{t \to \infty}\frac{1}{H(t,T)}
A_T(g|W|^{(\alpha+1)/\alpha}; t)<\infty,
\end{align*}
which contradicts \eqref{e2.21}. \smallskip

\noindent\emph{Case 2.} $y'(t)>0$ on $[T, \infty)$ for some $T\geq t_0$.
In this case, from \eqref{e2.9}, it follows \eqref{e2.10} holds for $t\geq T$.
Once again, we can compete the proof  by the procedure of the proof of Case 1.

\noindent\emph{Case 3.} $y'(t)<0$ on $[T, \infty)$ for some $T\geq t_0$.
The proof is exactly the same as for the same case in Theorem 2.1,
and hence is omitted.
\end{proof}


\begin{remark} \label{rmk2.1} \rm
 It is easy to check that condition (A2) can be replaced by
$$
\liminf_{t \to \infty}\int^t_{t_0}p(s)ds>-\infty,
$$
and still the conclusion of Theorems 2.1 and 2.2 hold.
\end{remark}

\begin{remark} \label{rmk2.2} \rm
 The results in this paper are presented in a form with a high
degree of generality, and thus they give many possibilities for
oscillation criteria with an appropriate choice of functions $H$
and $\rho$, we omit the details.
\end{remark}

\section{Examples}

In this section, we provide two examples to illustrate the results obtained in
this paper. Note that  criteria reported in the references do not apply to these
equations. For simplicity in these two examples, we take
$$
H(t,s)=(t-s)^2, \quad \rho(t)=1,
$$
then
$$
h(t,s)=\frac{2}{t-s}.
$$

\subsection*{Example 3.1}
 Consider the quasilinear differential equation
\begin{equation}
(t^{-\nu}|y'(t)|^{\alpha-1}y'(t))'
+t^{\lambda-1}( \lambda(2-\sin t)-t\cos t )|y(t)|^{\beta-1}y(t)=0,
\label{e3.1}
\end{equation}
for $t \geq t_0>0$, where $\nu, \lambda, \alpha, \beta$ are arbitrary positive
constants with $\alpha\neq \beta$, $\alpha\neq 2$, and for any $M>0$
$$
g(t)=\frac{\beta M(\nu+\alpha)}{\alpha^2}t^{\nu/\alpha}
\big[t^{(\nu+\alpha)/\alpha}-t_0^{(\nu+\alpha)/\alpha}\big]^{-1}.
$$
Then, for any $t\geq t_0$, we have
$$
\int^t_{t_0}p(s)ds=\int^t_{t_0}d [ s^{\lambda}(2-\sin s)]
=t^{\lambda}(2-\sin t)-k_1 \geq t^{\lambda}-k_1,
$$
where $k_1=t_0^{\lambda}(2-\sin t_0)$. Moreover
\begin{align*}
&\frac{1}{H(t,t_0)}A_{t_0}(p-\theta g^{-\alpha}\rho^{-(\alpha+1)}
|h|^{\alpha+1};t)\\
&= \frac{1}{(t-t_0)^2}\int^t_{t_0}\big\{(t-s)^2p(s)
-k_2\theta(t-s)^{1-\alpha}s^{-\nu}[s^{(\nu+\alpha)/\alpha}
-t_0^{(\nu+\alpha)/\alpha}]^{\alpha}\big\}ds\\
&= \frac{2}{(t-t_0)^2}\int^t_{t_0}(t-s)\int^s_{t_0}p(\tau)d{\tau}ds
-\frac{k_2\theta}{(t-t_0)^2}\int^t_{t_0}(t-s)^{1-\alpha}
s^{-\nu}\\
&\quad \big[s^{(\nu+\alpha)/\alpha}-t_0^{(\nu+\alpha)/\alpha}\big]^{\alpha}ds\\
&\geq\frac{2}{(t-t_0)^2}\int^t_{t_0}(t-s)(s^{\lambda}-k_1)ds
-\frac{k_2\theta}{(t-t_0)^2}\int^t_{t_0}(t-s)^{1-\alpha}s^{\alpha}ds\\
&\geq  \frac{2}{(t-t_0)^2}
\Big[\frac{t^{\lambda+2}}{(\lambda+1)(\lambda+2)}
-\frac{t\,t_0^{\lambda+1}}{\lambda+1}
+\frac{t_0^{\lambda+2}}{\lambda+2}-\frac{k_1 t^2}{2}+k_1 t\,t_0
+\frac{k_1 t_0^2}{2}\Big]\\
&\quad -\frac{k_2 \theta t^2}{(2-\alpha)(t-t_0)^2}
(1-\frac{t_0}{t})^{2-\alpha}.
\end{align*}
where $k_2=2^{\alpha+1}\big(\frac{\alpha^2}{\beta M(\nu+\alpha)}\big)^{\alpha}$.
Consequently,  \eqref{e2.6} is satisfied. Hence, \eqref{e3.1} is oscillatory by
Theorem 2.1.


\subsection*{Example 3.2}
Consider the quasilinear differential equation
\begin{equation}
(t^{\nu}|y'(t)|^{\alpha-1}y'(t))'
+( t^{\lambda}\cos t )|y(t)|^{\beta-1}y(t)=0,
\label{e3.2}
\end{equation}
for $t \geq t_0>0$, where $\nu, \lambda, \alpha, \beta$ are arbitrary
constants with $\lambda\leq 0$, $0<\alpha<2$, $\beta>0$, $\alpha\neq \beta$,
$\nu<\alpha$, and for any $M>0$
$$
g(t)=\frac{\beta M(\alpha-\nu)}{\alpha^2}t^{-\nu/\alpha}
\big[t^{(\alpha-\nu)/\alpha}-t_0^{(\alpha-\nu)/\alpha}\big]^{-1}.
$$
Moveover, for $t>s\geq T\geq t_0$, we have
\begin{align*}
&\limsup_{t \to \infty}\frac{1}{(t-T)^2}\int^t_T(t-s)^2
g^{-\alpha}(s)|h(t,s)|^{\alpha+1}ds\\
&= k_3\limsup_{t \to \infty}\frac{1}{(t-T)^2}
\int^t_T(t-s)^{1-\alpha}s^{\nu}
\big[s^{(\alpha-\nu)/\alpha}-t_0^{(\alpha-\nu)/\alpha}\big]^{\alpha}ds\\
&\leq  k_3\limsup_{t \to \infty}\frac{1}{(t-T)^2}
\int^t_T(t-s)^{1-\alpha}s^{\alpha}ds\\
&\leq  \frac{k_3}{2-\alpha}\limsup_{t \to \infty}
\frac{t^{\alpha}}{(t-T)^{\alpha}}=\frac{k_3}{2-\alpha},
\end{align*}
and
\begin{align*}
 \limsup_{t \to \infty}\frac{1}{(t-T)^2}\int^t_T(t-s)^2p(s)ds
&= \limsup_{t \to \infty}\frac{1}{(t-T)^2}
\int^t_T(t-s)^2s^{\lambda}\cos s\,ds\\
&\geq   -T^{\lambda}\sin T,
\end{align*}
where $k_3=2^{\alpha+1}\big(\frac{\alpha^2}{\beta M(\alpha-\nu)}\big)^{\alpha}$.
Let
$$
\phi(s)=\phi_1(s)-\theta\phi_2(s)=-s^{\lambda}\sin s-\varepsilon,
$$
where $\varepsilon=\theta k_3/(2-\alpha)$. Consider an integer $N$ such
that $2N\pi+\frac{5}{4}\pi\geq (1+\sqrt{2}\varepsilon)^{1/\lambda}$. Then, for all
integers $n\geq N$, we have
$$
\phi(s)\geq \frac{1}{\sqrt{2}}, \quad \forall
s \in [2n \pi+ \frac{5}{4}\pi, 2n \pi +\frac{11}{8}\pi],
$$
which implies
\begin{align*}
& \liminf_{t \to \infty}\frac{1}{(t-T)^2}\int^t_T(t-s)^2g(s)
(\phi_1(s)-\theta\phi_2(s))_+^{(\alpha+1)/\alpha}ds\\
&\geq \frac{k_4}{(t-T)^2}\sum_{n=N}^{\infty}
\int^{2n\pi+\frac{11}{8}\pi}_{2n\pi+\frac{5}{4}\pi}(t-s)^2s^{-\nu/\alpha}
[s^{(\alpha-\nu)/\alpha}-t_0^{(\alpha-\nu)/\alpha}]^{-1}ds\\
&\geq k_4 \sum_{n=N}^{\infty}
\int^{2n\pi+\frac{11}{8}\pi}_{2n\pi+\frac{5}{4}\pi}s^{-1}ds=\infty,
\end{align*}
where $k_4=\frac{\beta M(\alpha-\nu)}{\alpha^2}
(\frac{1}{\sqrt{2}})^{(\alpha+1)/\alpha}$. Hence,
by Theorem 3.2, \eqref{e3.2} is oscillatory.


\subsection*{Acknowledgement}
 The authors are grateful to the referee for her/his suggestions and comments
on the original manuscript.

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\end{document}