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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 101, 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-2011/101\hfil Oscillation theorems]
{Oscillation theorems for second-order neutral functional dynamic
 equations on time scales}

\author[C. Gao, T. Li, S. Tang, E. Thandapani\hfil EJDE-2011/101\hfilneg]
{Cunchen Gao, Tongxing Li, Shuhong Tang, Ethiraju Thandapani}
 % in alphabetical order

\address{Cunchen Gao \newline
College of Information Science and Engineering,
Ocean University of China, Qingdao, Shandong 266100,  China}
\email{ccgao@ouc.edu.cn}

\address{Tongxing Li \newline
School of Control Science and Engineering,
Shandong University, Jinan, Shandong 250061, China.\newline
School of Mathematical Science, University of Jinan,
Jinan, Shandong 250022,  China}
\email{litongx2007@hotmail.com}

\address{Shuhong Tang \newline
School of Information and Control Engineering, 
Weifang University, Weifang, Shandong 261061, China.\newline
College of Information Science and Engineering,
Ocean University of China, Qingdao, Shandong 266100,  China}
\email{wfxytang@163.com}

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

\thanks{Submitted March 21, 2011. Published August 10, 2011.}
\subjclass[2000]{34K11, 39A21, 34N05}
\keywords{Oscillation; neutral functional dynamic equation; \hfill\break\indent 
comparison theorem; time scales}

\begin{abstract}
 In this article, we obtain several comparison theorems for the
 second-order  neutral dynamic equation
 $$
 \Big(r(t)\big([x(t)+p(t)x(\tau(t))]^\Delta\big)^\gamma\Big)^\Delta
 +q_1(t)x^\lambda(\delta(t))+q_2(t)x^\beta(\eta(t))=0,
 $$
 where $\gamma,\lambda, \beta$ are ratios of positive
 odd integers. We compare such equation with the first-order
 dynamic inequalities in the sense that the absence of the
 eventually positive solutions of these first-order inequalities
 implies the oscillation of the studied equation.
\end{abstract}

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

\section{Introduction}

A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset
of the real numbers. The theory of time scales was introduced in
1988 by Hilger \cite{hilger} in order to unify continuous and
discrete analysis. Several authors have expounded on various
aspect of this new theory; see \cite{agarwal1, bohner1, bohner2}.


This article  concerns the oscillation  of solutions to the
second-order nonlinear neutral dynamic equation
\begin{equation}\label{101}
\Big(r(t)\big(\left[x(t)+p(t)x(\tau(t))\right]^\Delta
\big)^\gamma\Big)^\Delta
+q_1(t)x^\lambda(\delta(t))+q_2(t)x^\beta(\eta(t))=0
\end{equation}
on a time scale $\mathbb{T}$.

Since we are interested in oscillatory behavior of solutions we
will assume that the time scale $\mathbb{T}$ is not bounded above;
i.e., it is a time scale interval of the form
$[t_0,\infty)_\mathbb{T}:=[t_0,\infty)\cap\mathbb{T}$.

Below we assume that $\gamma,\lambda,\beta$ are ratios of positive
odd integers; $r,p,q_1, q_2$ are real-valued rd-continuous
functions; $r(t)>0$, $q_1(t)>0$, $q_2(t)>0$  for
$t\in[t_0,\infty)_\mathbb{T}$,
 $\int_{t_0}^\infty r^{-1/\gamma}(t)\Delta t=\infty$,
$\tau\in C_{rd}(\mathbb{T},\mathbb{T})$,
$\tau$ is strictly increasing and
$\tau([t_0,\infty)_\mathbb{T})=[\tau(t_0),\infty)_\mathbb{T}$,
$\delta\in C_{rd}(\mathbb{T},\mathbb{T})$,
$\eta\in C_{rd}(\mathbb{T},\mathbb{T})$,
$\lim_{t\to\infty}\delta(t)=\lim_{t\to\infty}\eta(t)=\infty$,
$\tau\circ\delta=\delta\circ\tau$
and $\tau\circ\eta=\eta\circ\tau$. We know from \cite{mad} that
$\tau\circ\sigma=\sigma\circ\tau$.

By a solution of \eqref{101},  we mean a nontrivial
real-valued function $x\in C_{rd}^1[T_x,\infty)_{\mathbb{T}}$,
$T_x\geq t_0$ which has the properties $x(t)+p(t)x(\tau(t))$
and
$r(t)\big([x(t)+p(t)x(\tau(t))]^\Delta\big)^\gamma$
are defined, and is $\Delta$-differentiable for $\mathbb{T}$, and
satisfies  \eqref{101} on $t\in[T_x,\infty)_{\mathbb{T}}$. The
solutions vanishing in some neighbourhood of infinity will be
excluded from our consideration. A solution $x$ of  \eqref{101}
is said to be oscillatory if it is neither eventually positive nor
eventually negative, otherwise it is nonoscillatory.
Equation \eqref{101} is called oscillatory if all its solutions are
oscillatory.

During the last few years, Ladde et al. \cite{Lad} summarized some
known oscillation criteria for differential equations. Tang and Liu
\cite{tang} investigated the oscillatory behavior of the first-order
nonlinear delay difference equation of the form
$$
x(n+1)-x(n)+p(n)x^\gamma(n-l)=0.
$$
 With the development of dynamic
equations on time scales, there has been much research activity
concerning the oscillation and nonoscillation of solutions of
non-neutral dynamic equations and neutral functional dynamic
equations on time scales, we refer the reader to the articles
[7--25], and the references cited therein. Agarwal and Bohner
\cite{RPA}, Bohner et al. \cite{mbbo}, \c{S}ahiner and Stavroulakis
\cite{sahiner1}, Braverman and B. Karpuz \cite{ebr}, and Zhang and
Deng \cite{binggen} studied the oscillation of first-order delay
dynamic equation on time scales
$$
x^\Delta(t)+p(t)x(\tau(t))=0.
$$
Agarwal et al. \cite{abs} considered the second-order delay dynamic
equation on time scales
$$
x^{\Delta\Delta}(t)+p(t)x(\tau(t))=0.
$$
Braverman and Karpuz \cite{ebr1} investigated the non-oscillation of
second-order delay dynamic equation
$$
(A_0x^\Delta)^\Delta(t)+\sum_{i\in[1,n]_\mathbb{N}}
A_i(t)x(\alpha_i(t))=f(t).
$$
We note that \cite{mad,RPA,mbbo,sahiner1,ebr}
obtained some sufficient conditions for the
nonexistence of eventually positive solutions of the first-order
dynamic inequality
$$
x^\Delta(t)+p(t)x(\tau(t))\leq0,
$$
where $\tau(t)<t$. For the oscillation of neutral dynamic
equations, Agarwal et al. \cite{agarwal2}, Erbe et al.
\cite{erbe}, \c{S}ahiner \cite{sahiner}, Saker \cite{saker1},
Saker et al. \cite{saker2}, Saker and O'Regan \cite{saker3},
Tripathy \cite{Tripathy}, Chen \cite{chen}, Zhang and Wang
\cite{shaoyan} and Wu et al. \cite{wu} investigated the
oscillatory nature of following neutral dynamic equation
\begin{equation}\label{102}
 \left(r(t)\big([x(t)+p(t)x(\tau(t))]^\Delta\big)^\gamma\right)^\Delta
+q(t)x^\gamma(\delta(t))=0.
\end{equation}


Clearly, \eqref{102} is a special case of  \eqref{101}.
However, there are few results to study the oscillation of
\eqref{101}. The purpose of this paper is to obtain some comparison
theorems for the oscillation of \eqref{101}. This paper is
organized as follows: In Section 2, we present the basic definitions
and the theory of calculus on time scales. In Section 3, we shall
establish some oscillation criteria for \eqref{101}.

In what follows, all functional inequalities considered in this
paper are assumed to hold eventually; that is, they are satisfied
for all sufficiently large  $t$.

\section{Preliminaries}

A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset
of the real numbers $\mathbb{R}$. Since we are interested in
oscillatory behavior, we suppose that the time scale under
consideration is not bounded above; i.e., it is a time scale
interval of the form $[t_0,\infty)_{\mathbb{T}}$. On any time
scale we define the forward and backward jump operators by
$$
\sigma(t):=\inf\{s\in\mathbb{T}|s>t\}, \ \ \text{and}\ \
\rho(t):=\sup\{s\in\mathbb{T}|s<t\}.
$$

A point $t\in \mathbb{T}$ is said to be left-dense if $\rho(t)=t$,
right-dense if $\sigma(t)=t$, left-scattered if $\rho(t)<t$, and
right-scattered if $\sigma(t)>t$. The graininess $\mu$ of the time
scale is defined by $\mu(t):=\sigma(t)-t$.

For a function $f:\mathbb{T}\to\mathbb{R}$ (the range
$\mathbb{R}$ of $f$ may actually be replaced by any Banach space),
the (delta) derivative is defined by
$$
f^\Delta(t)=\frac{f(\sigma(t))-f(t)}{\sigma(t)-t},
$$
if $f$ is continuous at $t$ and $t$ is right-scattered. If $t$ is
not right-scattered then the derivative is defined by
$$
f^\Delta(t)=\lim_{s\to t^+}\frac{f(\sigma(t))-f(s)}{t-s}
=\lim_{s\to t^+}\frac{f(t)-f(s)}{t-s},
$$
provided this limit exists.

A function $f:\mathbb{T}\to\mathbb{R}$ is said to be
rd-continuous if it is continuous at each right-dense point and if
there exists a finite left limit in all left-dense points. The set
of rd-continuous functions $f:\mathbb{T}\to\mathbb{R}$ is
denoted by $C_{rd}(\mathbb{T},\mathbb{R})$.


A function $f$ is said to be differentiable if its derivative exists.
The set of functions $f:\mathbb{T}\to\mathbb{R}$ that are
differentiable and whose derivative is rd-continuous function is
denoted by $C_{rd}^1(\mathbb{T},\mathbb{R})$.

The derivative and the shift operator $\sigma$ are related by the
formula
$$
f^\sigma(t)=f(\sigma(t))=f(t)+\mu(t)f^\Delta(t).
$$

Let $f$ be a real-valued function defined on an interval $[a,b]$.
We say that $f$ is increasing, decreasing, nondecreasing, and
non-increasing on $[a,b]$ if $t_1,\ t_2\in[a,b]$ and $t_2>t_1$
imply $f(t_2)>f(t_1),\ f(t_2)<f(t_1),\ f(t_2)\geq f(t_1)$ and
$f(t_2)\leq f(t_1)$, respectively. Let $f$ be a differentiable
function on $[a,b]$. Then $f$ is increasing, decreasing,
nondecreasing, and non-increasing on $[a,b]$ if $f^\Delta(t)>0$,
$f^\Delta(t)<0$, $f^\Delta(t)\geq0$, and $f^\Delta(t)\leq0$ for
all $t\in[a,b)$, respectively.

We will make use of the following product and quotient rules for the
derivative of the product $fg$ and the quotient $f/g$ (where
$g(t)g(\sigma(t))\neq0$) of two differentiable functions $f$ and $g$
\begin{gather*}
(fg)^\Delta(t)=f^\Delta(t)g(t)+f(\sigma(t))g^\Delta(t)
=f(t)g^\Delta(t) +f^\Delta(t)g(\sigma(t)), \\
\big(\frac{f}{g}\big)^\Delta(t)
=\frac{f^\Delta(t)g(t)-f(t)g^\Delta(t)}{g(t)g(\sigma(t))}.
\end{gather*}

For $a, b\in \mathbb{T}$ and a differentiable function $f$, the
Cauchy integral of $f^\Delta$ is defined by
$$
\int_a^bf^\Delta(t)\Delta t=f(b)-f(a).
$$
The integration by parts formula reads
$$
\int_a^bf^\Delta(t)g(t)\Delta
t=f(b)g(b)-f(a)g(a)-\int_a^bf^\sigma(t)g^\Delta(t)\Delta t,
$$
and infinite integrals are defined as
$$
\int_a^\infty f(s)\Delta
s=\lim_{t\to\infty}\int_a^tf(s)\Delta s.
$$


\section{Main Results}

In this section, we shall establish some comparison theorems for the
oscillation of \eqref{101}. Firstly, we give the following chain
rule on time scales which will play an important role in the proofs
of our results.

\begin{lemma}[\cite{bohner1}] \label{lem3.1}
Assume that $\sup \mathbb{T}=\infty$, and 
$v\in C_{rd}^1([t_0,\infty)_\mathbb{T})$
is a strictly increasing function and unbounded such that
$v([t_0,\infty)_\mathbb{T})=[v(t_0),\infty)_\mathbb{T}$. Then for
$x\in C_{rd}^1([t_0,\infty)_\mathbb{T},\mathbb{R})$, we have
\begin{equation}\label{32}
(x\circ v)^\Delta(t)=x^\Delta(v(t))v^\Delta(t)
\end{equation}
for $t\in[t_0,\infty)_\mathbb{T}$.
\end{lemma}

Below, we will give our results. For the sake of convenience, we
denote
\begin{gather*}
z(t)=x(t)+p(t)x(\tau(t)), \quad
Q_1(t)=\min\{q_1(t),q_1(\tau(t))\},\\
Q_2(t)=\min\{q_2(t),q_2(\tau(t))\},\quad
R(t)=\int_{t_0}^t\frac{1}{r^{1/\gamma}(s)}\Delta s,\\
H(t)=R(t)-R(t_1), \quad
Q_3(t)=Q_1(t)\int_{t_1}^{\sigma(t)}\frac{1}{r(s)} \Delta s, \\
Q_4(t)=Q_2(t)\int_{t_1}^{\sigma(t)}\frac{1}{r(s)} \Delta s,
\end{gather*}
for $t_1\geq t_0$ sufficiently large.

Without loss of generality we can deal only with the eventually
positive solutions of \eqref{101} in our proofs.

\begin{theorem} \label{thm3.2}
Assume that $\tau^\Delta(t)\geq \tau_0>0$,
$\lambda\leq1$ and $\beta\leq1$. Further, assume that there exists
a $p_0>0$ such that $0\leq p(t)\leq p_0^{\gamma/\lambda}<\infty$,
and $0\leq p(t)\leq p_0^{\gamma/\beta}<\infty$. If the first-order
neutral dynamic inequality
\begin{equation}\label{2931}
\begin{aligned}
&\Big(y(t)+\frac{p_0^\gamma}{\tau_0}y(\tau(t))\Big)^\Delta
+Q_1(t)H^\lambda(\delta(t))y^{\lambda/\gamma}(\delta(t))\\
&+Q_2(t)H^\beta(\eta(t))y^{\beta/\gamma}(\eta(t))\leq0
\end{aligned}
\end{equation}
has no eventually positive solution for all sufficiently large
$t_1$, then every solution of \eqref{101} is oscillatory.
\end{theorem}

\begin{proof}
 Assume that $x$ is an eventually positive solution
of \eqref{101}. Then we have
$\big(r(t)(z^\Delta(t))^\gamma\big)^\Delta<0$. It follows from
\eqref{101} and \eqref{32} that
\begin{equation}\label{301}
\left(r(t)(z^\Delta(t))^\gamma\right)^\Delta+q_1(t)x^\lambda(\delta(t))
+q_2(t)x^\beta(\eta(t))=0
\end{equation}
and
\begin{equation}\label{302}
\begin{aligned}
&\frac{p_0^\gamma}{\tau^\Delta(t)}\left(r(\tau(t))(z^\Delta
 (\tau(t)))^\gamma\right)^\Delta
+p_0^\gamma q_1(\tau(t))x^\lambda(\delta(\tau(t)))\\
&+p_0^\gamma
q_2(\tau(t))x^\beta(\eta(\tau(t)))=0.
\end{aligned}
\end{equation}
In view of $\tau^\Delta(t)\geq\tau_0>0$ and \eqref{302}, we see that
\[ % \label{303}
\frac{p_0^\gamma}{\tau_0}\left(r(\tau(t))
 (z^\Delta(\tau(t)))^\gamma\right)^\Delta
+p_0^\gamma q_1(\tau(t))x^\lambda(\delta(\tau(t)))+p_0^\gamma
q_2(\tau(t))x^\beta(\eta(\tau(t)))\leq0.
\]
Combining this inequality with \eqref{301}, we have
\begin{equation} \label{304}
\begin{aligned}
&\left(r(t)(z^\Delta(t))^\gamma\right)^\Delta
+\frac{p_0^\gamma}{\tau_0}\left(r(\tau(t))
 (z^\Delta(\tau(t)))^\gamma\right)^\Delta\\
& + q_1(t)x^\lambda(\delta(t))+p_0^\gamma
q_1(\tau(t))x^\lambda(\delta(\tau(t))) \\
& +q_2(t)x^\beta(\eta(t))+p_0^\gamma
q_2(\tau(t))x^\beta(\eta(\tau(t)))\leq0.
\end{aligned}
\end{equation}
If $\lambda\leq1$, from \cite[Lemma 2]{eth1}, we obtain
$$
x^\lambda(\delta(t))+p_0^\gamma x^\lambda(\delta(\tau(t)))\geq
[x(\delta(t))+p_0^{\gamma/\lambda}x(\delta(\tau(t)))]^\lambda\geq
z^\lambda(\delta(t)).
$$
Similarly,
$$
x^\beta(\eta(t))+p_0^\gamma x^\beta(\eta(\tau(t)))\geq
\left[x(\eta(t))+p_0^{\gamma/\beta}x(\eta(\tau(t)))\right]^\beta\geq
z^\beta(\eta(t)).
$$
Hence by \eqref{304}, we have
\begin{equation}\label{315}
\begin{aligned}
&\left(r(t)(z^\Delta(t))^\gamma\right)^\Delta
+\frac{p_0^\gamma}{\tau_0}\left(r(\tau(t))(z^\Delta(\tau(t)))^\gamma
\right)^\Delta\\
&+Q_1(t)z^\lambda(\delta(t))+Q_2(t)z^\beta(\eta(t))\leq0.
\end{aligned}
\end{equation}
It follows from \eqref{101} and
$\int_{t_0}^\infty\frac{1}{r^{1/\gamma}(t)}\Delta t=\infty$ that
$y(t)=r(t)(z^\Delta(t))^\gamma>0$ is decreasing. Thus, there exists
a $t_1\geq t_0$ such that
\begin{equation}\label{316}
z(t)\geq\int_{t_1}^t\frac{\big(r(s)
(z^\Delta(s))^\gamma\big)^{1/\gamma}}{r^{1/\gamma}(s)}
\Delta s\geq y^{1/\gamma}(t)\big(R(t)-R(t_1)\big).
\end{equation}
Then, setting $y(t)=r(t)(z^\Delta(t))^\gamma$ in \eqref{315} and
using \eqref{316}, one can see that $y$ is a positive solution of
inequality \eqref{2931}. This is a contradiction and the proof is
complete.
\end{proof}

\begin{theorem} \label{thm3.3}
Assume that $\tau^\Delta(t)\geq \tau_0>0$,
$\tau(t)\geq t$, $\lambda\leq1$ and $\beta\leq1$. Moreover, assume
that there exists a $p_0>0$ such that $0\leq p(t)\leq
p_0^{\gamma/\lambda}<\infty$, and $0\leq p(t)\leq
p_0^{\gamma/\beta}<\infty$. If the first-order dynamic inequality
\begin{equation}
\begin{aligned}
&u^\Delta(t) +\Big(\frac{\tau_0}{\tau_0+p_0^\gamma}
 \Big)^{\lambda/\gamma}
Q_1(t)H^\lambda(\delta(t))u^{\lambda/\gamma}(\delta(t))\\
&\quad + \Big(\frac{\tau_0}{\tau_0+p_0^\gamma}\Big)^{\beta/\gamma}
Q_2(t)H^\beta(\eta(t))u^{\beta/\gamma}(\eta(t))\leq0
\end{aligned} \label{9131}
\end{equation}
has no eventually positive solution for all sufficiently large
$t_1$, then every solution of \eqref{101} is oscillatory.
\end{theorem}

\begin{proof}
Assume that $x$ is a positive solution of \eqref{101}.
By the proof of Theorem \ref{thm3.2}, we find
$y(t)=r(t)(z^\Delta(t))^\gamma>0$ is decreasing and satisfies
\eqref{2931}. Let $u(t)=y(t)+p_0^\gamma y(\tau(t))/\tau_0$. From
$\tau(t)\geq t$, we have
$$
u(t)\leq \big(1+\frac{p_0^\gamma}{\tau_0}\big)y(t).
$$
Hence, we get that $u$ is a positive solution of \eqref{9131}. This
is a contradiction and the proof is complete.
\end{proof}

From Theorem \ref{thm3.3}, we have the following results.

\begin{corollary} \label{coro3.4}
Assume that $\delta(t)\leq\eta(t)$,
$\tau^\Delta(t)\geq \tau_0>0$, $\tau(t)\geq t$,
$\lambda=\beta\leq1$. Furthermore,  assume that there exists a
$p_0>0$ such that $0\leq p(t)\leq p_0^{\gamma/\lambda}<\infty$. If
the first-order dynamic inequality
$$
u^\Delta(t) +\frac{\tau_0^{\lambda/\gamma}}{(\tau_0+p_0^\gamma
)^{\lambda/\gamma}}
[Q_1(t)H^\gamma(\delta(t))+Q_2(t)H^\gamma(\eta(t))]u^{\lambda/\gamma}
(\eta(t))\leq0
$$
has no positive solution for all sufficiently large $t_1$, then
every solution of \eqref{101} is oscillatory.
\end{corollary}

\begin{proof}
Proceeding as in the proof of Theorem \ref{thm3.3}, $u$ is
decreasing and if $\delta(t)\leq\eta(t)$, then $u(\delta(t))\geq
u(\eta(t))$. Therefore, $u$ is a positive solution of the dynamic
inequality
$$
u^\Delta(t)
+\frac{\tau_0^{\lambda/\gamma}}{(\tau_0+p_0^\gamma)^{\lambda/\gamma}}
[Q_1(t)H^\gamma(\delta(t))+Q_2(t)H^\gamma(\eta(t))]
u^{\lambda/\gamma}(\eta(t))\leq0.
$$
This is a contradiction and the proof is complete.
\end{proof}

Similar to the proof of Corollary \ref{coro3.4}, we have the another
comparison result.

\begin{corollary} \label{coro3.5}
 Assume that $\delta(t)\geq\eta(t)$,
$\tau^\Delta(t)\geq \tau_0>0$, $\tau(t)\geq t$,
$\lambda=\beta\leq1$. Moreover, assume that there exists a $p_0>0$
such that $0\leq p(t)\leq p_0^{\gamma/\lambda}<\infty$. If the
first-order dynamic inequality
$$
u^\Delta(t)
+\frac{\tau_0^{\lambda/\gamma}}{(\tau_0+p_0^\gamma)^{\lambda/\gamma}}
[Q_1(t)H^\gamma(\delta(t))+Q_2(t)H^\gamma(\eta(t))]u^{\lambda/\gamma}
(\delta(t))\leq0
$$
has no positive solution for all sufficiently large $t_1$, then
every solution of \eqref{101} is oscillatory.
\end{corollary}

\begin{theorem} \label{thm3.6}
 Assume that $\tau(t)\geq t$, $\tau^\Delta(t)\geq
\tau_0>0$, $\gamma=1$, $\lambda\leq1$ and $\beta\leq1$. Further,
assume that there exists a $p_0>0$ such that $0\leq p(t)\leq
p_0^{1/\lambda}<\infty$, and $0\leq p(t)\leq
p_0^{1/\beta}<\infty$. If the first-order dynamic inequality
\begin{equation}\label{931}
\phi^\Delta(t)-\frac{\tau_0}{\tau_0+p_0}
Q_3(t)\phi^\lambda(\delta(t))-\frac{\tau_0}{\tau_0+p_0}
Q_4(t)\phi^\beta(\eta(t))\geq0
\end{equation}
has no eventually positive solution for all sufficiently large
$t_1$, then every solution of \eqref{101} is oscillatory.
\end{theorem}

\begin{proof}
Assume that $x$ is an eventually positive solution
of \eqref{101}. Then we have $\big(r(t)z^\Delta(t)\big)^\Delta<0$
 and $z^\Delta(t)>0$. Proceeding as in the proof of
Theorem \ref{thm3.2}, we
have
\begin{equation}\label{001}
 \left(r(t)z^\Delta(t)\right)^\Delta
+\frac{p_0}{\tau_0}\left(r(\tau(t))z^\Delta(\tau(t))\right)^\Delta
+Q_1(t)z^\lambda(\delta(t))+Q_2(t)z^\beta(\eta(t))\leq0.
\end{equation}
Integrating \eqref{001} from $t$ to $\infty$, we obtain
\begin{equation}\label{002}
r(t)z^\Delta(t)+\frac{p_0}{\tau_0}r(\tau(t))z^\Delta(\tau(t))
\geq\int_t^\infty
\left(Q_1(s)z^\lambda(\delta(s))+Q_2(s)z^\beta(\eta(s))\right)
\Delta s.
\end{equation}
Since $r(t)z^\Delta(t)$ is decreasing and $\tau(t)\geq t$,
it follows that
$$
\big(1+\frac{p_0}{\tau_0}\big)r(t)z^\Delta(t) \geq\int_t^\infty
\left(Q_1(s)z^\lambda(\delta(s))+Q_2(s)z^\beta(\eta(s))\right)
\Delta s.
$$
Integrating the last inequality from $t_1$ to $t$, from
\cite[Lemma 1]{karpuz}, we obtain
\begin{align*}
z(t)
&\geq \frac{\tau_0}{\tau_0+p_0}\int_{t_1}^t\frac{1}{r(u)}\int_u^\infty
\left(Q_1(s)z^\lambda(\delta(s))+Q_2(s)z^\beta(\eta(s))\right)
\Delta s\Delta u\\
&= \frac{\tau_0}{\tau_0+p_0}\int_{t_1}^t\left(Q_1(s)
z^\lambda(\delta(s))+Q_2(s)z^\beta(\eta(s))\right)
\int_{t_1}^{\sigma(s)}\frac{1}{r(u)} \Delta u\Delta s.
\end{align*}
Thus, we see that
$$
z(t)\geq\frac{\tau_0}{\tau_0+p_0}\int_{t_1}^t
\left(Q_3(s)z^\lambda(\delta(s))+Q_4(s)z^\beta(\eta(s))\right)
\Delta s.
$$
Denote the right hand side of the above inequality by $\phi(t)$.
Since $z(t)\geq \phi(t)$, we find that $\phi$ is a positive
solution of \eqref{931}.  This is a contradiction and the proof is
complete.
\end{proof}

From Theorem \ref{thm3.6}, we get the following result.

\begin{corollary} \label{coro3.7}
Assume that $\delta(t)\leq\eta(t)$,
$\tau(t)\geq t$, $\tau^\Delta(t)\geq \tau_0>0$, $\gamma=1$,
$\lambda=\beta\leq1$. Furthermore, assume that there exists a
$p_0>0$ such that $0\leq p(t)\leq p_0^{1/\lambda}<\infty$. If the
first-order dynamic inequality
$$
\phi^\Delta(t)-\frac{\tau_0}{\tau_0+p_0}
\big(Q_3(t)+Q_4(t)\big)\phi^\lambda(\delta(t))\geq0
$$
has no eventually positive solution for all sufficiently large
$t_1$, then every solution of \eqref{101} is oscillatory.
\end{corollary}

\begin{proof} Proceeding as in the proof of Theorem \ref{thm3.6},
$\phi$ is increasing and if $\delta(t)\leq\eta(t)$, then
$\phi(\delta(t))\leq \phi(\eta(t))$. Therefore, $\phi$ is a
positive solution of the dynamic inequality
$$
\phi^\Delta(t) -\frac{\tau_0}{\tau_0+p_0}
\big(Q_3(t)+Q_4(t)\big)\phi^\lambda(\delta(t))\geq0.
$$
This is a contradiction and the proof is complete.
\end{proof}

Similar to the proof of Corollary \ref{coro3.7}, we have another
comparison result.

\begin{corollary} \label{coro3.8}
 Assume that $\delta(t)\geq\eta(t)$,
$\tau(t)\geq t$, $\tau^\Delta(t)\geq \tau_0>0$, $\gamma=1$,
$\lambda=\beta\leq1$. Moreover, assume that there exists a $p_0>0$
such that $0\leq p(t)\leq p_0^{1/\lambda}<\infty$. If the
first-order dynamic inequality
$$
\phi^\Delta(t)-\frac{\tau_0}{\tau_0+p_0}
\big(Q_3(t)+Q_4(t)\big)\phi^\lambda(\eta(t))\geq0
$$
has no eventually positive solution for all sufficiently large
$t_1$, then every solution of \eqref{101} is oscillatory.
\end{corollary}

\begin{remark} \label{rmk3.9} \rm
 Assume that $\tau^\Delta(t)\geq\tau_0>0$ and
$\tau^{-1}\in C_{rd}(\mathbb{T},\mathbb{T})$, where $\tau^{-1}$ is
the inverse function of $\tau$. Similar to the methods of the
above, we can derive some comparison theorems for \eqref{101} when
$\tau(t)\leq t$, the details are left to the interested reader.

 Our results can be extended to the equation of the
general form
$$
\Big(r(t)\big([x(t)+p(t)x(\tau(t))]^\Delta\big)^\gamma\Big)^\Delta
+\sum_{i=1}^nq_i(t)x^{\lambda_i}(\delta_i(t))=0.
$$
\end{remark}

\subsection*{Acknowledgements}
The authors thank the anonymous referees for their suggestions which
improve the content of this article.


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\end{document}