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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 131, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/131\hfil Oscillation of solutions]
{Oscillation of solutions for third order
functional dynamic equations}

\author[E. M. Elabbasy, T.  S. Hassan \hfil EJDE-2010/131\hfilneg]
{Elmetwally M. Elabbasy, Taher S. Hassan}  % in alphabetical order

\address{Elmetwally M. Elabbasy \newline
Department of Mathematics, Faculty of Science\\
Mansoura University\newline
Mansoura, 35516, Egypt}
 \email{emelabbasy@mans.edu.eg}
\urladdr{http://www.mans.edu.eg/pcvs/10805/}

\address{Taher S. Hassan \newline
Department of Mathematics, Faculty of Science\\
Mansoura University\newline
Mansoura, 35516, Egypt}
\email{tshassan@mans.edu.eg}
\urladdr{http://www.mans.edu.eg/pcvs/30140/ }

\thanks{Submitted April 13, 2010. Published September 14, 2010.}
\subjclass[2000]{34K11, 39A10, 39A99}
\keywords{Oscillation; third order; functional dynamic equations;
time scales}

\begin{abstract}
 In this article we study the oscillation of solutions to
 the third order nonlinear functional dynamic equation
 $$
 L_{3}(x(t))+\sum_{i=0}^{n}p_i(t)\Psi_k{\alpha_ki}(x(h_i(t)))=0,
 $$
 on an arbitrary time scale $\mathbb{T}$. Here
 $$
 L_0(x(t))=x(t),\quad L_k(x(t))=\Big(\frac{[
 L_{k-1}x(t)]^{\Delta }}{a_k(t)}\Big)^{\gamma_kk}, \quad k=1,2,3
 $$
 with $a_1, a_2$  positive rd-continuous functions on
 $\mathbb{T}$ and $a_{3}\equiv 1$;
 the functions $p_i$ are nonnegative rd-continuous
 on $\mathbb{T}$ and not all $p_i(t)$ vanish in a neighborhood
 of infinity; $\Psi_k{c}(u)=|u|^{c-1}u$, $c>0$.
 Our main results extend known results and are illustrated by examples.
\end{abstract}

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

\section{Introduction}

The theory of time scales, which has recently received a  lot of
attention, was introduced by Stefan Hilger in his PhD dissertation
\cite{h}, written under the direction of Bernd Aulbach. Since then
a rapidly expanding body of literature has sought to unify,
extend, and generalize ideas from discrete calculus, quantum
calculus, and continuous calculus to arbitrary time scale
calculus. Recall that a time scale $\mathbb{T}$ is a nonempty,
closed subset of the reals, and the cases when this time scale is
the reals or the integers represent the classical theories of
differential and of difference equations. Many other interesting
time scales exist, and they give rise to many applications (see
\cite{bp}). Not only does the new theory of the so-called
``dynamic equations'' unify the theories of differential equations
and difference equations, but also extends these classical cases
to cases ``in between'', e.g., to the so-called $q$-difference
equations when ${\mathbb{T=}}q^{\mathbb{N}_0}$ (which has
important applications in quantum theory \cite{k}) and can be
applied on different types of time scales like
${\mathbb{T=}}h\mathbb{Z}$, ${\mathbb{T=N}}_0^{2}$ and $
\mathbb{T=H}_{n}$ the space of harmonic numbers. In this work a
knowledge and understanding of time scales and time scale notation
is assumed; for an excellent introduction to the calculus on time
scales, see Hilger \cite{h}, and Bohner and Peterson
\cite{bp,bp2}.

We are concerned with the oscillatory behavior of solutions to the
third order nonlinear functional dynamic equation
\begin{equation}
L_{3}(x(t))+\sum_{i=0}^{n}p_i(t)\Psi_k{\alpha
_i}(x(h_i(t)))=0,  \label{1.1}
\end{equation}
on an arbitrary time scale $\mathbb{T}$, where $L_0(x(t))
=x(t)$,
$$
L_k(x(t))=(\frac{[ L_{k-1}x(t)] ^{\Delta
}}{a_k(t)})^{\gamma_kk},\quad k=1,2,3,
$$
with $a_1, a_2$ are positive $rd$-continuous functions
on $\mathbb{T}$ and $a_{3}\equiv 1$, and $\gamma_kk$, $k=1,2$
are quotients of odd positive integers, $\gamma_k{3}=1$ and
$\alpha_k0=\gamma_1\gamma_k2$. We also assume
$\Psi_k{c}(u)=| u|^{c-1}u$, $c>0$
and $p_i$, $i=0,1,2,\dots ,n$ are nonnegative $rd$-continuous
functions on $\mathbb{T}$ such that not all  $p_i(t)$
vanish in a neighborhood of infinity. The functions
$h_i:\mathbb{T}\to \mathbb{T}$ satisfy
$\lim_{t\to \infty }h_i(t)=\infty$, $i=0,1,2,\dots ,n$. Since we
are interested in the oscillatory behavior of solutions near
infinity, we assume that $\sup \mathbb{T}=\infty $, and define the
time scale interval $ [t_0,\infty )_{\mathbb{T}}$ by
$[t_0,\infty )_{\mathbb{T} }:=[t_0,\infty )\cap \mathbb{T}$.
By a solution of \eqref{1.1} we mean a nontrivial real-valued
function $L_0x\in C_{rd}^{1}[T_{x},\infty )_{ \mathbb{T}}$,
$T_{x}\geq t_0$ which has the property that
$L_1x\in C_{rd}^{1}[T_{x},\infty )_{\mathbb{T}}$,
$L_2x\in C_{rd}^{1}[T_{x},\infty )_{\mathbb{T}}$ and $x(t)$ satisfies
equation \eqref{1.1} on $[T_{x},\infty )_{\mathbb{T}}$, where
$C_{rd}$ is the space of $rd$-continuous functions. The solutions
vanishing identically in some neighborhood of infinity will be
excluded from our consideration. A solution $x$ of \eqref{1.1} is
said to be oscillatory if it is neither eventually positive nor
eventually negative, otherwise it is nonoscillatory. Throughout
the paper, we define, for $ i=0,1,\dots ,n$ and $k=1,2$,
\begin{gather*}
A(s,u):=a_1(s)A_2^{1/\gamma_k1}(s,u),\quad A_k(
s,u):=\int_{u}^{s}a_k(u)\Delta u,
\\
\varphi_ki(t,u):=A_1(h_i(t),u)\phi
_{i,2}^{1/\gamma_k1}(t),\quad \phi_k{i,k}(t):=\int_{h_i(t)}^{\infty
}a_k(s)\Delta s,
\\
h(t):=\min \{ t,\ h_i(t);\ i=0,1,\dots ,n\} ,\quad
(\Phi ^{\Delta }(t))_{+}:=\max \{0,\Phi ^{\Delta }(t)\}.
\end{gather*}

In the previous few years, there has been an increasing interest
in obtaining sufficient conditions for the
oscillation/nonoscillation of solutions of different classes of
dynamic equations, we refer the reader to the papers
\cite{ag/bo3,ag/bo/gr1,ag/bo/gr/or2,Akin,BoHa,7,d,erbe1,erbe2,
erbe,d1,ha/er/pe2,ha/er/pe3,erhape10,erhape11,erhape12,EH,
er/ha/pe/sa3,soma,ha/er/pe1}
and the references cited therein. Regarding third order dynamic
equations, \cite{EA,EO,ha,erbee,erbeseveral} considered the third
order dynamic equation \eqref{1.1}, in particular case and under
quite restrictive conditions, for example, Erbe et al \cite{EA}
studied the dynamic equation \eqref{1.1}, when $n=0$,
$\gamma_1=\gamma_k2=1$, $\alpha_k0=1$ and $ h_0(t)\equiv t$
and established sufficient conditions which ensure the solution of
equation \eqref{1.1} is either oscillatory or tends to zero and
Erbe et al \cite{EO} extended the results which established in
\cite{EA}, when $\gamma_k2\geq 1$ is the quotient of odd positive
integers. Also, Hassan {\cite{ha} generalized the results which
were established in \cite{EO,EA}}, for the equation \eqref{1.1},
when $n=0$, $\gamma_k1=1$, $\alpha_k0=\gamma_k2$, $\gamma_2>0$
is the quotient of odd positive integers, and $h_0(t)\leq t$ and
$h_0^{\Delta }(t)\geq 0$, for $t\in \mathbb{T}$ and $h_0\circ
\sigma =\sigma \circ h_0$. A number of sufficient conditions for
oscillation were obtained for the cases when, for $k=1,2$,
\[
\int_{t_0}^{\infty }a_k(t)\Delta t=\infty .
\]
Erbe, Hassan and Peterson \cite{erbee} solved this problem for the
third order functional dynamic equation \eqref{1.1}, when $n=0$,
$\gamma_k1=1$, $\alpha_k0=\gamma_k2$, $\gamma_k2>0$ is the
quotient of odd positive integers and for both cases, for $k=1,2$,
\begin{equation}
\int_{t_0}^{\infty }a_k(t)\Delta t=\infty ,  \label{w}
\end{equation}
{and}
\begin{equation}
\int_{t_0}^{\infty }a_k(t)\Delta t<\infty .  \label{2.44}
\end{equation}
Recently, Erbe, Hassan and Peterson \cite{erbeseveral} extended
previous results for the third order dynamic equation \eqref{1.1}
under some restrictive conditions, $n=2$ and $h\circ \sigma
=\sigma \circ h$.

The fact that the condition $h\circ \sigma =\sigma \circ h$ is not
satisfied for some time scales, see \cite{erhape10}. The purpose
of this paper is to extend the oscillation criteria which are
established by \cite{erbeseveral}, {for the more general third
order functional dynamic equation with mixed arguments}
\eqref{1.1}, for several terms $n$ and without restrictive
condition $h\circ \sigma =\sigma \circ h$ and for both of the
cases \eqref{w} and \eqref{2.44}. We will still assume $\gamma
_1,\gamma_k2>0$ are the quotient of odd positive integers and,
hence our results will improve and extend results in the
\cite{EA,EO,ha,erbee,erbeseveral}, and many known results on
nonlinear oscillation.

\section{Main Results}

Throughout this paper, we assume that $\alpha_k1>\alpha_k2>\dots
>\alpha_k{m}>\alpha_k0>\alpha_k{m+1}>\dots >\alpha_k{n}>0$.
Before stating our main results, we begin with the following
lemmas which will play an important role in the proof of our main
results.

\begin{lemma}\label{lem1}
For each $n$-tuple $(\alpha_k1,\alpha_k2,\dots, \alpha_k{n})$,
there exists  $(\eta_k1,\eta_k2,\dots ,\eta_{n})$ with
$0<\eta_ki<1$ satisfying
\begin{equation}
\sum_{i=1}^{n}\alpha_ki\eta_ki=\alpha_k0,\quad
\sum_{i=1}^{n}\eta_i=1.  \label{ll1}
\end{equation}
\end{lemma}

The proof of the above lemma is the same as
\cite[Lemma 2.1]{ha/er/pe3}.

\begin{lemma} \label{lem2}
Let $a_1$ be nondecreasing and delta differentiable on
$[t_0,\infty )_{{\mathbb{T}}}$ and $x$ be a positive solution
of \eqref{1.1} such that
\begin{equation}
x^{\Delta }(t)>0\text{\quad and\quad }(L_1(
x(t)))^{\Delta }>0,\quad \text{for }t\geq t_0.
\label{0}
\end{equation}
Then, if
\begin{equation}
\sum_{i=0}^{n}\int_{t_0}^{\infty }p_i(t)h_i^{\alpha_ki}(t)\Delta
t=\infty ,  \label{2.1}
\end{equation}
we have
\[
x^{\Delta \Delta }(t)>0,\quad \big(\frac{x(t)}{t}\big)^{\Delta }<0\quad
\text{on }[t_0,\infty )_{{\mathbb{T}}}.
\]
\end{lemma}

\begin{proof}
Let $x$ be as in the statement of this lemma. Since
\begin{equation}
(L_1(x(t)))^{\Delta }=\frac{(
(x^{\Delta }(t))^{\gamma_k1})^{\Delta }}{a_1^{\gamma
_1\sigma }(t)}-\frac{(x^{\Delta }(t))^{\gamma
_1}(a_1^{\gamma_k1}(t))^{\Delta }}{
a_1^{\gamma_k1}(t)a_1^{\gamma_k1\sigma }(t)
}>0,\quad \text{for }t\geq t_0.  \label{2.3}
\end{equation}
Using the P\"{o}tzsche chain rule \cite[Theorem 1.90]{bp}, we have
\begin{equation}
\begin{aligned}
((x^{\Delta }(t))^{\gamma_k1})^{\Delta }
&= \gamma_k1\int_0^{1}[ x^{\Delta }(t)+h\mu (t)x^{\Delta \Delta }(t)
]^{\gamma_k1-1}\;dh\;x^{\Delta \Delta }(t)   \\
&= \gamma_k1x^{\Delta \Delta }(t)\int_0^{1}[ hx^{\Delta \sigma
}(t)+(1-h)x^{\Delta }(t)]^{\gamma_k1-1}dh.
\end{aligned} \label{2.5}
\end{equation}
and
\begin{equation}
\begin{aligned}
(a_1^{\gamma_k1}(t))^{\Delta }
&= \gamma_k1\int_0^{1}
[ a_1(t)+h\mu (t)a_1^{\Delta }(t)]^{\gamma
_1-1}\;dh\;a_1^{\Delta }(t)   \\
&= \gamma_k1a_1^{\Delta }(t)\int_0^{1}[ ha_1^{\sigma
}(t)+(1-h)a_1(t)]^{\gamma_k1-1}dh.
\end{aligned} \label{2.55}
\end{equation}
 Using  that $a_1$ is a nondecreasing and
$x$ is increasing, we obtain, from \eqref{2.3}, \eqref{2.5} and
\eqref{2.55} that $ x^{\Delta \Delta }(t)>0$ on $[t_0,\infty
)_{{\mathbb{T}}}$. Next, we show that $\big(\frac{x(t)}{t}\big)^{\Delta
}<0$. To see this, let $ U(t):=x(t)-tx^{\Delta }(t)$, then
$U^{\Delta }(t)=-\sigma (t)x^{\Delta \Delta }(t)<0$ for $t\in
[ t_0,\infty )_{{\mathbb{T}}}$. This implies that $U(t)$
is strictly decreasing on $[t_0,\infty )_{{\mathbb{T}} } $. We
claim $U(t)>0$ on $[t_0,\infty )_{{\mathbb{T}}}$. Assume not,
then there exists $t_1\geq t_0$ such that $U(t)<0$ on
$[t_1,\infty )_{\mathbb{T}}$. Therefore,
\begin{equation}
\big(\frac{x(t)}{t}\big)^{\Delta }=\frac{tx^{\Delta }(t)-x(t)}{t\sigma
(t)}=-\frac{U(t)}{t\sigma (t)}>0,\quad \text{for }t\in [ t_1,\infty
)_{{\mathbb{T}}}.  \label{2.8}
\end{equation}
Pick $t_2\in [ t_1,\infty )_{{\mathbb{T}}}$ so that $h_i(t)\geq
t_1$, for $t\geq t_2$ and $i=0,1,\dots ,n$. Then, for $t\geq t_2$
and $i=0,1,\dots ,n$, we have from \eqref{2.8} that
\[
\frac{x(h_i(t))}{h_i(t)}\geq \frac{x(t_1)}{t_1}=:c>0,\quad
\text{for $t\geq t_2$ and $i=0,1,\dots ,n$},
\]
so $x(h_i(t))\geq ch_i(t)$ for $t\geq t_2$ and $i=0,1,\dots ,n$.
Now by integrating both sides of the dynamic equation \eqref{1.1}
from $t_2$ to $ t$, we have
\[
L_2(x(t_2))-L_2(x(t)
)=\sum_{i=0}^{n}\int_{t_2}^{t}p_i(s)\Psi_k{\alpha_i}(x(h_i(s)))\Delta s.
\]
This implies, from \eqref{0} that
\begin{equation}
L_2(x(t_2))\geq
\sum_{i=0}^{n}\int_{t_2}^{t}p_i(s)\Psi_k{\alpha_ki}(x(h_i(s)))\Delta
s\geq \sum_{i=0}^{n}c^{\alpha_ki}\int_{t_2}^{t}p_i(s)h_i^{\alpha
_i}(s)\Delta s.  \label{2.9}
\end{equation}
Letting $t\to \infty $ we obtain a contradiction to assumption
\eqref{2.1}. Hence $U(t)=x(t)-tx^{\Delta }(t)>0$ on
$[t_0,\infty )_{{\mathbb{T}}}$. Consequently,
\begin{equation}
\big(\frac{x(t)}{t}\big)^{\Delta }=-\frac{U(t)}{t\sigma (t)}<0,\quad
t\in [ t_0,\infty )_{{\mathbb{T}}}.  \label{2.10}
\end{equation}
This completes the proof of Lemma \ref{lem2}.
\end{proof}

First, we establish oscillation criteria for  \eqref{1.1} when
\eqref{w} holds.

\begin{theorem}\label{thm11}
Let $\alpha_k0=\gamma_k1\gamma_k2$ and $h_i(t)$,
$i=0,1,2,\dots ,n$ be nondecreasing functions on
$[t_0,\infty )_{{\mathbb{T}}}$. Assume that \eqref{w} and
\begin{equation}
\int_{t_0}^{\infty }a_1(t)\Big(\int_{t}^{\infty }a_2(s)\Big(
\sum_{i=0}^{n}P_i(s)\Big)^{1/\gamma_k2}\Delta s\Big)
^{1/\gamma_k1}\Delta t=\infty ,  \label{21}
\end{equation}
where $P_i(s):=\int_{s}^{\infty }p_i(u)\Delta u$,
$i=0,1,\dots ,n$. Furthermore, suppose that, for all sufficiently large
$T_1$, there is $T>T_1$ such that $h(T)>T_1$ and
\begin{equation}
\limsup_{t\to \infty }Q_1(t)\Big(
\int_{T_1}^{h(t)}A(s,T_1)\Delta s\Big)
^{\alpha_k0}>1,  \label{v11}
\end{equation}
where
\[
Q_1(t):=P_0(t)
+\prod_{i=1}^{n}(\eta_ki^{-1}P_i(t)
)^{\eta_ki}.
\]
Then every solution of  \eqref{1.1} is either oscillatory
or tends to zero.
\end{theorem}

\begin{proof}
Assume \eqref{1.1} has a non-oscillatory solution $x$ on
$[t_0,\infty )_{\mathbb{T}}$. Then, without loss of
generality, there is a $t_1\in [ t_0,\infty
)_{\mathbb{T}}$, sufficiently large, such that $ x(t)>0$ and
$x(h_i(t))>0$ on $[t_1,\infty )_{\mathbb{T}}$, for
$i=0,1,2,\dots ,n$ and not all of the $p_i(t)$'$s$ are identically
zero on $[t_1,\infty )_{\mathbb{T}}$.  From \eqref{1.1}, we
have
\begin{equation}
L_{3}(x(t))=-\sum_{i=0}^{n}p_i(t)\Psi_k{\alpha
_i}(x(h_i(t)))<0,  \label{o}
\end{equation}
on$\ [t_1,\infty )_{{\mathbb{T}}}$. Then $L_2(x(t) )$ is
strictly decreasing on $[t_1,\infty )_{\mathbb{T}}$. Therefore,
$x^{\Delta }(t)$ and $(L_1(x( t)))^{\Delta }$ are eventually of
one sign. Then, there exists a sufficiently large $t_2\geq t_1$
so that $x^{\Delta }( t)$ and $L_1^{\Delta }(x(t))$ are of fixed
sign for $t\geq t_2$. Therefore, we consider the following four cases:
\begin{itemize}
\item[(i)] $x^{\Delta }(t)<0$ and $(L_1(x(t)))^{\Delta }<0$;
\item[(ii)] $x^{\Delta }(t)>0$ and $(L_1(x(t)))^{\Delta }<0$;
\item[(iii)] $x^{\Delta }(t)<0$ and $(L_1(x(t)))^{\Delta }>0$;
\item[(iv)] $x^{\Delta }(t)>0$ and $(L_1(x(t)))^{\Delta }>0$,
on $[t_2,\infty )_{{\mathbb{T}}}$.
\end{itemize}
 For  case (i), we have
\[
x(t)=x(t_2)+\int_{t_2}^{t}a_1(s)L_1^{1/\gamma
_1}(x(s))\Delta s\leq x(t_2)+L_1^{1/\gamma
_1}(x(t_2))\int_{t_2}^{t}a_1(
s)\Delta s.
\]
Hence by \eqref{w}, we have $\lim_{t\to \infty
}x(t)=-\infty $, which contradicts the fact that $x$ is a
positive solution of \eqref{1.1}. For the case (ii),
 from \eqref{o} we have
\begin{align*}
L_1(x(t))&= L_1(x(t_2)
)+\int_{t_2}^{t}a_2(s)L_2^{1/\gamma_k2}(x(s))\Delta s \\
&\leq L_1(x(t_2))+L_2^{1/\gamma_k2}(
x(t_2))\int_{t_2}^{t}a_2(s)\Delta s.
\end{align*}
Then, by \eqref{w}, we have $\lim_{t\to \infty}L_1(x(t))=-\infty $,
which contradicts $x^{\Delta }(t)>0$, for $t\geq t_2$.
For the case (iii), we have
$\lim_{t\to \infty }x(t)=c_0\geq 0$ and
$\lim_{t\to \infty }L_1(x(t))=c_1\leq 0$. If we assume
$c_0>0$, then $\Psi_k{\alpha_ki}(x(h_i(t)))\geq K$, for
$i=0,1,2,\dots ,n$ and $t\geq t_{3}\geq t_2$, where
$K:=\min \{ c_0^{\alpha
_i}:~i=0,1,2,\dots ,n\} $. Integrating equation \eqref{1.1} from
$t$ to $\infty $, we obtain
\begin{align*}
-L_2(x(t))&< -\sum_{i=0}^{n}\int_{t}^{\infty
}p_i(s)\Psi_k{\alpha_ki}(x(h_i(s)))\Delta s \\
&\leq -c_0\sum_{i=0}^{n}\int_{t}^{\infty }p_i(s)\Delta
s=-c_0\sum_{i=0}^{n}P_i(t),
\end{align*}
which implies
\[
-(L_1(x(t)))^{\Delta}
<-C_0a_2(t)\Big(\sum_{i=0}^{n}P_i(t)\Big)^{1/\gamma_k2},
\]
where $C_0:=K^{1/\gamma_k2}\geq 0$. Integrating this inequality
from $t$ to $\infty $, we obtain
\begin{align*}
L_1(x(t))&<-C_0\int_{t}^{\infty }a_2(
s)\Big(\sum_{i=0}^{n}P_i(s)\Big)^{1/\gamma_2}\Delta s+c_1\\
&\leq -C_0\int_{t}^{\infty }a_2(s)\Big(
\sum_{i=0}^{n}P_i(s)\Big)^{1/\gamma_k2}\Delta s.
\end{align*}
Finally, integrating the last inequality from $t_{3}$ to $t$, we obtain
\[
x(t)<-c_0\int_{t_{3}}^{t}a_1(s)\Big(\int_{s}^{\infty }a_2(
u)(\sum_{i=0}^{n}P_i(u))^{1/\gamma
_2}\Delta u\Big)^{1/\gamma_k1}\Delta s+x(t_{3}).
\]
Hence by \eqref{21}, we have $\lim_{t\to \infty
}x(t)=-\infty $, which contradicts the fact that $x$ is a positive
solution of \eqref{1.1}. Thus, we conclude that $\lim_{t\to \infty
}x(t)=0$. For the case (iv), integrating both sides of the dynamic
equation \eqref{1.1} from $t$ to $\infty $ and then using the
facts that $x(t)$ is strictly increasing and $h_i(t)$,
$i=0,1,2,\dots ,n$ are nondecreasing, we obtain
\begin{equation}
\sum_{i=0}^{n}\Psi_k{\alpha_ki}(x(h_i(t)))P_i(t)\leq
L_2(x(t)).  \label{11f}
\end{equation}
Since$\ L_2(x(t))$ is strictly decreasing on
$[t_2,\infty )_{{\mathbb{T}}}$, we obtain
\begin{align*}
L_1(x(t))
&> L_1(x(t))-L_1(x(t_2))
=\int_{t_2}^{t}a_2(s)L_2^{1/\gamma_k2}(x(s))\Delta s \\
&\geq L_2^{1/\gamma_k2}(x(t))\int_{t_2}^{t}a_2(s)\Delta s
=L_2^{1/\gamma_k2}(x(t))A_2(t,t_2).
\end{align*}
Hence
\[
x^{\Delta }(t)
\geq a_1(t)A_2^{1/\gamma_k1}(t,t_2)L_2^{1/\alpha_0}(x(t)).
\]
Similarly, we see that
\[
x(t)\geq L_2^{1/\alpha_k0}(x(t))
\int_{t_2}^{t}a_1(s)A_2^{1/\gamma_k1}(s,t_2)\Delta s,
\]
and so
\begin{equation}
x^{\alpha_k0}(t)\geq L_2(x(t))\Big(
\int_{t_2}^{t}A(s,t_2)\Delta s\Big)^{\alpha_k0}. \label{hh1}
\end{equation}
Pick $t_{3}>t_2$, sufficiently large, so that $h(t)>t_2$, for
$t\geq t_{3}$. Then from \eqref{hh1},  for $t\geq t_{3}$, we obtain
\begin{equation}
\begin{aligned}
\Psi_k{\alpha_k0}(x(h(t)))
&\geq L_2(x(h(t)))\Big(\int_{t_2}^{h(t)}A(s,t_2)\Delta s\Big)^{\alpha_k0}\\
&\geq L_2(x(t))\Big(\int_{t_2}^{h(t)}A(s,t_2)\Delta s\Big)^{\alpha_k0}.
\end{aligned} \label{00f}
\end{equation}
Using \eqref{00f} in \eqref{11f},  for $t\geq t_{3}$, we find that
\[
\sum_{i=0}^{n}\Psi_k{\alpha_ki}(x(h_i(t)))P_i(t)\leq \Psi
_{\alpha_k0}(x(h(t)))\Big(\int_{t_2}^{h(t)}A(
s,t_2)\Delta s\Big)^{-\alpha_k0},
\]
which yields from the fact $x^{\Delta }(t)>0$ on $[t_2,\infty
)_{{\mathbb{T}}}$ that
\begin{equation}
P_0(t)+\sum_{i=1}^{n}\Psi_k{\alpha_ki-\alpha
_0}(x(h(t)))P_i(t)<\Big(\int_{t_2}^{h(t)
}A(s,t_2)\Delta s\Big)^{-\alpha_k0}.  \label{&&}
\end{equation}
Using the arithmetic-geometric mean inequality see \cite[Page 17]{3}
\[
\sum_{i=1}^{n}\eta_kiu_i\geq \prod_{i=1}^{n}u_i^{\eta
_i},\quad \text{where }u_i\geq 0.
\]
  From Lemma \ref{lem1}, for $t\geq T$, we obtain
\begin{equation}
\begin{aligned}
\sum_{i=1}^{n}\Psi_k{\alpha_ki-\alpha_k0}(x(h(t)))P_i(t)
&=\sum_{i=1}^{n}\eta_ki(\eta_ki^{-1}\Psi_k{\alpha_ki-\alpha
_0}(x(h(t)))P_i(t)) \\
&\geq \prod_{i=1}^{n}(\eta_ki^{-1}P_i(
t))^{\eta_ki}\Psi_k{\eta_ki(\alpha_i-\alpha_k0)}(x(h(t)))\\
&\overset{\eqref{ll1}}{=} \prod_{i=1}^{n}(
\eta_ki^{-1}P_i(t))^{\eta_ki},
\end{aligned} \label{&}
\end{equation}
Using \eqref{&} in \eqref{&&}, we have
\[
Q_1(t)\Big(\int_{t_2}^{h(t)}A(
s,t_2)\Delta s\Big)^{\alpha_k0}<1,\quad \text{for }
t\geq t_{3}.
\]
Then
\[
\limsup_{t\to \infty }Q_1(t)\Big(
\int_{t_2}^{h(t)}A(s,t_2)\Delta s\Big)
^{\alpha_k0}\leq 1,
\]
which leads to a contradiction to \eqref{v11}.
\end{proof}

\begin{theorem} \label{thm4}
Assume that \eqref{w} and \eqref{21} hold.
Furthermore, suppose that, for all sufficiently large
$T\in [ t_0,\infty )_{{\mathbb{T}}}$, such that $h_i(T)>t_0$,
$i=0,1,\dots ,n$,
\begin{equation}
\sum_{i=0}^{n}\int_{t_0}^{\infty }p_i(t)A_1^{\alpha_ki}(
h_i(t),t_0)\Delta t=\infty .  \label{f3}
\end{equation}
Then every solution of equation \eqref{1.1} is either oscillatory
or tends to zero.
\end{theorem}

\begin{proof}
Assume \eqref{1.1} has a non-oscillatory solution $x$ on
$[t_0,\infty )_{\mathbb{T}}$. Then, without loss of
generality, there is a $t_1\in [ t_0,\infty
)_{\mathbb{T}}$, sufficiently large, such that $ x(t)>0$ and
$x(h_i(t))>0$ on $[t_1,\infty )_{\mathbb{T}}$, for
$i=0,1,2,\dots ,n$ and not all  $p_i(t)$ are identically
zero on $[t_1,\infty )_{\mathbb{T}}$. Therefore, as in the
proof of Theorem \ref{thm11}, we obtain there exists
$t_2\geq t_1$ so that
\[
(L_2(x(t)))^{\Delta }<0,\quad
(L_1(x(t)))^{\Delta }>0,\quad \text{on }
 [t_2,\infty )_{{\mathbb{T}}},
\]
and either $L_1(x(t))>0$ on $[t_2,\infty )_{\mathbb{T}}$ or
$\lim_{t\to \infty }x(t)=0$. Assume $L_1( x(t))>0$ on
$[t_2,\infty )_{{\mathbb{T}}}$. Since
$ (L_1(x(t)))^{\Delta }>0$ on $ [t_2,\infty )_{{\mathbb{T}}}$, then
\[
L_1(x(t))\geq L_1(x(t_2))=:c>0.
\]
Thus
\[
x(t)\geq x(t)-x(t_2)\geq
C\int_{t_2}^{t}a_1(s)\Delta s,
\]
where $C:=c^{1/\gamma_k1}$. Pick $t_{3}\geq t_2$ such that
$h_i(t)>t_2$, for $t\geq t_{3}$ and $i=0,1,\dots ,n$, then, for
$i=0,1,\dots ,n$,
\begin{equation}
x(h_i(t))>CA_1(h_i(t),t_2)\quad \text{on }
[t_{3},\infty )_{{\mathbb{T}}}.  \label{f1}
\end{equation}
It follows from \eqref{1.1} and \eqref{f1} that
\[
-L_{3}(x(t))= \sum_{i=0}^{n}p_i(t)\Psi_k{\alpha_i}(x(h_i(t)))
\geq \sum_{i=0}^{n}p_i(t)[ CA_1(h_i(t),t_2)]^{\alpha_ki}.
\]
Integrating both sides of the last inequality from $t_{3}$ to $t$,
we have
\begin{align*}
L_2(x(t_{3}))
&> \sum_{i=0}^{n}\int_{t_{3}}^{t}p_i(s)[ CA_1(h_i(
s),t_2)]^{\alpha_ki}\Delta s+L_2(x(
t))\\
&> \sum_{i=0}^{n}\int_{t_{3}}^{t}p_i(s)[ CA_1(h_i(
s),t_2)]^{\alpha_ki}\Delta s,
\end{align*}
which contradicts \eqref{f3}. This completes the proof.
\end{proof}

\begin{example} \label{exa1} \rm
Consider the third order dynamic equation \eqref{1.1},
for $t_0\geq 1$, where $0<\gamma_k1\leq 1$ and
$\gamma_k2=\frac{1}{\gamma_k1}$ are the
quotient of odd positive integers and $\alpha_ki$,
$i=0,1,\dots ,n$ are positive constants and we assume
$h_i(t)\leq t^{\gamma_k2}$, $i=0,1,\dots ,n$. Let
\[
a_1(t)=1,\quad a_2(t)=\frac{1}{t^{\gamma_k1}},\quad
p_i(t)=\frac{1}{th_i(t)},\quad i=0,1,\dots ,n.
\]
It is clear that conditions \eqref{w} hold, since
\[
\int_{t_0}^{\infty }\Delta t=\infty ,\quad
\int_{t_0}^{\infty }\frac{\Delta t}{t^{\gamma_k1}}=\infty ,\quad
\text{for } 0<\gamma_k1\leq 1,
\]
by \cite[Example 5.60]{bp2}. Note that
\begin{gather*}
\int_{s}^{\infty }p_i(u)\Delta u
= \int_{s}^{\infty }\frac{\Delta u}{
uh_i(u)}\geq \int_{s}^{\infty }\frac{\Delta u}{u^{\gamma_k2+1}}
\geq \frac{1}{\gamma_k2}\int_{s}^{\infty }\big(\frac{-1}{u^{\gamma
_2}}\big)^{\Delta }\Delta u
=\frac{1}{\gamma_k2}\frac{1}{s^{\gamma_2}},
\\
\int_{t}^{\infty }a_2(s)(\sum_{i=0}^{n}P_i(s))^{1/\gamma_k2}\Delta s
\geq \gamma_k0\int_{t}^{\infty }\frac{\Delta s}{s^{\gamma_k1+1}}
\geq \frac{\gamma_k0}{\gamma_k1}\int_{t}^{\infty }(\frac{-1}{
s^{\gamma_k1}})^{\Delta }\Delta s=\frac{\gamma_k0}{\gamma
_1t^{\gamma_k1}},
\end{gather*}
where $\gamma_k0:=(\frac{n+1}{\gamma_k2})^{1/\gamma_k2}$
and
\[
\int_{t_0}^{\infty }a_1(t)\Big(\int_{t}^{\infty }a_2(s)(
\sum_{i=0}^{n}P_i(s))^{1/\gamma_k2}\Delta s\Big)
^{1/\gamma_k1}\Delta t
\geq \gamma \int_{t_0}^{\infty }\frac{\Delta t}{t}
=\infty ,
\]
where $\gamma :=(\gamma_k0/\gamma_k1\big)^{1/\gamma_1}$, so that
condition \eqref{21} holds.
 To apply Theorem \ref{thm4}, it remains to prove that condition
\eqref{f3} holds. To see this, note that
\[
\sum_{i=0}^{n}\int_{t_0}^{\infty }p_i(t)\Big[
\int_{t_0}^{h_i(t)}a_1(s)\Delta s\Big]^{\alpha_i}\Delta t
\geq \int_{t_0}^{\infty }\big(\frac{1}{t}-\frac{t_0}{
th_0(t)}\big)\Delta t=\infty ,
\]
for those time scales, where $\int_{t_0}^{\infty }\frac{\Delta t}{
th_0(t)}<\infty $. Then, by Theorem \ref{thm4}, every solution of
equation \eqref{1.1} is either oscillatory or tends to zero.
\end{example}

By using Lemma \ref{lem2}, we obtain the following oscillation
criterion for \eqref{1.1}.

\begin{theorem}\label{thm1}
Let $\alpha_k0=\gamma_k1\gamma_k2$ and $a_1$ be
nondecreasing and delta differentiable on $[t_0,\infty )_{{\mathbb{T}}}$.
Assume that \eqref{w}, \eqref{2.1} and \eqref{21} hold.
Furthermore, suppose that there exists a positive
$\Delta -$differentiable function $
\phi (t)$ and, \textit{for all sufficiently large }$T_1$, there is
$T>T_1 $ such that
\begin{equation}
\limsup_{t\to \infty }\int_{T}^{t}\Big[ \phi (s)Q_2(s)-\frac{
((\phi ^{\Delta }(s))_{+})^{\alpha_k0+1}}{
(\alpha_k0+1)^{\alpha_k0+1}(\phi (s)A(s,T_1))
^{\alpha_k0}}\Big]\Delta s=\infty ,  \label{m2}
\end{equation}
where
\[
Q_2(t):=p_0(t)H_0(t)
+\prod_{i=1}^{n}(\eta_ki^{-1}p_i(t)H_i(t))^{\eta_ki},
\]
and, for $i=0,1,\dots ,n$,
\[
H_i(t):=\begin{cases}
1, & h_i(t)\geq t \\
(\frac{h_i(t)}{t})^{\alpha_ki}, & h_i(t)\leq t.
\end{cases}
\]
Then every solution of  \eqref{1.1} is either oscillatory
or tends to zero.
\end{theorem}

\begin{proof}
Assume \eqref{1.1} has a non-oscillatory solution $x$ on
$[t_0,\infty )_{\mathbb{T}}$. Then, without loss of
generality, there is a $t_1\in [ t_0,\infty
)_{\mathbb{T}}$, sufficiently large, such that $ x(t)>0$ and
$x(h_i(t))>0$ on $[t_1,\infty )_{\mathbb{T}}$, for
$i=0,1,2,\dots ,n$ and not all of the $p_i(t)$'$s$ are identically
zero on $[t_1,\infty )_{\mathbb{T}}$. Therefore, as in the
proof of Theorem \ref{thm11}, we obtain there exists $t_2\geq t_1$ so
that
\[
(L_2(x(t)))^{\Delta }<0,\quad
(L_1(x(t)))^{\Delta }>0,\quad \text{on } [t_2,\infty )_{{\mathbb{T}}},
\]
and either $L_1(x(t))>0$ on $[t_2,\infty )_{\mathbb{T}}$ or
$\lim_{t\to \infty }x(t)=0$. Assume $L_1( x(t))>0$ on
$[t_2,\infty )_{{\mathbb{T}}}$. Consider the Riccati substitution
\[
w(t)=\phi (t)\frac{L_2(x(t))}{x^{\alpha_0}(t)}.
\]
By the product rule and then the quotient rule
\begin{equation}
\begin{aligned}
w^{\Delta }(t)
&= \frac{\phi (t)}{x^{\alpha_k0}(t)}(L_2(x(t)))^{\Delta }
+\big(\frac{\phi (t) }{x^{\alpha_k0}(t)}\big)^{\Delta }L_2^{\sigma }(
x(t)) \\
&= \phi (t)\frac{L_{3}(x(t))}{x^{\alpha_k0}(t)}
+\Big(\frac{\phi ^{\Delta }(t)}{x^{\alpha_k0\sigma }(t)}-\frac{\phi
(t)(x^{\alpha_k0}(t))^{\Delta }}{x^{\alpha_k0}(t)x^{\alpha_k0\sigma
}(t)}\Big)L_2^{\sigma }(x(t)).
\end{aligned} \label{1.2}
\end{equation}
 From \eqref{1.2} and the definition of $w(t)$, we have,
for $t\geq t_2$
\[
w^{\Delta }(t)=-\phi (t)\sum_{i=0}^{n}p_i(t)
\frac{x^{\alpha_ki}(h_i(t))}{x^{\alpha_k0}(t)}
+\frac{\phi ^{\Delta }(t)}{\phi ^{\sigma }(t)}w^{\sigma
}(t)-\frac{\phi (t)(x^{\alpha_k0}(t))^{\Delta }}{\phi
^{\sigma }(t)x^{\alpha_k0}(t)}w^{\sigma }(t).
\]
Now, for a fixed $i$ and let $t$ be a fixed point in
$[t_2,\infty )_{\mathbb{T}}$. Then either $h_i(t)\leq t$
or $h_i(t)\geq t$. First, consider the case when $h_i(t)\geq t$.
Then, by using the fact that $x$ is strictly increasing, we
obtain $x(h_i(t))\geq x(t)$. Next, consider the case
when $h_i(t)\leq t$. Then, in view of Lemma \ref{lem2}, we obtain
$x(h_i(t))\geq \frac{h_i(t)}{t}x(t)$. It follows from the
definition of $H_i(t)$ that, for $t\geq t_2$,
\[
w^{\Delta }(t)\leq -\phi (t)\sum_{i=0}^{n}p_i(t)H_i(t)
x^{\alpha_ki-\alpha_k0}(t)+\frac{\phi ^{\Delta }(t)}{\phi ^{\sigma }(t)}
w^{\sigma }(t)-\frac{\phi (t)(x^{\alpha_k0}(t))^{\Delta }}{
\phi ^{\sigma }(t)x^{\alpha_k0}(t)}w^{\sigma }(t).
\]
As in the proof of Theorem \ref{thm11}, we obtain, for $t\geq t_2$,
\[
w^{\Delta }(t)\leq -\phi (t)Q_2(t)+\frac{\phi ^{\Delta }(t)}{\phi ^{\sigma
}(t)}w^{\sigma }(t)-\frac{\phi (t)(x^{\alpha_k0}(t))^{\Delta }
}{\phi ^{\sigma }(t)x^{\alpha_k0}(t)}w^{\sigma }(
t).
\]
Then, by the P\"{o}tzsche chain rule \cite[Theorem 1.90]{bp}, we obtain
\begin{align*}
(x^{\alpha_k0}(t))^{\Delta }
&= \alpha_k0\int_0^{1}[x(t)+h\mu (t)x^{\Delta }(t)]
^{\alpha_k0-1}dh\;x^{\Delta }(t)\\
&= \alpha_k0\int_0^{1}[ (1-h)x(t)
+hx^{\sigma }(t)]^{\alpha_k0-1}dh\;x^{\Delta }(
t)\\
&\geq \begin{cases}
\alpha_k0x^{(\alpha_k0-1)\sigma }(t)\;x^{\Delta }(
t),& 0<\alpha_k0\leq 1 \\
\alpha_k0x^{\alpha_k0-1}(t)\;x^{\Delta }(t)
,&  \alpha_k0\geq 1.
\end{cases}
\end{align*}
If $0<\alpha_k0\leq 1$, we have
\[
w^{\Delta }(t)\leq -\phi (t)Q_2(t)+\frac{\phi ^{\Delta }(t)}{\phi ^{\sigma
}(t)}w^{\sigma }(t)-\frac{\alpha_k0\phi (t)w^{\sigma }(
t)}{\phi ^{\sigma }(t)}\frac{x^{\Delta }(t)
}{x^{\sigma }(t)}\big(\frac{x^{\sigma }(t)}{
x(t)}\big)^{\alpha_k0},
\]
whereas if $\alpha_k0\geq 1$, we have
\[
w^{\Delta }(t)\leq -\phi (t)Q_2(t)+\frac{\phi ^{\Delta }(t)}{\phi ^{\sigma
}(t)}w^{\sigma }(t)-\frac{\alpha_k0\phi (t)w^{\sigma }(
t)}{\phi ^{\sigma }(t)}\frac{x^{\Delta }(t)
}{x^{\sigma }(t)}\frac{x^{\sigma }(t)}{x(
t)}.
\]
Using  that $x(t)$ is strictly increasing on
$[t_2,\infty )_{\mathbb{T}}$, we obtain that, for $\alpha_k0>0$,
\begin{equation}
w^{\Delta }(t)\leq -\phi (t)Q_2(t)+\frac{\phi ^{\Delta }(t)}{\phi ^{\sigma
}(t)}w^{\sigma }(t)-\frac{\alpha_k0\phi (t)w^{\sigma }(
t)}{\phi ^{\sigma }(t)}\frac{x^{\Delta }(t)
}{x^{\sigma }(t)}.  \label{w1}
\end{equation}
Then using  that $L_2(x(t))$ is strictly
decreasing on $[t_2,\infty )_{\mathbb{T}}$, we obtain that, for $t\geq
t_2 $,
\begin{equation}
\begin{aligned}
L_1(x(t))&> L_1(x(t))
-L_1(x(t_2))=\int_{t_2}^{t}a_2(s)L_2^{1/\gamma_k2}(x(s))\Delta s \\ &\geq L_2^{1/\gamma_k2}(x(t))
\int_{t_2}^{t}a_2(s)\Delta s
\geq L_2^{\sigma /\gamma_k2}(x(t))A_2(t,t_2).
\end{aligned} \label{p}
\end{equation}
  From \eqref{w1} and \eqref{p}, we obtain
\begin{equation}
w^{\Delta }(t)\leq -\phi (t)Q_2(t)+\frac{(\phi ^{\Delta }(t))
_{+}}{\phi ^{\sigma }(t)}w^{\sigma }(t)-\frac{\alpha_k0\phi
(t)A(t,t_2)}{\phi ^{\alpha \sigma }(t)}w^{\alpha
\sigma }(t),  \label{m}
\end{equation}
where $\alpha :=\frac{\alpha_k0+1}{\alpha_k0}$. Define $X\geq 0$ and
$Y\geq 0$ by
\[
X^{\alpha }:=\frac{\alpha_k0\phi (t)A(t,t_2)}{\phi ^{\alpha
\sigma }(t)}w^{\alpha \sigma }(t),\quad
Y^{\alpha -1}:=\frac{(\phi ^{\Delta }(t))_{+}}{\alpha (\alpha
_0\phi (t)A(t,t_2))^{1/\alpha }}.
\]
Then, using the inequality, see \cite{w},
\begin{equation}
\alpha XY^{\alpha -1}-X^{\alpha }\leq (\alpha -1)Y^{\alpha },  \label{mm}
\end{equation}
we obtain
\[
\frac{(\phi ^{\Delta }(t))_{+}}{\phi ^{\sigma }(t)}w^{\sigma
}(t)-\frac{\alpha_k0\phi (t)A(t,t_2)}{\phi
^{\alpha \sigma }(t)}w^{\alpha \sigma }(t)\leq
\frac{((\phi ^{\Delta }(t))_{+})^{\alpha_k0+1}}{
(\alpha_k0+1)^{\alpha_k0+1}(\phi (t)A(t,t_2))
^{\alpha_k0}}.
\]
  From the above inequality and \eqref{mm}, we obtain
\[
w^{\Delta }(t)\leq -\phi (t)Q_2(t)+\frac{((\phi ^{\Delta
}(t))_{+})^{\alpha_k0+1}}{(\alpha_k0+1)^{\alpha
_0+1}(\phi (t)A(t,t_2))^{\alpha_k0}}.
\]
Integrating both sides from $t_2$ to $t$,
\[
\int_{t_2}^{t}\Big[ \phi (s)Q_2(s)-\frac{((\phi ^{\Delta
}(s))_{+})^{\alpha_k0+1}}{(\alpha_k0+1)^{\alpha
_0+1}(\phi (s)A(s,t_2))^{\alpha_k0}}\Big]
\Delta s\leq w(t_2)-w(t)\leq w(t_2),
\]
which leads to a contradiction to \eqref{m2}.
\end{proof}

\begin{example}\label{exa2}\rm
Consider the third order nonlinear dynamic equation \eqref{1.1},
for $t_0\geq 1$, where $\gamma_k1$ and $\gamma_k2$ are the quotient of
odd positive integers and $\alpha_ki$, $i=0,1,\dots ,n$ are positive
constants with $\alpha_k1>\alpha_k2>\dots >\alpha_k{m}>\alpha_k0>\alpha
_{m+1}>\dots >\alpha_k{n}>0$ and also, we assume
$h_i(t)\geq t$, $i=0,1,\dots ,n$
such that \eqref{2.1} holds. Note that since $h_i(t)\geq t$,
$i=0,1,\dots ,n$ it follows that $H_i(t)\equiv 1$, $i=0,1,\dots ,n$.
Let
\begin{gather*}
a_1(t)=t^{1/\alpha_k0},\quad a_2(t)=\frac{1}{t^{1-1/\gamma_k2}}, \\
p_0(t)=\frac{\beta_k0}{t^{\alpha_k0+2}},\quad
p_i(t)=\frac{\beta_ki}{t^{\eta_ki+2}},\quad i=1,2,\dots ,n.
\end{gather*}
where $\eta_ki$, $i=1,2,\dots ,n$ and $\beta_ki$,
$i=0,1,\dots ,n$ are positive constants such that
$\alpha_k0\geq \eta_ki$, $i=1,2,\dots ,n$. It is clear
that conditions \eqref{w} hold, since
\[
\int_{t_0}^{\infty }a_1(t)\Delta t=\int_{t_0}^{\infty
}t^{1/\alpha_k0}\Delta t=\infty ,
\]
and
\[
\int_{t_0}^{\infty }a_2(t)\Delta t=\int_{t_0}^{\infty }
\frac{\Delta t}{t^{1-1/\gamma_k2}}=\infty ,
\]
by \cite[Example 5.60]{bp}. Therefore, we can find $T>T_1$
such that $ A_2(s,T_1)\geq 1$, for $s\geq T$. Also, using the
P\"{o}tzsche chain rule,
\begin{align*}
P_0(s)
&= \int_{s}^{\infty }p_0(u)\Delta u
=\beta_0\int_{s}^{\infty }\frac{\Delta u}{u^{\alpha_k0+2}} \\
&\geq \frac{\beta_k0}{\alpha_k0+1}\int_{s}^{\infty }(\frac{-1}{
u^{\alpha_k0+1}})^{\Delta }\Delta u
=\frac{\beta_k0}{\alpha_k0+1}\frac{1}{s^{\alpha_k0+1}},
\end{align*}
and
\begin{align*}
P_i(s)
&= \int_{s}^{\infty }p_i(u)\Delta u
=\beta_i\int_{s}^{\infty }\frac{\Delta u}{u^{\eta_ki+2}} \\
&\geq \frac{\beta_ki}{\eta_ki+1}\int_{s}^{\infty }(\frac{-1}{
u^{\eta_ki+1}})^{\Delta }\Delta u \\
&= \frac{\beta_ki}{\eta_ki+1}\frac{1}{s^{\eta_ki+1}} \\
&\geq \frac{\beta_ki}{\alpha_k0+1}\frac{1}{s^{\alpha_k0+1}},\quad
i=1,2,\dots ,n.
\end{align*}
Hence
\begin{align*}
\Big(\int_{t}^{\infty }a_2(s)\Big(\sum_{i=0}^{n}P_i(s)
\Big)^{1/\gamma_k2}\Delta s\Big)^{1/\gamma_k1}
&\geq \beta \Big(
\int_{t}^{\infty }\frac{\Delta s}{s^{\gamma_k1+1}}\Big)^{1/\gamma_k1}
\\
&\geq \frac{\beta }{\gamma_k1^{^{1/\gamma_k1}}}\Big(\int_{t}^{\infty
}(\frac{-1}{s^{\gamma_k1}})^{\Delta }\Delta s\Big)
^{1/\gamma_k1} \\
&= \frac{\beta }{\gamma_k1^{^{1/\gamma_k1}}}\frac{1}{t},
\end{align*}
where $\beta :=(\frac{1}{\alpha_k0+1}\sum_{i=0}^{n}\beta_ki)
^{1/\alpha_k0}$. Then
\begin{align*}
&\int_{t_0}^{\infty }a_1(t)\Big(\int_{t}^{\infty }a_2(s)(
\sum_{i=0}^{n}P_i(s))^{1/\gamma_k2}\Delta s\Big)
^{1/\gamma_k1}\Delta t \\
&= \frac{\beta }{\gamma_k1^{^{1/\gamma_k1}}}\int_{t_0}^{\infty }\frac{1
}{t^{1-1/\alpha_k0}}\Delta t=\infty ,
\end{align*}
so that condition \eqref{21} holds. Let us take
$\phi (t)=t^{\alpha_k0+1}$, then, by the P\"{o}tzsche chain rule
\[
\phi ^{\Delta }(t)=(\alpha_k0+1)\int_0^{1}(t+h\mu (t))^{\alpha
_0}dh\leq (\alpha_k0+1)(\sigma (t))^{\alpha_k0}.
\]
Now, we assume $\mathbb{T}$ is a time scale satisfying
$\sigma (t)\leq kt$, for some $k>0$, $t\geq T_k>T$. Note that
\begin{align*}
&\limsup_{t\to \infty }\int_{T}^{t}[ \phi (s)Q_2(s)-\frac{
((\phi ^{\Delta }(s))_{+})^{\alpha_k0+1}}{
(\alpha_k0+1)^{\alpha_k0+1}(\phi (s)A(s,T_1))
^{\alpha_k0}}]\Delta s \\
&\geq \limsup_{t\to \infty }\int_{T_k}^{t}[ \frac{\beta_k0
}{s}-\frac{k^{\alpha_k0(\alpha_k0+1)}}{s}]\Delta s \\
&\geq (\beta_k0-k^{\alpha_k0(\alpha_k0+1)})\limsup_{t\to
\infty }\int_{T_k}^{t}\frac{\Delta s}{s}=\infty ,
\end{align*}
if $\beta_k0>k^{\alpha_k0(\alpha_k0+1)}$ and hence \eqref{m2} holds.
We conclude that if $[ T,\infty )_{\mathbb{T}}$ is a time scale
where $\sigma (t)\leq kt$, for some $k>0$, $t\geq T_k$, then,
by Theorem \ref{thm1}, every solution of \eqref{1.1} is either
oscillatory or tends to zero if $\beta_k0>k^{\alpha_k0(\alpha_k0+1)}$.
\end{example}

In the following, we assume that \eqref{2.44} holds and establish
sufficient conditions which ensure that every solution $x(t)$ of
\eqref{1.1} is either oscillatory or tends to zero.
By Theorems \ref{thm11} and \ref{thm1} and
\cite[Theorem 2.1]{erbeseveral}, we obtain the following oscillation
criteria for equation \eqref{1.1}.

\begin{corollary} \label{coro1}
Let $\alpha_k0=\gamma_k1\gamma_k2$ and $h_i(t)$, $i=0,1,2,\dots ,n$ be
nondecreasing functions on $[t_0,\infty )_{{\mathbb{T}}}$. Assume that
\eqref{2.44} and \eqref{21} hold.
Furthermore, suppose that, for all sufficiently large
$T_1\in [ t_0,\infty )_{{\mathbb{T}}}$, there is $T>T_1$ such
that $h_i(T)>T_1$, $i=1,2,\dots ,n$,
\begin{equation}
\int_{T}^{\infty }a_1(t)\Big[ \int_{T}^{t}a_2(s)[
\sum_{i=1}^{n}\int_{T}^{s}p_i(u)\phi_k{i,1}^{\alpha_ki}(u)
\Delta u]^{1/\gamma_k2}\Delta s\Big]^{1/\gamma_k1}\Delta
t=\infty ,  \label{h1}
\end{equation}
and
\begin{equation}
\int_{T}^{\infty }a_2(t)\Big[ \sum_{i=1}^{n}\int_{T}^{t}p_i(u)\varphi
_i^{\alpha_ki}(u,T_1)\Delta u\Big]^{1/\gamma
_2}\Delta t=\infty .  \label{h2}
\end{equation}
If  condition \eqref{v11} holds, then every solution of \eqref{1.1} is
either oscillatory or tends to zero.
\end{corollary}

\begin{corollary} \label{coro2}
Assume that \eqref{2.44}, \eqref{21}, \eqref{h1} and \eqref{h2} hold.
If  \eqref{f3} holds, then
every solution of \eqref{1.1} is either oscillatory or tends to zero.
\end{corollary}

\begin{corollary} \label{coro3}
Let $\alpha_k0=\gamma_k1\gamma_k2$ and $a_1$ be nondecreasing and
delta differentiable on $[t_0,\infty )_{{\mathbb{T}}}$. Assume that
\eqref{2.44}, \eqref{2.1}, \eqref{21}, \eqref{h1} and \eqref{h2} hold.
If  \eqref{m2} holds, then every solution of \eqref{1.1} is either
oscillatory or tends to zero.
\end{corollary}

\begin{remark} \label{rmk1} \rm
If \eqref{21} is not satisfied, we have
sufficient conditions which ensure that every solution $x(t)$
of \eqref{1.1} oscillates or $\lim_{t\to \infty }x(t)$ exists as a
finite number.
\end{remark}

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\end{document}