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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 33, pp. 1--7.\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-2011/33\hfil Second-order impulsive DE's]
{Asymptotic behavior of second-order impulsive differential
equations}

\author[H. Liu, Q. Li \hfil EJDE-2011/33\hfilneg]
{Haifeng Liu, Qiaoluan Li} 

\address{Haifeng Liu \newline
 Department of Science and Technology, 
 Hebei Normal University,\newline
Shijiazhuang, 050016, China}
\email{liuhf@mail.hebtu.edu.cn}

\address{Qiaoluan Li \newline
College of Mathematics and Information Science,
Hebei Normal University,\newline
Shijiazhuang, 050016, China}
\email{qll71125@163.com}


\thanks{Submitted January 6, 2011. Published February 23, 2011.}
\thanks{Supported by grant L2009Z02 from the Key Foundation of Hebei
Normal University}
\subjclass[2000]{34K25, 34K45}
\keywords{Impulsive differential equation;
 asymptotic behavior; second-order}

\begin{abstract}
 In this article, we study the asymptotic behavior of
 all solutions of 2-th order nonlinear delay differential equation
 with impulses. Our main tools are impulsive differential
 inequalities and the Riccati transformation. We illustrate
 the results by an example.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

  Consider the impulsive differential equation
\begin{gather}
\big(r(t)(x'(t))^{\alpha}\big)'+p(t)(x'(t))^{\alpha}
+f(t, x(t-\delta))=0, \quad
t\ge t_0,\; t\neq t_k, \label{e1.1}\\
x(t_{k}^{+})=J_k(x(t_k)), \quad x'(t_{k}^{+})=I_k(x'(t_k)), \quad
k=1,2,3\dots, \label{e1.2}
\end{gather}
where $\alpha$ is the quotient of positive odd integers.

The theory  of impulsive
differential/difference equations is emerging as an important area
of  investigation, since it is much richer than  the corresponding
theory of differential/difference equations without impulsive
effects. Moreover, such equations may model several real world
phenomena \cite{l1}. There are many papers devoted to the oscillation
criteria of differential equations with impulses \cite{h1,l2,l3}
and to the asymptotic behavior of all solutions of differential
equations without impulses \cite{w1}.

Recently, Tang \cite{t1} studied the equation
\begin{gather*}
(r(t)x'(t))'+p(t)x'(t)+f(t, x(t-\delta))=0,
\quad t\neq t_k,\\
x(t_k^+)=J_k(x(t_k)), \quad  k=1,2,3\dots,\\
x'(t_{k}^{+})=I_{k}(x'(t_k)), \quad  k=1,2,3\dots.
\end{gather*}
He obtained sufficient conditions of asymptotic behavior
of all solutions of the equation.

Motivated by \cite{t1}, using impulsive differential inequality
and the Riccati transformation, we study the asymptotic
behavior of solutions of \eqref{e1.1}, \eqref{e1.2}.

\begin{definition} \label{def1} \rm
 For $\phi\in C([t_0-\delta,t_0], \mathbb{R})$,
a function $x:[t_0-\delta, +\infty)\to \mathbb{R}$ is called a
solution of \eqref{e1.1}, \eqref{e1.2} satisfying the initial
 value condition
$$
x(t)=\phi(t),\quad t\in [t_0-\delta,t_0]
$$
if the following conditions are satisfied:
\begin{itemize}
\item[(i)] $x(t)=\phi(t)$ for $t\in [t_0-\delta, t_0]$,
\item[(ii)] $x, x'$ are continuously differentiable for $t>t_0$,
$t\neq t_{k}$ ($k=1,2,\dots$)  and satisfy \eqref{e1.1},
\item[(iii)] $x(t_k^-)=x(t_k), x'(t_k^-)=x'(t_k)$, $k=1,2,\dots $ and
satisfy \eqref{e1.2}.
\end{itemize}
\end{definition}

As is customary, a solution of \eqref{e1.1}, \eqref{e1.2}
is said to be non-oscillatory if it is eventually positive or
eventually negative.
Otherwise, it will be called oscillatory.

\section{Main results}

In this paper, we  assume that the following conditions hold:
\begin{itemize}

\item[(H1)] $f$ is continuous on $[t_0,+\infty)\times \mathbb{R}$,
$xf(t, x)>0$ for $x\neq 0$, and
$\frac{f(t,\, x)}{g(x)}\geq h(t)$ for $x\neq 0$,
where $g(\gamma x)\geq \gamma g(x)$ for $\gamma>0$,
$x'g'(x)>0$, and  $h, r'$ are continuous on $[t_0, +\infty)$,
$h(t)\geq 0, r(t)>0$.

\item[(H2)] $p, J_k, I_k$ are continuous on $\mathbb{R}$ and
there exist positive numbers $a_{k}^{*}, a_k, b_{k}^{*}, b_k$
such that $a_{k}^{*}\leq \frac{I_k(x)}{x}\leq a_k,
b_{k}^{*}\leq \frac{J_k(x)}{x}\leq b_k$.

\item[(H3)] $\lim_{t\to \infty}\int_{t_j}^{t}\prod_{t_j<t_k<s}
\frac{a_{k}^{*}}{b_{k}}\exp(-\int_{t_j}^{s}
\frac{r'(\sigma)+p(\sigma)}{\alpha r(\sigma)}d\sigma)ds=+\infty$.

\item[(H4)]
\begin{align*}
&\sum_{m=1}^{n-1}\prod_{k=m}^{n-1}\prod_{l=0}^{m-1}
b_{j+k}a_{j+l}^{*}\int_{t_{j+m-1}}^{t_{j+m}}
\exp\Big(-\int_{t_j}^{u}\frac{r'(s)+p(s)}{\alpha
r(s)}ds\Big)du\\
&+\prod_{k=0}^{n-1}
a_{j+k}^{*}\int_{t_{j+n-1}}^{t_{j+n}}
\exp\Big(-\int_{t_j}^{u}\frac{r'(s)+p(s)}{\alpha
r(s)}ds\Big)du \to+\infty,\quad\text{as }n\to
\infty.
\end{align*}

\item[(H5)]
\[
\lim_{t\to \infty}\int_{t_0}^{t}\prod_{t_0<t_k<s}\frac{1}{c_k}
\exp(\int_{t_0}^{s}\frac{p(\sigma)}{r(\sigma)}d\sigma)
h(s)ds=+\infty,
\]
where
\[
c_k=\begin{cases}
a_{k}^{\alpha}, & t_k-\delta\neq t_j,\\
\frac{a_{k}^{\alpha}}{b_{j}^{*}}, & t_k-\delta=t_j.
\end{cases}
\]
\end{itemize}

In the following, we also  assume that  solutions
to \eqref{e1.1}, \eqref{e1.2} exist on $[t_0, +\infty)$.

\begin{lemma}[\cite{c1}] \label{lem1}
 Let the function $m\in PC^1(\mathbb{R}_+, \mathbb{R})$ satisfy
the inequalities
\begin{gather*}
m'(t)\leq p(t)m(t)+q(t), \quad t\neq t_k, \\
m(t_{k}^{+})\leq d_km(t_k)+b_k, \quad k=1,2,\dots,
\end{gather*}
where $p,q\in PC(R_+,R)$ and $d_k\geq 0,b_k$ are
constants, then
\begin{equation}
\begin{split}
m(t)&\leq m(t_0)\prod_{t_0<t_k<t}d_k
\exp\Big(\int_{t_0}^{t}p(s)ds\Big)
+\sum_{t_0<t_k<t}\Big(\prod_{t_k<t_j<t}d_j
\exp\Big(\int_{t_k}^{t}p(s)ds\Big)\Big)b_k\\
&\quad +\int_{t_0}^{t}\prod_{s<t_k<t}d_k
\exp\Big(\int_{s}^{t}p(\sigma )d\sigma\Big)q(s)d
s, \quad t\geq t_0.
\end{split}\label{e2.1}
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2}
Let $x$ be a solution of \eqref{e1.1}, \eqref{e1.2}.
Suppose that there exist some $T\geq t_0$ such that $x(t)>0,t\geq T$.
If {\rm (H1)--(H3)} are satisfied,
then $x'(t_k)>0$ and $x'(t)>0$ for $t\in (t_k, t_{k+1}]$,
where $t_k\geq T$, $k=1,2,\dots$.
\end{lemma}

\begin{proof}
 We first prove that $x'(t_k)>0$ for any $t_k\geq T$. If not,
there must exist some $j$ such that $x'(t_j)<0$, $t_j\geq T$ and
$x'(t_{j}^{+})=I_j(x'(t_j)) \leq a_{j}^{*}x'(t_j)<0$. Let
$$
x'(t_j)\exp\Big(\int_{t_0}^{t_j}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)
=: \beta <0.
$$
From \eqref{e1.1}, it is clear that
$$
\Big(x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)
\Big)'=-\frac{f(t, x(t-\delta))}{\alpha r(t)(x'(t))^{\alpha -1}}
\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).
$$
Since $\alpha$ is the quotient of positive odd integers,
$(x'(t))^{\alpha-1}>0$, we obtain
\begin{equation}
\Big(x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)
\Big)'<0. \label{e2.2}
\end{equation}
Hence, the function
$ x'(t)\exp(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds)$
is decreasing on $(t_j, t_{j+1}]$,
$$
x'(t_{j+1})\exp\Big(\int_{t_0}^{t_{j+1}}\frac{r'(s)+p(s)}{\alpha r(s)}
ds\Big)
\leq x'(t_{j}^{+})\exp\Big(\int_{t_0}^{t_{j}}\frac{r'(s)+p(s)}{\alpha
r(s)}ds\Big);
$$
i.e.,
$$
x'(t_{j+1})\exp\Big(\int_{t_0}^{t_{j+1}}\frac{r'(s)+p(s)}{\alpha r(s)}
ds\Big)\leq a_{j}^{*}\beta
$$
and
\begin{align*}
x'(t_{j+2})\exp\Big(\int_{t_0}^{t_{j+2}}\frac{r'(s)+p(s)}{\alpha r(s)}
ds\Big)
&\leq x'(t_{j+1}^{+})\exp\Big(\int_{t_0}^{t_{j
+1}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)\\
&\leq a_{j+1}^{*}a_{j}^{*}\beta.
\end{align*}
By induction, we obtain
$$
 x'(t_{j+n})\exp\Big(\int_{t_0}^{t_{j+n}}\frac{r'(s)+p(s)}{\alpha r(s)}
ds\Big)
\leq \prod_{k=0}^{n-1}a_{j+k}^{*}\beta,
$$
while for $t\in (t_{j+n}, t_{j+n+1}]$, we have
\begin{equation}
x'(t)\leq \prod_{t_j\leq t_k<t}a_{k}^{*}\beta
 \exp\Big(-\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).
\label{e2.3}
\end{equation}
 From the condition  $ x(t_{n}^{+})\leq b_nx(t_n), $ we have the
 impulsive differential inequality
\begin{gather*}
x'(t)\leq \prod_{t_j\leq t_k<t}a_{k}^{*}\beta
 \exp\Big(-\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big),
\quad t\neq t_k, \,k=j+1,j+2,\dots,\\
x(t_{k}^{+})\leq b_kx(t_k), \quad t=t_k,\; t\geq t_j.
\end{gather*}
Applying Lemma \ref{lem1}, we have
\begin{align*}
 x(t)
&\leq x(t_{j}^{+})\prod_{t_j<t_k<t}b_k+a_{j}^{*}\beta\int_{t_j}^{t}
\prod_{s<t_k<t}b_k\prod_{t_j<t_i<s}a_{i}^{*}
\exp\Big(-\int_{t_0}^{s}
\frac{r'(\sigma)+p(\sigma)}{\alpha r(\sigma)}d \sigma\Big)ds\\
&\leq \prod_{t_j<t_k<t}b_k\Big\{x(t_{j}^{+})+a_{j}^{*}\beta \int_{t_j}^{t}
\prod_{t_j<t_i<s}\frac{a_{i}^{*}}{b_i}
\exp\Big(-\int_{t_0}^{s}\frac{r'(\sigma)+p(\sigma)}
{\alpha r(\sigma)}d\sigma\Big)ds\Big\}. %\label{e2.4}
\end{align*}
Since $ x(t_k)>0$ for $t_k\geq T$, one can find that
the above inequality contradicts  (H3)
as $t\to\infty$, therefore,
$ x'(t_{k})\geq 0 (t\geq T)$.

By condition (H2), we have $x'(t_k^{+})\geq a_{k}^{*}x'(t_k)$
 for any $t_k\geq T$.
Because the function
 $x'(t)\exp(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds)$
is decreasing on $(t_{j+i-1}, t_{j+i}]$, we obtain
\[
x'(t)\exp(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds)>0
\]
for any $ t\in (t_{j+i-1}, t_{j+i}]$, which implies
$x'(t)\geq 0$ for $t\geq T$.
The proof is complete.
\end{proof}

\begin{theorem} \label{thm1}
If  {\rm (H1)-(H3), (H5)} are satisfied, then every solution
$x$ of \eqref{e1.1}, \eqref{e1.2} satisfies
$\liminf_{t\to\infty}|x(t)|=0$.
\end{theorem}

\begin{proof}
Let $x$ be a solution of \eqref{e1.1}-\eqref{e1.2}, and by
contradiction assume that
\[
\liminf_{t\to\infty}|x(t)|>0.
\]
 Without loss of generality, we may assume that
$x(t)>0$ on $(t_0, +\infty)$. By Lemma \ref{lem2}, $x'(t)>0$ for all
$t\geq t_0$. We use a Riccati transformation of the
form
\begin{equation}
V(t)=\frac{r(t)(x'(t))^{\alpha}}{g(x(t-\delta))}.\label{e2.5}
\end{equation}
Differentiating $V(t)$, we obtain
\begin{align*}
 V'(t)&= \frac{(r(t)(x'(t))^{\alpha})'g(x(t-\delta))-r(t)(x'(t))^{\alpha}g'(x(t-\delta))x'
(t-\delta)}{g^{2}(x(t-\delta))}
\\
&=\frac{-p(t)(x'(t))^{\alpha}-f(t, x(t-\delta))}{g(x(t-\delta))}
-\frac{x'(t-\delta)g'(x(t-\delta))} {r(t)(x'(t))^{\alpha}}V^2(t)\\
& \leq  -p(t)\frac{V(t)}{r(t)}-h(t).
\end{align*}
 From \eqref{e2.5} and (H1), it is clear that
\begin{align*}
 V(t_{k}^{+})
&= \frac {r(t_{k}^{+})(x'(t_{k}^{+}))^{\alpha}}
{g(x(t_{k}^{+}-\delta))} \\
&\leq \begin{cases}
\frac{r(t_{k})(x'(t_{k}))^{\alpha}a_{k}^{\alpha}}
{g(x(t_{k}-\delta))}=a_{k}^{\alpha}V(t_k)=c_kV(t_k),
&t_k-\delta \neq t_j, \\[3pt]
\frac{r(t_{k})(x'(t_{k}))^{\alpha}a_{k}^{\alpha}}{g(x(t_{j}^{+}))}
\leq \frac{a_{k}^{\alpha}}{b_{j}^{*}}V(t_{k})=c_{k}V(t_{k}),
&t_k-\delta = t_j,
\end{cases}
\end{align*}
where $c_k$'s  are defined in (H5). Applying Lemma \ref{lem1},
we  have
\begin{align*}
V(t)&\leq \prod_{t_0<t_k<t}c_k
\exp\Big(-\int_{t_0}^{t}\frac{p(s)}{r(s)}ds\Big)\\
&\quad\times \Big[V(t_0) -\int_{t_0}^{t}\prod_{t_0<t_k<s}
\frac{1}{c_k}\exp\Big(\int_{t_0}^{s}\frac{p(\sigma)}{r(\sigma)}
d\sigma\Big)h(s)ds\Big].
\end{align*}
By  (H5), the above inequality is impossible. The proof is complete.
\end{proof}

\begin{lemma} \label{lem3}
 Let $x$ be a solution of \eqref{e1.1}, \eqref{e1.2}.
Suppose that there exist some $T\geq t_0$ such that $x(t)>0$,
$t\geq T$. If (H1), (H2), (H4) are satisfied, then
$x'(t_k)>0$ and $x'(t)>0$ for
$t\in (t_k, t_{k+1}]$, where $t_k\geq T,
k=1,2,\dots$.
\end{lemma}

\begin{proof}
Firstly, for $x(t)>0$, $t\geq T$, we will prove that
$x'(t_k)>0$, for any $t_k\geq T$, $T\geq t_0$.
If not, there exist some $j$ such that $x'(t_j)<0$, $t_j\geq T$ and
$x'(t_{j}^{+})=I_j(x'(t_j)) \leq a_{j}^{*}x'(t_j)<0$.
 From \eqref{e1.1}, it is clear that
$$
\Big(x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)
\Big)'
=-\frac{f(t, x(t-\delta))}{\alpha r(t)(x'(t))^{\alpha -1}}
\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).
$$
Since $\alpha$ is the quotient of positive odd integers,
$(x'(t))^{\alpha-1}>0$, we obtain
$$
\Big(x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds
\Big)\Big)'<0.
$$
Hence, the function $x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{
\alpha r(s)}ds\Big)$
is decreasing on $(t_j, t_{j+1}]$,
$$
x'(t_{j+1})\exp\Big(\int_{t_0}^{t_{j+1}}\frac{r'(s)+p(s)}
{\alpha r(s)}ds\Big)
\leq x'(t_{j}^{+})\exp\Big(\int_{t_0}^{t_{j}}\frac{r'(s)+p(s)}{\alpha
r(s)}ds\Big)),
$$
i.e.,
$$
x'(t_{j+1})\leq a_{j}^{*}x'(t_j)\exp
\Big(-\int_{t_j}^{t_{j+1}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)
 $$
and
$$ x'(t_{j+2})\leq
a_{j+1}^{*}a_{j}^{*}x'(t_{j})\exp\Big(-\int_{t_j}^{t_{j
+2}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).
$$
By induction, we obtain
$$
x'(t_{j+n})
\leq \prod_{k=0}^{n-1}a_{j+k}^{*}x'(t_j)
\exp\Big(-\int_{t_j}^{t_{j+n}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).
$$
Because the function
$x'(t)\exp(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds)$
is decreasing on $(t_j, t_{j+1}]$, we have
\begin{equation}
x'(t)\leq a_{j}^{*}x'(t_j)
 \exp\Big(-\int_{t_j}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big),
\quad t\in (t_j,t_{j+1}].\label{e2.6}
\end{equation}
 Integrating \eqref{e2.6} from $m$ to $t$, we have
 $$
x(t)\leq x(m)+a_{j}^{*}x'(t_j)\int_{m}^{t}
\exp\Big(-\int_{t_j}^{u}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)du,
\quad t_j<m<t_{j+1}.
$$
 Let $t\to t_{j+1}$, $m\to t_{j}^{+}$. We have
\begin{align*}
 x(t_{j+1})
&\leq  x(t_{j}^{+})+a_{j}^{*}x'(t_j)\int_{t_j}^{t_{j+1}}
\exp\Big(-\int_{t_j}^{u}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)du\\
&\leq  b_jx(t_j)+a_{j}^{*}x'(t_j)\int_{t_j}^{t_{j+1}}
\exp\Big(-\int_{t_j}^{u}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)du,
 \end{align*}
and
\begin{align*}
 x(t_{j+2}) &\leq  x(t_{j+1}^{+})+a_{j+1}^{*}x'(t_{j+1})
\int_{t_{j+1}}^{t_{j+2}}\exp\Big(-\int_{t_{j+1}}^{u}\frac{r'(s)
 +p(s)}{\alpha r(s)}ds\Big))du\\
&\leq  b_{j+1}b_jx(t_j)+a_{j}^{*}b_{j+1}x'(t_j)
\int_{t_j}^{t_{j+1}}\exp\Big(-\int_{t_j}^{u}\frac{r'(s)
+p(s)}{\alpha r(s)}ds\Big)du\\
 &\quad +a_{j+1}^{*}a_{j}^{*}x'(t_j)\int_{t_{j+1}}^{t_{j+2}}
 \exp\Big(-\int_{t_{j}}^{u}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)du.
 \end{align*}
By induction, we have
\begin{align*}
 x(t_{j+n})
&\leq x'(t_j)\Big[\sum_{m=1}^{n-1}
 \prod_{k=m}^{n-1}\prod_{l=0}
 ^{m-1}b_{j+k}a_{j+l}^{*}\int_{t_{j+m-1}}^{t_{j+m}}
 \exp\Big(-\int_{t_{j}}^{u}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)du\\
 &\quad +\prod_{k=0}
 ^{n-1}a_{j+k}^{*}\int_{t_{j+n-1}}^{t_{j+n}}
\exp\Big(-\int_{t_{j}}^{u}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)du\Big]
+\prod_{k=0}^{n-1}b_{j+k}x(t_j). %\label{e2.7}
\end{align*}
 Since $x(t_k)>0\,(t_k\geq T)$, we  find that the above inequality
 contradicts condition (H4), therefore
 $x'(t_k)\geq 0$ for $t\geq T$.
 Further, for $t\in (t_j,t_{j+1}]$, we obtain
 $$
x'(t)\exp\Big(\int_{t_{0}}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)
\geq x'(t_{j+1})\exp
 \Big(\int_{t_{0}}^{t_{j+1}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)>0,
$$
 which implies $x'(t)>0$ for $t\geq T$. This completes the proof.
\end{proof}

Using Lemma \ref{lem3}, we have the following Theorem.

\begin{theorem} \label{thm2}
If  {\rm (H1), (H2), (H4), (H5)}
are satisfied, then every solution $x$ of \eqref{e1.1}, \eqref{e1.2}
satisfies $\liminf_{t\to \infty}|x(t)|=0$.
\end{theorem}

\subsection*{Example}
Consider
\begin{gather*}
\Big(t(x'(t))^{3}\Big)'-\big(x'(t)\big)^3
+\frac{1}{t^2}x(t-\frac{1}{3})=0, \quad
t\neq k, \; t\geq \frac{1}{2},\\
x'(k^{+})=\frac{k}{k+1}x'(k),\quad x(k^{+})=x(k), \quad
k=1,2,\dots.
\end{gather*}
Comparing with \eqref{e1.1}, \eqref{e1.2}, we see that
$r(t)=t$, $p(t)=-1$, $\alpha=3$, $\delta=1/3$,
$ t_{k+1}-t_k>1/3$ and $a_{k}=a_{k}^{*}=k/(k+1)$,
$b_k=b_{k}^{*}=1$. Obviously (H1), (H2) are satisfied,
\begin{align*}
&\lim_{t\to
\infty}\int_{t_j}^{t}\prod_{t_j<t_k<s}
\frac{a_{k}^{*}}{b_{k}}\exp\Big(-\int_{t_j}^{s}\frac{r'(\sigma)+p(\sigma)}{3
r(\sigma)}d\sigma\Big)ds\\
&>(j+1)\lim_{t \to \infty}\int_{t_{j}}^{t}\frac{ds}{s+1}=+\infty,
\end{align*}
and
\begin{align*}
&\lim_{t\to \infty}\int_{t_0}^{t}\prod_{t_0<t_k<s}
\frac{1}{c_k} \exp\Big(\int_{t_0}^{s}\frac{p(\sigma)}{r(\sigma)}
d\sigma\Big) h(s)ds\\
&=\lim_{t\to\infty}\int_{t_0}^{t}\prod_{t_0<t_k<s}
(\frac{1}{a_k})^{\alpha}\exp(-\ln s+\ln t_0)
\frac{1}{s^2}ds\\
&> \frac{1}{2}\lim_{t\to\infty}\int_{t_0}^{t}ds=+\infty.
\end{align*}
So (H3) and (H5) are satisfied.
 By Theorem \ref{thm1}, it is clear that every
solution of this equation satisfies
$\liminf_{t\to\infty}|x(t)|=0$.

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