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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 54, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/54\hfil Second-order damped differential equations]
{Oscillatory behavior for second-order damped
differential equation with nonlinearities including Riemann-Stieltjes
integrals}

\author[E. Tun\c{c},  H. Liu \hfil EJDE-2018/54\hfilneg]
{Ercan Tun\c{c}, Haidong Liu}

\address{Ercan Tun\c{c} \newline
Gaziosmanpasa University,
 Department of Mathematics, Faculty of Arts
and Sciences, 60240, Tokat, Turkey}
\email{ercantunc72@yahoo.com}

\address{Haidong Liu \newline
Qufu Normal University,
School of Mathematical Sciences,
273165, Qufu,  China}
\email{tomlhd983@163.com}

\thanks{Submitted March 17, 2017. Published Februay 22, 2018.}
\subjclass[2010]{34C10, 34C15}
\keywords{Forced oscillation; Riemann-Stieltjes integral; interval criteria;
\hfill\break\indent p-Laplacian; nonlinear differential equations}

\begin{abstract}
 In this article, we establish new oscillation criteria for forced second-order
 damped differential equations with nonlinearities that include
 Riemann-Stieltjes integrals. The results obtained here extend 
 related results reported in the literature, and can easily be 
 extended to more general equations of the type considered here.
 Two examples illustrate the results obtained here.
\end{abstract}

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

\section{Introduction}

This article concerns the oscillatory behavior of the forced
second order differential equation with a nonlinear damping term,
\begin{equation}\label{1.1}
\big( r(t)\phi _{\alpha }(x'(t))\big) '+p(t)\phi _{\alpha }(x'(t))+f(t,x)=e(t),
\quad t\geq t_{0}\geq0,
\end{equation}
with
\begin{equation}\label{1.2}
f(t,x)=q(t)\phi _{\alpha }(x(t))
+\int_{a}^{b}g(t,s)\phi_{\gamma(t,s)+\alpha-\alpha\beta(t)}(x(t))d\xi(s),
\end{equation}
where $a,b\in \mathbb{R}$ with $b\in (a,\infty )$, $\alpha >0$, and
 $\phi_{*}(u):=|u|^{*}\operatorname{sgn} u$.

In the remainder of this article  we assume that:
\begin{itemize}
\item [(i)] $r, p, q$ and $e$ $:[t_{0},\infty )\to \mathbb{R}$ are real valued
continuous functions with $r(t)>0$;

\item [(ii)] $g:[t_{0},\infty )\times [ a,b]\to \mathbb{R}$ is a real
valued continuous function;

\item [(iii)] $\beta :[t_{0},\infty )\to (0,\infty )$ and
$\gamma :[t_{0},\infty )\times [ a,b]\to \mathbb{R}$ are
real valued continuous function such that $\gamma (t, \cdot)$ is
strictly increasing
on $[a,b]$, and
\begin{equation}\label{1.3}
0<\gamma (t,a)<\alpha \beta (t)<\gamma (t,b) \quad\text{and}\quad
\alpha\beta (t)\leq \gamma (t,a)+\alpha, \quad \text{for }t\geq
t_{0};
\end{equation}

\item [(iv)] $\xi :[a,b]\to \mathbb{R}$ is a real valued strictly increasing
function.
\end{itemize}
Here $\int_{a}^{b}f(s)d\xi (s)$ denotes the Riemann-Stieltjes
integral of the function $f$ on $[a,b]$ with respect to $\xi $.

As usual, a nontrivial solution $x(t)$ of equation \eqref{1.1} is called
oscillatory if it has arbitrary large zeros, otherwise it is called
nonoscillatory.  Equation \eqref{1.1} is said to be oscillatory
if all its solutions are oscillatory.

We note that as special cases, when $\alpha=1$ and  $p(t)\equiv0$,
equation \eqref{1.1} reduces to the equation
\begin{equation}\label{1.4}
(r(t)x'(t))'+q(t)x(t)+\int_a^bg(t,s)\phi_{\gamma(t,s)+1-\beta(t)}(x(t))d\xi
(s)=e(t):
\end{equation}
when  $p(t)\equiv0$, $\beta(t)\equiv1$, $\gamma (t,s)=\gamma(s)$ and
$a=0$, equation \eqref{1.1} reduces to 
\begin{equation}\label{1.5}
\big( r(t)\phi _{\alpha }(x'(t))\big) '+q(t)\phi _{\alpha }(x(t))
+\int_{0}^{b}g(t,s)\phi_{\gamma(s)}(x(t))d\xi (s)=e(t);
\end{equation}
and when $\xi(s)$ is a step function, the integral term in the
equation \eqref{1.5} reduces to a finite sum and hence equation
\eqref{1.5} becomes 
\begin{equation}\label{1.6}
\left( r(t)\phi _{\alpha }(x'(t))\right) '+q(t)\phi _{\alpha }(x(t))
+\sum_{i=1}^nq_i(t)\phi _{\alpha_i }(x(t))=e(t).
\end{equation}

In recent years,  differential equations and variational problems
with variable exponent growth conditions have been investigated
extensively. We refer the reader to \cite{agza, aaz,
 kozh, lich, nas, sun, suwo1, sume1, sume2, won1}.
The study of such problems arise from nonlinear elasticity
theory and electrorheological fluids, see \cite{nas,won1}.
At the same time, some results on the oscillatory behavior of
solutions of equations with variable exponent growth conditions were
established in  \cite{lime, won2} and the references
therein. On the other hand, many authors have been interested in
differential  equations with nonlinearity given by
a Riemann-Stieltjes integral $\int_{a}^{b}f(s)d\xi (s)$.
Because the integral term becomes a finite sum when $\xi(s)$ is a step
function and a Riemann integral when $\xi(s)=s$. We refer to
\cite{hako,lime,suko} for more information. 
In particularly, Liu and Meng \cite{lime}
discussed equation \eqref{1.4}, Hassan and Kong \cite{hako} studied
equation \eqref{1.5}.

Motivated by the above, we will establish interval
oscillation criteria  for the general equation \eqref{1.1}
which involves variable exponent growth conditions. Our work is of
significance because equation \eqref{1.1} not only contains a
$\alpha$-Laplacian term but also contains a damping term and allows
nonlinear terms given by variable exponents. It is our belief that
the present paper will contribute significantly to the study of
oscillatory behavior of solutions of second order damped
differential equations with nonlinearities given by
Riemann-Stieltjes integrals.

The paper is organized as follows.
In Section 2 we establish interval oscillation criteria of both the
El-Sayed type and the Kong
type for equation \eqref{1.1}. In Section 3 we
apply our theory to two examples.

\section{Main results}

In the following, we denote by $L_{\xi }[a,b]$ the set of
Riemann-Stieltjes integrable functions on $[a,b]$ with respect to
$\xi $. We further assume that for any $t\in [ t_{0},\infty
)$, $\gamma (t,\cdot)$, $1/\gamma (t,\cdot)\in L_{\xi }[a,b]$.
To obtain our main results in this paper, we need the following
lemmas.

\begin{lemma}[\cite{hlp}]\label{l2.1}
 If $X$ and $Y$ are nonnegative and $\lambda >1$,
then
\begin{equation*}
\lambda XY^{\lambda -1}-X^{\lambda }\leq (\lambda -1)Y^{\lambda },
\end{equation*}
where equality holds if and only if $X=Y$.
\end{lemma}



The proofs of the following lemmas are similar to those of
\cite[Lemmas 2.1 and 2.2]{lime} and so the proofs will be
omitted.


\begin{lemma}\label{l2.2}
Assume that {\rm (iii)} and \eqref{1.3} hold.
Let $h=\sup\{s\in(a,b):\gamma(t,s)\leq \alpha\beta(t),\,t\in[t_0,\infty) \}$,
  and set
\begin{gather*}
m_1(t):=\int_h^b\frac{\alpha\beta^2(t)}{\gamma(t,s)}
\Big(\int_h^b\mathrm{d}\xi(s)\Big)^{-1}\mathrm{d}\xi(s), \quad  t\in[t_0,\infty), \\
m_2(t):=\int_a^h\frac{\alpha\beta^2(t)}{\gamma(t,s)}
\Big(\int_a^h\mathrm{d}\xi(s)\Big)^{-1}\mathrm{d}\xi(s), \quad t\in[t_0,\infty).
\end{gather*}
Then for any  function  $\theta$ satisfying
$\theta(t)\in (m_1(t),m_2(t))$ for $t\in[t_0,\infty)$, there exists a function
$\eta:[t_0,\infty)\times[a,b]\to (0, \infty)$ satisfying,
for any $t\in[t_0,\infty)$, $\eta(t,\cdot)\in L_\xi[a,b]$, such that
\begin{gather}\label{2.1}
\int_{a}^{b}\gamma (t,s)\eta (t,s)d\xi (s)=\alpha \beta
^{2}(t), \quad (t,s)\in [ t_{0},\infty )\times [ a,b], \\
\label{2.2}
\int_{a}^{b}\eta (t,s)d\xi (s)= \theta (t), \quad (t,s)\in
[ t_{0},\infty )\times [ a,b].
\end{gather}
\end{lemma}


\begin{lemma}\label{l2.3}
Let $\theta :[t_{0},\infty )\to (0,\infty )$ and
$\eta :[t_{0},\infty )\times [ a,b]\to (0,\infty )$ be
functions such that $\eta (t,\cdot)\in L_{\xi }[a,b]$  for any
$t\in [ t_{0},\infty )$ and \eqref{2.2} holds.
 Then, for any function $w:[t_{0},\infty )\times [
a,b]\to [ 0,\infty )$ satisfying, for any
$t\in[ t_{0},\infty )$, $w (t,\cdot)\in L_{\xi}[a,b]$, we have
\begin{equation}\label{2.3}
\int_{a}^{b}\eta (t,s)w(t,s)d\xi (s)\geq
\exp \Big(\frac{1}{\theta(t)} \int_{a}^{b}\eta (t,s)\ln [
\theta (t)w(t,s)] d\xi (s)\Big),
\end{equation}
where we use the convention that $\ln 0=-\infty $ and
$e^{-\infty}=0$.
\end{lemma}


Following El-Sayed \cite{say}, for $c,d\in [ t_{0},\infty )$
with $c<d$, we define the function class
 $\mathcal{E}(c,d):=\{u\in C^{1}[c,d]:u(c)=0=u(d),u\not\equiv0\} $.
Our first main result provides an oscillation criterion for equation
\eqref{1.1} of the El-Sayed type.

\begin{theorem}\label{t2.1}
Suppose that for any $T\geq t_{0}$, there exist $T\leq a_1<b_1\leq a_2<b_2$
such that for $i=1,2$,
\begin{gather}\label{2.4}
g(t,s)\geq 0\quad \text{for }(t,s)\in [ a_i,b_i]\times [ a,b],\\
\label{2.5}
(-1)^{i}e(t)\geq 0\quad \text{for }t\in [ a_i,b_i].
\end{gather}
Let $\theta$ be a function satisfying
$\theta(t)\in(m_1(t),\beta(t)]$ for $t\in[t_0,\infty)$, and
$\eta:[t_0,\infty)\times[a,b]\to (0,\infty)$  be a function
such that $1/\eta (t,\cdot)\in L_{\xi }[a,b]$ and
\eqref{2.1}-\eqref{2.2} hold. Suppose also that
for $i=1,2$, there exists a function $u_i\in \mathcal{E}(a_i,b_i)$ such that
\begin{equation}\label{2.6}
\int_{a_i}^{bi}[ \delta (t)Q(t)|u_i(t)| ^{\alpha +1}-\delta (t)r(t)|
u_i'(t)| ^{\alpha +1}] dt>0,
\end{equation}
where
\[
\delta (t):=\exp \Big(\int_{t_{0}}^{t}\frac{p(s)}{r(s)}ds\Big),
\]
and
\begin{equation}\label{2.7}
\begin{aligned}
Q(t)&= q(t)+\Big(\frac{\big(\beta^2(t)-\theta(t)\beta(t)
+\theta(t)\big)|e(t)|}{\beta^2(t)-\theta(t)\beta(t)
}\Big)^{\frac{\beta^2(t)-\theta(t)\beta(t)}{\beta^2(t)
 -\theta(t)\beta(t)+\theta(t)}} \\
&\quad\times \exp\Big(\frac{\theta(t)}{\beta^2(t)-\theta(t)\beta(t)+\theta(t)} 
 \Big[\ln\big(\beta^2(t)-\theta(t)\beta(t)  +\theta(t)\big) \\
&\quad +\frac{\int_a^b\eta(t,s)
\ln\frac{g(t,s)}{\eta(t,s)}\mathrm{d}\xi(s)}{\theta(t)}\Big]\Big).
\end{aligned}
\end{equation}
Here we use the convention that $\ln 0=-\infty $, $e^{-\infty }=0$,
and $0^{0}=1$ due to the fact that $\lim_{t\to
0^{+}}t^{t}=1$. Then equation \eqref{1.1} is oscillatory.
\end{theorem}


\begin{proof}
Assume that  \eqref{1.1} has an extendible solution $x(t)$
which is eventually positive or negative. Then, without loss of
generality, we may assume that there exists $t_1\in[t_{0},\infty)$
such that $x(t)>0$ for all $t\geq t_1$. When
$x(t)$ is an eventually negative, the proof follows the same way
except that the interval $[a_{2},b_{2}]$ instead of $[a_1,b_1]$
is used. Define the
function $w(t)$ by
\begin{equation} \label{2.8}
w(t)=\delta (t)\frac{r(t)\phi _{\alpha }(x'(t))}{\phi
_{\alpha }(x(t))}, \quad t\geq t_1.
\end{equation}
Then, in view of \eqref{1.1} and \eqref{2.8}, we obtain
\begin{equation}\label{2.9}
\begin{aligned}
&w'(t) \\
&= \delta '(t)\frac{r(t)\phi _{\alpha}(x'(t))}{\phi _{\alpha }(x(t))}
+\delta (t)\Big[ \frac{\left( r(t)\phi _{\alpha }(x'(t))\right) '}
 {\phi _{\alpha }(x(t))}
 -\frac{r(t)\phi _{\alpha }(x'(t))\left( \phi _{\alpha}(x(t))\right)'}
 {\left( \phi _{\alpha }(x(t))\right) ^{2}}\Big]  \\
&= \delta '(t)\frac{r(t)\phi _{\alpha }(x'(t))}{x^{\alpha
}(t)}-\delta (t)\frac{p(t)\phi _{\alpha }(x'(t))}{x^{\alpha }(t)}
-\delta (t)q(t) \\
&\quad -\delta (t)\int_{a}^{b}g(t,s)\big(x(t)\big) ^{\gamma
(t,s)-\alpha\beta (t)}d\xi (s)   
+\delta(t)\frac{e(t)}{x^{\alpha }(t)} \\
&\quad -\alpha \delta (t)r(t)\frac{\phi_{\alpha }(x'(t))x'(t)}{x^{\alpha +1}(t)}  \\
&= -\delta (t)q(t)-\delta (t)\int_{a}^{b}g(t,s)\left(x(t)\right)
^{\gamma (t,s)-\alpha\beta (t)}d\xi (s)
 +\delta (t)\frac{e(t)}{x^{\alpha }(t)} \\
&\quad -\alpha \delta (t)r(t)\frac{\phi _{\alpha }(x'(t))x'(t)}{x^{\alpha +1}(t)}  \\
&=-\delta (t)q(t)-\delta (t)\int_{a}^{b}g(t,s)\left(x(t)\right)
^{\gamma (t,s)-\alpha\beta (t)}d\xi (s)+\delta (t)\frac{e(t)}{x^{\alpha }(t)}
 \\
&\quad -\alpha \delta (t)r(t)\frac{| x'(t)| ^{\alpha +1}}{x^{\alpha +1}(t)}   \\
&=-\delta (t)q(t)-\delta (t)\int_{a}^{b}g(t,s)\left(x(t)\right)
^{\gamma (t,s)-\alpha\beta (t)}d\xi (s)+\delta (t)\frac{e(t)}{x^{\alpha }(t)}  \\
&\quad -\alpha \frac{| w(t)| ^{\frac{\alpha +1}{\alpha }}}
 {( \delta (t)r(t)) ^{1/\alpha }},
\end{aligned}
\end{equation}
for $t\geq t_1$.

From the assumption, there exists a nontrivial interval
$[a_1,b_1]\subset [ t_1,\infty )$ such that \eqref{2.4}
and \eqref{2.5} hold with $i=1$. Next, we consider two cases:
case (I)  $\theta(t)\equiv\beta(t)$, and case (II)
$\theta(t)\in(m_1(t),\beta(t))$.

Assume that case (I) holds. Then, in view of \eqref{2.4},
\eqref{2.5} and \eqref{2.9}, we see that, for  $t\in [a_1,b_1]$,
\begin{equation}\label{2.10}
w'(t)\leq -\delta (t)q(t)-\delta
(t)\int_{a}^{b}g(t,s)\big(x(t)\big) ^{\gamma
(t,s)-\alpha\beta
(t)}d\xi (s)-\alpha \frac{| w(t)| ^{\frac{\alpha +1}{
\alpha }}}{\left( \delta (t)r(t)\right) ^{1/\alpha }}.
\end{equation}
Clearly, from the assumption on $\eta$, we have that
\begin{equation}\label{2.11}
\int_{a}^{b}\eta (t,s)\left( \gamma (t,s)-\alpha\beta (t)\right) d\xi (s)=0.
\end{equation}
From \eqref{2.11} and Lemma \ref{l2.3}, we obtain, for $t\in [a_1,b_1]$,
\begin{align*}
&\int_{a}^{b}g(t,s)\big(x(t)\big) ^{\gamma (t,s)-\alpha\beta (t)}d\xi (s) \\
&=\int_{a}^{b}\eta (t,s)\eta ^{-1}(t,s)g(t,s)\left(x(t)\right)
^{\gamma (t,s)-\alpha\beta (t)}d\xi (s) \\
&\geq \exp \Big( \frac{1}{\beta (t)}\int_{a}^{b}\eta (t,s)\ln
[ \beta (t)\eta ^{-1}(t,s)g(t,s)\big(x(t)\big) ^{\gamma
(t,s)-\alpha\beta (t)}] d\xi (s)\Big) \\
&=\exp \Big( \frac{1}{\beta (t)}\int_{a}^{b}\eta (t,s)\ln
[\beta (t)\eta ^{-1}(t,s)g(t,s)] d\xi (s) \\
&\quad +  \frac{1}{\beta (t)} \int_{a}^{b}\eta (t,s)
 \ln [ \big( x(t)\big) ^{\gamma (t,s)-\alpha\beta (t)}] d\xi (s)\Big) \\
&=\exp \Big( \frac{1}{\beta (t)}\int_{a}^{b}\eta (t,s)\ln
[\beta (t)\eta ^{-1}(t,s)g(t,s)] d\xi (s) \\
&\quad + \frac{\ln x(t)}{\beta (t)} \int_{a}^{b}\eta (t,s)
 \left( \gamma (t,s)-\alpha\beta (t)\right) d\xi (s)\Big) \\
&=\exp \Big( \frac{1}{\beta (t)}\int_{a}^{b}\eta (t,s)\ln
\left[ \beta (t)\eta ^{-1}(t,s)g(t,s)\right] d\xi (s)\Big)\\
&=\exp \Big(\ln [\beta(t)]+ \frac{1}{\beta (t)}\int_{a}^{b}\eta (t,s)\ln
\left[ \eta ^{-1}(t,s)g(t,s)\right] d\xi (s)\Big).
\end{align*}
Using this in \eqref{2.10}, we see that, for $t\in [a_1,b_1]$,
\begin{equation}\label{2.12}
\begin{aligned}
w'(t)
&\leq -\delta (t)q(t)-\delta (t)\exp \Big(
\ln [\beta(t)] \\
&\quad + \frac{1}{\beta (t)}\int_{a}^{b}\eta (t,s)\ln
 [ \eta ^{-1}(t,s)g(t,s)] d\xi (s)\Big)
 -\alpha \frac{| w(t)| ^{\frac{\alpha +1}{\alpha }}}
 {( \delta (t)r(t)) ^{1/\alpha }}   \\
&= -\delta (t)Q(t)-\alpha \frac{| w(t)| ^{\frac{\alpha +1}{\alpha }}}
 {( \delta (t)r(t)) ^{1/\alpha }},
\end{aligned}
\end{equation}
where $Q(t)$ is defined by \eqref{2.7} with
$\theta(t)\equiv\beta(t)$.

Multiplying both sides of \eqref{2.12} by $|u_1(t)| ^{\alpha +1}$,
integrating from $a_1$ to $b_1$, and using integration by parts, we
obtain
\begin{equation}\label{2.13}
\begin{aligned}
&\int_{a_1}^{b_1}\delta (t)Q(t)| u_1(t)|^{\alpha +1}dt  \\
&\leq -\int_{a_1}^{b_1}|
u_1(t)| ^{\alpha +1}w'(t)dt-\alpha
\int_{a_1}^{b_1}| u_1(t)| ^{\alpha +1}\frac{
| w(t)| ^{\frac{\alpha +1}{\alpha }}}{(\delta(t)r(t)) ^{1/\alpha }}dt   \\
&=(\alpha +1) \int_{a_1}^{b_1}\phi _{\alpha
}(u_1(t))u_1'(t)w(t)dt-\alpha
\int_{a_1}^{b_1}| u_1(t)| ^{\alpha +1}\frac{| w(t)| ^{\frac{\alpha +1}{\alpha }}}
{(\delta(t)r(t)) ^{1/\alpha }}dt   \\
&\leq \int_{a_1}^{b_1}\big[ (\alpha +1)
| u_1(t)| ^{\alpha }| u_1'(t)| | w(t)| -\alpha |
u_1(t)| ^{\alpha +1}\frac{| w(t)|
^{\frac{\alpha +1}{\alpha }}}{( \delta (t)r(t)) ^{1/\alpha }}\big] dt.
\end{aligned}
\end{equation}
Applying Lemma \ref{l2.1} with
\begin{gather*}
X=\Big( \alpha \frac{| u_1(t)| ^{\alpha+1}}{( \delta
(t)r(t)) ^{1/\alpha }}| w(t)| ^{\frac{\alpha +1}{
\alpha }}\Big) ^{1/\lambda },\quad
\lambda =\frac{\alpha +1}{\alpha }, \quad
Y=\Big( \frac{\alpha ( \delta (t)r(t)) ^{\frac{1}{\alpha +1}}}{
\alpha ^{\frac{\alpha }{\alpha +1}}}| u_1'(t)| \Big) ^{\alpha },
\end{gather*}
we see that
\begin{equation*}
(\alpha +1) | u_1(t)| ^{\alpha}| u_1'(t)| | w(t)|
-\alpha | u_1(t)| ^{\alpha +1}\frac{|
w(t)| ^{\frac{\alpha +1}{\alpha }}}{( \delta (t)r(t)) ^{1/\alpha }}
\leq \delta (t)r(t)| u_1'(t)| ^{\alpha +1},
\end{equation*}
substituting this into \eqref{2.13} gives
\begin{equation*}
\int_{a_1}^{b_1}[ \delta (t)Q(t)|u_1(t)|^{\alpha +1}-\delta (t)r(t)
| u_1'(t)| ^{\alpha +1} ] dt\leq 0,
\end{equation*}
which contradicts \eqref{2.6} for $i=1$.

Next, assume that case (II) holds. From \eqref{2.2} and \eqref{2.5},
we have
\begin{equation}\label{2.14}
\begin{aligned}
&\delta (t)\int_a^bg(t,s)[x(t)]^{\gamma(t,s)-\alpha\beta(t)}\mathrm{d}\xi(s)
 -\delta (t)\frac{e(t)}{x^{\alpha }(t)} \\
&= \delta (t)\int_a^b \big[g(t,s)[x(t)]^{\gamma(t,s)-\alpha\beta(t)}
 -\frac{e(t)}{x^{\alpha }(t)}\frac{\eta(t,s)}{\theta(t)}\big]\mathrm{d}\xi(s) \\
&= \delta (t)\int_a^b\big[g(t,s)[x(t)]^{\gamma(t,s)-\alpha\beta(t)}
 +\frac{|e(t)|}{x^{\alpha }(t)}\frac{\eta(t,s)}{\theta(t)}\big]\mathrm{d}\xi(s) \\
&= \delta (t)\int_a^b\frac{\eta(t,s)}{\theta(t)}
 \big[\frac{\theta(t)}{\eta(t,s)}g(t,s)[x(t)]^{\gamma(t,s)-\alpha\beta(t)}
 +\frac{|e(t)|}{x^{\alpha }(t)}\big]\mathrm{d}\xi(s).
\end{aligned}
\end{equation}
If we let
\begin{gather}\label{2.15}
p=\frac{\theta(t)}{\beta^2(t)-\theta(t)\beta(t)+\theta(t)},\quad
q=\frac{\beta^2(t)-\theta(t)\beta(t)}{\beta^2(t)
 -\theta(t)\beta(t)+\theta(t)},\\
\label{2.16}
A=\frac{\beta^2(t)-\theta(t)\beta(t)+\theta(t)}{\eta(t,s)}g(t,s)[x(t)
 ]^{\gamma(t,s)-\alpha\beta(t)},  \quad
 B=\frac{1}{q}\frac{|e(t)|}{x^{\alpha }(t)},
\end{gather}
then from the Young inequality ($pA+qB\geq A^pB^q$, where $p+q=1$,
$p,q>0, A\geq0, B\geq0$),
we get
\begin{equation}\label{2.17}
\begin{aligned}
&\frac{\theta(t)}{\eta(t,s)}g(t,s)[x(t)]^{\gamma(t,s)
 -\alpha\beta(t)}+\frac{|e(t)|}{x^{\alpha }(t)} \\
&\geq \Big(\frac{\beta^2(t)-\theta(t)\beta(t)+\theta(t)}{\eta(t,s)}
 g(t,s)[x(t)]^{\gamma(t,s)-\alpha\beta(t)}\Big)^p
 \Big(\frac{1}{q}\frac{|e(t)|}{x^{\alpha }(t)}\Big)^q \\
&= \Big(\frac{\beta^2(t)-\theta(t)\beta(t)+\theta(t)}{\eta(t,s)}
 g(t,s)\Big)^p\Big(\frac{|e(t)|}{q}\Big)^q [x(t)]^{(\gamma(t,s)-\alpha\beta(t))p
 -q\alpha} \\
&= \Big(\frac{\beta^2(t)-\theta(t)\beta(t)+\theta(t)}{\eta(t,s)}
 g(t,s)\Big)^p\Big(\frac{|e(t)|}{q}\Big)^q [x(t)
 ]^{\frac{\gamma(t,s)\theta(t)-\alpha\beta^2(t)}{\beta^2(t)-\theta(t)
 \beta(t)+\theta(t)}}.
\end{aligned}
\end{equation}
By \eqref{2.1} and \eqref{2.2}, we get
\begin{equation}\label{2.18}
\int_a^b\eta(t,s)[\gamma(t,s)\theta(t)-\alpha\beta^2(t)]
\mathrm{d}\xi(s)\equiv0, \quad  \text{for any }  t\in [t_0,\infty).
 \end{equation}
From \eqref{2.14}-\eqref{2.18} and Lemma \ref{l2.3}, we see that,
for $t\in[a_1, b_1]$,
\begin{align*}
&\delta(t)\int_a^bg(t,s)[x(t)]^{\gamma(t,s)-\alpha\beta(t)}\mathrm{d}\xi(s)
 -\delta(t)\frac{e(t)}{x^{\alpha }(t)} \\
&\geq\delta(t)\int_a^b\frac{\eta(t,s)}{\theta(t)}
 \Big(\frac{\beta^2(t)-\theta(t)\beta(t)+\theta(t)}{\eta(t,s)}g(t,s)\Big)^p
 \Big(\frac{|e(t)|}{q}\Big)^q \\
&\times [x(t)]^{\frac{\gamma(t,s)\theta(t)-\alpha\beta^2(t)}
 {\beta^2(t)-\theta(t)\beta(t)+\theta(t)}}\mathrm{d}\xi(s) \\
&\geq\delta(t)\exp\Big(\frac{1}{\theta(t)}
 \int_a^b\eta(t,s)\ln \Big[\Big(\frac{\beta^2(t)-\theta(t)\beta(t)+\theta(t)}
 {\eta(t,s)}g(t,s)\Big)^p \\
&\quad\times \Big(\frac{|e(t)|}{q}\Big)^q 
 [x(t)]^{\frac{\gamma(t,s)\theta(t)-\alpha\beta^2(t)}{\beta^2(t)
 -\theta(t)\beta(t)+\theta(t)}}\Big]\mathrm{d}\xi(s)\Big) \\
&=\delta(t)\exp\Big(\frac{1}{\theta(t)}
  \int_a^b\eta(t,s)\ln \Big[\Big(\frac{\beta^2(t)
 -\theta(t)\beta(t)+\theta(t)}{\eta(t,s)} g(t,s)\Big)^p 
  \Big(\frac{|e(t)|}{q}\Big)^q \Big] \mathrm{d}\xi(s) \Big) \\
&\times \exp\Big(\frac{1}{\theta(t)}
 \int_a^b\eta(t,s)\Big[\frac{\gamma(t,s)\theta(t)-\alpha\beta^2(t)}{\beta^2(t)
 -\theta(t)\beta(t)+\theta(t)}\Big]\ln x(t) \mathrm{d}\xi(s)\Big) \\
&=\delta(t)\exp\Big(\frac{1}{\theta(t)}
 \int_a^b\eta(t,s)\ln \Big[\Big(\frac{\beta^2(t)-\theta(t)\beta(t)
 +\theta(t)}{\eta(t,s)}g(t,s)\Big)^p\Big(\frac{|e(t)|}{q}\Big)^q \Big]
 \mathrm{d}\xi(s)\Big)  \\
&\times\exp\Big(\frac{1}{\theta(t)}
\frac{\ln x(t)}{\beta^2(t)-\theta(t)\beta(t)
 +\theta(t)}\int_a^b\eta(t,s)\Big[\gamma(t,s)\theta(t)-\alpha\beta^2(t)\Big] 
 \mathrm{d}\xi(s)\Big) \\
&=\delta(t)\exp\Big(\frac{p}{\theta(t)}
 \int_a^b\eta(t,s)\ln \Big[\frac{\beta^2(t)-\theta(t)\beta(t)
 +\theta(t)}{\eta(t,s)}g(t,s)\Big]\mathrm{d}\xi(s) \\
&\quad +\frac{1}{\theta(t)} \ln\big(\frac{|e(t)|}{q}\big)^q
 \int_a^b\eta(t,s)\mathrm{d}\xi(s)\Big) \\
&=\delta(t)\exp\Big(\frac{p}{\theta(t)}
 \int_a^b\eta(t,s)\Big[\ln\big(\beta^2(t)-\theta(t)\beta(t)+\theta(t)\big)
 +\ln\frac{g(t,s)}{\eta(t,s)} \Big]\mathrm{d}\xi(s) \\
&\quad +\ln\Big(\frac{|e(t)|}{q}\Big)^q\Big) \\
&=\delta(t)\Big(\frac{|e(t)|}{q}\Big)^q\exp\Big(\frac{p}{\theta(t)}
\ln\big(\beta^2(t)-\theta(t)\beta(t)
+\theta(t)\big)\int_a^b\eta(t,s)\mathrm{d}\xi(s) \\
&\quad +\frac{p}{\theta(t)}
\int_a^b\eta(t,s) \ln\frac{g(t,s)}{\eta(t,s)}\mathrm{d}\xi(s)\Big) \\
&=\delta(t)\Big(\frac{\big(\beta^2(t)-\theta(t)\beta(t)
+\theta(t)\big)|e(t)|}{\beta^2(t)-\theta(t)\beta(t)}
\Big)^{\frac{\beta^2(t)-\theta(t)\beta(t)}{\beta^2(t)-\theta(t)\beta(t)
+\theta(t)}}  \\
&\quad\times \exp\Big(\frac{\theta(t)}{\beta^2(t)-\theta(t)\beta(t) +\theta(t)}
 \Big[\ln\big(\beta^2(t)-\theta(t)\beta(t)+\theta(t)\big)\\
&\quad +\frac{1}{\theta(t)} \int_a^b\eta(t,s) \ln\frac{g(t,s)}{\eta(t,s)}
 \mathrm{d}\xi(s) \Big]\Big).
\end{align*}
Then from \eqref{2.9} and above inequality, we have
\begin{equation} \label{2.19}
\begin{aligned}
\omega'(t)
&\leq-\delta(t)q(t)-\delta(t)\Big(\frac{\big(\beta^2(t)-\theta(t)\beta(t)
 +\theta(t)\big)|e(t)|}{\beta^2(t)-\theta(t)\beta(t)}
 \Big)^{\frac{\beta^2(t)-\theta(t)\beta(t)}{\beta^2(t)
 -\theta(t)\beta(t)+\theta(t)}}  \\
&\quad \times \exp\Big(\frac{\theta(t)}{\beta^2(t)-\theta(t)\beta(t)
 +\theta(t)}\Big[\ln\big(\beta^2(t)-\theta(t)\beta(t)+\theta(t)\big) \\
&\quad +\frac{1}{\theta(t)}\int_a^b\eta(t,s) \ln\frac{g(t,s)}{\eta(t,s)}\mathrm{d}\xi(s)
 \Big]\Big) -\alpha \frac{| w(t)| ^{\frac{\alpha +1}{\alpha }}}
 { ( \delta (t)r(t)) ^{1/\alpha }} \\
&=-\delta(t)Q(t)-\alpha \frac{| w(t)| ^{\frac{\alpha +1}{\alpha }}}
 { ( \delta (t)r(t)) ^{1/\alpha }}\,,
\end{aligned}
\end{equation}
where $Q(t)$ is defined by \eqref{2.7} with
$\theta(t)\in(m_1(t),\beta(t))$. The rest of the proof is similar to
that of case (I) and hence is omitted. This completes  the proof of
Theorem \ref{t2.1}.
\end{proof}


Following Philos \cite{phi} and Kong \cite{kon}, we say that for any
$a,b\in \mathbb{R}$ with $a<b$, a function $H(t,s)$ belongs to a
function class  $\mathcal{H}(a,b)$, denoted by $H\in
\mathcal{H}(a,b)$, if $H\in C(\mathbb{D},[0,\infty))$, where 
$\mathbb{D}=\{ (t,s):b\geq t\geq s\geq a\} $, which
satisfies
\begin{equation*}
H(t,t)=0,\quad H(b,s)>0,\quad H(s,a)>0 \quad \text{for } b>s>a,
\end{equation*}
and $H(t,s)$ has continuous partial derivative $\partial
H(t,s)/\partial t$ and $\partial
H(t,s)/\partial s$ on $[a,b]\times[a,b]$ such that
\begin{gather*}
\frac{\partial H}{\partial t}(t,s)=(\alpha +1)h_1(t,s)
 H^{\frac{\alpha}{\alpha +1}}(t,s), \\
\frac{\partial H}{\partial s}(t,s)=(\alpha +1)h_{2}(t,s)
 H^{\frac{\alpha }{\alpha +1}}(t,s),
\end{gather*}
where $h_1,h_{2}\in L_{loc}(D,\mathbb{R})$.

Our next result uses the function class $\mathcal{H}(a,b)$ to
establish an oscillation criterion for equation \eqref{1.1} of the
Kong-type.

\begin{theorem}\label{t2.2} 
Suppose that for any $T\geq t_{0}$, there exist
nontrivial subinterval $[a_1,b_1]$ and $[a_{2},b_{2}]$ of
$[T,\infty )$ such that \eqref{2.4} and \eqref{2.5} hold for
$i=1,2$. Let $\theta$ and $\eta$ be functions defined as in Theorem
\ref{t2.1} such that $1/\eta (t,\cdot)\in L_{\xi }[a,b]$ and
\eqref{2.1}-\eqref{2.2} hold.
Suppose also that for $i=1,2$, there exists $c_i\in (a_i,b_i)$ and 
$H_i\in \mathcal{H}(a_i,b_i)$ such that
\begin{equation}\label{2.20}
\begin{aligned}
&\frac{1}{H_i(c_i,a_i)}\int_{a_i}^{c_i}
 [ \delta (s)Q(s)H_i(s,a_i)-\delta (s)r(s)|
 h_{i1}(s,a_i)| ^{\alpha +1}] ds   \\
&+\frac{1}{H_i(b_i,c_i)}\int_{c_i}^{b_i}
 [\delta (s)Q(s)H_i(b_i,s)-\delta (s)r(s)|
 h_{i2}(b_i,s)| ^{\alpha +1}] ds>0,
\end{aligned}
\end{equation}
where $\delta(t)$ and $Q(t)$ are as in Theorem \ref{t2.1}. Then
equation \eqref{1.1} is oscillatory.
\end{theorem}

\begin{proof} 
Proceeding as in the proof of Theorem \ref{t2.1}, we again arrive at 
\eqref{2.12} and \eqref{2.19}. In view of  \eqref{2.12} and
\eqref{2.19}, we see that
\begin{equation}\label{2.21}
w'(t) \leq -\delta (t)Q(t)-\alpha \frac{| w(t)| ^{\frac{\alpha +1
}{\alpha }}}{( \delta (t)r(t)) ^{1/\alpha }},\quad  t\in [ a_1,b_1].
\end{equation}
Multiplying both sides of  \eqref{2.21}, with $t$ replaced by $s$,
by $H_1(s,a_1)$ and integrating from $a_1$ to $c_1$, we see that
\begin{equation*}
\int_{a_1}^{c_1}\delta (s)Q(s)H_1(s,a_1)ds
\leq -\int_{a_1}^{c_1}H_1(s,a_1)w'(s)ds-\alpha
\int_{a_1}^{c_1}H_1(s,a_1)\frac{| w(s)| ^{\frac{\alpha +1}{\alpha }}}
 {( \delta (s)r(s)) ^{1/\alpha }}.
\end{equation*}
Integrating by parts, we obtain
\begin{equation}\label{2.22}
\begin{aligned}
&\int_{a_1}^{c_1}\delta (s)Q(s)H_1(s,a_1)ds \\
&\leq -H_1(c_1,a_1)w(c_1)+\int_{a_1}^{c_1}(\alpha
+1)|h_{11}(s,a_1)|H^{\frac{\alpha }{\alpha +1}}_1(s,a_1)|w(s)|ds   \\
&\quad -\alpha \int_{a_1}^{c_1}H_1(s,a_1)\frac{|
w(s)| ^{\frac{\alpha +1}{\alpha }}}{( \delta
(s)r(s)) ^{1/\alpha }}ds.
\end{aligned}
\end{equation}
Applying Lemma \ref{l2.1} with
\begin{gather*}
X=\Big( \alpha \frac{H_1(s,a_1)| w(s)| ^{\lambda }}
 {( \delta (s)r(s)) ^{1/\alpha }}\big) ^{1/\lambda }, \quad 
 \lambda =\frac{\alpha +1}{\alpha }, \quad
 Y=\Big( \frac{\alpha ( \delta (s)r(s)) ^{\frac{1}{\alpha +1}}}{
\alpha ^{\frac{\alpha }{\alpha +1}}}|
h_{11}(s,a_1)| \Big) ^{\alpha },
\end{gather*}
we see that
\begin{align*}
&(\alpha +1)|h_{11}(s,a_1)|H^{\frac{\alpha }{\alpha
+1}}_1(s,a_1)|w(s)|-\alpha
H_1(s,a_1)\frac{| w(s)| ^{\frac{\alpha +1}{\alpha }}}
{( \delta (s)r(s)) ^{1/\alpha }} \\
&\leq \delta (s)r(s)| h_{11}(s,a_1)| ^{\alpha +1},
\end{align*}
substituting this into \eqref{2.22}, we obtain
\begin{equation*}
\int_{a_1}^{c_1}[ \delta
(s)Q(s)H_1(s,a_1)-\delta (s)r(s)|
h_{11}(s,a_1)| ^{\alpha +1}] ds
\leq -H_1(c_1,a_1)w(c_1)
\end{equation*}
or
\begin{equation}\label{2.23}
\frac{1}{H_1(c_1,a_1)}\int_{a_1}^{c_1}[ \delta
(s)Q(s)H_1(s,a_1)-\delta (s)r(s)|
h_{11}(s,a_1)| ^{\alpha +1}] ds\leq -w(c_1).
\end{equation}
Similarly, multiplying both sides of \eqref{2.21}, with $t$ replaced
by $s$, by $H_1(b_1,s)$ and integrating it from $c_1$ to
$b_1$, and then applying Lemma \ref{l2.1}, we see that
\begin{equation}\label{2.24}
\frac{1}{H_1(b_1,c_1)}\int_{c_1}^{b_1}[ \delta
(s)Q(s)H_1(b_1,s)-\delta (s)r(s)|
h_{12}(b_1,s)| ^{\alpha +1}] ds\leq w(c_1).
\end{equation}
Combining \eqref{2.23} and \eqref{2.24},  we arrive at
\begin{align*}
&\frac{1}{H_1(c_1,a_1)}\int_{a_1}^{c_1}
[\delta (s)Q(s)H_1(s,a_1)-\delta (s)r(s)|
h_{11}(s,a_1)| ^{\alpha +1}] ds \\
&+\frac{1}{H_1(b_1,c_1)}\int_{c_1}^{b_1}
[\delta (s)Q(s)H_1(b_1,s)-\delta (s)r(s)|
h_{12}(b_1,s)| ^{\alpha +1}] ds\leq 0
\end{align*}
which contradicts \eqref{2.20} for $i=1$, and completes the proof.
\end{proof}

\begin{remark} \label{r3.1} \rm
When $p(t)\equiv0$, $\beta (t)\equiv1$, $\alpha=1$, $a=0$ and 
$\gamma (t,s)=\gamma(s)$, Theorems \ref{t2.1} and \ref{t2.2} reduce to 
\cite[Theorems 2.1 and 2.2]{suko}.
When $p(t)\equiv0$, $\beta (t)\equiv1$, $a=0$ and
 $\gamma (t,s)=\gamma (s)$,  Theorems \ref{t2.1} and \ref{t2.2} reduce to 
\cite[Theorems 2.1 and 2.2]{hako}.
When $p(t)\equiv0$  and  $\alpha =1$,   Theorems \ref{t2.1} and \ref{t2.2}
reduce to \cite[Theorems 2.1 and 2.2]{lime}.
\end{remark}



\section{Examples}

In this section, we will work out two numerical examples to illustrate our
 main results. Here we use the convention that $\ln0=-\infty$ 
and $e^{-\infty}= 0$.

\begin{example} \label{exam:3.1} \rm
Consider equation \eqref{1.1} with
$\alpha=2$, $r(t)=1$, $p(t)=0$,  $q(t)=\lambda\sin 4t$ with
$\lambda>0$ is a constant, $a=1$, $b=3$, $\gamma(t,s)=se^{-t}$,
$g(t,s)\equiv1$, $\beta(t)=e^{-t}$, $\xi(s)=s$, and 
$e(t)=-f(t)\cos 2t$ with $f\in C[0,\infty)$ is any nonnegative function. For any
$T\geq0$, we choose $k\in\mathbb{Z}$ large enough that $2k\pi\geq T$
and let $a_1= 2k\pi$, $a_2=b_1=2k\pi+\frac{\pi}{4}$, and
$b_2=2k\pi+\frac{\pi}{2}$. Then, \eqref{2.5}  and \eqref{2.6} hold,
and we have $m_1(t)=2\ln \frac{3}{2}e^{-t}$ and $m_2(t)=2\ln
2e^{-t}$. With
\begin{gather*}
\theta(t)=\delta e^{-t},\quad  \delta\in(2\ln(3/2),1],\quad
p=\frac{\delta-2\ln(3/2)}{4\ln2-2\ln 3}, \\
\eta(t,s)= \begin{cases}  
2pe^{-t}/s,  &(t,s)\in[0,\infty)\times[1,2),\\
 2(1-p)e^{-t}/s,  &(t,s)\in[0,\infty)\times[2,3],
\end{cases} 
\end{gather*}
it is easy to verify that \eqref{2.1}  and \eqref{2.2} hold. Letting
$u_i(t)=\sin4t$ for $t\in[a_i, b_i]$, $i=1,2$, and from the
definition of $Q(t)$, we see that
\begin{align*}
Q(t)
&=\lambda\sin 4t+\big[\big(1+\frac{\delta e^t}{1-\delta}\big)f(t)|\cos2t|
 \big]^{\frac{1-\delta}{1-\delta+\delta e^t}}  \\
&\quad \times\exp\Big(\frac{\delta e^t}{1-\delta+\delta
e^t}\big[\ln\big(e^{-2t}-\delta e^{-2t}+\delta
e^{-t}\big)-\frac{e^t}{\delta}\int_1^3\eta(t,s)\ln\eta(t,s)\mathrm{d}s\big]\Big)\\
&=:  F(\lambda,\delta,t),
\end{align*}
from this and $\delta(t)=1$, we obtain
\begin{gather*}
\int_{a_1}^{b_1}\delta(t)Q(t)|u_1(t)|^3\mathrm{d}t
 =\int_0^{\pi/4}\widetilde{F}(\lambda,\delta,t)\sin^34t\mathrm{d}t, \\
 \int_{a_2}^{b_2}\delta(t)Q(t)|u_2(t)|^3\mathrm{d}t
 =-\int_{\frac{\pi}{4}}^{\pi/2}\widetilde{F}
 (\lambda,\delta,t)\sin^34t\mathrm{d}t,
\end{gather*}
where
\begin{align*}
\widetilde{F}(\lambda,\delta,t)
&=\lambda\sin 4t+\Big[\Big(1+\frac{\delta e^{t+2k\pi}}{1-\delta}\Big)
f(t+2k\pi)|\cos2t|\Big]^{\frac{1-\delta}{1-\delta+\delta e^{t+2k\pi}}}   \\
&\quad\times \exp\Big(\frac{\delta e^{t+2k\pi}}{1-\delta+\delta e^{t+2k\pi}}
 \Big[\ln\big(e^{-2(t+2k\pi)}-\delta e^{-2(t+2k\pi)}+\delta e^{-(t+2k\pi)}\big) \\
&\quad -\frac{e^{t+2k\pi}}{\delta}\int_1^3\eta(t+2k\pi,s)
 \ln\eta(t+2k\pi,s)\mathrm{d}s\Big]\Big),
\end{align*}
and
$$
\int_{a_i}^{b_i}\delta(t)r(t)|u_i^{\prime}(t)|^3\mathrm{d}t
=\int_{a_i}^{b_i}64|\cos^34t|\mathrm{d}t=\frac{64}{3}.
$$
Thus, by Theorem \ref{t2.1} we see that \eqref{1.1} is oscillatory
if
$\int_0^{\pi/4}\widetilde{F}(\lambda,\delta,t)
\sin^34t\mathrm{d}t> 64/3$
and $-\int_{\pi/4}^{\pi/2}\widetilde{F}(\lambda,\delta,t)
\sin^34t\mathrm{d}t>64/3$.
\end{example}

\begin{example}\label{exam:3.2} \rm
Consider equation \eqref{1.1} with $\alpha=3/2$, $r(t)=1$,
$p(t)=1$, $q(t)=\lambda\sin t$ with $\lambda>0$ is a constant,
$a=1$, $b=3$, $\gamma(t,s)=s(\cos \frac{t}{2}+\frac{3}{2})$,
$g(t,s)\equiv1$, $\beta(t)=\cos \frac{t}{2}+\frac{3}{2}$,
$\xi(s)=s$, and $e\in C[0,\infty)$ be any function satisfying
$(-1)^ie(t)\geq 0$ on $[a_i, b_i]$ for  $i=1,2$. For any $T\geq0$,
we choose $k\in\mathbb{Z}$ large enough that $2k\pi\geq T$ and let
$a_1= 2k\pi$, $a_2=b_1=2k\pi+\frac{\pi}{4}$,
$b_2=2k\pi+\frac{\pi}{2}$, $c_1=2k\pi+\frac{\pi}{8}$ and
$c_2=2k\pi+\frac{3\pi}{8}$. Then, it is easy to see that \eqref{2.5}
and \eqref{2.6}  hold, and
 $m_1(t)=\ln 2(\cos \frac{t}{2}+\frac{3}{2})$ and
$m_2(t)=3\ln \frac{3}{2}(\cos \frac{t}{2}+\frac{3}{2})$. With
\begin{gather*}
\theta(t)=\delta (\cos \frac{t}{2}+\frac{3}{2}),\quad
\delta\in(\ln 2,1],\quad 
p=\frac{\delta-\ln 2}{3\ln\frac{3}{2}-\ln 2},\\
\eta(t,s)=\begin{cases}
  3p (\cos \frac{t}{2}+\frac{3}{2})/s, 
  &(t,s)\in[0,\infty)\times[1,3/2),\\
 (1-p)(\cos \frac{t}{2}+\frac{3}{2})/s,
  &(t,s)\in[0,\infty)\times[/3/2,3], 
\end{cases}
\end{gather*}
we see that \eqref{2.1}  and \eqref{2.2} are valid, and from the
definition of $Q(t)$, we obtain
\begin{align*}
Q(t)
&=\lambda\sin t+\Big[\Big(1+\frac{\delta }{(1-\delta)
 (\cos \frac{t}{2}+\frac{3}{2})}\Big)|e(t)|\Big]^{\frac{(1-\delta)
 (\cos \frac{t}{2}+\frac{3}{2})}{(1-\delta)(\cos \frac{t}{2}+\frac{3}{2})+\delta}}\\
&\quad \times  \exp\Big(\frac{\delta}{(1-\delta)(\cos
\frac{t}{2}+\frac{3}{2})+\delta }\Big[\ln\big((\cos
\frac{t}{2}+\frac{3}{2})^2-\delta (\cos
\frac{t}{2}+\frac{3}{2})^2 \\
&\quad +\delta (\cos \frac{t}{2}+\frac{3}{2})\big)
 -\frac{1}{\delta (\cos \frac{t}{2}
 +\frac{3}{2})}\int_1^3\eta(t,s)\ln\eta(t,s)\mathrm{d}s\Big]\Big).
\end{align*}
If we choose $H(t,s)=(t-s)^{5/2}$, then $h_1(t,s)=1$,
$h_2(t,s)=-1$. Since $\delta (t)=e^t$,  by Theorem \ref{t2.2}, we see
that \eqref{1.1} is oscillatory if
\[
\int_{2k\pi}^{2k\pi+\frac{\pi}{8}}Q(s)e^s(s-2k\pi)^{5/2}\mathrm{d}s
+\int_{2k\pi+\frac{\pi}{8}}^{2k\pi+\frac{\pi}{4}}Q(s)e^s(2k\pi+\pi/4-s)^{5/2}
\mathrm{d}s>e^{2k\pi}(e^{\pi/4}-1),
\]
and
\begin{align*}
&\int_{2k\pi+\frac{\pi}{4}}^{2k\pi+\frac{3\pi}{8}}Q(s)e^s(s-2k\pi-\pi/4)^{5/2}
 \mathrm{d}s
+\int_{2k\pi+\frac{3\pi}{8}}^{2k\pi+\frac{\pi}{2}}Q(s)e^s(2k\pi+\pi/2-s)^{5/2}
 \mathrm{d}s \\
&>e^{2k\pi}(e^{\pi/2}-e^{\pi/4}).
\end{align*}
\end{example}

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