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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 54, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/54\hfil Existence of non-oscillatory solutions]
{Existence of non-oscillatory solutions for second-order advanced
half-linear \\ differential equations}

\author[A. Cheng, Z. Xu\hfil EJDE-2013/54\hfilneg]
{Aijun Cheng, Zhiting Xu}  % in alphabetical order

\address{Aijun Cheng \newline
School of Mathematical Sciences, South China Normal University,
Guangzhou 510631,  China}
\email{chengaijun@163.com}

\address{Zhiting Xu \newline
School of Mathematical Sciences, South China Normal University,
Guangzhou 510631,  China}
\email{xuzhit@126.com}

\thanks{Submitted August 4, 2012. Published February 21, 2013.}
\subjclass[2000]{34C10, 34C55}
\keywords{Oscillation; nonoscillation; half-linear;
 advanced differential equation}

\begin{abstract}
 In this article, we  establish the necessary and sufficient conditions for existence of non-oscillatory
 solutions for the second-order advanced half-linear differential equation
 \begin{equation*}
 \big(r(t)|x'(t)|^{\alpha-1}x'(t)\big)'+p(t)|x(h(t)\big)|^{\alpha-1}x(h(t))=0,\quad
  t\geq t_0.
 \end{equation*}
 The obtained results generalize some well-known theorems in the literature
\end{abstract}

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

\section{Introduction}

 Consider the second-order  advanced  half-linear differential equation
\begin{equation}\label{e1.1}
\Big(r(t)|x'(t)|^{\alpha-1}x'(t)\Big)'+p(t)|x(h(t)\big)|^{\alpha-1}x(h(t))=0,
\quad t\geq t_0,
\end{equation}
where $\alpha>0$ is a constant, $r\in C^1([t_0,\infty),\mathbb{R}^+)$ with
 $\int^{\infty}_{t_0}r^{-1/\alpha}(t)dt=\infty$,
$p\in C([t_0,\infty)$, $[0,\infty))$ with $p(t)\not\equiv0$, and 
$h\in C([t_0, \infty),\mathbb{R})$ with $t\leq h(t)$.

By a solution to  \eqref{e1.1}  we mean a function 
$x\in C^1([T_x,\infty), \mathbb{R})$,
$T_x\geq t_0$, such that 
$r|x'|^{\alpha-1}x'\in C^1([T_x,\infty), \mathbb{R})$ and 
$x$ satisfies \eqref{e1.1} for all $t\geq T_x$. 
Solutions of  \eqref{e1.1} vanishing in some neighborhood of infinity will be
excluded from our consideration. A solution of  \eqref{e1.1} is said to be 
oscillatory if it has arbitrarily
large zeros, and otherwise it is said to be non-oscillatory.  
Equation \eqref{e1.1} is called oscillatory
if all its solutions are oscillatory. 
Similarly, it is called non-oscillatory if all its solutions are non-oscillatory.

Equation \eqref{e1.1} can be considered as the natural generalization of 
the  linear differential equation
\begin{equation}\label{e1.2}
\big(r(t)x'(t)\big)'+p(t)x(t)=0,
\end{equation}
or of the half-linear differential equation
\begin{equation}\label{e1.3}
\Big(r(t)|x'(t)|^{\alpha-1}x'(t)\Big)'+p(t)|x(t)|^{\alpha-1}x(t)=0
\end{equation}
and of the advanced differential equation
\begin{equation}\label{e1.4}
\big(r(t)x'(t)\big)'+p(t)x(h(t))=0,\quad t\leq h(t),
\end{equation}

The oscillation and nonoscillation of \eqref{e1.2}-\eqref{e1.3} has been
extensively investigated from various viewpoints
during the previous 60 years, see for example  the monographs \cite{AGR, DR} 
and the references therein. 
To motivate the formulation of our main results, we wish to quote the 
following known non-oscillation results.  

\begin{theorem}[{\cite[p. 379]{W}}] \label{thm1.1}
 Equation \eqref{e1.2} has a nonoscillatory solution if and only if there
 is a positive differentiable function $\varphi(t)$ defined on 
$[t_1,\infty)$, $t_1\geq t_0$, such that
\begin{equation*}
\varphi'(t)+\frac{\varphi^2(t)}{r(t)}\leq -p(t),\quad t\geq t_1.
\end{equation*}
\end{theorem}

\begin{theorem}[{\cite[Theorem 2.1]{Y}}] \label{thm1.2}
 Assume that
\begin{equation*}
\int^{\infty}_t\frac{ds}{r(s)}=\infty \quad \text{and}\quad
0\leq \int^{\infty}_tp(s)ds<\infty, \quad t\in [t_0,\infty)
\end{equation*}
hold. Define a sequence of function  $\{\upsilon_n(t)\}^{\infty}_0$ as follows:
\begin{gather*}
\upsilon_0(t)=\int^{\infty}_tp(s)ds, \quad
\upsilon_1(t)=\int^{\infty}_t\frac{\upsilon^2_0(s)}{r(s)}ds,\\
\upsilon_{n+1}(t)=\int^{\infty}_t\frac{[\upsilon_0(s)+\upsilon_n(s)]^2}{r(s)}ds,
\quad t\in [t_0,\infty), \quad n=1,2,\dots.
\end{gather*}
Then  \eqref{e1.2} is non-oscillatory if and only if there exists
$t_1\geq t_0$ such that
\begin{equation*}
\lim_{n\to\infty}\upsilon_n(t)=\upsilon(t)<\infty \quad \text{for }t\geq t_1.
\end{equation*}
\end{theorem}

Recently, Yang and Lo \cite{YL}  extended Theorem \ref{thm1.2} to  \eqref{e1.3}, 
see \cite[Theorem 1]{YL}.
On the other hand, in 1991, Lu \cite{L} extended Theorem \ref{thm1.1}
to  \eqref{e1.4}. More precisely, Lu proved the following theorem.

\begin{theorem}[{ \cite[Lemma 2]{L}}] \label{thm1.3}
 Equation \eqref{e1.4} has a nonoscillatory solution if and only if there
is a positive differentiable function $\varphi(t)$ defined on $[t_1,\infty)$, 
$t_1\geq t_0$, such that
\begin{equation*}
\varphi'(t)+\frac{\varphi^{2}(t)}{r(t)}\leq-p(t)\exp\Big(\int^{h(t)}_{t}
\frac{\varphi(s)}{r(s)}ds\Big),\quad t\geq t_1.
\end{equation*}
\end{theorem}

For related works for \eqref{e1.2}, see. e.g., \cite{HI,KN, LY,TY}.

Inspired by  \cite{L, W, Y,YL}, in this article, we extend the results by 
Lu \cite{L}, Wintner \cite{W}, Yan \cite{Y}, and Yang and Lo \cite{YL} to
the Equation \eqref{e1.1}.
We establish necessary and sufficient conditions for existence of
non-oscillatory solutions to \eqref{e1.1}. 
Using these results, we further establish oscillation criteria
for \eqref{e1.1}.
The obtained results 
generalize some well-known theorems in the literature.

\section{Main results}

\begin{theorem}\label{thm2.1}
 If
\begin{equation} \label{e2.1}
\int^{\infty}_{t_0}p(s)ds=\infty,
\end{equation}
then  \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
 Suppose to the contrary that  \eqref{e1.1} has a non-oscillatory solution $x(t)$.
We assume that $x(t)>0$ and $x(h(t))>0$ for $t\geq t_1\geq t_0$.
A similar proof is done if we assume $x(t)< 0$ on $[t_1,\infty)$.
Since $p(t)\geq 0$ on $[t_1,\infty)$,  
$\big(r(t)|x'(t)|^{\alpha-1}x'(t)\big)'\leq 0$,
hence, $r(t)|x'(t)|^{\alpha-1}x'(t)$ is non-increasing on $[t_1,\infty)$,
therefore, $x'(t)$ is eventually of constant sign. 
If $x'(t)<0$ for $t\geq t_1$, then
$$
r(t)|x'(t)|^{\alpha-1}x'(t)\leq r(t_1)(-x'(t_1))^{\alpha-1}x'(t_1)=:-c<0\,.
$$
It follows that
\begin{equation*}
x(t)\leq x(t_1)-c^{1/\alpha}\int^t_{t_1}\frac{ds}{r^{1/\alpha}(s)}\to-\infty \quad \text{as}\,\,\,
t\to\infty,
\end{equation*}
which contradicts $x(t)> 0$. Thus, $x'(t)>0$ for $t\geq t_1$. Let
\begin{equation} \label{e2.2}
w(t)=\frac{r(t)|x'(t)|^{\alpha-1}x'(t)}{|x(t)|^{\alpha-1}x(t)}.
\end{equation}
Obviously, $w(t)>0$, and $r(t)(x'(t))^\alpha=w(t)(x(t))^\alpha$; i.e.,
\begin{equation}\label{e2.3}
\frac{x(h(t))}{x(t)}=\exp\Big(\int^{h(t)}_{t}\big(\frac{w(s)}{r(s)}\big)
^{1/\alpha}ds\Big).
\end{equation}
Then, from \eqref{e1.1} and  \eqref{e2.3}, we obtain
\begin{equation} \label{e2.4}
w'(t)+\alpha\frac{(w(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)}
+p(t)\exp\Big(\alpha\int^{h(t)}_{t}\big(\frac{w(s)}{r(s)}
\big)^{1/\alpha}ds\Big)=0,
\end{equation}
consequently,
\begin{equation*}
w'(t)+p(t)\leq 0.
\end{equation*}
Integrating the above inequality from $t_1$ to $t$ $(t > t_1)$, we have
\begin{equation*}
w(t)\leq w(t_1)-\int^t_{t_1}p(s)ds\to-\infty \quad \text{as }  t\to+\infty,
\end{equation*}
which contradicts $w(t)> 0$.
\end{proof}

According to Theorem \ref{thm2.1}, we can furthermore restrict our attention to the case:
\begin{equation} \label{e2.5}
\int^{\infty}_tp(s)ds<\infty.
\end{equation}
For convenience, we define $P(t)=\int^{\infty}_tp(s)ds$ for 
$t\geq t_0$. Firstly, we give the following Lemma.

\begin{lemma}\label{lem2.1}
Let \eqref{e2.5} hold. Suppose that  \eqref{e1.1} has a nonoscillatory solution $x(t)\neq 0$ for
$t\geq t_1\geq t_0$, and let $w(t)$ be defined by \eqref{e2.2}. 
Then the following statements hold for $t\geq t_1$:
\begin{gather} \label{e2.6}
w(t)>0,\quad \lim_{t\to\infty}w(t)=0, \\
 \label{e2.7}
\int^{\infty}_{t}\frac{(w(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds<\infty,\\
 \label{e2.8}
I(t)=\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}
\big(\frac{w(\tau)}{r(\tau)}\big)^{{1/\alpha}}d\tau\Big)ds< \infty,\\
 \label{e2.9}
w(t)=\alpha\int^{\infty}_{t}\frac{(w(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds
+I(t).
\end{gather}
\end{lemma}

\begin{proof} 
Assume that $x(t)>0$ on $[t_1,\infty)$. A similar argument holds
if we assume $x(t)<0$ on $[t_1,\infty)$. 
Proceeding as in the proof of Theorem \ref{thm2.1},
we know $x'(t)> 0$ for $t\geq t_1$. 
 Hence,  $w(t)> 0$ for $t \geq t_1$, and \eqref{e2.4} holds and
\begin{equation*}
w'(t)\leq -\alpha \frac{(w(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)}.
\end{equation*}
It follows that
\begin{equation*}
\frac{1}{w^{1/\alpha}(t)}-\frac{1}{w^{1/\alpha}(t_1)}
\geq\int^{t}_{t_1}\frac{1}{r^{1/\alpha}(s)}ds
\to\infty \quad \text{as }  t\to+\infty,
\end{equation*}
thus $\lim_{t\to\infty}w(t)=0$. Integrating \eqref{e2.4} from 
$t$ to $T$ ($T\geq t\geq t_1$), we have
\begin{equation} \label{e2.10}
w(T)-w(t)+\alpha\int^{T}_{t}\frac{(w(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds
+\int^{T}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}
\big(\frac{w(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds=0.
\end{equation}
Let $T \to\infty$, then from \eqref{e2.10} it follows that
\begin{align*}
w(t)=\alpha\int^{\infty}_{t}\frac{(w(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds
+I(t),\quad t\geq t_1.
\end{align*}
Hence, \eqref{e2.9} holds. Furthermore, \eqref{e2.7} and \eqref{e2.8} hold.
\end{proof}

\begin{theorem}\label{thm2.2}
Let \eqref{e2.5} hold.  Equation \eqref{e1.1} is non-oscillatory if and only 
if there exist $t_1\geq t_0$ and $\varphi(t)\in C^1([t_1, \infty),\mathbb{R}^+)$ 
such that
\begin{equation} \label{e2.11}
\varphi'(t)+\alpha\frac{(\varphi(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)}+p(t)
\exp\Big(\alpha\int^{h(t)}_{t}
\big(\frac{\varphi(s)}{r(s)}\big)^{1/\alpha}ds\Big)\leq 0,\quad t\geq t_1.
\end{equation}
\end{theorem}

\begin{proof} The ``only if" part. Let $x(t)$ be a non-oscillatory solution 
of  \eqref{e1.1}. 
Assume that $x(t)>0$ and $x(h(t))>0$ for $t\geq t_1$. 
Then, by Lemma \ref{lem2.1}, the function
$w(t)\in C^1([t_1, \infty), \mathbb{R}^+)$ defined by \eqref{e2.2} 
satisfies \eqref{e2.9}. Differentiation of \eqref{e2.9} shows that
$w(t)$ is a solution of \eqref{e2.11} on $[t_1, \infty)$.

The ``if" part. 
 It follows from \eqref{e2.11} that $\varphi'(t)<0$, 
hence $\varphi(t)$ is decreasing and is bounded from below;
consequently, its limit exists, namely,
$\lim_{t\to\infty}\varphi(t)=d\geq 0$. Next, we prove that
$d=0$.
Indeed, it follows from \eqref{e2.11} that
\begin{equation*}
\varphi'(t)\leq -\alpha\frac{(\varphi(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)}.
\end{equation*}
Dividing  both sides of the above inequality by $(\varphi(t))^{(\alpha+1)/\alpha}$, and integrating from
$t$ to $T$, then we obtain
\begin{equation*}
\frac{1}{\varphi^{1/\alpha}(T)}-\frac{1}{\varphi^{1/\alpha}(t)}\geq\int^{T}_{t} 
\frac{ds}{r^{1/\alpha}(s)},
\end{equation*}
letting $T\to\infty$ in the above, we have
$\lim_{T\to\infty}\varphi(T)=0$. 
Then integrating \eqref{e2.11} from $t$ to $\infty$, we have
\begin{equation*}
\alpha\int^{\infty}_{t}\frac{(\varphi(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds
+\int^{\infty}_{t}p(s)
\exp\Big(\alpha\int^{h(s)}_{s}\big(\frac{\varphi(\tau)}
{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds\leq \varphi(t),\quad t\geq t_1,
\end{equation*}
which implies that for $t\geq t_1$,
\begin{equation*}
\int^{\infty}_{t}\frac{(\varphi(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds
<\infty,\quad
\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}
\big(\frac{\varphi(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds<\infty.
\end{equation*}
Define the following mapping
\begin{equation} \label{e2.12}
(Ly)(t)=\alpha\int^{\infty}_{t}\frac{(y(s))^{(\alpha+1)/\alpha}}
{r^{1/\alpha}(s)}ds+
\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}
\big(\frac{y(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds,
\end{equation}
for $t\geq t_1$. Let
\begin{equation*}
x_0(t)\equiv 0, \quad x_n(t)=L(x_{n-1}(t)), \quad n=1,2,3,\dots.
\end{equation*}
It is easy to show that
\begin{equation*}
x_0(t)\leq x_1(t)\leq \dots\leq x_n(t)\leq \dots\leq \varphi(t).
\end{equation*}
Hence
\begin{equation*}
\lim_{n\to\infty}x_n(t)=u(t)\leq\varphi(t).
\end{equation*}
 By \eqref{e2.12}, we have
\[
x_n(t)=\alpha\int^{\infty}_{t}\frac{(x_{n-1}(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds+
\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}
\big(\frac{x_{n-1}(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds,
\]
for $t\geq t_1$.
By  Levi's monotone convergence theorem, and letting  $n\to\infty$ in the 
above equation, we obtain
\begin{equation} \label{e2.13}
u(t)=\alpha\int^{\infty}_{t}\frac{(u(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds+
\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}
\big(\frac{u(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds,\quad t\geq t_1.
\end{equation}
Set
\begin{equation*}
x(t)=\exp\Big(\int^{t}_{t_1}\big(\frac{u(\tau)}{r(\tau)}
\big)^{1/\alpha}d\tau\Big),\quad t\geq t_1\,.
\end{equation*}
Then
\begin{equation} \label{e2.14}
u(t)=\frac{r(t)(x'(t))^{\alpha}}{(x(t))^{\alpha}}\,.
\end{equation}
By  \eqref{e2.13} and \eqref{e2.14}, we have
\begin{equation*}
(r(t)(x'(t))^{\alpha})'+p(t)(x(h(t)))^{\alpha}=0;
\end{equation*}
i.e.,
\begin{equation*}
(r(t)|x'(t)|^{\alpha-1}x'(t))'
+p(t)|x(h(t))|^{\alpha-1}x(h(t))=0,\quad t\geq t_1.
\end{equation*}
Thus, $x(t)$ is a non-oscillatory solution of  \eqref{e1.1}.
\end{proof}

\begin{corollary}\label{coro2.1}
Let \eqref{e2.5} hold. If $h(t)\equiv t$, then  \eqref{e1.1} is non-oscillatory 
if and only if there exist $t_1\geq t_0$, and 
$\varphi(t)\in C^1([t_1, \infty), \mathbb{R}^+\big)$ such that
\begin{equation*}
\varphi'(t)+\alpha\frac{\varphi^{(\alpha+1)/\alpha}(t)}{r^{1/\alpha}(t)}
+p(t)\leq 0,\quad t\geq t_1,
\end{equation*}
\end{corollary}

We remark that for  \eqref{e1.4}, Theorem \ref{thm2.2} and
 Corollary \ref{coro2.1} reduce to
\cite[Lemma 2]{L}  and \cite[Corollary 1]{L}, respectively.


Let \eqref{e2.5} hold. Define a sequence of functions 
$\{\upsilon_n(t)\}_0^\infty$  as follows (if they exist):
\begin{equation} \label{e2.15}
\begin{gathered}
\upsilon_0(t)=P(t)=\int^{\infty}_{t}p(s)ds, \\
\upsilon_{n+1}(t)=\alpha\int^{\infty}_{t}
\frac{(\upsilon_n(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds
+\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}
\big(\frac{\upsilon_n(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds,
\end{gathered}
\end{equation}
for $n=0,1,2,\dots$, $t\geq t_1$.
Clearly, $\upsilon_0(t)\geq 0$ and $\upsilon_1(t)\geq \upsilon_0(t)$.
By induction, we obtain
\begin{equation} \label{e2.16}
\upsilon_{n+1}(t)\geq \upsilon_n(t),\quad n=0,1,2\dots;
\end{equation}
i.e., the  sequence $\{\upsilon_n(t)\}^\infty_0$
is nondecreasing on $[t_0,\infty)$.

\begin{theorem} \label{thm2.3}
Let \eqref{e2.5} hold. Then   \eqref{e1.1} is non-oscillatory if and only if 
there exists $t_1\geq t_0$ such that
$\{\upsilon_n(t)\}_{0}^{\infty}$ exists and converges; i.e.,
\begin{equation} \label{e2.17}
\lim_{n\to\infty} \upsilon_{n}(t)=\upsilon(t)<\infty,\quad t\geq t_1.
\end{equation}
\end{theorem}

\begin{proof} The ``only if" part. 
Suppose that $x(t)$ is a non-oscillatory solution of  \eqref{e1.1}.
Without loss of generality, we  assume that $x(t)>0$ and $x(h(t))>0$ 
on $[t_1,\infty)$. Let $w(t)$ be defined by \eqref{e2.2},
by Lemma \ref{lem2.1}, we obtain \eqref{e2.9}, which follows
\begin{equation*}
w(t)\geq\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}\big(\frac{w(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds
\geq \int^{\infty}_{t}p(s)ds=\upsilon_0(t)\geq 0.
\end{equation*}
By \eqref{e2.9} again, we have
\begin{equation*}
w(t)\geq \alpha\int^{\infty}_{t}
\frac{(\upsilon_0(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds
+\int^{\infty}_tp(s)\exp\Big(\alpha\int^{h(s)}_s\big(\frac{\upsilon_0(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds
=\upsilon_1(t).
\end{equation*}
By induction, we obtain
\begin{equation} \label{e2.18}
w(t)\geq \upsilon_n(t)\geq 0,\,\, n=0,1,2\dots,\quad t\geq t_1.
\end{equation}
It follows from \eqref{e2.16} and \eqref{e2.18} that \eqref{e2.17} holds.

The ``if" part. 
Assume that the function sequence $\{\upsilon_n(t)\}_0^{\infty}$ 
exists and converges.
It follows from \eqref{e2.16} and \eqref{e2.17} that
\begin{equation*}
0\leq \upsilon_n(t)\leq \upsilon(t),\quad n=1,2,\dots,\,\,\, t\geq t_1.
\end{equation*}
By Levi's monotone convergence theorem for \eqref{e2.15}, we obtain
\begin{equation*}
\upsilon(t)=\alpha\int^{\infty}_{t}
\frac{(\upsilon(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds
+\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s}
\big(\frac{\upsilon(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds.
\end{equation*}
Consequently,
\begin{equation*}
\upsilon'(t)+\alpha\frac{(\upsilon(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)}
+p(t)\exp\Big(\alpha\int^{h(t)}_{t}\big(\frac{\upsilon(s)}{r(s)}\big)^{1/\alpha}ds\Big)=0,\,\,\, t\geq t_1.
\end{equation*}
Then, by Theorem \ref{thm2.2},  \eqref{e1.1} is non-oscillatory.
\end{proof}

As a consequence of Theorem \ref{thm2.3}, we have the following result.

\begin{theorem} \label{thm2.4}
Let \eqref{e2.5} hold. Then  \eqref{e1.1} is oscillatory  if one of the 
following conditions holds:
\begin{itemize}
\item[(1)] There exists an integer $m$ such that 
$\upsilon_n(t)$ is defined for
$n=1,2,\dots,m-1$, but $\upsilon_m(t)$ does not exist;

\item[(2)] $\{\upsilon_n(t)\}^\infty_0$ is defined for $n=1,2,\dots $,
but for arbitrarily large $T\geq t_0$, there exists $t^*>T$ such that
$\lim_{n\to\infty}\upsilon_n(t^*)=\infty$.
\end{itemize}
\end{theorem}

\begin{corollary}\label{coro2.2}
Let \eqref{e2.5} hold. Assume that there exists  
$R(t)\in C^1([t_0, \infty),\mathbb{R}^+\big)$ with
$R'(t)=r^{-1/\alpha}(t)$, and there exists 
$\lambda_0 >\alpha^\alpha/(\alpha+1)^{\alpha+1}$
such that for all sufficiently large $t$,
\begin{equation} \label{e2.19}
R^\alpha(t)P(t) \geq \lambda_0.
\end{equation}
Then  \eqref{e1.1} is oscillatory .
\end{corollary}

\begin{proof}
It follows from \eqref{e2.19} that $\upsilon_0(t)\geq\lambda_0R^{-\alpha}(t)$, 
which implies, by \eqref{e2.15},
\begin{equation*}
\upsilon_1(t)\geq \upsilon_0(t)+\alpha \lambda_0^{(\alpha+1)/\alpha}
\int^{\infty}_t\frac{d R(s)}{R^{\alpha+1}(s)}
\geq\frac{\lambda_1}{R^\alpha(t)},\quad
\lambda_1=\lambda_0+\lambda_0^{(\alpha+1)/\alpha}>\lambda_0.
\end{equation*}
By induction, we can show that
\begin{equation*}
\upsilon_{n+1}(t)\geq \frac{\lambda_{n+1}}{R^\alpha(t)}\quad \text{and}\quad
\lambda_{n+1}=\lambda_0+\lambda_n^{(\alpha+1)/\alpha}>\lambda_n, \quad
\text{for }n=1,2, \dots.
\end{equation*}
Now we claim that $\lim_{n\to\infty}\lambda_n=\infty$. 
Otherwise, as $\lambda_n$ is monotone increasing,
we must have $\lim_{n\to\infty}\lambda_n=\lambda<\infty$, and $\lambda>0$
satisfies the equation $\lambda=\lambda_0+\lambda^{(\alpha+1)/\alpha}$. 
Note that $\lambda_0 >\alpha^\alpha/(\alpha+1)^{\alpha+1}$, 
then, by H\"{o}lder inequality, we have
\begin{align*}
\lambda=\lambda_0+\lambda^{(\alpha+1)/\alpha}>&\frac{\alpha+1}{\alpha}
\Big[\frac{1}{\alpha+1}\big(\frac{\alpha}{\alpha+1}\big)^{\alpha+1}
+\frac{\alpha}{\alpha+1}\lambda^{(\alpha+1)/\alpha}\Big]\\
\geq& \frac{\alpha+1}{\alpha}\frac{\alpha}{\alpha+1}\lambda=\lambda,
\end{align*}
which is impossible. Hence, the claim is true. Consequently,
$\lim_{n\to\infty}\upsilon_n(t)=\infty$. 
 Thus, by Theorem \ref{thm2.4} (2), Equaton \eqref{e1.1} is oscillatory.
\end{proof}

\begin{corollary}\label{coro2.3}
Let \eqref{e2.5} hold. Assume that there exists $\gamma_0>(\alpha+1)^{-(\alpha+1)/\alpha}$ such that
for all sufficiently large $t$,
\begin{equation} \label{e2.20}
\int^{\infty}_t\frac{ P^{(\alpha+1)/\alpha}(s)}{r^{1/\alpha}(s)}ds\geq\gamma_0P(t).
\end{equation}
Then  \eqref{e1.1} is oscillatory.
\end{corollary}


\begin{proof} It follows from \eqref{e2.15} and \eqref{e2.20} that
\begin{equation*}
\upsilon_1(t)\geq \gamma_1 P(t), \quad \gamma_1=1+\alpha\gamma_0>1.
\end{equation*}
Assume that $\upsilon_n(t)\geq \gamma_n P(t)$, then, by \eqref{e2.15} 
again and induction, we have
\begin{equation*}
\upsilon_{n+1}(t)\geq \gamma_{n+1} P(t), \quad
\gamma_{n+1}=1+\alpha\gamma_0\gamma^{(\alpha+1)/\alpha}_n,\quad n=1,2,\dots.
\end{equation*}
We now claim that
\begin{equation} \label{e2.21}
\gamma_{n+1}>\gamma_n, \quad n=1,2, \dots.
\end{equation}
Indeed, in view of the fact that $\gamma_1>1$ and $(\alpha+1)/\alpha>1$, we have
\begin{equation*}
r_2=1+\alpha\gamma_0\gamma^{(\alpha+1)/\alpha}_1>1+\alpha\gamma_0=\gamma_1.
\end{equation*}
Moreover, we have
\begin{equation*}
r_3=1+\alpha\gamma_0\gamma^{(\alpha+1)/\alpha}_2
>1+\alpha\gamma_0\gamma_1^{(\alpha+1)/\alpha}=\gamma_2.
\end{equation*}
Hence, by induction, we can show that \eqref{e2.21} holds. Then, by
an argument similar to the proof of Corollary \ref{coro2.2}, we can
prove $\lim_{n\to\infty}\lambda_n=\infty$;
consequently $\lim_{n\to\infty}\upsilon_n(t)=\infty$.
It follows from Theorem \ref{thm2.4} (2) that  \eqref{e1.1} is oscillatory.
 \end{proof}

\begin{theorem} \label{thm2.5}
Let \eqref{e2.5} hold.  If  \eqref{e1.1} has a nonoscillatory solution, then
\begin{equation} \label{e2.22}
\lim_{t\to\infty}\upsilon(t)\exp\Big(\alpha\int^t_{t_1}
\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big)<\infty,
\end{equation}
where $\upsilon(t)$ satisfies \eqref{e2.17}.
\end{theorem}

\begin{proof}
Suppose $x(t)\neq 0$ is a nonoscillatory solution of  \eqref{e1.1} 
for $t\geq t_1$.
Let $w(t)$ be defined by \eqref{e2.2}, it follows from \eqref{e2.4}
 and \eqref{e2.9} that
\begin{align*}
-w'(t)&=\alpha \frac{(w(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)}
+p(t)\exp\Big(\alpha\int^{h(t)}_{t}\big(\frac{w(s)}{r(s)}\big)^{1/\alpha}ds\Big)
\\
&\geq \alpha \frac{(w(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)}
=\alpha w(t)\big(\frac{w(t)}{r(t)}\big)^{1/\alpha} \\
&\geq \alpha w(t)\big(\frac{P(t)}{r(t)}\big)^{1/\alpha},
\end{align*}
hence,
\begin{equation} \label{e2.23}
w(t)\leq w(t_1)\exp\Big(-\alpha\int^{t}_{t_1}
\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big).
\end{equation}
On the other hand, by induction, we have $w(t)\geq\upsilon_n(t)$, 
$n=0,1,2,\dots$.
Combining this with \eqref{e2.23}, we obtain
\begin{equation} \label{e2.24}
\upsilon_n(t)\exp\Big(\alpha\int^t_{t_1}\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big)
\leq w(t_1),\quad n=1,2,\dots.
\end{equation}
Note that from Theorem \ref{thm2.3}, it follows that 
$\lim_{n\to\infty}\upsilon_n(t)=\upsilon(t)$,
then by \eqref{e2.24}, we have
\begin{equation*}
\lim_{n\to\infty}\upsilon_n(t)\exp\Big(\alpha\int^{t}_{t_1}
\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big)
=\upsilon(t)\exp\Big(\alpha\int^{t}_{t_1}
\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big)\leq w(t_1),
\end{equation*}
and then we obtain the desired inequality \eqref{e2.22}.
\end{proof}

As a direct consequence of Theorem \ref{thm2.5}, we obtain the following theorem.

\begin{theorem} \label{thm2.6}
Let \eqref{e2.5} hold, and $\upsilon_n(t)$ be defined for $n=1,2,\dots,m$. 
If one of the following conditions holds:
\begin{itemize}
\item[(1)] $\lim_{t\to\infty}\upsilon_m(t)\exp\Big(\alpha\int^{t}_{t_0}
\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big)=\infty$,

\item[(2)] Condition \eqref{e2.17} holds, and
 $\lim_{t\to\infty}\upsilon(t)\exp\Big(\alpha\int^{t}_{t_0}
\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big)=\infty$,
\end{itemize}
then  \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{theorem} \label{thm2.7}
Let \eqref{e2.5}  hold and
\begin{equation} \label{e2.25}
\lim_{t\to\infty}\int^t_{t_0}\exp\Big(-\alpha\int^s_{t_0}
\big(\frac{P(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds<\infty\,.
\end{equation}
If there exists $m\geq 1$ such that
\begin{equation} \label{e2.26}
\lim_{t\to\infty}\int^t_{t_0}\upsilon_m(s)ds=\infty,
\end{equation}
then  \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Assume that $x(t)\neq 0$ is a non-oscillatory solution of  \eqref{e1.1} 
for $t\geq t_1$.
Let $w(t)$ be defined by \eqref{e2.2},  similar to the proof of 
Theorem \ref{thm2.5},  we have
\begin{equation} \label{e2.27}
\upsilon_m(t)\leq w(t_1)\exp\Big(-\alpha\int^{t}_{t_1}
\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big),\quad  m\geq 1.
\end{equation}
Integrating \eqref{e2.27} from $t_1$ to $t$, and then letting $t\to\infty$
makes \eqref{e2.25} contradict \eqref{e2.26}.
Hence,  \eqref{e1.1} is oscillatory.
\end{proof}

\section{Examples}
In this section, we will give some examples to illustrate our main results.


\begin{example} \label{examp3.1} \rm
 Consider the equation
\begin{equation} \label{e3.1}
\Big(\frac{1}{t}|x'(t)|^{-1/2}x'(t)\Big)'
+\frac{3\lambda}{2t^{5/2}}|x(3t)|^{-1/2}x(3t)=0,\quad t\geq t_0,
\end{equation}
where
\begin{equation*}
\alpha=\frac{1}{2}, \quad r(t)=\frac{1}{t}, \quad 
h(t)=3t,\quad \lambda> 0.
\end{equation*}
Then $R^{1/2}(t)P(t)=\lambda\sqrt3/3$.
By Corollary \ref{coro2.2}, if there exists 
$\lambda_0 > 2\sqrt{3}/3$ such
that $\lambda\geq \sqrt{3}\lambda_0$, i.e., $\lambda> 2$, 
then  \eqref{e3.1} is oscillatory.
\end{example}

\begin{example} \label{examp3.2}\rm Consider the equation
\begin{equation} \label{e3.2}
\big(t|x'(t)|x'(t)\big)'+\frac{k}{t^{2}}|x(2t)|x(2t)=0,\quad t\geq t_0,
\end{equation}
where
$\alpha=2$, $r(t)=t$, $h(t)=2t$,  $k> 0$.
Then $P(t)=k/t$. 
If  $k>1/27$, then \eqref{e3.2} is oscillatory.
Indeed, note that there exists $\gamma_0\in (\frac{\sqrt3}{9},\sqrt{k})$, then
\begin{equation*}
\int^{\infty}_t\frac{ P^{1+1/\alpha}(s)}{r^{1/\alpha}(s)}ds
=\frac{k^{3/2}}{t}=\frac{k}{t}\sqrt{k}\geq
\gamma_0\frac{k}{t}> \frac{P(t)}{(\alpha+1)^{(\alpha+1)/\alpha}},
\end{equation*}
for all sufficiently large $t$. Hence, by Corollary \ref{coro2.3}, 
the conclusion holds.
\end{example}


\begin{example} \label{examp3.3}\rm
 Consider the equation
\begin{equation} \label{e3.3}
\Big(\frac{1}{\sqrt{t}}|x'(t)|^{1/2}x'(t)\Big)'
+\frac{k}{t^{5/2}}\Big(\frac{3}{2}+\frac{3}{2\ln{t}}
+\frac{1}{\ln^2t}\Big)|x(2t)|^{1/2}x(2t)=0,\quad t\geq 1,
\end{equation}
where
\begin{equation*}
k>0, \quad \alpha=\frac{3}{2},\quad r(t)=\frac{1}{\sqrt{t}},\quad h(t)=2t,
\quad p(t)=\frac{k}{t^{5/2}}(\frac{3}{2}+\frac{3}{2\ln{t}}+\frac{1}{\ln^2{t}}).
\end{equation*}
 Note that
\begin{align*}
\upsilon_0(t)=P(t)=\frac{k}{t^{3/2}}(1+\frac{1}{\ln{t}}), \quad
\upsilon_1(t)>\frac{9k^{5/3}}{7t^{7/6}}+\frac{k}{t^{3/2}}(1+\frac{1}{\ln{t}}).
\end{align*}
Then
\begin{align*}
&\lim_{t\to\infty}\upsilon_1(t)\exp\Big(\alpha\int^t_1
\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big)\\
&\geq\lim_{t\to\infty}\Big(\frac{9k^{5/3}}{7t^{7/6}}
+\frac{k}{t^{3/2}}\big(1+\frac{1}{\ln{t}}\big)\Big)\exp\Big(\frac{3}{2}
\int^{t}_{1}\big(\frac{k(1+\frac{1}{\ln{s}})}{s}\big)^{2/3}ds\Big)\\
&\geq\lim_{t\to\infty}\Big(\frac{9k^{5/3}}{7t^{7/6}}
+\frac{k}{t^{3/2}}\Big)\exp\Big(\frac{3}{2}\int^{t}_{1}
\big(\frac{k}{s}\big)^{2/3}ds\Big)\\
&\geq\lim_{t\to\infty}\frac{k_1}{t^{3/2}}e^{k_2t^{1/3}}=\infty,
\end{align*}
where $k_1=k e^{-9/2k^{2/3}}$ and $k_2=9k^{2/3}/2$.
Thus, Theorem \ref{thm2.6} (1) is satisfied for $m=1$. Hence  \eqref{e3.3} 
is oscillatory.
\end{example}

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\end{document}