\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 $$\label{e1.1} \Big(r(t)|x'(t)|^{\alpha-1}x'(t)\Big)'+p(t)|x(h(t)\big)|^{\alpha-1}x(h(t))=0, \quad t\geq t_0,$$ 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 $$\label{e1.2} \big(r(t)x'(t)\big)'+p(t)x(t)=0,$$ or of the half-linear differential equation $$\label{e1.3} \Big(r(t)|x'(t)|^{\alpha-1}x'(t)\Big)'+p(t)|x(t)|^{\alpha-1}x(t)=0$$ and of the advanced differential equation $$\label{e1.4} \big(r(t)x'(t)\big)'+p(t)x(h(t))=0,\quad t\leq h(t),$$ 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 $$\label{e2.1} \int^{\infty}_{t_0}p(s)ds=\infty,$$ 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 $$\label{e2.2} w(t)=\frac{r(t)|x'(t)|^{\alpha-1}x'(t)}{|x(t)|^{\alpha-1}x(t)}.$$ Obviously, $w(t)>0$, and $r(t)(x'(t))^\alpha=w(t)(x(t))^\alpha$; i.e., $$\label{e2.3} \frac{x(h(t))}{x(t)}=\exp\Big(\int^{h(t)}_{t}\big(\frac{w(s)}{r(s)}\big) ^{1/\alpha}ds\Big).$$ Then, from \eqref{e1.1} and \eqref{e2.3}, we obtain $$\label{e2.4} w'(t)+\alpha\frac{(w(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)} +p(t)\exp\Big(\alpha\int^{h(t)}_{t}\big(\frac{w(s)}{r(s)} \big)^{1/\alpha}ds\Big)=0,$$ 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: $$\label{e2.5} \int^{\infty}_tp(s)ds<\infty.$$ 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 $$\label{e2.10} w(T)-w(t)+\alpha\int^{T}_{t}\frac{(w(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds +\int^{T}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s} \big(\frac{w(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds=0.$$ 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 $$\label{e2.11} \varphi'(t)+\alpha\frac{(\varphi(t))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(t)}+p(t) \exp\Big(\alpha\int^{h(t)}_{t} \big(\frac{\varphi(s)}{r(s)}\big)^{1/\alpha}ds\Big)\leq 0,\quad t\geq t_1.$$ \end{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 $$\label{e2.12} (Ly)(t)=\alpha\int^{\infty}_{t}\frac{(y(s))^{(\alpha+1)/\alpha}} {r^{1/\alpha}(s)}ds+ \int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s} \big(\frac{y(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds,$$ 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 $$\label{e2.13} u(t)=\alpha\int^{\infty}_{t}\frac{(u(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds+ \int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s} \big(\frac{u(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds,\quad t\geq t_1.$$ 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 $$\label{e2.14} u(t)=\frac{r(t)(x'(t))^{\alpha}}{(x(t))^{\alpha}}\,.$$ 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): $$\label{e2.15} \begin{gathered} \upsilon_0(t)=P(t)=\int^{\infty}_{t}p(s)ds, \\ \upsilon_{n+1}(t)=\alpha\int^{\infty}_{t} \frac{(\upsilon_n(s))^{(\alpha+1)/\alpha}}{r^{1/\alpha}(s)}ds +\int^{\infty}_{t}p(s)\exp\Big(\alpha\int^{h(s)}_{s} \big(\frac{\upsilon_n(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds, \end{gathered}$$ 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 $$\label{e2.16} \upsilon_{n+1}(t)\geq \upsilon_n(t),\quad n=0,1,2\dots;$$ 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., $$\label{e2.17} \lim_{n\to\infty} \upsilon_{n}(t)=\upsilon(t)<\infty,\quad t\geq t_1.$$ \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 $$\label{e2.18} w(t)\geq \upsilon_n(t)\geq 0,\,\, n=0,1,2\dots,\quad t\geq t_1.$$ 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 $t$, $$\label{e2.19} R^\alpha(t)P(t) \geq \lambda_0.$$ 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 $t$, $$\label{e2.20} \int^{\infty}_t\frac{ P^{(\alpha+1)/\alpha}(s)}{r^{1/\alpha}(s)}ds\geq\gamma_0P(t).$$ 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 $$\label{e2.21} \gamma_{n+1}>\gamma_n, \quad n=1,2, \dots.$$ 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 $$\label{e2.22} \lim_{t\to\infty}\upsilon(t)\exp\Big(\alpha\int^t_{t_1} \big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big)<\infty,$$ 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, $$\label{e2.23} w(t)\leq w(t_1)\exp\Big(-\alpha\int^{t}_{t_1} \big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big).$$ 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 $$\label{e2.24} \upsilon_n(t)\exp\Big(\alpha\int^t_{t_1}\big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big) \leq w(t_1),\quad n=1,2,\dots.$$ 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 $$\label{e2.25} \lim_{t\to\infty}\int^t_{t_0}\exp\Big(-\alpha\int^s_{t_0} \big(\frac{P(\tau)}{r(\tau)}\big)^{1/\alpha}d\tau\Big)ds<\infty\,.$$ If there exists $m\geq 1$ such that $$\label{e2.26} \lim_{t\to\infty}\int^t_{t_0}\upsilon_m(s)ds=\infty,$$ 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 $$\label{e2.27} \upsilon_m(t)\leq w(t_1)\exp\Big(-\alpha\int^{t}_{t_1} \big(\frac{P(s)}{r(s)}\big)^{1/\alpha}ds\Big),\quad m\geq 1.$$ 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 $$\label{e3.1} \Big(\frac{1}{t}|x'(t)|^{-1/2}x'(t)\Big)' +\frac{3\lambda}{2t^{5/2}}|x(3t)|^{-1/2}x(3t)=0,\quad t\geq t_0,$$ 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 $$\label{e3.2} \big(t|x'(t)|x'(t)\big)'+\frac{k}{t^{2}}|x(2t)|x(2t)=0,\quad t\geq t_0,$$ 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 $$\label{e3.3} \Big(\frac{1}{\sqrt{t}}|x'(t)|^{1/2}x'(t)\Big)' +\frac{k}{t^{5/2}}\Big(\frac{3}{2}+\frac{3}{2\ln{t}} +\frac{1}{\ln^2t}\Big)|x(2t)|^{1/2}x(2t)=0,\quad t\geq 1,$$ 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 $k_2=9k^{2/3}/2$. 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