\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 244, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/244\hfil Oscillation for third-order differential equations] {Oscillation and Property B for third-order differential equations with advanced arguments} \author[B. Bacul\'ikov\'a, J. D\v{z}urina \hfil EJDE-2016/244\hfilneg] {Blanka Bacul\'ikov\'a, Jozef D\v{z}urina} \address{Blanka Bacul\'ikov\'a \newline Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Ko\v{s}ice, Letn\'a 9, 04200 Ko\v{s}ice, Slovakia} \email{blanka.baculikova@tuke.sk} \address{Jozef D\v{z}urina \newline Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Ko\v{s}ice, Letn\'a 9, 04200 Ko\v{s}ice, Slovakia} \email{jozef.dzurina@tuke.sk} \thanks{Submitted March 15, 2016. Published September 7, 2016.} \subjclass[2010]{34C10, 34K11} \keywords{Third-order functional differential equation; Property B; \hfill\break\indent advanced argument} \begin{abstract} We establish sufficient conditions for the third-order nonlinear advanced differential equation \begin{equation*} \Big(a(t)[\big(b(t)y'(t)\big)']^{\gamma}\Big)'-p(t)f(y(\sigma(t)))=0 \end{equation*} to have property B or to be oscillatory. These conditions are based on monotonic properties and estimates of non-oscillatory solutions, and essentially improve known results for differential equations with deviating arguments and for ordinary differential equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} We consider the nonlinear third-order differential equation with advanced argument \begin{equation}\label{eE} \Big(a(t)[\big(b(t)y'(t)\big)']^{\gamma}\Big)'-p(t)f(y(\sigma(t)))=0. \end{equation} In the sequel we will assume: \begin{itemize} \item[(H0)] $\gamma$ is quotient of odd positive integers. \item[(H1)] $a(t), b(t), p(t)\in C([t_0,\infty))$, $ \sigma(t) \in C^1([t_0,\infty))$, $a(t), b(t), p(t)$ are positive, $\sigma'(t)>0$, $\sigma(t)\geq t$. \item[(H2)] $f(u)\in C(\mathbb R)$, $uf(u)>0$ for $u\neq 0$, $f(uv)\geq f(u)f(v)$ for $uv>0$, $f$ is nondecreasing. \item[(H3)] $\int_{t_0}^{\infty}\frac{1}{a^{1/{\gamma}}(t)} \mathrm{d} t=\infty$, $\int_{t_0}^{\infty}\frac{1}{b(t)}\mathrm{d} t=\infty$. \end{itemize} By a solution of \eqref{eE}, we mean a function $y(t) \in C^{1}([T_{y},\infty ))$, $T_{y}\geq t_{0}$, that satisfies \eqref{eE} on $[T_{y},\infty)$. We consider only those solutions $y(t)$ of \eqref{eE} that satisfy $\sup \{|y(t)|:t \geq T\}>0$ for all $T\geq T_{y}$. We assume that \eqref{eE} possesses such a solution. A solution of \eqref{eE} is called oscillatory if it has arbitrarily large zeros on $[T_{y},\infty)$ and otherwise, it is called to be nonoscillatory. An equation is said to be oscillatory if all its solutions are oscillatory. The study of oscillatory properties of third and higher order linear ordinary differential equations began as far back in the pioneering work of Kneser \cite{Kn}. A new impetus to investigations in this direction was given by the works of Chanturia and Kiguradze \cite{KCH}. Their results concern property B for the linear differential equation \begin{equation}\label{Ko} y'''(t)-q(t)y(\sigma(t))=0 \end{equation} with $\sigma(t)\equiv t$. By property B of \eqref{Ko} it is meant the situation when every positive solution $y(t)$ of \eqref{Ko} is strongly increasing, i.e. $$ y'(t)>0,\quad y''(t)>0,\quad y'''(t)>0. $$ Over the previous few decades, oscillation theory and asymptotic behavior of differential equations related to \eqref{eE}, have drawn extensive attention and the significant body of relevant literature has been devoted to this topic (see \cite{agar}--\cite{Ti}) Especially, in the earlier article \cite{Kop} Koplatadze et al presented excellent criteria for the qualitative properties of solutions of binomial differential equation with deviating argument. In this article, we extend their technique that yields property B of \eqref{Ko} to \eqref{eE}. Here, we derive new monotonic properties of nonoscillatory solutions of \eqref{eE} that permit us to achieve new sufficient conditions for \eqref{eE} to have property B or to be oscillatory. Our results essentially improve many known results not only for differential equations with deviating arguments but for ordinary differential equations as well. As in oscillation theory, all functional inequalities considered are assumed to hold eventually; that is, they are satisfied for all $t$ large enough. \section{Preliminaries} We begin with the structure of possible nonoscillatory solutions of \eqref{eE} which follows from an analogy of Kiguradze \cite{KCH} lemma and canonical form of studied equation. We introduce the following classes of nonoscillatory (let us say positive) solutions of \eqref{eE}: $$ y(t)\in \mathscr{N}_1 \iff y'(t)>0,\;(b(t)y'(t))'<0,\; \Big(a(t)[(b(t)y'(t))']^{\gamma}\Big)'>0 $$ and $$ y(t)\in \mathscr{N}_3 \iff y'(t)>0,\,(b(t)y'(t))'>0,\; (a(t)[(b(t)y'(t))']^{\gamma})'>0, $$ eventually. \begin{lemma}\label{lem1} Assume that $y(t)$ is an eventually positive solution of \eqref{eE}, then $y(t)\in \mathscr{N}_1$ or $y(t)\in \mathscr{N}_3$. \end{lemma} Now, we derive some important monotonic properties and estimates of nonoscillatory solutions, that will be applied in our main results. To simplify our notation, let us denote \begin{gather*} A(t)=\int_{t_{*}}^t\frac{1}{a^{1/{\gamma}}(s)}\,\mathrm{d} s,\quad B(t)=\int_{t_{*}}^t\frac{1}{b(s)}\,\mathrm{d} s,\\ C(t)=\int_{t_{*}}^t\frac{1}{b(u)}\int_{t_{*}}^{u}\frac{1}{a^{1/{\gamma}}(s)} \,\mathrm{d} s\mathrm{d} u, \quad P(t)=\frac{1}{a^{1/{\gamma}}(t)}[\int_{t}^{\infty} p(s)\,\mathrm{d}{s}]^{1/{\gamma}}. \end{gather*} for $t_{*}$ is large enough. \begin{lemma}\label{est1} Let $y(t)\in\mathscr{N}_3$ be a positive solution of \eqref{eE} and \begin{equation}\label{con1} \int_{t_{*}}^{\infty}p(s)f(C(\sigma(s)))\mathrm{d} s=\infty. \end{equation} Then $y(t)/C(t)$ is eventually increasing. \end{lemma} \begin{proof} Assume, that $y(t)$ is a positive solution of \eqref{eE} satisfying $y(t)\in\mathscr{N}_3$ eventually, let us say for $t\geq t_{*}$. We claim that \eqref{con1} implies \begin{equation}\label{lim} \lim_{t\to\infty}a^{1/\gamma}(t)(b(t)y'(t))'=\infty. \end{equation} If not, then $$ \lim_{t\to\infty}a^{1/\gamma}(t)(b(t)y'(t))'=2\ell>0 $$ and since $a^{1/\gamma}(t)(b(t)y'(t))'$ is increasing, we have $$ a^{1/\gamma}(t)(b(t)y'(t))'>\ell, $$ eventually. An integration of the last inequality leads to $$ b(t)y'(t)\geq \ell A(t), $$ which implies $y(t)\geq \ell C(t) $ or \begin{equation}\label{2el} f(y(\sigma(t)))\geq f(\ell)f(C(\sigma(t))). \end{equation} On the other hand, integrating \eqref{eE} from $t_*$ to $\infty$, one gets $$ (2\ell)^\gamma\geq\int_{t_*}^{\infty} p(s)f(y(\sigma(s)))\,\mathrm{d}{s}, $$ which in view of \eqref{2el} yields $$ (2\ell)^\gamma\geq f(\ell)\int_{t_*}^{\infty} p(s)f(C(\sigma(s)))\,\mathrm{d}{s}. $$ This contradicts \eqref{con1} and we conclude that \eqref{lim} holds. Now, using that $a^{1/\gamma}(t)(b(t)y'(t))'$ is increasing, we see that for all $t\geq t_1>t_*$, \begin{align*} b(t)y'(t) &=b(t_1) y'(t_1)+\int_{t_{1}}^t a^{1/\gamma}(s) \frac{(b(s)y'(s))'}{a^{1/\gamma}(s)}\,\mathrm{d} s\\ & \leq b(t_1) y'(t_1)+a^{1/\gamma}(t)(b(t)y'(t))'\int_{t_{1}}^t \frac{1}{a^{1/\gamma}(s)}\,\mathrm{d} s\\ &= b(t_1) y'(t_1)-a^{1/\gamma}(t)(b(t)y'(t))'\int_{t_{*}}^{t_1} \frac{1}{a^{1/\gamma}(s)}\,\mathrm{d} s\\ &\quad+a^{1/\gamma}(t)(b(t)y'(t))'\int_{t_{*}}^t \frac{1}{a^{1/\gamma}(s)}\,\mathrm{d} s. \end{align*} By \eqref{lim}, this implies $$ b(t)y'(t)\leq a^{1/\gamma}(t)(b(t)y'(t))' \int_{t_{*}}^t \frac{1}{a^{1/\gamma}(s)}\,\mathrm{d} s $$ for all $t$ large enough, let us say $t\geq t_2>t_1$, and therefore $$ \Big(\frac{b(t)y'(t)}{A(t)}\Big)' = \frac{(b(t)y'(t))'A(t)-b(t)y'(t)\frac{1}{a^{1/{\gamma}}(t)}}{A^2(t)}\geq0. $$ Thus, $\frac{b(t)y'(t)}{A(t)}$ is increasing for $t\geq t_2>t_{*}$. Then this fact yields \begin{equation}\label{2.3} \begin{split} y(t)&= y(t_2)+ \int_{t_{2}}^t\frac{A(u)b(u)y'(u)}{b(u)A(u)}\mathrm{d} u \\ &\leq y(t_2)+ \frac{b(t)y'(t)}{A(t)}\int_{t_{2}}^t\frac{A(u)}{b(u)}\,\mathrm{d} u\\ &=y(t_2)-\frac{b(t)y'(t)}{A(t)}\int_{t_{*}}^{t_2}\frac{A(u)}{b(u)} \,\mathrm{d} u+\frac{b(t)y'(t)}{A(t)}\int_{t_{*}}^t\frac{A(u)}{b(u)}\,\mathrm{d} u. \end{split} \end{equation} On the other hand, by L'Hospital rule $$ \lim_{t\to\infty}\frac{b(t)y'(t)}{A(t)} = \lim_{t\to\infty}a^{1/\gamma}(t)(b(t)y'(t))'=\infty $$ and so in view of \eqref{2.3}, there exists $t_3>t_2$ such that $$ y(t)\leq \frac{b(t)y'(t)}{A(t)}\int_{t_{*}}^t\frac{A(u)}{b(u)}\,\mathrm{d} u, \quad t\geq t_3. $$ Consequently, $$ \Big(\frac{y(t)}{C(t)}\Big)' = \frac{y'(t)C(t)-y(t)A(t)\frac{1}{b(t)}}{C^2(t)}\geq0, $$ which implies that $y(t)/C(t)$ is eventually increasing. The proof is complete. \end{proof} \begin{lemma}\label{est2} Let $y(t)\in\mathscr{N}_1$ be a positive solution of \eqref{eE}. Then $y(t)/B(t)$ is eventually decreasing. \end{lemma} \begin{proof} Assume, that $y(t)$ is an eventually positive solution of \eqref{eE} satisfying $y(t)\in\mathscr{N}_1$ for $t\geq t_{*}$. Then $b(t)y'(t)$ is decreasing and we see that $$ y(t)\geq \int_{t_{*}}^t b(s)y'(s)\frac{1}{b(s)}\,\mathrm{d} s\\ \geq b(t)y'(t)\int_{t_{*}}^t\frac{1}{b(s)}\,\mathrm{d} s. $$ This implies $$ \Big(\frac{y(t)}{B(t)}\Big)'= \frac{y'(t)B(t)-y'(t)\frac{1}{b(t)}}{A^2(t)}\leq 0, \quad t\geq t_{*}. $$ Thus, $y(t)/B(t)$ is eventually decreasing and the proof is complete. \end{proof} \begin{remark} \label{rmk2} \rm For $a(t)=b(t)\equiv1$ and $\gamma=1$ Lemmas \ref{est1} and \ref{est2} reduce to the results by Koplatadze et al. So we extended their result from linear differential equations to nonlinear equations with the extra factor $b$. \end{remark} \section{Criteria for property B} Now, we provide several criteria for the class $\mathscr{N}_1$ of \eqref{eE} to be empty. In the literature such case is referred to as \emph{property B} of \eqref{eE}. \begin{theorem} \label{cr1} Assume that \begin{equation}\label{bbb1} \int_{t_*}^{\infty}\frac{1}{b(v)} \int_{v}^{\infty} \frac{1}{a^{1/\gamma}(u)} \Big[\int_{u}^{\infty} p(s)\,\mathrm{d}{s}\mathrm{d}{u} \Big]^{1/\gamma}\,\mathrm{d}{v}=\infty, \end{equation} and \begin{equation}\label{cond} \lim_{u\to\pm\infty} \frac{u}{f^{1/\gamma}(u)}=K_1<\infty. \end{equation} If \begin{align*} %\label{blanka} &\limsup_{t\to\infty} \Big\{ f^{1/\gamma}(\frac{1}{B(\sigma(t))}) \int_{t_*}^{t} f^{1/\gamma}(B(\sigma(s)))B(s)P(s)\,\mathrm{d}{s} \\ &+\int_{t}^{\sigma(t)} B(s)P(s)\,\mathrm{d}{s} + B(\sigma(t))\int_{\sigma(t)}^{\infty} P(s)\,\mathrm{d}{s}\Big\}>K_1, \end{align*} then \eqref{eE} has property B. \end{theorem} \begin{proof} Assume on the contrary, that \eqref{eE} possesses an eventually positive solution $y(t)\in \mathscr{N}_1$, $t\geq t_{*}$. Integration \eqref{eE} twice from $t$ to $\infty$ yields \begin{align*} b(t)y'(t) &\geq \int_{t_*}^{\infty}\frac{1}{a^{1/\gamma}(s)} \Big[\int_{s}^{\infty} p(x)f(y(\sigma(x)))\,\mathrm{d}{x} \Big]^{1/\gamma}\,\mathrm{d}{s}\\ &\geq\int_{t_*}^{\infty}f^{1/\gamma}(y(\sigma(s))) \frac{1}{a^{1/\gamma}(s)}\Big[\int_{s}^{\infty} p(x)\,\mathrm{d}{x} \Big]^{1/\gamma}\,\mathrm{d}{s}\\ &=\int_{t_*}^{\infty}f^{1/\gamma}(y(\sigma(s)))P(s)\,\mathrm{d}{s}, \end{align*} where we have used the monotonicity of $f(y(\sigma(t)))$. Integrating the last inequality from $t_{*}$ to $t$ and then changing the order of integration, one obtains \begin{align*} y(t) &\geq \int_{t_*}^{t} \frac{1}{b(u)} \int_{u}^{\infty}f^{1/\gamma}(y(\sigma(s)))P(s)\,\mathrm{d}{s} \mathrm{d}{u}\\ &=\int_{t_*}^{t} f^{1/\gamma}(y(\sigma(s)))P(u)B(u)\,\mathrm{d}{u} +B(t) \int_{t}^{\infty} f^{1/\gamma}(y(\sigma(s)))P(s)\,\mathrm{d}{s}. \end{align*} Therefore, \begin{align*} y(\sigma(t)) & \geq \int_{t_*}^{t} f^{1/\gamma}(y(\sigma(s)))P(u)B(u)\,\mathrm{d}{u} \\ &\quad +\int_{t}^{\sigma(t)} f^{1/\gamma}(y(\sigma(s)))P(u)B(u)\,\mathrm{d}{u} +B(\sigma(t)) \int_{\sigma(t)}^{\infty} f^{1/\gamma}(y(\sigma(s)))P(s)\,\mathrm{d}{s}. \end{align*} Using that $y(t)$ is increasing and $y(t)/B(t)$ is decreasing, we have \begin{equation}\label{zara} \begin{split} y(\sigma(t)) &\geq f^{1/\gamma}\Big(\frac{y(\sigma(t))}{B(\sigma(t))}\Big) \int_{t_*}^{t} f^{1/\gamma}(B(\sigma(s)))P(u)B(u)\,\mathrm{d}{u} \\ &\quad +f^{1/\gamma}(y(\sigma(t)))\int_{t}^{\sigma(t)} P(u)B(u)\,\mathrm{d}{u} \\ &\quad +f^{1/\gamma}(y(\sigma(t)))B(\sigma(t)) \int_{\sigma(t)}^{\infty} P(s)\,\mathrm{d}{s}. \end{split} \end{equation} That is, \begin{align*} \frac{y(\sigma(t))}{f^{1/\gamma}(y(\sigma(t)))} &\geq f^{1/\gamma}(\frac{1}{B(\sigma(t))})\int_{t_*}^{t} f^{1/\gamma}(B(\sigma(s)))P(u)B(u)\,\mathrm{d}{u} \\ &\quad +\int_{t}^{\sigma(t)} P(u)B(u)\,\mathrm{d}{u} +B(\sigma(t)) \int_{\sigma(t)}^{\infty} P(s)\,\mathrm{d}{s}. \end{align*} It follows from \eqref{bbb1} that $y(t)\to\infty$ as $t\to\infty$. Taking $\limsup$ as $t\to\infty$ on both sides of the previous inequality, we are led to a contradiction with the assumptions of the theorem. The proof is complete. \end{proof} \begin{theorem}\label{cr2} Assume that \begin{gather}\label{jjj1} \int_{t_*}^{\infty}\frac{1}{a^{1/\gamma}(u)} \Big[\int_{u}^{\infty} p(s)f(B(\sigma(s)))\,\mathrm{d}{s} \Big]^{1/\gamma}\,\mathrm{d}{u} =\infty, \\ \label{jcond} \lim_{u\to0} \frac{u}{f^{1/\gamma}(u)}=K_2<\infty. \end{gather} If \begin{align*} &\limsup_{t\to\infty} \Big\{ \frac{1}{B(\sigma(t))} \int_{t_*}^{t} f^{1/\gamma}(B(\sigma(s)))B(s)P(s)\,\mathrm{d}{s} \\ &\quad +\frac{f^{1/\gamma}(B(\sigma(t)))}{B(\sigma(t))} \int_{t}^{\sigma(t)} B(s)P(s)\,\mathrm{d}{s} + f^{1/\gamma}(B(\sigma(t)))\int_{\sigma(t)}^{\infty} P(s)\,\mathrm{d}{s}\Big\} >K_2, \end{align*} then \eqref{eE} has property B. \end{theorem} \begin{proof} Assume that \eqref{eE} possesses an eventually positive solution $y(t)\in \mathscr{N}_1$, $t\geq t_{*}$. By Lemma \ref{est2}, function $y(t)/B(t)$ is decreasing and we shall prove that \eqref{jjj1} implies \begin{equation}\label{mmm} \lim_{t\to\infty} \frac{y(t)}{B(t)}=0. \end{equation} On the contrary assume that $\lim_{t\to\infty} y(t)/B(t)=\ell>0$. Then $y(t)/B(t)\geq\ell$; therefore $$ f(y(\sigma(t)))=f\Big(\frac{y(\sigma(t))}{B(\sigma(t))}B(\sigma(t)) \Big)\geq f(\ell)f(B(\sigma(t))). $$ Moreover, integrating \eqref{eE} twice yields \begin{equation}\label{fff} \begin{split} b(t_*)y'(t_*) &\geq \int_{t_*}^{\infty}\frac{1}{a^{1/\gamma}(u)} \Big[\int_{u}^{\infty} p(s)f(y(\sigma(s)))\,\mathrm{d}{s} \Big]^{1/\gamma}\,\mathrm{d}{u}\\ &\geq f(\ell)\int_{t_*}^{\infty}\frac{1}{a^{1/\gamma}(u)} \Big[\int_{u}^{\infty} p(s)f(B(\sigma(s)))\,\mathrm{d}{s} \Big]^{1/\gamma}\,\mathrm{d}{u}. \end{split} \end{equation} This contradict the assumptions of the theorem; we conclude that \eqref{mmm} holds. On the other hand, setting $$ z(t)=\frac{y(\sigma(t))}{B(\sigma(t))}, $$ condition \eqref{zara} and (H2) imply \begin{align*} \frac{z(t)}{f^{1/\gamma}(z(t))} &\geq \frac{1}{B(\sigma(t))}\int_{t_*}^{t} f^{1/\gamma}(B(\sigma(s)))P(u)B(u)\,\mathrm{d}{u} \\ &\quad +\frac{f^{1/\gamma}(B(\sigma(t)))}{B(\sigma(t))}\int_{t}^{\sigma(t)} P(u)B(u)\,\mathrm{d}{u} +f^{1/\gamma}(B(\sigma(t))) \int_{\sigma(t)}^{\infty} P(s)\,\mathrm{d}{s}. \end{align*} Taking the $\limsup$ as $t\to\infty$ on both sides of the previous inequality, we have a contradiction with the assumptions of our theorem. The proof is complete. \end{proof} Now we apply the criteria obtained to superlinear, sublinear and half-linear cases of \eqref{eE}, where $\delta $ is quotient of odd positive integers. \begin{corollary}\label{cca} Let \eqref{bbb1} hold and \begin{align*} &\limsup_{t\to\infty} \Big\{ B^{-\delta/\gamma}(\sigma(t)) \int_{t_*}^{t} B^{\delta/\gamma}(\sigma(s))B(s)P(s)\,\mathrm{d}{s} \\ &\quad +\int_{t}^{\sigma(t)} B(s)P(s)\,\mathrm{d}{s} + B(\sigma(t))\int_{\sigma(t)}^{\infty} P(s)\,\mathrm{d}{s}\Big\} >0, \end{align*} then the superlinear differential equation \begin{equation*}\label{eEd} \big[a(t)\big(b(t)(y'(t))^{\gamma}\big)'\big]'-p(t)y^{\delta}(\sigma(t))=0,\quad \delta>\gamma. \end{equation*} has property B. \end{corollary} \begin{corollary} \label{ccl} Let \eqref{bbb1} hold and \begin{align*} &\limsup_{t\to\infty} \Big\{ B^{-1}(\sigma(t))\int_{t_*}^{t} B(\sigma(s))B(s)P(s)\,\mathrm{d}{s} \\ &\quad +\int_{t}^{\sigma(t)} B(s)P(s)\,\mathrm{d}{s} + B(\sigma(t))\int_{\sigma(t)}^{\infty} P(s)\,\mathrm{d}{s}\Big\} >1, \end{align*} then the halflinear differential equation \begin{equation}\label{eEH} [a(t)(b(t)(y'(t))^{\gamma})']'-p(t)y^{\gamma}(\sigma(t))=0. \end{equation} has property B. \end{corollary} \begin{corollary}\label{ccb} Let \eqref{jjj1} hold. If \begin{align*} &\limsup_{t\to\infty} \Big\{ \frac{1}{B(\sigma(t))} \int_{t_*}^{t} B^{\delta/\gamma}(\sigma(s))B(s)P(s)\,\mathrm{d}{s} \\ &+\frac{B^{\delta/\gamma}(\sigma(t))}{B(\sigma(t))}\int_{t}^{\sigma(t)} B(s)P(s)\,\mathrm{d}{s} + B^{\delta/\gamma}(\sigma(t))\int_{\sigma(t)}^{\infty} P(s)\,\mathrm{d}{s}\Big\} >0, \end{align*} then the sublinear differential equation \begin{equation}\label{eESB} \big[a(t)\big(b(t)(y'(t))^{\gamma}\big)'\big]' -p(t)y^{\delta}(\sigma(t))=0,\quad \gamma>\delta. \end{equation} has property B. \end{corollary} Note that corollaries \ref{cca}--\ref{ccb} essentially improve the results known for \eqref{Ko}. \section{Oscillation} Our previous results concern property B of \eqref{eE}. To achieve oscillation, we need to eliminate also the class $\mathscr{N}_3$. \begin{theorem}\label{cr3} Let the assumptions of \eqref{con1} hold. Assume that \begin{equation}\label{conddd} \lim_{u\to\pm\infty} \frac{u}{f^{1/\gamma}(u)}=K_3<\infty. \end{equation} If \begin{equation*} \limsup_{t\to\infty} \frac{1}{C(\sigma(t))}\int_{t}^{\sigma(t)}\frac{1}{b(v)} \int_{v}^{t} \frac{1}{a^{1/\gamma}(u)} \Big[\int_{u}^{t} p(s)f(C(\sigma(s)))\,\mathrm{d}{s}\mathrm{d}{u} \big]^{1/\gamma}\,\mathrm{d}{v}>K_3, \end{equation*} then the class $\mathscr{N}_3=\emptyset$ for \eqref{eE}. \end{theorem} \begin{proof} Assume that \eqref{eE} possesses an eventually positive solution $y(t)\in \mathscr{N}_3$, $t\geq t_{*}$. An integration of \eqref{eE} from $s$ to $t0$ and $\lambda>1$. Simple computation shows that $$ A(t)\sim 4t^{1/4},\quad B(t)\sim \frac{3t^{2/3}}2{},\quad C(t)\sim \frac{48t^{11/12}}{11}. $$ By Corollary \ref{ccl}, condition \begin{equation}\label{ex1} \frac32(\frac{36}{11})^{3}a^3(3+\ln\lambda)>1, \end{equation} guarantees property B of \eqref{eEx}. On the other hand, by Theorem \ref{cr4}, condition \begin{equation}\label{ex2} \begin{split} &\frac{3}{2}a^3\lambda^{11/12}[(4-\frac{12}{11}) \ln^3\lambda-(3.4^2-\frac{3.12^2}{11^2})\ln^2\lambda\\ &+(3.4^3-\frac{6.12^3}{11^3})\ln\lambda - 6.4^4(1-\frac{1}{\lambda^{1/4}}) +\frac{6.12^4}{11^4}(1-\frac{1}{\lambda^{11/12}})] >1, \end{split} \end{equation} guarantees that $\mathscr{N}_3=\emptyset$ for \eqref{eEx}, By Theorem \ref{cr4}, Equation \eqref{eEx} is oscillatory if both conditions \eqref{ex1} and \eqref{ex2} are satisfied. Thus, in particular when $\lambda=2$, \begin{gather*} a>0.17269\; \Rightarrow\; \text{property B of \eqref{eEx}},\\ a>4.2262\; \Rightarrow\; \text{oscillation of \eqref{eEx}}. \end{gather*} Note that Koplatadze' criteria cannot be used nor for examination of property B nor for oscillation of \eqref{eEx}. \end{example} Although our results are oriented for advanced differential equations, Corollaries 1--3 improve Chanturia's tests [11] for property B of ordinary differential equation without deviating argument which read as follows: If $$ \limsup_{t\to\infty} t\int_t^\infty sp(s)\,\mathrm{d}{s} > 2, $$ then \eqref{Ko} has property B. Note that for the Euler equation $$ y'''(t)-\frac{p}{t^3}y(t)=0 $$ Chanturia's criterion for property B requires $p>2$, while Corollary \ref{cca} requires only $p>1$. 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