\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 38, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/38 \hfil Comparison theorems for neutral equations] {Comparison theorems for third-order neutral differential equations} \author[Z. Do\v{s}l\'a, P. Li\v{s}ka \hfil EJDE-2016/38\hfilneg] {Zuzana Do\v{s}l\'a, Petr Li\v{s}ka} \address{Zuzana Do\v{s}l\'a \newline Department of Mathematics and Statistics, Masaryk University, Kotlsk 2, Brno, 611 37, Czech Republic} \email{dosla@math.muni.cz} \address{Petr Li\v{s}ka \newline Department of Mathematics and Statistics, Masaryk University, Kotlsk 2, Brno, 611 37, Czech Republic} \email{xliska@math.muni.cz} \thanks{Submitted September 30, 2015. Published January 26, 2016.} \subjclass[2010]{34K40, 34C10} \keywords{Oscillation of solutions; neutral equation; functional equation} \begin{abstract} We establish comparison theorems for the oscillation of solutions to third-order neutral differential equations via linear ordinary and delay differential equations. Several applications illustrate the role of the deviating argument in the differential operator. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} The recent monograph \cite{PADHIBOOK} is devoted to the various aspects of differential equations of third order. In particular, Chapter 6 concerns the oscillation of delay differential equations. Motivated by these results and recent ones for delay and neutral differential equations \cite{AGR,DZUBA,DZUBA2012,DTT,TONXING}), we study the relationship between ordinary, delay and neutral differential equations. In this article we study the third-order neutral differential equation \begin{equation} \label{EP} \Big(\frac{1}{p(t)}\Big(\frac{1}{r(t)}\big[x(t)+a(t)x\big(\gamma(t)\big) \big]'\Big)'\Big)' + q(t)f\big(x\big(\delta(t)\big)\big)=0, \end{equation} where $t\geq t_0$. We make the following assumptions: \begin{itemize} \item[(i)] $p(t)$, $r(t)$, $q(t)$, $a(t)$, $\gamma(t)$, $\delta(t) \in C[t_0, \infty)$, $p(t)$, $r(t)$, $q(t)$, $\gamma(t)$, $\delta(t)$ are positive for $t\geq t_0$, \item[(ii)] $\int_{t_0}^{\infty}p(t)\,\mathrm{d}t = \int_{t_0}^{\infty}r(t)\,\mathrm{d}t=\infty$, \item[(iii)] $\gamma(t)\leq t$, $\lim_{t\to\infty}\gamma(t)=\infty$, \item[(iv)] $\lim_{t\to\infty}\delta(t)=\infty$, \item[(v)] $0\leq a(t)\leq a_0<1$ for $t\geq t_0$, \item[(vi)] $f\in C(\mathbb{R},\mathbb{R})$, $f(0)=0$ and $f(v)v>0$ for $v\neq0$. \end{itemize} It is convenient to set, for each solution $x$ of \eqref{EP}, \begin{equation}\label{OZNU} u(t)=x(t)+a(t)x\big(\gamma(t)\big)\,. \end{equation} For this function we define the functions \[ u^{[0]} = u, \quad u^{[1]} = \frac{1}{r(t)}u', \quad u^{[2]} = \frac{1}{p(t)}\Big(\frac{1}{r(t)}u'\Big)' =\frac{1}{p(t)}\big(u^{[1]}\big)' \] that are called quasiderivatives of $u$. To simplify notation, we set \[ L_3(\cdot)=\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{p(t)} \frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{r(t)}\frac{\mathrm{d}}{\mathrm{d}t}(\cdot)\,. \] Assumption (ii) implies that operator $L_3$ is in the so-called canonical form. A solution $x$ of \eqref{EP} is said to be \emph{proper} if it is defined on the interval $[t_0,\,\infty)$ and satisfies the condition \[ \sup \{|x(s)|\colon t\leq s < \infty \} > 0 \quad\text{for all } t \geq t_0. \] A proper solution is called \emph{oscillatory} or \emph{nonoscillatory} according to whether it does or does not have arbitrarily large zeros. Equation \eqref{EP} covers not only the linear ordinary differential equations (ODE when $a(t)=0$, $\delta(t)=t$) but also the functional differential equations (FDE when $a(t)=0$). It is natural to try to investigate the relationship between \eqref{EP} and the corresponding linear ODE or FDE. The oscillation theory of these equations was deeply studied by many authors; in the case of the ODEs we refer reader to \cite{CDMcomp,CDMNA,CDMV,ERBE,HANAN} and the monograph \cite{KIG}, in the case of the FDEs we refer to \cite{AGR,GAPT,MOJSEJ} and the monograph \cite{GYORI}. Recently, a considerable attention has been paid to the asymptotic theory of the neutral differential equations, see e.g. \cite{DZUBA2,DZUBA,DZUBA2012,DTT,TONXING} and the monograph \cite[Section 10.4--10.6]{GYORI}. Oscillatory properties of the third-order neutral differential equations are usually described \cite{DZUBA2,DZUBA,DTT,TONXING} in the sense corresponding to the so-called property A. Therefore, motivated by the classical definition of property A for the higher order ordinary differential equations by Kiguradze \cite{KIG} and its extension for the functional equations by Kusano and Naito \cite{KN}, we introduce the following definition of property A for equation \eqref{EP}. \begin{definition} \label{def1} \rm Equation \eqref{EP} is said to have \textit{property A} if any proper solution $x$ of \eqref{EP} is oscillatory or satisfies \[ \lim_{t\to\infty} x(t)=0. \] \end{definition} Some authors (e.g. \cite{TONXING}) use a different terminology and instead of using property A, they say that equation \eqref{EP} is \textit{almost oscillatory}. Our aim here is to give comparison theorems for \eqref{EP} via the linear ordinary or functional differential equations of the form \begin{equation}\label{klinear} L_3 y(t)+ k q(t)y\big(\delta(t)\big)=0, \end{equation} where $\delta(t)\leq t$ and $k$ is a suitable constant. These results enable us to obtain oscillation criteria for \eqref{EP} from those given for \eqref{klinear}. We refer to \cite[Sections 6.2-6.3]{PADHIBOOK}, where numerous criteria for the oscillation of \eqref{klinear} can be found. We will give a special attention to the case when the differential operator $L_3$ is symmetric, i.e. $p(t)=r(t)$, prototype of that is the linear neutral equation \begin{equation*} \Big(x(t)+a(t)x\big(\gamma(t)\Big)''' + q(t)x\big(\delta(t)\big)=0. \end{equation*} Our main tool for the comparison method is the linearization technique. Therefore in Sections 2 and 3 we recall basic properties of linear equation \eqref{klinear}. Section 3 also contains some new results for the FDEs. In Section 4 properties of nonoscillatory solutions of \eqref{EP} are given. Our main results are stated in Section 5. Section 6 presents some applications. \section{Preliminaries: Linear ODE} Consider the third-order linear differential equation \begin{equation}\label{LTO} \Big(\frac{1}{p(t)}\Big(\frac{1}{r(t)}x'(t)\Big)'\Big)' + q(t)x(t)=0. \end{equation} For completeness, we summarize basic results concerning the oscillatory behaviour of \eqref{LTO}, which we will need in our later consideration. It is well-known (see for instance \cite{KN}) that all nonoscillatory solutions $x$ of \eqref{LTO} can be divided into the two classes: \begin{gather*} \mathcal{N}_0 = \big\{ x \text{ solution of }\eqref{LTO}, \exists T_x \colon x(t)x^{[1]}(t) < 0,\; x(t)x^{[2]}(t) > 0 \text{ for } t \geq T_x \big\}\\ \mathcal{N}_2 = \{ x \text{ solution of }\eqref{LTO}, \exists T_x \colon x(t)x^{[1]}(t) > 0,\; x(t)x^{[2]}(t) > 0 \text{ for } t \geq T_x \big\}. \end{gather*} \begin{definition} \rm Equation \eqref{LTO} is said to have \textit{property A} if every proper solution $x$ of \eqref{LTO} is oscillatory or satisfies \[ \bigl|x^{[i]}(t)\bigr|\downarrow0\quad\text{as } t\to\infty,\quad i=0,1,2. \] Equation \eqref{LTO} is said to have \emph{property $\bar{A}$} if any proper solution $x$ of \eqref{LTO} is oscillatory or belongs to $\mathcal{N}_0$. \end{definition} \begin{theorem}[{\cite[Theorem 5]{CDMV}}] \label{THA} If \begin{equation}\label{oldconvergent} \int_{t_0}^{\infty}q(t)\int_{t_0}^t r(s) \int_{t_0}^s p(v)\,\mathrm{d}v\,\mathrm{d}s\,\mathrm{d}t < \infty, \end{equation} then all solutions of \eqref{LTO} are nonoscillatory. \end{theorem} \begin{theorem}[{\cite[Lemma 2.2]{CDMNA}}] \label{OLD_nonempty_B} Equation \eqref{LTO} has \textit{property $\bar{A}$} if and only if it has at least one oscillatory solution. \end{theorem} \begin{theorem}[{\cite[Theorem 2.2]{CDMNA}}] \label{OLD_property} Equation \eqref{LTO} has property A if and only if it has at least one oscillatory solution and \begin{equation}\label{PODMINKA} \int_{t_0}^{\infty}q(t)\int_{t_0}^t p(s) \int_{t_0}^s r(v)\,\mathrm{d}v\,\mathrm{d}s\,\mathrm{d}t = \infty. \end{equation} \end{theorem} From Theorems \ref{THA}--\ref{OLD_property} we obtain the following results. \begin{proposition}\label{OLD_nonempty} The class $\mathcal{N}_0$ is not empty for \eqref{LTO}. If \eqref{oldconvergent} holds, then $\mathcal{N}_2$ is not empty for \eqref{LTO}. \end{proposition} \begin{proof} The first part follows from results of Hartman and Wintner \cite[p. 506]{HARTMAN}. The second part follows from Theorems \ref{THA} and \ref{OLD_nonempty_B}. \end{proof} \begin{proposition}\label{eq0} Consider equation \eqref{LTO}, where $p(t)=r(t)$ for large $t$. Then \eqref{LTO} has property A if and only if it has property $\bar{A}$. \end{proposition} \section{Functional differential equations} Consider the linear functional differential equation \begin{equation}\label{FDE} L_3x(t)+q(t)x\big(\delta(t)\big)=0. \end{equation} The classification of nonoscillatory solutions of \eqref{FDE} and definitions of property A and $\bar{\text{A}}$ are the same as for equation \eqref{LTO}. We recall the comparison theorem for the functional differential equations stated in \cite[Theorem 2]{KN}. We reformulate it in a slightly different form, which will be useful for our purpose. Consider the third-order linear functional differential equations \begin{equation}\label{Lmajor} L_3 y(t)+ q_1(t)y\big(\delta_1(t)\big)=0 \end{equation} and \begin{equation}\label{Lminor} L_3 z(t)+ q_2(t)z\big(\delta_2(t)\big)=0 \end{equation} where $q_1(t)\geq q_2(t)>0$ and $\lim_{t\to\infty}\delta_1(t)=\lim_{t\to\infty}\delta_2(t)=\infty$. \begin{proposition}\label{KNCOMP} Assume \[ \delta_1(t)\geq \delta_2(t) \quad \text{and}\quad q_1(t)\geq q_2(t)\quad \text{for }t\geq t_1. \] \begin{itemize} \item[(a)] If there exists a solution $y\in \mathcal{N}_2$ of \eqref{Lmajor}, then there exists a solution $z\in \mathcal{N}_2$ of \eqref{Lminor}. \item[(b)] If there exists a solution $y\in \mathcal{N}_0$ of \eqref{Lmajor} such that $\lim_{t\to\infty}|y(t)|>0$, then there exists a solution $z\in \mathcal{N}_0$ of \eqref{Lminor} such that $\lim_{t\to\infty}|z(t)|>0$. \end{itemize} \end{proposition} \begin{proposition}\label{exist} If $\delta(t)\leq t$ and \begin{equation}\label{NEPODMINKA} \int_{t_0}^{\infty}q(t)\int_{t_0}^t p(s) \int_{t_0}^s r(v)\,\mathrm{d}v\,\mathrm{d}s\,\mathrm{d}t <\infty, \end{equation} then equation \eqref{FDE} has a solution $x\in \mathcal{N}_0$ such that $\lim_{t\to\infty}|x(t)|>0$. \end{proposition} \begin{proof} By Theorem \ref{OLD_property} and Proposition \ref{OLD_nonempty}, equation \eqref{LTO} has a solution $x$ in the class $\mathcal{N}_0$ such that $\lim_{t\to\infty}|x(t)|>0$. Now the conclusion follows from Proposition \ref{KNCOMP}-b). \end{proof} By Proposition \ref{exist} we have that if the delay equation \eqref{FDE} has property A, then equation \eqref{LTO} has also property A. Under the additional conditions the delay equations can be compared with ODE (without delay). \begin{proposition}[{\cite[Theorem 8]{KN}}] \label{eq} Let $|t-\delta(t)|$ be bounded and let functions $p(t)$, $r(t)$ be non-increasing for $t\in [t_0,\infty)$. Then equation \eqref{FDE} has property A if and only if equation \eqref{LTO} has property A. \end{proposition} Our next theorem extends Proposition \ref{eq0} for the functional differential equations with the symmetrical operator \begin{equation}\label{S} \Big(\frac{1}{p(t)}\Big(\frac{1}{p(t)}x'(t)\Big)'\Big)' + q(t)x\big(\delta(t)\big)=0 \end{equation} and complements some results from \cite[Chapter 6]{PADHIBOOK}. \begin{theorem}\label{thm1} Consider equation \eqref{S} and assume that $\delta(t)\leq t$. Then the following statements are equivalent: \begin{itemize} \item[(a)] $\mathcal{N}_2=\emptyset$, i.e. \eqref{S} has property $\bar{A}$; \item[(b)] every solution is oscillatory or tends to zero as $t\to\infty$, i.e. \eqref{S} has property A. \end{itemize} \end{theorem} \begin{proof} "(b)$\Rightarrow$ (a)": It is immediate. "(a)$\Rightarrow$ (b)": Assume by contradiction that there exists a solution $x\in \mathcal{N}_0$ of \eqref{S} such that $\lim_{t\to\infty} x(t)=c>0$. Consider the linear equation \begin{equation}\label{S1} \Big(\frac{1}{p(t)}\Big(\frac{1}{p(t)}y'(t)\Big)'\Big)' + q(t)\frac{x(\delta(t))}{x(t)}y(t)=0. \end{equation} Then $y=x$ is a solution of \eqref{S1}. By Theorem \ref{OLD_property}, we have \[ \int_{t_0}^{\infty}q(t)\frac{x(\delta(t))}{x(t)}\int_{t_0}^t p(s) \int_{t_0}^s p(v)\,\mathrm{d}v\,\mathrm{d}s\,\mathrm{d}t < \infty. \] Obviously, $\lim_{t\to\infty}\frac{x(\delta(t))}{x(t)}=1$, so \[ \int_{t_0}^{\infty}q(t)\int_{t_0}^t p(s) \int_{t_0}^s p(v)\,\mathrm{d}v\,\mathrm{d}s\,\mathrm{d}t < \infty. \] By Theorem \ref{THA}, the linear equation \begin{equation}\label{S2} \Big(\frac{1}{p(t)}\Big(\frac{1}{p(t)}z'(t)\Big)'\Big)' + q(t)z(t)=0 \end{equation} does not have oscillatory solutions. Therefore it has a solution $z\in \mathcal{N}_2$ by Proposition \ref{OLD_nonempty}. Consider the linear equation \begin{equation}\label{S3} \Big(\frac{1}{p(t)}\Big(\frac{1}{p(t)}v'(t)\Big)'\Big)' + q(t)\frac{z(t)}{z(\delta(t))}v\big(\delta(t)\big)=0. \end{equation} Then $v=z$ is a solution of \eqref{S3}. Since $z$ is increasing and $\delta(t)\leq t$, we have \[ \frac{z(t)}{z(\delta(t))} \geq 1 \quad \text{ for large }t. \] By the comparison theorem for the functional differential equation (Proposition \ref{KNCOMP}), equation \eqref{S} has a solution $x\in \mathcal{N}_2$, a contradiction. \end{proof} \section{Neutral nonlinear equation - basic properties} In this section we study properties of nonoscillatory solutions of \eqref{EP}. \begin{lemma} \label{lem1} Let $x$ be a nonoscillatory solution of \eqref{EP} and let $u$ be defined by \eqref{OZNU}. Then $u$, $u^{[1]}$, $u^{[2]}$ are monotone for large $t$. \end{lemma} \begin{proof} Set $y=u^{[1]}$ and $z=u^{[2]}$. Then $x$ is a solution of \eqref{EP} if and only if $(u, y, z)$ is a solution of the system \begin{gather*} u'(t)=r(t)y(t)\\ y'(t)=p(t)z(t)\\ z'(t)=-q(t)f\big(x\big(\delta(t)\big)\big). \end{gather*} From the last equation we see that $z'$ is of one sign for large $t$ and so $z$ is of one sign as well. Using this fact we obtain from the second equation that the same is true for $y'$. Similarly, we obtain from the first equation that $u'$ is also of one sign. Therefore $u$, $u^{[1]}$ and $u^{[2]}$ are monotone. \end{proof} \begin{lemma}\label{NEROVNOST1} Let $x$ be a solution of \eqref{EP} and let $u$ be defined by \eqref{OZNU}. If either $u(t)>0$ and $u^{[1]}(t)>0$ or $u(t)<0$ and $u^{[1]}(t)<0$ for $t\geq T$, then \begin{equation}\label{TH1ESTI} (1-a_0)|u(t)|\leq |x(t)| \leq |u(t)| \end{equation} for $t\geq T$. \end{lemma} \begin{proof} Assume that $u(t)>0$ and $u^{[1]}(t)>0$ for $t\geq T$. Since $\gamma(t)\leq t$ and $u$ is an increasing function, we have $x\big(\gamma(t)\big)\leq u\big(\gamma(t)\big)\leq u(t)$. Hence \[ x(t)=u(t)-a(t)x\big(\gamma(t)\big) \geq u(t)-a_0x\big(\gamma(t)\big)\geq u(t)-a_0u\big(\gamma(t)\big)\geq u(t)(1-a_0). \] The proof for $u(t)<0$ and $u^{[1]}(t)<0$ for $t\geq T$ is similar and is omitted. \end{proof} \begin{lemma}\label{TRIDICI} Let $x$ be a nonoscillatory solution of \eqref{EP} and let $u$ be defined by \eqref{OZNU}. Then there are only two possible classes of solutions \begin{gather*} \mathcal{N}_0 = \big\{ x \text{ solution}, \exists T_x\colon u(t)u^{[1]}(t)<0,\; u(t)u^{[2]}(t)>0 \text{ for } t \geq T_x \big\},\\ \mathcal{N}_2 = \big\{ x \text{ solution}, \exists T_x\colon u(t)u^{[1]}(t)>0,\; u(t)u^{[2]}(t)>0 \text{ for } t \geq T_x \big\}. \end{gather*} \end{lemma} \begin{proof} Without loss of generality we may assume that there exists $t_1$ such that $x\big(\delta(t)\big)>0$, $x(t)>0$ for $t\geq t_1$. Then $u(t)\geq x(t)>0$ and from \eqref{EP}, \[ \big(u^{[2]}(t)\big)'=-q(t)f\big(x\big(\delta(t)\big)\big)<0,\quad t\geq t_1. \] Therefore $u^{[2]}$ is decreasing and there exists $t_2\geq t_1$ such that there are two possibilities, either $u^{[2]}(t)<0$ or $u^{[2]}(t)>0$ for $t \geq t_2$. Assume that $u^{[2]}(t)<0$ for $t\geq t_2$. Then there exists a constant $M>0$ such that \[ u^{[2]}(t)\leq-M<0. \] Integrating this inequality from $t_2$ to $t$ we obtain \[ u^{[1]}(t)\leq u^{[1]}(t_2) - M \int_{t_2}^t p(s)\,\mathrm{d}s. \] Letting $t\to\infty$ and using the fact that $\int_{t_0}^{\infty}p(t)\,\mathrm{d}t=\infty$, we obtain $u^{[1]}(t) \to-\infty$, i.e. $u^{[1]}(t)<0$ eventually. Proceeding by the same way and using the fact that $\int_{t_0}^{\infty}r(t)\,\mathrm{d}t=\infty$, we obtain $u(t)\to-\infty$, a contradiction. Thus $u^{[2]}(t)>0$ and $u^{[1]}$ is increasing for $t\geq t_2$. Therefore there are two possibilities, either $u(t)>0$, $u^{[1]}(t)<0$, $u^{[2]}(t)>0$, or $u(t)>0$, $u^{[1]}(t)>0$, $u^{[2]}(t)>0$. \end{proof} \begin{lemma}\label{LIMITNIN2} Let $x$ be a solution of \eqref{EP} from the class $\mathcal{N}_2$. Then \[ \lim_{t\to\infty}|x(t)|=\lim_{t\to\infty}|u(t)|=\infty. \] \end{lemma} \begin{proof} Let $x \in\mathcal{N}_2$. Without loss of generality we may assume that $x$ is eventually positive, i.e. there exists $T\geq t_0$ such that $x(t)>0$, $u(t)>0$, $u^{[1]}(t)>0$ and $u^{[2]}(t)>0$ for $t\geq T$. As $u^{[1]}$ is positive and increasing function there exists $K>0$ such that $u^{[1]}(t)\geq K$ for large $t$. Integrating this inequality from $T$ to $t$ we obtain \[ u(t)\geq u(T)+K\int_{T}^tr(s)\,\mathrm{d}s. \] Letting $t\to\infty$ and using the fact that $\int_{t_0}^{\infty}r(t)\,\mathrm{d}t=\infty$, we obtain $u(t)\to\infty$. By Lemma \ref{NEROVNOST1}, $x(t)\geq (1-a_0)u(t)$. From this it follows that $x(t)\to\infty$. \end{proof} \begin{proposition}\label{LIMITNI} Let $x$ be a solution of \eqref{EP} from the class $\mathcal{N}_0$. Then \[ \lim_{t\to\infty}u^{[i]}(t)=0\quad \text{for } i=1,2 \] and \begin{equation}\label{nonzero} \liminf_{t\to\infty}|x(t)|>0 \quad \Longleftrightarrow \quad \lim_{t\to\infty}|u(t)|>0. \end{equation} Moreover, if \eqref{PODMINKA} holds, then \begin{equation}\label{LIMCHOV} \lim_{t\to\infty}x(t)=\lim_{t\to\infty}u(t)=0. \end{equation} \end{proposition} \begin{proof} Assume that $x\in\mathcal{N}_0$. Without loss of generality we may assume that $x$ is eventually positive, i.e. $u(t)>0$, $u^{[1]}(t)<0$, $u^{[2]}(t)>0$ for $t\geq T_x$. Since $u$ is positive, there exists $\lim_{t\to\infty}u^{[i]}(t)=\ell _i$, $i=0,1,2$. First, assume that $\ell_1<0$. Then $u'(t)\leq \ell_1 r(t)$. Integrating from $T_x$ to $t$ and letting $t\to\infty$ we obtain a contradiction with the positivity of $u$. In the similar manner we can see that $\ell_2=0$. If $\ell=\ell_0>0$, then for any $\varepsilon>0$ we have $l+\varepsilon >u\big(\gamma(t)\big)>l$ for large $t$, and choosing $0<\varepsilon<\frac{l(1-a_0)}{a_0}$ we obtain the lower estimate \begin{equation}\label{ODHAD} x(t)=u(t)-a(t)x\big(\gamma(t)\big) > l - a_0u\big(\gamma(t)\big) > l-a_0(l+\varepsilon)=k(l+\varepsilon)>kl, \end{equation} where $k=\frac{l-a_0(l+\varepsilon)}{l+\varepsilon}>0$, i.e. $\liminf_{t\to\infty}|x(t)|>0$. The vice versa in \eqref{nonzero} follows from \eqref{OZNU}. To prove \eqref{LIMCHOV}, assume by contradiction that $\ell=\ell_0>0$. From \eqref{ODHAD} and in view of the fact that $f$ is continuous, there exists $K$ such that \[ f\big(x\big(\delta(t)\big)\big)\geq K \] for large $t$. Hence from equation \eqref{EP} it follows that \[ \Big(u^{[2]}(t)\Big)' \leq -q(t)K. \] Integrating this inequality two times from $t$ to $\infty$ we obtain \[ -u^{[1]}(t)\geq K\int_t^{\infty} p(v)\int_v^{\infty}q(s)\,\mathrm{d}s\,\mathrm{d}v. \] Integrating from $t_1$ to $t$ we obtain \[ -u(t)+u(t_1)\geq K \int_{t_1}^{t}r(w)\int_w^{\infty} p(v) \int_v^{\infty}q(s)\,\mathrm{d}s\,\mathrm{d}v\,\mathrm{d}w. \] Letting $t\to\infty$ we obtain \[ \int_{t_1}^{\infty}r(w)\int_w^{\infty} p(v)\int_v^{\infty}q(s)\,\mathrm{d}s\,\mathrm{d}v\,\mathrm{d}w <\infty. \] Changing the order of the integration we obtain the contradiction with condition \eqref{PODMINKA}. Therefore $l=0$ and the inequality $0\leq x(t) \leq u(t)$ implies that $\lim_{t\to\infty}x(t)=0$. \end{proof} \section{Main results: Comparison theorems} We state comparison theorems under the assumption that \begin{equation}\label{LIMSUP} \limsup_{|v|\to\infty}\frac{v}{f(v)}<\infty. \end{equation} Set \[ S_f=\limsup_{v\to\infty}\frac{v}{f(v)}. \] Our first theorem is based on the comparison with the linear ordinary differential equations and holds for the advanced argument $\delta(t)\geq t$. \begin{theorem}\label{thm2} Assume that \eqref{LIMSUP} holds and $\delta(t)\geq t$. \begin{itemize} \item[(i)] If $S_f>0$ and the linear ODE \begin{equation}\label{Linear} L_3y(t)+ \frac{1-a_0}{S_f}\, q(t)\,y(t)=0 \end{equation} has property A, then equation \eqref{EP} has also property A. \item[(ii)] If $S_f=0$, i.e. $\lim_{|v|\to\infty}\frac{f(v)}{v}=\infty$, and for some $K>0$ the linear ODE \begin{equation}\label{Superlin} L_3y(t)+ K q(t)y(t)=0 \end{equation} has property A, then equation \eqref{EP} has also property A. \end{itemize} \end{theorem} \begin{proof} (i) Let \eqref{Linear} have property A and let $x$ be a solution of \eqref{EP} such that $x(t)>0$ for $t\geq t_1$, $t_1\geq t_0$ and $u(t)$ be defined by \eqref{OZNU}. Assume by contradiction that $x\in\mathcal{N}_2$. Then $u$ is nondecreasing and so $u(t)\leq u\big(\delta(t)\big)$. Using Lemma \ref{NEROVNOST1} we obtain the following estimate \begin{equation}\label{LinearEST} 1-a_0\leq \frac{x\big(\delta(t)\big)}{u\big(\delta(t)\big)} \leq \frac{x\big(\delta(t)\big)}{u(t)}. \end{equation} Consider the equation \begin{equation}\label{linearized} \Big(\frac{1}{p(t)}\Big(\frac{1}{r(t)}y'(t)\Big)'\Big)' + q(t)\frac{f\big(x\big(\delta(t)\big)\big)}{u(t)}y(t)=0. \end{equation} This equation has a solution $y=u$ satisfying $y(t)>0$, $y^{[1]}(t)>0$, $y^{[2]}(t)>0$ for large $t$, i.e. $y$ is a solution of \eqref{linearized} from the class $\mathcal{N}_2$. Since $S_f>0$, we can make the following estimate \[ f(v)\geq\frac{v}{S_f} \quad \text{for large }v. \] By Lemma \ref{LIMITNIN2}, we have that $x(t)\to\infty$ as $t\to\infty$, so from here and \eqref{LinearEST} there exists $T\geq t_1$ such that \[ q(t)\frac{f\big(x\big(\delta(t)\big)\big)}{u(t)} \geq q(t)\frac{x\big(\delta(t)\big)}{S_fu(t)} \geq q(t)\frac{1-a_0}{S_f}. \] Since \eqref{Linear} has property A, $\mathcal{N}_2=\emptyset$ for \eqref{Linear}. Consequently, by Proposition \ref{KNCOMP}, $\mathcal{N}_2=\emptyset$ for \eqref{linearized}, a contradiction. Now assume that $x\in\mathcal{N}_0$. Since \eqref{Linear} has property A, we have according to Theorem \ref{OLD_property} that \eqref{PODMINKA} holds. By Proposition \ref{LIMITNI}, $\lim_{t\to\infty}x(t)=0$. \smallskip (ii). We proceed by a similar way as before. Let \eqref{Superlin} have property A for some $K>0$. First, assume that equation \eqref{EP} has a solution $x\in\mathcal{N}_2$ such that $x(\delta(t))>0$ for $t\geq t_1$ and $u$ is defined by \eqref{OZNU}. Consider the linear delay equation \begin{equation}\label{linearizedf} L_3 y(t)+ q(t)\frac{f\big(x\big(\delta(t)\big)\big)}{u(t)}y(t)=0. \end{equation} This equation has a solution $y=u$ from the class $\mathcal{N}_2$. By Lemma \ref{LIMITNIN2}, $\lim_{t\to\infty}x(t)=\infty$. Since $S_f=0$, we have \[ \frac{f\big(x\big(\delta(t)\big)\big)}{x\big(\delta(t)\big)}\geq\frac{K}{(1-a_0)} \] for large $t$. From here and \eqref{LinearEST} \[ \frac{f\big(x\big(\delta(t)\big)\big)}{u\big(t\big)}= \frac{f\big(x\big(\delta(t)\big)\big)}{x\big(\delta(t)\big)} \frac{x\big(\delta(t)\big)}{u(t)}\geq \frac{K}{1-a_0}(1-a_0)=K. \] Thus equation \eqref{linearizedf} is a majorant of \eqref{Superlin}. Since $\mathcal{N}_2=\emptyset$ for \eqref{Superlin}, we have by Proposition \ref{KNCOMP} that $\mathcal{N}_2=\emptyset$ for \eqref{linearizedf}, a contradiction. If $x\in\mathcal{N}_0$, then by the same argument as in the proof of (i) we obtain \eqref{PODMINKA}, which implies that $\lim_{t\to\infty}x(t)=0$. \end{proof} Our second theorem is established for the delay argument $\delta(t)\leq t$. \begin{theorem}\label{thm3} Assume that \eqref{LIMSUP} holds and $\delta(t)\leq t$. \begin{itemize} \item[(i)] If $S_f>0$ and the linear delay equation \begin{equation}\label{EQCOMP2} L_3y(t)+\frac{1-a_0}{S_f}q(t)y\big(\delta(t)\big)=0 \end{equation} has property A, then equation \eqref{EP} has also property A. \item[(ii)] If $S_f=0$, i.e. $\lim_{|v|\to\infty} f(v)/v=\infty$, and for some $K>0$ the linear delay equation \[ L_3y(t)+ K q(t)y(t)=0 \] has property A, then equation \eqref{EP} has also property A. \end{itemize} \end{theorem} \begin{proof} (i) Let \eqref{EQCOMP2} have property A and let $x$ be a solution of \eqref{EP} such that $x(t)>0$ for $t\geq t_1$, $t_1\geq t_0$ and $u(t)$ be defined by \eqref{OZNU}. Assume by contradiction that $x\in\mathcal{N}_2$, and consider the delay equation \begin{equation}\label{EQCOMPH2} L_3y(t) + q(t)\frac{f\big(x\big(\delta(t)\big)\big)}{u\big(\delta(t)\big)} y\big(\delta(t)\big)=0. \end{equation} This equation has a solution $y=u$ satisfying $y(t)>0$, $y^{[1]}(t)>0$, $y^{[2]}(t)>0$ for large $t$, i.e. $y$ is the solution of \eqref{EQCOMPH2} from the class $\mathcal{N}_2$. By the same argument as in the proof of Theorem \ref{thm2}-(i) we obtain \[ \frac{f\big(x\big(\delta(t)\big)\big)}{u\big(\delta(t)\big)} \geq \frac{1-a_0}{S_f}\,. \] Now by Proposition \ref{KNCOMP}, $\mathcal{N}_2=\emptyset$ for \eqref{EQCOMPH2}, a contradiction. Assume that $x\in\mathcal{N}_0$, $x(t)>0$ for large $t$ and assume by contradiction that $\lim_{t\to\infty}u(t)=\ell>0$. Then there exists $c_1>0$ such that $x\big(\delta(t)\big)\geq c_1$ for large $t$. Now, $f$ being continuous, we can assume that there exists $c_2>0$ such that \begin{equation}\label{10} \frac{f\big(x\big(\delta(t)\big)\big)}{u(t)}\geq c_2 \end{equation} for large $t$. Consider the linear equation \[ L_3z(t) + q(t)\frac{f\big(x\big(\delta(t)\big)\big)}{u(t)}z(t)=0. \] This equation has a solution $z=u$ which tends to a nonzero constant. Hence by Theorem \ref{OLD_property}, \[ \int_{t_0}^{\infty}q(t) \frac{f\big(x\big(\delta(t)\big)\big)}{u(t)} \int_{t_0}^{t} p(s)\int_{t_0}^{s}r(v)\,\mathrm{d}v\,\mathrm{d}s\,\mathrm{d}t<\infty. \] From \eqref{10} we conclude that \[ \int_{t_0}^{\infty}q(t)\int_{t_0}^{t} p(s)\int_{t_0}^{s}r(v)\,\mathrm{d}v\,\mathrm{d}s\,\mathrm{d}t<\infty. \] Applying Proposition \ref{exist} to \eqref{EQCOMP2} we obtain that equation \eqref{EQCOMP2} has a solution $y\in\mathcal{N}_0$ such that $\lim_{t\to\infty}|y(t)|>0$. This is a contradiction with the fact that \eqref{EQCOMP2} has property A. (ii) The proof is similar to the proof of Theorem \ref{thm2}-(ii) and is omitted. \end{proof} Now we complete Theorem \ref{thm3} for the neutral equation with the symmetric operator. \begin{corollary}\label{Coreq} If the linear ODE \begin{equation}\label{L1} y'''(t)+(1-a_0)q(t)y(t)=0 \end{equation} has an oscillatory solution, then the neutral equation \begin{equation}\label{EPG} \big(x(t)+a(t)x\big(\gamma(t)\big)\big)''' + q(t)x(t-\sigma)=0, \quad \sigma>0 \end{equation} has property A. \end{corollary} \begin{proof} By Theorem \ref{OLD_nonempty_B} equation \eqref{L1} has property $\bar{{\rm A}}$ and by Proposition \ref{eq0} it has property A. Therefore by Proposition \ref{eq} the delay equation \[ y'''(t)+(1-a_0)q(t)y(t-\sigma)=0 \] has property A. Using Theorem \ref{thm3} with $S_f=1$ we obtain the assertion. \end{proof} \begin{remark} \label{rmk1} \rm Equation \eqref{EPG} with $\gamma(t)=t-\tau$, $\tau>0$, has been considered in \cite{GYORI}. Corollary \ref{Coreq} extends \cite[Theorem 10.4.1]{GYORI} for $n=3$, where it was proved that \eqref{EPG} has property A provided $\int^\infty q(t)\,\mathrm{d}t=\infty$. \end{remark} \begin{corollary}\label{MainCor} Let $\delta(t)\leq t$. If the linear delay equation \begin{equation}\label{SD} \Big(\frac{1}{p(t)}\Big(\frac{1}{p(t)}y'\Big)'\Big)' +(1-a_0)q(t)y\big(\delta(t)\big)=0 \end{equation} has property $\bar{A}$, then the neutral equation \begin{equation}\label{NeutralS} \Big(\frac{1}{p(t)}\Big(\frac{1}{p(t)}\big[x(t)+a(t)x\big(\gamma(t)\big)\big]' \Big)'\Big)' + q(t)x\big(\delta(t)\big)=0. \end{equation} has property A. \end{corollary} \begin{proof} By Theorem \ref{thm1}, we have that \eqref{SD} has property A and using Theorem \ref{thm3} with $S_f=1$ we obtain the assertion. \end{proof} \subsection*{Open problem} As far as the class $\mathcal{N}_0$ is concerned, it is always nonempty for equation \eqref{LTO}, while it can be empty for equation \eqref{FDE} with $\delta(t)0$, $\delta(t)\frac{2}{3(1-a_0)\sqrt{3}}\,. \end{equation*} Consider the corresponding linear ODE \[ y'''(t)+(1-a_0)\frac{k}{t^3}y(t)=0. \] It is well-known \cite{HANAN} that if $(1-a_0)k >\frac{2}{3\sqrt{3}}$ then this equation has an oscillatory solution, and it has property A. Applying Theorem \ref{thm2} we obtain the conclusion. \end{example} \begin{example} \label{Ex2} \rm Consider the neutral equation \[ \Big(x(t)+\frac{1}{2}x\big(\gamma(t)\big)\Big)''' + \frac{k}{t^3}x(t-c)=0, \quad c\in \mathbb{R}. \] This equation has the property A for every $k>4/(3\sqrt{3})$. Indeed, the case $c\leq 0$ follows from Example \ref{Ex1} and the case $c>0$ follows from Corollary \ref{Coreq}. If we apply \cite[Theorem 2.7]{DTT} we obtain that this equation has property A for $k>1$. Hence we can say our result improves the one mentioned there. \end{example} Now consider the linear neutral equation \begin{equation}\label{LinearEPR} \Big(\frac{1}{p(t)}\Big(\frac{1}{r(t)}\big[x(t)+a(t)x\big(\gamma(t)\big)\big]' \Big)'\Big)' + q(t)x(t)=0. \end{equation} \begin{corollary} \label{Cor1} Let \eqref{PODMINKA} and at least one of the following conditions hold: \begin{itemize} \item[(i)] \begin{equation*} \int_{t_0}^{\infty}q(t)\int_{t_0}^t r(s)\,\mathrm{d}s=\infty, \end{equation*} \item[(ii)] \begin{equation*} \limsup_{t\to\infty}\int_{t_0}^{t} p(s)\,\mathrm{d}s \int_{t}^{\infty}q(s)\frac{\int_{t_0}^s r(u) \int_{t_0}^u p(v)\,\mathrm{d}v\,\mathrm{d}u}{\int_{t_0}^{s} p(u)\,\mathrm{d}u}\mathrm{d}s>\frac{1}{1-a_0}. \end{equation*} \end{itemize} Then equation \eqref{LinearEPR} has property A. \end{corollary} \begin{proof} Either condition (i) or (ii) ensures that the corresponding linear equation \begin{equation}\label{newex} L_3y(t)+ \frac{1}{1-a_0} q(t) y(t)=0 \end{equation} has an oscillatory solution, see \cite[Theorem 8]{CDMV} or \cite[Lemma 2.2]{KIG}, respectively. Moreover, \eqref{PODMINKA} ensures that \eqref{newex} has property A. Applying Theorem \ref{thm2} we obtain the conclusion. \end{proof} \begin{example}[{\cite[Example 3.1]{TONXING}}] \label{Ex3} \rm Consider the neutral equation \begin{equation}\label{ex1b} \Big(t\Bigl(x(t)+a_0\,x\Bigl(\frac{t}{2}\Bigr)\Bigr)''\Big)' + \frac{k}{t^2}x(t)=0, \end{equation} where $a_0\in [0,1)$. Applying Corollary \ref{Cor1}-(i) we obtain that this equation has property A for any $k>0$. Observe that applying \cite[Theorem 2.1]{TONXING} or \cite[Corollary 3]{DZUBA} we obtain that \eqref{ex1b} has property A for $k>(4l(1-a_0))$ for some $l\in(1/4,1)$, or $k>2/(1-a_0)$, respectively. \end{example} Now consider the neutral delay equation \begin{equation}\label{LinearEP} \Big(x(t)+a(t)x\big(\gamma(t)\big)\Big)''' + q(t)x\big(\delta(t)\big)=0, \quad \delta(t)\frac{2}{1-a_0}, \end{equation*} \item[(ii)] $\delta(t)\frac{1}{1-a_0}, \end{equation*} \end{itemize} \end{corollary} \begin{example}\label{Ex4} \rm Consider the equation \begin{equation*} %\label{exPeq1} \Big(x(t)+a(t)x\big(\gamma(t)\big)\Big)''' + \frac{k}{t^3}x(\mu t)=0. \end{equation*} where $0<\mu<1$. By Corollary \ref{C3}-(i), this equation has property A for \begin{equation*} %\label{ex3pod} k>\frac{4}{(1-a_0)\mu^4}. \end{equation*} \end{example} \begin{example} \label{Ex5} \rm Consider the equation \begin{equation*} %\label{LEP} \Big(x(t)+a(t)x\big(\gamma(t)\big)\Big)''' + \frac{k}{t^3}x^{\lambda}(\mu t)=0, \end{equation*} where $\lambda>1$ is a quotient of odd positive integers and $0<\mu<1$. Using Example \ref{Ex4} with $a_0=0$ and Theorem \ref{thm3}-(ii) we obtain that this equation has property A for any $k>0$. \end{example} \subsection*{Acknowledgements} The authors are supported by Grant P201/11/0768 of the Czech Science Foundation. \begin{thebibliography}{10} \bibitem{AGR} R. P. Agarwal, S. R. Grace, D. O'Regan; On the oscillation of certain functional differential equations via comparison methods, \textit{J. Math. Anal. Appl.}, \textbf{286}(2003), 577-602. \bibitem{DZUBA2} B. Bacul\'i­kov\'a, J. D\v{z}urina; On the asymptotic behavior of a class of third order nonlinear neutral differential equations, \textit{Cent. Eur. J. Math.}, \textbf{8}(2010), 1091--1103. \bibitem{DZUBA} B. Bacul'i­kov\'a, J. D\v{z}urina; Oscillation of third-order neutral differential equations, \textit{Math. Comput. Model.}, \textbf{52}(2010), 215--226. \bibitem{DZUBA2012} B. Bacul\'i­kov\'a, J. D\v{z}urina; Oscillation theorems for higher order neutral differential equations, \textit{Appl. Math. Comput.}, \textbf{219}(2012), 3769--3778. \bibitem{CDM2} M. Cecchi, Z. Do\v{s}l\'a, M. Marini; Asymptotic behavior of solutions of third order delay differential equations, \textit{Arch. Math. (Brno)} \textbf{33} (1997), 99--108. \bibitem{CDMcomp} M. Cecchi, Z. Do\v{s}l\'a, M. Marini; An equivalence theorem on properties A, B for third order differential equations, \textit{Annali Mat. Pura Appl.}, \textbf{173} (1997), 373--389. \bibitem{CDMNA} M. Cecchi, Z. Do\v{s}l\'a, M. Marini; On nonlinear oscillations for equations associated to disconjugate operators, \textit{Nonlinear Anal.} \textbf{30} (1997), 1583--1594. \bibitem{CDMV} M. Cecchi, Z. Do\v{s}l\'a, M. Marini, G. Villari; On the qualitative behavior of solutions of third order differential equations, \textit{J. Math. Anal. Appl.}, \textbf{197} (1996), 749--766. \bibitem{DTT} J. D\v{z}urina, E. Thandapani, S. Tamilvanan; Oscillation of solutions to third-order half-linear neutral differential equations, \textit{Electron. J. Differential Equations} \textbf{29} (2012), 1--9. \bibitem{ERBE} L. Erbe; Existence of oscillatory solutions and asymptotic behavior of solutions of a third order linear differential equations, \textit{Pacific J. Math.}, \textbf{64} (1976), 369--385. \bibitem{GAPT} S. R. Grace, R. P. Agarwal, R. Pavani, E. Thandapani; On the oscillation of certain third order nonlinear functional differential equations, \textit{Appl. Math. Comput.}, \textbf{202}(2008), 102--112. \bibitem{GYORI} I. Gyori, G. Ladas; \textit{Oscillation theory of delay differential equations with applications}, Clarendon Press, Oxford, 1991. \bibitem{HANAN} M. Hanan; Oscillation criteria for third order differential equations, \textit{Pacific J. Math.} \textbf{11}(1961), 919--944. \bibitem{HARTMAN} P. Hartman; Ordinary differential equations, 2\,nd ed., Birkhauser, Boston, 1982. \bibitem{KIG} I. T. Kiguradze, T. A. Chanturia; Asymptotic properties of solutions on nonautonomous ordinary differential equations, Kluwer Academic Publishers, Dordrecht, 1993. \bibitem{KN} T. Kusano, M. Naito; Comparison theorems for functional differential equations with deviating arguments, \textit{J. Math. Soc. Japan}, \textbf{33}(1981), No.~3, 509--532. \bibitem{MOJSEJ} I. Mojsej; Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument, \textit{Nonlinear Anal.}, \textbf{68}(2008), 3581--3591. \bibitem{TONXING} T. Li, C. Zhang, G. Xing; Oscillation of third-order neutral delay differential equations, \textit{Abstr. Appl. Anal.}, \textbf{2012}(2012), 1--11. \bibitem{PADHIBOOK} S. Padhi, S. Pati; \textit{Theory of third-order differential equations}, Springer, New Delhi, 2014. \end{thebibliography} \end{document}