\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 271, pp. 1--11.\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-2013/271\hfil Decay of non-oscillatory solutions] {Decay of non-oscillatory solutions for a system of neutral differential equations} \author[H. \v{S}amajov\'a, E. \v{S}p\'anikov\'a, J. G. Dix \hfil EJDE-2013/271\hfilneg] {Helena \v{S}amajov\'a, Eva \v{S}p\'anikov\'a, Julio G. Dix} \address{Helena \v{S}amajov\'a \newline Department of Applied Mathematics, Faculty of Mechanical Engineering, University of \v{Z}ilina, J. M. Hurbana 15, 010 26 \v{Z}ilina, Slovakia} \email{helena.samajova@fstroj.uniza.sk} \address{Eva \v{S}p\'anikov\'a \hfill\break Department of Applied Mathematics, Faculty of Mechanical Engineering, University of \v{Z}ilina, J. M. Hurbana 15, 010 26 \v{Z}ilina, Slovakia} \email{eva.spanikova@fstroj.uniza.sk} \address{Julio G. Dix \hfill\break Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666, USA} \email{jd01@txstate.edu} \thanks{Submitted June 18, 2013. Published December 16, 2013.} \subjclass[2000]{34K11, 34K25, 34K40} \keywords{Neutral differential equation; oscillatory solution; \hfill\break\indent non-oscillatory solution; asymptotic properties of solutions} \begin{abstract} In this article we study the non-oscillatory solutions for a system of neutral functional differential equation. We give sufficient conditions for all non-oscillatory solutions to tend to zero as $t$ approaches infinity. Our results are illustrated with an example. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this article we study the non-oscillatory solutions to the homogeneous system of neutral differential equations \begin{equation} \label{e1} \begin{gathered} \big[|y_1(t)-a(t)y_1(g(t))|^{\beta-1} \big(y_1(t)-a(t)y_1(g(t))\big)\big]' = p_1(t)y_2(t), \\ y_i'(t) = p_i(t)y_{i+1}(t), \quad i=2,3,\dots,n-1,\\ y_n'(t) = \sigma p_n(t)f(y_1(h(t))), \quad t\geq t_0, \end{gathered} \end{equation} where $\beta$ is a positive constant, $n \geq 3$, $\sigma =\pm 1$, and $a,g,h,f,p_i$ are continuous functions that satisfy the condition specified below. Asymptotic properties of solutions to systems of functional differential equations with deviating arguments have been studied by many authors; see for example the references in this article and their references. When the coefficients $p_i$ are positive, \eqref{e1} can be written as $n$-order differential equation. In which case there are many results available, including for non-homogeneous and more general equations; see for example \cite{Di,Mi1}. The existence of oscillatory solutions to \eqref{e1}, with $\beta=1$, and such that $\lim_{t \to \infty} y_i(t) = 0$ or $\lim_{t \to \infty} |y_i(t)| = \infty$ were established in \cite{SaSp}. The existence of non-oscillatory solutions to \eqref{e1} has been shown among others by Maru\v{s}iak \cite{Ma2} and Erbe-Kong-Zhang \cite{ErKoZh}. Non-oscillatory solutions for equation of the type \eqref{e1}, with $\beta=1$, have been grouped in to classes by Maru\v{s}iak \cite{Ma1}. The authors in \cite{SpSa2} expanded this classification, and used it for showing that if the function $y_1(t)$ is bounded, then non-oscillatory solutions decay to zero as $t \to \infty$. The goal in this article is to show the decay of non-oscillatory solutions, without such assumption. The results here are different from those in \cite{Di} in the sense that our coefficients $p_i$ are allowed to have zeros, and our delayed arguments $g(t)$ and $h(t)$ are allowed to exceed $t$, while in \cite{Di} the delay arguments are bounded by $t$. However, in \cite{Di} the differential equation has a forcing term that \eqref{e1} does not have. In this article, as in \cite{SpSa2}, we use the following assumptions: \begin{itemize} \item[(A1)] the coefficient $ a : [t_0,\infty ) \to (0,\infty ] $ is a continuous function; \item[(A2)] the advanced arguments $ g,h : [t_0,\infty ) \to \mathbb{R}$ are continuous and strictly increasing functions, with $\lim_ {t \to \infty } g(t) = \infty$ and $ \lim _{t \to \infty }h(t) = \infty $; \item[(A3)] The coefficients $ p_i : [t_0,\infty ) \to [0,\infty )$ are continuous functions, $p_n$ is not identically zero in any neighborhood of infinity, and $\int_{t_0}^\infty p_i(t)\,dt =\infty$ for $i=1,2,\dots, n-1$; \item[(A4)] the function $ f:\mathbb{R} \to \mathbb{R} $ is continuous, with $uf(u) > 0$ for $ u \neq 0 $, and there is a positive constant $M$ such that $|f(u)|\geq M|u|^\beta$. \end{itemize} The inverse of the functions in (A2) will be denoted by $g^{-1}(t)$ and $h^{-1}(t)$. For simplifying of notation, we define the function \begin{equation}\label{ez} z_1(t) = |y_1(t) - a(t)y_1(g(t))|^{\beta-1}\big(y_1(t) - a(t)y_1(g(t))\big). \end{equation} Note if $\beta$ is the quotient of odd integers, then $|x|^{\beta-1}x= x^\beta$. Also note that for $\beta>0$ and $x$ a differentiable function, $(|x|^{\beta-1}x)'$ and $x'$ have the same sign. This is proved by considering the possible signs of $x$. A function $ y=(y_1, \dots, y_n)$ is a solution of \eqref{e1} if there is a $t_1\geq t_0 $ such that $y$ is continuous for $t\geq \min \{t_1, g(t_1), h(t_1) \}$; the functions $z_1(t)$ and $y_i(t)$, $i=2,3, \dots, n $ are continuously differentiable on $[t_1,\infty)$; and $y$ satisfies \eqref{e1} on $[t_1, \infty )$. In this article, we consider only solutions that are eventually non-trivial; i.e., solutions such that $$ \sup_{t\geq t_1} \max_{1\le i\leq n} |y_i(t)| > 0 \,. $$ A solution is non-oscillatory if there exist $i$ and $T_y \geq t_0$ such that $y_i(t)\neq 0$ for all $t \geq T_y$. Otherwise, a solution $y$ is said to be oscillatory. \section{Preliminaries} Our first lemma is a simplified version of \cite[lemma 2.1]{Di}, \cite[lemma 5.2.1]{LaLaZh}, \cite[lemma 2.2]{SpSa2}. \begin{lemma} \label{lem1} Let $ y=(y_1, \dots, y_n)$ be a solution of \eqref{e1}. Assume that {\rm (A3)} holds and $y_n'(t)$ is eventually of one sign. \begin{itemize} \item[(i)] There exists $t_2\geq t_0$ such that $z_1,y_2,\dots,y_n$ are monotonic and of constant sing on $[t_2, \infty )$. \item[(ii)] There exists an index $\ell$ such that $z_1,y_2,\dots,y_\ell$ have the same sign, and $y_\ell(t)y_{\ell+1}(t)<0$, $y_{\ell+1}(t)y_{\ell+2}(t)<0$, \dots, $y_n(t)y_n'(t)<0$. When $z_1(t)y_2(t)<0$ we set $\ell=1$, and when $z_1,y_2,\dots,y_n'$ have the same sign, we set $\ell=n+1$. \end{itemize} \end{lemma} \begin{lemma} \label{lem2} Under the assumptions on Lemma \ref{lem1}, for $s,t,x_k \in [t_2,\infty)$, we have: If there is a $k$ in $\{2,3,\dots, n\}$ for which $y_k(t)y_{k+1}(t)<0$ (with $y_{n+1}=y_n'$), then \begin{equation} \label{eJ} \begin{aligned} &|y_k(x_k)|\\ &\geq \int_{x_k}^t p_k(x_{k+1}) \int_{x_{k+1}}^t \dots \int_{x_{n-1}}^t p_{n-1}(x_n) \int_{x_n}^t |y_n'(x_{n+1})|\, dx_{n+1}\dots dx_{k+1}\\ &:=J_{n+1-k}(x_k,t; p_k, \dots, p_{n-1},1;|y_n'|), \quad \forall x_k\leq t\,. \end{aligned} \end{equation} Note that $t$ can be arbitrarily large, thus we can use the limit as $t\to\infty$. Also note that when $k=1$, the above estimate has the form $|z_1(x_1)|\geq J_n(\dots)$. If there is a $k$ in $\{2,3,\dots, n+1\}$ for which $z_1(t),y_2(t),\dots, y_k(t)$ have the same sign (with the convention $y_{n+1}=y_n'$), then \begin{equation} \label{eI} \begin{aligned} |z_1(t)| &\geq \int_s^t p_1(x_2) \int_s^{x_2}p_2(x_3) \dots \int_s^{x_{k-1}} p_{k-1}(x_k)|y_k(x_k)|\,dx_k\dots dx_2\\ &:=I_{k-1}(s,t;p_1,\dots,p_{k-1};|y_k|)\quad \forall s \leq t\,. \end{aligned} \end{equation} When $k=\ell$ as defined by Lemma \ref{lem1}, and $2\leq \ell\leq n+1$, we have \begin{equation} \label{eIJ} \begin{aligned} &|z_1(t)|\\ &\geq \int_s^t p_1(x_2) \int_s^{x_2} \dots \int_s^{x_{\ell-1}} p_{\ell-1}(x_\ell) \int_{x_\ell}^t p_\ell(x_{\ell+1}) \\ &\quad \times \int_{x_{\ell+1}}^t p_{\ell+1}(x_{\ell+2}) \int_{x_{\ell+2}}^t \dots \int_{x_{n-1}}^t p_{n-1}(x_n) \int_{x_n}^t |y_n'(x_{n+1})|\, dx_{n+1}\dots dx_2\\ &= I_{\ell-1}\Big(s,t;p_1,\dots,p_{\ell-1}; J_{n+1-\ell}\big(x_\ell,t; p_\ell, \dots, p_{n-1},1;|y_n'|\big)\Big), \quad \forall s\leq t\,. \end{aligned} \end{equation} \end{lemma} \begin{proof} Assuming that $y_m $ and $y_{m+1}$ have opposite signs, we have \begin{align*} |y_m(x_m)| &=|y_m (t)|+ \int_{x_m}^t p_m(x_{m+1}) |y_{m+1}(x_{m+1})|\,dx_{m+1}\\ &\geq \int_{x_m}^t p_m(x_{m+1}) |y_{m+1}(x_{m+1})|\,dx_{m+1}. \end{align*} Inequality \eqref{eJ} follows by applying this inequality for $y_k, y_{k+1}, \dots, y_n$ (with the convention $y_{n+1}=y'_n$). Now assume that $z_1$ and $y_2$ are of the same sign. Then \[ |z_1(t)|=|z_1(s)|+ \int_s^t p_1(x_2)|y_2(x_2)|\,dx_2 \geq \int_s^t p_1(x_2)|y_2(x_2)|\,dx_2. \] Using this inequality for $y_2,y_3, \dots y_k$, we obtain \eqref{eI}. When $k=\ell$ in the two inequalities above, we have \eqref{eIJ}. \end{proof} The functionals similar to $I_k$ and $J_k$ have been defined recursively in \cite{Ma1,SpSa2}. \begin{lemma}[{\cite[Lemma 2.2]{JaKu}}] \label{lem3} Assume {\rm (A1)--(A2)} hold, $g(t)>t$, and $$ 1 \leq a(t) \quad \text{for } t\geq t_0. $$ Let $y_1(t)$ be a continuous non-oscillatory solution to the functional inequality $$ y_1(t)[y_1(t)-a(t)y_1(g(t))] > 0 $$ defined in a neighborhood of infinity. Then $y_1(t)$ is bounded. Moreover, if there exist a constant $a_*$ such that $$ 1 < a_* \leq a(t), \quad \forall t\geq t_0, $$ then $\lim_ {t \to \infty } y_1(t) = 0$. \end{lemma} \begin{lemma}[{\cite[Lemma 2.1]{JaKu}}] \label{lem4} Assume {\rm (A1)--(A2)} hold, $g(t) t$ for $t\geq t_0$, and $$ \liminf _ {t\to \infty } \int_t^{\delta (t)} q(s)\,\mathrm{d}s > \frac{1}{e} . $$ Then the functional inequality $$ x'(t) - q(t)x(\delta (t)) \geq 0,\quad t\geq t_0 $$ has no eventually positive solution, and the functional inequality $$ x'(t) - q(t)x(\delta (t)) \leq 0,\quad t\geq t_0 $$ has no eventually negative solution. \end{lemma} The next Lemma can be proved as in \cite[Lemma 2]{Ma1}. \begin{lemma}\label{lem7} Let $ y = (y_1,y_2,\dots, y_n )$ be a non-oscillatory solution of \eqref{e1}, and let $\lim_{t \to \infty} |z_1(t)| = L_1$, $\lim_{t \to \infty} |y_k(t)| = L_k$ for $k= 2, \dots, n$. For $k \geq 2$, \begin{equation}\label{e13} L_k > 0 \implies L_i = \infty, \quad\text{for } i= 1, \dots, k-1\,. \end{equation} For $1 \leq k < n$, \begin{equation}\label{e14} L_k < \infty \implies L_i = 0, \quad\text{for } i=k+1, \dots, n. \end{equation} \end{lemma} \section{Main results} \begin{theorem}\label{thm1} Assume {\rm (A1)--(A4)}, and let $\sigma=(-1)^n$. Also assume the following conditions hold: there exist constants $a_*, a^*$ such that \begin{gather}\label{t11} 1 < a_* \leq a(t)\leq a^*, \quad\text{for } t\geq t_0;\\ \label{t12} t < g(t)< h(t) \quad \text{for } t\geq t_0; \end{gather} for all $k$ in $\{3,4,\dots, n\}$, the functionals defined by \eqref{eJ}-\eqref{eI} satisfy \begin{gather}\label{t13} \begin{aligned} \limsup_{s \to \infty} I_{k-1}\Big(&s, g^{-1}(h(s)) ;p_1,\dots,p_{k-1}; J_{n+1-k}\big(x_k, g^{-1}(h(s)); p_k, \\ &\dots, p_n; \frac{M}{a^\beta(g^{-1}(h))}\big)\Big) > 1\,; \end{aligned}\\ \label{t14} \liminf_{s \to \infty} \int _s ^{g^{-1}(h(s))} p_1(x_2) J_{n-1}\Big(x_2,\infty; p_2,\ldots, p_n;\frac{M}{a^\beta(g^{-1}(h))}\Big)\,dx_2 > \frac {1}{e}\,. \end{gather} Then every non-oscillatory solution of \eqref{e1} decays to zero; i.e., $\lim_ {t\to \infty} y_i(t) =0$ for $i=1,2, \dots, n$. \end{theorem} \begin{proof} Let $y$ be a non-oscillatory solution of \eqref{e1}. Note that if $y(t)$ is solution of \eqref{e1}, then $-y(t)$ is also a solution; therefore, we assume that $y_1(t)$ is positive, without loss of generality. Then by (A4), $y_n'$ is one sign and, by Lemma \ref{lem1}, each of the functions $z_1$, $y_2$, \dots is of one sign (positive or negative); thus we have only the following cases: \medskip \noindent\textbf{Case 1p:} $z_1(t)>0$ for all $t\geq t_2$, and no restriction on $y_2,y_3,\dots$. Since $z_1(t)$ is positive so is $y_1(t)-a(t)y_1(g(t))$. By Lemma \ref{lem3}, $\lim_{t\to \infty}y_1(t)=0$. Then $\lim_{t\to \infty}z_1(t)=0$, because $a$ is bounded. Then by Lemma \ref{lem7}, $\lim_{t\to \infty}y_i(t)=0$ for $i=1,2,\dots,n$. \medskip \noindent \textbf{Case 1n2p:} $z_1(t)<0$, $y_2(t)>0$ for all $t\geq t_2$, and no restriction on $y_3,y_4,\dots$. Then by \eqref{eJ} we have $\ell=1$, and $y_2,y_4,y_6,\dots$ are positive, while $y_3,y_5,y_7,\dots$ are negative. However, (A4), the choice $\sigma=(-1)^n$, and the fact that $y_1>0$ do not allow this case to happen. See Theorem \ref{thm2} below. \medskip \noindent \textbf{Case 1n2n3n:} $z_1(t)<0$, $y_2(t)<0$ $y_3(t)<0$ for all $t\geq t_2$, and no restriction on $y_4,y_5,\dots$. Then $\ell\geq 3$ in Lemma \ref{lem1}. By \eqref{eIJ} \begin{equation} \label{ennn} \begin{aligned} z_1(t)&\leq -I_{\ell-1}\Big(s,t;p_1,\dots,p_{\ell-1}; J_{n+1-\ell}\big(x_\ell,t;p_\ell,\dots,p_n; |\sigma f(y_1(h))|\big)\Big)\\ &\leq I_{\ell-1}\Big(s,t;p_1,\dots,p_{\ell-1}; J_{n+1-\ell}\big(x_\ell,t;p_\ell,\dots,p_n; -M(y_1(h))\big)\Big) \end{aligned} \end{equation} for all $s\leq t$. Since $z_1(t)$ is negative so is $y_1(t)-a(t)y_1(g(t))$. Therefore, $(-z_1(t))^\beta =a(t)y_1(g(t))-y_1(t)< a(t)y_1(g(t))$, and $z_1(t)>-a^\beta(t)y_1^\beta (g(t))$. Then for $t=g^{-1}(h(x_{n+1}))$, \begin{equation} \label{ez/a} -y_1^\beta \big(g^{-1}(h(x_{n+1}))\big) < \frac{z_1(g^{-1}(h(x_{n+1}))}{a^\beta\big(g^{-1}(h(x_{n+1}))\big)}\,. \end{equation} Applying this inequality and that $z_1$ is non-decreasing, in \eqref{ennn}, we have \begin{align*} &z_1(t)\\ &\leq z_1\big(g^{-1}(h(s))\big) I_{\ell-1}\Big(s,t;p_1,\dots,p_{\ell-1}; J_{n+1-\ell}\big(x_\ell,t;p_\ell,\dots,p_n; M/a^\beta(g^{-1}(h)) \big)\Big)\,. \end{align*} Since $g(t)0$ for all $t\geq t_2$, and no restriction on $y_4,y_5,\dots$. Using \eqref{eJ} for $y_2$, we obtain \[ y_2(s)\leq -J_{n-1}(s,t;p_2,\dots,p_n;|\sigma f(y_1(h))|) \leq J_{n-1}(s,t;p_2,\dots,p_n;- y_1^\beta(h)M) \] Applying \eqref{ez/a} and that $z_1$ is non-increasing we have \[ y_2(s)\leq z_1\big(g^{-1}(h(s))\big) J_{n-1}\big(s,t;p_2,\dots,p_n;M/a^\beta(g^{-1}(h))\big)\quad \forall s\leq t; \] therefore, \[ y_2(s)\leq z_1\big(g^{-1}(h(s))\big) J_{n-1}\big(s,\infty;p_2,\dots,p_n;M/a^\beta(g^{-1}(h))\big)\,. \] Multiplying by $p_1(s)$ in both sides, we note that $z_1$ is a negative solution of the differential inequality \[ z_1'(s)- z_1\big(g^{-1}(h(s))\big)p_1(s) J_{n-1}\big(s,\infty;p_2,\dots,p_n;M/a^\beta(g^{-1}(h))\big)\leq 0\,. \] Since $g(s)\frac{1}{e} \] which is provided by \eqref{t14}. The fact that $z_1$ is negative and is a solution of the differential inequality contradicts Lemma \ref{lem5}. Therefore, this case can not happen. The proof is complete. \end{proof} Next we remove the condition $\sigma=(-1)^n$ in Theorem \ref{thm1}, at the cost of restricting the coefficient $p_n$. \begin{theorem}\label{thm2} Assume {\rm (A1)--(A4)}, \eqref{t11}--\eqref{t12}, and that $p_n$ is bounded below by a positive constant. Then every non-oscillatory solution of \eqref{e1} decays to zero; i.e., $\lim_ {t\to \infty} y_i(t) =0$ for $i=1,2, \dots, n$. \end{theorem} \begin{proof} Let $y$ be a non-oscillatory solution of \eqref{e1}, and without loss of generality assume that $y_1(t)$ is positive. The proofs of the various cases are the same as in Theorem \ref{thm1}, except for one case. \noindent \textbf{Case 1n2p:} $z_1(t)<0$, $y_2(t)>0$ for all $t\geq t_2$. Then by \eqref{eJ} we have $\ell=1$. First we show that $\liminf_{t\to\infty}y_1(t)=0$. The function $y_n$ being monotonic and having its derivative with opposite sign imply the existence of $\lim_{t\to\infty}y_n(t)$. From \eqref{e1} it follows that \[ \int_{t_2}^\infty p_n(t)\sigma f(y_1(h(t)))\,dt < \infty\,. \] Recall that $|f(y)|\geq M| y|^\beta$ and that $p_n$ is bounded below by a positive constant. Using a contradiction argument, we can show that $\liminf_{t\to\infty}y_1(t)=0$. Then by (A2), \[ \liminf_{t\to\infty}y_1(g(t))=0, \quad \liminf_{t\to\infty}y_1(h(t))=0\,. \] Next we show that $\lim_{t\to\infty}z_1(t)=0$. Since $z_1$ is negative and non-decreasing, there exists $L_1$ such that $0\geq L_1 =\lim_{t\to\infty}z_1(t)>-\infty$. Let $\{t_k\}$ be a sequence such that \[ \lim_{k\to\infty}y_1(g(t_k))= \liminf_{t\to\infty}y_1(g(t))=0\,. \] Since $z_1(t)$ is negative so is $y_1(t)-a(t)y_1(g(t))$. From $y_1$ being positive, \[ -(-z_1(t_k))^{1/\beta} +a(t_k)y(g(t_k)) = y_1(t_k) > 0. \] In the limit as $k\to\infty$, and using that $a$ is bounded function, we have \[ 0\geq -(-L_1)^{1/\beta}+0=\liminf_{k\to\infty}y_1(g(t_k))\geq 0\,. \] Thus $L_1=0$. Next we show that $y_1(g)$ is bounded from above, which implies $y_1$ being bounded from above. Suppose that $y_1(g)$ is unbounded, then there is a sequence $\{t_k\}$ such that $\lim_{k\to\infty}y_1(g(t_k))=\infty$, and $y_1(g(s))\leq y_1(g(t_k))$ for all $s\leq t_k$. Since $g$ is strictly increasing, $y_1(g(s))\leq y_1(g(t_k))$ for all $s$ for which $g(s)\leq g(t_k)$. By (A2), for each $t_k$, there exists an $s$ such that $t_k=g(s)$. Then by \eqref{t12}, $t_k1$, and \[ \liminf_{k\to\infty}y_1(t_k) \leq \limsup_{k\to\infty}y_1(t_k) \leq \limsup_{t\to\infty}y_1(t) =\alpha \] From $z_1(t)$ and $y_1(t)-a(t)y_1(g(t))$ being negative, we have \[ -(-z_1(t_k))^{1/\beta}+a(t_k)y(g(t_k)) = y(t_k), \] which by taking the limit inferior yields \[ 0+a_*\alpha \leq \liminf_{k\to\infty}y_1(t_k) \leq \alpha\,. \] Since $a_*>1$, the only choice for $\alpha$ is being zero. Therefore, $\lim_{t\to\infty}y_1(t)=0$. By Lemma \ref{lem7}, $\lim_{t\to\infty}y_i(t)=0$ for $i=2,3,\dots, n$, which completes the proof. \end{proof} \begin{theorem}\label{thm3} Assume {\rm (A1)--(A4)}, and let $\sigma=(-1)^{n+1}$. Also assume the following conditions: there exist a constant $a^*$ such that \begin{gather}\label{t31} 0 < a(t)\leq a^*<1, \quad\text{for } t\geq t_0;\\ \label{t32} g(t) \frac {1}{e}\,. \end{gather} Then every non-oscillatory solution of \eqref{e1} decays to zero; i.e., $\lim_ {t\to \infty} y_i(t) =0$ for $i=1,2, \dots, n$. \end{theorem} \begin{proof} Let $y$ be a non-oscillatory solution of \eqref{e1}, and without loss of generality, assume that $y_1(t)$ is positive. Then by (A4), $y_n'$ is one sign and, by Lemma \ref{lem1}, each of the functions $z_1$, $y_2$, \dots is of one sign (positive or negative); thus we have only the following cases: \medskip \noindent\textbf{Case 1n:} $z_1(t)<0$ for all $t\geq t_2$, and no restriction on $y_2,y_3,\dots$. Since $z_1(t)$ is negative, so is $y_1(t)-a(t)y_1(g(t))$. By Lemma \ref{lem4}, $\lim_{t\to \infty}y_1(t)=0$. Then $\lim_{t\to \infty}z_1(t)=0$, because $a$ is bounded. Then by Lemma \ref{lem7}, $\lim_{t\to \infty}y_i(t)=0$ for $i=1,2,\dots,n$. \medskip \noindent \textbf{Case 1p2n:} $z_1(t)>0$, $y_2(t)<0$ for all $t\geq t_2$, and no restriction on $y_3,y_4,\dots$. Then by \eqref{eJ} we have $\ell=1$, and $y_2,y_4,y_6,\dots$ are negative, while $y_3,y_5,y_7,\dots$ are positive. However, (A4), the choice $\sigma=(-1)^{n+1}$, and the fact that $y_1>0$ do not allow this case to happen. See Theorem \ref{thm4} below. \medskip \noindent \textbf{Case 1p2p3p:} $z_1(t)>0$, $y_2(t)>0$ $y_3(t)>0$ for all $t\geq t_2$, and no restriction on $y_4,y_5,\dots$. Then $\ell\geq 3$ in Lemma \ref{lem1}. By \eqref{eIJ} \begin{equation} \label{eppp} \begin{aligned} z_1(t) &\geq I_{\ell-1}\Big(s, h(s) ;p_1,\dots,p_{\ell-1}; J_{n+1-\ell}\big(x_\ell,t; p_\ell, \dots, p_n; |\sigma f(y_1(h))|\big)\Big)\\ &\geq I_{\ell-1}\Big(s, h(s) ;p_1,\dots,p_{\ell-1}; J_{n+1-\ell}\big(x_\ell,t; p_\ell, \dots, p_n; M y_1^\beta(h) \big)\Big) \end{aligned} \end{equation} for all $s\leq t$. Since $z_1(t)$ is positive, so is $y_1(t)-a(t)y_1(g(t))$. Using the inequality $z_1^{1/\beta}(t)=y_1(t)-a(t)y_1(g(t))< y_1(t)$, we have $z_1(t)< y_1^\beta (t)$. Using that $z_1$ is non-decreasing, in \eqref{eppp}, we have \[ z_1(t)\geq z_1(h(s)) I_{\ell-1}\Big(s, h(s) ;p_1,\dots,p_{\ell-1}; J_{n+1-\ell}\big(x_\ell,t; p_\ell, \dots, p_n; M \big)\Big)\,. \] Since $s0$, $y_2(t)>0$, $y_3(t)<0$ for all $t\geq t_2$, and no restriction on $y_4,y_5,\dots$. Using \eqref{eJ} for $y_2$, we obtain \[ y_2(s)\geq J_{n-1}(s,t;p_2,\dots,p_n;|\sigma f(y_1(h))|) \geq J_{n-1}(s,t;p_2,\dots,p_n; y_1(h)M)\,. \] Using that $z_1(t)$ and $y_1(t)-a(t)y_1(g(t))$ are positive, we have the inequalities $(z_1(t))^{1/\beta}=y_1(t)-a(t)y_1(g(t))< y_1(t)$ and $z_1(t)< y_1^\beta(t)$. Using that $z_1$ is non-decreasing we have \[ y_2(s)\geq z_1(h(s)) J_{n-1}\big(s,t;p_2,\dots,p_n;M)\big)\quad \forall s\leq t\,; \] therefore, \[ y_2(s)\geq z_1(h(s)) J_{n-1}\big(s,\infty;p_2,\dots,p_n;M)\big)\,. \] Multiplying by $p_1(s)$ in both sides, we note that $z_1$ is a positive solution of the differential inequality \[ z_1'(s)- z_1(h(s)) J_{n-1}\big(s,\infty;p_2,\dots,p_n;M\big)\geq 0\,. \] Since $s\frac{1}{e}\,, \] which is provided by \eqref{t34}. The fact that $z_1$ is positive and is a solution of the differential inequality contradicts Lemma \ref{lem5}. Therefore, this case can not happen. The proof is complete. \end{proof} Next we remove the condition $\sigma=(-1)^n$ in Theorem \ref{thm3}, but we need to restrict the coefficient $p_n$. \begin{theorem} \label{thm4} Assume {\rm (A1)--(A4)}, \eqref{t31}--\eqref{t32}, and that $p_n$ is bounded below by a positive constant. Then every non-oscillatory solution of \eqref{e1} decays to zero; i.e., $\lim_ {t\to \infty} y_i(t) =0$ for $i=1,2, \dots, n$. \end{theorem} \begin{proof} Let $y_1$ be a non-oscillatory solution of \eqref{e1}, and without loss of generality assume that $y_1(t)$ is positive. The proofs of the various cases are the same as in Theorem \ref{thm3}, except for one case. \noindent \textbf{Case 1p2n:} $z_1(t)>0$, $y_2(t)<0$ for all $t\geq t_2$. Then by \eqref{eJ} we have $\ell=1$. Since $z_1(t)$ is positive, so is $y_1(t)-a(t)y_1(g(t))$. The proof of $\liminf_{t\to\infty}y_1(t)=0$ is the same as in Theorem \ref{thm3}. Next we show that $\lim_{t\to\infty}z_1(t)=0$. Since $z_1$ is positive and non-increasing, $\lim_{t\to\infty}z_1(t)$ exists. From the inequalities $z_1^{1/\beta}(t)=y(t)-a(t)y(g(t)) < y_1(t_k)$ and $z_1(t) 0\,. \] In the limit as $k\to\infty$, the left-hand side approaches zero, while the right-hand side approaches $+\infty$. This contradiction implies $y_1$ being bounded from above. Next we show that $\limsup_{t\to\infty}y_1(t)=0$. Let \[ \alpha:=\limsup_{t\to\infty}y_1(t)=\limsup_{t\to\infty}y_1(g(t))\geq 0, \] and let $\{t_k\}$ be a sequence such that $\lim_{k\to\infty}y_1(t_k)=\alpha$. Note that $\lim_{k\to\infty}z_1(t_k)=0$, $\limsup_{k\to\infty}a(t_k)\leq a^*<1$, and \[ \limsup_{k\to\infty}y_1(g(t_k)) \leq \limsup_{t\to\infty}y_1(g(t)) =\alpha\,. \] From $z_1(t)$ and $y(t)-a(t)y(g(t))$ being positive, we have $y_1(t_k)= z_1^{1/\beta}(t_k)+a(t_k)y(g(t_k)) $, which by taking in the limit superior, yields \[ \alpha=\lim_{k\to\infty}y_1(t_k)\leq 0+a^*\limsup_{k\to\infty}y_1(gt_k)) \leq a^*\alpha\,. \] Since $a^*<1$, the only choice for $\alpha$ is being zero. Therefore, $\lim_{t\to\infty}y_1(t)=0$. By Lemma \ref{lem7}, $\lim_{t\to\infty}y_i(t)=0$ for $i=2,3,\dots, n$, which completes the proof. \end{proof} \begin{example} \label{examp1} \rm To illustrate Theorem \ref{thm1}, we set $a(t)= 2$, $\beta=1$, $f(y)=y$, $g(t)= 4t$, $h(t)=8t$, $M=1$, $n=5$, $p_1(t)= t$, $p_2(t)= 3t$, $p_3(t)= 5t$, $p_4(t)= 7t$, $p_5 =36 t^{-9}$, and $\sigma=(-1)^5=-1$. Then for $t\geq 1$, a solution of \eqref{e1} has the form \[ y_1(t)=2t^{-1},\quad z_1(t)= y_2(t)= - 1 ^{-3},\quad y_3(t)= t ^{-5},\quad y_4(t)= -t ^{-7},\quad y_5(t)= t ^{-9}. \] Note that $z_1(t)=t^{-1}$, $g^{-1}(h(s))=2s$ and $p_5(x_6) M/a^\beta(g^{-1}(h(x_6)))= 18x_6^{-9}$. Then \begin{align*} &\int_s^{2s} x_2\int_{x_2}^\infty 3x_3\int_{x_3}^\infty 5x_4\int_{x_4}^\infty 7x_5\int_{x_5}^\infty 18x_6^{-9}\,dx_6\dots\,dx_2\\ &=\frac{18(1)(3)(5)(7)}{(2)(4)(6)(8)}\ln(2) > 1/e \end{align*} which satisfies \eqref{t14}. To check \eqref{t13}, we compute the expression \[ I_{k-1}\Big(s, 2s ;p_1,\dots,p_{k-1}; J_{6-k}\big(x_k, 2s; p_k, \\ \dots, p_5; 1/2\big)\Big) \] which has the following values: 2.00296 for $k=2$, 6.17293 for $k=3$, 14.8507 for $k=4$, and 34.7885 for $k=5$. Clearly all the conditions for Theorem \ref{thm1} are satisfied and the solution decays to zero as $t\to\infty$. \end{example} We remark that the results in Theorems \ref{thm1}--\ref{thm4} when the coefficient $a(t)$ crosses, or approaches, the value $1$ remains an open question. On the other hand, Theorems \ref{thm1}--\ref{thm4} can easily be extended to difference equation and to time scales; see the extensions indicated in \cite{Di}. \subsection*{Acknowledgments} The first two authors gratefully acknowledge the Scientific Grant Agency (VEGA) of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences for supporting this work under Grant No. 1/1245/12. \begin{thebibliography}{99} \bibitem{ErKoZh} L. H. Erbe, Q. Kong, B. G. Zhang; \emph{Oscillation Theory for Functional Differential Equations}, Marcel Dekker, New York, Basel, Hong Kong, 1995. \bibitem{Di} J. G. Dix; \emph{Oscillation of solutions to a neutral differential equation involving an n-order operator with variable coefficients and a forcing term} Differ. Equ. Dynamic Systems DOI 10.1007/s12591-013-0160-z. \bibitem{DoOl1} B. Dorociakov\'a, M. Kubjatkov\'a, R. Olach; \emph{Existence of positive solutions of neutral differential equations}, Abstract and applied analysis, (2012), ID 307968, 14 pages. \bibitem{DoOl2} B. Dorociakov\'a, A. Najmanov\'a, R. Olach; \emph{Existence of nonoscillatory solutions of first-order neutral differential equations}, Abstract and applied analysis, (2011), ID 346745, 9 pages. \bibitem{JaKu} J. Jaro\v s, T. Kusano: \emph{On a class of first order nonlinear functional differential equations of neutral type}, Czechoslovak Math. J. \textbf{40}, (115), (1990), 475-490. \bibitem{LaLaZh} G. S. Ladde, V. Lakshmikantham, B. G. Zhang; \emph{Oscillation theory of differential Equations with deviating arguments}, Marcel Dekker, Inc., New York and Basel, 1987. \bibitem{Ma1} P. Maru\v siak; \emph{Oscillatory properties of functional differential systems of neutral type}, Czechoslovak Math. J. \textbf{43}, (118), (1993), 649-662. \bibitem{Ma2} P. Maru\v siak; \emph{On unbonded nonoscillatory solutions of systems of neutral differential equations}, Czechoslovak Math. J., \textbf{42}, (117), (1992), 117--128. \bibitem{Mi1} B. Mihal\'\i kov\'a; \emph{Asymptotic behaviour of solutions of two-dimensional neutral differential systems}, Czechoslovak Math. J. \textbf{53}, (128), (2003), 735-741. \bibitem{Mih1} T. Mih\'aly; \emph{On the oscillatory and asymptotic properties of solutions of systems of neutral differential equations}, Nonlinear Analysis, \textbf{66}, (2007), 2053-2063. \bibitem{Ol} R. Ol\'ah; \emph{Oscillation of differential equation of neutral type}, Hirosh. Math. J., \textbf{25}, (1995), 1-10. \bibitem{ReSm} J. Rebenda, Z. \v Smarda; \emph{Stability and asymptotic properties of a system of functional differential equations with nonconstant delays}, Applied mathematics and Computation 219, (2013), 6622-6632. \bibitem{Sa1} H. \v Samajov\'a; \emph{Neutral two-dimensional functional differential systems -- asymptotic properties of solutions}, Studies of the University of \v Zilina, Math. Series, Vol. \textbf{19}, (2005), 49-56. \bibitem{Sp1} E. \v Sp\'anikov\'a; \emph{Oscillation of differential systems of neutral type}, Czechoslovak Math. J. \textbf{55}, (130), (2005), 263-271. \bibitem{Sp2} E. \v Sp\'anikov\'a; \emph{Asymptotic properties of solutions to three-dimensional functional differential systems of neutral type}, Electronic Journal of Differential Equations, \textbf{2005}, No. 47, (2005), 1-9. \bibitem{Sp3} E. \v Sp\'anikov\'a; \emph{Asymptotic properties of solutions of differential systems with deviating arguments}, \v Casopis pro p\v estov\'an\'i matematiky, Vol. \textbf{115}, No.2, (1990), 178-191. \bibitem{SpSa1} E. \v Sp\'anikov\'a, H. \v Samajov\'a; \emph{Asymptotic behaviour of non-oscillatory solutions of neutral differential systems}, Studies of the University of \v Zilina, Math. Series, Vol. \textbf{17}, (2003), 147-152. \bibitem{SpSa2} E. \v Sp\'anikov\'a, H. \v Samajov\'a; \emph{Asymptotic properties of solutions to n-dimensional neutral differential systems}, Nonlinear Analysis, \textbf{71}, (2009), 2877-2885. \bibitem{SaSp} H. \v Samajov\'a, E. \v Sp\'anikov\'a; \emph{On asymptotic behaviour of solutions to n-dimensional systems of neutral differential equations}, Abstract and applied analysis, (2011), ID 791323, 19 pages. \bibitem{St} S. Stan\v ek; \emph{Oscillation behaviour of solutions of neutral delay differential equations}, \v Casopis pro p\v estov\'an\'i matematiky, Vol. \textbf{115}, No.1, (1990), 92-99. \end{thebibliography} \end{document}