\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 47, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/47\hfil Asymptotic properties of solutions] {Asymptotic properties of solutions to three-dimensional functional differential systems of neutral type} \author[E. \v Sp\'anikov\'a\hfil EJDE-2005/47\hfilneg] {Eva \v Sp\'anikov\'a} \address{ Eva \v Sp\'anikov\'a \hfill\break Department of Appl. Mathematics, University of \v Zilina, J. M. Hurbana 15, 010 26 \v Zilina, Slovakia} \email{eva.spanikova@fstroj.utc.sk} \date{} \thanks{Submitted February 12, 2005. Published April 27, 2005.} \thanks{Supported by grant VEGA No. 2/3205/23 of Scientific Grant Agency of Ministry of Education \hfill\break\indent of Slovak Republic and Slovak Academy of Sciences} \subjclass[2000]{34K25, 34K40} \keywords{Differential system of neutral type; asymptotic properties of solutions} \begin{abstract} In this paper, we study the behavior of solutions to three-dimensional functional differential systems of neutral type. We find sufficient conditions for solutions to be oscillatory, and to decay to zero. The main results are presented in three theorems and illustrated with one example. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} We consider neutral functional differential systems $$\label{1} \begin{gathered} \big[y_1(t)-a(t)y_1(g(t))\big]'=p_1(t)y_2(t), \\ y_2'(t)=p_2(t)y_3(t), \\ y_3'(t)=-p_3(t)f(y_1(h(t))), \quad t\geq t_0 . \end{gathered}$$ The following conditions are assumed: \begin{itemize} \item[(a)] $a : [t_0,\infty ) \to (0,\infty ]$ is a continuous function; \item[(b)] $g: [t_0,\infty ) \to \mathbb{R}$ is a continuous and increasing function and $\lim _ {t \to \infty } g(t) = \infty$; \item[(c)] $p_i : [t_0,\infty ) \to [0,\infty )$, $i=1,2,3$ are continuous functions; $p_3$ not identically equal to zero in any neighbourhood of infinity, $\int^\infty p_j(t)\,dt =\infty$, $j=1,2$; \item[(d)] $h : [t_0,\infty ) \to \mathbb{R}$ is a continuous and increasing function and $\lim _{t \to \infty }h(t) = \infty$; \item[(e)] $f:\mathbb{R} \to \mathbb{R}$ is a continuous function, $uf(u) > 0$ for $u \not = 0$ and $|f(u)|\geq K|u|$, where $K$ is a positive constant. \end{itemize} For $t_1 \geq t_0$, we define $$\widetilde t_1 = \min \{t_1, g(t_1), h(t_1) \}\,.$$ A function $y=(y_1,y_2,y_3)$ is a solution of the system (\ref{1}) if there exists a $t_1\geq t_0$ such that $y$ is continuous on $[\widetilde t_1, \infty )$, $y_1(t)-a(t)y_1(g(t)),y_i(t)$, $i=2,3$ are continuously differentiable on $[t_1,\infty)$ and $y$ satisfies (\ref{1}) on $[t_1, \infty )$. Denote by $W$ the set of all solutions $y=(y_1, y_2,y_3)$ of the system (\ref{1}) which exist on some ray $[T_y,\infty) \subset [t_0,\infty )$ and satisfy $$\sup \big\{ \sum_{i=1}^3 |y_i(t)|:t \geq T \big\} > 0 \quad \hbox {for any } T \geq T_y.$$ A solution $y \in W$ is considered to be non-oscillatory if there exists a $T_y \geq t_0$ such that every component is different from zero for $t \geq T_y$. Otherwise a solution $y \in W$ is said to be oscillatory. The purpose of this article is to study asymptotic properties of solutions to the three-dimensional functional differential systems of neutral type (\ref{1}) and also the special asymptotic properties of solutions whose first component is bounded. The asymptotic and oscillatory properties of solutions to differential systems with deviating arguments has been studied for example in the papers \cite{FoWe, IvMa, KiKu, Mi2, Sp, SpSa}. For a $y_1(t)$, we define $$\label{2} z_1(t) =y_1(t)-a(t)y_1(g(t)).$$ \vskip 2mm \noindent Denote \begin{gather*} P_1(s,t)= \int_t^s p_1(x)\,dx ,\quad P_{1,2}(s,t)= \int_t^s p_1(v) \int_t^v p_2(x) \, dx\,dv,\\ P_2(s,t)= \int_t^s p_2(x)\,dx ,\quad P_{2,1}(s,t)= \int_t^s p_2(v) \int_t^v p_1(x) dx\,dv, \quad s\geq t \geq t_0. \end{gather*} \section{Classification of non-oscillatory solutions} \begin{lemma}[{\cite[Lemma 1]{Ma2}}] \label{le1} Let $y \in W$ be a solution of \eqref{1} with $y_1(t) \neq 0$ on $[t_1, \infty )$, $t_1 \geq t_0$. Then $y$ is non-oscillatory and $z_1(t), y_2(t), y_3(t)$ are monotone on some ray $[T,\infty)$, $T\geq t_1$. \end{lemma} Let $y \in W$ be a non-oscillatory solution of (\ref{1}). From (\ref{1}) and (c) it follows that the function $z_1(t)$ from (\ref{2}) has to be eventually of constant sign, so that either $$\label{3} y_1(t) z_1(t)>0$$ or $$\label{4} y_1(t) z_1(t)<0$$ for sufficiently large $t$. Assume first that (\ref{3}) holds. From \cite[Lemma 4]{Ma2} it follows the statement in Lemma \ref{le2}. \begin{lemma}\label{le2} Let $y = (y_1,y_2,y_3 ) \in W$ be a non-oscillatory solution of \eqref{1} on $[t_1,\infty )$ and that \eqref{3} holds. Then there exists a $t_2\geq t_1$ such that for $t\geq t_2$ either $$\label{5} \begin{gathered} y_1(t)z_1(t) >0 \\ y_2(t)z_1(t) <0 \\ y_3(t)z_1(t)>0 \end{gathered}$$ or $$\label{6} \begin{gathered} y_i(t)z_1(t) >0 ,\quad i=1,2,3. \end{gathered}$$ \end{lemma} Denote by $N_1^+$ the set of non-oscillatory solutions of (\ref{1}) satisfying (\ref{5}), and by $N_3^+$ the non-oscillatory solutions of (\ref{1}) satisfying (\ref{6}). Now assume that (\ref{4}) holds. With the aid of the Kiguradze's Lemma is easy to prove Lemma \ref{le3}. \begin{lemma}\label{le3} Let $y = (y_1,y_2,y_3 ) \in W$ be a non-oscillatory solution of \eqref{1} on $[t_1,\infty )$ and \eqref{4} holds. Then there exists a $t_2\geq t_1$ such that for $t\geq t_2$ either $$\label{7} \begin{gathered} y_1(t)z_1(t) <0 \\ y_2(t)z_1(t) > 0 \\ y_3(t)z_1(t) < 0 \end{gathered}$$ or $$\label{8} \begin{gathered} y_1(t)z_1(t) <0 \\ y_i(t)z_1(t) >0 ,\quad i=2,3. \end{gathered}$$ \end{lemma} Denote by $N_2^-$ the sets of non-oscillatory solutions of (\ref{1}) satisfying (\ref{7}), and by $N_3^-$ the non-oscillatory solutions of (\ref{1}) satisfying (\ref{8}). Denote by $N$ the set of all non-oscillatory solutions of (\ref{1}). Obviously by Lemmas \ref{le2} and \ref{le3}, we have $$\label{9} \quad N=N_1^+ \cup N_3^+ \cup N_2^- \cup N_3^- .$$ \begin{lemma}\label{le4} Suppose that $a(t)$ is bounded on $[t_2,\infty )$ and $y \in W$ be a non-oscillatory solution of the system (\ref{1}) with $y_1(t)$ bounded on $[t_2,\infty )$, $t_2 \geq t_0$. Then $$y\in N_1^+ \cup N_2^- .$$ \end{lemma} \begin{proof} We must show that the set $N_3^+\cup N_3^-$ is empty. Let $y \in W$ be a non-oscillatory solution of (\ref{1}) with $y_1(t)$ bounded on $[t_2,\infty )$ and $y\in N_3^+\cup N_3^-.$ Without loss of generality we suppose that $y_1(t)>0$ on $[t_2, \infty)$. Because $a(t)$ and $y_1(t)$ are bounded, $z_1(t)$ is bounded on $[t_3, \infty)$, where $t_3\geq t_2$ is sufficiently large. If $y\in N_3^+\cup N_3^-$ then a function $|y_2(t)|$ is nondecreasing and $$|y_2(t)|\geq M,\quad 0s\geq t_3. From (\ref{10}) and (c) we have \lim _ {t \to \infty } |z_1(t)| = \infty . This contradicts the fact that z_1(t) is bounded and N_3^+\cup N_3^- =\emptyset . The proof is complete. \end{proof} \begin{lemma}[{\cite[Lemma 2.2]{JaKu}}] \label{le5} In addition to the conditions \textrm{(a)} and \textrm{(b)} suppose that$$ 1 \leq a(t) \quad \hbox {for } t\geq t_0. $$Let y_1(t) be a continuous non-oscillatory solution of the functional inequality$$ y_1(t)[y_1(t)-a(t)y_1(g(t))]>0 $$defined in a neighbourhood of infinity. Suppose that g(t)>t for t\geq t_0. Then y_1(t) is bounded. If, moreover,$$ 1<\lambda _{\star}\leq a(t),\quad t\geq t_0 $$for some positive constant \lambda _{\star}, then \lim _ {t \to \infty } y_1(t) = 0. \end{lemma} \begin{lemma}[{\cite[Lemma 4]{Ol}}] \label{le6} Assume that q: [t_0,\infty )\to [0,\infty ) and \delta : [t_0,\infty )\to \mathbb{R} are continuous functions, \lim _{t \to \infty }\delta (t) = \infty, \delta (t) > t for t\geq t_0, and$$ \liminf _ {t\to \infty } \int_t^{\delta (t)} q(s)\,ds > \frac{1}{e} . $$Then the functional inequality$$ x'(t) - q(t)x(\delta (t)) \geq 0,\quad t\geq t_0, $$has no eventually positive solution, and$$ x'(t) - q(t)x(\delta (t)) \leq 0,\quad t\leq t_0 $$has no eventually negative solution. \end{lemma} \section{Oscillation theorems} \begin{theorem}\label{th1} Suppose that \begin{gather}\label{11} a(t) \hbox { is bounded for t\geq t_0,}\\ \label{12} g(t)1, \\ \label{14} \liminf _{t \to \infty} \int _t ^{g^{-1}(h(t))} p_1(v)\int _v^{\alpha(v)} {K P_2(u,v) p_3(u) \,du\,dv\over {a(g^{-1}(h(u)))}} > \frac {1}{e}, \end{gather} where g^{-1}(t) is the inverse function of g(t). Then every solution y = (y_1, y_2, y_3)\in W of \eqref{1} with y_1(t) bounded is oscillatory. \end{theorem} \begin{proof} Let y \in W be a non-oscillatory solution of (\ref{1}) with y_1(t) bounded. From Lemma~\ref{le4} we have y\in N_1^+\cup N_2^- on [t_2,\infty). Without loss of generality we may suppose that y_1(t) is positive for t\geq t_2. \noindent I) Let y\in N_1^+ on [t_2,\infty). In this case $$\label{15} y_1(t)>0,\ \ z_1(t)>0,\ \ y_2(t)<0,\ \ y_3(t)>0 \quad \hbox {for } t\geq t_2.$$ Integrating \int _t^s P_{2,1}(u,t)y'_3(u)\,du by parts with  f(u)= P_{2,1}(u,t), g(u)=y_3(u), and one gets$$ \int_t^s P_{2,1}(u,t)~y'_3(u)\,du = P_{2,1}(s,t)~y_3(s) - \int_t^s P_{1}(u,t)~y'_2(u)\,du. $$Integrating by parts again with  f(u)= P_{1}(u,t), g(u)=y_2(u), we have $$\label{doda4} \int _t^s P_{2,1}(u,t)~y'_3(u)\,du = P_{2,1}(s,t)~y_3(s) - P_{1}(s,t)~y_2(s) + z_1(s) - z_1(t)\,.$$ This equation implies $$\label{16} z_1(t)=z_1(s)-P_1(s,t)y_2(s)+P_{2,1}(s,t)y_3(s)- \int _t^s P_{2,1}(u,t)y'_3(u)\,du,$$ for s>t\geq t_2. From (\ref{16}) in regard to (\ref{15}), (e) and the third equation of (\ref{1}), we get $$\label{17} z_1(t)\geq \int _t^s KP_{2,1}(u,t)p_3(u) y_1(h(u))\,du, \quad s>t\geq t_2.$$ Since z_1(t)\leq y_1(t)\  for t\geq t_2, it follows that $$\label{18} z_1(h(t))\leq y_1(h(t))\quad \hbox {for} \quad \ t\geq t_3,$$ where t_3\geq t_2  is sufficiently large. Combining (\ref{17}) and (\ref{18}) we have$$ z_1(t)\geq \int _t^s K P_{2,1}(u,t)p_3(u) z_1(h(u))\,du ,\quad s>t\geq t_3. $$Putting s=h^{-1}(t)  and using the monotonicity of z_1(h(u)) from the previous inequality we obtain \begin{gather*} z_1(t)\geq z_1(t)\int _t^{h^{-1}(t)} KP_{2,1}(u,t)p_3(u) \,du ,\quad\ t\geq t_3 ;\\ 1\geq \int _t^{h^{-1}(t)} KP_{2,1}(u,t)p_3(u) \,du ,\quad t\geq t_3, \end{gather*} which contradicts (\ref{13}) and N_1^+ =\emptyset. \smallskip \noindent(II) Let y\in N_2^- on [t_2,\infty). In this case $$\label{19} y_1(t)>0,\quad z_1(t)<0,\quad y_2(t)<0,\quad y_3(t)>0 \quad \hbox {for} \quad t\geq t_2.$$ Integrating \int _t^s P_2(u,t)y'_3(u)\,du by parts we derive the integral identity $$\label{20} y_2(t)=y_2(s)-P_2(s,t)y_3(s) +\int _t^s P_2(u,t)y'_3(u)\,du, \quad s>t\geq t_2.$$ From (\ref{20}) with regard to (\ref{19}), (e) and the third equation of (\ref{1}) we get $$\label{21} y_2(t)\leq -\int _t^s KP_2(u,t)p_3(u) y_1(h(u))\,du, \quad s>t\geq t_2.$$ Because \ z_1(t)> -a(t)y_1(g(t)) for t\geq t_2 it follows \begin{gather} z_1(g^{-1}(h(t)))> -a(g^{-1}(h(t))) y_1(h(t)) ; \nonumber\\ \label{22} -y_1(h(t))<{z_1(g^{-1}(h(t)))\over a(g^{-1}(h(t)))}\quad \hbox {for } t\geq t_2. \end{gather} Combining (\ref{21}) and (\ref{22}), we have$$ y_2(t)\leq \int _t^s {K P_2(u,t)p_3(u) z_1(g^{-1}(h(u)))\,du \over a(g^{-1}(h(u)))} ,\quad s>t\geq t_2. $$Multiplying the last inequality by p_1(t), using the first equation of (\ref{1}) and the monotonicity of  z_1(g^{-1}(h(t))) we get$$ z'_1(t) \leq \Big[ p_1(t)\int _t^s {KP_2(u,t)p_3(u)\,du \over a(g^{-1}(h(u)))} \Big] z_1(g^{-1}(h(t))) ,\quad s>t\geq t_2. $$Let s=\alpha (t) and so$$ z'_1(t) - \Big[ p_1(t)\int _t^{\alpha (t)} {KP_2(u,t)p_3(u)\,du \over a(g^{-1}(h(u)))} \Big] z_1(g^{-1}(h(t)))\leq 0,\quad t\geq t_2. $$By Lemma \ref{le6} and condition (\ref{14}), the last inequality has no eventually negative solution, which is a contradiction and N_2^-=\emptyset. The proof is complete. \end{proof} \begin{theorem}\label{th2} Suppose that \begin{gather}\label{23} 1<\lambda _{\star}\leq a(t)\leq c, \quad \mbox{for t\geq t_0 and some constants \lambda _{\star}, c};\\ \label{24} t< g(t)0  for t\geq t_2 and so \lim _ {t \to \infty }y_2(t) = 0. Let \lim _ {t \to \infty } y_3(t) = P,\ 00. \end{gathered} In this example a(t)=2, g(t)=3t, h(t)=9t, p_1(t)=p_2(t)=t,  p_3(t)=45 t^{-5}, f(t)=t, K=1, P_2(u,v)={1\over 2} (u^2-v^2). We chose \alpha (t)=2t  and calculate the condition (\ref{14}) as follows$$ \liminf _{t \to \infty} {45\over 4} \int _t ^{3t} v\int _v^{2v} (u^2-v^2) u^{-5}\,du\,dv ={405\over 256} \ln 3 . $$All conditions of Theorem \ref{th2} are satisfied. Then for every non-oscillatory solution y\in W of (\ref{30}) with y_1(t) bounded, it holds$$\lim _ {t \to \infty } y_1(t) = \lim _ {t \to \infty } y_2(t) = \lim _ {t \to \infty } y_3(t) =0. $$For instance functions$$ y_1(t)={1\over t} ,\quad y_2(t)={-1\over 3t^3} ,\quad y_3(t)={1\over{ t^5}} ,\quad t\geq t_0$$are such a kind of solutions. \begin{theorem}\label{th3} Suppose that \begin{gather}\label{31} 1<\lambda _{\star}\leq a(t), \quad \mbox{t\geq t_0 for some constant \lambda _{\star}}; \\ \label{32} \limsup _{t \to \infty} \int _{h^{-1}(g(t))}^t {K P_{1,2}(t,u)p_3(u)\,du\over {a(g^{-1}(h(u)))}} >1\,. \end{gather} and (\ref{14}), (\ref{24}) and (\ref{25}) hold. Then for every non-oscillatory solution y\in W of \eqref{1}, it holds \lim _ {t\to \infty} y_i(t) =0, i=1,2,3. \end{theorem} \begin{proof} Let y \in W be a non-oscillatory solution of (\ref{1}). From (\ref{9}) we have y\in N_1^+ \cup N_3^+ \cup N_2^- \cup N_3^-  on [t_2,\infty). Without loss of generality we may suppose that y_1(t) is positive for t\geq t_2. \smallskip \noindent (I) Let y\in N_1^+ on [t_2,\infty). In this case (\ref{15}) holds. By Lemma \ref{le5} it follows that \lim _ {t \to \infty } y_1(t) = 0. We prove, that \lim _ {t \to \infty } y_2(t) =\lim _ {t\to \infty } y_3(t) = 0 indirectly analogously as in the case (I) of the proof of Theorem~\ref{th2}. \smallskip \noindent (II) Let y\in N_3^+ on [t_2,\infty). In this case $$\label{33} y_1(t)>0,\quad z_1(t)>0,\quad y_2(t)>0,\quad y_3(t)>0 \quad \hbox {for } t\geq t_2.$$ In this case,$$ y_2(t)\geq M,\quad 0s\geq t_2. From (\ref{34}) and (c) we have $\lim _ {t \to \infty } z_1(t) = \infty$ and the function $z_1(t)$ is unbounded. From (\ref{2}), (\ref{24}), (\ref{31}) we have $$y_1(t)>a(t)y_1(g(t))>y_1(g(t)),$$ which implies that $y_1(t)$ is bounded, but $\ z_1(t)0,\quad z_1(t)<0,\quad y_2(t)<0,\quad y_3(t)<0 \quad \hbox {for } t\geq t_2. By interchanging the order of integrating in$P_{1,2}(t,u)$, we have $$\int _s^t P_{1,2}(t,u)y'_3(u)\,du = \int _s^t \Big( \int _u^t p_{2}(x) \int _x^t p_1(v)\,dv\,dx \Big) y'_3(u)\,du$$ and integrating$ \int _s^t P_{1,2}(t,u)y'_3(u)\,du $by parts with$f(u)= \int _u^t p_{2}(x) \int _x^t p_1(v)\,dv\,dx$,$g(u)=y_3(u)$, we get $$\int _s^t P_{1,2}(t,u)y'_3(u)\,du = - P_{1,2}(t,s) y_3(s) + \int_s^t P_{1}(t,u) y'_2(u)\,du.$$ Integrating by parts again with$f(u)= P_{1}(t,u)$,$g(u)=y_2(u)$, one gets $$\int _s^t P_{1,2}(t,u)y'_3(u)\,du = - P_{1,2}(t,s) y_3(s) - P_{1}(t,s) y_2(s) - z_1(s) + z_1(t).$$ From the equation about, we derive the integral identity $$\label{36} z_1(t)=z_1(s)+P_1(t,s)y_2(s)+P_{1,2}(t,s)y_3(s)+ \int _s^t P_{1,2}(t,u)y'_3(u)\,du,$$ for$t>s\geq t_2$. From (\ref{36}) in regard to (\ref{35}), (e) and the third equation of (\ref{1}) we get $$\label{37} -z_1(t)\geq \int _s^t KP_{1,2}(t,u)p_3(u) y_1(h(u))\,du, \quad t>s\geq t_2.$$ Since$z_1(t)\geq -a(t)y_1(g(t))$for$t\geq t_2$it follows that $$y_1(g(t))\geq {z_1(t)\over -a(t)}\quad \hbox {for } t\geq t_2.$$ From the above inequality we have $$\label{38} y_1(h(t))\geq {z_1(g^{-1}(h(t)))\over -a(g^{-1}(h(t)))} , \quad t\geq t_2.$$ Combining (\ref{37}) and (\ref{38}) we have $$-z_1(t)\geq \int _s^t {-K P_{1,2}(t,u)p_3(u) z_1(g^{-1}(h(u)))\,du \over a(g^{-1}(h(u)))} , \quad t>s\geq t_2.$$ Putting$s=h^{-1}(g(t)) $and using the monotonicity of$z_1(g^{-1}(h(u)))$from the last inequality we get $$-z_1(t)\geq -z_1(t)\int _{h^{-1}(g(t))}^t {KP_{1,2}(t,u)p_3(u) \,du \over a(g^{-1}(h(u)))} ,\quad t\geq t_3\,,$$ where$t_3\geq t_2$is sufficiently large and $$1\geq \int _{h^{-1}(g(t))}^t {KP_{1,2}(t,u)p_3(u) \,du \over a(g^{-1}(h(u)))}, \quad t\geq t_3,$$ which contradicts \eqref{32} and$N_3^- =\emptyset$. The proof is complete. \end{proof} \begin{thebibliography}{99} \bibitem{FoWe} {I. Foltynska and J. Werbowski:} \emph{On the oscillatory behaviour of solutions of system of differential equations with deviating arguments}, Colloquia Math. Soc. J. B. {\bf30}, Qualitative theory of Diff. Eq. Szeg\'ed, (1979), 243-256. \bibitem{IvMa} { A. F. Ivanov and P. Maru\v siak:} \emph{Oscillatory properties of systems of neutral differential equations}, Hiroshima Math. J. {\bf 24} (1994), 423-434. \bibitem{JaKu} { J. Jaro\v s and T. Kusano:} \emph{On a class of first order nonlinear functional differential equations of neutral type}, Czechoslovak Math. J. {\bf 40}, (115) (1990), 475-490. \bibitem{KiKu} { Y. Kitamura and T. Kusano:} \emph{On the oscillation of a class of nonlinear differential systems with deviating argument}, J. Math. Annal. and Appl.{\bf 66}, (1978), 20-36. \bibitem{Ma1} { P. Maru\v siak:} \emph{Oscillation criteria for nonlinear differential systems with general deviating arguments of mixed type}, Hiroshima Math. J. {\bf 20} (1990) 197-208. \bibitem{Ma2} { P. Maru\v siak:} \emph{Oscillatory properties of functional differential systems of neutral type}, Czechoslovak Math. J. {\bf 43}, (118) (1993), 649-662. \bibitem{Mi1} { B. Mihal\'\i kov\'a:} \emph{A note on the asymptotic properties of systems of neutral differential equations}, Proceedings of the International Scientific Conference of Mathematics, University of \v Zilina, (2000), 133-139. \bibitem{Mi2} { B. Mihal\'\i kov\'a:} \emph{Asymptotic behaviour of solutions of two-dimensional neutral differential systems}, Czechoslovak Math. J. 53, (128) (2003), 735-741. \bibitem{Ol} { R. Olach:} \emph{Oscillation of differential equation of neutral type}, Hirosh. Math. J. 25 (1995), 1-10. \bibitem{Sp} { E. \v Sp\'anikov\'a :} \emph{Oscillatory properties of the solutions of three-dimensional differential systems of neutral type}, Czechoslovak Math. J. 50, (125) (2000), 879-887. \bibitem{SpSa} { E. \v Sp\'anikov\'a and H. \v Samajov\'a:} \emph{Asymptotic behaviour of the nonoscillatory solutions of differential systems of neutral type},$2^{nd}\$ International Conference APLIMAT 2003, Bratislava, (2003), 671-676. \end{thebibliography} \end{document}