\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
\begin{equation} \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}
\end{equation}
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
\begin{equation}\label{2}
 z_1(t) =y_1(t)-a(t)y_1(g(t)).
\end{equation}
\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
\begin{equation}\label{3}
 y_1(t) z_1(t)>0
\end{equation}
or
\begin{equation}\label{4}
 y_1(t) z_1(t)<0
\end{equation}
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
\begin{equation}\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}
\end{equation}
or
\begin{equation}\label{6}
\begin{gathered}
y_i(t)z_1(t) >0 ,\quad i=1,2,3.
\end{gathered}
\end{equation}
\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
\begin{equation}\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}
\end{equation}
or
\begin{equation}\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{equation}
\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
\begin{equation}\label{9}
\quad  N=N_1^+ \cup N_3^+ \cup N_2^- \cup N_3^- .
\end{equation}

\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
$$
|y_2(t)|\geq M,\quad  0<M= \hbox{const.} \quad
\hbox {for } t\geq t_3.
$$
Integrating the first equation of
(\ref{1}) from $s$ to $t$ and using the last inequality we get
\begin{equation}\label{10}
 |z_1(t)| - |z_1(s)| \geq M
 \int _s^tp_1(u)du,\quad t>s\geq t_3.
\end{equation}
 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)<h(t)<t<\alpha (t) \quad \hbox {for $t\geq t_0$,}
\end{gather}
 where  $\alpha: [t_0,\infty)\to \mathbb{R} $ is a
continuous function,
\begin{gather}\label{13}
\limsup _{t \to \infty} \int _t^{h^{-1}(t)}K P_{2,1}(u,t)p_3(u)\,du >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
\begin{equation}\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.
\end{equation}
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
\begin{equation}\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)\,.
\end{equation}
This equation implies
\begin{equation}\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,
\end{equation}
for $s>t\geq t_2$. From (\ref{16}) in regard to (\ref{15}),
(e) and the third equation of (\ref{1}), we get
\begin{equation}\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.
\end{equation}
Since $z_1(t)\leq y_1(t)\ $ for $t\geq t_2$, it follows that
\begin{equation}\label{18}
 z_1(h(t))\leq y_1(h(t))\quad \hbox {for} \quad \ t\geq t_3,
 \end{equation}
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
\begin{equation}\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.
\end{equation}
Integrating $\int _t^s P_2(u,t)y'_3(u)\,du$ by
parts we derive the integral identity
\begin{equation}\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.
\end{equation}
 From (\ref{20}) with regard to (\ref{19}), (e) and the third equation
of (\ref{1}) we get
\begin{equation}\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.
\end{equation}
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)<h(t) \quad \hbox {for }t\geq t_0 ; \\
\label{25}
t<\alpha (t),\quad  \hbox {where $\alpha: [t_0,\infty)\to \mathbb{R}$
is a continuous function}
\end{gather}
and (\ref{14}) holds. Then for every non-oscillatory solution,
$y\in W$ of \eqref{1} with $y_1(t)$ bounded, we have $\lim _ {t
\to \infty} y_i(t) =0$, $i=1,2,3$.
\end{theorem}

\begin{proof}
Let $y \in W$ be a non-oscillatory solution of (\ref{1}) 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 $t\geq t_2$. \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.

 Let $\lim _ {t \to \infty } y_2(t) = -S,\ 0<S=\hbox
{const.}$ Then
\begin{equation}\label{26}
y_2(t)\leq -S,\quad t\geq t_2.
\end{equation}
Integrating the first equation of (\ref{1}) from $t_2$ to $t$ and
using (\ref{26}) we get
\begin{equation}\label{27}
z_1(t)-z_1(t_2) \leq -S\int _{t_2}^t p_1(s)\,ds,\quad t\geq t_2.
\end{equation}
 From this inequality and (c) we have
$\lim _ {t \to \infty } z_1(t) = -\infty $ which  contradicts
$z_1(t)>0 $ for $t\geq t_2$ and so $\lim _ {t \to \infty }y_2(t) = 0$.

 Let $\lim _ {t \to \infty } y_3(t) = P,\ 0<P=\hbox{const.}$ Then
\begin{equation}\label{28}
y_3(t)\geq P,\quad t\geq t_2.
\end{equation}
Integrating the second equation of (\ref{1}) from $t_2$ to $t$ and
using (\ref{28}) we get
\begin{equation}\label{29}
 y_2(t)-y_2(t_2) \geq P\int _{t_2}^t p_2(s)\,ds,\quad
t\geq t_2.
\end{equation}
 From (\ref{29}) and (c) we have
$\lim _ {t \to \infty } y_2(t) = \infty $ and that contradicts
$y_2(t)<0   $ for $t\geq t_2$ and so $\lim _ {t \to \infty }
y_3(t) = 0. $ \smallskip

\noindent (II) Let $y\in N_2^-$ on $[t_2,\infty)$. Analogously as
in the case (II) of the proof of Theorem~\ref{th1} we can show
that $N_2^-=\emptyset$. The proof is complete.
\end{proof}


\subsection*{Example} 
Consider the system
\begin{equation}\label{30}
\begin{gathered}
  \big[y_1(t)-2y_1(3t)\big]'=t y_2(t), \\
y_2'(t)=t y_3(t),\\
y_3'(t)=- 45 t^{-5} y_1(9t),
\quad t\geq t_0>0.
\end{gathered}
\end{equation}
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
\begin{equation}\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.
\end{equation}
In this case,
$$
y_2(t)\geq M,\quad  0<M= \hbox {const.} \quad
\hbox {for }  t\geq t_2.
$$
Integrating the first equation of (\ref{1}) from $s$ to $t$ and using
the last inequality we get
\begin{equation}\label{34}
  z_1(t) - z_1(s) \geq M  \int _s^tp_1(u)du,\quad t>s\geq t_2.
 \end{equation}
 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 $\ z_1(t)<y_1(t)$
which is a contradiction. $N_3^+ =\emptyset$. \smallskip

\noindent (III) Let $y\in N_2^-$ on $[t_2,\infty)$. Analogously as
in the case (II) of the proof of Theorem~\ref{th1} we can show that
$N_2^- =\emptyset$.\smallskip

\noindent (IV) Let $y\in N_3^-$ on $[t_2,\infty)$. In this case
\begin{equation}\label{35}
 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.
\end{equation}
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
\begin{equation}\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,
\end{equation}
for $t>s\geq t_2$. From (\ref{36}) in regard to (\ref{35}), (e)
and the third equation of (\ref{1}) we get
\begin{equation}\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.
\end{equation}
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
\begin{equation}\label{38}
 y_1(h(t))\geq {z_1(g^{-1}(h(t)))\over -a(g^{-1}(h(t)))} ,
\quad  t\geq t_2.
\end{equation}
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}

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\end{document}