\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 185, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/185\hfil Oscillation of solutions]
{Oscillation of solutions to second-order nonlinear differential
equations of generalized Euler type}

\author[A. Aghajani, D. O'Regan, V. Roomi \hfil EJDE-2013/185\hfilneg]
{Asadollah Aghajani, Donal O'Regan, Vahid Roomi}  % in alphabetical order

\address{Asadollah Aghajani \newline
School of Mathematics, Iran University of Science and Technology,
P.O. box 16846-13114
Narmak, Tehran, Iran}
\email{aghajani@iust.ac.ir, Tel +9821-73225491, Fax +9821-73225891}

\address{Donal O'Regan \newline
School of Mathematics, Statistics and Applied Mathematics,
National University of Irland, Galway, Irland}
\email{donal.oregan@nuigalway.ie, Tel +353091 524411, Fax +353091 525700}

\address{Vahid Roomi \newline
School of Mathematics, Iran University of Science and Technology,
Narmak, Tehran, Iran}
\email{roomi@iust.ac.ir}

\thanks{Submitted May 30, 2013. Published August 10, 2013.}
\subjclass[2000]{34C10, 34A12}
\keywords{Oscillation; nonlinear differential equations; Lienard system}

\begin{abstract}
 We are concerned with the oscillatory behavior of the solutions of
 a generalized Euler differential equation where the nonlinearities
 satisfy smoothness conditions which guarantee the uniqueness of
 solutions of initial value problems, however, no conditions of
 sub(super) linearity are assumed. Some implicit necessary and
 sufficient  conditions and some explicit sufficient conditions are
 given for all nontrivial solutions of this equation to be
 oscillatory or nonoscillatory. Also, it is proved that solutions
 of the equation are all oscillatory or all nonoscillatory and
 cannot be both.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The Euler equation arises in some practical problems in  physics
and engineering. In particular there are many results in the
literature on the existence of oscillatory, periodic and almost
periodic solutions of Euler equation.

The oscillation problem for second-order nonlinear differential
equations has been studied in many papers;
see the references in this article.
In this article, we consider the second-order nonlinear differential 
equation of generalized Euler type
\begin{equation}\label{1}
t^{2}\ddot{u} + f(u)t \dot{u} + g(u) = 0\quad t > 0,
\end{equation}
and give some implicit necessary and sufficient conditions and
some explicit sufficient conditions for all nontrivial solutions
of this equation to be oscillatory. Here, $f(u)$ and $g(u)$
satisfy smoothness conditions which guarantee the uniqueness of
solutions of initial value problems and
\begin{equation}\label{2}
ug(u) > 0 \quad\text{if } u \ne 0.
\end{equation}

We suppose that all solutions of \eqref{1} are continuable in the
future. A nontrivial solution of \eqref{1} is said to be 
\emph{oscillatory} if it has arbitrarily large zeros. Otherwise, the
solution is said to be \emph{nonoscillatory}.
For brevity, equation \eqref{1} is called oscillatory 
(respectively nonoscillatory) if all nontrivial solutions are oscillatory 
(respectively nonoscillatory). Because of Sturm's separation theorem, the
solutions of second order linear differential equations are either
all oscillatory or all nonoscillatory, but not both. Thus, we can
classify second order linear differential equations into two
types. However, the oscillation problem for \eqref{1} is not easy,
because $g(u)$ and $f(u)$ are nonlinear.

If we let $f(u)=0$ and $g(u)=\lambda u$, then \eqref{1} is called
Euler differential equation. In this case, the number $1/4$ is
called the oscillation constant and it is the lower bound for all
nontrivial solutions of \eqref{1} to be oscillatory. In fact, it
is well known that if $\lambda>1/4$, then all nontrivial solutions
of \eqref{1} are oscillatory and otherwise they are
nonoscillatory. Other results on the oscillation constant for
linear differential equations can be found in \cite{g2,g3,g4,h1,k1,n1}
and the references cited therein.

Several authors consider oscillation of solutions of second-order
ordinary differential equations and some results can be found in
\cite{g1,g2,m1,s1,w2}.
Wong \cite{w2} studied the equation
\begin{equation}\label{3}
u''+a(t)g(u) = 0, \quad t > 0,
\end{equation}
which includes the Emden Fowler differential equation. He used
Sturm's comparison theorem and proved the following two
theorems:

\begin{theorem} \label{thmA}
Assume that $a(t)$ is continuously differentiable
and satisfies
\begin{equation}\label{4}
t^2a(t)\geq 1,
\end{equation}
for $t$ sufficiently large, and that there exists a $\lambda$ with
$\lambda>1/4$ such that
\begin{equation}\label{5}
\frac{g(u)}{u} \ge \frac{1}{4} + \frac{\lambda }{(\log |u|)^{2}},
\end{equation}
for $|u|$ sufficiently large. Then all nontrivial solutions of
\eqref{3} are oscillatory.
\end{theorem}

\begin{theorem} \label{thmB}
 Assume that $a(t)$ is continuously differentiable and satisfies
\begin{equation}\label{6}
0\leq t^2a(t)\leq 1,
\end{equation}
for $t$ sufficiently large and
\begin{equation}\label{7}
A(t):=\frac{a'(t)}{2a^\frac{3}{2}(t)}+1=o(t) \quad\text{as }
t\to \infty.
\end{equation}
If, in addition, $A(t)\leq 0$ and there exists a $\lambda$ with
$0<\lambda\leq1/16$ such that
\begin{equation}\label{8}
\frac{g(u)}{u} \le \frac{1}{4} + \frac{\lambda }{(\log |u|)^{2}},
\end{equation}
for $u>0$ or $u<0$, $|u|$ sufficiently large, then all nontrivial
solutions of \eqref{3} are nonoscillatory.
\end{theorem}

The oscillation problem for \eqref{1} (when $f(u)=0$) has been
solved  when
\begin{equation*}
\limsup_{|u|\to \infty}\frac{g(u)}{u}<\frac{1}{4} \ \ or \
\ \liminf_{|u|\to \infty}\frac{g(u)}{u}>\frac{1}{4}.
\end{equation*}

In \cite{a7} the authors gave sufficient conditions for all
nontrivial solutions of \eqref{1} to be oscillatory (when $f(u)=0$)
which can be applied in the case:
\begin{equation*}
\liminf_{|u| \to \infty}\frac{g(u)}{u}\leq\frac{1}{4}\leq
\limsup_{|u|\ \to \infty}\frac{g(u)}{u}.
\end{equation*}
In the next section, we will introduce a Li\'{e}nard system which
is equivalent to \eqref{1}. To study the oscillation problem for
\eqref{1} the significant point is to find conditions for deciding
whether all orbits intersect the vertical isocline $y=F(u)$ and
the $y$-axis in the equivalent Li\'{e}nard system.


\section{Equivalent Li\'{e}nard System and Property of $(X^+)$}

The change of variable $t = e^{s}$ reduces \eqref{1} to the
equation
\begin{equation}\label{16}
\ddot {u} + \dot {u} (f(u)-1)+ g(u) = 0, \quad s \in {\mathbb{R}},
\end{equation}
where  $\dot {} = \frac{d}{ds}$. Equation \eqref{16} is usually
studied by means of an equivalent plane differential system. The
most common one is
\begin{equation}\label{17}
\begin{gathered}
 \dot{u} =y-F(u) \\
 \dot{y} = -g(u),
\end{gathered}
\end{equation}
where $F(u)=\int_0^u f(\eta)d\eta-u$. This system is of
Li\'{e}nard type. Hereafter we denote $s$ by $t$ again.


\begin{definition} \label{def2.1} \rm
System \eqref{17} has property $(X^{+})$ in the right half plane
(resp. in the left half plane), if for every point
$P=(u_{0},y_{0})$ with $y_{0}>F(u_{0})$ and $u_{0}\geq0$ (resp.
$y_{0}<F(u_{0})$ and $u_{0}\leq0$), the positive semitrajectory of
\eqref{17} passing through $P$ denoted by
$\gamma^+(P):=\{(u(t),y(t))|t>t_0, \ (u(t_0),y(t_0))=P\}$ crosses
the vertical isocline $y=F(u)$.
\end{definition}

Several interesting sufficient conditions for property $(X^{+})$
have been presented in \cite{a2,a5,a6,a7,g3,g4,h1,v2}.
To study the oscillation problem for \eqref{1} we must
find conditions for deciding whether all orbits intersect the
vertical isocline $y=F(u)$ in the equivalent Li\'{e}nard system
\eqref{17} or not.
Let
\begin{equation*}
G(u)=\int_{0}^{u}g(\xi)d\xi.
\end{equation*}


Recently, in \cite{a6} the authors presented some sufficient
conditions for property $(X^{+})$ in the right and left half-plane
for system \eqref{17}.

\begin{theorem}[{\cite[Theorem 2.3]{a2}}] \label{thmC}
 Assume that
$G(+\infty)=+\infty$. Then, system \eqref{17} has property
$(X^{+})$ in the right half-plane if
\begin{equation}\label{33}
\limsup_{u\to+\infty}
\Big(\int_{b}^{u}\Big(\frac{F(\eta)g(\eta)}{(2G(\eta))^{3/2}}
+\frac{g(\eta)}{G(\eta)}\Big)d\eta
+\frac{F(u)}{\sqrt{2G(u)}}\Big)  =+\infty,
\end{equation}
for some $b>0$.
\end{theorem}

\begin{theorem}[{\cite[Theorem 4.3]{a2}}] \label{thmD}
 Assume that
$G(-\infty)=+\infty$. Then, system \eqref{17} has property
$(X^{+})$ in the left half-plane if
\begin{equation}\label{34}
\liminf_{u\to-\infty} \int_{u}^{b}\Big(
-\frac{F(\eta)g(\eta)}{(2G(\eta))^{3/2}}+\frac{g(\eta)}{G(\eta)}\Big)d\eta
+\frac{F(u)}{\sqrt{2G(u)}} =-\infty,
\end{equation}
for some $b<0$.
\end{theorem}

The following theorem, which is a modification of Theorem \ref{thm2.1} in
\cite{a5}, gives a necessary and sufficient condition for system
\eqref{17} to have property $(X^{+})$ in the right half-plane.

\begin{theorem}[\cite{g4}]\label{thm2.1}
System \eqref{17} fails to have property $(X^{+})$ in the right
half-plane if and only if there exist a constant $b\geq0$ and a
function $\varphi\in C^1([b,+\infty))$ such that $\varphi(u)>0$
for $u\geq b$ and
\begin{equation}\label{19}
g(u)\leq-\varphi(u)(F'(u)+\varphi'(u)),\quad\text{for } u\geq b.
\end{equation}
\end{theorem}

Similarly, the following analogous result is obtained with respect
to property $(X^{+})$ in the left half-plane for system
\eqref{17}.

\begin{theorem} \label{thm2.2}
System \eqref{17} fails to have property $(X^{+})$ in the left
half-plane if and only if there exist a constant $b\leq0$ and a
function $\varphi\in C^1((-\infty,b])$ such that $\varphi(u)>0$,
for $u\leq b$ and
\begin{equation}\label{21}
g(u)\geq-\varphi(u)(-F'(u)+\varphi'(u)), \quad\text{for } u\leq b.
\end{equation}
\end{theorem}

\begin{definition} \label{def2.2} \rm
Equation \eqref{1} has property $(X^{+})$ in the right half
plane (resp. in the left half plane), if system \eqref{17},
which is equivalent with \eqref{1}, has property $(X^{+})$ in
the right half plane (resp. in the left half plane).
\end{definition}

Thus we have the following two theorems.

\begin{theorem} \label{thm2.3}
Equation \eqref{1} fails to have property $(X^{+})$ in the right
half-plane if and only if there exist a constant $b\geq0$ and a
function $\varphi\in C^1([b,+\infty))$ such that $\varphi(u)>0$
for $u\geq b$ and
\begin{equation}\label{22}
g(u)\leq-\varphi(u)(f(u)-1+\varphi'(u)),\quad\text{for } u\geq b.
\end{equation}
\end{theorem}

\begin{theorem} \label{thm2.4}
Equation \eqref{1} fails to have property $(X^{+})$ in the left
half-plane if and only if there exist a constant $b\leq0$ and a
function $\varphi\in C^1((-\infty,b])$ such that $\varphi(u)>0$,
for $u\leq b$ and
\begin{equation}\label{23}
g(u)\geq-\varphi(u)(1-f(u)+\varphi'(u)), \ \ \ for \ \ u\leq b.
\end{equation}
\end{theorem}

\section{Explicit Conditions for Property of $(X^+)$}

In this section we present some explicit sufficient conditions for
equation \eqref{1} having property $(X^{+})$ in the right
half-plane.

\begin{corollary} \label{coro3.1}
Suppose that $G(+\infty)=+\infty$ and
$\liminf_{u\to \infty}f(u)>1$. Then  equation \eqref{1} has property
 $(X^{+})$ in the right half-plane.
\end{corollary}

\begin{proof}
 By way of contradiction, we suppose that equation
\eqref{1} fails to have property $(X^{+})$ in the right
half-plane. Therefore by Theorem \ref{thm2.3} there exist a constant
$b\geq 0$ and a function $\varphi\in C^1([b,+\infty))$ such that
$\varphi(u)>0$ for $u\geq b$ and
\begin{equation}\label{eq 3.1}
g(u)\leq-\varphi(u)(f(u)-1+\varphi'(u)).
\end{equation}
However, since $\liminf_{u\to \infty}f(u)>1$ there exists
$b'>0$ such that $f(u)>1$ for $u>b'$. Now from \eqref{eq 3.1} for
$u>max\{b, b'\}=b''$ we have
\[
-g(u)\geq \varphi(u) \varphi'(u).
\]
By integration we have
\[
-G(u)+G(b'') \geq \frac{\varphi^2(u)}{2}-\frac{\varphi^2(b'')}{2},
\]
for $u$ sufficiently large. This is a contradiction since
$G(+\infty)=+\infty$ and the proof is complete.
\end{proof}

\begin{corollary} \label{coro3.2}
Suppose that $\limsup_{u\to \infty}f(u)=\lambda <1$. Then
equation \eqref{1} fails to have property $(X^{+})$ in the right
half-plane if
\begin{equation}\label{eq 2.10}
\limsup_{u\to \infty}\frac{g(u)}{u} <
\frac{(1-\lambda)^2}{4}.
\end{equation}
\end{corollary}

\begin{proof}
By \eqref{eq 2.10} there exist $\beta$ with $\lambda
< \beta < 1$ and $b>0$ such that
\begin{equation*}
\frac{g(u)}{u} < \frac{(1-\beta)^2}{4} \quad\text{for } u\geq b.
\end{equation*}
However, since $\limsup_{u\to \infty}f(u)=\lambda$ there
exists a $b'>0$ such that $f(u)<\beta$ for $u>b'$. Now let
$\varphi(u)=(\frac{1-\beta}{2})u$. For $u\geq \max\{b, b'\}$ we
have
\begin{align*}
\frac{-\varphi(u)(f(u)-1+\varphi'(u))}{u}
&=-\frac{(1-\beta)}{2}
\big(-1+f(u)+\frac{1-\beta}{2} \big) \\
& \geq -\frac{(1-\beta)}{2}
\big(-1+\beta+\frac{1-\beta}{2} \big)\\
&= \frac{(1-\beta)^2}{4}> \frac{g(u)}{u}.
\end{align*}
From Theorem \ref{thm2.3} this equation fails to have property $(X^+)$ in
the right half-plane.
\end{proof}

\begin{example} \label{examp3.1} \rm
Consider  \eqref{1} with
\begin{gather*}
f(u)=\beta \sin u + \frac{1}{u^2+1}, \quad  \beta<1,\\
g(u)=u^{\gamma}\tan^{-1} u+\alpha u, \quad
\alpha<\frac{(1-\beta)^2}{4}, \;  0<\gamma<1.
\end{gather*}
Since  $\limsup_{u\to \infty}f(u)=\beta<1$ and
\[
\limsup_{u\to \infty}\frac{g(u)}{u}
=\limsup_{u\to \infty}\frac{tan^{-1} u}{u^{1-\gamma}}+\alpha
=\alpha<\frac{(1-\beta)^2}{4},
\]
by  Corollary \ref{coro3.2} this equation does not have property $(X^+)$ in
the right half-plane.
\end{example}

\begin{corollary} \label{coro3.3}
Suppose that $\limsup_{u\to \infty}f(u)<\beta <1$. Then
\eqref{1} fails to have property $(X^{+})$ in the right
half-plane, if there exist $\lambda$ with
 $0\leq \lambda < (1-\beta)^2/16$ and a $k>0$ such
\begin{equation}\label{eq 2.11}
\frac{g(u)}{u} \leq
\frac{(1-\beta)^2}{4}+\frac{\lambda}{(log(ku))^2},
\end{equation}
for $u$ sufficiently large.
\end{corollary}

\begin{proof}
 Define $\varphi(u)=\frac{1-\beta}{2} u+\frac{\alpha
u}{log(ku)}$ where $ k>0 $ and $\alpha$ is a constant such that
$\lambda<\frac{1-\beta}{2}\alpha-\alpha^2<\frac{(1-\beta)^2}{16}$,
(notice that $\max_{\alpha\in
\mathbb{R}}(\frac{1-\beta}{2}\alpha-\alpha^2)=\frac{(1-\beta)^2}{16}$).
For $u$ sufficiently large we have
\begin{align*}
&\frac{-\varphi(u)(f(u)-1+\varphi'(u))}{u}\\
&=\Big(\frac{1-\beta}{2}+\frac{\alpha }{\log(ku)}
\Big)\Big(1-\frac{1-\beta}{2} -\frac{\alpha \log(ku)
-\alpha}{(\log(ku))^2}-f (u)\Big) \\
&\geq\Big(\frac{1-\beta}{2}+\frac{\alpha }{\log(ku)}
\Big)\Big(1-\frac{1-\beta}{2}-\frac{\alpha
\log(ku)-\alpha}{(\log(ku))^2}-\beta\Big)\\
& =\frac{(1-\beta)^2}{4}+
\frac{\frac{1-\beta}{2}\alpha-\alpha^2}{(\log(ku))^2}
+\frac{\alpha^2}{(\log(ku))^3}\\
& > \frac{(1-\beta)^2}{4}+\frac{\lambda}{(\log(ku))^2}\geq\frac{g(u)}{u}.
\end{align*}
From Theorem \ref{thm2.3} this equation fails to have property $(X^+)$ in
the right half-plane.
\end{proof}

\begin{corollary} \label{coro3.4}
Suppose that $\limsup_{u\to \infty}f(u)=0$. Then
\eqref{1} fails to have property $(X^{+})$ in the right
half-plane, if there exist $\lambda$ with $0<\lambda <1/16$,
and a $k>0$ such that
\begin{equation}\label{28}
\frac{g(u)}{u} \leq \frac{1}{4}+\frac{\lambda}{(\log(ku))^2},
\end{equation}
for $u$ sufficiently large.
\end{corollary}

\begin{corollary} \label{coro3.5}
Suppose that there exists a constant $b>0$ such that
$g(u)$ is in $C^1([b,+\infty))$. Then \eqref{1} fails to have
property $(X^{+})$ in the right half-plane, if
\begin{equation*}
g'(u)\leq -f(u),
\end{equation*}
for $u$ sufficiently large.
\end{corollary}

\begin{proof}
 For $u$ sufficiently large let $\varphi(u)=g(u)$.
Note that from \eqref{2},
$g'(u)\leq -f(u)$ implies
$g(u)\leq -g(u)(f(u)-1+g'(u))$,
so
\[
g(u)\leq -\varphi(u)(f(u)-1+\varphi'(u)),
\]
for $u$ sufficiently large. Thus, if $g'(u)\leq -f(u)$ by
Theorem \ref{thm2.3} this equation fails to have property $(X^+)$ in the right
half-plane.
\end{proof}

In \cite{a5} the authors proved the following lemma.

\begin{lemma} \label{lem3.1}
Let $\lambda >1/16$ and $k>0$. Then there does not exist a
function $\varphi(u)$ such that $\varphi \in C^1(\mathbb{R})$,
$\varphi(u)> 0$ for $u$ sufficiently large, and
\begin{equation}\label{30}
\varphi(u)(1-\varphi'(u)) \geq \frac{1}{4}u+\frac{\lambda
u}{(\log(ku))^2},
\end{equation}
for $u$ sufficiently large.
\end{lemma}
From the above lemma, immediately we have the following result.

\begin{lemma} \label{lem3.2}
Let $\lambda >1/16$, $k>0$ and $f(u)\geq 0$ for $u$
sufficiently large. Then there does not exist a function
$\varphi(u)$ such that $\varphi \in C^1(\mathbb{R})$,
$\varphi(u)> 0$ for $u$ sufficiently large, and
\begin{equation}\label{32}
\varphi(u)(1- f(u) - \varphi'(u)) \geq \frac{1}{4}u+\frac{\lambda
u}{(\log(ku))^2},
\end{equation}
for $u$ sufficiently large.
\end{lemma}

\begin{corollary} \label{coro3.6}
Let $\lambda >\frac{1}{16}$, $k>0$ and $f(u)\geq 0$ for $u$
sufficiently large. Then equation $\eqref{1}$ has property
$(X^{+})$ in the right half-plane if
\begin{equation}\label{31}
\frac{g(u)}{u} \geq \frac{1}{4}+\frac{\lambda}{(\log(ku))^2},
\end{equation}
for $u$ sufficiently large.
\end{corollary}

\begin{remark} \label{rmk3.6} \rm
Corollaries \ref{coro3.1}, \ref{coro3.2}, \ref{coro3.3},
\ref{coro3.4}, \ref{coro3.5} and \ref{coro3.6} can be formulated for
 property $(X^+)$ in the left half-plan.
\end{remark}

We say that system \eqref{17} has property $(Z_1^{+})$ (resp.,
$(Z_3^{+})$) if there exists a point $P(u_{0},y_{0})$ with
$y_{0}=F(u_0)$ and $u_0\geq 0$ (resp.,  $u_0\leq0$)  such that the
positive semitrajectory of \eqref{17}  starting at $P$  approaches
the origin through only the first (resp., third) quadrant. Now we
consider the following theorem and corollary concerning property
$(Z_1^{+})$  which was established in \cite{a4}.

\begin{theorem}[\cite{a4}] \label{thm3.1}
Suppose that there exists a $\delta>0$ such that $F(u)>0$ for
$0<u<\delta$. If for every $ k\in[0,1] $ there exists a constant
 $ \gamma_{k}>0$  such that
\begin{equation}\label{e3.3}
\liminf_{u\to 0^+}\Big(\frac{\int_0^u
\frac{g(\eta)}{F(\eta)}d\eta}{(1-k+\gamma_{k})
(k+\gamma_{k})F(u)}\Big)>1,
\end{equation}
then \eqref{17} fails to have property $(Z_1^{+})$.
\end{theorem}

\begin{corollary}[\cite{a4}] \label{coro3.7}
Suppose that there exists a $\delta>0$ such that $F(u)>0$ for
$0<u<\delta$. If there exists a  $ \xi$ with $ 1<\xi\leq 2 $ such
that
\begin{equation}\label{eq 3.4}
\liminf_{u\to 0^+}\Big(\frac{\int_0^u
\frac{g(\eta)}{F(\eta)}d\eta}{\xi F(u)}\Big)>1,
\end{equation}
then the system \eqref{17} fails to have property $(Z_1^{+})$.
\end{corollary}

Using Theorem \ref{thm3.1}, we prove  the following  lemma about the
asymptotic behavior of system \eqref{17}.

\begin{lemma} \label{lem3.3}
For each point $P(p,F(p))$ with $p>0$, the positive semitrajectory
of \eqref{17} starting at $P$ crosses the negative $y$-axis if one
of the following conditions hold.
\begin{itemize}
\item[(i)]    There exists a $\delta>0$ such that
$F(u)<0$ for $0<u<\delta$ or $F(u)$ has an infinite number of
positive zeroes clustering at $u=0$.

\item[(ii)]   If there exists a $\delta>0$ such
that $F(u)>0$ for $0<u<\delta$ and the conditions of Theorem
\ref{thm3.1} hold.
\end{itemize}
\end{lemma}

\begin{proof}
Suppose that there exists a point $P(p,F(p))$ with
$p>0$ such that the positive semitrajectory of \eqref{17} starting
at $P$ does not intersect the negative $y$-axis. Let $(u(t),y(t))$
be a solution of \eqref{17} defined on an interval $[t_0,\infty)$
with $(u(t_0),y(t_0))=P$. Then the solution $(u(t),y(t))$
corresponds to the positive semitrajectory of \eqref{17} starting
at $P$. At $t=t_0$ we have
\[
\dot{u}(t_0)=0 ,\quad \dot{y}(t_0)=-g(p)<0.
\]
So, this positive trajectory enters the region
$D:=\{(u,y): y<F(u),\,  u>0\}$. Taking into account the vector field of
\eqref{17} we have
\begin{equation*}
\dot{u}<0 \quad\text{in }  D.
\end{equation*}
Thus, by the assumption that this trajectory does not intersect
the negative $y$-axis we have
\begin{equation*}
0<u(t)\leq u(t_0) \quad\text{for }t\geq t_0.
\end{equation*}
On the other hand, $\dot{y}<0$ in $D$. So, if
$y(t)$ does not approach $-\infty$ it must tends to $0$ (note that the
origin is the only equilibrium point of \eqref{17}), which is
impossible in both cases (i) and (ii). Therefore,
\begin{equation*}
y(t)\to-\infty  \quad\text{as } t\to +\infty.
\end{equation*}
Hence, it follows from the first equation of \eqref{17}  that
\begin{equation*}
\dot{u}(t)\to-\infty \quad\text{as } t\to\infty,
\end{equation*}
and therefore, there exists a $t_1>t_0$ such that
\begin{equation*}
\dot{u}(t)\leq -1 \quad\text{for } t\geq t_1.
\end{equation*}
Integration  leads to
\begin{equation*}
-u(t_1)<u(t)-u(t_1)\leq t_1-t\to-\infty \quad\text{as }
t\to\infty.
\end{equation*}
This is a contradiction,  the proof is complete.
\end{proof}

Turning our attention to the left half-plane and using
the change of variables $(u,y)\to (-u,-y)$, we can
formulate Theorem \ref{thm3.1} and Corollary \ref{coro3.7} for
property $(Z_3^{+})$ as follows.

\begin{theorem}\label{thm3.2}
Suppose that there exists a $\delta>0$ such that $F(u)<0$ for
$-\delta<u<0$. If for every $ k\in[0,1] $ there exists a constant
$ \gamma_{k}>0 $ such that
\begin{equation}\label{eq 3.5}
\liminf_{u\to 0^-}\Big(\frac{\int_0^u
\frac{g(\eta)}{F(\eta)}d\eta}{(1-k+\gamma_{k})
(k+\gamma_{k})F(u)}\Big)>1,
\end{equation}
then the system \eqref{17} fails to have property $(Z_3^{+})$.
\end{theorem}

\begin{corollary}\label{coro3.8}
Suppose that there exists a $\delta>0$ such that $F(u)<0$
for $-\delta<u<0$. If there exists a  $ \xi$ with $ 1<\xi\leq 2 $ such
that
\begin{equation}\label{eq 3.6}
\liminf_{u\to 0^-}\Big(\frac{\int_0^u
\frac{g(\eta)}{F(\eta)}d\eta}{\xi F(u)}\Big)>1,
\end{equation}
then  \eqref{17} fails to have property $(Z_3^{+})$.
\end{corollary}

\begin{lemma} \label{lem3.4}
For each point $P(-p,F(-p))$ with $p>0$, the positive
semitrajectory of \eqref{17} starting at $P$ crosses the positive
$y$-axis if one of the following conditions hold.
\begin{itemize}
\item[(i)]  There exists a $\delta>0$ such that
$F(u)>0$ for $-\delta<u<0$ or $F(u)$ has infinite number of
negative zeroes clustering at $u=0$.

\item[(ii)]
 If there exists a $\delta>0$ such
that $F(u)<0$ for $-\delta<u<0$ and the conditions of Theorem
\ref{thm3.2} hold.
\end{itemize}
\end{lemma}

\section{Oscillation Theorems}

In this section we present implicit necessary and sufficient
condition for all nontrivial solutions of this system to be
oscillatory or nonoscillatory. To do this, we need the following
two lemmas. In the proof of these lemmas we follow the ideas used
in \cite[Lemmas 4.1 and 4.2]{s3}.

\begin{lemma} \label{lem4.1}
Every solution of \eqref{17} is unbounded, except the zero
solution, if $uF(u)<0$.
\end{lemma}

\begin{proof}
By way of contradiction, we suppose that there
exists a bounded solution $(u(\zeta),y(\zeta))$ of \eqref{17}
initiating at $\zeta=\zeta_0>0$; that is,
\begin{equation}\label{24}
|u(\zeta)|+|y(\zeta)| \leq A \quad\text{for } \zeta \geq \zeta_0,
\end{equation}
with  $A>0$. Then the solution $(u(\zeta),y(\zeta))$ circles
clockwise around the origin.
Let $\zeta_i>\zeta_0$ and $b_i>0$ with
\begin{equation*}
(u(\zeta_i),y(\zeta_i))=(0,b_i) \quad\text{for } i=1, 2.
\end{equation*}
Define a Liapunov function
\begin{equation*}
V(u,y)=\frac{1}{2}y^2 +G(u).
\end{equation*}

From $uF(u)<0$ and \eqref{2}, we have
\begin{equation*}
\dot{V}_{\eqref{17}}(u,y)=-F(u)g(u)>0 \quad\text{if } u \neq 0.
\end{equation*}
If $\zeta_1<\zeta_2$, then $b_1<b_2$. In fact, we have
\begin{equation*}
\frac{1}{2}b_1^2=V(u(\zeta_1),y(\zeta_1))<V(u(\zeta_2),y(\zeta_2))
=\frac{1}{2}b_2^2.
\end{equation*}
Thus, if this solution starts at a point $(0,b_1)$ on the
$y$-axis, it circles clockwise around the origin and intersects
the $y$-axis again at a point $(0,b_2)$ with $b_2>b_1$. Thus, it
can not tend to the origin. Now by Poincare-Bendixson theorem
\cite{c2} its $\omega$-limit set is a periodic orbit. Therefore,
we can conclude that, there exists a simple closed curve $C$
surrounding the origin such that
\begin{equation}\label{25}
\operatorname{dist}\big\{(u(\zeta),y(\zeta)),C \big\}\to 0 \quad
\text{as }\zeta\to \infty.
\end{equation}
Let $\delta_0>0$ be sufficiently small and define
\begin{equation*}
M=\max \big\{g(u) : \delta_0 \leq u \leq A \big\}, \quad
m=\min \big\{-F(u)g(u) : \delta_0 \leq u \leq A \big\}.
\end{equation*}
By \eqref{25} the solution $(u(\zeta),y(\zeta))$ does not stay in
$\big\{(u,y) : |u| <\delta_0 \big\}$. Hence, using the fact that
$(u(\zeta),y(\zeta))$ circles clockwise around the origin and
tends to $C$, there exist sequences $\{t_n \}$ and $\{t'_n \}$
with $\zeta_0<t_n<t'_n<t_{n+1}$ and $t_n\to\infty$ as
$n\to\infty$ such that
\begin{equation*}
u(t_n)=u(t'_n)=\delta_0, \quad y(t_n)>\delta_0,
y(t'_n)<-\delta_0
\end{equation*}
and
$u(\zeta)>\delta_0$ for $t_n<\zeta<t'_n$.
We have
\begin{equation*}
-2\delta_0>y(t'_n)-y(t_n)=-\int_{t_n}^{t'_n}g(u(\zeta))d\zeta
\geq-M(t'_n-t_n),
\end{equation*}
and therefore, for $\zeta>t'_n$,
\begin{align*}
V(u(\zeta),y(\zeta))- V(u(\zeta_0),y(\zeta_0))
&= -\int_{\zeta_0}^\zeta F(u(\eta))g(u(\eta))d\eta  \\
&\geq  \sum_{k=1}^n \int_{t_k}^{t'_k} -F(u(\zeta))g(u(\zeta))d\zeta \\
&\geq m \sum_{k=1}^n (t'_k - t_k)\geq \frac{2m \delta_0}{M}n,
\end{align*}
which tends to $\infty$ as $n\to\infty$. Thus,
$V(u(\zeta),y(\zeta)) \to +\infty$ as $\zeta \to +\infty$.
This contradicts \eqref{24} and completes the proof.
\end{proof}

Consider the  Liapunov function
\begin{equation*}
V(u,y)=\frac{1}{2}y^2 +G(u)
\end{equation*}
and consider the curve
\begin{equation*}
V(u,y)= V(u_0,y_0),
\end{equation*}
where $u_0>0$.

It is obvious that if $uF(u)<0$, $F'(u)\leq 0$ and from  \eqref{2}
that $V(u,F(u))$ is increasing for $u>0$ and decreasing for $u<0$,
and $V(0,0)=0$. Thus, the equation
\begin{equation*}
V(u,F(u))= V(u_0,y_0)
\end{equation*}
has exactly two roots. Therefore, there exist two points of
intersection of this curve with the curve $y=F(u)$.

Let $(-a,F(-a))$ and $(-b,F(-b))$ be the intersection points,
where $a>0$ and $b>0$. Define
\begin{equation*}
S=\big\{(u,y): -a\leq u \leq c , V(u,y)\leq V(u_0,y_0) \big\}
\end{equation*}
in which $c=\max \{b,u_0 \}$. Then it is clear that $S$ is a
bounded set. Lemma \ref{lem4.1} shows that every solution of \eqref{17}
starting in $S\setminus \{ 0 \}$ does not remain in $S$. Note that
\begin{equation*}
\dot{V}_{\eqref{17}}(u,F(u))= -F(u)g(u)>0 \quad\text{if } u\neq 0.
\end{equation*}
Then we also see that every solution of \eqref{17} starting in
$S^c$, the complement of $S$ in $\mathbb{R}^2$, stays in $S^c$ for
all future time.
Therefore, we have the following lemma.

\begin{lemma} \label{lem4.2}
Suppose
\begin{equation}\label{e3.3b}
uF(u)<0, \quad  F'(u)\leq 0.
\end{equation}
Then every solution of \eqref{17} starting in
$S\setminus \{0 \}$ enters $S^c$ which is a positive invariant set with respect to \eqref{17}.
\end{lemma}

\begin{remark} \label{rmk4.1} \rm
Lemma \ref{lem4.2} holds even if the condition $F'(u)\leq 0$ is replaced by
$F'(u)<\frac{-g(u)}{F(u)}$.
\end{remark}

The main theorem of this section is as follows.

\begin{theorem} \label{thm4.1}
Suppose \eqref{e3.3} holds. All nontrivial solutions of
\eqref{17} are nonoscillatory if there exist a constant $R>0$
and a function $\varphi \in C^1(\mathbb{R}-(-R,R))$ such that
\begin{equation}\label{26}
\varphi(|u|)>0 , \quad\text{and}\quad
\frac{g(u)}{u}\leq \frac{-\varphi(|u|)(F'(u)+\varphi'(|u|))}{|u|},
\end{equation}
for $u>R$ or $u<-R$. Otherwise, all nontrivial solutions of
\eqref{17} are oscillatory. Therefore, the solutions of
second-order nonlinear differential equation \eqref{1} are all
oscillatory or all nonoscillatory and cannot be both.
\end{theorem}

\begin{proof}
 First suppose that \eqref{26} does not hold. Then
 \eqref{17} which is equivalent to \eqref{1} has property
$(X^+)$ in the right and left half-plan. Thus, it follows from
Lemmas \ref{lem3.3} and \ref{lem3.4} that every solution of \eqref{17} keeps on
rotating around the origin except the zero solution. Hence, all
nontrivial solutions of \eqref{1}
are oscillatory.

Now suppose that \eqref{26} holds. Then by Theorems 
\ref{thm2.1} or \ref{thm2.2},
system \eqref{17} fails to have property $(X^+)$ in the right or
left half-plane. We consider only the case that \eqref{17} fails
to have property $(X^+)$ in the right half-plane, because the
proof in the other case is similar. Hence, there exists a point
$P_0(u_0,y_0)$ with $u_0\geq 0$ and $y_0>F(u_0)$ such that
positive semitrajectory of \eqref{17} runs to infinity without
intersecting the
curve $y=F(u)$.

Suppose \eqref{17} has an oscillatory solution. Let $\gamma^+(Q)$
be the positive semitrajectory which corresponds to the
oscillatory solution of \eqref{1} starting from point $Q$. By
virtue of Lemma \ref{lem4.2} we see that $\gamma^+(Q)$ eventually goes
around the set $S$ infinitely many times. Therefore, it crosses
the half-line $\{(u,y): u=u_0 \ and \ y>y_0 \}$ at a point
$P_1(u_0,y_1)$ with $y_1>y_0$. From the uniqueness of solution for
the initial value problem, it turns out that
\begin{itemize}
\item[(i)] $\gamma^+(Q)$ coincides with $\gamma^+(P_1)$ except for the arc
$QP_1$

\item[(ii)] $\gamma^+(P_1)$ lies above $\gamma^+(P_0)$
\end{itemize}
Hence, $\gamma^+(Q)$ runs to infinity without crossing the curve
$y=F(u)$. This contradicts the fact that $\gamma^+(Q)$ circles the
set $S$ and completes the proof.  
\end{proof}

\begin{example} \label{examp4.1} \rm
Consider \eqref{17} with
\[
F(u)=-u^3-u \sin^2u,\quad 
g(u)=\alpha |u|^m \operatorname{sgn}(u)+\sin u ,
\]
for $  |u| $  sufficiently  large  with $ \alpha>1 $ and $ m\in \mathbb{R} $.
It is obvious that $F(u)$ satisfies \eqref{e3.3b} and for
$u>0$,
\[
\frac{g(u)}{u} =\alpha u^{m-1}+\frac{\sin u}{u}.
\]
On the other hand, for $u>0$ by choosing $\varphi(u)=u^\beta$ with
$\beta<3$ we have
\begin{equation*}
\frac{-\varphi(|u|)(F'(u)+\varphi'(|u|))}{|u|}=3 u^{\beta+1} +
u^{\beta-1}\sin^2u + u^{\beta}\sin 2u - \beta u^{2\beta-2}.
\end{equation*}
Therefore, if $m<\beta+2$ or if $m=\beta+2$ and $\alpha<3$, then
there exist a constant $R>0$ such that for $u>R$ condition
\eqref{26} holds. Thus by Theorem \ref{thm4.1} all nontrivial solutions
of this system are nonoscillatory.
\end{example}

\begin{remark} \label{rmk4.2} \rm
Theorem \ref{thm4.1} holds even if the condition $F'(u)\leq 0$ replaced by
$F'(u)<-g(u)/F(u)$.
\end{remark}

The following result follows from the above theorem.

\begin{theorem} \label{thm4.2}
All nontrivial solutions of \eqref{1} are nonoscillatory if
there exist a constant $R>0$ and a function $\varphi \in
C^1(\mathbb{R}-(-R,R))$ such that
\begin{equation}\label{27}
\varphi(|u|)>0 \quad\text{and}\quad 
\frac{g(u)}{u}\leq \frac{-\varphi(|u|)((f(u)-1)+\varphi'(|u|))}{|u|},
\end{equation}
for $u>R$ or $u<-R$. Otherwise, all nontrivial solutions of
\eqref{1} are oscillatory. Therefore, the solutions of
second-order nonlinear differential equation \eqref{1} are all
oscillatory or all nonoscillatory and cannot be both.
\end{theorem}

\begin{remark} \label{rmk4.3} \rm
Corollaries 3.1, 3.2, 3.3, 3.4, 3.5 and 3.6 can be formulated for
the solutions of equation $\eqref{1}$ to be nonoscillatory or
oscillatory.
\end{remark}

The following corollaries of Theorems \ref{thm4.1} and \ref{thm4.2} are very useful
in applications.

\begin{corollary} \label{coro4.1}
Suppose that all nontrivial solutions of \eqref{1} with $g_1$
are oscillatory (resp. nonoscillatory). If
\begin{equation*}
\frac{g_2(u)}{u}>\frac{g_1(u)}{u}, \quad \text{(resp. 
$\frac{g_2(u)}{u}<\frac{g_1(u)}{u}$,)}
\end{equation*}
for $|u|>R$ with a sufficiently large $R$, then all nontrivial
solutions of \eqref{1} with $g_2$ are oscillatory (resp.
nonoscillatory).
\end{corollary}

\begin{corollary} \label{coro4.2}
Suppose that all nontrivial solutions of \eqref{1} with $f_1$
are oscillatory (resp. nonoscillatory). If
\begin{equation*}
f_1(u)>f_2(u),\quad\text{(resp. $f_1(u)<f_2(u)$,)}
\end{equation*}
for $|u|>R$ with a sufficiently large $R$, then all nontrivial
solutions of \eqref{1} with $f_2$ are oscillatory (resp.
nonoscillatory).
\end{corollary}



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\end{document}