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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 165, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/165\hfil Continuability and boundedness of solutions]
{Continuability and boundedness of solutions to nonlinear
second-order differential equations}

\author[L. Wang, R. McKee, L. Usyk\hfil EJDE-2010/165\hfilneg]
{Lianwen Wang, Rhonda McKee, Larysa Usyk}

\address{Lianwen Wang \newline
Department of Mathematics and Computer Science\\
University of Central Missouri\\
Warrensburg, MO 64093, USA}
\email{lwang@ucmo.edu}

\address{Rhonda McKee \newline
Department of Mathematics and Computer Science\\
University of Central Missouri\\
Warrensburg, MO 64093, USA}
\email{mckee@ucmo.edu}

\address{Larysa Usyk \newline
Department of Mathematics and Computer Science\\
University of Central Missouri\\
Warrensburg, MO 64093, USA}
\email{lmu58230@ucmo.edu}


\thanks{Submitted February 27, 2010. Published November 17, 2010.}
\subjclass[2000]{34C11, 34C12}
\keywords{Nonlinear differential equations; second order;
boundedness; \hfill\break\indent
monotonicity; continuability; asymptotic properties}

\begin{abstract}
 Continuability, boundedness, and monotonicity of solutions for a
 class of second-order nonlinear differential equations are
 discussed. It is proved that all solutions are eventually
 monotonic and can be extended to infinity under some natural
 assumptions. Moreover, necessary and sufficient conditions for
 boundedness of all solutions are established. The results obtained
 have extended and improved some analogous existing ones.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In this article we consider the continuability, boundedness, and
monotonicity of solutions for the second-order nonlinear
differential equation
\begin{equation}\label{e1}
[p(t)h(x(t))f(x'(t))]' = q(t)g(x(t)),\quad t \ge a.
\end{equation}
The behavior, such as continuability, boundedness, monotonicity,
osciallation, and asymptoticity, of solutions to second-order
differential equations
\begin{gather}\label{e2}
[p(t)x'(t)]' = q(t)g(x(t)),\quad t \ge a, \\
[p(t)h(x(t))x'(t)]' = q(t)g(x(t)),\quad t \ge a,
\end{gather}
both are special cases of \eqref{e1} with $f(r) = r$, has been
extensively discussed by many authors; see, e.g.,
\cite{cmv89,cmv95, che91, m84, ny92} and references therein.
In the case of $f(r)= \Phi_p(r) = |r|^{p-2}r$,
$p> 1$, the so-called $p$-Laplacian operator, that are for
half-linear equations
\begin{equation}\label{e4}
[p(t)\Phi_p(x'(t))]' = q(t)\Phi_p(x(t)),\quad  t \ge a,
\end{equation}
Emden-Fowler type equations
\begin{equation}\label{e5}
[p(t)\Phi_p(x'(t))]' = q(t)\Phi_\beta(x(t)),\quad  t \ge a,
\end{equation}
and more general equations
\begin{equation}\label{e6}
[p(t)\Phi_p(x'(t))]' = q(t)g(x(t)),\quad t \ge a.
\end{equation}
A considerable effort has been devoted to the study
of continuability, boundedness, monotonicity, and asymptoticity
of solutions due to their various applications;
see for example \cite{cdkm07, cdmv06, cdm05, cdm00, cdm01, dr05,
ku00, ku01, l02, m93, mnu02, t99}.

 Wang \cite{w09}  discussed properties of solutions for   \eqref{e1}
with general increasing functions $f$ in the case $h \equiv 1$;
the results in \cite{w09} extended and improved many results
obtained for  \eqref{e4}, \eqref{e5}, and \eqref{e6}. However,
we discovered that nontrivial functions $h$ play an important
role to the continuability, boundedness, and monotonicity for
solutions of \eqref{e1}. The main contribution of this
article is to address the role of the functions $h$ in the
discussion of the continuability and boundedness for solutions
of \eqref{e1}; see Theorems \ref{th3}, \ref{th4}, and \ref{th7}.
For example, from \cite{kc93} we know that the assumption (H3)
(see Section 2) can not be omitted for the continuability and
boundedness of solutions to Emden-Fowler equation \eqref{e5},
 but (H3) does not hold in the case $\beta > p$. This situation
can be improved by introducing a nontrivial function $h$ in the
differential operator; consider a simple differential equation
 \begin{equation}\label{e7}
[t^2h(x)x']' = \frac{1}{t^2}\Phi_3(x),\quad t \ge 1.
\end{equation}
we can not use the results in \cite{kc93} to decide the
 continuability and  boundedness of all solutions for \eqref{e7}
in the case $h \equiv 1$ since (H3) is obviously invalid.
However, with $h(r) = r^2 + 1$, (H3) holds since
$$
\int_{1}^{\infty} \frac{dr}{f^{-1}(z(r))}
= \int_{1}^{\infty} \frac{(r^2+1)dr}{r^3} =\infty
$$
and
$$
\int_{-\infty}^{-1} \frac{dr}{f^{-1}(z(r))}
= -\int_{-\infty}^{-1} \frac{(r^2+1)dr}{r^3} = -\infty.
$$
By Theorem \ref{th3} all solutions of \eqref{e7} can be extended
to $[1, \infty)$. Moreover, it is easy to verify that
\begin{equation*}
J_1 =\int_1^{\infty}\Big(\frac{1}{t^2}\int_1^t\frac{1}{s^2}ds\Big)\,dt
< \infty,\quad
J_2 =-\int_1^{\infty}\Big(\frac{1}{t^2}\int_1^t\frac{1}{s^2}ds\Big)\,dt
> -\infty.
\end{equation*}
It follows from Theorem \ref{th7} that all solutions of \eqref{e7}
are bounded on $[1, \infty)$.

The results obtained in this article generalize, complement, or
improve some analogous ones existing in the literature.
By solution of \eqref{e1} we mean a differentiable function $x$
such that $p(t)h(x(t))f(x'(t))$ is differentiable and satisfies
\eqref{e1} on $[a, \alpha_x)$, $\alpha_x \le \infty$, the maximum
existence interval of $x$. A solution $x$ of \eqref{e1} is said to
be eventually monotonic if there exists a $t_1 \ge a$ such that it
is monotonic on $[t_1, \alpha_x)$.

In this article we consider only nontrivial solutions of \eqref{e1},
in other words,  solutions that are not identically equal to zero
on their existence interval.

Throughout this article, we assume that
\begin{itemize}
\item[(H)]
\begin{itemize}
\item $p(t), q(t): [a,\infty) \to \mathbb{R}$ are continuous
and $p(t) > 0$ and $q(t) > 0$;
\item $h(r):  \mathbb{R} \to \mathbb{R}$ is continuous
and $h(r) > 0$;
\item $g(r): \mathbb{R} \to \mathbb{R}$ is continuous
and $rg(r) > 0$ for $r \neq 0$;
\item $f(r): \mathbb{R} \to \mathbb{R}$ is continuous,
increasing, and $rf(r)> 0$ for $r \neq 0$.
\end{itemize}

\item[(H1)] There exists a constant $M_1 > 0$ such that
$$
|f^{-1}(uv)| \le M_1 |f^{-1}(u)||f^{-1}(v)|, \quad \forall u, v
\in \mathbb{R}.
$$
\end{itemize}

\begin{remark} \label{rmk1.1} \rm
Assumption (H1) holds for $f(r) = \Phi_p(r)$ with $M_1 = 1$.
In fact, we have
\begin{equation}\label{equal}
f^{-1}(uv) =  f^{-1}(u)f^{-1}(v), \quad \forall u, v
\in \mathbb{R}.
\end{equation}
\end{remark}

\begin{remark} \label{rmk1.2} \rm
Let
\begin{equation*}
f(r) =
\begin{cases}
r, & |r| \le 1,\\
\sqrt[3]{r}, & |r| > 1.
\end{cases}
\end{equation*}
Then
\begin{equation*}
f^{-1}(r) =
\begin{cases}
r, & |r| \le 1,\\
r^3, & |r| > 1.
\end{cases}
\end{equation*}
It is easy to see that (H1) holds with $M_1 = 1$, but \eqref{equal}
does not hold in this case.
\end{remark}

We will prove that the monotonicity and boundedness properties of
solutions to \eqref{e1} can be characterized by means of the
convergence of the following two integrals
\begin{gather*}
 J_1 := \int_a^\infty f^{-1}\Big(\frac{1}{p(t)}\int_a^t
q(s)ds\Big)dt,\\
J_2 := \int_a^\infty
f^{-1}\Big(-\frac{1}{p(t)}\int_a^t q(s)ds\Big)dt.
\end{gather*}

This article is organized as follows: Section 1 is the
introduction. The background, motivation, and the main
contributions of the paper are briefly addressed in this section.
Continuability of solutions is discussed in Section 2. Section 3
deals with the existence of class $A$ and class $B$ solutions. In
Section 4, necessary and sufficient conditions for boundedness of
all solutions are established. Also, several
examples and remarks are provided in this section to compare our results with some
known results in the literature.

\section{Continuability of Solutions}

In this section we discuss the continuability of solutions to
\eqref{e1}. First of all, we give two lemmas that will be
used later on. The first lemma is a minor extension of Proposition 1 in \cite{m84}.

\begin{lemma}\label{th1}
If $x(\cdot)$ is a solution of \eqref{e1}
with maximal existence interval $[a,\alpha_x)$, $0 < \alpha_x \leq
\infty$, then $x(\cdot)$ is eventually monotonic.
\end{lemma}

\begin{proof}
 Let $F(t) = p(t)h(x(t))f(x'(t))x(t)$.
Note that $F(t)$ is continuous on $[a,\alpha_x)$ and
$F'(t) = q(t)g(x(t))x(t) + p(t)h(x(t))f(x'(t))x'(t) \geq 0$,
then $F(t)$ is nondecreasing on $[a, \alpha_x)$.
The rest part of the proof is omitted.
\end{proof}

\begin{lemma}\label{th2}
If a solution $x(\cdot)$ of \eqref{e1} is bounded on every
finite subinterval of $[a, \alpha_x)$, the maximal existence interval,
then $\alpha_x = \infty$.
\end{lemma}

\begin{proof}
Assume that $\alpha_x$ is a finite number. By
Lemma~\ref{th1} there exists a $b \ge a$ such that $x(t)$ is
monotone on $[b, \alpha_x)$. We assume $x(t) > 0$, $t \in [b,
\alpha_x)$, without loss of generality. Since $x(t)$ is bounded on
any finite subinterval of $[b, \alpha_x)$, then
$\lim_{t\to\alpha_x-}x(t)$ exists finitely and
$\lim_{t\to\alpha_x-}x'(t) = \infty$. Integrating \eqref{e1} from
$a$ to $t$ implies
$$
p(t)h(x(t))f(x'(t)) = p(a)h(x(a))f(x'(a)) +
\int_{a}^tq(s)g(x(s))ds.
$$
Hence,
\begin{align*}
\lim_{t\to\alpha_x-}p(t)h(x(t))f(x'(t))
&= p(a)h(x(a))f(x'(a)) +
\int_{a}^{\alpha_x}q(s)g(x(s))ds \\
&:= A \in (-\infty, \infty).
\end{align*}
Define
$H(t) = p(t)h(x(t))f(x'(t))$.
Then
$$
x'(t) = f^{-1}\Big( \frac{H(t)}{p(t)h(x(t))}\Big).
$$
The continuity of $f^{-1}(r)$ implies
$$
\lim_{t\to\alpha_x-}x'(t)=  f^{-1}\Big(
\frac{A}{p(\alpha_x)h(x(\alpha_x))}\Big) < \infty.
$$
This is a contradiction. Therefore, $\alpha_x = \infty$ and
the proof is complete.
\end{proof}

It follows from Lemmas~\ref{th1} and \ref{th2} that all
solutions of \eqref{e1} except the trivial solution can be
divided into two classes:
\begin{gather*}
\begin{aligned}
A = \{& x  \text{ solution of \eqref{e1} defined on }
 [a,\alpha_x): x(t)x'(t) > 0  \\
&\text{in a left neighborhood of } \alpha_x\},
\end{aligned}\\
B = \{x  \text{ solution of \eqref{e1} defined on }
 [a,\infty): x(t)x'(t) < 0 \text{ for } t \ge a\}.
\end{gather*}

It is well-known that for some equations of type \eqref{e1},
class $A$ solutions are not continuable at infinity;
 see \cite{kc93} for the discussion of the binomial equations
of type $x'' = q(t)|x|^{\gamma}\operatorname{sgn} x$.

In the next we consider the continuability of solutions to
\eqref{e1}. Let

\begin{itemize}
\item[(H2)] $g(r)$ is nondecreasing
for $|r| \ge m$ where $m > 0$ is a real number.
\item[(H3)]
$$\int_{1}^{\infty} \frac{dr}{f^{-1}(z(r))} = \infty, \quad
\int_{-\infty}^{-1} \frac{dr}{f^{-1}(z(r))} = -\infty,
$$
where $z(r) = g(r)/h(r)$.
\end{itemize}

\begin{theorem}\label{th3}
Under assumptions {\rm (H2),(H3)},  all solutions of \eqref{e1}
can be extended to [$a,\infty$).
\end{theorem}

\begin{proof}
 We consider class $A$ solutions only since all class
$B$ solutions can be extended to
 [$a, \infty$). Let $x(\cdot)$ be a class $A$ solution of \eqref{e1}
and without loss of generality we assume that $x(t) > 0$ and
$x'(t) > 0$ for all $t \in [b,\alpha_x)$.
If $\alpha_x < \infty$, by Lemma \ref{th2},
 $x(t) \to \infty$ as $t \to \alpha_x-$. Hence, there exists
a real number $d > b$ such that
 $x(t)\geq m$ for $d \leq t < \alpha_x$.

Integrating \eqref{e1} from $d$ to $t$  we have
$$
p(t)h(x(t))f(x'(t)) = p(d)h(x(d))f(x'(d))
 + \int_d^tq(s)g(x(s))ds.
$$
It follows from (H2) that
 \begin{align*}
f(x'(t)) &= \frac{p(d)h(x(d))f(x'(d))}{p(t)h(x(t))}
 + \frac{1}{p(t)h(x(t))}\int_d^tq(s)g(x(s))ds\\
 &\leq\frac{p(d)h(x(d))f(x'(d))}{p(t)h(x(t))}
 + \frac{g(x(t))}{p(t)h(x(t))}\int_d^tq(s)ds\\
&=
\frac{g(x(t))}{p(t)h(x(t))}\Big(\frac{p(d)h(x(d))f(x'(d))}{g(x(t))}
+\int_d^tq(s)ds\Big ) \\
&\leq
\frac{g(x(t))}{p(t)h(x(t))}\Big(\frac{p(d)h(x(d))f(x'(d))}{g(x(d))}
+\int_d^tq(s)ds \Big).
\end{align*}
Since $q(t) > 0$, we can choose $k > 1$ and $t_1 \geq d $ such
that for $t\geq t_1$,
$$
\frac{p(d)h(x(d))f(x'(d))}{g(x(d))} +\int_d^tq(s)ds \leq
k\int_d^tq(s)ds.
$$
Then
\begin{gather*}
f(x'(t)) \leq \frac{kg(x(t))}{p(t)h(x(t))}\int_d^t q(s)ds,
\\
x'(t) \leq f^{-1}\Big(\frac{kg(x(t))}{p(t)h(x(t))}\int_d^tq(s)ds\Big).
\end{gather*}
Taking into account (H1) we have
\begin{align*}
x'(t)
&\leq f^{-1}\Big(kz(x(t))\frac{1}{p(t)}\int_d^tq(s)ds\Big)\\
&\leq
M_1^2f^{-1}(k)f^{-1}\big(z(x(t))\big)f^{-1}
\Big(\frac{1}{p(t)}\int_d^tq(s)ds\Big).
\end{align*}
Dividing both sides by $f^{-1}(z(x(t)))$ and integrating from
$t_1$ to $t$ we have
\begin{equation}\label{4}
\int_{x(t_1)}^{x(t)}\frac{dr}{f^{-1}(z(r))} \leq M_1^2f^{-1}(k)
\int_{t_1}^tf^{-1}\Big(\frac{1}{p(s)}\int_d^{s}q(\sigma)d\sigma\Big)ds.
\end{equation}
Letting $t \to \alpha_x-$, we have
$$
\int_{x(t_1)}^{\infty}\frac{dr}{f^{-1}(z(r))} \leq M_1^2f^{-1}(k)
\int_{t_1}^{\alpha_x}f^{-1}\Big(\frac{1}{p(s)}\int_d^{s}q(\sigma)d\sigma
\Big)ds < \infty,
$$
which is a contradiction to (H3). Therefore, all
solutions of \eqref{e1} can be extended to [$a, \infty$).
 \end{proof}

Without the monotonic condition (H2), we have the following
theorem.
Let
\begin{itemize}
\item[(H4)] There exists a constant $M_2 > 0$ such that
 $|g(r)| \le M_2$ for all $r \in \mathbb{R}$.
\item[(H5)]
$$\int_{1}^{\infty} \frac{dr}{f^{-1}(\frac{1}{h(r)})} = \infty, \quad
\int_{-\infty}^{-1} \frac{dr}{f^{-1}(\frac{1}{h(r)})} =
\infty.
$$
\end{itemize}

\begin{theorem}\label{th4}
Under assumptions {\rm (H4), (H5)}, all solutions of \eqref{e1}
can be extended to [$a, \infty$).
\end{theorem}

\begin{proof}
Similar to the proof of Theorem~\ref{th3}, we consider
positive class $A$ solutions only. Let $x$ be a positive class $A$
solution of \eqref{e1} with maximum existence interval $[a,
\alpha_x)$ such that $x(t) > 0$ and $x'(t)
> 0$ for all $t \in [b,\alpha_x)$.
If $\alpha_x < \infty$, then $x(t) \to \infty$ as $t
\to \alpha_x-$. From
$$
p(t)h(x(t))f(x'(t)) = p(b)h(x(b))f(x'(b)) +\int_{b}^tq(s)g(x(s))ds
$$
and (H4) we have
\[
p(t)h(x(t))f(x'(t)) \le p(b)h(x(b))f(x'(b)) + M_2\int_{b}^tq(s)ds.
\]
 Choosing $k > 1$ and $t_1\geq b$ such that for $t\geq t_1$
$$
p(b)h(x(b))f(x'(b)) + M_2\int_{b}^tq(s)ds \leq k\int_{b}^tq(s)ds.
$$
By (H1), we have
\[
x'(t)  \le M^2_1f^{-1}(k)f^{-1}\Big(\frac{1}{h(x(t))}\Big)
f^{-1}\Big(\frac{1}{p(t)}\int_{b}^tq(s)ds\Big).
\]
Dividing both sides by $f^{-1}\big(\frac{1}{h(x(t))}\big)$,
integrating from $t_1$ to $t$, and letting $t \to \alpha_x-$, we
have
$$
\int_{x(t_1)}^{\infty}\frac{dr}{f^{-1}\Big(\frac{1}{h(r)}\Big)} \leq M_1^2f^{-1}(k)
\int_{t_1}^{\alpha_x}f^{-1}\Big(\frac{1}{p(s)}\int_d^{s}q(\sigma)d\sigma
\Big)ds < \infty,
$$
which is a contradiction to (H5). Therefore, $x(\cdot)$
 can be extended to infinity.
\end{proof}

\section{Existence of Class $A$ \& $B$ Solutions}

In this section we discuss the existence of class $A$ and class $B$
solutions of \eqref{e1}. We assume that \eqref{e1} has a
unique solution for any initial conditions $(x(a), x'(a))$ with
$x(a) \neq 0$.

\begin{theorem}\label{th5}
Equation \eqref{e1} has both positive and negative class $A$
solutions.
\end{theorem}

\begin{proof}
Let $x$ be the solution of \eqref{e1} with initial
conditions $x(a)> 0$ and $x'(a) > 0$. From the proof of
Lemma~\ref{th1} we have $F(t) > 0$ for
$t \in [a, \alpha_x)$ because of $F(a) > 0$ in this case.
Therefore, $x(t)x'(t) > 0$ for $t \in [a, \alpha_x)$ and $x$
is a positive class $A$ solution. Similarly, let $\tilde{x}$
be the solution of \eqref{e1} with initial conditions
$\tilde{x}(a) < 0$ and $\tilde{x}'(a) < 0$. We can show that
$\tilde{x}$ is a negative class $A$ solution.
\end{proof}

Now, we discuss sufficient conditions for the existence of class $B$
solutions. The following simple lemma is needed for the proof.

\begin{lemma}\label{sl}
If $x$ is a solution of \eqref{e1} on $[t_1, t_2]$ such that
$x(t_1) = x(t_2) = 0$, then $x(t) = 0$ for all $t \in [t_1, t_2]$.
\end{lemma}

Notice that if otherwise $x(t_1) = x(t_2) = 0$, then
$F(t_1) = F(t_2) = 0$ which contradicts the monotonicity of $F$.
Let \begin{itemize}
\item[(H2A)] $g(r)$ is nondecreasing on $(-\infty, \infty)$.
\item[(H6)] There exists $r_0 > 0$ such that
$$
\int_0^{\pm r_0}\frac{dr}{f^{-1}(z(r))} = \infty.
$$
\end{itemize}

\begin{theorem}\label{t4}
Assume {\rm (H2A), (H6)}. Then
\begin{itemize}
\item[(a)] Equation \eqref{e1} has both positive and negative
class $B$ solutions.
\item[(b)] Equation \eqref{e1} has no
solution which is nonzero but eventually identically equal to zero.
\end{itemize}
\end{theorem}

\begin{proof}
(a) We prove that \eqref{e1} has a positive solution, the case of
negative solution is similar. Assume $x_0 > 0$. The solution of
\eqref{e1} with initial conditions $x(a) = x_0$ and $x'(a) = c$ is
denoted by $x(t) := x(t, c)$ that has the form
$$
x(t) = x_0 + \int_a^t f^{-1}\Big(\frac{p(a)h(x_0)f(c)}{p(s)h(x(s))}+
\frac{1}{p(s)h(x(s))}\int_a^s
q(\sigma)g(x(\sigma))d\sigma\Big)ds.
$$
Define two sets $U$ and $L$ as
\begin{gather*}
U = \{c \in \mathbb{R}: \text{ there exists some
$\bar t \ge a$  such that } x'(\bar t, c) > 0\},\\
L = \{c \in \mathbb{R}: \text{ there exists some $\bar t \ge a$
 such that } x(\bar t, c) < 0\}.
\end{gather*}
Then $U \cap L = \emptyset$. Clearly, $U \neq \emptyset$.
With the same argument as Theorem 4 in \cite{w09} we are able
to prove that $U$ is open.
Next we show $L \neq \emptyset$. Define
$M_2 := \max_{0 \le r \le x_0}h(r) > 0$ and
$M_3 := \max_{a \le t \le a+1}p(t) > 0$. Let
\begin{equation}\label{c}
c < f^{-1}\Big(\frac{M_2M_3f^{-1}(-x_0)-g(x_0)
\int_a^{a+1}q(s)ds}{p(a)h(x_0)}\Big).
\end{equation}
We claim $x'(t, c) < 0$ for $a \le t \le a+1$.
Otherwise, there exists $t_1 \in (a, a+1]$ such that
$x'(t_1, c) = 0$ and $x'(t, c) < 0$ for $t \in [a, t_1)$.
It follows from \eqref{c} that
\begin{align*}
0 &= p(t)h(x(t_1, c))f(x'(t_1, c)) = p(a)h(x_0)f(c)
+ \int_a^{t_1}q(s)g(x(s, c))ds\\
&\le p(a)h(x_0)f(c) + g(x_0)\int_a^{a+1}q(s)ds < 0.
\end{align*}
This is a contradiction and hence $x(t, c)$ is decreasing on
$[a,a+1]$.

If there exists a $b \in (a, a+1]$ such that $x(b, c) < 0$, then
$c \in L$ and $L \neq \emptyset$. Otherwise, $x(t)\ge 0$ on $[a,
a+1]$. Hence, we have from \eqref{c} that
\begin{align*}
x(a+1, c) &= x_0 +
\int_a^{a+1}f^{-1}\Big(\frac{p(a)h(x_0)f(c)}{p(t)h(x(t))}+
\frac{1}{p(t)h(x(t))}\int_a^t q(s)g(x(s))ds\Big)dt\\
&\le  x_0 + \int_a^{a+1}f^{-1}\Big(\frac{p(a)h(x_0)f(c) +
g(x_0)\int_a^{a+1}q(s)ds}{M_2M_3}\Big)dt < 0.
\end{align*}
This shows that $c \in L$. Clearly, $L$ is open. Therefore,
$\mathbb{R}-(U\cup L) \neq \emptyset$. For any $c \in
\mathbb{R}-(U\cup L)$, $x(t, c)$ is a non-increasing nonnegative
solution on $[a, \infty)$. We will show that $x(t, c) > 0$ on $[a,
\infty)$. If not, there exists $t_0 > a$ such that $x(t_0) = 0$
and $x(t) = 0$ for $t \ge t_0$ and $x'(t_0) = 0$. Note that for $t
\in [a, t_0]$ we have
\begin{align*}
x'(t)
&= f^{-1}\Big(-\frac{1}{p(t)h(x(t))}\int_t^{t_0} q(s)g(x(s))ds\Big)\\
&\ge M_1f^{-1}(z(x(t)))f^{-1}\Big(-\frac{1}{p(t)}\int_t^{t_0}
q(s)ds\Big).
\end{align*}
Dividing both sides by $f^{-1}(z(x(t)))$ and integrating from $a$
to $t_0$, we have
\begin{equation*}
\int_a^{t_0}\frac{x'(t)}{f^{-1}(z(x(t)))}dt \ge
M_1\int_a^{t_0}f^{-1}\Big(-\frac{1}{p(t)}\int_t^{t_0} q(s)ds\Big)dt.
\end{equation*}
That is,
\begin{equation*}
\int_0^{x_0}\frac{1}{f^{-1}(z(r))}dr \le
-M_1\int_a^{t_0}f^{-1}\Big(-\frac{1}{p(t)}\int_t^{t_0}
q(s)ds\Big)dt < \infty,
\end{equation*}
a contradiction to (H6).  Therefore, $x(t) > 0$ for $t \ge a$.
Note that $x'(t) \le 0$ for $t \ge a$. It follows from equation
\eqref{e1} and $x(t) > 0$  that $x'(t) \neq 0$ for $t \ge a$.
Hence,  $x'(t) < 0$ for $t \ge a$ and $x \in B$.

The proof of part (b) follows from the end part of the proof of
part (a).
\end{proof}

\section{Boundedness of Solutions}

In this section we discuss the boundedness of all solutions of
\eqref{e1}, some necessary and sufficient conditions are obtained.

\begin{theorem}\label{th7}
Assume {\rm (H2), (H3)}. Then all positive (negative) solutions of
\eqref{e1} are bounded if and only if $J_1 < \infty$
($J_2 >-\infty$).
\end{theorem}

\begin{proof} We consider positive solutions only since the
case of negative solutions can be handled similarly.

\emph{Necessity}. Let $x(\cdot)$ be a positive bounded class $A$
solution. Then $x(t)
> 0$ and $x'(t) > 0$ for $t\geq b > a$ and
$\lim_{t\to\infty}x(t)= l \in (0, \infty)$. By the Extreme Value
Theorem we have $L_1:=\min_{x(b)\leq r\leq l}g(r) > 0$. Hence
\begin{align*}
p(t)h(x(t))f(x'(t))&= p(b)h(x(b))f(x'(b)) +
\int_{b}^tq(s)g(x(s))ds \geq L_1 \int_{b}^tq(s)ds.
\end{align*}
Since $x(\cdot)$ is continuous and bounded and $h(r)$ is
continuous,  $h(x(\cdot))$ is bounded. Let $h(x(t)) \leq K$
for $t \in [a, \infty)$. Then
\begin{gather*}
Kp(t)f(x'(t)) \geq p(t)h(x(t))f(x'(t)) \geq L_1
\int_{b}^tq(s)ds, \\
\frac{K}{L_1}f(x'(t)) \geq \frac{1}{p(t)}\int_{b}^tq(s)ds.
\end{gather*}
By (H1) we have
\begin{align*}
f^{-1}\Big(\frac{1}{p(t)}\int_{b}^tq(s)ds\Big) &\leq
f^{-1}\Big(\frac{K}{L_1}f(x'(t))\Big) \leq
M_1f^{-1}\Big(\frac{K}{L_1}\Big)x'(t).
\end{align*}
Integrating from $b$ to $t$ and letting $t \to \infty$ we have
\[
J_1 = \int_{b}^{\infty}
f^{-1}\Big(\frac{1}{p(t)}\int_{b}^tq(s)ds\Big)dt \leq
M_1f^{-1}\Big(\frac{K}{L_1}\Big)(l-x(b)) < \infty.
\]

\emph{Sufficiency}.  We will prove by contradiction. Let $x(\cdot)$
be a unbounded class $A$ solution of \eqref{e1}.
Then $x(t) > 0$ and $x'(t)> 0$ on $[b,\infty)$, and there
exists a real number  $d \ge b$ such that
$x(t)\geq m$ for $d \leq t < \infty$. Similar to
the proof of Theorem~\ref{th3}, we have the inequality
 \begin{equation*}
\int_{x(t_1)}^{x(t)}\frac{dr}{f^{-1}(z(r))} \leq M_1^2f^{-1}(k)
\int_{t_1}^tf^{-1}\Big(\frac{1}{p(s)}\int_d^{s}q(\sigma)d\sigma\Big)ds.
\end{equation*}
Letting $t \to \infty$ and noting that $x(\infty) = \infty$, we
have
$$
\int_{x(t_1)}^{\infty}\frac{dr}{f^{-1}(z(r))} \leq M_1^2f^{-1}(k)
\int_{t_1}^{\infty}f^{-1}\Big(\frac{1}{p(s)}\int_{b}^{s}q(\sigma)d\sigma\Big)ds
\le M_1^2f^{-1}(k)J_1 < \infty.
$$ This is a contradiction to (H3).
Therefore, $x$ is bounded.
\end{proof}

\begin{corollary} \label{cor4.2}
Assume {\rm (H2), (H3)}.
If  \eqref{e1} has a positive
(negative) bounded class $A$ solution, then all positive (negative)
solutions in class $A$ are bounded. On the other hand, if
\eqref{e1} has an unbounded positive (negative) class $A$ solution,
then all positive (negative) solutions in class $A$ are unbounded.
\end{corollary}

\begin{theorem}\label{th9}
Assume {\rm (H4), (H5)}. Then all positive (negative) solutions of
\eqref{e1} are bounded if and only if
$J_1 <  \infty$ ($J_2 >-\infty$).
\end{theorem}

\begin{proof} We prove only the case of positive solutions, since
the argument is similar for negative solutions.

\emph{Necessity}. Let $x(\cdot)$ be a positive bounded class $A$
solution, i.e., $x(t)> 0$ and $x'(t) > 0$ for $t \in [b, \infty)$.
Then $\lim_{t\to\infty}x(t)= l_1 \in (0, \infty)$. By the Extreme Value
Theorem we have $L_2:=\min_{x(b)\leq r\leq l_1}g(r) > 0$. Hence
\begin{align*}
p(t)h(x(t))f(x'(t))&= p(b)h(x(b))f(x'(b)) +
\int_{b}^tq(s)g(x(s))ds \geq L_2 \int_{b}^tq(s)ds.
\end{align*}
Similar to the proof of Theorem~\ref{th7}, let $h(x(t)) \leq K$
for $t \in [b, \infty)$. We have
$$
\frac{K}{L_2}f(x'(t)) \geq
\frac{1}{p(t)}\int_{b}^tq(s)ds.
$$
Hence
\begin{align*}
f^{-1}\Big(\frac{1}{p(t)}\int_{b}^tq(s)ds\Big) \leq
M_1f^{-1}\Big(\frac{K}{L_2}\Big)x'(t).
\end{align*}
Integrating from $b$ to $t$ and letting $t \to \infty$, we have
\begin{align*}
J_1 = \int_{b}^{\infty}
&f^{-1}\Big(\frac{1}{p(t)}\int_{b}^tq(s)ds\Big)dt \le
M_1f^{-1}\Big(\frac{K}{L_2}\Big)(l_2-x(b)) < \infty.
\end{align*}

\emph{Sufficiency}. Assume that $x(\cdot)$ is any positive class $A$
solution. It follows from
$$
p(t)h(x(t))f(x'(t)) = p(b)h(x(b))f(x'(b))
+ \int_{b}^tq(s)g(x(s))ds
$$
and (H4) that
\[
p(t)h(x(t))f(x'(t)) \le p(b)h(x(b))f(x'(b)) +
M_2\int_{b}^tq(s)ds.
\]
Similar to the proof of Theorem~\ref{th4}, we have
$$
 \int_{x(t_1)}^{\infty}\frac{dr}{f^{-1}\Big(\frac{1}{h(r)}\Big)}
\leq M_1^2f^{-1}(k)
\int_{t_1}^{\infty}f^{-1}\Big(\frac{1}{p(t)}\int_d^tq(s)ds
\Big)dt \le M_1^2f^{-1}(k)J_1 < \infty.
$$
Therefore, $x(\cdot)$ is bounded and the proof is complete.
\end{proof}

\begin{corollary} \label{coro4.4}
Let {\rm (H4)} and {\rm(H5)} hold. If \eqref{e1} has a positive
(negative) bounded class $A$ solution, then all positive (negative)
solutions in class $A$ are bounded. On the other hand, if \eqref{e1}
has an unbounded positive (negative) class $A$ solution,
then all positive (negative) solutions in class $A$ are unbounded.
\end{corollary}

\begin{remark} \label{rmk5.1} \rm
The condition (H3) of Theorem~\ref{th7} is sharp. For example,
consider the following equation
\begin{equation}\label{exa}
(t^6(x^2(t)+1)(x'(t))^3)' = \frac{162}{t^4}(3x^7(t)+2x^5(t)),
\quad t \geq 1,
\end{equation}
where $p(t)=t^6$, $h(r)=r^2+1$, $f(r) = r^3$, $g(r) =
3r^7+2r^5$, $q(t)=\frac{162}{t^4}$.
Clearly, conditions (H), (H1), and (H2) are satisfied.  By simple
computation,
we have
$$
\int_{1}^{\infty} \frac{dr}{f^{-1}(z(r))}<
\infty,\quad \int_{-\infty}^{-1} \frac{dr}{f^{-1}(z(r))} > -\infty.
$$
This shows that (H3) does not hold. We claim $J_1 < \infty$ and
$J_2 > -\infty$. Indeed,
\[
J_1 =\int_{1}^{\infty}{\sqrt[3]{\frac{54}{t^6}\Big(-\frac{1}{t^3}+1\Big)}}\,dt
\le 3\sqrt[3]{2} \int_{1}^{\infty}\frac{1}{t^2}dt = 3\sqrt[3]{2},
\]
and
\[
J_2 =-\int_{1}^{\infty}{\sqrt[3]{\frac{54}{t^6}\Big(-\frac{1}{t^3}+1\Big)}}\,dt
\ge -3\sqrt[3]{2} \int_{1}^{\infty}\frac{1}{t^2}dt =
-3\sqrt[3]{2}.
\]
However, $x(t)=t^3$ is a positive unbounded class $A$ solution of
\eqref{exa} on $[1,\infty)$ and $x(t)=-t^3$ is a negative
unbounded class $A$ solution on $[1,\infty)$. Therefore,
Theorem~\ref{th7} fails without (H3).
\end{remark}

\begin{remark} \label{rmk5.2} \rm
The condition (H5) of Theorem~\ref{th9} is sharp. For example,
consider the following equation
\begin{equation}\label{exa-1}
\Big(\frac{t^4}{x^4(t)+1}(x'(t))^3\Big)' =
\frac{4t^3}{(t^2+1)^2}\,g(x(t)), \quad t \geq 1,
\end{equation}
where $p(t)=t^4$, $h(r)=\frac{1}{r^4+1}$, $f(r) = r^3$, $q(t) =
\frac{4t^3}{(t^2+1)^2},$ and
$$g(r)=
\begin{cases}
1, & r \ge 1,\\
|r|, & |r| \le 1,\\
-1, & r \le -1.
\end{cases}
$$
Clearly, conditions (H), (H1), and (H4) are satisfied. We claim
$J_1 < \infty$ and $J_2
> -\infty$. Indeed,
\begin{align*}
J_1 &=
\int_{1}^{\infty}{\sqrt[3]{\frac{1}{t^4}\Big(\frac{t^4}{t^4+1}-\frac{1}{2}\Big)}}\,dt
 < \int_{1}^{\infty}{\sqrt[3]{\frac{1}{t^4+1}}}\,dt < \infty,
\end{align*}
and
\begin{align*}
J_2 &=
-\int_{1}^{\infty}{\sqrt[3]{\frac{1}{t^4}\Big(\frac{t^4}{t^4+1}-\frac{1}{2}\Big)}}\,dt
> -\int_{1}^{\infty}{\sqrt[3]{\frac{1}{t^4+1}}}\,dt > -\infty.
\end{align*}
However, (H5) does not hold since
$$
\int_{1}^{\infty} \frac{dr}{f^{-1}(\frac{1}{h(r)})}=\\
\int_{1}^{\infty}\sqrt[3]{\frac{1}{r^4+1}}\,dr <
\infty,
$$
and
$$
\int_{-\infty}^{-1} \frac{dr}{f^{-1}(\frac{1}{h(r)})} =\\
\int_{-\infty}^{-1}\sqrt[3]{\frac{1}{r^4+1}}\,dr < \infty.
$$
It is easy to check that $x(t)=t$ is a positive unbounded class $A$
solution of \eqref{exa} on $[1,\infty)$ and $x(t)=-t$ is a
negative unbounded class $A$ solution on $[1,\infty)$. Therefore,
Theorem~\ref{th7} fails without (H5).
\end{remark}

\begin{remark} \label{rmk5.3} \rm
Theorem~\ref{th3} and Theorem~\ref{th7} generalize
\cite[Theorem 1]{m84} since (H3) reduces to (iii)
of \cite{m84} if $f(r) =r$. Moreover, the differentiability
of $p(\cdot)$ and $h(\cdot)$
is not required as of \cite{m84}. Theorems \ref{th3}, \ref{th5}, and
\ref{th7}  generalize \cite[Theorem 8]{cdm01}.
Moreover, under (H2), Theorems \ref{th3}, \ref{th5}, and
\ref{th7} improve \cite[Theorem 8]{cdm01} since (H3) improves
\cite[(22)]{cdm01}; see the discussion in \cite{w09}.
Theorem \ref{th3} generalizes \cite[Theorem 3.9]{mnu02}.
Theorems \ref{th3}, \ref{th5}, and \ref{th7}
generalize \cite[Theorem 1]{w09}. Theorem~\ref{t4} generalizes
\cite[Theorem 2.1]{mnu02}. Under
(H2A), Theorem~\ref{t4} improves \cite[Theorem 6]{cmv89}
since \cite[(hp)]{cmv89} is replaced by a weaker condition (H6).
\end{remark}

\subsection*{Acknowledgments}
The authors wish to thank the anonymous referee for the
valuable suggestions and comments which
have resulted in a great improvement of this paper.

\begin{thebibliography}{10}
\bibitem{cdkm07} M. Cecchi, Z. Do\v{s}l\'{a}, I. Kiguradze,
and M. Marini;
On nonnegative solutions of singular boundary-value problems
for Emden-Fowler-type differential systems, Differential
Integral Equations,  20  (2007), 1081--1106.

\bibitem{cdm10}
M. Cecchi, Z. Do\v{s}l\'{a}, and M. Marini;
 On second order differential equations with nonhomogeneous
$\Phi$-Laplacian, Boundary Value Problems, Volume 2010 (2010),
Article ID 875675.

\bibitem{cdmv06} M. Cecchi, Z. Do\v{s}l\'{a}, M. Marini,
and I. Vrko\v{c}; Integral conditions for nonoscillation of second
order nonlinear differential equations,   Nonlinear Anal.,
64  (2006), 1278--1289.

\bibitem{cdm05} M. Cecchi, Z. Do\v{s}l\'{a}, and M. Marini;
 Half-linear differential equations with oscillating coefficient,
Differential Integral Equations,  18  (2005), 1243--1256.

\bibitem{cdm00} M. Cecchi, Z. Do\v{s}l\'{a} and M. Marini;
 On the dynamics of the generalized Emden-Fowler equations,
Georgian Math. J., 7 (2000), 269--282.

\bibitem{cdm01} M. Cecchi, Z. Do\v{s}l\'{a}, and M. Marini;
 On nonoscillatory solutions of differential equations with
$p$-Laplacian, Adv. Math. Sci. Appl., 11 (2001), 419--436.

\bibitem{cmv89} M. Cecchi, M. Marini, and G. Villari;
 On the monotonicity property for a certain class of second order
differential equations, J. Differential equations, 82 (1989), 15--27.

\bibitem{cmv95} M. Cecchi, M. Marini, and G. Villari;
 On some classes of continuable solutions of a nonlinear
differential equations, J. Differential equations, 118 (1995), 403--419.

\bibitem{che91} S. Chen, Q. Huang, and L. H. Erbe;
Bounded and zero-convergent solutions of a class of Stieltjes
integro-differential equations,
Proc. Amer. Math. Soc., 113 (1991), 999--1008.

\bibitem{dr05} O. Do\v{s}l\'{y} and P. \v{R}eh\'{a}k;
\emph{Half-linear Differential Equations}, North-Holland,
Mathematics Studies 202, Elsevier 2005.

\bibitem{ku00} K. Kamo and H. Usami;
 Asymptotic forms of positive solutions of
second-order quasilinear ordinary differential equations, Adv.
Math. Sci. Appl., 10 (2000), 673--688.

\bibitem{ku01} K. Kamo and H. Usami;
Asymptotic forms of positive solutions of
second-order quasilinear ordinary differential equations with
sub-homogeneity, Hiroshima Math. J., 31 (2001), 35--49.

\bibitem{kc93} I. Kiguradze and A. Chanturia;
 \emph{Asymptotic properties of solutions of nonautonomous ordinary
differential equations}, Kluwer Acad. Publ. G., Dordrecht, 1993.

\bibitem{l02} W. Li;
 Classification schemes for positive solutions of nonlinear
differential systems, Mathematical and Computer Modelling,
36 (2002), 411--418.

\bibitem{m84} M. Marini;
 Monotone solutions of a class of second order nonlinear
differential equations, Nonlinear Anal., 8 (1984), 261--271.

\bibitem{m93} J. D. Mirzov;
 \emph{Asymptotic properties of solutions of the
systems of nonlinear nonautonomous ordinary differential
equations}, (Russian), Maikop, Adygeja Publ. 1993. English
translation: Folia, Mathematics 14, Masaryk University Brno 2004.

\bibitem{mnu02} M. Mizukami, M. Naito, and H. Usami;
Asymptotic behavior of
solutions of a class of second order quasilinear ordinary
differential equations,  Hiroshima Math. J., 32 (2002),  51--78.

\bibitem{ny92} Y. Nagabuchi and M. Yamamoto;
On the monotonicity of solutions for a class of nonlinear
differential equations of second order, Math.
Japonica, 37 (1992) 665--673.

\bibitem{t99} T. Tanigawa;
 Existence and asymptotic behavior of positive
solutions of second order quasilinear differential equations,
Advances in Math. Sci. and Appl., 9 (1999), 907--938.

\bibitem{w09} L. Wang;
 On monotonic solutions of systems of nonlinear second
order differential equations, Nonlinear Anal., 70 (2009), 2563--2574.

\end{thebibliography}
\end{document}