\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 105, pp. 1--8.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/105\hfil Riccati transformation]
{A Riccati technique for proving oscillation of a half-linear equation}
\author[P. \v Reh\'ak \hfil EJDE-2008/105\hfilneg]
{Pavel \v Reh\'ak}
\address{Pavel \v Reh\'ak \newline
Institute of Mathematics, Academy of Sciences,
\v Zi\v zkova 22, CZ61662 Brno, Czech Republic}
\email{rehak@math.muni.cz}
\thanks{Submitted May 12, 2008. Published August 6, 2008.}
\thanks{Supported by grants KJB100190701 from the Grant Agency of ASCR,
201/07/0145 from \hfill\break\indent
the Czech Grant Agency, and AV0Z010190503 from
the Institutional Research Plan}
\subjclass[2000]{34C10}
\keywords{Half-linear differential equation; Riccati technique;
oscillation criteria}
\begin{abstract}
In this paper we study the oscillation of solutions to
the half-linear differential equation
$$
(r(t)|y'|^{p-1}\mathop{\rm sgn} y)'+c(t)|y|^{p-1}\mathop{\rm sgn} y=0,
$$
under the assumptions $\int^\infty r^{1/(1-p)}(s)\,ds<\infty$,
$r(t)>0$, $p>1$.
Our main tool is a Riccati type transformation for using the
so called ``function sequence technique''.
This method leads to new and to known oscillation and comparison
results. We also give an example that illustrates our results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
The Riccati type transformation plays an important role in qualitative
theory of the half-linear differential equation
\begin{equation} \label{e}
(r(t)\Phi(y'))'+c(t)\Phi(y)=0,
\end{equation}
where $r$ and $c$ are continuous functions
on $[a,\infty)$ with $r(t)>0$, and $\Phi(u)=|u|^{p-1}\mathop{\rm sgn} u$
with $p>1$.
Monograph \cite{book} presents a systematic and compact treatment
of the qualitative theory of the above equation.
Recall that \eqref{e} can be viewed at least in three ways:
(1) as a natural generalization of a linear differential equation,
(2) as a differential equation with one dimensional $p$-Laplacian,
(3) as a special case of a generalized Emden-Fowler (quasilinear)
differential equation.
If there exists a positive solution $y$ of \eqref{e} on some interval
$[t_0,\infty)$,
then the function $w=r\Phi(y'/y)$ satisfies the generalized Riccati
differential equation
\begin{equation} \label{riccati}
w'+c(t)+(p-1)r^{1-q}(t)|w|^q=0
\end{equation}
on $[t_0,\infty)$. Here $q$ is the conjugate number to $p$; i.e.,
$1/p+1/q=1$.
A nontrivial solution of \eqref{e}
is said to be oscillatory if it has zeros
of arbitrary large value, and non-oscillatory otherwise.
An equation is said to be oscillatory if all its solutions are
oscillatory, and non-oscillatory otherwise.
Note that one solution of \eqref{e} is oscillatory
if and only if every solution of \eqref{e} is oscillatory,
which follows from the Sturm type separation result. Further,
if the generalized Riccati differential inequality
$w'+c(t)+(p-1)r^{1-q}(t)|w|^q\le 0$ is solvable on some interval
$[t_0,\infty)$, then \eqref{e} is non-oscillatory.
Methods based on these relations are referred as the
Riccati technique. There are several refinements of this idea:
Using a weighted Riccati type substitution;
working with integral, instead of differential, Riccati type equations
and inequalities
using a function sequence technique;
finding effective estimates for solutions of Riccati type equations;
etc. See for example \cite[Sections 2.2, 5.5]{book}.
It is known that many oscillation and asymptotical results for
\eqref{e} substantially depend on the convergence or the
divergence of the integral $\int^\infty r^{1-q}(s)\,ds$.
In contrast to the linear case, a suitable transformation
satisfactorily transfering one case into the other is not available for
\eqref{e} and hence it is often necessary to examine these cases
separately -- by using different approaches.
Note that usually the case with the convergent integral is more difficult
than the convergent case, which can be modelled according to the
case $r(t)\equiv 1$.
We study the convergent case; i.e., we assume that
\begin{equation} \label{1}
\int^\infty r^{1-q}(s)\,ds<\infty.
\end{equation}
The principal aim of this paper is to establish the so-called
function sequence technique for \eqref{e} under condition
\eqref{1}, and then to show some applications of this method.
The function sequence techniques for \eqref{e} with
$\int^\infty r^{1-q}(s)\,ds=\infty$ were studied
in \cite{book,HIKT,li-yeh}.
For this article \cite{kusano} is a useful reference.
This paper is organized as follows. In the next section we present
a modification of the Riccati technique involving a Riccati type
integral inequality. These relations are then utilized in
Section~\ref{S:3} to show the equivalence between nonoscillation
of \eqref{e} and convergence of certain function sequence. In the
last section, we apply this method to derive Hille-Nehari type
oscillation criteria and a Hille-Wintner type comparison theorem
for equation \eqref{e}. We also give an example of an equation
which, in particular, can be proved to be oscillatory using our
new results, but other known criteria are inapplicable.
\section{Modified Riccati Type Inequality}
We start with showing that in the relation between non-oscillation
of \eqref{e} and solvability of \eqref{riccati}, under condition
\eqref{1}, the Riccati type differential equation or inequality
can be replaced by certain Riccati type integral equation or
inequality. For the first time, it was observed in \cite{kusano}.
Here we recall this result, we add some refinements, and also give
two new proofs. Denote
$$
R(t):=\int_t^\infty r^{1-q}(s)\,ds
$$
and
\begin{align*}
\mathcal{S}(u)(t)&:=\int_t^\infty R^p(s)c(s)\,ds+p\int_t^\infty
r^{1-q}(s)R^{p-1}(s) u(s)\,ds\\
&\quad +(p-1)\int_t^\infty r^{1-q}(s)R^p(s)|u(s)|^q ds\,.
\end{align*}
\begin{theorem} \label{T:1}
(i) Assume $c(t)\ge0$ for large $t$. If \eqref{e} is
non-oscillatory, then $\int^\infty R^p(s)c(s)\,ds<\infty$ and there
is $w$ satisfying $R^{p-1}(t)w(t)\ge -1$ and
$R^p(t)w(t)=\mathcal{S}(w)(t)$ for large $t$. Moreover,
$\limsup_{t\to\infty}R^{p-1}(t)w(t)\le 0$.
(ii) Assume that $\infty>\int_t^\infty R^p(s)c(s)\,ds\ge 0$ for
large $t$. If there is $w$ satisfying $R^{p-1}(t)w(t)\ge -1$ and
$R^p(t)w(t)\ge\mathcal{S}(w)(t)$ for large $t$, then \eqref{e} is
non-oscillatory.
\end{theorem}
\begin{proof}
(i) See \cite{kusano} or \cite[Section~2.2]{book}.
(ii) Set
$v(t)=R^{-p}(t)\mathcal{S}(w)(t)$. For convenience we skip the argument
$t$ sometimes in the computations. Differentiating the equality
$R^pv=\mathcal{S}(w)$ we get
\begin{equation} \label{wv}
0=R^pv'+R^pc-pR^{p-1}v^{1-q}v+pR^{p-1}r^{1-q}w+(p-1)r^{1-q}|R^{p-1}w|^q.
\end{equation}
We will show that
\begin{equation} \label{wwvv}
pR^{p-1}r^{1-q}w+(p-1)r^{1-q}|R^{p-1}w|^q\ge pR^{p-1}r^{1-q}v+(p-1)r^{1-q}|R^{p-1}v|^q.
\end{equation}
Observe that the function
\begin{equation} \label{incr}
x\to px+(p-1)|x|^q\text{ is strictly increasing for $x\ge-1$}.
\end{equation}
From $R^pv=\mathcal{S}(w)\le R^p w$, we have $v\le w$.
We know $R^{p-1}w\ge-1$. Next we show that also $R^{p-1}v\ge-1$.
From $v=R^{-p}\mathcal{S}(w)$, we have that $R^{p-1}v\ge-1$ if and only
if $\mathcal{S}(w)\ge-R$, i.e., $\int_t^\infty R^p(s)c(s)\,ds+
\int_t^\infty
r^{1-q}(s)[pR^{p-1}(s)w(s)+(p-1)|R^{p-1}(s)w(s)|^q+1]\,ds\ge 0$.
But the above inequality is satisfied because $\int_t^\infty
R^p(s)c(s)\,ds\ge 0$ and
$pR^{p-1}w+(p-1)|R^{p-1}w|^q+1\ge-p+(p-1)+1=0$ which follows from
\eqref{incr} and $R^{p-1}w\ge-1$. Hence, $R^{p-1}v\ge-1$ which
together with \eqref{incr} and $v\le w$ yields \eqref{wwvv}. Using
\eqref{wwvv} in \eqref{wv} we obtain $0\ge
R^pv'+R^pc+(p-1)r^{1-q}R^p|v|^q$, or $0\ge
v'+c+(p-1)r^{1-q}|v|^q$. Thus \eqref{e} is non-oscillatory.
\end{proof}
\begin{remark} \label{rmk1}\rm
(i) The part (ii) of the theorem was proved in \cite{kusano} using
a different technique,
based on the Schauder-Tychonov fixed point theorem,
under the stronger assumptions $c(t)\ge 0$ and $R^{p-1}(t)w(t)$ is bounded.
A closer examination of that proof shows that these assumptions actually are not needed.
Later, in this paper, we present another proof of the part (ii) of the theorem, which arises out as a by-product when deriving
the function sequence technique.
(ii) From Theorem~\ref{T:1} (i), we immediately get the following simple criterion:
If $c(t)\ge0$ and $\int^\infty R^p(s)c(s)\,ds=\infty$, then \eqref{e} is oscillatory.
(iii) We conjecture that in the part (i) of the theorem,
the condition $c(t)\ge 0$ can be relaxed, e.g., to $\int_t^\infty R^p(s)c(s)\,ds\ge 0$.
\end{remark}
\section{Function Sequence Technique} \label{S:3}
We are in a position to establish the function sequence
technique for \eqref{e} under condition \eqref{1}.
Denote
\begin{gather*}
H(t)=R^{-p}(t)\int_t^\infty R^p(s)c(s)\,ds,\\
\mathcal{G}(u)(t)=R^{-p}(t)\int_t^\infty
r^{1-q}(s)[pR^{p-1}(s)u(s)+(p-1)|R^{p-1}(s)u(s)|^q]\,ds.
\end{gather*}
Observe that $H+\mathcal{G}(u)=R^{-p}\mathcal{S}(u)$. Further, $-R^{1-p}$ is a
fixed point for $\mathcal{G}$, and for $u$ with $uR^{p-1}\ge-1$,
$\mathcal{G}(u)$ is increasing with respect to $u$, which follows from
\eqref{incr}. Define the sequence $\{\varphi_k(t)\}$ as follows
$$
\varphi_0=-R^{1-p},\quad \varphi_{k+1}=H+\mathcal{G}(\varphi_k),\quad
k=0,1,2,\dots.
$$
It is easy to see that $\varphi_{k+1}\ge\varphi_k$, $k=0,1,2,\dots$,
provided $H\ge0$.
\begin{theorem} \label{T:seq}
Let $c(t)\ge 0$ for large $t$.
Equation \eqref{e} is non-oscillatory if and only if there exists
$t_0\in[a,\infty)$ such that
$\lim_{k\to\infty}\varphi_k(t)=\varphi(t)$ for $t\ge t_0$, i.e.,
$\{\varphi_k(t)\}$ is
well defined and pointwise convergent.
\end{theorem}
\begin{proof}
\textit{Only if part:} If \eqref{e} is non-oscillatory then there
is a function $w$ satisfying $R^{p-1}(t)w(t)\ge -1$ and
$R^p(t)w(t)=\mathcal{S}(w)(t)$ for large $t$, say $t\ge t_0$, by
Theorem~\ref{T:1}. In fact, instead of the equality
$R^p(t)w(t)=\mathcal{S}(w)(t)$ we may take the inequality
$R^p(t)w(t)\ge\mathcal{S}(w)(t)$, and the proof works as well. See also
Remark~\ref{R:2} (i), why this is useful. For convenience we skip
the argument $t$ sometimes in the computations. Since $w\ge
-R^{1-p}$, we have $w\ge\varphi_0$. Further
$\varphi_1=H+\mathcal{G}(\varphi_0)=H+\varphi_0\ge\varphi_0$ and
$\varphi_1=H+\mathcal{G}(\varphi_0)\le H+\mathcal{G}(w)=w$. Hence,
$\varphi_0\le\varphi_1\le w$ and $R^{p-1}\varphi_1\ge-1$.
Similarly, $w=H+\mathcal{G}(w)\ge H+\mathcal{G}(\varphi_1)=\varphi_2\ge
H+\mathcal{G}(\varphi_0)=\varphi_1$, hence,
$\varphi_0\le\varphi_1\le\varphi_2\le w$. By induction,
$\varphi_k\le\varphi_{k+1}\le w$ for $k=0,1,2,\dots$. Hence,
$\lim_{k\to\infty}\varphi_k(t)=\varphi(t)$.
\textit{If part:} If $\lim_{k\to\infty}\varphi_k(t)=\varphi(t)$,
then from the monotonicity of $\{\varphi_k\}$ it follows
$\varphi_k\le\varphi$ and $R^{p-1}\varphi_k\ge-1$ for
$k=0,1,2,\dots$, on $[t_0,\infty)$. Applying the Lebesgue monotone
convergence theorem in $\varphi_{k+1}=H+\mathcal{G}(\varphi_k)$, we get
$\varphi=H+\mathcal{G}(\varphi)$, or $R^p\varphi=\mathcal{S}(\varphi)$. Now it
is easy to see that $\varphi$ solves the generalized Riccati
equation \eqref{riccati}, and thus \eqref{e} is non-oscillatory.
\end{proof}
\begin{remark} \label{R:2} \rm
(i) A closer examination of the proof shows that, as a by-product,
we have obtained another proof of Theorem~\ref{T:1} (ii). Indeed,
if $w$ satisfies $R^{p-1}w\ge -1$ and $R^pw\ge\mathcal{S}(w)$, then
$\lim_{k\to\infty}\varphi_k(t)=\varphi(t)$, which implies
non-oscillation of \eqref{e}.
(ii) In the if part, $c(t)\ge 0$ can be relaxed to $\int_t^\infty R^p(s)c(s)\,ds\ge 0$.
We conjecture that this is possible also in the only if part.
(iii) The approximating sequence $\{\varphi_k\}$ is not the only one that is available.
Another possibility is, for instance, the sequence $\{\psi_k\}$, defined by
$\psi_0=\mathcal{G}(H-R^{1-p})$ and $\psi_{k+1}=\mathcal{G}(H+\psi_k)$.
\end{remark}
\begin{corollary} \label{C:1}
Let $c(t)\ge 0$ for large $t$.
Equation \eqref{e} is oscillatory if and only if either
\begin{itemize}
\item[(i)]
there is $m\in\mathbb{N}$ such that $\varphi_k$ is defined for $k=1,2,\dots,m-1$, but
$\varphi_m$ does not exists, i.e.,
$$
\int_t^\infty r^{1-q}(s)[pR^{p-1}(s)\varphi_{m-1}(s)+
(p-1)|R^{p-1}(s)\varphi_{m-1}(s)|^q]\,ds=\infty,
$$
or
\item[(ii)]
$\varphi_k$ is defined for $k=1,2,\dots$, but for arbitrarily large
$t_0\ge a$, there is $t_\ast\ge t_0$
such that $\lim_{k\to\infty}\varphi_k(t_\ast)=\infty$.
\end{itemize}
\end{corollary}
\section{Applications}
In this section we show how the function sequence technique can
be applied. By means of this method, we establish oscillation and
comparison results for \eqref{e}; some of them are known, some of
them are new or improving known ones. We start with modified
Hille-Nehari type criteria.
\begin{theorem} \label{T:RS}
Let $c(t)\ge0$ for large $t$. If
\begin{equation} \label{RS>0}
\limsup_{t\to\infty}R^{-1}(t)\mathcal{S}(\varphi_k)(t)>0
\end{equation}
for some $k\in\mathbb{N}\cup\{0\}$, then \eqref{e} is oscillatory.
\end{theorem}
\begin{proof}
If equation \eqref{e} is non-oscillatory, then as in the proof of
Theorem~\ref{T:seq}, we have $\varphi_k(t)\le w(t)$,
$k=0,1,2,\dots$ for large $t$. Moreover, $R^{-1}(t)\mathcal{S}(w)(t)\le
R^{p-1}(t)w(t)$ for large $t$ and
$\limsup_{t\to\infty}R^{p-1}(t)w(t)\le 0$ by Theorem~\ref{T:1}.
Hence,
\begin{align*}
\limsup_{t\to\infty}R^{-1}(t)\mathcal{S}(\varphi_k)(t)
&\le\limsup_{t\to\infty}R^{-1}(t)\mathcal{S}(w)(t)\\
&\le\limsup_{t\to\infty}R^{p-1}(t)w(t)\le0,
\end{align*}
which contradicts \eqref{RS>0}.
\end{proof}
Taking $k=0$ in the previous theorem, we have the following statement,
which was established also in \cite{kusano}.
\begin{corollary} \label{C:2}
Let $c(t)\ge0$ for large $t$. If
$$
\limsup_{t\to\infty}R^{-1}(t)\int_t^\infty R^p(s)c(s)\,ds>1,
$$
then \eqref{e} is oscillatory.
\end{corollary}
\begin{theorem} \label{T:HN1}
Let $c(t)\ge0$ for large $t$. If
\begin{equation} \label{osc}
\liminf_{t\to\infty}R^{-1}(t)\int_t^\infty R^p(s)c(s)\,ds>q^{-p},
\end{equation}
then \eqref{e} is oscillatory.
\end{theorem}
\begin{proof}
Condition \eqref{osc} can be rewritten as
\begin{equation} \label{osc1}
\int_t^\infty R^p(s)c(s)\,ds\ge\gamma R(t)
\end{equation}
for large $t$, say $t\ge t_0$, where $\gamma>q^{-p}$.
Then
\begin{equation} \label{phi1}
\varphi_1(t)=H(t)+\mathcal{G}(\varphi_0)(t)\ge R^{-p}(t)\gamma R(t)-R^{1-p}(t)
=\gamma_1 R^{1-p}(t),
\end{equation}
$t\ge t_0$, where $\gamma_1=\gamma-1$.
Note that $\gamma_1>-1$ and $R^{p-1}(t)\varphi_1(t)>-1$.
Hence, in view of \eqref{incr}, \eqref{osc1}, and \eqref{phi1},
$\varphi_2(t)=H(t)+\mathcal{G}(\varphi_1)(t)\ge\gamma R^{1-p}(t)+R^{-p}(t)
\int_t^\infty r^{1-q}(s)[p\gamma_1+(p-1)|\gamma_1|^q]\,ds=\gamma_2 R^{1-p}(t)$, where
$\gamma_2=\gamma+p\gamma_1+(p-1)|\gamma_1|^q$.
Since $\gamma_1>-1$, we have $\gamma_2>\gamma-p+p-1=\gamma-1=\gamma_1$ by \eqref{incr},
and so $\gamma_2>\gamma_1>-1$ and $R^{p-1}(t)\varphi_2(t)>-1$.
Arguing as above, by induction,
\begin{equation} \label{phik}
\varphi_k(t)\ge\gamma_k R^{1-p}(t),\quad k=1,2,\dots,
\end{equation}
where $\{\gamma_k\}$ is defined by
\begin{equation} \label{gamma}
\gamma_{k+1}=\gamma+p\gamma_k+(p-1)|\gamma_k|^q,\quad k=1,2,\dots.
\end{equation}
Moreover, $\gamma_{k+1}>\gamma_k>-1$, $k=1,2,\dots$.
Hence the limit $\lim_{k\to\infty}\gamma_k=L
\in(-1,\infty)\cup\{\infty\}$ exists. We claim that $L=\infty$.
If not, then \eqref{gamma}
yields
\begin{equation} \label{L-eq}
|L|^q+L+\gamma/(p-1)=0.
\end{equation}
We show that this equation has no solution in $(-1,\infty)$. We
distinguish two cases. If $L\in[0,\infty)$, then
$|L|^q+L+\gamma/(p-1)\ge\gamma/(p-1)>0$, a contradiction. To show
that also $L\in(-1,0)$ is impossible, it is sufficient to examine
the problem $x=g(x;\lambda)$, $x\in(-1,0)$, where
$g(x;\lambda)=\lambda+px+(p-1)|x|^q$ and $\lambda$ is a parameter.
It is easy to see that $-q^{1-p}$ is a fixed point of
$g(\cdot;q^{-p})$, and the parabola-like curve $x\to g(x;q^{-p})$
touches the line $x\to x$ at $x=-q^{1-p}$. Since $\gamma>q^{-p}$,
the problem $x=g(x;\gamma)$ has no solution in $(-1,0)$. But this
problem is equivalent to \eqref{L-eq}, and so
$\lim_{k\to\infty}\gamma_k=\infty$. Hence, from \eqref{phik}, we
have $\lim_{k\to\infty}\varphi_k(t)=\infty$ for $t\ge t_0$.
Equation \eqref{e} is oscillatory by Corollary~\ref{C:1}.
\end{proof}
\begin{theorem} \label{T:HN2}
Let $c(t)\ge0$ for large $t$. If
\begin{equation} \label{nonosc}
R^{-1}(t)\int_t^\infty R^p(s)c(s)\,ds\le q^{-p}\quad \text{for
large $t$},
\end{equation}
then \eqref{e} is non-oscillatory.
\end{theorem}
\begin{proof}
Condition \eqref{nonosc} can be rewritten as $\int_t^\infty
R^p(s)c(s)\,ds\le\delta R(t)$ for large $t$, say $t\ge t_0$, where
$0<\delta\le q^{-p}$. Similarly as in the previous part, with a
wide utilization of \eqref{incr}, we get
\begin{equation} \label{phidelta}
\varphi_k(t)\le\delta_k R^{1-p}(t),\quad k=1,2,\dots,
\end{equation}
where $\{\delta_k\}$ is defined by
\begin{equation} \label{delta}
\delta_{k+1}=\delta+p\delta_k+(p-1)|\delta_k|^q,\quad k=1,2,\dots
\end{equation}
and $\delta_1=\delta-1$. Moreover, $\delta_{k+1}>\delta_k>-1$,
$k=1,2,\dots$. We claim that $\{\delta_k\}$ converges. Consider
the fixed point problem $x=g(x;\lambda)$, where $g$ is defined as
above. In addition to the already mentioned properties of $g$, we
remark that $g(\cdot;\lambda)$ has the minimum at $x=-1$,
$g(-1;\lambda)=\lambda-1$, and
$g:[-1,-q^{1-p}]\to[q^{-p}-1,-q^{1-p}]$. Hence, if we choose
$x_1=q^{-p}-1$, then the approximating sequence
$x_{k+1}=g(x_k;q^{-p})$ is strictly increasing and converges to
$-q^{1-p}$. Consequently, $\{\delta_k\}$ defined by \eqref{delta}
with $\delta_1=\delta-1$ converges as well, and permits
$\delta_k\le x_k<-q^{1-p}$. Thus $\{\varphi_k\}$ converges by
\eqref{phidelta}, and so \eqref{e} is non-oscillatory by
Theorem~\ref{T:seq}.
\end{proof}
\begin{remark} \label{rmk3} \rm
Theorems \ref{T:HN1} and \ref{T:HN2} were proved also in
\cite{kusano}, using a different technique. See also
\cite[Section~2.3.1]{book}.
\end{remark}
Now we give an example of an equation involving parameters which,
in particular, can be proved to be oscillatory using our new
results, but other known criteria are inapplicable.
\begin{example} \label{exa1} \rm
Let $r(t)=t^{(1-q)t}(1+\log t)^{q-1}$ and
$c(t)=t^{pt}[\lambda t^{-t}(1+\log t)+\gamma t^{-t}(1+\log t)\sin t+\gamma t^{-t}\cos t]$
in equation \eqref{e}, where $\lambda>\gamma>0$.
It is easy to see that $c(t)>0$ for large $t$ and $R(t)=t^{-t}$.
Further,
\begin{align*}
&R^{-1}(t)\int_t^\infty R^p(s)c(s)\,ds\\
&=t^t\int_t^\infty\big[\lambda s^{-s}(1+\log s)
+\gamma s^{-s}(1+\log s)\sin s +\gamma
s^{-s}\cos s\big]\, ds\\
&=t^t(\lambda t^{-t}+\gamma t^{-t}\sin t)\\
&=\lambda+\gamma \sin t.
\end{align*}
If $\lambda+\gamma\le q^{-p}$, then \eqref{e} is non-oscillatory by Theorem~\ref{T:HN2}.
If $\lambda-\gamma>q^{-p}$, then \eqref{e} is oscillatory by Theorem~\ref{T:HN1}.
Thus next we assume $\lambda-\gamma\le q^{-p}$ and $\lambda+\gamma>1$. Then Theorem~\ref{T:HN1}
cannot be applied, but \eqref{e} is oscillatory by Corollary~\ref{C:2}.
Now assume that $\lambda+\gamma\le 1$ and $\lambda+\gamma+f(\lambda+\gamma-1)>0$,
where
$f(x)=px+(p-1)|x|^q$. Then Corollary~\ref{C:2} cannot be applied, but \eqref{e} is oscillatory by
Theorem~\ref{T:RS} with $k=1$. Indeed, this follows from the equality
\begin{align*}
R^{-1}(t)\mathcal{S}(\varphi_1)(t)
&=\lambda+\gamma\sin t+t^t\int_t^\infty
s^{-s}(1+\log s)\big[p(\lambda+\gamma\sin s-1)\\
&\quad +(p-1)|\lambda+\gamma\sin s-1|^q\big]\,ds.
\end{align*}
It is easy to see that the sets of $\lambda$'s and $\gamma$'s,
which satisfy these requirements, are nonempty.
Using Theorem~\ref{T:RS} with $k\ge 2$ we can similarly handle the
cases where $\lambda+\gamma+f(\lambda+\gamma-1)$
is nonpositive, but is not ``too negative''.
\end{example}
Next we prove a Hille-Wintner type comparison theorem.
Along with \eqref{e} consider the equation
\begin{equation} \label{e2}
(r(t)\Phi(x'))'+\tilde c(t)\Phi(x)=0,
\end{equation}
where $\tilde c$ is continuous on $[a,\infty)$.
\begin{theorem} \label{thm6}
Let $c(t)\ge0$ and
\begin{equation} \label{cond-HW}
\int_t^\infty R^p(s)c(s)\,ds\ge\int_t^\infty R^p(s)\tilde c(s)\,ds\ge 0
\end{equation}
for large $t$. If \eqref{e} is non-oscillatory, then
\eqref{e2} is non-oscillatory.
\end{theorem}
\begin{proof}
If \eqref{e} is non-oscillatory, then $\{\varphi_k\}$ is well defined
and $\lim_{k\to\infty}\varphi_k(t)=\varphi(t)$
by Theorem~\ref{T:seq}. The following computations hold for large $t$.
From condition \eqref{cond-HW}, we have
$H(t)\ge R^{-p}(t)\int_t^\infty R^p(s)\tilde c(s)\,ds=:\tilde
H(t)$. Then $\varphi_1(t)=H(t)+\mathcal{G}(\varphi_0)(t)\ge\tilde H(t)
+\mathcal{G}(\varphi_0)(t)=:\tilde\varphi_1(t)$. Clearly,
$\tilde\varphi_1(t)\ge\varphi_0(t)=:\tilde\varphi_0(t)$. By
induction, $\varphi_{k+1}(t)\ge\tilde
H(t)+\mathcal{G}(\tilde\varphi_k)(t)=:\tilde\varphi_{k+1}(t)$,
$k=0,1,2,\dots$. Moreover, $\tilde\varphi_k(t)\le\varphi(t)$ and
$\tilde\varphi_k(t)\le\tilde\varphi_{k+1}(t)$, $k=0,1,2,\dots$.
Consequently, \eqref{e2} is non-oscillatory by Theorem~\ref{T:seq}
and Remark~\ref{R:2} (ii).
\end{proof}
\begin{remark} \label{rmk4} \rm
(i) This theorem was established also in \cite{kusano} by direct using
of the Riccati technique.
See also \cite[Section~2.3.1]{book}.
Notice however that here we do not require $\tilde c$ to be nonnegative.
(ii)
Under the conditions of the theorem, oscillation of
\eqref{e2} implies oscillation of \eqref{e}.
(iii) From Hille-Nehari type criteria (Theorem~\ref{T:HN1} and
Theorem~\ref{T:HN2}) we get that the generalized Euler differential equation
\begin{equation} \label{ee}
(r(t)\Phi(y'))'+\lambda r^{1-q}(t)R^{-p}(t)\Phi(y)=0
\end{equation}
is oscillatory if and only if $\lambda>q^{-p}$. Note that
$y=R^{(p-1)/p}$ is a nonoscillatory solution of \eqref{ee} with
$\lambda=q^{-p}$. Observe that, conversely, knowing this result,
Theorems \ref{T:HN1} and \ref{T:HN2} can be alternatively obtained
by the Hille-Wintner type result comparing equation \eqref{e} with
equation \eqref{ee}. Similar but a little bit more complicated
approach to establish these theorems was used in \cite{kusano}:
The proofs there are based on a knowledge of oscillation behavior
of certain generalized Euler differential equation (which a
special case of \eqref{ee}), Hille-Wintner type comparison
theorem, and a transformation of independent variable. At any
rate, we believe that the approach via the function sequence
technique has an advantage over this comparison method in cases
where a transformation is not available or guessing a solution is
difficult. This may concern, e.g., a discrete counterpart of
\eqref{e}, a half-linear difference equation.
\end{remark}
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\textit{Half-linear Differential Equations}, Elsevier,
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\bibitem{HIKT} H. Hoshino, R. Imabayashi, T. Kusano, T. Tanigawa;
On second order half-linear oscillations, \textit{Adv. Math. Sci. Appl.}
\textbf{8} (1998), 199--216.
\bibitem{kusano} T.~Kusano, Y.~ Naito; Oscillation and nonoscillation
criteria for second order quasilinear differential equations,
\textit{Acta. Math. Hungar.} \textbf{76} (1997), 81--99.
\bibitem{li-yeh} H. J. Li, C. C. Yeh; Nonoscillation of half-linear
differential equations, \textit{Publ. Math. (Debrecen)}
\textbf{49} (1996), 327--334.
\end{thebibliography}
\end{document}