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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 28, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/28\hfil Decaying oscillatory solutions]
{Criteria and estimates for decaying oscillatory solutions for some second-order
  quasilinear ODEs}

\author[Tadie \hfil EJDE-2017/28\hfilneg]
{Tadie}

\address{Tadie \newline
Mathematics Institut \\
Universitetsparken 5 \\
2100  Copenhagen, Denmark}
\email{tadietadie@yahoo.com}

\dedicatory{Dedicated to the family  Teku  Kuate Kamguem Ebenizer}

\thanks{Submitted September 14, 2016. Published January 24, 2017.}
\subjclass[2010]{34C10, 34K15}
\keywords{Oscillation criteria for ODE and Estimates of decaying solutions}

\begin{abstract}
 Oscillation criteria for the solutions of quasilinear second order ODE
 are revisited.  In our early works \cite{t6, t7}, we obtained basic 
 oscillation criteria for
 $$
 \big\{ \phi_\alpha(u'(t))\big\}' + \alpha c(t) \phi_\beta(u(t)) =0
 $$
 by estimating of the diameters of the nodal sets of the solutions.
 The focus of this work is to estimate the decay of the oscillatory solutions.
 Let $u$ be a strongly oscillatory solution, $(t_m)$ the increasing sequence
 of zeros of $u'$, and $D_m$ the nodal set of $u$ that contains $t_m$.
 We  estimate $|u(t_m)|_\infty:=\max_{t\in D_m} |u(t)|$ and
 the diameter of $D_m$ as $m\to \infty$.
\end{abstract}

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

\section{Introduction}

 For some constants $b, \beta,  q,  c_0, \alpha >0 $ and
$\phi_\gamma(S):= |S|^{\gamma - 1} S $ (with $\gamma>0$),
we consider problems of the type
\begin{equation} \label{e1.1}
\begin{gathered}
 \big\{ \phi_\alpha(u'(t))\big\}' + \alpha c(t) \phi_\beta(u(t)) =0 , \quad  t>0; \\
 u(0)=0, \quad  u'(0)=b>0,
\end{gathered}
\end{equation}
where  $c \in C^1(\mathbb{R}^+, (c_0, \infty))$, with $ c'>0$  and
 $c(t) =O(t^q)$ as $t\to\infty$.
We will review some oscillation criteria for such equations and
establish estimates of the decay of oscillatory solutions of
\eqref{e1.1}.

\begin{definition} \rm
A function $u$ is said to be oscillatory if it has a zeros in every exterior
domain $\Omega_T := (T,\infty)$ with $T\geq 0$.
A function $u$ is said to be strongly oscillatory if its zeros are isolated,
or if it has  nodal sets in every $\Omega_T$.
A nodal set of a function $v$ is an interval $D(v)=[t, s] $ such that
$v(t)=v(s)=0 $ and $v\neq 0 $ in $(t,s)$.
For the function  $v^+(t)=\max\{0,v(t)\}$, the nodal set $D(v^+)=[t, s] $ 
is such that $v(t)=v(s)=0$ with $v> 0 $ in $(t,s)$.

An equation (or a problem) is said to be oscillatory in $\Omega_T$ if its
bounded and non-trivial solutions belong to  $C^2(\Omega_T)$ and are
strongly  oscillatory.
\end{definition}

For a strongly oscillatory function $u$,
$[ D(u)]$ will denote the set of nodal sets of $u$.
In this case there are two increasing sequences $(x_k)$ and $(t_k)$ such that
$x_k<t_k<x_{k+1}$,  $u(x_k)=0$, and  $u'(t_k)=0$.
We denote $D_k:=D_k(u)=[x_k, x_{k+1}]$
as a nodal set of $u$. We denote $|u(t_k)|:= \max_{D_k(u)} |u(t)|$.
For $a,b \in \mathbb{R}$, we define $a \wedge b:= \min \{ a, b \}$.

Our main result for problem \eqref{e1.1} reads as follows.

\begin{theorem} \label{thmA}
For all $b,\alpha,  \beta, c_0>0$,  any non-trivial and bounded solution of
\eqref{e1.1} is strongly oscillatory in $[0, \infty)$. With the corresponding
 elements as defined above.
as $ m\to \infty$, with $\beta_\ast:= \alpha \wedge \beta$, we have
\begin{gather}
\frac{\pi_\alpha}{[c(x_{m+1})]^{1/(\beta_\ast+1)}}
<| x_{m+1} - x_m |< \frac{\pi_\alpha}{[c(x_m)]^{1/(\beta_\ast+1)}}, \label{e1} \\
|u(t_m)|_\infty  \leq {\rm const.}  [c( t_m)]^{-1/(\beta_\ast+1)}
= {\rm const.} [t_m ]^{-q/(\beta_\ast+1)}, \label{e2}
\end{gather}
where
$$
\pi_\alpha:=\frac{2\pi}{(\alpha+1) \sin [\pi/(\alpha+1)]}\,.
$$
\end{theorem}

The result in \cite{t4} is limited to estimates \eqref{e1} of the diameters
of the nodal sets, for the case $\alpha=\beta>0$.

Now we present some Picone-type formulae which will be used throughout this article.
 To start, for $w, y \in C^1(\mathbb{R}, \mathbb{R})$  and $\gamma>0$ we define
(see  e.g. \cite{j2,j1})
\begin{equation} \label{e1.2}
\zeta_\gamma (w,y):= |w'|^{\gamma+1}  - (\gamma +1) w' \phi_\gamma
\big(\frac wy y' \big) +  \gamma \big|\frac wy  y'\big|^{\gamma+1}
\end{equation}
which is strictly positive and is null only if there exists
$\mu \in \mathbb{R}$ such that  $w \equiv \mu y$.

Let $ C,  C_1,  \alpha, \beta>0$ and
$ w,  z, u \in C^1(\mathbb{R}) $, respectively, be solutions in $\mathbb{R}^+$,
for
\begin{gather*}
\big\{ \phi_\alpha(u'(t))\big\}' + c(t)\alpha  \phi_\beta(u(t)) =0 ; \\
\big(\phi_\alpha(z')\big)' + C \alpha \phi_\alpha(z)=0 , \\
\big(\phi_\alpha(w')\big)' + C_1 \alpha \phi_\beta(w)=0.
\end{gather*}
Using  that for $\gamma>0 $ and $S, T \in \mathbb{R}$,
\[
S\phi_\gamma'(S)= \gamma \phi_\gamma (S), \quad
S\phi_\gamma(S)=|S|^{\gamma+1},\quad
\phi_\gamma( ST )=\phi_\gamma(S) \phi_\gamma(T),
\]
wherever $u\neq 0$, we have
\begin{equation} \label{e1.3}
\begin{gathered}
\big[ z\phi_\alpha(z') - z\phi_\alpha(\frac zu u') \big]'
=\zeta_\alpha (z,u) + \alpha  |z|^{\alpha+1}
\big\{ c(t) |u|^{\beta-\alpha} - C \big\}; \\
\begin{aligned}
\big[ w\phi_\alpha(w') - w\phi_\alpha(\frac wu u') \big]'
&=\zeta_\alpha (w,u) + \alpha   |w|^{\beta+1}
\big\{ c(t)\big |\frac uw \big |^{\beta-\alpha} - C_1 \big\}\\
& =\zeta_\alpha (w,u) + \alpha   |w|^{\alpha+1}
\big\{ c(t)\big| u \big |^{\beta-\alpha}
 - C_1 \big|w \big|^{\beta - \alpha} \big\}.
\end{aligned}
\end{gathered}
\end{equation}

Note that:
\begin{itemize}
\item[(1)] For  $\mu >0$, if  the function $Z(t):= \mu z(t)$ is used
 in \eqref{e1.3}(i),
$ \big(\phi_\alpha(Z')\big)' + C \alpha \phi_\alpha(Z)=0 $
  and \eqref{e1.3}(i) remains the same with $Z$ replacing $z$.

\item[(2)]  But if $W(t):= \mu w(t)$ then
\[
\big(\phi_\alpha(W')\big)' + \mu^{\alpha- \beta}   C_1 \alpha \phi_\beta(W)=0
\]
 and \eqref{e1.3}(ii) with $W$ holds with $C_1$ replaced by
$\mu^{\alpha-\beta} C_1$.

\item[(3)] The Picone-type formulae in \eqref{e1.3}  will be the main
tools in this work. In fact as the formulae make sense only wherever
$u\neq 0$ ($w \neq 0)$,  if the right-hand side of the formula happens to
be strictly positive in  a set $ D $ then  the integration over
$D$  would give 0 at the left and a strictly positive value at the right
 if $u\neq 0$ ($w\neq 0$) in $D$ and $u|_{\partial D}=0$.
 Therefore if the right-hand side of \eqref{e1.3} is strictly positive on
a set $D$, we cannot have $u$ $(w) \neq 0$ inside $D$ and
 $u|_{\partial D} =0$, implying that $u$ has to have a zero inside such a $D$.
\end{itemize}

Now we study equations with  positive constant coefficients.

\begin{theorem} \label{thm1.2}
For each $k, \theta ,  \beta >0 $, any bounded and non-trivial  solution $u$
of the problem
\begin{equation} \label{e1.4}
\big\{ \phi_\theta(u')\big\}' + k \theta \phi_\beta(u)=0, \quad t>0; \quad
u(0)=0; \quad u'(0)=A>0
\end{equation}
is  oscillatory  and
\begin{equation} \label{e1.5}
\begin{gathered}
(\beta+1)|u'(t)|^{\theta+1}  + k (\theta+1)|u(t)|^{\beta+1}
=(\beta+1)  A^{\theta+1} \quad\forall t>0, \\
u(T)=0 \Rightarrow |u'(T)|=A \quad \forall T>0 \\
u'(S)=0 \Rightarrow |u(S)|= \big[\frac{ (\beta+1)A^{\theta+1}}{k (\theta+1)}
\big]^{\frac 1{\beta+1} }  \quad \forall S>0,\\
\text{which implies }
\max_{\mathbb{R}^+} |u|=  \big[\frac{(\beta+1) A^{\theta+1}}{k(\theta+1)}
\big]^{\frac 1{\beta+1} }
\text{ and } \max_{\mathbb{R}^+}   |u'| = A.
\end{gathered}
\end{equation}
When $\beta=\theta>0$, \eqref{e1.5}(iv) reads
$$
 \max_{\mathbb{R}^+} |u|= \big[\frac 1k  \big]^{\frac 1{\beta+1} } A \quad
\text{and}\quad \max_{\mathbb{R}^+}   |u'| = A.
$$
\end{theorem}


\begin{proof}
That this problem is oscillatory has been established in \cite{t6, t7}
but for self-contained purpose we show it
 using a method relevant to the present work.
Let $u\in C^2(\mathbb{R}^+)$ be a non-trivial and bounded solution
of \eqref{e1.2}.
Then
\begin{align*}
\big(\phi_\theta(u')\big)'
& = \Big(\big[u'^2\big]^{(\theta -1)/2} u' \Big)'\\
&= u'' \Big(\big[u'^2\big]^{(\theta -1)/2}\Big)
+ u'\Big( (u'^2 )^{(\theta -1)/2} \Big)' \\
&= u'' \Big(|u'|^{\theta-1} \Big)
+ (\theta -1) u'' |u'|^{\theta -1}\\
&= \theta u'' |u'|^{\theta-1}
\end{align*}
  and
\[
 u' \big( \phi_\theta(u') \big)'
=\frac \theta 2 (u'^2)' \big(u'^2 \big)^{(\theta -1)/2}
=\frac \theta {(\theta +1)} \big( |u'|^{\theta+1} \big)'.
\]
Similarly
\[
\theta k u' \phi_\beta (u) = \theta k u'u |u|^{\beta-1}
= \frac \theta 2 k (u^2)' \big(u^2 \big)^{(\beta -1)/2}
= \frac \theta{ (\beta+1)} k \big(|u|^{\beta+1} \big)' .
\]
The two inequalities above  lead to
\begin{equation} \label{e1.6}
\big\{ (\beta+1)|u'|^{\theta+1}  + k (\theta+1)|u |^{\beta+1} \big\}'=0
\end{equation}
and \eqref{e1.5}(i)  follows. \eqref{e1.5}(ii) to \eqref{e1.5}(v)
follow immediately.

Assume that $u>\nu>0$ in some $\Omega_T$.
Then with $k$ replacing $c(t)$,  in \eqref{e1.3}(i),
$k|u|^{\beta-\alpha} - C \geq k\nu^{\beta-\alpha} -C >0 $
if we take $C$ small enough. With this,
 the integration over $D(z^+)\subset \Omega_T$ would lead to a contradiction
as the left hand side would be   zero and the right strictly positive.

If in such an $\Omega_T$ $ u>0$ and $u\searrow 0$ then (i) or (ii)
would  be violated.
Therefore $u$ has to have  a zero in any $\Omega_T$.
\end{proof}

\begin{corollary} \label{coro1.3}
Let $ A_1,  A_2, \theta, \beta>0$ $\theta\geq \beta $.
Let $u_1$ and $u_2$, respectively, be oscillatory  solutions  for
\begin{equation} \label{e1.7}
\big\{ \phi_\theta(u_i')\big\}' + k_i \theta \phi_\beta(u_i)=0, \quad t>0;
\quad u(0)=0; \quad u'(0)=A_i>0
\end{equation}
 with
\[
 \frac{A_1^{\theta+1}}{k_1} < \frac{A_2^{\theta+1}}{k_2}.
\]
Let $D(u_i^+)$ denote a nodal set of $u_i^+$, and assume that
$D(u_1^+)\cap D(u_2^+) \neq \emptyset$.
If $R \in D:= D(u_1^+)\cap D(u_2^+)$ with $u'_1(R)=u'_2(R)=0$, then
\begin{equation} \label{e1.8}
\max_{D(u_1^+)} u_1 :=u_1(R)  > \max_{D(u_2^+)} u_2 :=u_2(R) .
\end{equation}
Let $ u_1,  u_2 ,  u_3 $, respectively, be non-trivial oscillatory solutions for
\begin{equation} \label{e1.9}
 \big\{ \phi_\theta(u_i')\big\}' + k_i \theta \phi_\beta(u_i)=0,\;  t>0; \quad
u(0)=0; \quad u'(0)=A>0,
\end{equation}
where $k_1>k_2>k_3>0$.
Then if there is $S >0 $ such that for some $ D(u_1^+)$,  $D(u_2^+) $ and $D(u_3^+)$,
\[
 S \in  D(u_1^+) \cap D(u_2^+) \cap D(u_3^+), \quad
D(u'_i (S)=0, \quad\text{for } i=1,2,3,
\]
then
\[
 \max_{D(u_3^+)}  u_3^+(t) = u_3(S) \leq \max_{D(u_2^+)}  u_2^+(t)
= u_2(S) \leq \max_{D(u_1^+)}  u_1^+(t) = u_1(S).
\]
\end{corollary}

The proof of the above corollary follows straight from \eqref{e1.5}(iv).


\begin{remark} \label{rmk1a} \rm
(1)  It  is easy to show that when the coefficient of $\phi_\beta$ is a positive
constant, the solutions are periodic.

(2)  There are two transformations which could be used in some proofs:
\begin{itemize}
\item[(i)] For any  oscillatory function $u$, and $ \lambda>0$,
 the associated function $ u_\lambda (t):= \lambda u(t)$ is also  oscillatory,
 having exactly the same zeros as $u$ but with
$ |u_\lambda |_\infty =\lambda |u|_\infty $ and
 $|u'_\lambda|_\infty =\lambda |u'|_\infty$.

\item[(ii)] For $ \xi \in \mathbb{R}$, the translated function
$ U_\xi(t):= u(t+\xi) $ would be also oscillatory as $u$ and the curve
$ (t,  U_\xi (t) ) $ would be that  of $u$, slit  alongside the $t$-axis forward
(if $\xi<0$) or backward (if $ \xi>0$).
\end{itemize}

(3)  Let $u$ and $v$, respectively, be oscillatory solutions  of
\begin{gather*}
\big(\phi_\alpha(u')\big)' + c(t)\phi_\beta(u)=0; \quad  t>0; \\
\big(\phi_\alpha(v')\big)' + C\phi_\beta(v)=0,\;  t>0; \quad v(0)=0, \quad v'(0)=b>0.
\end{gather*}
If some of their nodal sets satisfy $ D(u^+)\cap D(v^+)\neq \emptyset $,  and
$ R\in D(u^+)$ satisfies  $ u'(R)=0$,
then $ \xi $ can be chosen such that the transformed
$W(t):= v(t+\xi) $ has the same singularity $R$ in the resulted $ D(W^+)$
i.e. $W'(R)= u'(R)=0$.

In summary if
(a) $D(u^+)\cap D(v^+)\neq \emptyset $ and
(b) $u$ has a zero  inside $D(v^+) $, then
there is $ (\xi,  \lambda )\in \mathbb{R} \times \mathbb{R}^+ $ such that for
some $ R\in D(u^+)$, then the function
$V(t):= \lambda v(t+\xi)$  satisfies $V'(R)=u'(R)=0 $ and
$|V|_\infty=\lambda |v|_\infty$.
\end{remark}

\section{Equations with increasing and unbounded coefficients}

It is known that if   $ c(t) $ is increasing  and unbounded  then if
$(x_n)_{n\in \mathbb{N}} $ denotes the increasing
successive zeros of the oscillatory  solution $z$ of
\begin{equation} \label{e2.1}
 \big(\phi_\alpha(z')\big)' + \alpha c(t)\phi_\alpha(z)=0 ,
\end{equation}
then
\[
  |x_{n+1} - x_n|=O\big(  \pi_\alpha [c(x_n)]^{-1/(\alpha +1)} \big)
\]
for large $n$. In fact  as for large $ m\in \mathbb{N}$,
$ c(x_m) \leq  c(t) \leq  c(x_{m+1})  $,   inside
$ D_{m}:=[x_{m},  x_{m+1}] $; from \cite{t4}, with
$ C(x):=[c(x)]^{1/(\alpha+1)}$, we have
\begin{equation} \label{e2.2}
\frac{\pi_\alpha}{C(x_{m+1})}
\leq |x_{m+1}  -  x_{m}| \leq  \frac{\pi_\alpha}{C(x_m)}.
 \end{equation}

\begin{lemma} \label{lem2.1}
For some $  c_0>0$ let $c\in C^1(\mathbb{R},  (c_0,  \infty) )$  be  increasing,
and let $\alpha , \beta>0$.
Then any non-trivial and  bounded solution  $u$ of
\begin{gather*}
\big\{ \phi_\alpha(u'(t))\big\}' + c(t)\alpha  \phi_\beta(u(t)) =0 , \quad  t>0; \\
u(0)=0, \quad  u'(0)=b>0
\end{gather*}
is   oscillatory.
\end{lemma}


\begin{proof}
The  oscillatory character of the equations have been established in our
early papers  \cite{t4, t7}
but for later use purpose, we provide  some slightly different
 proofs using  Picone-type formulae.

(1) Assume that  $ \alpha \geq \beta>0$.
Let $u$ be such a solution and  with  some $C>0$. Let   $z$ be an  oscillatory
solution of
$$
 \big(\phi_\alpha(z')\big)' + C \alpha \phi_\alpha(z)=0; \quad t>0 .
$$

If we suppose that $ u>\mu>0 $ in some $ \Omega_S  $ then
$  c(t)|u(t)|^{\beta - \alpha} > c(t) \mu^{\beta - \alpha} $  for  $t>S$ and
 the right-hand side of \eqref{e1.3}(i)  is eventually  strictly positive
in $\Omega_S$.

Assume that $ u>0  $ in some $\Omega_T$ for some $T>0$ and $ u \searrow 0$ as
$t\to \infty$.
Still  because $ 0<\beta\leq \alpha$, the function $c(t)|u(t)|^{\beta - \alpha} $
is unbounded in $\Omega_T$ and  the right-hand side of \eqref{e1.3}(i)
is  eventually strictly positive in $\Omega_S$ for large $S>T$.
In those cases,  the right-hand side of \eqref{e1.3}(i)  is strictly
positive in any such a $  D(z^+) \subset \Omega_T$.
Thus the assumption cannot stand; $u$ has a zero in any $\Omega_T$.

(2) Assume that $ \beta>\alpha>0$.
For  a  constant   $C>0$ and an  oscillatory solution $z$ of
$$
\big(\phi_\alpha(w')\big)' + \alpha C \phi_\beta(w)=0, \; t>0;
\quad w(0)=0, \quad w'(0)=b>0
$$
wherever $ u\neq 0$ in some  interval $D$, \eqref{e1.3}(ii) holds
(with $C$ instead of $C_1$ ).

As $C$ is constant, from \eqref{e1.5}, $w^+$ has a constant maximum value
in any nodal set  $D(w^+)$ which is
\begin{equation*}
 |w|_\infty:= |w|_{C(D(w^+))}=\max_{D(w^+)}|w|
=  \big[\frac{(\beta+1) b^{\alpha+1}}{(\alpha+1)C}\big]^{\frac 1{(\beta+1)} }.
\end{equation*}
We see that the smaller $ b :=w'(0)  $ is, the smaller  $|w|_\infty$ will be.

If there exists $\nu>0$ such that $u>\nu $ in $\Omega_R$ then as $c$ is unbounded,
the right-hand side of \eqref{e1.3}(ii) is eventually strictly positive in
any nodal set $D(w^+) \subset \Omega_S  $ for large enough $ S>R$ as we would have
\[
\big\{ \frac{c(t)}C \big |\frac uw \big |^{\beta-\alpha} - 1 \big\} >
\big\{ \frac{c(t)}C \big |\frac u{|w|_\infty} \big |^{\beta-\alpha} - 1 \big\}
\]
 with an unbounded $c(t)$.

 Assume that $ u>0  $ and $ u $ decreases to zero at $\infty$ in some
$ \Omega_T$, with $T>0$.
Then for any $ R>T  $ and $ J_R:=[R,  2R]$, we define
$ \nu := u(2R):= \min_{J_R} [u^+]$.
We take $ C:=c(R):=C_1$ and $R>T$ so big that $w(R)=O( R^{-q/(\beta+1)}  )$.
With such a large $ c(R)  $, $w^+$ has many nodal sets $D(w^+)$ in $J_R$ and with
$ b $ small enough,  $ |w|_\infty < \nu $  and
 $ \big\{ \frac{c(t)}{C(R)}  \big|\frac \nu{w} \big|^{\beta - \alpha} - |  \big\}>0$
 in many of them.

The integration over such a $D(w^+)$ of \eqref{e1.3}(ii) would lead to a
 contradiction as the left hand side would give 0
and the right strictly positive. Thus $ u>0 $ cannot hold in any $\Omega_T$.
This, as above,  completes the oscillatory character of $u$.
\end{proof}



\begin{theorem} \label{thm2.2}
(1) Let $u$ and $z$, respectively, be oscillatory solutions of
\begin{equation*}
 \big\{ \phi_\alpha(u'(t))\big\}' + c(t)\alpha  \phi_\beta(u(t)) =0,  \quad
\big(\phi_\alpha(z')\big)' + m \alpha \phi_\alpha(z)=0, \quad t>0
\end{equation*}
 where  for some $ c_0>0$, $c\in C^1(\mathbb{R},  (c_0,  \infty) )$ is an
increasing  and unbounded  function and $ \alpha \geq \beta >0$.

Assume that  there are two overlapping nodal sets $ D(z^+) $ and $ D(u^+)$
such that
\begin{itemize}
\item[(i)] thee exists $R \in D(z^+)\cap D(u^+)$ such that $z'(R)=u'(R)=0$;

\item[(ii)]  $u$ has a zero inside $D(z^+)$  and
$ \big\{c(t) |u|^{\beta - \alpha} - m\big\} > 0 $ in $ D(z^+)$.
\end{itemize}
Then $ D(u^+) \subset D(z^+)  $ whence
\begin{equation} \label{e2.3}
\operatorname{diam} \big[ D(u^+) \big] \leq \operatorname{diam}
\big[  D(z^+) \big] = O\big( \big[\frac 1m \big]^{1/(\alpha +1)}\big).
\end{equation}

(2) Also if $ 0<\alpha<\beta  $ instead of $z$ the solution $w$ of
\[
\big(\phi_\alpha(w')\big)' + m \alpha \phi_\beta(w)=0, \quad t>0
\]
is used, then under the conditions (i) and (ii)  the results hold with $w$
replacing $z$ with the  following changes:
$  \big\{c(t) \big| \frac uw \big|^{\beta - \alpha} - m\big\} > 0 $ in
$ D(w^+)  $ and we have
\begin{equation} \label{e2.4}
\operatorname{diam} \big[ D(u^+) \big] \leq \operatorname{diam}
\big[  D(w^+) \big] = O\big( \big[\frac 1m \big]^{1/(\beta +1)}\big).
\end{equation}
\end{theorem}

\begin{proof}
Let  $  D(z^+):= [t_1,  t_2] $  and $D(u^+):= [x_1,  x_2]  $ with
 $ t_1 < x_1 < R < t_2$. We claim that
$ R<x_2 < t_2$.

Otherwise if $ u>0 $ in $ (R,  t_2) $ the integration of \eqref{e1.3}(i)
(where $m=C$) over $(R,t_2)$ leads to an absurdity as unlike the right-hand side,
the left would be zero. Thus $x_2$ has to be between $R$ and $t_2$  and using
\eqref{e2.2}, it leads to  \eqref{e2.3}.

For the case of $w$ we just use \eqref{e1.3}(ii) instead of \eqref{e1.3}(i).
\end{proof}

As a prelude for the next results we have the following Lemma;

\begin{lemma} \label{lem2.3}
For the strongly oscillatory solution $u$ of
\begin{equation} \label{e2.5}
  \big(\phi_\alpha(u')\big)' +  \alpha c(t) \phi_\beta(u)=0, \quad t> 0;
\quad u(0)=0,  u'(0)=b>0
\end{equation}
define the increasing sequences $ (T_k) $ and $ S_k)$ such that
\begin{itemize}
\item[(1)]  for all $n\in \mathbb{N}$,
$[T_n,  T_{n+1}]:=D_n \in \big[ D(u^+)\big]$, $S_n \in D_n$;
 $u'(S_n)=0$;

\item[(2)]  $c_n(t) = c(t)$  for $t\in (0,  T_n]$  and
$ c_n(t)=c(T_n)$  for $t\geq T_n$.
\end{itemize}
For any $n$, let  $u_n$ and $z_n$, respectively, be the solutions of
\begin{gather*}
\big(\phi_\alpha(u')\big)' +  \alpha c_n(t) \phi_\beta(u)=0 ,\\
\big(\phi_\alpha(z')\big)' +  \alpha c(T_n) \phi_\beta(z)=0; \quad z(0)=0,\quad
  z'(0)=u'(T_n).
\end{gather*}
Then $ u_n \equiv z_n $ in  $\Omega_{T_n}  $ and  with
$\beta_\ast:=\max \{ \alpha,  \beta\}$, as
$ n\to \infty$,
\begin{equation} \label{e2.6}
|u_n|_{D(u_n^+)} = z_n(S_n)
=  \Big[\frac{(\beta+1)
u'(T_n)^{\theta+1}}{c(T_n)(\theta+1)} \Big]^{\frac 1{\beta_\ast+1}}
 = O\big( [T_n]^{-q/(\beta_\ast +1)}\big).
\end{equation}
\end{lemma}

\begin{proof}
The identity $u_n \equiv z_n$ in $\Omega_{T_n}  $  is due to the fact that
the two satisfy  the same initial values at $T_n$.
In fact if $w$ and $v$ are two $C^2(\Omega_T) $ solutions for
$$
\big(\phi_\alpha (u') \big)' + \alpha c(t) \phi_\beta(u)=0; \quad u(T)= 0 ,
\quad u'(T)=b>0
$$
then without loss of generality we assume that $ u'>v'>0$ in some $ (T,  \tau)$.

From $ \phi_\alpha(w')'= \alpha  \frac{w''}{w'} \phi_\alpha (w')$
 (as $ S\phi'_\alpha ( S)= \alpha \phi_\alpha(S) $), and from their equations
\[
v'u'' - u' v''= c(t) |u'v'|^{1-\alpha}\big[v^\beta |u'|^{\alpha -1}
- u^\beta |v'|^{\alpha -1}\big]:=c(t)|u'v'|^{1-\alpha}\Gamma(u,v),
\]
$\Gamma(u,v)=0 $ at $T$ and remains strictly positive as long as $v'>0$.
Therefore as long as $v'>0$, $\frac {u'}{v'} $ is increasing  as
 $ v'u'' - u' v'' =(v')^2 \big(\frac {u'}{v'} \big)'$. But from these formulae,
 $v'$ should not be zero while $u'> 0$. Thus $v'$ and $u'$ have the same
first zero after $T$ which is a contradiction. \eqref{e2.6} follows from
 \eqref{e2.3} and \eqref{e2.4}.
\end{proof}

\section{Estimates for some decaying oscillatory solutions}

Now we  take for  oscillatory functions $ z:=z_R  $
which will a fortiori  depend upon the function $u$ through  their bounded
coefficients. Namely we will use $z$, a solution of
$$
\big\{ \phi_\alpha(z')\big\}' + \alpha C \phi_\beta(z)=0; \; t>0;\quad z(0)=0;
\quad u'(0)=b>0
$$
where $C$ will be the value of $c$ at some point $R>0$.

\begin{theorem} \label{thm3.1}
Let  $R,  c_0,    \beta,  \alpha>0  $ and
 $ c \in C^1( \mathbb{R}^+,  ( c_0,  \infty)) $  be unbounded
and increasing.  Then if $u$ and $ z:=z_R $ are, respectively,  two non-trivial
oscillatory  solutions of
\begin{equation}  \label{e3.1}
\begin{gathered}
\big\{ \phi_\alpha(u'(t))\big\}' + c(t)\alpha  \phi_\beta(u(t)) =0 , \\
\big(\phi_\alpha(z')\big)' + c(R) \alpha \phi_\beta(z)=0, \quad t>0;
\quad  z(0)=0;\;  z'(0)=b>0.
\end{gathered}
\end{equation}
 Then  there is $R_1>0$ such that $ u $ has a zero inside any nodal set
$D(z^+) \subset \Omega_R $ for all $R>R_1$.
\end{theorem}

\begin{proof}
Let $u$ and $z$ be such  oscillatory solutions. We saw that any multiplication of
 $z$ by a positive $ \lambda>0 $ would not affect  any
$ D(z) $ but only that $ |\lambda z|_\infty =
\lambda |z|_\infty $. Also for all $T>0$, there are a multitude of
$ D(z^+) $ and $D(u^+)$ inside $\Omega_T$.

(1)  Suppose that $\beta>\alpha>0.$
Let $ T_1>0 $ be such that  $ c(t)>1$ for all $t>T_1$.
Assume that there exists $T>T_1$ such that for all $R>T  $ there is a nodal set
 $D(z_R^+):= D_1(z^+)\subset \Omega_R  $  such that $ u>0$ in $ D_1(z^+)$.


We take $T_1$ big enough for $ J_R:=[R,  2R]  $ to contain many nodal sets of
$z^+$ including $ D_1(z^+)  $
which is guaranteed by the fact that bigger $R$ is, the smaller
$\operatorname{diam}(D(z_R^+))$ is.

If for some $ \nu>0$,  $| u|^{\beta - \alpha}>\nu^{\beta - \alpha}>0 $ in
$D_1(z^+)$, then,  in  $D_1(z^+):=D(Z^+)$, the function $ Z(t)=: \nu z(t)$ satisfies
\begin{equation} \label{e3.2}
\begin{gathered}
\big(\phi_\alpha(Z')\big)' + \nu^{\beta - \alpha} c(R) \alpha \phi_\beta(Z)=0,
\quad t>0,  \\
\big[ Z\phi_\alpha(Z') - Z\phi_\alpha(\frac Zu u') \big]'
=\zeta_\alpha (Z,u) + \alpha  |Z|^{\alpha+1}
\big\{ c(t) |u|^{\beta-\alpha} -  \nu^{\beta - \alpha} c(R) \big\}>0 .
\end{gathered}
\end{equation}
The integration over $D(Z^+)$ of \eqref{e3.2} provides a contradiction.
Therefore the assumption cannot be true and $u$ has to have a zero in $D_1(z^+)$.

(2)  Assume that $ \alpha \geq \beta >0$.
For this case \eqref{e1.3}(i) is used instead of \eqref{e3.2}, and
the same conclusion is obtained.
\end{proof}


\begin{corollary} \label{coro3.2}
(1)  Let  $u$ and $z$ be the two solutions in \eqref{e3.1}  where $ C>0$ is arbitrary.  Let two of their nodal sets,
Let $D(u^+)$ and $D(z^+)$, be such that $u$ has a zero in $D(z^+)$   and
$  S \in D(u^+) $ is the singularity of $u^+$ therein.
 Then  there is $ \xi \in \mathbb{R} $ such that the  translated function
$ Z(t):= z(t+\xi)$ satisfies
$$
Z'(S)=u'(S)=0 , \quad  D(u^+) \subset D(Z^+), \quad
\operatorname{diam}  D(u^+) \leq \operatorname{diam}  D(z^+).
$$
(2) Moreover, for $t$ large enough,
\begin{equation}
\max_{D(u^+)}  u^+ := |u|_{D(u^+)} \leq   \max_{D(Z^+)}  Z^+
:= |Z^+|_{D(Z^+)} = |z^+|_{D(z^+)}.
\end{equation}
\end{corollary}

\begin{proof}
(1) This follows from  Theorem \ref{thm2.2} and Theorem \ref{thm3.1}.
(2) follows from Lemma \ref{lem2.3}.
\end{proof}

\begin{proof}[Proof of the Theorem \ref{thmA}]
Any such a solution of \eqref{e1.1} is strongly oscillatory  by
 Lemma \ref{lem2.1}  and  \cite{t4, t7}.
The estimates follow from Theorem \ref{thm2.2},  Theorem \ref{thm3.1}
  and Corollary \ref{coro3.2}.
\end{proof}


\section{An application}
For a restoring $h\in C( \mathbb{R}) $
(i.e. $ \forall y\in \mathbb{R} \setminus \{0\}, yh(y)>0 $)
consider the problem
\begin{equation} \label{e4.1}
\big\{ \phi_\alpha(u')\big\}' + \alpha c(t) h(u)=0,\;  t>0; \quad u(0)=0,
\quad u'(0)=b>0,
\end{equation}
where $ \alpha,  \beta,  q>0$ and $c$ being as before  and
 for small $ S>0$, $h((s) =O\big(S^\beta \big)$.

For the strongly oscillatory solution $z$ of
 $ \big\{ \phi_\alpha(z')\big\}' + \alpha C \phi_\alpha(z)=0$,  $t>0 $,
 and $w$ of $ \big\{ \phi_\alpha(w')\big\}' + \alpha C \phi_\beta(w)=0$ ,
 wherever $u\neq 0$, we have
\begin{equation}\label{e4.2}
\begin{gathered}
\big[z\phi_\alpha(z') - z\phi_\alpha( \frac zu u'  ) \big]'
=\zeta_\alpha(z,u) + \alpha C|z|^{\alpha+1}
\big\{ \frac{c(t)h(u)}{C \phi_\alpha (u)} - 1 \big\}, \\
\big[w\phi_\alpha(w') - w\phi_\alpha( \frac wu u'  ) \big]'
=\zeta_\alpha(w,u) + \alpha |w|^{\alpha+1}
 \big\{ \frac{c(t)h(u)}{C \phi_\alpha (u)} - |w|^{\beta-\alpha} \big\}
\end{gathered}
\end{equation}
As $h$ is a restoring function, we can define the function
$h_1 \in C(\mathbb{R}^+,  \mathbb{R}^+) $ by
$ h(S):= h_1(S^2)S$ for all
$S\in \mathbb{R}$  and define
$ H_1(t):=\int_0^t  s h_1(s^2)ds$
such that  equation \eqref{e4.1} reads
\begin{equation} \label{e4.3}
\big(\phi_\alpha(u')\big)' + \alpha c(t) h_1(u^2)u=0, \; t>0; \quad u(0)=0,
\quad u'(0)=b>0.
\end{equation}
Thus, similar to Theorem \ref{thm1.2}, we have the following result.

\begin{lemma} \label{lem4.1}
With $h_1$ defined in \eqref{e4.3}, $\forall C,  \alpha,  b,  \beta>0$ the problem
$$
\big(\phi(u')\big)' + \alpha C h_1(u^2)u=0, \; t>0; \quad u(0)=0, \quad u'(0)=b
$$
is strongly oscillatory. furthermore  and for its solution $u$, and all $t>0$,
we have
\begin{equation}\label{e4.4}
\begin{gathered}
2|u'(t)|^{\alpha+1} + (\alpha+1) C H_1(u^2(t))  = 2b^{\alpha+1}, \\
u(S)=0 \text{ and }u'(T)=0 \;\Longrightarrow\;
|u'(S)|=  b \text{ and } |u(T)|
= \big[ H_1^{-1}\big( \frac{2b^{\alpha+1}}{(\alpha+1)C} \big)\big]^{1/2}.
\end{gathered}
\end{equation}
\end{lemma}

\begin{proof}
From $ \big(\phi(u')\big)' + \alpha C h_1(u^2)u=0$,
$ u' u''\phi_\alpha'(u') + \alpha C h_1(u^2)uu'
=\alpha u'' \phi_\alpha(u') + \alpha \frac C2  (u^2)' h_1(u^2)= 0 $
 thus
\[
\frac 12 (u'^2)' \big( u'^2 \big)^{ \frac{\alpha -1}2}
 +  \frac C2  (u^2)' h_1(u^2)
=\big[ \frac 1{\alpha+1} |u'|^{\alpha+1} + \frac C2 H_1(u^2) \big]'=0
\]
leading to \eqref{e4.4}(i). Then \eqref{e4.4}(ii) follows as well.
The oscillation of the solution is obtained as  for the  Theorem \ref{thm1.2}.
\end{proof}

\begin{theorem} \label{thm4.2}
For $c_0, \alpha,  \beta ,  q>0$, let $ h_1 \in C(\mathbb{R},  [0,  \infty)  ) $ with
$ h_1(S^2)S =O( s^\beta ) $ for small $S>0$
and $ c\in C^1(\mathbb{R},  (c_0,  \infty)) $ with $c' >0 $ and
$  c(t)= O(t^q) $  as $ t \to \infty$. Then
 any non-trivial and bounded solution of
\begin{equation} \label{e4.5}
\big(\phi_\alpha(u')\big)' + \alpha c(t) h_1(u^2)u =0, \; t>0; \quad u(0)=0;
 \quad u'(0)=b>0
\end{equation}
is strongly oscillatory.

(1) Moreover for any $ R>0$ let $z:=z_R$ be  a non-trivial oscillatory solution of
\[
\big(\phi_\alpha(z')\big)' + c(R) \alpha \phi_\alpha(z)=0; \quad t>0 .
\]
Then for $S>0$ large enough, the oscillatory solution $u$ of \eqref{e4.5}
has a zero in any nodal set $  D(z_R^+)
\subset \Omega_S  $ for $ R>S$.

(2) Consequently  as $ t\to \infty $, for $ \beta_\ast:= \alpha \wedge \beta $
the solution in \eqref{e4.5} has the estimates
\begin{equation}\label{e4.6}
\begin{gathered}
|u(t)| \leq  {\rm const.}  [t]^{\frac{-q}{\beta+1}}
:={\rm const.}\big[\frac 1{c(t)}\big]^{1/(\beta_\ast+1)}, \\
 \operatorname{diam}( D(u^+) )
=O\Big(  \big[\frac 1{c(t)}\big]^{1/(\beta_\ast +1)}   \Big).
\end{gathered}
\end{equation}
\end{theorem}

\begin{proof}
(1) For some $C>0$ let $z$ be a strongly oscillatory solution to
\[
\big\{ \phi_\alpha(z')\big\}' + \alpha C \phi_\alpha(z)=0 .
\]
Then  \eqref{e4.2}(i) with  $h(u)$ replaced by $h_1(u^2)u $ becomes
\[
 \big[z\phi_\alpha(z') - z\phi_\alpha( \frac zu u'  ) \big]'
=\zeta_\alpha(z,u) + \alpha C|z|^{\alpha+1}
 \big\{ \frac{c(t)h_1(u^2)u}{C \phi_\alpha (u)} - 1 \big\}.
\]
If we assume that $u>\nu>0$ in some $\Omega_R$, then
\[
\zeta_\alpha(z,u) + \alpha C|z|^{\alpha+1}
 \big\{ \frac{c(t)h_1(u^2)u}{C \phi_\alpha (u)} - 1 \big\}
>\zeta_\alpha(z,u) + \alpha C|z|^{\alpha+1} \big\{ c(t)G(\nu) - 1 \big\}
\]
with
\[
 G(\nu):= \inf_{u\geq \nu}  \frac{c(t)h_1(u^2)u}{C \phi_\alpha (u)}.
\]
Because $c(t)$ is unbounded, $\{ c(t)G(\nu) - 1\}$ is eventually
strictly positive.
Assume that  that $ u>0$ and decreases to zero in  some $\Omega_S$.

(a) Case where $ \alpha>\beta>0$.
For very large $ R>S $, as $  u\searrow 0$,
\[
  \big\{ \frac{c(t)h_1(u^2)u}{C \phi_\alpha (u)} - 1 \big\}
> {\rm const.}\big[\frac{c(t)}C  |u|^{\beta - \alpha} - 1\big]>0
\]
eventually and the integration over $D(z)$ of \eqref{e4.2}(i) leads to
a contradiction.

(b) Let $ \beta \geq \alpha>0$ and $w$ the oscillatory solution  in \eqref{e4.2}(ii).
Assume that $ u>0 $ in some $\Omega_S$.
We use  $h_1(u^2)u$ instead of $h(u)$ there.
For $T>0$, We define $J_T:=(T,  2T)$
  and $ \nu:=\nu(T)= \inf_{J_T} \frac{ h_1(u^2)u}{C \phi_\alpha (u)}$.
 We take $ R>S$ so large that $ w^+ $ has many nodal sets in $J_R$  and
$ c(t)>C$ there.
We  choose $b=w'(0)$ such that $( w^+)^{\beta - \alpha} < \nu(R) $.
Then in $J_R$,
$$
\big\{ \frac{c(t)h_1(u^2)u}{C \phi_\alpha (u)} - |w|^{\beta-\alpha} \big\}>0,
$$
and  integration over $D(w)$ of \eqref{e4.2}(ii) leads to a contradiction.
 Therefore $u$ cannot remain positive throughout any $\Omega_T$.

Assume that there is $ T>0$ such that for all $R>T$, there is a nodal set
$D(z_R^+) :=D_R \subset J_R$
such that for some $\mu>0$,  $ u>\mu $ on $D_R$.  We remind that
$ c(t)\geq c(R)$ for all $t>R$. Then similar to (a) and (b) above, we see that
as we can make $ z_R^+$ arbitrary small in $J_R$, we cannot find $ T $ and
$ \mu>0$ such that the assumption holds.

(2)  The estimates are obtained through  the Corollary \ref{coro3.2},
 keeping in mind that as
$h_1(\tau^2) \leq {\rm const.} \tau^\beta$,  we have
$H_1(\tau)\leq {\rm const.} \tau^{\beta+1}$.
\end{proof}

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\end{document}