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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 146, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/146\hfil Sturm-Picone type theorems]
{Sturm-Picone type theorems for second-order nonlinear
differential equations}

\author[A. Tiryaki\hfil EJDE-2014/146\hfilneg]
{Aydin Tiryaki}  % in alphabetical order

\address{Aydin Tiryaki \newline
Department of Mathematics and Computer Sciences, 
Faculty of Arts and Sciences, Izmir University,
35350 Uckuyular, Izmir, Turkey}
\email{aydin.tiryaki@izmir.edu.tr}

\thanks{Submitted March 5, 2014. Published June 20, 2014.}
\subjclass[2000]{34C10, 34C15}
\keywords{Comparison theorem; Sturm-Picone theorem;  half-linear;
\hfill\break\indent  second order differential equations; singular equation;
 variational lemma}

\begin{abstract}
 The aim of this article is to give Sturm-Picone type theorems for the
 pair of second-order nonlinear differential equations
 \begin{gather*}
 (p_1(t)|x'|^{\alpha-1}x')'+q_1(t)f_1(x)=0  \\
 (p_2(t)|y'|^{\alpha-1}y')'+q_2(t)f_2(y)=0,\quad t_1<t<t_2
 \end{gather*}
 in both regular and singular cases. Our results include some
 earlier results and generalize the well-known comparison theorems
 given by Sturm \cite{Sturm}, Picone \cite{Picone}  and Leighton
 \cite{Leighton} which play a key role in the qualitative behaviour
 of the solutions.
\end{abstract}

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

\section{Introduction}

In 1836 the first important comparison theorem was given by Sturm
\cite{Kreith, Swanson}, which deals with a pair of linear ordinary
differential equations
\begin{gather}
\big(p_1(t)x'\big)'+q_1(t)x=0,  \label{1.1} \\
\big(p_2(t)y'\big)'+q_2(t)y=0 \label{1.2}
\end{gather}
 on a bounded interval $(t_1, t_2)$ where $p_1$, $q_1$,
$p_2$, $q_2$ are real-valued continuous functions and $p_1(t)>0$,
$p_2(t)>0$ on $[t_1, t_2]$. In this celebrated paper, Sturm
\cite{Sturm} proved the following remarkable result.

\begin{theorem}[Sturm's Comparison Theorem] \label{thm1.1}
Suppose $p_1(t)=p_2(t)$ and
$q_1(t)>q_2(t)$, $\forall t \in (t_1, t_2)$. If there exists a
nontrivial real solution $y$ of \eqref{1.2} such that
$y(t_1)=0=y(t_2)$, then every real solution of \eqref{1.1} has
at least one zero in $(t_1, t_2)$.
\end{theorem}

In 1909, Picone \cite{Picone} modified Sturm's theorem as follows.

\begin{theorem}[Sturm-Picone Theorem] \label{thm1.2}
 Suppose that $p_2(t)\geq p_1(t)$ and
$q_1(t)\geq q_2(t)$, for all $t \in (t_1, t_2)$. If there
exists a nontrivial real solution $y$ of \eqref{1.2} such that
$y(t_1)=0=y(t_2)$, then every real solution of \eqref{1.1}
unless a constant multiple of $y$ has at least one zero in
$(t_1, t_2)$
\end{theorem}

Note that Theorem \ref{thm1.2} is a special case of Leighton's theorem
\cite{Leighton}. For a detailed study and earlier developments of
this subject, we refer the reader to the books \cite{Kreith,
Swanson}. Sturm-Picone theorem is extended in several directions,
see \cite{Ahmad} and \cite{Ahmad2} for linear systems,
\cite{Muller} for nonselfadjoint differential equations,
\cite{TyagiR} for implicit differential equations,
\cite{Dosly, JarosandKusano, Li} for half-linear equations,
\cite{Allegretto2} for degenerate elliptic equations, \cite{Zhang}
for linear equations on time scales. There is also a good amount
of interest in the qualitative theory of partial differential
equations to determine whether the given equation is oscillatory
or not and Sturm-Picone theorem, also plays an important role in
this direction. For earlier developments, we refer to
\cite{Picone, Sturm, Swanson}, and for recent developments we
refer to Yoshida's book \cite{Yoshida}. Sturm comparison theorem
for the half-linear elliptic equation and Picone type identities
have been studied in, for example, \cite{Allegretto, Allegretto2,
Dosly, Fisnarova, JarosKusanoYoshida, JarosKusanoYoshida2, Tadie,
Yoshida2}.

When some or all of $p_1$, $q_1$, $p_2$, $q_2$ are not continuous
at $t_1$ or $t_2$ or at $t_1$ and $t_2$ both, where the
possibility that the interval is unbounded is not excluded, then
\eqref{1.1}, \eqref{1.2} are called singular differential
equations. Analog of Theorems  \ref{thm1.1}, \ref{thm1.2} and other
 related theorems
for singular differential equations have been obtained earlier
(see \cite{Swanson}). Recently, in \cite{Aharonov}, Sturm's
theorem for a pair of singular linear differential equations was
proved assuming that the solution of minorant equation is
principal at both end points of the interval. Very recently, Tyagi
\cite{Tyagi} studied a pair of second order nonlinear differential
equations
\begin{gather}
(p_1(t)x')'+q_1(t)f_1(x)=0,  \label{1.3}\\
(p_2(t)y')'+q_2(t)f_2(y)=0,\quad t_1<t<t_2 \label{1.4}
\end{gather}
under suitable sufficient conditions.  He gave the
generalization of these theorems to \eqref{1.3} and \eqref{1.4} for
regular and singular cases. Tyagi's paper \cite{Tyagi} is the
first generalization of Sturm-Picone theorem by establishing a
nonlinear version of Leighton's variational Lemma. In the linear
case, Tyagi's results reduce to the celebrated Sturm-Picone and
Leighton theorems.
     But it is obvious that Tyagi's
result does not work for the half-linear case.Our aim is to give an
answer for this case. As far as our understanding goes, there is
no generalization of Leighton-type theorems for nonlinear
differential equations that contain the half-linear equation.

In this paper motivated by the ideas in \cite{Tyagi}, extending
Tyagi's results, we prove a nonlinear analogue for Leighton's
theorem and we give a generalization to Sturm-Picone theorem by
establishing a suitable nonlinear version of Leighton's
variational lemma which contain the half-linear and also the
linear equations. Our results also include the singular case.


\section{Regular Sturm-Picone theorem for nonlinear equations}

Let us consider a pair of second-order nonlinear ordinary
differential equations
\begin{gather}
\ell{x}:=\big(p_1(t)|x'|^{\alpha-1}x'\big)'+q_1(t)f_1(x)=0, \label{2.1}\\
L{y}:=\big(p_2(t)|y'|^{\alpha-1}y'\big)'+q_2(t)f_2(y)=0,\quad t_1<t<t_2 \label{2.2}
\end{gather}
where $p_1$, $p_2\in C^{1}\big([t_1,t_2], (0,\infty)\big)$,
$q_1$, $q_2\in C \big([t_1,t_2], R \big)$, $f_1$, $f_2\in C
\big(R, R \big)$, $\alpha$ is a real positive constant, $l$ and
$L$ are differential operators or mappings whose domains consist
of all real-valued functions $x\in C^{1}[t_1,t_2]$, such that
$p_1|x'|^{\alpha-1}x'$ and $p_2|x'|^{\alpha-1}x' \in
C^{1}[t_1,t_2]$, respectively.
In what follows, we assume the following hypotheses with respect
to functions $f_1$ and $f_2$:
\begin{itemize}
\item[(H1)] Let $f_1\in C^{1}(R,R)$ and there exist
    $\alpha_1    > 0$, $\alpha_0>0$ such that 
    $\alpha_{0}|x|^{\alpha-1}\leq  f_1'(x)\neq 0$ and $\alpha_1|x|^{\alpha-1}x \geq
    f_1(x)\neq 0$, for all $0 \neq x \in\mathbb{R}$, and $f_1(0)=0$, $f'_1(0)\geq 0$.


\item[(H2)] Let $f_2 \in C(R,R)$ and there exist
    $\alpha_2$, $\alpha_3 \in (0,\infty)$ such that
    $\alpha_{3}|y|^{\alpha+1}\leq
    f_2(y)y\leq\alpha_2|y|^{\alpha+1}$, for all $0\neq y   \in\mathbb{R}$.
\end{itemize}


\begin{remark} \label{rmk2.1}\rm
Assumption (H1) motivates us to take the
nonlinearities of the form
$$
f_1(x)=|x|^{\alpha-1}x\big(1\mp\text{ a nonlinear part}\big)
$$
where nonlinear part is decaying at $\infty$.
\end{remark}

\begin{remark} \label{rmk2.2} \rm
Assumption  (H2) simply says that
$\frac{f_2(y)}{|y|^{\alpha-1}y}$ is bounded, for all $0\neq y\in\mathbb{R}$.
\end{remark}

We begin with a lemma and the definition of some concepts, needed in this
article.

\begin{lemma}[\cite{Hardy, JarosKusanoYoshida}] \label{lem2.1} \rm
Define $\varphi (u):=|u|^{\alpha-1}u$, $\alpha>0$. If $x$, $y\in\mathbb{R}$ then
$$
x\varphi(x)+ \alpha y \varphi(y)-(\alpha+1)x\varphi(y) \geq 0
$$
where equality holds if and only if $x=y$.
\end{lemma}

Let  $U$ be the set of all real valued functions
$u \in C^{1}[t_1, t_2]$, such that $u(t_1)=u(t_2)=0$, where
$t_1$ and $t_2$ are consecutive zeros of $u$.
Also define the functionals $j$ and $J: U\to R$ by
\begin{equation} \label{2.3}
\begin{gathered}
j(u)=\int_{t_1}^{t_2} \{p_1(t)|u'(t)|^{\alpha+1}-C_1q_1(t)|u(t)|^{\alpha+1} \}dt
 \\
J(u)=\int_{t_1}^{t_2} \{p_2(t)|u'(t)|^{\alpha+1}-(\alpha_2q_2^{+}(t)
-\alpha_{3}q_2^{-}(t))|u(t)|^{\alpha+1}\}dt
\end{gathered}
\end{equation}
where $ C_1=(\frac{\alpha_0}{\alpha_1{\alpha}})^{\alpha}\alpha_1$,
$q_2^{+}=\max\{q_2,0\}$ and $q_2^{-}=\max\{-q_2,0\}$.
The variation $V(u)$ is defined as
\begin{equation}
V(u)=J(u)-j(u). \label{2.4}
\end{equation}

\begin{theorem}[Leighton's variational type lemma] \label{thm2.1}
Suppose that there exists a \\
function $u \in U$, not identically zero in any open subinterval
of $(t_1,t_2)$ such that  $j(u)\leq 0$. If $x$ is a nontrivial
solution of \eqref{2.1} such that {\rm (H1)} holds, then $x$
has a zero in $(t_1,t_2)$ except possibly when
$|u|^{\alpha}=|Kf_1(x)|$ for some nonzero constant $K$.
\end{theorem}

\begin{proof}
Assume on the contrary that the statement is false. Let $x(t)\neq 0$
for every $t \in (t_1,t_2)$. We observe that the following
equality is valid on $(t_1,t_2)$:
 %\label{2.5}
\begin{align*}
& (\frac{\alpha u(t)
\varphi(u(t))}{f_1(x(t))}p_1(t)\varphi (x'(t)))'\\
&=\frac{\alpha u(t)
\varphi(u(t))}{f_1(x(t))}(-q_1(t)f_1(x(t)))+p_1(t)\varphi(x'(t))(\frac{\alpha
u(t) \varphi(u(t))}{f_1(x(t))})'\\
&= -\alpha q_1(t)u(t)\varphi(u(t))+p_1(t)\varphi(x'(t))
\Big[{\frac{\alpha(\alpha+1)u'(t)\varphi(u(t))}{f_1(x(t))}}\\
&\quad -\frac{\alpha u(t)\varphi(u(t))x'(t)f'_1(x(t))}{f_1^{2}(x(t))}\Big] \\
&= -\alpha q_1(t)u(t)\varphi(u(t))-p_1(t)\frac{|f_1(x(t))|^{\alpha-1}}
{(f'_1(x(t)))^{\alpha}}
\Big\{\alpha^{\alpha+1} u'(t)\varphi(u'(t))\\
&\quad -\alpha(\alpha+1)\varphi\Big(\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}\Big){u'(t)} \\
&\quad +\alpha{\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}}\varphi
 \Big(\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}\Big)
-\alpha^{\alpha+1} u'(t)\varphi(u'(t))\Big\}.
\end{align*}
Using Lemma \ref{lem2.1} with $ x=\alpha u' (t)$ and
$ y=\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}$ and
hypothesis (H1), we obtain
\begin{align*} % 2.6
&\big(\frac{\alpha u(t) \varphi(u(t))}{f_1(x(t))}p_1(t)\varphi (x'(t))\big)'\\
&\leq -\alpha q_1(t)|u(t)|^{\alpha+1}+\alpha^{\alpha+1} p_1(t){
\frac{(\alpha_1)^{\alpha-1}}{\alpha_{0}^\alpha}}|u'(t)|^{\alpha+1}\\
&\quad-p_1(t)\frac{|f_1(x(t))|^{\alpha-1}}{(f'_1(x(t)))^{\alpha}}
\Big[|\alpha u'(t)|^{\alpha+1}+\alpha |\frac{u(t)x'(t)f'_1(x(t))}
{f_1(x(t))}|^{\alpha+1} \\
&\quad -(\alpha+1)\alpha u'(t)\varphi (\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))})\Big].
\end{align*}
This implies
\begin{equation} \label{2.6}
\begin{aligned}
&p_1(t)|u'(t)|^{\alpha+1}-C_1q_1(t)|u(t)|^{\alpha+1}\\
&\geq C_1\Big( \frac{u(t)\varphi(u(t))}{f_1(x(t))} p_1(t)\varphi (x'(t))\Big)'\\
&\quad+\frac{C_1}{\alpha}p_1(t)\frac{|f_1(x(t))|^{\alpha-1}}{(f'_1(x(t)))^{\alpha}}
\Big\{|\alpha u'(t)|^{\alpha+1}+\alpha |\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}|^{\alpha+1}
\\
&\quad-\alpha(\alpha+1)u'(t)\varphi\Big( \frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}
\Big) \Big\}.
\end{aligned}
\end{equation}
Integrating  over $(t_1,t_2)$, it follows that
\begin{equation} \label{2.7}
\begin{aligned}
&\int_{t_1}^{t_2}(p_1(t)|u'(t)|^{\alpha+1}-C_1q_1(t)|u(t)|^{\alpha+1})dt\\
&\geq  C_1(\frac{|u(t)|^{\alpha+1} p_1(t) \varphi(x'(t))}{f_1(x(t))})
 \mid_{t_1}^{t_2}+\frac{C_1}{\alpha}\int_{t_1}^{t_2}p_1(t)
 \frac{|f_1(x(t))|^{\alpha-1}}{(f'_1(x(t)))^{\alpha}}
 \Big\{ |\alpha u'(t)|^{\alpha+1}\\ 
&\quad+\alpha |\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}|^{\alpha+1}
-\alpha(\alpha+1)u'(t)\varphi(\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}) \Big\}dt.
\end{aligned}
\end{equation}
Now, there are three cases for the behavior of $x(t)$ at $t_1$ and
$t_2$.
\smallskip


\noindent \textbf{Case 1.}
 If both $x(t_1)\neq 0$ and $x(t_2)\neq 0$, then
it follows from \eqref{2.7} and $u \in U$ that $j(u)\geq 0$ and from
Lemma \ref{lem2.1}
\begin{align*}
&\int_{t_1}^{t_2}p_1(t)\frac{|f_1(x(t))|^{\alpha-1}}{(f'_1(x(t)))^{\alpha}}
\{|\alpha u'(t)|^{\alpha+1}+\alpha
|\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}|^{\alpha+1}\\
&-\alpha(\alpha+1)u'(t)\varphi\Big(
\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}\Big) \}dt=0
\end{align*}
 if and only if
$$
\alpha u'(t)-\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}\equiv 0.
$$
This implies 
$$
|u(t)|^\alpha=|Kf_1(x(t))|,\quad \forall t \in (t_1,t_2)
$$
and for some constant $K$. Since $t_1$ and $t_2$ are
consecutive zeros of $u$, this implies that $u(t)\neq 0$ for all
$t \in (t_1,t_2)$. So $K$ is a non-zero constant. Using this fact,
we obtain $j(u)>0$, which leads a contradiction. This
contradiction shows that $x$ vanishes at least once in
$(t_1,t_2)$.
\smallskip


\noindent \textbf{Case 2.}
 If both $x(t_1)= 0$ and $x(t_2)=0$, then
$x'(t_1)\neq 0$ and $x'(t_2)\neq 0$. It follows from the fact that
zeros of a nontrivial solution of \eqref{2.1} are simple, which can be
proved as follows. Indeed we prove only the case $x(t_1)= 0$.
Assume on the contrary that $x'(t_1)= 0$. We take $x(t)>0$ on
$(t_1, t_2)$ in the case $x(t)<0$ on $(t_1, t_2)$ is similar and
hence omitted. It follows from \eqref{2.1} that
$$
x'(t)=\varphi^{-1}\{-\frac{1}{p_1(t)}\int_{t_1}^{t}q_1(s)f_1(x(s))ds\},
$$
where $\varphi^{-1} (s)= |s|^{{\frac{1}{\alpha}}-1}s$ is
the inverse function of $\varphi$. Since $x(t_1)= 0$ and 
$p_1 \in C^{1}([t_1,t_2],(0, \infty))$,
\begin{align*}
x(t)&=\int_{t_1}^{t}\varphi^{-1}\Big(-\frac{1}{p_1(\xi)}
 \int_{a}^{\xi}q_1(s)f_1(x(s))ds\Big)d\xi \\
&\leq (t-t_1)\varphi^{-1}\Big(M \int_{t_1}^{t}|q_1(s)||f_1(x(s))|ds\Big)
\end{align*}
for $t_1\leq t \leq t_2$, where
 $$
M=\max \big\{\frac{1}{p_1(t)}: t_1\leq t \leq t_2 \big\}
$$
Hence
$$
\varphi(x(t))\leq (t-t_1)^{\alpha} M \int_{t_1}^{t}|q_1(s)|f_1(x(s))|ds \quad
\text{for }t_1\leq t \leq t_2 .
$$
Using (H1), it follows from the Gronwall inequality that
$\varphi(x(t)) =0$ for each $t\in [t_1, t_2]$. This implies that
$x(t)=0$ on $(t_1,t_2)$, which contradicts the hypothesis $x(t)>0$
on $(t_1,t_2)$. Then if $x(t_1)=0$, by L'Hospital's Rule,
considering (H1), assuming $x'(t_1)>0$,
$$
\lim_{t\to t_1^{+}} \varphi \big(\frac{u(t)}{x(t)}\big)
=\varphi \big(\lim_{t\to t_1^{+}} {\frac{u'(t)}{x'(t)}}\big)<\infty
$$
and
\begin{align*}
\lim_{t\to t_1^{+}} \frac{u(t)}{\alpha_1}\varphi
 \big(\frac{u(t)}{x(t)} \big)p_1(t)\varphi(x'(t)) 
&\leq\lim_{t\to t_1^{+}} \frac{u(t)\varphi (u(t)) p_1(t)
\varphi(x'(t))}{f_1(x(t))}\\
&\leq \lim_{t\to t_1^{+}} \frac{\alpha}{\alpha_{0}}u(t)
\varphi \big(\frac{u(t)}{x(t)} \big)p_1(t)\varphi(x'(t)) ,
\end{align*}
we have
$$
\lim_{t\to t_1^{+}} \frac{u(t)\varphi (u(t)) p_1(t)\varphi(x'(t))}{f_1(x(t))}=0.
$$
Similarly,
$$
\lim_{t\to t_2^{-}} \frac{u(t)\varphi (u(t)) p_1(t)\varphi(x'(t)) }{f_1(x(t))}=0, 
$$
if $x(t_2)=0$.

Therefore, we obtain from \eqref{2.7} that $j(u)\geq 0$ and hence we obtain
a contradiction $j(u)> 0$ unless $|f_1(x)|$ is a constant
multiple of $|u|^{\alpha}$.
\smallskip

\noindent\textbf{Case 3.}
 If $x(t_1)= 0$ and $x(t_2)\neq 0$ or
$x(t_1)\neq 0$, $x(t_2)= 0$, then as in the proof of Case 1, it is
obvious that $j(u)>0$ which leads a contradiction and hence $x$
vanishes at least once in $(t_1,t_2)$. This completes the
proof.
\end{proof}

From Theorem \ref{thm2.1} we have the following result which is an
extension of Leighton's Theorem for \eqref{2.1} and \eqref{2.2}.

\begin{theorem} \label{thm2.2}
Let {\rm (H1), (H2)} hold. If there exists a nontrivial real
solution $y$ of $L_{y}=0$ in $(t_1,t_2)$ such that
$y(t_1)=y(t_2)=0$ and $V(y)\geq 0$, then every nontrivial
solution $x$ of $\ell x=0$ has one of the following properties:
\begin{itemize}
\item[(i)] $x$ has a zero in $(t_1,t_2)$ or,
\item[(ii)] $|f_1(x)|$ is a nonzero constant multiple of
    $|y|^{\alpha}$.
\end{itemize}
\end{theorem}

\begin{proof}
Since $y(t_1)=0=y(t_2)$ and $Ly(t)=0$, by  applying
Green's identity, we have
\begin{gather*}
y(t)\Big(p_2(t)|y'(t)|^{\alpha-1}y'(t)\Big)'+q_2(t)f_2(y(t))y(t)=0,\\
\begin{aligned}
\Big(p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\Big)'
&=y(t)\Big(p_2(t)|y'(t)|^{\alpha-1}y'(t)\Big)'+|y'(t)|^{\alpha+1}p_2(t)\\
&=-q_2(t)f_2(y(t))y(t)+|y'(t)|^{\alpha+1}p_2(t).
\end{aligned}
\end{gather*}
Integrating both side from $t_1$ and $t_2$, we obtain
\begin{equation}
\int_{t_1}^{t_2} \Big(q_2(t)f_2(y(t))y(t)-p_2(t)|y'(t)|^{\alpha+1}\Big)dt=0. 
\label{2.9}
\end{equation}
In view of (H2), one can see that
\begin{equation}
\int_{t_1}^{t_2} \{ (q_2(t)f_2(y(t))y(t)
-\Big(\alpha_2q_2^{+}(t)-\alpha_{3}q_2^{-}(t)\Big)|y(t)|^{\alpha+1}\}dt \leq 0
 \label{2.10}
\end{equation}
By \eqref{2.9} and \eqref{2.10}, we have $J(y)\leq 0$.
Since $V(y)\geq 0$ this implies that
$$
j(y)\leq J(y) \leq 0
$$
and hence by an application of Theorem \ref{thm2.1} every
nontrivial solution $x$ of $\ell x=0$ has at least one zero in
$(t_1,t_2)$ except possibly when $|f_1(x(t))|$ is a nonzero
constant multiple of $|y(t)|^{\alpha}$. This completes the proof.
\end{proof}


\begin{remark} \label{rmk2.3}\rm
 If the condition $V(y)\geq 0$  is
strengthened to $V(y)>0$, conclusion  (ii) of Theorem \ref{thm2.2} does not hold.
\end{remark}

From Theorem \ref{thm2.2} we immediately have the following Corollary which
is an extension of Sturm-Picone Comparison Theorem for the
equations \eqref{2.1} and \eqref{2.2}.



\begin{corollary} \label{coro2.1}
Let {\rm (H1)} and {\rm (H2)} hold. Suppose there exists a nontrivial
solution $y$ of $Ly=0$ in  $ (t_1, t_2)$ such that
$y(t_1)=0=y(t_2)$ if $p_2(t)\geq p_1(t)$ and
$$
C_1q_1(t)-\Big( \alpha_2q_2(t)-(\alpha_{3}-\alpha_2)q_2^{-}(t)\Big)\geq 0
$$
for every $t \in (t_1,t_2)$, then every nontrivial
solution $x$ of $\ell x=0$ has at least one zero in $(t_1,t_2)$
unless $|f_1(x)|$ is a nonconstant multiple of $|y|^{\alpha}$.
\end{corollary}

From Theorem \ref{thm2.1}, Theorem \ref{thm2.2} and Corollary 
\ref{coro2.1} we easily obtain
the following results which are straight forward extensions of the
variational Lemma, Leighton's theorem and the celebrated
Sturm-Picone theorem from \cite{Kreith, Leighton, Sturm, Swanson}
valid for linear second order equations to the case of half-linear
equations.

\begin{corollary} \label{coro2.2}
Let $f_1(x)=|x|^{\alpha-1}x$ in \eqref{2.1} if
$$
\int_{t_1}^{t_2} \{p_1(t)|u'(t)|^{\alpha+1}-q_1(t)|u(t)|^{\alpha+1} \}dt \leq 0,
$$
 where $u \in U$, not identically zero in any open
subinterval of $(t_1, t_2)$, then every nontrivial solution
$x$ of \eqref{2.1} has a zero in $(t_1, t_2)$ except possibly when
$u=Kx$ for some nonzero constant $K$.
\end{corollary}

\begin{corollary} \label{coro2.3}
Let us consider  equations \eqref{2.1} and \eqref{2.2} with
$f_1(u)=|u|^{\alpha-1}u=f_2(u)$. Suppose there exists a
nontrivial solution $y$ of $Ly=0$ in $(t_1, t_2)$ such that
$y(t_1)=0=y(t_2)$. If
$$
\int_{t_1}^{t_2} \{ (p_2(t)-p_1(t))|y'(t)|^{\alpha+1}
+ (q_1(t)-q_2(t))|y(t)|^{\alpha+1}\}dt\geq 0,
$$
then every nontrivial solution $x$ of $\ell x=0$ has at least one zero in
$(t_1,t_2)$ except possibly it is a constant multiple of $y$.
\end{corollary}

\begin{corollary} \label{coro2.4}
Consider the equations \eqref{2.1} and \eqref{2.2} with
$f_1(u)=|u|^{\alpha-1}u=f_2(u)$. Let $p_2(t)\geq p_1(t)$
and $q_1(t)\geq q_2(t)$ for every $t \in (t_1,t_2)$. If there
exists a nontrivial solution $y$ of $Ly=0$ in $(t_1,t_2)$ such
that $y(t_1)=0=y(t_2)$, then any nontrivial solution $x$ of
$\ell x=0$ either has a zero in $(t_1,t_2)$ or it is a nonzero
constant multiple of $y$.
\end{corollary}

Note that the Corollaries \ref{coro2.2}--\ref{coro2.4} were also obtained 
by Jaros and Kusano \cite{JarosandKusano}. But their proofs depend on the
Picone-type and Wirtinger-type inequalities. Corollary \ref{coro2.3} was
also obtained by Li and Yeh \cite{Li} using different way.

\section{Singular Sturm-Picone theorem for nonlinear equations}

In this section, we consider the second-order nonlinear singular
equations
\begin{gather}
\ell_{s}x:=\Big(p_1(t)|x'|^{\alpha-1}x'\Big)'+q_1(t)f_1(x)=0  \label{3.1}\\
L_{s}y:=\Big(p_2(t)|y'|^{\alpha-1}y'\Big)'+q_2(t)f_2(y)=0 \quad t_1<t<t_2, \label{3.2}
\end{gather}
where $p_1,p_2 \in C\big((t_1,t_2), (0,\infty)\big)$,
$q_1,q_2\in C\big((t_1,t_2), R\big)$, some or
all of $p_1$, $p_2$, $q_1$, $q_2$ may not be continuous at
$t_1$ or $t_2$ or at $t_1$ and $t_2$ both, where the possibility
that the interval is unbounded is not excluded. Let 
$f_1, f_2 \in C(R,R)$, $\ell_{s}$ and $L_{s}$ are differential operators or
mappings whose domains consists of all real-valued functions 
$x \in C^{1}(t_1,t_2)$ such that $p_1|x'|^{\alpha-1}x'$ and
$p_2|x'|^{\alpha-1}x' \in C^{1}(t_1,t_2)$ respectively.


We begin with the following quadratic functionals corresponding to
\eqref{3.1} and \eqref{3.2} respectively. For $t_1<\xi<\eta<t_2$, let
\begin{gather}
j_{\xi\eta}(u)
=\int_{\xi}^{\eta} \{p_1(t)|u'(t)|^{\alpha+1}-C_1q_1(t)|u(t)|^{\alpha+1}\}dt 
 \label{3.3} \\
J_{\xi\eta}(u)=\int_{\xi}^{\eta}\{p_2(t)|u'(t)|^{\alpha+1}-(\alpha_2q_2^{+}(t)
 -\alpha_{3}q_2^{-}(t))|u(t)|^{\alpha+1}\}dt. \label{3.4}
\end{gather}
Let us define $j_{s}(u)=\lim_{\xi\to t_1^{+},
\eta\to t_2^{-}}{j_{\xi\eta}(u)}$ and
$J_{s}(u)=\lim_{\xi\to t_1^{+}, \eta\to
t_2^{-}}{J_{\xi\eta}(u)}$ whenever the limit exists.

The domains $D_{j_{s}}$ of $j_{s}$ and $D_{J_{s}}$ of $J_{s}$ are
defined to be the set of all real-valued continuous functions 
$u \in C^{1}(t_1, t_2)$ such that $j_{s}(u)$ and $J_{s}(u)$
exist.
Let us define
\begin{equation}
A_{t_1t_2}[u,x]=\lim_{t\to t_2^{-}}\frac{\alpha u(t)\varphi(u(t))p_1(t)
\varphi(x'(t))}{f_1(x(t))}-\lim_{t\to t_1^{+}}
\frac{\alpha u(t)\varphi(u(t))p_1(t)\varphi(x'(t))}{f_1(x(t))} \label{3.5}
\end{equation}
 whenever the limits on the right-hand side exist. The
variation $V_{s}(u)$ is defined as
\begin{equation}
V_{s}(u)=J_{s}(u)-j_{s}(u);  \label{3.6}
\end{equation}
i.e.,
\[
V_{s}(u)=\int_{t_1}^{t_2}\{\Big(p_2(t)-p_2(t)\Big)|u'(t)|^{\alpha+1}
+(C_1q_1(t)-(\alpha_2q_2^{+}(t)-\alpha_{3}q_2^{-}(t)))|u(t)|^{\alpha+1}\}dt
\]
with domain $D:=D_{j_{s}}\cap D_{J_{s}}$.
We begin with the singular version of Leighton's variational type
lemma for \eqref{3.1}.

\begin{theorem} \label{thm3.1}
Suppose there exists a function $u \in D$, not identically zero in
any open interval subinterval of $(t_1, t_2)$ such that
$j_{s}(u)\leq 0$.  If $x$ is a nontrivial solution of \eqref{3.1}
such that the hypotheses (H1) holds  and $A_{t_1t_2}[u,x]
\geq 0$, then $x$ has a zero in $(t_1,t_2)$ unless $|f_1(x)|$ is
a nonzero constant multiple of $|u|^{\alpha}$.
\end{theorem}

\begin{proof}
Assume for the sake of contradiction that equation \eqref{3.1} has
a nonzero, nontrivial solution on $(t_1,t_2)$. Along the same
lines of proof of Theorem \ref{thm2.1}, we see that the inequality
\eqref{2.6} holds on $(t_1,t_2)$. An integration of \eqref{2.6}
over $(\xi, \eta)$ yields
\begin{align*}
j_{\xi\eta}(u)
&\geq C_1\frac{u(t)\varphi(u(t))p_1(t)\varphi(x'(t))}{f_1(x(t))}
\mid_{\xi}^{\eta}+\frac{C_1}{\alpha} \int _{\xi}^{\eta} p_1(t) 
\frac{|f_1(x(t))|^{\alpha+1}}{(f'_1(x(t)))^{\alpha}}
\Big\{ |\alpha u'(t)|^{\alpha+1}\\
&\quad +\alpha |\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}|^{\alpha+1}
-\alpha(\alpha+1)u'(t)\varphi\Big(\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}
\Big)\Big\}dt\,.
\end{align*}
Letting $\xi\to t_1^{+}$, $\eta \to t_2^{-}$
and using $A_{t_1t_2}[u,x]\geq 0$ we obtain
\begin{equation} \label{3.7}
\begin{aligned}
j_{s}(u)
&\geq \frac{C_1}{\alpha} \int _{t_1}^{t_2} p_1(t) 
\frac{|f_1(x(t))|^{\alpha+1}}{(f'_1(x(t)))^{\alpha}}
\Big\{ |\alpha u'(t)|^{\alpha+1}
+\alpha |\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}|^{\alpha+1}\\
&\quad -\alpha(\alpha+1)u'(t)\varphi\Big(\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}\Big)
 \Big\}dt 
\end{aligned}
\end{equation}
and
\begin{align*}
&\int _{t_1}^{t_2} p_1(t)
\frac{|f_1(x(t))|^{\alpha+1}}{(f'_1(x(t)))^{\alpha}} 
\Big\{|\alpha u'(t)|^{\alpha+1}\\
&+\alpha |\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}|^{\alpha+1}
-\alpha(\alpha+1)u'(t)\varphi\Big(\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}\Big)\Big\}dt=0
\end{align*}
if and only if
$$
|\alpha u'(t)|^{\alpha+1}+\alpha
|\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}|^{\alpha+1}
-\alpha(\alpha+1)u'(t)\varphi\Big(\frac{u(t)x'(t)f'_1(x(t))}{f_1(x(t))}\Big) 
\equiv 0
$$
According to Lemma \ref{lem2.1}, this implies 
$$
|u(t)|^{\alpha}=|Kf_1(x(t))|\quad\text{for every } t\in (t_1,t_2)
$$
and for some nonzero constant $K$. 
Using this fact, we have $j_{s}(u)>0$ which leads to a contradiction. 
This contradiction shows that
$x$ vanishes at least once in $(t_1,t_2)$. This completes the
proof.
\end{proof}

As in Section 2, from Theorem \ref{thm3.1} plays an important role to
establish the following result which is an extension of Leighton's
theorem for equations \eqref{3.1} and \eqref{3.2} for the singular
case.

\begin{theorem} \label{thm3.2}
Suppose that there exists a nontrivial real solution $y \in D$ of
$L_{s}y=0$ in $(t_1, t_2)$. Let $x$ be any nontrivial solution
of $\ell_{s}x=0$. Let also (H1) and (H2). If
$A_{t_1t_2}[y,x]\geq 0$,
$$
\lim_{t\to t_1^{+}}p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\geq 0,\quad
\lim_{t\to t_2^{-}}p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\leq 0
$$ 
and $V_{s}(y)> 0$,
then $x$ has at least one zero in $(t_1, t_2)$. If the condition
$V(y)>0$ is weakened to $V_{s}(y)\geq 0$ the same conclusion holds
unless $|f_1(x)|$ is a nonzero constant multiple of
$|y|^{\alpha}$.
\end{theorem}

From Theorem \ref{thm3.2}, we have the following corollary which is the
extension of Sturm-Picone comparison theorem for equations \eqref{3.1} and
\eqref{3.2}.


\begin{corollary} \label{coro3.1}
Suppose that there exists a nontrivial real solution $y \in D$ of
$L_{s}y=0$ in $(t_1, t_2)$. Let $x$ be any nontrivial solution
of $\ell_{s}x=0$. Let also (H1) and (H2). If
$A_{t_1t_2}[y,x]\geq 0$, $ p_2(t)\geq p_1(t)$,
\begin{gather*}
\lim_{t\to t_1^{+}}p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\geq 0 , \quad
\lim_{t\to t_2^{-}}p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\leq 0, \\
C_1q_1(t)-(\alpha_2q_2(t)-(\alpha_{3}-\alpha_2)q_2^{-}(t))\geq 0\quad
 \forall t \in (t_1, t_2),
\end{gather*} 
then $x$ has at least one zero in $(t_1, t_2)$ unless
$|f_1(x)|$ is a nonzero constant multiple $|y|^{\alpha}$.
\end{corollary}

Finally the results in Theorems \ref{thm3.1}--\ref{thm3.2}
 and Corollary \ref{coro3.1} which
are nonlinear extensions of the variational lemma, Leighton's
theorem and Sturm-Picone theorem respectively, can also be given
for the singular half-linear case as in the following:

\begin{corollary} \label{coro3.2}
Let $f_1(x)=|x|^{\alpha-1}x$ in \eqref{3.1}. Suppose that there
exists a function $u \in D_{j_{s}}$, not identically zero in any
open subinterval of $(t_1,t_2)$ such that $j_{s}(u)\leq 0$. If
$x$ is a nontrivial solution of \eqref{3.1} such that
$A_{t_1t_2}[u,x]\geq 0$, then $x$ has a zero in $(t_1, t_2)$
except possibly when $u=Kx$ for some nonzero constant $K$.
\end{corollary}

\begin{corollary} \label{coro3.3}
Let us consider  equations \eqref{3.1} and \eqref{3.2} with
$f_1(u)=|u|^{\alpha-1}u=f_2(u)$. Suppose that there exists a
nontrivial real solution of $y \in D$ of $L_{s}y=0$. Let $x$ be
any nontrivial solution of $\ell_{s}x=0$. If
$V_{s}(y)\geq 0$, $ A_{t_1t_2}[y,x]\geq 0$ and
$$
\lim_{t\to
t_1^{+}}p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\geq 0 ,\quad 
\lim_{t\to t_2^{-}}p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\leq 0 ,
$$ 
then $x$ has at least one zero in $(t_1, t_2)$ unless $x$ is a nonzero
constant multiple of $y$.
\end{corollary}

\begin{corollary} \label{coro3.4}
Consider the equations \eqref{3.1} and \eqref{3.2} with
$f_1(u)=|u|^{\alpha-1}u=f_2(u)$. Suppose that there exists a
nontrivial real solution $y \in D$ of $L_{s}y=0$. Let $x$ be any
nontrivial solution of  $\ell_{s}(x)=0$. If
$A_{t_1t_2}[y,x]\geq 0$, $p_2(t)\geq p_1(t)$,
$q_1(t)\geq q_2(t)$ for all $t \in (t_1,t_2)$, and
$$
\lim_{t\to t_1^{+}}p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\geq 0,\quad 
\lim_{t\to t_2^{-}}p_2(t)y(t)|y'(t)|^{\alpha-1}y'(t)\leq 0 ,
$$
then any nontrivial solution $x$ of $\ell_{s}x=0$ either
has a zero in $(t_1, t_2)$ or it is a nonzero constant multiple of
$y$.
\end{corollary}

\subsection*{Acknowledgments}
 This manuscript is dedicated to the memory of honorable
Necdet Do\u{g}anata, the founder of Izmir University.
The author would like to thank the
anonymous referee for his/her useful comments and suggestions.

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\end{document}