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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 154, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/154\hfil Sturm-Picone type theorems]
{Sturm-Picone type theorems for nonlinear differential systems}

\author[A. Tiryaki \hfil EJDE-2015/154\hfilneg]
{Aydin Tiryaki}

\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 10, 2015. Published June 11, 2015.}
\subjclass[2010]{34C10, 34C15}
\keywords{Comparison theorem; Sturm-Picone theorem;
\hfill\break\indent  half-linear equations; nonlinear differential systems}

\begin{abstract}
 In this article, we establish a Picone-type inequality for a pair of
 first-order nonlinear differential systems. By using this inequality,
 we give Sturm-Picone type comparison theorems for these systems and a
 special class of second-order half-linear equations with damping term.
\end{abstract}

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

\section{Introduction}

Let $\alpha>0$ and define $\varphi_{\alpha} (s)=|s|^{\alpha-1}s$
if $s\neq 0$ and $\varphi_{\alpha} (0)=0$. By comparing with the zeros
of the first component of the solution of the system
\begin{equation} \label{d1}
\begin{gathered}
x'=a(t)x+b(t)\varphi_{1/\alpha}(y)  \\
y'=-c(t)\varphi_{\alpha}(x)-d(t)y
\end{gathered}
\end{equation}
we would like to obtain
some information about the existence and distribution of zeros of
the first component of the solution of the system
\begin{equation} \label{d2}
\begin{gathered}
u'=A(t)u+B(t)\varphi_{1/\alpha}(v) \\
v'=-C(t)\varphi_{\alpha}(u)-D(t)v
\end{gathered}
\end{equation}
 where $a, A, b, B, c, C, d$ and $D$ are continuous
real-valued functions on a given interval $I$ and $b(t)>0$ and
$B(t)>0$ in $I$. The existence and uniqueness of the solution of
the initial and boundary value problems for \eqref{d1} (or
\eqref{d2}) were considered by  Elbert \cite{Elbert} and
Mirzov \cite{Mirzov1976, Mirzov1979}.

We have the following special cases, considering, for example the
second system:

If $A(t)\equiv D(t)$ in $I$, then \eqref{d2} is the nonlinear
Hamiltonian system
\[ %\label{d3}
u'=\frac{\partial H}{\partial v}, \;\;v'=- \frac{\partial H}{\partial u}
\]
 where
\begin{equation}\label{d4}
H(t;u,v)=\frac{1}{\alpha+1} C(t)|u|^{\alpha+1}+A(t)uv
+\frac{\alpha}{\alpha+1}B(t)|v|^{1+\frac{1}{\alpha}}.
\end{equation}

When $A(t)\equiv 0$ in $I$, the system \eqref{d2} is equivalent
to the scaler second-order half-linear equation
\begin{equation}\label{d5}
(P(t)\varphi_{\alpha}(u'))'+R(t)\varphi_{\alpha}(u')+Q(t)\varphi_{\alpha}(u)=0
\end{equation}
where the coefficient functions are
\[
P(t)\equiv B(t)^{-\alpha}, \quad R(t)=D(t)B(t)^{-\alpha}, \quad Q(t)=C(t).
\]
If $A(t)\equiv 0$ and $D(t)\equiv 0$ in $I$, then \eqref{d5}
reduced to the half-linear Sturm-Liouville equation
\begin{equation}\label{d6}
(P(t)\varphi_{\alpha}(u'))'+Q(t)\varphi_{\alpha}(u)=0.
\end{equation}
Moreover, if we take the transformation
\begin{equation} \label{ektransformations}
\begin{gathered}
u=h(t)W   \\
v=\frac{1}{h(t)}z
\end{gathered}
\end{equation}
where $h'(t)=A(t)h(t)$, i.e
$h(t)=\exp\big(\int_{t_{0}}^{t} A(s)ds \big)$ in system \eqref{d2}
with $A(t)\equiv D(t)$, is equivalent for any $r \in C^{1}(I)$
\begin{equation} \label{ek17}
\Big(P_{1}(t)\varphi_{\alpha}(W')\Big)'+R_{1}(t)\varphi_{\alpha}(W)+Q_{1}(t)\varphi_{\alpha}(W)=0
\end{equation}
 where the coefficient function are
\begin{gather*}
P_{1}(t)=r(t), \quad
R_{1}(t)=(\alpha+1)r(t)A(t)-r'(t)-\alpha r(t) \frac{B'(t)}{B(t)},\\
Q_{1}(t)=r(t)C(t)B^{\alpha} (t).
\end{gather*}

It is not difficult to see that if we choose $r(t)=B^{-\alpha}(t)$
we get the scalar second-order half-linear equation
\begin{equation}
\Big(P(t)\varphi_{\alpha}(W')\Big)'+(\alpha+1)R(t)\varphi_{\alpha}(W')
+Q(t)\varphi_{\alpha}(W)=0 \label{ek18}
\end{equation}
where $P$, $R$ and $Q$ are defined as in \eqref{d5}.

Most of the classical results in oscillation theory are formulated
for the solutions of the self-adjoint Sturm-Liouville equations of
the form
\begin{gather}
-(p_{1}(x)u')'+p_{0}(x)u=0, \label{11} \\
-(P_{1}(x)v')'+P_{0}(x)v=0, \label{12}
\end{gather}
where $p_{0}$, $p_{1}$, $P_{0}$, $P_{1}$ are real valued
continuous functions and $p_{1}$ and $P_{1}$ are positive on an
appropriate interval. The starting point for this theory is the
well known comparison theorem for  Sturm \cite{Sturm} discovered
in 1836.

\begin{theorem}[Sturm Comparison Theorem] \label{thm11}
Suppose that $p_{1}(x)\equiv P_{1}(x)$ and \break $P_{0}(x)\leq p_{0}(x)$ and
$P_{0}(x)\neq p_{0}(x)$ for $x \in [x_{1}, x_{2}]$.
If $x_1$ and $x_2$ are consecutive zeros of a
nontrivial real solutions $u$ of \eqref{11}, then every real
solution of $v$ of \eqref{12} has a zero in $(x_{1}, x_{2})$.
\end{theorem}

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

\begin{theorem}[Sturm-Picone Theorem] \label{thm12}
 Suppose that $0<P_{1}(x)\leq p_{1}(x)$ and
$P_{0}(x)\leq p_{0}(x) $ for $[x_1, x_2]$. If $x_1$ and $x_2$ are
consecutive zeros of a nontrivial real solutions $u$ of
\eqref{11}, then every real solution of $v$ of \eqref{12} has
one of the following properties:
\begin{itemize}
\item[(i)]  $v(x)$ has a zero in $(x_1,x_2)$ 
\item[(ii)]  $v(x)$ is a constant multiple of $u(x)$.
\end{itemize}
\end{theorem}

Note that Theorem \ref{thm12} is a special case of Leighton's theorem
\cite{Leighton1962}. For a detailed study and earlier developments of
this subject, we refer the reader to the books \cite{Kreith1973, Swanson}.

The original proof by Picone was based on the identity
\begin{equation}
\frac{d}{dx}\big[\frac{u}{v}(vp_{1}u'-uP_{1}v' )\big]
=(p_{0}-P_{0})u^{2}+(p_{1}-P_{1})u'^{2}+P_{1}\big(u'-\frac{u}{v}v'
\big)^{2} \label{13}
\end{equation}
 which holds for all real valued functions $u$ and $v$
defined on $[x_1,x_2]$ such that $u$, $v$, $p_{1}u'$ and $P_{1}v'$
are differentiable on $[x_1,x_2]$ and $v(x)\neq 0$ for
$x \in [x_1,x_2]$.

The identity \eqref{13} has been a useful tool not only in
comparing equations \eqref{11} and \eqref{12} but also in
establishing Wirtinger type inequalities for the second-order
linear ordinary differential equation and lower bounds for the
eigenvalues of the associated eigenvalue problems and was
generalized to high-order ordinary differential operators as well
as the partial differential operators of the elliptic type
\cite{Allegretto and Huang98, Dosly2002, JarosandKusano1999,
JarosKusanoYoshida2002, Kreith1970, Yoshida2008, Yoshida2009}.


Sturm-Picone theorem is extended in several directions, see,
Ahmad and  Lazer \cite{Ahmad1} and  Ahmad \cite{Ahmad2} for
linear systems,  M\"{u}ller-Pfeiffer \cite{Muller} for
non-selfadjoint differential equations, Tyagi and V. Raghavenda
\cite{TyagiandRaghavendra2008} for implicit differential equations, W.
Allegretto \cite{Allegretto2001} for degenerate elliptic equations,
Zhang and Sun \cite{Zhang} for linear equations on time
scales, Jaro\v{s} and Kusano \cite{JarosandKusano1999}
for half linear equations, \cite{Tiryaki1, Tyagi2013} for
nonlinear equations. 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. In
this direction, Sturm-Picone theorem plays an important role.
\cite{Allegretto2000, Allegretto2001, Dosly2002, Fisnarova,
JarosKusanoYoshida2000, JarosKusanoYoshida2002, Swanson, Tadie, Tiryaki2,
Yoshida2008, Yoshida2009}.


In 1999, Jaros and Kusano  \cite{JarosandKusano1999} generalized
Picone's identity \eqref{13} to the class of nonlinear
second-order differential equations
\begin{gather}
\Big(p_{1}(x)\varphi (u')\Big)'+p_{0}(x)\varphi (u)=0, \label{144} \\
\Big(P_{1}(x)\varphi (v')\Big)'+P_{0}(x)\varphi (v)=0, \label{155}
\end{gather}
 where $\varphi (s):= |s|^{\alpha-1}s$, $\alpha>0$,
$p_1$, $p_0$, $P_1$, $P_0$ are defined as before. The above
equations are also called half-linear or sometimes homogeneous of
degree $\alpha$. They established a suitable Picone-type identity
as follows
\begin{equation}
\begin{split}
&\frac{d}{dt}\{\frac{u}{\varphi(v)}\left(\varphi(v)p_{1}\varphi(u')
-\varphi(u)P_{1}\varphi(v')\right)\}\\
&=\left(p_{1}-P_{1}\right)|u'|^{\alpha+1}
+\left(P_{0}-p_{0}\right)|u|^{\alpha+1}\\
&+P_{1} \big[|u'|^{\alpha+1}+\alpha |\frac{uv'}{v}|^{\alpha+1}-(\alpha+1)u'\varphi
\Big(\frac{uv'}{v}\Big)\big].
\end{split}  \label{166}
\end{equation}

Using the above identity, they obtained the following
comparison results which is extension of Theorem \ref{thm12} to the class
of half linear equations \eqref{144} and \eqref{155}.

\begin{theorem}[\cite{JarosandKusano1999}] \label{thm13}
 Suppose that $0<P_{1}(x)\leq p_{1}(x)$ and
$p_{0}(x)\leq P_{0}(x)$ for $x \in [x_{1}, x_{2}]$.
If $x_{1}$, $x_{2}$ are consecutive zeros of a nontrivial real solution $u$ of
\eqref{144}, then every solution $v$ of (1.8) has a zero in
$(x_{1}, x_{2})$ except possibly it is a constant multiple of $u$.
\end{theorem}

While qualitative theory of scalar cases are well-developed, only
little is known about the general systems, particularly in the
case where $a(t)\neq 0$, $A(t)\neq 0$ or $a(t)\neq d(t)$,
$A(t)\neq D(t)$ in $I$ (for some results concerning the case
$\alpha=1$ see \cite{Kreith1970, Kreith1973}).
Elbert \cite{Elbert}, proved that if $b(t)>0$
and $B(t)>0$ on $I$ and $(x, y)$ is a solution of \eqref{d2}
such that the function $x(t)$ has consecutive zeros at
$t_{1}, t_{2} \in I$ and \eqref{d1} is a Sturmian majorant for \eqref{d2}
in the sense that
\begin{equation} \label{ineq}
[ B(t)-b(t)] |\xi|^{\alpha+1}+\big[A(t)-a(t)-\frac{d(t)-D(t)}{\alpha} \big]
\xi \varphi_{\alpha}(\eta)+\frac{C(t)-c(t)}{\alpha}|\eta|^{\alpha+1}\geq 0,
\end{equation}
 for all $\xi, \eta \in R$ and $t \in I$, then for
any solution $(u, v)$ of \eqref{d1} the first component
$u(t)$ has at least one zero in $(t_{1}, t_{2})$.

Note that the inequality \eqref{ineq} holds for $\xi, \eta \in R \setminus \{0\}$
if $B(t)>b(t)$, $C(t)>c(t)$, and
\[
\big(B(t)-b(t)\big)\big(C(t)-c(t)\big)^{\alpha+1}
\geq \big(\frac{\alpha}{\alpha+1} \big)^{\alpha+1}
 |\alpha\big(A(t)-a(t)\big)-\left(d(t)-D(t)\right)|^{\alpha+1}.
\]

Elbert proved his result by means of the generalized
Pr\"ufer transformation. In the particular case $a(t) \equiv A(t)
\equiv D(t) \equiv 0$, Elbert's criterion reduces to the half-linear
generalization of the classical Sturm-Picone comparison theorem due
to Mirzov  \cite{Mirzov1976}.

Recently, Jaro\v{s} studied the system \eqref{d2} under
suitable sufficient conditions. He established Picone-type
identity for the nonlinear system of the form \eqref{d2} and
applied it to derive Wirtinger type inequalities. He also gave
some results to obtain information about the existence and
distribution of zeros of the first component of the solution of
\eqref{d2}. Indeed the following result is interesting.

\begin{theorem}[\cite{Jaros2013}] \label{thm14}
 If for some nontrivial $C^{1}$-function $x$ defined on
$[t_{1}, t_{2}]$ and satisfying $x(t_{1})=x(t_{2})=0$, the
condition
$$
J(x)=\int_{t_{1}}^{t_{2}} \Big[ B(t)^{-\alpha}|x'-\frac{\alpha A(t)+D(t)}{\alpha+1}
 x|^{\alpha+1}-c(t)|x|^{\alpha+1}  \Big] dt\leq 0
$$
 holds, then for any solution $(u,v)$ of \eqref{d2} the
first component $u(t)$ either has a zero in $(t_{1}, t_{2})$ or is
a constant multiple of $x(t)\exp \big( \int_{t_{0}}^{t}
\frac{A(s)-D(s)}{\alpha+1} ds \big)$ for some $t_{0} \in I$
\end{theorem}

We would like to obtain some
information about the existence and distribution of the zeros of
the first component of the solution of \eqref{d2} by comparing
with the zeros of the first component of the solution of
\eqref{d1} and obtain sufficient conditions for the case including
$B(t)\geq b(t)$ and $C(t)\geq c(t)$.

Note that our results, that are formulated in terms of the continuous
function $a(t)=\frac{\alpha A(t)+D(t)}{(\alpha+1)}$ yield a variety
of comparison results. Even if we reduce our consideration to the special cases of
$a(t)$ mentioned above, our results seem to be new.

\section{Picone-type inequality and Leightonian comparison theorems}

Let
\begin{equation} \label{d21}
\Phi_{\alpha}(\xi, \eta):= \xi \varphi_{\alpha}(\xi)
+\alpha \eta  \varphi_{\alpha}(\eta)-(\alpha+1)\xi \varphi_{\alpha}(\eta).
\end{equation}
 for $\varepsilon, \eta  \in R$ and $\alpha>0$. From the Young inequality, 
it follows that
$\Phi_{\alpha}(\xi,\eta)\geq 0$ for all $\xi,\eta \in R$, and the
equality holds if and only if $\xi= \eta$.
The Picone-type inequality in the following lemma is of basic
importance for our main results, it may be verified directly by
differentiation.

\begin{lemma}[Picone-type inequality] \label{lem21}
 Suppose that $(u,v)$ is a solution of
\eqref{d2} such that $u(t)\neq 0$ in $I$. If there exists a
solution $(x,y)$ of \eqref{d1}, then
\begin{equation}
\begin{aligned}
&\frac{d}{dt} \Big[\frac{x}{\varphi_{\alpha}(u)} \\
\Big( \varphi_{\alpha}(u)y-\varphi_{\alpha}(x)v \Big)  \Big]
&\geq \Big[C(t)-c(t)-\frac{1}{\alpha+1}|a(t)-d(t)| \Big] x \varphi_{\alpha}(x)  \\
&\quad +\Big[b(t)-\frac{b^{\alpha+1}(t)}{B^{\alpha}(t)}
 -\frac{\alpha}{\alpha+1}|a(t)-d(t)| \Big]y \varphi_{1/\alpha}(y)\\
&\quad +B^{-\alpha}(t)\Phi_{\alpha} (b(t)\varphi_{1/\alpha}(y),
  B(t)\frac{x}{u}\varphi_{1/\alpha}(v))  \\
&\quad -\Big[(\alpha+1)a(t)-\alpha A(t)-D(t) \Big]x \varphi_{\alpha} 
\Big(\frac{x}{u}\Big)v. 
\end{aligned}\label{d22}
\end{equation}
\end{lemma}

We begin with the following functionals $V_{\sigma\tau}$ and
$M_{\sigma\tau}$ defined for $t_{1}<\sigma<\tau<t_{2}$ and
solutions $(x,y)$ of \eqref{d1} and $(u,v)$ of \eqref{d2} with
$u(t)\neq 0$ in $I$ by
\begin{equation} \label{d23}
\begin{aligned}
&V_{\sigma\tau}(x)\\
&= \int_{\sigma}^{\tau} \{\Big[C(t)-c(t)-\frac{1}{\alpha+1}|a(t)-d(t)| 
\Big]|x|^{\alpha+1}  \\
&\quad +\Big[b(t)- \frac{b^{\alpha+1}(t)}{B^{\alpha}(t)}
 -\frac{\alpha}{\alpha+1}|a(t)-d(t)| \Big] b^{-(\alpha+1)}(t) 
|x'-a(t)x|^{(\alpha+1)} \} dt
\end{aligned}
\end{equation}
and
$$
M_{\sigma \tau}[x; u, v]=\int_{\sigma}^{\tau} B^{-\alpha}(t) 
\Big(\Phi_{\alpha} (x'-ax, B(t)\frac{x}{u}\varphi_{1/\alpha}(v)\Big) dt.
$$
From Lemma \ref{lem21}, by using the definition of
$V_{\sigma\tau}(x)$ we have the following lemma.

\begin{lemma} \label{lem22}
Let $(x,y)$ and $(u,v)$ be solutions of \eqref{d1} and \eqref{d2}
respectively such that $u(t)\neq 0$ in $I$ and let $[\sigma,
\tau]\subset I$. Then for the first component $x(t)$ of the
solution of \eqref{d1}, the following inequality holds:
\begin{equation} \label{d24}
\begin{aligned}
&\big[\frac{x}{\varphi_{\alpha}(u)} (\varphi_{\alpha}(u)y-\varphi_{\alpha}(x)v) \big] 
\big|_{\sigma}^{\tau} \\
&\geq V_{\sigma\tau}(x)-\int_{\sigma}^{\tau} \big[(\alpha+1)a(t)-\alpha A(t)-D(t) 
\big]x \varphi_{\alpha} \Big(\frac{x}{u}\Big)v dt.
\end{aligned}
\end{equation}
Moreover, the inequality holds in \eqref{d24} if and only if
\begin{equation} \label{d25}
x'=\Big(a(t)+B(t)\frac{\varphi_{1/\alpha}(v)}{u} \Big)x.
\end{equation}
\end{lemma}

\begin{proof}
Integrating \eqref{d22} from $\sigma$ to $\tau$ and using positive
semidefiniteness of the form $\Phi_{\alpha}$, we obtain
\begin{equation} \label{d26}
\begin{aligned}
&\big[\frac{x}{\varphi_{\alpha}(u)} (\varphi_{\alpha}(u)y-\varphi_{\alpha}(x)v) \big] 
\big|_{\sigma}^{\tau} \\
&\geq V_{\sigma\tau}(x)+M_{\sigma\tau}(x; u, v) 
-\int_{\sigma}^{\tau} \big[(\alpha+1)a(t)-\alpha A(t)-D(t) 
\big]x \varphi_{\alpha} \Big(\frac{x}{u}\Big)v dt \\
&\geq V_{\sigma\tau}(x)-\int_{\sigma}^{\tau} 
\big[(\alpha+1)a(t)-\alpha A(t)-D(t) \big]x \varphi_{\alpha} 
\Big(\frac{x}{u}\Big)v dt
\end{aligned}
\end{equation}
which gives \eqref{d24}. The equality
obviously holds in \eqref{d24} if and only if
$\Phi_{\alpha}
(x'-ax, B(t)\frac{x}{u}\varphi_{1/\alpha}(v)) =0$ in
$[\sigma, \tau]$ which is equivalent with the condition
\eqref{d25}.
\end{proof}

From Lemma \ref{lem22}, we easily obtain the variation $V(x)$
and $M(x; u, v)$ if we assume the existence of the limits
\begin{equation} \label{d27}
V(x)=\lim_{\sigma\to t_{1}^{+}, \tau \to t_{2}^{-} } V_{\sigma\tau}(x), \quad
M(x; u,v)=\lim_{\sigma\to t_{1}^{+}, \tau \to t_{2}^{-} } M_{\sigma\tau}(x; u, v).
\end{equation}
Now define the domains $D_{V}$ and $D_{M}$ of $V$ and $M$ respectively, 
to be sets of all real-valued solutions of \eqref{d1} such
that $V(x)$ and $M(x; u, v)$ exist. Also for the
solution $x \in D_{V} \cap D_{M}$ of \eqref{d1} and the solution
$(u,v)$ of \eqref{d2} with $u(t)\neq 0$ in $I=(t_{1}, t_{2})$, we
denote
\begin{equation} \label{d28}
\begin{gathered}
S_{1}(x; u,v)=\lim_{t\to t_{1}^{+}}\Big[x{\varphi_{\alpha}
 \Big( \frac{x'-a(t)x}{b(t)}\Big)-x\varphi_{\alpha} 
 \Big( \frac{x}{u}\Big)v} \Big]  \\
S_{2}(x; u,v)=\lim_{t\to t_{2}^{-}} 
 \Big[x{\varphi_{\alpha}\Big( \frac{x'-a(t)x}{b(t)}\Big)-x\varphi_{\alpha} 
 \Big( \frac{x}{u}\Big)v} \Big]
\end{gathered}
\end{equation}
 whenever the limits in \eqref{d28} exist.

\begin{theorem} \label{thm21}
Let $(x,y)$ and $(u,v)$ be solutions of \eqref{d1} and \eqref{d2}
respectively with $u(t) \neq 0$ in $I$ satisfying
\begin{equation} \label{d29}
\Big[(\alpha+1)a(t)-\alpha A(t)-D(t) \Big]\frac{v}{\varphi_{\alpha}(u)} \leq 0
\end{equation}
 in $I$. Then the solution $x \in D_{V} \cap D_{M}$ of
\eqref{d1} for which the limits in \eqref{d28} exist, the
inequality
\begin{equation} \label{d210}
S_{2}(x ; u,v)-S_{1}(x; u,v) \geq V(x)
\end{equation}
 holds. Furthermore if $[(\alpha+1)a(t)-\alpha
A(t)-D(t)]\frac{v}{\varphi_{\alpha}(u)} = 0$ in $I$, then the
equality in \eqref{d210} occurs if and only if $x(t)$ is a
solution of \eqref{d25}.
\end{theorem}

As an  immediate consequence of the above theorem we have the
following result.

\begin{corollary} \label{coro21}
Let $(x,y)$ and $(u,v)$ be solutions of \eqref{d1} and \eqref{d2}
respectively with $u(t) \neq 0$ in $I$ and
\begin{equation} \label{d211}
\big[(\alpha+1)a(t)-\alpha A(t)-D(t)\big]\frac{v}{\varphi_{\alpha}(u)} = 0
\end{equation}
in $I$. Then for every solution $x \in D_{V} \cap D_{M}$
of \eqref{d1} for which both limits in \eqref{d28} exists, the
inequality \eqref{d210} is valid. Moreover, the inequality holds
in \eqref{d210} if and only if
$$
x(t)=K u(t) \exp \Big( \int_{t_{0}}^{t} (a(s)-A(s))ds \Big)
$$
for some constants $K\neq 0$ and $t_{0} \in I$.
\end{corollary}

In the case where $a(t)\equiv A(t)\equiv D(t)$ in $I$
the condition \eqref{d29} is trivially satisfied. Clearly, in this special case, 
the equality in \eqref{d210} is satisfied if and only if $x(t)$ is a 
constant multiple of $u(t)$.

Another way, to guarantee the equality in \eqref{d29} is to
choose
\begin{equation} \label{ek214}
a(t)=\frac{\alpha A(t)+D(t)}{\alpha+1}.
\end{equation}
By choosing $a(t)$ this way, we have the following
important results.

\begin{corollary} \label{coro22}
If $(x,y)$ and $(u,v)$ are solutions of \eqref{d1} and \eqref{d2}
respectively with $u(t)\neq 0$ in $I$ and $x \in D_{V} \cap D_{M}$
is such that the limits in \eqref{d28} exist and satisfy $S_{2}(x;
u,v)\geq 0$, $S_{1}(x; u,v)\leq 0$, then
\begin{equation} \label{d212}
\begin{aligned}
 V(x)&=\int_{t_{1}}^{t_{2}} \Big\{
\Big[C(t)-c(t)-\frac{1}{\alpha+1}|\frac{ \alpha A(t)+D(t)}{\alpha+1}-d(t)|
\Big]|x|^{\alpha+1} \\
&\quad+ \Big[b(t)-\frac{b^{\alpha+1}(t)}{B^{\alpha}(t)}
 -\frac{\alpha}{\alpha+1}|\frac{ \alpha A(t)+D(t)}{\alpha+1}-d(t)|  \Big] \\
&\quad \times B^{-(\alpha+1)} |x'-\frac{ \alpha A(t)
 +D(t)}{\alpha+1}x|^{\alpha+1}  \Big\} dt \leq 0
\end{aligned}
\end{equation}
Furthermore, the equality in \eqref{d212} is satisfied if and only if
\[ %\label{d213}
x(t)=Ku(t)\exp\Big(-\int_{t_{0}}^{t} \frac{(A(s)-D(s))}{\alpha+1} ds \Big)
\]
for some $t_{0} \in I$.
\end{corollary}

\begin{corollary} \label{coro23}
Let $V(x)$ be defined as in \eqref{d212}. If $(x, y)$ is a solution of 
\eqref{d1} satisfying $x(t_{1})=x(t_{2})=0$, the condition
$V(x) \geq 0$
 holds, then for any solution $(u,v)$ of \eqref{d2} the
first component $u(t)$ has one of the following properties:
\begin{itemize}
\item[(i)] $u$ has a zero in $(t_{1}, t_{2})$ or,

\item[(ii)] $u$ is a nonzero constant multiple of
        $x(t)\exp\big(\int_{t_{0}}^{t}
        \frac{(A(s)-D(s))}{\alpha+1} ds \big)$,  for some
        $t_{0} \in I$.
\end{itemize}
\end{corollary}

\begin{remark} \label{remk1} \rm
If the condition $V(x)\geq 0$ is strengthened to $V(x)>0$,
conclusion (ii) of Corollary \ref{coro23} does not hold.
\end{remark}

From Corollary \ref{coro23} we immediately have the following
result which is an extension of Sturm-Picone Comparison Theorem of
the systems \eqref{d1} and \eqref{d2}.

\begin{theorem} \label{thm22}
Suppose there exists a nontrivial solution $(x,y)$ of \eqref{d1}
in $(t_{1}, t_{2})$ such that $x(t_{1})=x(t_{2})=0$. If
\begin{equation} \label{ek215}
C(t)\geq c(t)+\frac{1}{\alpha+1} \Big|\frac{\alpha A(t)+D(t)}{\alpha+1}-d(t)  \Big|
\end{equation}
and
\[
b(t) \geq \frac{b^{\alpha+1}(t)}{B^{\alpha}(t)}
+\frac{\alpha}{\alpha+1}\Big| \frac{\alpha A(t)+D(t)}{\alpha+1}-d(t) \Big|
\]
for every $t \in (t_{1}, t_{2})$, then the first
component $u(t)$ of every nontrivial solution $(u,v)$ of
\eqref{d2} has at least one zero in $(t_{1}, t_{2})$ unless $u$ is
a nonzero constant multiple of
\begin{eqnarray*}
x(t)\exp\Big(\int_{t_{0}}^t \frac{A(s)-D(s)}{\alpha+1} ds  \Big).
\end{eqnarray*}
\end{theorem}

\begin{remark}\label{remk2} \rm
Note that when $a(t)\equiv d(t)\equiv A(t)\equiv D(t)$, the case \eqref{ek214} 
is already satisfied, hence we can obtain special cases of the above results 
given in Corollary \ref{coro22}-\ref{coro23} and Theorem \ref{thm22}.
\end{remark}

Now we consider a class of second-order half-linear equations with
damping term:
\begin{gather} \label{x1}
\Big(b^{-\alpha}(t)\varphi_{\alpha}(w') \Big)'
 +(\alpha+1)b^{-\alpha}(t)D(t)\varphi_{\alpha}(w')+c(t)\varphi_{\alpha}(w)=0,\\
\label{x2}
\Big(B^{-\alpha}(t)\varphi_{\alpha}(W') \Big)'
 +(\alpha+1)B^{-\alpha}(t)D(t)\varphi_{\alpha}(W')+C(t)\varphi_{\alpha}(W)=0
\end{gather}
Note that Equation \eqref{x2} is the same as Equation (1.8), which is obtained from \eqref{d2}, Equation \eqref{x1} can be obtained from \eqref{d1} using similar transformations.
From Remark \ref{remk2}, we immediately have the following
theorem which is straightforward Sturm-Picone comparison result
for the above damped half-linear equations.

\begin{theorem} \label{thm23}
Suppose that there exists a nontrivial real solution $w$ of
\eqref{x1} in $(t_{1}, t_{2})$ such that $w(t_{1})=0=w(t_{2})$. If
$B(t)\geq b(t)$ and $C(t)\geq c(t)$, then every nontrivial
solution $W$ of \eqref{x2} either has a zero in $(t_{1}, t_{2})$
or it is a nonzero constant multiple of $w$.
\end{theorem}

\begin{remark} \label{rmk3} \rm
Note that, Theorem \ref{thm23} is a partial answer to the open problem
given in \cite{Tiryaki-Oinarov-Ramazanova}.
\end{remark}

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\end{document}