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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 28, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/28\hfil Liapunov-type integral inequality]
{Liapunov-type integral inequalities for certain
 higher-order differential equations}

\author[S. Panigrahi\hfil EJDE-2009/28\hfilneg]
{Saroj Panigrahi}

\address{Saroj Panigrahi \newline
Department of Mathematics and Statistics,
University of Hyderabad, Hyderabad 500 046, India}
\email{spsm@uohyd.ernet.in, panigrahi2008@gmail.com}

\thanks{Submitted October 19, 2008. Published February 5, 2009.}
\thanks{Supported by National Board of Higher
 Mathematics, Department of Atomic Energy, India}
\subjclass[2000]{34C10}
\keywords{Liapunov-type inequality; oscillatory solution; disconjugacy;
\hfill\break\indent
higher order differential equations}

\begin{abstract}
 In this paper, we obtain Liapunov-type integral inequalities
 for certain nonlinear, nonhomogeneous differential equations
 of higher order with without any restriction on the zeros of
 their higher-order derivatives of the solutions by using elementary
 analysis. As an applications of our results, we show that oscillatory
 solutions of the equation converge to zero as $t\to \infty$.
 Using these inequalities, it is also shown  that
 $(t_{m+ k} - t_{m})  \to \infty $ as
 $m \to \infty$, where $1 \le k \le n-1$ and $\langle t_m \rangle $ is an
 increasing sequence of zeros of an oscillatory solution of
 $ D^n y + y f(t, y)|y|^{p-2} = 0$, $t \ge 0$, provided that
 $W(., \lambda)  \in L^{\sigma}([0, \infty), \mathbb{R}^{+})$,
 $1 \le \sigma \le \infty$ and for all $\lambda > 0$. A criterion for
 disconjugacy of nonlinear homogeneous equation is obtained in an
 interval $[a, b]$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}


\section{Introduction}

The Russian mathematician A. M. Liapunov \cite{Liapunov} proved the
following remarkable inequality:
    If $y(t)$ is a nontrivial solution of
\begin{equation} \label{e1}
y'' + p(t)y = 0,
\end{equation}
with $y(a) = 0 =y(b)$ ($a < b$) and $y(t) \neq 0 $ for
$ t \in (a, b)$, then
\begin{equation} \label{e2}
\frac{4}{b - a} < \int_{a}^{b} |p(t)|dt,
\end{equation}
where  $ p \in L_{\rm loc}^1$.
This inequality provides a lower bound
for the distance between consecutive zeros of $y(t)$.
If $p(t) = p > 0$, then \eqref{e2} yields
\[
(b - a ) > 2/{\sqrt p}.
\]
In \cite{Hartman2}, the inequality \eqref{e2} is strengthened to
\begin{equation} \label{e3}
\frac{4}{b - a} < \int_{a}^{b} p_{+}(t)dt,
\end{equation}
where $p_{+}(t) = \max\{p(t), 0\}$. The inequality \eqref{e3} is the best
possible in the sense that if the constant 4 in \eqref{e3} is replaced by
any larger constant, then there exists an example of \eqref{e1} for which
\eqref{e3} no longer holds (see \cite[p. 345]{Hartman2}, \cite{Kowng}).
 However, stronger results were obtained in \cite{Brown,Kowng}.
In \cite{Kowng} it is shown that
\[
\int_{a}^{c} p_{+}(t) dt > \frac {1}{c - a}\quad \text{and}\quad
\int_{c}^{b} p_{+}(t) dt > \frac{1}{b - c},
\]
where $c \in (a, b) $ such that $y'(c) = 0$. Hence
\[
\int_{a}^{b} p_{+}(t)dt > \frac{1}{c - a} + \frac{1}{b - c} =
\frac{(b - a)}{(c - a)(b - c)} \ge \frac{4}{b - a}.
\]
   In \cite[Corollary 4.1]{Brown}, the authors obtained
\[
\frac {4}{b - a} < \big| \int_{a}^{b}p(t)dt \big|
\]
from which \eqref{e2} can be obtained. The inequality finds applications in
the study of boundary value problems. It may be used to provide a
lower bound on the first positive proper value of the
Sturm-Liouville problems
\begin{gather*}
y''(t) + \lambda q(t)y = 0 \\
y(c) = 0 = y(d)\quad (c < d)
\end{gather*}
and
\begin{gather*}
y''(t) + ( \lambda + q(t))y = 0\\
y(c) = 0 = y(d)\quad  (c < d)
\end{gather*}
by letting $p(t)$ to denote ${\lambda}q(t)$ and $\lambda + q(t)$
respectively in \eqref{e2}. The disconjugacy of \eqref{e1} also
depends on \eqref{e2}.
Indeed, equation \eqref{e1} is said to be disconjugate if
\[
\int_{a}^{b} |p(t)|dt \le 4/(b - a).
\]
Equation \eqref{e1} is said to be disconjugate on $[a, b]$ if no
non-trivial solution of \eqref{e1} has more than one zero.
Thus \eqref{e2} may be
regarded as a necessary condition for conjugacy of \eqref{e1}.
Inequality \eqref{e2} has lots of applications in eigenvalue
problems, stability, etc. A number of proofs are known and
generalizations and improvements have also been given
(see \cite{Hartman2,Levin,patula,Willet,Wong}.
 Inequality \eqref{e3} was generalized to the condition
\begin{equation} \label{e4}
\int_{a}^{b} (t-a)(b-t)p_{+}(t)dt > (b-a)
\end{equation}
by Hartman and Wintner \cite{Hartman}. An alternate proof of the inequality
\eqref{e4}, due to Nihari \cite{Nihari}, is given in
\cite[Theorem 5.1 Ch XI]{Hartman2}. For
the equation
\begin{equation} \label{e5}
y''(t) + q(t)y' + p(t)y = 0,
\end{equation}
where $p, q \in C([0, \infty), R)$, Hartman and Wintner \cite{Hartman}
established the inequality
\begin{equation} \label{e6}
\int_{a}^{b} (t-a)(b-t)p_{+}(t)dt +
\max\Big\{\int_{a}^{b}(t-a)|q(t)|, \int_{a}^{b}(b-t)|q(t)|dt\Big\}
> (b-a)
\end{equation}
which reduces to \eqref{e4} when $q(t) = 0$. In particular, \eqref{e6} implies the
\emph{de la vallee Poussin inequality} \cite{Reid}. In \cite{Galbrith},
Galbraith has shown that if $a$ and $b$ are successive zeros of \eqref{e1}
with $p(t) \ge 0 $ a linear function, then
\[
(b-a)\int_{a}^{b}p(t)dt \le \pi^{2}.
\]
This inequality provides an upper bound for two successive zeros of
an oscillatory solution of \eqref{e1}.
 Indeed, if $p(t) = p > 0$, then $(b-a) \le \pi/(p)^{1/2}$.
Fink \cite{Fink},  obtained both upper and
lower bounds of $(b-a)\int_{a}^{b}p(t)dt$, where $p(t) \ge 0$ is
linear. Indeed, he showed that
\[
{\frac{9}{8}}\lambda_{0}^{2} \le (b-a)\int_{a}^{b}p(t)dt \le \pi^{2}
\]
and that these are the best possible bounds, where $\lambda_{0}$ is the
first positive zero of $J_{1/3}$ and $J_{n}$ is the Bessel
function. The constant ${\frac{9}{8}}\lambda_{0}^{2} = 9.478132\dots $
and $\pi^{2} = 9.869604\dots $, so that it gives a delicate test for
the spacing of the zeros for linear $p$.
Fink \cite{Fink2} investigated the behaviour of the functional
$(b-a)\int_{a}^{b}p(t)dt$, where $p$ is in a certain class of sub or
supper functions. Eliason \cite{Eliason,Eli1}  obtained upper and lower bounds
of the functional $(b-a)\int_{a}^{b}p(t)dt$, where $p(t)$ is
concave or convex.  St Marry and Eliason \cite{Marry} considered the
same problem for \eqref{e5}.  Bailey  and Waltman \cite{Baily} applied
different techniques to obtain both upper and lower bounds for the
distance between two successive zeros of solution of \eqref{e5}. They also
considered nonlinear equations. In a recent paper, Brown and
Hinton \cite{Brown} used Opial's inequality to obtain lower bounds for the
spacing of the zeros of a solution of \eqref{e1} and lower bounds of the
spacing $\beta - \alpha$, where $y(t)$ is a solution of \eqref{e1}
satisfying $y(\alpha) = 0 = y'(\beta)$ and
$y'(\alpha) = 0 = y(\beta) (\alpha < \beta)$.

 Inequality \eqref{e2} is generalized to second order nonlinear
differential equation by Eliason \cite{Eli1}, to delay differential
equations of second order in
\cite{Eli2,Eli3} and  by Dahiya and Singh \cite{Dahya},
and to higher order differential  equation by Pachpatte \cite{Pach}.
In a recent work \cite{Parhi1}, the authors have obtained a Liapunov-type
inequality for third order differential equations of the form
\begin{equation} \label{e7}
y''' + p(t)y = 0,
\end{equation}
where $p \in L_{\rm loc}^1$. The inequality is used to study many
interesting properties of the zeros of an oscillatory solution of
\eqref{e7} (see \cite[Theorems 5, 6]{Parhi1}). Indeed, Pachpatte derived
Liapunov-type inequalities for the equation of the form
\begin{equation} \label{e8}
\begin{gathered}
D^n[ r(t)D^{n-1}(p(t)g(y'(t))) ] + y(t) f(t,y(t)) = Q(t), \\
D^n[ r(t)D^{n-1}(p(t)h(y(t))y'(t)) ] + y(t) f(t,y(t)) = Q(t), \\
D^n[ r(t)D^{n-1}(p(t)h(y(t))g(y'(t))) ] + y(t) f(t,y(t)) = Q(t),
\end{gathered}
\end{equation}
under appropriate conditions, where $n \ge 2$ is an integer and
$ D= d^n/dt^n$. It is clear that the results in \cite{Pach} are not
applicable to odd order equations. Furthermore, he has taken the
restriction on the zeros of higher order derivatives
\cite[Theorem 1]{Pach}. We may observe that in \cite[p.530, Example]{Pach},
$y'''(3\pi/4) \ne 0$ because $y'''(t) = 2e^{-t}(\cos t - \sin t)$.
On the other hand, $y'''(\pi/4) = 0$ but $ \pi/4  \notin  (\pi/2, 3\pi/2) $
 and $y'''(5\pi/4) = 0$ but $5\pi/4 < \pi$. Although this example does
not illustrate \cite[Theorem 1]{Pach}, it has motivated us to remove the
restriction on the zeros of higher order derivatives of the solution
of \eqref{e5}.

The objective of this paper is to obtain Liapunov-type integral
inequality for the nth-order differential equation
\begin{equation} \label{e9}
\Big(\frac {1}{r_{n-1}(t)} \dots
\Big(\frac{1}{r_{2}(t)}\Big(\frac{1}{r_1(t)}|y'(t)|^{p-2}y'(t)\Big)'\Big)'
\dots  \Big)' + |y(t)|^{p-2}f(t,y(t))y = Q(t),
\end{equation}
under appropriate assumptions on $ r_{i}(t)$, $1\le i\le n-1, f$ and
$Q$. Here $n\ge 2, p > 1$ are even and odd integers. In this work we
remove this restriction on the zeros of higher order derivatives.
Further, we show that every oscillatory solution of \eqref{e9} converges to
zero as $ t\to \infty$ with the help of Liapunov-type inequality. We
also generalize a theorem of Patula  \cite[Theorem 2]{patula} to higher
order equations. A criteria for diconjugacy of nonlinear homgeneous
equation is obtained in an interval $[a, b]$ by the help of the
inequality.

\section{Main results}

Equation \eqref{e9} may be written as
\begin{equation} \label{e10}
D^ny + y f(t, y)|y(t)|^{p-2} = Q(t),
\end{equation}
where $ n \ge 2$ is an integer,
\[
Dy = \frac{1}{r_1(t)}|y'(t)|^{p-2}y'(t),\quad
D^{i}y = \frac{1}{r_{i}(t)}(D^{i-1}y)',
\]
$ 2 \le i \le n $, and $ r_{n}(t) \equiv 1$. We assume that
\begin{itemize}
\item[(C1)] $r_{i}: I \to \mathbb{R}$ is continuous and
$r_{i}(t) > 0, 1 \le i \le n - 1 $ and $Q:I \to \mathbb{R} $
is continuous, where $I$ is a real interval.

\item[(C2)] $f : I\times \mathbb{R}\to \mathbb{R}$ is continuous such
that $ |f(t, y)| \le W(t, |y|)$, where $W : I \times \mathbb{R^ {+}}
\to \mathbb{R}^{+}$ is continuous, $W(t, u) \le W(t, v)$ for $0 \le
u \le v$ and $\mathbb{R}^{+} = [0, \infty]$.

\end{itemize}
We define
\begin{align*}
&E(t,r_{2}(t), r_{3}(s_{2}), \dots  ,r_{n-1}(s_{n-2});z(s_{n-1}) ) \\
&= r_{2}(t) \int_{\alpha_1}^{t}r_{3}(s_{2})\
\int _{\alpha_{2}}^{s_{2}}r_{4}(s_{3}) \dots\\
&\quad \int_{\alpha_{n-3}}^{s_{n-3}}r_{n-1}(s_{n-2})
\int_{\alpha_{n-2}}^{s_{n-2}}z(s_{n-1})ds_{n-1}ds_{n-2}\dots  ds_{2},
\end{align*}
where $z(t)$ is a real valued continuous function defined on
$[a, b] \subset I(a < b)$ and
$ \alpha_1, \alpha_{2},\dots ,\alpha_{n-2} $
are suitable points in $[a, b]$, and
\begin{align*}
&\overline{E}(t,r_{2}(t), r_{3}(s_{2}), \dots ,
  r_{n-1}(s_{n-2});z(s_{n-1}) ) \\
&= r_{2}(t) \Big|\int_{\alpha_1}^{t}r_{3}(s_{2})\Big|
\int _{\alpha_{2}}^{s_{2}}r_{4}(s_{3}) \dots
\Big|\int_{\alpha_{n-3}}^{s_{n-3}}r_{n-1}(s_{n-2})
\Big|\int_{\alpha_{n-2}}^{s_{n-2}}z(s_{n-1})ds_{n-1}
\Big|ds_{n-2}\Big|\\
&\quad \dots \Big|ds_{2}\Big|.
\end{align*}

\begin{theorem} \label{thm1}
Suppose that {\rm (C1)-(C2)} hold.
Let $\alpha_1, \alpha_{2}, \dots , \alpha_{n-2}  \in
[a,b]$, where $ \alpha_1, \alpha_{2},\dots , \alpha_{n-2}$ are the
zeros of $D^{2}y(t), D^{3}y(t), \dots ,D^{n-2}y(t), D^{n-1}y(t)$
respectively, $ [a, b] \subset I(a < b) $ and $y(t)$ is a nontrivial
solution of \eqref{e10} with $y(a) = 0 = y(b)$. If $c$ is a point in
$(a, b)$ where $|y(t)|$ attains maximum and
$ M = \max\{|y(t)| : t\in [a,b]\} = |y(c)|$, then
\begin{equation} \label{e11}
\begin{aligned}
1 &< \big(\frac {1}{2}\big)^p\Big(\int_{a}^{b}(
r_1(s_1))^{1/(p-1)}ds_1\Big)^{p-1}\Big(\int _{a}^{b}
\big[\overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}), \dots  ,
 r_{n-1}(s_{n-2});\\
&\quad  W(s_{n-1}, M))
+  \frac {1}{M^{p-1}}\overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}),
 \dots  , r_{n-1}(s_{n-2});|Q(s_{n-1})|)\big]
ds_1\Big),
\end{aligned}
\end{equation}
for $ n \ge 3 $ and
\begin{equation} \label{e12}
1 < \big(\frac {1}{2}\big)^p\Big(\int_{a}^{b}(
r_1(t))^{1/(p-1)}dt\Big)^{p-1} \Big[\int_{a}^{b} W(t, M) dt +
\frac {1}{M^{p-1}} \int_{a}^{b} |Q(t)| dt\Big],
\end{equation}
for $n = 2$.
\end{theorem}

\begin{proof} Let $ n \ge 3 $. Integrating \eqref{e10} from
$\alpha_{n-2}$ to $t \in [a, b]$, we obtain
\begin{align*}
&D^{n-1}y(t) + \int_{\alpha_{n-2}}^{t}y(s_{n-1})f(s_{n-1},
y(s_{n-1}))|y(s_{n-1})|^{p-2}ds_{n-1} \\
&= \int_{\alpha_{n-2}}^{t}
Q(s_{n-1})ds_{n-1};
\end{align*}
that is,
\begin{align*}
&(D^{n-2}y(t))' + r_{n-1}(t)
\int_{\alpha_{n-2}}^{t}y(s_{n-1})f(s_{n-1}, y(s_{n-1}))
|y(s_{n-1})|^{p-2}ds_{n-1} \\
&= r_{n-1}(t) \int_{\alpha_{n-2}}^{t} Q(s_{n-1}) ds_{n-1}.
\end{align*}
Further integration from $\alpha_{n-3}$ to $t \in [a, b]$ yields
\begin{align*}
&D^{n-2}y(t) \\
&+\int_{\alpha_{n-3}}^{t} r_{n-1}(s_{n-2})\Big(
\int_{\alpha_{n-2}}^{s_{n-2}}y(s_{n-1})
f(s_{n-1}, y(s_{n-1}))|y(s_{n-1})|^{p-2}ds_{n-1}\Big)ds_{n-2} \\
&= \int_{\alpha_{n-3}}^{t} r_{n-1}(s_{n-2})\Big(
\int_{\alpha_{n-2}}^{s_{n-2}} Q(s_{n-1}) ds_{n-1}\Big)ds_{n-2}.
\end{align*}
Proceeding as above we obtain
\begin{align*}
& D^{2}y(t) +  \int_{\alpha_1}^{t}r_{3}(s_{2})
\int _{\alpha_{2}}^{s_{2}}r_{4}(s_{3}) \dots  \\
& \int_{\alpha_{n-3}}^{s_{n-3}}r_{n-1}(s_{n-2})
\int_{\alpha_{n-2}}^{s_{n-2}}y(s_{n-1})f(s_{n-1},
y(s_{n-1}))|y(s_{n-1})|^{p-2}ds_{n-1}ds_{n-2} \dots  ds_{2},
\\
&= \int_{\alpha_1}^{t}r_{3}(s_{2}) \int
_{\alpha_{2}}^{s_{2}}r_{4}(s_{3}) \dots
\int_{\alpha_{n-3}}^{s_{n-3}}r_{n-1}(s_{n-2})
\int_{\alpha_{n-2}}^{s_{n-2}}Q(s_{n-1})ds_{n-1}ds_{n-2}\dots ds_{2};
\end{align*}
that is,
\begin{align*}
&(Dy(t))' +  E(t, r_{2}(t), r_{3}(s_{2}), \dots ,r_{n-1}(s_{n-2});
y(s_{n-1})f(s_{n-1}, y(s_{n-1}))
|y(s_{n-1})|^{p-2})\\
& = E(t, r_{2}(t), r_{3}(s_{2}), \dots ,r_{n-1}(s_{n-2}); Q(s_{n-1})).
\end{align*}
Hence
\begin{equation} \label{e13}
\begin{aligned}
|(Dy(t))'| &\le  M^{p-1}\overline{E}(t, r_{2}(t), r_{3}(s_{2}), \dots ,
 r_{n-1}(s_{n-2}); W(s_{n-1}, M)) \\
&\quad   +  \overline{E}(t, r_{2}(t), r_{3}(s_{2}), \dots ,
r_{n-1}(s_{n-2}); | Q(s_{n-1})|).
\end{aligned}
\end{equation}
Since
\begin{gather*}
M = |y(c)| = \Big|\int_{a}^{c} y'(s_1) ds_1\Big| \le
\int_{a}^{c}|y'(s_1)| ds_1,\\
M = |y(c)| = \Big|\int_{c}^{b} y'(s_1) ds_1\Big| \le
\int_{c}^{b}|y'(s_1)| ds_1,
\end{gather*}
it follows that
\[
2M \le \int_{a}^{b} |y'(s_1)|ds_1.
\]
First, using H\"olders inequality with indices $p$ and
$p/(p-1)$ and then integrating by parts we obtain
\begin{equation}
\begin{aligned}
M^p &\le \big(\frac {1}{2}\big)^p\Big(\int_{a}^{b}|y'(s_1)|ds_1\Big)^p
 \\
& = \big(\frac {1}{2}\big)^p\Big(\int_{a}^{b}(r_1(s_1))^{1/p}
(r_1(s_1))^{-1/p} |y'(s_1)|ds_1\Big)^p \\
&\le \big(\frac {1}{2}\big)^p\Big(\int_{a}^{b}(r_1(s_1))^{1/(p-1)}ds_1
\Big)^{p-1}
\Big(\int_{a}^{b}(r_1(s_1))^{-1} |y'(s_1)|^pds_1\Big) \\
&= \big(\frac {1}{2}\big)^p\Big(\int_{a}^{b}(r_1(s_1))^{1/(p-1)}
ds_1\Big)^{p-1}
\Big([(r_1(s_1))^{-1}|y'(s_1)|^{p-2}y'(s_1) y(s_1)]_{a}^{b} \\
&\quad - \int_{a}^{b}[(r_1(s_1))^{-1}|y'(s_1)|^{p-2}y'(s_1)]' y(s_1)
ds_1\Big)\\
&= - \big(\frac {1}{2}\big)^p\Big(\int_{a}^{b}(r_1(s_1))^{1/(p-1)}
ds_1\Big)^{p-1} \int_{a}^{b}(Dy)'(s_1)y(s_1)ds_1
\\
&\le \big(\frac {1}{2}\big)^p\Big(\int_{a}^{b}(r_1(s_1))^{1/(p-1)}
ds_1\Big)^{p-1}
\int_{a}^{b}|(Dy)'(s_1)||y(s_1)|ds_1.
\end{aligned} \label{e14}
\end{equation}
Using  \eqref{e13},
\begin{align*}
M^p &< \big(\frac{1}{2}\big)^p\Big(\int_{a}^{b}(r_1(s_1))
 ^{1/(p-1)}ds_1\Big)^{p-1}\\
&\quad\times \Big[M^p\int_{a}^{b}\overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}),
\dots ,r_{n-1}(s_{n-2}); W(s_{n-1}, M))ds_1\\
&\quad + M \int_{a}^{b}\overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}),
\dots ,r_{n-1}(s_{n-2});| Q(s_{n-1})|)ds_1\Big];
\end{align*}
that is,
\begin{align*}
1 &< \big(\frac{1}{2}\big)^p\Big(\int_{a}^{b}(r_1(s_1))^{1/(p-1)}
 ds_1\Big)^{p-1}\\
&\quad\times \Big[\int_{a}^{b}\overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}),
 \dots ,r_{n-1}(s_{n-2}); W(s_{n-1}, M))ds_1\\
 &\quad + \frac{1}{M^{p-1}} \int_{a}^{b}\overline{E}(s_1, r_{2}(s_1),
 r_{3}(s_{2}), \dots , r_{n-1}(s_{n-2});| Q(s_{n-1})|)ds_1\Big].
\end{align*}
When $ n = 2 $,  \eqref{e10} has the form
\[
(Dy)'(t) + y(t) f(t, y(t))|y(t)|^{p-2} = Q(t).
\]
Hence \eqref{e14} yields
\[
M^p <
\big(\frac{1}{2}\big)^p\Big(\int_{a}^{b}(r_1(s_1))^{1/(p-1)}ds_1\Big)^{p-1}\Big[\int_{a}^{b}|y(t)|^p
|f(t, y(t))|dt + \int_{a}^{b}|y(t)||Q(t)|dt \Big];
\]
that is,
\[
1 <
\big(\frac{1}{2}\big)^p\Big(\int_{a}^{b}(r_1(t))^{1/(p-1)}dt\Big)^{p-1}\Big[\int_{a}^{b}W(t,M)
dt
 + \frac{1}{M^{p-1}}\int_{a}^{b}|Q(t)|dt\Big ].
\]
Thus the proof is complete.
\end{proof}

\subsection*{Remarks}
If $r_{i}(t) = 1; i = 1, 2, \dots , n-1$; $p = 2$;
$f(t, y) = p(t)$ and $n = 2, 3$; then inequalities \eqref{e12}
and \eqref{e11} reduce respectively, to the inequalities \eqref{e2} and
\[
\int_{a}^{b}|p(t)|dt > 4/(b-a)^{2}.
\]
This inequality provides a lower bound of the distance between
consecutive zeros of the solution $y(t)$. For the various
applications of this inequality one can see \cite{Parhi1}.

 Liapunov-type integral inequalities for \eqref{e8} can be obtained
under suitable assumptions on $g$ and $h$.

 If $r_{i}(t) = 1$; $i = 1, 2, \dots ,n-1$; $n = 3$, $p = 2$,
$f(t, y) = q(t)|y(t)|^{\beta - 1}$ and $Q(t) = 0$, then \eqref{e10}
reduces to
\begin{equation} \label{e15}
y'''(t) + q(t)|y(t)|^{\beta - 1}y = 0,\quad t \ge 0,
\end{equation}
where $\beta$ is a positive constant and $q : [0, \infty) \to
[0,\infty) $ is a continuous function is called an
\emph{Emden-Fowler} equations of third order. If $y(t)$ is a solution
of \eqref{e15} with $y(a) = 0 = y(b)$, $(a < b)$ and $y(t) \ne 0$ for
 $t \in (a, b)$,
then the spacing between zeros of solutions of
\eqref{e15} may be computed by using \eqref{e11}.

\begin{example} \label{exa1} \rm
Consider
\begin{equation} \label{e16}
y'''(t) + y^{2}(t) = \sin^{2}t - \cos t,\quad t \ge 0.
\end{equation}
Clearly, $y(t) = \sin t$ is a solution of \eqref{e16} with
$ y(0) = 0 = y(\pi)$, $y''(0) = 0 = y''(\pi)$.
$M =  \max_ {t \in [0, \pi]} | \sin t |= 1$.
 From Theorem \ref{thm1} it follows that
\[
1 < \frac{\pi}{4}\int_{0}^{\pi}[\overline{E}(s_1, r_{2}(s_1),
W(s_{2}, M)) + \frac{1}{M} \overline{E}(s_1,
r_{2}(s_1),|Q(s_{2})|)]ds_1,
\]
where
\begin{gather*}
\overline{E}(s_1, r_{2}(s_1), W(s_{2}, M)) =
\Big|\int_{0}^{s_1}M ds_{2}\Big|
= \begin{cases}
s_1, & s_1 > 0,\\
 - s_1, & s_1 < 0,
\end{cases}
\\
\overline{E}(s_1, r_{2}(s_1),|Q(s_{2})|)
= \Big|\int_{0}^{s_1}\Big|\sin^{2}s_{2} - \cos s_{2}\Big| ds_{2}\Big|
= \begin{cases}
2s_1, & s_1 > 0 ,\\
 - 2s_1, & s_1 < 0 .
\end{cases}
\end{gather*}
Hence
\begin{gather*}
\int_{0}^{\pi}\overline{E}(s_1, r_{2}(s_1), W(s_{2}, M))ds_1
=\begin{cases}
\pi^{2}/2, & s_1 > 0,\\
 - \pi^{2}/2, & s_1 < 0,
\end{cases}
\\
\int_{0}^{\pi}\overline{E}(s_1, r_{2}(s_1),|Q(s_{2})|)ds_1 =
\begin{cases}
\pi^{2}, &  s_1 > 0 ,\\
 - \pi^{2}, & s_1 < 0 .
\end{cases}
\end{gather*}
As $\overline{E} > 0$, then $s_1 > 0$ and
\begin{gather*}
\int_{0}^{\pi}\overline{E}(s_1, r_{2}(s_1), W(s_{2}, M))ds_1 =
\pi^{2}/2,
\\
\int_{0}^{\pi}\overline{E}(s_1, r_{2}(s_1),|Q(s_{2})|)ds_1 =
\pi^{2}.
\end{gather*}
Thus by Theorem \ref{thm1},
$1 < 3\pi^{3}/8 $ or $ 3\pi^{3} > 8 $, which is obviously true.
\end{example}

\begin{theorem} \label{thm2}
Suppose that {\rm (C1)-(C2)} hold.
 Let $ \alpha_1, \alpha_{2},\dots , \alpha_{n-3},
\alpha_{n-2}$ be the zeros of $D^{2}y(t), D^{3}y(t),
\dots ,D^{n-2}y(t), D^{n-1}y(t)$ respectively, in
$ [a, b] \subset I(a < b)$, where $y(t)$ is a nontrivial solution of
\[
D^ny + y f(t, y)|y(t)|^{p-2} = 0
\]
with $y(a) = 0 = y(b)$. If $c$ is a point in $(a, b)$, where
$|y(t)|$ attains a maximum, then the point `$c$' cannot be very close
to `$a$' as well as `$b$'.
\end{theorem}

\begin{proof}
Let $ M = \max\{|y(t)|: t\in [a, b]\} = |y(c)|$. Then
$y'(c) = 0$. Since
\[
y(c) = \int_{a}^{c}y'(t)dt,
\]
using H\"olders inequality with indices $p$ and $p/(p-1)$ and
then integrating by parts we obtain
\begin{align*}
&M^p \\
&\le \big(\frac {1}{2}\big)^p\Big(\int_{a}^{c}|y'(t)|dt\Big)^p\\
& = \big(\frac {1}{2}\big)^p\Big(\int_{a}^{c}r_1(t)^{1/p}
  r_1(t)^{-1/p}|y'(t)|dt\Big)^p \\
&\le \big(\frac {1}{2}\big)^p\Big(\int_{a}^{c}r_1(t)^{1/(p-1)}\Big)^{p-1}
 \Big(\int_{a}^{c} r_1(t)^{-1}|y'(t)|^pdt\Big) \\
&= \big(\frac {1}{2}\big)^p\Big(\int_{a}^{c}r_1(t)^{1/(p-1)}\Big)^{p-1}
 \Big(\Big[r_1(t)^{-1}|y'(t)|^{p-2}y'(t)y(t)\Big]_{a}^{c}
 - \int_{a}^{c}(Dy)'(t)y(t)dt\Big) \\
&\le  \big(\frac {1}{2}\big)^p\Big(\int_{a}^{c}r_1(t)^{1/(p-1)}\Big)^{p-1}
\Big(\int_{a}^{c}|(Dy)'(t)||y(t)|dt \Big).
\end{align*}
Proceeding as Theorem \ref{thm1} we obtain
\[
|(Dy)'(t)| \le M^{p-1} \overline{E}(t, r_{2}(t), r_{3}(s_{2}),
\dots ,r_{n-1}(s_{n-2}); W(s_{n-1}, M)).
\]
Hence
\begin{align*}
1 &< \big(\frac {1}{2}\big)^p\Big(\int_{a}^{c}r_1(t)^{1/(p-1)}\Big)^{p-1}\\
&\quad\times \Big(\int_{a}^{c}\overline{E}(t, r_{2}(t), r_{3}(s_{3}),
\dots ,r_{n-1}(s_{n-2}); W(s_{n-1}, M)) dt\Big);
\end{align*}
that is,
\begin{equation} \label{e17}
\begin{aligned}
&\Big[\Big(\int_{a}^{c}r_1(t)^{1/(p-1)}\Big)^{p-1}\Big]^{-1} \\
&< \big(\frac{1}{2}\big)^p \Big(\int_{a}^{c}\overline{E}(t,
r_{2}(t), r_{3}(s_{2}), \dots ,r_{n-1}(s_{n-2}); W(s_{n-1}, M))dt\Big)
<\infty.
\end{aligned}
\end{equation}
Thus `$c$' cannot be very close to `$a$' because
\[
\lim _{c \to
a+}\Big[\Big(\int_{a}^{c}r_1(t)^{1/(p-1)}\Big)^{p-1}\Big]^{-1}
= \infty.
\]
Next we have to show that `$c$' cannot be very close to  `$b$'.
Since
\[
|y(c)| = \Big|\int_{a}^{c}y'(t)dt\Big|,
\]
then proceeding as above to obtain
\begin{align*}
M^p &\le \big(\frac {1}{2}\big)^p\Big(\int_{c}^{b}|y'(t)|dt\Big)^p
\\
&= \big(\frac {1}{2}\big)^p\Big(\int_{c}^{b}r_1(t)^{1/(p-1)}\Big)^{p-1}
\Big(\Big[\int_{c}^{b}
r_1(t)^{p-1}|y'(t)|^{p-2}y'(t)y(t)\Big]_{c}^{b}\\
&\quad - \int_{c}^{b}(Dy)'(t)y(t)dt\Big) \\
&\le  \big(\frac {1}{2}\big)^p\Big(\int_{c}^{b}r_1(t)^{1/(p-1)}\Big)^{p-1}
\int_{c}^{b}|(Dy)'(t)||y(t)|dt \\
&< M^p \big(\frac {1}{2}\big)^p\Big(\int_{c}^{b}r_1(t)^{1/(p-1)}\Big)^{p-1}\\
&\quad \times \Big(\int_{c}^{b} \overline{E}(t, r_{2}(t), r_{3}(s_{2}),
\dots ,r_{n-1}(s_{n-2});W(s_{n-1}, M))dt\Big).
\end{align*}
Hence
\begin{align*}
&\Big[\Big(\int_{c}^{b}r_1(t)^{1/(p-1)}\Big)^{p-1}\Big]^{-1} \\
&<\big(\frac{1}{2}\big)^p \Big(\int_{c}^{b}\overline{E}(t,
r_{2}(t), r_{3}(s_{2}), \dots ,r_{n-1}(s_{n-2}); W(s_{n-1}, M))dt\Big)
<\infty.
\end{align*}
Thus `$c$' cannot be very close to `$b$' because
\[
\lim _{c \to b-}\Big[\Big(\int_{c}^{b}r_1(t)^{1/(p-1)}\Big)^{p-1}\Big]^{-1}
= \infty.
\]
This completes the proof of the theorem.
\end{proof}

We remark that Theorem \ref{thm2} need not hold if
$\alpha_{i} \notin [a, b] $ for some $ i \in \{1, 2, \dots , n-2\}$.

\section{Applications}

    In this section we present some of the applications of
the Liapunov-type inequality obtained in Theorem \ref{thm1} to study
the asymptotic behaviour of oscillatory solution of \eqref{e10}.

\noindent\textbf{Definition.}
   A solution $y(t)$ of \eqref{e10} is
said to be \emph{oscillatory} if there exists a sequence
$<t_{m}>\subset [0,\infty)$ such that  $y(t_{m}) = 0$, $m \ge 1$ and
$t_{m} \to \infty $ as
$ m \to \infty $.

\begin{theorem} \label{thm3}
Suppose that {\rm (C1)-(C2)} hold. Let
$W(t,\lambda) \in L^{\sigma}([0,\infty), \mathbb{R}^{+})$
for all $\lambda > 0$, where $ 1 \le \sigma < \infty$. Let
$r_{i}(t) \le K $ for $t \ge 0 $ and $1 \le i \le n - 1$, where
$K > 0 $ is a constant. If $< t_{m} > $ is an increasing sequence
of zeros of an oscillatory solution $y(t)$ of
\[
D^ny + yf(t, y)|y(t)|^{p-2} = 0 \quad t \ge 0,
\]
such that $\alpha_1, \alpha_{2}, \dots ,\alpha_{n-2} \in (t_{m}, t_{m
+ k})$, $1 \le k \le n-1$, for every large $ m $, then
$ (t_{m + k} - t_{m}) \to \infty$, as $ m \to \infty$, where
$\alpha_1, \dots ,\alpha_{n-2} $ are the zeros of
$ D^2y(t)$, $D^{3}y(t)$, \dots, $D^{n-2}y(t)$, $D^{n-1}y(t)$,
respectively.
\end{theorem}

\begin{proof}
If possible, let there exist a subsequence $ \langle t_{m_{i}} \rangle $
of $ \langle t_{m} \rangle$ such that $(t_{m_{i}+k} - t_{m_{i}})\le M $
for every $i$, where $ M > 0 $ is a constant. Let
$M_{m_{i}} = \max\{|y(t)|: t \in [t_{m_{i}}, t_{m_{i}+k}]\} =
|y(s_{m_{i}})|$, where $s_{m_{i}} \in (t_{m_{i}}, t_{m_{i}+k})$.
Since $ W(t, \lambda) \in L^{\sigma}([0, \infty), \mathbb{R}^{+})$ for all
$\lambda> 0 $, then
\[
\int_{0}^{\infty}W^{\sigma}(t, \lambda)dt < \infty, \quad
\text{for all } \lambda > 0.
\]
Hence
\[
\int_{t}^{\infty}W^{\sigma}(t, \lambda) dt \to 0\quad\text{as }
t \to \infty.
\]
Thus, for $1 < \sigma <\infty$, we may have
\[
\int_{t_{m_i}}^{\infty}W^{\sigma}(t, \lambda)dt  <  [K^{n-1}M^{n - 1
+ \frac{1}{\mu}}]^{-1}
\]
for large $i$, where $\frac{1}{\mu} + \frac{1}{\sigma} = 1$. From
\eqref{e17} we obtain
\[
\Big[\int_{t_{m_i}}^{s_{i}}((r_1(t)^{1/(p-1)})^{p-1}\Big]^{-1} <
\big(\frac{1}{2}\big)^p K^{n-2}\big(t_{m_{i}+k} -
t_{m_{i}}\big)^{n-2} \int_{t_{m_{i}}}^{t_{m_{i}+k}}W(t,
M_{m_{i}})dt;
\]
that is,
\[
1 < \big(\frac{1}{2}\big)^p K^{n-1}\big(t_{m_{i}+k}
- t_{m_{i}}\big)^{n-1} \int_{t_{m_{i}}}^{t_{m_{i}+k}}W(t,
M_{m_{i}})dt.
\]
The use of  H\"older's inequality yields
\begin{align*}
1 &< \big(\frac{1}{2}\big)^pK^{n-1}\big(t_{m_{i}+k} -
t_{m_{i}}\big)^{n-1} \big(t_{m_{i}+k} - t_{m_{i}}\big)^{1/\mu}
\Big[\int_{t_{m_{i}}}^{t_{m_{i}+k}}W^{\sigma}(t, M_{m_{i}})dt\Big]
^{1/\sigma} \\
&\le \big(\frac{1}{2}\big)^pK^{n-1}\big(t_{m_{i}+k} -
t_{m_{i}}\big)^{n-1 + \frac {1}{\mu}}
\Big[\int_{t_{m_{i}}}^{\infty}W(t, M_{{m_{i}}})dt\Big]^{1/\sigma}
 \\
&< \big(\frac{1}{2}\big)^pK^{n-1}M^{{n-1} + {\frac{1}{\mu}}}
\Big[K^{n-1}M^{n-1+{\frac{1}{\mu}}}\Big]^{-1} = {\frac
{1}{2^p}}, .
\end{align*}
a contradiction. For $\sigma = 1$, we can choose $i$ large enough
such that
\[
\int_{t_{m_{i}}}^{\infty}W(t, M_{m_{i}}) < [ K^{n-1}M^{n-1}]^{-1}
\]
and
\begin{align*}
1 &< \big(\frac{1}{2}\big)^pK^{n-1}(t_{{m_{i} + k}} -
t_{{m_{i}}})^{n-1}\int_{t_{m_{i}}}^{t_{m_{i} + k }}
W(t, M_{m_{i}})dt\\
&< \big(\frac{1}{2}\big)^p K^{n-1} M^{n-1}[K^{n-1} M^{n-1}]^{-1}
= {\frac{1}{2^p}},{\hspace*{0.55in}}
\end{align*}
a contradiction.
 Hence the Theorem is proved.
\end{proof}

\begin{theorem} \label{thm4}
 Suppose that {\rm (C1)-(C2)} hold with $I = [0, \infty)$.
Let there exist a continuous function
$H: I \to \mathbb{R}^{+}$ such that $W(t,L) \le H(t)$ for every constant
$ L > 0 $. Let
\[
\int_{0}^{\infty}r_1(t)^{1/(p-1)}ds_1 < \infty.
\]
If
\begin{gather*}
\int_{0}^{\infty}\overline{E}(t, r_{2}(t), r_{3}(s_{2}), \dots ,
r_{n-1}(s_{n-2}); |Q(s_{n-1})|)dt < {\infty}, \\
\int_{0}^{\infty}\overline{E}(t, r_{2}(t), r_{3}(s_{2}), \dots ,
r_{n-1}(s_{n-2}); H(s_{n-1}))dt
 < \infty,
\end{gather*}
for $ n \ge 3 $, and
\[
\int_{0}^{\infty}H(t)dt < \infty, \quad
\int_{0}^{\infty}|Q(t)|dt < \infty
\]
for $n = 2$;
then every oscillatory solution of \eqref{e10} converges to zero
 as $t \to \infty$.
\end{theorem}

\begin{proof}
Let $y(t)$ be an oscillatory solution of \eqref{e10} on
$[T_{y} , \infty),T_{y} \ge 0 $. Hence
$\lim  inf_{t \to  \infty}|y(t)| = 0 $. To complete the proof of the
theorem it is sufficient to
show that $lim sup_ {t \to  \infty}|y(t)| = 0$. If possible, let
$lim sup_{t \to \infty}|y(t)| = \lambda > 0$. Choose
$0 < d < \lambda/2$. From the given assumptions it follows that it is
possible to choose a large $T_{0} > 0 $ such that, for
$t \ge T_{0}$,
\begin{gather*}
\int_{t}^{\infty}r_1(s_1)^{1/(p-1)}ds_1 < 2^{p/(p-1)}, \\
\int_{t}^{\infty}\overline{E}(s_1, r_{2}(s_1),
r_{3}(s_{2}),\dots ,r_{n-1}(s_{n-2}); |Q(s_{n-1})|)ds_1 < d^{p-1}, \\
\int_{t}^{\infty}\overline{E}(s_1, r_{2}(s_1),
r_{3}(s_{2}),\dots ,r_{n-1}(s_{n-2}); H(s_{n-1}))ds_1 < 1
\end{gather*}
for $ n \ge 3 $, and
\[
 \int_{t}^{\infty}H(s) ds < d^{p-1}, \quad
 \int_{t}^{\infty}|Q(s)|ds < d \]
for $ n = 2 $. Since $y(t)$ is
oscillatory, we can find a $t_1 > T_{0}$ such that $y(t_1) = 0$.
Let $T_{0}^{*} > t_1 $ be such that $ \alpha_1, \alpha_{2}, \dots ,
\alpha_{n-3}, \alpha_{n-2} \in [t_1, T_{0}^{*}]$, where
$\alpha_1, \alpha_{2}, \dots , \alpha_{n-3}, \alpha_{n-2} $ are the
zeros, respectively, of $ D^{2}y(t), \dots ,D^{n-2}y(t)$. Further,
$\limsup_{t \to \infty}|y(t)| > 2d$ implies that we can find a
$T^{**} > t_1$ such that $\sup\{|y(t): t \in [t_1, T_{0}^{**}]\}
> d$. Let $T_1 = max\{T_{0}^{*}, T_{0}^{**} \}$.
 Let $t_{2} > T_1$
 such that $y(t_{2}) = 0$. Let $M = \max\{|y(t)|: t\in [t_1, t_{2}]\}$,
then $ M > d$. From Theorem \ref{thm1} we obtain \eqref{e11}
 for $n \ge 3$ and \eqref{e12} for $ n = 2$, with $ a = t_1$ and
$b = t_{2}$. Hence, For $ n \ge 3$,
\begin{align*}
1 &< \big(\frac {1}{2}\big)^p\Big(\int_{t_1}^{\infty}((
r_1(s_1))^{1/(p-1)}ds_1\Big)^{p-1}\\
&\quad\times \int_{t_1}^{\infty} \Big[\overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}),
 \dots , r_{n-1}(s_{n-2}); H(s_{n-1})) \\
&\quad +  \frac {1}{M^{p-1}}\overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}),
 \dots , r_{n-1}(s_{n-2});|Q(s_{n-1})|)\Big] ds_1\\
&< \big(\frac{1}{2}\big)^p\big(2^{p/(p-1)}\big)^{p-1}\big[1 +
\big(\frac{d}{M}\big)^{p-1}\big] < 2,
\end{align*}
a contradiction. Hence $\limsup _{t\to \infty}|y(t)| = 0$.
Thus the proof of the theorem is complete.
\end{proof}

\begin{example} \label{exa2} \rm
 Consider
\begin{equation} \label{e18}
(e^{t}(e^{t}y^{2}y')')' + y^{3} = e^{-4t}(8cos^{3}t + 13 sin^{3}t +
10 \cos t - 6 \sin t ) + e^{-6t} \sin^{3}t,
\end{equation}
where $t \ge 0$.
Thus $ r_1(t) = e^{-t}$, $r_{2}(t) = e^{-t}$, $f(t, y) = 1$,  and
hence $H(t) = 1$. Clearly, $y(t) = e^{-2t}\sin t$ is a solution of
\eqref{e18} with $y(0) = 0 $ and $(e^{t}y^{2}(t)y'(t))' = 0$ for
$t = 0,\pi$. Hence $ \alpha_1 = 0, \pi$. Let $ \alpha_1 = 0$.
 Since
\begin{gather*}
\overline{E}(s_1, r_{2}(s_1); H(s_{2})) = s_1e^{-s_1}\quad
\text{for } s_1 > 0,
\\
\overline{E}(s_1, r_{2}(s_1); |Q(s_{2})|) \le 38
s_1e^{-s_1}\quad \text{for } s_1 > 0,
\end{gather*}
it follows that
\begin{gather*}
\int_{0}^{\infty}\overline{E}(s_1, r_{2}(s_1); H(s_{2}))ds_1 =
1, \\
\int_{0}^{\infty}\overline{E}(s_1, r_{2}(s_1); |Q(s_{2})|)ds_1
\le 38.
\end{gather*}
Again taking $\alpha_1 = \pi$, we obtain
\begin{gather*}
\overline{E}(s_1, r_{2}(s_1); H(s_{2})) = (s_1 -
\pi)e^{-s_1}\quad\text{for } s_1 > \pi,\\
\overline{E}(s_1, r_{2}(s_1); |Q(s_{2})|) \le 38( s_1 -
\pi)e^{-s_1}\quad\text{for } s_1\,.
> \pi,
\end{gather*}
Then
\begin{gather*}
\int_{\pi}^{\infty}\overline{E}(s_1, r_{2}(s_1); H(s_{2}))ds_1
= e^{-\pi}, \\
\int_{\pi}^{\infty}\overline{E}(s_1, r_{2}(s_1);
|Q(s_{2})|)ds_1 \le 38e^{-\pi}.
\end{gather*}
From Theorem \ref{thm4} it follows that every oscillatory solution of \eqref{e18}
tends to zero as t tends to infinity.
\end{example}

\begin{theorem} \label{thm5}
 If
\begin{equation} \label{e19}
\begin{aligned}
&\big(\frac{1}{2}\big)^p\Big(\int_{a}^{b}r_1(s_1)^{1/(p-1)}ds_1\Big)^{p-1}\\
&\times \int_{a}^{b} \overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}),
\dots ,r_{n-1}(s_{n-2}); |p(s_{n-1})|ds_1 \le 1,
\end{aligned}
\end{equation}
then
\begin{equation} \label{e20}
D^ny + p(t)y|y|^{p-2} = 0
\end{equation}
is disconjugate on $[a, b]$, where $p(t)$ is a real-valued continuous
function on $[a, b]$.
\end{theorem}

\noindent\textbf{Definition.}
Equation \eqref{e20} is said to be
disconjugate in $[a, b]$ if no non-trivial solution of \eqref{e20}
has more than $n-1$ zeros (counting multiplicities).

\begin{proof}[Proof of Theorem \ref{thm5}]
Indeed, if \eqref{e20} is not disconjugate on $[a, b]$,
then it admits a nontrivial solution $y(t)$ has $n$ zeros in
$[a, b]$. Let these zeros be given by $ a \le a_1 < a_{2} <\dots < a_{n-1}
< a_{n} \le b$. Then $D^{2}y(t), D^{3}y(t), \dots ,D^{n-1}y(t)$ have
zeros in $[a_1, a_{n}]$. From Theorem \ref{thm1}, it follows that
\begin{align*}
1 &< \big(\frac{1}{2}\big)^p\Big(\int_{a_1}^{a_n}
  r_1(s_1)^{1/(p-1)}ds_1\Big)^{p-1}\\
&\quad\times \int_{a_1}^{a_{n}} \overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}), \dots ,
r_{n-1}(s_{n-2}); |p(s_{n-1}|)ds_1\\
&\le
\big(\frac{1}{2}\big)^p\Big(\int_{a}^{b}r_1(s_1)^{1/(p-1)}ds_1\Big)^{p-1}\\
&\quad\times \int_{a}^{b} \overline{E}(s_1, r_{2}(s_1), r_{3}(s_{2}),
\dots ,r_{n-1}(s_{n-2}); |p(s_{n-1}|)ds_1,
\end{align*}
a contradiction. Hence \eqref{e20} is disconjugate on $[a, b]$.
\end{proof}

\noindent\textbf{Remark.}
 If $r_{i}(t) = 1; i = 1, 2, \dots ,n - 1$; $p = 2, n = 3$,
then \eqref{e19} reduces to
\[
\int_{a}^{b}|p(t)|dt \le 4/(b-a)^{2}.
\]
Thus the above inequality may be regarded as a sufficiency
condition for the  disconjugacy of the equation \eqref{e7}.

As a final remark, we note that the
results obtained in this paper generalize the results by
Pachpatte \cite{pach}.

\subsection*{Acknowledgements} The author would like to
thank the anonymous referee for his/her valuable suggestions.

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\end{thebibliography}

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