\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 139, pp. 1--14.\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-2017/139\hfil Lyapunov-type inequalities]
{Lyapunov-type inequalities for third-order linear differential equations}
\author[M. F. Akta\c{s}, D. \c{C}akmak \hfil EJDE-2017/139\hfilneg]
{Mustafa Fahri Akta\c{s}, Devrim \c{C}akmak}
\dedicatory{Dedicated to the memory of Ayd{\i}n Tiryaki (June 1956 - May 2016)}
\address{Mustafa Fahri Akta\c{s} \newline
Gazi University,
Faculty of Sciences,
Department of Mathematics,
06500 Teknikokullar, Ankara, Turkey}
\email{mfahri@gazi.edu.tr}
\address{Devrim \c{C}akmak \newline
Gazi University,
Faculty of Education,
Department of Mathematics Education,
06500 Teknikokullar, Ankara, Turkey}
\email{dcakmak@gazi.edu.tr}
\thanks{Submitted January 5, 2017. Published May 24, 2017.}
\subjclass[2010]{34C10, 34B05, 34L15}
\keywords{ Lyapunov-type inequalities; Green's Functions;
\hfill\break\indent three-point boundary conditions}
\begin{abstract}
In this article, we establish new Lyapunov-type inequalities for third-order
linear differential equations
\[
y'''+q( t) y=0
\]
under the three-point boundary conditions
\[
y( a) =y( b) =y( c) =0
\]
and
\[
y( a) =y''( d) =y( b) =0
\]
by bounding Green's functions $G(t,s)$ corresponding to appropriate boundary
conditions. Thus, we obtain the best constants of Lyapunov-type inequalities
for three-point boundary value problems for third-order linear differential
equations in the literature.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
Lyapunov \cite{lyapunov} obtained the remarkable result:
If $ q\in C( [0,\infty ),\mathbb{R})$ and $y$ is a nontrivial solution of
\begin{equation}
y''+q( t) y=0 \label{7}
\end{equation}
under the Dirichlet boundary conditions
\begin{equation}
y( a) =y( b) =0 \label{m1}
\end{equation}
and $y( t) \not\equiv 0$ for $t\in ( a,b) $, then
\begin{equation}
\frac{4}{b-a}\leq \int_{a}^{b}| q( s) | ds\,. \label{6}
\end{equation}
Thus, this inequality provides a lower bound for the distance between two
consecutive zeros of $y$. The inequality \eqref{6} is the best possible in
the sense that if the constant 4 in the left hand side of \eqref{6} is
replaced by any larger constant, then there exists an example of \eqref{7}
for which \eqref{6} no longer holds
(see \cite[p. 345]{Hartman:1982}, \cite[p. 267]{Kelley}).
In this paper, our aim is to obtain the best constants of
Lyapunov-type inequalities for third-order linear differential equations
with three-point boundary conditions. The above result of
Lyapunov has found many applications in areas like eigenvalue problems,
stability, oscillation theory, disconjugacy, etc. Since then, there have
been several results to generalize the above linear equation in many
directions; see the references.
There are various methods used to obtain Lyapunov-type inequalities for
different types of boundary value problems. One of the most useful methods
is as follows: Nehari \cite{Nehari} started with the Green's function of the
problem \eqref{7} with \eqref{m1}, which is
\begin{equation}
G( t,s) =-\begin{cases}
\frac{( t-a) ( b-s) }{b-a}, & a\leq s\leq t, \\[4pt]
\frac{( s-a) ( b-t) }{b-a}, & t\leq s\leq b,
\end{cases} \label{55}
\end{equation}
and he wrote
\begin{equation}
y( t) =\int_{a}^{b}G( t,s) q( s) y(
s) ds. \label{56}
\end{equation}
Then by choosing $t=t_{0}$, where $|y(t)|$ is maximized and canceling out
$|y(t_{0})|$ on both sides, he obtained
\begin{equation}
1\leq \max_{a\leq t\leq b} \int_{a}^{b}|G(t,s)|| q(s) | ds. \label{57}
\end{equation}
Note that if we take the absolute maximum value of the function
$|G(t,s)|$ for all $t,s\in [ a,b] $ in \eqref{57}, then we obtain the
inequality \eqref{6}. Following the ideas of these papers, this method has
been applied in a huge number of works to different second and higher order
ordinary differential equations with different types of boundary conditions.
We see that by bounding the Green's function $G(t,s)$ in various ways, we
can obtain the best constants in the Lyapunov-type inequalities in other
differential equations with associated boundary conditions as well. Thus, we
obtain the best constants of the Lyapunov-type inequalities for three-point
boundary value problems for third-order linear differential equations by
using the absolute maximum values of the Green's functions $G(t,s)$ in the
literature.
In this article, we consider the third-order linear differential equation of
the form
\begin{equation}
y'''+q( t) y=0, \label{1}
\end{equation}
where $q\in C( [0,\infty ),\mathbb{R}) $ and $y( t) $ is a real solution of
\eqref{1} satisfying the three-point boundary conditions
\begin{equation}
y( a) =y( b) =y( c) =0 \label{200}
\end{equation}
and
\begin{equation}
y( a) =y''( d) =y( b) =0
, \label{z7}
\end{equation}
$a,b,c,d\in \mathbb{R}$ with $a**0\,. \label{169}
\end{gather}
\end{theorem}
\begin{theorem}[{\cite[Theorem 2.2]{Dhar1}}] \label{thmM}
Assume that $y( t) $ is a nontrivial solution of \eqref{174}
with \eqref{175}.
\begin{itemize}
\item[(a)] If $y(b)=0$ for $b\in ( a,c) $
and $y( t) \neq 0$ for $t\in [a,b)\cup (b,c]$, then one of the
following holds:
\begin{itemize}
\item[(i)] $\frac{( 2n-1) !2^{2n}}{(c-a)^{2n}S_{n}}<\int_{a}^{c}q^{-}( s) ds$
\item[(ii)] $\frac{( 2n-1) !2^{2n}}{(c-a)^{2n}S_{n}}<\int_{a}^{c}q^{+}( s) ds$
\item[(iii)] $\frac{( 2n-1) !2^{2n}}{(c-a)^{2n}S_{n}}<\int_{a}^{b}q^{-}( s) ds
+\int_{b}^{c}q^{+}(s) ds$.
\end{itemize}
\item[(b)] If $y(a)=0$\textit{\ and }$y( t) \neq 0$
for $t\in (a,c]$, then
\begin{equation}
\frac{( 2n-1) !2^{2n}}{(c-a)^{2n}S_{n}}<\int_{a}^{c}q^{+}(
s) ds. \label{167}
\end{equation}
\item[(c)] If $y(c)=0$ and $y( t) \neq 0$
for $t\in \lbrack a,c)$, then
\begin{equation}
\frac{( 2n-1) !2^{2n}}{(c-a)^{2n}S_{n}}<\int_{a}^{c}q^{-}(s) ds, \label{166}
\end{equation}
where $q^{-}( t) $, $q^{+}( t) $,
and $S_{n}$ are given in \eqref{161}, \eqref{160}, and
\eqref{170}, respectively.
\end{itemize}
\end{theorem}
In this paper, we use Green's functions to obtain the best constants of
Lyapunov-type inequalities for the problems \eqref{1} with \eqref{200} or
\eqref{z7} in the literature. In addition, we obtain lower bounds for the
distance between two points of a solution of the problems \eqref{1}
with \eqref{200} or \eqref{z7}.
\section{Some preliminary lemmas}
We state important lemmas which we will use in the proofs of our main
results. In the following lemma, we construct Green's function for the third
order nonhomogeneous differential equation
\begin{equation}
y'''=g( t) \label{m34}
\end{equation}
with the three-point boundary conditions \eqref{200} inspired by Murty and
Sivasundaram \cite{Murty} as follows.
\begin{lemma} \label{lem2.1}
If $y( t) $ is a solution of \eqref{m34} satisfying $y( a) =y( b) =y( c) =0$
with $a**