\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 $a<b<c$\ and $a\leq d\leq b$ are three points 
and $y( t) \not\equiv 0$ for $t\in ( a,b) \cup ( b,c) $ and 
$t\in( a,b) $, respectively.

Some of the recent studies about Lyapunov-type inequalities for third and
higher order boundary value problems are as follows:
In 1999, Parhi and Panigrahi \cite{ParhiPanigrahi3} established the
inequalities similar to the classical Lyapunov inequality \eqref{6} for the
third-order linear differential equation \eqref{1} under the three-point
boundary conditions \eqref{200} and \eqref{z7} as follows.

\begin{theorem}[{\cite[Theorem 2]{ParhiPanigrahi3}}] \label{thmA} 
If $y( t) $ is a nontrivial solution of the problem \eqref{1} with \eqref{200}, 
then 
\begin{equation}
\frac{4}{(c-a)^{2}}<\int_{a}^{c}| q(s)|\, ds\,.  \label{199}
\end{equation}
\end{theorem}

\begin{theorem}[{\cite[Theorem 1]{ParhiPanigrahi3}}] \label{thmB}  
If $y( t)$ is a nontrivial solution of the problem \eqref{1} with \eqref{z7}, 
then 
\begin{equation}
\frac{4}{(b-a)^{2}}<\int_{a}^{b}| q(s)| \,ds\,.  \label{209}
\end{equation}
\end{theorem}

In 2010, Yang et al.\ \cite{YangKimLo} extended the inequality \eqref{209}
for the  third-order linear differential equation
\begin{equation}
( r_2( t) ( r_{1}( t) y'))') '+q(t)y=0.  \label{280}
\end{equation}

\begin{theorem}[{\cite[Theorem 1]{YangKimLo}}] \label{thmC} If 
$y( t) $ is a nontrivial solution of \eqref{280} satisfying the conditions
\begin{gather}
y( a) =y( b) =0,  \label{5} \\
( r_{1}( t) y'(t)) ' \text{ has a zero }d\in \lbrack a,b],  \label{85}
\end{gather}
then
\begin{equation}
\begin{aligned}
&\min_{a\leq t_{0}\leq b}\Big[ \Big( \int_{a}^{t_{0}}r_{1}( s)
ds\int_{a}^{t_{0}}r_2( s) ds\Big) ^{-1}
+\Big(\int_{t_{0}}^{b}r_{1}( s) ds\int_{t_{0}}^{b}r_2( s)ds\Big) ^{-1}\Big]\\
&<2\int_{a}^{b}| q(s)| \,ds\,,
\end{aligned}\label{268}
\end{equation}
 where $| y(t_{0})| =\max \{|y(t)| :a\leq t\leq b\}$.
\end{theorem}

In 2013, Kiselak \cite{Kiselak} extended the Lyapunov-type inequalities from
linear differential equation to the  third-order half-linear
differential equation
\begin{equation}
\begin{aligned}
&\Big( \frac{1}{r_2(t)}\Big| \Big( \frac{1}{r_{1}(t)}\big|y'(t)|
^{p_{1}-1}y'(t)\Big) '| ^{p_2-1}
\Big( \frac{1}{r_{1}(t)}| y'(t)| ^{p_{1}-1}y'(t)\Big) '\Big) '\\
&+q(t)| y(t)| ^{p_3-1}y(t)=0, 
\end{aligned} \label{253}
\end{equation}
where $0<p_{1},p_2$ and $p_3=p_{1}p_2$.

\begin{theorem}[{\cite[Theorem 2.1]{Kiselak}}] \label{thmD} 
If $y( t)$ is a nontrivial solution of \eqref{253}
satisfying the conditions \eqref{5}, and
\begin{equation}
\Big( \frac{1}{r_{1}(t)}| y'(t)|
^{p_{1}-1}y'(t)\Big) '\text{  has a zero }d\in \lbrack a,b],  \label{294}
\end{equation}
then 
\begin{equation}
\begin{aligned}
&\min_{a\leq t_{0}\leq b}\Big[ \Big( \int_{a}^{t_{0}}r_{1}^{1/p_{1}}(s) ds
\Big( \int_{a}^{t_{0}}r_2^{1/p_2}( s) ds\Big)
^{1/p_{1}}) ^{-1} \\
&+ \Big( \int_{t_{0}}^{b}r_{1}^{1/p_{1}}( s) ds\Big(
\int_{t_{0}}^{b}r_2^{1/p_2}( s) ds\Big) ^{1/p_{1}}\Big)
^{-1}\Big] 
<2\Big( \int_{a}^{b}| q(s)| ds\Big)^{1/p_3},
\end{aligned}\label{261}
\end{equation}
 where $| y(t_{0})| =\max \{|y(t)| :a\leq t\leq b\}$.
\end{theorem}

\begin{theorem}[{\cite[Theorem 2.2]{Kiselak}}] \label{thmE} 
If $y( t) $ is a nontrivial solution of the problem \eqref{253} with 
\eqref{200}, then 
\begin{equation}
\begin{aligned}
&\min_{t_{0}\in [ a,c] }\Big[ \Big(
\int_{a}^{t_{0}}r_{1}^{1/p_{1}}( s) ds(
\int_{a}^{t_{0}}r_2^{1/p_2}( s) ds) ^{1/p_{1}}\Big)^{-1}\\
&+\Big( \int_{t_{0}}^{c}r_{1}^{1/p_{1}}( s) ds\Big(
\int_{t_{0}}^{c}r_2^{1/p_2}( s) ds\Big) ^{1/p_{1}}\Big)
^{-1}\Big] \\
&<2\Big( \int_{a}^{c}| q(s)| ds\Big)^{1/p_3},
\end{aligned}  \label{198}
\end{equation}
 where $| y(t_{0})| =\max \{|y(t)| :a\leq t\leq c\}$.
\end{theorem}

In 2014, Dhar and Kong \cite{Dhar} obtained the following Lyapunov-type
inequalities for third-order half-linear differential equation \eqref{253}.

\begin{theorem}[{\cite[Theorem 2.5]{Dhar}}] \label{thmF} 
If $ y( t) $ is a nontrivial solution of \eqref{253} with \eqref{5} and 
\eqref{294}, then 
\begin{equation}
\begin{aligned}
&2^{p_2+p_3}\Big(\int_{a}^{b}r_{1}^{-1/p_{1}}( s) ds\Big)
^{-p_3}\Big( \int_{a}^{b}r_2^{-1/p_2}( s) ds\Big)
^{-p_2}\\
&<\int_{a}^{d}q^{-}(s)ds+\int_{d}^{b}q^{+}(s)ds, 
\end{aligned}  \label{65}
\end{equation}
where
\begin{gather}
q^{-}( t) =\max \{ -q( t) ,0\},  \label{161} \\
q^{+}( t) =\max \{ q( t) ,0\} . \label{160}
\end{gather}
\end{theorem}

From \eqref{161} and \eqref{160}, it is easy to see that 
$-q^{-}(t)\leq q(t)\leq q^{+}(t)$, $q^{-}(t)\leq |q(t)|,q^{+}(t)\leq |q(t)|$, and 
$|q(t)|=q^{+}(t)+q^{-}(t)$ for $t\in \lbrack a,b]$.

\begin{theorem}[{\cite[Theorem 2.6]{Dhar}}] \label{thmG} If 
$y( t)$ is a nontrivial solution of \eqref{253} with 
\eqref{200}, then 
\begin{equation}
\begin{aligned}
&2^{p_2+p_3}\Big( \int_{a}^{b}r_{1}^{-1/p_{1}}( s) ds\Big)^{-p_3}
\Big( \int_{a}^{b}r_2^{-1/p_2}( s) ds\Big)^{-p_2} \\
&<\max_{a\leq d\leq b} \Big[ \int_{a}^{d}q^{-}(s)ds+
\int_{d}^{b}q^{+}(s)ds\Big] , 
\end{aligned} \label{194}
\end{equation}
where $d\in \lbrack a,b]$  is given in \eqref{294}, or
\begin{equation}
\begin{aligned}
&2^{p_2+p_3}\Big(\int_{b}^{c}r_{1}^{-1/p_{1}}( s) ds\Big)
^{-p_3}\Big( \int_{b}^{c}r_2^{-1/p_2}( s) ds\Big)^{-p_2}\\
&<\max_{b\leq d\leq c}\Big[ \int_{b}^{d}q^{-}(s)ds+
\int_{d}^{c}q^{+}(s)ds\Big] , 
\end{aligned} \label{64}
\end{equation}
where $d\in \lbrack b,c]$  is given in \eqref{294}.
As a result,
\begin{equation}
\begin{aligned}
&2^{p_2+p_3}\Big( \int_{a}^{c}r_{1}^{-1/p_{1}}( s) ds\Big)^{-p_3}
\Big( \int_{a}^{c}r_2^{-1/p_2}( s) ds\Big)^{-p_2} \\
&<\max_{a\leq d\leq c} \Big[ \int_{a}^{d}q^{-}(s)ds
+\int_{d}^{c}q^{+}(s)ds\Big] ,
\end{aligned}  \label{63}
\end{equation}
where $d\in \lbrack a,c]$, 
$q^{-}( t) $,  and $q^{+}( t) $  are given in \eqref{294}, \eqref{161}
 and \eqref{160}, respectively.
\end{theorem}

In 2016, Dhar and Kong \cite{Dhar2} obtained the following result for
third-order linear differential equation \eqref{1}.

\begin{theorem}[{\cite[Theorem 2.1]{Dhar2}}] \label{thmH} 
If $y( t)$ is a nontrivial solution of \eqref{1} with \eqref{200}, 
then one of the following holds:
\begin{itemize}
\item[(a)] $2<\int_{a}^{c}( s-a) ( c-s)q^{-}( s) ds$

\item[(b)] $2<\int_{a}^{c}( s-a) ( c-s)q^{+}( s) ds$

\item[(c)] $2<\int_{a}^{b}( s-a) ( b-s)q^{-}( s) ds+\int_{b}^{c}( s-b) ( c-s)
q^{+}( s) ds$,
\end{itemize}
where $q^{-}( t) $  and $q^{+}(t) $ are given in \eqref{161} and
 \eqref{160}, respectively.
\end{theorem}

In 2003, Yang \cite{Yang} obtained the Lyapunov-type inequalities for the
following $( 2n+1) $-th order differential equations
\begin{equation}
y^{(2n+1)}+q(t)y=0  \label{333}
\end{equation}
for $n\in\mathbb{N}$ and $n$-th order differential equations
\begin{equation}
y^{(n)}+q(t)y=0  \label{197}
\end{equation}
for $n\geq 2$, $n\in \mathbb{N}$, as follows.

\begin{theorem}[{\cite[Theorem 1]{Yang}}] \label{thmI} 
If $y(t)$ is a nontrivial solution of \eqref{333} satisfying the conditions
\begin{equation}
y^{(i)}(a)=y^{(i)}(b)=0  \label{263}
\end{equation}
for $i=0,1,\dots ,n-1$  and
\begin{equation}
y^{( 2n) }(t)\text{ has a zero }d\in ( a,b),  \label{296}
\end{equation}
then
\begin{equation}
\frac{n!2^{n+1}}{(b-a)^{2n}}<\int_{a}^{b}| q(s)| ds\,.
\label{264}
\end{equation}
\end{theorem}

\begin{theorem}[{\cite[Theorem 2]{Yang}}] \label{thmJ} 
If $y(t)$  is a solution of \eqref{197} satisfying the conditions
\begin{equation}
y(a)=y(t_2)=\dots =y(t_{n-1})=y(b)=0,  \label{196}
\end{equation}
where $a=t_{1}<t_2<\dots <t_{n-1}<t_{n}=b$ $y(t)\neq 0$
 for $t\in (t_{k},t_{k+1})$, $k=1,2,\dots ,n-1$,
then 
\begin{equation}
\frac{( n-2) !n^{n-1}}{(b-a)^{n-1}( n-1) ^{n-2}}
<\int_{a}^{b}| q(s)| \,ds \,. \label{195}
\end{equation}
\end{theorem}

In 2010, \c{C}akmak \cite{Cakmak} obtained the following Lyapunov-type
inequality for  problem \eqref{197} with \eqref{196} by fixing the fault
in Theorem \ref{thmJ} given by Yang \cite{Yang}.

\begin{theorem}[{\cite[Theorem 1]{Cakmak}}] \label{thmK} 
If $y(t)$ is a nontrivial solution of \eqref{197} with \eqref{196}, then
\begin{equation}
\frac{( n-2) !n^{n}}{(b-a)^{n-1}( n-1) ^{n-1}}
<\int_{a}^{b}| q(s)| \,ds \,. \label{22}
\end{equation}
\end{theorem}

Recently, Dhar and Kong \cite{Dhar1} obtained Lyapunov-type inequalities for
odd-order linear differential equations
\begin{equation}
y^{(2n+1)}+( -1) ^{n-1}q(t)y=0  \label{174}
\end{equation}
for $n\in \mathbb{N}$.

\begin{theorem}[{\cite[Theorem 2.1]{Dhar1}}] \label{thmL} 
If $y( t) $ is a nontrivial solution of \eqref{174} satisfying the conditions
\begin{equation}
y^{(i+1)}(a)=y^{(i+1)}(c)=0  \label{175}
\end{equation}
for $i=0,1,\dots ,n-1$  and
\begin{equation}
y(b)=0\quad \text{for }b\in [ a,c] ,  \label{172}
\end{equation}
then 
\begin{equation}
\frac{( 2n-1) !2^{2n}}{(c-a)^{2n}S_{n}}<\int_{a}^{c}|q(s)| \,ds \,, \label{171}
\end{equation}
 where
\begin{gather}
S_{n}=\sum_{j=0}^{n-1}\sum_{k=0}^{j}2^{2k-2j}
\binom{n-1+j}{j}
\binom{j}{k} B( n+1,n+k-j),  \label{170} \\
B(\alpha,\beta) =\int_{0}^{1}z^{\alpha -1}( 1-z)
^{\beta -1}dz\quad  \text{the Beta function for }\alpha ,\beta >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 $a<b<c$  and $y( t) \not\equiv 0$
 for $t\in ( a,b) \cup ( b,c) $, then
\begin{equation}
y( t) =\int_{a}^{c}G_{c}( t,s) g( s) ds,  \label{193}
\end{equation}
 where, for $t\in [ a,b] $,
\begin{equation}
G_{c}( t,s) =\begin{cases}
G_{c1}( t,s) =  \frac{( s-a) ^{2}( b-t)
( c-t) }{2( b-a) ( c-a) }, & a\leq s<t\leq b<c \\[4pt]
G_{c2}( t,s) =  \frac{( t-a) ^{2}( b-s)
( c-s) }{2( b-a) ( c-a) } \\
 +\frac{(s-t) ( t-a) [ ( b-s) ( c-a)
+( c-s) ( b-a) ] }{2( b-a) (c-a) }, & a\leq t\leq s<b<c \\[4pt]
G_{c3}( t,s) =  \frac{( c-s) ^{2}( b-t)( t-a) }{2( c-a) ( c-b) }, &  a<t<b<s<c
\end{cases} \label{35}
\end{equation}
and for $t\in [ b,c] $,
\begin{equation}
G_{c}( t,s) =\begin{cases}
G_{c4}( t,s) =  -\frac{( s-a) ^{2}( t-b)
( c-t) }{2( b-a) ( c-a) },  &  a<s<b<t<c\\[4pt]
G_{c5}( t,s) =  -\frac{( c-t) ^{2}( s-b)
( s-a) }{2( c-a) ( c-b) } \\
 -\frac{(t-s) ( c-t) [ ( s-b) ( c-a)
+( s-a) ( c-b) ] }{2( c-a) (c-b) }, & a<b<s\leq t\leq c \\
G_{c6}( t,s) =  -\frac{( c-s) ^{2}( t-b)
( t-a) }{2( c-a) ( c-b) },  &  a<b\leq t<s\leq c.
\end{cases} \label{34}
\end{equation}
\end{lemma}

\begin{proof}
Integrating  \eqref{1} from $a$ to $t$ to find $y$, we obtain
\begin{gather}
y''( t) =d_2+\int_{a}^{t}g( s) \,ds, \label{m31} \\
y'( t) =d_{1}+d_2( t-a) +\int_{a}^{t}(t-s) g( s) \,ds,  \label{m32} \\
y( t) =d_{0}+d_{1}( t-a) +d_2\frac{( t-a)
^{2}}{2}+\int_{a}^{t}\frac{( t-s) ^{2}}{2}g( s) \,ds\,.  \label{m33}
\end{gather}
Thus, the general solution of \eqref{1} is \eqref{m33}.

Now, by using the boundary conditions \eqref{200}, we find the constants 
$d_{0}$, $d_{1}$, and $d_2$. Thus, $y(a)=0$ implies $d_{0}=0$ and
$y(b)=y(c)=0$ imply
\begin{gather}
d_{1}=\int_{a}^{c}\frac{( c-s) ^{2}( b-a) }{2(
c-a) ( c-b) }g( s) \,ds
-\int_{a}^{b}\frac{(b-s) ^{2}( c-a) }{2( c-b) ( b-a) }
g( s) \,ds , \label{192} \\
d_2=\int_{a}^{b}\frac{( b-s) ^{2}}{( c-b) (
b-a) }g( s) \,ds-\int_{a}^{c}\frac{( c-s) ^{2}}{
( c-a) ( c-b) }g( s)\, ds.  \label{m39}
\end{gather}
Substituting the constants $d_{0}$, $d_{1}$, and $d_2$ in the general
solution \eqref{m33}, we obtain
\begin{equation}
\begin{aligned}
y( t) 
&=\int_{a}^{t}\Big( \frac{( c-s) ^{2}(
t-a) ( b-t) }{2( c-a) ( c-b) }-\frac{
( b-s) ^{2}( t-a) ( c-t) }{2(b-a) ( c-b) }+\frac{( t-s) ^{2}}{2}\Big)g( s) ds \\
&\quad +\int_{t}^{b}( \frac{( c-s) ^{2}( b-t) (t-a) }{2( c-a) ( c-b) }
 -\frac{( b-s)^{2}( t-a) ( c-t) }{2( b-a) (c-b) }) g( s) ds \\
&\quad +\int_{b}^{c}\frac{( c-s) ^{2}( b-t) ( t-a)}{2( c-a) ( c-b) }g( s) ds
\end{aligned} \label{m36}
\end{equation}
for $t\in [ a,b] $ and
\begin{align*}
y( t) &=\int_{a}^{b}\Big( \frac{( c-s) ^{2}(
b-t) ( t-a) }{2( c-a) ( c-b) }
-\frac{( b-s) ^{2}( t-a) ( c-t) }{2(b-a) ( c-b) }+\frac{( t-s) ^{2}}{2}\Big)
g( s) ds \\
&\quad + \int_{b}^{t}( \frac{( c-s) ^{2}( b-t)
(t-a) }{2( c-a) ( c-b) }+\frac{( t-s)^{2}}{2}) g( s) ds \\
&\quad + \int_{t}^{c}\frac{( c-s) ^{2}( b-t) ( t-a)
}{2( c-a) ( c-b) }g( s) ds 
\end{align*}  %\label{37}
for $t\in [ b,c] $. This completes the proof.
\end{proof}

Consider the function $G_{c}( t,s) $ for $t\in [ a,b] $. It is easy to see that
\begin{equation}
\begin{aligned}
0\leq G_{c1}( t,s) 
&\leq G_{c1}( s) 
=\frac{(s-a) ^{2}( b-s) ( c-s) }{2( b-a)( c-a) } \\
&\leq G_{c1}^{\ast }( s) 
=\frac{(s-a) ^{2}( c-s) ^{2}}{2( b-a) ( c-a) }
\end{aligned} \label{191}
\end{equation}
for $a\leq s<t\leq b<c$. Since the function $G_{c1}^{\ast }(s)$ takes the
maximum value at $\frac{a+c}{2}$, i.e.
\begin{equation}
G_{c1}^{\ast }(s)
\leq \max_{a\leq s\leq b} G_{c1}^{\ast }(s) 
=G_{c1}^{\ast }( \frac{a+c}{2})
=\frac{(c-a) ^{3}}{32( b-a) }.  \label{190}
\end{equation}
Thus, $0\leq G_{c1}( t,s) \leq G_{c1}^{\ast }( s) \leq
\frac{( c-a) ^{3}}{32( b-a) }$ for $a\leq s<t\leq b$.
Now, we consider $0\leq G_{c2}( t,s) $ for $a\leq t\leq s<b<c$. 
Let $g_{c1}( t,s) =\frac{( t-a) ^{2}(b-s) ( c-s) }{2( b-a) ( c-a) }$ and
$g_{c2}( t,s) =\frac{( s-t) ( t-a) [( b-s) ( c-a) +( c-s) ( b-a)] }
{2( b-a) ( c-a) }$. We know that
\begin{equation}
\max_{a\leq t\leq s<b} G_{c2}( t,s) 
\leq \max_{a\leq t\leq s<b} g_{c1}( t,s)  
+\max_{a\leq t\leq s<b} g_{c2}( t,s) .  \label{189}
\end{equation}
Thus, we find the maximum value of the functions 
$g_{c1}( t,s) $ and $g_{c2}( t,s) $. It is easy to see that from \eqref{190}, 
\[
0\leq g_{c1}( t,s) \leq G_{c1}^{\ast }( s) 
\leq \frac{ ( c-a) ^{3}}{32( b-a) }
\]
 for $a\leq t\leq s<b$. Now,
we find the absolute maximum of $g_{c2}( t,s) $. 
$g_{c2}(t,s) $ takes its maximum value at the point 
\[
(t_{0},s_{0}) =\Big( \frac{4a^{2}-ab-ac-2bc}{3( 2a-b-c) },
\frac{ab+ac+2a^{2}-4bc}{3( 2a-b-c) }\Big),
\]
 and its maximum value is 
\[
g_{c2}( \frac{4a^{2}-ab-ac-2bc}{3( 2a-b-c) },
\frac{ab+ac+2a^{2}-4bc}{3( 2a-b-c) })
=\frac{4}{27}(\frac{( c-a) ( b-a) }{2a-b-c}) ^{2}.
\]
 Thus, we have
\begin{equation}
G_{c}( t,s)
 \leq \min \Big\{ \frac{( c-a) ^{3}}{32( b-a) },\frac{( c-a) ^{3}}{32( b-a) }
+\frac{4}{27}\Big( \frac{( c-a) ( b-a) }{2a-b-c}\Big) ^{2}\Big\}
=\frac{( c-a) ^{3}}{32( b-a) }
\label{188}
\end{equation}
for $a\leq t$, $s<b$ \cite{Agarwal, Yang}. Similarly, we obtain
\begin{equation}
0\leq G_{c3}( t,s) \leq G_{c3}( s) 
=\frac{(c-s) ^{2}( s-a) ^{2}}{2( c-a) ( c-b) }
\leq \frac{( c-a) ^{3}}{32( c-b) }  \label{300}
\end{equation}
for $a<t<b<s<c$. Therefore, we have
\begin{equation}
G_{c}( t,s) \leq 
\begin{cases}
\frac{( c-a) ^{3}}{32( b-a) }, & a\leq t,s\leq b,\\
\frac{( c-a) ^{3}}{32( c-b) }, & t<b<s<c
\end{cases} \label{299}
\end{equation}
for $t\in [ a,b] $. Similarly, it is easy to see that we have
\begin{equation}
| G_{c}( t,s) | \leq \begin{cases}
\frac{( c-a) ^{3}}{32( b-a) }, & a<s<b<t, \\
\frac{( c-a) ^{3}}{32( c-b) }, & b\leq t,s\leq c,
\end{cases} \label{186}
\end{equation}
for $t\in [ b,c] $.

Now, we give another important lemma. In the following lemma, we construct
Green's function for the third order nonhomogeneous differential equation 
\eqref{m34} with the three-point boundary conditions \eqref{z7} inspired by
Moorti and Garner \cite{Moorti} as follows.

\begin{lemma}[{\cite[Table 1]{Moorti}}] \label{lem2.2}
If $y( t) $  is a solution of \eqref{m34}
 satisfying $y( a) =y''( d) =y(b) =0$  with $a\leq d\leq b$  and 
$y( t) \not\equiv 0$ for $t\in ( a,b) $, then
\begin{equation}
y( t) =\int_{a}^{b}G_{d}( t,s) g( s) \,ds,  \label{d6}
\end{equation}
holds, where for $d<s$,
\begin{equation}
G_{d}( t,s) =\begin{cases}
G_{d1}( t,s) =  \frac{( t-a) ( b-s) ^{2}}{
2( b-a) }, & a\leq t\leq s\leq b, \\[4pt]
G_{d2}( t,s) =  \frac{( s-a) ( b-t) ^{2}}{2( b-a) } \\
+\frac{( t-s) ( b-t) (b+s-2a) }{2( b-a) }, & a\leq s<t\leq b,
\end{cases}  \label{d7}
\end{equation}
and for $s\leq d$,
\begin{equation}
G_{d}( t,s) 
=\begin{cases}
G_{d3}( t,s) 
=  -\frac{( t-a) ^{2}( b-s)}{2( b-a) } \\
 -\frac{( s-t) ( t-a) (2b-a-s) }{2( b-a) }, & a\leq t\leq s\leq b, \\[4pt]
G_{d4}( t,s) =  -\frac{( s-a) ^{2}( b-t)}{2( b-a) }, & a\leq s<t\leq b\,.
\end{cases}  \label{d8}
\end{equation}
\end{lemma}

Consider the function $G_{d}( t,s) $ for $d<s$. It is easy to see that
\begin{equation}
0\leq G_{d1}( t,s) \leq G_{d1}( s)
 =\frac{(s-a) ( b-s) ^{2}}{2( b-a) }  \label{225}
\end{equation}
for $a\leq t\leq s\leq b$. Since the function 
$G_{d1}(s)$ takes the maximum value at $\frac{2a+b}{3}$, i.e.
\begin{equation}
G_{d1}(s)
\leq \max_{a\leq s\leq b} G_{d1}( s)
=G_{d1}( \frac{2a+b}{3}) =\frac{2( b-a) ^{2}}{27}.  \label{224}
\end{equation}
Thus, $0\leq G_{d1}( t,s) \leq G_{d1}( s) \leq \frac{2( b-a) ^{2}}{27}$ 
for $a\leq t\leq s\leq b$. Now, we consider $0\leq G_{d2}( t,s) $
 for $a\leq s<t\leq b$. Let $g_{d1}(t,s) =\frac{( s-a) ( b-t) ^{2}}{2(
b-a) }$ and $g_{d2}( t,s) =\frac{( t-s) (b-t) ( b+s-2a) }{2( b-a) }$.
 We know that
\begin{equation}
\max_{a\leq s<t\leq b} G_{d2}( t,s) 
\leq \max_{a\leq s<t\leq b} g_{d1}( t,s) 
+\max_{a\leq s<t\leq b} g_{d2}( t,s) .  \label{223}
\end{equation}
Thus, we find the maximum value of the functions $g_{d1}( t,s) $
and $g_{d2}( t,s) $. It is easy to see that from \eqref{224}, 
$0\leq g_{d1}( t,s) \leq G_{d1}( s) \leq \frac{2(b-a) ^{2}}{27}$
for $a\leq s<t\leq b$. Now, we find the absolute
maximum of $g_{d2}( t,s) $. $g_{d2}( t,s) $ takes its
maximum value at\ the point 
\[
( t_{0},s_{0}) =( \frac{2a+b}{3},\frac{4a-b}{3}) ,
\]
 and its maximum value is 
\[
g_{d2}( \frac{2a+b}{3},\frac{4a-b}{3}) =\frac{4( b-a) ^{2}}{27}.
\]
 Thus, 
\begin{equation}
G_{d}( t,s) \leq \min \big\{ \frac{2( b-a) ^{2}}{27},
\frac{2( b-a) ^{2}}{27}+\frac{4( b-a) ^{2}}{27}
\big\} 
=\frac{2( b-a) ^{2}}{27}  \label{222}
\end{equation}
for $d<s$. Similarly, it is easy to see that we have
\begin{equation}
| G_{d}( t,s) | \leq \frac{2( b-a)^{2}}{27}  \label{221}
\end{equation}
for $s\leq d$.

\begin{remark} \label{rmk2.1}\rm
It is easy to see that if we take $d=a$ or 
$d=b$ in Lemma \ref{lem2.2}, the problems \eqref{1} with \eqref{z7} become 
 two-point boundary value problems.
\end{remark}

\section{Main results}

Now, we give one of main results of this paper.

\begin{theorem} \label{thm3.1}
If $y( t) $ is a nontrivial solution of the problem \eqref{1}
 with \eqref{200}, then 
\begin{equation}
C\leq \int_{a}^{c}| q( s) | \,ds\,,  \label{185}
\end{equation}
 where 
\[
C=\min \big\{ \frac{32( c-b) }{( c-a)^{3}},
\frac{32( b-a) }{( c-a) ^{3}}\big\}.
\]
\end{theorem}

\begin{proof}
Let $y( a) =y( b) =y( c) =0$ where $a,b,c\in\mathbb{R}$ with $a<b<c$ are 
three points, and $y$ is not identically zero on $(a,b) \cup ( b,c) $. 
From \eqref{193}, \eqref{299}, and \eqref{186}, we obtain
\begin{align}
| y( t) | &\leq \int_{a}^{c}| G_{c}( t,s) | | y'''( s) | ds \label{125} \\
&\leq \int_{a}^{b}\frac{( c-a) ^{3}}{32( b-a) }|y'''( s) | ds
+\int_{b}^{c}\frac{( c-a) ^{3}}{32( c-b) }| y'''( s) | ds
\leq \frac{1}{C}\int_{a}^{c}|y'''( s) | ds.  \label{18}
\end{align}
From  \eqref{1} and  inequality \eqref{18}, we obtain
\begin{equation}
| y'''( t) | 
=|q( t) | | y( t) | 
\leq\frac{| q( t) | }{C}\int_{a}^{c}| y'''( s) | ds.  \label{14}
\end{equation}
Integrating from $a$ to $c$ both sides of \eqref{14}, we obtain
\begin{equation}
\int_{a}^{c}| y'''( s) | ds
\leq \frac{1}{C}\int_{a}^{c}| y'''(s) | ds\int_{a}^{c}| q( s) | ds.  \label{15}
\end{equation}
Next, we prove that
\begin{equation}
0<\int_{a}^{c}| y'''( s)| ds.  \label{16}
\end{equation}
If \eqref{16} is not true, then we have
\begin{equation}
\int_{a}^{c}| y'''( s) |ds=0.  \label{17}
\end{equation}
It follows from \eqref{18} and \eqref{17} that $y(t)\equiv 0$ for 
$t\in( a,c) $, which contradicts with \eqref{200} since 
$y(t) \neq 0$ for all $t\in ( a,c) $. Thus, by using \eqref{16}
 in \eqref{15}, we obtain  inequality \eqref{185}.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
It is easy to see that in the special cases,  inequality \eqref{185} is 
sharper than \eqref{199}, \eqref{198},
\eqref{63}, \eqref{195}, \eqref{22}, and \eqref{171} in the sense that
they follow from \eqref{185}, but not conversely. 
Therefore, our result improves Theorems \ref{thmA}, \ref{thmE}, \ref{thmG},
\ref{thmH}, and \ref{thmJ}--\ref{thmM}, 
in the special cases. In fact, the Lyapunov-type inequality \eqref{185} 
is the best possibility for  problem \eqref{1} with \eqref{200} 
in the sense that the constant $32$ in
the left hand side of \eqref{185} cannot be replaced by any larger constant.
\end{remark}

Note that we can rewrite  Green's functions $G_{c}( t,s) $
given in \eqref{35} and \eqref{34} as follows:
 for $t\in [ a,b] $,
\begin{equation}
0\leq G_{c}( t,s) 
\leq \begin{cases}
G_{c1}( s) =  \frac{( s-a) ^{2}( b-s)
( c-s) }{2( b-a) ( c-a) },  &  a\leq s<t\leq b \\[4pt]
G_{c2}( s) =  \frac{( s-a) ^{2}( b-s)
( c-s) }{2( b-a) ( c-a) } \\
+\frac{(s-a) ^{2}\left[ ( b-s) ( c-a) +( c-s)
( b-a) \right] }{2( b-a) ( c-a) }, & a\leq t\leq s<b \\[4pt]
G_{c3}( s) =  \frac{( c-s) ^{2}( s-a) ^{2}
}{2( c-a) ( c-b) } , & b<s<c
\end{cases}   \label{298}
\end{equation}
and for $t\in [ b,c] $,
\begin{equation}
| G_{c}( t,s) | 
\leq  \begin{cases}
G_{c4}( s) =  \frac{( s-a) ^{2}( c-s) ^{2}}{2( b-a) ( c-a) } , & a<s<b \\[4pt]
G_{c5}( s) =  \frac{( c-s) ^{2}( s-b)( s-a) }{2( c-a) ( c-b) } \\
 +\frac{(c-s) ^{2}[ ( s-b) ( c-a) +( s-a)
( c-b) ] }{2( c-a) ( c-b) }, & b<s\leq t\leq c \\[4pt]
G_{c6}( s) =  \frac{( c-s) ^{2}( s-b)
( s-a) }{2( c-a) ( c-b) } , & b\leq t<s\leq c\,.
\end{cases}  \label{297}
\end{equation}
Thus,  for $t\in \left[ a,b\right] $, we have
\begin{equation}
0\leq G_{c}( t,s) \leq \max_{a\leq s\leq c} 
\big\{G_{c1}( s) ,G_{c2}( s) ,G_{c3}( s) \big\}
=\max_{a\leq s\leq c} \big\{ G_{c2}( s) ,G_{c3}(s) \big\}   \label{266}
\end{equation}
and for $t\in [ b,c] $,
\begin{equation}
| G_{c}( t,s) | \leq \max_{a\leq s\leq c}
\big\{ G_{c4}( s) ,G_{c5}( s) ,G_{c6}(s) \big\} 
=\max_{a\leq s\leq c} \big\{ G_{c4}(s) ,G_{c5}( s) \big\} .  \label{265}
\end{equation}
Thus, from \eqref{266} and \eqref{265}, we obtain
\begin{equation}
| G_{c}( t,s) | \leq G_{c\ast }( s)
=\max_{a\leq s\leq c}\big\{ G_{c2}( s) ,G_{c3}(s) \big\} \text{  or  }
\max_{a\leq s\leq c}\big\{G_{c4}( s) ,G_{c5}( s) \big\}   \label{235}
\end{equation}
for $s\in [ a,c] $ \cite{Agarwal, Yang}.

The proof of the following result proceeds as in Theorem \ref{thm3.1}
 by using \eqref{235} instead of \eqref{125} and hence it is omitted.

\begin{theorem} \label{thm3.2}
If $y( t) $ is a nontrivial solution of the problem \eqref{1} with \eqref{200}, 
then 
\begin{equation}
1\leq \int_{a}^{c}G_{c\ast }( s) | q( s)
| \,ds\,,  \label{234}
\end{equation}
 where $G_{c\ast }( s) $ is given in \eqref{235}.
\end{theorem}

\begin{theorem} \label{thm3.3}
If $y( t) $ is a nontrivial solution of the problem \eqref{1}
 with \eqref{200}, then 
\begin{equation}
1\leq \int_{a}^{c}| G_{c}( t_{0},s) || q( s) | \,ds\,,  \label{179}
\end{equation}
where $| y(t_{0})| =\max \{|y(t)| :a\leq t\leq c\}$.
\end{theorem}

\begin{proof}
Let $y( a) =y( b) =y( c) =0$ where $a,b,c\in\mathbb{R}$ with $a<b<c$ are 
three points, and $y$ is not identically zero on 
$(a,b) \cup ( b,c) $. Pick $t_{0}\in (a,c)$ so that 
$| y(t_{0})| =\max \{| y(t)| :a\leq t\leq c\}$. From  \eqref{1} and 
 \eqref{193}, we obtain
\begin{equation}
\begin{aligned}
| y( t_{0}) | 
&=|\int_{a}^{c}G_{c}( t_{0},s) [ -q( s) y(s)] ds| \\
&\leq \int_{a}^{c}| G_{c}( t_{0},s) || q( s) | | y( s)| ds 
\end{aligned} \label{12}
\end{equation}
and hence
\begin{equation}
| y( t_{0}) | 
\leq | y(t_{0}) | \int_{a}^{c}| G_{c}( t_{0},s)| | q( s) | ds.  \label{13}
\end{equation}
Dividing both sides by $| y(t_{0})| $, we obtain 
inequality \eqref{179}.
\end{proof}

Now, we give other main results of this paper under three-point boundary
conditions \eqref{z7}. The proofs of following results are similar to that
of Theorems \ref{thm3.1}--\ref{thm3.3} and hence they are omitted.

\begin{theorem} \label{thm3.4}
If $y( t) $ is a nontrivial solution of  problem \eqref{1} with \eqref{z7}, 
then
\begin{equation}
\frac{27}{2( b-a) ^{2}}\leq \int_{a}^{b}| q(s) | \,ds\,.  \label{m21}
\end{equation}
\end{theorem}

\begin{remark} \label{rmk3.2} \rm
It is easy to see in the special cases that the inequality \eqref{m21}
 is sharper than \eqref{209}, \eqref{268}, \eqref{261}, \eqref{65}, 
and \eqref{264} in the sense that they
follow from \eqref{m21}, but not conversely. 
Therefore, our result improves Theorems \ref{thmB}--\ref{thmD}, \ref{thmF}, 
and \ref{thmI} in the special cases. 
In fact, the Lyapunov-type inequality \eqref{m21} is the best possibility 
for the problem \eqref{1} with \eqref{z7} in the sense that the 
constant $\frac{27}{2}$  in
the left hand side of \eqref{m21} cannot be replaced by any larger constant.
\end{remark}

From \eqref{d7} and \eqref{d8}, it is easy to see that for $d<s$,
\begin{equation}
G_{d}( t,s) \leq \begin{cases}
G_{d1}( s) =  \frac{( s-a) ( b-s) ^{2}}{2( b-a) }, & a\leq t\leq s \\[4pt]
G_{d2}( s) =  \frac{( s-a) ( b-s) ^{2}}{2( b-a) } 
 +\frac{( b-s) ^{2}( b+s-2a) }{2( b-a) } , & a\leq s\leq t
\end{cases}  \label{41}
\end{equation}
and for $s\leq d$,
\begin{equation}
| G_{d}( t,s) |
 \leq \begin{cases}
G_{d3}( s) =  \frac{( s-a) ^{2}( b-s) }{2( b-a) } 
 +\frac{( s-a) ^{2}( 2b-a-s) }{2( b-a) } , &  a\leq t\leq s\leq b \\
G_{d4}( s) =  \frac{( s-a) ^{2}( b-s) }{
2( b-a) } , &  a\leq s<t\leq b\,.
\end{cases} \label{42}
\end{equation}
Therefore, we have
\begin{equation}
| G_{d}( t,s) | \leq G_{d\ast }( s)
=\max_{a\leq s\leq b} \{ G_{d2}( s) ,G_{d3}(s) \} .  \label{k4}
\end{equation}

\begin{theorem} \label{thm3.5}
If $y( t) $ is a nontrivial solution of the problem 
\eqref{1} with \eqref{z7}, then 
\begin{equation}
1\leq \int_{a}^{b}G_{d\ast }( s) | q( s)
|\, ds\,,  \label{204}
\end{equation}
 where $G_{d\ast }( t) $ is given in \eqref{k4}.
\end{theorem}

\begin{theorem} \label{thm3.6}
If $y( t) $ is a nontrivial solution of the problem 
\eqref{1} with \eqref{z7}, then 
\begin{equation}
1\leq \int_{a}^{b}| G_{d}( t_{0},s) |
| q( s) | \,ds\,,  \label{11}
\end{equation}
 where $| y(t_{0})| =\max \{|y(t)| :a\leq t\leq b\}$.
\end{theorem}

We may adopt a different point of view and use \eqref{185} or \eqref{m21} to
obtain an extension of the following oscillation criterion due originally to
Liapounoff (cf. \cite{Borg}): $y''(t)$ and $y''( t) y^{-1}( t) $ are 
continuous for $a\leq t\leq b$, with $y( a) =y( b) =0$, then
\begin{equation}
\frac{4}{b-a}<\int_{a}^{b}| y''( s) y^{-1}( s) | ds.  \label{m81}
\end{equation}
Thus, \eqref{185} or \eqref{m21} leads to the following extension: 
If $ y'''(t)$ and $y'''( t) y^{-1}( t) $ are continuous for $a\leq t\leq c$ 
or $a\leq t\leq b$, $y( t) $ has three points including $a$, $b$, $c$, and $d$, 
then
\begin{equation}
C\leq \int_{a}^{c}| y'''( s)y^{-1}( s) | ds\text{ or }\frac{27}{2( b-a)
^{2}}\leq \int_{a}^{b}| y'''( s)
y^{-1}( s) | \,ds.  \label{178}
\end{equation}

Now, we give another application of the obtained Lyapunov-type inequalities
for the  eigenvalue problem
\begin{equation}
y'''+\lambda h( t) y=0  \label{51}
\end{equation}
under three points boundary conditions \eqref{200} or \eqref{z7}. 
Thus, if there exists a nontrivial solution $y( t) $ of linear homogeneous
problem \eqref{51}, then we have
\begin{equation}
\frac{C}{\int_{a}^{c}| h( s) | ds}\leq \lambda
\quad \text{or}\quad
\frac{27}{2( b-a) ^{2}\int_{a}^{b}| h(s) | ds}\leq \lambda .  \label{m80}
\end{equation}

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