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\AtBeginDocument{{\noindent\small
Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
{\em Electronic Journal of Differential Equations},
Conf. 19 (2010), pp. 151--159.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document} \setcounter{page}{151}
\title[\hfilneg EJDE-2010/Conf/19/\hfil Third order three-point BVP]
{Positive solutions to a nonlinear third order three-point boundary
value problem}
\author[J. R. Graef, L. Kong, B. Yang\hfil EJDE/Conf/19 \hfilneg]
{John R. Graef, Lingju Kong, Bo Yang} % in alphabetical order
\address{John R. Graef \newline
Department of Mathematics,
University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA}
\email{John-Graef@utc.edu}
\address{Lingju Kong \newline
Department of Mathematics,
University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA}
\email{Lingju-Kong@utc.edu}
\address{Bo Yang\newline
Department of Mathematics and Statistics,
Kennesaw State University, Kennesaw, GA 30144, USA}
\email{byang@kennesaw.edu}
\thanks{Published September 25, 2010.}
\subjclass[2000]{34B15}
\keywords{Boundary value problems; existence of positive
solutions; \hfill\break\indent
nonexistence of positive solutions; nonlinear
equations; third order problem; \hfill\break\indent
three point boundary conditions}
\begin{abstract}
We consider a third order three point boundary value problem.
Some upper and lower estimates for positive solutions of the
problem are proved. Sufficient conditions for the existence
and nonexistence of positive solutions for the problem are
obtained. An example is included to illustrate the results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
Recently, second and higher order multi-point boundary value
problems have attracted a lot of attention. In 2004, Henderson
\cite{H} considered the second order three point boundary value
problem
\begin{equation} \label{ee00}
\begin{gathered}
u''(t)+f(u(t))=0,\quad 0\le t\le 1, \\
u(0)=u(p)-u(1)=0.
\end{gathered}
\end{equation}
In 2006, Graef and Yang \cite{GY-2} studied the third order
nonlocal boundary value problem
\begin{gather}
u'''(t) = g(t)f(u(t)), \quad 0\le t\le 1, \label{ee1} \\
u(0) = u'(p) = u''(1) = 0. \label{ee2}
\end{gather}
For some other results on third order boundary value problems
we refer the reader to the papers \cite{AD,GY,GSZ,HT,M,W}.
%
Motivated by these works, in this paper we consider the third
order three point nonlinear boundary value problem
\begin{gather}
u'''(t) = g(t)f(u(t)), \quad 0\le t\le 1, \label{bhu}\\
u(0)= u(p) -u(1) = u''(1) = 0. \label{rfv}
\end{gather}
%
To our knowledge, the problem \eqref{bhu}--\eqref{rfv} has not
been considered before. The boundary conditions \eqref{rfv}
are closely related to some other boundary conditions.
Firstly, \eqref{rfv} contains \eqref{ee00} as a part.
We also note that $u(p)-u(1)=0$ implies that there exists
$\beta\in(p,1)$ such that $u'(\beta)=0$, and therefore the
boundary conditions \eqref{rfv} imply
\[
u(0)=u'(\beta)=u''(1)=0.
\]
Hence, boundary conditions \eqref{rfv} are closely related
to the conditions \eqref{ee2}. If we let $p\to 1^-$,
then \eqref{rfv} ``tends to''
\begin{equation}
u(0) = u'(1) = u''(1) = 0, \label{rfvvv}
\end{equation}
which are often referred to as the (1,2) focal boundary conditions.
In this paper, we are interested in the existence and nonexistence
of positive solutions of the problem \eqref{bhu}--\eqref{rfv}.
%
By a {\it positive solution}, we mean a solution $u(t)$ to
the boundary value problem such that $u(t)>0$ for $00$ such that $(F_0+\epsilon)B(1+p)^2/4p <1$.
Then there exists $H_1 >0 $ such that
$$
f(x)\leq (F_0+\epsilon)x \ \ {\rm for}\ \ 00$ such that
$$
(f_\infty-\delta)\int_c^{1-c} G(2p/(1+p),s) g(s) a(s)\,ds >1.
$$
Now, there exists $ H_3 >0 $ such that
$f(x)\geq (f_\infty-\delta)x$ for $x\geq H_3$.
Let $H_2=H_1 + {H_3}/{c}$.
If $u\in P$ with $\|u\| = H_2$, then for $c\le t\le 1-c$, we have
$$
u(t)\geq\min\{t,1-t\}\|u\| \ge c H_2\ge H_3.
$$
So, if $u\in P$ with $\|u\| = H_2$, then
%
\begin{align*}
(Tu)(2p/(1+p))
& \geq
\int_c^{1-c} G(2p/(1+p),s) g(s) f(u(s))\,ds
\\
& \geq
\int_c^{1-c} G(2p/(1+p),s) g(s) (f_\infty-\delta)u(s)ds
\\
& \ge
(f_\infty-\delta) \|u\| \int_c^{1-c} G(2p/(1+p),s) g(s) a(s)\,ds
\\
& >
\|u\|
\\
&\ge
u(2p/(1+p)),
\end{align*}
%
which means $Tu\not\le u$. So, if we let
$\Omega_2=\{ u\in X: \|u\|0$ such that
$$
A(f_0-\epsilon)>1.
$$
There exists $H_1 >0$ such that $f(x)\ge (f_0-\epsilon)x$ for
$x\geq H_1$.
If $u\in P$ with $\|u\| = H_1$, then
%
\begin{align*}
(Tu)(2p/(1+p))
& \ge
\int_0^1 G(2p/(1+p),s) g(s) f(u(s))\,ds
\\
& \ge
\int_0^1 G(2p/(1+p),s) g(s) (f_0-\epsilon)u(s)ds
\\
& \ge
(f_0-\epsilon) \|u\| \int_0^1 G(2p/(1+p),s) g(s) a(s)\,ds
\\
& >
\|u\|
\\
&\ge
u(2p/(1+p)),
\end{align*}
%
which means $Tu\not\le u$. So, if we let
$\Omega_1=\{ u\in X\,|\ \|u\|0$ such that $f(x)\le (F_{\infty}+\delta) x$
for $x\geq H_3$.
If we let $M = \max_{0\leq x\leq H_3}f(x)$, then
$f(x)\le M + (F_{\infty}+\delta)x$ \text{for} $x\ge 0$. Let
\[
K = M \int_0^1 G(2p/(1+p),s)g(s)ds+1,
\]
and let $H_2 = H_1+K \big(\frac{4p}{(1+p)^2}-(F_{\infty}+\delta)B
\big)^{-1}$.
%
Now for each $u\in P$ with $\|u\| = H_2$, we have
\begin{align*}
(Tu)(2p/(1+p))
& =
\int_0^1 G(2p/(1+p),s) g(s) f(u(s))\,ds
\\
& \le
\int_0^1 G(2p/(1+p),s) g(s) (M+(F_{\infty}+\delta)u(s))\,ds
\\
& <
K + (F_{\infty}+\delta)\int_0^{1} G(2p/(1+p),s) g(s) u(s)\,ds
\\
& \le
K + (F_{\infty}+\delta)\|u\| \int_0^{1} G(2p/(1+p),s) g(s) \,ds
\\
&\le
K + (F_{\infty}+\delta)B\|u\|
\\
& \le
\Big(\frac{4p}{(1+p)^2} - (F_{\infty}+\delta)B\Big) H_2
+ (F_{\infty}+\delta)B H_2
\\
&=
\frac{4p}{(1+p)^2} \|u\|
\\
&\le
u(2p/(1+p)),
\end{align*}
%
which means $ Tu\not\ge u $.
%
So, if we let $\Omega_2=\{ u\in X\ |\ \|u\|0$ for $0x$ for
all $x\in(0,+\infty)$, then the problem \eqref{bhu}--\eqref{rfv}
has no positive solutions.
\end{theorem}
We conclude the paper with an example.
\begin{example} \rm
Consider the third-order boundary-value problem
\begin{gather}
u'''(t)=g(t)f(u(t)), \quad 00$ is a parameter.
We easily see that $F_0=f_0=\lambda$ and
$F_{\infty}=f_{\infty}=3\lambda$.
Calculations indicate that
$$
A=\frac{5268393409}{216850636800},\quad
B=\frac{33611}{1229312}.
$$
From Theorem \ref{tt6} we see that if
$$
13.7203 \approx\frac{ 1}{ 3A } < \lambda
< \frac{48}{49B}\approx 35.8282,
$$
then problem \eqref{rfv2}--\eqref{rfv3} has at least one
positive solution.
From Theorems \ref{tt8} and \ref{tt9}, we see that if
$$
\lambda<\frac{ 16 }{49B}\approx 11.9427 \quad\text{or}\quad \lambda>\frac{1}{A}\approx 41.1607,
$$
then problem \eqref{rfv2}--\eqref{rfv3} has no positive solutions.
This example shows that our existence and nonexistence
conditions work very well.
\end{example}
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\bibitem{GY-2} {J. R. Graef} and B. Yang
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\end{thebibliography}
\end{document}