\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 118, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/118\hfil Multiple positive solutions]
{Multiple positive solutions for a third-order three-point BVP with
sign-changing Green's function}
\author[J.-P. Sun, J. Zhao\hfil EJDE-2012/118\hfilneg]
{Jian-Ping Sun, Juan Zhao} % in alphabetical order
\address{Department of Applied Mathematics,
Lanzhou University of Technology,
Lanzhou, Gansu 730050, China}
\email[Jian-Ping Sun]{jpsun@lut.cn}
\email[Juan Zhao]{z\_1111z@163.com}
\thanks{Submitted May 29, 2012. Published July 14, 2012.}
\subjclass[2000]{34B10, 34B18}
\keywords{Third-order three-point boundary-value problem; \hfill\break\indent
sign-changing Green's function; positive solution}
\begin{abstract}
This article concerns the third-order three-point
boundary-value problem
\begin{gather*}
u'''(t)=f(t,u(t)),\quad t\in [0,1], \\
u'(0)=u(1)=u''(\eta)=0.
\end{gather*}
Although the corresponding Green's function
is sign-changing, we still obtain the existence of at least
$2m-1$ positive solutions for arbitrary positive integer
$m$ under suitable conditions on $f$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction}
Third-order differential equations arise from a variety of
areas of applied mathematics and physics, e.g., in the deflection of
a curved beam having a constant or varying cross section, a
three-layer beam, electromagnetic waves or gravity driven flows and
so on \cite{4}.
Recently, the existence of single or multiple positive solutions to
some third-order three-point boundary-value problems (BVPs for
short) has received much attention from many authors. For example,
in 1998, by using the Leggett-Williams fixed point theorem, Anderson
\cite{0} proved the existence of at least three positive solutions
to the problem
\begin{gather*}
-x'''(t)+f(x(t))=0,\quad t\in [0,1],\\
x(0)=x'(t_2)=x''(1)=0,
\end{gather*}
where $t_2\in [\frac{1}{2},1)$. In 2003, Anderson
\cite{1.1} obtained some existence results of positive solutions for
the problem
\begin{gather*}
x'''(t)=f(t,x(t)),\quad t_1\leq t\leq t_3,\\
x(t_1)=x'(t_2)=0,\quad \gamma x(t_3)+\delta x''(t_3)=0.
\end{gather*}
The main tools used were the Guo-Krasnosel'skii and Leggett-Williams
fixed point theorems. In 2005, Sun \cite{9} studied the existence of
single and multiple positive solutions for the singular BVP
\begin{gather*}
u'''(t)-\lambda a(t)F(t,u(t))=0,\quad t\in (0,1),\\
u(0)=u'(\eta)=u''(1)=0,
\end{gather*}
where $\eta\in [\frac{1}{2},1)$, $\lambda$ was a positive
parameter and $a(t)$ was a nonnegative continuous function defined
on $(0, 1)$. His main tool was the Guo-Krasnosel'skii fixed point
theorem. In 2008, by using the Guo-Krasnosel'skii fixed point
theorem, Guo, Sun and Zhao \cite{3} obtained the existence of at
least one positive solution for the problem
\begin{gather*}
u'''(t)+h(t)f(u(t))=0,\quad t\in (0,1),\\
u(0)=u'(0)=0,\quad u'(1)=\alpha u'(\eta ),
\end{gather*}
where $0<\eta <1$ and $1<\alpha <1/\eta $. For more results
concerning the existence of positive solutions to third-order
three-point BVPs, one can refer to \cite{3.1,3.2,13,12,10,11}.
It is necessary to point out that all the above-mentioned works are
achieved when the corresponding Green's functions are positive,
which is a very important condition. A natural question is that
whether we can obtain the existence of positive solutions to some
third-order three-point BVPs when the corresponding Green's
functions are sign-changing. It is worth mentioning that Palamides
and Smyrlis \cite{7} discussed the existence of at least one
positive solution to the singular third-order three-point BVP with
an indefinitely signed Green's function
\begin{gather*}
u'''(t)=a(t)f(t,u(t)),\quad t\in (0,1),\\
u(0)=u(1)=u''(\eta)=0,\quad \eta\in (\frac{17}{24},1).
\end{gather*}
Their technique was a combination of the Guo-Krasnosel'skii fixed
point theorem and properties of the corresponding vector field. The
following equality
\begin{equation}
\max_{t\in[0,1]}\int_0^1G(t,s)a(s)f(s,u(s))ds
=\int_0^1\max_{t\in[0,1]} G( t,s)a(s)f(s,u(s))ds \label{0}
\end{equation}
played an important role in the process of their proof.
Unfortunately, the equality \eqref{0} is not right. For a
counterexample, one can refer to our paper \cite{16}.
Motivated greatly by the above-mentioned works, in this
paper we study the following third-order three-point BVP
\begin{equation}
\begin{gathered}
u'''(t)=f(t,u(t)),\quad t\in [0,1], \\
u'(0)=u(1)=u''(\eta)=0,
\end{gathered}\label{1.1}
\end{equation}
where $f\in C([0,1]\times[0,+\infty),\ [0,+\infty))$ and
$\eta\in(\frac{1}{2},1)$. Although the corresponding Green's function
is sign-changing, we still obtain the existence of at least
$2m-1$ positive solutions for arbitrary positive integer
$m$ under suitable conditions on $f$.
In the remainder of this section, we state some fundamental
concepts and the Leggett-Williams fixed point theorem \cite{2}.
Let $E$ be a real Banach space with cone $P$. A map $\sigma:
P \to (-\infty,+\infty)$ is said to be a concave functional
if
\[
\sigma(tx+(1-t)y)\geq t\sigma(x)+(1-t)\sigma(y)
\]
for all $x,y\in P$ and $t\in [0,1]$. Let $a$ and $b$ be two numbers
with $0a\}\neq \emptyset$ and
$\sigma(Ax)>a$ for $x\in P(\sigma,a,b)$;
\item[(ii)] $\| Ax\|a$ for $x\in P(\sigma,a,c)$ with $\| Ax\|>b$.
\end{itemize}
Then $A$ has at least three fixed points $x_1,x_2,x_{3}$ in
$\overline{P_c}$ satisfying
\[
\| x_1\|d,\ \sigma(x_{3})\frac{a}{H_2},\quad t\in [1-\theta,\theta],\;
u\in[a,\frac{a}{\theta^{*}}],\label{3.2} \\
f(t,u)<\frac{c}{H_1},\quad t\in [0,\eta],\; u\in[0,c]. \label{3.3}
\end{gather}
Then \eqref{1.1} has at least three positive solutions $u$,
$v$ and $w$ satisfying
\[
\|u\| a\}\neq \emptyset$
and $\sigma(Au) > a$ for all $u \in P(\sigma, a, \frac{a}{\theta^{*}})$.
In fact, the constant function
$\frac{a+\frac{a}{\theta^{*}}}{2}$ belongs to
$\{u \in P(\sigma,a,\frac{a}{\theta^{*}}): \sigma(u) > a\}$.
On the one hand, for $u\in P(\sigma,a,\frac{a}{\theta^{*}})$, we
have
\begin{equation}
a\leq\sigma(u)=\min_{t\in[1-\theta,\theta]}u(t)\leq u(t)
\leq \|u\|\leq \frac{a}{\theta^{*}} \label{3.35}
\end{equation}
for all $t\in [1-\theta,\theta]$.
Also, for any $u\in P$ and $t\in [1-\theta,\theta]$, we
have
\begin{align*}
&\int_0^{1-\theta} G(t,s)f(s,u(s))ds+\int_\theta^\eta G(t,s)f(s,u(s))ds
+\int_\eta^1 G(t,s)f(s,u(s))ds\\
&\geq \int_0^{1-\theta} (1-t)s f(s,u(s))ds
-\int_\eta^1 \frac{(1-s)^{2}}{2}f(s,u(s))ds\\
&\geq f(\eta,u(\eta))[\int_0^{1-\theta} (1-t)s ds
-\int_\eta^1 \frac{(1-s)^{2}}{2}ds]\\
&\geq f(\eta,u(\eta))[\int_0^{1-\theta} (1-t)s ds
-\int_\theta^1 \frac{(1-s)^{2}}{2}ds]\\
&= f(\eta,u(\eta))[\frac{(1-t)(1-\theta)^{2}}{2}-\frac{(1-\theta)^{3}}{6}]\\
&\geq f(\eta,u(\eta))[\frac{(1-\theta)(1-\theta)^{2}}{2}
-\frac{(1-\theta)^{3}}{6}]\\
&= f(\eta,u(\eta))\frac{(1-\theta)^{3}}{3}
\geq 0,
\end{align*}
which together with \eqref{3.2} and \eqref{3.35} implies
\begin{align*}
\sigma(Au)
&= \min_{t\in[1-\theta,\theta]}\int_0^1 G(t,s)f(s,u(s))ds\\
&\geq \min_{t\in[1-\theta,\theta]}\int_{1-\theta}^\theta G(t,s)f(s,u(s))ds\\
&> \frac{a}{H_2}\min_{t\in[1-\theta,\theta]}\int_{1-\theta}^\theta G(t,s)ds
= a
\end{align*}
for $u\in P(\sigma,a,\frac{a}{\theta^{*}})$.
Finally, we verify that if $u \in P(\sigma, a, c)$ and
$\| Au\| > a/\theta^{*}$ , then $\sigma(Au) > a$.
To see this, we suppose that $u \in P(\sigma, a, c)$ and
$\| Au\| > a/\theta^{*}$. Then it follows from $Au \in P$
that
\[
\sigma(Au)= \min_{t\in[1-\theta,\theta]}(Au)(t)
\geq \theta^{*}\| Au\| > a.
\]
To sum up, all the hypotheses of the Leggett-Williams fixed
point theorem are satisfied. Therefore, $A$ has at least three fixed
points; that is, \eqref{1.1} has at least three positive
solutions $u, v$ and $w$ satisfying
\[
\|u\|\frac{a_j}{H_2},\quad t\in [1-\theta,\theta],\;
u\in[a_j,\frac{a_j}{\theta^{*}}],\; 1\leq j \leq
m-1.\label{3.5}
\end{gather}
Then \eqref{1.1} has at least $2m-1$ positive solutions in
$\overline{P_{d_{m}}}$.
\end{theorem}
\begin{proof}
We use induction on $m$. First, for $m = 1$, we know from
\eqref{3.4} that $A :\overline{P_{d_1}}\to P_{d_1}$.
Then it follows from Schauder fixed point theorem that
\eqref{1.1} has at least one positive solution in
$\overline{P_{d_1}}$.
Next, we assume that this conclusion holds for $m = k$. To
show that this conclusion also holds for $m=k+1$, we suppose that
there exist numbers $d_i$ $(1\leq i\leq k+1)$ and $a_j$
$(1\leq j \leq k)$ with
$0\frac{a_j}{H_2},\quad t\in [1-\theta,\theta],\;
u\in[a_j,\frac{a_j}{\theta^{*}}],\; 1\leq j \leq k.\label{3.7}
\end{gather}
By assumption, \eqref{1.1} has at least $2k-1$ positive
solutions $u_i\ (i = 1,2,\dots,2k-1)$ in $\overline{P_{d_k}}$.
At the same time, it follows from Theorem \ref{thm3.1},
\eqref{3.6} and \eqref{3.7} that \eqref{1.1} has at least three positive
solutions $u, v$ and $w$ in $\overline{P_{d_{k+1}}}$
such that
\[
\|u\|