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\AtBeginDocument{{\noindent\small {\em
Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 112, 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 (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/112\hfil Multiple positive solutions]
{Multiple Positive solutions for nonlinear third-order three-point
boundary-value problems}
\author[L.-J. Guo, J.-P. Sun and Y.-H. Zhao\hfil EJDE-2007/112\hfilneg]
{Li-Jun Guo, Jian-Ping Sun, Ya-Hong Zhao} % in alphabetical order
\address{Department of Applied Mathematics \\
Lanzhou University of Technology \\
Lanzhou, Gansu, 730050, China}
\email[L.-J. Guo]{school520@lut.cn}
\email[J.-P. Sun (Corresponding author) ]{jpsun@lut.cn}
\email[Y.-H. Zhao]{zhaoyahong88@sina.com}
\thanks{Submitted April 18, 2007. Published August 18, 2007.}
\thanks{Supported by the NSF of Gansu Province of China}
\subjclass[2000]{34B10, 34B18}
\keywords{Third-order boundary value problem; positive solution;
\hfill\break\indent
three-point boundary value problem; existence; cone; fixed point}
\begin{abstract}
This paper concerns the nonlinear third-order three-point
bound\-ary-value 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 <\frac 1\eta $. First,
we establish the existence of at least three positive
solutions by using the well-known Leggett-Williams fixed point theorem.
And then, we prove the existence of at least $2m-1$ positive
solutions for arbitrary positive integer $m$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
Third-order differential equations arise in a variety of different 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{g1}. Recently,
third-order boundary value problems (BVPs for short) have received much
attention. For example, \cite{d1,f1,h1,l2,y2} discussed some third-order
two-point BVPs, while \cite{a1,a2,m1,s1,y1} studied some third-order three-point
BVPs. In particular, Anderson \cite{a1} obtained some existence results of
positive solutions for the BVP
\begin{gather}
x'''(t)=f(t,x(t)),\quad t_1\leq t\leq t_3, \label{0.1}\\
x(t_1)=x'(t_2)=0,\quad \gamma x(t_3)+\delta x''(t_3)=0 \label{0.2}
\end{gather}
by using the well-known Guo-Krasnoselskii fixed point theorem \cite{g2,k1}
and Leggett-Williams fixed point theorem \cite{l1}. In 2005, the author in
\cite{s1} established various results on the existence of single and multiple
positive solutions to some third-order differential equations satisfying the
following three-point boundary conditions
\begin{equation}
x(0)=x'(\eta )=x''(1)=0, \label{0.3}
\end{equation}
where $\eta \in [\frac 12,1)$. The main tool in \cite{s1} was the
Guo-Krasnoselskii fixed point theorem.
Recently, motivated by the above-mentioned excellent works, we
\cite{g3} considered the third-order three-point BVP
\begin{gather}
u'''(t)+h(t)f(u(t))=0,\quad t\in (0,1), \label{1.1}\\
u(0)=u'(0)=0,\quad u'(1)=\alpha u'(\eta ),
\label{1.2}
\end{gather}
where $0<\eta <1$. By using the Guo-Krasnoselskii fixed point theorem, we
obtained the existence of at least one positive solution for the BVP
\eqref{1.1}--\eqref{1.2} under the assumption that $1<\alpha <\frac 1\eta $
and $f $ is either superlinear or sublinear.
In this paper, we will continue to study the BVP \eqref{1.1}--\eqref{1.2}.
First, some existence criteria for at least three positive solutions to the
BVP \eqref{1.1}--\eqref{1.2} are established by using the well-known
Leggett-Williams fixed point theorem. And then, for arbitrary positive
integer $m$, existence results for at least $2m-1$ positive solutions are
obtained.
In the remainder of this section, we state some fundamental concepts and the
Leggett-Williams fixed point theorem.
Let $E$ be a real Banach space with cone $P$. A map
$\sigma :P\to[0,+\infty )$ is said to be a nonnegative continuous
concave functional on $P $ if $\sigma $ is continuous and
\[
\sigma (tx+(1-t)y)\geq t\sigma (x)+(1-t)\sigma (y)
\]
for all $x$, $y\in P$ and $t\in [0,1]$. Let $a$, $b$ be two numbers such
that $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, \quad
\sigma (x_3)\frac{d_1}C,\quad u\in [ d_1,\frac{d_1}\gamma ] , \label{2} \\
f(u)<\frac cD,\quad u\in [ 0,c] . \label{2.1}
\end{gather}
Then the BVP \eqref{1.1}--\eqref{1.2} has at least three positive solutions.
\end{theorem}
\begin{proof}
Let the Banach space $E=C[ 0,1] $ be equipped
with the norm
\[
\| u\| =\max_{0\leq t\leq 1}| u(t)| .
\]
We denote
\[
P=\{ u\in E: u(t)\geq 0,\; t\in [ 0,1] \}.
\]
Then, it is obvious that $P$ is a cone in $E$. For $u\in P$, we define
\[
\sigma (u)=\min_{t\in [ \frac \eta \alpha ,\eta ] }u(t)
\]
and
\begin{equation}
Au(t)=\int_0^1G(t,s)h(s)f(u(s))ds,\quad t\in [0,1]. \label{3.1}
\end{equation}
It is easy to check that $\sigma $ is a nonnegative continuous concave
functional on $P$ with $\sigma (u)\leq \| u\| $ for $u\in P$ and
that $A:P\to P$ is completely continuous and fixed points of $A$ are
solutions of the BVP \eqref{1.1}--\eqref{1.2}.
We first assert that if there exists a positive number $r$ such that
$f(u)<\frac rD$ for $u\in [ 0,r] $, then $A:\overline{P_r}
\to P_r$.
Indeed, if $u\in \overline{P_r}$, then for $t\in [0,1]$,
\begin{align*}
(Au)(t)&=\int_0^1G(t,s)h(s)f(u(s))ds\\
&<\frac rD\int_0^1G(t,s)h(s)ds\\
&\leq \frac rD\max_{t\in [0,1]}\int_0^1G(t,s)h(s)ds=r.
\end{align*}
Thus, $\| Au\| d_1\} \neq \emptyset $ and $\sigma (Au)>d_1$ for all $u\in
P(\sigma ,d_1,d_1/\gamma )$.
In fact, the constant function
\[
\frac{d_1+d_1/\gamma }2\in \{ u\in P(\sigma ,d_1,d_1/\gamma
):\sigma (u)>d_1\} .
\]
Moreover, for $u\in P(\sigma ,d_1,d_1/\gamma )$, we have
\[
d_1/\gamma \geq \| u\| \geq u(t)\geq \min_{t\in [ \frac \eta
\alpha ,\eta ] }u(t)=\sigma (u)\geq d_1
\]
for all $t\in [ \frac \eta \alpha ,\eta ] $. Thus, in view of
(\ref{2}), we see that
\begin{align*}
\sigma (Au) &=\min_{t\in [ \frac \eta \alpha ,\eta ]
}\int_0^1G(t,s)h(s)f(u(s))ds\\
&\geq \min_{t\in [ \frac \eta \alpha ,\eta
] }\int_{\frac \eta \alpha }^\eta G(t,s)h(s)f(u(s))ds \\
&> \frac{d_1}C\min_{t\in [ \frac \eta \alpha ,\eta ] }\int_{\frac
\eta \alpha }^\eta G(t,s)h(s)ds=d_1
\end{align*}
as required.
Finally, we assert that if $u\in P(\sigma ,d_1,c)$ and
$\|Au\| >d_1/\gamma $, then $\sigma (Au)>d_1$.
To see this, we suppose that $u\in P(\sigma ,d_1,c)$ and
$\| Au\| >d_1/\gamma $, then, by Lemma \ref{lem2.2} and
Lemma \ref{lem2.3}, we have
\begin{align*}
\sigma (Au)&=\min_{t\in [ \frac \eta \alpha ,\eta ]
}\int_0^1G(t,s)h(s)f(u(s))ds\\
&\geq \gamma \int_0^1g(s)h(s)f(u(s))ds\geq \gamma
\int_0^1G(t,s)h(s)f(u(s))ds
\end{align*}
for all $t\in [ 0,1] $. Thus
\[
\sigma (Au)\geq \gamma \max_{t\in [ 0,1]
}\int_0^1G(t,s)h(s)f(u(s))ds=\gamma \| Au\| >\gamma \frac{d_1}
\gamma =d_1.
\]
To sum up, all the hypotheses of the Leggett-Williams theorem are satisfied.
Hence $A$ has at least three fixed points, that is, the BVP
\eqref{1.1}--\eqref{1.2} has at least three positive solutions $u$, $v$,
and $w$ such that
\[
\| u\| d_0, \quad \min_{t\in [ \frac \eta \alpha ,\eta ] }w(t)\frac{a_j}C,\quad u\in [ a_j,\frac{a_j}\gamma ] ,\quad
1\leq j\leq m-1. \label{4.2}
\end{gather}
Then, the BVP \eqref{1.1}--\eqref{1.2} 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 (\ref{4.1}) that
$A:\overline{P_{d_1}} \to P_{d_1}$, then, it follows from Schauder
fixed point theorem that the BVP \eqref{1.1}--\eqref{1.2} has at
least one positive solution in $\overline{P_{d_1}}$.
Next, we assume that this conclusion holds for $m=k$. In order to prove 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}C,\quad u\in [ a_j,\frac{a_j}\gamma ] ,\; 1\leq j\leq k.
\label{6}
\end{gather}
By assumption, the BVP \eqref{1.1}--\eqref{1.2} 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}, (\ref{5}) and (\ref{6}) that the BVP
\eqref{1.1}--\eqref{1.2} has at least three positive solutions
$u$, $v$, and $w$ in $\overline{P_{d_{k+1}}}$ such that
\[
\| u\| d_k, \quad \min_{t\in [ \frac \eta \alpha ,\eta ] }w(t)