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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 $0<a<b$ and $\sigma $ be a nonnegative continuous concave functional on
$P$. We define the following convex sets
\begin{gather*}
P_a=\{ x\in P: \| x\| <a\} , \\
P(\sigma ,a,b)=\{ x\in P: a\leq \sigma (x),\; \|x\| \leq b\} .
\end{gather*}

\begin{theorem}[Leggett-Williams fixed point theorem] \label{thm1.1}
 Let $A :\overline{P_c}\to \overline{P_c}$ be completely continuous and
$\sigma $ be a nonnegative continuous concave functional on $P$ such that
$\sigma (x)\leq \| x\| $ for all $x\in \overline{P_c}$. Suppose
that there exist $0<d<a<b\leq c$ such that
\begin{itemize}
\item[(i)] $\{ x\in P(\sigma ,a,b):\sigma (x)>a\} \neq
\emptyset $ and $\sigma (Ax)>a$ for $x\in P(\sigma ,a,b)$;

\item[(ii)]  $\| Ax\| <d$ for $\|x\| \leq d$;

\item[(iii)] $\sigma (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
\[
\| x_1\| <d, \quad a<\sigma (x_2),\quad \|x_3\| >d, \quad
\sigma (x_3)<a.
\]
\end{theorem}

\section{Preliminary Lemmas}

In this section, we present several important lemmas whose proof can be
found in \cite{g3}.

\begin{lemma} \label{lem2.1}
Let $\alpha \eta \neq 1$. Then for $y\in C[0,1] $, the BVP
\begin{gather}
u'''(t)+y(t)=0,\quad t\in (0,1),  \label{(2.1)} \\
u(0)=u'(0)=0,\quad u'(1)=\alpha u'(\eta ) \label{2.2}
\end{gather}
has a unique solution $u(t)=\int_0^1G(t,s)y(s)ds$, where
\begin{equation}
G(t,s)=\frac 1{2(1-\alpha \eta )}\begin{cases}
(2ts-s^2)(1-\alpha \eta )+t^2s(\alpha -1), & s\leq \min \{\eta ,t\}, \\
t^2(1-\alpha \eta )+t^2s(\alpha -1), & t\leq s\leq \eta ,\\
(2ts-s^2)(1-\alpha \eta )+t^2(\alpha \eta -s), & \eta \leq s\leq t, \\
t^2(1-s), & \max \{\eta ,t\}\leq s
\end{cases}
  \label{2.30}
\end{equation}
is called the Green's function.
\end{lemma}

For convenience, we denote
\begin{equation}
g(s)=\frac{1+\alpha }{1-\alpha \eta }s(1-s), \quad s\in [ 0,1].  \label{2.05}
\end{equation}
For the Green's function $G(t,s)$, we have the following two lemmas.

\begin{lemma} \label{lem2.2}
Let $1<\alpha <\frac 1\eta $. Then for any
$(t,s)\in [0,1]\times [0,1]$,
\[
0\leq G(t,s)\leq g(s).
\]
\end{lemma}

\begin{lemma} \label{lem2.3}
 Let $1<\alpha <\frac 1\eta $. Then for any
 $(t,s)\in [ \frac \eta \alpha ,\eta ] \times [0,1]$,
\[
\gamma g(s) \leq G(t,s),
\]
where $0<\gamma =\frac{\eta ^2}{2\alpha ^2(1+\alpha )}\min \{\alpha
-1,1\}<1 $.
\end{lemma}

\section{Main results}

In the remainder of this paper, we assume that the following conditions are
satisfied:
\begin{itemize}
\item[(A1)] $1<\alpha <\frac 1\eta $;

\item[(A2)] $f\in C([0,\infty ),[0,\infty ))$;

\item[(A3)] $h\in C([0,1],[0,\infty ))$ and is not identical zero on $[ \frac
\eta \alpha ,\eta ] $.
\end{itemize}
For convenience, we let
\begin{gather*}
D=\max_{t\in [ 0,1] }\int_0^1G(t,s)h(s)ds, \\
C=\min_{t\in [ \frac \eta \alpha ,\eta ] }
\int_{\frac \eta \alpha }^\eta G(t,s)h(s)ds.
\end{gather*}

\begin{theorem} \label{thm3.1}
Assume that there exist numbers $d_0$, $d_1$ and $c$
with $0<d_0<d_1<\frac{d_1}\gamma <c$ such that
\begin{gather}
f(u)<\frac{d_0}D,\quad u\in [ 0,d_0] ,  \label{1} \\
f(u)>\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, $\| Au\| <r$, that is, $Au\in P_r$. Hence, we have shown
that if (\ref{1}) and (\ref{2.1}) hold, then $A$ maps $\overline{P_{d_0}}$
into $P_{d_0}$ and $\overline{P_c}$ into $P_c$.

Next, we assert that $\{ u\in P(\sigma ,d_1,d_1/\gamma )
:\sigma (u)>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
\[
\| u\| <d_0,\quad d_1<\min_{t\in [ \frac \eta \alpha ,\eta ] }v(t),
\quad \| w\| >d_0, \quad \min_{t\in [ \frac \eta \alpha ,\eta ] }w(t)<d_1.
\]
\end{proof}

\begin{theorem} \label{thm3.2}
Let $m$ be an arbitrary positive integer.
Assume that there exist numbers $d_i$ ($1\leq i\leq m$) and $a_j$
($1\leq j\leq m-1$) with
$0<d_1<a_1<\frac{a_1}\gamma <d_2<a_2<\frac{a_2}\gamma <\dots
<d_{m-1}<a_{m-1}<\frac{a_{m-1}}\gamma <d_m$ such that
\begin{gather}
f(u)<\frac{d_i}D,\quad u\in [ 0,d_i] ,\quad 1\leq i\leq m,
\label{4.1}
\\
f(u)>\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
$0<d_1<a_1<\frac{a_1}\gamma <d_2<a_2<\frac{a_2}\gamma
<\dots <d_k<a_k<\frac{a_k}\gamma <d_{k+1}$ such that
\begin{gather}
f(u)<\frac{d_i}D,\quad u\in [ 0,d_i] ,\; 1\leq i\leq k+1, \label{5} \\
f(u)>\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
\[
\| u\| <d_k, \quad a_k<\min_{t\in [ \frac \eta \alpha,\eta ] }v(t),
 \quad \| w\| >d_k, \quad \min_{t\in [ \frac \eta \alpha ,\eta ] }w(t)<a_k.
\]
Obviously, $v$ and $w$ are different from $u_i$ ($i=1,2,\dots,2k-1$).
Therefore, the BVP \eqref{1.1}--\eqref{1.2} has at least $2k+1$ positive
solutions in $\overline{P_{d_{k+1}}}$, which shows that this conclusion also
holds for $m=k+1$.
\end{proof}

\noindent
\textbf{Example 3.3.} We consider the  BVP
\begin{gather}
u'''(t)+24f(u(t))=0,\quad t\in (0,1), \label{10} \\
u(0)=u'(0)=0,\quad u'(1)=\frac 32u'(\frac 12), \label{11}
\end{gather}
where
\[
f(u)=\begin{cases}
\frac{u^2+1}{28}, & u\in [ 0,\frac 12] , \\
\frac{275}{56}u-\frac{135}{56}, & u\in [ \frac 12,1] , \\
2u^{\frac 14}+\frac 12, & u\in [ 1,90] , \\
\frac{u-90}{20}(160\cdot 110^{\frac 18}-2\cdot 90^{\frac 14}-\frac
12)
+2\cdot 90^{\frac 14}+\frac 12, & u\in [90,110], \\
160u^{\frac 18}, & u\in [110,\infty).
\end{cases}
\]
A simple calculation shows that
\[
D=11,\quad C=\frac{11}{27}, \quad \gamma =\frac 1{90}.
\]
Let $m=3$. If we choose
\[
d_1=\frac 12,\quad d_2=90.1,\quad d_3=11000,\quad a_1=1, \quad
a_2=110,
\]
then the conditions (\ref{4.1}) and (\ref{4.2}) are satisfied.
Therefore, it follows from Theorem \ref{thm3.2} that the BVP
\eqref{10}--\eqref{11} has at least five positive solutions.

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\end{document}