\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 159, pp. 1--10.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/159\hfil Positive solutions of fourth-order
four-point BVP]
{Multiple  positive solutions of fourth-order
 four-point boundary-value problems with changing sign coefficient}

\author[Z. Fang, C. Li, C. Bai \hfil EJDE-2008/159\hfilneg]
{Zheng Fang, Chunhong Li, Chuanzhi Bai}  

\address{Zheng Fang  \newline
School of Science\\
Jiangnan University \\
Wuxi, Jiangsu 214122,  China}
\email{fangzhengm@yahoo.com.cn}

\address{Chunhong Li \newline
Department of  Mathematics \\
Huaiyin Teachers College\\
Huaian, Jiangsi 223300,  China}
\email{lichshy2006@126.com}

\address{Chuanzhi Bai \newline
Department of  Mathematics \\
Huaiyin Teachers College\\
Huaian, Jiangsi 223300,  China}
\email{czbai8@sohu.com}


\thanks{Submitted September 3, 2008. Published December 3, 2008.}
\subjclass[2000]{34B10, 34B15}
\keywords{Four-point boundary-value problem; positive solution;
\hfill\break\indent fixed point on a cone}

\begin{abstract}
 In this paper, we investigate the existence of multiple positive
 solutions of the fourth-order four-point boundary-value problems
 \begin{gather*} 
 y^{(4)}(t) = h(t) g(y(t), y''(t)), \quad  0 < t < 1, \\
 y(0) = y(1) = 0, \\
 a y''(\xi_1) - b y'''(\xi_1) = 0, \quad  c y''(\xi_2) + d y'''(\xi_2) = 0,
 \end{gather*}
 where $0 < \xi_1 < \xi_2 < 1$. We show the existence of three
 positive solutions by applying the Avery and Peterson fixed point
 theorem in a cone, here $h(t)$ may change sign on $[0, 1]$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

 Recently, several authors have studied  the existence of positive solutions
 to boundary-value problems for fourth-order
differential equations. For details; see, for example,
\cite{b2,h1,l1,l2,y1,z1,z2,z3}.  Zhong, Chen and Wang
\cite{z3} investigated the fourth-order nonlinear ordinary
differential equation
\begin{equation}
 y^{(4)}(t) - f(t, y(t), y''(t)) = 0,
\quad   0 \leq t \leq 1, \label{e1.1}
\end{equation}
 with the four-point boundary conditions
 \begin{equation}
\begin{gathered}
y(0) = y(1) = 0, \\
a y''(\xi_1) - b y'''(\xi_1) = 0, \quad  c
y''(\xi_2) + d y'''(\xi_2) = 0,
\end{gathered}
\label{e1.2}
\end{equation}
 where  $f \in C([0, 1] \times [0, \infty) \times (- \infty, 0],
[0, \infty))$, $a, b, c, d$ are nonnegative constants, and $0 \leq
\xi_1 < \xi_2 \leq 1$. Some results on the existence of at least one
positive solution to BVP \eqref{e1.1}-\eqref{e1.2} are obtained by using the
Krasnoselskii fixed point theorem.
Their key result reads as follows.

\begin{lemma}[{\cite[Lemma 2.2]{z3}}] \label{lem1.1}
 If $\alpha = ad + bc + ac(\xi_2 -
\xi_1) \not= 0$ and $h(t) \in C[\xi_1, \xi_2]$, then the
boundary-value problem
\begin{equation*}
\begin{gathered}
u^{(4)}(t) = h(t), \quad  0 < t < 1, \\
u(0) = u(1) = 0, \\
 a u''(\xi_1) - b u'''(\xi_1) =0,
\quad c u''(\xi_2) + d u'''(\xi_2) = 0
\end{gathered}
\end{equation*}
 has a unique solution
 \begin{equation}
 u(t) = \int_0^1 G_1(t, s)
\int_{\xi_1}^{\xi_2} G_2(s, \tau) h(\tau) \,d\tau\, ds, \label{e1.3}
\end{equation}
where
\begin{equation} \label{e1.4}
\begin{gathered}
 G_1(t, s) = \begin{cases}
s(1 - t), & 0 \leq s \leq t \leq 1, \\
t(1 - s), & 0 \leq t < s \leq 1,
 \end{cases}\\
G_2(t, s) = \frac{1}{\alpha} \begin{cases}
(a(s - \xi_1) + b)(d + c(\xi_2 - t)), &  s < t  \leq 1, \  \xi_1 \leq s \leq \xi_2,\\
(a(t - \xi_1) + b)(d + c(\xi_2 - s)), & 0 \leq t \leq s, \  \xi_1
\leq  s \leq \xi_2.
\end{cases}
\end{gathered}
 \end{equation}
\end{lemma}

Unfortunately this lemma is wrong. Indeed, by
\cite[Lemma 2.1]{b1}, expression \eqref{e1.3} should be replaced by
\begin{equation}
\begin{aligned}
 u(t) & = \int_0^1 G_1(t, s) \int_t^{\xi_1} (\tau -
s) h(\tau) \,d\tau\,ds \\
& \quad  + \frac{1}{\delta} \int_0^1 G_1(t, s)
\int_{\xi_1}^{\xi_2} (a(\xi_1 - s) - b)(c(\xi_2 - \tau) + d)
h(\tau) \,d\tau\,ds,
\end{aligned}  \label{e1.5}
\end{equation}
 where  $\delta = a d + b c + a c (\xi_2 - \xi_1) >0$.
So the conclusions in \cite{z3} should be reconsidered.
 If $f(t, y(t), y''(t))$ in \eqref{e1.1}
 are replaced by $h(t) g(y(t), y''(t))$,  then \eqref{e1.1}  reduces to
\begin{equation}
y^{(4)}(t) - h(t) g(y(t), y''(t)) = 0, \quad  0 \leq
t \leq 1, \label{e1.6}
\end{equation}
 where $h \in C[0, 1]$ and $g \in C([0, \infty) \times (- \infty,
0], [0, \infty))$.


To the authors' knowledge, no one has studied the existence of
positive solutions for problem \eqref{e1.6}, \eqref{e1.2}  using
the assumption that $h(t)$ changes sign. Hence, the aim of this
paper is to investigate  the existence of  positive
solutions of the BVP \eqref{e1.6} and \eqref{e1.2}  by using a
triple positive fixed-point theorem of Avery and Peterson in
\cite{a1}.

\section{Preliminaries}

Let $E = \{y \in C^2[0, 1]:y(0) = y(1) = 0\}$. Then we have the
following lemma.

\begin{lemma}[\cite{z3}] \label{lem2.1}   For $y \in E$, we get
$$
 \|y\|_{\infty} \leq \|y^{\prime}\|_{\infty} \leq
\|y''\|_{\infty}, $$
 where  $\|y\|_{\infty} =
\sup_{t \in [0, 1]}|y(t)|$.
\end{lemma}

By Lemma \ref{lem2.1}, $E$ is a Banach space with the norm
$\|y\|= \|y''\|_{\infty}$.
We define the operator $T : E \to E$ by
 \begin{equation}
 T y(t) = \int_0^1 G_1(t, s) (Q y)(s) ds,  \label{e2.1}
 \end{equation}
 where $G_1(t, s)$ as in \eqref{e1.4}, and
\begin{equation}
\begin{aligned}
 (Q y)(s) & = \int_{\xi_1}^s (\tau - s) h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau  \\
 & \quad  + \frac{1}{\delta} \int_{\xi_1}^{\xi_2} (b - a(\xi_1 -
s))(c(\xi_2 - \tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau.
\end{aligned}  \label{e2.2}
\end{equation}
 Here,  $\delta = a d + b c + a c (\xi_2 - \xi_1) > 0$.

 From  \cite[Lemma 2.1]{b1}, we easily know that $u(t)$ is a solution of the
 four-point boundary-value problem  \eqref{e1.6}, \eqref{e1.2} if and only if $u(t)$
 is a fixed point of the operator $T$.

It is rather straightforward to show that
\begin{equation}
 0 \leq G_1(t, s) \leq G_1(s, s), \quad 0 \leq t, s \leq 1,
\label{e2.3}
\end{equation}
 and
\begin{equation}
  G_1(t, s) \geq  \omega G_1(s, s), \quad t \in [\omega, 1 -
\omega], \quad s \in [0, 1],
  \label{e2.4}
  \end{equation}
 where
\begin{equation}
 0 < \omega < \min \{\xi_1, 1-\xi_2\} <  \frac{1}{2}. \label{e2.5}
 \end{equation}

For the convenience of the reader, we present some definitions
from the cone theory in Banach spaces.

\subsection*{Definition}
  The map $\alpha$ is said to be a
nonnegative continuous concave functional on a cone $P$ of a real
Banach space $E$ provided that $\alpha: P \to [0, \infty)$ is
continuous and
$$  \alpha(tx +(1-t)y) \ge t \alpha(x) + (1-t)\alpha(y),
\quad \forall x, y \in P, \quad 0 \le t \le 1.  $$
 Similarly, we say the map $\beta$ is a nonnegative continuous
 convex functional on a cone $P$ of a real Banach space
 $E$ provided that $\beta: P \to [0, \infty)$ is continuous and
 $$ \beta(tx +(1-t)y) \le t \beta(x) + (1-t)\beta(y),
\quad  \forall x, y \in P, \quad 0 \le t \le 1.
$$

Let $\gamma$ and $\theta$ be nonnegative continuous convex
functionals on $P$, $\alpha$ be a nonnegative continuous concave
functional on $P$, and $\psi$ be a nonnegative continuous
functional on $P$. Then for positive real numbers $a, b, c$, and
$d$, we define the following convex sets:
\begin{gather*}
P(\gamma, d)  = \{x \in P : \gamma(x) < d\}, \\
P(\gamma, \alpha, b, d)  = \{x \in P : b
 \le \alpha(x), \gamma(x) \le d \}, \\
P(\gamma, \theta, \alpha, b, c, d)  = \{x \in P : b \le
\alpha(x), \theta(x) \le c, \gamma(x) \le d \},  \\
R(\gamma, \psi, a, d)  = \{x \in P : a \le
\psi(x), \gamma(x) \le d\}.
\end{gather*}
The following fixed-point theorem due to Avery and Peterson is
fundamental in the proof of our main result.

\begin{lemma}[\cite{a1}] \label{lem2.2}
Let $P$ be a cone in a real
Banach space $E$. Let $\gamma$ and $\theta$ be nonnegative
continuous convex functionals on $P$, $\alpha$ be a nonnegative
continuous concave functional on $P$, and $\psi$ be a nonnegative
continuous functional on $P$ satisfying $\psi(\lambda x) \le
\lambda \psi(x)$ for $0 \le \lambda \le 1$, such that for some
positive numbers $M$ and $d$,
 $$
\alpha(x) \le \psi(x) \quad \text{and}\quad \|x\| \le M \gamma(x),
$$
for all $x \in \overline{P(\gamma, d)}$. Suppose
$T: \overline{P(\gamma, d)} \to \overline{P(\gamma, d)}$
is completely continuous and there exist
positive numbers $a, b$, and $c$ with $a < b$ such that
\begin{itemize}
\item[(i)]  $\{x \in P(\gamma, \theta, \alpha, b, c, d) :
\alpha(x)
>b\}     \neq \empty$ and $\alpha(Tx) >b $ for
$x \in P(\gamma, \theta, \alpha, b, c, d)$;

\item[(ii)]  $\alpha(Tx) >b$ for $x \in P(\gamma,  \alpha, b,
    d)$ with $\theta(Tx) >c$;

 \item[(iii)]  $0 \not\in R(\gamma, \psi, a, d)$ and $\psi(Tx) <a$
    for $ x \in R(\gamma, \psi, a, d) $ with $\psi(x) =a$.
 \end{itemize}
 Then $T$ has at least three fixed points $x_1, x_2, x_3 \in
\overline{P(\gamma, d)}$, such that
$$ \gamma(x_i) \le d   \ \text{for }  i=1, 2, 3,  \quad  b <
\alpha(x_1), \quad  a < \psi(x_2) \ {\rm with}  \ \alpha(x_2) < b,
  \quad  \psi(x_3) < a. $$
  \end{lemma}

  \section{Main result}

Define the cone $P \subset E = \{y \in C^2[0, 1]:y(0) = y(1) =
0\}$  by
 $$
 P = \{y \in  E :  y(t) \geq 0, \;  y \text{ is  concave  on }  [0, 1]\}.
$$

 Let the nonnegative, increasing,  continuous functionals $\gamma$,
  $\psi$, $\theta$ and $\alpha$ be
 $$
\gamma(y) = \max _{0 \leq t \leq 1}|y''(t)|, \quad
\psi(y) = \theta(y) = \max _{0 \leq t \leq 1}|y(t)|, \quad
\alpha(y)=  \min _{\omega \le t\le 1 - \omega}|y(t)|.
$$
We make the following assumptions:
\begin{itemize}
\item[(H1)]  $g : [0, \infty) \times (-\infty, 0] \to [0, \infty)$
is continuous;

\item[(H2)]  $h \in C[0, 1]$,  $h(t) \leq 0$, $\forall t \in [0,
\xi_1]$,  $h(t) \geq 0$, $\forall t \in [\xi_1, \xi_2]$, $h(t)
\leq 0$, $\forall t \in [\xi_2, 1]$,  and $h(t)$ is not identically zero
on any subinterval of $[0, 1]$.
 \end{itemize}

\begin{lemma} \label{lem3.1}
   Assume  that {\rm (H1)--(H2)} hold. If $b \geq a
\xi_1$ and $d \geq c (1 - \xi_2)$, then $T : P \to P$ is
completely continuous.
\end{lemma}

\begin{proof}  For each $t \in [0, 1]$, we consider three
cases:

\noindent Case 1: $t \in [0, \xi_1]$. For any $y \in P$, we have from
\eqref{e2.2}, (H1), (H2) and $b \geq a \xi_1$ that
\begin{equation}
\begin{aligned}
 (Q y)(t) & = \int_t^{\xi_1} (t - \tau) h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau  \\
& \quad  + \frac{1}{\delta} \int_{\xi_1}^{\xi_2} (b - a \xi_1 + a
t)(c(\xi_2 - \tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \geq 0.
\end{aligned} \label{e3.1}
\end{equation}

\noindent Case 2: $t \in [\xi_1, \xi_2]$. For each $y \in P$, we have from
(H1), (H2) and \eqref{e2.2} that
\begin{equation}
\begin{aligned}
 (Q y)(t) & = \int_{\xi_1}^t (\tau - t) h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau  \\
 & \quad  + \frac{1}{\delta} \int_{\xi_1}^t (b - a \xi_1 + a t)(c(\xi_2 -
\tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau  \\
  & \quad  + \frac{1}{\delta} \int_t^{\xi_2} (b - a \xi_1 + a t)(c(\xi_2 -
\tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \\
& = \frac{1}{\delta} \int_{\xi_1}^t (b + a(\tau - \xi_1))(c (\xi_2
- t) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \\
 & \quad + \frac{1}{\delta} \int_t^{\xi_2} (b + a(t - \xi_1))(c(\xi_2 -
\tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \geq
0.
\end{aligned}
  \label{e3.2}
  \end{equation}

\noindent Case 3: $t \in [\xi_2, 1]$. For any $y \in P$, we have from (H1),
 (H2), \eqref{e2.2} and $d \geq (1 - \xi_2) c$ that
\begin{equation}
\begin{aligned}
 (Q y)(t) & = \int_{\xi_1}^{\xi_2} (\tau - t) h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau + \int_{\xi_2}^t (\tau - t)
h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau \\
 & \quad  + \frac{1}{\delta} \int_{\xi_1}^{\xi_2} (b - a \xi_1 + a
t)(c(\xi_2 - \tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \\
 & =  \frac{1}{\delta}
\int_{\xi_1}^{\xi_2}(b + a(\tau - \xi_1))(d - c(t - \xi_2))
h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau  \\
 & \quad  + \int_{\xi_2}^t (\tau - t) h(\tau)
 g\big(y(\tau), y''(\tau)\big) d \tau \geq 0.
\end{aligned}
  \label{e3.3}
  \end{equation}
  Thus, from \eqref{e3.1}-\eqref{e3.3}, we get
  \begin{equation}
 (Q y)(t) \geq 0,  \quad  t \in [0, 1]. \label{e3.4}
 \end{equation}
Therefore, by \eqref{e2.1}, $G_1(t, s) \geq 0$ and \eqref{e3.4},
we obtain
\begin{equation}
 (T y)(t) \geq 0,  \quad  t \in [0, 1]. \label{e3.5}
 \end{equation}
Obviously, we have  $(T u)(0) = (T u)(1) = 0$, and
 $$ (T u)''(t) = - (Q u)(t) \leq 0, \quad
t \in [0, 1].  $$
 Hence, $T : P \to P$.  Moreover,  it is easy to
check by the Arzera-Ascoli theorem that the operator $T$ is
completely continuous.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
 By $\delta = a d + b c + a c (\xi_2 - \xi_1)> 0$, $b \geq a \xi_1$
and $d \geq c(1 - \xi_2)$, we have $b > 0$
and $d > 0$.
\end{remark}

For convenience of notation, we set
\begin{gather}
 M = \int_0^{\xi_1} - \tau h(\tau) d \tau + \int_{\xi_2}^1 -(1 -
\tau) h(\tau) d\tau + \big(\xi_2 - \xi_1 + \frac{b
d}{\delta}\big) \int_{\xi_1}^{\xi_2} h(\tau) d\tau, \label{e3.6}\\
m = \min \{m_1, m_2\}, \label{e3.7}
\end{gather}
where
 \begin{equation}
 \begin{gathered}
m_1  = \frac{b d}{\delta} \int_{\xi_1}^{\xi_2} h(\tau) d \tau
\int_{\xi_1}^{\xi_2} G_1 (\omega, s)ds, \\
m_2  = \frac{b d}{\delta} \int_{\xi_1}^{\xi_2} h(\tau) d \tau
\int_{\xi_1}^{\xi_2} G_1 (1 - \omega, s)ds.
\end{gathered}
  \label{e3.8}
  \end{equation}
We are now in a position to present and prove our main results.

\begin{theorem} \label{thm3.3}
  Let $b \geq a \xi_1$ and $d \geq c(1 - \xi_2)$. Assume
{\rm (H1)--(H2)} hold. Suppose there exist constants $0 < p < q <
\min \{\omega, \frac{1}{8}\} r$ such that

 \begin{itemize}
\item[(H3)]  $ g(u, v) \le r/M$,  for $(u, v) \in [0, r]
\times [- r, 0]$,

\item[(H4)]  $g(u, v) > q/m$,  for
 $(u, v) \in [q,q/\omega] \times [- r, 0]$,

\item[(H5)]  $ g(u, v) < 8 p/M$,  for
 $(u, v) \in [0, p] \times [- r, 0]$,
\end{itemize}
 where $M, m$ are as in \eqref{e3.6}-\eqref{e3.7},
then  \eqref{e1.6}, \eqref{e1.2} has at
least three positive solutions $y_1$, $y_2$, and $y_3$ such that
\begin{gather*}
\max _{0 \leq t \leq 1} |y_i''(t)|  \le r, \quad
 \text{for }  i=1, 2, 3;  \\
\min _{\omega \le t \le 1 - \omega}|y_1(t)| > q; \quad
p < \max _{0 \leq t \leq 1}|y_2(t)|; \\
\min_{\omega \le t \le 1-\omega}|y_2(t)| < q; \quad
 \max _{0 \leq t \leq 1} |y_3(t)| < p.
\end{gather*}
\end{theorem}

\begin{proof}   From Lemma \ref{lem3.1},  $T : P
\to P$  is completely continuous.  We now show that all the
conditions of Lemma \ref{lem2.2}  are satisfied.

If $y \in \overline {P(\gamma, r)}$, then  $ \gamma(y) = \max_{0
\leq t \leq 1}|y''(t)| \le r$. By Lemma
\ref{lem2.1}, we have $\max_{0 \leq t \leq 1}|y(t)| \le r$, then
assumption (H3)  implies  $g(y(t), y''(t)) \le r/M$. On the other hand,
from \eqref{e3.1}-\eqref{e3.3},
we have
\begin{equation}
\begin{aligned}
 \max _{0 \leq t \leq \xi_1} (Q y)(t) & \leq \int_0^{\xi_1} -
\tau h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau \\
 & \quad  + \frac{1}{\delta} \int_{\xi_1}^{\xi_2} b (c(\xi_2 - \tau) + d)
h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau  \\
&  \leq \int_0^{\xi_1} - \tau h(\tau)
 g\big(y(\tau), y''(\tau)\big) d \tau  \\
& \quad  + \frac{1}{\delta} b(c(\xi_2 - \xi_1) + d)
\int_{\xi_1}^{\xi_2} h(\tau) g\big(y(\tau), y''(\tau)\big)
d\tau,
\end{aligned}
\label{e3.9} \end{equation}
\begin{equation}
\begin{aligned}
 \max _{\xi_1 \leq t \leq \xi_2} (Q y)(t) & \leq
\frac{1}{\delta} \int_{\xi_1}^t (b + a(t - \xi_1))(c (\xi_2 -
\tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \\
& \quad + \frac{1}{\delta} \int_t^{\xi_2} (b + a(t -
\xi_1))(c(\xi_2 - \tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \\
& = \frac{1}{\delta} \int_{\xi_1}^{\xi_2} (b + a(t -
\xi_1))(c(\xi_2 - \tau) + d) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \\
& \leq \frac{1}{\delta} (b + a(\xi_2 - \xi_1))(c(\xi_2 - \xi_1) +
d) \int_{\xi_1}^{\xi_2} h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau,
\end{aligned}
\label{e3.10} \end{equation}
 and
\begin{equation}
\begin{aligned}
 \max _{\xi_2 \leq t \leq 1} (Q y)(t) & \leq \frac{1}{\delta}
\int_{\xi_1}^{\xi_2} d (b + a(\tau - \xi_1)) h(\tau) g\big(y(\tau), y''(\tau)\big) d\tau \\
& \quad  + \int_{\xi_2}^1  - (1 - \tau) h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau \\
& \leq \frac{1}{\delta} d (b + a(\xi_2 - \xi_1))
\int_{\xi_1}^{\xi_2} h(\tau) g\big(y(\tau), y''(\tau)\big)
d\tau \\
& \quad  + \int_{\xi_2}^1  - (1 - \tau) h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau.
\end{aligned}
\label{e3.11} \end{equation}
 By \eqref{e3.9}-\eqref{e3.11}, we get
\begin{align}
 \gamma(T y)
& = \max _{t \in [0, 1]}|(T y)''(t)| = \max _{t \in [0, 1]}|(Q y)(t)| \notag \\
& = \max \Big\{\max _{0 \leq t \leq \xi_1}|(Q y)(t)|, \max _{\xi_1
\leq t \leq \xi_2}|(Q y)(t)|,  \max _{\xi_2
\leq t \leq 1}|(Q y)(t)|\Big\} \notag \\
& \leq \int_0^{\xi_1} - \tau h(\tau) g\big(y(\tau), y''(\tau)\big)
d \tau + \int_{\xi_2}^1 - (1 - \tau)
h(\tau)g\big(y(\tau), y''(\tau)\big) d \tau  \notag \\
& \quad +  \frac{1}{\delta} (b + a(\xi_2 - \xi_1))(c(\xi_2 -
\xi_1) + d) \int_{\xi_1}^{\xi_2}  h(\tau)
 g\big(y(\tau), y''(\tau)\big) d\tau \label{e3.12} \\
& \leq \frac{r}{M}  \Big(\int_0^{\xi_1} - \tau h(\tau) d
\tau + \int_{\xi_2}^1 - (1 - \tau) h(\tau)d\tau + \big(\xi_2 -
\xi_1 + \frac{b d}{\sigma}\big) \int_{\xi_1}^{\xi_2} h(\tau)
d\tau\Big) \notag \\
& = \frac{r}{M}  M = r. \notag
\end{align}
  Hence, $T: \overline {P(\gamma, r)}
\to \overline {P(\gamma, r)}$.

To check condition (i) of Lemma \ref{lem2.2}, we choose
$y(t) = q/\omega$, $0 \le t \le 1 $. It is easy to see that
$y(t)= q/\omega \in P(\gamma, \theta, \alpha, q,q/\omega, r)$ and
$\alpha(y)=q/\omega> q$, and so
$\{y \in P(\gamma, \theta, \alpha, q, q/\omega, r)
 : \alpha(y) > q\} \neq \emptyset$.
Hence, if $y \in P(\gamma, \theta, \alpha, q, q/\omega,r)$,
then $q \le y(t) \le q/\omega$, $ - r \leq y''(t) \le 0 $
for $\omega \le t \le 1 - \omega$. From
assumption (H4), we have $g(y(t), y''(t))> b/m$ for
$\omega \le t \le 1 - \omega$, and by the
definitions of $\alpha$ and the cone $P$, we  distinguish
two cases as follows:

\noindent Case (1): $\alpha(T y)= (T y)(\omega)$.  By \eqref{e3.4} and
\eqref{e3.2}, we have
\begin{align*}
& \alpha(T y) \\
&=(Ty)(\omega) = \int_0^1 G_1 (\omega, s) (Q y)(s) ds \\
 & >  \int_{\xi_1}^{\xi_2} G_1 (\omega, s) (Q y)(s) ds\\
 & \geq \frac{1}{\delta}
\int_{\xi_1}^{\xi_2} G_1 (\omega, s)ds \Big[\int_{\xi_1}^s b d
h(\tau) g\big(y(\tau), y''(\tau)\big) d \tau +
\int_s^{\xi_2} b d h(\tau) g\big(y(\tau), y''(\tau)\big) d
\tau \Big] \\
& = \frac{b d}{\delta} \int_{\xi_1}^{\xi_2} G_1 (\omega, s)ds
\int_{\xi_1}^{\xi_2} h(\tau) g\big(y(\tau), y''(\tau)\big) d
\tau  \\
& \geq \frac{b d}{\delta} \frac{q}{m} \int_{\xi_1}^{\xi_2}
G_1(\omega, s)ds \int_{\xi_1}^{\xi_2} h(\tau) d \tau \\
 &= \frac{q}{m} \cdot m_1 \geq q.
\end{align*}

\noindent Case (2): $\alpha(T y)= (T y)(1 - \omega)$.  Similarly, we obtain
\begin{align*}
 \alpha(T y)&=(T y)(1 - \omega)
   >  \int_{\xi_1}^{\xi_2} G_1 (1 - \omega, s) (Q y)(s) ds \\
   & \geq \frac{b d}{\delta} \frac{q}{m} \int_{\xi_1}^{\xi_2} G_1 (1
- \omega, s)ds \int_{\xi_1}^{\xi_2} h(\tau) d \tau  \\
& = \frac{q}{m} \cdot m_2 \geq q.
\end{align*}
 i.e.,
$$
\alpha(T y) > q, \quad   \forall  y \in P(\gamma,
\theta, \alpha, q, \frac{q}{\omega}, r).
$$
  This show that
condition (i) of Lemma \ref{e2.2}  is satisfied.  Secondly,  we
have
 $$
\alpha(T y) = \min _{\omega \leq t \leq
1-\omega} |(Ty)(t)| \geq  \omega  \|Ty\|_{\infty} = \omega
\theta(T y) >  \omega \frac{q}{\omega} = q,
$$
  for all $y \in P(\gamma, \alpha, q, r)$ with $\theta(T y) > q/\omega$.
Thus, condition (ii) of Lemma \ref{lem2.2} is satisfied.

We finally show that (iii) of Lemma \ref{lem2.2} also holds.
Clearly, as $\psi(0)=0 < p$, there holds that $0 \not\in R(\gamma,
\psi, p, r)$. Suppose that $y \in R(\gamma, \psi, p, r)$ with
$\psi(y) = p$. Then, by (H5) and \eqref{e3.12}, we get
\begin{align*}
 \psi(T y)
&= \max_{0 \leq t \leq 1}|(T y(t)|
 = \max_{0 \leq t \leq 1} \int_0^1 G_1(t, s) (Q y)(s)ds \\
&= \max _{0 \leq t \leq 1} \Big|\int_0^{\xi_1} G_1(t, s)
(Q y)(s) ds  + \int_{\xi_1}^{\xi_2} G_1(t, s) (Q y)(s) ds \\
&\quad + \int_{\xi_2}^1 G_1(t, s) (Q y)(s) ds \Big| \\
& \leq \max _{0 \leq t \leq 1} \Big[\max _{0 \leq s
\leq \xi_1}(Q y)(s) \int_0^{\xi_1} G_1(t, s) ds + \max
_{\xi_1 \leq s \leq \xi_2}(Q y)(s) \int_{\xi_1}^{\xi_2}
G_1(t, s) ds   \\
& \quad  + \max _{\xi_2 \leq s \leq 1}(Q y)(s)
\int_{\xi_2}^1 G_1(t, s)
 ds \Big] \\
& \leq \max \big\{\max _{0 \leq s \leq \xi_1}(Q y)(s), \
\max _{\xi_1 \leq s \leq \xi_2}(Q y)(s), \  \max
_{\xi_2 \leq s \leq 1} (Q y)(s) \big\} \max _{0
\leq t \leq 1} \int_0^1 \! G_1(t, s) ds \\
& \leq \max _{0 \leq t \leq 1} \int_0^1 G_1(t, s) ds
\Big[\int_0^{\xi_1} - \tau h(\tau) g\big(y(\tau),
 y''(\tau)\big) d \tau \\
&\quad + \int_{\xi_2}^1 - (1 - \tau) h(\tau)
  g\big(y(\tau), y''(\tau)\big) d \tau  \\
& \quad +  \frac{1}{\delta} (b + a(\xi_2 - \xi_1))(c(\xi_2 -
\xi_1) + d) \int_{\xi_1}^{\xi_2} h(\tau) g\big(y(\tau), y''(\tau)\big)
 d\tau \Big] \\
& \leq \max _{0 \leq t \leq 1} \int_0^1 G_1(t, s) ds \cdot
\frac{8 p}{M} \Big[\int_0^{\xi_1} - \tau h(\tau) d \tau +
\int_{\xi_2}^1 - (1 - \tau) h(\tau)d\tau  \\
& \quad +   \big(\xi_2 - \xi_1 + \frac{b d}{\sigma}\big)
\int_{\xi_1}^{\xi_2} h(\tau) d\tau\Big] \\
& =  \frac{1}{8} \cdot \frac{8 p}{M}\cdot M = p.
\end{align*}
 So, condition (iii) of Lemma \ref{lem2.2} is satisfied. Therefore, an
application of Lemma \ref{lem2.2}  imply the boundary-value
problem \eqref{e1.6}, \eqref{e1.2}  has at least three positive
solutions $y_1, y_2$, and $y_3$ such that
\begin{gather*}
\max _{0 \leq t \leq 1}|y_i''(t)|  \le r, \quad
 \text{for } i=1, 2, 3;  \quad   \min _{\omega \le t \le
1 - \omega}|y_1(t)| > q; \\
 p < \max _{0 \leq t \leq
1}|y_2(t)|, \quad   \min_{\omega \le t \le
1-\omega}|y_2(t)| < q; \quad  \max _{0 \leq t \leq 1}
|y_3(t)| < p.
\end{gather*}
 The proof is complete. \end{proof}


Now, we give an example to demonstrate our result.
Consider the  fourth-order four-point boundary-value problem
\begin{gather}
 y^{(4)}(t) = h(t) g(y(t), y''(t)), \quad 0 < t < 1,  \label{e3.13} \\
\begin{gathered}
 y(0) = y(1) = 0, \\
 y''(\frac{1}{3}) - y'''(\frac{1}{3}) = 0,
  \quad   y''(\frac{2}{3}) + y'''(\frac{2}{3}) = 0,
\end{gathered} \label{e3.14}
\end{gather}
where $\xi_1 = \frac{1}{3}$, $\xi_2 = \frac{2}{3}$, $h(t) = 9
\pi \sin (3t-1)\pi$,  and
\[
 g(u, v) = \begin{cases}
\frac{u^2}{2} - (\frac{v}{150})^3, &
 0 \leq u \leq 1, \ v \leq 0, \\
11 \sqrt[4]{u-1} - (\frac{v}{150})^3 + \frac{1}{2}, &
 1 < u \leq 9, \ v \leq 0, \\
11 \sqrt[4]{8} + \frac{1}{2} - (\frac{v}{150})^3, &
 u > 9, \ v \leq 0.
\end{cases}
\]
 It is easy to check that the functions $h$ and $g$ satisfy (H1) and (H2).
Set $\omega = 1/3$. It follows from a
direct calculation that
\begin{align*}
 M &= 9 \pi \Big[\int_0^{1/3} - \tau \sin (3 \tau - 1)\pi d \tau
+  \int_{2/3}^1  -(1 - \tau)\sin (3 \tau - 1)\pi d\tau
\\
& \quad  + \frac{16}{21} \int_{1/3}^{2/3} \sin (3 \tau -
1)\pi d\tau\Big] \\
& = \frac{46}{7},
 \end{align*}
 and
$$
m = 9 \pi \cdot \frac{3}{7}
\int_{1/3}^{2/3} \sin (3 \tau - 1)\pi d \tau \cdot \min
\Big\{\int_{1/3}^{2/3} G (\frac{1}{3}, s)ds, \int_{1/3}^{2/3} G
(\frac{2}{3}, s)ds\Big\} = \frac{2}{7}.
$$
 Choose $p = 1$, $q = 3$ and $r = 130$, then we have
\begin{gather*}
  g(u, v) \leq 1.151 < 1.21 = \frac{8
p}{M}, \quad \text{for }  0 \leq u \leq 1, \ - 130 \leq v \leq 0; \\
  g(u, v) \geq 14.232 > 10.5 =
\frac{q}{m}, \quad \text{for }  3 \leq u \leq 9, \ - 130 \leq v
\leq 0; \\
g(u, v) \leq 19.651 < 19.78 =
\frac{r}{M}, \quad \text{for }  0 \leq u \leq 130, \ - 130 \leq v
\leq 0.
\end{gather*}
 Noticing that $b > \xi_1 a$ and $d > (1 - \xi_2)c$
hold, then all conditions of  Theorem \ref{thm3.3} hold. Hence, by
Theorem \ref{thm3.3}, BVP \eqref{e3.13}, \eqref{e3.14} has at
least three positive solutions $y_1$, $y_2$ and $y_3$ such that
\begin{gather*}
\max _{0 \leq t \leq 1} |y_i''(t)|  \le 130, \quad \text{for  } i=1, 2, 3;  \quad
\min _{\frac{1}{3} \le t \le \frac{2}{3}}|y_1(t)| > 3; \\
1 < \max _{0 \leq t \leq 1}|y_2(t)|, \quad
\min_{\frac{1}{3} \le t \le \frac{2}{3}}|y_2(t)| < 3 \quad
\max _{0 \leq t \leq 1} |y_3(t)| < 1.
\end{gather*}

\subsection*{Acknowledgements}
The authors are grateful to the anonymous referee for his or
her useful suggestions. This research was supported by grant 10771212
from the National Natural Science Foundation of China,
and by grant 06KJB110010  from the  Natural Science
Foundation of  Jiangsu Education Office.


\begin{thebibliography}{00}

\bibitem{a1}  R. Avery, A. Peterson; \emph{Three positive fixed points
of nonlinear operators on order Banach spaces}, Comput. Math.
Appl. 42 (2001) 313-322.

\bibitem{b1}  C. Bai, D. Yang,  H. Zhu; \emph{ Existence of solutions
for fourth order differential equation with four-point boundary
conditions},  Appl. Math. Lett. 20(2007) 1131-1136.

\bibitem{b2}  Z. Bai, H. Wang;  \emph{On positive solutions of some
nonlinear fourth-ordr beam equations},  J. Math. Anal. Appl. 270
(2002) 357-368.

\bibitem{h1}  Z. Hao, L. Liu, L. Debnath; \emph{A necessary and
sufficiently condition for the existence of positive solution of
fourth-order singular boundary-value problems}, Appl. Math. Lett.
16 (2003) 279-285.

\bibitem{l1}   Y. Li;
\emph{Positive solutions of fourth-order boundary-value problems
with two parameters}, J. Math. Anal. Appl. 281
(2003) 477-484.

\bibitem{l2}   B. Liu;  \emph{Positive solutions of fourth-order two
point boundary-value problems},  Appl. Math. Comput. 148 (2004)
407-420.

\bibitem{y1}   D. Yang, H. Zhu, C. Bai;  \emph{Positive solutions for
semipositone fourth-order two-point boundary-value problems},
Electronic Journal of Differential Equations, Vol. 2007(2007), No.
16, 1-8.

\bibitem{z1}   X. Zhang, L. Liu, H. Zou;  \emph{Positive solutions of
fourth-order singular three point eigenvalue problems}, Appl.
Math. Comput.  189 (2007) 1359-1367.

\bibitem{z2}  M. Zhang, Z. Wei;  \emph{Existence of positive solutions
for fourth-order m-point boundary-value problem with variable
parameters},  Appl. Math. Comput. 190 (2007) 1417-1431.

\bibitem{z3}  Y. Zhong, S. Chen, C. Wang;  \emph{Existence results for
a fourth-order ordinary differential equation with a four-point
boundary condition},  Appl. Math. Lett. 21 (2008) 465-470.

\end{thebibliography}
\end{document}