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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 154, pp. 1--8.\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-2007/154\hfil Existence of positive solutions]
{Existence of positive solutions for the symmetry three-point
 boundary-value problem}

\author[Q. Ma \hfil EJDE-2007/154\hfilneg]
{Qiaozhen Ma}

\address{Qiaozhen Ma \newline
 College of Mathematics and Information Science,
 Northwest Normal University,\newline
 Lanzhou, Gansu, 730070, China}
\email{maqzh@nwnu.edu.cn}

\thanks{Submitted November 24, 2006. Published November 16, 2007.}
\thanks{Supported by grants 10671158 from NSFC,  10626042
 from the Mathematical Tianyuan \hfill\break\indent
 Foundation, and 3ZS061-A25-016 from the Natural Sciences Foundation of Gansu, and \hfill\break\indent  NWNU-KJCXGC-03-40}
\subjclass[2000]{34B10}
\keywords{Positive solution; three-point boundary value problem}

\begin{abstract}
 In this paper, we show the existence of single
 and multiple positive solutions for the symmetry three-point boundary
 value problem under suitable conditions by using classical fixed point
 theorem in cones.
\end{abstract}

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

 \section{Introduction}

 Since Gupta \cite{g2} studied three-point boundary value problems for
 the nonlinear ordinary differential equation, many classical
 results have been obtained by using Leray-Schauder continuation
 theorem, nonlinear alternatives of Leray-Schauder and coincidence
 degree theory. For more information, we refer the reader to
 \cite{f1,g2,m1,m2} and reference  therein.
 The study of multi-point boundary-value problems for linear
 second-order differential equations was initiated by II'in and
 Moiseev \cite{i1}.
 While the multi-point boundary value problem arise in the
 different areas of applied mathematics and physics. For instance,
 many problems in the theory of elastic stability can be handled
 as a multi-point problem \cite{t1}. Therefore, it's necessary to
 continue to extend and investigate.

 Ma \cite{m1}, by using fixed-point index theorems and Leray-Schauder
 degree and upper and lower solutions, considered the multiplicity
 of positive solutions of the problem
\begin{gather}
u''+\lambda h(t)f(u)=0, \quad t \in (0, 1), \label{e1.1}\\
u(0)=0, \quad u(1)=\alpha u(\eta), \label{e1.2}
\end{gather}
where $0<\eta<1$, $0<\alpha<1/\eta$, assuming that $f\in
C([0,\infty), [0,\infty))$, $h\in C([0,1), [0,\infty))$,
 and $f$ is superlinear. In the present paper, we study the existence
of single and multiple
positive solutions to nonlinear symmetry three-point boundary
value problem
\begin{gather}
 u''+\lambda a(t)f(u)=0, \quad t \in (0, 1), \label{e1.3} \\
 u(0)=\beta u(\eta), \quad u(1)=\alpha u(\eta). \label{e1.4}
\end{gather}
Where $\lambda>0$ is a positive parameter, $\alpha>0$, $\beta>0$,
$0<\eta<1$.

Clearly, problem \eqref{e1.3}-\eqref{e1.4} is more generic than
\eqref{e1.1}-\eqref{e1.2}, that is to say, our problem is
\eqref{e1.1}-\eqref{e1.2} for $\beta=0$. Moreover,
\eqref{e1.3}-\eqref{e1.4} is transformed immediately into the
classical Dirichlet problem for $\alpha=\beta=0$. And when
$\beta=0$, $\alpha=1$, $\eta\to 1$ problem
\eqref{e1.3}-\eqref{e1.4} is changed into the mixed boundary value
problem. In addition, our results will be obtained under
conditions that do not require $f$ to be either superlinear or
sublinear. In short, our problem gives a frame to these problems
under more generic conditions. We make the following assumptions.
\begin{itemize}
\item[(i)] $a\in C([0,1], [0,+\infty))$ and there exists $x_0 \in [0,1]$
such that $a(x_0)>0$.

\item[(ii)] $f \in C([0,+\infty), [0,+\infty))$ and there exist
nonnegative constants in the extended reals, $f_0$, $f_\infty$,
such that
$$f_0=\lim_{u\to 0+}\frac{f(u)}{u}, \quad
 f_\infty=\lim_{u\to \infty}\frac{f(u)}{u}.$$

\item[(iii)] $f(0)>0$, for $t \in[0,1]$.

\end{itemize}

\begin{remark} \label{rmk1.1} \rm
 It is easy to see that if (iii) holds, then
there exist two constants $a, b \in (0, \infty)$, such that
$0<f(u)\leq b$, for $u\in [0, a]$.
\end{remark}

The key tool in our approach is the following Krasnoselskii's
fixed point theorem in a cone.

\begin{theorem}[\cite{g1}] \label{thm1.1}
 Let $E$ be a Banach space and $K\subset E$ be a cone in $E$.
Suppose that $\Omega_1, \Omega_2$ are
bounded open subset of $K$ with $0\in \Omega_1,
\bar{\Omega}_1\subset \Omega_2$, and $A: K\to K$ is a
completely continuous operator such that either
\begin{gather*}
\|Aw\|\leq \|w\|, \quad w \in \partial\Omega_1, \quad
\|Aw\|\geq \|w\|, \quad w \in \partial\Omega_2, \quad\text{or}\\
\|Aw\|\geq \|w\|, \quad w \in \partial\Omega_1, \quad \|Aw\|\leq
\|w\|, \quad w \in \partial\Omega_2.
\end{gather*}
Then $A$ has a fixed point in $\bar{\Omega}_2\setminus \Omega_1$.
\end{theorem}

\section{Preliminary Lemmas}

\begin{lemma}[\cite{l1}] \label{lem2.1}
Let $\beta \neq \frac{1-\alpha\eta}{1-\eta}$. Then, for $y\in C[0,1]$,
boundary-value problem
\begin{gather}
u''+y(t)=0,\quad t\in (0,1),  \label{e2.1}\\
u(0)=\beta u(\eta), \quad u(1)=\alpha u(\eta). \label{e2.2}
\end{gather}
has a unique solution
\begin{align*}
u(t)=&-\int_{0}^{t}(t-s)y(s)ds+\frac{(\beta-\alpha)t-\beta}
 {(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}(\eta-s)y(s)ds\\
&+\frac{(1-\beta)t+\beta\eta}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)y(s)ds.
\end{align*}
\end{lemma}

\begin{lemma}[\cite{l1}] \label{lem2.2}
Let $0<\alpha<1/\eta$, $0<\beta<\frac{1-\alpha\eta}{1-\eta}$.
Then, for $y\in C[0,1]$, and $y\geq 0$, the unique solution of
problem \eqref{e2.1}-\eqref{e2.2} satisfies
$$
u(t) \geq 0, ~t\in[0,1].
$$
\end{lemma}

\begin{lemma}[\cite{l1}] \label{lem2.3}
Let $0<\alpha<\frac{1}{\eta}$,
$0<\beta<\frac{1-\alpha\eta}{1-\eta}$. Then, for $y\in C[0,1]$,
and $y\geq 0$, the unique solution of problem \eqref{e2.1}-\eqref{e2.2} satisfy
$$\min_{t\in [0,1]}u(t) \geq \gamma \|u\|,$$
where
$$
\gamma=\min\{\frac{\alpha(1-\eta)}{1-\alpha\eta}, \alpha\eta,
\beta\eta, \beta(1-\eta)\}.$$
\end{lemma}

Note that $u=u(t)$ is a solution of \eqref{e1.3}-\eqref{e1.4}, if and only if
\begin{equation}
\begin{aligned}
u(t)=&\lambda[-\int_{0}^{t}(t-s)a(s)f(u(s))ds+\frac{(\beta-\alpha)t-\beta}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}(\eta-s)a(s)f(u(s))ds\\
&+\frac{(1-\beta)t+\beta\eta}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(u(s))ds]:=A_\lambda u(t).
\end{aligned} \label{e2.3}
\end{equation}

Define a cone $K$ in the Banach space $C[0,1]$,
$$
K=\{u: u\in C[0,1],\;  u \geq 0,\;  \min_{t\in[0,1]}u(t)\geq \gamma
\|u\|\}.
$$
By Lemmas \ref{lem2.2} and \ref{lem2.3}, we know that $A_\lambda K \subset
K$ and it is not hard to verify that $A_\lambda: K\to K$ is
a completely continuous.

\section{Main Results}

Throughout this paper, we shall use the following notation
$$
A=\frac{1+\beta(1+\eta)}{(1-\alpha\eta)-\beta(1-\eta)}\int_{0}^{1}(1-s)a(s)ds,
\quad
B=\frac{\beta(1-\eta)}{(1-\alpha\eta)-\beta(1-\eta)}\int_{0}^{\eta}sa(s)ds.$$

Here and  below we assume that $\alpha\eta<1$.

 \begin{theorem} \label{thm3.1}
Suppose that (i)-(ii) hold. Then we have
\begin{enumerate}
\item If $Af_0<\gamma Bf_\infty$, then for each $\lambda\in
(\frac{1}{\gamma Bf_\infty}, \frac{1}{Af_0})$, the problem
\eqref{e1.3}-\eqref{e1.4} has at least one positive solution.

\item If $f_0=0$ and $f_\infty=\infty$, then for any $\lambda\in
(0,\infty)$, the problem \eqref{e1.3}-\eqref{e1.4} has at least one positive
solution.

\item If $f_\infty=\infty$, $0<f_0<\infty$, then for each
$\lambda\in (0,\frac{1}{Af_0})$, the problem \eqref{e1.3}-\eqref{e1.4} has at
least one positive solution.

\item If $f_0=0$, $0<f_\infty<\infty$, then for each $\lambda\in
(\frac{1}{\gamma Bf_\infty}, \infty)$, the problem \eqref{e1.3}-\eqref{e1.4} has
at least one positive solution.

\end{enumerate}
\end{theorem}

\begin{proof}
 Since the proof of (2)-(4) is similar to the proof of (1), we only prove (1).
Let $\lambda\in (\frac{1}{\gamma Bf_\infty}, \frac{1}{Af_0})$, and
choose $\varepsilon>0$ such that
\begin{equation}
\frac{1}{\gamma B(f_\infty -\varepsilon)}\leq
\lambda\leq\frac{1}{A(f_0+\varepsilon)}. \label{e3.1}
\end{equation}
By the definition of $f_0$, there exists $H_1>0$ such that
$f(x)\leq(f_0+\varepsilon)x$ for $x\in[0,H_1]$. Let $u\in K$ with
$\|u\|=H_1$, by \eqref{e2.3} and \eqref{e3.1}, we conclude that
\begin{equation}
\begin{aligned}
A_\lambda u(t)
 &\leq \frac{\lambda\beta t}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}(\eta-s)a(s)f(u(s))ds\\
&\quad +\frac{\lambda(t+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(u(s))ds\\
&\leq \frac{\lambda\beta }{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(u(s))ds\\
&\quad +\frac{\lambda(1+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(u(s))ds\\
&=\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(u(s))ds\\
&\leq\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)(f_0+\varepsilon)u(s)ds\\
&\leq\lambda A(f_0+\varepsilon)\|u\|\leq \|u\|.
\end{aligned}
\label{e3.2}
\end{equation}
As a result, $\|A_\lambda u\| \leq \|u\|$.  Let
$\Omega_1=\{u\in K: \|u\|<H_1\}$, then
\begin{equation}
\|A_\lambda u\| \leq \|u\|, \quad \text{for }  u\in K\cap
\partial \Omega_1. \label{e3.3}
\end{equation}
Again thanks to the definition of $f_\infty$, there exists
$\hat{H}_2>0$ such that $f(x)\geq (f_\infty-\varepsilon)x$, for
every $x\in [\hat{H}_2, \infty)$. Denote
$H_2=\max\{2H_1, \frac{\hat{H}_2}{\gamma}\}$,
 $\Omega_2=\{u\in K: \|u\|<H_2\}$.

If $u\in K$ with $\|u\|=H_2$, then
$\min_{t\in[0,1]}u(t)\geq\gamma\|u\|\geq \hat{H}_2$. It leads to
\begin{equation}
\begin{aligned}
A_\lambda u(0)&=-\frac{\lambda\beta
}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}(\eta-s)a(s)f(u(s))ds\\
&\quad +\frac{\lambda\beta \eta}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(u(s))ds\\
&\geq-\frac{\lambda\beta }{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}(\eta-s)a(s)f(u(s))ds\\
&\quad +\frac{\lambda\beta \eta}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}(1-s)a(s)f(u(s))ds\\
&=\frac{\lambda\beta(1- \eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}sa(s)f(u(s))ds\\
&\geq\frac{\lambda\beta(1- \eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}sa(s)(f_\infty-\varepsilon)u(s)ds\\
&\geq\lambda\gamma B(f_\infty-\varepsilon)\|u\| \geq \|u\|.
\end{aligned}
\label{e3.4}
\end{equation}
Consequently, $\|A_\lambda u\|\geq \|u\|$ for
$u\in K \cap \partial\Omega_2$.

Thus, according to the first condition of Theorem \ref{thm1.1}, $A_\lambda$
has a fixed point $u(t)$ with $H_1\leq \|u\| \leq H_2$ in $K \cap
(\bar{\Omega}_2\setminus\Omega_1)$.
\end{proof}

 \begin{theorem} \label{thm3.2}
Suppose that (i)-(ii) hold. Then we have
\begin{enumerate}
\item If $Af_\infty<\gamma Bf_0$, then for each $\lambda\in
(\frac{1}{\gamma Bf_0}, \frac{1}{Af_\infty})$, the problem
\eqref{e1.3}-\eqref{e1.4} has at least one positive solution.

\item If $f_0=\infty$ and $f_\infty=0$, then for any $\lambda\in
(0,\infty)$, the problem \eqref{e1.3}-\eqref{e1.4} has at least one positive
solution.

\item If $f_\infty=\infty$, $0<f_0<\infty$, then for each
$\lambda\in (0,\frac{1}{Af_\infty})$, the problem \eqref{e1.3}-\eqref{e1.4} has
at least one positive solution.

\item If $f_0=0$, $0<f_\infty<\infty$, then for each $\lambda\in
(\frac{1}{\gamma Bf_0}, \infty)$, the problem \eqref{e1.3}-\eqref{e1.4} has at
least one positive solution.
\end{enumerate}
\end{theorem}

\begin{proof} Since the proof of (2)-(4) is similar to the proof of (1),
we only prove (1).
Let $\lambda\in (\frac{1}{\gamma Bf_0}, \frac{1}{Af_\infty})$, and
choose $\varepsilon>0$ such that
\begin{equation}
\frac{1}{\gamma B(f_0 -\varepsilon)}\leq
\lambda\leq\frac{1}{A(f_\infty+\varepsilon)}. \label{e3.5}
\end{equation}
 By the definition of $f_0$, there exists $H_3>0$ such that
$f(x)\geq(f_0-\varepsilon)x$ for $x\in[0,H_3]$. Let $u\in K$ with
$\|u\|=H_3$ such that $\min_{t\in[0,1]}u(t)\geq\gamma\|u\|$.
Similar to the estimates of \eqref{e3.4}, we obtain
\begin{equation}
\begin{aligned} A_\lambda u(0)&\geq\frac{\lambda\beta(1-
\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}sa(s)f(u(s))ds\\
&\geq\frac{\lambda\beta(1- \eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}sa(s)(f_0-\varepsilon)u(s)ds\\
&\geq\lambda\gamma B(f_0-\varepsilon)\|u\| \geq \|u\|.
\end{aligned}
\label{e3.6}
\end{equation}
Hence, it follows that $\|A_\lambda u\|\geq \|u\|$. Set
$\Omega_1=\{u\in K: \|u\|<H_3\}$, we claim
 $$
\|A_\lambda u\|\geq \|u\|,\quad \text{for } u\in K\cap
\partial\Omega_1.
$$
Again in line with the definition of $f_\infty$, there exists
$\tilde{H}_4$ such that $f(x)\leq (f_0+\varepsilon)x$, for
$x\in[\tilde{H}_4, \infty)$. We discuss two possible cases:.

\subsection*{Case 1}
 Suppose that $f$ is bounded, that is, there exists a
positive constant $M_1$ such that $f(x)\leq M_1$ for all $x\in [0,
\infty)$. Set $H_4=\max\{2H_3, \lambda M_1A\}$. If $u\in K$ with
$\|u\|=H_4$, similar to \eqref{e3.2}, we obtain
\begin{equation}
\begin{aligned} A_\lambda u(t)
 &\leq\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(u(s))ds\\
&\leq\lambda M_1A\leq H_4= \|u\|.
\end{aligned}
\label{e3.7}
\end{equation}
Thus, by setting $\Omega_2=\{u\in K: \|u\|<H_4\}$, we get
$$
\|A_\lambda u\|\leq \|u\|, \quad \text{for}
\quad u\in K\cap \partial\Omega_2.
$$

\subsection*{Case 2}
 Suppose that $f$ is unbounded, we choose $H_4>\max\{2H_3,
\gamma^{-1}\tilde{H}_4\}$ such that $f(x)\leq f(H_4)$, for
 $x\in [0,H_4]$. Let $u\in K$ with $\|u\|=H_4$, we have
\begin{equation}
\begin{aligned}
A_\lambda u(t)
 &\leq\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(u(s))ds\\
&\leq\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f(H_4)ds\\
&\leq\lambda A(f_\infty+\varepsilon) H_4 \leq \|u\|.
\end{aligned} \label{e3.8}
\end{equation}
Let $\Omega_2=\{u\in K: \|u\|<H_4\}$, this yields
$$
\|A_\lambda u\| \leq \|u\|, \quad \text{for }  u\in K \cap
\partial\Omega_2.
$$
As a result, from the above estimates and by Theorem \ref{thm1.1}, it
follows that $A_\lambda$ has a fixed point $u \in K \cap
(\bar{\Omega}_2\setminus \Omega_1)$.
\end{proof}

 \begin{theorem} \label{thm3.3}
Suppose that (i)-(ii) are true. In addition, assume
that there exist two positive constants $H_5, H_6$ with
$H_5<\gamma H_6$ and $AH_6\leq BH_5$ such that
\begin{enumerate}
\item $f(x)\leq \frac{H_5}{\lambda A}$, $\forall x\in [0,H_5]$,

\item $f(x)\geq \frac{H_6}{\lambda B}$, $\forall x\in [\gamma H_6,
H_6]$.
\end{enumerate}
Then problem \eqref{e1.3}-\eqref{e1.4} has at least one positive
solution $u^*\in K$ with $H_5\leq \|u^*\|\leq H_6$.
\end{theorem}

The proof is similar to the proofs of Theorems \ref{thm3.1}
 and \ref{thm3.2}, so we omit it.

\begin{theorem} \label{thm3.4}
 Suppose that (i)-(iii) hold, moreover,
$f_\infty=\infty$. Then there exists a positive constant
$\Lambda_1$ such that problem \eqref{e1.3}-\eqref{e1.4} has at least two
positive solutions for $\lambda$ small enough.
\end{theorem}


\begin{proof} From (3) of theorems \ref{thm3.1} and \ref{thm3.2},
 we can see that
\eqref{e1.3}-\eqref{e1.4} has a positive solution $u_1$ satisfying
\begin{equation}
\|u_1\|\geq H, \label{e3.9}
\end{equation}
where $H$ is a suitable constant for $\lambda\in (0,\mu^*)$, and
$\mu^*=\min\{\frac{1}{Af_0}, \frac{1}{Af_\infty} \}$.

To find the second positive solution of \eqref{e1.3}-\eqref{e1.4}, we set
\begin{equation}
f^*(u)= \begin{cases}
f(u), &\text{for }  u\in [0,a],\\
f(a), &\text{for } u\in [a, \infty),
\end{cases}  \label{e3.10}
\end{equation}
then $0<f^*(u)\leq b$ for $u\in[0, \infty)$, where $a, b$ are
given in remark \ref{rmk1.1}.

Now we consider the auxiliary equation
\begin{equation}
u''+\lambda a(t) f^*(u)=0, \quad t\in (0,1) \label{e3.11}
\end{equation}
with the boundary value conditions
\begin{equation}
u(0)=\beta u(\eta), \quad u(1)=\alpha u(\eta). \label{e3.12}
\end{equation}
It is easy to check that \eqref{e3.11}-\eqref{e3.12} is equivalent
to the fixed point equation $u=F_\lambda u$, where
\begin{align*}
F_\lambda u(t)
&:=\lambda[-\int_{0}^{t}(t-s)a(s)f^*(u(s))ds \\
&\quad +\frac{(\beta-\alpha)t-\beta}{(1-\alpha\eta)-\beta(1-\eta)}
 \int_{0}^{\eta}(\eta-s)a(s)f^*(u(s))ds\\
&\quad +\frac{(1-\beta)t+\beta\eta}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f^*(u(s))ds].
\end{align*}
Clearly, $F_\lambda: K\to K$ is completely continuous and
$F_\lambda (K) \subset K$. Set
\begin{gather}
H_7=\min\{\frac{H}{2}, a\}, \label{e3.13} \\
\Lambda=
\min\{H_7[\frac{(1+\beta+\beta\eta)M}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)ds]^{-1}, \mu^*\} \nonumber
\end{gather}
and fix $\lambda \in (0,\Lambda)$, where
$M=\max\{f^*(u): 0\leq u\leq H_7\}$.

Choose $\Omega_3=\{u\in C[0,1]: \|u\|<H_7\}$, then for
$u\in K\cap\partial\Omega_3$, we have
\begin{equation}
\begin{aligned} F_\lambda u(t)
 &\leq\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)f^*(u(s))ds\\
&\leq\frac{\lambda
M(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}(1-s)a(s)ds\\
&\leq H_7.
\end{aligned} \label{e3.14}
\end{equation}
Therefore, $\|F_\lambda u\| \leq \|u\|$,  for $u\in K \cap
\partial\Omega_3$.

From (iii) we know that $\lim_{u\to 0+}\frac{f^*(u)}{u}=+\infty$.
This means that there exists a
constant $H_8$ $(H_8<H_7)$ such that $f^*(u) \geq \rho u$ for
$u\in[0,H_8]$, where
$$
\frac{\lambda\rho\beta\gamma(1-\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{1}sa(s)ds \geq 1.
$$
Also
\begin{equation}
\begin{aligned}
F_\lambda u(0)&\geq\frac{\lambda\beta(1-
\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}sa(s)f^*(u(s))ds\\
&\geq\frac{\lambda\beta(1- \eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}sa(s)\rho u(s)ds\\
&\geq
\frac{\lambda\rho\beta\gamma(1-\eta)}{(1-\alpha\eta)-\beta(1-\eta)}
\int_{0}^{\eta}sa(s)ds \|u\| \geq \|u\|.
\end{aligned}
\label{e3.15}
\end{equation}
Thus, we may let $\Omega_4=\{u\in C[0,1]: \|u\|<H_8\}$, so that
$\|F_\lambda u\|\geq \|u\|$, for  $u\in K\cap \partial\Omega_4$.

By the second part of Theorem \ref{thm1.1}, it follows that
\eqref{e3.11}-\eqref{e3.12}
has a positive solution $u_2$ satisfying
\begin{equation}
H_8\leq \|u_2\|\leq H_7. \label{e3.16}
\end{equation}
Combining with \eqref{e3.10}, \eqref{e3.13}, we obtain that $u_2$ is also a
solution of \eqref{e1.3}-\eqref{e1.4}.

In other words, from \eqref{e3.9} and \eqref{e3.16} we show
that \eqref{e1.3}-\eqref{e1.4} has two
distinct positive solutions $u_1$ and $u_2$ for $\lambda \in (0,\Lambda_1)$.
\end{proof}

\begin{theorem} \label{thm3.5}
Suppose that (i)-(iii) hold, furthermore,
$f_0=f_\infty=0$. Then the problem \eqref{e1.3}-\eqref{e1.4} has at least two
positive solutions for $\lambda $ large enough.
\end{theorem}

Proof is the same as that of Theorem \ref{thm3.4}, we omit it.


\begin{thebibliography}{00}

\bibitem{f1} W. Feng and J .R. L.Webb;
\emph{Solvability of a three-point nonlinear
boundary value problem at resonance}.  Nonlinear Analysis(TMA),
1997, 30(6), 3227-3238.

\bibitem{g1} D. Guo and V.Lakshmikantham;
\emph{Nonlinear problems in abstract cones}.
Academic Press, San Diego, 1988.

\bibitem{g2} C. P. Gupta;
\emph{Solvability of three-point nonlinear boundary value
problem for a second order ordinary differential equation.}
J. Math. Anal. Appl., 168(1992), 540-551.

\bibitem{i1} V. A. II'in, E. I.  Moiseev;
\emph{Nonlocal boundary value problem
of the second kind for a Sturm-Liouville operator}. Differential
Equations, 1987, 23(8), 979-987.

\bibitem{l1} H. Luo and Q. Ma;
\emph{positive solutions to a generalized
second-order three-point boundary value problem on time scales}.
Electronic Journal of Differential Equations. 2005, 17, 1-14.

\bibitem{m1} R. Ma;
\emph{Multiplicity of positive solutions for second-order
three-point boundary value problem}. Comp. Math. Appl. 40(2000),
193-204.

\bibitem{m2} R. Ma and Q. Ma;
\emph{Positive solutions for semipositone m-point
boundary value problem. Acta Mathematica Sinica}, English series.
2004, Vol.20, No.2, 273-282.

\bibitem{t1} S. Timoshenko; \emph{Theory of elastic stability}.
McGraw-Hill, New York, 1961.

\end{thebibliography}

\end{document}