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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 150, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/150\hfil Triple solutions]
{Triple solutions for multi-point boundary-value problem with
$p$-Laplace operator}

\author[H. Li, Y. Liu\hfil EJDE-2009/150\hfilneg]
{Haitao Li, Yansheng Liu}  % in alphabetical order

\address{Haitao Li \newline
Department of Mathematics, 
Shandong Normal University, Jinan, 250014, China}
\email{haitaoli09@gmail.com}

\address{Yansheng Liu \newline
Department of Mathematics, 
Shandong Normal University, Jinan, 250014, China}
\email{yanshliu@gmail.com}

\thanks{Submitted April 22, 2009. Published November 25, 2009.}
\thanks{Supported by grants 209072 from the Key Project of
 Chinese Ministry of Education, \hfill\break\indent
 and  J08LI10 from the Science
 and by Technology Development Funds of Shandong \hfill\break\indent
 Education Committee}
\subjclass[2000]{34B10, 34B15}
\keywords{Triple solutions; $p$-Laplace operator; fixed point theorem;
\hfill\break\indent  multi-point boundary-value problem}

\begin{abstract}
  Using a fixed point theorem due to Avery and Peterson,
  this article shows the existence of solutions for multi-point
  boundary-value problem with $p$-Laplace operator and parameters.
  Also, we present an example to illustrate the results obtained.
\end{abstract}

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

\section{Introduction }

 During the previous two decades, boundary-value problems for
second-order differential equations with $p$-Laplace operator have
been extensively studied and a lot of excellent results have been
established by using fixed point index theory, upper and lower
solution arguments, fixed point theorem like Leggett-Williams
multiple fixed point theorem and so on (see
\cite{g1,g2,h1,h2,h3,j1,l1,m1,w1,w2,w3,w4} and
references therein). For example,  Ma, Du and Ge \cite{m1} studied
the following boundary-value problem (BVP, for short) with
$p$-Laplace operator
\begin{gather*}
(\varphi_{p}(u'(t)))' + q(t)f(t, u(t))=0,\quad  t\in (0, 1);\\
u'(0)=\sum_{i=1}^{n}\alpha_{i}u'(\xi_{i}),
u(1)=\sum_{i=1}^{n}\beta_{i}u(\xi_{i}),
\end{gather*}
where $\varphi _{p}(s)=| s| ^{p-2}s$, $p>1$,
$\varphi _{p}^{-1}=\varphi_{q}$,
$ \frac{1}{p}+\frac{1}{q}=1$, and
$0<\xi_{1}<\xi_{2}<\dots <\xi_{n}<1$. The nonlinearity $f$ is
not depending on $u'$.
Using the upper and lower solutions method, they obtained sufficient
conditions for the existence of one positive solution.

Lv, O'Regan and Zhang \cite{l1} considered the following boundary-value problem
(BVP) with $p$-Laplace operator
\begin{gather*}
(\varphi_{p}(y'(t)))' + q(t)f(y(t))=0,\quad  t\in [0, 1];\\
y(0)=y(1)=0.
\end{gather*}
By Leggett-Williams multiple fixed point theorem, they provided
sufficient conditions for the existence of multiple (at least three)
positive solutions.

 Recently  Ji, Tian and Ge \cite{j1} studied the following boundary-value problem,
in which the nonlinearity contains $u'$,
\begin{equation} \begin{gathered}
(\varphi_{p}(u'(t)))' + \lambda f(t, u(t), u'(t))=0,\quad  t\in (0, 1);\\
u'(0)=\sum_{i=1}^{n}\alpha_{i}u'(\xi_{i}), \quad
u(1)=\sum_{i=1}^{n}\beta_{i}u(\xi_{i}).
\end{gathered} \label{e1.1}
\end{equation}
Applying Krasnosel'skii fixed point theorem, they obtained the
existence of at least one positive solution.

Wang and Ge \cite{w3} studied the  multi-point
boundary-value problem
\begin{gather*}
(\varphi_{p}(u'(t)))' + q(t)f(t, u(t), u'(t))=0,\quad  t\in (0, 1);\\
u(0)=\sum_{i=1}^{n}\alpha_{i}u(\xi_{i}), \quad
u(1)=\sum_{i=1}^{n}\beta_{i}u(\xi_{i}).
\end{gather*}
Using the fixed point theorem due to Avery and Peterson, they
provided sufficient conditions for the existence of multiple
positive solutions.

Motivated by \cite{j1,w3}, we  investigate \eqref{e1.1}. We study
boundary value conditions that are different from those in
\cite{l1,w3}. We obtain three solutions by the fixed point theorem
due to Avery and Peterson, which is different from the methods in
\cite{j1,l1,m1}. To the best of our knowledge, \eqref{e1.1} has not
been studied via this fixed point theorem.

This article is organized as follows. Section 2 gives some preliminaries.
Section 3 is devoted to the existence of triple solutions for \eqref{e1.1}.
Finally an example is shown  to illustrate the  results obtained.
Now, we give some notation which will be used later.

Let $X=C^{1}[0, 1]$ be a Banach space with the norm
\[
\| u\|=\max\big\{\max_{t\in [0, 1]}|u(t)|,
\max_{t\in [0, 1]}|u'(t)|\big\}.
\]
A function $u(t)$ is called a
positive solution of  \eqref{e1.1} if $u\in X$, satisfies
\eqref{e1.1} and $u(t)>0$ for $t\in (0, 1)$. Let
\begin{gather*}
 C^{\ast}[0, 1]=\{u\in X: u(t)\geq 0, u'(t)\leq 0, u'(t)
\text{ is nonincreasing for } t\in[0, 1]\},\\
P=\{u\in X: u(t)\geq 0, u'(t)\leq 0, u'(t)
\text{ is concave on }t\in[0, 1]\}.
\end{gather*}
It is easy to see $P$ is a cone of $X$.

 In this paper, we assume the following hypotheses:
\begin{itemize}
\item[(H1)] $\alpha_{i}, \beta_{i}\geq 0$,
 $0<\sum_{i=1}^{n}\alpha_{i}$, $\sum_{i=1}^{n}\beta_{i}<1$.

\item[(H2)] $f\in C([0, 1]\times [0, +\infty)\times (-\infty, 0], [0,
 +\infty))$.
 \end{itemize}

\section{Preliminaries}

 In this section, we  provide some background
definitions  from the study of cone in Banach spaces; see for
example \cite{g3}.

Let $(E,\|\cdot\|)$ be a real Banach space. A nonempty, closed,
 convex set $P\subseteq E$ is said to be a cone provided the
following two conditions are satisfied:
\begin{itemize}
\item[(a)] if $y\in P$ and $\lambda\geq 0$, then $\lambda y\in P$;

\item[(b)] if $y\in P$ and $-y\in P$, then $y=0$.
\end{itemize}

If $P\subseteq E$ is a cone, we denote the order induced by $P$ on
$E$ by $\leq$, that is, $x\leq y$ if and only if $y-x\in P$.


A 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
$$
\alpha (tx+(1-t)y)\geq t\alpha (x)+(1-t)\alpha (y)
$$
for all $x, y\in P$ and $0\leq t\leq 1$.

Similarly, we say a 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
$$
\beta (tx+(1-t)y)\leq t\beta (x)+(1-t)\beta (y)
$$
for all $x,y\in P$ and $0\leq t\leq 1$.

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\leq\alpha (x), \gamma
(x)\leq d\},\\
P(\gamma, \theta, \alpha, b, c, d)=\{x\in P| b\leq\alpha (x),
\theta (x)\leq c, \gamma (x)\leq d\},
\end{gather*}
and a closed set
\[
R(\gamma, \psi, a, d)=\{x\in P| a\leq\psi (x), \gamma (x)\leq d\}.
\]

The following fixed point theorem is fundamental in
the proofs of our main results.

\begin{lemma}[\cite{a1}] \label{lem2.1}
 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 $\psi (\lambda x)\leq \lambda\psi (x)$
for $0\leq \lambda\leq 1$, such that for some positive numbers
$L$ and $d$,
\[
\alpha (x)\leq \psi (x)\quad\text{and}\quad
\| x\|\leq L\gamma (x), \forall x\in \overline{P(\gamma, d)}.
\]
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[(S1)] $\{x\in P(\gamma, \theta, \alpha, b, c, d)| \alpha
(x)>b\}\neq \emptyset$ and $\alpha (Tx)>b$ for $x\in P(\gamma,
\theta, \alpha, b, c, d)$

\item[(S2)] $\alpha (Tx)>b$ for $x\in P(\gamma, \alpha, b, d)$ with
$\theta (Tx)>c$

\item[(S3)] $0\notin R(\gamma, \psi, a, d)$ 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
$\gamma (x_{i})\leq d$ for $i=1, 2, 3$;
$b<\alpha (x_{1})$;
$a<\psi (x_{2})$ with $\alpha (x_{2})<b$;
$\psi (x_{3})<a$.
\end{lemma}


To prove the main results in this paper, we will employ the
following lemmas.

\begin{lemma}[\cite{j1}] \label{lem2.2}
Assume {\rm (H1)-(H2)}, and let
\[
k=\frac{\varphi_{p}(\sum_{i=1}^{n}\alpha_{i})}
{1-\varphi_{p}(\sum_{i=1}^{n}\alpha_{i})}.
\]
For $x\in C^{\ast}[0, 1]$, if $u(t)$ is a solution of the problem
\begin{gather*}
(\varphi_{p}(u'(t)))' + \lambda f(t, x(t), x'(t))=0,\quad  t\in (0, 1);\\
u'(0)=\sum_{i=1}^{n}\alpha_{i}u'(\xi_{i}), \quad
u(1)=\sum_{i=1}^{n}\beta_{i}u(\xi_{i}),
\end{gather*}
then
\begin{equation}
\begin{aligned}
u(t)&=-\frac{\sum_{i=1}^{n}\beta_{i}\int_{\xi_{i}}^{1}\varphi_{q}
(Ax-\int_{0}^{s}\lambda f(r, x(r), x'(r))dr)ds}
{1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad -
\int_{t}^{1}\varphi_{q}(Ax-\int_{0}^{s}\lambda f(r, x(r),
x'(r))dr)ds,
\end{aligned} \label{e2.1}
\end{equation}
where $Ax\in [-k\lambda \int_{0}^{1}f(s, x(s), x'(s))ds, 0]$
is unique and satisfies
\begin{equation}
\varphi_{q}(Ax)=\sum_{i=1}^{n}\alpha_{i}
\varphi_{q}(Ax-\int_{0}^{\xi_{i}}\lambda f(s, x(s), x'(s))ds),
\label{e2.2}
\end{equation}
\end{lemma}

Define the operator $T$ by
\begin{align*}
(Tu)(t)&=-\frac{\sum_{i=1}^{n}\beta_{i}\int_{\xi_{i}}^{1}\varphi_{q}(Au-\int_{0}^{s}\lambda
f(r, u(r), u'(r))dr)ds}{1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad -
\int_{t}^{1}\varphi_{q}(Au-\int_{0}^{s}\lambda f(r, u(r),
u'(r))dr)ds.
\end{align*}
 Then by Lemma \ref{lem2.2} it is easy to see $u(t)$ is
a solution of \eqref{e1.1} if and only if $u(t)=(Tu)(t)$.

\begin{lemma}[\cite{j1}] \label{lem2.3}
 For each $\lambda>0$, the operator $T: P\to P$ is completely continuous.
\end{lemma}

Now we give an important property of $Ax$ defined by \eqref{e2.1}.

\begin{lemma} \label{lem2.4}
Assume {\rm (H1)} holds. Then for each
$x\in C^{\ast}[0, 1]$, $\tau\in (0, \xi_{1})$,
\begin{equation}
\frac{\varphi_{p}(\sum_{i=1}^{n}\alpha_{i})}{1-\varphi_{p}
(\sum_{i=1}^{n}\alpha_{i})}\int_{\tau}^{\xi_{1}}\lambda
f(r, x(r), x'(r))dr\leq -Ax\leq k\int_{0}^{1}\lambda f(r,
x(r), x'(r))dr.
\label{e2.3}
\end{equation}
\end{lemma}

\begin{proof}
 By \eqref{e2.1}, we have
\begin{align*}
\varphi_{q}(Ax)
&= \sum_{i=1}^{n}\alpha_{i}\varphi_{q}(Ax-\int_{0}^{\xi_{i}}
 \lambda f(s, x(s), x'(s))ds)\\
&\geq \sum_{i=1}^{n}\alpha_{i}\varphi_{q}(Ax-\int_{0}^{1}
 \lambda f(s, x(s), x'(s))ds),
\end{align*}
and
\begin{align*}
\varphi_{q}(Ax)
&=\sum_{i=1}^{n}\alpha_{i}\varphi_{q}(Ax-\int_{0}^{\xi_{i}}
 \lambda f(s, x(s), x'(s))ds)\\
&\leq \sum_{i=1}^{n}\alpha_{i}\varphi_{q}(Ax-\int_{\tau}^{\xi_{1}}
 \lambda f(s, x(s), x'(s))ds).
\end{align*}
From the increasing property of $\varphi_{q}$ and the two
inequalities above, it is easy to get the conclusion.
\end{proof}

Set
\begin{gather*}
m:=\frac{2^{\frac{1}{q-1}}+1}{\xi_{1}}, \quad
l:=\frac{\frac{(m+1)\xi_{1}}{2^{\frac{1}{q-1}}}+\xi_{1}^{-m}}{2},\\
N:=\frac{\sum_{i=1}^{n}\beta_{i}(1-\xi_{i})
 +(1-\sum_{i=1}^{n}\beta_{i})(1-\xi_{1})}{1-\sum_{i=1}^{n}\beta_{i}}.
\end{gather*}
Choose an $\tau\in (0, \xi_{1})$ such that
$l^{-1/m}<\tau<\xi_{1}$, and define the functionals:
\begin{equation}
\gamma (x)=\psi (x):=\| x\|, \quad
\theta (x):=\max_{t\in [0, \tau]}| x'(t)|, \quad
\alpha (x):=\min_{t\in[\tau, \xi_{1}]}x(t),\quad
 \forall x\in P.
\label{e2.4}
\end{equation}
Then it is easy to get the following lemma.

\begin{lemma} \label{lem2.5}
The four functionals defined by \eqref{e2.4} satisfy
Lemma \ref{lem2.1}. In addition, for each $x\in P$,
$\theta (x)=-x'(\tau)$, $\alpha (x)=x(\xi_{1})$,
$\gamma (x)=\psi(x)=x(0)$.
\end{lemma}

\section{Main Results}

 First we state the following hypotheses to be used in this article.
\begin{itemize}
\item[(H3)]  There exists a positive constant $H$ such that
$$
f(t, u, v)<lt^{m}\varphi_{p}(| u|+| v|),
$$
for $t\in [0,1]$ and $(u, v)\in \mathbb{R}^2$ satisfying $0\leq |u|+| v|\leq H$.

\item[(H4)] There exist positive constants $b, d$ such that
\begin{gather*}
\max\{\frac{1}{1-\xi_{1}},\, \frac{1}{2l^{q-1}},\,
\frac{1}{N}\}b<d\leq \frac{1}{2}H, \\
f(t, u, v)>\varphi_{p}(b),\quad \text{for } (t, u, v) \in [\tau,
\xi_{1}]\times [b, d]\times [-d, 0].
\end{gather*}
\end{itemize}
 Now we are ready to state our main results.

\begin{theorem} \label{thm3.1}
 Assume {\rm (H1)-(H4)}. Let
$$
M=\frac{1-\sum_{i=1}^{n}\beta_{i}\xi_{i}}
{(1-\sum_{i=1}^{n}\beta_{i})\varphi_{q}(1-\varphi_{p}
(\sum_{i=1}^{n}\alpha_{i}))}.
$$
Then for each $\lambda$ satisfying
\begin{equation}
\frac{1}{\xi_{1}M^{\frac{1}{q-1}}}\leq \lambda
\leq \frac{1}{\frac{1}{m+1}2^{\frac{1}{q-1}}lM^{\frac{1}{q-1}}},
\label{e3.1}
\end{equation}
and $a\in (0, b)$, Equation \eqref{e1.1} has at least three solutions
$x_{1}(t), x_{2}(t), x_{3}(t)$ satisfying
\begin{itemize}
\item[(i)] $\| x_{i}\|\leq d,\ i=1, 2, 3$;

\item[(ii)] $b<\min\{| x_{1}(t)|| t\in [0, \tau]\}$;

\item[(iii)] $\| x_{2}\|>a,\ \min\{x_{2}(t)| t\in [0,\tau]\}<b$;

\item[(iv)] $\| x_{3}\|<a$.
\end{itemize}
\end{theorem}

\begin{proof}
We divide the proof of this theorem in four steps.

\textbf{Step 1.} Let us show $T: \overline{P(\gamma, d)}\to
\overline{P(\gamma, d)}$.
In fact, for any $u\in \overline{P(\gamma, d)}$, it is not difficult
to see
\begin{equation}
\| Tu\|=\max\{(Tu)(0), -(Tu)'(1)\}.\label{e3.2}
\end{equation}
 From \eqref{e2.3}, \eqref{e3.1}, and (H3), we obtain
\begin{align*}
(Tu)(0)
&= -\frac{\sum_{i=1}^{n}\beta_{i}\int_{\xi_{i}}^{1}\varphi_{q}(Au-\int_{0}^{s}\lambda
 f(r, u(r), u'(r))dr)ds}{1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad - \int_{0}^{1}\varphi_{q}(Au-\int_{0}^{s}\lambda f(r, u(r),
 u'(r))dr)ds\\
&\leq -\frac{\sum_{i=1}^{n}\beta_{i}\int_{\xi_{i}}^{1}\varphi_{q}(-k\int_{0}^{1}\lambda f(r, u(r),u'(r))dr-\int_{0}^{s}\lambda
 f(r, u(r), u'(r))dr)ds}{1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad - \int_{0}^{1}\varphi_{q}(-k\int_{0}^{1}\lambda f(r,
 u(r),u'(r))dr-\int_{0}^{s}\lambda f(r, u(r),
 u'(r))dr)ds\\
&\leq
 \lambda^{q-1}\frac{1-\sum_{i=1}^{n}\beta_{i}\xi_{i}}
{(1-\sum_{i=1}^{n}\beta_{i})\varphi_{q}(1-\varphi_{p}
(\sum_{i=1}^{n}\alpha_{i}))}\varphi_{q}\Big(\int_{0}^{1}
 f(r, u(r),u'(r))dr\Big)\\
&<  (\frac{1}{m+1})^{q-1}2\lambda^{q-1}l^{q-1}M\| u\|\\
&\leq  (\frac{1}{m+1})^{q-1}2\lambda^{q-1}l^{q-1}Md
\leq  d
\end{align*}
and
\begin{align*}
 -(Tu)'(1)
&= -\varphi_{q}(Au-\int_{0}^{1}\lambda
 f(r, u(r), u'(r))dr)\\
&\leq  \varphi_{q}(\int_{0}^{1}\lambda
 f(r, u(r), u'(r))dr+\int_{0}^{s}\lambda f(r,
u(r), u'(r))dr)\\
&\leq \lambda^{q-1}\frac{1}{\varphi_{q}(1-\varphi_{p}
 (\sum_{i=1}^{n}\alpha_{i}))}\varphi_{q}(\int_{0}^{1}
 f(r, u(r),u'(r))dr)\\
&< (\frac{1}{m+1})^{q-1}2\lambda^{q-1}l^{q-1}
 \frac{1}{\varphi_{q}(1-\varphi_{p}(\sum_{i=1}^{n}\alpha_{i}))}\| u\|
\leq  d.
\end{align*}
Thus $\| Tu\|=\max\{(Tu)(0), -(Tu)'(1)\}\leq d$. Hence
$T: \overline{P(\gamma, d)}\to \overline{P(\gamma,d)}$.

\textbf{Step 2.} Check condition (S1) of Lemma \ref{lem2.1}.
Choose an integer $w>0$ such that
$\max\{\frac{1}{1-\xi_{1}},\frac{1}{N}\}<w\leq \frac{d}{b}$.
Set $u(t)=wb(1-t)$. Then
$$
b<\theta (u)=wb,\ \gamma (u)=wb\leq d,\ b<\alpha (u)=wb(1-\xi_{1})<wb.
$$
Therefore, $u(t)=wb(1-t)\in P(\gamma, \theta, \alpha, b, wb, d)$,
and $\alpha (u)>b$. This guarantees that
$\{u\in P(\gamma, \theta,
\alpha, b, wb, d)| \alpha (u)>b\}\neq \emptyset$.
For any $u\in P(\gamma, \theta, \alpha, b, wb, d)$, it is easy to
see
$$
b\leq u(t)\leq d,\quad
-d\leq u'(t)\leq 0,\quad \forall t\in [\tau, \xi_{1}].
$$
Thus by (H4), $f(t, u(t),u'(t))>\varphi_{p}(b)$.

By Lemma \ref{lem2.2} and Lemma \ref{lem2.3}, it is not difficult to see
\begin{align*}
\alpha(Tu)&=(Tu)(\xi_{1})\\
&= -\frac{\sum_{i=1}^{n}\beta_{i}\int_{\xi_{i}}^{1}
 \varphi_{q}(Au-\int_{0}^{s}\lambda
 f(r, u(r), u'(r))dr)ds}{1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad - \int_{\xi_{1}}^{1}\varphi_{q}(Au-\int_{0}^{s}\lambda f(r, u(r),
  u'(r))dr)ds\\
&\geq \frac{\sum_{i=1}^{n}\beta_{i}\int_{\xi_{i}}^{1}
 \varphi_{q}(k\int_{0}^{\xi_{1}}\lambda f(r, u(r),u'(r))dr
 +\int_{0}^{s}\lambda
 f(r, u(r), u'(r))dr)ds}{1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad + \int_{\xi_{1}}^{1}\varphi_{q}(k\int_{0}^{\xi_{1}}\lambda
 f(r,u(r),u'(r))dr+\int_{0}^{s}\lambda f(r, u(r), u'(r))dr)ds\\
&\geq \lambda^{q-1}\frac{1-\sum_{i=1}^{n}
 \beta_{i}\xi_{i}}{(1-\sum_{i=1}^{n}\beta_{i})\varphi_{q}
 (1-\varphi_{p}(\sum_{i=1}^{n}\alpha_{i}))}\varphi_{q}
 (\int_{\tau}^{\xi_{1}} f(r, u(r),u'(r))dr)\\
&> \lambda^{q-1}\xi_{1}^{q-1}Mb
\geq b.
\end{align*}
This shows that condition (S1) of Lemma \eqref{e2.1} is satisfied.

\textbf{Step 3.} Examine (S2) of Lemma \ref{lem2.1}.
For any $u\in P(\gamma, \alpha, b, d)\ \text{with}\ \theta (Tu)>wb$,
we know
\begin{equation}
\theta (Tu)=-(Tu)'(\tau)
=\varphi_{q}\Big(\int_{0}^{\tau}\lambda f(r,
u(r), u'(r))dr-Au\Big)>wb.
\label{e3.3}
\end{equation}
Therefore by \eqref{e2.3} and \eqref{e3.3},
\begin{align*}
 \alpha (Tu)
&=(Tu)(\xi_{1})\\
&= -\frac{\sum_{i=1}^{n}\beta_{i}
\int_{\xi_{i}}^{1}\varphi_{q}(Au-\int_{0}^{s}\lambda
f(r, u(r), u'(r))dr)ds}{1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad - \int_{\xi_{1}}^{1}\varphi_{q}(Au-\int_{0}^{s}\lambda f(r, u(r),
u'(r))dr)ds\\
&\geq  \frac{\sum_{i=1}^{n}\beta_{i}\int_{\xi_{i}}^{1}\varphi_{q}
\Big(k\int_{0}^{\tau}\lambda f(r, u(r),u'(r))dr-Au\Big)ds}
 {1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad + \int_{\xi_{1}}^{1}\varphi_{q}(k\int_{0}^{\tau}\lambda f(r,
u(r),u'(r))dr-Au)ds\\
&= \frac{\sum_{i=1}^{n}\beta_{i}(1-\xi_{i})
 +(1-\sum_{i=1}^{n}\beta_{i})(1-\xi_{1})}{1-\sum_{i=1}^{n}\beta_{i}}
 \varphi_{q}(\int_{0}^{\tau}\lambda f(r, u(r), u'(r))dr-Au)\\
&> Nwb
> b.
\end{align*}
Thus, condition (S2) of Lemma \eqref{e2.1} is satisfied.

\textbf{Step 4.} Finally we show (S3) of Lemma \ref{lem2.1} holds.
Since $\psi (0)=0<a$, we know $0\notin R(\gamma, \psi, a, d)$.
For each $u\in R(\gamma, \psi, a, d)$,
$\psi (u)=\|u\|=a$, by \eqref{e2.3}, \eqref{e3.1}, and (H3),
we obtain
\begin{align*}
(Tu)(0)
&= -\frac{\sum_{i=1}^{n}\beta_{i}\int_{\xi_{i}}^{1}\varphi_{q}
 (Au-\int_{0}^{s}\lambda
 f(r, u(r), u'(r))dr)ds}{1-\sum_{i=1}^{n}\beta_{i}}\\
&\quad - \int_{0}^{1}\varphi_{q}(Au-\int_{0}^{s}\lambda f(r, u(r),
u'(r))dr)ds\\
&\leq \lambda^{q-1}\frac{1-\sum_{i=1}^{n}
 \beta_{i}\xi_{i}}{(1-\sum_{i=1}^{n}\beta_{i})
 \varphi_{q}(1-\varphi_{p}(\sum_{i=1}^{n}\alpha_{i}))}
 \varphi_{q}\Big(\int_{0}^{1}
 f(r, u(r),u'(r))dr\Big)\\
&<  (\frac{1}{m+1})^{q-1}2\lambda^{q-1}l^{q-1}M\| u\|\\
&=  (\frac{1}{m+1})^{q-1}2\lambda^{q-1}l^{q-1}Ma
\leq  a
\end{align*}
and
\begin{align*}
-(Tu)'(1)
&= -\varphi_{q}(Au-\int_{0}^{1}\lambda
  f(r, u(r), u'(r))dr)\\
&\leq  \varphi_{q}(k\int_{0}^{1}\lambda f(r,
u(r), u'(r))dr+\int_{0}^{s}\lambda f(r,u(r), u'(r))dr)\\
&\leq \lambda^{q-1}\frac{1}{\varphi_{q}(1-\varphi_{p}
(\sum_{i=1}^{n}\alpha_{i}))}\varphi_{q}\Big(\int_{0}^{1}
f(r, u(r),u'(r))dr\Big)\\
&< (\frac{1}{m+1})^{q-1}2\lambda^{q-1}l^{q-1}\frac{1}{\varphi_{q}
(1-\varphi_{p}(\sum_{i=1}^{n}\alpha_{i}))}\| u\|
\leq  a.
\end{align*}
Therefore,
 $$
\psi (u)=\| u\|=\max\{(Tu)(0), -(Tu)'(1)\}<a.
$$
So condition (S3) of Lemma \eqref{e2.1} is satisfied.
Thus an application of Lemma \ref{lem2.1} implies that the boundary value
 problem \eqref{e1.1} has at
least three solutions $x_{1}(t), x_{2}(t), \ x_{3}(t)$
satisfying (i)--(iv).
\end{proof}

We remark that in Theorem \ref{thm3.1}, the two solutions
$x_{1}(t)$ and $x_{2}(t)$ are positive, while $x_{3}(t)$
may be the trivial solution.


\subsection{Example}
 Consider the differential equation
\begin{equation}
\begin{gathered}
(\varphi_{p}(u'(t)))' + \lambda f(t, u(t), u'(t))=0,\quad  t\in (0, 1);\\
u'(0)=\sum_{i=1}^{2}\alpha_{i}u'(\xi_{i}),\quad
u(1)=\sum_{i=1}^{2}\beta_{i}u(\xi_{i}),
\end{gathered}\label{e3.4}
\end{equation}
where $p=3/2$, $q=3$,
$\alpha_{1}=\alpha_{2}=\beta_{1}=\beta_{2}=1/4$,
$\xi_{1}=0.9$, $\xi_{2}=0.95$,
$$
f(t, u, v)=1.8t^{10(\sqrt{2}+1)/9} \sqrt{u+|v|},\quad
 (t, u, v)\in [0, 1]\times[0, +\infty)\times(-\infty, 0].
$$
Choose $l= 1.835055448$, $m=10(\sqrt{2}+1)/9$, $H=20000$,
$d=10000$, $b=100$, $a=50$, $\tau=0.88$, then by simple
calculations, it is easy to show (H1)-(H4) are satisfied.
Therefore, by Theorem \ref{thm3.1},
for $9\sqrt{430}/430 \leq\lambda\leq1.461370837$,
 Equation \eqref{e3.4} has at least three solutions.


\subsection*{Acknowledgements}
The authors want to thank the anonymous referee for the suggestions 
on the paper.

\begin{thebibliography}{00}

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

\bibitem{g1}  W. Ge;
\emph{Boundary value problems for nonlinear differential
equations}, Science Press, Beijing, 2007 (in Chinese).

\bibitem{g2}  W. Ge, J. Ren;
\emph{An extension of Mawhin's continuation
theorem and its applications to boundary value problems with a
$p$-Laplacian},  Nonl. Anal., {\bf 58} (2004), 477-488.

\bibitem{g3} D. Guo, V. Lakshmikantham;
\emph{Nonlinear problems in abstract cones}, Academic Press,
New York,  1988.

\bibitem{h1} X. He, W. Ge;
\emph{Existence of positive solution for a
one-dimensional $p$-Laplacian boundary value problem}, Acta.
Mathematica. Sinica., \textbf{46}(4)\rm(2003), 805-810 (in Chinese).

\bibitem{h2} X. He, W. Ge;
\emph{Twin positive solutions for the
one-dimensional $p$-Laplacian boundary value problems},  Nonl.
Anal., \textbf{56} (7) (2004), 975-984.

\bibitem{h3} S. Hong;
\emph{Triple positive solutions of three-point boundary
value problems for $p$-Laplacian dynamic equations on time scales},
 Comput. Math. Appl., \textbf{206}(2007), 967-976.

\bibitem{j1} D. Ji, Y. Tian, W. Ge;
\emph{The existence of positive solution
of multi-point boundary value problem with a $p$-Laplace operator},
Acta. Mathematica. Sinica., \textbf{52} (1)\rm(2009), 1-8 (in
Chinese).

\bibitem{l1} H. Lv, D. O'Regan, C. Zhang;
\emph{Multiple positive solutions for the one dimensional singular
$p$-Laplacian},  Appl. Math. Comput., \textbf{133} (2002), 407-422.

\bibitem{m1} D. Ma, Z. Du, W. Ge;
\emph{Existence and iteration of monotone
positive solutions for multi-point boundary value problem with
$p$-Laplacian operator},  Comput. Math. Appl., \bf 50\rm(2005),
729-739.

\bibitem{w1} J. Wang;
\emph{The existence of positive solutions for the
one-dimensional $p$-Laplacian}, Proc. Amer. Appl.,
\textbf{125} (8)\rm(1997), 2275-2283.

\bibitem{w2} Y. Wang, W. Ge;
\emph{M-point boundary value problem for second
order nonlinear differential equation},  J. Anal. Anal.,
\textbf{85} (2006), 659-667.

\bibitem{w3} Y. Wang, W. Ge;
\emph{Existence of triple positive solutions for
multi-point boundary value problems with a one dimensional
$p$-Laplacian},  Comput. Math. Appl., \textbf{54} (2007), 793-807.

\bibitem{w4}  Y. Wang, C. Hou;
\emph{Existence of multiple positive solutions
for one-dimensional $p$-Laplace},  J. Math. Anal. Appl.,
\textbf{315} (2006), 144-153.

\end{thebibliography}

\end{document}