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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 24, 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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/24\hfil Positive solutions]
{Positive solutions for second-order m-point boundary-value
problems with nonlinearity depending on the first derivative}
\author[L. Yang,  X. Liu,  C. Shen\hfil EJDE-2006/24\hfilneg]
{Liu Yang, Xiping Liu, Chunfang Shen}  

\address{College of Science, University of Shanghai for Science
and Technology, Shanghai 200093, China}
\email[Liu Yang]{xjiangfeng@163.com}
\email[Chunfang Shen]{minirose1982@163.com}
\email[Xiping Liu]{xipingliu@163.com}

\date{}
\thanks{Submitted June 6, 2005. Published February 23, 2006.}
\thanks{Supported by the Foundation of Educational Commission of Shanghai}
\subjclass[2000]{34B10, 34B15}
\keywords{Boundary value problem; positive solution; cone;
          fixed point theorem}

\begin{abstract}
 We consider multiplicity of positive solutions for
 second-order $m$-point boundary-value problems, with the first
 order derivative involved in the nonlinear term.
 Using a fixed point theorem, we show  the existence of at least
 three positive solutions. By giving an example we illustrate
 the main result of the article.
\end{abstract}

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

\section{Introduction}

 Multi-point boundary-value problems for ordinary differential equations
arise in different areas of applied mathematics and physics. For
example, the vibrations of a guy wire of uniform cross-section and
composed of N parts of different densities can be set up as a
multi-point boundary-value problem,many problems in the theory of
elastic stability can be handled as multi-point boundary-value
problems too.Recently, the existence and multiplicity of positive
solutions for nonlinear ordinary differential equations and
difference equations have received a great deal of attentions.To
identify a few,we refer the reader to \cite{a1,c1,m1,m2,m3} and references
therein.  Ma and Wang \cite{m4} obtained the existence of one positive
solution for more general three-point boundary-value problem
\begin{gather}
u''(t)+a(t)u'(t)+b(t)u(t)+h(t)f(u)=0,\quad t\in
(0,1),\label{e1.1}\\
u(0)=0,\quad u(1)=\alpha u(\eta),\quad 0<\eta <1,\label{e1.2}
\end{gather}
under the assumption that $f$ is either suplinear or sublinear, and
that the following conditions are satisfied:
\begin{itemize}
\item[(H1)] $f\in C([0,+\infty),[0,+\infty))$
\item[(H2)] $h\in C([0,1],[0,+\infty))$ and there exists
$x_{0}\in (0,1)$ such that $h(x_{0})>0$
\item[(H3)] $a\in C[0,1]$, $b\in C([0,1],(-\infty,0])$
\item[(H4)] $0<\alpha \phi_{1}(\eta)<1$, where
$\phi_{1}$ is the unique solution of the linear problem
\begin{gather}
\phi_{1}''(t)+a(t)\phi_{1}'(t)+b(t)\phi_{1}(t)=0,\quad t\in (0,1),\label{e1.3}\\
\phi_{1}(0)=0,\quad \phi_{1}(1)=1.\label{e1.4}
\end{gather}
\end{itemize}
In \cite{m4}, the authors used a fixed point theorem for a mapping defined on
Banach spaces with cones, by  Guo and Krasnosel'skii \cite{g1}.
However all the above works about positive solutions were done under
the assumption that the first order derivative $x'$ is not involved
in the nonlinear term. On the other hand, to the best of our knowledge,
there are very few work considering the multiplicity
of positive solutions with dependence on derivatives.

In this paper, we consider the existence of at least three positive
solutions for the equation
\begin{equation}
x''(t)+a(t)x'(t)+b(t)x(t)+f(t,x(t),x'(t))=0,\quad t\in (0,1),\label{e1.5}
\end{equation}
subject to the boundary conditions
\begin{equation}
x(0)=0,\quad x(1)=\sum^{m-2}_{i=1} \alpha_{i} x(\xi_{i}),\label{e1.6}
\end{equation}
or to the boundary conditions
\begin{equation}
x'(0)=0, \quad x(1)=\sum^{m-2}_{i=1} \alpha_{i} x(\xi_{i}),\label{e1.7}
\end{equation}
where $\xi_{i}\in(0,1)$, $\alpha_{i}>0$, $i=1,2,\dots,m-2$ are given
constants.

The interest in triple solutions evolved from the Leggett-Williams
fixed point theorem \cite{l1}. When $x'$ does not appear in
nonlinear term there are results about several nonlinear ordinary
differential equations, obtained by the Leggett-Williams fixed
point theorem; see \cite{g2,h1}. Recently Avery and Peterson
\cite{a2}, Bai and Ge \cite{b1} generalized the fixed point
theorem of Leggett-Williams by using theorem of fixed point index
and Dugundji extension theorem. As applications of the results in
\cite{b1,b2}, it has been obtained the existence of triple
positive solutions of the boundary-value problem
\begin{gather}
 x''(t)+a(t)f(t,x(t),x'(t))=0,\quad 0<t<1,\label{e1.8}\\
 x(0)=x(1)=0,\quad\text{or}\quad x(0)=x'(1)=0.\label{e1.9}
\end{gather}
By using the main results of \cite{b1,m4}, we give some simple criteria
for the existence of multiple positive
solutions for problem \eqref{e1.5} subject to \eqref{e1.6} or \eqref{e1.7}.

\section{Background definitions and preliminaries}

 For the convenience of the reader,we present here the necessary
definitions from cone theory in Banach spaces. This
definitions can be found in the literature.

\subsection*{Definitions}
Let $E$ be a real Banach space over $\mathbb{R}$.
A nonempty convex closed set
$P\subset E$ is said to be a cone provided that
(i) $au\in P$, for all $u\in P$, $a\geq 0$, and
(ii) $u,-u\in P$ implies $u=0$.

Note that every  cone $P\subset E $ induces an ordering in $E$ given
 by $x\leq y$ if $y-x\in P$.

 An operator is called completely continuous if it is continuous and
maps bounded sets  into precompact sets.

 The map $\alpha$ is  said  to  be a nonnegative continuous convex
functional  on cone $P$ of a  real Banach space E provided that
$ \alpha :P\to [0,+\infty)$ is continuous and
$$
\alpha (tx+(1-t)y)\leq t\alpha(x)+(1-t)\alpha(y),\quad \forall
 x,y \in P\;  t\in [0,1].
$$

 The map $\beta$ is  said to be a nonnegative continuous concave functional
on cone $P$ of a real  Banach space $E$ provided that
$\beta :P\to [0,+\infty)$ is continuous and
$\beta(tx+(1-t)y)\geq t\beta(x)+(1-t)\beta(y)$, for all $x,y\in P$ and
$ t\in [0,1]$.

 Suppose $\theta,\gamma:P\to [0,+\infty)$ are two nonnegative continuous
convex functionals satisfying
\begin{equation}
\|x\|\leq k\max\{\theta(x),\gamma(x)\} \quad\text{for }
 x\in P,\label{e2.1}
\end{equation}
where $k$ is a positive constant,and
\begin{equation}
\Omega=\{x\in P|\theta(x)<r,\gamma(x)<L\}\neq\emptyset,
\quad r>0,\; L>0.\label{e2.2}
\end{equation}
 Let $r>a>0,L>0$ be given, $\gamma,\theta:P\to [0,+\infty)$ be nonnegative
continuous convex functionals
satisfying \eqref{e2.1} and \eqref{e2.2}, $\alpha$ be a nonnegative
continuous concave functional on $P$. Define the following convex sets:
\begin{gather*}
P(\gamma,L;\theta,r)=\{x\in P|\gamma(x)<L,\theta(x)<r\},\\
\overline{P}(\gamma,L;\theta,r)=\{x\in P|\gamma(x)\leq L,\theta(x)\leq r\},
\\
P(\gamma,L;\theta,r;\alpha,a)=\{x\in P|\gamma(x)<L,\theta(x)<r,\alpha(x)>a\},
\\
\overline{P}(\gamma,L;\theta,r;\alpha,a)=\{x\in P|\gamma(x)\leq
L,\theta(x)\leq r,\alpha(x)\geq a \}.
\end{gather*}

\begin{lemma} \label{lem2.1}
Let E be a Banach space,$P\subset E$ be a cone and $r_{2}\geq c>b>r_{1}>0$,
$L_{2}\geq L_{1}>0$ be given. Assume that $\gamma,\theta$ are nonnegative
continuous convex functionals on P such that \eqref{e2.1}, \eqref{e2.2}
are satisfied. $\alpha$ is a nonnegative continuous concave functional
on $P$ such that $\alpha(x)\leq \theta(x)$ for all
$x\in \overline{P}(\gamma,L_{2};\theta,r_{2})$ and let
$T:\overline{P}(\gamma,L_{2};\theta,r_{2})\to \overline{P}(\gamma,L_{2};
\theta,r_{2})$ be a completely continuous operator. Suppose
that
\begin{itemize}
\item[(S1)] The set $\{x\in \overline{P}(\gamma,L_{2};\theta,c;\alpha,b):
\alpha(x)>b\}$ is not empty, and
$\alpha(Tx)>b$ for $x$ in $\overline{P}(\gamma,L_{2};\theta,c;\alpha,b)$;

\item[(S2)] $\gamma(Tx)<L_{1}$, $\theta(Tx)<r_{1}$ for all
$x\in \overline{P}(\gamma,L_{1};\theta,r_{1})$;

\item[(S3)] $\alpha(Tx)>b$, for all
$x\in \overline{P}(\gamma,L_{2};\theta,r_{2};\alpha,b)$ with
$\theta(Tx)>c$.
\end{itemize}
Then $T$ has at least three fixed points $x_{1},x_{2},x_{3}$ in
$\overline{P}(\gamma,L_{2};\theta,r_{2})$. Further,
\begin{gather*}
x_{1}\in P(\gamma,L_{1};\theta,r_{1});\quad
x_{2}\in \{\overline{P}(\gamma,L_{2};\theta,r_{2};\alpha,b):\alpha(x)>b\},
\\
x_{3}\in \overline{P}(\gamma,L_{2};\theta,r_{2})\setminus(\overline{P}
(\gamma,L_{2};\theta,r_{2};\alpha,b)\cup \overline{P}
(\gamma,L_{1};\theta,r_{1})).
\end{gather*}
\end{lemma}

\section{Positive solutions of \eqref{e1.5}, \eqref{e1.6}}

 To state the main results of this section,we need the following lemma,
which was established by Ma and Wang \cite{m4}.

\begin{lemma} \label{lem3.1}
Assume that (H3) holds. Let $\phi_{1},\phi_{2}$ be  solutions of
\eqref{e1.3}, \eqref{e1.4}, and
\begin{gather}
\phi_{2}''(t)+a(t)\phi_{2}'(t)+b(t)\phi_{2}(t)=0,\quad
t\in (0,1),\label{e3.1}\\
\phi_{2}(0)=1,\quad \phi_{2}(1)=0.\label{e3.2}
\end{gather}
Then
 $\phi_{1}$ is strictly increasing and
 $\phi_{2}$ is strictly decreasing on [0,1].
\end{lemma}

 Inspired by \cite{m4}, we state following lemma which can be regard as a natural
extension.

\begin{lemma} \label{lem3.2}
Suppose (H3) and
\begin{equation} \label{H5}
0<\sum_{i=1}^{m-2}\alpha_{i} \phi_{1}(\xi_{i})<1.
\end{equation}
Then the problem
\begin{gather}
x''(t)+a(t)x'(t)+b(t)x(t)+y(t)=0,\quad t\in(0,1)\label{e3.3}\\
x(0)=0,\quad x(1)=\sum^{m-2}_{i=1} \alpha_{i}x(\xi_{i}),\label{e3.4}
\end{gather}
is equivalent to integral equation
\begin{equation}
x(t)=\int^{1}_{0}G(t,s)p(s)y(s)ds+A \phi_{1}(t),\label{e3.5}
\end{equation}
where
\begin{gather}
A=\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1}\alpha_{i}\xi_{i}}
\int^{1}_{0}G(\xi_{i},s)p(s)y(s)ds,\label{e3.6}
\\
p(t)=\exp\Big(\int^{t}_{0}a(s)ds\Big), \quad
\rho=\phi'_{1}(0),   \label{e3.7} \\
G(t,s)=\frac{1}{\rho}\begin{cases}
\phi_{1}(t)\phi_{2}(s) &{s\geq t}\\
\phi_{1}(s)\phi_{2}(t) &{s\leq t},
\end{cases} \nonumber\\
u(t)\geq 0\quad\mbox{if } y(t)\geq 0. \nonumber
\end{gather}
\end{lemma}

The proof of this lemma is very similar to a proof in \cite{m4}, so we omit
it here.
 Let
$$
q(t)=\min\{\frac{\phi_{1}(t)}{|\phi_{1}|_{0}},
\frac{\phi_{2}(t)}{|\phi_{2}|_{0}}\},\quad t\in [0,1]
$$
where $|y(t)|_{0}=\max|y(t)|,t\in [0,1]$.

The following Lemma was also
established by Ma and Wang.

\begin{lemma} \label{lem3.3}
Suppose (H3) and \eqref{H5} are satisfied, $y\in C[0,1],y\geq 0$,
then the solution of  \eqref{e3.3}-\eqref{e3.4} satisfies
\begin{equation}
u(t)\geq |u|_{0}q(t),t\in [0,1].\label{e3.8}
\end{equation}
\end{lemma}
Thus, for any $\delta \in [0,1/2]$, there exists $\lambda$ such that
\begin{equation}
u(t)\geq \lambda|u|_{0},t\in [\delta,1-\delta],\label{e3.9}
\end{equation}
where $\lambda=\min\{q(t):t\in [\delta,1-\delta]\}$.
Let
\begin{gather*}
M=\max_{0\leq t\leq 1} \int^{1}_{0}G(t,s)p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)ds;
\\
N=\max_{0\leq t\leq 1}|\int^{1}_{0}\frac{\partial G(t,s)}{\partial t}p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}'(t)}{1-\sum^{m-2}_{i=1}\alpha_{i}
\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)ds|;
\\
m=\min_{\delta\leq t\leq 1-\delta}\int^{1-\delta}_{\delta}G(t,s)p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(\delta)}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{1}(\xi_{i})}\int^{1-\delta}_{\delta}G(\xi_{i},s)p(s)ds.
\end{gather*}
 To present our main results, we assume there exist constants
$r_{2}\geq \frac{b}{\lambda}>b>r_{1}>0$, $L_{2}\geq L_{1}>0$ such that
$\frac{b}{m}<\min\{\frac{r_{2}}{M},\frac{L_{2}}{N}\}$ and the
following assumptions hold:
\begin{itemize}
\item[(A1)]
 $f(t,u,v)\in C([0,1]\times[0,+\infty)\times R,[0,+\infty))$;
\item[(A2)]   $f(t,u,v)< \min\{r_{1}/M,L_{1}/N\},(t,u,v)\in
[0,1]\times [0,r_{1}]\times [-L_{1},L_{1}]$;
\item[(A3)] $f(t,u,v)> b/m,(t,u,v)\in [\delta,1-\delta]\times
[b,b/\lambda]\times [-L_{2},L_{2}]$;
\item[(A4)] $f(t,u,v)\leq \min\{r_{2}/M,L_{2}/N\},(t,u,v)
\in [0,1]\times [0,r_{2}]\times [-L_{2},L_{2}]$.
\end{itemize}

\begin{theorem} \label{thm3.1}
Under assumption {\rm (A1)-(A4), (H3)}, \eqref{H5},
Problem \eqref{e1.5}-\eqref{e1.6} has at
least three positive solutions $x_{1},x_{2},x_{3}$ satisfying
\begin{equation}
\begin{gathered}
\max_{0\leq t\leq1}x_{1}(t)\leq r_{1},\max_{0\leq t\leq1}|x'_{1}(t)|
\leq L_{1};\\
b<\min_{\delta\leq t\leq 1-\delta}x_{2}(t)\leq \max_{0\leq t
\leq 1}x_{2}(t)\leq r_{2},\max_{0\leq t\leq 1}|x'_{2}(t)|\leq L_{2};
\\
\max_{0\leq t\leq 1} x_{3}(t)\leq \frac{b}{\lambda},\max_{0\leq t\leq 1}
|x'_{3}(t)|\leq L_{2}.
\end{gathered} \label{e3.10}
\end{equation}
\end{theorem}

\begin{proof}
Problem \eqref{e1.5}-\eqref{e1.6} has a solution $x=x(t)$ if and only
if $x$ solves the operator equation
$$
x(t)=\int^{1}_{0}G(t,s)p(s)f(s,x(s),x'(s))ds+A \phi_{1}(t)=(Tx)(t),
\quad 0<t<1.
$$
Let $X=C^{1}[0,1]$ be endowed with the ordering $x\leq y$ if
$x(t)\leq y(t)$ for all $t\in [0,1]$, and the maximum norm
$$
\|x\|=\max\big\{\max_{0\leq t\leq 1}|x(t)|,\max_{0\leq t\leq 1}|x'(t)|
\big\}.
$$
Define the cone $P\subset X$ by
$$
P=\{x\in X|x(t)\geq 0,\min_{\delta\leq t\leq 1-\delta}x(t)\geq
\lambda|x(t)|_{0},t\in [0,1]\}.
$$
Define functionals
\begin{gather*}
\gamma(x)=\max_{0\leq t\leq 1}|x'(t)|,\quad
\theta(x)=\max_{0\leq t\leq 1}|x(t)|, \\
\alpha(x)=\min_{\delta\leq t\leq 1-\delta}|x(t)|,for\ x\in X.
\end{gather*}
Then $\gamma,\theta: P\to [0,+\infty)$ are nonnegative continuous
convex functionals satisfying \eqref{e2.1} and \eqref{e2.2};
$\alpha$ is nonnegative continuous concave functional with
$\alpha(x)\leq \theta(x)$ for all $x\in X$.

 Now we verify that all the conditions of Lemma \ref{lem2.1} are satisfied.
 If $x\in \overline{P}(\gamma,L_{2};\theta,r_{2})$,
then $\gamma(x)\leq L_{2},\theta(x)\leq r_{2}$ and assumption
$(A_{4})$ implies
$$
f(t,x(t),x'(t))\leq \min\{\frac{L_{2}}{N},\frac{r_{2}}{M}\},
$$
consequently
\begin{align*}
\theta(Tx)&=\max_{0\leq t\leq 1}[\int^{1}_{0}G(t,s)p(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(t)}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)f(s,x(s),x'(s))ds]\\
&\leq \frac{r_{2}}{M}\max_{0\leq t\leq 1}[\int^{1}_{0}G(t,s)p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(t)}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)ds]\\
&\leq \frac{r_{2}}{M}[\max_{0\leq t\leq 1}\int^{1}_{0}G(t,s)p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)ds]\\
&\leq \frac{r_{2}}{M}M=r_{2}.
\end{align*}
Also
\begin{align*}
\gamma(Tx)&=\max_{0\leq t\leq 1}|\int^{1}_{0}
 \frac{\partial G(t,s)}{\partial t}p(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}'(t)}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)f(s,x(s),x'(s))ds|\\
&\leq \frac{L_{2}}{N}\max_{0\leq t\leq 1}|\int^{1}_{0}
 \frac{\partial G(t,s)}{\partial t}p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}'(t)}{1-\sum^{m-2}_{i=1}
 \alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)ds|\\
&\leq \frac{L_{2}}{N}N=L_{2}.
\end{align*}
Hence,$T:\overline{P}(\gamma,L_{2};\theta,r_{2})\to
\overline{P}(\gamma,L_{2};\theta,r_{2})$ and $T$ is completely
continuous on $[0,1]$.
In the same way, if $x\in \overline{P}(\gamma,L_{1};\theta,r_{1})$,
then assumption (A2) yields
$$
f(t,x(t),x'(t))<\min\{\frac{L_{1}}{N},\frac{r_{1}}{M}\},\quad
0\leq t\leq 1.
$$
As in the argument above,we can obtain that
$T:\overline{P}(\gamma,L_{1};\theta,r_{1})\to P(\gamma,L_{1};\theta,r_{1})$.
Therefore, condition (S2) of Lemma \ref{lem2.1} is satisfied.

 To check condition (S1) of Lemma \ref{lem2.1}, we choose
$x(t)=\frac{b}{\lambda}=c$.
It's easy to see
$x(t)=\frac{b}{\lambda}\in \overline{P}(\gamma,L_{2};\theta,c;\alpha,b)$
and $\alpha(\frac{b}{\lambda})>b$.
So $\{x\in \overline{P}(\gamma,L_{2};\theta,c;\alpha,b)|\alpha(x)>b)\}
\neq \emptyset$.
If $x\in P(\gamma,L_{2};\theta,c;\alpha,b)$, we have
$b\leq x(t)\leq \frac{b}{\lambda},|x'(t)|<L_{2}$ for
$\delta\leq t\leq 1-\delta$. From the assumption (A2), we have
$$
f(t,x(t),x'(t))> \frac{b}{m}.
$$
By the definition of $\alpha$ and the cone $P$,
\begin{align*}
\alpha(Tx)&=\min_{\delta\leq t\leq 1-\delta}[\int^{1}_{0}
G(t,s)p(s)f(s,x(s),x'(s))ds \\
&\quad +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(t)}{1-\sum^{m-2}_{i=1}
 \alpha_{i}\phi_{1}(\xi_{i})}\int^{1}_{0}G(\xi_{i},s)p(s)f(s,x(s),x'(s))ds]\\
&> \frac{b}{m}\min_{0\leq t\leq 1}[\int^{1-\delta}_{\delta}G(t,s)p(s)ds
 +\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(t)}{1-\sum^{m-2}_{i=1}
 \alpha_{i}\phi_{1}(\xi_{i})}\int^{1-\delta}_{\delta}G(\xi_{i},s)p(s)ds]\\
&\leq \frac{b}{m}[\min_{0\leq t\leq 1}\int^{1-\delta}_{\delta}G(t,s)p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{1}(\delta)}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{1}(\xi_{i})}\int^{1-\delta}_{\delta}G(\xi_{i},s)p(s)ds]\\
&\geq \frac{b}{m}m=b.
\end{align*}
Then $\alpha(Tx)>b$, for all
$x\in \overline{P}(\gamma,L_{2};\theta,c;\alpha,b)$.
This shows that condition (S1) of lemma \ref{lem2.1} is also satisfied.
Finally we show (S3) holds too.
 Suppose $x\in \overline{P}(\gamma,L_{2};\theta,r_{2};\alpha,b)$
with $\theta (Tx)>\frac{b}{\lambda}$. Then,by the definition of
$\alpha$ and $Tx\in P$, we have
$$
\alpha(Tx)=\min_{\delta\leq t\leq 1-\delta}|(Tx)(t)|
\geq \lambda\cdot \max_{0\leq t\leq 1}|(Tx)(t)|
=\lambda\cdot\theta(Tx)
=\lambda\cdot\frac{b}{\lambda}=b.
$$
So condition (S3) of lemma \ref{lem2.1} is satisfied. Therefore,
Lemma \ref{lem2.1} yields that problem \eqref{e1.5}-\eqref{e1.6} has at least
three positive solutions $x_{1},x_{2},x_{3}$ in
$\overline{P}(\gamma,L_{2};\theta,r_{2})$ and \eqref{e3.10} is satisfied.
\end{proof}

\begin{remark} \label{rmk4.1} \rm
In Lemma \ref{lem2.1}, we need only
$T:\overline{P}(\gamma,L_{2};\theta,r_{2})\to \overline{P}(\gamma,L_{2};
\theta,r_{2})$; therefore, condition
(A1) can be substituted with the weaker condition
\begin{itemize}
\item[(C1)]
$f \in C([0,1]\times [0,r_{2}] \times [-L_{2},L_{2}],[0,+\infty))$.
\end{itemize}
 From the proof of Theorem \ref{thm3.1}, it is easy to see that, if conditions
like (A1)-(A4) are appropriate combined,
we can obtained an arbitrary number of positive solutions for this problem.
\end{remark}

\begin{corollary} \label{coro3.1}
Suppose condition (A1) is satisfied and there exist constants
$0<r_{1}<b_{1}<b_{1}/\lambda\leq r_{2}<b_{2}<b_{2}/\lambda\dots
\leq r_{n},0<L_{1}\leq L_{2}\leq \dots \leq L_{n-1},n\in N$,
such that
$$
b_{i}/m\leq \min\{\frac{r_{i+1}}{M},\frac{L_{i+1}}{N}\}
$$
If the following two conditions are satisfied then problem
\eqref{e1.5}-\eqref{e1.6} admits at least $2n-1$ positive solutions.
\begin{itemize}
\item[(E1)] $f(t,u,v)< \min\{\frac{r_{i}}{M},\frac{L_{i}}{N}\}$,
$(t,u,v)\in [0,1]\times [0,r_{i}]\times [-L_{i},L_{i}]$, $1\leq i\leq n$;

\item[(E2)] $f(t,u,v)> \frac{b_{i}}{m}$,
$(t,u,v)\in [\delta,1-\delta]\times [b_{i},\frac{b_{i}}{\lambda}]
\times [-L_{i+1},-L_{i+1}],1\leq i\leq n-1$.
\end{itemize}
\end{corollary}

\begin{proof} When $n=1$, it follows from condition (E1) that
$$
T:\overline{P}(\gamma,L_{1};\theta,r_{1})\to P(\gamma,L_{1};\theta,r_{1})
\subseteq \overline{P}(\gamma,L_{1};\theta,r_{1}).
$$
Then by Schauder's fixed-point theorem,
$T$ has at least one fixed point
$x_{1}$ in $P(\gamma,L_{1};\theta,r_{1})$.
When $n=2$, it is clear that Theorem \ref{thm3.1} holds.
Then we can obtain at least three positive solutions
$x_{2},x_{3},x_{4}$. Along
this way, we can complete the proof by the induction method.
\end{proof}

\section{Positive solutions of \eqref{e1.5}, \eqref{e1.7}}

 In this section we study problem \eqref{e1.5}, \eqref{e1.7}.
The method and existence results are remarkable
analogous to those in section 3. First, we give some Lemmas.
 Suppose $\phi_{3}$ is the unique solution of the
linear boundary-value problem
\begin{gather}
\phi_{3}''(t)+a(t)\phi_{3}'(t)+b(t)\phi_{1}(t)=0,\quad
 t\in (0,1),\label{e4.1}\\
\phi_{3}'(0)=0,\quad \phi_{3}(1)=1. \label{e4.2}
\end{gather}
satisfying
\begin{equation} \label{H6}
 0<\sum^{m-2}_{i=1}\alpha_{i}\phi_{3}(\xi_{i})<1.
\end{equation}
Then problem
\begin{gather}
x''(t)+a(t)x'(t)+b(t)x(t)+y(t)=0,t\in(0,1)\label{e4.3}\\
x'(0)=0,\hspace*{4em}x(1)=\sum^{m-2}_{i=1}
\alpha_{i} x(\xi_{i}),\label{e4.4}
\end{gather}
is equivalent to integral equation
\begin{equation}
x(t)=\int^{1}_{0}G_{1}(t,s)p(s)y(s)ds+A_{1} \phi_{3}(t),\label{e4.5}
\end{equation}
where
\begin{gather}
A_{1}=\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1}\alpha_{i}\phi_{3}(\xi_{i})}
\int^{1}_{0}G_{1}(\xi_{i},s)p(s)y(s)ds,\label{e4.6}
\\
 p(t)=\exp(\int^{t}_{0}a(s)ds),\quad \rho_{1}=-\phi_{3}(0)\phi'_{2}(0),
\label{e4.7}
\\
G_{1}(t,s)=\frac{1}{\rho_{1}}
\begin{cases}
\phi_{3}(t)\phi_{2}(s)&{1\geq s\geq t\geq 0}\\
\phi_{3}(s)\phi_{2}(t)&{0\leq s\leq t\leq 1},
\end{cases}
\nonumber\\
x(t)\geq 0\quad \text{if }y(t)\geq 0. \nonumber
\end{gather}
 Let (H3), \eqref{H6} be satisfied, substituting $\phi_{1}(t)$ with
$\phi_{3}(t)$, we can obtain a similar result as in lemma \ref{lem3.3}.
Let
\begin{gather*}
M_{1}=\max_{0\leq t\leq 1} \int^{1}_{0}G_{1}(t,s)p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{3}(\xi_{i})}\int^{1}_{0}G_{1}(\xi_{i},s)p(s)ds;\\
N_{1}=\max_{0\leq t\leq 1}|\int^{1}_{0}\frac{\partial
G_{1}(t,s)}{\partial t}p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{3}'(t)}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{3}(\xi_{i})}\int^{1}_{0}G_{1}(\xi_{i},s)p(s)ds|;\\
m_{1}=\min_{\delta\leq t\leq 1-\delta}\int^{1-\delta}_{\delta}
G_{1}(t,s)p(s)ds
+\frac{\sum^{m-2}_{i=1}\alpha_{i}\phi_{3}(\delta)}{1-\sum^{m-2}_{i=1}
\alpha_{i}\phi_{3}(\xi_{i})}\int^{1-\delta}_{\delta}G_{1}(\xi_{i},s)p(s)ds.
\end{gather*}
 Analogous to Theorem \ref{thm3.1}, using results established above, it is
not difficult to show that problem \eqref{e1.5},\eqref{e1.7} has at least
three positive solutions.

\begin{theorem} \label{thm4.1}
 Suppose conditions (H3), \eqref{H6}, (C1) are satisfied and there
exist constants $r_{2}\geq \frac{b}{\lambda_{1}}>b>r_{1}>0$,
$L_{2}\geq L_{1}>0$ such that
$\frac{b}{m_{1}}<\min\{\frac{r_{2}}{M_{1}},\frac{L_{2}}{N_{1}}\}$
and the following assumptions hold:
\begin{itemize}
\item[(A5)] $f(t,u,v)< \min\{r_{1}/M_{1},L_{1}/N_{1}\},(t,u,v)\in [0,1]
\times [0,r_{1}]\times [-L_{1},L_{1}]$;
\item[(A6)] $f(t,u,v)> b/m_{1},(t,u,v)\in [\delta,1-\delta]\times
[b,b/\lambda_{1}]\times [-L_{2},L_{2}]$;
\item[(A7)] $f(t,u,v)\leq \min\{r_{2}/M_{1},L_{2}/N_{1}\},(t,u,v)
\in [0,1]\times [0,r_{2}]\times [-L_{2},L_{2}]$.
\end{itemize}
Then problem \eqref{e1.5}, \eqref{e1.7} has at least three positive
solutions $x_{1},x_{2},x_{3}$ satisfying
\begin{equation} \label{e4.8}
\begin{gathered}
\max_{0\leq t\leq1}x_{1}(t)\leq r_{1}, \quad
\max_{0\leq t\leq1}|x'_{1}(t)|\leq L_{1};\\
b<\min_{\delta\leq t\leq 1-\delta}x_{2}(t)\leq \max_{0\leq t\leq 1}
x_{2}(t)\leq r_{2}, \quad
\max_{0\leq t\leq 1}|x'_{2}(t)|\leq L_{2};\\
\max_{0\leq t\leq 1} x_{3}(t)\leq \frac{b}{\lambda_{1}}, \quad
\max_{0\leq t\leq 1} |x'_{3}(t)|\leq L_{2}.
\end{gathered}
\end{equation}
\end{theorem}

Further we can establish following multiplicity results of problem
\eqref{e1.5}, \eqref{e1.7}.

\begin{corollary} \label{coro4.1}
Suppose condition (A1) is satisfied and there exist constants
$0<r_{1}<b_{1}<b_{1}/\lambda\leq r_{2}<b_{2}<b_{2}/\lambda\dots
\leq r_{n}$, $0<L_{1}\leq L_{2}\leq \dots \leq L_{n-1}$, $n\in N$,
such that
$$
b_{i}/m\leq \min\{\frac{r_{i+1}}{M},\frac{L_{i+1}}{N}\}
$$
If the following two conditions are satisfied
then problem \eqref{e1.5}, \eqref{e1.7} admits at least $2n-1$ positive
solutions:
\begin{itemize}
\item[(E3)] $f(t,u,v)< \min\{\frac{r_{i}}{M_{1}},\frac{L_{i}}{N_{1}}\}$,
$(t,u,v)\in [0,1]\times [0,r_{i}]\times [-L_{i},L_{i}]$, $1\leq i\leq n$;
\item[(E4)] $f(t,u,v)> \frac{b_{i}}{m_{1}}$,
$(t,u,v)\in [\delta,1-\delta]\times [b_{i},\frac{b_{i}}{\lambda_{1}}]
\times [-L_{i+1},-L_{i+1}]$, $1\leq i\leq n-1$
\end{itemize}
\end{corollary}

\section{Examples}
In this section we present an example to illustrate our
main results. Consider the boundary-value problem
\begin{equation}
\begin{gathered}
x''(t)-x(t)+f(t,x(t),x'(t))=0,\quad 0<t<1\\
x(0)=0,x(1)=e^{1/2}x(1/2),
\end{gathered} \label{e5.1}
\end{equation}
where
\[
f(t,u,v)=\frac{1}{5}\begin{cases}
e^{t}+u^{4}+(\frac{v}{1000})^{3}&{0\leq u\leq 5}\\
e^{t}+625+(\frac{v}{1000})^{3}&{u>5}
\end{cases}
\]
Considering lemma \ref{lem3.1}, \ref{lem3.2}, we obtain
\begin{gather*}
\phi_{1}(t)=\frac{e^{1+t}-e^{1-t}}{e^{2}-1},\quad
\phi_{1}'(0)=\frac{2e}{e^{2}-1}, \\
\phi_{2}(t)=\frac{e^{2-t}-e^{t}}{e^{2}-1}, \quad
p(t)=1,\quad \delta=\frac{1}{4},\quad
\lambda=\frac{e^{\frac{5}{4}}-e^{\frac{3}{4}}}{e^{2}-1},
\\
G(t,s)=\frac{1}{2e(e^{2}-1)} \begin{cases}
(e^{1+t}-e^{1-t})(e^{2-s}-e^{s})&{s\geq t}\\
(e^{1+s}-e^{1-s})(e^{2-t}-e^{t})&{s\leq t},\\
\end{cases}
\end{gather*}
\begin{align*}
M&=\max_{0\leq t\leq 1} \int^{1}_{0}G(t,s)ds
  +\frac{e^{1/2}}{1-e^{1/2}\phi_{1}(1/2)}\int^{1}_{0}G(1/2,s)ds\\
 &= \frac{(e+1-2e^{1/2})(1+e^{1/2}+e^{3/2})}{e+1}.
\end{align*}
\begin{align*}
m&=\min_{\frac{1}{4}\leq t\leq \frac{3}{4}}
 \int_{\frac{1}{4}}^{\frac{3}{4}}G(t,s)ds+
\frac{e^{1/2}\phi_{1}(1/4)}{1-e^{1/2}\phi_{1}(1/2)}
\int_{\frac{1}{4}}^{\frac{3}{4}}G(1/2,s)ds \\
&=\frac{1}{2}-e^{1/2}+\frac{e^{7/4}-e^{5/4}}{e-1}.
\end{align*}
\begin{align*}
N&=\max_{0\leq t\leq 1}|\int^{1}_{0}\frac{\partial G(t,s)}{\partial t}ds+
\frac{e^{1/2}\phi_{1}'(t)}{1-e^{1/2}\phi_{1}(1/2)}\int^{1}_{0}G(1/2,s)ds|\\
&=\frac{(e^{1/2}-1)(e^{2}+1)}{e-1}.
\end{align*}
Choose $r_{1}=1$, $r_{2}=1000$, $b=4$, $L_{1}=10,L_{2}=1000$,
then $\min\{\frac {r_{1}}{M},\frac{L_{1}}{N}\}=\frac{1}{M}$,
$\min\{\frac{r_{2}}{M},\frac{L_{2}}{N}\}=\frac{1000}{N}$.
We can check that conditions (C1), (H3), \eqref{H5} are satisfied
and that $f(t,u,v)$ satisfies
\begin{gather*}
f(t,u,v)<\frac{1}{M},\quad\mbox{for }
(t,u,v)\in [0,1]\times [0,1]\times [-10,10];\\
f(t,u,v)>\frac{4}{m}, \quad\mbox{for }
 (t,u,v)\in [\frac{1}{4},\frac{3}{4}]\times [4,\frac{4}{\lambda}]
\times [-1000,1000];\\
f(t,u,v)<\frac{1000}{N},\quad\mbox{for }
 (t,u,v)\in [0,1]\times [0,1000]\times [-1000,1000].
\end{gather*}
Then all assumptions of Theorem \ref{thm3.1} hold. Thus,  \eqref{e5.1} has at
least three positive solutions $x_{1},x_{2},x_{3}$ satisfying
\begin{gather*}
\max_{0\leq t\leq 1}x_{1}(t)\leq 1,\quad
\max_{0\leq t\leq 1}|x'_{1}(t)|\leq 10;\\
4<\min_{\frac{1}{4}\leq t\leq \frac{3}{4}}x_{2}(t)\leq
\max_{0\leq t\leq 1}x_{2}(t)\leq 1000,\quad
\max_{0\leq t\leq 1}|x'_{2}(t)|\leq 1000;\\
\max_{0\leq t\leq 1}x_{3}(t)\leq \frac{4}{\lambda}, \quad
\max_{0\leq t\leq 1}|x'_{3}(t)|\leq 1000\,.
\end{gather*}

\begin{remark} \label{rmk5.1} \rm
We see that the nonlinear term  involves the first order derivative
and can it change sign.
The early results for multiplicity of positive solutions, to the author's
best knowledge, are not applicable to the problem above. Meanwhile,
as the nonlinear term does not satisfy the suplinear or sublinear
condition even if the nonlinear term is $f(t,u,v)=f(u)$,
we can not obtain even one positive solution of this problem from
 \cite{m4}.
\end{remark}

\begin{thebibliography}{99}

\bibitem{a1} R. I. Avery, C. J. Chyan, J. Henderson.
\emph{Twin solutions of boundary-value problems for ordinary differential
equations and finite difference equations}, Comput. Math. Appl,
42(2001), 695-704.
\bibitem{a2} R. I. Avery and A. C. Peterson.
\emph{Three positive fixed points of nonlinear operators on an ordered
Banach space}, Comput. Math. Appl. 208(2001), 313-322.

\bibitem{b1} Bai, Zhanbing; Ge, Weigao. \emph{Existence of three positive
solutions for some second-order boundary value problems},
Comput. Math. Appl. 48(2004), 699-707.
\bibitem{b2} Bai, Zhanbing; Wang, Yifu; Ge, Weigao.
\emph{Triple positive solutions for a class of two-point boundary-value
problems}, Electronic Journal of Differential Equations. Vol 2004(2004)
no. 6, 1-8.

\bibitem{c1} Wing-Sum Cheung, \emph{Jingli Ren.Positive solutions
for m-point boundary-value problems}, J. Math. Anal. Appl. 303(2005),
565-575.

\bibitem{g1} Guo, Dajun;  Lakshmikantham, V.
\emph{Nonlinear Problems in Abstract Cones}, San Diego: Academic Press, 1988.
\bibitem{g2} Guo, Yanping; Liu, Xiujun;  Qiu, Jiqing.
\emph{Three positive solutions for higher m-point boundary-value problems},
J. Math. Anal. Appl, 289(2004), 545-553.

\bibitem{h1} He, Xiaoming;  Ge, Weigao. \emph{Triple solutions for
second-order three-point boundary-value problems}, J. Math. Anal. Appl,
268(2002), 256-265.

\bibitem{l1} R. W. Leggett and L. R. Williams.
\emph{Multiple positive fixed points of nonlinear operators on ordered
Banach spaces}, Indiana Univ. Math. J. 28(1979), 673-688.

\bibitem{m1} Ma, Ruyun.
\emph{Positive solutions for a nonlinear three-point boundary-value
problem}, Electronic Journal of Differential Equations, Vol.
1999 (1999), no. 34, 1-8.
\bibitem{m2} Ma, Ruyun.
\emph{Positive solutions for second-order three-point boundary-value
problems,Applied Mathematics Letters}, 14 (2001), 1-5.
\bibitem{m3} Ma, Ruyun. \emph{Positive solutions of a nonlinear m-point
boundary-value problem}, Comput. Math. Appl. 42 (2001), 755-765.
\bibitem{m4} Ma, Ruyun; Wang, Haiyan. \emph{Positive solutions of nonlinear
three-point boundary-value problems}, J. Math. Anal. Appl. 279(2003),
1216-1227.


\end{thebibliography}

\end{document}